&EPA
US EM Oki a\ R-KWtr i«1 [
          United States
          Environmental Protection
          Agency
             Office of Research and
             Development
             Washington DC 20460
EPA/600/R-01/070
September 2001
Development of
Recommendations and
Methods to Support
Assessment of Soil Venting
Performance and Closure
               100 200  300
                       400 500  600

                       Distance from well (cm)
                     700  800 900  1000

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                                                       EPA/600/R-01/070
                                                       September 2001
DEVELOPMENT OF RECOMMENDATIONS AND METHODS TO SUPPORT
   ASSESSMENT OF SOIL VENTING PERFORMANCE AND CLOSURE

                               by

                        Dominic C. DiGiulio
             Subsurface Protection and Remediation Division
             National Risk Management Research Laboratory
                       Ada, Oklahoma 74820

                               and

                          Ravi Varadhan
                 (formerly with Dynamac Corporation)
                      Johns Hopkins University
                       School of Public Health
                        Baltimore, Maryland
                      Contract No. 68-C4-0031
                          Project Officer
                        Dominic C. DiGiulio
             Subsurface Protection and Remediation Division
             National Risk Management Research Laboratory
                       Ada, Oklahoma 74820
     NATIONAL RISK MANAGEMENT RESEARCH LABORATORY
            OFFICE OF RESEARCH AND DEVELOPMENT
           U. S. ENVIRONMENTAL PROTECTION AGENCY
                     CINCINNATI, OHIO 45268

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NOTICE

       The U.S. Environmental Protection Agency through its Office of Research and
Development funded and managed the research described here through in-house efforts and
under Contract No. 68-C4-0031 to the Dynamac Corporation. It has been subjected to the
Agency's peer and administrative review and has been approved for publication as an EPA
document.

       All data generated by EPA staff were subjected to the analytical Quality Assurance Plan
for ManTech Environmental Research Services Corp. Results of field-based studies and
recommendations provided in this document have been subjected to extensive external and
internal peer and administrative review. This report provides technical recommendations, not
policy guidance.  It is not issued as an EPA Directive, and the recommendations of this report are
not binding on enforcement actions carried out by the U.S. EPA or by the individual States of the
United States of America. Neither the United States  Government nor the authors accept any
liability or responsibility resulting from the use of this document. Implementation of the
recommendations of the document, and the interpretation of the results provided through that
implementation, are the sole responsibility of the user.

       Computer programs described within this report support analysis of gas flow and solute
transport in porous media. Neither the United States Government nor the authors accept liability
or responsibility resulting from the use of computer programs contained within this document.
Use of these codes is the sole responsibility of the user. Mention of trade names or commercial
products  does not constitute endorsement or recommendation for use.
                                           11

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FOREWORD

       The U.S. Environmental Protection Agency is charged by Congress with protecting the
Nation's land, air, and water resources. Under a mandate of national environmental laws, the
Agency strives to formulate and implement actions leading to a compatible balance between
human activities and the ability of natural systems to support and nurture life. To meet these
mandates, EPA's research program is providing data and technical support for solving
environmental problems today and building a science knowledge base necessary to manage our
ecological resources wisely, understand how pollutants affect our health, and prevent or reduce
environmental risks in the future.

       The National Risk Management Research Laboratory is the Agency's center for
investigation of technological and management approaches for reducing risks from threats to
human health and the environment.  The focus of the Laboratory's research program is on
methods for the prevention and control of pollution to air, land, water, and subsurface resources;
protection of water quality in public water systems; remediation of contaminated sites and
ground water; and prevention and control of indoor air pollution. The goal of this research effort
is to catalyze development and implementation of innovative, cost-effective environmental
technologies;  develop scientific and engineering information needed by EPA to support
regulatory and policy decisions; and provide technical support and information transfer to ensure
effective implementation of environmental regulations and strategies.

       The overall purpose of the report is to improve the "state of the art" and "state of the
science" of soil venting application. Results  of field-based research and comprehensive and
detailed literature reviews on gas flow and vapor transport are provided to form the basis and
defense of recommendations to improve site  characterization, design, and monitoring practices in
support of venting application.
                                        teph^n G. Schmelling, Acting Dire^tc
                                       Subsurface Protection and Remedfatiofi Division
                                       National Risk Management ReseaidAaboratory
                                           in

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ABSTRACT

       Soil venting, which includes gas extraction and/or gas injection, is the primary method
used in the United States to remove volatile organic compounds (VOCs) from unsaturated
subsurface porous media. The widespread use of venting is due to its above ground simplicity of
operation and proven ability to remove contaminant mass inexpensively relative to competing
technologies. However, there is little consistency in assessment of performance of the
technology in U.S. EPA or State regulatory programs. Design, monitoring, and closure schemes
are typically based on empirical methods with little consideration given to fundamental physical,
chemical, and biological processes controlling mass removal during venting operation.  This
results in the technology not being utilized to its fullest potential, nor its limitations being well
understood.

       The purpose of this document was to improve the current "state of the art" and "state of
the science" of soil venting application. This purpose was accomplished by attainment of three
specific objectives.  The first objective was to develop a regulatory approach to assess closure
including measures to ensure consistency in ground-water and vadose zone remediation. The
second objective was to provide comprehensive and detailed literature reviews on gas flow and
vapor transport.  These reviews formed the basis of recommendations and methods to improve
venting design and monitoring. The third objective was to perform research to improve various
aspects of venting application.

       To fulfill the first objective, a  strategy was proposed to base closure on regulatory
evaluation of four components: (1) site characterization, (2) design, (3) monitoring, and (4) mass
flux to and from ground water. Concurrent consideration of each component results in
converging lines of reason or a "weight of evidence" approach thereby increasing the likelihood
of correctly assessing the suitability for closure. Each component is interrelated requiring
continuous evaluation during the life of the project. A vadose zone paradigm was developed to
dynamically link the performance of ground-water remediation to vadose zone remediation.
Substantial progress in remediating ground water translates to increasingly stringent soil venting
closure requirements, while lack of progress in remediating ground water translates into less
stringent closure requirements.

       To fulfill the second objective, a detailed and comprehensive literature review describing
laboratory and numerical experiments performed to elucidate and quantify rate-limited mass
transport process during venting was provided.  This literature review provided the basis for
development of an innovative method of design based on attainment of a critical pore-gas
velocity in contaminated media.  A critical pore-gas velocity is defined as a pore-gas velocity
which results in slight deviation from equilibrium conditions. Selection of a pore-gas velocity to
support venting design requires consideration of rate-limited gas-NAPL, gas-water, and solids-
water mass exchange on a pore-scale and rate-limited mobile-immobile gas exchange on a field-
scale. Results of single- and multiple-well gas pressure and pore-gas velocity simulations
demonstrated that a design based on attainment of a critical pore-gas velocity is superior to the
                                           IV

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commonly used empirical method of radius of influence (ROI) testing. These simulations clearly
showed that the ROI is an ill-defined entity and that ROI-based designs do not guarantee
sufficient gas flow in porous media for timely remediation.

       This literature review also revealed that observation of low asymptotic vapor
concentrations in effluent gas is not a sufficient condition to demonstrate progress in mass
removal from contaminated soils. Different mass retention and rate coefficients produce very
similar effluent curves suggesting that effluent and vapor probe concentration data can not be
used to estimate mass transport coefficients and remaining VOC mass. Thus, the ubiquitous
empirical method of assessing venting performance based  on observation of asymptotic vapor
concentrations is questionable.

       The third objective was fulfilled by conducting research in the following areas:
(1) linearization of the gas flow equation, (2) one-dimensional steady-state gas flow with
slippage, (3) two-dimensional steady-state gas flow and permeability estimation in a domain
open to the atmosphere, (4) two-dimensional steady-state gas flow and permeability estimation in
a semi-confined domain, (5) two-dimensional transient gas flow and permeability estimation, (6)
radius of influence versus critical pore-gas velocity based venting  design, (7) modification of a
gas extraction well to minimize water-table upwelling, (8) rate-limited vapor transport with
diffusion modeling, (9) respiration testing, and (10) one-dimensional, analytical, vadose zone
transport modeling to assess mass flux to and from the capillary fringe.  A number of public
domain FORTRAN codes were developed to achieve these research objectives. Example input
and output files and the source code for each program are contained within the appendices.

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                  VI

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                               TABLE OF CONTENTS


Notice	ii

Foreword	iii

Abstract	iv

List of Symbols	xii

List of Figures	xvii

List of Tables	xxv

Acknowledgments	xxvi

Executive  Summary	xxvii

1.     Introduction	1

2.     Proposed Approach for Assessment of Performance and Closure of Venting
      Systems	4

      2.1    Limitations in the Use of Vapor Concentration Asymptotes	4
      2.2    Limitations of Rebound Testing	5
      2.3    Estimation of Soil Concentration from Soil-Gas Concentration	6
      2.4    VadoseZone Solute Transport Modeling	9
      2.5    Proposed Approach for Assessment of Performance and Closure	10
      2.6    Mass Flux Assessment	11

3.     Gas Flow in Porous Media - Fundamental Principles	14

      3.1    Compressibility	14
      3.2    Viscosity	14
      3.3    Gas Potential Function	16
      3.4    Gas Slippage	17
      3.5    Visco-Inertial Effects	19
      3.6    Relative Permeability	20
      3.7    Components of Specific Discharge	24
      3.8    Continuity Equation for Gas Flow	25
      3.9    Linearization of Gas Flow Equation	26
      3.10  Conclusions	28
                                          vn

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                               TABLE OF CONTENTS
4.      One-Dimensional Gas Flow and Permeability Estimation	32

       4.1    Permeameter Design	32
       4.2    Formulation of a One-Dimensional Steady-State Gas Flow Equation With
             Gas Slippage	33
       4.3    Analysis of Effect of Gas Slippage on One-Dimensional Pressure and
             Pore-Gas Velocity Simulation	36
       4.4    Minipermeameter Testing	50
       4.5    Formulation of a One-Dimensional Transient Gas Flow Equation with Gas
             Slippage in a Semi-Infinite Domain	52
       4.6    Formulation of a One-Dimensional Transient Gas Flow Equation With Gas
             Slippage in a Finite Domain	53
       4.7    Formulation of One-Dimensional Gas Flow Equations Without Gas Slippage
             in a Finite Domain with Time-Dependent Boundary Conditions	54
       4.8    Formulation of the Pseudo-Steady-State Radial Gas Flow Equation	56
       4.9    Formulation of the Radial Transient Confined Gas Flow Equation	57
       4.10   Conclusions	58

5.      Two-Dimensional, Axisymmetric, Steady-State Gas Permeability Estimation,
       Pore-Gas Velocity Calculation, and Streamline Generation in a Domain Open
       to the Atmosphere	60

       5.1    Model Formulation	60
       5.2    Computational Method	63
       5.3    Site Description	64
       5.4    Results and Discussion	64
       5.5    Conclusions	71

6.      Two-Dimensional, Axisymmetric, Steady-State Gas Flow,  Pore-Gas Velocity
       Calculation, and Permeability Estimation in a Semi-Confined Domain	72

       6.1    Model Formulation	73
       6.2    Computational Method	75
       6.3    Site Description	75
       6.4    Results and Discussion	76
       6.5    Conclusions	78

7.      Two-Dimensional, Axisymmetric, Transient Gas Flow and Permeability Estimation
       Analysis	80
                                         Vlll

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                               TABLE OF CONTENTS
       7.1    Model Formulation for Line-Source Open to the Atmosphere	81
       7.2    Model Formulation for Line-Source with Leaky Confined Flow	82
       7.3    Model Formulation for a Finite-Radius Well with Leaky Confined Flow	83
       7.4    Sensitivity Analysis of Finite-Radius, Transient Solution	87
       7.5    Description of Field Site	89
       7.6    Results and Discussion	98
       7.7    Conclusions	99

7.      VOC Retention in Porous Media	105

       8.1    Vapor Concentration	105
       8.2    Vapor Pressure	105
       8.3    Henry's Law Constant	109
       8.4    Gas-Water Interfacial Partitioning	109
       8.5    Gas-Solids Partitioning	Ill
       8.6    Solids-Water Partitioning	Ill
       8.7    NAPL-Gas/Water Partitioning	112
       8.8    Advective-Dispersive Equation for Vapor Transport	114
       8.9    Dispersion	115

8.      Rate-Limited Vapor Transport: Selection of Critical Pore-Gas Velocities to
       Support Venting Design	119

       9.1    Definition of Critical Pore-Gas Velocity	119
       9.2    Approaches to Estimating Critical Pore-Gas Velocities	120
       9.3    Rate-Limited NAPL-Gas Exchange	124
       9.4    Rate-Limited Gas-Water Exchange (First-Order Kinetics)	131
       9.5    Rate-Limited Gas-Water Exchange (Aggregate-Diffusion Modeling)	135
       9.6    Rate-Limited Solids-Water Exchange	138
       9.7    Combined Rate-Limited Gas-Water and Solids-Water Exchange	143
       9.8    Rate-Limited Mobile-Immobile Gas Exchange	144
       9.9    Combined Rate-Limited Mobile-Immobile Gas and Gas-Water Exchange	147
       9.10   Conclusions	147

10.    Limitations of ROI Evaluation for Soil Venting Design: Simulations to Support
       Venting Design Based on Attainment of Critical Pore-Gas Velocities in
       Contaminated Media	149

       10.1   Critical Pore-Gas Velocity Estimation for Test Site	150
                                          IX

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                               TABLE OF CONTENTS
       10.2   Single-Well Simulations	152
       10.3   Multi-Well Simulations	155
       10.4   Design Strategy	164
       10.5   Conclusions	164

11.     Use of a Combined Air Injection/Extraction (CIE) Well to Minimize Vacuum
       Enhanced Water Recovery	166

       11.1   Field Methods and Materials	166
       11.2   Results and Discussion	174
       11.3   Conclusions	176

12.     Monitoring Strategies to Assess Concentration and Mass Reduction	178

       12.1   VOC Monitoring at the Vapor Treatment Inlet and Individual Wellheads
             Using a Portable PID or FID	178
       12.2   Observation of Evaporation/Condensation Fronts	182
       12.3   Vapor Effluent Asymptotes	186
       12.4   Flow Variation and Interruption Testing	188
       12.5   Use of Flow Interrupt!on Testing to Estimate Soil Concentration	189
       12.6   Use of Gas Phase Partitioning Tracer Testing to Assess Residual NAPL
             Removal During Venting Application	192
       12.7   Soil Sample Collection, Storage, and Analysis	198
       12.8   Conclusions	201

13.     Assessment of Rate-Limited Vapor Transport with Diffusion Modeling	204

       13.1   Model Formulation	204
       13.2   Site Description	206
       13.3   Results and Discussion	207
       13.4   Conclusions	209

14.     Use of Respiration Testing to Monitor Subsurface Aerobic Activity	210

       14.1   Discussion of Favorable Conditions for In-Situ Biodegradation	210
       14.2   Respiration Testing	214
       14.3   Site Description	216
       14.4   Results and Discussion	220
       14.5   Conclusions	221

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                              TABLE OF CONTENTS
15.    Evaluation of Mass Exchange to the Atmosphere and Ground Water	236

      15.1   Modeling Approach	236
      15.2   Model Formulation	238
      15.3   Representation of the Initial Condition	242
      15.4   Selection of the Upper Boundary Condition	243
      15.5   Selection of Lower Boundary Conditions	245
      15.6   Derivation of Analytical Solutions for First-Type, Time-Dependent
             Lower Boundary Condition	246
      15.7   Derivation of Analytical Solutions for the Zero-Gradient Lower Boundary
             Condition	252
      15.8   Example Simulations	255
      15.9   Conclusions	270

References	272

Appendix A (MFROAINV)	299
Appendix B (SAIRFLOW)	310
Appendix C (MFRLKINV)	318
Appendix D (TFRLK)	331
Appendix E (FRLKINV)	342
Appendix F (MAIRFLOW)	357
Appendix G (Vapor Diffusion)	369
Appendix H (VFLUX)	373
                                        XI

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LIST OF SYMBOLS


Latin

A     area [L2]
A^    gas-water interfacial area [L2L"3]
Cia    concentration at the gas-water interface [ML"3]
Cg     concentration in gas phase [ML"3]
Cg*    equilibrium vapor concentration [ML"3]
Csat   saturated vapor concentration [ML"3]
Cn     NAPL phase concentration [ML"3]
Cs     solids phase concentration [MM"1]
Csl    solids concentration in equilibrium domain [MM"1]
Cs2    solids concentration in rate-limited domain [MM"1]
CT     total soil concentration [ML"3]
Cw    concentration in bulk water or intra-aggregate space [ML"3]
C™'   saturated aqueous concentration [ML"3]
D     dimensionless depth (dv/d)
Dg     molecular gas  diffusivity [L2!"1]
Dy     dispersion coefficient [L2!"1]
Dm    mechanical dispersion coefficient [L2!"1]
Dw    molecular aqueous diffusion coefficient [L2!"1]
F      fraction of instantaneous sorption sites [-]
G0     geometric shape factor of minipermeameter [-]
H     Henry's Law Constant [-]
AHV   heat of vaporization [ML2T"2mol"1]
AHvb   heat of vaporization at boiling point [ML2T"2mol"1]
I       inertial flow factor [L"1]
J0     zero-first-order Bessel functions of the first kind
Jj     first-order Bessel functions of the first kind
K0     zero-order modified Bessel functions of the second kind
Kj     first-order modified Bessel functions of the second kind
Kd     equilibrium sorption coefficient [L3M"J]
KF     Fishtine dipole moment factor [-]
Kia    adsorption coefficient between gas and gas-water interface [L]
Kng    NAPL - gas partition coefficient [-]
Knw    NAPL - water partition coefficient [-]
Koc    organic carbon - water partition coefficient [L3M"J]
Kow    octanol-water partition coefficient [L3M"J]
L     dimensionless lengh (dL/d)
Lc     characteristic contamination distance [L]
                                            xn

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LIST OF SYMBOLS
Mg    molecular weight of gas mixture [Mmole"1]
M;    molecular weight of compound i [Mmole"1]
Mn    average molecular weight of NAPL [Mmole"1]
P      gas pressure [ML"1!"2]
P     average pressure [ML"1!"2]
PD    prefix denominator [-]
Pe    Peclet number [-]
Pv     vapor pressure [ML"1!"2]
Qm    mass flow rate [M!"1]
Qv    volumetric flow rate [L3!"1]
01     universal gas constant [ML2/!2molK]
R     retardation factor [-]
RD    retardation factor used for prefix denominator [-]
Re    Reynolds number [-]
Rs     retardation factor used for bicontinuum sorption kinetics [-]
Sg     gas saturation [-]
Sn     NAPL saturation [-]
Sg e    gas saturation at emergence point [-]
Sgs    maximum gas saturation [-]
Sg    effective gas saturation [-]
Sw    water saturation [-]
Sw    effective water saturation [-]
ASvb   entropy of vaporization at boiling point [ML2!"3mor1]
!      temperature [K]
!b     boiling point [K]
!c     critical point [K]
U     Kirckhoff transformed pressure
Vb    borehole storage volume [L3]
Vm    molar gas volume [L^mol"1]
Y0    zero-order Bessel function of the second kind
Yj    first-order Bessel function of the second kind
Z      gas compressibility factor [-]
a      radius or aggregate radius [L]
b      gas slippage or Klinkenberg parameter [ML"1!"2]
c      geometric shape factor: spherical is 15, planar is 3
as     total acceleration in Lagrangian framework [L!"2]
d      depth or thickness of a domain [L]
d'     thickness of semi-confining layer [L]
d50    mean grain diameter [L]
                                          Xlll

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LIST OF SYMBOLS
dL     depth to lower portion of sand pack [L]
dm     molecular diameter [L]
dp     pore diameter [L]
du     depth to upper portion of sand pack [L]
dS     length of cylinder in potential formulation [L]
e      (k/kzf
e'     k'd/kzd'
f      friction factor [-]
fg     mass fraction of contaminant in gas phase [MM"1]
f;      mass fraction of contaminant accumulated at gas-water interface [MM"1]
fn     mass fraction of contaminant in NAPL phase [MM"1]
fs      mass fraction of contaminant on solid phase [MM"1]
fw     mass fraction of contaminant in water phase [MM"1]
foc     fraction of organic carbon content [MM"1]
g      gravitational constant (980 cm s"2)
h      capillary pressure [L]
hd     empirical Brooks-Corey fitting parameter [L]
k     gas permeability tensor [L2]

kt     intrinsic permeability tensor [L2]

k'     gas permeability of semi-confining layer [L2]
kj     radial gas permeability [L2]
k,.g     relative permeability to gas [-]
kl(r)    intrinsic radial permeability to gas [L2]
kx     longitudinal gas permeability [L2]
kl(x)    intrinsic one-dimensional permeability to gas [L2]
kz     vertical component of gas permeability [L2]
kl(z)    intrinsic vertical permeability to gas [L2]
mm    molecular mass [M]
m     numerical index in summation series, number of capillary tubes, or empirical van
       Genuchten water retention parameter []
n      numerical index in summation series, number of molecules [-], or empirical van
       Genuchten water retention parameter
q      volumetric specific discharge vector [LT"1]
qr     radial component of specific discharge vector [LT"1]
qz     vertical component of specific discharge vector [LT"1]
qm     mass specific discharg vector [ML^T"1]
r      radial coordinate [L]
TJ      radius of influence [L]
r;      inner radius of tip seal of minipermeameter [L]
                                           xiv

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LIST OF SYMBOLS
r0      outer radius of tip seal of minipermeameter [L]
rw     radius of well or filter pack [L]
tc      critical time necessary to remove 99% of contaminant mass [T]
v      average linear pore-gas velocity [LT"1]
vc(ng)   critical pore-gas velocity for NAPL-gas exchange [LT"1]
vc(gw)   critical pore-gas velocity for gas-water exchange [LT"1]
v'c(a-gw) critical pore-gas velocity for gas-water exchange using aggregate diffusion [LT"1]
       from Gierke et al. (1992)
vc(a-gw)  critical pore-gas velocity for gas-water exchange using aggregate diffusion [LT"1]
       from Ng and Mei (1996)
vc(D-gw) critical pore-gas velocity for gas-water exchange using prefix denominator [LT"1]
w     average gas velocity in direction of flow in thin layer adjacent to pore wall [LT"1]
y      distance from a pore or capillary wall [L]
y     average dist.  from a pore wall in which the last molecular collision occurred [L]
z      vertical coordinate [L]

Greek

<&     gas potential [L]
<&f     gas potential loss due to friction [L]
S      spreading coefficient
a      empirical van Genuchten water retention fitting parameter []
6C     characteristic diffusion path length or radius [L]
KC     Boltzmann constant (1.38066 x 10"23 JK"1)
X      mean free path of gas molecules [L] or empirical Brooks-Corey water retention fitting
       value
Agg     mobile gas-immobile gas first order rate [T"1]
Agw    gas-water first-order rate coefficient [T"1]
A'gw   gas-water first-order rate coefficient defined by Gierke et al. (1992) [T"1]
Ang     NAPL-gas first order rate coefficient [T"1]
Xws     water-solid first order rate coefficient [T"1]
K      first-order degradation rate [T"1]
|j.g     dynamic gas viscosity [MTL"2]
cogw    gas-water Damkohler  number [-]
cosw    solids-water Damkohler  number [-]
pb     bulk density of soil [ML"3]
pg     gas density [ML"3]
pj      density of compound j [ML"3]
pn     NAPL density [ML"3]
ps     particle density [ML"3]

                                            xv

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LIST OF SYMBOLS
a^    gas-NAPL interfacial tension
o^    gas-water interfacial tension
onw    NAPL-water interfacial tension
t|      porosity [L3L"3]
(j>      pressure squared [M2L"2T"4]
4>!     pressure squared at radius of influence [M2L"2T"4]
T      gas phase shearing stress [ML"1!"2]
Tg     gas phase tortuosity factor [-]
TW     water phase tortuosity factor [-]
8g     mobile volumetric gas content or inter-aggregate gas-filled porosity [L3L"3]
9n     volumetric NAPL content [L3L"3]
8W     volumetric water content or intra-aggregate porosity [L3L"3]
Xi      mole fraction of organic compound i in NAPL
Yi      activity coefficient of organic compound i in NAPL [-]
T\I      streamfunction for gas flow
i)      kinematic gas viscosity [L2!"1]
                                           xvi

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                                  LIST OF FIGURES
Figure 2.1    Schematic of remediation zones for mass flux assessment ................................... 12

Figure 3.1    Percent deviation from Kidder's (1957) second-order perturbation solution
              for PJ/PQ = 0.01 to 0.4 for (a) 0 = PP0 in addition to $ = P0 and
                    = P0  alone [[[ 29
Figure 3.2    Percent Deviation from Kidder's (1957) second-order perturbation solution
              for PJ/PQ = 0.5 to 0.9 for (a) 0 = PP0 in addition to J0 = P0 and

                    = P  alone [[[ 30
Figure 4.1   Normalized pressure computation as a function of mass flow rate, distance,
             and gas slippage factor (k,.gkl(x) = 1.0 x 10"08 cm2) .................................................. 36
Figure 4.2   Normalized pressure computation as a function of mass flow rate, distance,
             and gas slippage (k,.gkl(x) = 1.0 x 10"09 cm2) [[[ 37
Figure 4.3   Normalized pressure computation as a function of mass flow rate, distance,
             and gas slippage (k,.gkl(x) = 1.0 x 10"10 cm2) [[[ 37
Figure 4.4   Error in pressure computation as a function of mass flow rate and distance
             when gas slippage is neglected (k,.gkl(x) = 1.0 x 10"08cm2) ....................................... 38
Figure 4.5   Error in pressure computation as a function of mass flow rate and distance
             when gas slippage is neglected (k,.gkl(x) = 1.0 x 10"09cm2) ....................................... 39
Figure 4.6   Error in pressure computation as a function of mass flow rate and distance
             when gas slippage is neglected (k,.gkl(x) = 1.0 x 10"10cm2) ....................................... 39
Figure 4.7   Pore-gas velocity computation as a function of mass flow rate, distance, and
             gas slippage ( 6g = 0.1, k,.gkl(x) = 1.0 x 10'08 cm2) [[[ 40
Figure 4.8   Pore-gas velocity computation as a function of mass flow rate, distance, and
             gas slippage (6g = 0.1, kjt^ = 1.0 x 10'09 cm2) [[[ 40
Figure 4.9   Pore-gas velocity computation as a function of mass flow rate, distance, and
             gas slippage (6g = 0.1, kjt^ = 1.0 x 1040 cm2) [[[ 41
Figure 4.10  Error in pore-gas velocity computation as a function of mass flow rate and
             distance when gas slippage is neglected (8g = 0.1, k,.gkl(x) = 1.0 x  10"08 cm2) ........ 42
Figure 4.11  Error in pore-gas velocity computation as a function of mass flow rate and
             distance when gas slippage is neglected (8g = 0.1, k,.gkl(x) = 1.0 x 10"09 cm2) ......... 42
Figure 4.12  Error in pore-gas velocity computation as a function of mass flow rate and
             distance when gas slippage is neglected (8g = 0.1, k,.gkl(x) = 1.0 x 10"10 cm2) ......... 43
Figure 4.13  Pressure computation as a function of mass flow rate, distance, and gas

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                                   LIST OF FIGURES
Figure 4.16   Error in pressure computation as a function of mass flow rate and distance
              when gas slippage is neglected (k,.gkl(x) = 1.0 x 10"08cm2) ....................................... 45
Figure 4.17   Error in pressure computation as a function of mass flow rate and distance
              when gas slippage is neglected (k,.gkl(x) = 1.0 x 10"9cm2) ........................................ 46
Figure 4.18   Error in pressure computation as a function of mass flow rate and distance
              when gas slippage is neglected (1^^ = l.Ox 10"10cm2) ...................................... 46
Figure 4.19   Pore-gas velocity computation as a function of mass flow rate, distance,
              and gas slippage (6g = 0.1, k,.gkl(x) = 1.0 x 10'08 cm2) ............................................. 47
Figure 4.20   Pore-gas velocity computation as a function of mass flow rate, distance,
              and gas slippage (6g = 0.1, krgkl(x) = 1.0 x 10'09 cm2) .............................................. 47
Figure 4.21   Pore-gas velocity computation as a function of mass flow rate, distance,
              and gas slippage (6g = 0.1, k,.gkl(x) = 1.0 x 10'10 cm2) .............................................. 48
Figure 4.22   Error in pore-gas velocity computation as a function of mass flow rate and
              distance when gas slippage is neglected (8g = 0.1, k,.gkl(x) = 1.0 x 10"08 cm2) ......... 48
Figure 4.23   Error in pore-gas velocity computation as a function of mass flow rate and
              distance when gas slippage is neglected (8g = 0.1, k^k^ = 1.0 x 10"09 cm2) ......... 49
Figure 4.24   Error in pore-gas velocity computation as a function of mass flow rate and
              distance when gas slippage is neglected (8g = 0.1, k,.gkl(x) = 1.0 x 10"10 cm2) ......... 49
Figure 5.1    Gas permeability testing schematic at the USCG Station, Traverse City, MI ....... 65
Figure 5.2    Observed versus predicted pressure for test . Applied flow and vacuum
              75. 1 1 g/s (13 1 .8 scfm) and 0.877 atm respectively.  Estimated k^k^ =
              6.1x ID'07 cm2, kj^/kjc^ = 1.2, RMSE = 6.50 x 10'04 atm ............................... 67
Figure 5.3    Observed (+) versus simulated (dashed lines) pressure differential
              (cm water) for test 1 [[[ 68
Figure 5.4.    Calculated pressure differential (cm of water - dashed lines), pore-gas
              velocity (cm/s - dash-dot lines), and streamlines (solid lines) for test ................... 69
Figure 5.5    Pore-gas velocity along a transect (centerline of well) as a function of
              applied flow and radial distance using average k,. and k^k^/k^k^ ratio .............. 70
Figure 7.1.    Pressure response as a function of time for finite-radius (FR) and line  source
              solutions (LS) (d = 600 cm, dv/d = 0.683, dL/d = 0.783, z/d = 0.733, rw/d =
              0.017, 6g = 0.1, krgkl(r)  = 1.0 x ID'9 cm2, k^/k^k^ = 1.0, Qm= 1.0 g/s,
              Vb = 15000 cm3, k^k^yd' = 1.0 x 10'11 cm) [[[ 89
Figure 7.2.    Error in pressure computation as a function of time for finite-radius (FR)
              and line source solutions (LS) (d = 600 cm, dv/d = 0.683, dL/d = 0.783, z/d =
              0.733, rw/d = 0.017, 6g = 0.1, k^ = 1.0 x  10'9 cm2, k^/k^k^  = 1.0,
              Qm= 1.0 g/s, Vb = 15000 cm3, kjc^/d' = 1.0 x  10'11 cm) .................................... 90
Figure 7.3    Pressure response as a function of time and vertical elevation for finite-radius
              (FR) and line source solutions (LS) (d = 600 cm, dv/d = 0.683,dL/d = 0.783,
              r = rw, 6g = 0.1, krgkl(r) = 1.0 x ID'9 cm2, kjt^/ig^ = 1.0, Qm= 1.0 g/s,

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                                  LIST OF FIGURES
Figure 7.4   Error in pressure computation as a function of time and vertical elevation
             for finite-radius (FR) and line source solutions (LS) (d=600 cm, dv/d =
             0.683, dL/d = 0.783, r = rw, 6g = 0.1, kjt^ =1.0 x 10'9 cm2, k^/k^k^ =
             1.0, Qm=1.0 g/s, Vb = 15000 cm3, kjt^/d^ 1.0 x 10-ncm	91
Figure 7.5   Pressure response as a function of time and borehole storage (d = 600 cm,
             du/d = 0.683, dL/d = 0.783, z/d = 0.733, r = rw, 6g = 0.1, k^ = 1 x 10'9 cm2,
             ig
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                                  LIST OF FIGURES
              1.60 x 10-9 cm2, k^/krgk^ = 3.50, kjc^/d' = 8.20 x 10'11 cm, 6g = 0.010,
              Vb=l 1,560 cm3, RMSE = 4.655 x 10'3 atm)	102
Figure 7.19   Observed versus fitted pressure response for Test 4 in MWE-02-03 (Qm =
              0.722 g/s, k^ = 1.60 x ID'9 cm2, klgkl(r)/klgkl(z) = 4.29, kjc^/d' =
              2.44 x ID'12 cm, 6g = 0.011, Vb = 9,199 cm3, RMSE = 6.35 x 10'3 atm)	103
Figure 10.1   Observed vacuum as a function of logarithmically transformed radial
              distance at tested wells for ROI determination. Line represent result of
              linear regression with correlation coefficients R2 as provided	149
Figure 10.2   Vacuum (inches of water) (dashed lines) and pore-gas velocity (cm/s)
              (solid lines) plots for tested wells	154
Figure 10.3   Multi-well simulation using permeability and leakance data from well W-5
              (k, = 6.87 x ID'7 cm2, k/kz = 1.01, leakance = 2.24 x 10'09 cm). Depth = 15
              feet. Dashed lines denotes vacuum (inches of water), solid lines denote
              pore-gas velocity (cm/s)	157
Figure 10.4   Multi-well simulation using permeability and leakance data from well W-l
              (k, = 2.90 x 10'7 cm2, k/kz =1.17, leakance = 9.29 x 10'10 cm). Depth = 15
              feet. Dashed lines denotes vacuum (inches of water), solid lines denote
              pore-gas velocity (cm/s)	158
Figure 10.5   Multi-well simulation using permeability and leakance data from well W-7
              (k, = 1.42 x ID'7 cm2, k/kz = 0.98, leakance = 3.62 x 10'11 cm). Depth = 15
              feet. Dashed lines denotes vacuum (inches of water), solid lines denote
              pore-gas velocity (cm/s)	159
Figure 10.6   Multi-well simulation using permeability and leakance data from well W-6
              (k,. = 1.81  x 10'7 cm2, k/kz = 0.74, leakance = 8.62 x 10'11 cm). Depth = 15
              feet. Dashed lines denotes vacuum (inches of water), solid lines denote
              pore-gas velocity cm/s)	160
Figure 10.7   Multi-well simulation using permeability and leakance data from well W-3
              (k, = 1.80 x ID'7 cm2, k/kz = 4.29, leakance = 4.04 x 10'10 cm). Depth = 15
              feet. Dashed lines denotes vacuum (inches of water), solid lines denote
              pore-gas velocity (cm/s)	161
Figure 10.8   Vacuum and pore-gas velocity at the ground water interface using
              permeability and leakance data from well W-5 (k,. = 6.87 x 10"7 cm2, k/kz =
              1.01, leakance = 2.24 x 10'09 cm). Depth = 15 feet.  Total flow = 1000 scfm,
              wells W-l, 3, 5, 6 = 175 scfm, wells W-4, 7 = 150 scfm	162
Figure 10.9   Vacuum and pore-gas velocity at the ground water interface using
              permeability and leakance data from well W-5 (k,. = 6.87 x 10"7 cm2, k/kz =
              1.01, leakance = 2.24 x 10'09 cm). Depth = 15 feet.  Total flow = 400 scfm,
              16 wells with 25 scfm at each well	163
Figure 11.1   Locations of dual vapor extraction wells (VE-series), observation wells
              (OB-series), sparging well (SP-1) and vapor probe clusters (CP-series) used
                                           xx

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                                  LIST OF FIGURES
             in study at Site 8, Vance AFB	167
Figure 11.2  Cross-sectional schematic of DVE wells, observation wells, vapor probe
             clusters and soil conditions encountered during drilling	168
Figure 11.3.  Photograph of DVE wells (wells with tubes sticking up - DVE-1 in
             foreground), observation wells, vapor probe clusters, air water separation
             tank (cylindrical tank with label), GAC units, and frac tank for contaminated
             water storage (rectangular shaped structure)	169
Figure 11.4  Schematic of a dual vacuum extraction (DVE) well used at Site 8,Vance,
             AFB	170
Figure 11.5  Schematic of sparging well used at Site 8,Vance, AFB	171
Figure 11.6  Schematic of an observation well used at Site 8,Vance, AFB	172
Figure 11.7  Schematic of a vapor probe cluster used at Site 8, Vance, AFB	173
Figure 11.8  Schematic of modified observation well 3 to enable combined air injection
             and extraction at Site 8, Vance, AFB	174
Figure 11.9.  Photograph of modified observation well, OB-3	175
Figure 11.10 Water recovery during vacuum extraction, vacuum extraction and sparging,
             and combined air extraction/injection at OB-3	176
Figurell.ll  Pressure differential measurements in vapor probes from DVE well and the
             combined air injection/extraction well at OB-3	177
Figure 13.1  Concentration (dry wt.) profile in clay lense	206
Figure 13.2  Best fit line of TCE vapor concentration profile as a function of time at the
             gas extraction well	207
Figure 13.3.  Simulated average TCE concentration in clay lense as  a function of time
             and moisture content	208
Figure 14.1  Dissolved combined benzene, toluene, ethylbenzene, and xylene (BTEX)
             concentrations before venting at Elizabeth City USCGBase, January, 1993	217
Figure 14.2  Total (wet) Petroleum Hydrocarbon (TPH) profile in 70E	218
Figure 14.3  Total (wet) Petroleum Hydrocarbon (TPH) profile in 70D	218
Figure 14.4  Oxygen depletion test at vapor probe 70A.  Solid and dashed lines denote
             linear regression for zero- and first-order kinetics respectively	222
Figure 14.5  Oxygen depletion test at vapor probe 70B. Solid and dashed lines denote
             linear regression for zero- and first-order kinetics respectively	222
Figure 14.6  Oxygen depletion test at vapor probe 70C. Solid and dashed lines denote
             linear regression for zero- and first-order kinetics respectively not using zero
             concentration data point	223
Figure 14.7  Oxygen depletion test at vapor probe 70D. Solid line denotes linear regression
             for zero-order kinetics not using last data point at zero concentration	223
Figure 14.8  Oxygen depletion test at vapor probe 70E. Solid and dashed lines denote
             linear regression for zero- and first-order kinetics respectively excluding last
             data point at zero concentration	224

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                                   LIST OF FIGURES
Figure 14.9   Oxygen depletion test at vapor probe 70F.  Solid and dashed lines denote
              linear regression for zero- and first-order kinetics respectively excluding
              last data point at zero concentration	224
Figure 14.10  Oxygen depletion test at vapor probe TOG.  Solid and dashed lines denote
              linear regression for zero- and first-order kinetics respectively	225
Figure 14.11  Oxygen depletion test at vapor probe 70H.  Solid and dashed lines denote
              linear regression for zero- and first-order kinetics respectively excluding
              last data point at zero concentration	225
Figure 14.12  Oxygen depletion test at vapor probe 701. Solid and dashed lines denote
              linear regression for zero- and first-order kinetics respectively excluding
              last data point at zero concentration	226
Figure 14.13  Oxygen depletion test at vapor probe 70J. Solid and dashed lines denote
              linear regression for zero- and first-order kinetics respectively excluding
              last data point at zero concentration	226
Figure 14.14  Oxygen depletion test at vapor probe 70K.  Solid and dashed lines denote
              linear regression for zero- and first-order kinetics respectively excluding
              last data point at zero concentration	227
Figure 14.15  Oxygen depletion test at vapor probe 70M. Solid and dashed lines denote
              linear regression for zero- and first-order kinetics respectively excluding
              last data point at zero concentration	227
Figure 14.16  Oxygen depletion test at vapor probe 70N.  Solid and dashed lines denote
              linear regression using and not using last data point at zero concentration
              respectively	228
Figure 14.17  Oxygen depletion test at vapor probe 70P.  Solid and dashed lines denote
              linear regression for zero- and first-order kinetics respectively excluding
              last data point at zero concentration	228
Figure 14.18  Oxygen depletion test at vapor probe 70Q.  Solid and dashed lines denote
              linear regression for zero- and first-order kinetics respectively excluding
              last data point at zero concentration	229
Figure 14.19  Oxygen depletion test at vapor probe 70R.  Solid and dashed lines denote
              linear regression for zero- and first-order kinetics respectively excluding
              last data point at zero concentration	229
Figure 14.20  Oxygen depletion test at vapor probe 70S.  Solid and dashed lines denote
              linear regression for zero- and first-order kinetics respectively	230
Figure 14.21  Oxygen depletion test at vapor probe 70T.  Solid line denotes linear
              regression for zero-order kinetics	230
Figure 14.22  Oxygen depletion test at vapor probe 70U.  Solid and dashed lines denote
              linear regression for zero- and first-order kinetics respectively excluding
              last data point at zero concentration	231
Figure 14.23  Oxygen depletion test at vapor probe 70V.  Solid and dashed lines denote
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                                  LIST OF FIGURES
             linear regression for zero- and first-order kinetics respectively excluding
             last data point at zero concentration	231
Figure 14.24 Oxygen depletion test at vapor probe 70X.  Solid and dashed lines denote
             linear regression for zero- and first-order kinetics respectively excluding
             last data point at zero concentration	232
Figure 15.1  Hypothetical initial soil concentration profile	257
Figure 15.2  Hypothetical ground-water concentration profile	258
Figure 15.3  Mass flux to ground water as a function of time, infiltration rate, and water
             saturation for a first-type, time-dependent (TD) lower boundary condition.
             NAPL absent, no degradation	258
Figure 15.4  Mass flux to ground water as a function of time and water saturation for
             first-type, time-dependent (TD) and zero-gradient (ZG) lower boundary
             conditions.  NAPL absent, no degradation, qw = 0.035 cm/d	259
Figure 15.5  Mass fraction of initial mass to lost to ground water as a function of time and
             water saturation for first-type, time-dependent (TD) and zero-gradient lower
             boundary conditions. NAPL absent, no degradation, qw = 0.035 cm/d	260
Figure 15.6  Average total soil concentration as a function of time and water saturation
             for first-type, time-dependent (TD) and zero-gradient (ZG) lower boundary
             conditions.  NAPL absent, no degradation, qw = 0.035 cm/d	261
Figure 15.7  Total soil concentration as a function of time (0.5 - 10.0 years) and depth
             for first-type, time-dependent (TD) and zero-gradient (ZG) lower boundary
             conditions.  Water saturation = 0.9, NAPL absent, no degradation, qw =
             0.035 cm/d	261
Figure 15.8  Total soil concentration as a function of time (15-50 years) and depth for
             first-type, time-dependent (TD) and zero-gradient (ZG) lower boundary
             conditions.  Water saturation = 0.9, NAPL absent, no degradation, qw =
             0.035 cm/d	262
Figure 15.9  Mass flux to the capillary fringe as a function of time and degradation
             half-life for first-type, time-dependent (TD) and zero-gradient (ZG) lower
             boundary conditions. Water saturation = 0.9, NAPL absent, qw = 0.035
             cm/d	263
Figure 15.10 Mass fraction of initial mass lost to degradation as a function of time and
             degradation half-life for first-type, time-dependent (TD) and zero-gradient
             (ZG) lower boundary conditions. Water saturation = 0.9, NAPL absent,
             qw = 0.035 cm/d	263
Figure 15.11 Mass flux to ground water as a function of time and NAPL saturation for
             first-type, time-dependent (TD) and zero-gradient (ZG) lower boundary
             conditions.  Water saturation = 0.7, no degradation, qw = 0.035 cm/d	265
Figure 15.12 Mass fraction of initial mass lost to ground water as a function of time and
             NAPL saturation for first-type, time-dependent (TD) and zero-gradient (ZG)
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                                  LIST OF FIGURES
             lower boundary conditions. Water saturation = 0.7, no degradation, qw =
             0.035 cm/d	265
Figure 15.13 Average total soil concentration as a function of time and NAPL saturation
             for first-type, time-dependent (TD) and zero-gradient (ZG) lower boundary
             conditions. Water saturation = 0.7, no degradation, qw = 0.035 cm/d	266
Figure 15.14 Mass flux to ground water as a function of time and water saturation
             (0.7 - 0.9) for first-type, time-dependent (TD) and zero-gradient (ZG) lower
             boundary conditions. NAPL absent, no degradation, qw = 0.035 cm/d	266
Figure 15.15 Average total soil concentration as a function of time for the low initial soil
             concentration profile for first-type, time-dependent (TD) and zero-gradient
             (ZG) lower boundary conditions.  NAPL absent, no degradation, qw =
             0.035 cm/d	267
Figure 15.16 Total soil concentration as a function of time (0.5-10 years) and depth for
             low initial soil concentration profile and first-type, time-dependent (TD)
             and zero-gradient (ZG) lower boundary conditions. Water saturation = 0.9,
             NAPL absent, no degradation, qw = 0.035 cm/d	268
Figure 15.17 Total soil concentration as a function of time (15-50 years) and depth for
             low initial soil concentration profile and first-type, time-dependent (TD)
             and zero-gradient (ZG) lower boundary conditions.Water saturation = 0.9,
             NAPL absent, no degradation, qw=0.035 cm/d	268
Figure 15.18 Soil-water concentration as a function of time (15-50 years) and depth for
             low initial soil concentration profile for first-type, time-dependent (TD)
             and zero-gradient (ZG) lower boundary conditions. Water saturation = 0.9,
             NAPL absent, no degradation, qw = 0.035 cm/d	269
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                                  LIST OF TABLES
Table 3.1     Summary of capillary pressure - relative permeability models (From
             Duryetal., 1999)	23
Table 5.1     Test results USCG Station, Traverse City, MI Table 5.1	66
Table 5.2     Gas permeability test summary for USCG Station, Traverse City, MI	66
Table 6.1.a   Summary of well and vapor probe depths	76
Table 6.1.b   Data for gas permeability estimation	77
Table 6.2.a   Summary of gas permeability estimation for domain open to atmosphere
             (high leakance)	79
Table 6.2.b   Summary of gas permeability estimation for semi-confined domain	79
Table 7.1     Ten best parameter estimation fits for test 1 inMWE-02-03	100
Table 7.2     Ten best parameter estimation fits for test 2 in MWE-02-03	101
Table 7.3     Ten best parameter estimation fits for test 3 in MWE-02-03	102
Table 7.4     Ten best parameter estimation fits for test 4 in MWE-02-03	103
Table 9.1     Summary of column and sand tank studies where rate-limited vapor
             transport has been evaluated	122
Table 9.2     Correlation of solids-water mass transfer coefficients with soil-water
             partition coefficients	141
Table 10.1    Input parameters for estimation of pore-gas velocity	151
Table 10.2    Radii of Influence (ft) as a function of observed vacuum	152
Table 10.3    Well spacing (ft)	152
Table 11.1    Soil Textural Analysis from OB-2	167
Table 12.1    Maximum concentration and associated IPs of VOCs detected during pilot
             scale testing at the Picillo Farm Site	180
Table 12.2    Summary of tracers used for vadose zone partitioning tracer studies	197
Table 13.1    Input for diffusion modeling	208
Table 14.1    Summary of zero- and first-order oxygen depletion rates at Elizabeth City
             USCG Base	220
Table 15.1    Summary of VFLUX Model Capabilities	238
Table 15.2    Input for modeling	256
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ACKNOWLEDGMENTS
       The authors express their appreciation for helpful comments received from the following
reviewers of this document:

Mr. Dave Becker
Hazardous, Toxic, and Radioactive Waste
  Center of Expertise
U.S. Army Corps of Engineers
Omaha, Nebraska

Dr. Mark Brusseau
Soil, Water, and Environmental Science Department and
Hydrology and Water Resources Department
University of Arizona
Tucson, Arizona

Dr. Paul C. Johnson
Department of Civil and Environmental Engineering
College of Engineering and Applied Sciences
Arizona State University
Tempe, Arizona

Dr. Dave Kreamer
Water Resources Management Graduate Program
University of Nevada, Las Vegas
Las Vegas, Nevada

Ms. Michelle  Simon
U.S. Environmental Protection Agency
National Risk Management Research Laboratory
Cincinnati,  Ohio

The authors also wish to express appreciation to:

Ms. Anna Krasko
U.S. Environmental Protection Agency
New England Region
Boston, Massachusetts

for assistance  in developing a number of portions of this document and for help in the field.
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EXECUTIVE SUMMARY

       Soil venting, soil vacuum extraction (SVE), and bioventing are terms commonly used to
describe in-situ technologies in which gas flow is induced in subsurface unconsolidated or
consolidated unsaturated media for the purpose of volatilizing or biodegrading organic
chemicals.  Gas (typically air) is injected and/or extracted from one or more wells causing a
pressure differential and subsequent advective gas flow.  Removal of volatile organic compounds
(VOCs) is achieved by non-aqueous phase liquid (NAPL) evaporation, NAPL dissolution,
desorption, air-water partitioning, biodegradation, and abiotic degradation. Removal of semi-
volatile organic compounds is achieved partly through abiotic processes but primarily through
biodegradation due to the effective delivery of oxygen.

       Soil vacuum extraction (SVE), defined as including only gas extraction, is considered a
presumptive remedy (a detailed technology screening process is not necessary for
implementation) in the U.S. Environmental Protection Agency's (U.S. EPA) Superfund program.
Bioventing, commonly defined as in-situ gas circulation designed to maximize contaminant mass
removal through biodegradation while minimizing air flow rates and operating costs especially
those associated with above-ground vapor treatment, is characterized by relatively "low" pore-
gas velocities or pore-volume  exchange rates. However, similar pore volume exchange rates or
pore-gas velocities however may be optimal for operation under conditions of rate-limited mass
transport.  This observation and the fact that during venting, simultaneous volatilization and
biodegradation occurs whether or not the latter process is intended suggests that the commonly
used definition of bioventing is non-unique.  In this document, an operational term, venting, is
used to define gas injection and/or extraction without specific reference to biodegradation or any
other subsurface processes. It is considered a general term encompassing SVE and bioventing.

       Soil venting has become the primary method used in the United States to remove VOCs
from unsaturated subsurface media. In 1997,  SVE was applied or planned to be applied at 27%
of Superfund sites. Since this statistic does not include gas injection or combined gas extraction
and injection, venting application at Superfund sites likely exceeds 30%. The popularity and
widespread use of venting is due to its simplicity of operation and proven ability to remove
contaminant mass inexpensively compared to other technologies.

       Despite the common use of venting in the Superfund program, there is little consistency
in approach to assessment of performance and closure. Assessment of the technology's
performance and eventual decisions on closure are based primarily on negotiations between
responsible parties and regulators.  In this process there  is widespread use and reliance on
empirical methods as opposed to an emphasis on understanding fundamental physical,  chemical,
and biological processes controlling mass removal during the venting operation. This results in
the technology not being used to its fullest potential, nor its limitations being well understood.

       The overall purpose of the work described in this document was to improve the "state of
the art" and "state of the science" of soil venting application. This purpose was accomplished by
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attainment of three specific objectives. The first objective was to develop an overall regulatory
approach to assess venting performance and closure including measures to ensure consistency in
ground-water and vadose zone remediation. The second objective was to provide comprehensive
and detailed literature reviews on gas flow and vapor transport. These reviews formed the basis
of recommendations and methods to improve venting design and monitoring.  The third objective
was to perform research to improve various aspects of venting application.  This document
should benefit regulators, practitioners, and research scientists needing detailed knowledge of
soil venting application.

      Research conducted to support soil venting application consisted of analysis of: (1)
linearization of the gas flow equation, (2) one-dimensional steady-state gas flow with slippage,
(3) two-dimensional steady-state gas flow and permeability estimation in a domain open to the
atmosphere, (4) two-dimensional steady-state gas flow and permeability estimation in a semi-
confined domain, (5) two-dimensional transient gas flow and permeability estimation, (6) radius
of influence versus critical pore-gas velocity based venting design, (7) modification of a gas
extraction well to minimize water-table upwelling, (8) rate-limited vapor transport with diffusion
modeling, (9) respiration testing, and (10) one-dimensional, analytical, vadose zone transport
modeling to assess mass flux to and from the capillary fringe.  A number of FORTRAN codes
were developed to achieve these research objectives.  Example input and output files and the
source code for each program are contained within the appendices.

Concise summaries of each section except section 1  (introduction) are presented below.

       Section 2

      It is clear that an environmentally protective, flexible, technically achievable, and
consistently applied approach for assessment of performance and closure of venting systems is
needed.  Any approach used to assess performance of a venting system should encourage good
site characterization, design, and monitoring practices since mass removal can be limited by poor
execution of any of these components. Also, any approach used to assess closure of a venting
system must link ground-water remediation to vadose zone remediation since the two are
interrelated.  A strategy is proposed in section 2 for assessment of venting performance and
closure based on regulatory evaluation of (1) site characterization, (2) design,  (3) performance
monitoring, and (4) mass flux to and from ground water.  These components form converging
lines of evidence regarding performance and closure. Evaluation is on a pass/fail basis since this
type of critique provides the greatest flexibility in decision making. Failure in evaluation of one
or more component(s) results in overall venting closure failure. Such a "weight of evidence"
approach increases the likelihood of correctly assessing the performance of a venting system and
its suitability for closure. Each component is interrelated and requires continuous evaluation
during the life of the project.  Approval of individual components occurs concurrently at the
perceived end  of the project.

      This closure approach encourages and will likely in some cases force good site
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characterization, design, and monitoring practices. If evaluation of all these factors individually
supports closure, then it is likely that closure is indeed appropriate. If one or more of the
components does not support closure, then it is likely that either closure is inappropriate or that a
conscious decision must be made to accept a limiting condition. For instance, in the presence of
significant rate-limited vapor transport, closure would not proceed if there was potential for
substantial continued mass flux to ground water.  In this case, venting may need to be applied
indefinitely at low pore-gas velocities or in a pulsed mode.  Regulators however could decide to
turn off the system if indefinite ground-water remediation or containment is anticipated or if the
remaining mass moving into the ground water will not impact a potential receptor. Regardless of
the situation, assessment of these components enables flexible, organized, and informed decision
making.

       The most common method of assessing whether to initiate, continue, or cease venting
application or any other in-situ vadose zone remediation technology, involves periodic collection
of soil samples for comparison with compound-specific, concentration-based standards which are
often based on the results of vadose zone modeling.  Remediation is considered complete when
vadose zone modeling indicates that these standards will be attained in the future. Both soil-
based and leachate-based standards are typically stringent because it is required that soils be
remediated to the extent that mass flux to ground water not result in aqueous concentrations
exceeding maximum contaminant levels (MCLs).  However, at  sites where it is unlikely that
ground-water remediation will result in attainment of MCLs, the need for soil or soil-water based
vadose zone remediation standards that assume attainment of MCLs becomes questionable.
Stringent remediation standards and other factors such as mass transport limitations often make
attainment of venting closure difficult at many sites.

       Soil remediation goals must reflect the realities of ground-water remediation.  This was
accomplished by partitioning subsurface remediation areas into  three  distinct zones for
performance evaluation purposes. The first zone is bounded on the upper end by the soil surface
and on the lower end by the  seasonally high water table. This zone consists of consistently
unsaturated media (above the region of water table fluctuation) where mass flux to and from
ground water occurs through infiltration and diffusion, and mass flux from ground water to the
vadose zone occurs through diffusion.

       The second zone consists of periodically de-saturated media due to water-table
fluctuation or dewatering. Often, it will consist of a highly contaminated "smear" zone
containing residual NAPL where venting is combined with dewatering to remove contaminant
mass from a localized region.  This zone is bounded  on the upper end by the seasonally high
water table and on the lower end by the maximum depth to ground water during dewatering.

       The third zone is bounded on the upper end by a ground-water level with or without
dewatering and on the lower end by the targeted depth of ground-water remediation. It represents
media that remains saturated during venting. Ground-water concentrations in this zone vary
temporally and determine compliance in the first two zones.  The second zone is in compliance
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when ground-water concentrations are less than or equal to ground-water concentrations in the
third zone.  This ensures that remediation of ground water within a smear zone will be attempted
to levels consistent with maximum levels of deeper contamination.  This ensures consistency in
ground-water remediation efforts and avoids reduction of ground-water concentrations in the
second zone to levels lower than what would occur through vertical recontamination from the
third zone.

       In the case of a light non-aqueous phase spill where ground-water concentrations within a
"smear" zone are at much higher levels than beneath the "smear" zone, low concentrations in the
third zone forces aggressive dewatering and venting application in the second zone.  In the case
of a dense non-aqueous phase spill where ground-water concentrations may be at high levels
deep within an aquifer, remediation within the dewatered region or second zone proceeds only to
the degree at which it is consistent with remediation in deeper regions or the third zone. The first
zone is in compliance with the second zone when mass flux to ground water is close to zero or
the direction of mass flux is primarily from ground water to the vadose zone (i.e. mass flux due
to vapor diffusion from contaminated ground water exceeds mass flux due to vapor diffusion and
infiltration from the vadose zone at the ground-water - vadose zone boundary). This ensures
aggressive venting application in the vadose zone when ground water has very low levels of
contamination, and less aggressive venting application when vapor diffusion will result in
recontamination of cleansed soils. Thus, venting performance in the first and second zones is
dynamically linked to the performance of ground-water remedial efforts in the third zone.
Substantial progress in remediating ground water translates to increasingly stringent soil venting
performance standards, while lack of progress  in remediating ground water translates into less
stringent soil remediation requirements.

       Section 3

       In section 3, two methods of linearization of the gas flow equation were evaluated.  The
governing equation for gas flow is by definition a non-linear partial differential equation because
the dependent variable, 4> or pressure squared,  is expressed in  a form other than unity. To solve
this equation analytically, linearization of the AJ> term is necessary. The most simplistic approach
is to let v'4> equal atmospheric pressure (Patm) or some other constant pressure and to let 4> = PPatm,
thereby removing consideration of compressibility. A second  approach is to let •/$ equal Patm but
still solve for 4>. A third approach is to replace v'4> with a prescribed time-varying function which
in some manner reflects the rate of change of initial pressure distribution. For analytical model
development and derivations in this dissertation, only the first two approaches are practical.

       Error incurred in pressure computation using the first and second approaches was
evaluated using Kidder's (1957) second-order  perturbation solution to the one-dimensional
transient gas flow equation. Maximum error occurred at low absolute pressures and were much
less when solving directly for 4> (~6 %) than when compressibility was neglected (-80%).  It is
often  assumed that compressibility is of minor importance during vacuum extraction because
near atmospheric pressure is commonly encountered in sandy  soils.  However, in recent years,
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vacuum extraction has been increasingly applied in lower permeability soils such as silts and
glacial till, where at least in the vicinity of gas extraction wells, low absolute pressure (less than
0.5 atmospheres) occurs.

       This analysis demonstrated that solving directly for 4> is adequate for solving gas flow
problems while the § =  PPatm approach incurs too much error and thus should not be used for
venting application.  In section 3, figures were generated illustrating error in pressure
computation as a function of linearization method.  These figures eliminate the need to solve
Kidder's (1957) lengthy perturbation solution when assessing error due to linearization of the gas
flow equation.

       Section 4

       In  section 4, an analytical solution was derived for one-dimensional, steady-state gas flow
incorporating the effect of gas slippage. This solution was  then used to assess the impact of gas
slippage on one-dimensional pressure and pore-gas velocity simulation as a function of flow rate
and gas permeability. Simulations during gas extraction demonstrated that neglecting gas
slippage results in underestimation of absolute pressure, whereas during air injection, neglecting
gas slippage results in overestimation of absolute pressure.  In both cases, pressure differential
from atmospheric pressure was overestimated when neglecting gas slippage. For both gas
extraction and injection, variation in pressure was greatest at the point of extraction  or injection
and at lower gas permeability values. However, the magnitude of error in pressure computation
for gas injection was far lower than for gas extraction because of higher absolute pressures
during gas injection.  Even at the lowest gas permeability, maximum error for pressure
computation during gas injection did not exceed 6%, whereas during gas extraction, error
exceeded 80%. Thus, if gas slippage is neglected, gas injection should result in a better estimate
of gas permeability than gas extraction.

       Computed pore-gas velocities during gas injection were much lower than during gas
extraction at the same flow rates because of increased gas density. Error in pore-gas velocity
computation neglecting gas slippage during gas injection was much lower than during gas
extraction (13% versus over 600%).  Similar to gas extraction, error during gas injection
increased with decreased permeability.  However, unlike gas extraction, error increased with
distance from the point of injection.

       These findings have important implications for laboratory- and field-scale  gas flow and
vapor transport studies.  For vapor transport column studies, it is apparent that lower and better
controlled pore-gas velocity profiles can be attained by gas injection compared to gas extraction.
At the field-scale, neglecting gas slippage may result in significant error in gas permeability
estimation and severe error in pore-gas velocity computation during gas extraction.  The impact
of gas slippage on field-scale gas permeability estimation and venting design requires further
investigation.
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       Section 5

       In section 5, data collected from the U.S. Coast Guard Station in Traverse City, Michigan
was used to demonstrate gas permeability estimation, pore-gas velocity calculation, and
streamline generation in a domain open to the atmosphere. An analytical solution was derived
for computation of streamlines to a finite-radius well.  For gas permeability estimation, random
guesses constrained with  decreasing intervals of radial and vertical permeability and analysis of
root mean square errors (RMSE) were used to ensure attainment of global versus  local minimum
values.  Confidence in permeability estimation was demonstrated by providing plots of observed
versus simulated pressure response. Plots of pore-gas velocity as a function of distance and flow
rate were provided to provide a preliminary estimate of soil venting well spacing. Through
discussion and example, a partial list of elements necessary for sound gas permeability estimation
was provided. These include: (1) placement of narrowly screened pressure monitoring points
close to a discretely screened gas extraction well to capture the vertical component of gas flow,
(2) testing at several flow rates to establish reproducibility, (3) analysis of error using  the RMSE
of observed versus simulated pressure response, and (4) plots of observed versus  simulated
pressure response to provide confidence is permeability estimation.

       Section 6

       In section 6, data collected from a Superfund Site on the Atlantic Coastal Plain was used
to demonstrate difficulties encountered when attempting to use radius of influence (ROI)-based
pressure data and the pseudo-steady-state radial flow equation for gas permeability estimation.
There are two primary concerns with use of radial flow equations and ROI pressure data for gas
permeability estimation.  First, one-dimensional transient and pseudo-steady-state radial flow
equations may result in higher estimates of radial gas permeability compared to solutions
allowing vertical leakage because for a given flow rate, confined flow or semi-confined flow
with low leakance (vertical gas permeability of a semi-confining layer divided by its thickness)
results in a higher pressure differential compared to unconfmed flow.  Strict confined  conditions
rarely if ever occur in the field.  Even at sites where a concrete or asphalt cap exists, small cracks
or flow through a gravel subbase can result in significant recharge from the atmosphere. Second,
ROI test data may not be  suitable for estimation of anisotropy, leakance, and subsequent
computation of axisymmetric cylindrical (r,z) or Cartesian (x, y, z) components of pore-gas
velocity because: (1) gas monitoring wells are often located so far away from gas  extraction wells
that resulting pressure differentials are too low to distinguish from atmospheric pressure, (2)
sensitivity of both radial and vertical gas permeability estimation decreases considerably with
increasing radial distance from a gas extraction well, and (3) vacuum extraction wells and
monitoring points are screened over large portions of the vadose zone providing vertically
integrated measurements  of pressure differential as opposed to point values required for vertical
permeability and leakance estimation using analytical axisymmetric cylindrical or three-
dimensional numerical models.

       Gas permeability testing at the Atlantic Coastal  Plain site indicated that use of the
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pseudo-steady-state radial flow equation provided a slight but consistent overestimation of radial
permeability compared to solutions which allowed recharge at the upper boundary. However,
little if any correlation between increased leakance and overestimation of radial permeability was
observed. A comparison of radial gas permeability estimates for unconfined and semi-confined
domains revealed that radial permeability estimation was relatively insensitive to leakance.
Estimation of vertical gas permeability expressed in the ratio of k/kz though was very sensitive to
leakance and illustrated the difficulty in resolving the correlated effect of leakance and anisotropy
with ROI pressure data. The assumption of high leakance or a domain open to the atmosphere
resulted in higher k/kz ratios compared to semi-confined conditions.

       Section 7

       In section 7, a sensitivity analysis and field application of an analytical solution for finite-
radius, transient testing gas permeability estimation was presented. Line-source/sink analytical
solutions are currently used exclusively for transient, axisymmetric, two-dimensional gas flow
analysis and parameter estimation. The implicit assumption in use of these solutions is that
borehole storage effects are insignificant except in the immediate vicinity of the wellbore.  The
radii away from a well and borehole storage volume over which borehole storage effects become
important though have not been investigated for gas flow. Thus, use of pressure monitoring
points very close to a gas extraction or injection well may introduce some unquantified  error in
gas permeability estimation. Borehole storage effects would be expected to be of greatest
significance in single-interval, transient testing.  Similar to slug testing in ground-water
hydrology, single-interval testing provides permeability estimation over a relatively small volume
of subsurface media thereby providing a mechanism for assessing physical heterogeneity or
spatial variability in permeability on a scale much smaller than full scale field-scale tests.

       Simulations comparing transient finite-radius and line source/sink solutions during gas
injection revealed that the line-source/sink solution resulted in a more rapid rise in pressure at the
wellbore at early time but then leveled off to a lower normalized pressure at late time compared
to the finite-radius solution.  The former effect was due to lack of a delayed response from
borehole storage.  The latter effect was likely due to the fact that for the line source/sink solution
simulated pressure response into the formation at a distance equivalent to the wellbore radius.
Lower pressure differential response at late time for the line-source/sink solution would likely
lead to gas permeability overestimation for a given flow rate. At the wellbore, pressure was
overestimated at early time for the line source/sink solution and underestimated over well shut-in
by as much as 10% . Thus, for single-interval, transient testing, borehole storage effects can
become a moderate source of error if not properly accounted for.

       Under conditions of these  simulations, error due to use of a line-source/sink solution
disappeared at a relatively short distance from the wellbore.  Thus, under typical testing
conditions when a gas extraction or injection well is not used as the pressure monitoring point,
transient gas permeability estimation using a line-source/sink solution would result in little error.
A comparison of line-source/sink versus finite-radius simulations at the wellbore at different
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vertical elevations revealed little error (less than 1.5%) associated with using a line-source/sink
solution. Thus, for simulations conducted here, line-source/sink solutions appear appropriate for
single well, multi-interval testing. However, conditions specified here are not reflective all
potential testing conditions.  Thus, when observation points are close to a gas extraction or
injection well, it may be useful to utilize a transient, finite-radius solution to assess the
importance of finite-radius and wellbore effects.

       A sensitivity analysis of the finite-radius solution revealed that in single-interval testing,
increased borehole storage volume results in an increased delayed response to steady-state
conditions. Increased gas-filled porosity results in a somewhat similar transient response to
increased borehole storage volume but the transient effects are much more prolonged. Single-
interval simulations also demonstrated that higher k/kz ratios prolong attainment of steady-state
conditions and result in a higher normalized pressure response compared to lower k/kz ratios.
The effect of leakance on transient pressure response at the wellbore however appeared to be
minor. Finally, single-interval simulations demonstrated that lower gas permeability
significantly prolongs attainment of steady-state conditions and results in a much higher pressure
differential.

       The usefulness of transient, single-interval gas permeability testing was then
demonstrated at the Picillo Farm Superfund Site in  Rhode Island. Testing revealed that single-
interval transient gas permeability testing is an improvement over single-interval steady-state gas
permeability testing in that the former provides an estimate of gas-filled porosity and does not
rely on attainment of steady-state conditions which  is sometimes questionable for low
permeability media (small pressure change in time after initial pressure or vacuum application).
The method though provided little confidence in leakance and k/kz estimation.  Thus, resolution
of anisotropy and leakance requires multiple interval testing.

       Section 8

       In section 8, a basic description of volatile organic compound (VOC) retention in porous
media is provided to assist the reader in understanding subsequent sections in mass transport.  An
understanding of VOC retention in porous media is critical to designing, monitoring, and closing
venting systems.  Each phase in unsaturated porous media can contribute to the retention and
mass transport of VOCs.  Phases present are solid minerals, organic matter, bulk water,
nonaqueous phase liquids (NAPL), and gas. Partitioning mechanisms related to these phases are
solid mineral - water, solid mineral - gas, organic matter - water, water - gas, NAPL - water, and
NAPL - gas.

       Section 9

       In section 9, a detailed and comprehensive literature review describing laboratory and
numerical experiments conducted to elucidate and quantify rate-limited mass transport process
during venting was provided. This review was used to support the development of a method of
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venting design based on specification of a critical pore-gas velocity in subsurface porous media.
A critical pore-gas velocity is defined as a pore-gas velocity which results in slight deviation
from equilibrium conditions.  Selection of a pore-gas velocity to support venting design requires
consideration of rate-limited gas-NAPL, gas-water, and solids-water mass exchange on a pore-
scale and rate-limited mobile-immobile gas exchange on a field-scale. During rate-limited mass
transfer, as pore-gas velocity increases, vapor concentration decreases. However, an increased
pore-gas velocity still results in an increased mass removal rate and hence shorter remediation
time because of an increased concentration gradient between equilibrium and nonequilibrium
vapor and soil-water concentrations. Thus, there is a trade-off between selection of a design
pore-gas velocity and remediation time.

       Five methods were described that could be used separately or concurrently to select a
critical pore-gas velocity for venting application. The approach most suitable for use by
practitioners appears to be use of algorithms and Damkohler numbers. However, a limitation of
using Damkohler numbers and algorithms to estimate critical pore-gas velocities is  that mass
transport processes are considered independently when in reality complex interactions between
these processes dictate overall mass removal. Also, there are additional complications to
consider in selection of critical pore-gas velocities. Virtually all research on rate-limited vapor
transport and development of algorithms for pore-gas velocity  determination originated from
controlled sand tank and laboratory column studies where field-scale heterogeneity  was not a
factor. Thus, optimal pore-gas velocities in the field are likely to be very site-specific and
somewhat smaller than indicated by Damkohler numbers, algorithms, and published studies.
Also, characteristic lengths of contamination used in Damkohler numbers and algorithms are not
constant but decrease in time due to NAPL and sorbed mass removal thereby reducing calculated
optimal pore velocities.  Finally, the cost of well installation and venting operation (e.g.,
electricity, vapor treatment) should be considered since achievement of a minimum target pore-
gas velocity at some sites may be prohibitively expensive.  Thus, it is apparent that  selection of a
critical pore-velocity is not a straightforward process but involves some degree of judgement and
iterative reasoning (gas flow modeling to see what pore-gas velocities can be achieved given
certain well spacing and flow rates). However, while there are limitations to the use of
algorithms and Damkohler numbers in selecting a design pore-gas velocity, these numbers
provide a starting point after which more sophisticated analysis can proceed if desired.

       Section 10

       In section 10, data was utilized from a Superfund site (described in section 6) to describe
limitations of ROI evaluation in more detail than had been done previously  and to demonstrate an
alternative method of design based on specification and attainment of a critical pore-gas velocity
in contaminated subsurface media.  Information was utilized from section 9 to provide the basis
for  selection of a critical or design pore-gas velocity (0.01 cm/s) for use at this site.  ROI
evaluation is by far the most common method used for venting design in the United States. In
practice, ROIs are determined by plotting vacuum as a function of logarithmically transformed
radial distance and applying linear regression to extrapolate to  a distance at which a specified
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vacuum would be observed. For SVE design, overlapping circles of ROIs for individual wells
are then drawn on a site map to indicate an "effective" remediation area.

       Results of single-well simulations demonstrated several important deficiencies in ROI
evaluation. First, cross-sectional vacuum profiles were very curvilinear especially under
conditions of high leakance values and k/kz ratios clearly illustrating that the ROI is an ill-
defined entity. Second, an ROI-based design resulted in subsurface pore-gas velocities lower
than 0.01 cm/s especially at low leakance values and high k/kz ratios. Third, attainment of a 0.01
cm/s pore-gas velocity could not be guaranteed by simply raising the magnitude of subsurface
vacuum (e.g., from 0.05 to 0.1 inches of water). Observed vacuum was a function of boundary
conditions (e.g., leakance), applied mass flow, anisotropy, permeability, and geometry of
screened intervals requiring at a minimum, two-dimensional simulation.

       Multi-well simulations illustrated additional deficiencies in ROI-based design practices.
It was evident that as leakance decreased, a velocity profile for a given total flow rate became
more uniform but vacuum required to maintain a pore-gas velocity of 0.01 cm/s increased
significantly - a relationship that would not be apparent from ROI testing.  In one simulation, a
pore-gas velocity of 0.01 cm/s could not be achieved despite vacuum levels exceeding 3.5 inches
of water whereas in other portions of the site a pore-gas velocity of 0.01 cm/s was achieved at
less than 1.5 inches of vacuum.  This reinforces previous observations with single-well
simulations that the magnitude of a vacuum level in subsurface media is not an indication of
effective gas flow contrary to assumptions made in ROI testing.  It was apparent that the
suitability of using a specified vacuum  levels of 0.05 or 0.1 inches of water, typical of ROI
testing, to ensure adequate gas circulation decreased with increasing k/kz ratios  and decreasing
leakance values.  ROI-based designs then are more likely to be appropriate as k/kz ratios
approach 1.0 or less and when significant leakance occurs.

       Finally, it was demonstrated that when attempting to achieve a critical design pore-gas
velocity, it is far  more efficient from an energy and vapor treatment perspective  to install
additional wells rather than pump existing wells at a higher flow rate. In one simulation, the total
flow rate from 6  wells would have had  to exceeded 1000 scfm to meet a pore-gas velocity of 0.01
cm/s throughout  contaminated soils. When the total number of wells was increased from 6 to 16
however, a pore velocity of 0.01 cm/s was achieved throughout the entire contaminated region at
only 400 scfm. In addition, the use of additional vapor extraction wells resulted in a much more
uniform pore-gas velocity throughout the contaminated area.

       Section 11

       In section 11,  the concept of combining gas injection and extraction (CIE) in a single well
to avoid water-table upwelling or to reduce ground-water recovery during vacuum extraction was
described. Field testing was conducted at Vance Air Force Base located in northwestern
Oklahoma. Typically, when vacuum is applied to a well screened in or near the water table,
water-level rise occurs within the well which often reduces or  completely obstructs gas flow.
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Gas injection on the other hand results in soil-water movement away from a screened interval.
Thus, it is logical that injection of pressurized gas or at least maintenance of atmospheric
pressure at the base of a vacuum extraction well would reduce water recovery.

       At Vance AFB, injection of air at the base of a gas extraction well reduced water recovery
by greater than 90% compared to dual vapor extraction. Vacuum measurements in soil during
combined air injection and extraction were similar to vacuum application alone for all but the
deepest probes. Vacuum measurement in shallow probes indicated that combined air injection
and extraction did not result in uncontrolled air flow and hence vapor migration toward the
surface.

       Section 12

       In this section, a number of recommendations were provided for VOC monitoring of
venting sites. Flow and total vapor concentration monitoring using a PID or FID or other
suitable instrument should be performed at the vapor treatment or blower inlet because it
provides  an estimate of VOC mass removal rate and total VOC mass removed as a function of
time for an entire venting system. Flow and vapor concentration monitoring using a PID or FID
or other suitable instrument should be performed at individual gas extraction wellheads to
determine the most heavily contaminated portions of a site requiring perhaps the most
monitoring, and to enable potential optimization of the vapor treatment system (e..g.,
prioritization of venting operation at highly contaminated areas, vapor stream richness or
leanness for catalytic oxidation unit overheating or supplemental fuel requirement). When
utilizing an FID or PID for VOC screening, the user must be aware of fundamental differences in
detector operation, ionization potential of target compounds, and gas matrix effects.  The high
ionization potential of many common VOCs will result in nondetection using a conventional
10.6 or even a 11.7 eVPID lamp.  The high halogen content of many common VOCs will result
in underestimation or nondetection of VOCs using an FID. Gas matrix effects such as humidity,
carbon dioxide, and alkane (especially methane) dramatically decrease PID response.  Gas matrix
effects combine in a nonlinear manner resulting in a decreased PID response far greater than
what would be expected from additive effects.  PIDs are also much more prone to nonlinearly in
response  compared to FIDs.  Gas matrix and linearity problems however can be overcome
through the use of serial dilution techniques.

       Observation of evaporation fronts in individual extraction well effluent gas and
evaporation/condensation fronts in well-placed vapor probe clusters could potentially provide
valuable information on the progress of venting remediation of NAPL contaminated soils. Under
equilibrium conditions, individual components of a NAPL propagate through soil in a sequence
of evaporation-condensation fronts at speeds proportional  to their vapor pressure. Evaporation-
condensation fronts have likely not been reported during field application because of nonuniform
gas flow (bypass flow as opposed to through-flow conditions, fully three-dimensional flow as
opposed to one-dimensional or radial flow, spatial variability of NAPL distribution, common
monitoring practices (determination of total VOC with a PID or FID as opposed to identification
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and quantification of individual compounds), and rate-limited transport.  Under equilibrium
conditions in the field, sharp evaporation-condensation fronts are not likely to be discernable but
the vapor concentration of lighter components should decrease monotonically with time
compared to heavier components which should increase for some time and then decrease.  In
contrast, under rate-limited transport conditions, extensive extensive effluent tailing should be
observed with vapor concentrations profiles remaining fairly similar. Thus, use of GC/MS
monitoring in gas extraction well and vapor probe clusters may reveal patterns of component
removal from NAPL, NAPL removal itself, and rate-limited vapor transport.

       There is no way to directly relate observation of an asymptote to an environmental benefit
and little potential to ensure consistency in decision making. Vapor concentration of an
asymptote at a highly contaminated site may be much higher than at a lesser contaminated site
leading perhaps to less stringent remediation at the highly contaminated site. Observation of an
effluent asymptote may be related to venting design (e.g., well screening and spacing) or gas flow
patterns separate or in addition to rate-limited vapor transport.  Since both design considerations
and rate-limited vapor transport can cause extensive effluent tailing and ineffectual removal of
accessible contaminant mass, it is critical to first rigorously assess venting design,
implementation, and associated monitoring prior to concluding that rate-limited vapor transport
significantly constrains mass removal. An asymptote can be observed in gas extraction wells
while much of the contaminant mass remains in soils. Observation of low asymptotic vapor
concentrations in effluent gas is not a sufficient condition to demonstrate progress in mass
removal from contaminated soils.  Different mass retention and rate coefficients can produce
very similar effluent curves suggesting that effluent and vapor probe concentration data can not
be used to estimate mass transport coefficients and VOC concentration in the aqueous and solids
phase. Thus, effluent and vapor probe concentration data is insufficient to  quantitatively assess
the progress of soil venting remediation.

       Some indicators of rate-limited transport in the field may be decreased vapor
concentration with increased pore-gas velocity (flow variation) and extensive vapor
concentration tailing accompanied by vapor rebound after cessation of operation (flow
interruption). In both NAPL and non-NAPL contaminated soils, flow variation provides a simple
yet powerful test of the validity of local equilibrium. For NAPL contaminated soils however,
results of flow interruption results can be confusing if the propagation of evaporation-
condensation fronts are not considered. One potential use of flow interruption or rebound data is
to use vapor concentrations and partition coefficients to estimate total soil concentration.
However, there is presently so much uncertainty and error inherent in estimation that calculations
can at best only provide qualitative insight into soil concentration levels.

       Section 13

       In section 13, data collected from Norton AFB in California demonstrated that diffusion
modeling can provide a simple yet powerful way of assessing rate-limited vapor transport in
discrete lenses of low permeability. Vacuum extraction had been applied at the site for two
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years. VOCs in gas extraction wells and vapor probes had approached low asymptotic levels.
Lithologic examination of cores and sample analysis though revealed much higher levels of
VOCs remaining in lenses of fine-grained material compared to surrounding coarser-grained
material. This observation indicated that mass removal from fine-grained material was likely
limited by diffusion.

       The contaminant profile within a fine-grained lense was used to establish the initial
condition for diffusion modeling. Effluent concentrations at the gas extraction well were used to
establish first-type, time-dependent boundary conditions.  At the moisture saturation at the time
of sampling (95%), simulations indicated that in excess of 80 years would be required to reach a
trichloroethylene (TCE) concentration of 10 ug/kg (remediation goal) because the high moisture
content resulted in a low effective diffusion coefficient. Vadose zone transport simulations
conducted by a remedial contractor however indicated that mass flux of remaining TCE in soils
to ground water would be insignificant.  These simulations and simulations conducted  for
diffusion modeling were used then to  support venting closure at this site.

       Section 14

       In section 14, the results of respiration testing and core sampling during venting
application at the U.S. Coast Guard Base in Elizabeth City, North Carolina were described.
Since measurement of biodegradation rates at a field scale is difficult and expensive, indicators
of aerobic microbial activity such as oxygen depletion and carbon dioxide production are
commonly used to demonstrate microbial activity. During venting application, it has become
common practice to use zero-order oxygen consumption rates to directly estimate hydrocarbon
degradation rates. Problems associated with this practice are discussed at length in section 14.

       Data from respiration tests were fitted to zero- and first-order kinetic  relationships. The
significance of distinguishing zero- and first-order kinetics for oxygen consumption is that
oxygen depletion is constant for the former and a function of oxygen concentration for the latter.
If the oxygen depletion rate is a function of oxygen concentration, then oxygen may become a
limiting factor below some concentration level above zero. In this site study, discernment of
zero- versus first-order  oxygen  depletion kinetics, was achieved at some but not all monitored
locations due to too few data points at critical times. In areas of high contamination, oxygen
depletion followed zero-order kinetics.  Soil sampling at the test site after 18 months of venting
revealed removal of BTEX compounds to very low concentrations but poor removal of higher
molecular weight and lower volatility compounds.

       Section 15

       In section 15, a one-dimensional, analytical, unsaturated zone VOC transport model
termed "VFLUX" was derived and described. One-dimensional analytical  modeling may be
appropriate under some conditions for regulatory decision making. However, even at this level
of modeling, a sensitivity analysis of how model input and selected boundary conditions affect
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model output and decision making is critical. VFLUX was used to assess the effect of water
saturation, NAPL saturation, degradation-half-life, and selection of boundary conditions at the
unsaturated zone - capillary fringe interface on model output and regulatory decision making in
regard to vadose zone remediation, specifically venting initiation or closure.

       Results of simulations revealed that selection of lower boundary conditions and highly
sensitive input parameters such as water saturation, NAPL saturation, and degradation half-life
significantly affected model output and decision making whether to initiate or cease venting
application, especially when soil concentrations were low.  The first-type boundary condition
allowed mass transfer across the unsaturated zone - capillary fringe interface by infiltration and
diffusion. The zero-gradient lower boundary condition only allowed mass transfer across the
interface by infiltration with soil-water concentrations independent of boundary conditions. For
highly contaminated soils, the first-type boundary conditions resulted in higher magnitude but
shorter duration pulses of mass to the capillary fringe and a higher cumulative mass fraction to
the capillary fringe compared to the zero-gradient lower boundary condition.

       For highly contaminated soils, regardless of which boundary condition is chosen, model
simulations clearly indicated a need to initiate or continue venting application.  For soils at low
contaminant levels however, use of the first-type boundary  condition resulted in the primary
direction of mass flux being from the capillary fringe to the unsaturated zone questioning the
need to initiate or continue venting.  The zero-gradient boundary condition on the other hand
resulted in sustained but low magnitude mass flux to the capillary fringe perhaps prompting a
decision to initiate or continue venting.  Thus, the two boundary conditions resulted in
contradictory decisions.

       In reality, neither the first-type nor zero-gradient appeared to adequately represent mass
transfer across the unsaturated zone - capillary fringe interface.  Use of the first-type boundary
condition appeared to overestimate mass flux to the capillary fringe when soils were highly
contaminated while use of the zero-gradient boundary condition did not incorporate the effect of
capillary fringe contamination  on unsaturated zone VOC transport. Use of a third-type boundary
condition does not remedy this situation since mass flux is a desired output of modeling and a
diffusive boundary layer would eliminate the contribution of infiltration to mass flux. One
solution to this problem would be to use numerical modeling and directly incorporate a capillary
fringe and variable saturation with depth.  In this way, a first-type, time-dependent boundary
condition could still be utilized but the effect of vapor diffusion across the interface would be
greatly dissipated and the linkage between vadose zone and ground-water remediation would
remain intact.  However, analytical screening models such as VFLUX should still be adequate for
decision making when elevated levels of contamination indicate potential for prolonged mass
flux to ground water using both types of boundary conditions.
                                            xl

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1      INTRODUCTION

       Soil venting, soil vacuum extraction (SVE), and bioventing are terms commonly used to
describe in-situ technologies in which gas flow is induced in subsurface unconsolidated or
consolidated unsaturated media for the purpose of volatilizing or biodegrading organic
chemicals.  Gas (typically air) is injected and/or extracted from one or more wells causing a
pressure differential and subsequent advective gas flow.  Removal of volatile organic compounds
(VOCs) is achieved by non-aqueous phase liquid (NAPL) evaporation, NAPL dissolution,
desorption, air-water partitioning, biodegradation, and abiotic degradation. Removal of semi-
volatile organic compounds is achieved partly through abiotic processes but primarily through
biodegradation due to the effective delivery of oxygen.  At Superfund and RCRA sites,
contaminated gas effluent from extraction wells is often treated above ground prior to discharge
to the atmosphere.

       Soil vacuum extraction (SVE), defined as including only gas extraction, is considered a
presumptive remedy (a detailed technology screening process is not necessary for
implementation) in the U.S. Environmental Protection Agency's (U.S. EPA) Superfund program.
Bioventing, commonly defined as including gas extraction and/or injection, describes in-situ gas
circulation designed to maximize contaminant mass removal through biodegradation as opposed
to volatilization. Bioventing systems are characterized by relatively "low" pore-gas velocities or
pore-volume exchange rates. However,  relatively low pore-gas velocities may be optimal for
operation under conditions of rate-limited mass transport. This observation and the fact that
during venting, simultaneous volatilization and biodegradation occurs whether or not the latter
process is intended suggests that the definition bioventing is non-unique.  In this document, an
operational term, venting, is used to define gas injection and/or extraction without specific
reference to biodegradation or any other subsurface processes. It is considered a general term
encompassing SVE and bioventing.

       Soil venting has become the primary method used in the United States to remove VOCs
from unsaturated subsurface media. In 1997,  SVE was applied or planned to be applied at 27%
of Superfund sites (Kovalick, 1999). Since this  statistic does not include gas injection or
combined gas extraction and injection, venting application at Superfund sites likely exceeds
30%.  Recently, the technology has been used to control vapor migration to structures and hence
reduce potential indoor air exposure. The popularity and widespread use of venting is due to its
simplicity of operation and proven ability to remove contaminant mass inexpensively compared
to other in-situ technologies.  The technology can be implemented with minimal site disturbance
and with standard readily available equipment (e.g., PVC pipe, blowers).

       Despite the common use of venting in the Superfund program, there is little consistency
in approach to assessment of performance and closure.  This problem exists for most if not all
subsurface remedial technologies.  A recent Inspector General's audit submitted to Congress
(USEPA, 1998a) pointed out that:

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       "EPA is not consistently using a scientifically-based, systematic planning process to take
       actions at Superfund hazardous waste sites... [Because of a lack of preferred systematic
       planning processes] "the Agency completed Superfund actions without known quality
       data for decision making and without sufficiently documenting important decision
       criteria...Until a scientifically-based, systematic planning process is implemented
       consistently nationwide, [EPA] can not be sure that decision makers have the information
       needed to make the best decisions about response actions at Superfund sites."

       Assessment of performance of venting systems varies widely among and within U.S.
EPA's regional offices and their corresponding states. Eventual decisions on closure are
commonly based on site-specific negotiations with responsible parties (e.g., industry, military
agencies) and their supporting consultants and attorneys. Assessment of performance and
closure becomes contingent upon the goals of those involved, their corresponding education and
experience, and the availability of information necessary for informed data analysis and decision
making. There is often a lack of understanding or even awareness of published research relevant
to characterizing, designing, and monitoring venting sites by those tasked with conducting such
studies. This is evidenced by the widespread use and reliance on empirical methods to design
and evaluate venting performance as opposed to placing an emphasis on understanding
fundamental physical, chemical, and biological processes controlling mass removal during the
venting operation.  This results in the technology not being utilized to its fullest potential, nor its
limitations being well understood. Examples of empiricism are radius of influence (ROI) testing
to support venting design and the use of effluent asymptotes to justify venting cessation. With
regard to availability of information necessary for informed data analysis and decision making, a
set of minimum site-specific data requirements is rarely outlined. However, even when state and
federal agencies are clear on data objectives, data collection typically becomes a contentious
process due to costs associated with site characterization and system monitoring. Thus, decisions
are commonly made based on available data "at hand" fueling speculation and debate as to what
the data actually means. These problems build throughout the life of a project, eventually
climaxing when attempts are made to shut the system down.

       Quite often closure negotiations consider the perceived effectiveness of a remediation
system more so than the actual  effectiveness of the system. In other words, a regulatory agency
is likely to grant closure or allow the termination of SVE operation if they think the responsible
party has done the best they could with the technology.  Thus, if the parties involved do not
understand the technology, then it might be misapplied or prematurely terminated.  Thus, poorly
designed and operated SVE systems may lead to quicker closures/system terminations than well-
designed systems.

       Assessment of soil venting performance and closure has become critically important in
U.S. EPA  regional  offices because of lack of guidance in evaluating venting systems and the  fact
that many  systems are operating without a definable or achievable endpoint. This issue is
particularly important in California which has far more venting sites  than any other state, many
of which are closed military bases that are being converted to civilian use. The issue of

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struggling with measures of "success" in subsurface remediation was recently highlighted in an
editorial by Conant and Rao (2000). They point out that

       "In subsurface remediation, policy and technical concerns are inextricably linked in an
       iterative deb ate... Whether the desired goals can be achieved is not necessarily apparent at
       the outset...Restoration to very low or background levels cannot be achieved at a large
       percentage of sites, at least not in a time frame or at costs expected by any of the
       stakeholders.  Gradual recognition of this fact has caused a return, whether in site-specific
       deliberations or the formulation of policy or guidance, to the broader discussion of
       appropriate, attainable remediation goals...[While] the public is understandably reluctant
       to abandon the ultimate cleanup objectives...parties responsible for site cleanup are
       equally reluctant to undertake open-ended actions when they can"t predict the likely
       results with any reasonable certainty, and when they have no assurance about what may
       be required of them in the future... [Our] experience has also taught us much about the
       complexity of the subsurface systems in which remediation technologies are applied,
       allowing for improvements in technology design and the monitoring strategies critical to
       measuring their success...It has also caused us to reconsider the ultimate uses and
       appropriateness of data collected."

       The purpose of this document is to address the problems described above by developing
recommendations and methods to support assessment of soil venting performance and closure.
This goal is to be accomplished by attainment of three objectives.  The first objective is to
develop an overall regulatory approach to assess performance and closure of venting systems
including measures to ensure consistency in ground-water and vadose zone remediation. The
second objective is to provide comprehensive and detailed literature reviews on gas flow and
vapor transport. These reviews form the basis of recommendations and methods to improve
venting design and monitoring. The third objective is to summarize research conducted to
improve various aspects of venting application. Literature reviews and technical
recommendations provided in this document are very detailed but not prescriptive. Thus, the
user of the document retains considerable flexibility.  A number of case studies were included the
document to enhance understanding of important principles.

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2.     PROPOSED APPROACH FOR ASSESSMENT OF PERFORMANCE AND
       CLOSURE OF VENTING SYSTEMS

2.1    Limitations in the Use of Vapor Concentration Asymptotes

       Cessation of venting application is often proposed based on attainment of an asymptote, a
concentration level during asymptotic conditions, or some subjectively determined mass removal
rate (e.g., pounds per day) from gas extraction wells.  There are several concerns with this
approach to assessment of performance and closure of venting systems.

       First, there is  no way to directly relate observation of an asymptote to environmental
benefit and little potential to ensure consistency in decision making.  Vapor concentration of an
asymptote at a highly contaminated site may be much higher than at a lesser contaminated site
leading perhaps to less stringent remediation at the highly contaminated site.

       Second, observation of an effluent asymptote may be related to venting design (e.g., well
spacing) or operating conditions (e.g. flow rate) separate or in addition to rate-limited vapor
transport.  Rathfelder et al. (1991) demonstrated that effluent tailing can be caused by gas flow
dynamics. Hypothetical simulations in a  homogeneous, two-dimensional axisymmetric flow
domain with NAPL under equilibrium conditions resulted in decreasing effluent concentrations
that could be construed as tailing. Direct recharge from the atmosphere and subsequent vertical
flow resulted in clean retreating evaporation fronts both vertically above and radially distant from
a gas extraction well. This caused the formation of a wedge-shaped contamination zone whereby
some continually decreasing fraction of gas flow intersected contaminated soil leading to the
appearance of an asymptote.  Shan et al. (1992) demonstrated that in a domain open to the
atmosphere, most gas flow to an extraction well originates from atmospheric recharge in the
immediate vicinity of the well. Thus, after some period of vacuum operation, vapor
concentrations from gas extraction wells become largely reflective of soil and contaminant
conditions in the immediate vicinity of the wells while higher concentrations more distant from
wells are largely diluted. Effluents in gas extraction wells then approach low concentrations and
an asymptote while significant and accessible mass may remain in soils between gas extraction
wells.  This situation  is exacerbated by placing venting wells too far apart - a condition typical of
ROI-based designs. A similar situation occurs in stratified soils where gas flow occurs primarily
in layers of greater gas permeability.  Analogous to the first situation, mass removal  occurs more
rapidly in soils receiving greater gas flow. Vapor concentrations in gas extraction wells become
largely reflective of soil and contaminant conditions in these layers while higher vapor
concentrations from lenses of lower permeability are largely diluted.  Rathfelder et al. (1991)
simulated effluent tailing when a gas extraction well was placed across two stratigraphic layers
having an order of magnitude difference in permeability.  In this case as before,  effluent tailing
and apparent rate-limited mass removal was caused by variation in pore-gas velocity, not by rate-
limited NAPL-gas exchange. Lowered local gas permeability in zones containing pure phase
residual would cause  the same effect.  Other examples of design factors potentially causing the
observation of an effluent asymptote include inadequate lowering of the water table, inadequate

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dewatering of perched water table zones, and gas extraction wells not placed in or near source
areas.

       Third, changing mole fractions of individual components in NAPL may also cause the
appearance of an asymptote in vapor effluent. Baehr et al. (1989) simulated mass removal of
gasoline in medium-grained sieved sand. Experimental and simulated mass flux profiles
matched well suggesting attainment of equilibrium conditions. The decrease in total mass flux
with time though exhibited apparent tailing due to selective removal of more volatile
hydrocarbons. This behavior is in contrast with column and sand tank studies using single
component residually contaminated soils which show no reduction in mass flux until residual
NAPL has evaporated.  In a field setting, it would be easy to mistake tailing due for selective
volatilization of a multi-component NAPL for rate-limited behavior.

       Fourth, several groups of investigators (Fisher et al., 1996; Ng and Mei, 1996; and
Armstrong et al., 1994) have clearly demonstrated in laboratory column, sand tank, and
numerical experiments that observation of low asymptotic vapor concentrations in effluent gas is
a necessary but not sufficient condition to demonstrate progress in mass removal from
contaminated soils. An asymptote due to rate-limited mass exchange can be observed in gas
extraction wells while much of the contaminant mass remains in soils. Ng and Mei (1996)
simulated the time variation in effluent vapor and aqueous soil aggregate concentration.  Effluent
vapor concentration dropped rapidly with time to the point of having indistinguishable values,
however the decay of the aqueous phase was significantly slower. Since sorptive equilibrium
exchange was assumed within the aggregates, the aqueous phase was indicative of mass  removal
within aggregates.  Thus, effluent vapor phase monitoring provided little insight on the rate of
concentration reduction in soil. Armstrong et al.  (1994) developed a numerical one-dimensional
model which incorporated first-order gas-water and solids-water rate-limited mass exchange to
simulate vapor effluent data obtained by McClellan and Gillham (1992). Virtually identical fits
were obtained to effluent curves for both continuous and pulsed schemes using two parameter
sets. Simulated remaining mass however differed sharply with the second parameter set resulting
in far greater mass retention. Their sensitivity analysis and calibration of data showed that
different mass retention and rate coefficients can  produce very similar effluent curves suggesting
that effluent and vapor probe concentration data can not be used to estimate mass transport
coefficients and VOC concentration in the aqueous and solids phase.  They concluded that
effluent concentration data is insufficient to quantitatively assess the progress of soil venting
remediation.  Thus, it appears that effluent and vapor probe concentration data can only be used
in a qualitative sense to assess performance.

2.2    Limitations of Rebound Testing

       Another common method of assessing the performance and closure of venting systems is
flow interruption or rebound testing. Presumably, minimal rebound or lack  of rebound after
some period of system cessation indicates remediation success and subsequent site closure.
Alternatively, extensive effluent tailing and rebound after system cessation indicates rate-limited

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mass exchange and little benefit for continued venting operation. Thus, rebound testing is
invariably used to justify closure. However, as previously discussed, significant mass removal
can occur over an extended period time even in the presence of substantial rate-limited mass
exchange.

       In NAPL contaminated soils, results of flow interruption testing can be confusing if
propagation of evaporation-condensation fronts are not considered. Hayden et al. (1994) provide
a good example of this. They introduced gasoline at a NAPL saturation value of 2% in two
sandy soils and monitored column effluent concentrations of benzene, m-xylene, p-xylene, and
naphthalene.  Flow interruption testing during the initial period of venting application (1 hour)
resulted in a decrease in benzene concentration and an increase in m-xylene, p-xylene and
toluene effluent concentration. Flow interruption testing at 3 hours resulted in a decrease in
benzene and toluene effluent concentration and an increase in m-xylene and p-xylene. Hayden et
al. (1994) explain that reconciliation of this behavior is possible only if one considers
evaporation-condensation fronts. During initial venting operation, high vapor pressure
compounds such as benzene would be removed first from the intake portion of the column and
increase in mole fraction and vapor concentration downstream as the evaporation front moves
through the column. Cessation of venting then would cause a decrease at the effluent end of the
soil column and an increase in the intake end of the column as vapor diffusion becomes the
dominant transport mechanism. Upon re-start, effluent concentrations would actually drop. For
compounds like m-xylene and p-xylene, mole fractions and vapor concentrations would initially
increase at the inlet of the soil column.  Thus,  during flow interruption, vapor diffusion would
cause an increase in vapor concentration downstream resulting in an increase in vapor
concentration upon start-up. Thus, in Hayden et al.'s (1994) column experiments, increases in
m-xylene and p-xylene upon start-up was not likely the result of rate-limited mass transport, but
propagation of evaporation and condensation fronts.  In a field setting, flow interruption testing
would provide evidence of rate-limited transport for a particular compound only after its
evaporation front exited the soil column. Otherwise, flow interruption testing could provide
confounding results.

2.3    Estimation of Soil Concentration from Soil-Gas Concentration

       Another common use of flow interruption data is to use vapor concentrations and
partition coefficients to estimate total soil concentration.  This would provide an indication of
soil concentration reduction, or through extrapolation, mass reduction over time for specific
compounds. Knowledge of soil concentration reduction is helpful if site closure is based on
attainment of compound specific soil standards.  Monitoring of VOC concentrations in vapor
probes for this purpose would be preferable to vapor extraction wells because they draw soil gas
from a much smaller volume of soil and thus are less likely to be affected by decreased vapor
concentration due to large scale heterogeneity. Viewed superficially, the idea appears to have
merit because  soil sample collection is expensive, especially when drilling at extensive depths
and in soils containing large rocks (e.g., glacial till). Drilling can cause disruption of venting
operation and manufacturing processes at active operating facilities. Also, drilling often results

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in the generation of a substantial amount of cuttings which must be disposed of as hazardous
waste. It can also be argued that soil sample collection is by nature "hit" or "miss" because
sample volume is small compared to the volume of soil affected by soil-gas collection.  Thus, if
contaminant mass remains in soil, there is a higher probability of detection with soil-gas
sampling than soil-solids sampling. Finally, soil sample collection is obviously not appropriate
for venting operation in fractured rock or other consolidated material.

       Johnson and Kreamer (1994) showed that soil vapor concentrations were able to be
correlated with soil concentrations in a sand tank study.  However, other investigators (Stephens,
1995; Hahne and Thomsen, 1991; Kerfoot, 1991) have reported poor correlation between soil-
gas and matrix concentration.  Stephens (1995) reported poor correlation between
trichloroethylene (TCE)  concentrations in soil gas and soil matrix samples collected during
drilling at approximately the same depth.  He states that the more frequent detection of TCE in
soil-gas compared to the soil matrix was due at least partially to much lower detection limits
achieved for soil gas compared to the soil  matrix.  This statement is sensible when one considers
that the vapor detection limit for TCE using EPA Method TO-14 is > 0.1 ppbv which is
equivalent to 4.28 x 10"04 ug/kg using equilibrium partition coefficients and the assumption of an
organic carbon content of 0.1%, porosity of 0.4, and a volumetric water content of 0.2.  This
equivalent soil concentration is far below the EPA  Superfund Contract Laboratory detection limit
of 0.05 ug/kg. Stephens (1995) states that another factor likely causing poor correlation between
trichloroethylene (TCE)  concentrations in soil gas and soil matrix samples is the fact that the soil
matrix is far more heterogeneous with respect to contaminant distribution compared to soil-gas.
Another likely reason for poor correlation between soil gas and soil matrix samples is that
significant VOC loss can occur during soil sample collection and storage resulting in reported
VOC concentrations lower than expected (Siegrist and Jenssen, 1990).

       Significant uncertainty in  estimation of parameters making up a soil-gas partition
coefficient and difficulty in measuring a steady-state vapor concentration in the field makes
estimation of total soil concentration from vapor measurements challenging. Under equilibrium
conditions, the relationship between a nonionic organic compound concentration in soil and soil
gas is a function of a compound's chemical properties (Henry's Law Constant, organic-carbon-
water partition coefficient, and NAPL-water or NAPL-air partition coefficient), fluid distribution
(gas, water, and NAPL fluid saturations), and soil properties (  soil  organic carbon content,
porosity,  and particle density).  If separate phase NAPL is available from a site, batch laboratory
studies can used to directly determine NAPL-air or NAPL-water partition coefficients for
compounds of concern.  If free NAPL is not available, NAPL-water or NAPL-air partition
coefficients can be estimated from a compound of interest's vapor pressure and molecular weight
and mole fractions, densities, and molecular weights of individual  compounds in the NAPL. At
many sites though, there are so many compounds in NAPL contaminated soil that accurate mole
fraction identification and quantification is impossible. An average NAPL density and molecular
weight which can be used in place of individual NAPL constituent information if the mole
fractions  of dominant chemical classes in the NAPL can be quantified. However, equations used
to estimate NAPL-water or NAPL-air partition coefficients reveal that these partition coefficients

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will not be constant but change as a function of changing mole fractions of various compounds.

       Since a compound's organic-carbon-water partition coefficient, Henry's Law Constant,
vapor pressure, molecular weight can be found in the literature or estimated for many
compounds, a soil-gas partition coefficient has 7 primary unknown parameters - gas, water, and
NAPL saturation, particle density, porosity, average NAPL density, and average molecular
weight. The parameters describing NAPL are of greatest interest because analysis of mass
fractions reveals that the majority of contaminant mass is partitioned in the NAPL phase even at
very low NAPL saturation (e.g., 0.5%).  Error in estimation of NAPL saturation would
dramatically affect estimates of soil concentration and contaminant mass.  There is presently no
standard method of estimating NAPL saturation in-situ in soil but gas-phase partitioning tracer
testing shows promise. In-situ water saturation would need to be estimated with the use of a
neutron probe or other downhole device. Gas saturation could be estimated from knowledge of
porosity and volumetric water content. Each  estimation procedure involves error, which from an
analysis of variance, propagates non-linearly.  Finally, there is no way to assess the accuracy of
estimation of soil-concentration from soil-gas measurements because the volume of soil being
affected by advective gas flow to a probe is likely far greater that the volume of soil typically
sampled for laboratory analysis. Collection of soil samples for soil-gas estimation comparison
also necessitates consideration of spatial variability which  can be significant even on a small
scale.

       Estimation of soil concentration also requires measurement of steady-state soil-gas
concentration. In a well designed venting system, mass removal will  eventually limited by
combined liquid and vapor diffusion from less permeable soil regions not receiving direct gas
flow. In a poorly designed system, mass removal will be limited by these factors plus  inadequate
gas flow in regions still amenable to significant advective phase mass transport.  In the former
case, at the cessation of venting, vapors diffuse slowly from immobile regions into mobile
domains where advective gas flow occurs. When a soil's moisture, organic carbon, or NAPL
content is high, time for attainment of equilibrium conditions can be excessive. Thus, when
collecting vapor samples for total soil concentration or mass estimation purposes, a correction
factor is necessary for estimation of steady-state vapor concentration. Assuming that
concentration is caused by vapor diffusion, estimation of steady-state vapor concentration will
involve the use of some type of a diffusion-based model. The diffusion coefficient will
incorporate the 7 unknown parameters of the  soil-gas partition coefficient plus three additional
unknown parameters - tortuosity in the gas and liquid phases and the diffusion path length.
Tortuosity values could be estimated from fluid saturation  and porosity. It is assumed that
literature values  can be found for molecular gas and water  diffusion coefficients.  Since there are
likely to be numerous localized sources of vapors in contaminated soils, including contaminated
ground water, the diffusion path length will represent some averaged  distance.  In practice, it
would be necessary to fit a diffusion-based equation to vapor concentration versus time data
during vapor rebound to estimate a lumped parameter incorporating the diffusion path length.
This will of require the collection of several gas samples over time at each vapor sampling point
followed by gas chromatography analysis. Another practical problem is devising an appropriate

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methodology of vapor sampling.  The collection of vapor samples may cause local dis-
equilibrium, thus later vapor sampling results would be biased by earlier sampling efforts.

       It is apparent that analysis of rebound testing and estimation of soil concentration from
soil-gas measurement is a difficult process. It requires a substantial amount of work, expense
(GC analysis of vapor), and thorough  knowledge of subsurface conditions.  Even under the best
of conditions, appreciable estimation error can be expected. While drilling is expensive and
disruptive, it still offers the best opportunity to evaluate vertical subsurface contamination
profiles.  While rebound or flow interruption testing could be a fruitful area of research, its use at
present for quantitative assessment of venting performance and eventual closure does not appear
defensible. Soil-gas monitoring during venting operation and periods of shut-down though might
provide valuable qualitative insight into progress towards remediation and aid in timing of soil
sample collection.

2.4     Vadose Zone Solute Transport Modeling

       Probably the most common method of assessing whether to initiate, continue, or cease
venting application or any other in-situ vadose zone remediation technology, involves periodic
collection of soil samples for comparison with compound-specific, concentration-based standards
which  are often based on the results of vadose zone modeling.  A variation of this approach, is
attainment of "drinkable leachate" or soil-water concentrations adjusted for a "dilution-
attenuation factor" at the vadose zone - ground-water interface. Remediation is considered
complete when vadose zone solute transport modeling indicates that these standards will be
attained in the future.  Both soil-based and leachate-based standards are typically stringent
because it is required that soils be remediated to the extent that mass flux to ground water not
result in aqueous concentrations exceeding maximum contaminant levels (MCLs). The National
Research Council (1994), however states that:

        "regulatory  agencies should recognize that ground water restoration to health-based goals
       is impracticable with existing technologies at a large number of sites [and that]
       ...improvements are needed...to manage the large number of sites where technical
       limitations may present problems."

At sites where it is unlikely that ground-water remediation will result in attainment of MCLs, the
need for soil or soil-water based vadose zone remediation standards that assume attainment of
MCLs  becomes questionable. Stringent remediation  standards and other factors such as mass
transport limitations often make attainment of venting closure difficult at many  sites.

       Mathematical models that simulate soil-water movement to ground water are often used
to determine  soil concentration-based remediation standards. These models vary in complexity
from simple and conservative algebraic water-balance equations that neglect degradation and
volatilization to more process descriptive finite-difference and finite-element numerical codes.
Lack of data to support spatial discretization of soil properties (e.g., hydraulic conductivity,

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porosity, bulk density, moisture content, total organic carbon content) and contaminant
distribution, however, commonly limits the use of more complex models.

       Considering that venting is often applied to remove contaminant mass from variably
saturated media for ground-water protection, it would seem that only an analysis of mass flux to
and from ground water is necessary for assessment of soil venting operation and closure.
Unfortunately, vadose zone subsurface fate and transport processes are  not understood well
enough to permit the  sole use of deterministic models for decision making.  Though a basic
understanding of the physics, chemistry, and biology of many fundamental subsurface processes
exists, spatial and temporal distribution of contaminants in the subsurface are the end result of a
vast number of processes whose complex interactions can not be effectively simulated.  Thus,
mass-flux modeling should only be one of several tools used to evaluate operation and closure of
venting sites.

2.5    Proposed Approach for Assessment of Performance and Closure

       It is clear that an environmentally protective, flexible, technically  achievable, and
consistently applied approach for assessment of performance and closure  of venting systems is
needed. Any approach used to assess performance of a venting system  should encourage good
site characterization,  design, and monitoring practices since mass removal can be limited by poor
execution of any of these components.  Also, any approach used  to assess closure of a venting
system must link ground-water remediation to vadose zone remediation since the two are
interrelated.

       The following is a description of a general strategy proposed for assessment of
performance and closure of soil venting systems. This discussion is followed by a proposed
scheme for linking ground-water and vadose zone remediation. There is no regulatory
requirement that EPA or State regulators adhere to this strategy or recommendations
given in this report.  The strategy is based on four components considered integral to successful
venting application: (1) site characterization, (2) design, (3) performance monitoring, and (4)
mass flux to and from ground water. These four components form converging lines of reasoning
or a preponderance of evidence regarding performance and closure.  Evaluation is on a pass/fail
basis since this type of critique provides the greatest flexibility in decision making.  Failure in
evaluation of one or more component(s) results in overall venting closure failure. Such a
"weight of evidence" approach greatly increases the likelihood of correctly assessing the
performance of a venting system and its suitability for closure. The use of converging lines of
evidence has become popular in evaluation of subsurface remedial technologies, especially
bioremediation, as subsurface fate and transport processes are too complex to allow definitive
assessment of progress. U.S. EPA's "Technical Protocol for Evaluating Natural Attenuation of
Chlorinated Solvents in Ground Water" (USEPA, 1998b) most recently adopted this convergent
lines of evidence approach.

       Since each component is interrelated and requires continuous evaluation during the life of
                                           10

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the project, regulatory approval of individual components must occur concurrently and not until
the perceived end of the project. Premature approval of a closure component would limit later
corrective action. For example, performance monitoring may reveal deficiencies in design.
Premature approval of a design then could limit implementation of measures needed for
successful remediation. Notice of deficiencies in evaluation of individual components however
could occur at any time since in many cases it will be apparent early on that deficiencies exist.
For instance, during the site characterization phase, U.S. EPA and State regulators may be
unconvinced that the vertical and lateral extent of contamination is sufficiently delineated for
venting application. In this case, responsible parties would be notified up-front that venting
closure will not be attained until this situation is rectified.  Thus, this closure approach
encourages and likely  in some cases forces good site characterization, design, and monitoring
practices.

       If evaluation of all these factors individually supports closure, then it is likely that closure
is indeed appropriate.  If one or more of the components does not support closure, then it is likely
that either closure is inappropriate or that a conscious decision must be made to accept a limiting
condition. For instance, in the presence of significant rate-limited vapor transport, closure would
not proceed if there was potential for substantial continued mass flux to ground water. In this
case, venting may need to be applied indefinitely at low pore-gas velocities or in a pulsed mode.
Regulators however could decide to turn off the system if indefinite ground-water remediation or
containment is anticipated or if the remaining mass moving into the ground water will not impact
a potential receptor. Regardless of the situation, assessment of these components  enables
flexible, organized, and informed decision making.

2.6    Mass Flux Assessment

       Evaluation of mass flux to and  from ground water is vital in integrating venting
application with progress in ground-water remediation. Soil remediation goals must reflect the
realities of ground-water remediation.  As illustrated in Figure 2.1, subsurface remediation areas
are partitioned into three distinct zones for performance evaluation purposes. Zone 1 is bounded
on the upper end by the soil surface and on the lower end by the seasonally high water table.
Zone  1 consists of consistently unsaturated media (above the region  of water table fluctuation)
where mass flux to ground water occurs through a combination of infiltration and diffusion or by
diffusion alone.  Mass flux from ground water to the vadose zone occurs through diffusion.
Hughes et al. (1992) demonstrated that ground-water contamination  can occur solely by vapor
diffusion from residually  contaminated soil by diffusion across the capillary fringe or by
fluctuation in the water table which traps vapors in the zone of saturation.

       Zone 2 consists of periodically  de-saturated media due to water-table fluctuation or
dewatering. Often, it will consist of a highly contaminated "smear" zone containing residual
NAPL where venting is combined with dewatering to remove contaminant mass from a localized
region. Zone 2 is bounded on the upper end by the seasonally high water table and on the lower
end by the maximum depth to ground water during dewatering.  The base of zone 2 will be
                                            11

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variable depending on targeted dewatering depths throughout a site area. The maximum depth of
zone 2 will be dictated by the vertical profile of contamination and feasibility of dewatering to a
depth of interest. If a smear zone exists in highly permeable media where dewatering is
Figure 2.1 Schematic of remediation zones for mass flux assessment

technically infeasible or requires large pumping rates, zone 2 can be thought of as the region
where another source control technology such as sparging is used. If dewatering is not to be used
in conjunction with venting, then zone 1 is directly linked to zone 3.

       Zone 3 is bounded on the upper end by a ground-water level with or without dewatering
and on the lower end by the targeted depth of ground-water remediation. It represents media that
remains saturated during venting. Ground-water concentrations in zone 3 vary temporally and
determine compliance in zones 1  and  2. Zone 2 is in compliance when ground-water
concentrations are less than or equal to ground-water concentrations in zone 3. This ensures that
remediation of ground water within a  smear zone will be attempted to levels consistent with
maximum levels of deeper contamination. This ensures consistency in ground-water remediation
efforts and avoids reduction of ground-water concentrations in zone 2 to levels lower than what
would occur through vertical recontamination from zone 3.  Implementation of this approach
then requires adequate vertical profiling of ground-water concentrations in zone 2 and 3.  This
can be achieved by installation of short-screened multi-well clusters, multi-port monitoring wells,
or diffusive samplers in fully screened wells.
                                           12

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       In the case of a light non-aqueous phase spill where ground-water concentrations within a
"smear" zone are at much higher levels than beneath the "smear" zone, low concentrations in
zone 3 forces aggressive dewatering and venting application in zone 2. In the case of a dense
non-aqueous phase spill where ground-water concentrations may be at high levels deep within an
aquifer, remediation within the dewatered region or zone 2 proceeds only to the degree at which
it is consistent with remediation in deeper regions or zone 3.  Zone 1 is in compliance with zone
2 when mass flux to ground water is close to zero or the direction of mass flux is primarily from
ground water to the vadose zone (i.e. mass flux due to vapor diffusion from contaminated ground
water exceeds mass flux due to vapor diffusion and infiltration from the vadose zone at the
ground-water - vadose zone boundary). This ensures aggressive venting application in the
vadose zone when ground water has very low levels of contamination, and less aggressive
venting application when vapor diffusion will result in recontamination of cleansed soils.  Thus,
venting performance for zones 1 and 2 is  dynamically linked to the performance of ground-water
remedial efforts in zone 3. Substantial progress in remediating ground water translates to
increasingly stringent soil venting performance standards, while lack of progress in remediating
ground water translates into less stringent soil remediation requirements.  Regardless of the
strategy chosen to close an venting site, there must be a link between ground-water and soils
remediation.  Remediation in the unsaturated zone should not proceed independently of ground-
water conditions. Unfortunately, this aspect of closure is missing at most sites.
                                           13

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3.     GAS FLOW IN POROUS MEDIA - FUNDAMENTAL PRINCIPLES

       Systematic investigation of gas flow in porous and consolidated media originated in the
petroleum industry to support characterization and development of natural gas reservoirs (Muskat
and Botset, 1931). Gas exchange between soils and the atmosphere has been of interest to soil
and agricultural scientists for decades. More recently, hydrologists, soil scientists, and
environmental engineers have been interested in gas flow in porous media to support design of
venting systems for remediation of contaminated soils containing VOCs or other compounds
capable or aerobic biodegradation (Johnson et al., 1990a,b; Baehr and Hult, 1991; Shan et al.,
1992; Baehr and Joss,  1995). Most recently, gas flow studies have been conducted to evaluate
potential gaseous movement at proposed high-level radioactive waste sites (Ahlers et al., 1995,
1999; Shan et al. 1999) and barometric pressure propagation in soils (Shan, 1995, Rojstaczer and
Tunks, 1995).  Analysis of gas flow and permeability estimation in subsurface media requires a
thorough knowledge of fundamental gas flow principles. The purpose of this section is to
provide a review of background information necessary to formulate and apply analytical and
numerical solutions to gas flow.

3.1    Compressibility

       Compressibility is one of two primary factors differentiating gas flow from liquid flow.
The second factor, gas slippage along pore walls, will be discussed later. Katz et al. (1959) were
among the first investigators to derive analytical solutions for gas flow by utilizing the ideal gas
law to express gas density. This approach however was found to be of limited use in the
petroleum industry because of deviation from the ideal gas law at high pressures typical of gas
reservoirs and  large pressure variation near production wells.

       At pressures  typical of soil venting operation (less than atmospheric pressure up to
perhaps 1.5 atm), deviation from the ideal gas law for gases of interest (primarily O2 and N2) will
be insignificant and the ideal gas law can be used to estimate gas density, p  [M L"3] by
where Mg is the molecular weight of a gas mixture [M mole"1] (atmospheric air = 28.8 g mole"1),
P is gas pressure [M L"1 T"2] (1 atm = 1013250 g cm"1 s"2), 91 is the universal gas constant [M L2
T"2 mol"1 K"1] (8.3143 x 107 g cm2mole"1 s"2 K"1), and T is temperature [K].  Equation (3.1) is
utilized to estimate gas density in virtually all analytical and numerical codes supporting venting
application.

3.2    Viscosity

       When a shearing stress is applied to a fluid, it deforms continuously. During laminar
flow, adjacent layers (lamina) of fluid molecules move at different velocities with changes

                                           14

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gradually occurring from lamina to lamina. Velocity increases from zero (for non-slip
conditions) at a planar stationary surface to a maximum value at the center of a tube or pore.
Each layer experiences frictional force called viscous drag from adjacent slower-moving lamina.
This effect is manifested as resistance to flow. According to Newton's Law for laminar flow,
resistance to gas flow is expressed by

 F           dv
— = T = -fi —                                                                  (3.2)
 A         g dy

where F = a force applied along a lamina of flowing fluid [M L T"2], A [L2] = the area of contact
along adjacent lamina, T = gas phase shearing stress along adjacent lamina [M L"1 T"2], |j.g = a
constant of proportionality called dynamic gas viscosity [M T"1 L"1] (|j.g for atmospheric gas at
20°C = 1.76 x 10"4 g cm"1 s"1), v = gas velocity in lamina [L T"1], and y = a distance from a fixed
surface [L].  Viscosity sometimes appears in fluid flow problems in the form
                                                                                  (3.3)
where ug is called kinematic viscosity [L2 T"1]. Fluids for which shearing stress (T) is linearly
related to the rate of shearing strain (dv/dy) are designated as Newtonian fluids.  All gases are
Newtonian fluids.

       Gas viscosity is largely independent of pressure up to 10 atmospheres (Bird, Stewart, and
Lightfoot, 1960).  For gases at low density, viscosity increases with increasing temperature
whereas viscosity decreases with increasing temperature in liquids. If gas molecules are
conceptualized as hard spheres without molecular interaction, gas viscosity can be estimated
from momentum transfer principles by (Bird, Stewart, and Lightfoot, 1960)
                                                                                  (   }
where mm = molecular mass [M], KC = Boltzmann constant (1.38066 x 10"23 J K"1) , and dm =
molecular diameter [L].  From equation (3.4), gas viscosity theoretically increases with
temperature to the !/2 power. At low density typical of atmospheric conditions however, Bird,
Stewart, and Lightfoot (1960) state that gas viscosity increases with temperature to the 0.6 to 1.0
power of temperature because of molecular interactions.

       Gas viscosity is also a function of component mixture.  Bird, Stewart, and Lightfoot
(1960) provide an expression to estimate the viscosity of a multi-component gas mixture
                                            15

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                                                                                  (3.5)
              Q
           j    *j
         J=l
where
                                                                                  (3.6)
                          ^	


and n = number of chemical species, and %; and %j, (j.; and |j.j, and M; and Mj are gas phase mole
fraction, viscosities, and molecular weights of species i andj, respectively. Bird, Stewart, and
Lightfoot (1960) state that equation (3.5) has been able to reproduce measured values of viscosity
of nonpolar gas mixtures within 2%. This equation however is not appropriate for polar (e.g.,
H2O, NH3) molecules because of angle dependent force fields existing between such molecules.
Analytical codes require the assumption of constant gas viscosity for simulation of gas flow and
estimation of gas permeability. However, during the course of venting, gas composition can
change from a mixture of methane, carbon dioxide, nitrogen, and hydrocarbon vapors to a
mixture more representative of atmospheric  air. Numerical vapor transport codes such as
MISER (Abriola et al., 1996) utilize equation (3.5) to account for variable gas composition.

3.3     Gas Potential Function

       Using a force balance acting on an ideal (non-viscous) compressible fluid within a control
volume, Bernoulli's Equation can be derived to express the components of potential (<&) [L] for
gas flow

      i    i
                                dP
                                      = constant                                   (3-7)
                         g
where the first, second, and third terms represent inertial, gravitational, and pressure components
in terms of energy per unit weight and Pref vref, and zref, are reference pressure, velocity, elevation
respectively.  During gas permeability estimation and flow analysis both the velocity and
elevation contributions to total potential are typically assumed to be negligible.  The assumption
of a negligible component of velocity potential may not be accurate in the immediate vicinity of
an extraction or injection well. The assumption of negligible elevation potential may not be
accurate for chlorinated contaminant laden  gas present at many hazardous waste sites under low
gradient conditions. Advective gas transport can be induced by density as well as pressure
gradients. Density-driven flow may be of concern because of the relatively high vapor pressures
and molecular weights of many common volatile contaminants. Density-driven flow will be
most significant in soils that have high gas  permeability in the vicinity of nonaqueous phase
liquid present either as residual or mobile phase (Falta et al., 1989; Sleep and Sykes, 1989; and


                                           16

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Mendoza and Frind,  1990a,b). In addition, factors which reduce gas-phase concentration, such as
partitioning to pore water and sorption, will reduce the significance of density-driven flow (Falta
etal. 1989).

       In real  fluids, mechanical energy loss due to friction occurs during viscous fluid
movement through soil pores, thus, potential does not remain constant.  Fluid flow occurs from
regions of higher to lower energy and is accompanied by an irreversible transformation of
mechanical to  thermal energy through the mechanism of fluid friction (Hubbert, 1940). Fluid
potential loss (<&f) can be expressed as
                                                                                   (3.8)
where dp is pore diameter [L], L [L] is path length, and ftj) is an expression a function of
viscosity. Thus, frictional fluid potential loss increases linearly with increasing shear stress
(viscosity) and distance of travel, increases linearly with decreasing pore diameter, and increases
with the square of pore-gas velocity. Since gravitational and inertial components of gas potential
are considered negligible in most gas flow models, the difference in gas potential between any
two points in space is simply the integral term in equation (3.7).

3.4    Gas Slippage

       The second factor differentiating gas flow from liquid flow is gas slippage along pore
walls.  In contrast to liquid flow, gas velocity at the pore wall can be significant. Klinkenberg
(1941) derived an expression for specific discharge (qx) [L T"1] incorporating gas slippage
                                                                                   p.9)
            p)dx

where kx = gas permeability at infinite gas pressure (high pressure where the effect of gas
slippage is negligible) [L2], P = mean pressure [M L"1 T"2], and b = a gas slip factor [M L"1 T"2]
(expressed in terms of pressure). A general formulation of equation (3.9)
        1
1, =	
(3.10)
where q; = Darcy specific discharge vector [L T"1], and k = gas permeability tensor [L2] is
commonly used in the literature to solve partial differential equations involving gas slippage
(e.g., Kaluarachchi, 1995).  Notice that in equation (3.10) mean pressure has been replaced by
pressure at a point.  In column studies though, the mean pressure (average of pressure at ends of
column) formulation is retained. When the gas slippage factor is set to zero, equation (3.10)

                                            17

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reverts to the more commonly expressed equation for specific discharge
Since values of b are always positive, equation (3.10) reveals that gas slippage results in specific
discharge greater than would be predicted from equation (3.11) and that specific discharge
increases with increasing values of the slip flow factor and decreasing pressure.  As pressure goes
to infinity, equation (3.10) reverts to equation (3.11). Another way of looking at equation (3.10)
is that gas slippage flow results in an enhancement in apparent or observed gas permeability by a
factor of (1+b/P).

       Klinkenberg (1941) derived the expression


      -^                                                                         (3.12)
relating the slip flow factor, b, to the mean free path of gas molecules (K) [L], pore diameter, and
a proportionality constant (c). From equation (3.12), the gas slip factor and hence gas slippage
should increase with decreased pore diameter.  Thus, the slip factor should be larger for less gas
permeable materials and smaller for more permeable material.  Laboratory studies by Reda
(1987) and Persoff and Hulen (1996) suggest that gas slippage can be important in fine-grained
media.  Baehr and Hult (1991) demonstrated through calculations that omission of gas slippage
can result in pressure errors greater than 10% for soils having intrinsic permeabilities of less than
10"9 cm2. Massmann (1989) related the relative importance of slip flow for low pressure systems.
He calculated that materials with a pore radii greater that 0.001 mm would exhibit minimal
effects of slip flow.

       The constant b would be expected to decrease with increasing water saturation, especially
in poorly sorted media, since only the largest pores would be available to conduct gas flow.
Stonestrom and Rubin (1989b) reported that the slip correction factor ( b IP ) decreased linearly
with increasing water content in experiments with Oakley sand. However in studies with
consolidated sandstones, Estes and Fulton (1956) reported fairly constant slip factors over a wide
range of water contents as did Detty (1992) with Berino loamy fine sand. Thus, experimentally,
the effect of water saturation on gas slippage is ambiguous.

       From equation (3.12), gas slippage should also increase with increased mean free path.
Mean free path is defined as the average distance traveled between consecutive collisions.
Conceptually, the mean free path of molecules in  a gas state at very high pressure should be
equivalent to the mean free path of molecules in a liquid state.  Bird, Stewart, and Lightfoot
(1960) provide an equation for estimating mean free path incorporating relevant variables
                                            18

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where n equals number of molecules and dm equals molecular diameter [L]. Equation (3.13)
indicates that mean free path should decrease with the square of molecular diameter and number
of gas molecules.  Thus, gas slippage should increase with decreasing molecular diameter and
gas density.

       The traditional method used to determine the Klinkenberg or gas slippage factor in
laboratory cores is to plot apparent gas permeability (dependent variable) versus the reciprocal of
average pressure from steady-state tests. Using linear regression, the high pressure gas
permeability (dependent variable intercept) and gas slip factor (slope divided by intrinsic
permeability) can  be calculated.  Once the gas slip factor is known for a soil type, pressure and
pore-gas velocity  simulations can be conducted. Wu  et al. (1998) however state that the
traditional method of estimating the gas slip factor may not be appropriate because the pressure
profile is not linear and therefore apparent permeability varies along the length of the column.
This observation relates back to whether to express pressure in equation as an average or point
pressure. The impact of gas slippage on one-dimensional analysis of gas flow will be assessed in
detail in section 4 of this document.

3.5     Visco-Inertial Effects

       Equation (3.10) or Darcy's Law is widely accepted as valid under conditions  of viscous
laminar flow. At  high pore-gas velocities however, visco-inertial effects or non-laminar flow
may become important. During non-laminar flow, higher pressure loss is observed for a given
mass flux than expected.  Departure from laminar flow is often indicated by the presence of high
Reynolds numbers (Re) [dimensionless] defined by
                                                                                 (3.14)
      *****
where qm equals specific mass flux [ML^T"1] and 8g = volumetric gas content [dimensionless].
Yu (1985) conducted column experiments to test the validity of Darcy's law for gas flow through
various-sized sands and showed that Darcy's Law was valid for Re < 6.

       One equation used to assess deviation from  laminar flow originally presented by
Forchheimer in 1901 is
  dx     k       *  *

where I is an inertial flow factor [L"1]. This appears to be Darcy's law without gas slippage with

                                           19

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an additional term added on the right hand side of the equation to represent kinetic energy loss
resulting from high velocity flow.  In experiments with Berino loamy sand, Detty (1992) found
that I increased two orders of magnitude as sample saturation increased.  He states that the
increased inertial flow corresponding to decreased gas permeability agrees with values reported
by Keelan (1989). Detty (1992) also reported that data analysis using the Rawlins and
Schellhardt (1936) equation indicated increased inertial flow at increased pressure gradients and
gas flux as sample saturation increased. Detty (1992) concluded though that plots using
Rawlins's and Schellhardt's (1936) and Forchheimer's equations provide a qualitative rather
quantitative indication of visco-inertial flow. That is, the plots can be used to determine whether
non-laminar flow is occurring or not but can not determine the actual degree of non-laminar
flow.

       The importance of visco-inertial flow in laboratory column and field-scale application of
venting is unclear at present. However, it is apparent that if visco-inertial flow does occur, the
effect would likely be underestimation of gas permeability since non-laminar flow increases
energy loss relative to laminar flow.

3.6    Relative Permeability

       Intuitively, it is apparent that the presence of water or a NAPL phase will interfere with
the flow of gas. Quantitatively, this permeability  reduction is expressed in terms of a relative
permeability factor, k^ [-], which varies from zero to one. The gas permeability tensor is related
to relative permeability by
                                                                                  (3.16)
where  kl is the intrinsic permeability tensor [L2].


       Gas flow in soils is generally described in terms of constitutive relationships between
capillary pressure, gas-water saturation, and permeability for gas and water (Brooks and Corey,
1966; Stonestrom and Rubin, 1989a,b; Springer et al., 1995, Fischer et al., 1996; and Garbesi et
al., 1996). However, in contrast to unsaturated water permeability, relatively few experimental
studies have been conducted to evaluate the relationship between gas permeability, fluid
saturation, and capillary pressure.  In unsaturated media, gas is considered the nonwetting phase
and water the wetting phase. In moist soil, water forms a continuous phase. A substantial
portion of the gas phase though can be isolated in discontinuous, immobile domains entrapped by
the aqueous phase (Bear,  1972; Stonestrom and Rubin, 1989a,b; Fisher et al., 1997).  For
modeling purposes, it is usually assumed that the gas phase is continuous up to the maximum
water phase saturation (or residual gas phase saturation).  During laboratory drainage
experiments however, large discrepancies have been observed between residual gas-phase
saturation and the "emergence point" of gas-phase permeability (Stonestrom and Rubin, 1989a,b;

                                           20

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Fisher et al., 1996, 1997; Dury et al., 1999). Stonestrom and Rubin (1989a,b) denote saturations
at which gas permeability disappears and reappears during wetting and drainage experiments as
extinction and emergence points respectively.  These points delimit a range of saturation for
wetting and drainage hysteresis branches over which both phases are continuous (i.e. conduct
fluid flow). For the aqueous phase, the distinction between residual saturation and emergence
point has been considered negligible (Brooks and Corey, 1964; Corey, 1986).

       A plausible explanation of zero gas permeability at non-zero gas saturation is that gas
permeability is directly related to the continuity of gas-filled pores connecting boundaries (Dury
et al., 1999; White et al., 1972; Stonestrom and Rubin, 1989a) of which gas or water saturation
are secondary indicators. Muskat (1949) proposed that relative permeability of a phase goes to
zero when the  phase's continuity is broken and that point might occur at saturations higher than
residual saturation.  Dury et al. (1999) state that extinction and emergence of gas flow above
residual saturation is a phenomenon that arises through the transition from the microscopic to the
macroscopic scale.  On the microscopic scale, the three-dimensional structure of pore space can
be explicitly accounted whereas on the macroscopic scale, spatial  averaging and use of empirical
terms such as the extinction and emergence point must be used to describe processes occurring
on a microscopic scale.

       Pore-scale network models can be used to explicitly account for the three-dimensional
connectivity of pores. This is accomplished by representing the pore space as a lattice of pore
bodies or sites which are connected to one another by pore throats or bonds. Probability
distribution functions can then be used to generate distributions of bond and site radii.  As
explained by Fisher and Celia (1999) though, drainage and imbibition in bonds or sites are
controlled not  only by size but also by location and accessibility in a three-dimensional lattice.
Fisher and Celia (1999) were able to predict extinction and emergence points for gas flow in a
quartz sand mixture using a pore-scale network model. They state that as opposed to empirical
retention and capillary bundle models, the discontinuity of the gas phase at high wetting
saturation is inherent in network models and thus has significant conceptual advantages.

       In principle, no additional manipulation of lattice topology should be required for the
prediction of permeabilities after adjustment to retention data. However, in practice, generation
of an appropriate network of pores to match retention data and predict permeabilities is difficult.
Development of methods to more accurately describe liquid retention and permeability
relationships is an active area of research. For instance Tuller et al. (1999) devised a new model
considering both capillary and adsorptive contributions to liquid retention in angular pore space.
Or and Tuller (1999) then developed a statistical framework for upscaling the  pore size model  to
represent a sample size.

       While pore-scale network models have long-term potential use, the empirical capillary
pressure - water saturation relationships of Brooks and Corey (1964) and van Genuchten (1980)
combined with the capillary bundle models of Burdine (1953) and Mualem (1976) are most often
used to estimate gas and water permeability. With reseating, these models can account for gas
                                           21

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entrapment or discontinuities. Lenhard and Parker (1987) partitioned nonwetting phases into
continuous and discontinuous phases to develop a relationship for relative nonwetting phase
permeability based on the capillary pressure - saturation relationship of van Genuchten (1980)
and capillary bundle model of Mualem (1976).  Luckner et al. (1989) adapted the van Genuchten
(1980) and Mualem (1976) equations by reseating nonwetting phase saturation to residual
nonwetting phase saturation. Fisher et al. (1996) then extended Luckner et al.'s (1989) approach
to scale effective nonwetting phase saturation to the emergence point of nonwetting phase
conductance.

       Using Fisher et al.'s (1996) approach, effective gas saturation ( Sv ) [-] can be defined by
                                                                  &


       S~S                                                                     (3.17)
                                                                                  ^     '
      o     _ o
       g,max    g,e
where Sg is gas saturation (volume of gas-filled pore space divided by total pore space) [-], Sge is
the gas phase emergence point saturation [-], and Sgmax is a maximum gas-phase saturation [-].
Similarly, effective water saturation ( Sw ) [-] can be defined by
      „     _ „
               w,e
where Sw is water saturation (volume of water-filled pore space divided by total pore space) [-],
Swe is the water-phase emergence point saturation [-], and Swmax is maximum water-phase
saturation [-]. However, with respect to the wetting phase, the difference between residual
saturation and emergence point has been considered to be negligible and irrelevant (Brooks and
Corey, 1964).

       In contrast to unsealed gas and water saturation values, effective or scaled gas and water
saturations do not in general add up to one. Combining the retention models of Brooks and
Corey (1964)
                h>hd                                                             (3.19)


^=1    h
-------
Sw= \\ + (ah)"
                                                                (3.21)
respectively where a, A, hd, m, and n are empirical parameters and h is capillary pressure given in
units of length and the models of Burdine (1953)
  rg
f—/ f-
J  /j2CrW  J,
                            *
               -h(x)
                   ^ '
                                                                               (3.22)
andMualem(1976)
             /2
            
-------
       Dury et al. (1999) state that the term raised to the powers of A and B represent capillarity
while the term raised to the power (j. represents connectivity and tortuosity of the pores.  They
found that after scaling to an emergence point water saturation for drainage experiments, that
selection of the permeability model was decisive for fitting experimental data.  Prediction of gas
permeability did not appear to be very sensitive to the choice of the retention model.  Best fits to
experimental data were found when (j. was optimized. Dury et al. (1999) point out that the
connectivity - tortuosity term |j. is not based on theoretical  considerations. The value of 0.5 used
for the Mualem (1976) was based on empirical evidence from a variety of hydraulic conductivity
data where n ranged from -0.5 to 2.5 for different soils. No comparable studies have been
conducted for gas permeability. The value of 2.0 used for the Burdine (1953) is also empirically
based. Luckner et al. (1989) arbitrarily choose a value of 0.33 while Parker and Lenhard (1987)
and Fisher et al. (1996) both used a value of (J. = 0.5 by analogy with hydraulic conductivity.
Thus, there is no theoretical or practical reason to confine selection of |J. to 0.5 or 2.0.  Detty
(1992) demonstrated that the relative permeability of gas is greater than water for the same
saturation because water is the wetting fluid and occupies the smaller pores.

3.7    Components of Specific Discharge

       Specific discharge is  a macroscopic quantity defined as the volume of discharge of fluid
over some representative area per time. It is a vector in three-dimensional space in Cartesian
coordinates and in two-dimensional axisymmetric cylindrical coordinates. Darcy's law given in
terms of its gas permeability tensor (expressed as a matrix) can be described by:
 q
^l(xx)  ^l(xy)  "-z(xz)
IT     If      If
\(yx)  Kl(yy)  Kl(yz)
                     kl(zx)
             kl(zz)
       dPIdx
       dPIdy
       dPIdz
                                         (3.26)
or

             I+*.
                     kl(zr)
       -r(rz)

       l(zz)
dPIdr
dPIdz
                                         (3.27)
In analytical gas permeability testing, it is commonly assumed that site-specific coordinates are
aligned with the principal axes of gas permeability whereby equations (3.26) and (3.27) become
          rg
flK
                p
 l(xx)
 0

 0
  0

W)
  0
  0

  0

kl(zz)
dPIdx
dPIdy
dPIdz
                                                                                  (3.28)
or
                                           24

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                    k,
                     'l(rr)
                           kl(zz)
dPIdr
                                               (3.29)
dPIdz                                        V    '
However, this is done for mathematical convenience, specifically to allow development of
analytical solutions. Actual principal directions of gas permeability may be quite different.  This
would become readily apparent if one injected a gas tracer and noted movement in a direction not
parallel with the pressure gradient.

3.8    Continuity Equation for Gas Flow

       The continuity equation for gas flow can be expressed by
                                                                                 (330)
    dt
The assumption of constant volumetric gas content, viscosity, temperature, and molecular weight
yields
If we let 4> = P2 then


                     (    A ^ a/A  I
                                                                                 (3.32)
which is the governing partial differential equation commonly used for analytical and numerical
gas flow analysis.

       Baehr and Hult (1991) state that natural areal temperature variations can be neglected
over the scale of a pneumatic test.  Also, temperature variations due to energy transport
associated with induced gas movement will be negligible as a result of the high thermal capacity
of natural sediments and low-energy content of subsurface gas.  The assumption of constant
volumetric gas content however may impart error under aggressive operating conditions (e.g.
high vacuum or pressure) in soils having a high moisture content. Redistribution of gas and
water during pneumatic testing  can change the original spatial distribution of gas permeability.
Vacuum extraction will cause water to move towards the well while gas injection will cause
water to move away from the well. Thus, gas extraction and gas injection pneumatic tests may
provide different estimates of pneumatic permeability.
                                           25

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3.9    Linearization of Gas Flow Equation

       Equation (3.32) is by definition a non-linear partial differential equation. To solve this
equation analytically, linearization of the •/$ term is necessary.  The first and most simplistic
approach is to let v(j) equal atmospheric pressure (Patm) or some other constant pressure and to let
p2 = p*patmj thereby solving for P not P2 and removing consideration of compressibility.  Johnson
et al. (1988) used this linearization to solve the Theis equation for gas flow.  A second approach
is to let v(j) equal to a constant pressure but still solve for § or P2.  Baehr and Hult (1988) and
Falta (1996) let v(j) = Patm.  Falta (1996) states that with this approximation, the gas is assumed to
be compressible with a constant compressibility factor of l/Patm. A third approach is to replace
v'4> with a prescribed time-varying function which in some manner reflects the rate of change of
initial pressure distribution (Wu et al., 1998).  Since the first and second approaches are easy to
implement and commonly used for analytical and numerical solution of the gas flow equation, it
is useful  to evaluate error associated with these two linearization methods to determine whether
use of the third approach is necessary.

       Error incurred using the first and second approaches can be evaluated using Kidder's
(1957) perturbation solution to the one-dimensional transient gas flow equation
  kx   dt    dx \   dx

subject to

 P(jt,0) = P0,     0°»

Kidder's (1957) second-order perturbation solution is summarized as follows


                                                                                   (3.37)
  —   = 1 - aGJ0 - a^ - aGJ2
where
                                                                                   (3.38)
                                            26

-------
SJQ=\-erf(g)                                                                    (3.39)


                                                                                  (3.40)
        n      8^/2      2n         \6nilz
                                                                                  (3.41)

        (2- = P0 linearized form of the one-dimensional transient gas flow equation
for the boundary conditions described above. Linearization by letting 4> = P*P0 and v(j) = P0
results in
-- = l-flr'fiT0                                                                     (3.44)
-M)

where


tf' = A-3-.                                                                      (3.45)
       Kidder' s (1957) second-order perturbation solution and the dimensionless variable, q, are
used to compare the accuracy of the •/$ = P0 combined with P2 = P*P0 to the •/$ = P0 approach to
linearization. Results of simulations are summarized in Figures 3.1 and 3.2. Deviation from
Kidder' s second-order perturbation solution is defined here as
                                           27

-------
Deviation from second order solution =
                                        p   p2nd-order
                                          p
                                           2nd—order
(3.46)
       As P!/PO (ratio of pressure at x = 0 to pressure at infinite distance) approaches zero, the v(t>
= P0 approach to linearization provides a much more accurate approximation to P/P0 (ratio of
pressure at a prescribed distance and time to pressure at infinite distance) for various values of c,
compared to the P2 = P*P0 approach to linearization. It is noteworthy that the maximum
deviation from Kidder's (1957) second-order perturbation solution when solving directly for 4>
and linearizing ,/ by v(t> = P0 is less than 6 %. Error approaching 6% occurs as the ratio of P^
approaches zero or as the pressure at x = 0 approaches a perfect vacuum. The v(t> = P0 combined
with P2 = P*P0 approach to linearization appears acceptable only when the ratio of P^ is greater
than 0.5. However, even under these conditions, solving directly for 4> and linearizing v(t> by v(t> =
P0 approach to linearization is clearly superior.

       This analysis demonstrates that solving directly for (t> and linearizing ,/ by AJ> = P0
approach to linearization is adequate for solving gas flow problems associated with venting
application. This approach to linearization will be used for equation development throughout the
remainder of this document.

3.10   Conclusions

1.      At pressures typical of soil venting operation (less than  atmospheric pressure up to
       perhaps 1.5 atm), deviation from ideality for gases of interest (primarily O2 and N2) will
       be insignificant and the ideal gas law can be used to estimate gas density.

2.      Gas viscosity is largely independent of pressure up to 10 atmospheres. At low density,
       gas viscosity increases with temperature to the 0.6 to  1.0 power of temperature because of
       molecular interactions. Gas viscosity of a nonpolar multicomponent mixture can be
       estimated to within 2% from individual gas mole fractions, viscosities, and molecular
       weights.

3.      Inertial and gravitational components of Bernoulli's equation are typically neglected in
       gas flow analysis. The assumption of a negligible component of inertial potential though
       may be inaccurate in the immediate vicinity of an extraction or injection well. The
       assumption of negligible gravitational potential may be  inaccurate for chlorinated
       contaminant laden gas present near a NAPL source area under low gradient conditions.

4.      Frictional fluid potential loss is proportional to shear stress and distance of travel,
       increases to the square of pore-gas velocity, and increases linearly with decreasing pore
       diameter.  The last factor explains why gas potential dissipates over short distances in fine

                                            28

-------
           g
           o
           C/5
           ^H
           
-------
^o
-t—>
J3
"o
C/3
u*
O
-O
O
T3
O
O
/FV^Pc
                                                    xM)
Figure 3.2  Percent deviation from Kiddefs (1957) second-order perturbation solution for P/Pg = 0.5 to
0.9 for (a) (J) = PP0  in addition to J(j) = P^ and (b) J(j) = P0 alone
                                                   30

-------
       grained soils such as silts and clays.

5.      A reduction in molecular diameter, pore diameter, and gas density should increase slip
       flow.  Apparent gas permeability should be approximately a linear function of reciprocal
       pressure. Thus, in dry or water free media, apparent permeability extrapolated to the
       reciprocal of infinite pressure should give intrinsic (fluid independent) permeability.
       Since the gas slippage factor, b, is inversely proportional to the pore radii, it should be
       small  for high gas permeable material and larger for low gas permeable materials. Also,
       the gas slippage factor would be expected to decrease with increasing water saturation
       since only the largest pores would conduct gas flow. However with the exception of one
       study, experimental evidence does not support this statement.

6.      At high pore-gas velocities, visco-inertial effects or non-laminar (non-Darcy) flow may
       become important. For one-dimensional flow, Forchheimer's equations may provide a
       qualitative indication of visco-inertial flow.

7.      Gas permeability is directly related to the continuity of gas-filled pores. A substantial
       portion of the gas phase is often isolated in discontinuous, immobile domains entrapped
       by the aqueous phase. Water saturation at which gas permeability disappears during
       wetting and reappears during drainage are referred to as  extinction and emergence points
       respectively. These points delimit a range of saturation for wetting and drainage
       hysteresis branches over which both phases are continuous. With reseating to incorporate
       gas entrapment, empirical capillary pressure - water saturation relationships of Brooks
       and Corey (1964) and van Genuchten (1980) combined with capillary bundle models  of
       Burdine (1953) and Mualem (1976) can be used to estimate gas permeability from
       capillary pressure and water saturation information.

8.      The gas flow equation is by definition a non-linear partial differential equation. To solve
       this equation analytically, linearization of the v(t> term is  necessary.  The first and  most
       simplistic approach is to let •/$ equal atmospheric pressure (Patm) or some  other constant
       pressure and to let P2 = P*Patm, thus solving for P not P2. A second approach is to let v(t>
       equal  to a constant pressure but still solve for § or P2. Comparison with Kidder's (1957)
       demonstrate that the latter approach is much more accurate at pressure differential
       exceeding 0.5 atmospheres.
                                            31

-------
4.     ONE-DIMENSIONAL GAS FLOW AND PERMEABILITY ESTIMATION

       While not representative of fully three-dimensional field-scale gas flow, one-dimensional
flow analysis can provide valuable insight into fundamental gas flow processes such as the effect
of compressibility and gas slippage on pressure and pore-gas velocity profiles. Analysis of one-
dimensional gas flow is especially important in designing and interpreting laboratory column
vapor mass transport studies where knowledge of pore-gas velocity profiles becomes critical.
Because field-scale tests provide estimation of gas permeability over an integrated volume of
porous media too large to discern small discrete layers of less permeable materials (e.g., lenses of
silt and clay), gas permeability estimation in "minimally" disturbed samples could be useful  in
determining small-scale gas permeability variation in subsurface media. Testing with
undisturbed or minimally disturbed soil cores though provides only an estimate of vertical
permeability. However, variation in horizontal and vertical permeability of a soil core may be an
order of magnitude or less than variation in permeability between distinct stratigraphic lenses.
For example, variation in gas permeability between a sand and clay would be expected to be
much greater than variation in horizontal and vertical permeability within the clay or sand core.

4.1     Permeameter Design

       One-dimensional permeability estimation most often involves testing in laboratory-scale
columns or permeameters.  Basic features that permeameters should exhibit for accurate gas
permeability estimation include:  (1) a sample-holding device capable of forming a tight seal
between the sample and container wall without significantly altering the geometry or physical
properties of the  sample; (2) end caps and filters which form a tight seal with the sample but do
not restrict gas flow; and (3) a method to vary moisture content uniformly throughout the sample.
Guidance for conducting gas permeability tests in permeameters such as the American Petroleum
Institute's "Recommended Practice for Core-Analysis Procedure RP-40" (API, 1960) and
American Society for Testing and Material's Method D4525-90 (ASTM,  1990) entitled
"Standard Test Method for Permeability of Rocks by Flowing Air" is largely intended for rock
cores.  The preferred apparatus in ASTM Method D4525-90 is to have a rock specimen confined
and compressed by an elastomer sleeve to avoid bypassing of air along the sample wall. In the
Hassler (1944) permeameter, a rubber diaphragm inflated with high pressure liquid or air is used
to seal the sample.  In the Fancher (1933) permeameter, an elastomer bushing and compression
yoke is used to confine and seal a rock specimen. In these methods, the rock specimen must be
sufficiently hard  and consolidated to resist cracking or other deformation since gas permeability
estimation is extremely sensitive to bypass flow. Another approach, more appropriate for
unconsolidated media, is to confine a sample with a rigid bushing or tube such as would occur
during core collection with a shelby tube.  This method however does not allow direct
manipulation of moisture content as would be necessary for evaluation of gas permeability as a
function of moisture content.

       Corey (1986) developed an apparatus to determine gas permeability as a function of
moisture content by placing a sample in an annulus between a water-wetted ceramic cylinder and
                                           32

-------
a rubber sleeve. The ceramic cylinder was used to ensure a constant and uniform moisture
content in the sample during testing. Short circuiting of gas flow and poor contact with the
ceramic cylinder along the sample wall boundaries was prevented by applying pneumatic
pressure (50 to 100 kPa) across the rubber sleeve. Stonestrom (1987) modified Corey's (1986)
design by adding concentrically grooved ceramic end plates to allow simultaneous water
desaturation and air flow through the sample and peripheral features for co-determination of
trapped gas, water-retention characteristics, and air and water permeability functions. Springer et
al. (1998) describe an apparatus referred to as a "soil-air permeameter" in which soils are packed
in a 1-bar rated ceramic cylinder to allow water extraction and uniform water content and
capillary pressure throughout the column.  This device has the capability to evaluate gas
permeability as both a function of capillary pressure and water content.

       When attempting to evaluate small-scale gas permeability variation in the field, use of
permeameters such as those described by Corey (1986), Stonestrom (1987), and Springer et al.
(1988) provide the opportunity of determining capillary pressure - gas permeability curves but
require repacking of soil samples thereby compromising pneumatic sample integrity. Use of
shelby tubes during  core collection allows collection of minimally disturbed cores, but
elimination of short-circuiting along the tube wall can not be guaranteed. Use of both
permeameters and shelby tubes measure only the vertical  component of the permeability tensor.
Thus, it would appear that  all laboratory-scale methods of measuring gas permeability have
notable limitations.  However, there is currently no other direct means to estimate  gas
permeability on the  scale of centimeters in the field.  Field-scale testing integrates gas
permeability estimation on the scale of m3 or larger.

4.2    Formulation of a One-Dimensional Steady-State Gas  Flow Equation With  Gas Slippage

       Because of small sample size and hence very rapid transient response, estimation of gas
permeability with permeameters or soil columns is typically accomplished using steady-state
solutions. The governing equation for one-dimensional, steady-state gas flow incorporating gas
slippage is
                                                                                  (4.1)
The boundary condition for equation (4.1) when gas is injected or extracted is represented by


                                                                                  (4.2)
                dx
                    x=0
MA
where Qm = mass flow rate [M T"1]. The boundary condition where gas exits the column is
represented by constant pressure

                                           33

-------
(L) = L .                                                                        (4.3)

In the literature, flow rates are often expressed in terms of volume and sometimes in terms of
mass. The mixing of units causes confusion.  In this document, all flow units are expressed in
terms of mass to maintain consistency and to emphasize the point that analytical solutions
presented here demand a constant mass flux not volumetric gas flow into or out of the formation
or column. The relationship between volumetric flow (Qv) and mass flow (Qm) is given by:
                                                                                  (4.4)
Negative values of Qm denote gas extraction while positive values of Qm denote gas injection.

       The Kirchhoff transformation can be used to remove the non-linearity associated with gas
slippage (Kaluarachchi, 1995) by letting
                                                                                  (4-5)
Integration of equation (4.5) results in

                                                                                  (4.6)


Expressing  in terms of P2 and use of the quadratic equation on (4.6) results in
P = -b + Jb2+Pr2ef+2bPref+U .                                                    (4.7)


Differentiation of U in equation (4.6) results in


                                                                                  (4.8)
 dx   I    ^(f) \ dx

Equation (4. 1) now becomes

                                                                                  ,.n.
                =°                                                               (4-9)
with boundary conditions:
                                           34

-------
U(L) = 0                                                                         (4.10)

and

,  ,   dU
dx
                     MA
                                                                                 (4.11)
Integration of equation (4.9) provides a solution

       J^9tT//^           o
-------
4.3    Analysis of Effect of Gas Slippage on One-Dimensional Pressure and Pore-Gas Velocity
       Simulation
       Equations (4.13), (4.14), and (4.16) were used to evaluate the effect of flow rate on
pressure and pore-gas velocity computation with and without gas slippage as a function of
distance in a hypothetical 30 cm long soil column having diameter of 10 cm. Gas extraction
occurs at x = 0. Simulations were conducted using gas permeabilities of 1 .0 x 10"08 cm2,
1.0 x 10"09 cm2, and 1.0 x 10"10 cm2. Mass flow rates were adjusted to maintain a constant
Qn/kj.gk^ ratio. From equation (4.14), when this ratio is held constant, non-slip-corrected
pressure as a function of distance must be identical. This is not the case however for slip-
corrected gas flow since b varies with gas permeability.

       Figures 4. 1 through 4.3 illustrate the effect of flow rate on pressure computation as a
function of distance with and without gas slippage during gas extraction. For each figure, the
disparity between slip-corrected and non-slip-corrected computed pressure increases with
increasing flow rate and decreases with increased distance from the extraction point.  At higher
flow rates, neglecting gas slippage results in underestimation of absolute pressure and
overestimation of pressure differential from atmospheric pressure. These effects become more
severe as gas permeability and pressure is reduced.
l.U
0.9 -
0.8 -
0.7 -


06-

s
e? 0.5 -
t±
0.4 -



0.3 -

02-

0.1 -
n n
	 • 	 '

•••-'•'~'-'-~'''~"" ^ ~ '•''" "" -• '' '••*'
-*'"*• — <••'*'••''
- *^ ' **'' *'' ''^
.•* is" ^ • ' , . "*

.•*' '^ *''*.•'''
V"" x-'.x1'"
X** /'
X"" X*


,- /
<• f-


t


	 • "

•"'





	 Qm
	 Qm

	 Qm

„
vm
	 Qm

	 Qm
	 Qm
	 Qm








= - 0.03 g/s, b = 0
= - 0.03 g/s, b = 0.052 atm

.03 g/S, D — 0


.03 g/s, b — O.ODZ atm
= - 0.07 g/s, b = 0

= - 0.07 g/s, b = 0.052 atm
= - 0.09 g/s, b = 0 atm
= - 0.09 g/s, b = 0.052 atm
                0
10
20
25
30
                                                15
                                          Distance (cm)
Figure 4.1 Normalized pressure computation as a function of mass flow rate, distance, and gas
slippage factor (kj^, = 1.0 x 10'08 cm2).
                                           36

-------
   0.6 -
   0.5 -
   0.4 -
   0.3 -
   0.2 -
   0.1 -
   0.0
                	Qm = - 0.001 g/s, b = 0
                	Qm = - 0.001 g/s, b = 0.129 atm
                	Qm = -0.003 g/s, b = 0
                	Qm = - 0.003 g/s, b = 0.129 atm
                	Qm = -0.005 g/s, b = 0
                	Qm = - 0.005 g/s, b = 0.129 atm
                	Qm = -0.007 g/s, b = 0
                	Qm = - 0.007 g/s, b = 0.129 atm
                	Qm = - 0.009 g/s, b = 0 atm
                	Qm = - 0.009 g/s, b = 0.129 atm
       0
10
20
                                                                         25
                       30
                                                  15
                                            Distance (cm)
Figure 4.2 Normalized pressure computation as a function of mass flow rate, distance, and gas
slippage (l^gkl(x) = 1.0 x 10-09 cm2).
fin
l.U
09 -

0.8 -
0.7 -

U.b -
0.5 -
0.4 -

0.3 -


0.2 -
0.1 -

.

-— ~~i 	 ._•-—•"'""
..-—•"""""" •sf;S':S' :"~" *•-'
...-"*-*""*""*"" ,--:^:^ ".-
-• •*-" ^-~ "*" •• **"
. ' '' ^ -" ^ '•'"'''
/'
/•'
/•
/





•
— JT- "•-"- •"••"'"" 	 	
" ,..-*---*":* .^
'i-.f'-'-*''*"^-"'
-""'


	 Qm =
	 Qm =
	 Qm —
	 Qm =

	 Qm =
	 Qm =
	 Qm =
	 Qm =
	 ======
-^•'^•".'^••••'.^
j»*'"^ ..-•'''
. • *"*



- 0.0003 g/s, b
- 0.0003 g/s, b
- O.OOOD g/S, u
- 0.0005 g/s, b

- 0.0007 g/s, b
- 0.0007 g/s, b
- 0.0009 g/s, b
- 0.0009 g/s, b







= 0
= 0

= 0

= (
= 0
= 0
= 0
f^:Sr#-
'•*" '






.3 16 atm

.3 16 atm


.3 16 atm
atm
.3 16 atm
        0
                                       10
20
                                   25
                       30
                                                  15
                                            Distance (cm)
Figure 4.3  Normalized pressure computation as a function of mass flow rate, distance, and gas
slippage (ig^, = 1.0 x ID'10 cm2).
                                      37

-------
       Figures 4.4 through 4.6 illustrate error in pressure calculation due to gas slippage. Error
is calculated by
Error in pressure calculation =
                                    P(b * 0)
                                                                                   (4.17)
Error is negative since absolute pressure for slip-corrected flow is always greater than for non-
slip-corrected flow under vacuum application.  Error in pressure computation due to gas slippage
increases with gas flow and decreased permeability. Note that error in pressure computation
exceeded 60%, 70%, and 80% at gas permeability values of 1.0 x 10'08, 1.0 x r09, and
1.0 x 10~10 cm2 respectively.  The magnitude of these errors may necessitate consideration of gas
slippage for laboratory and field-scale estimation of gas permeability in soils having when
vacuum is applied. As will be discussed, consideration of gas slippage may  not be necessary for
estimation of gas permeability when gas is injected or pressure is applied.

       Figures 4.7 through 4.9 illustrate the effect of flow rate on calculated pore-gas velocity as
a function of distance assuming a volumetric gas content of 0.1 during air extraction. At the
point of extraction, pore-gas velocity increases for both slip-corrected and non-slip-corrected gas
flow because of lower gas density.  However, at higher flow rates and low absolute pressures
near the extraction point, calculated pore-gas velocities for non-slip corrected flow are much
higher than for slip-corrected flow.
u.u
xO
.1 -20.0 -
-f-»
JS
"1 -30.0 -
U
1 -40.0 -
8
_, rr\ r\
g OU.U
W -60.0 -
- /u.u
(
/'
f
/
/
/
i
i
i
i
i

(jn\ - u.ui g/s
• 	 Qm = - 0.03 g/s
	 Qm — 0.05 g/s
	 Qm = - 0.07 g/s
	 Qm = - 0.09 g/s
) 5 10 15 20 25 3
Distance (cm)
Figure 4.4 Error in pressure computation as a function of mass flow rate and distance when gas
slippage is neglected (krgkl(x) =\.0x 10"08cm2).
                                            38

-------
             -10.0 -
          §  -20.0
             -30.0
             -40.0 -
             -50.0 H
             -60.0 H
             -70.0 -:
             -80.0
                                                        -Qm = -0.001 g/s
                                                        • Qm = - 0.003 g/s
                                                        -Qm = -0.005 g/s
                                                        • Qm = - 0.007 g/s
                                                        • Qm = - 0.009 g/s
                   0
                                                            25
                    30
                              5          10         15         20
                                              Distance (cm)
Figure 4.5  Error in pressure computation as a function of mass flow rate and distance when gas
slippage is neglected (krgkl(x) = l.Ox 10"09cm2).
             -10.0 -
           g -20.0 H
          -J—»
          I -30.0 H
          ~3
          O
           u -40.0 H
           00
          fin
          ^c
          o
          W
             -50.0 -
-60.0 -
             -70.0 -

	Qm = -0.0001 g/s
	Qm = -0.0003 g/s
	Qm = -0.0005 g/s
	Qm = -0.0007 g/s
	Qm = -0.0009 g/s
                                        10         15         20
                                              Distance (cm)
                                                            25
                    30
Figure 4.6  Error in pressure computation as a function of mass flow rate and distance when gas
 slippage is  neglected (k,.gkl(x) = l.Ox 10"10cm2).
                                              39

-------
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45 0 -

40.0 -


S^^ n
J J.U
o
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'o
-i 25.0 -
>>
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£ 15.0 -
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fin
10.0 -
5.0 -



i
i
\



\
\
\
\
\
\
X.

~-^ """"•••-»'.'"""•--

""""•--—-_ 	
'•-'.-:-
-.-.-:--.:-.:-^-.-:..Sfis


Qm

Qm
	 Qm
	 Qm


m
	 Qm
	 Qm
	 Qm
	 Qm
	 Qm


"""""•—---—". "^ —
•r: — ::=: — -'=: = : —
•---•-••-----•--•--•*----*-- = *





= 0.03 g/s, b = 0
= 0.03 g/s, b = 0.052 atm



= 0.05 g/s, b = 0.052 atm
= 0.07 g/s, b = 0
= 0.07 g/s, b = 0.052 atm
= 0.09 g/s, b = 0
= 0.09 g/s, b = 0.052 atm


- -__ . ___
= : = :=::=::=--=: = ::
= « = « = »=.«,.= ,„,»=.„, 	

10         15         20
     Distance (cm)
                                                                          25
                                              30
Figure 4.7  Pore-gas velocity computation as a function of mass flow rate, distance, and gas
slippage ( 6g = 0.1, kjt^ =\.0x lO'08 cm2).
             9.0
             8.0 -
          ^ 7.0 -
          c«
          Jl 6.0 H
             4.0 -
             3.0 -
             2.0 -
             1.0 --
             0.0
               	Qm = - O.OOlg/s, b = 0
               	Qm = - 0.001 g/s, b = 0.129 atm
               	Qm = - 0.003 g/s, b = 0
               	Qm = - 0.003 g/s, b = 0.129 atm
               	Qm = - 0.005 g/s, b = 0
               	Qm = - 0.005 g/s, b = 0.129 atm
               	Qm = -0.007 g/s, b = 0
               	Qm = - 0.007 g/s, b = 0.129 atm
               	Qm = -0.009 g/s, b = 0
               	Qm = - 0.009 g/s, b = 0.129 atm
                 0
10
                      20
25
30
                                                   15
                                             Distance (cm)
Figure 4.8  Pore-gas velocity computation as a function of mass flow rate, distance, and gas
slippage (6g = 0.1, kk(x) =\.0x 10'09 cm2).
                                              40

-------


t/5
g
>
<5
O




.80 -
0.70 -

.60 -
0.40 -
0.30 -
0.20 -
.10



1
1
\
\
\
\
\
\
\
	 	 ".~:~:~:~
1 i i


	 Qm =
	 Qm =

	 Qm —
	 Qm =
	 Qm =
	 Qm =
	 Qm =


1


- 0.0003 g/s, b
- 0.0003 g/s, b
- 0.0005 g/S, u
- 0.0005 g/S, D
- 0.0007 g/s, b
- 0.0007 g/s, b
- 0.0009 g/s, b
- 0.0009 g/s, b

= = =•• — . — .—



= 0
= 0.3 16 atm

.316 atm
= 0
= 0.3 16 atm
= 0
= 0.3 16 atm

	
1
                 0
10
20
                                                                      25
30
                                                 15
                                           Distance (cm)
Figure 4.9 Pore-gas velocity computation as a function of mass flow rate, distance, and gas
slippage (6g = 0.1, kjc^ =l.0x 10'10 cm2).
       Figures 4.10 through 4.12 illustrate error in pore-gas velocity calculation for slip-
corrected and non-slip-corrected gas flow. Error is defined by

                                         /7.  /-\\  -.//.  , /~\\
Error in pore — gas velocity calculation = -
                                             v(b * 0)
                                                                                   (4.18)
Positive values of error denote that pore-gas velocity is overestimated during vacuum application
when slip flow is neglected. The magnitude of error increases with decreased absolute pressure
and decreased gas permeability. As illustrated in Figure 4.12, at very low absolute pressure (near
extraction point) and gas permeability, error exceeds 600%. This finding has important
implications for laboratory column studies in silt or lower permeability media in which rate-
limited vapor transport needs to be evaluated as a function of pore-gas velocity.  The application
of vacuum and neglecting gas slippage can  result in a huge error in pore-gas velocity
computation.

       Now consider what happens during gas injection.  As illustrated in Figures 4.13 through
4.15, simulated absolute pressure profiles incorporating gas slippage are lower than profiles
neglecting gas slippage. During gas extraction, neglecting slippage results in underestimation of
absolute pressure whereas during gas injection, neglecting gas slippage results in overestimation
of absolute pressure. In both cases though, pressure differential from atmospheric pressure is
                                            41

-------
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\
\
\
\
\.
-•- 	 __., 	 '""'•' 	

i i i i
) 5 10 15 20
Qi i i i~\ f\ 1 rs /c-
111 — U.U1 g/S
	 Qm = 0.03g/s


	 Qm = 0.07g/s
	 Qm = 0.07g/s
















|
25 3
                                           Distance (cm)
Figure 4.10 Error in pore-gas velocity computation as a function of mass flow rate and distance
when gas slippage is neglected (6g = 0.1, k,.gkl(x) = 1.0 x  10"08 cm2).
JJU
« 300 -
o
J 250 -
03
0
•§ 200 -
o
^ 150 -

-------

N=
a
o
-^
cs
3
o
£>
o
J2
>
o
a
O
PH
q
^H
O
b
U-l
/uu

600 -


500 -

400 -


300 -

200 -



100 -


n

	 Qm =
	 Qm =
• ,^
i ^m
\ 	 Qm =
I 	 Qm =
i
\

i
\
\
\
\

X
""•-..
"""•-. *""""*--^.
"~" •""• - -— """" "•—-
	 	 r.'.r. 	 — :=-.—; 	 ..


-0.0001 g/s
- 0.0003 g/s


- 0.0007 g/s
- 0.0009 g/s














0
10
20
25
30
                                                 15
                                            Distance (cm)
Figure 4.12 Error in pore-gas velocity computation as a function of mass flow rate and distance
when gas slippage is neglected (8g = 0.1, k,gkl(x) =l.0x 10"10 cm2).

overestimated.  For both gas extraction and injection, variation in pressure is greatest at the point
of extraction or injection and lowest gas permeability values.

       As indicated in Figures 4.16 through 4.18 however, the magnitude in error in pressure
computation for gas injection is far lower than for gas extraction because of higher absolute
pressures during gas injection. Even at the lowest gas permeability, maximum error does not
exceed 6.0% whereas during gas extraction, error exceeded 80%. Thus, if gas slippage is
neglected, gas injection should result in a better estimate of gas permeability than gas extraction.
Fortunately, most permeameter testing is conducted by gas injection.

       Figures  4.19 through 4.21 provide an analysis of pore-gas velocity as a function of flow
rate and slip conditions during gas injection.  Pore-gas velocities during gas injection are much
lower than during gas extraction at the same flow rate because of increased gas density. Thus, it
may be preferable to use gas injection during laboratory column studies as opposed to gas
extraction to achieve lower and better controlled pore-gas velocities.

       As illustrated in Figures 4.22 through 4.24, error in pore-gas velocity computation
neglecting gas slippage during gas injection is much lower than during gas extraction. Similar to
gas extraction though, error increases with decreased permeability. However, unlike gas
extraction, error increases with distance from the point of injection. In any event, under worst
case conditions, error during gas injection is less than 13% where it exceeded 600% during gas
extraction simulations.
                        43

-------
   1.4!

   1.40 -

   1.35 -

   1.3(

 | 1.2!

£  1.20-

   1.15 -

   1.10 -

   1.05
   1.00
                                                    — Qm = 0.01 g/s,b = 0
                                                    — Qm = 0.01 g/s, b = 0.052 atm
                                                    --•Qm = 0.03 g/s, b = 0
                                                    --•Qm = 0.03 g/s, b = 0.052 atm
                                                    ---Qm = 0.05 g/s, b = 0
                                                    --- Qm = 0.05 g/s, b = 0.052 atm
                                                    -• Qm = 0.07g/s,b = 0
                                                    - • Qm = 0.07 g/s, b = 0.052 atm
                                                    -- Qm = 0.09 g/s,b = 0 atm
                                                    — Qm = 0.09 g/s, b = 0.052 atm
         0
                    5
10
                                                                 20
25
30
                                                     15
                                               Distance (cm)
Figure 4.13 Pressure computation as a function of mass flow rate, distance, and gas slippage
        = l.Ox 10-°8cm2).
    1.45
   1.40 -

   1.35 -

   1.30

a 1.25 -
PH
    1.00
                                                 	Qm = 0.001 g/s,b = 0
                                                 	Qm = 0.001 g/s, b = 0.129 atm
                                                 	Qm = 0.003 g/s, b = 0
                                                 	Qm = 0.003 g/s, b = 0.129 atm
                                                 	Qm = 0.005 g/s, b = 0
                                                 	Qm = 0.005 g/s, b = 0.129 atm
                                                 	Qm = 0.007 g/s, b = 0
                                                 	Qm = 0.007 g/s, b = 0.129 atm
                                                 	Qm = 0.009 g/s, b = 0
                                                 	Qm = 0.009 g/s, b = 0.129 atm
         0
                                                                  25
                                               30
                              5           10         15          20
                                                Distance (cm)
Figure 4.14 Pressure computation as a function of mass flow rate, distance, and gas slippage
    cl(x)= l.Ox 10-°9cm2).
                                      44

-------








G
PH
PH




1

1

1

1

1

1

1
1
1
1
45 -,

40 -

35 -

30 -

25 -

20 -

15 -
10 -
05 -
nn

•-x
'•»..
"••v..
fci:*t. 'v'-
*^ '**" »
* *A * x .
" *^" * *^" * *™ " **" -
"'"•-- . ^ ' -^ " "^ . ^ ' -



	 Qm =
	 Qm =

	 Qm =
	 Qm =
	 Qm =
	 Qm =
	 Qm =
"- ^- ^-^ ^"^' ^'*^,
"""••-.._ """--.7"--^ ^"'-x. " s . ,
---~__^ """•••---...'' '" "'-->-^ ^>XC X--
	 ~"~~--~^ ~~~~^
=======^:

0

0

0001 g/s

0003 g/s
0.0003 g/s
0.0005 g/s
0
0
0005 g/s
0007 g/s
0.0007 g/s
0
0




=3
0009 g/s
0009 g/s



:;^:
~ i • .

,b

,b
,b
,b
,b
,b
,b
,b
,b




=K




= 0
= 0

= 0
= 0




316 atm
3 16 atm

= 0.316 atm
= 0
= 0



_'v..
BK^M

3 16 atm




^^^^^LS&W
       0
                                                           25
                      30
                           5          10         15         20
                                           Distance (cm)
Figure 4.15 Pressure computation as a function of mass flow rate, distance, and gas slippage.
   1.2
   1.0 -
^o
J3 0.8
^o
13
U
 u 0.6
 C/5
 $
£
_g
 o
& 0.2 -\
0.4 -
                                                       	Qm = 0.01 g/s
                                                       	Qm = 0.03 g/s
                                                       	Qm = 0.05 g/s
                                                       	Qm = 0.07 g/s
                                                       	Qm = 0.09 g/s
'\   v.
   \   X
         X
                                                            \
                                                           \\
       0
10
20
                                                            25
                       30
                                                 15
                                            Distance (cm)
Figure 4.16 Error in pressure computation as a function of mass flow rate and distance when gas
slippage is neglected (kigkl(x) = l.Ox 10"08cm2).
                                    45

-------
                                                                  	Qm = 0.001 g/s
                                                                  	Qm = 0.003 g/s
                                                                  	Qm = 0.005 g/s
                                                                  	Qm = 0.007 g/s
                                                                  	Qm = 0.009 g/s
                                      10         15         20
                                            Distance (cm)
                                25
                     30
Figure 4.17  Error in pressure computation as a function of mass flow rate and distance when gas
slippage is neglected (krgkl(x) = 1.0 x 10"9cm2).
             7.0
                                                                  	Qm = 0.0003 g/s
                                                                      Qm = 0.0005 g/s
                                                                  	Qm = 0.0007 g/s
                                                                  	Qm = 0.0009 g/s
                 0
10
20
25
30
                                                  15
                                            Distance (cm)
Figure 4.18 Error in pressure computation as a function of mass flow rate and distance when gas
slippage is neglected (kigkl(x)= 1.0 x 10"10 cm2).
                                             46

-------





00
a

o
jo
on
^H
O
fin




i n n -,
1U.U
9.0 -

8.0 -
7.0 -
6.0 -
s n
4.0 -
•3 0
2.0 -
n n
	 (jm — u.uig/s, D - u 	 (^m - u.ui g/s, D - U.UD/ atm
	 Qm = 0.03 g/s, b = 0 	 Qm = 0.03 g/s, b = 0.052 atm
	 Qm — 0.05g/s, b — 0 ...... Qm _ Q 05 g/s> ^ _ Q.052 atm
	 Qm = 0.07 g/s, b = 0 	 Qm = 0.07 g/s, b = 0.052 atm
	 Qm = 0.09 g/s, b = 0 	 Qm = 0.09 g/s, b = 0.052 atm

..-•-•"" !l--
	 ,_ — •- — __„.—— ••
	 -::;:—"—-"- 	
= ::=:: 	 . 	 ; : 	 '
-: = : = : = : 	
.-------^-;^:.-.'i::i::—--—: "•—--— i-— --— -- —--—--•"•-••


                              10
                                                 20
           25
           30
                                                     15
                                               Distance (cm)
Figure 4.19 Pore-gas velocity computation as a function of mass flow rate, distance, and gas
slippage (Q. = 0.1, k,.gkl(x) = 1.0 x lO'08 cm2).
 on
 o
JO
>
 on
6
 o
             	Qm = 0.001 g/s, b = 0
             	Qm = 0.003 g/s, b = 0
             	Qm = 0.005 g/s, b = 0
             	Qm = 0.007 g/s, b = 0
             	Qm = 0.009 g/s, b = 0
                                          	Qm = 0.001 g/s, b = 0.129 atm
                                          	Qm = 0.003 g/s, b = 0.129 atm
                                          	Qm = 0.005 g/s, b = 0.129 atm
                                          	Qm = 0.007 g/s, b = 0.129 atm
                                          	Qm = 0.009 g/s, b = 0.129 atm
1.0 -r
0.9 -
0.8 -
0.7 -	
0.6 -
0.5 -
0.4 -.:::::---	
0.3 -	
0.2 -
0.1
0.0
         0
                           10
20
25
                                                                                        30
                                                      15
                                                Distance (cm)
Figure 4.20 Pore-gas velocity computation as a function of mass flow rate, distance, and gas
slippage (6g = 0.1, k,.gkl(x) = 1.0 x lO'09 cm2).
                                     47

-------
              0.10 -r
          o
          o
          o
          
          6  •

           2
          w
-••" .-^" ^.-*
.•*"' .**'

II
) 5 10 15 20

	 Qm = 0.03 g/s
	 Qm = 0.07 g/s
	 Qm = 0.07 g/s
i
25 3
                                              Distance (cm)
Figure 4.22 Error in pore-gas velocity computation as a function of mass flow rate and distance
when gas slippage is neglected (8g = 0.1, k,gkl(x) = 1.0 x 10"08 cm2).
                                               48

-------
              13.0
           §  12.0 H
          j3  11.0-
           oj
          U
          •-S  10.0 -
           o
.£
 o

>    9.0 -
           s
           o
           o
8.0

7.0

6.0
                                                        	Qm = 0.001 g/s
                                                        	Qm = 0.003 g/s
                                                        	Qm = 0.005 g/s
                                                        	Qm = 0.007 g/s
                                                        	Qm = 0.009 g/s
                         10         15         20
                              Distance (cm)
                                                                         25
                                                                          30
Figure 4.23  Error in pore-gas velocity computation as a function of mass flow rate and distance
when gas slippage is neglected (6g = 0.1, k,.gkl(x) = 1.0 x 10"09 cm2).
               13
          §  12.8 H
         j5  12.6 -
          03
         U
         •t  12-4 H
 CD
 l/l
 S
O
          o
          fin
         W
             12.2 -
               12-
             11.6
                                                                         . = 0.0001 g/s
                                                        	Qm = 0.0003 g/s

                                                        	Qm = 0.0007 g/s
                                                        	Qm = 0.0009 g/s
                  0
                              10
                                               20
25
30
                                                   15
                                             Distance (cm)
Figure 4.24  Error in pore-gas velocity computation as a function of mass flow rate and distance
when gas slippage is neglected (8g = 0.1, k^k^ = 1.0 x 10"10 cm2).
                                             49

-------
4.4    Minipermeamter Testing

       Minipermeameter testing provides a rapid, nondestructive, inexpensive means of
obtaining gas permeability estimates in exposed consolidated media or surface soils. One
potential application for venting design is estimation of vertical gas permeability of a leaky
confining layer (e.g., asphalt or concrete cap). Basic operation involves compression of a tip seal
against rock or other test media through which gas is injected. Pressure differential and flow rate
are then recorded to estimate gas permeability. Characteristics of the resulting flow field and
sample support (volume of media affected) are dictated by the dimensions of the tip seal. To
date, most minipermeameter studies have been performed using a single, small (about 0.31 cm
inner radius) tip seal (Tidwell and Wilson, 1997).  Originally developed in the petroleum
industry (Eijpe and Weber,1971) for rapid field and laboratory acquisition of permeability data,
the gas minipermeameter has found widespread use in characterizing the spatial  distribution of
permeability in outcrops as well as rock cores and slabs.  Studies have been performed on a wide
range of geologic media including eolian sandstones (Chandler et al., 1989; Goggin et al.,
1988b), fluvial sandstones  (Dreyer et al.,  1990; Davis et al., 1993), carbonates (Kittridge et al.,
1990) and volcanic tuffs (Fuller and Sharp, 1992). Three basic minipermeameter designs exist: a
nonsteady, or pressure decay (Jones, 1992), high volume steady-state (e.g., Sharp et al., 1993),
and low volume steady-state (Davis et al., 1994).  Davis et al.'s (1994) method was designed to
avoid particle movement in weakly lithified rocks and unconsolidated media typical of
subsurface hydrology investigations. They used a glass syringe containing a piston falling under
its own weight to inject air and demonstrated that the piston was capable of providing a small but
constant pressure and flow rate over a period of several seconds. Using fabricated sandstone
cores, Davis et al. (1994) found that their minipermeameter apparatus compared well with a
minipermeameter using a constant supply of air. Their apparatus however underestimated gas
permeability compared to one-dimensional core testing using the ASTM D4525  method. They
state that this may be due to a scale effect. According to the analysis of Goggin  et al. (1988a),
the volume affected by gas flow is roughly a hemisphere with a radius four times the internal tip
seal radius.  For Davis et al.'s (1994) apparatus, this suggests that the volume affected is
approximately 7.4 cm3 whereas gas flowed through 100 cm3 in one-dimensional column studies.

       Tidwell and Wilson (1997) used a gas minipermeameter termed "multisupport
permeameter" (MSP) to investigate permeability upscaling.  They state that one  of the major
problems in aquifer/petroleum reservoir characterization stems from technological constraints
that limit the measurement of material properties to sample support (volume of media sampled)
much smaller than those that can be accommodated in current predictive models. This disparity
in support requires measured data to be averaged or upscaled to yield effective properties at the
computational grid block scale. Tidwell and Wilson (1997) state that quantitative investigation
of permeability upscaling requires that measurement at different supports be consistent in four
basic ways.  First multisupport permeability data must be collected from the same physical
sample, thus requiring the measurement technique to be nondestructive.  Second, near exhaustive
sampling is required at each support to avoid errors induced by sparse data effects. For this
reason, large suites of data must be collected at each sample support, requiring measurements to
                                           50

-------
be rapid and inexpensive.  Third, measurements must be sensitive to slight changes in
permeability at all sample supports.  Thus, measurement error must be small and consistent.
Fourth, measurements must be consistent in terms of flow geometry, boundary conditions, and
computational techniques as to provide a uniform basis for comparison.  Also, as upscaling
studies need to be performed on a wide variety of geologic media, the technique must have a
large dynamic permeability range and ability to discern heterogeneities occurring at different
length scales.  They state that the gas minipermeameter meets these basic criteria.

       Tidwell and Wilson (1997) and others use an equation developed by Goggin et al.
(1988a) to estimate gas permeability
                  ,_
krklr= -    Ll -                                                     (4.19)
                   P2  P2
                      ~  atm
where P0 = pressure applied at the tip seal/rock interface, r; = inner radius of tip seal [L], r0 =
outer radius of tip seal [L], and G0(r0/r;) = geometric shape factor [-].  In this model, complexities
associated with the flow field geometry are summarized by G0(r0/r;), a single, analytically
derived, dimensionless function that varies according to the ratio of the outer tip seal radius to
the inner tip seal radius.  It is apparent that this equation is similar to the one-dimensional,
steady-state gas flow equation when PL is replaced by Patm and (A/L) is replaced by r{GQ(rJr^.  For
minipermeameter measurements subject to a homogenous, isotropic, semi-infinite half-space, the
bounding surface of resultant gas flow field is approximated by a hemisphere (Goggin et al.,
1988a). Tidwell and Wilson (1997) state that the effective radius of the bounding surface is
governed primarily by the inner radius of the tip  seal while the geometry of the diverging flow
paths beneath the tip seal is controlled by the ratio r0/r;. They define the effective radius in terms
of the radial distance at which the minipermeameter response is no longer sensitive to media
heterogeneities.  Numerical experiments  conducted by Goggin et al. (1988a) indicate that the
effective radius can vary from 2.5r; to 4 r; depending on whether a 10% or 5%  change in
minipermeameter response respectively is used as the defining criteria.  Thus,  variation in r; may
cause variation in permeability estimation and variation in r0/r; may lead variation in permeability
estimation even when r; is maintained constant.  For this reason, Tidwell and Wilson (1997) used
different inner radii (0.15, 0.31, 0.63, 1.27, and 2.54) in their investigations but maintained a
constant r0/r; ratio of 2. Also, during their study, Tidwell and Wilson  (1997) noted a consistent
exponential decrease in permeability with increasing tip seal compression.  They state that as
compression stress is increased, the seal deforms into surface irregularities preventing gas flow
from  short-circuiting between the rock-seal interface.  Thus, compression stress tests must be
determined for each tip seal prior to permeability estimation. Acceptable compression then is the
point where permeability estimation shows little decrease with increased compression.
                                           51

-------
4.5    Formulation of a One-Dimensional Transient Gas Flow Equation with Gas Slippage in a
       Semi-Infinite Domain

       Derivation of an analytical solution for one-dimensional transient gas flow in a semi-
infinite domain might be useful to evaluate pressure propagation as a function of time in semi-
infinite media. The governing equation for one-dimensional transient gas flow is given by
       dt    dx
A solution can be derived when equation (4.20) is subject to equation (4.2) and
      = at
    , V) = atm •
Use the Kirckhoff transformation modifies equation (4.21) to
 //

        dt
                        du
                     W17
Allowing gas permeability to be constant,  •/$ = Patm and

a =
                                                                                 (4.20)
                                                                                 (4.21)


                                                                                 (4.22)
                                                                                 (4.23)
                                                                                 (4.24)
results in

 dU     d2U
                                                                                 (4.25)
with boundary and initial conditions:
 dx
where
                                                                                 (4.26)
                                           52

-------
                                                                                  (4.27)
 lim U = 0
 X—>°»
[7(jc,0) = 0.
                                                                   (4.28)
                                                                   (4.29)
Transforming the governing equation and boundary conditions to a Laplace Domain in time (s)
and use of the inverse Laplace Transform from Abramowitz and Stegun (1970)
                                                                                  (4.30)
where k is a constant results in
2j-e"to—?=erfc\   *
                                                                                  (4.31)
where pressure is as defined in equation (4.7) and Patm is substituted for Pref. Equation (4.31) was

checked by plugging it into (4.25) and observing no residual.

4.6    Formulation of a One-Dimensional Transient Gas Flow Equation With Gas Slippage in a
       Finite Domain

       Now consider a finite domain where equation (4.20) is subject to equation (4.22),
equation (4.2) at x = L, and

0(0,0 = 0flftB                                                                      (4.32)

Again, use of the Kirckhoff transformation results in equation (4.25), with initial condition
equation (4.29), a flux boundary at  x = L similar to equation (4.26), and
[7(0,0 =
The solution is given by Carslaw and Jaeger (1959), p. 113 as:
                                                                   (4.33)
                                           53

-------
                                            •                                     .....
                                           sin  - - - -                         (4.34)
                ;r2     (211 + 1)2                     2L

where equation (4.7) expresses the solution in terms of pressure.  This solution could have
potential in permeameter testing of low permeable media where pressure transducers are placed
along the axis of flow.

4.7    Formulation of One-Dimensional Transient Gas Flow Equations Without Gas Slippage in
       a Finite Domain With Time-Dependent Boundary Conditions

       Analytical solutions for one-dimensional, transient gas flow with time-dependent
boundary conditions are useful when using variation in barometric pressure to estimate gas
permeability in one or more layers. The method is based upon the observation that when
atmospheric pressure changes  at land surface, gas moves to or from the vadose zone to maintain
a pressure balance between gas in the soil and the atmosphere.  The rate of gas movement and the
resultant rate of pressure change at depth are affected by both the gas permeability and gas-filled
porosity of materials in the vadose zone (Weeks, 1978).

       Movement of gas to and from the vadose zone due to variation in barometric pressure
was first analyzed by Buckingham (1904). He presented an equation for the attenuation of the
amplitude and  phase lag of a periodic atmospheric pressure wave at any depth in a homogeneous
layer bounded below by an impermeable boundary (e.g., water table).  Later Stallman (1967) and
Stallman and Weeks (1969) measured variation in barometric pressure and pressure variation at
depth to determine in situ vertical gas permeability.  Their method was based on the  assumption
that the unsaturated materials comprised a single homogeneous layer.  Using the same
assumptions, Roza et al. (1975) used an analytical solution and the principle of superposition to
determine gas permeability of material comprising several nuclear chimneys (vertical sections of
bedrock containing rubble caused by subsurface nuclear explosions) at the Nevada Test Site.
The nuclear chimney rubble was assumed to consist of a homogeneous unit extending to infinity
below land surface.  Although the assumption that air movement  can occur to infinite depth did
not accurately represent actual boundary conditions, computed gas permeabilities compared well
with those determined by numerical analysis of air injection data. Weeks (1978) used the
methods of Stallman (1967) and Weeks and Stallman (1969) to estimate the gas permeability of
discrete layers. He determined the gas permeability of each layer through trial and error. He
ignored gas compressibility, assuming that it would result in insignificant error due to the
relatively small magnitude of barometric pressure variations.

       Consider  the one-dimensional gas flow equation without gas slippage (since pressure
differential is very low)
                                          54

-------
            ..
       at          dx


subject to an initial condition allowing pressure variation with distance
                                                                                    (4.36)
a no flux boundary at the water table (x = 0)
       = 0                                                                          (4.37)
 3**=o


and time-dependent pressure variation at the surface (x = L)


(f>(L,t} = f(t)                                                                       (4.38)


 The solution is given by Carslaw and Jaeger (1959), p. 104 as:
   c, t) = — > e     4L     cos  —
         T *—*                     IT
         -*-'   r\                \     ^-Lj
           n=0                ^-
                    e   4L     f(t ')dt '+    f(x ') cos
                                V;        ^ V  J
                                                         2L
where without gas slippage
                                                                                    (4.39)
If pressure variation is allowed at both boundaries then


0(0,0 = /i (0                                                                       (4.41)


0(L,t) = f2(t)                                                                       (4.42)


The solution for this situation is given by Carslaw and Jaeger (1959), p. 104 as:
                                             55

-------
               -an n t     ,     x
                ^<,nfel
                                                                                  (4.43)
Shan (1995) used these two solutions to present two scenarios for testing: (1) a domain
consisting of a region between the water table and the soil surface or at some depth in soil, and
(2) a domain between any two points within the soil.  The latter scenario provides a method to
evaluate pneumatic permeability in discrete layers. Both methods require a minimum of three
measurement points; one on each boundary and one between boundaries.

4.8    Formulation of the Pseudo-Steady-State Radial Gas Flow Equation

       The pseudo-steady-state radial flow equation is the equation most commonly used by
venting practitioners to estimate gas permeability in the field. The governing ordinary
differential equation  describing  radial, pseudo-steady-state, confined gas flow is expressed as:

  r\
 d d)   1 d(b                                                                       ,, „ „,
—Jf +	31 = 0                                                                   (4.44)
 dr    r dr

with boundary conditions:

0fa) = &                                                                         (4.45)
                                                                                  (4.46)
       rw    -<" g    rgnl(r)


The solution for the pseudo- steady- state flow equation is easily derived and given by:
                                   r,>r                                          (4.47)
           Mgxdkrgkl(r}

where ^ = pressure squared at ^ [(ML"1!"2)2], rj= arbitrary radial distance (radius of influence or
ROI) at which a constant pressure is prescribed [L], r = radial distance from well center line [L],
rw = radius of well and filter pack [L], d = thickness of domain [L], and kj.= radial pneumatic
permeability [L2].

       Notice that in equation (4.47), when Qm is negative (air extraction), $ goes to infinity as


                                            56

-------
the radius goes to infinity. When Qm is positive (air injection), 4> goes to negative infinity as the
radius goes to infinity.  Thus, the prefix "pseudo" reflects the fact that there is no true steady-
state solution to the radial flow equation unless a constant pressure boundary (i.e., ROI) is
specified since flow is assumed to occur under non-leaky or confined conditions. The limitations
of using the pseudo-steady-state radial flow equation for gas permeability estimation will be
discussed in detail in section 6 where a ROI data set from an actual site is used to estimate gas
permeability.

4.9    Formulation of the Radial Transient Confined Gas Flow Equation

       The transient, radial flow equation is expressed as:
                                                                                   (448)
with boundary conditions:

         atm                                                                       (4.49)
               mg
rhm—!- = -- - —                                                           (4.50)
  '->« dr     nkrgkl(r}dMg

A solution to transient, radial flow equation using these boundary conditions, Laplace
Transforms and the assumption that AJ> = Patm gives:
                            ,    --                                                 (4.51)
              2ndkrgkl(r}Mg  J  T

where


u =	^—^	                                                                   (4.52)
    4krgkl(r}tPatm

The solution to equation (4.48) given by Johnson et al. (1988,1990a,b) is:


                                                                                   (4.53)
where u is a previously defined and P' is a gauge pressure. The solution is obtained in P

                                            57

-------
instead of P2 because Johnson et al. (1988, 1990a,b) linearized the radial, transient equation by
letting P2=P*Patm instead of explicitly solving for P2. As discussed in section 3.10, this
linearization approach can lead to significant error when pressure differential exceeds 0.5
atmospheres. The integral on the right-hand-side of the equation is the well known exponential
integral.  When u < 0.01, it can be approximated by:
        = -0.57721- In u                                                           (4.54)
Regardless of the linearization procedure, derivations of both equations (4.51) and (4.53) require
homogeneous conditions in addition to: (1) strict radial flow, (2) completely impermeable upper
and lower boundaries (no leakage from the surface), (3) no wellbore storage (line sink).  Beckett
and Huntley (1994) modified the Hantush and Jacob  (1955) equation for transient radial flow in a
semi-confined domain by applying the linearization P2= P*Patm

4.10   Conclusions

1.      While not representative of fully three-dimensional field-scale gas flow, one-dimensional
       testing and  analysis can provide insight into fundamental processes such as the effect of
       compressibility and gas slippage on pressure and pore-gas velocity profiles. Analysis of
       one-dimensional gas flow is especially important in designing and interpreting laboratory
       column vapor mass transport studies where knowledge of pore-gas velocity profiles
       becomes critical. Because field-scale tests provide estimation of gas permeability over an
       integrated volume of porous media too large to discern small discrete layers of less
       permeable materials (e.g., lenses of silt and clay), gas permeability estimation in
       minimally disturbed or reconstructed samples could be useful in determining small-scale
       gas permeability variation in subsurface media.

2.      When attempting to evaluate small-scale gas permeability variation in the field, use of
       permeameters such as those described by Corey (1986), Stonestrom (1987), and Springer
       et al. (1988) provide the opportunity of determining capillary pressure - gas permeability
       curves but require repacking of soil samples thereby compromising pneumatic sample
       integrity. Use of shelby tubes during core collection allows collection of  "minimally"
       disturbed cores, but elimination of short-circuiting along the tube wall can not be
       guaranteed  during testing. It would appear that while all laboratory-scale methods
       involve some degree of compromise in sample integrity, there is currently no other means
       to estimate  gas permeability on the scale of centimeters in the field.

3.      The governing equation for one-dimensional, steady-state gas flow incorporating gas
       slippage was derived and used to assess the impact of gas slippage on one-dimensional
       pressure and pore-gas velocity simulation. During gas extraction, neglecting gas slippage
       results in underestimation of absolute pressure whereas during air injection, neglecting
       gas slippage results in overestimation of absolute pressure.  In both cases though, pressure

                                            58

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differential from atmospheric pressure is overestimated when neglecting gas slippage.
For both gas extraction and injection, variation in pressure is greatest at the point of
extraction or injection and at lower gas permeability values.  However, the magnitude in
error in pressure computation for gas injection was far lower than for gas extraction
because of higher absolute pressures during gas injection.  Even at the lowest gas
permeability, maximum error for gas injection did not exceed 6.0% whereas during gas
extraction, error exceeded 80%. Thus, if gas slippage is neglected, gas injection should
result in a better estimate of gas permeability than gas extraction.  Pore-gas velocities
during gas injection were much lower than during gas extraction at the same flow rates
because of increased gas density. Error in pore-gas  computation neglecting gas slippage
during gas injection was much  lower than during gas extraction.  Similar to gas
extraction, error during gas injection increased with decreased permeability. However,
unlike gas extraction, error increased with distance from the point of injection.
Maximum pore-gas velocity computation error during simulations was less than 13%
whereas it exceeded 600% during gas extraction simulations.  Thus,  for vapor transport
column studies, it would seem  that  lower and better controlled pore-gas velocity profiles
can be attained through gas injection.
                                     59

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5.     TWO-DIMENSIONAL, AXISYMMETRIC, STEADY-STATE GAS
       PERMEABILITY ESTIMATION, PORE-GAS VELOCITY CALCULATION,
       AND STREAMLINE GENERATION IN A DOMAIN OPEN TO THE
       ATMOSPHERE

       Field-scale estimation of gas permeability and subsequent computation of pore-gas
velocity profiles are critical elements of sound soil venting design. Many venting practitioners
however are unaware of equations and data interpretation methods appropriate for gas
permeability estimation and pore-gas velocity computation. In this section, the governing
equation for unconfined (open to the atmosphere), steady-state, axisymmetric cylindrical flow is
presented along with an analytical solution derived by Baehr and Hult (1991).  Data collected
from a U.S. Coast Guard Station in Traverse City, Michigan is used to demonstrate gas
permeability estimation, pore-gas velocity calculation, and streamline generation in a domain
open to the atmosphere.

5.1    Model Formulation

       The governing equation for unconfined (open to the atmosphere), steady-state,
axisymmetric cylindrical flow is:
When subject to the following boundary conditions:

(r,Q} = atm                                                                   (5.2)

     >(r,z) = (t>atm                                                                (5.3)
   -(r,d) = 0                                                                   (5.4)
 dz
                 0
-------
Baehr and Hult (1991) provide a solution as follows
          M  n rw&  kl(r)(dL-du}
where

m = n — 1 / 2,
                                   I1
                                          cos
                                                      - cos
                                      I
                                                     i  --
                                                d    [_ d   e
                                                IV-   1IHV I  1  .  \
                                                Knl —	sin
I krgkl(r)
                                                             (5.9)
and z = vertical distance below land surface [L], dv = vertical distance to top of sand pack or well
screen [L], dL = vertical distance to bottom of sand pack or well screen [L], k,.gkl(z) = vertical gas
permeability [L2], K0 = zero-order modified Bessel functions of the second kind, Kx = first-order
modified Bessel functions of the second kind, atm = atmospheric pressure squared [(ML"1!"2)2].
Shan et al. (1992) solved equation (5.1) by approximating the well as a line sink of uniform
strength. They argue that while this approximation leads to errors in simulating the pressure
distribution in the immediate vicinity of a well screen, these errors are not expected to be large.
Baehr and Hull's (1991) finite-radius solution though is generally preferred because it is more
accurate near a wellbore, easy to program, and is utilized in the public domain gas permeability
code AIR2D (Joss and Baehr, 1997). However, further work on steady-state analysis of gas flow
which shows promise for field application continues (Warrick and Rojano, 1999)

       Radial specific discharge, qr [LT"1], is calculated by
         krgkl(r)
       2   fig
                                                                                   (5.10)
where
       Mgnkrgkl(r) (dL - dv }rw
                 cos
                                           mndn
                                           - —
                                             d
                             - cos
 (mndL
\  d
                           (mn rw
                         1  ~^	
                           I  d  e ,
                                            r ( mn r |  .  fmnz }
                                            -i\ —— I sin |	I
               d  e
Vertical specific discharge, qz [LT"1], is calculated by
d  )
                                                                                   (5.11)
         krgkl(z)
                    dz
                                                                                   (5.12)
                                            61

-------
where
                                    cos
                                      7 ^     ( mnd,
                                      — - cos 	-
                                                 d
( rrnt r \
 	cos
{_ d  ej
                                                                               mnz \
                                                                               	
                                                                                e  )
                                                                                  (5.13)
Pore-velocity is calculated by dividing specific discharge by the volumetric gas content
conducting air flow.  For venting design though, it is the norm, q [LT"1], of the vector pair that is
of interest which is calculated by
                                                                                  (5.14)
       In a domain open to the atmosphere, streamfunction computation is helpful to visualize
the fraction of flow to a well as a function of radial distance. In homogeneous isotropic media,
Cauchy-Riemann equations can be used to generate orthogonal families of equipotential and
streamline curves. In anisotropic media, equipotential and streamline curves are not orthogonal
but equations relating potential and streamfunction can be expressed by (Bearl972, p.235):
                                                                                  (5.15)
 dr   rkrgkl(r)     dz
  i ,   ,    \l/2
1  Krgkl(r} I    9^
r \ Jf  Jf
r I KrgKl(z)
                                                                                  (5.16)
Solving for the streamfunction i); results in
         M  n k  kl(r)(dL -dv)rw n=l m
                                          cos
                                                      - cos
                                                               d

                                                                 mn
                                                                                cos -
                                                                                     mnz
                                                                  d e)    ^  d
        +const
                                                                                  (5.17)

Examination of this equation reveals that the streamfunction is undefined at r = 0 since

 lim^1(r) = oo.                                                                   (5.18)
                                           62

-------
Also, i|/(r,b) equals a constant streamline along the lower impermeable boundary [cos(rmi)=0] or
capillary fringe which will be set to zero. Normalized streamfunction values can now be
calculated by dividing by the calculated streamfunction by the maximum streamfunction value.
The difference in normalized streamfunction values then is equivalent to percent flow to the
venting well.  The analytical solution presented here for computation of the streamfunction is for
a finite-radius well which differs from Shan's et al. (1992) solution for a line source/sink.

5.2    Computational Method

       In the public domain program AIR2D (Joss and Baehr, 1997) a non-linear data fitting
algorithm
            N                    2
minimize  £ (^,zJ-F(^,zj)                                                (5.19)
            2=1
is used to estimated kjc^ and k,.gkl(z). The method serves to minimize the objective function,/
where  P(rt , zi ) is the estimated pressure and P(rt , zt ) is the observed pressure at the observation
point. Unlike linear fitting least squares fitting however, non-linear least squares fitting does not
guarantee the minimum least squares solution. Non-linear fitting algorithms sometimes find a
local minimum rather than a global minimum which represents the true least squares solution.
Thus, good starting estimates are essential for non-linear algorithms. A different approach to
estimation of kj.gkl(r) and kj.gkl(z) was pursued here to circumvent, to some degree, problems with
finding a global minium. Random guesses (groups of 5000) constrained within decreasing
intervals of k^k^ and k,.gkl(z) were used until the lowest root mean squared error (RMSE) defined
RMSE = \-	                                                 (5.20)
was attained and the best 10 to 100 random guesses for k,.gkl(r) and k,.gkl(z) were constant. For
instance, the first 5000 guesses were constrained for k^k^ values ranging from 1.0 x 10"10 to 1.0
x 10"06 cm2 with k^k^/k^k^ ratios allowed to vary from 0.1 to 10.0. Examination of the RMSE
for the top 100 guesses quickly revealed that by far the best guesses centered around a variation
of k,. between 1.0 x lO'07 to 1.0 x lO'06 cm2 with kj^/kjc^ ratios between 1.0 and 5.0. The next
5000 guesses were then constrained within these limits to further narrow the estimation range.
This procedure was continued 3 more times for a total of 25,000 guesses until estimates of k^k^
and kj.gk^/kj.gk^ ratios and RMSE for top 100 guesses were identical. Using this procedure, each
guess can be individually examined to ensure that the set of lowest RMSE values do not give
significantly different k^k^ and kj.gk^/kj.gk^ estimates.  While somewhat tedious (taking 10 to
15 minutes for each flow rate in this application), this approach provided considerable  insight
                                           63

-------
into the parameter estimation process and confidence in parameter estimates.  Huang et al. (1999)
used a trial-and-error approach for estimation of gas permeability and also argued that this type
of approach offers the modeler insight into the parameter estimation process.

       A FORTRAN program, MFROAINV, was written to facilitate parameter estimation
computations.  The source code and sample input and output files are included in Appendix A.
Another FORTRAN program, S AIRFLOW was written to facilitate computation of pore-gas
velocity and streamlines for one well in a domain open to the atmosphere. The source code and
sample input and output files are included in Appendix B.

5.3     Site Description

       A series of gas permeability tests were conducted at the U.S. Coast Guard Aviation
Station in Traverse City, Michigan in April, 1990 to aid in design of a soil venting project.  The
Coast Guard reported that contamination of soil and ground water had been caused by an
accidental release of an estimated 132,000 liters of aviation fuel in 1969.  The spill created a
ground-water plume about 80 meters wide and 300 meters long.  Previous soil sampling efforts
revealed that most of the mass from the spill was confined to a narrow band above and below the
water table.  Soils were characterized as well-sorted, coarse-grained sand with an mean grain size
of 0.35 mm. At the time of testing, the water table was located 5.2 meters below land surface.

       Wells and vapor probe clusters were installed into 15 cm diameter open boreholes drilled
using hollow-stem augering.  Two wells constructed of 12.5 cm diameter, schedule 40 PVC pipe
were placed 6.1 meters apart. Factory slotted screens were set at 3.9 to 4.2 m below a
topographically flat ground surface. As illustrated in Figure 5.1, vapor probes clusters were
placed along a line between the two wells. Each vapor probe cluster consisted of three 5 cm
long, 6.35 mm diameter stainless steel screens connected to the surface by 6.35 mm copper
tubing and quick connectors. Probes in each cluster were placed at depths of 0.9, 2.7, and 4.2 m
below ground surface. Wells and probe clusters were completed by backfilling native sand into
open boreholes. Pressure gauges were installed on both wells to allow applied pressure or
vacuum monitoring at the wellhead.  Pressure differential in vapor probes were measured with
magnehelic gauges.  The lowest pressure differential that could be accurately measured with the
magnehelic gauges was 0.13 cm or 0.05 inches of water or 0.00097 atm as wind appeared to
affect readings below that level.  Flow was measured with pitot tubes installed upgradient of
wells.

5.4     Results and Discussion

       Seven vacuum extraction tests were conducted to estimate radial and vertical gas
permeability. Table 5.1 provides a summary of applied flow and pressure response while Table
5.2 provides a summary of radial and vertical gas permeability estimates for each test.  Figure 5.2
illustrates observed versus simulated pressure response and a corresponding correlation
coefficient (R2) using linear regression for test 1. The other 6 tests showed similar responses and
                                           64

-------
Blower ^x-BaU
Infection j^s^ Valve waii p»o«aiifp
i ^gg Well ricaSUIV
(•

4 bch Schedule 40/
PVCPipe — '

Schedule 40
WC Pipe

•






=








Pressure Pro






K Cluster

OD Copper
Tubing




Quick
Connector






? "\f .





=

m

Blower
Extraction


i
0.3m
T
Figure 5.1 Gas permeability testing schematic at the USCG Station, Traverse City, MI

correlation coefficients. Plots of observed versus simulated pressure response did not visually
indicate the presence of any systematic bias. Generally, as indicated by R2 values in Table 5.2,
observed versus simulated pressure response was good. Figure 5.3 illustrates a two-dimensional
cross-sectional  schematic of observed versus simulated pressure differential for test 1. Again, the
other 6 tests showed a similar response. Observed versus simulated pressure differential showed
close agreement providing additional confidence that gas permeability was correctly estimated
and that relatively homogeneous conditions existed. The low standard deviation and 95%
confidence interval (CI) for both k,.gkl(r) and kjc^/kjc^ ratio indicate reproducibility of results
over a  wide span of flow rates.  There was no observable trend in permeability estimation as a
function of applied flow and vacuum as would be expected from Klinkenberg (1941) effects
(increased permeability with increased vacuum) or soil-water movement toward the well
(decreased permeability with increased vacuum).  Klinkenberg (1941) effects were not expected
because of the high permeability of these sandy soils.  Soil-water movement toward the well due
to vacuum application was not expected to be significant because the sand appeared to be coarse-
grained and well sorted meaning that fairly low capillary pressure would be needed to drain these
soils to a low soil-water content. Examination of drill cuttings during well and probe installation
appeared to indicate relatively homogeneous sand deposits. There was no indication of change in
texture with depth. Thus k^k^/k^k^ ratios near 1.0 were expected and reasonable.
       Figure 5.4 illustrates a two-dimensional, cross-sectional plot of pressure differential,

                                           65

-------
Table 5.1 Test results USCG Station, Traverse City, MI
r
(cm)

152.4
152.4
152.4
304.8
304.8
304.8
457.2
457.2
457.2
609.6
609.6
609.6
914.4
914.4
914.4
z
Testl
Test 2
(cm) (75. 11 g/s) (66.57 g

P
(atm)
91.44 0.99803
274.
426.
32 0.99434
72 0.98574
91.44 0.99877
274.
426.
32 0.99680
72 0.99459
91.44 0.99926
274.
426.
32 0.99828
72 0.99730
91.44 0.99931
274.
426.
32 0.99877
72 0.99867
91.44 0.99961
274.
426.
32 0.99941
72 0.99921
P (atm)
0.99803
0.99434
0.98598
0.99877
0.99680
0.99434
0.99914
0.99828
0.99730
0.99924
0.99872
0.99865
0.99961
0.99948
0.99924
Test3
Test 4
i/s) (56.90 g/s) (47.23 g/s)
P (atm)
0.99852
0.99508
0.98820
0.99902
0.99730
0.99508
0.99939
0.99840
0.99791
0.99963
0.99926
0.99926
0.99970
0.99951
0.99934
P
0.
0.
0.
0.
0.
0.
0.
0.
(atm)
99877
99607
99066
99926
99779
99607
99951
99877
0.99828
0.
0.
0.
0.
0.
0.
99948
99911
99907
99970
99956
99946
Tests
Test 6
Test?
(38.69 g/s) (32.43 g/s) (32.43 g/s)
P (atm)
0.99926
0.99730
0.99361
0.99951
0.99877
0.99730
0.99951
0.99902
0.99862
0.99961
0.99936
0.99936
0.99975
0.99970
0.99966
Table 5.2 Gas permeability test summary for USCG Station, Traverse
Test
Flow
Pressure k^k^ krgk-i(Z) kl(r)/l
li(z)
RMSE
R2
P
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
City,

(atm)
99939
99779
99496
99963
99902
99791
99956
99921
99882
99970
99951
99951
99988
99988
99975
MI

P (atm)
0.99963
0.99840
0.99607
0.99951
0.99902
0.99813
0.99970
0.99936
0.99904
0.99975
0.99961
0.99956
0.99988
0.99988
0.99975


at well

1
2
3
4
5
6
7
(g/s)
75.11
66.57
56.90
47.23
38.69
32.43
32.43
(atm)
0.877
0.877
0.897
0.911
0.941
0.951
0.961
(cm2) (cm2)
6.
5.
5.
5.
7.
7.
1.
Average 7.
10E-07 4.
59E-07 4.
27E-07 4.
89E-07 1.25
24E-07 1.32
61E-07 1.14



92E-07 4.41E-07 1.34
50E-07 4.
97E-07 1.51

65E-07 5.47E-07 1.40
12E-06 5.
03E-07 4.
StdDev 2.05E-07 5.


95%
CI 1
52E-07 4.
75E-07 1.95
91E-07 1.42
50E-08 0.26
07E-08 0.19




(atm)
6.50E-04
6.06E-04
5.24E-04
3.70E-04
2.75E-04
2.06E-04
1.45E-04
3.97E-04
2.00E-04


0.97
0.96
0.97
0.97
0.96
0.97
0.98




































                                           66

-------
            1.002 -|

            1.000 -

            0.998 -

         Is, 0.996 -

         1  0.994 -
         W
         ^  0.992 -
         "3
         •g  0.990 -
         s
            o.

            0.986 -

            0.984
                                                       IT = 0.9673
                                                                              1.002
                                                                              75.1 1 g/s
                                                                                   = 1.2,
                0.984   0.986  0.988  0.990  0.992  0.994  0.996  0.998  1.000
                                       Observed Pressure (atm)

Figure 5.2 Observed versus predicted pressure for test. Applied flow and vacuum
(131.8 scfm) and 0.877 atm respectively. Estimated k,gkl(r) = 6.1x 10"07 cm2,
RMSE = 6.50x 10-°4atm
pore-gas velocity (assuming a volumetric gas content of 0.2) , and streamlines for test 1 .  Each
streamline originates at ground surface and terminates at the well screen.  By convention, the top
streamline, extending from r = 0, z = 0 to the well screen has a streamline value of one while the
bottom streamline, extending from r = °°, z = d, and up the well screen has a value of zero.  This
convention was used here and by Falta et al.  (1993) but is the reverse of that used by Shan et al.
(1992).  In Figure 5.4, the area between pairs of streamlines represents 10% of flow going to the
well.  Iso-contours for pressure differential, velocity, and streamlines are all highly curvilinear
due to the constant pressure boundary at the  surface and partial penetration of the gas extraction
well.  Streamlines indicate that most gas recharge occurs in the immediate vicinity of the well
(e.g.,  70% of gas recharge occurs within 8 meters of the well).  Pore-gas velocities are very high
throughout the tested domain because of high gas permeability and applied flow rate (75. 1 1  g/s
or 131.8 scfm).  These types of plots are useful for visualizing flow to a well. For instance,
Wilson et al. (1988) and Shan et al. (1992) used plots of gas  pressure and streamlines to argue
that little gas flow occurs below a well screen suggesting that the screened intervals should
always be placed below a zone of contamination.

       Figure 5.5 illustrates pore-gas velocity along a vertical transect (depth at center point of
well)  as a function of radial distance and applied flow using average measured k,. and the
kj.gk^/kj.gk^ values. Note that pore-gas velocity is on a logarithmic scale to encompass orders of
magnitude decrease with increasing radial distance. An increase in mass flow at small applied
                                            67

-------
 g -100	
 o
 c/3 -i
§

    200	-
   -300—
£
   -400-
 C/5
• i—H
Q
   -500-
                                                                                       0.41
            \   \
 /  '  I  »
 /  i I  I
1  T!  !
                 1
\ \  5.6

 \  *   \
 \  »   \
 I  '   I
                                      \    \
                                            I   1
                                            I    I
                                                             l.Vt
                                                             1  >  "

\     I
               100     200
                                 300      400     500     600
                                         Distance from well (cm)
                                                     700
                                                                                       0.81

                                                800     900     1000
     Figure 5.3 Observed (+) versus simulated (dashed lines) pressure differential (cm water) for test 1

-------

-500
             100      200      300     400      500      600     700      800      900     1000
                                        Distance from well (cm)
        Figure 5.4.  Calculated pressure differential (cm of water - dashed lines), pore-gas velocity (cm/s - dash-dot
        lines), and streamlines (solid lines) for test 1.

-------
   0
                                                                   	0.57g/s(l scfm)
                                                                   	2.85 g/s (5 scfm)
                                                                   	5.70 g/s (10 scfm)
                                                                   	11.40 g/s (20 scfm)
                                                                   	17.1 g/s (30 scfm)
                                                                         •22.8 g/s (40 scfm)
                                                                         • 28.5 g/d (50 scfm)
200     400
600     800     1000    1200    1400    1600    1800    2000
      Distance from well (cm)
Figure 5.5 Pore-gas velocity along a transect (centerline of well) as a function of applied flow and radial
distance using average k,. and k^k^/k^k^ ratio

-------
flow rates (from 0.57 to 2.85 g/s) resulted in a significant increase in pore-gas velocity
throughout the modeled domain. An increase in flow at high flow rates (22.8 to 28.5 g/s)
however resulted in little additional increase in pore-gas velocity since most recharge, as
indicated by Figure 5.4, is in the immediate vicinity of the well. Pore-gas velocity plots are
useful to estimate initial well spacing based a desired flow rate and pore-velocity.  For instance,
at a flow rate of 2.85 g/s, attainment of a pore-gas velocity of 0.01 cm/s may require an
approximate well spacing of about 200 cm (2 x  100 cm) whereas at a flow rate of 11.4 g/s, an
approximate well spacing of 400 cm might be acceptable.  However, because of pressure
superposition effects, multiple well simulations would be necessary to more accurately space
wells as a function of desired flow rate. It is obvious from Figure 5.5 though that there is a
tradeoff between well spacing and flow rate as one would  intuitively expect.

5.5    Conclusions

1.     Data from a site was used to demonstrate radial and vertical gas permeability estimation
       in  soils open to  the atmosphere. Through discussion and example, a partial list of
       elements necessary for sound gas permeability estimation was provided: placement of
       narrowly screened pressure monitoring points close to a discretely screened gas extraction
       well to capture the vertical component of gas flow, testing at several flow rates to
       establish reproducibility, analysis of error using the RMSE of observed versus simulated
       pressure response, and plots of observed  versus simulated pressure response to provide
       confidence is permeability estimation.

2.     An illustration of how information on gas permeability can be used to generate plots of
       pore-gas velocity as a function  of distance and flow rate, essential in venting design, was
       provided.  Field-testing and data interpretation methods outlined here in addition to
       available software programs such as AIR2D (Joss and Baehr, 1997) and GASSOLVE
       (Falta, 1996) should be useful to practitioners tasked with estimating gas permeability.
                                            71

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6.     TWO-DIMENSIONAL, AXISYMMETRIC, STEADY-STATE GAS FLOW,
       PORE-GAS VELOCITY CALCULATION, AND PERMEABILITY ESTIMATION
       IN A SEMI-CONFINED DOMAIN

       In section 4, the governing equation for one-dimensional, pseudo-steady-state radial flow
was presented. As previously discussed, the prefix "pseudo" reflects the fact that there is no true
steady-state solution to the radial flow equation unless a constant pressure boundary (i.e., ROI) is
specified since flow is assumed to occur under non-leaky or confined conditions.  There are two
primary concerns with use of the pseudo-steady-state radial flow equation and ROI data for gas
permeability estimation. First, one-dimensional, transient and steady-state radial flow equations
may result in higher estimates of radial gas permeability compared to solutions allowing vertical
leakage.  Strict confined conditions rarely if ever occur in the field.  Even at  sites where a
concrete or asphalt cap exists, small cracks or flow through a gravel subbase can result in
significant recharge from the atmosphere.  Beckett and Huntley (1994) examined vertical leakage
at five sites having asphalt and concrete caps and found that "vertical leakage was the rule, rather
than the exception". Beckett and Huntley (1994) compared estimates of radial gas permeability
using the Theis equation (1935)  (fully confined flow) and the Hantush-Jacob equation (1955)
(radial flow with vertical leakage).  Both equations were modified for gas flow by invoking
linearization of the dependent variable, P2 = P*Patm, thereby removing consideration of
compressibility.  Johnson et al. (1988, 1990a,b) describe this linearization for the Theis equation.
Beckett and Huntley (1994) found that overestimation of radial gas permeability increased with
increased leakage at the surface. At one site, use of the Theis equation (1935) compared to the
Hantush-Jacob equation (1955) resulted in overestimation of radial gas permeability by a factor
of 20.  This is likely due to the fact that for a given flow rate, confined flow or semi-confined
flow with low leakance results in a higher pressure differential compared to unconfmed flow.

       Second, and perhaps more importantly though, is that ROI test data may not be suitable
for estimation of anisotropy, leakance, and subsequent computation of axisymmetric cylindrical
(r,z) or Cartesian (x, y, z) components of pore-gas velocity. During ROI testing, gas monitoring
wells are often located so far away from gas  extraction wells that resulting pressure differentials
are too low to distinguish from atmospheric pressure.  Chen (1999) demonstrated that sensitivity
of both radial and vertical gas permeability estimation decreases considerably with increasing
radial distance from a gas extraction well.  When pressure differential approaches zero, gas flow
is no longer sensitive to radial and vertical gas permeability values.  Also, during ROI testing,
vacuum extraction wells and monitoring points are screened over large portions of the vadose
zone providing vertically integrated measurements of pressure differential as opposed to point
values required for vertical permeability and leakance estimation using analytical axisymmetric
cylindrical or three-dimensional  numerical models. Chen (1999) demonstrated that screening a
gas extraction well deep within and over a relatively small  portion of the modeled domain
enhances vertical gas permeability estimation.  Gas extraction wells used for ROI evaluation are
typically screened over a large portion of the vadose zone.  ROI monitoring practices all stem
from a one-dimensional visualization of gas flow and attempting to "physically" locate a radius
where pressure differential is near atmospheric pressure.
                                           72

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       In this section, the governing equation for two-dimensional, axisymmetric, steady-state
gas flow in a semi-confined domain will be presented along with an analytical solution derived
by Baehr and Joss (1995). This solution will be applied to a site where radius of influence (ROI)
testing and the pseudo-steady-state radial flow equation were used to estimate gas permeability.

6.1    Model Formulation

       The governing partial differential equation for axisymmetric cylindrical gas flow is
k  kl(r} \Z-Z-+ -ZZ- \ + kkl(z)^-j- = Q                                              (6.1)
       ^ dr   r  dr J         3z


with boundary conditions:


k  kt,z) —^- =  rg     (6 — (f)atm)     r>rw,z = 0                                     (6.2)



30   n                T                                                          /,; Ox
	 =0     r>rM;,  2 = «                                                          (°-3)


lim   = oo


90                       ,                                                       /^ ^x
—1- = 0     r = rw,  Q           L
  r    xkrgkl(r}Mg (dL -du)rw


|^ = 0     r = rw,  dLatm = atmospheric pressure squared [(ML"1!"2)2], z = depth below semi-confining layer
[L], d = depth below semi-confining layer to ground water [L], dv = depth below semi-confining
to top of sand pack [L], dL = depth below semi-confining layer to base of sand pack [L], k^k^ =
vertical gas permeability below semi-confining layer [L2], k^k^ = vertical gas permeability in
semi-confining layer [L2], and d' = thickness of semi-confining layer [L]. Baehr and Joss (1995)
incorporate the leakance term (k^k^/d') in boundary condition (6.2) as opposed to adding a
leakance term in the governing partial differential equation as previously done by Baehr and Hult
(1991). Baehr and Joss (1995) state that boundary condition (6.2) is an approximation of the
conservation of mass principle.  It is obtained by using a finite-difference approximation of

                                            73

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specific discharge across the thickness of the layer of lower permeability and equating the
resultant expression to the vertical component of mass flow defined in the domain as z
approaches zero. Baehr and Joss (1995) state that this is a more rigorous approach than adding a
leakance term to the governing partial differential equation. They did not however provide an
analysis of error incurred by using the stated less rigorous approach.  The solution provided by
Baehr and Joss (1995) for  axi-symmetric, cylindrical, steady-state, leaky confined flow is given
by:
                                                                                   (6.8)
where
                                     .                                             (6.9)
                                     *
                             If  If
                             KrgKl(z)
       .  (qn(d-du)
      sm  - - - -  -sin
                                                                                   (6.10)
and qn are positive solutions (n = 1,2,3...) to


tan (
-------
The norm or magnitude of the specific discharge vector is as previously defined in section 5.
Pore-gas velocities are defined in terms of the norm of the specific discharge vector since it
represents the overall magnitude of pore-gas velocity through soil and reduces complications
incurred with analyzing two- and three-dimensional vector components of flow.

6.2    Computational Method

       When attempting to estimate three parameters (k,.gkl(r), k,.gkl(z), and leakance) with few
pressure measurements (some of which are of minimum usefulness because of near atmospheric
pressure response), convergence to a local minimum and estimation non-uniqueness are major
concerns. To circumvent these problems to some degree, the estimation approach described in
section 5 was pursued here.  A series of random guesses was applied (groups of 5000)
constrained within decreasing intervals of k^k^, k,.gkl(z), and leakance until the lowest root mean
squared error (RMSE) was attained for a set of k^k^, k,.gkl(z), and leakance estimates. In this
way, variation in parameter estimations for the same or similar RMSE values were assessed.
Little variation in parameter estimates would presumably be an indication of a satisfactory
estimation.  A FORTRAN program, MFRLKINV, was written in to facilitate computations. The
source code and sample input and output files for this program are included in Appendix C.
Huang et al. (1999) state that the inverse problem is usually ill posed and model structure error,
which is difficult to estimate, often dominates other errors. As will be discussed, this
observation proved valid at this site when attempting to compare parameter estimates from open
and semi-confined domains.

6.3    Site Description

       To demonstrate how ROI testing constrains gas permeability estimation, an attempt was
made to estimate radial and vertical gas permeability and leakance through a semi-confined layer
at a field site where ROI testing was used for venting design.  The field site is located in an urban
area in sandy soils along the Atlantic Coastal Plain. An asphalt layer (parking lot) is present on
the site surface. Tables 6.1.a summarizes depths of screened intervals of gas extraction (W -
series) and monitoring (v - series) wells.  Table 6.1 .b summarizes flow and vacuum response
during ROI testing. Depth to ground water at gas extraction wells varied from about 12 to 15
feet below ground surface. Midpoint monitoring depths adjusted for water-table elevation were
used as "point" values for gas permeability estimation. It was assumed that the thickness of the
capillary fringe was insignificant compared to the thickness of the tested domain for sandy soils
at the site. It is apparent from Table 6.1 .b that monitoring points were placed at distant radii in an
attempt to locate the radius at which vacuum approached 0.01 inches  of water  even though most
vacuum dissipation and thus highest pressure gradients occurred  within 15 feet of each well.
Magnehelic gauges were used to measure vacuum. As evident in Table 6. l.b, many readings
were at or below 0.05 inches of water.  Cho and DiGiulio (1992) found when using magnehelic
gauges they could not distinguish vacuum readings below 0.05 inches of water from ambient
atmospheric pressure using due to wind velocity induced pressure variation in the vicinity of the
gauges.  Random variation in vacuum readings at or below 0.05 inches of water during testing at
                                           75

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Table 6.1.a Summary of well and vapor probe depths

well     top of bottom of   depth    Midpoint
      sandpack  sandpack  to water Monitoring.
            (ft)       (ft)       (ft)	(ft)
v-1
v-2
v-3
v-4
v-5
v-6
v-7
v-8
v-9
v-10
v-12
W-l
W-3
W-4
W-5
W-6
W-7
4.5
5.0
5.0
4.5
4.5
5.0
5.0
5.0
5.0
5.0
5.0
6.8
6.3
6.3
7.0
6.5
5.8
9.5
10.0
10.0
9.5
9.5
10.0
10.0
10.0
10.0
10.0
10.0
20.5
18.6
16.3
24.0
14.5
15.7











14.05
14.78
12.80
15.24
12.28
11.86
7.0
7.5
7.5
7.0
7.0
7.5
7.5
7.5
7.5
7.5
7.5
10.4
10.5
9.6
11.1
9.4
8.8
W-l (31.3 ft - 66.5 ft), and W-6 (63.2 ft - 75.3 ft) support this observation.

6.4    Results and Discussion

       Estimates of radial gas permeability using the pseudo-steady-state radial flow equation
and linear regression were compared with radial gas permeability estimated using an analytical
solution developed by Baehr and Joss (1995) for steady-state, two-dimensional, axisymmetric air
flow in a semi-confined domain.  The presence of an asphalt layer does not guarantee semi-
confined conditions as a sand sub-base or fractures in the asphalt may result in significant gas
recharge near an extraction well.  Thus, gas permeability was estimated under conditions of high
leakance (i.e., unconfined) and semi-confined domains as selection of the upper boundary
condition can significantly affect vertical gas permeability estimation or anisotropy.

       Table 6.2.a and 6.2.b provide a summary of gas permeability estimates for testing at each
well assuming unconfined and semi-confined domains.  Radial permeability estimated using the
pseudo-steady-state radial flow equation is compared with radial permeability estimated using
equation (6.8) under unconfined (high leakance) and semi-confined upper boundary conditions.
In all cases but one (W-7 unconfined), use of the pseudo-steady-state radial flow equation
appeared to provide a slight but consistent overestimation of radial permeability compared to
solutions which allowed recharge at the upper boundary. However, little if any correlation
between increased leakance and overestimation of radial permeability was observed. A
comparison of radial gas permeability estimates for unconfined and semi-confined domains

                                           76

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Table 6.1.b Data for gas permeability estimation
W-l

11 scfm (6.26 g/s)
Ob s point distance
(ft)
v-10
v-9
v-8
v-4
v-5
W-3

W-4
9 scfm (5.
Obs point
v-2
v-1
v-12
v-3
v-5
v-6
W-6
15
31.3
49.5
56.8
59.9
66.5


.12 g/s)
distance
(ft)
15
30.2
45.9
50.3
70.5
85

11 scfm (6.26 g/s)
Obs point distance
(ft)
v-4
v-7
v-6
W-7
v-5
W-l
W-3
v-1 7
14.7
31.7
33.1
41
48.3
63.2
63.2
75 3

vacuum
(in water)
0.24
0.04
0.01
0.05
0.02
0.01


vacuum
(in water)
0.41
0.15
0.04
0.25
0.01
0.02

vacuum
(in water)
0.8
0.49
0.3
0.16
0.15
0.05
0.02
005
W-3
20 scfm (1
Obs point
v-5
v-12
v-6
v-4
W-7
v-7
W-6
W-5

1.3 8 g/s)
distance
(ft)
15
36
40.6
48.5
51.9
52
63.4

29 scfm (16.50 g/s)
Obs point distance
(ft)
v-8
v-9
v-10
W-3


W-7
15
32.3
49.6
51.6



24 scfm (13. 66 g/s)
Obs point distance
(ft)
v-6
v-4
W-6
v-5
v-12
W-3
W-4
15
36
40.9
41.5
42.3
51.8
88.2

vacuum
(in water)
1.2
0.31
0.33
0.27
0.16
0.1
0.12

vacuum
(in water)
0.24
0.06
0.01
0.01



vacuum
(in water)
2.9
1.5
1.1
1.1
1.1
0.24
0.02
revealed that radial permeability estimation was relatively insensitive to leakance.

       As evident in Tables 6.2.a and 6.2.b however, estimation of vertical gas permeability
expressed in the ratio of k^k^/k^k^ was very sensitive to selected boundary conditions and
illustrates the difficulty in resolving the correlated effect of leakance and anisotropy with ROI
test data. Use of (6.8) with high leakance or a domain open to the atmosphere resulted in higher
            ratios compared to semi-confined conditions.  While higher leakance and
            ratios result in similar RMSE values compared to lower leakance and k^
                                           11

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ratios, subsurface pressure and pore-gas velocity patterns for the two scenarios are quite different
and can not be resolved without vertical profiling of vacuum or pressure data.  Thus, accurate
estimation of vertical permeability and leakance requires the use of small (relative to the
thickness of the simulated domain) screened intervals for extraction wells and pressure
observation points and placement of observation points relatively close to gas extraction wells.
These criteria were not met at this site nor would be met at any site where ROI testing was
performed to design a venting system. Given this situation, we are faced with the task of
attempting to salvage ROI data  for use in vertical permeability and leakance estimation. At this
site, with the exception of well W-4, RMSE of estimates for gas permeability were slightly better
for the semi-confined compared to the unconfined upper boundary condition. Estimation of
leakance at well W-4 varied widely for the same RMSE of 1.83 x  10"4 (atm) which was probably
due to field-scale heterogeneity (measurements of 0.04 and 0.25 inches of water at 45.9 and 50.3
feet respectively). Given this and the observation that k^k^yk^k^ values for the semi-confined
domain are lower and perhaps more credible for sandy soils, use of the semi-confined  upper
boundary condition would appear reasonable.  However as previously mentioned, because of the
nature of ROI testing, two-dimensional cross-sectional diagrams for each test condition can not
be plotted to compare simulated versus observed pressure response to demonstrate confidence in
vertical permeability and leakance estimation.

6.5    Conclusions

1.      This section provided an example of gas permeability and leakance estimation in a semi-
       confined  domain and illustrated difficulties encountered when attempting to use ROI-
       based data for parameter estimation.

2.      Use of the pseudo-steady-state radial flow equation appeared to provide a  slight but
       consistent overestimation of radial permeability compared to solutions which allowed
       recharge at the upper boundary. However, little if any correlation between increased
       leakance  and overestimation of radial permeability was observed.  A comparison of radial
       gas permeability  estimates for unconfined and semi-confined domains revealed that radial
       permeability estimation  was  relatively insensitive to leakance. Estimation of vertical gas
       permeability expressed in the ratio of k^k^/k^k^ though was very sensitive to leakance
       and illustrated the difficulty in resolving the correlated effect of leakance and anisotropy
       with ROI test data. The assumption of high leakance or a domain open to the atmosphere
       resulted in higher k^k^/k^k^ ratios compared to semi-confined conditions.
                                           78

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Table 6.2.a  Summary of gas permeability estimation for domain open to atmosphere (high leakance)
well
W-l
W-3
W-4
W-5
W-6
W-7
Table
well
W-l
W-3
W-4
W-5
W-6
W-7
krgkl(r)*(cm2)
9.65 x ID'7
3.02 x ID'7
5.74 x ID'7
1.78xlO-6
3.16xlO-7
1.99xlO-7
n krgkl(r)**(cm2)
19 3.20xlO-7(3.02xlO-7- 3.38x10'
113 1.86 x ID'7 (1.82 x ID'7 - 1.90 x 10''
235 4.98 x ID'7 (4.73 x 10'7 - 5.23 x 10"
1 7.28 x ID'7
91 2.42 x ID'7 (2.38 x 10'7 - 2.45 x 10'
18 2.03 x ID'7 (1.99 x 10'7 - 2.06 x 10"
6.2.b Summary of gas permeability estimation for
krgkl(r)*(cm2)
9.65 x ID'7
3.02 x ID'7
5.74 x ID'7
1.78xlO-6
3.16xlO-7
1.99xlO-7
krgkl(r)+(cm2)
20 2.90 x ID'7 (2.80 x 10'7- 3.00 x 10'7)
45 1.80 x ID'7 (1.75 x 10'7 - 1.85 x 10'7)
1 1 4.69 x ID'7 (4.28 x 10'7 - 5.09 x 10'7)
1 6.87 xlO-7
6 1.81xlO-7(1.73xlO-7-1.88xlO-7)
4 1.42xlO-7(1.41xlO-7-1.43xlO-7)
krsklM */krsklM * * krsklM * */krJclfz) * * RMSE** (atm)
7) 2.85-3.20 2.76(2.55 -2.97) 4.87xlQ-5
7) 1.59-1.66 6.92(6.71-7.12) 1.23 xlO'4
7) 1.10-1.21 21.93(19.55-24.31) l.SSxlO'4
2.45 2.47 l.OSxlO-5
7) 1.29-1.33 12.15(11.94-12.63) 1.73xlQ-4
•7) 0.97-1.00 24.17(23.55-24.79) 6.94xlQ-4
semi-confined domain
k Jt */k J( k Jf +/k J( + leakance (cm)
3.31-3.45 1.17(0.85-1.49) 9.29 x ID'10 (7.07 x lO'10- 1.15 x 10'9)
1.63-1.73 4.29(3.07-5.50) 4.04 x lO'10 (2.18 x 10'10 - 5.89 x lO'10)
1.13-1.34 14.52(5.80-23.23) 4.23 x 10'8 (1.28 x 10'10 - 8.45 x 10'8)
2.6 1.01 2.24 xlO-9
1.68-1.83 0.74(0.52-0.96) 8.62 x 10'11 (8.24 x 10'" - 9.00 x 10'")
1.39-1.41 0.98(0.91-1.05) 3.62 x 10'11 (3.61 x 10'" - 3.63 x 10'")







RMSE+(atm)
4.85 x ID'5
1.22xl04
1.83xlO-4
9.10xlO-6
1.60xl04
5.67 xlO4
* pitimntprl iiiina fvmatinn f4 ^4^ anH linear rparp
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7.     TWO-DIMENSIONAL, AXISYMMETRIC, TRANSIENT GAS FLOW AND
       PERMEABILITY ESTIMATION ANALYSIS

       Transient gas permeability testing is not commonly performed during site characterization
activities at sites in which soil venting is planned.  The reason for may be because transient
pressure response often occurs quicker than field technicians can make enough useful manual
measurements.  However, it may also have to do with the need for additional equipment (e.g.,
pressure transducers) and more sophisticated and time consuming data analysis. Transient
testing though has some distinct advantages over steady-state testing. First, attainment of steady-
state conditions can take hours or even days in low permeability media when the distance from
extraction or injection well to a monitoring point is extensive. If gas extraction is used, a lengthy
testing period may require use of vapor treatment equipment resulting in additional costs. If gas
injection is used, prolonged uncontrolled release of VOCs from subsurface media to the
atmosphere may create a health and safety issue.  Short testing times, alleviates both of these
problems.  Second, after a relatively short period of initial pressure rise or drop, pressure change
can be slow, especially in soils having very low gas permeability,  making discernment of steady-
state conditions difficult. Premature cessation of testing will likely result in overestimation of
gas permeability since the utilized pressure differential is less than the true pressure differential
for a given flow rate under steady-state conditions. Third, transient testing enables estimation of
gas-filled porosity. Knowledge of gas-filled porosity is necessary for pore-velocity calculation to
support venting design and monitoring activities. Fourth, the large number of data points
collected during transient testing may enable more accurate parameter estimation. During
steady-state testing, there are rarely more than 10 monitoring points for data analysis. Transient
testing allows use of a hundred or more pressure measurements at each monitoring point.
Finally, since periodic pressure measurement must be performed anyway to observe attainment
of steady-state conditions it would appear wasteful not to increase pressure measurement
frequency and utilize this data for  quantitative analysis considering high  personnel and drilling
costs incurred to support gas permeability testing.

       At present, only line-source/sink  analytical solutions exist for transient, axisymmetric,
two-dimensional gas flow analysis and parameter estimation.  The implicit assumption in use of
these solutions is that borehole storage effects are insignificant except in the immediate vicinity
of the wellbore.  The radii away from a well and borehole storage volume over which borehole
storage effects become important though have not been investigated for gas flow. Thus, use of
pressure monitoring points very close to  a gas extraction or injection well may introduce some
unquantified error in gas permeability  estimation.

       Borehole storage effects would be expected to be of greatest significance in single-
interval, transient testing.  Single-interval testing provides permeability estimation over a
relatively small volume of subsurface media thereby  providing a mechanism for assessing
physical heterogeneity or spatial variability in permeability on a scale much smaller than full
field-scale tests. The scale of single-interval tests however would still be larger than that
obtained with collection of discrete soil core and subsequent laboratory testing. While there are
                                            80

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no published studies of single-interval, transient, gas permeability testing, single-interval,
transient, hydraulic testing is common in ground-water hydrology (i.e., slug testing). Joss et al.
(1992) conducted single-interval, steady-state, gas permeability testing. With single-interval,
steady-state testing however parameter estimation is limited to one parameter, radial gas
permeability, since only one pressure point can be used for each flow rate.  Transient, single-
interval testing would enable the use of many pressure measurements as a function of time
thereby allowing estimation of radial permeability, vertical permeability, leakance, and gas-filled
porosity.

       The purpose of this section is to: (1) summarize available transient, axisymmetric, two-
dimensional, source/sink analytical solutions, (2) introduce a solution for a finite-radius solution
incorporating borehole storage, (3) conduct a sensitivity analysis for the finite-radius solution,
and (4) evaluate the usefulness of transient, single-interval,  finite-radius testing at a field site.

7.1    Model Formulation for Line-Source Open to the Atmosphere

       The governing equation describing two-dimensional, axisymmetric gas flow in
anisotropic, homogeneous media in a domain open to the atmosphere is (Baehr and Hult, 1991)
When subject to the initial condition:

 = atm-   r>rw,  0 = atm'   r>rw,  z = 0;   t >0                                                     (7.3)


^ = 0;   r>rw,  z = d;   t>0                                                      (7.4)
 dz

 lim  = ((>atm     00                                                    (7.5)



                                         00                            (7.6)
      dr    Mgnkrgkl(r}(dL-dv]

where
                                            81

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n(z)=  1;  dv
-------
                                                     .   r>r..
                                                                           ,>0.  (7.13)
When subject to the initial condition (7.2) and boundary conditions (7.4), (7.5), (7.6), and (7.7) in
addition to boundary condition:
-    = 0;  r>rw,   z = 0;   t>0
                                                                                  (7.14)
Baehr and Hull's (1989) provide a solution as follows:
    "atm
where u is as defined in (7.10) and
                                   sin
                                                .  (nndj
                                          ,   rsm —r
                                         d   )     {   d
                                                                                  (7.15)
          k k
                <
                                                                                  (7.16)
    =?
               k k
               KrgKl(z')

              krgkl(r^
                                                                                  (7.17)
w\
=  —ev
                                                                                  (7.18)
This solution is also coded in GASSOLVE (Falta, 1996)

7.3    Model Formulation for a Finite-Radius Well with Leaky-Confined Flow

       Varadhan and DiGiulio (2001) recently derived an analytical solution for two-
dimensional, transient flow in a semi-confined domain with gas extraction or injection from a
finite-radius well. Equation (7.1) is used as the governing partial differential equation describing
two-dimensional, axisymmetric transient gas flow in anisotropic, homogeneous media. Initial
                                           83

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condition (7.2) and boundary conditions (7.4), and (7.5) are used in addition to


"aT=  rV.(z)(^~^);   r>r^'  z = 0'  r>0                                     (7-19)

which is the boundary condition used by Baehr and Joss (1995) for a semi-confined domain. The
condition at the well (r = rw) taking wellbore storage into consideration may be expressed as
where Dz is as defined in (7.7), Qm(t) is a time-dependent mass flow pump function [MT1]
(negative when gas is extracted, positive when gas is injected), and Vb = wellbore storage volume
[L3]. The first term in equation (7.20) denotes the mass rate of gas entering the well from the soil
domain.  The second term denotes the rate of change of mass inside the wellbore. The wellbore
storage term includes the volume of the pipe and associated tubing to the pump plus the air-filled
porosity contained within the sandpack.  For this reason, the wellbore storage term was expressed
explicitly as a volume specified by the user instead of as a function of the dimensions of the well
as is commonly done. It was assumed that the gas volume inside the wellbore remains constant
and that the change in mass is due only to change in gas pressure.

       Varadhan and DiGiulio (2001) express the governing partial differential equation and
associated initial and boundary conditions in a non-dimensional form as follows:
                                                                                (7.21)
          ar   a;?2   R dR      az

0> = 0;  R>1,  0 < Z < 1,   r = 0                                                  (7.22)

ao
 az
 ao
    = BO;  R>1,  Z = 0,  r>0                                                 (7.23)
    = 0;   R>1,   Z = l,  r>0                                                   (7.24)
 az    '

                                     R = l,   00                    (7.25)
                                          84

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°=  T^'   D^<~L                                                       (7.26)
           0;      0 < Z < D,   L < Z < 1

The dimensionless variables are defined by


0=-^--1                                                                       (7.27)
                                                                                  (728)
                           ^;    L=^j-                                        (7.29)
                            d           d
                                                                                  (7.30)
B=  /g/(Z),,                                                                    (7.31)
T =	^	                                                                   (7.32)
    ^(L-D)
                                                                                  (7.33)
where the pump function Qm(t) = Q0g(T); Q0 (MT"1) is the maximum of the absolute value of
Qm(t), and -1  < g(T) < 1, for all T >0.

       Equation (7.21) and boundary condition (7.25) are nonlinear because of the radical term
on the left hand side of the equation.  To solve equation (7.21) analytically, linearization of the
radial term is necessary. As discussed in section 3, the easiest method of linearization with little
loss of accuracy is to let (1+ $)'/2 or P/Patm equal some constant. Baehr and Hult (1988) and Falta
(1996) let P equal atmospheric pressure (Patm) or (1+ $)'/2 = 1.  Falta (1996) states that with this
approximation, gas is assumed to be compressible with a constant compressibility factor of
                                           85

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       Varadhan and DiGiulio (2001) developed an exact analytical solution to equation (7.21)
subject to (7.22)-(7.26)
                                                                                 (7.34)
where
                                                                                 (7.35)
R(R,T)=-
          2K
2_
n
           2Y-
                                     -i
                                (4)
H(r,(^2/L2-42)
                                                      udu
                                                                                 (7.36)
 E = wJ  M  - F
The eigenvalues Kn are obtained as the n-th positive roots of
The roots of ^n are determined by
                                                                       (7.37)


                                                                       (7.38)
                                                                                 (7.39)
                                                                                 (7.40)
J0 and Jj are zero- and first-order Bessel functions of the first kind, Y0 and Yt are zero-and first-
order Bessel function of the second kind respectively.  The function H(T,A), where A is a
dummy variable, depends on the pumping function g(T) as follows
                                                                                 (7.41)
                                           86

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When the pumping rate is constant


H(T,A} = — (\-e~ATY                                                           (7.42)


       Computation of $(R,Z,T) is difficult because the integrand in (7.36) is slowly decaying
and oscillatory. Furthermore, when the Fourier summation in (7.34) is slowly convergent,
evaluation of (7.36) must be performed numerous times. Thus, while accurate, the solution is
computationally inefficient. This led Varadhan and DiGiulio (2001) to pursue the use of a
numerical inversion algorithm in place of (7.36).  Specifically, transformation in time (T) using p
as the Laplace  transform variable yielded
    ,D = rnvLaplace _                                                       (7.43)
                    \conK,(con} + PTK,(con}\


where

           „         9   o\1/2
                            .                                                     (7.44)
       Varadhan and DiGiulio (2001) used the Gaver-Stehfest algorithm (Stehfest, 1970) to
invert (7.43). The Gaver-Stehfest algorithm has been widely used in well hydraulics and found
to be efficient and acceptably accurate when a constant pumping function is used (Moench and
Ogata, 1981; Dougherty and Babu, 1984; and Butler and Liu, 1993). Davies and Martin (1979)
however found that the Gaver-Stehfest algorithm does not perform well for a sinusoidal pumping
function. Varadhan and DiGiulio (2001) used a sinusoidal pumping function to check the
performance of the Gaver-Stehfest algorithm for this application and found that the algorithm did
indeed perform poorly and that performance deteriorated with time. Thus, while the Gaver-
Stehfest algorithm is computationally more efficient than Varadhan and DiGiulio's (2001) exact
analytical solution, it can not be used for sinusoidal or other oscillatory functions.  In practice
however, a constant or step function is usually specified. Under these conditions, the Gaver-
Stehfest algorithm performs well and substantially speeds  up computation compared to the exact
solution.

7.4    Sensitivity Analysis of Finite-Radius. Transient Solution

       Varadhan and DiGiulio (2001) did not perform a sensitivity analysis for their analytical
solution nor provide an example application. A sensitivity analysis is provided here using input
representative of conditions present at a field site where the solution will be tested.  A
FORTRAN program (TFRLK) was written to solve the forward solution of Varadhan and
DiGiulio's  (2001) solution. Example input and output files and the source code is provided in
Appendix D. Figures 7.1 and 7.2 provide a comparison of normalized pressure (P/Patm) and error

                                           87

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as a function of time and normalized radial distance (r/d) for finite-radius (FR) and line-source
(LS) solutions in a semi-confined domain having a thickness of 600 cm, borehole storage volume
of 15000 cm3, volumetric gas content of 0.1, and leakance of 1.0 x  10"11 cm2. Error is defined as
           P  ^^    P
          P          P
error —    atm        * atm
(LS)-^-(FR)
                                                                      (7.45)
               p
        v       atm           s

As illustrated in Figure 7.1, during pumping at the wellbore (r/d = 0.017), the line-source/sink
solution simulates a more rapid rise in pressure at early time but then levels off to a lower
normalized pressure at late time compared to the finite-radius solution.  The former effect is due
to lack of a delayed response from borehole storage. The latter effect is likely due to the fact that
the simulated distance r/d = 0.017 is the wellbore radius for the finite-radius solution and
approximately 10 cm into the formation for the line source/sink solution.  Lower pressure
differential response at late time for the line-source/sink solution would likely lead to gas
permeability overestimation for a given flow rate. As illustrated in Figure 7.2, at the wellbore,
pressure is overestimated at early time for the line source/sink solution and underestimated over
well shut-in by as much as 10% for the given mass flow rate.  Thus, when conducting single-
interval, transient testing, borehole storage effects become a significant source of error if not
properly accounted  for.  Under conditions of these simulations, error due to use of a line-
source/sink solution disappears at r/d = 0.150 or 90 cm (about 3 feet).  Thus, under typical testing
conditions when a gas extraction or injection well is not used as the pressure monitoring point
and the closest observation point is greater than 150 cm away, transient gas permeability
estimation using a line-source/sink solution would result in little error.

       Figures 7.3 and 7.4 provide a comparison of line-source/sink and finite-radius simulations
at the wellbore but at different vertical elevations to assess error associated with using the line-
source solution for testing at different intervals in multi-level wells. It appears from Figure 7.4
that there is little error (less than 1.5%) associated with using a line-source/sink solution under
conditions specified here. However, conditions specified here are not representative of all
potential testing conditions.  Thus, when observation points are close to a gas extraction or
injection well, it may be useful assess the importance of finite-radius and wellbore effects.

       Figure 7.5 illustrates the effect of borehole storage on the transient pressure response at
the borehole during single-interval testing. Increased borehole storage volume results in an
increased delayed response to steady-state conditions.  Figure 7.6 illustrates the effect of
volumetric gas content on transient pressure response at the borehole during single-interval
testing. Increased gas-filled porosity results in a somewhat similar transient response to
increased borehole storage volume but the transient effects are much more prolonged.  If
borehole storage volume if unknown or can only be roughly estimated because of uncertainties in
the filter pack volume and volumetric gas content in the filter pack, similarity  in response of
borehole storage volume and volumetric gas content create the possibility of non-unique

                                            88

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parameter estimation due to correlated behavior.  Figure 7.7 illustrates the effect o
ratio or anisotropy on transient pressure response at the wellbore. Higher k^k^/k^k^ ratios
prolong attainment of steady-state conditions and result in a higher normalized pressure response
compared to lower krgkl(r)/k1.gkl(z) ratios. Figure 7.8 illustrates the effect of leakance on transient
pressure response at the wellbore. This effect, at least under the conditions specified in single-
interval testing, was minor. As demonstrated in section 6 however, when conducting gas
permeability testing on a larger scale, simulated pressure response becomes very sensitive to
estimated leakance and the correlated effect of leakance and kjc^/kjc^ ratios becomes
apparent.  Finally,  the effect of radial permeability estimation is illustrated in Figure 7.9. Lower
gas permeability significantly prolonged attainment of steady-state conditions and resulted in a
much higher pressure differential.

7.5    Description of Field-Site
       The concept of transient single-interval gas permeability testing was tested at the Picillo
Farm Superfund Site in Coventry, Rhode Island.  Unsaturated and saturated subsurface media at
the site are heavily contaminated with chlorinated and non-chlorinated cycloalkanes, alkanes,
alkenes, phenolic, and aromatic hydrocarbons as a result drum and bulk liquid disposal in 1977.
At least 10,000 drums and an undetermined volume of bulk liquid wastes containing industrial
solvents, PCBs, paint sludges, plasticizers, explosives, resins, and still bottoms were disposed of
in several trench areas throughout an eight-acre portion of the site. Between 1980 and 1982, U.S.
EPA removed drums and remaining drum fragments offsite for disposal.  In 1993, U.S. EPA
                                                               	r/d = 0.017 (FR)
                                                               	r/d = 0.017 (LS)
                                                               	r/d = 0.050 (FR)
                                                               	r/d = 0.050 (LS)
                                                               	r/d = 0.150 (FR)
                                                               	r/d = 0.150 (LS)
                       50
100
150
  200
Time (s)
250
300
350
400
Figure 7.1.  Pressure response as a function of time for finite-radius (FR) and line source
solutions (LS) (d = 600 cm, dv/d = 0.683, dL/d = 0.783, z/d = 0.733, r^d = 0.017, 6g = 0.1, kj
= 1.0 x ID'9 cm2, kjt^/ig^ = 1.0, Qm= 1.0 g/s, Vb = 15000 cm3, kj^/d' = 1.0 x 10'11 cm).
                                            89

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           15.0
           10.0 -
         o
        W
                                                                 	r/d = 0.017
                                                                 	r/d = 0.050
                                                                 	r/d = 0.150
          -10.0 -
          -15.0
                       50      100     150     200    250
                                            Time (s)
                                                             300
350
400
Figure 7.2.  Error in pressure computation as a function of time for finite-radius (FR) and line
source solutions (LS) (d = 600 cm, dv/d = 0.683, dL/d = 0.783, z/d = 0.733, rw/d = 0.017, 6g = 0.1,
igtl(r) = 1.0 x 10-9 cm2, kjc^/igc^ =  1.0, Qm= 1.0 g/s, Vb = 15000 cm3, ig^/d' = 1.0 x ID'11
cm).
            1.12
            1.10 -
            1.08 -
            1.06 -
            1.04 -
            1.02
            1.00
                                                                 z/d = 0.558 (FR)
                                                                 z/d = 0.558 (LS)
                                                             	z/d = 0.383 (FR)
                                                             	z/d = 0.383 (LS)
                                                             	z/d = 0.203 (FR)
                                                             	z/d = 0.203 (LS)
                 0
                       1000   2000    3000   4000    5000   6000   7000    8000
                                            Time (s)
Figure 7.3 Pressure response as a function of time and vertical elevation for finite-radius (FR)
and line source solutions (LS) (d = 600 cm, dv/d = 0.683,dL/d = 0.783, r = rw, 6a = 0.1, k^k^ =
1.0 x ID'9 cm2, ig^/igc^,  = 1.0, Qm= 1.0 g/s, Vb = 15000 cm3, k,.gkl(z,/d'  = LO x 10'11 cm).
                                            90

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             1.5 -i
             1.0 -
             0-5 H
             0.0
            -0.5 -
            -1.0 J
	z/d = 0.558
	z/d = 0.383
	z/d = 0.203
                 0      1000    2000   3000   4000    5000   6000   7000    8000
                                           Time (s)

Figure 7.4 Error in pressure computation as a function of time and vertical elevation for finite-
radius (FR) and line source solutions (LS) (d=600 cm, dL/d = 0.683, dL/d = 0.783, r = rw, 8g = 0.1,
k,.gkl(r) =1.0 x ID'9 cm2, kjc^/kjc^ =1.0, Qm=1.0 g/s, Vb = 15000 cm3, kjc^./d^ 1.0 x lO^cm
                                                              -Vb = 5000cm3
                                                           	Vb=15000cm3
                                                           	Vb = 25000 cm3
                                                           	Vb = 35000cm3
                                                           	Vb = 50000 cm3
              0    100   200   300   400    500   600   700   800   900   1000
                                          Time (s)
Figure 7.5 Pressure response as a function of time and borehole storage (d = 600 cm, dv/d =
0.683, dL/d = 0.783, z/d = 0.733, r = rw, 80 = 0.1, kj^, = 1 x 10'9 cm2, k^/kjc^ = 1.0, Qm =
1.0 g/s, ig^^/d'= 1.0x10-" cm).
                                           91

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                                                            vol. gas content = 0.01
                                                         	vol. gas content = 0.05

                                                            vol. gas content = 0.1

                                                         	vol. gas content = 0.2

                                                         	vol. gas content = 0.3
           1.0
0
500
1000
                                                           3000    3500    4000
                                    1500    2000    2500
                                          Time (s)
Figure 7.6 Pressure response as a function of time and volumetric gas content (8g) (d = 600 cm,
du/d = 0.683, dL/d = 0.783, z/d = 0.733, r = rw, krgkl(r) = 1 x 10'9 cm2, kjc^/k^k^ = 1.0, Qm= 1.0
g/s, Vb= 15000 cm3, kjt^/d' = 1.0 x lO'11 cm)
             1.9
          S  1.5 -
         1500   2000   2500
               Time (s)
                                       3000   3500   4000
                       500    1000
Figure 7.7 Pressure response as a function of time and l^gk^/k^k^ (or k/kz) ratio (d = 600 cm,
du/d = 0.683, dL/d = 0.783, z/d = 0.733, r = rw, k^., =1.0 x 10'9 cm2, 6g =0.1, Qm = 1.0 g/s, Vb =
15000 cm3, ^k^/d' = 1.0 x lO'11 cm)
                                            92

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                                                             	k'/d'= l.Oe-llcm
                                                             	k'/d' = 1.0e-12cm
                                                                 k'/d' = 1.0e-13cm
                                                             	k'/d' = 1.0e-14cm
                                                             	k'/d'= 1.0e-15cm
                0      500    1000    1500    2000    2500    3000    3500    4000
                                             Time (s)
Figure 7.8 Pressure response as a function of time and k^k^/d^or k'/d') (d = 600 cm, dLJ/d =
0.683, dL/d = 0.783, z/d = 0.733, r = rw, kjt^ = 1.0 x 10'9 cm2, lc,.gkl(r)/k,.gkl(z) = 1.0, 6g = 0.1, Qm =
l.Og/s, Vb= 15000 cm3)
2.4 -i

2.2 -

2.0 -

1.8 -

1.6 -

1.4 -

1.2
                                                                  -kr= 1.0E-09cm2
                                                               	kr = 3.0E-10cm2
                                                               	kr= 1.0E-10cm2
                                                               	kr = 5.0E-ll cm2
                                                               	kr = 3.0E-ll cm2
             1.0
                0      1000    2000    3000    4000    5000    6000    7000    8000
                                             Time (s)
Figure 7.9 Pressure response as a function of time and k^k^ (or k,.) (d = 600 cm, dv/d = 0.683,
dL/d = 0.783, z/d = 0.733, r = rw, 60 = 0.1, k^/k^ = 1.0, Qm= 0.10 g/s, Vb = 15000 cm3,
                                             93

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issued a Record of Decision (ROD) to the responsible parties mandating the use of soil vacuum
extraction and ground-water pump and treat to remove remaining contaminants in unsaturated
and saturated subsurface media.  The original well design for transient, single-interval testing at
the Picillo Farm site was to install 5 cm (2-inch) boreholes with roto-sonic drilling followed by
direct-push of 5 cm i.d., 7.6 cm o.d. (3 - inch) heavy gauge aluminum vertically slotted pipe.
Roto-sonic drilling was selected as the preferred drilling method because of the presence of large
cobbles and boulders encountered during previous drilling efforts. Aluminum was chosen for
well construction because of its compatibility with neutron probe counts. Prior to gas
permeability testing, the borehole was to be logged with a neutron probe to correlate moisture
content with gas permeability estimates and potentially develop in-situ water-gas permeability
curves.  The use of aluminum as well casing however was abandoned in favor of vertically
slotted 5 cm i.d. drill rod high strength carbon steel illustrated in Figure 7.10 because of concern
with structural integrity and that casing would have to be pushed through boulders having a cored
diameter slightly smaller that the outside diameter of the pipe. It was hoped that a tight fit
between the carbon steel pipe and borehole wall, nearly continuous vertical slots, and straddle
packers would enable gas permeability estimation over small intervals on the order of 10 cm.
Figure 7.10 Photograph of vertically slotted 5 cm i.d. high tensile carbon steel pipe initially
installed for gas permeability testing at the Picillo Farm Superfund Site.
                                            94

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                                                         Locked well cover
                                                    4" diameter PVC pipe
           25.5'
                    1.5'concrete
                    1.5'bentonite
                       2.0' sand
                    1.5' bentonite
1.0 screen
                       2.0' sand ilTITil   iTHfTll 1.0' screen
                    1.5' bentonite
                       2.0' sand
1.0' screen
                   1.5'bentonite
                       2.0' sand
1.0' screen
                   1.5' bentonite
                       3.5' sand I            1 2.5' screen
                   1.5' bentonite
                       3.5' sand
2.5' screen
                               Well Schematic
Figure 7.11 Typical well installation details for MWE wells installed at the Picillo Farm
Superfund site.

However development of the well revealed severe problems with siltation and plugging of the
vertical slots.  During well development, a large quantity (20 liters) of a very fine sand was
retrieved.  Sounding with a water meter at the base of the well revealed that this material
continued to flow into the well throughout the development process. Based on these
observations, this well construction method was abandoned in favor of a more conventional but
less desirable method illustrated in Figure 7.11.  Actual completion intervals for sandpack and
bentonite seals varied by about ± 8 cm (3 inches).

      Dry granular bentonite was poured from the surface through the PVC well and borehole
                                          95

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annular space and hydrated in place through the addition of water.  Coarse sand (grade #0) was
deposited in a similar manner with the volume for each sandpack interval recorded to assist in
borehole storage estimation.  Wells and factor slotted screens (0.002 inch) were constructed of 4-
inch internal diameter, schedule-40 PVC pipe. The installation of six discrete screened intervals
allows measurement of gas permeability over more  intervals than typically present (2 or 3) at
hazardous waste sites but each screened interval and sandpack is fixed in place and covers a
larger vertical interval (approximately 60 cm) than desired. Nevertheless, given the difficult
drilling and subsurface conditions present at this site, this well installation method likely
represents the limit of conventional well construction.

       In 1998, U.S. EPA's Office of Research and Development,  installed a test plot consisting
of 13 multi-level wells within the west trench portion of the site.  The wells were spaced
approximately 150 cm (5 feet) apart.  The purpose of the test plot is to gain a better
understanding of gas flow and rate-limited mass transfer on a field-scale during soil venting
application. Roto-sonic drilling was used to create  8-1/4 inch diameter boreholes for well
installation. Soil cores were carefully logged with detailed observation of the vertical
dimensions of lenses of silt and clay.  Soil cores revealed a high heterogeneous subsurface
environment consisting of trench backfill, sandy glacial  till, and glaciofluvial sand, silt, and clay
deposits. One soil core consisted entirely of sausage-shaped, polymer-like, white solid waste.
Another soil core resulted in retrieval of a drum fragment containing a semi-solid dark brown to
red viscous material. Chemical analysis of both waste materials revealed very high
concentrations of chlorinated and non-chlorinated solvents.

       During single-interval transient gas permeability testing, a 1.2 g/s capacity diaphragm
pump was used to inject air into a riser pipe containing a 1.25-inch internal diameter, 6-inch,
schedule-80 PVC screen separated by two 2-foot long inflatable packers. As illustrated in Figure
7.12, flow was measured and controlled at surface using a 150-mm flowmeter (± 2% accuracy)
containing  a high precision control valve.  Air injection temperature was measured with a type-J
thermocouple and a thermocouple thermometer (±0.5%  of reading  accuracy).  Air injection
pressure was measured with a digital  manometer (± 1.0% of range accuracy) and magnehelic
gauges. As illustrated in Figure 7.13, a quick-connect was used at the top of the packer riser pipe
to prevent release of gas pressure during well shut-in. Pressure as a function of time at the gas
injection interval and interval immediately above the injection interval was measured with
wireless, self contained, 7/8-inch o.d. pressure transducers (Levelogger - Solinst Canada -
accurate to 0.2% full scale).  Compensation of pressure loss through subsurface piping and
surface tubing was avoided by measurement of pressure in this manner.  Pressure readings were
taken every 0.5  seconds during testing. Rubber couplings were used between sections of PVC
pipe to ensure minimal loss of pressure at the pipe threads.  As illustrated in Figure 7.14, the
wireless pressure transducer data was located between two pneumatic packers.  After testing,
data was downloaded into a laptop personal computer using an optical reader.

       Single-interval, transient testing was demonstrated in well MWE-02 in a screened interval
(MWE-02-03) having an annular sandpack extending from 298 to 354 cm below the base of an
                                            96

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asphalt cap. Depth to the water table was 521 cm below the base of the asphalt cap.  Values of
radial permeability, vertical permeability and leakance were allowed to vary freely, but borehole
storage volume was constrained between 7500 and 12500 cm3. Based on the volume of the
annular sandpack, assumption of a volumetric gas content of 0.3 in the sandpack, and above and
below ground piping and tubing, a borehole storage volume of 10,000 cm3 was calculated.
Borehole storage volume was allowed to vary 25% to incorporate uncertainty in volumetric gas
content and annular sandpack volume estimation.
Figure 7.12  Flow, pressure, and temperature measurement at the surface during gas permeability
estimation in well MWE-02
                                          97

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7.6    Results and Discussion

       The approach to parameter estimation described in sections 5 and 6 was applied to
parameter estimation here. A series of random guesses was applied (groups of 5000) constrained
within decreasing intervals of k,.gkl(r), k^k^, leakance, borehole storage volume, and gas-filled
porosity until the lowest root mean squared error (RMSE) was attained for a set of estimates. In
this way, variation in parameter estimations for the same  or similar RMSE values were assessed.
A program was written in FORTRAN (FRLKTNV) to facilitate computations.  Example input
and output files and the source code for this program is provided in Appendix E.  Tables 7.1
through 7.4 provide a summary of the 10 best fit estimates (based on lowest RMSE) for each of
Figure 7.13 Close-up of quick-connect at top of 3.2 cm i.d., sch-80 PVC pipe to prevent gas
pressure release at shut-in
Figure 7.14 Illustration of wireless pressure transducer clamped between pneumatic packers for
use in single-interval and two-interval, transient gas permeability testing in MWE wells
                                           98

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 four tests conducted atMWE-02-03. Figures 7.16 through 7.19 illustrate observed versus
simulated pressure response for the fit in each test having the lowest RMSE. Values of radial
permeability for the 10 best fits in all 4 tests provided very consistent results demonstrating
reproducibility in testing at least for radial permeability estimation.  Values of leakance though
varied over an order of magnitude for the 10 best fits demonstrating uncertainty in this parameter.
Ratios of krgkl(r)/krgkl(z) also varied widely again demonstrating uncertainty in this parameter.
Uncertainty in leakance and k^k^/k^k^ ratios occur because these parameters are correlated and
single-interval testing provides no vertical resolution in pressure measurement.  Estimates of gas-
filled porosity however were very consistent despite modest variability in borehole storage
volume demonstrating reproducibility in estimation of gas-filled porosity.
Figure 7.15 Photograph of downloading wireless pressure transducer data into a laptop
computer using an optical reader.

7.7    Conclusions

1.      During pumping at the wellbore, the line-source/sink solution simulated a more rapid rise
       in pressure at early time but then leveled off to a lower normalized pressure at late time
       compared to the finite-radius solution. The former effect was due to lack of a delayed
       response from borehole storage.  The latter effect was likely due to the fact that for the
       line source/sink solution, simulation extended into the formation at a distance equivalent
       to the wellbore radius. Lower pressure differential response at late time for the line-
       source/sink solution would likely lead to gas permeability overestimation for a given flow
                                            99

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          1.25
          1.20 -
          1.15 -
          1.10 -
          1.05
          1.00
               0
50
100          150
    Time (s)
200
250
Figure 7.16 Observed versus fitted pressure response for Test 1 in MWE-02-03 (krgkl(r) = 1.54 x
1C'9 cm2, krgk^/kjt^ = 5.86, k,.gkl(z,/d' = 1.51 x 10'12 cm, 6g = 0.011, Vb = 10,260 cm3, RMSE =
3.121x 10-3atm)
Table 7.1 Ten best parameter estimation fits for test 1 in MWE-02-03
k,.gkl(r)
(cm2)
.54E-09
.56E-09
.78E-09
.60E-09
.46E-09
.68E-09
.77E-09
.58E-09
.38E-09
.44E-09
(cm2)
2.62E-10
3.51E-10
2.12E-10
3.28E-10
3.13E-10
2.89E-10
2.33E-10
3.29E-10
6.02E-10
3.51E-10
JVk^
5.86
4.45
8.40
4.87
4.68
5.82
7.59
4.80
2.29
4.12
(cm)
1.51E-12
1.84E-11
3.25E-11
7.24E-11
1.85E-12
4.51E-12
4.97E-11
8.80E-11
2.25E-11
1.84E-12
eg
0.011
0.014
0.011
0.014
0.012
0.011
0.010
0.012
0.013
0.011
vb
(cm3)
10260
12280
12330
12360
10360
12030
12190
11760
11630
9955
RMSE
(atm)
3.121E-03
3.171E-03
3.230E-03
3.233E-03
3.243E-03
3.296E-03
3.322E-03
3.369E-03
3.403E-03
3.403E-03
                                          100

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         1.30
         1.00
              0
50
100          150
    Time (s)
200
250
Figure 7.17  Observed versus fitted pressure response for Test 2 in MWE-02-03 (krgkl(r) = 1.81 x
ID'9 cm2, k^k^/kjc^, = 9.02, k^./d' = 7.20 x 10'12 cm, 6g = 0.008, Vb = 11,790 cm3, RMSE =
3.23x 10-3atm)
Table 7.2 Ten best parameter estimation fits for test 2 in MWE-02-03
(cm2)
1
1
1
1
1
1
1
1
1
1
.81E-09
.76E-09
.76E-09
.76E-09
.80E-09
.81E-09
.74E-09
.83E-09
.71E-09
.72E-09
(cm2)
2
2
2
2
2
2
2
1
2
2
.01E-10
.31E-10
.23E-10
.35E-10
.12E-10
.OOE-10
.30E-10
.86E-10
.56E-10
.47E-10
9
7
7
7
8
9
7
9
6
6
.02
.63
.91
.52
.46
.03
.58
.81
.70
.98
Z) k,.gkl(z.)/d
(cm)
7.20E-12
6.14E-12
6.46E-11
7.40E-12
6.80E-12
3.74E-11
5.05E-11
4.27E-12
5.80E-11
8.23E-12

0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
eg
.008
.008
.008
.008
.008
.009
.008
.008
010
.009
vb
(cm3)
11790
11860
11860
11640
11510
11480
11670
11560
11710
11880
RMSE
(atm)
3
3
3
3
3
3
3
3
3
3
.23E-03
.23E-03
.27E-03
.29E-03
.30E-03
.37E-03
.37E-03
.37E-03
.39E-03
.42E-03
                                         101

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          1.40 !
          1.35 -
          1.05
          1.00
               0
50
100
150
200
250
                                            Time (s)
Figure 7.18 Observed versus fitted pressure response for Test 3 in MWE-02-03 (krgkl(r) = 1.60 x
1C'9 cm2, krgk^/kjt^ = 3.50, k,.gkl(z,/d' = 8.20 x 10'11 cm, 6g = 0.010, Vb = 11,560 cm3, RMSE =
4.655 x ID'3 atm)
Table 7.3 Ten best parameter estimation fits for test 3 in MWE-02-03
k,.gk^(r)
(cm2)










.60E-09
.92E-09
.47E-09
.82E-09
.73E-09
.71E-09
.40E-09
.78E-09
.92E-09
.74E-09
krgk,.(Z) I
-------
         1.45
               0
50
100          150
    Time (s)
200
250
Figure 7.19 Observed versus fitted pressure response for Test 4 in MWE-02-03 (Qm = 0.722 g/s,
k,.0kl(r) = 1.60 x ID'9 cm2, k^yk^k^ = 4.29, kjt^./d' = 2.44 x 10'12 cm, 6g = 0.011, Vb = 9,199
cm3, RMSE = 6.35 x ID'3 atm)

Table 7.4 Ten best parameter estimation fits for test 4 in MWE-02-03











(cm2)
.60E-09
.65E-09
.46E-09
.57E-09
.64E-09
.66E-09
.63E-09
.68E-09
.68E-09
.49E-09
\f \f
^S'P if 7}
(cm2)
3.72E-10
4.28E-10
7.35E-10
3.78E-10
2.89E-10
4.41E-10
5.29E-10
2.68E-10
3.92E-10
7.42E-10
r \r l\r «
T"O 1 (ry ^H"ff 1
4.29
3.86
1.98
4.17
5.68
3.76
3.08
6.26
4.28
2.00
(cm)
2.44E-12
1.74E-11
3.02E-11
1.15E-12
1.03E-12
1.18E-11
9.05E-11
1.93E-11
1.98E-11
1.83E-11
eg
0.011
0.016
0.010
0.013
0.011
0.018
0.011
0.010
0.020
0.013
vb
(cm3)
9199
10370
10870
9155
8925
10140
11310
8069
9777
10230
RMSE
(atm)
6.35E-03
6.47E-03
6.53E-03
6.55E-03
6.55E-03
6.66E-03
6.69E-03
6.73E-03
6.78E-03
6.81E-03
                                          103

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       rate. Thus, for single-interval, transient testing, borehole storage effects can become a
       significant source of error if not properly accounted for.

2.      Under conditions of these simulations, error due to use of a line-source/sink solution
       disappeared at a relatively short distance from the wellbore. Thus, under typical testing
       conditions when a gas extraction or injection well is not used as the pressure monitoring
       point, transient gas permeability estimation using a line-source/sink solution would result
       in little error.

3 .      A comparison of line-source/sink versus finite-radius simulations at the wellbore at
       different vertical elevations revealed little error (less than 1.5%) associated with using a
       line-source/sink solution. Thus, for simulations conducted here, line-source/sink
       solutions appear appropriate for single well, multi-interval testing. However, conditions
       specified here are not reflective all potential testing conditions. Thus, when observation
       points are close to a gas extraction or injection well, it may be useful to utilize a transient,
       finite-radius solution to assess the importance of finite-radius and wellbore effects.

4.      Single-interval simulations demonstrated that increased borehole storage volume results
       in an increased delayed response to steady-state conditions. Increased gas-filled porosity
       results in a somewhat similar transient response to increased borehole storage volume but
       the transient effects are much more prolonged.
5.      Single-interval simulations demonstrated that higher k^k^/k^k^ ratios prolong
       attainment of steady-state conditions and result in a higher normalized pressure response
       compared to lower k^k^/k^k^ ratios.  The effect of leakance on transient pressure
       response at the wellbore however appeared to be minor.

6.      Single-interval simulations demonstrated that lower gas permeability significantly
       prolongs attainment of steady-state conditions and results in a much higher pressure
       differential.

7.      Finally, testing at the Picillo Farms Superfund Site revealed that single-interval transient
       gas permeability testing is an improvement over single-interval steady-state gas
       permeability testing in that the former provides an estimate of gas-filled porosity and does
       not rely on attainment of steady- state conditions which is sometime questionable for low
       permeability media (small pressure change in time after initial pressure or vacuum
       application).  The method though provided little confidence in leakance and k/kz
       estimation necessitating testing in two or more vertically spaced intervals for which
       existing analytical solutions are available and appropriate.
                                            104

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8      VOC RETENTION IN POROUS MEDIA

       An understanding of volatile organic compound (VOC) retention in porous media is
critical to designing, monitoring, and closing venting systems. Each phase in unsaturated porous
media can contribute to the retention and mass transport of VOCs. Phases present are solid
minerals, organic matter, bulk water, nonaqueous phase liquids (NAPL), and gas. Partitioning
mechanisms related to these phases are solid mineral - water, solid mineral - gas, organic matter -
water, water - gas, NAPL - water, and NAPL - gas.

8.1    Vapor Concentration

       Vapor concentration can be expressed in terms of volume of VOC per volume of gas
(e.g., parts per million volume or ppmv), or mass per volume (e.g., ug/cm3).  Since from the Ideal
Gas Law, volumetric concentration is equivalent to molar concentration (moles of VOC per
moles of gas), conversion of from units of ppmv to ug/cm3 can be accomplished by
                                                                                (8.1)
where M; (g/mole), P(atm), 01 = 0.08205 (L atm/K mole), and T (°K). For instance, at 20°C
(293.15°K) and atmospheric pressure, 100 ppmv benzene (M; = 78.1 1 g/mole) is equivalent to
0.3247 ug/cm3.

8.2    Vapor Pressure

       Vapor pressure can be defined as the gas pressure exerted by the vapor of a compound in
equilibrium with its pure condensed liquid or solid phase. At a given temperature, a number of
molecules in the condensed phase will acquire sufficient energy to overcome molecule-molecule
attractions in the condensed phase and escape to the vapor phase. Meanwhile in the vapor phase,
continuous collisions of vapor molecules with the surface of the condensed phase will cause a
return of some molecules to the condensed phase. The most commonly measured vapor pressure
is the boiling point (Tb), which is the temperature at which vapor pressure is equal to 1
atmosphere.

       Since molecules in the gas phase at or below a pressure of one atmosphere are separated
by relatively large distances and thus have little molecular interaction, the extent to which
compounds escape to the vapor phase is largely controlled by molecular interactions in the
condensed phase.  In the liquid phase,  the summation of molecular interactions which cause
molecule-molecule attraction is represented by the heat of vaporization (AHV) [ML2T"2mor1].
Large heats of vaporization are associated with high boiling points. When attempting to estimate
vapor pressures for compounds not having tabulated or referenced values, it is helpful to
understand the nature of molecular interactions in the condensed phase.  Schwarzenbach et al.
(1993) provide a discussion on van der Waals, dipole-dipole and hydrogen bonding interactions

                                          105

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which cause increased attraction between like molecules. All compounds experience van der
Waals forces.  Even nonpolar molecules such as alkanes, which exhibit a time-average smooth
distribution of electrons throughout their structure, experience instantaneous displacement of
electrons such that momentary electron-rich and electron poor structural regions develop. This
momentary redistribution of charges triggers a displacement of electrons in neighboring alkane
molecules causing molecular attraction.  Vapor pressure decreases and the heat of vaporization
increases with increasing alkane size or carbon number because the summation of van der Waals
forces or attraction is directly related to molecular size.  Van der Waals forces are also
responsible for decreasing vapor pressure with increasing size of alkyl groups on alkylbenzenes.

       Schwarzenbach et al. (1993) explain that dipole interactions result from electron-
attracting or electronegative properties of various types of atoms included in organic compounds
(H, C, S, I
-------
760 mmHg.  The latter method is also less prone to error compared to the former method. Vapor
pressure estimation techniques are valuable for compounds having low vapor pressures since
experimental data is scare or may be unreliable.

       The Antoine Equation has the general form


]nPv=A2	^—                                                               (8.3)
        2   T-C2

where


A2=   B*                                                                      (8.4)
  L
                                                                                (8.5)


and C2 is estimated by Thomson's Rule or

C2 =-18.0 + 0.19r6                                                              (8.6)

AHvb is the heat of vaporization at the boiling point [ML2T"2mor1]. Substituting equations (8.3)
and 8.4) into equation (8.2) yields
       &Hvb<
1         1
                          -c2)   ft-c2)
                                                                                (8.7)
In these equations Pv, Tb, and AHvb are expressed in units of atmospheres, Kelvin, and cal/mole.
The constant 91 is given as 1.987 cal mol"1 K-l. The ratio AHvb/Tb can be evaluated using a
method introduced by Fishtine (1963):
                                                                                (8.8)
where ASvb is the entropy of vaporization at the boiling point and KF is derived from a
consideration of dipole moments of polar and nonpolar molecules.  Grain (1990) provides a table
of KF for various compound classes. Equation (8.7) is a modification of the well known
relationship of constant entropy of vaporization developed by Trouton in 1884.

       From the Antoine Equation, plots of In Pv (y-axis) versus the reciprocal of temperature (x-

                                          107

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axis) yield a straight line within a temperature range where phase change does not occur (liquid
to solid). If the melting point of a compound lies within the temperature range considered, the
vapor pressure curves shows a break at this transition temperature.  Below the melting point, a
compound vaporizes from the solid phase. This process is known as sublimation. The vapor
pressure of a solid can be approximated by extrapolating the vapor pressure of an imaginary
liquid (subcooled liquid) at the given temperature with modification using the entropy of melting,
however, the second method of vapor pressure estimation which was developed for solids is
more reliable (Crane, 1990).

       The second method incorporates the temperature dependence of the heat of vaporization.
Vapor pressure can be estimated by
ln/>
_^L 3-—   -2m\3-—\   In  —
                                                                                (8.9)
The value of m in equation (8.8) depends on the physical state of the compound at the
temperature of interest. For solids, Grain (1990) recommends the following values
 T
— >0-6;
           m = 0.36
                                                                (8.10)
0.6 > — >0.5;   m = 0.80
                                                                (8.11)
— <0.5;
                                                                (8.12)
When the boiling point is only available at a reduced pressure (Pj), Grain (1990) recommends
estimation of vapor pressure by
                  )m      /       \rn-l   /  \
                         (    IT I      \ T
                    -2m 3-—    In  —
                             T1         T1
                         V    71  y      V7ly
                                                                                 (8.13)
The ratio AH^/Tj can then be evaluated by:
           [8.75 + 9t (In 7] - In/I)]
                                                                (8.14)
8.3    Henry's Law Constant
                                          108

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       Partitioning from the gas phase to the aqueous phase can be expressed by

Cg=HCw                                                                        (8.15)

where Cw = aqueous concentration [ML"3] and H = is the "dimensionless" Henry's law constant.
The "dimensionless" Henry's Law constant" is not really dimensionless but has units of [vol-
liquid/vol-gas].  Henry's law constants are often given in units of atm-m3/mole which can be
converted to a dimensionless form by
           8.205x1 o~5r
                                                                                  (8.16)
where H is Henry's law constant expressed in atm-m3/mole and T = temperature (°K). When a
laboratory derived value can not be found, a Henry's Law Constant can be estimated by:


          CsfMl
#=16.04^ - -                                                                 (8.17)
                                                                                  ^    '
where Csatw, and Csatg are the saturation aqueous (solubility) and vapor phase (can be derived
from vapor pressure) concentrations respectively [ML"3]. For the equation above, Csatw, and Csatg
are expressed in units of mg/1 and mmHg respectively.  M; is the molecular weight [M/mole] of
the compound of interest.

8.4    Gas-Water Interfacial Partitioning

       In the absence of NAPL, retention of VOCs in porous media has generally been
considered to include adsorption on mineral surfaces and partitioning in soil organic matter and
water.  However, Pennell  et al. (1992), Hoffet al. (1993), Conklin et al. (1995), and Brusseau et
al. (1997) found in laboratory experiments that retention was significantly greater than
attributable to sorption and partitioning into bulk water.  They attributed this additional retention
to adsorption at the gas-water interface. For example, Hoffet al.  (1993) estimated that 48-56%
of the observed retention of straight-chain alkanes by sandy aquifer material was due to gas-water
interfacial adsorption. Pennell et al. (1992) reported that accumulation at the gas-water interface
accounted for about 50% of total p-xylene retention in unsaturated soils and clay minerals.
Conklin et al. (1995) evaluated gas-water interfacial adsorption of benzene  and p-xylene in desert
sand having a low organic carbon content (0.097%) and sorption capacity (saturated soil-water
partition  coefficients for benzene and p-xylene 0.0380 and 0.240 cm3g"1 respectively) at
volumetric water contents of 0.10 and 0.18. Retardation coefficients estimated from column gas
flow studies revealed that 34% and 61% of p-xylene was retained at the gas-water interface at
volumetric water contents of 0.1 and 0.18 respectively while  less than 10%  of benzene was
retained for either volumetric water content. Brusseau et al. (1997) evaluated gas-water
                                           109

-------
interfacial adsorption of methane, heptane, TCE, and benzene in glass beads (0.35 - 0.73 mm
diameter), silica sand, and aquifer material (< 2mm fraction consisting of 89.5% sand, 4.1% silt,
6.4% clay, and 0.03% organic carbon) at volumetric water contents of 0.119, 0.094, and 0.16
respectively.  Retardation coefficients estimated from column gas flow studies revealed that,
2.6%, 67.2%, and 83.5% of methane, TCE, and heptane respectively was retained at the gas-
water interface in the silica sand system.  In the glass bead system, 61.6% to 73.5%  of TCE and
53.5% to 60.6% of benzene was retained at the gas-water interface. In the aquifer material
system, 29.1% to 47.6% of TCE was retained at the gas-water interface.  These experiments
indicate that retention at the gas-water interface may be important in soils with low organic
carbon or absent of NAPL.

         The magnitude of interfacial partitioning is largely controlled by the specific interfacial
area Aja [L"J]and the Kia = the interfacial sorption coefficient [L].  Partitioning from the gas phase
to the gas-water interface can be expressed by:

C,a=K,aCg                                                                       (8.18)

where Cg = gas  concentration [ML"3], and Cia = concentration at the gas-water interface [ML"2].
Gvirtzman and Roberts (1991) state that maximum interfacial area should occur when water
films are relatively thin and pendular rings form across adjacent mineral grains. As soil moisture
content decreases further and approaches monolayer water coverage on mineral surfaces, Aja will
approach the specific surface area of the porous medium.  Costanza and Brusseau (2000) state
that physical soil characteristics such as soil texture, pore size distribution and wetting/drying
cycles are factors that should affect the geometry of the soil water in porous media and thus are
considerations in assessing Aja. Gas-water interfacial area is important not  only in determining
retention at the interface but in gas-water mass exchange kinetics.  It is expected that thin water
films on mineral surfaces would have minimal mass-transfer rate limitations while aqueous
diffusion limited mass transfer may be important for water-filled pores. Experimental evidence
indicates that vapor adsorption at the gas-water interface may be considered instantaneous
relative to other transport processes (Lorden et al. 1998) however there has been no  systematic
investigation to verify this assumption.

      Karger et al. (1971), Pennell  et al. (1992), and Hoff et al. (1993) established  that
increased hydrophobicity leads to greater retention at the gas-water interface.  However, other
physicochemical properties besides hydrophobicity determine retention at the gas-water interface.
As Costanza and Brusseau (2000) explain, at a given vapor pressure, a low molecular weight
compound such as methane will show much less interfacial adsorption than a higher molecular
weight compound such as octane.  This is attributed to the higher volatility  of methane. As
saturation vapor pressure  is  approached however, the trend is reversed where more volatile
compounds like methane begin to exhibit greater total interfacial adsorption compared to
compounds like octane. This is due to the much larger saturation vapor pressures of lower
molecular weight compounds.  Vapor pressure is proportional to the number of molecules in the
vapor phase  and subsequently, to the number of collisions with the gas-water interface. Larger

                                           110

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saturation vapor pressures of lower molecular weight compounds provide a greater probability of
adsorption. Thus, less volatile compounds will dominate interfacial adsorption at lower vapor
concentrations or partial pressures while more volatile compounds will dominate interfacial
adsorption near saturation vapor pressure.  Costanza and Brusseau (2000) state however that
even in the presence of bulk NAPL, it would be rare for vapor concentrations to approach
saturation levels.

       The observation that greater hydrophobicity leads to greater retention at the gas-water
interface appears to contradict the fact the interfacial partition coefficients are higher for more
hydrophilic compounds. More polar compounds such as aromatics and chlorinated hydrocarbons
tend to have higher interfacial partition coefficients compared to nonpolar compounds. More
polar compounds though tend to partition strongly into the bulk water phase thereby reducing
concentration at the gas-water interface. Thus water content becomes important in determining
concentration at the interface.  Costanza and Brusseau (2000) provide a summary of Kia
estimation methods and a table of literature values of Kia for selected compounds.

8.5    Gas-Solids Partitioning

       Partitioning coefficients have only been found to be relatively independent of the degree
of saturation when gravimetric water contents exceed 2-5%.  Direct sorption from the gas phase
to the mineral surfaces may be neglected in most cases except under very arid conditions (Chiou
and Shoup, 1985).  In extremely dry soil-water systems, solid-vapor adsorption has been  shown
to be related to the mineral surface and moisture content of the soil (McCarty et al., 1981; Goss
and Eisenreich, 1996). However, adsorption is no longer affected by moisture content upon
coverage of the internal mineral surface by liquid water (Chiou and Shoup, 1985). Gas-solids
partitioning will not likely be an important process for soil venting systems in a temperate
environment.  The importance of gas-solids partitioning during venting in an arid environment is
unknown.

8.6    Solids-Water Partitioning

       Partitioning from the aqueous phase to the solid phase can be expressed in the form of a
linear Freundlich isotherm

Cs=KdCw                                                                       (8.19)

where Cs = solids concentration [MM"1] and Kd = the sorption partition coefficient [L/'M"1]. The
mechanism underlying sorption of nonionic organic compounds from aqueous solution onto
aquifer solids, sediments and soils is postulated to be hydrophobic partitioning into a stationary
organic phase, such as soil organic matter (Karickhoff et al., 1979; Means et  al., 1981; Chiou et
al., 1979,1983; Gschwend and Wu, 1985). The soil-water partition coefficient (Kd) can be
directly determined by laboratory testing or estimated by the empirical relationship
                                           111

-------
Kd~Kocfoc                                                                      (8.20)

where Koc = organic-carbon-water partition coefficient (L^M"1) and foc = fraction of organic
carbon [MM"1]. This relationship is generally recognized as being valid when foc > 0.001
(McCarty et al., 1981; Schwarzenbach and Westall, 1981; Karickhoff, 1984). For soils with low
organic carbon content (foc < 0.1%), sorption to mineral grains may be dominant (Piwoni and
Banner] ee, 1989) requiring direct measurement of soil-water partition coefficients. Organic
carbon-water partition coefficients are often estimated from octanol-water partition coefficients
or water solubility.

       A compound's Koc is typically estimated from a compound's water solubility or octanol-
water partition coefficient. The applicability however of using octanol-water partition coefficient
and aqueous solubility data to estimate Koc has been questioned however because some organic
compounds have similar aqueous solubility but very different octanol-water partition coefficients
(Ellgehausen et al. 1981, Mingelgrin and Gerstl, 1983, Olsen and Davis, 1990). A more
fundamental approach using molecular topology, specifically a parameter called the first-order
connectivity index which describes the size and structure of an organic molecule, has been
suggested to estimate Koc (Koch, 1983; Sabljic, 1987, 1989; Sabljic and Protic, 1982).  Fetter
(1992) states that molecular topology has several advantages over the use of octanol-water
partition coefficient and water solubility regression equations for estimation of an organic
compound's Koc value: (1) there is a theoretical basis to molecular topology for nonpolar organic
compounds; (2) the literature contains a wide range of experimentally derived values for octanol-
water partition coefficients and aqueous solubility values for nonpolar organic compounds where
there is no way to know which values are correct; (3) some compounds with similar aqueous
solubility values have quite different octanol-water partition coefficients; (4) there are  a number
of competing regression equations involving octanol-water and water solubility; (5) regression
equations involving solubility and octanol-water coefficients to estimate Koc were devised strictly
for nonpolar organic compounds. Another advantage of using molecular topology is that the
direct correspondence between molecular structure and molecular connectivity makes  its possible
to better understand the underlying mechanisms of sorption on a molecular level (Sablijc, 1989).
Hu et al. (1995) estimated sorption mass transfer coefficients direct from molecular connectivity
indexes.

8.7     NAPL-Gas/Water Partitioning

       Equilibrium exchange between NAPL and the aqueous phase can be expressed by

           ^                                                                     (8.21)
where Y; and %; are the activity coefficient [-] and mole fraction [-] of component i in the NAPL
mixture respectively. Equilibrium exchange between NAPL and the gas phase can be expressed
by
                                           112

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Cg = 7lXlCsgat                                                                    (8.22)

thus,

cg = ylZlcsgat = riXiHC? = HCW .                                                 (8.23)

A NAPL-gas partition coefficient, Kng [-] can defined by


^=FL                                                                        (8'24)
       cg

where Cn = non-aqueous phase liquid concentration [ML"3].  Since Cn can be expressed as


                                                                                 (8.25)
and
where Pv is the vapor pressure of an organic compound. Kng can be expressed as a function of
vapor pressure or water solubility
           7=1                 7=1


This equation can be simplified by
  JT-S   _ _ r-\ n f      ._- _- ^~, cs-tt -,                                                         ^    *
   *
where Mn and pn are the mean molecular weight [Mmol"1] and density [ML"3] of the NAPL
mixture respectively. Alternatively, a NAPL-water partition coefficient (Knw) can be expressed
as
                                           113

-------
             M,
                                                                                 (8.29)
             7=1
                 Pi
which is consistent with formulations of Lee et al.(1992) and Cline et al. (1992) for NAPL-water
partitioning when the activity coefficient equals unity. The activity coefficient for a compound in
a mixture of structurally similar compounds is close to unity.  This has been shown for mixtures
of aromatic compounds like benzene and toluene as well as saturated alkane constituents like
hexane and octane (Leinonen et al 1973 and Sanemasa et al. 1987). Mixtures of alkanes and
aromatic constituents, however show some deviation from ideality that can result in somewhat
higher aqueous concentrations than predicted.

8.8     Advective-Dispersive Equation for Vapor Transport
       Total retention of VOCs in porous media can be described by
CT =cK0K+ Amcm +cw0w + Pbcs + cn en
                                                                   (8.30)
where CT = total soil concentration [ML ], 8g = volumetric gas content [L L"3], 8W = volumetric
water content [L3L"3], 8n = volumetric NAPL content [L3L"3], and pb = bulk density [ML"3].  Thus,
under equilibrium conditions, the total soil concentration may be estimated from gas
concentration by
    =C
     ^
                                      Sirs,
                                                                   (831)
       The first generation of models designed to simulate gas-phase advective transport of
volatile organic chemicals were based on the assumption of equilibrium or ideal transport
(Johnson et al., 1988; Wilson et al., 1987, 1988; Baehr et al., 1989).  That is rate-limited mobile-
immobile gas (physical heterogeneity), gas-water, solids-water, and gas-NAPL exchange is
insignificant or does not occur.  The advective-dispersive equation including a storage term at the
air-water interface is given by:
    dt
dt
                      ac,
                 ac.
dx,
                                                                                 (8.32)
where D;j = dispersion coefficient tensor [L T ]. Invoking the assumption of local equilibrium
                                          114

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for the gas-water interface, gas-bulk water and solid-water phases results in the commonly
expressed advective-dispersive equation
     p              ^         p
R — L=n___L  _v — L                                                    (8.33)
    dt      1J dxt ( dXj }    dxt

where the retardation coefficient R [-] equals

R = l+^a4a.+Md_+J^+ - 9tr^ -
         eg      eKu    #H
          8      8       8
                                     7=1
Equation (8.34) is only valid when the volumetric NAPL content remains constant and mole
fractions remain the same such as in the case of a recalcitrant waste oil. Using R, mass fractions
in gas (fs), gas-water interface (fia), water (fw), solids (fs), and NAPL (fn) can be calculated as
  w
  w   H90R
                                                                                  (8-36)
                                                                                  (8.37)
 f = PbKd                                                                        (8.38)
  °
8.9    Dispersion

       When the advective-dispersion equation using the local equilibrium assumption is used to
fit effluent breakthrough curves, rate-limited vapor transport results in breakthrough curves
which give the "appearance" of increased dispersion.  The dispersion coefficient however should
not be used as a correction term to account for rate-limited mass transport. This has been
attempted repeatedly with failure.  To explicitly account for this rate-limited mass transport, the
contribution of molecular diffusion and mechanical dispersion to dispersive flux must first be

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quantified. The dispersion coefficient is defined as:


£> = — + £>mv                                                                    (8.40)
where Dg = free-gas molecular diffusivity [L2T4], Tg = tortuosity factor of gas filled porosity [-],
and Dm = mechanical dispersion coefficient [L], explicitly accounts for the combined effects of
longitudinal molecular diffusion and mechanical dispersion. On the pore-scale (typical of a soil
column study), mechanical dispersion is believed to be caused by: (1) fluid moving faster in the
center of pores than along edges, (2) some fluid particles traveling along longer flow paths
compared to other particles, and (3) larger pores conducting more fluid than smaller pores
(Fetter, 1992). On a macroscopic or field scale, mechanical dispersion is much larger and
appears to be caused primarily by spatial variation in hydraulic conductivity (Fetter 1992, and
Domenico and Schwartz, 1990). Since mechanical dispersion is combined with diffusion in the
advective-dispersive equation, the contribution of mechanical dispersion to overall dispersion is
often not clear.  Pfannkuch (1962) conducted a series of column experiments to determine
dispersion as a function of Peel et number (Pe) which he defined as
                                                                                  (8.41)

where d50 is the mean grain size [L] and Dw is the aqueous molecular diffusion coefficient [L2!"1].
Pfannkuch (1962) plotted Peclet numbers versus dispersion coefficients normalized to free
aqueous diffusion coefficients to determine the affect of pore velocity on mechanical dispersion.
He observed that at high pore velocities or Peclet numbers when mechanical dispersion became
dominant over aqueous diffusion, dispersion normalized for aqueous diffusion versus Peclet
numbers plotted as a straight line with 45 degree angle indicating that pore velocity directly
determined mechanical dispersion.  Mechanical dispersivity then is considered a characteristic
property in a medium and in a three dimension system, a tensor similar to hydraulic conductivity.

       A number of ground- water field studies however have demonstrated that dispersion is
scale dependent. In many of these studies physical heterogeneity and mechanical dispersion at
the macroscopic scale appeared to have contributed significantly to overall dispersion. Values of
dispersion are in general are two or more orders of magnitude greater in field studies compared to
column experiments (Domenico and Schwartz, 1990). Stochastic theory (Gelhar, 1979) predicts
that macroscale dispersivity should approach an constant or asymptotic value. These large values
may however not be appropriate for simulation at the scale of interest.  This makes selection of a
mechanical dispersion coefficient difficult for model simulation. Domenico and Schwartz (1990)
state that one way to explain spatially variable dispersivities is in relation to some representative
elemental volume (REV).  With the spreading of a solute, eventually a transition occurs from the
microscale to the macroscale.  The implicit assumption is that at each scale an REV exists where
dispersivity is constant.  In the zone of transition between scales, it may be difficult to define a


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value for dispersivity however. As a consequence, small changes in volume could result in
significant variability. Also, mass may not be normally distributed as classical theory would
predict resulting in undefmable dispersivity values.  Thus, in field-scale gas transport studies,
mechanical dispersivity may be largely an undefmable input parameter requiring sensitivity
analysis for each application to determine its importance in  model output and decision making.

       Diffusion in the gas phase is considerably more complex than in the aqueous phase. Gas-
phase diffusion can occur not only by molecular diffusion but also by Knudsen diffusion and
nonequimolar diffusion (Thorstenson and Pollock, 1989). Knudsen diffusion and nonequimolar
diffusion are all molecular diffusion processes arising from  random kinetic energy and motion of
the molecules. They only differ in that specific conditions impose constraints and conditions on
directions and rates of diffusion. Equations based on Pick's law are not adequate for
representing diffusion in systems where contributions of Knudsen and nonequimolar diffusion
are significant.  In these systems, the "dusty gas" model is more appropriate (Thorstenson and
Pollock, 1989). The contribution of Knudsen diffusion however will be negligible except for
very fine grained materials (Thorstenson and Pollock, 1989).  Equations based on Pick's law are
also not appropriate for systems where the concentration in  the gas phase is not dilute or where
evaporative fluxes  (mass transfer from organic liquid to the gas phase) occur (Baehr and Bruell,
1990). The assumption of dilute concentrations in the gas phase will be valid except when large
sources of chemicals with large vapor pressures are present.  During soil venting application, a
detailed description of diffusion is probably unnecessary because the contribution of advective
flux will overwhelm that of diffusive flux when pressure differences are on the order of 1%
(Thorstenson and Pollock, 1989). Hence the use of equations based on Pick's law to describe
diffusive flux is probably adequate in most cases. Molecular diffusion occurs from random
molecular motion due to the thermal kinetic energy of individual molecules. This motion causes
random spreading and mixing.  The coefficient describing this motion is much larger in the gas
phase compared to the liquid phase because molecules are much farther apart in the gas phase.
Dg depends on the molecular weight of the chemical diffusing in soil gas and on soil temperature
and pressure.

       To directly  assess the contribution of molecular diffusion and mechanical dispersion to
dispersive flux during gas flow without the interfering effects of physical heterogeneity or rate-
limited sorption, Popovicova and Brusseau (1997) conducted of a series of laboratory column
experiments using carefully packed 0.59 mm dry glass beads to create a homogeneous media.
Methane was used  as a nonsorbing conservative tracer.  Experiments were  conducted at gas
velocities ranging from 0.1 to 3.33 cm/s. Breakthrough curves for methane were symmetrical
indicating no rate-limiting behavior.  Peclet numbers, which arise during non-dimensionalization
of the advective dispersion equation were defined as


Pe = —                                                                          (8.42)


and determined through a non-linear, least squares optimization program.  At high gas velocities

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when mechanical dispersion is the primary cause of dispersion, Peclet numbers approach L/Dm
because Dg/Tg« Dmv.  Popovicova and Brusseau (1997) found that Peclet numbers were a
function of gas velocity only at velocities less than 0.33 cm/s indicating that molecular diffusion
was significant at velocities less than 0.33 cm/s.  At gas velocities less than 0.33 cm/s,
Popovicova and Brusseau (1997) assumed that molecular diffusion was the only process
contributing to dispersion.  Use of a literature value for the free air molecular diffusion for
methane (0.28 cm2/s) enabled calculation of a tortuosity for the homogeneous media. Fitted total
dispersion values then allowed determination of the percent contribution of diffusion and
mechanical dispersion to total dispersion as a function of gas velocity.  They found that the
contribution of molecular diffusion to total dispersion was less than 3% for gas velocities greater
than 2.5 cm/s.  Comparison of these results with those reported by Brusseau (1993) shows a
difference of approximately three orders of magnitude between gas and water velocities at which
the contribution of both longitudinal molecular diffusion and mechanical dispersion was 50%.
This difference is expected since vapor-phase diffusion coefficients are approximately four
orders of magnitude greater than aqueous diffusion coefficients. Thus, under typical field
conditions where pore-gas velocities away from the immediate vicinity of a gas extraction well
range from 0.001 to 0.1 cm/s,  molecular diffusion will dominate the dispersion term.

       During field-scale gas  displacement experiments, low pore-gas  velocities and high
dispersion coefficients due to  gaseous molecular diffusion will lead to low Peclet numbers.
Brusseau et al. (1989) demonstrated that low Peclet numbers produce breakthrough curve
asymmetry similar to nonideal transport. They used Brenner's solution to the advective-
dispersive equation which is solved for a flux-type inlet boundary condition and finite zero-
gradient outlet boundary condition, to assess the effect of Peclet numbers when R = 1 on
generation of asymmetrical breakthrough curves.  They noted that as the Peclet number increases
in homogeneous material, breakthrough curves approach a symmetrical, sigmoidal shape.
However, for Peclet numbers  less than approximately 10, breakthrough curves are noticeably
asymmetrical.  While this analysis was conducted for solute transport, this finding is important
for gas transport because under some column experimental conditions and likely under typical
field conditions,  the Peclet number will  often be less than 10. Thus dispersion due to molecular
diffusion could easily be mistaken for nonequilbrium behavior if not properly accounted for.
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9.     RATE-LIMITED VAPOR TRANSPORT: SELECTION OF CRITICAL PORE-
       GAS VELOCITIES TO SUPPORT VENTING DESIGN

       The effectiveness of soil venting as a method to remove VOCs from unsaturated media
basically relies on two factors: 1) the ability to move gas or air directly through contaminated
media by advection and 2) rate-limited mass exchange.  Anecdotal evidence from field studies
and published studies (Krishnayya et al., 1988; Croise et al., 1989; Ho and Udell, 1992; Kearl et
al., 1991) indicate that physical heterogeneity on a large-scale such as lenses or layers of low
permeability, perched water-table conditions, or regions of local saturation can severely limit the
effectiveness of venting application.  Reduced mass extraction due to physical heterogeneity can
be overcome to some degree by improved venting design (e.g., increasing well spacing or
improved placement of screened intervals). For instance, in the mid-western United States (e.g.,
Nebraska) it is common to have a thick layer of lower permeability loess overlaying a higher
permeability layer of sand. When both media are contaminated, much better advective gas flow
can be achieved in the loess by installing separate gas extraction wells in this material as opposed
to screening wells across both stratigraphic layers whereby most gas flow would occur  in the
sand layer.  It is impractical  and in most cases infeasible though to selectively screen extraction
wells in thin discontinuous lenses of silt or clay because of physical dimensions and inability to
locate these lenses in the field.  Because of the latter consideration, physical heterogeneity on this
scale is not commonly explicitly described in numerical models but instead represented by first-
order rate constants.

       A growing number of published studies indicate that rate-limited gas-NAPL, gas-water,
and water-solids mass exchange on a small, pore, or microscopic scale can also significantly
limit the effectiveness of venting application. Laboratory column and sand tank studies such as
those reported by McClellan and Gillham (1992) and Ng et al. (1999) have demonstrated that
volatilization from NAPL in contact with the mobile gas phase ("free" NAPL) initially controls
mass removal during venting operation.  Following free NAPL removal, transfer of VOCs to the
mobile gas phase is controlled by mobile- immobile gas (e.g., macropores such as that described
by Popovicova and Brusseau, 1997, 1998), intraaggregate aqueous diffusion (Gierke et al., 1992;
Ng and Mei, 1996), interparticle aqueous diffusion (Grathwohl and Reinhard,  1993; Conklin et
al., 1995; Fisher et al., 1996), solids-water (Croise et al., 1994; Lorden et al., 1998), or  a
combination of gas-water and solids-water (Brusseau, 1991) mass exchange.

       The discussion that follows on selection of pore-gas velocities to support venting design
then considers rate-limited gas-NAPL, gas-water, and solids-water mass exchange on a pore
scale and rate-limited mobile-immobile gas exchange on field-scale where physical heterogeneity
can not be explicitly represented or overcome by improved venting design.

9.1    Definition of Critical Pore-Gas Velocity

       During rate-limited dominated mass transfer, as pore-gas velocity increases, vapor
concentration decreases resulting in decreased "efficiency" in mass removal. Efficiency is
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defined as mass removed per volume of extracted gas.  However, an increased pore-gas velocity
still results in an increased mass removal rate (mass removed per time) and hence shorter
remediation time because of an increased concentration gradient between equilibrium and
nonequilibrium vapor and soil-water concentrations. Thus, there is a trade-off between selection
of a design pore-gas velocity and remediation time. The importance of maintaining a significant
concentration gradient during rate-limited dominated vapor transport was demonstrated by
Armstrong et al. (1994) during analysis of pulsed versus continuous venting operation.  In pulsed
pumping, the strategy is to extract gas only until the tailed portion of the effluent curve is
reached.  The gas extraction well is then shut-in to allow subsurface gas-phase concentrations to
rebound. Pulsed pumping is intended to provide lower energy and vapor treatment costs
compared too continuous operation. The length of on and off periods is dependent on kinetic
mass transport coefficients and pore-gas velocity.  In order to evaluate mass removal efficiency
under a pulsed regime, Armstrong et al. (1994) investigated two pumping scenarios under gas-
water and solids-water rate-limited mass exchange. The pumping schedules and corresponding
pore-gas velocities were as follows: (1) 2 hours on at 0.14 cm/s, 8 hours off; and 2)  1 hour on at
0.14 cm/s, 4 hours off; each giving an average pore-gas velocity over the full cycle of 0.03 cm/s.
These scenarios were compared to continuous pumping at 0.03 cm/s and 0.14 cm/s. As
expected, continuous pumping at a rate at 0.14 cm/s resulted in the highest mass removal rate.
Continuous pumping rate at 0.03 cm/s was more effective than the 1-4 hour pulsed pumping
cycle at 0.14 cm/s.  The slowest mass removal rate was the 2-8 hour cycle at 0.14 cm/s.
Overall, the continuous but slow pumping rate or pore-gas velocity gave the best mass removal
performance. Armstrong et al. (1994) state that this is because continuous pumping maximizes
the concentration gradient for diffusive mass transfer for longer periods of time.

       Armstrong et al. (1994) simulated the time hypothetically required to achieve 99% mass
removal under continuous pumping as a function of pore-gas velocity. This relationship was
highly nonlinear. The curve had two distinct parts, a slow part in which gas residence time was
long allowing partitioning processes time to equilibrate and a fast part in which mass transport
kinetics controlled mass removal. For the slow part, the limiting value corresponding to zero
pore-gas velocity was the time required for 99% mass removal by diffusion.  Applying even a
low pore-gas velocity in  this scenario resulted in a rapid decrease in remediation time compared
to mass removal by vapor diffusion. Armstrong et al. (1994) selected a design or "critical" pore-
gas velocity at the point or general location where the two portions of the curve met. A critical
pore-gas velocity then is defined as a pore-gas velocity which results in slight deviation from
equilibrium conditions.

9.2     Approaches to Estimating Critical Pore-Gas Velocities

       There are conceivably at least five ways that could be used separately or concurrently to
select a critical pore-gas velocity for venting application. The  most rigorous approach would be
to conduct site-specific field-scale tests to elucidate and quantify immobile gas  - mobile gas
(macropores, layers of different permeability), water-gas, solids-water, and NAPL-gas rate-
limiting processes.  However little research has been performed in this area, thus published field-
                                          120

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scale procedures are lacking. Development of field-scale procedures to identify and quantify
rate-limited processes during venting constitutes a definitive research need.

       A second method would be to conduct laboratory-scale tests with extrapolation of results
to the field scale.  A variety of procedures for conducting laboratory column tests can be found in
the literature. However with the exception of just a few studies, methodologies have been
designed to isolate or evaluate just one and at most two rate-limiting process in clean sieved
sand, glass beads, or constructed media (e.g., fired clay aggregates) in which contaminants are
artificially introduced into soil columns.  These laboratory experiments have been valuable in
understanding fundamental mass transport processes but are of limited use for direct application
at hazardous waste sites.

       A third approach would be to use fictitious but "representative" soils data and mass
transport coefficients to conduct numerical modeling simulating rate-limited behavior.
Rathfelder et al. (1995) state that of the three-phase (NAPL, water, gas) models which have been
applied to venting application, only the models of Sleep and Sykes (1989), Rathfelder et al.
(1991), and Lingineni and Dhir (1992) consider rate-limited NAPL-gas, gas-water, and water-
solids mass exchange. Recently, Abriola et al. (1996) developed a two-dimensional, finite-
element, two-phase flow, public domain model  (MISER) which incorporates these rate-limited
transport processes using first-order rate coefficients that would be suitable for estimation of a
critical pore-gas velocity. The model is robust but complicated, requiring a considerable
investment in time for proper use.

       A fourth approach would be to  review available literature in which rate-limited vapor
transport has been observed for similar VOCs and soils (including moisture content) and select a
pore-gas velocity lower than that which resulted in deviation from the local equilibrium
assumption.  Table 9.1 provides a summary of published studies on rate-limited NAPL-gas, gas-
water, and solids-water mass exchange during gas advection. However, while a review of
available literature provides valuable insight into rate-limited mass transport processes,  it is
unlikely to find a situation which matches site-specific conditions well.  Also, pore-gas velocities
used for laboratory column studies are  often at least an order of magnitude higher (0.1 to 10
cm/s) than typical field application (0.01  cm/s or less).  In addition, the characteristic transport
length in which gas must pass through  to observe rate-limited behavior is much smaller for a
laboratory-scale study than would be expected in the field.

       A fifth approach would be to utilize Damkohler numbers, which represent the ratio
between reaction time and advection time and recently published algorithms for rate-limited gas-
water, solids-water, and NAPL-gas mass transfer to calculate a critical pore-gas velocity. Bahr
and Rubin (1987) concluded that the local equilibrium assumption is applicable if Damkohler
numbers exceed  100. Jennings and Kirkner (1984) obtained good approximations to the local
equilibrium assumption for Damkohler numbers as low as 10. Brusseau (1991) states that the
dynamic range for Damkohler numbers is between 0.01 and 100 with values over 10 approaching
the local equilibrium assumption.  Thus, based on these analyses, it appears reasonable to
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Table 9.1  Summary of column and sand tank studies where rate-limited vapor transport has been evaluated
pore-gas vel.
(cm/s)
0.178* -0.703*
2.07*, 4.47*,
10.92*
0.055
0.636
0.016
0.144-0.545
0.009-0.018
0.01,0.017,
0.020
0.061
0.1-0.2
0.015* -0.076*
7.83*
media description
moist Borden sand, foc = 0.00017- 0.0008, S0 = 0.088, Sw
= 0.25
moist medium-grained sieved sand (avg. grain size =
0.90 mm), S0 = 0.075, Sw = 0.22,
uniform sand with aggregated porous media (fired clay)
(data from Gierke et al., 1991)
desert soil (data from Roberts, 1990)
Yolo loam (data from Brown and Rolston, 1980)
moist sorted sand (median grain size 0.25-0.5 mm)
moist clean quartz sand (grain sizes in the range of 0.08-
1.2mm)
moist coarse nonaggregated Ottawa sand (Sw = 0.26), dry
aggregated porous media (fired clay)
aggregated porous media (fired clay)
silt loam, foc=0.025, water content 0.0, 0. 16, 0.22 by
weight
two sandy soils having foc (0.001, 0.0165) and saturation
(0.08, 0.43) values respectively. S0 = 0.02
three mixtures of dry sand with average particle diameter
of 1.2, 0.25, and 0.04 mm
*ng
(s-1)
not rate-limited
not rate-limited
NAPL absent
NAPL absent
NAPL absent
NAPL absent
NAPL absent
NAPL absent
NAPL absent
rate-limiting at
16% water
content
rate-limited
behavior unclear
not rate-limited
V
(s-1)
4.0 x 10'5 (after
evaporation of NAPL)
not evaluated
3.06 xlO'4
2.78 x 10'2
1.28 x 10-3
3.0xlO'4-5.5 xlO'4
5.0xlO-6-1.0xlO-3
not rate-limited
aggregate diffusion
not evaluated
not evaluated
not applicable
"•ws
(s-1)
5.0 xlO'5
(after evaporation of
NAPL)
not evaluated
no sorption
2.78 x 10'3
8.33 x 10-4
no sorption
no sorption
no sorption
no sorption
not evaluated
not evaluated
not evaluated
author(s)
Armstrong et al.
(1994) and
McClellan and
Gillham(1992)
Baehr et al.
(1989)
Brusseau(1991)
Brusseau(1991)
Brusseau(1991)
Conklin et al.
(1995)
Fisher et al.
(1996,1998)
Gierke et al.
(1992)
Gierke et al.
(1992)
Harper et al.
(1998)
Hayden et al.
(1994)
Ho and Udell
(1992)
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pore-gas vel.
(cm/s)
1.5*
0.002-0.161
9.35*,7.73*
7.52*, 30.00*,
50.15*
0.29, 0.50
0.0074
0.053*, 3.143*
0.167-4.50
0.27*
0.35
media description
dry sand having average particle diameter of 0.25 mm
coarse nonaggregated Ottawa sand (0.59 - 0.84 mm dia.)
with initial uniform moisture content, countercurrent gas
and water flow
dry and moist Monterey sand, mean grain size 0.25mm,
S0 = 17% , Sw = 0 & 16%
dry glass beads (0.36-mm dia), S0 = 0. 13
dry fine-grained sand
unspecified soil
unspecified soil from Superfund site
moistened glass beds (0.59 cm dia)
dry glass beads (0.50-1.0 mm dia) and dry Borden sand
(0.25-0.50 mm dia)
moist fine to medium sand, foc=0.004, Sw=0.24
*ng
(s-1)
not rate-limited
NAPL absent
not-rate limited
not rate-limited
NAPL absent
rate-limited-
aggregate
diffusion
modeling
NAPL absent
NAPL absent
not rate-limited
NAPL absent
gw
(s-1)
not applicable
equilibrium
not evaluated
not applicable
not applicable
aggregate diffusion
modeling
not evaluated
3.15xlO-3-6.96 x
io-2
not applicable
rate-limited
"•ws
(s-1)
not evaluated
no sorption
not evaluated
not evaluated
4.14xlO-7-5.59xlO-4
aggregate diffusion
modeling
1.15xlO-6,2.27xlO-6
1.24 xlO'2- 2.43x10-'
tailing after
evaporation of NAPL
rate-limited
author(s)
Hoetal. (1994)
Imhoff and Jaffe
(1994)
Liang and Udell
(1999)
Lingineni and
Dhir (1992)
Lorden et al.
(1988)
Ngetal. (1999)
and Ostendorf et
al. (1997)
Nadim et al.
(1997)
Popovicova and
Brusseau (1998)
Rathfelder et al.
(1991) and
Bloes et al.
(1992)
Wehrle and
Brauns (1994)
calculated from information provided in publication
ng     NAPL - gas first-order rate constant
^     gas - water first-order rate constant
roc     solids - water first-order rate constant
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estimate critical pore-gas velocity based on Damkohler numbers of 10. A limitation using
Damkohler numbers and algorithms to estimate critical pore-gas velocities is that mass transport
processes are considered independently when in reality complex interactions between these
processes dictate overall mass removal. It would appear that a combination of the fourth and
fifth approaches is the easiest and most practical method of estimating critical pore-gas velocities
for most venting practitioners until guidance for conducting laboratory and field-scale testing can
be provided.

       There are however additional complications to consider in selection of critical pore-gas
velocities. Virtually all research on rate-limited vapor transport and development of algorithms
for pore-gas velocity determination originated from controlled sand tank and laboratory column
studies where field-scale heterogeneity was not a factor. Thus, optimal pore-gas velocities in the
field are likely to be very site-specific and somewhat smaller than indicated by Damkohler
numbers, algorithms, and published studies. Also, characteristic lengths of contamination used
in Damkohler numbers and algorithms are not constant but decrease in time due to NAPL and
sorbed mass removal thereby reducing calculated optimal pore velocities.  Finally, the cost of
well installation and venting operation (e.g., electricity, vapor treatment) should be considered
since achievement of a minimum target pore-gas velocity at some sites may be prohibitively
expensive.  Thus,  it is apparent that selection of a critical pore-velocity is not a straightforward
process but involves some degree of judgement and iterative reasoning (gas flow modeling to see
what pore-gas velocities can be achieved given certain well spacing and flow rates).
Nevertheless, venting design based on attainment of critical pore-gas velocities  is superior to the
most commonly used soil venting design method, radius of influence (ROI) testing, which will
be discussed in sectionlO. A design based on attainment of critical pore-gas velocities in
contaminated media involves consideration of fundamental subsurface gas flow and mass
transport processes whereas a design based on ROI testing simply ensures attainment of vacuum
in a contaminated region.

       The fourth and fifth approaches to estimating a critical pore-gas velocity are considered
here by providing a critical literature review of laboratory studies, Damkohler numbers, and
algorithms published on rate-limited vapor transport. A large body of literature is developing on
rate-limited vapor transport with contributions primarily from environmental engineering, soil
and ground-water science. There exists no comprehensive detailed review that  evaluates the
impact of various  nonequilibrium factors on vapor transport, their relative significance, and
various modeling  approaches used to describe rate-limited vapor transport.  The following
discussion provides such a review and provides additional insight into how an understanding of
rate-limited vapor transport process can improve current practices in design. Operationally, rate-
limited mass transfer exchange will be separated into four categories:  (1) gas-water, (2) solids-
water, (3) mobile-immobile gas and (4) gas-NAPL exchange.

9.3    Rate-Limited NAPL-Gas Exchange

        An awareness of NAPL distribution in porous media under unsaturated  conditions is
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helpful in understanding rate-limited NAPL-gas exchange. In natural unsaturated porous media,
NAPL is often distributed in two phases: "free" and "trapped" (Parker and Lenhard, 1987). Free
NAPL is considered a thin film or lense in direct contact with mobile gas, while trapped NAPL is
considered as discontinuous blobs or ganglia occluded within the bulk-water phase.  NAPL
entrapment in two-phase (NAPL-water) systems has been extensively studied.  Typically, NAPL
is the nonwetting phase while water is the wetting phase.  The immiscible displacement of NAPL
by water results in a fraction of the NAPL being snapped off and left behind as blobs or ganglia
in an otherwise water-filled pore space (Chatzis et al., 1983; Wilson and Conrad, 1984). In a
NAPL, water, gas system, fluid distribution is much more complex and may depend on the
spreading coefficient of the NAPL.  The spreading coefficient (2) is defined as:

    = agw-anw-agn                                                            (9.1)

where: o^ = gas-water interfacial tension, onw = NAPL-water interfacial tension, and ogn = gas -
NAPL interfacial tension.  NAPLs exhibiting low interfacial tensions with water and gas have
positive spreading coefficients and are theorized to spread as films on residual water. NAPLs
exhibiting high interfacial tensions with water and gas have negative spreading coefficients and
are theorized to exist as entrapped coalesced lenses or ganglia along air-water interfaces.
Petroleum hydrocarbons generally have positive spreading coefficients while many halogenated
solvents have negative spreading coefficients. NAPLs with positive spreading coefficients are
expected to have a much higher NAPL-gas contact area and hence a high rate of mass transfer
compared to nonspreading NAPLs.  Soils contaminated with spreading NAPLs are  expected to
have lower residual saturations compared to nonspreading NAPLs.  Blunt et al. (1995) however
concluded that contrary to accepted theory, both spreading and nonspreading NAPLs may form
thin films along water interfaces in three-phase water-wet porous media.  They suggest that even
nonspreading NAPLs may form films from thinning of previously thick NAPL layers during
NAPL drainage.  This is supported by Hayden and Voice (1993) who observed the existence of
continuous NAPL films, as well as pendular rings and V-shaped wedges, for a nonspreading
NAPL entrapped in a water-wet sandy aquifer material. In experiments with two spreading
NAPLs, styrene and toluene, and one nonspreading NAPL PCE, Wilkins et al. (1995) noted that
spreading coefficients had minimal influence on interphase NAPL-gas mass transfer indicating
similarities in pore-scale NAPL distribution.  Residual NAPL saturations were comparable for all
three NAPLs ranging between 4 and 5%.  These findings suggest that the influence of spreading
coefficients in NAPL-gas mass exchange is ambiguous.

       Wilkins et al. (1995) present a conceptual model for NAPL volatilization from
unsaturated porous media.  The wetting fluid, water, preferentially occupies the smallest pores,
filling pore bodies and creating water films along soil particles.  The invasion and drainage of
NAPL produces intermediate wetting NAPL films along air-water interfaces, and in
heterogeneous media, complex geometric patterns of residual NAPL. As a result, all residual
NAPL may not be readily  accessible to the mobile gas phase.  Initially, mass transfer in NAPL
contaminated soils is rapid due to the presence of NAPL films exposed directly to soil  gas which
provide a large surface area to volume ratio.  Over extended periods of time however, NAPL

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films are removed and the remaining NAPL mass exists as pendular rings, wedges surrounding
aqueous pendular rings, filled pore throats, and NAPL isolated between aqueous wedges or
within pore throats surrounded by water.  Increases in fluid saturation and natural pore structure
heterogeneity may further restrict the mobility of the gas phase, limiting access to entrapped
NAPL. When NAPL is surrounded by water, mass removal becomes limited by aqueous
diffusion in a manner similar to that for saturated zone processes.  Thus, according to Wilkins et
al.'s (1995) conceptualization, NAPL-gas mass transport should be constrained by increasing
water content. This observation is important, since virtually all through-flow (gas flow directly
through NAPL contaminated soils) laboratory column studies in which NAPL has been placed in
dry porous media, have not shown rate-limited NAPL-gas mass transfer. Rate-limited NAPL-gas
mass transfer has only been demonstrated in water wet soils, but this observation has not been
consistent.

       McClellan and Gillham (1992) conducted one-dimensional through-flow experiments in a
steel box to evaluate mass transfer between uniformly mixed residual TCE and air. They mixed
free phase TCE with moist Borden sand, resulting in residual NAPL and water saturation of 8.8%
and 25% respectively. Effluent vapor  concentrations during the first 72 hours of operation were
relatively constant and at or near saturation (35,000 to 36,000 ppmv). During a 12 hour period,
pore-gas  velocity was increased from 0.18 to 0.63 cm/s (calculated from information provided by
authors).  Relatively constant concentrations were observed in the effluent indicating that local
equilibrium had been maintained at the higher pore-gas velocity.  Flow rate was directly
proportional to mass flux rate. A sharp decline in vapor concentration (about 1.3 orders of
magnitude) though was observed at approximately 100 hours. McClellan and Gillham (1992)
believe that the sharp decline in concentration correspond to the time when the last liquid or
residual TCE was removed and the kinetics of aqueous and sorbed phase mass removal became
dominant. After this time, increased flow rate caused a reduction in vapor concentration.
Further evidence of rate-limited transport was seen during flow interruption. The system was
turned off for a period of one hour. Upon restarting the system, the effluent concentration
increased significantly compared to effluent concentrations just prior to shut down but then
quickly decreased to pre-test effluent concentrations. A second flow interruption test was then
conducted with a shut down time of only 35 minutes resulting in a smaller rebound concentration
upon restart.  McClellan and Gillham (1992) attempted to assess the mass remaining in the steel
box at the end of the study by using partition coefficients with an average soil gas concentration
(16 ppmv) after a 19 hour shut down period.  They concluded that in excess of 99% of
contaminant mass had been removed prior to observation of rate-limited vapor transport.

       Baehr et al. (1989) simulated mass removal of gasoline under through-flow conditions in
pre-moistened medium-grained sieved sand having a NAPL saturation of 7.5% and water
saturation of 22.4%. Experimental and simulated mass flux profiles matched well suggesting
attainment of equilibrium conditions at a pore velocities of 2.07, 4.47, and 10.92 cm/s (calculated
based on information provided by authors).  They state that the ability of model  simulations to
describe experimental results conducted at three flow rates supports their hypothesis that local
equilibrium conditions prevailed during mass removal.  A plot of hydrocarbon mass removed as
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a function of volume of air passed through their columns revealed no dependence on pore-gas
velocity. Analysis of chromatograms indicated a loss of lower molecular weight compounds
with time.

       Rathfelder et al. (1991) and Bloes et al. (1992) simulated mass removal of a NAPL
mixture of benzene, toluene, and TCE in dry glass beads and dry Borden sand under through-
flow conditions.  Distinguishable evaporation  fronts of benzene, TCE, and toluene  eluted in
sequential order as expected from Raoult's law. Experimental and predicted effluent vapor
concentrations matched well during evaporation of NAPL suggesting attainment of equilibrium
conditions at a pore velocity of 0.27 cm/s (calculated based on information provided by authors).
Some tailing in the glass beads and Borden sand was evident however at low vapor
concentrations.

        To simulate rate-limited gas-NAPL exchange, the NAPL storage term in the advective-
dispersive equation (8.18) can be expressed by
      » - o  3
       ~
(9.2)
where Ang is a NAPL - gas first-order rate constant [T1]. Rathfelder et al. (1991) conducted
hypothetical simulations of xylene removal in dry soils with a NAPL saturation of 1.0% using Ang
values ranging from 2.23 x 10"6 to 2.23 x 10"5 s"1 in a radial flow system with an initial
contaminated radius of 10 m. Simulations showed that Ang rates had little effect on time for
complete mass removal as long as equilibrium vapor concentrations were established prior to
reaching the gas extraction well. As Ang was reduced, the travel distance over which saturated
vapor concentrations were established increased causing the evaporation front to become
disperse.

       Ho et al. (1994) conducted through-flow column studies in a dry sand with a multi-
component NAPL mixture of benzene, toluene, and o-xylene. Using a local equilibrium model
they successfully simulated evaporation fronts in the gas effluent.  Similar to Rathfelder et al.
(1991), they were able to predict the successive rise in effluent vapor concentration of heavier
VOCs upon depletion of lighter VOCs.

       Liang and Udell (1999)  successfully simulated mass removal of a NAPL mixture of
benzene, toluene, and o-xylene  in dry and moist (water saturation = 15%) Monterey  sand (mean
particle diameter = 0.25 mm) in through-flow experiments using the assumption of local
equilibrium. Pore-gas velocities for the moist and dry sand experiments were 9.35 and 7.73 cm/s
respectively (calculated based on information provided by authors). Similar to Rathfelder et al.
(1991) and Ho et al. (1994), they observed sharp evaporation fronts eluting in order of volatility
or vapor pressure (benzene, toluene, o-xylene respectively). Breakthrough curves and the
amount of time required to recover the compounds in both cases were nearly identical. Liang and
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Udell (1999) concluded that at least in their experiments, the presence of residual water had
virtually no effect on NAPL-gas mass transfer. Liang and Udell (1999) also conducted bypass
experiments where gas flow in a glass column occurred only on the soil surface. In these
experiments, extensive effluent tailing was observed.

       Lingineni and Dhir (1992) introduced ethyl alcohol into dry 0.36-mm glass beads to
achieve a residual NAPL saturation of 13% and observed temperature reduction along the axis of
flow due to evaporation under through-flow conditions. They state that the latent heat of
vaporization required to  evaporate an organic liquid is partially provided by the incoming air,
while the rest is absorbed from the porous media.  After the fluid is completely evaporated, soil
temperatures rise due to  heat transfer from air to soils at higher temperature.  Simulations of
temperature and evaporation fronts at pore-gas velocities of 7.52, 30.00, and 50.15 cm/s
(calculated based on information provided by authors) matched experimental data suggesting
attainment of equilibrium conditions.

       Hayden et al. (1994) introduced gasoline at NAPL saturation value of 2% in two sandy
soils (less than 5% silt and clay) having organic carbon  content of 0.1% and 1.65% and water
saturation values of 8.0% and 43.0% respectively and monitored effluent concentrations of
benzene, m,p-xylene, and naphthalene. They conducted flow variation and interruption testing to
evaluate NAPL-gas rate-limited exchange. For the high organic content and water saturation
soil, flow variation resulted in pore-gas velocities of 0.076 and 0.020 cm/s (calculated based on
information provided by authors) while for the low organic carbon content and water saturation
soil, flow variation resulted in calculated pore-gas velocities of 0.057 and 0.015 cm/s  (calculated
based on information provided by authors). Flow variation and interruption testing revealed that
rate-limited behavior was not a function of moisture content and only occurred at very low
component concentrations. Hayden et al. (1994) state that possible explanations of observed
rate-limited behavior at low mole fractions or concentrations include:  1) heterogenous flow
patterns and NAPL distribution, 2) rate-limited gas-water  or solids-water exchange, 3) rate-
limited gas-NAPL exchange, and 4) a combination of all or some of the factors above. They
state that small-scale physical heterogeneity or permeability variation may result in unequal flow
distribution causing some NAPL containing pore regions to experience much more gas
movement than others. Mole fractions of components having high vapor pressure would then be
higher in NAPL containing regions of low gas exchange compared to more weathered NAPL in
regions of high gas exchange.  This would cause tailing, and an increase in effluent vapor
concentration upon reduction or stoppage of gas flow.  It would also explain rate-limited
behavior for compounds having high volatility while effluent monitoring of lower volatility
compound show no rate-limited behavior. However, another viable explanation of rate-limited
transport for compounds having high volatility would be almost complete mass removal from
NAPL with remaining mass present in soil-water, humic material, and solids where rate-limited
behavior is well documented in the literature. Thus, it is difficult to elucidate the cause of rate-
limited behavior of compounds having high volatility at low concentrations in NAPL
contaminated soil.  It would appear then that Hayden et al.'s (1994) study can not be used to
document rate-limited NAPL-gas exchange.
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       Harper et al. (1998) observed rate-limiting behavior in a silt loam using toluene and a
mixture of toluene and m-xylene as a NAPL source at a gravimetric moisture content of 22% but
not under air-dried conditions or at a gravimetric moisture content of 16%.  At a gravimetric
moisture content of 22% using a binary mixture of toluene and m-xylene, no distinct evaporation
fronts were discernable.  Effluent concentrations displayed rapid and extended tailing. In
comparison to other published studies (e.g., Hayden et al., 1994), rate-limited behavior for NAPL
components was present at high mole fractions or soil concentrations. Harper et al. (1998) state
that the abrupt transition from equilibrium to nonequilibrium conditions with small changes in
water content (16% to 22%) suggests a nonlinear dependence of mass transfer on soil water
content.  Harper et al. (1998) are among the first investigators to use non-sandy  soils or glass
beads to investigate NAPL-gas mass transfer and the first investigators to definitively
demonstrate reduced NAPL-gas mass transfer with increased moisture content.  As they point
out, venting is often applied in silty soils, yet few investigators use silty or clayey soils for
laboratory studies.

       Of particular concern is that in contrast to typical published column studies where NAPL
is placed in dried soils or in moist soils after pre-wetting, NAPL contaminated soils at a field site
are subject to numerous wetting and drying cycles. This suggests that NAPL distribution on a
pore-scale in natural soils may be more heterogeneous and NAPL-gas exchange may be more
rate-limited than in artificially contaminated soils typically used in published studies.  Ostendorf
et al. (1997) provide one of the few published studies evaluating rate-limited NAPL-gas
exchange in "naturally" contaminated media. Ng et al. (1999) developed an aggregate diffusion
model similar to Ng and Mei (1996) and utilized Ostendorf et al's. (1997) laboratory column
based data to  simulate mass removal of individual compounds. In Ng et al.'s (1999) model,
intraaggregate space is filled with trapped NAPL, water, and gas while interaggregate space is
filled with mobile gas and  a thin film of NAPL surrounding aggregate particles. NAPL
constituents in intraaggregate  space move to interaggregate space by NAPL dissolution and
aqueous diffusion. Calibration of their model to vapor transport along the axis of the column of
one of the most volatile constituents, 2-methylpentane, resulted in an estimated trapped NAPL
fraction of 70%  Simulated versus observed vapor concentrations of compounds having relatively
high vapor pressures compared well. There were some discrepancies however for compounds
having lower  pressures which they attribute to not including enough lower volatility components
in the simulated NAPL mixture. The important observation for venting design purposes however
is observed deviation from equilibrium at a relatively low pore-gas velocity (0.0074 cm/s)
reinforcing concern that NAPL-gas exchange in  "naturally" contaminated soils may be of larger
magnitude than in artificially contaminated soils.

       Wilkins et al. (1995) conducted a series of soil column experiments to evaluate NAPL-
gas rate-limited mass transfer. They packed air-dried sieved soils in columns to obtain four
uniform sand size fractions and two graded sand mixtures. By  achieving macroscopically
homogeneous unsaturated porous media systems, they were able to study NAPL-gas mass
transfer without the interfering affects of physical heterogeneity. Relatively uniform soil
moisture and NAPL distribution was maintained in soils throughout the columns.  Single
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component NAPLs (styrene, toluene, and perchloroethylene) were used for experimentation.
       Parameters considered for describing mass transfer process included (1) mean grain size
(d50), (2) grain size uniformity index, (3) NAPL residual saturation, (4) water saturation, (5) and
effective air filled porosity. Multiple linear regression analyses were performed to determine the
best fit model for experimental data.  Wilkins et al. (1995) observed that mean grain size
provided the most significant statistical correlation describing NAPL-air mass transfer.
NAPL-vapor phase mass transfer decreased with decreased mean grain size. Correlation between
the uniformity index and mass transfer was significantly weaker. A model based on the residual
NAPL saturation resulted in a negative correlation, indicating an inverse relationship between
mass transfer and the entrapped NAPL content.  However, the statistical significance of this
model was considerably less than the mean grain size model. Wilkins et al. (1995) explain that
entrapped NAPL preferentially occupies medium-sized pores while forming films along
gas-water interfaces. Water, the  wetting fluid, preferentially occupies smaller pores inhibiting
NAPL entry while gas, a nonwetting fluid occupies the largest pores. Thus the medium-sized
pores, which correspond to the mean grain size, may control NAPL distribution and have the
greatest effect on mass transfer.  A decrease in mean grain size or an increase in residual fluid
saturation would be expected to increase gas phase restriction to pore spaces and reduce
interphase mass transfer.

        Based on experimental observations, Wilkins et al. (1995) derived an empirical equation
for estimation of the NAPL-gas mass transfer coefficient
3    iA-0.42 n°-38 0.62 rO.44
V=1°    Dg  V   d50
                                                                                   (9.3)
Equation (9.3) can be used to estimate a critical pore-gas velocity for NAPL-gas exchange, v
                                                                                     c(ng)-
 vc(ng)
                     Cn
                                                                                  (9.4)
where Lc is a characteristic contamination length [L] and Cg* is equilibrium vapor concentration
[ML"3].  For a slight deviation from equilibrium conditions (Cg/Cg*) = 0.99, vc(ng), can be
estimated by:
                      2•63
               50   c
 vc(ng)
           12.110'
                                                                                  (9.5)
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9.4    Rate-Limited Gas-Water Exchange (First-Order Kinetics)

       Armstrong et al. (1994) explain the use of a first-order rate coefficient to quantitatively
describe rate-limited gas-water exchange. On a conceptual basis, it is assumed that diffusive
mass transfer between the gas and water phase takes place through an aqueous boundary layer.
The driving force for mass transport then is a gradient between the average aqueous phase
concentration and an equilibrium aqueous concentration at the gas-water interface.  To simulate
rate-limited gas-water exchange, the water storage term in the advective-dispersive equation
(8.18) can be expressed by
                                                                                  (9-6)
where K^ is a lumped first-order gas-water mass transfer coefficient [T"1] which is the product
specific interfacial area [L"1] between gas and the aqueous phase and a gas-water mass transfer
coefficient [LT1].  In the boundary layer model, K^ depends on the compound's aqueous
diffusion coefficient and thickness of the boundary layer. The thickness however is not a
measurable quantity since the process represents a variety of diffusion pathlengths in
heterogeneous porous media.  The gas-water mass transfer coefficient then has meaning only as a
lumped parameter representing the integral of all physical nonequilibrium processes affecting
movement of contaminants between the aqueous and gaseous phase including diffusion with the
soil moisture and dead-end pore effects.

       Conklin et al. (1995) evaluated rate-limited gas-water exchange of benzene and p-xylene
in desert sand having a median grain size of 0.25 to 0.50 mm  and organic carbon content of
0.097% at volumetric water contents of 10% and 18%. Conklin et al. (1995) observed significant
interfacial gas-water adsorption for p-xylene but not for benzene.  In experiments using pore-gas
velocities varying from 0.139 to 0.545 cm/s, breakthrough curves for benzene indicated tailing
only at a volumetric water content of 18% while breakthrough curves for p-xylene exhibited
tailing at both volumetric water contents with tailing of p-xylene more pronounced than benzene
at a volumetric water content of 18%.  Conklin et al. (1995) state that since tailing was absent in
breakthrough curves for benzene at a volumetric water content of 10%, it is unlikely that mass
transport was limited by solids-water exchange.  They noted that a first-order kinetics model
could fit the initial but not the long tailing portion of the desorption curve of benzene at a
volumetric water content of 18% suggesting that more than one rate-limiting process was
occurring, possibly in series. The  average mass transfer coefficients fitting the initial and long
tailing portions of the desorption curve for benzene were (5.8 ± 2.4) x 10"3  and (5.5 ± 2) x 10"4 s ~l
respectively.  The mass transfer coefficients describing long tailing were consistent with
calculations for sorption retarded aqueous diffusion assuming a diffusion path length equivalent
to the particle radius. Thus, they concluded that lower mass transfer coefficients likely described
aqueous intraparticle diffusion while higher mass transfer coefficients may have represented film
diffusion.  Similar results were found for p-xylene where the average mass transfer coefficients
fitting the initial portion of the desorption curve ranged from (3.7 ± 0.8) x 10"3 and (4. 1 ± 2.0) x

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10"2 s 4 at moisture contents of 18 and 10% respectively and (3.0 ± 2) x 10"4 s 4 for the long
tailing portion of the desorption curve.  Conklin et al's. (1995) experimental results indicate that
soil-water content can significantly affect gas-water rate constants and hence the time scale of
remediation.  Higher moisture content results in more void space between and within particles
filled with water thereby increasing the diffusion path length in the aqueous phase. Thus, design
practices such as installation of a synthetic liner or asphalt cap to minimize infiltration or lower
soil moisture content may increase gas-water mass exchange and thereby reduce remediation
time.

       Fisher et al. (1996) conducted tank studies with clean quartz sand (grain sizes in the range
of 0.08-1.2 mm) to determine if mass removal during gas advection could be affected solely by
rate-limited mass transfer in interparticle soil water (water between soil particles). They
conducted soil-water partitioning batch studies and observed negligible sorption of tested volatile
organic compounds.  Fisher et al. (1996) also sought to evaluate whether departure from local
equilibrium due to gas-water limited mass transfer could be described by a first-order kinetics
approach and the effect of water saturation and physiochemical properties (especially H and Kd)
on volatile organic compound mass removal. Volatile organic compounds 1,1,1-trichloroethane
(1,1,1-TCA), 1,1,2-trichloroethane (1,1,2-TCA), perchloroethene (PCE),  and trichloroethene
(TCE) were tested. Differences in mass removal of 1,1,1-TCA and 1,1,2-TCA were of particular
interest because the dimensionless Henry's law constant of the former and latter vary over an
order of magnitude, 0.73  and  0.04 respectively.  Four venting experiments were performed
according to saturation profiles in the tank. Water content varied vertically throughout the tank
providing nonuniform water contents.  VOCs were introduced into the tank by gas diffusion or
through saturation of sands containing dissolved constituents.  Average linear velocities in the
sand tank varied from 0 (no interconnected gas-filled porosity) up to 0.02 cm/s.  Relative vapor
concentrations were plotted against a dimensionless time (T) defined as
      vt
T = ——                                                                          (9.7)
    RDL

where Lisa reference length  [L] chosen as the distance between two monitoring points,  and RD
is a retardation factor derived from the local equilibrium assumption and defined as


                                                                                   (9.8)
                      .
         0gH   0gH

Relative vapor concentration was plotted on a logarithmic scale to effectively illustrate
concentration reduction over two orders of magnitude. All tests were characterized by a steep
initial decline in vapor concentration followed by a long tail.  A sharp transition in relative vapor
concentration reduction occurred at T = 2.  For T < 2, relative vapor concentrations decreased
linearly (indicative of equilibrium conditions), while for T > 2, tailing with eventual constant
relative concentration occurred over time. Fisher et al. (1996) state that the similarity of all
experimental curves suggests that vapor concentration reduction was controlled by the same
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variables affecting T, namely, water saturation, linear velocity, and Henry's law constant and that
different slopes of curves in the range of T < 2 and T > 2 indicate two different processes in mass
removal. They hypothesize that the steep initial decrease in gas concentration likely corresponds
to removal of VOCs from water films surrounding particle surfaces, which is not likely to be
rate-limited, whereas tailing may have been caused by removal from wedges.  While the extent
and geometry of the films and wedges depend on water saturation, in general, the mean distance
from the gas-water interface must be greater for wedges than for films.  The tailing processes
indicates that advective transport in the gas phase must have proceeded much faster than mass
transfer into the gas phase. This finding means that even at remediation sites where the soil
material is seemingly homogeneous, low in  sorptive capacity, without secondary porosity, and at
low water saturation, soil venting operation  will likely be affected by local nonequilibrium.
Significant departure from equilibrium for 1,1,1-TCA removal was noted at velocities as low as
0.00914 cm/s.

       In their study, Fisher et al. (1996) found gas flow emergence and extinction points
occurred at water saturations of 0.6 and 0.7 respectively. As discussed in section 3, the
importance of minimizing soil-water content during venting is reinforced  by the observation that
advective gas flow does not occur until water saturation decreases to an "emergence point". In
initially dry media, advective gas flow does  not cease until water saturation increases to an
"extinction point". Emergence and extinction points refer to water saturations in which
interconnected gas voids are able to conduct advective  flow. A number of investigators
(Stonestrom and Rubin, 1989a,b; Fisher et al., 1997; and Dury et al. 1999) have documented the
presence of trapped, nonconducting gas during capillary pressure - saturation and relative
permeability measurement. Drainage requires a lower water saturation for gas flow to occur
compared to wetting.  The obvious  implication for venting design and monitoring is that
subsurface media must be significantly desaturated for  advective gas transport to occur. Because
water saturation emergence points are lower than extinction points due to hysteresis, this is
particularly important during water table lowering or dewatering for venting application.  Proper
design and monitoring strategies then must involve determination of minimum gas saturations in
specific media to induce advective gas flow and periodic in-situ monitoring to ensure that these
gas saturations are maintained. When soils are dewatered to allow application of soil venting,
water level monitoring in piezometers alone is not adequate to demonstrate sufficient
desaturation to conduct gas flow. Piezometers do not provide useful information in estimating
the thickness of the capillary fringe and the water saturation profile above the capillary fringe.

       Relationships to estimate critical pore-gas velocities for gas-water exchange will now be
explored.  Brusseau (1991) states that if liquid-liquid mass transfer is the rate controlling step,
then Agw can be related to the free-liquid diffusion coefficient by
where Dw = free-aqueous molecular diffusivity [L2T"J], TW = tortuosity factor of water filled

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porosity [-], c = a shape factor (15 for a sphere, 3 for a plane), and 6C = a characteristic diffusion
path length (e.g., radius of a sphere). A Damkohler number describing gas-water can be
expressed by (Brusseau, 1991, Popovicova and Brusseau, 1998)
Using a value of 10 for the gas-water Damkohler number, a critical pore-gas velocity for gas-
water exchange, vc(gw)  can be estimated by

           D  ft T
      —     w w  c
       An alternative to using Damkohler numbers to estimate a critical pore-gas velocity for
gas-water exchange, vc(gw) is to use the concept of a prefix denominator. Fisher et al. (1998) used
a procedure termed "Separation of the Kinetically Influenced Term" (SKIT), developed by Bahr
and Rubin (1997), to derive a dimensionless "prefix denominator" (PD) defined by
                                                                                  (912)
which provides a measure of the degree of departure from local equilibrium. The SKIT
procedure consists of deriving equivalent dimensionless formulations of an equilibrium and
kinetics model and identifying the additional term(s) in the kinetics model. Fisher et al. (1998)
state that compared to a gas-water Damkohler number, use of the prefix denominator as an
indicator of nonequilibrium has the advantage that it includes factors which represent gas and
water saturation and physico-chemical properties such as a Henry's law constant and the solids-
water partition coefficient. They found however that PD values tend to decrease with time due to
the fact that mass transfer coefficients describing gas-water exchange decrease with increasing
time.  Fisher et al. (1998) state that a critical velocity using the prefix denominator to optimize
field application of venting can be expressed by:
           A,-,.,,L.,l\r)tl 0 _
Vc(D-gw)
Since gas-water mass transfer approaches equilibrium conditions as the PD values approaches 10,
Fisher et al. (1998) recommend using a PD value of 10 to compute vc(I>gw).  Assuming a PD value
of 10, a spherical shaped rate-limited domain, and equations (9.8), (9.9), and (9.13), a critical
gas-water velocity using the prefix denominator can be estimated by:
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                        +0w+pbKd).                                            (9.14)
9.5    Rate-Limited Gas-Water Exchange (Aggregate Diffusion Modeling)

       Early development of the aggregate diffusion modeling took place in the chemical
engineering field. Rao and co-workers (Nkedi-Kizza et al., 1982; Rao et al. 1980a,b) applied
these models to describe sorption nonequilibrium during solute transport. Rate-limited sorption
has been modeled by using diffusion equations based on Pick's law (Wu and Gschwend, 1986)
and by using chemical reaction equations (Selim et al. 1976; Cameron and Klute, 1977).  When
using intraaggregate diffusion as the only causative factor for nonideality, accessibility to
sorption sites becomes rate-limited but rate-limited solids-water exchange itself is not explicitly
considered.  Wu and Gschwend (1986) discuss a model involving retarded intraparticle liquid
diffusion where sorption at the sorbent-solution interface is assumed to be instantaneous.

       Rao et al. (1982) assessed whether solute diffusion from nonspherical aggregates could be
approximated by diffusion from equivalent spheres. They showed that nonspherical aggregates
could be represented by equivalent spherical aggregates whose radii are  such that the sphere
volume is equal to the volume of the nonspherical aggregate. Rasmuson (1985) extended a
physical diffusion model developed for spherical aggregates to slabular and cylindrical shapes.  A
form factor was employed to adjust the intraaggregate diffusion equation to the selected shape.
van Genuchten (1985) developed a method that extended the aggregate diffusion equation to
more general conditions involving aggregates with nonspherical geometries. The method is
based on the use of a geometry-dependent shape factor that can be used to transform an aggregate
of a given shape and size into an equivalent sphere with similar diffusion characteristics.

       Rao et al. (1982) described a method whereby a range of spherical aggregates can be
reduced to a single, equivalent aggregate size.  The method involves computing the equivalent
radius from the volume-weighted radii of each aggregate-size class. Rao et al. (1982)
demonstrated that the equivalent aggregate size could provide a good approximation of diffusion
in actual mixed-sized medium. Nkedi-Kizza et al. (1982) observed that a distribution of
aggregate sizes from 0.2 to 0.47 cm in diameter was well represented by an equivalent spherical
aggregate size.  Combining the shape transformations of van Genuchten (1985) with the
aggregate size distribution transformation of Rao et al. (1982), porous medium consisting of
different sizes and shapes of aggregates can be transformed in equivalent spherical aggregates of
uniform size. In this manner, it may be possible to extend aggregate diffusion modeling to field-
scale application.

       Gierke et al.  (1992) used 30 cm long, 5 cm i.d. glass columns packed with a uniform
Ottawa sand and a manufactured aggregated porous soil material (APSM) to evaluate vapor
transport in highly permeable nonaggregated and aggregated soils. Toluene vapor transport was
observed in the sand and APSM under dry and wet conditions (27% and 67% water saturation).
Sorption under wet conditions was assumed negligible because in previous studies (Gierke et al.,

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1990) TCE sorbed neither to the sand in a water-saturated column nor to the APSM in an
aqueous batch tests. For model development, Gierke et al. (1992) divided mass transport into
two zones: (1) mobile gas and (2) aggregates composed of immobile water and solid soil
particles. Gas was assumed to be a continuous phase. A Freundlich isotherm was included in
model development to describe non-linear vapor-mineral surface sorption in dry experiments.
One-dimensional gas flow was assumed to be steady and incompressible. Experiments with dry
sand, moist sand, and dry APSM at pore-gas velocities of 0.017, 0.020, 0.01 cm/s showed no
rate-limited behavior.  Experiments with moist APSM however showed deviation of ideality at a
pore-gas velocity of 0.061 cm/s indicating that intraaggregate diffusion controlled mass transport
under these conditions.

       Gierke et al. (1992) provide an equivalency relationship describing when dispersion, film
transfer, and intraaggregate diffusion have similar impacts on the breakthrough curve of a
linearly sorbed organic vapor
                                                                                 (9.15)
       ^   ug)       ^   ug)

where


Ed=  Dw6>wL 2                                                                  (9.16)
     ^—                                                                       (9.17)
      3v
                                                                                 (9.18)
They state that theoretically this equation is valid for linear isotherms, Pe > 40, and Ed > 13.33.
In practice however, it can be used for values of Pe and Ed at least as low as 1 . Parker and
Valocchi (1986) used a moment analysis to derive an identical relationship.  Gierke et al. (1992)
state that this equation can be used to guide specification of an appropriate pore-gas velocity for
venting application. They state that the chosen velocity should be low enough that Ed and St are
large in comparison to Pe. Decreasing pore-gas velocity increases Ed and St and decreases Pe.
However, the value of Pe though should be as high as possible to minimize the impact of gas
diffusion on vapor transport.  Gierke et al. (1992) provide an equation to estimate the gas-water
mass transfer coefficient that would simulate the observed impact of intraaggregate diffusion
                                          136

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    =15    *>*>   .                                                                (9.19)
Note that equation (9.19) differs from Agw defined in equation (9.9) by H8g in the denominator.
Parker and Valocchi (1986) developed a relationship similar to equation (9.19) for equating first-
order kinetics to spherical diffusion under saturated conditions.  The motivation for development
of these relationships is that numerically, intraaggregate diffusion requires more computational
effort than first-order mass transfer.  Gierke et al. (1992) state that for this reason, intraaggregate
diffusion has not been included in field-scale models.

       Ng and Mei (1996) simulated compressible gas flow and vapor transport within
aggregated media in a radial coordinate system.  They conceptualized soil as a periodic array of
spherical aggregates and explicitly simulated mass transfer from water saturated aggregates to
interaggregate pore gas.  They neglected the presence of water and NAPL within interaggregate
space and assumed: intraaggregate water to be immobile, negligible film resistance at aggregate-
macropore interfaces,  and equilibrium sorptive exchange between soil and aqueous phases inside
aggregates.  With this  approach, solute concentration gradients within the aggregates or immobile
regions is allowed. This is in contrast to the one-or two-site first-order rate models. Ng and Mei
(1996) used  high Peclet numbers (100-1000)  for simulations assuming a radial extent of
contamination near an SVE wells on the order of 10 meters, gas velocity from 0.001 to 0.01 cm/s
and gas diffusion coefficients on the order of 0.01 cm2/s. At high Peclet numbers, macroscale
diffusion becomes insignificant compared to advection in the vapor phase.

       Ng and Mei (1996) derived two dimensionless parameters, a and E,, to examine
rate-limited vapor transport in the absence of NAPL.


o =                                                                              (9.20)
and


 £_}_"• \    <* / • *   n- j v     &,                                                    (921)


where


 Da = —=	z—z	=•                                                      (9.22)


where ps = particle density [ML"3].


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       The parameter o is the ratio of aggregate diffusion time which governs how fast
compounds are removed from aggregates to global advection time. Local equilibrium exists
when diffusion in aggregates is rapid or o »1.  E, is equal to the ratio of mass partitioned in
aggregates to mass in the vapor phase per bulk volume of soil in the case of local equilibrium.  A
higher value of E, means greater retardation of vapor transport.  When o »1 and E, « 1, vapor
concentration is well predicted by local equilibrium theory. Ng and Mei (1996) examined the
time necessary to reduce vapor and aqueous concentrations to 1% of their initial values in
interaggregate and intraaggregate domains for values of a ranging from 0.001 to 10 and three
values of  E, (0.01, 1, 100).  Since equilibrium sorptive exchange is assumed in aggregates,
reduction to 1% of initial aqueous concentration is equivalent to 99% mass removal in
aggregates. Based on the results of their simulations, Ng and Mei (1996) suggest that in the
absence of NAPL and kinetically limited sorptive exchange, a critical pore-gas velocity occurs
when mass removal is only slightly limited by diffusion in aggregates.  They suggest that this
occurs at:

(1 + 
-------
equilibrium - a short initial phase of fast uptake, followed by an extended period of much slower
uptake.  Generally, about 50% of sorption occurs within the first few minutes to hours, with the
remainder occurring over periods of days or months (Jaffe and Ferrara, 1983; Wu and Gschwend,
1986; Karickhoff, 1980; McCall and Agin, 1985; Oliver, 1985; Ball and Roberts, 1985; and
Curtis et al., 1986).  Diffusion within soil organic matter has been proposed as a plausible
mechanism responsible for rate-limited sorption (Bouchard et al., 1988; Brusseau and Rao,
1989a; Brusseau et al., 1991; Karickhoff and Morris, 1985, Lee et al. 1988).  Soil organic matter
is essentially a three-dimensional network or randomly oriented polymer chains having a
relatively open, flexible structure perforated with voids (Khan,  1978; Schnitzer, 1978). Brusseau
(1994) summarizes experimental observations which support the hypothesis that sorbate
diffusion into organic matter is a causative factor in rate-limited sorption.

       The simplest approach in modeling sorption kinetics is to utilize a one-site, first-order
model in which the sorption rate is taken as a function of the concentration difference between
the sorbed and solution phases.  To simulate rate-limited solids-water exchange, the solids
storage term in the advective-dispersive equation can be expressed by:


pb^ = 0^(cw-^-}                                                         (9.25)
where Kws = water-solids first-order mass transfer coefficient [T1]. The one-site, first-order
sorption rate model however has failed to fit experimental data well in both solute
(Schwarzenbach and Westall, 1981; Rao and Jessup, 1983; and Wu and Gschwend, 1986) and
vapor (Grathwohl and Reinhard, 1993; Wehrle and Brauns,  1994; and Croise et al., 1994)
transport experiments. Croise et al. (1994) found that the solids-water mass transfer rate
decreased exponentially with time during gas extraction. Kaleris and Croise (1997) state that if
a conservative estimate of remediation time is needed, a first-order kinetic approach can be
considered  provided a low mass transfer rate is assumed.

       Selim et al. (1976) and Cameron and Klute (1977) were among the first investigators to
use a first-order bicontinuum or two-site sorption model where sorption is assumed to be
instantaneous for a fraction of the sorbent and rate-limited for the remainder. Two-site modeling
has described sorption kinetics well in controlled laboratory batch and column  studies. In
bicontinuum or two-site modeling, solids-water mass exchange can be represented in the
equilibrium domain by

              -                                                                 (9.26)
and in the rate-limited domain by


                      v-Csl]                                                      (9.27)


                                           139

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where F equals fraction of instantaneous sorption sites [-], Csl equals solids concentration
in equilibrium domain [MM"1], and Cs2 equals solids concentration in rate-limited domain
[MM"1].  The one-site model is a special case of the two-site model where F = 0 (all sites are rate-
limited). In the two-site sorption model, equilibrium and rate-limited sites can be conceptualized
as occurring in series or parallel. The former conceptualization is consistent with the viewpoint
that soil organic matter is a polymeric like substance where sorption to the surface is rapid
followed by liquid diffusion into the soil organic matter matrix.  Mathematically however, both
series and parallel sorption site conceptualizations are indistinguishable (Karickhoff, 1980;
Karickhoff and Morris, 1985).

       Popovicova and Brusseau (1998) derived a bicontinuum  solids-water Damkohler
number (cosw) for gas flow expressed by


(OSW=^LRS                                                                     (9.28)
        v

where


R =-^(\-F}Kj.                                                              (9.29)
  -S    y-l T T V     / 6i                                                                ^    '
Assuming that rate-limited solids-water mass exchange approaches equilibrium conditions as cos
approaches 10, a solids-water critical pore-gas velocity, vc(sw) [LT1] can be estimated by


                       l-F                                                        (9.30)
       Use of equation (9.30) to assess vc(sw) requires estimation of Aws and F.  Brusseau and Rao
(1989a) state that intra-sorbent mass transfer is a function of three factors: (1) aqueous diffusivity
of the diffusing species, (2) resistance to diffusion associated with the sorbent matrix, and (3)
diffusion path length. For nonionic, low polarity organic compounds, the soil-water partition
coefficient (Kd) is predominately influenced by the compound's organic carbon-water partition
coefficient (Koc), and the fraction of organic carbon content (foc).  The Koc is a function of the size
and molecular structure of the solute and therefore related to diffusivity while foc is related to the
diffusion path length. For any given polymer (humic material) and solvent (water), the diffusion
coefficient in a polymer decreases exponentially with increasing molecular weight or size of the
diffusing molecule. This is especially true for polymer-sorbate systems where the size of the
diffusing molecule is similar to the size of the polymer mesh (Brusseau et al., 1991) as is the
case with humic material.  Hence, if a diffusion mechanism is responsible for rate-limited
sorption, an inverse relationship between Xsw and Kd is expected.  Brusseau and Rao (1989a,b)
collected rate-limited sorption data from a number of investigators, plotted the logarithms of
sorption rate constants and soil-water partition coefficients, and observed an inverse relationship

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consistent with previous observations of Karickhoff (1980,1984), and Nkedi-Kizza et al. (1989).
These and other correlations between \w - Kd are summarized in Table 9.2. These correlation
equations then can be used to estimate Ksw.

       Brusseau et al. (1991) investigated the effect of compound structure on sorption kinetics
to assess when deviation from Xsw - Kd correlations would be expected. Brusseau and Rao (1991)
found that sorbates containing single or multiple substitutions of methyl or chlorine groups
exhibited Xws - Kd relationships similar to unsubstituted aromatic hydrocarbons. Also,
chlorinated alkenes/alkanes exhibited Kws - Kd relationships similar to unsubstituted and
substituted aromatics.  In addition, neutral species of ionogenic compounds exhibited Xws - Kd
behavior similar to nonionic compounds. For these compounds, sorbate structure had a minor
impact on the nature of rate-limited sorption as compared to Kd values. Brusseau and Rao (1991)
state that the Kws - Kd correlation for simple molecules such as benzene and chlorobenzene
represents a "standard-state" behavior, with deviations from the standard state being a function of
the relative degree of additional sorbate-sorbent interactions. Sorbates with complex structures
such as some pesticides may have additional constraints to diffusion and specific sorbate-sorbent
interactions. For instance, Brusseau et al. (1991) evaluated the impact of single or multiple

Table 9.2 Correlation of solids-water mass transfer coefficients with soil-water partition
coefficients
 Equation (hr"1)
Compound Class
Reference
 log Kws = 0.301 -0.668 log Kd
 (r2 = 0.95)
wide variety of compounds with
log Kd varying from -1.5 to 5.5
Brusseau and Rao
(1989b)
 log Kws = 0.960 -0.836 log Kd
 (r2 = 0.99)

 log ^=1.18-0.885 log Kd
 (r2 = 0.99)

 log Aws = 0.944-0.830 log Kd
 (r2 = 0.97)

 log Aws = 0.56-0.47 log Kd
  (r2 = 0.85)
 log Kws = 0.75 -0.63 log Kd
  (r2 = 0.99)

 log Aws = 0445-0.86 log Kd
  (r2 = 0.87)
chlorobenzenes


chlorobenzenes


polynuclear aromatics


chloro-aliphatics, aromatics,
chloro-aromatics, and
chlorophenols
Brusseau and Rao
(1989b)

Brusseau et al. (1990)
Brusseau and Rao
(1989b)

Brusseau and Rao
(1991)
aromatics and methyl-aromatics     Brusseau et al. (1991)
aromatics, chloroalkenes,
polynuclear aromatics
Huetal. (1994)
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additions of methyl groups on the benzene molecule using toluene (methylbenzene), o-xylene (o-
dimethylbenzene), p-xylene (p-dimethylbenzene), and trimethylbenzene.  The addition of single
methyl functional groups did not result in behavior deviating from the behavior exhibited by
unsubstituted molecules.  They also evaluated the impact of adding a single n-alkyl group on the
benzene molecule by using ethylbenzene, n-propylbenzene, n-butylbenzene, and n-hexylbenzene.
In this case, addition of an alkyl chain of three or more carbon units did alter the expected Xsw -
Kd correlation. Brusseau et al. (1991) argue that the presence of three- or four-carbon chain n-
alkyl groups provides an increased  opportunity for a diffusing molecule to become entangled
with the polymer chain.

       Estimation of the fraction of sorbent in instantaneous sorption occurs, F, does not appear
as straightforward as estimation of  Asw Values of F appear to vary with chemical and sorbent
ranging from 0.12 to 0.52 for a combination of 24 chemicals and 11 sorbents (Karickhoff and
Morris, 1985; Lee et al., 1988; and  Brusseau et al., 1991). Lee et al. (1988) reported a median
value of 0.17 for F in the transport of trichloroethene and p-xylene in columns packed with
Borden and Lula sandy low-organic aquifer material. Brusseau et al (1991) observed a linear
increase in F with increasing log Kow (octanol-water partition coefficient) for a number of
compounds. Brusseau et al. (1991) state that since Kow is directly related to the molecular size of
a solute, this behavior is consistent with the intra-organic matter diffusion model.  They state that
the polymeric mesh may act as a molecular sieve, whereby larger sorbate molecules are excluded
from some portion of the internal volume because of size constraints. Thus the size of the rate-
limited domain may vary with sorbate size.  Molecular hindrance would cause F to increase
assuming that the instantaneous fraction  of sorption sites remains constant.  Lee et al. (1991)
found an increase in F with increasing log Kd for ionized chlorophenols. For pentachlorophenol
(PCP). They observed that F values decreased linearly with increasing methanol addition.  Lee  et
al. (1992)  argue that since organic cosolvents are known to cause swelling of the organic matter
matrix, this increase may be primarily in interior of the organic matter matrix or within the  rate-
limited domain.  They also noted that F increases with increasing ionic strength of the solute
containing solution.  They explain that this is most likely a result of a tightening of the organic
matter matrix with increasing divalent cation concentration.

       Recently, there have been efforts to generalize the sorbent surface into parallel classes of
reaction sites with each site having its own unique rate constant. For instance, Lorden et al.
(1998) describe a y-distribution model that generalizes two-site sorption heterogeneity into a
continuum of sorption sites that are grouped into different classes according to a y probability
density function for both equilibrium  and rate parameters. They conducted  gas flow column
studies to evaluate the performance of the y-distribution model against the more commonly used
two-site, first-order and spherical diffusion models.   Laboratory columns consisted of packed air
dried fine-grained sand (93% sand,  6.3% silt) having an organic carbon content of 0.91%.
Gravimetric measurements indicated a gas-filled porosity of 52%, and moisture content of 1.6%
by weight.  Lorden et al. (1998) state that at this  moisture content, soil particles were covered by
approximately 18 molecular layers  of water (based on measured BET surface area of 3.3 m2/g
and a surface area of 10.8 x 10"20 m2 for a molecule of water). Soils were maintained at 90%
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relative humidity (RH) for all transport experiments.  TCE was injected into the columns at 90%
(472.5 mg/1) and 10% (52.5 mg/1) of vapor saturation to simulated flow close to aNAPL source.
Flow rates were chosen so that resulting pore-gas velocities ranged between 0.1 to 0.8 cm/s.
Equilibrium sorption experiments were conducted at 90% RH over a wide range of vapor
concentrations revealing a non-linear isotherm.  Lorden et al. (1998) however state that
concentration dependency on the partition coefficient probably did not have a significant affect
on the ability of the y-distribution model to fit ETC data because of the strong departure from
equilibrium. The only cause of asymmetric effluent breakthrough curves during experimentation
was postulated to be from sorption kinetics since symmetric breakthrough curves were observed
using a conservative tracer (4% by volume methane in nitrogen). Particular emphasis was placed
on the relative ability of each model to fit the slow tailing or low concentration portion of the
breakthrough curves. C/C0 was plotted on a log scale to facilitate visual observation of fit at low
vapor concentrations.

       BTCs revealed that when the equilibrium sorption partition coefficient remained fixed at
a laboratory measured value, both the two-site model and  spherical diffusion model were unable
to fit effluent C/Co concentrations below 0.1 and 0.01 respectively while the y-distribution
model was able to successfully fit data through C/Co concentrations of 0.001. When the
partition coefficient was included as a fitting parameter, the quality of fit improved significantly
for the spherical diffusion model but remained poor for the two-site first-order model. The two-
site first-order model tended to provide sharp and straight concentration declines as opposed to a
smooth transition in concentration with pore volume exchanges.  Also, mass transport
coefficients increased with velocity for the spherical diffusion and two-site model when the
partition coefficient was included as a fitting parameter. There is no way to directly use Lorden
et al.'s work to estimate a critical pore-gas velocity for solids-water exchange, but it is important
to realize that there may be limitations to two-site and spherical diffusion modeling in assessing
mass transfer.

9.7    Combined Rate-Limited  Gas-Water and Solids-Water Exchange

       Wehrle and Brauns (1994)  describe column experiments conducted in moist fine to
medium sand with foc = 0.4% and water saturation = 24%. Vapor effluent curves and flow
interruption tests displayed significant rate-limited mass transport behavior at a pore-gas velocity
of 0.35 cm/s. Vapor concentration increased over two orders of magnitude during flow
interruption testing. A comparison of tests with and without organic  carbon removed revealed
less but still significant rate-limited behavior with organic carbon removed. Thus, vapor
transport was likely limited by both gas-water and solids-water exchange. At the end of the
experiments, 97% of contaminant mass had been removed after approximately  50,000 pore
volumes.

       McClellan and Gillham (1992) reported effluent tailing and significant concentration
rebound after flow interruption tests. Vapor concentrations in the effluent persisted even after
22,650 pore volumes of air had passed through soils.  Armstrong et al. (1994) developed a
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numerical one-dimensional model incorporating first-order gas-water and solids-water rate-
limited mass exchange to simulate vapor effluent data obtained by McClellan and Gilham
(1992).  Virtually identical fits were obtained to effluent curves for both continuous and pulsed
schemes using two parameter sets: 1) Kd and Agw values of 2.1 cm3/g and 2 x 10"5 s"1 respectively
(equilibrium sorption) and 2) Kd, K^, and Kws values of 10.1 cm3/g, 4 x 10"5 s"1, and 5 x 10"5 s"1
respectively.  Simulated remaining mass however differed sharply with the second parameter set
resulting in far greater mass retention. Their sensitivity analysis and calibration of data showed
that different rate constants and mass retention characteristics can produce very similar effluent
curves suggesting that effluent and vapor probe concentration data can not be used to estimate
mass transport coefficients and VOC concentration in the aqueous and solids phase.

      Brusseau (1991) developed a model which included a mobile and immobile domain with
rate-limited solids-water exchange in both domains and simulated experimental results of
Roberts (1990) and Brown and Rolston (1980) using independent derived data (i.e.simulation
without curve fitting). Deviation from local equilibrium was evident in all three studies where
pore-gas velocities ranged from 0.016 to 0.635 cm/s. Goltz and Oxley (1994) modified
Brusseau's (1991) one-dimensional formulation to enable simulation of radial flow using a
numerical Laplace transform inverse routine.  Huang and Goltz (1999) provided: (1) an exact
solution to Brusseau's (1991) formulation in radial coordinates for Peclet numbers less than  12;
(2) an exact solution to  Brusseau's (1991) formulation in radial coordinates for infinite Peclet
numbers (no dispersion); and (3) an approximate solution to Brusseau's (1991) formulation in
radial coordinates over  a wide range of Peclet numbers.

       To assess combined gas-water and  solids-water rate-limited exchange during gas flow
without the interfering effects of physical heterogeneity, Popovicova and Brusseau (1998) packed
0.59 mm moistened glass beads to create a homogeneous media with a volumetric water content
of 11.9%.  Methane was used as a nonsorbing conservative tracer while trichloroethene, benzene,
and toluene were used as sorbing tracers. Experiments were conducted at gas velocities ranging
from 0.1 to 3.33  cm/s. Breakthrough curves for methane were symmetrical for all tested pore-gas
velocities indicating absence of rate-limited mobile - immobile gas exchange. Breakthrough
curves for trichloroethene and benzene however were noticeably asymmetric. In addition, the
degree of tailing increased with increasing velocity.  Peclet numbers obtained by applying the
ideal advective-dispersive model to trichlorethene and benzene transport were lower than those
obtained for methane under the same experimental conditions indicating increased dispersion
from rate-limited behavior.  Values of gas-water and solids-water Damkohler numbers less than
10 indicated a significant degree of nonequilibrium caused by diffusion in water and rate-limited
sorption.

9.8   Rate-Limited Mobile-Immobile Gas Exchange

       The mobile-immobile domain concept has been used for solute transport analysis for
many years with a large number of associated publications. Coats and Smith (1964) were among
the first investigators to perform mobile-immobile domain modeling for nonsorbing solutes.
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Later, van Genuchten and Wierenga (1976) extended the model for sorbing solutes. In mobile-
immobile domain formulations, rate-limited mass exchange is represented by a two-site model
where a concentration gradient between the mobile and immobile domain and a first-order rate
constant controls mass exchange. When using the two-site, first-order rate modeling approach,
an average immobile-region concentration is assumed in contrast to the aggregate diffusion
model where a concentration gradient within the immobile domain is allowed.

       For soil venting application, gas-filled regions where advective flow occurs is defined as
the mobile domain while gas- and water-filled regions where advective flow does not occur is
defined as the immobile domain. Mass exchange from water-filled regions has been previously
discussed in the context of gas-water mass transfer using first-order rate constants or spherical
diffusion modeling.  Immobile domains have been conceptualized as lenses or layers of low
permeability media surrounded by higher permeability media, intra-aggregate porosity, dead-end
pores, bulk water, surface films, and matrix porosity of fractured media. In stratified media, gas
will flow primarily through layers of higher gas permeability. Vapors in highly permeable strata
will be extracted first by advection.  As the contaminant from these strata becomes depleted,
diffusion from the lower permeable strata will control mass removal.

       Krishnayya et al. (1988), Croise et al. (1989), Ho and Udell (1991), and Kearl et al.
(1991) demonstrated the effect of layered porous media on gas transport.  Rathfelder et al. (1991)
demonstrated that effluent tailing can occur because of gas flow being channeled in stratigraphic
layers of lesser contamination but greater permeability.

       Kearl et al. (1991) conducted gas transport experiments in a column packed with layers of
sand and silty clay with moisture contents of 2 and 10% respectively. They concluded that
diffusion between layers of different permeability controlled transport and removal of
contaminants at gas velocities ranging from 2 to 45 cm/s.  These pore-gas velocities though are
very high compared to typical pore-gas velocities expected in the field (0.001 to 0.1 cm/s).

       Kaleris and Croise (1999) used a well-mixed reservoir (WMR) model to evaluate the
effect of variation in permeability and retardation factor in a two-layer system.  They compared
simulation results of their WMR model with a finite-element numerical model to assess
conditions under which the WMR model was valid.  Ho and Udell (1992) considered cases in
which gas flow occurred only through a layer  of high permeability media with contaminant
removal from a layer of low permeability media occurring only through diffusion from a NAPL
source.

       To assess the effect of physical heterogeneity due only to mobile and immobile gas mass
transfer without the interfering effects rate-limited gas-water and rate-limited solids-water
exchange during soil venting, Popovicova and Brusseau (1997) conducted of a series of
laboratory column experiments using dry 0.59 mm glass beads and a centered 5 mm o.d. stainless
steel screen to represent a macropore.  Methane was used as a nonsorbing conservative tracer
while trichloroethene, benzene, and toluene were used as sorbing tracers. Experiments were
                                          145

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conducted at gas velocities ranging from 0.1 to 3.33 cm/s. In the heterogeneous media, increased
tailing or ETC asymmetry for methane was evident with increasing velocity.  At low pore-gas
velocities (e.g., 0.1 cm/s) BTCs were symmetrical. Popovicova and Brusseau (1997) explain that
as the residence time of the vapor decreases with increasing velocity, the departure from
equilibrium increases due to insufficient time available for mass exchange between macropore
and micropore domains.  To compare contributions of molecular diffusion, mechanical
dispersion, and micropore-macropore exchange to overall dispersion, a mechanical dispersivity
value was estimated from experiments previously conducted without the macropore at pore-gas
velocities in excess of 1.67 cm/s.  At a pore-gas velocities greater than 2.0 cm/s, micropore-
macropore exchange contributed greater than 50% of total dispersion.  At a pore gas velocity of
3.33 cm/s, micropore-macropore exchange contributed greater than 80% of total dispersion.
ETC asymmetry appeared to increase for sorbing tracers in the heterogeneous system. However,
when the micropore-macropore first-order exchange coefficients were adjusted to incorporate
differences in the free-gas molecular diffusion coefficients, this increased asymmetry was
effectively accounted for and thus was not due to sorption kinetics.

       In this experimental system, it is likely that advective transport occurred in the micropore
as well as the macropore domain. Thus, breakthrough asymmetry could have been influenced by
pore-gas velocity variation as well as from diffusive micropore-macropore exchange.  If
significant advective flow occurred in the micropore region, then fairly high pore-gas velocities
would be required to induce breakthrough curve asymmetry. This is not analogous to a field
situation where micropore-macropore or stratigraphic layer exchange may be completely
constrained by diffusion. Thus, much lower pore-gas velocities could induce breakthrough
asymmetry due to physical heterogeneity.

       A Damkoher number for mobile gas - immobile gas transfer can be expressed by
where X^ is an immobile gas - mobile gas first-order mass transfer coefficient [T1]. By analogy
to Brusseau's (1991) estimation of "k^, Agg could be estimated by

Thus, assuming that immobile gas - mobile gas exchange approaches equilibrium conditions as
(ogg approaches 10 and a spherical shaped immobile domain, a critical pore-gas velocity for
immobile gas - mobile gas exchange can be estimated by
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                                                                                 (9'33)
9.9    Combined Rate Limited Mobile-Immobile Gas and Gas-Water Exchange
       Popovicova and Brusseau (1998) conducted of a series of laboratory column experiments
similar to Popovicova and Brusseau (1997) except that 0.59 mm glass beads were saturated and
allowed to drain to achieve a volumetric gas porosity of 9.9% prior to conducting experiments.
As before, a centered 5 mm o.d. stainless steel screen was added to their column to represent a
macropore and methane was used as a nonsorbing conservative tracer while trichloroethene,
benzene, and toluene were used as sorbing tracers. Experiments were conducted at gas velocities
ranging from 0.1 to 3.33 cm/s.  Their model formulation allowed for retention at the gas-water
interface, assumed instantaneous sorption, and allowed consideration of rate-limited: (1)  mobile-
immobile gas exchange,  (2) mobile gas-mobile water exchange, and (3) immobile gas-immobile
water exchange.

       Transport of methane through the heterogeneous system exhibited asymmetric
breakthrough which increased with pore velocity. Analysis of data indicated that advective flux
was not confined to the macropore. Gas advection also in occurred in the glass bead domain.
The magnitude of their mobile - immobile gas Damkohler numbers (0.88-2.83) indicated
significant rate-limited mobile-immobile gas exchange. Popovicova and Brusseau (1998) state
that generally, diffusive mass transfer in the gas phase is considered relatively rapid, especially
compared to aqueous-phase diffusion. However, the results of their experiments illustrate that
gas-phase mass transfer can be rate limited at the pore-gas velocities used for their column
studies. The transport of TCE exhibited greater breakthrough asymmetry and decreasing Peclet
numbers compared to methane at similar pore gas-velocities indicating the presence of additional
rate-limiting processes which they attributed to diffusion within immobile water and rate-limited
sorption.

9.10   Conclusions

1 .      Selection of pore-gas velocities to support venting design requires consideration of rate-
       limited gas-NAPL, gas-water, and solids-water mass exchange on a pore-scale and rate-
       limited mobile-immobile gas exchange on a field-scale. During rate-limited dominated
       mass transfer, as pore-gas velocity increases, vapor concentration decreases. However, an
       increased pore-gas velocity still results in an  increased mass removal rate and hence
       shorter remediation time because of an increased concentration gradient between
       equilibrium and nonequilibrium vapor and soil-water concentrations. Thus, there is a
       trade-off between selection of a design pore-gas velocity and remediation time.

2.      Five ways that could be used separately or concurrently to select a critical pore-gas
       velocity for venting application were described.  The approach most suitable for use by
       practitioners appears to be use of algorithms  and Damkohler numbers.  As mentioned

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though, a limitation using Damkohler numbers and algorithms to estimate critical pore-
gas velocities is that mass transport processes are considered independently when in
reality complex interactions between these processes dictate overall mass removal.  Also,
there are additional complications to consider in selection of critical pore-gas velocities.
Virtually all research on rate-limited vapor transport and development of algorithms for
pore-gas velocity determination originated from controlled sand tank and laboratory
column studies where field-scale heterogeneity was not a factor. Thus, optimal pore-gas
velocities in the field are likely to be very site-specific and somewhat smaller than
indicated by Damkohler numbers, algorithms, and published studies. Also, characteristic
lengths of contamination used in Damkohler numbers and algorithms are not constant but
decrease in time due to NAPL and sorbed mass  removal thereby reducing calculated
optimal pore velocities. Finally, the cost of well installation and venting operation (e.g.,
electricity, vapor treatment) should be considered since achievement of a minimum target
pore-gas velocity at some sites may be prohibitively expensive.  Thus, it is apparent that
selection of a critical pore-velocity is not a straightforward process but involves some
degree of judgement and iterative reasoning (gas flow modeling to see what pore-gas
velocities can be achieved given certain well spacing and flow rates). However, while
there are limitations to the use of algorithms and Damkohler numbers in selecting a
design pore-gas velocity, these numbers provide a starting point after which more
sophisticated analysis can proceed if desired.
                                    148

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10.    LIMITATIONS OF ROI EVALUATION FOR SOIL VENTING DESIGN:
       SIMULATIONS TO SUPPORT VENTING DESIGN BASED ON ATTAINMENT
       OF CRITICAL PORE-GAS VELOCITIES IN CONTAMINATED MEDIA

       By far, the most common method used for venting design in the United States is radius of
influence (ROI) evaluation. In a review of over 100 Superfund and Resource Conservation and
Recovery Act (RCRA) sites at U.S. EPA's Office of Research and Development, National Risk
Management and Research Laboratory in Ada, Oklahoma, only one documented case could be
found where a non-ROI method had been used for venting design. ROI testing involves
evaluating the maximum radial extent of induced subsurface vacuum to some arbitrarily
specified level (typically 0.01 to 0.1 inches of water) in single well tests.  As illustrated in Figure
10.1, in practice, ROIs are determined by plotting vacuum as a function of logarithmically
transformed radial distance and applying linear regression to extrapolate to the distance at which
a specified vacuum would be observed.  Vacuum monitoring points are often located significant
distances from gas extraction wells in an attempt to "physically" locate ROIs for given flow
rates.  For SVE design, overlapping circles of ROIs for individual wells are then drawn on a site
map to indicate an "effective" remediation area.

       Johnson et al. (1990a,b) first discussed ROI assessment as a method for spacing SVE
                                                      W-l (ROI = 64 ft, R2 = 0.82)
                                                      W-3 (ROI = 64 ft., R2 = 0.95)
                                                      W-4 (ROI = 85 ft., R2 = 0.72)
                                                      W-5 (ROI = 51 ft., R2 = 0.97)
                                                      W-6 (ROI = 69 ft., R2 = 0.93)
                                                      W-7 (ROI = 78 ft., R2 = 0.94)
           0 -\
             10
100
                                         Distance (ft)
Figure 10.1  Observed vacuum as a function of logarithmically transformed radial distance at
tested wells for ROI determination. Line represent result of linear regression with correlation
coefficients R2 as provided.

                                         149

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wells.  The U.S. EPA (1991) later incorporated ROI measurement into guidance for SVE design.
The implicit assumption in ROI-based designs is that observation of subsurface vacuum ensures
sufficient gas flow in contaminated soils for timely remediation.  However, measurement of
vacuum at best only ensures containment of contaminant vapors (Johnson and Ettinger, 1994).
This is even in question when the magnitude of applied vacuum in soil is comparable to pressure
differential caused by variation in barometric pressure and/or fluctuation of the water table.
Diurnal barometric pressure changes in soil can be on the order of a few mbar (Massmann  and
Farrier, 1992) where 0.1 inch water vacuum is equivalent to 0.25 mbar. Cho and DiGiulio
(1992) illustrated how specific discharge decreases dramatically with radial distance from gas
extraction and injection wells and recommended that venting design be based on subsurface gas
flow analysis instead of ROI testing.  Johnson (1988) described how the addition of 13 extraction
wells within the ROI of other extraction wells increased blower VOC concentration by 4000
ppmv and 40 kg/day. Johnson (1988) concluded that the ROI was not an effective design
parameter for locating extraction wells and that operation costs could be reduced by increasing
the number of extraction wells as opposed to pumping at higher rates with fewer wells.

       In this section, data was utilized from a Superfund site (described in section 6) where ROI
testing was conducted by a remedial contractor to describe limitations of ROI evaluation in detail
and to demonstrate an alternative method of design based on specification and attainment of
critical pore-gas velocities in contaminated subsurface media.  Other than ROI assessment, little
has been published on alternative quantitative design criteria for venting systems (Sawyer and
Kamakoti, 1998).  Information was utilized from section 9 to provide the basis for selection of a
minimum or critical pore-gas velocity for use at this site. Using single-well gas flow simulations,
an assessment was made whether a selected critical pore-gas velocity could be achieved at
measured ROI's.  A series of multiple-well gas flow simulations were then conducted to assess
how variation in anisotropy and leakance affect three-dimensional vacuum and pore-gas velocity
profiles and determination of an ROI. Finally, an assessment was made when attempting to
achieve a critical design pore-gas velocity, whether it is more efficient to install additional wells
or pump existing wells at a higher flow rate.

10.1    Critical Pore-Gas Velocity Estimation for Test Site

       At the test site described in section 6, preliminary single- and multiple-well gas flow
modeling indicated that a critical pore-gas velocity of 0.01 cm/s was easily achievable.  Many of
the soils related input parameters required by Damkohler numbers and algorithms were not
available and thus had to be estimated from literature values summarized in Table 10.1.  A
critical pore-gas velocity for gas-water exchange using a  gas-water Damkohler number of 10 was
calculated to be 2.70 cm/s.  A critical pore-gas velocity for gas-water exchange using a prefix
denominator value of 10 was calculated to be 5.23 cm/s.  Using relationship developed by Ng
and Mei (1996) for aggregate diffusion, a critical gas-water pore-gas velocity of 5.80 cm/s was
calculated.  Use of a solids-water Damkohler number of 10 and assumption that F = 0 resulted in
computation of a critical pore-gas velocity for solids-water exchange at 2.22 cm/s.  Information
was not available to calculate a critical pore-gas velocity for mobile-immobile gas exchange but
                                          150

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Table 10.1 Input parameters for estimation of pore-gas velocity
      Description                    Value       Units   Ref.
sw
11
Ps
^s
*w
So
d50
LC
water saturation
porosity
particle density
gas phase tortuosity
water phase tortuosity
aggregate radius
mean grain size
0.26
0.33
2.65
2.9
1.4
0.035
0.08
radius of contaminated region 800
Kd soil-water partition coefficient 0.300
H
°g
DW
11.
12.
13.
14.
Henry's Law Constant
gas diffusion coefficient
water diffusion coefficient
0.26
0.082
l.lx 10-°5
[-] 1
[-] 1
g/cm3 1
[-] 1
[-] 1
cm 1
cm 2
cm 3
cm3/g 4
[-] 1
cm2/s 1
cm2/s 1
Value used by Gierke et al. (1992) for sand
Value used for Wilkins et al. (1995) for graded sand mixture
Based on 53 foot (average distance between closest wells) well spacing
Assumed based on Koc of 300 cm3/g for toluene and foc = 0.001
at pore-gas velocity of 0.01 cm/s, an immobile gas region having a radius greater that 61 cm
would be necessary to limit mass removal. The relationship developed by Wilkins et al. (1995)
using a C/C0 vapor concentration of 0.99 yielded an extremely high and unrealistic pore-velocity
for deviation from equilibrium conditions. At a pore-gas velocity of 0.01 cm/s, a characteristic
contamination of length of less that 1 cm would be necessary to achieve near equilibrium
conditions in NAPL contaminated soils. Thus, use of the information provided in Table 10.1
without consideration of field-scale heterogeneity indicates that a fairly high pore-gas velocity
could be utilized at this site. However, given the potential for overestimation of critical pore-gas
velocities due to field-scale heterogeneity, overestimation of the characteristic length of
contamination, and  observation during gas flow modeling that pore-gas velocities on the
magnitude of 1 cm/s would require very close well spacing, a critical pore-gas velocity of 0.01
cm/s was chosen for use at this site.  Using similar reasoning, a design pore-gas velocity of 0.01
cm/s was also  selected for use at the Picillo Farms Superfund site containing glacialfluvial
outwash and glacial till (Woodward-Clyde and Envirogen, 1998) based on three-dimensional


                                           151

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numerical gas flow modeling with the public domain model AIR3D (Joss and Baehr, 1995).

10.2   Single-Well Simulations

       Results from ROI testing are summarized in Table 10.2 for vacuum levels of 0.1, 0.05,
and 0.0 inches of water and graphically displayed in Figure 10.1 with associated linear regression
correlation coefficients.  It is apparent that monitoring points were placed at distant radii in an
attempt to "physically" locate the radius at which vacuum approached 0.01 inches of water even
though most vacuum dissipation and thus highest pressure gradients occurred within 15 feet of
each well. Using a ROI-based design, wells would be spaced 60 - 146, 80 - 150, and 128 - 170
feet apart for ROI vacuum of 0.1, 0.05, and 0.0 inches of water respectively.  Yet as shown in
Table 10.3 and Figures 10.3 through 10.8, adjacent wells were typically placed 43 to 65 feet
apart. Thus, well spacing at this site is conservative for an ROI-based design.

       With the exception of well W-4, Figure 10.2 illustrates two-dimensional profiles of
calculated vacuum and vector norm pore-gas velocities using the arithmetic mean of gas

Table 10.2 Radii of Influence (ft) as a function of observed vacuum
 well    0.1 in.   0.05 in.   atm.
         water   water     press.
W-l
W-3
W-4
W-5
W-6
W-7
32
57
55
30
58
73
45
60
65
40
64
75
64
64
85
51
69
78
Table 10.3 Well spacing (ft)

W-l
W-3
W-4
W-5
W-6
W-7
W-l
0
67
123
65
66
94
W-3
67
0
59
53
64
53
W-4
123
59
0
79
118
89
W-5
65
53
79
0
104
106
W-6
66
64
118
104
0
43
W-7
94
53
89
106
43
0
                                           152

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permeability and leakance values in Table 6.2b. Calculations for well W-4 were not conducted
because the range of leakance and anisotropy values were too wide to allow selection of
representative values.  Vacuum profiles are curvilinear especially near wells due to partial
penetration of gas extraction wells, aniostropy, and leakance of air from the surface. The
definition of a ROI however implies a constant pressure differential at a given radius. As k/kz
ratios increase, vacuum contours become increasing curvilinear until eventually as illustrated for
well W-3, there is too much vertical variation to assign a vacuum level to any specific radial
distance. If a vacuum level of 0.05 inches of water had been used to designate a ROI for W-3,
the ROI would have varied from 54 to 78 feet. Thus, the ROI is an ill-defined and nondescript
term for anisotropic soils or where there is significant leakance or partially penetrating gas
extraction wells. This reality however is missed during ROI testing because gas monitoring
wells are typically screened over an extensive portion of the vadose zone thereby integrating
pressure or vacuum measurements and giving the impression that pressure differential is only a
function of radial distance.

       Recognizing, that ROI measurement does not incorporate vertical variation in pressure
differential, Chen and Gosselin (1998) proposed using an area of influence or zone of influence
(ZOI) to describe the effectiveness of a gas extraction well.  This area or zone then would be  a
two-dimensional surface over which "effective" gas flow would occur. While an improvement
over the ROI concept, the ZOI concept still requires arbitrary specification of a pressure
differential which again does not guarantee attainment of a design critical pore-gas velocity.  For
instance, vacuum readings of 0.05 inches of water at wells W-l, W-3, W-5, W-6, and W-7,
resulted in approximate computed pore-gas velocities of 0.002 - 0.003, 0.0007 - 0.001, 0.005 -
0.007, 0.0007, and less that 0.001 cm/s respectively.  Vacuum readings of 0.1 inches of water
(typically the highest vacuum that practitioners use to establish a ROI) at wells W-l, W-3, W-5,
W-6, and W-7 resulted in approximate computed pore-gas velocities of 0.005 - 0.007, 0.0015 -
0.003, 0.01 - 0.02, 0.0015, and less than 0.001 cm/s respectively. In only one case did a vacuum
of 0.1 inches of water result in attainment of 0.01 cm/s or higher (W-5). For wells W-l, W-3, W-
5, W-6, and W-7, vacuum measurements of approximately 0.25, 0.5, 0.1, 0.5, and 1.0 inches  of
vacuum respectively would be necessary to ensure attainment of a pore-gas velocity of 0.01 cm/s.

       The results of single-well simulations demonstrate several important deficiencies in ROI
evaluation.  First, as previously discussed, cross-sectional two-dimensional simulations clearly
illustrate that the ROI is an ill-defined entity. Vacuum profiles are very curvilinear especially
under conditions of high leakance values and k/kz ratios.  Second, an ROI-based design, may
result in subsurface pore-gas velocities too low for optimal gas circulation especially at low
leakance values and high k/kz ratios. Third, attainment of critical pore-gas velocities can not be
guaranteed by simply raising the magnitude of vacuum during venting application from say 0.05
inches of water to some other  arbitrarily specified level (e.g., 0.1 inches of water).  Observed
vacuum is a function of boundary conditions (e.g., leakance), applied mass flow, anisotropy,
permeability, and geometry of screened intervals.  In some cases, relatively high vacuum (e.g.,
greater than 1 inch of water) must be maintained between wells to ensure sufficient subsurface
gas flow.
                                           153

-------
Jl
&
Q
 -5-


-10-1
  °-G \  *\
    \ \  N
m  \\\   \
 B?  ,|,   \
                      \V
               o
               o
                                         o
                                         8
0
10
20
                                      40      50      60

                                      Distance from W-l (ft)
                                                              70
                                                                   80
                                                                         I
                                                                        90
tfn

^


Q
    -5-


   -10-
      0       10       20       30      40      50      60

                                      Distance from W-3 (ft)
                                                              70
                                                                   80
                                                                         I
                                                                        90
    -5-
                N\    \ N
                       *<
   -15
            JL
                          111
                                                                           ~~r~
                                                                           90
       10      20      30      40       50       60

                               Distance from W-5 (ft)
                                                              70
                                                                      80
Q -




10-


III
~~" ' \
~~-^ *^\
°2\\ ^ \
\ \ \
to o ° Z ;
P 'f 'S 1 8

\
o
In
= 1 ?
2 *

*
P
&


T
1 1



^ i
= P SN ?
* ' 12?
^ w u, |-


III

o

^






i i §
§ 1 §

i
1 1
0 10 20 30 40 50 60 70 80 90
                                      Distance from W-6 (ft)
Q -


-5-
10-

-^ \ \
~~^. *• \
- \ \ \
^ ^ %
1 M i

0 10
\ \
L*J t-J
1 \ \ 1 \
4* o o o o
\ i p | s '^ 8 'a
M 1 r
i l
"il
\
0
^
1? *
ii i i i i i i
20 30 40 50 60 70 80 90
                                      Distance from W-7 (ft)


Figure 10.2 Vacuum (inches of water) (dashed lines) and pore-gas velocity (cm/s) (solid lines)
plots for tested wells.
                                          154

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10.3   Multi-Well Simulations

       To assess expected performance of a fully operational system, mean gas permeability and
leakance values estimated at wells W-l, 3, 5, 6, and 7 were utilized to conduct a series of three-
dimensional multi-well simulations. Steady-state vacuum and pore-gas velocities (vector norm)
were simulated in three-dimensional space by applying superposition theory to (6.8) and
converting radial coordinates to Cartesian coordinates.  A FORTRAN program was written,
MAIRFLOW, to compute a three-dimensional gas pressure and pore-gas velocity field from
multiple gas injection or extraction wells.  The code also allows particle tracking using the
Simple Runge-Kutta (second-order) method. Sample input and output files in addition to the
source code are located in Appendix F.

       A depth of 15 feet was chosen for all multiple-well simulations because most
contaminants in the vadose zone are located near the water table.  An analytical modeling
approach was appropriate at this site since distinct stratigraphic layers were not encountered
necessitating numerical analysis of vacuum and pore-gas velocity profiles. As illustrated in
Figure 10.3, isotropic conditions (k/kz = 1.01) and high leakance (2.24 x 10"09 cm) resulted in
significant recharge from the surface causing poor areal coverage of the 0.01 cm/s pore velocity
contour and low vacuum  (0.1 - 0.15 inches of water) between extraction wells.  A similar k/kz
ratio (1.17) and slightly lower leakance (9.29 x 10"10 cm) depicted in Figure 10.4, resulted in
minimal improved areal coverage of the 0.01 cm/s pore velocity contour but somewhat higher
vacuum (0.2 - 0.5 inches of water) between extraction wells. Again nearly isotropic conditions
(k/kz ratio = 0.98) but significantly lower leakance (3.62 x 10"11 cm) shown in Figure 10.5
resulted in dramatic improvement in the areal coverage of the 0.01 cm/s pore velocity contour
and much higher vacuum (3.0 - 4.0 inches of water) between extraction wells. It is evident that
as leakance decreases, a velocity profile for a given total flow rate becomes more uniform but
vacuum required to maintain a critical design pore-gas velocity increases significantly - a
relationship not apparent  from ROI testing.  Notice the  large area in the left center portion of
Figure 10.5 where a pore-velocity of 0.01 cm/s was not achieved despite vacuum levels
exceeding 3.5  inches of water whereas in other portions of the site a pore-velocity of 0.01 cm/s
was achieved at less than 1.5 inches of vacuum.  This reinforces previous observations with
single-well simulations that the magnitude of a vacuum level in subsurface media is not an
indication of effective gas flow contrary to assumptions made in ROI testing. When the k/kz
ratio is reduced to 0.74 and leakance increased to 8.62 x 10"10 cm as illustrated in Figure 10.6, the
areal coverage of the 0.01 cm/s contour decreased as expected because of increased vertical flow
and recharge from the surface.  A comparison of anisotropy and leakance values estimated from
testing at wells W-l (k/kz =1.17, k'/b' = 9.29 x  1040 cm) and W-3  (k/kz = 4.29, k'/b'= 4.04  x
10"10 cm) illustrated in Figure 10.4 and 10.7  respectively reveals that for similar leakance values,
higher k/kz ratios significantly increase the areal coverage of a  critical design pore-gas velocity
and increase vacuum between wells. A comparison of anisotropy and leakance values estimated
from testing at wells W-6 (k/kz = 0.74, k'/b' = 8.62 x 10'11 cm) and W-3 (k/kz = 4.29, k'/b' =
4.04 x 10"10 cm) illustrated in Figure 10.6 and 10.7 respectively reveals fairly similar pore-gas
velocity and vacuum profiles. This indicates that at least at the targeted depth of 15 feet,
                                           155

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decreased leakance largely made up for a decreased k/kz ratio. However, inspection of Figure
10.2 for W-3 reveals that increased k/kz ratios result in much more curvilinear flow and the
similarity in pore-gas velocity and vacuum profiles is limited to this depth. Regardless of
specific k/kz ratios and leakance values used for simulation however, it is apparent that the
suitability of using vacuum levels of 0.05 or 0.1 inches of water, typical of ROI testing and ROI-
based designs, to ensure  adequate gas circulation decreases with increasing k/kz ratios and
decreasing leakance values. As illustrated in Figures 10.2 and 10.3 for well W-5, ROI-based
designs are more likely to be appropriate as k/kz ratios approach 1.0 or less and when significant
leakance occurs. The latter two situations though are in direct conflict with assumptions made in
derivation of pseudo-steady-state equation, the equation in which ROI-testing is based.

       The effect of well screen placement on vacuum and pore-gas velocity profiles  while not
evaluated here was demonstrated by DiGiulio (1998c) and Shan et al. (1992).  Specifically,
DiGiulio (1988c) showed in simulations assuming a domain open to the atmosphere, isotropic
conditions,  k,. = 1.5 x 10"7 cm2 and flow = 10 scfm that a pore-gas velocity of 0.01 cm/s could not
be achieved directly below a well screened 0.1 to 0.3 times the length of domain below land
surface even though a pressure differential between 0.3 to 0.5 inches of water vacuum was
present.

       Finally, when attempting to achieve a critical design pore-gas velocity, it is more efficient
from a vapor treatment perspective to install additional wells rather than pump existing wells at a
higher flow rate. As illustrated in Figure 10.8, using permeability and leakance information from
well W-5, total flow from existing wells would have to exceed 1000 scfm to meet a pore-gas
velocity of 0.01 cm/s at a depth of 15 feet throughout most of the contaminated region. Note
however though that there was a small region between wells W-l and W-3 were this design
velocity could not be achieved. As illustrated in Figure 10.9, when the total number of wells was
increased from 6 to  16, nearly a nearly continuous pore velocity of 0.01 cm/s was achieved
throughout the entire contaminated region at only 400 scfm.  Depending on the depth  and media
through which drilling occurs, capital costs associated with well construction may be more than
offset by savings in vapor treatment costs. Another benefit of installing additional vapor
extraction wells is that pore-gas velocity is much more uniform throughout a contaminated area.
When only  6 wells were  used to  simulate venting operation at 1000 scfm, there were large
regions in which pore-gas velocities exceeded 0.05 and 0.1 cm/s. When the number of wells was
increased to 16, pore-gas velocity was maintained below 0.05 cm/s throughout the vast majority
of the contaminated region. At lower pore-gas velocities, vapor transport would be less affected
by mass transport limitations. Thus, an increased number of wells would result in more
concentrated vapor stream, which is advantageous from a vapor treatment perspective.
                                           156

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     180-
       0
         0     20     40
60    80    100    120
         Distance (ft)
140   160    180
Figure 10.3 Multi-well simulation using permeability and leakance data from well W-5 (k,. =
6.87 x 10'7 cm2, k/kz = 1.01, leakance = 2.24 x 10'09 cm).  Depth = 15 feet. Dashed lines denotes
vacuum (inches of water), solid lines denote pore-gas velocity (cm/s).
                                        157

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 C/5
       0
         0     20     4cT   60     80     100    120    140    160   180
                                       Distance (ft)
Figure 10.4 Multi-well simulation using permeability and leakance data from well W-l (k,. =
2.90 x 10'7 cm2, k/kz =1.17, leakance = 9.29 x 1040 cm).  Depth = 15 feet.  Dashed lines denotes
vacuum (inches of water), solid lines denote pore-gas velocity (cm/s)
                                        158

-------
    180

    160

    140-


 ^ 12°"

 |  100
 B
 S    80

      60

      40

      20

       0
         0     20     40     60     80    100   120   140   160   180

                                     Distance (ft)

Figure 10.5 Multi-well simulation using permeability and leakance data from well W-7 (k,. =
1.42 x 10'7 cm2, k/kz = 0.98, leakance = 3.62 x 1041 cm). Depth = 15 feet. Dashed lines denotes
vacuum (inches of water), solid lines denote pore-gas velocity (cm/s).
                                      159

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     100
 C/5
       o
         0     20     40     60     80     100    120   140    160    180
                                       Distance (ft)
Figure 10.6 Multi-well simulation using permeability and leakance data from well W-6 (k,. =
1.81 x 10'7 cm2, k/kz = 0.74, leakance = 8.62 x 1041 cm).  Depth = 15 feet. Dashed lines denotes
vacuum (inches of water), solid lines denote pore-gas velocity (cm/s).
                                        160

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 -t—>
 C/5
       o
0      20
60
80     100   120    140    160
 Distance (ft)
                                                                           180
Figure 10.7 Multi-well simulation using permeability and leakance data from well W-3 (k,. =
1.80 x 10'7 cm2, kykz = 4.29, leakance = 4.04 x 1040 cm). Depth = 15 feet. Dashed lines denotes
vacuum (inches of water), solid lines denote pore-gas velocity (cm/s)
                                         161

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     180
     100
 C/5
         0     20    40     60     80     100    120    140   160    180

                                      Distance (ft)

Figure 10.8 Vacuum and pore-gas velocity at the ground water interface using permeability and
leakance data from well W-5 (k, = 6.87 x 10'7 cm2, k/kz = 1.01, leakance = 2.24 x 10'09 cm).
Depth = 15 feet.  Total flow = 1000 scfm, wells W-l, 3,5,6 = 175 scfm, wells W-4, 7 = 150
scfm
                                       162

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    180


    160


    140-


    120
 g   100
 C/5
 • i-H
 Q
      80
      60
      40-
      20-
       0
        0     20     40    60     80    100    120    140    160    180

                                      Distance (ft)

Figure 10.9 Vacuum and pore-gas velocity at the ground water interface using permeability and
leakance data from well W-5 (k, = 6.87 x 10'7 cm2, kykz = 1.01, leakance = 2.24 x 10'09 cm).
Depth =15 feet.  Total flow = 400 scfrn, 16 wells with 25 scfm at each well.
                                       163

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10.4   Design Strategy

       Using attainment of a critical pore-gas velocity as a design criteria, optimal well spacing,
screened interval placement, and flow rate selection could be determined using three-
dimensional analytical or numerical models. Numerical analysis can be very useful when distinct
stratigraphic layers or complex boundary conditions are present at a  site. Simulations could be
conducted iteratively (trial and error) or perhaps more efficiency through the use of optimization
routines such as those described by Sawyer and Kamakoti (1998), and Sun et al. (1996).  The
trial and error approach while tedious and time consuming allows greater insight into site-
specific conditions determining design while the latter approach is more computationally
efficient and allows easier evaluation of economic factors important to design. Ideally,
information on spatial variability of radial and vertical gas permeability would be available to
support three-dimensional numerical gas flow modeling.  Since gas permeability is a function of
moisture content, spatial variability of gas permeability for any given geologic environment will
be far greater than for saturated hydraulic conductivity determined for ground-water
investigations. Unfortunately, this level of detail necessary to support numerical modeling is
invariably absent at field sites.

10.5   Conclusions

1.      By far, the most common method used for venting design in the United States is radius of
       influence (ROI) evaluation.  In practice, ROIs  are determined by plotting vacuum as a
       function of logarithmically transformed radial  distance and applying linear regression to
       extrapolate to the distance at which a specified vacuum would be observed. For SVE
       design, overlapping circles of ROIs for individual wells  are then drawn on a site map to
       indicate an "effective" remediation area.  In this section, data was utilized from a
       Superfund site (described in section 6) where ROI testing was conducted by a remedial
       contractor to describe limitations of ROI evaluation in more detail than has been done
       previously and to demonstrate an alternative method  of design based on specification and
       attainment of critical pore-gas velocities in contaminated subsurface media. Information
       was utilized from  section 9 to provide the basis for selection  of a minimum or critical
       pore-gas velocity for use at this site.

2.      The results of single-well simulations demonstrated several important deficiencies in ROI
       evaluation. First, cross-sectional two-dimensional simulations clearly illustrate that the
       ROI is an ill-defined entity.  Vacuum profiles were very curvilinear especially under
       conditions of high leakance values and k/kz ratios. Second, an ROI-based design,
       resulted in subsurface pore-gas velocities too low for optimal gas circulation especially at
       low leakance values and high k/kz ratios.  Third, attainment of critical pore-gas velocities
       could not be guaranteed by simply raising the magnitude of vacuum during venting
       application from say 0.05 inches of water to some other  arbitrarily specified level (e.g.,
       0.1 inches of water).  Observed vacuum was a  function of boundary conditions (e.g.,
       leakance), applied mass flow, anisotropy, permeability, and geometry of screened
                                           164

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       intervals.

3.      Multi-well simulations illustrated additional deficiencies in ROI-based design practices.
       It was evident that as leakance decreased, a velocity profile for a given total flow rate
       became more uniform but vacuum required to maintain a critical design pore-gas velocity
       increased significantly - a relationship that would not be apparent from ROI testing.  In
       one simulation, a design pore-velocity of 0.01 cm/s could not be achieved despite vacuum
       levels exceeding 3.5 inches of water whereas in other portions of the site  a pore-velocity
       of 0.01 cm/s was achieved at less than 1.5 inches of vacuum. This reinforces previous
       observations with single-well simulations that the magnitude of a vacuum level in
       subsurface media is not an indication of effective gas flow contrary to assumptions made
       in ROI testing. It was apparent that the suitability of using vacuum levels of 0.05 or 0.1
       inches of water, typical of ROI testing and ROI-based designs, to ensure adequate gas
       circulation decrease with increasing k/kz ratios and decreasing leakance values. ROI-
       based designs are more likely to be appropriate as k/kz ratios approach 1.0 or less and
       when significant leakance occurs.

4.      Finally, it was demonstrated that when attempting to achieve a critical design pore-gas
       velocity, it is more  efficient from a vapor treatment perspective to install  additional wells
       rather than pump existing wells at a higher flow rate. In one simulation, it was
       demonstrated that the total flow rate from 6 wells would have to exceed 1000 scfm to
       meet a pore-gas velocity of 0.01 cm/s. When the total number of wells was increased
       from 6 to 16 however a pore velocity of 0.01 cm/s was achieved throughout the entire
       contaminated region at only 400 scfm.  The use of additional vapor extraction wells
       resulted in a much more uniform pore-gas velocity throughout the contaminated area.
                                           165

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11.    USE OF A COMBINED AIR INJECTION/EXTRACTION (CIE) WELL TO
       MINIMIZE VACUUM ENHANCED WATER RECOVERY

       The application of vacuum decreases energy potential in soils and hence induces water
movement toward the screened interval of a gas extraction well.  Increased moisture content
results in reduced gas permeability in the vicinity of the gas extraction.  Also, when vacuum is
applied to a gas extraction well screened in or near the water table, water-level rise occurs within
the well (upwelling) reducing or completely obstructing gas flow.  Gas  injection on the hand,
results in soil-water movement away from and hence increased gas permeability in the vicinity of
screened interval.

       Since, the application of vacuum causes water-table upwelling,  soil-water flow, and
decreased gas permeability in the immediate vicinity of a gas extraction well, it is logical that
injection of pressurized gas or at least maintenance of atmospheric pressure at the base of a
vacuum extraction well would reduce water recovery.  Gas injection however poses its own
problems. Since gas  injection is inherently more effective than gas extraction there is a
possibility of uncontrolled advective vapor transport away from the base of a combined air
injection and extraction well. Also, subsurface vacuum would be reduced perhaps leading to less
effective capture of vapors.  The concept of combined  air injection and extraction (CIE) was
evaluated  during a pilot study at Vance Air Force Base located in northwestern Oklahoma.  The
information presented in this section is also discussed in DiGiulio (1995).

11.1   Field Methods and Materials

       Testing was conducted in July and August of 1994 at Site-8 on Vance Air Force Base
(AFB) located in northwestern Oklahoma.  Soils and ground-water are  contaminated with
petroleum hydrocarbons, trichloroethene  (TCE) and vinyl chloride. Five  leaking underground
storage tanks were removed in January, 1989. Four of the tanks stored  lubricating oil, diesel fuel,
and kerosene. The remaining tank stored waste  oils and  solvents.  Contaminated soils in the
immediate vicinity of the tanks were excavated and replaced with fill consisting primarily of silt.
After compaction of the fill,  the site was completely paved with a 25 cm  concrete slab.

       Figure 11.1 illustrates the location of wells and probe clusters installed by the U.S. Army
Corps of Engineers. MW-8-4 was installed during a previous investigation.  All boreholes were
drilled to a diameter of 20.3 cm using hollow-stem augering techniques.  As illustrated in Figure
11.2, numerous isolated saturated or near saturated regions, often associated with a sandstone-
gravel/silt mixture, were identified by visual and manual examination of continuous split-spoon
cores during drilling.  As outlined in Table 11.1, the results of particle size tests indicate that in
the study area, residuum above weathered shale  and sandstone consists of silt loam soils. Soil
textural analysis was  conducted by the U.S. Army Corps of Engineers' Southwestern Division
Laboratory located in Dallas, Texas.
                                           166

-------
                    MW-8-4
                  Scale  «- i m ->
              © Dual vapor extraction well
              X Observation well
              A Monitoring well
              %. Vapor probe cluster
              • Sparge well
                        VE-28
                        CP-3 *
                        OB-3 x
                        CP-2 *
                        OB-2 x
                        CP-1 *
                        VE-l e
                        OB-l x
SP-1
Figure 11.1  Locations of dual vapor extraction wells (VE-series), observation wells (OB-series),
sparging well (SP-1) and vapor probe clusters (CP-series) used in study at Site 8, Vance AFB
Table 11.1 Soil Textural Analysis from OB-2
Depth (m)
0.0-0.61
1.52-3.05
4.57-6.10
4.57-6.10
Sand
6
2
4
4
Silt
65
77
76
73
Clay
29
21
20
23
USDA Soil
Textural Class
silt loam
silt loam
silt loam
silt loam
Moisture
Content (% by
mass)
28.9
16.0
20.8
21.1
                                          167

-------
                                             q
                                             CO
q
vd
                             I I 1 | 1 I I I |  I 1 1  I  1 t  I  I  I  I  I  I  I  I  I
                                               II II II II M .1. LI I M M I
Figure 11.2 Cross-sectional schematic of DVE wells, observation wells, vapor probe clusters

and soil conditions encountered during drilling.
                                           168

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       Figure 11.3 illustrates equipment used during testing.  Water was extracted from dual
vapor extraction (DVE) wells (tubes sticking up), passed through an air-water separation tank
(smaller cylindrical tank), and pumped into a 190,000 liter rectangular storage container.
Contaminated vapors were treated with granular activated carbon (GAC) (larger cylindrical tank)
prior to discharge to the atmosphere. Figure 11.4 illustrates construction details of DVE wells
used to extract air and water during vapor extraction. Combined extraction of air and water was
maintained by insertion of a 2.54 cm outside diameter tube  at the base of a 10.16 cm outside
diameter well from which vacuum from a common source was applied to both the tube and well
annulus.  Air and water extraction was facilitated by the use of "bleed" air at the top of the well.
Operationally, when water-table upwelling occurs during vacuum extraction, the inner tube is
supposed to remove water to prevent screen obstruction.  DVE is often applied below the water-
table to produce a cone of depression and enable simultaneous air and ground-water extraction.
The mass flow rate of extracted air was measured using 2.54 cm annubars and manometers to
determine pressure differential.  Gas flow measurement though was complicated by constant
surging of water flow and erratic variation of manometers.  "Bleed" air flow rate was measured
using a Dwyer flowmeter.  Extracted gas temperature was measured using Cole-Parmer 6.35 mm
NPT pipe thermocouples and a Cole-Parmer digital thermometer.
Figure 11.3.  Photograph of DVE wells (wells with tubes sticking up - DVE-1 in foreground),
observation wells, vapor probe clusters, air water separation tank (cylindrical tank with label),
GAC units, and frac tank for contaminated water storage (rectangular shaped structure)
                                          169

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       Figure 11.5 illustrates construction details of sparging well SP-1.  Mass air flow was
measured using a Dieterich standard annubar DNT-10 in a 2.54 cm steel diameter pipe with a
Dwyer Capsehelic gauge to determine pressure differential.  Static injection pressure was
determined using a Dwyer Magnehelic gauges while temperature was measured using Cole-
Parmer pipe thermocouples and a Cole-Parmer digital thermometer. Mass flow was calculated
using software provided by MD Controls, a distributor of Dieterich annubars.  Sparging during
DVE operation was conducted to determine whether injection of air and subsequent decrease in
vacuum near the water-table around SP-1 would decrease water recovery at the DVE wells.
                     2"x1 1/4" BUSHING

                      4"x2" BUSHING

                       4"x4"x2" TEE

                   TEMPERATURE GAUGE
                   (0-220T)	
             !" BALL VALVE (TYP)
                     1/2" BALL VALVE
                  VACUUM GAUGE
                  (0-30"Hg)
                  6" SO. STEEL
                  PROTECTIVE CASING
                  W/HINGED  LOCKABLE
                  TOP
                                                   4" OIA. STAINLESS
                                                   STEEL RISER
                                                            EXISTING CONCRETE
                                                            PAVEMENT

                                                     3 FT. OF CEMENT GROUT
  OIA. SCH 80
PVC INTAKE LINE
                      •PITOT TUBE VAPOR
                      FLOW INDICATOR
                      CONNECTION (SADDLE
                      W/ FEM. NPT AND
                      PLUG)
TO VAPOR/
LIQUID SEPARATOR
                      VAPOR SAMPLE PORT
                                                  1  1/4" DIA. PVC
                                                  GROUNDWATER
                                                  EXTRACTION TUBE

                                                  1  FT. OF BENTONITE SEAL
                                                  5.06 FT. OF 4"  DIA. SS RISER

                                                 -15.79 FT. OF 4" DIA. SS
                                                  WIRE-WRAPPED  SCREEN WITH
                                                  0.010-INCH OPENINGS
                                                  18 FT. OF SAND PACK

                                                  BOTTOM PLUG
                          NOT TO SCALE
Figure 11.4 Schematic of a dual vacuum extraction (DVE) well used at Site 8,Vance, AFB
                                               170

-------
                             2" TEE
                       PRESSURE GAUGE
                       (0-150 PSI)
2"x1 1/4" BUSHING
W/THREADEO PLUG
                      6" SO. STEEL
                      PROTECTIVE CASING
                      W/H1NGEO LOCKABLE
                      TOP
                                                   -11.3 FT. OF CEMENT GROUT
                                                             EXISTING CONCRETE
                                                             PAVEMENT
                                                    17.16 FT. OF 2" DIA PVC RISER



                                                    1 FT. OF 8ENTONITE SEAL

                                                    4.7 FT. OF SAND PACK

                                                    1.02 FT. PVC SLOTTED 2" DIA
                                                    SCREEN W/0.01" OPENINGS

                                                    BOTTOM PLUG
                           NOT TO SCALE
Figure 11.5 Schematic of sparging well used at Site 8, Vance, AFB

Figure 11.6 illustrates typical construction of observation wells.  The primary purpose of these
wells was to monitor water levels during DVE and sparging application.  Figure 11.7 illustrates
construction details of vapor probe clusters used to monitor subsurface pressure and vacuum.
Stainless-steel, 6.35 mm diameter, 27 mm long wire screen probes obtained from the Geoprobe
Corporation were used to monitor pressure and vapor concentrations.  Probes were connected to
6.35 mm outside diameter copper tubing and Swagelok quick connects using compression
fittings. The shallowest probe in each cluster was connected to probes by 3.18 mm outside
diameter stainless-steel tubing.  Cole Partner thermocouples were placed adjacent to probes at
each depth to measure soil temperature.

       Figure 11.8 illustrates construction details of observation well OB-3 modified for
                                             171

-------
                         4"x2" REDUCER BUSHING
                         W/2" THREADED PLUG-
                     HOSE BARB
                     CONNECTION W/
                     BALL VALVE-
                  S'' SQ. STEEL
                  PROTECTIVE CASING
                  W/HINGED  LOCKABLE
                  TOP
4" DIA. PVC RISER
W/MALE THREADS
4" DIA. STAINLESS STEEL
RISER W/FEMALE THREADS

     EXISTING CONCRETE
     PAVEMENT
                                                   CEMENT/
                                                   BENTONITE GROUT
                                                   BENTONITE SEAL

                                                   4" DIA. SS RISER

                                                   4" DIA. SS WIRE-WRAPPED SCREEN

                                                   SAND PACK
                                                   BOTTOM PLUG
                    NOT TO SCALE
Figure 11.6  Schematic of an observation well used at Site 8,Vance, AFB

combined air injection and extraction (CIE). This observation well was modified in the field
when it became apparent that DVE wells were producing water at rate (6 liters per minute) much
higher than that encountered during well development.  It appeared that vacuum from DVE wells
was causing water-table upwelling while water extraction during well development consisted of
perched water. Depth to the true water table however was never identified, thus this observation
is speculative.  The purpose of the modification was to figure out a way to extract gas at the
targeted depth while minimizing ground-water extraction. Modification consisted of insertion of
a 2.54 cm schedule 40 PVC pipe into the 10.16 cmoutside diameter observation well OB-3.
Numerous small-diameter holes were manually drilled over a length of 60 cm at the base of the
                                             172

-------
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                            Modification of OB - 3
                Compressor
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   Threaded bushing

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Figure 11.8 Schematic of modified observation well 3 to enable combined air injection and
extraction at Site 8, Vance, AFB

11.2   Results and Discussion

       Water recovery rates were estimated by periodic purging (twice per day) of an air/water
separator. During the four week test period, over 128,700 liters of contaminated water were
produced by the two DVE wells. Bleed air was needed continuously in both wells to maintain air
extraction.  Figure 11.10 illustrates water recovery rates from operation of dual vacuum
extraction alone, dual vacuum extraction in conjunction with sparging, and operation of
combined air injection and extraction (CIE) in OB-3. Sparging in conjunction with DVE resulted
in a measurable reduction in water recovery compared to the operation of DVE alone.  Combined
air injection and extraction resulted in a further reduction in water recovery.  Compared to DVE
operation alone, water recovery was reduced over 90% with the CIE well.
                                          174

-------
Figure 11.9.  Photograph of modified observation well, OB-3

       The reduction in water recovery from the CTE well could be caused by two mechanisms:
(1) elimination of water table upwelling, and (2) reduction in vacuum enhanced recovery of
perched water. While the location of the true water-table was never discerned, previous drilling
activities suggest its presence in the underlying weathered shale and sandstone. The application
of positive pressure at the base of the CIE well would eliminate a water potential gradient for
water-table upwelling.  In fact, water recovery during combined air injection and extraction was
similar (0.4 liters per minute) to water recovery  in observation wells during well development
prior to DVE operation (i.e., no applied vacuum). The design of the CIE well however would
also result in less vacuum within the lower annulus of the well and thus reduce water recovery
from perched water or soil-water movement. Similar water recovery rates prior to DVE
operation and CIE operation appear to indicate that water-table upwelling was either the major or
a contributing causative factor in high water recovery.
                                           175

-------
             8.0

             7.0 -

             6.0 -

             5.0-
          
-------
       within and near a gas extraction well which often reduces or completely obstructs gas
       flow. Gas injection on the hand, results in soil-water movement away from and hence
       increased gas permeability in the vicinity of screened interval. Since, the application of
       vacuum causes water-table upwelling, soil-water flow, and decreased gas permeability in
       the immediate vicinity of a gas extraction well, it is logical that injection of pressurized
       gas or at least maintenance of atmospheric pressure at the base of a vacuum extraction
       well would reduce water recovery.

       The concept of combined air injection and extraction (CIE) was evaluated during a pilot
       study at Vance Air Force Base located in northwestern Oklahoma.  Injection of air at the
       base of a gas extraction well reduced water recovery by 90% compared to dual vapor
       extraction. Vacuum measurements in soil from combined air extraction and injection
       were similar for all but the deepest probes.  Vacuum measurement in shallow probes
       indicated that combined air injection and extraction did not result in uncontrolled air flow
       toward the surface.

       Additional research is necessary to improve the design of the CIE well and assess it
       applicability at other sites.
     fc
     ta
     o
     >
CP- CP- CP- CP- CP- CP-
1-1 1-2 1-3 1-4 2-1 2-2
200 -i
100 -
-100 -

-200 -


-300 -

/inn


n







CP-
2-3



•

D VE-2 (Pw = 0.55 atm, Qm = 0.62 g/s)



• VE-1 (Pw = 0.60 atm, Qm = 3.05 g/s) and
VE-2 (Pw = 0.56 atm, Qm = 0.45 g/s)



• OB-3 (Pw = 0.57 atm, Qm = 3.40 g/s and
Pw= 1.56 atm, Qm =
= 4.58 g/s)

^iil







CP-
2-4
















-











CP-
3-1











CP-
3-2











CP- CP-
3-3 3-4


I






















Figurell.ll  Pressure differential measurements in vapor probes from DVE well and the
combined air injection/extraction well at OB-3.
                                           177

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12.    MONITORING STRATEGIES TO ASSESS CONCENTRATION AND MASS
       REDUCTION

       During venting operation, VOC monitoring using a portable photoionization detector
(PID) or flame ionization detector (FID) at the vapor treatment or blower inlet, gas extraction
wells, and vapor probes invariably becomes the primary method of assessing venting
performance. Vapor analysis using gas chromatography (GC) or gas chromatography/mass
spectroscopy (GC/MS) is commonly viewed as being too expensive. Other methods of
performance assessment such as collection and analysis of soil samples are often viewed as too
disruptive and expensive. As will be discussed, monitoring the vapor treatment inlet, gas
extraction wells, and vapor probes with a portable PID or FID provides useful information to
assess the performance of a venting  system. However, monitoring of evaporation-condensation
fronts of individual components can provide valuable information in assessing venting
performance in NAPL contaminated soils. This type of analysis requires GC or GC/MS analysis
of vapor at individual gas extraction wells and vapor probe clusters between wells. Because of
rate limited vapor transport effects, there is considerable uncertainty as to what decreasing vapor
concentrations in extraction wells and vapor probes actually represent. Vapor concentration
asymptotes in gas extraction wells may or may not be indicative of rate-limited vapor transport.
Flow variation testing may reveal the presence of rate-limited vapor transport in NAPL and non-
NAPL contaminated soils.  Flow interruption (rebound) testing to evaluate rate-limited vapor
transport however may only be applicable in non-NAPL contaminated soils.  Thus, vapor
concentration monitoring to assess venting performance is not straightforward. There are many
complications, seldom appreciated, which lead to a false sense of accomplishment in venting
remediation. Ultimately, assessment of concentration and mass reduction may necessitate soil
sample collection and analysis.

12.1   VOC Monitoring at the Vapor Treatment Inlet and Individual Wellheads Using a Portable
       PID or FID

       Flow and vapor concentration monitoring at the vapor treatment or blower inlet provides
a direct estimate of the VOC mass removal rate and total VOC mass removed as a function of
time for the entire venting system. Monitoring of carbon dioxide concentration has sometimes
been used to estimate the mass removal rate and total mass removed as a function  of time by
biodegradation. Flow, CO2, and vapor monitoring at individual wellheads has  been used to
estimate mass removal by volatilization and biodegradation at specific locations throughout a site
in the same way as that used at the vapor treatment or blower inlet.  Flow and vapor
concentration monitoring at individual wellheads can be used to determine the  most heavily
contaminated portions of a site requiring perhaps the most intensive monitoring and to optimize
vapor treatment. For instance, if a vapor treatment system has insufficient capacity to treat gas
flow from all operating wells, flow and concentration information form each well  can be used to
initially operate wells yielding the greatest mass flux. VOC monitoring at the wellhead enables
prioritization of venting operation at high contaminated areas.  If catalytic oxidation is used to
treat gas effluent, a vapor stream too rich in VOCs may cause overheating of the catalyst while a
                                          178

-------
vapor stream too lean in VOCs may require addition of supplemental fuel. Selectively operating
gas extraction wells may enable minimization of vapor treatment costs. Flow and vapor
monitoring at individual wellheads then allows assessment of mass flux from various regions of
a site and enhances flexibility in vapor treatment operation.

       VOC monitoring at the vapor treatment inlet and gas extraction wellhead is typically
performed using a PID or FID. PIDs utilize photons from ultraviolet (UV) light to remove the
outermost electron from many organic and some inorganic compounds. Ions formed are
collected on a charged plate producing a current proportional to compound concentration.  The
ion current is then amplified and displayed on the meter as parts per million volume (ppmv).
Electrons subsequently return to positively charged ions reforming the original vapor allowing
non-destructive VOC detection. The energy required for ionization is measured in electron volts
(eV). Ionization occurs when the energy supplied by a UV lamp exceeds a compound's
ionization potential (IP).  UV lamps are commercially available in 8.3, 9.5, 10.6 and 11.7 eV,
with the 10.6 eV lamp being the most common.  The 11.7 eV lamp extends the range of
compounds detected with a PID but is seldom used because it is relatively expensive, requires
constant maintenance and frequent replacement.  The  11.7 eV lamp has a window made of
lithium fluoride which is degraded by UV light, is very hygroscopic, and readily absorbs water
from air even when not in use. This causes the window to swell and decreases the amount of
light transmitted through the window.  PIDs are very sensitive to aromatic hydrocarbons (e.g.,
benzene, toluene, ethylbenzene, xylene isomers) and chlorinated compounds with carbon-carbon
double bonds (e.g., trichloroethylene, perchoroethylene).

       FIDs use ionization as the method of detection much the same as PIDs except that
ionization is accomplished by the use of a hydrogen flame. A pump draws sample gas from a
hand-held probe and passes it to a mixing chamber which is fed with pure hydrogen gas from a
small cylinder through a regulator.  The mixture is ignited, producing organic ions and electrons
which generate a current through the gas.  FIDs respond to organic compounds that burn or
ionize in the presence of a hydrogen and air flame (i.e., compounds containing large numbers of
carbon-hydrogen bonds). The magnitude of ionization is proportional to the number of carbon
atoms within the sample. FID like PID response though varies among various compounds
requiring the use of correction factors to estimate a specific vapor concentration from a
calibration gas concentration.  FID response is a function of carbon content, arrangement of
carbon atoms in a molecule, and the presence and arrangement of non-carbon functional groups.
FIDs are very sensitive to petroleum hydrocarbons (especially methane), less sensitive to
compounds having carbonyl, alcohol, halogen, and amine functional groups, and relatively
insensitive to heavily halogenated compounds such as carbon tetrachloride.

       Table 12.1 contains the maximum concentration (determined with GC/MS analysis) and
associated IPs of VOCs detected during pilot scale testing at the Picillo Farm Superfund Site. It
is apparent that use of a 10.6 eV lamp would result in  detection of only 2  of the 6 target VOCs
occurring in the highest concentration. However, FID response to the other 4 compounds would
also be expected to be poor given the heavily halogenated nature of these compounds.  Thus,
                                          179

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Table 12.1. Maximum concentration and associated IPs of VOCs detected during pilot scale
testing at the Picillo Farm Site.
Compound Highest vapor phase
cone, (ppmv)
toluene
1,1,1 -trichloroethane
chloroform
methylene chloride
hexane
trichlorofluoromethane (Freon 11)
trichloroethene
total xylenes
methylpentanes
benzene
ethylbenzene
methylcyclopentanes
methylhexanes
tetrachloroethene
methylcyclohexanes
cyclohexane
heptane
methylheptanes
l,2,2-trichloro-l,2,2-trifluoroethane (Freon 113)
1 ,2-dichloroethane
1,1-dichloroethane
dimethylcyclohexanes
pentane
1,2-dichloroethene (total)
dichlorofluoro methane (Freon 21)
styrene
dimethylcyclopentanes
2-butanone (MEK)
trimethylbenzenes (total)
acetone
chlorobenzene
1,1-dichloroethene
1,3-dichloropropene (total)
4-ethyltoluene
chloroethane
bromomethane
vinyl chloride
carbon tetrachloride
bromodichloromethane
1 , 1 ,2-trichloroethane
chloromethane
dibromochloromethane
1 ,2-dichloropropane
1 , 1 ,2,2-tetrachloroethane
bromoform
1 ,2-dichlorobenzene
3500
2500
1100
911
810
800
730
720
340
320
317
280
180
173
170
160
150
145
140
89
80
75
75
74
57
53
30
17
13
13
12
12
10
8
6
6
6
5
5
5
5
5
5
4
4
3
IP*
(eV)
8.82
11.00
11.37
11.32
10.13
11.77
9.47
8.44 - 8.56
10.06 - 10.08
9.25
8.77
9.85
?
9.32
9.64
9.86
9.92
9.84
11.99
11.04
11.06
9.41-9.98
10.35
9.66
12.39
8.43
9.92-9.95
9.51
8.27 - 8.48
9.71
9.06
9.79
?
7
10.97
10.54
9.99
11.47
10.6
11.00
8.90
10.59
10.87
11.10
10.48
9.06
* CRC Handbook of Chemistry and Physics, 73rd Edition, D.R. Lide (Ed.), CRC Press (1993).

                                         180

-------
screening with a portable PID or FID may result in underestimation of total VOC concentration
and mass removal at this site.

       There are other considerations in PID and FID use. Robbins et al. (1990a) found
decreased response with decreased flow rate for both a tested FID and PID, although the PID was
particularly sensitive to decreased flow rate. PID response was found to be sensitive even to the
presence of a slight vacuum in sampling tedlar bags. This problem was overcome by placing a
weight on the bag during measurement.  The weight induced a slight positive pressure in the bag
and resulted in increased consistency in PID readings. Robbins et al. (1990a) state that response
as a function of flow rate would introduce non-linearity in the correlation between measured and
actual concentration.  PID manufacturers state that response is essentially independent of gas
flow rate as long as it is sufficient to satisfy the pump demand. If a gas bag or sample jar is used
having sufficient volume, the instrument will draw gas at its normal flow rate.

       The range of linear response is generally greater for FIDs compared to PIDs. For instance
through the use of a serial dilution technique, Robbins et al. (1990a) demonstrated a linear
response to benzene, methane, isobutylene, and vapors from unleaded gasoline at concentrations
up to 1000 ppmv using an FID. Comparison of FID and PID response to benzene at
concentrations up to 1000 ppmv revealed nonlinearity (underestimation of concentration) of the
PID response above 125 ppmv. Comparison of FID and PID response to gasoline vapors also
revealed nonlinearity of the PID response above 125 ppmv. These observations indicate that
PIDs may underestimate vapor concentration when  elevated levels of contamination are present.

       In comparison to FID response, PID response can be significantly quenched by high
moisture, methane or alkane, and CO2 content. Robbins et al. (1990a) evaluated the effect of
relative humidity (0 to 90%) on FID and PID response to benzene. The FID showed no
discernable effect while the PID response markedly decreased at relative  humidity as low as
10%. Since soil gas exits a gas extraction well at or nearlOO% relative humidity, PIDs likely
systematically underestimate vapor concentration. Methane and alkanes  also have a quenching
effect on PID response.  Robbins et al. (1990a) observed a decreased PID response to isobutylene
in the presence of elevated levels of methane (2000 - 12000 ppmv) and butane (500 - 3000
ppmv).  Methane has an IP of 12.51 eV and thus can not be detected with a PID. FIDs though
respond strongly to methane. Methane if present in significant concentrations  in effluent gas,
would cause a higher FID compared to PID response.  Robbins et al. (1990a) also evaluated FID
and PID response to benzene in the presence of carbon dioxide concentrations  varying from 0 to
30%. PID response decreased with increasing carbon dioxide concentration while FID response
increased slightly as carbon dioxide concentrations rose. Neither FIDs or PIDs ionize carbon
dioxide, thus both responses are due to matrix or gas composition effects.

       Robbins et al. (1990a) demonstrated that when a variety of gas compositional  conditions
cause a decreased PID or FID response, the overall effect is not additive but instead equal to the
product of each response factor.  Since PIDs are much more sensitive to gas compositional
effects compared to FIDs, high humidity, methane, alkane, and carbon dioxide concentrations
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could cause PID response to be a small fraction of expected response based on calibration of a
dry, clean calibration gas. Also, since response factors due to gas compositional effects are
generally non-linear and gas composition varies from sample to sample, estimation of actual total
vapor concentration (to a calibrated hydrocarbon) from measured values becomes intractable
unless serial dilution as explained by Robbins et al. (1990a,b) is performed. Serial dilution
allows evaluation of gas composition on instrument response, calculation of response factors, and
extrapolation to initial gas sample concentrations.  Robbins et al. (1990a) demonstrated that
without serial dilution, PID readings are unreliable and generally underestimate total vapor
concentration.

12.2   Observation of Evaporation/Condensation Fronts

       Observation of evaporation fronts in individual extraction well effluent gas and
evaporation/condensation fronts in well-placed vapor probe clusters could potentially provide
valuable information on the progress of venting remediation of NAPL contaminated soils.
Removal of NAPL under equilibrium conditions in single component laboratory-scale through-
flow (a term commonly used to denote absence of bypass-flow or where gas is forced directly
through contaminated soils) experiments is characterized by a sharp evaporation front or very
rapid decline in effluent concentration. For instance, during sand tank studies in a moist Borden
sand, McClellan and Gillham (1992) observed a sharp decline in TCE vapor concentration (about
1.3 orders of magnitude) after an extensive period of essentially constant concentration.  They
state that this abrupt decrease in vapor concentration corresponded to the time when the last
liquid or residual TCE was removed and the kinetics of aqueous and sorbed phase mass removal
became dominant.

       Abrupt decreases in effluent vapor concentration has also been observed in multi-
component NAPL contaminated soils but is complicated by changing mole-fractions of
individual components throughout the soil column during gas through-flow. Rathfelder et al.
(1991) conducted through-flow column studies in dry glass beads and dry Borden sand with a
multi-component NAPL mixture of benzene, toluene, and TCE.  Using a local equilibrium
model, they successfully simulated sharp evaporation fronts eluting in sequential order of
decreased vapor pressure (benzene, TCE, toluene respectively) as expected from Raoult's law.
Ho et al. (1994) conducted through-flow column studies in a dry sand with a multi-component
NAPL mixture of benzene, toluene, and o-xylene.  Similar to Rathfelder et al. (1991), they
successfully simulated successive evaporation fronts in the gas effluent using a local equilibrium
model.  Ho et al. (1994) found that individual components of a NAPL propagate with separate
evaporation fronts at speeds proportional to their vapor pressure. This is consistent with
mathematical analysis of Zaidel  and Zazovsky (1999) who found that the velocity and sequence
of evaporation-condensation fronts can be predicted by a component's vapor pressure even at
low NAPL  saturations.  Hayden et al. (1994) found mass removal was controlled by the presence
of NAPL and component vapor pressures even at high organic carbon contents (1.65%) At very
low NAPL  saturations however, gas-water and solids-water partitioning must be taken into
account in addition to NAPL-gas partitioning.
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       Ho et al. (1994) observed and successfully simulated evaporation-condensation fronts
along the length of a soil column. Condensation of less volatile components, as measured by
total concentration analysis, occurred downstream of evaporation fronts. For instance, total
concentrations of toluene and o-xylene increased significantly at the location of the benzene
evaporation front. After evaporation of benzene, concentrations of o-xylene concentrations
increased significantly at the toluene evaporation front.  As clean gas sweeps through the
upstream portion of NAPL contaminated soil, more volatile components are preferentially
volatilized thereby reducing the mole fractions of more volatile compounds and increasing the
mole fractions of less volatile compounds. This increases vapor concentrations of less volatile
compounds which are transported downstream to areas where their mole fractions are lower,
causing condensation and increased concentrations above initial levels. Upstream evaporation
and downstream condensation occur through a soil column until all but the last component is
removed. They observed that a sharp decrease in effluent concentration occurs when a
compound's evaporation front has reached the exit and near complete removal from the soil
column has occurred. Ho et al. (1994)  also observed that effluent vapor concentrations remain
fairly constant prior to the first evaporation front reaching the exit.  Based on this observation,
they state that mole fractions of individual compounds in a NAPL mixture can be calculated from
knowledge  of a compound's vapor pressure and initial concentration.

       Ostendorf et al.  (1997) carried out a venting experiment on an intact core taken from a
site having  soils contaminated with weathered fuel.  Along the axis of flow, the vapor
concentration of lighter components decreased monotonically with time compared to heavier
components which increased for some time and then decreased.  Ng et al. (1999) successfully
simulated Ostendorf et al's. (1997) data using a vapor transport model coupling macroscale
advective-dispersive vapor transport with microscale aggregate diffusion. In their model, the
macroscale region consisted of mobile gas and free NAPL while aggregates consisted of water,
gas, and trapped NAPL. All transport to and from microscale and macroscale regions was
assumed to be controlled by aqueous diffusion.

       Ng et al. (1999) then conducted hypothetical simulations of propagation of evaporation
fronts and downstream  condensation in aggregated and nonaggregated soils contaminated by a
NAPL mixture of benzene, toluene, and o-xylene in  a radial flow system. For aggregated soils,
the o-xylene vapor concentration exhibited a downstream increase with time approaching its
saturated vapor concentration similar to nonaggregated soils.  However, in contrast to
nonaggreated soils, a sharp evaporation front was not observed.  Vapor concentrations of o-
xylene decayed slowly and exhibited extensive tailing due to  slow release from aggregates. Ng et
al. (1999) observed that in aggregated media, mass removal of compounds having lower
volatility is prolonged a disproportionate period of time compared to compounds having higher
volatility. In simulations with benzene, toluene, and o-xylene, Ng et al. (1999)  observed that
during condensation, the feeding of o-xylene into aggregates lasted substantially longer than
benzene and toluene.  While benzene and toluene were being removed, o-xylene continued to be
stored in aggregates.  This trend only stopped when the macropore o-xylene vapor concentration
began to drop or when free NAPL saturation vanished at a macroscale position. The o-xylene
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aqueous concentration inside aggregates then exhibited a discontinuous gradient.  In the outer
regions of aggregates, o-xylene diffused outward while in the inner region the o-xylene aqueous
concentration continued to rise in response to lower o-xylene concentrations in the interior.
Thus, in multi-component NAPL contaminated soils in a field setting, time for mass removal of
compounds having lower volatility or vapor pressure will be extensive.  Ng et al.'s (1999) radial
flow simulations also show that even under nonequilibrium conditions,  the time at which the
vapor concentration of the compound with the lowest vapor pressure drops abruptly is the time in
which the free NAPL is completely volatilized. Under nonequilibrium conditions, this abrupt
decrease in vapor concentration is accompanied by extensive tailing.

       The pattern of a sharp decline in effluent concentration at removal or evaporation of a
NAPL component observed in through-flow experiments though has not been reported during
field application in NAPL contaminated soils. This is most likely because of nonuniform gas
flow (bypass flow as opposed to through-flow conditions), fully three-dimensional flow as
opposed to radial flow as simulated by Ng et al., 1999), spatial variability of NAPL distribution,
common monitoring practices (determination of total VOC with a PID or FID as opposed to
identification and quantification of individual compounds), and rate-limited vapor transport.
Given the first two factors, even under equilibrium conditions, NAPL removal during field
operation is more likely to be characterized by a gradual rather than sharp reduction in total and
individual component concentration. Ho and Udell (1992) conducted a series of sand tank
experiments which demonstrate this point well.  They used three sorted dry sands to assess
deviation from equilibrium conditions in homogeneous and two-layer soil systems containing
one and two-component NAPL formulations.  During the first experiment, a single component
NAPL (toluene) was injected into a 0.25 mm mean diameter sand to achieve a residual saturation
of 12%. Effluent concentration remained relatively constant for the first 20 minutes but at only
60% of its saturated value.  Based on the cross-sectional area of NAPL contaminated soils
coming in contact with gas flow, approximately 70% of the saturated concentration should have
been achieved.  The presence of NAPL though likely reduced gas permeability in contaminated
soils causing some bypass flow. Effluent concentrations  then rapidly declined over the next 40
minutes with residual NAPL visually disappearing (dyed red). A sharp  evaporation front though
was not observed.  Ho and Udell (1992) state that the gradual decrease in effluent concentration
could have been caused by reduced gas-NAPL contact area, decreased critical length  for
equilibration, or reduced mass transfer coefficients as the toluene evaporated. They argue
however that since the length of contaminated media required for equilibration was likely short,
by the time the NAPL contaminated area had shrunk to a size with violated equilibrium
conditions, much of the contaminant had already been removed.  Thus,  equilibrium conditions
within NAPL contaminated soils likely prevailed with decreasing concentrations due solely to
decreasing gas flow contact area. This experiment was repeated using a mixture of toluene and
o-xylene at similar mole fractions at a residual saturation of 17%.  Toluene effluent concentration
decreased in a manner similar to the single component experiment but as its vapor concentration
decreased, the vapor concentration of o-xylene increased as predicted by Raoult's law.  At
exhaustion of toluene the o-xylene concentration remained constant for a short time before
decreasing in a manner similar to toluene.  Discernable but not sharp evaporation fronts were
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observed.

       Ho and Udell (1992) then injected free-phase toluene into a lower permeability layer
overlain by a higher permeability layer (14:1 ratio). Pore-gas velocities (calculated using
formation provided by the authors) in the higher and lower permeability regions were 10.96 and
0.86 cm/s respectively. Slow recession of toluene residual occurred at the upgradient gas flow
end similar to the unlayered soil system indicating significant gas flow through the layer of lower
permeability. Thus, mass removal in the lower permeability lense was still controlled by
advection.  In contrast to unlayered tests though, effluent concentrations showed a prolonged,
steady and somewhat linear decrease in effluent concentration over time.  Repeating this
experiment with a toluene - o-xylene mixture indicated a continued decrease in o-xylene
concentration throughout the experiment as opposed to an increase in o-xylene concentration as
toluene evaporated.  The o-xylene effluent concentration however did not decrease as rapidly as
the toluene concentration and eventually a cross-over point occurred where o-xylene
concentration became higher than toluene concentration. While evaporation fronts were not
discernable in this experiment, it was still apparent that preferential volatilization of toluene was
occurring.

       Finally, free-phase toluene and  a toluene and o-xylene mixture was injected into a lower
permeability layer overlain by a higher permeability layer with a 160:1 ratio. Pore-gas velocities
(calculated using information provided by the authors) in the higher and lower permeability
regions were 5.1 and 0.03 cm/s respectively. Comparison of dyed NAPL residual removal with
an experiment where the low permeability layer was overlain by air, indicated that mass removal
was controlled by diffusion. In contrast to all previous experiments, extensive effluent tailing
was observed.  While a cross-over point between toluene and o-xylene effluent concentration
occurred, the concentration profiles were very similar.

       Liang and Udell (1999) conducted a series of bypass-flow experiments with toluene and
o-xylene where gas flow in a glass column was allowed to occur above but not through NAPL
contaminated soils.  This would occur in the field for instance when air flow occurs around a silt
or sand lense. They demonstrated that  an evaporation front slowly penetrates into the bypassed
media and that a zone of variable liquid mole fractions grows in length ahead of the front because
of continual evaporation and diffusion to the region of advective flow. The mole fractions of
higher volatility compounds are lower than lower volatility compounds at the evaporation front
but the liquid mole fractions farther away from the evaporation front remain at initial values. As
a result, the measured mass ratio of compounds in effluent gas during bypass flow stays close to
the initial mass ratio of compounds in the effluent gas. This effluent pattern is very different
from increasing ratios of lower volatility compounds seen during advectively dominated flow.

       Studies by Rathfelder et al. (1991), Ho and Udell (1992), Ho et al. (1994), Liang and
Udell (1999), and Ng et al. (1999) make a strong case for monitoring individual NAPL
components as opposed to just monitoring total hydrocarbons in gas extraction well effluent and
vapor probes to assess the progress of remediation. At the  start of venting, component ratios are
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indicative of initial mole fraction ratios of residual NAPL providing valuable insight into the
composition of residual NAPL in contaminated soils. During the period in which mass removal
is dominated by advection, there could be a series of muted evaporation fronts in gas extraction
wells where compounds with high vapor pressures show steady decline and compounds with
lower vapor pressures show a temporary increase in vapor concentration prior to dropping in
concentration. Patterns of increasing and subsequent decreasing vapor concentrations of lower
volatility compound could be indicative of removal of these compounds from NAPL. A decrease
in compounds having the lowest vapor pressure could be indicative of evaporation of accessible
NAPL.  Monitoring of vapor concentrations in well placed vapor probe clusters could provide
evidence of the movement of evaporation-condensation fronts radially toward gas extraction
wells and again provide evidence of successful removal of NAPL during venting application.  In
a complex multi-component NAPL, a mixture of high, medium, and low vapor pressure
compounds could be targeted for monitoring in individual well  gas effluent and in vapor probe
clusters. As NAPL-gas exchange becomes  dominated by diffusion or bypass flow, component
concentrations in the effluent gas and vapor probes should return to ratios close to what was
initially observed  again reinforcing the need to estimate initial NAPL mole fractions.

12.3   Vapor Effluent Asymptotes

       Often, cessation of venting application is proposed based on attainment of an asymptote,
a concentration level during asymptotic  conditions, or some subjectively determined mass
removal rate (e.g., pounds per day) from gas extraction wells. However, there is no way to
directly relate observation of an asymptote to  environmental benefit and little potential to ensure
consistency in decision making.  Vapor  concentration of an asymptote at a highly contaminated
site may be much  higher than at a lesser contaminated site leading perhaps to less stringent
remediation at the highly contaminated site. Observation of an  effluent asymptote may be related
to venting design  (e.g., well spacing) or  operating conditions (e.g. flow rate) separate or in
addition to rate-limited vapor transport.  Rathfelder et al. (1991) demonstrated that effluent
tailing in NAPL contaminated soil can be caused by flow dynamics. Hypothetical simulations in
a homogeneous, one-dimensional, confined radial flow system with a NAPL and water saturation
of 1% and 15% respectively showed that under equilibrium conditions, the bulk  of contaminant
volatilization occurred along a clean retreating radial evaporation front as air flow is forced
directly through contaminated soil.  During two-dimensional axisymmetric flow however, direct
recharge from the atmosphere and subsequent vertical flow results in clean retreating evaporation
fronts both vertically above and radially distant from a gas extraction well.  This caused the
formation of a wedge-shaped contamination zone whereby some continually decreasing fraction
of gas flow intersected contaminated soil leading to decreasing  effluent concentrations that could
be construed as tailing.  Shan et al. (1992) demonstrated that in a domain open to the atmosphere,
most gas flow to an extraction well originates from atmospheric recharge in the immediate
vicinity of the well. Thus, after some period of vacuum operation, vapor concentrations from gas
extraction wells become largely reflective of soil and contaminant conditions in the immediate
vicinity of the wells while higher concentrations more distant from wells are largely diluted.
Effluents in gas extraction wells then approach low concentrations and an asymptote while
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significant and accessible mass may remain in soils between gas extraction wells. This situation
is exacerbated by placing venting wells too far apart - a condition typical of radius of influence
based designs discussed in sectionlO.

       A similar situation occurs in stratified soils where gas flow occurs primarily in layers of
greater gas permeability.  Analogous to the first situation, mass removal occurs more rapidly in
soils receiving greater gas flow and vapor concentrations in gas extraction wells become largely
reflective of soil and contaminant conditions in these layers while higher vapor concentrations
from lenses of lower permeability are largely diluted. Rathfelder et al. (1991) simulated effluent
tailing when a gas extraction well was placed across two stratigraphic layers having an order of
magnitude difference in permeability. In this case as before, effluent tailing and apparent rate-
limited mass removal was caused by variation in pore-gas velocity, not by rate-limited NAPL-gas
exchange. Lowered local gas permeability in zones containing pure phase residual would cause
the same effect.  Other examples of design factors potentially causing the observation of an
effluent asymptote include inadequate lowering of the water table, inadequate dewatering of
perched water table zones, and gas extraction wells not placed in or near source areas.

       Changing mole fractions of individual components in NAPL may also cause the
appearance of an asymptote in vapor effluent.  Baehr et al. (1989) simulated mass removal of
gasoline in medium-grained sieved sand. Experimental and simulated mass flux profiles
matched well suggesting attainment of equilibrium conditions.  The decrease in total mass flux
with time and exhibited apparent tailing due to selective removal of more volatile hydrocarbons.
This behavior is in contrast with column and sand tank studies using single component residually
contaminated soils which show no reduction in mass flux until residual NAPL has evaporated.
In a field setting, it would be easy to mistake tailing due to selective volatilization of a multi-
component NAPL for rate-limited behavior.

       Since both design considerations and rate-limited vapor transport can cause extensive
effluent tailing and ineffectual removal of accessible contaminant mass, it is critical to first
rigorously assess venting design, implementation, and  associated monitoring prior to concluding
that rate-limited vapor transport significantly constrains mass removal. If gas flow and design
considerations can be eliminated as a causative factors in the observation of asymptotes, then the
possibility of rate-limited mass exchange must be considered. Controlled laboratory experiments
(Armstrong et al., 1994; Conklin et al., 1995; Fisher et al., 1996, Gierke et al.,  1992; Grathwohl
et al., 1990; McClellan and Gillham, 1990) have shown that very long effluent tailing can occur
as a result of rate-limited  mass exchange indicating that soils venting may be inefficient in
achieving complete removal of VOCs. Several groups of investigators (Fisher et al., 1996; Ng
and Mei, 1996;  and Armstrong et al.,  1994) have clearly demonstrated in laboratory columns,
sand tank studies, and mathematical analysis that an asymptote can be observed in gas extraction
wells while much of the contaminant mass remains in soils and that observation of low
asymptotic vapor concentrations in effluent gas is not a sufficient condition to demonstrate
progress in mass removal from contaminated soils.
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       Ng and Mei (1996) demonstrated the inadequacy of assessing the effectiveness of a SVE
system using only effluent vapor monitoring.  They simulated the time variation in effluent vapor
and aqueous aggregate concentration. Effluent vapor concentration dropped rapidly with time to
the point of having indistinguishable values however the decay of the aqueous phase was
significantly slower.  Since sorptive equilibrium exchange was assumed within the aggregates,
the aqueous phase was indicative of mass removal within aggregates. Thus, effluent vapor phase
monitoring provided little insight on the rate of concentration reduction in soil.

       Armstrong et al. (1994) developed a numerical one-dimensional model which
incorporated first-order gas-water and solids-water rate-limited mass exchange to simulate vapor
effluent data obtained by McClellan and Gillham (1992).  Virtually identical fits were obtained to
effluent curves for both continuous and pulsed schemes using two parameter sets: 1) Kd and X^
values of 2.1 cm3/g and 2 x 10"5 s"1 respectively (equilibrium sorption) and 2) Kd, Agw, and Kws
values of 10.1 cm3/g, 4 x 10"5 s"1, and 5 x 10"5 s"1 respectively.  Simulated remaining mass
however differed sharply with the second parameter set resulting in far greater mass retention.
Their sensitivity analysis and calibration of data showed that different mass retention and rate
coefficients can produce very similar effluent curves suggesting that effluent and vapor probe
concentration data can not be used to estimate mass transport coefficients and VOC
concentration in the aqueous and solids phase.  Thus, effluent and vapor probe concentration data
is insufficient to quantitatively assess the progress of soil venting remediation. When coupled
with uncertainty in initial mass distribution, it appears that effluent and vapor probe
concentration data can only be used in a qualitative sense to assess performance.

12.4   Flow Variation and Interruption Testing

       Results of controlled column and sand tank studies reveal that some indicators of rate-
limited transport may be decreased vapor concentration with increased pore-gas velocity (flow
variation) and extensive vapor concentration tailing accompanied by vapor rebound after
cessation of operation (flow interruption).  In lab oratory-scale gas flow or miscible displacement
studies, flow variation and interruption is commonly used to determine the validity of the local
equilibrium assumption. If no change vapor or solute effluent concentration occurs during an
increase, decrease, or start-up upon periodic stoppage of flow, then it is assumed that the local
equilibrium assumption holds.  In both NAPL and non-NAPL contaminated soils, flow variation
could provide a simple yet powerful test of the validity of local equilibrium. For NAPL
contaminated soils however, results of flow interruption testing could be confusing if the
propagation of evaporation-condensation fronts are not considered.  Hayden et al. (1994) provide
a good example of this. They introduced gasoline at NAPL saturation value of 2% in two sandy
soils (less than 5% silt and clay) having organic carbon content of 0.1% and 1.65% and water
saturation values of 8% and 43% respectively and monitored effluent concentrations of benzene,
m,p-xylene, and naphthalene.  For the high organic content and water saturation soil, flow
variation resulted in pore-gas velocities (calculated from information provided) being varied
between 0.076 and 0.020 cm/s while for the low organic carbon content and water saturation soil,
flow variation resulted in calculated pore-gas velocities (calculated from  information provided)
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being varied between 0.057 and 0.015 cm/s. Flow variation had no effect benzene, m,p-xylene,
and naphthalene effluent concentrations during initial period of venting indicating the validity of
the local equilibrium assumption.  However after a prolonged period of venting, benzene effluent
concentration decreased abruptly (indicating elution of the benzene evaporation front) and
variation in flow rate caused a decrease in benzene concentration with an increase in flow rate
and an increase in benzene with a decrease in flow rate.  During this time period m,p-xylene,
toluene, and naphthalene effluent concentration remained unaffected by flow variation. Thus, at
low soil or NAPL benzene concentrations, mass removal appeared to become rate-limited and
the method of flow variation worked well in discriminating between rate-limited and non-rate-
limited transport.  The results of flow interruption testing though were more difficult to discern.
Flow interruption testing during the initial period of venting application (1 hour) resulted in a
decrease in benzene concentration and an increase in m,p-xylene  and toluene effluent
concentration. Flow interruption testing  at 3 hours resulted in a decrease in benzene and toluene
effluent concentration and an increase in m,p-xylene. Reconciliation  of this behavior is possible
only if one considers evaporation-condensation fronts. During initial venting operation, high
vapor pressure compounds such as benzene would be removed first from the intake portion of the
column and increase in mole fraction and vapor concentration downstream as the evaporation
front moves through the column. Cessation of venting then would cause a decrease at the
effluent end of the soil column and an increase in the intake end of the column as vapor diffusion
becomes the dominant transport mechanism. Upon re-start, effluent concentrations would drop.
For compounds like m, p-xylene, the mole fraction and vapor concentration would initially
increase at the inlet of the soil column. Thus during flow interruption, vapor diffusion would
cause an increase in vapor concentration  downstream resulting in an increase in vapor
concentration upon start-up.  Thus, the increase in m, p-xylene upon start-up  was  likely not the
result of rate-limited mass transport, but propagation of evaporation and condensation fronts. In
a field setting, flow interruption testing would provide evidence of rate-limited  transport for a
particular  compound only after its  evaporation front exited the soil column.  This  might be
accomplished by sampling an array of vapor probes situated between  a relative  clean-gas
boundary and a extraction well. Otherwise, flow interruption testing could provide confounding
results.

12.5   Use of Flow Interruption Testing to Estimate  Soil Concentration

       One potential use of flow interruption or rebound data is to use vapor concentrations and
partition coefficients to estimate total soil concentration. Monitoring of VOC concentrations in
vapor probes for this purpose is preferable to vapor extraction wells because probes draw soil gas
from a much smaller volume of soil and thus are less likely to be affected by  vapor concentration
variation due to large scale heterogeneity.  Viewed superficially, the idea appears  to have merit
because soil sample collection can be expensive, especially when drilling at extensive depths and
in soils containing large rocks (e.g., glacial till).  Drilling can cause disruption of venting
operation and manufacturing processes at active operating facilities. Also, drilling often results
in the generation of a substantial amount of cuttings which must be disposed of as hazardous
waste. Other benefits of vapor measurement to estimate total soil concentration are that volatile
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organic compound (VOC) loss occurs during sample collection, storage, and analysis.  VOC loss
during sample collection can be significant for highly volatile compounds in soils with low
organic carbon and moisture content.  Vinyl  chloride is often not detected in soils samples even
when high concentrations are detected in soil gas. It can be argued that soil sample collection is
by nature "hit" or "miss" because sample volume is small compared to the volume of soil
affected by soil-gas collection.  Thus, if contaminant mass remains in soil, there is a higher
probability of detection with soil-gas sampling than soil-solids  sampling. Finally, soil sample
collection is obviously not appropriate for venting operation in  fractured rock or other
consolidated material.

       Several investigators have attempted to correlate soil-gas concentrations with total soil
concentrations. Stephens (1995) reported poor correlation between trichloroethylene (TCE)
concentrations in soil gas and soil matrix samples collected during drilling at approximately the
same depth. TCE was detected in 36 of 39 soil-gas samples but only 10 of 83 soil matrix
samples with a detection limit of 50 ug/kg. Stephens (1995) states that the more frequent
detection of TCE in soil-gas compared to the soil matrix is probably partially due to the much
lower detection limits achieved for soil gas compared to the soil matrix and the fact that the soil
matrix is far more heterogeneous with respect to contaminant distribution compared to soil-gas.
Soil matrix samples with detectable TCE, the concentrations reported by the analytical laboratory
were far below those that would be predicted based on equilibrium with measured TCE
concentrations in adjacent soil-gas making reasonable assumptions regarding soil moisture, bulk
density, and organic carbon content.  Stephens (1995) states that if vapor phase equilibrium
between soil-gas and soil matrix is assumed, then the discrepancy most likely reflects the
shortcomings of soil matrix analytical procedures which often result in reported VOC
concentrations lower than expected as reported by Siegrist and  Jenssen (1990). Hahne and
Thomsen (1991) and Kerfoot (1991) also report poor correlation between soil-gas and matrix
concentration.

       Significant uncertainty in estimation  of parameters making up a soil-gas partition
coefficient and difficulty in  measuring a steady-state vapor concentration in the field makes
estimation of total soil concentration from vapor measurements challenging. Under equilibrium
conditions, the relationship between a nonionic organic compound concentration in soil and soil
gas is a function of: (1) a gas-water partition coefficient (Henry's Law Constants), an organic-
carbon-water partition coefficient, and NAPL-water or NAPL-gas partition coefficients, (2) gas,
water, and NAPL fluid saturations, and (3) soil organic carbon  content, porosity, and particle
density. If separate phase NAPL is available from a site, batch  laboratory studies can used to
directly determine NAPL-gas or NAPL- water partition coefficients for compounds of concern.
If free NAPL is not available, NAPL-water or NAPL-gas partition coefficients can be estimated
from a compound of interest's vapor pressure and molecular weight and mole fractions, densities,
and molecular weights of individual compounds in the NAPL.  At many sites though, there are so
many compounds in NAPL  contaminated soil that accurate mole fraction identification and
quantification is impossible. An average NAPL  density and molecular weight can be used in
place of individual NAPL constituent information if the mole fractions of dominant  chemical
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classes in the NAPL can be quantified. However, equations used to estimate NAPL-water or
NAPL-air partition coefficients reveal that these partition coefficients will not be constant but
change as a function of changing mole fractions of various compounds. Since a compound's
organic-carbon-water partition coefficient, Henry's Law Constant, vapor pressure, molecular
weight can be found in the literature or estimated for many compounds, a soil-gas partition
coefficient has 7 primary unknown parameters - gas, water, and NAPL saturation, particle
density, porosity, average NAPL density, and average NAPL molecular weight. The parameters
describing NAPL are of greatest interest because analysis of mass fractions reveals that the
majority of contaminant mass is partitioned in the NAPL phase even at very low NAPL
saturation (e.g., 0.5%). Error in estimation of NAPL saturation would dramatically affect
estimates of soil concentration and contaminant mass.  There is presently no standard method of
estimating NAPL saturation in-situ in soil but partitioning tracer testing shows promise.  In-situ
water saturation would need to be  estimated with the use of a neutron probe or other downhole
device.  Gas saturation could be estimated from knowledge of porosity and volumetric water
content.  Each estimation procedure involves error, which from an analysis of variance,
propagates non-linearly. Finally, there is no way to  assess the accuracy of estimation of soil-
concentration from soil-gas measurements because the volume of soil being affected by
advective gas flow to a probe is likely greater that the volume of soil typically sampled for
laboratory analysis.  Collection of soil  samples  for soil-gas estimation comparison also
necessitates consideration of spatial variability which can be significant even on a small scale.

       Estimation of soil concentration also requires measurement of steady-state soil-gas
concentration. In a well designed venting system, mass removal will eventually limited by
combined liquid and vapor diffusion from less permeable soil regions not receiving direct
airflow.  In a poorly designed system, mass removal will be limited by these factors plus
inadequate air flow in regions still amenable to significant advective phase mass transport. In the
former case, at the cessation of venting, vapors diffuse slowly from immobile regions into mobile
domains where  advective gas flow occurs. When a  soil's moisture, organic carbon, or NAPL
content is high,  time for attainment of equilibrium conditions can be excessive.  Thus, when
collecting vapor samples for total soil concentration or mass estimation purposes, a correction
factor is necessary for estimation of steady-state vapor concentration. Assuming that
concentration is caused by vapor diffusion, estimation of steady-state vapor concentration will
involve the use  of some type of a diffusion-based model.  The diffusion coefficient will
incorporate the  7 unknown parameters of the soil-gas partition  coefficient plus three additional
unknown parameters - tortuosity in the gas and liquid phases and the diffusion path length.
Tortuosity values could be estimated from fluid saturation and  porosity. Literature values can be
found for molecular gas and water diffusion coefficients. Since there are likely to be numerous
localized sources of vapors  in contaminated soils, including contaminated ground water, the
diffusion path length will represent some averaged distance.  In practice, it will  be necessary to
fit a diffusion-based equation to vapor concentration versus time data during vapor rebound to
estimate  a lumped parameter incorporating the diffusion path length.  This will  require the
collection of several gas samples over time at each vapor sampling point followed by gas
chromatography analysis. Another practical problem is devising an appropriate methodology of
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vapor sampling.  The collection of vapor samples may cause local dis-equilibrium, thus later
vapor sampling results would be biased by earlier sampling efforts.

       It is apparent that defensible estimation of soil concentration from soil-gas measurement
is a difficult process. It requires a substantial amount of work, expense (GC analysis of vapor),
and thorough knowledge of subsurface conditions. Even under the best of conditions, substantial
estimation error can be expected.  While drilling and subsequent soil sampling is expensive and
disruptive, it may still offer the best opportunity to evaluate vertical subsurface contamination
profiles.  Soil-gas monitoring during venting operation and periods of shut-down however might
provide insight on the timing of soil sample collection.

12.6   Use of Gas Phase Partitioning Tracer Testing to Assess Residual NAPL Removal During
       Venting Application

       Partitioning tracer tests have been widely used since the 1970's in the petroleum industry
to estimate residual oil saturation and assess the effectiveness of enhanced oil recovery (Allison
et al., 1991;  Cooke,  1971; Deans,  1978, Ferreira et al., 1992; Lichtenberger, 1991;  Sheely, 1978;
Sheely and Baldwin, 1982; Tang,  1992;  Tang and Harker, 1991a,b; and Tomich et al., 1973).
Jin et al. (1995) state that single-well partitioning tracer tests have been conducted at over 300 oil
field reservoirs throughout the world in a wide variety of geologic settings. More recently,
aqueous-phase partitioning tracers have been used in ground-water (Annable et al., 1998;
Brusseau et al., 1999; Cain et al., 2000, Jin et al.,  1997; Nelson and Brusseau, 1996; Young et al.,
1999; Wilson and Mackay,  1995)  and vadose zone (Deeds et al., 1999; Mariner et al., 1999;
Whitley et al., 1999) systems. In the environmental field, partitioning tracer studies have been
used to: (1) locate subsurface NAPL pools, lenses, and residual, (2) estimate NAPL saturation,
and (3) provide a means of assessing the performance of NAPL removal during remediation.
Most recently, Nelson et al. (1999) used gas-phase partitioning tracers in the field to estimate in-
situ water content. Brusseau et al. (1997) had  previously examined the potential of using gas-
phase tracers for measuring soil-water content and gas-water interfacial area in column studies.

       The theoretical bases for the retention of dissolved or gaseous compounds by immobile,
immiscible liquid phases has been described by Jin et al. (1995) and Brusseau et al. (1997).
Basically, when tracers having different NAPL-fluid partition coefficients are injected into the
subsurface, nonpartitioning tracers remain in and travel with the velocity of the transporting fluid
phase while  partitioning tracers move back and forth (in response to concentration gradients)
between stationary NAPL and the transporting fluid. During gas-phase partitioning tracer tests in
the vadose zone, it is commonly assumed that  water and NAPL are immobile during the short
duration of the test.  The net result is that transport of partitioning tracers lags behind or is
retarded compared to nonpartitoning tracers. The magnitude of retardation is primarily
dependent upon NAPL-fluid partition coefficients and saturation of NAPL in subsurface systems.
For a given partitioning tracer,  the greater the chromatographic separation in breakthrough
curves, the greater the NAPL saturation.
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       The ratio of residence times or average travel times of partitioning and nonpartitioning
tracers can be expressed by
R = L                                                                          (12.1)
     *n

where R [-] is a retardation factor and t  [T] andtn [T] are travel times for partitioning and

nonpartitioning tracers respectively.  The retardation described in terms of fluid saturation can be
expressed by
                                                                                 (122)
        TjS     TjSH   SH    S
                                 g
Use of Raoult's and Henry's Laws allows the NAPL-gas partition coefficient (Kng) [-] to be
approximated in a number of ways


                                 ^-   "'"--.                             (1«)
                                v.p V
                                fi* v* n

           7=1    J

       Tracer travel times can be estimated by comparing specific points of respective
breakthrough curves (Cooke, 1971; Deans  1971; Tang 1992, 1995) or by fitting data to various
forms of the advective-dispersive equation. The most widely used and preferred method
however is computation of first temporal moments because this method is less sensitive to rate-
limited mass transfer and is not dependent  on a suite of assumptions necessary in modeling
analysis. Temporal moments are defined by Jin et al., 1995)


_    fcw*  ,.

and
_   (tCc(t)dt
     *
       Cc(t}dt
tc=-f --                                                                 (12.5)
where ts [T] is the tracer injection duration, tf [T] is the duration of the test, and Cp(t) and Cc(t)

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represent partitioning and conservative tracer concentrations respectively [ML"3] as a function of
time.  The moment method is susceptible to error when breakthrough curves are incomplete due
to premature termination of field test or when concentrations fall below detection limits. These
conditions occur when mass transfer constraints become extreme (substantial fraction of
recovered tracer mass in tail of response curve).  Application of high flow rates exacerbate
effluent tailing and thus  should be avoided. The response curve tail though can be extrapolated
to zero concentration by fitting a straight line to late time data on a semilog plot (Skopp, 1984
and Pope et al., 1994) to improve the accuracy of moment estimation. Deeds et al. (1999)
reported that in all but a  few of 72 gas-phase tracer studies conducted at a site, extrapolation had
only a minor effect on NAPL saturation estimation.  In most cases, greater error originated from
estimation of partition coefficients.

       Average NAPL saturation (Sn) can be calculated by (Deeds et al., 1999)
                                                                            (126)
t(Kng-l) + tp
      c
This equation differs from computation of Sn in saturated systems by inclusion of the factor (1-
Sw).  If water saturation is estimated using a gas-water partition coefficient, equation (12.16)
must be solved simultaneously with (Deeds et al., 1999)
      tw-tc)(l-Sn)
      = - '- - ^.                                                             (12.7)
The pore volume, Vp [L3] (8g +8W + 8n), swept by the tracers is given by (Deeds et al., 1999)
y
  p
where Qv is the flow rate of subsurface gas through NAPL.  Qv is not necessarily the volumetric
flow rate of an extraction/injection well pair because of gas recharge and discharge from and to
the atmosphere.  The total volume of NAPL detected, Vn is given by (Deeds et al., 1999)
                                                                                  (12.9)
                 »g
       Nonpartitioning and partitioning tracers should ideally: (1) be initially absent from or at a
very low concentration in subsurface media, (2) be readily available and relatively inexpensive,
(3) be non-toxic at concentrations employed, (4) be resistant to mass loss (e.g., degradation), and

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(5) have a low analytical detection limit.  A low detection limit enables tracer detection over a
scale of three to four magnitudes and injection at relatively low concentrations. Addition of
tracers at low concentrations ensures that the transport of tracers will not be affected by changes
in fluid density.  Perfluorcarbons generally meet all of these criteria and thus have been widely
used for gas-phase partitioning tracer studies. Perfluorcarbons however have been identified as
powerful reactants in global warming leading to restricted use. The mass injected for field-scale
testing however is insignificant compared to industrial use and it can be argued that testing yields
a net environmental benefit. Thus, concern over global warming should not be a limiting factor
for testing. Table (12.2) summarizes specific tracers used for gas-phase tracer studies.

       Partitioning tracers used for laboratory and field tests should have NAPL-gas partition
coefficients large enough so that NAPL-gas partitioning dominates tracer retention. This
minimizes consideration of competing retention processes such as interfacial gas-water
partitioning, gas-bulk water partitioning, and sorption. NAPL-gas partition coefficients should
also be large enough to ensure that breakthrough curves of partitioning tracers are clearly
distinguishable from nonpartitioning tracers but small enough such that the length of the test
remains practical.  For field-scale studies, Whitley et al. (1999) suggest that retardation factors
should be greater than 1.2 and less than 4. If an approximate range of NAPL saturations to be
encountered is known, partitioning tracers can be selected to provide retardation factors within a
desired range. Since a range of NAPL saturations is frequently unknown (indeed the purpose of
testing is often to determine NAPL saturation), it is useful to use multiple partitioning tracers
with different partition coefficients providing a range of potential retardation. The potential
presence of a very low NAPL saturation may necessitate the use of a gas tracer having  a fairly
high NAPL-gas partition coefficient. When multiple partitioning tracers are used, travel-time
analysis can be conducted using various pairs of partitioning tracers  as well as partitioning -
nonpartitioning tracers.  This approach can increase the robustness of the results.  Vapor
transport simulations can be very useful  in design of a partitioning tracer test (Deeds et al., 1999;
Nelson et al., 1999).  The tracer breakthrough concentrations must be in a detectable range
throughout the course of the test.  Simulations provide an estimate of the injected tracer mass
necessary to yield these  concentrations and an estimate of the  duration of the test. However,
despite pretest analytical or numerical simulation of breakthrough  curves, proper  selection of
gas-phase tracers may require actual testing and retesting.

       NAPL-gas partition coefficients can be directly determined by batch equilibrium
partitioning tests. However, NAPL composition changes significantly during venting
application resulting in spatial and temporal variation in NAPL-gas partition coefficients. VOCs
will be removed from NAPL in order of their vapor pressures eventually leaving higher
molecular weight lower volatility compounds as NAPL residual. Using phase equilibria theory,
Deeds et al. (1999) calculated a decrease in perfluorocarbon partition coefficients ranging from
15.9 to 22.0% resulting  from NAPL weathering.  They state that variation in tracer gas partition
coefficients can result in substantial error in NAPL saturation estimation. Thus, variation in gas-
phase tracer partition coefficients as a function of NAPL weathering or composition should be
assessed during batch gas-phase partitioning tracer studies.
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       Partitioning tracer tests typically consist of simultaneous injection of several tracers with
different NAPL-fluid partition coefficients at one or more injection wells and subsequent
measurement of tracer concentrations at one or more extraction or monitor wells.  A number of
well configurations could be used for testing.  One well configuration involves the use of a single
extraction well and a non-pumping or passive injection well where the passive well is used to
introduce the gaseous tracer solution into the flow field created by the extraction well which is
used for sampling. A second configuration involves use of a single pumping injection well and
one or more monitoring wells surrounding an injection well. A third configuration involves use
of a combination of pumping injection and extraction wells. If multi-level sampling devices are
used during testing, a three-dimensional distribution of NAPL saturation can be developed.
When partitioning tracer tests are used to assess the performance of remedial actions at NAPL-
contaminated sites, initial tracer tests are conducted to establish baseline conditions followed by
periodic testing throughout the time of active remediation.  By virtue of sampling  a much larger
volume of the subsurface in comparison to soil cores, the partitioning tracer method has a much
greater chance of detecting NAPL saturation.  The sensitivity of the partitioning tracer method
for NAPL detection is a function of the area of influence of the tracer test (swept volume),
amount of NAPL in the swept volume, the NAPL-gas partition coefficient of the tracer, and
constraining factors such as rate-limited mass transfer and physical heterogeneity.

       During testing, mass exchange between partitioning tracers and NAPL is assumed to be
instantaneous. However, this assumption is valid only when the time required for tracer
diffusion within NAPL regions is short comparted to advective transport time. When a
partitioning tracer initially comes in contact with NAPL, a concentration gradient  exists between
exterior and interior regions of the NAPL region. This causes the partitioning tracer to diffuse
into interior regions of the NAPL zone. When the tracer pulse is followed by tracer-free gas, the
concentration gradient reverses causing partitioning tracer movement back toward exterior
regions of the NAPL zone. If the time of the tracer pulse is insufficient to allow the tracer to
fully penetrate the NAPL zone prior to injection of clean gas, an inward concentration gradient
will still exist in the interior of the NAPL zone. This  could significantly delay recovery of a
partitioning tracer to a gas extraction well and give the impression of mass loss. Recovery of a
partitioning tracer gas could also be caused by failure to capture all injection well  flow lines (e.g.,
escape of tracer gas to the  atmosphere). However, analysis of mass recovery of conservative
tracers correct for this possibility.

       Measured residual  saturation values may underestimate true saturation values because of
bypass flow (gas flow around a NAPL zone due to reduced relative permeability),  rate-limited
mass transfer,  and mass loss.  Thus, residual NAPL saturation values obtained with partitioning
tracer tests are, at least initially, considered underestimates of actual values.  When interpreting
results of partitioning tracer tests, it is important to realize that residual saturation  values
represent "global"  or average values within a tested domain. A small localized mass of NAPL
could have a small impact on retardation as measured at an extraction well and thus the signal
from this mass could be lost in normal noise associated with field data. This effect though can be
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Table 12.2. Summary of tracers used for vadose zone partitioning tracer studies
 Tracer
Use
Investigators
 methane (CH4)                         conservative


 argon                                 conservative

 sulfur hexafluoride (SF6)                conservative

 trifluoromethane (CF3)                 NAPL-gas

 carbon tetrafluoride (CF4)               NAPL-gas

 difluoromethane (CF2H2)                gas-water/
                                       NAPL-gas

 bromochlorodifluoromethane            gas-water
 (CBrClF2)

 dibromodifluoromethane (CBr2F2)       gas-water

 hexafluoroethane (C2F6)                NAPL-gas

 octafluoropropane (C3F8)                NAPL-gas

 octafluorocyclobutane (C4F8)            NAPL-gas


 dodecafluorodimethylcyclobutane       NAPL-gas
 (C6F12)

 perfluoromethylcyclohexane (C7F14)     NAPL-gas

 perfluoro-l,3-dimethylcyclohexane      NAPL-gas
 (C8F16)

 perfluoro-l,3,5-trimethylcyclohexane    NAPL-gas
 (C9F18)

 perfluorodecalin (C10F18)                NAPL-gas
                Deeds et al. (1999), Mariner et al.
                (1999), Nelson etal. (1999)

                Whitleyetal. (1999)

                Mariner et al. (1999)

                Whitleyetal. (1999)

                Whitleyetal. (1999)

                Mariner etal. (1999)


                Nelson etal. (1999)


                Nelson etal. (1999)

                Whitleyetal. (1999)

                Whitleyetal. (1999)

                Mariner et al. (1999), Whitley et
                al. (1999)

                Mariner et al. (1999), Whitley et
                al. (1999)

                Deeds etal. (1999)

                Deeds et al. (1999), Mariner et al.
                (1999), Whitleyetal. (1999)

                Deeds et al. (1999), Mariner et al.
                (1999), Whitleyetal. (1999)

                Deeds etal. (1999)
minimized by using closely spaced wells where the swept volume is small. Another
complicating effect in tracer test evaluation is variable flow conditions. For instance Cain et al.
(2000) noted that variation in ground-water elevation and flow caused considerable variation in
tracer test results since different NAPL regions were accessed for different hydrodynamic
situations.
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12.7 Soil Sample Collection. Storage, and Analysis

       Effective June 13, 1997, EPA limited use of SW-846, Method 5030 to water samples and
methanol extracts of solids (e.g., soils, sediments, and solid waster) samples. EPA established
Method 5035 to provide an alternative to Method 5030 for low level (< 200 ug/kg) soil analysis
and to incorporate chemical preservation and sample storage techniques to limit volatilization
and biodegradation of organic compounds during sample storage and handling of both low and
high concentration (>200 ug/kg) samples. Method  5035 can be used in conjunction with any
appropriate gas chromatography (GC) procedure, including, but not limited to Methods 8015
(GC/FID), 8021 (GC/PID), and 8260.

       For low concentration samples, 1 gram of sodium bisulfate and 5 ml of organic-free
reagent water are added to glass sample 40-ml vials prior to sample collection to form an acid
solution (pH  < 2) thereby reducing or preventing biodegradation during sample storage.  Sodium
bisulfate should not be used however if a sample contains high levels of carbonates which may
result in sample effervescence and subsequent loss  of VOCs.  A clean magnetic stirring bar is
also added to each vial as a means of stirring the sample unless sonication or another mechanical
means is to be used. A sample (approximately 5 g) is then placed in a vial, sealed with a septum
lined screw-cap, and stored at 4°C until analysis. Several  techniques can be used to transfer a
sample to the relatively narrow opening of the low concentration sample vial. These include
devices such as the EnCore™ sampler, the Purge-and-Trap Soil Sampler™, or a cut plastic
syringe. At the laboratory, the entire vial is placed unopened into a closed-system purge-and-trap
system. Immediately before analysis, organic-free reagent water, surrogates, and internal
standards are added without opening the sample vial.  The vial containing the sample is then
heated to 40°C and the VOCs are purged using an inert gas into a trap.  When purging is
complete, the trap is heated and backflushed with helium to desorbed the trapped sample
components into a GC for analysis using the appropriate method, for analysis. Since the sample
is never exposed to the atmosphere after sampling,  losses  of VOCs during sample transport,
handling, and analysis is considered negligible.  Since samples can be contaminated by diffusion
of VOCs (particularly methylene chloride and fluorocarbons) through the septum seal of a
sample vial during shipment and storage, a trip blank containing organic-free reagent water
should be carried through sampling, handling, and storage protocols to serve as a check on
contamination.

       In Method 5035,  there is an option to preserve high concentration samples in the field
with an organic solvent.  In the procedure, 10-ml of the organic solvent is added to 40-ml vials
prior to sample collection.  If the sample is oil-free, a water-miscible solvent such as methanol or
polyethylene glycol  or (PEG) can be  used. If the sample contains oil, a water immiscible solvent
such as hexadecane  can be used if the oil is  insoluble in a  water miscible solvent (e.g., methanol,
PEG).  In the laboratory, an aliquot from the vial is introduced into the GC system.  If the vial
contains oily waste or soil, Method 5030 is used if the oil  is soluble in a water-miscible solvent
and Method 3585 is used if the oil is soluble in a water-immiscible solvent. If the solubility of
the  oil in water miscible or immiscible solvents  is unknown, the sample should be stored without
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the use of any solvents. If the oil is insoluble in methanol, PEG, and hexadecane, the use of other
solvents should be investigated. In any event, the sample should be stored at 4°C prior to
analysis.

       There are two problems however associated with use of methanol as a preservative
though. First, the use of methanol introduces a significant dilution factor that raise the method
quantitation limit. The exact dilution factor depends on the masses of solvent and sample, but
generally exceeds 1000. Thus, when low sample concentrations are suspected, duplicate
samples should be collected for use of the low concentration preservation technique in addition
to methanol preservation. The second problem is that addition of methanol is likely to cause the
sample to fail the ignitability characteristic test.

       Studies conducted by Liikala et al. (1996) and Siegrist and Jennsen (1990) have
demonstrated conclusively that volatilization  during soil sample collection, handling, and storage
imparts a significant negative bias on VOC analytical results and that this bias can be minimized
by sample preservation in methanol. Liikala et al. (1996) demonstrated that some VOC loss can
occur even in soil samples preserved with methanol. Liikala et al. (1996) spiked 10 ml of
methanol contained in 40-ml amber vials with bromochloromethane, chloroform, carbon
tetrachloride, m-dichlorobenzene, trichloroethylene, 1,2-dibromomethane, 1,1,2,2-
tetrachloroethane, and l,2-dibromo-3-chloropropane. A number of vials were kept refrigerated at
4°C in the laboratory for 82 days while other vials were transported on ice to a field site where
2.5 to 9.0 g of soil was added to vials.  These vials were then stored at 4°C for 82 days prior to
analysis.  Mean recoveries from vials containing methanol stored in the laboratory ranged from
80.7% for choroform to 88.0% for bromochloromethane.  Mean recoveries from vials containing
methanol transported to the field ranged from 74.0% for trichloroethylene to 80.3% for
bromochloromethane.

       Liikala et al. (1996) assessed the commonly used bulk storage method (minimal
headspace) with methanol preservation for 50 soil sample pairs at a site contaminated by
chlorinated and petroleum solvents.  For bulk storage, subsamples were collected using stainless
steel trowels and tightly packed into 4-oz wide mouth jars with Teflon-lined screw-top lids.  For
methanol preservation, 3.67 to 11.4 grams of soil were placed in 40-ml amber vials containing 10
ml of purge-and-trap grade methanol. All samples were  stored on ice.  For the bulk method
analysis,  15g aliquot of soil sample was placed into a purge vessel. Five ml of laboratory grade
organic free water was then transferred to the vessel, heated to 40°C, and purged with ultra-high-
purity helium. Tenax traps were then desorbed at 225°C for 4 minutes with the sample
transferred to a gas chromatograph. In the methanol method of analysis, a 2.5-ml aliquot was
removed from the 40-ml vial, placed into a 50-ml flask, and diluted to 50-ml with boiled Milli-Q
water. A second VOA vial was then filled to zero headspace and used for direct injection into a
purge-and-trap system similar to the bulk storage method. Analysis revealed that benzene
concentrations using methanol storage were 1-2 orders of magnitude greater than the bulk storage
method for 26 of 34 sample pairs. Toluene concentrations using methanol storage were 1-3
orders of magnitude greater than bulk storage for 29 of 43 sample pairs. Ethylbenzene
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concentrations using methanol storage were 1 order of magnitude greater than bulk storage for
only 7 of 40 sample pairs. Concentrations of m+p-xylene using methanol storage were 1-2
orders of magnitude greater than bulk storage for 18 of 41 samples. Finally, concentrations of o-
xylene using the methanol method were 1-2 orders of magnitude greater than bulk storage for
only 8 of 41 samples. Liikala et al. (1996) state that considering that vapor pressures at 20C for
benzene, toluene, ethylbenzene, and m,o,p-xylene are 76, 22,7, and 5 mmHg respectively, it is
likely that losses occurred from volatilization during sample transport, storage, and handling .

       Siegrist and Jenssen (1990) compared VOC retention from five soil sample collection and
storage methods:

       (1) undisturbed samples in glass bottles with high headspace (1.5 cm i.d., 10 cm long core
       extruded directly into a Teflon-sealed, 128 ml glass bottle with headspace volume about
       85% of container volume),

       (2) undisturbed samples in glass bottles with low headspace (3.0 cm i.d.,  10 cm long core
       extruded into a Teflon-sealed, 128 ml glass bottle with headspace volume about 40% of
       container volume),

       (3) undisturbed samples immersed in methanol in glass bottles (3.0  cm i.d., 10  cm long
       cores extruded into a Teflon-sealed, 300 ml glass bottle with 100 ml reagent grade
       methanol and headspace volume about 40% of container volume),

       (4) disturbed samples in glass bottles with low headspace (3.0 cm i.d., 10 cm long cores
       emptied in 7 - 10 aliquots with a stainless steel spoon and deposited in a Teflon-sealed
       128 ml glass bottle with a headspace volume about 40% of container volume),

       (5) disturbed samples in empty plastic bags with low headspace (40 ml soil sample
       emptied in 7 - 10 aliquots with a stainless steel spoon and deposited into a 500  ml
       laboratory-grade plastic bag with zip closure and a headspace volume about 40%  of
       container volume).

With the exception of methanol containing samples, VOC analyses of soil samples involved
extracting a 10 g of sample with 10 ml of 2-propanol and 4 ml pentane and then extracting it
again with 5 ml of 2-propanol and 4 ml pentane prior to GC injection. VOC analyses of  soil
samples containing methanol involved removing  a 4 ml aliquot of methanol from the sample vial
followed by addition of 2 ml water and 2 ml pentane. The pentane extract was removed  and the
procedure repeated. The two pentane extracts were then combined prior to GC injection.

       The undisturbed soil samples preserved with methanol consistency yielded the  highest
VOC concentrations.  Negative bias was greatest  for lack of methanol immersion (up to 81%),
followed by headspace volume (up to 17%), soil disturbance (up to 15%). Negative bias for
sample storage in plastic bags was greatest (up to 100%). In general, sampling bias declined with
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increasing sorption affinity and lower Henry's Law constant. Siegrist and Jenssen (1990) state
that the principal mechanism underlying negative bias for compounds evaluated (methylene
chloride, 1,2-dichloroethane, 1,1,1-trichloroethane, trichloroethylene, toluene, and
chlorobenzene) was likely volatilization. They state further that analyses of VOCs in soils
having relatively low sorption affinity, particularly where concentrations are anticipated in the
range of cleanup action levels, may require collection of undisturbed samples with infield
immersion in methanol.

12.8   Conclusions

       Research on development of field-scale methods to elucidate and quantify rate-limited
vapor transport is needed to allow stakeholders at particular sites to make informed decisions on
modification of design, operation, and eventual closure. At present, it is difficult to even
ascertain whether vapor transport is occurring in the field let alone attempt to elucidate and
quantify the process. Under rate-limited conditions, mass removal is dictated by kinetic rate
constants which are for the most part unknown at venting sites. However, if design and
implementation factors can be eliminated as major contributing causes of reduced mass flux and
extensive vapor concentration tailing, it is possible that mass removal is constrained by rate-
limited vapor transport.

       Given discussion in this section, the following recommendations can be provided for
VOC monitoring of venting sites.

1.      Flow and total vapor concentration monitoring using a PID or FID or other suitable
       instrument should be performed at the vapor treatment or blower inlet because it provides
       an estimate of VOC  mass removal rate and total VOC mass removed as a function of time
       for an entire venting system.

2.      Flow and vapor concentration monitoring using a PID or FID or other suitable instrument
       should be performed at individual gas extraction wellheads to determine the most heavily
       contaminated portions of a site requiring  perhaps the most monitoring, and to enable
       potential optimization of the vapor treatment system (e..g., prioritization of venting
       operation at highly contaminated areas, vapor stream richness or leanness for catalytic
       oxidation unit overheating or supplemental fuel requirement).

3.      When utilizing an FID or PID for VOC screening, the user must be aware of fundamental
       differences in detector operation, ionization potential of target compounds, and gas
       matrix effects. The high ionization potential of many common VOCs will result in
       nondetection using a conventional  10.6 or even a 11.7 eV PID lamp. The high halogen
       content of many common VOCs will result in underestimation or nondetection of VOCs
       using an FID.  Gas matrix effects such as humidity, carbon dioxide, and alkane
       (especially methane) dramatically decrease PID response. Gas matrix effects combine in
       a nonlinear manner resulting in a decreased PID response far greater than what would be
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       expected from additive effects.  PIDs are also much more prone to nonlinearly in
       response compared to FIDs. Gas matrix and linearity problems however can be overcome
       through the use of serial dilution techniques.

4.      Observation of evaporation fronts in individual extraction well effluent gas and
       evaporation/condensation fronts in well-placed vapor probe clusters could potentially
       provide valuable information on the progress of venting remediation of NAPL
       contaminated soils. Under equilibrium conditions, individual components of a NAPL
       propagate through soil in a sequence of evaporation-condensation fronts at speeds
       proportional to their vapor pressure. Evaporation-condensation fronts have likely not
       been reported during field application because of nonuniform gas flow (bypass flow as
       opposed to through-flow conditions, fully three-dimensional flow as opposed to one-
       dimensional or radial flow, spatial variability of NAPL distribution, common monitoring
       practices (determination of total VOC with a PID or FID as opposed to identification and
       quantification of individual compounds), and rate-limited transport. Under equilibrium
       conditions in the field, sharp evaporation-condensation fronts are not likely to be
       discernable but the vapor concentration of lighter components should decrease
       monotonically with time compared to heavier components which should increase for
       some time and then decrease. In contrast, under rate-limited transport conditions,
       extensive extensive effluent tailing should be observed with vapor concentrations profiles
       remaining fairly similar.  Thus,  use of GC/MS monitoring in gas extraction well  and
       vapor probe clusters may reveal patterns of component removal from NAPL, NAPL
       removal itself, and rate-limited  vapor transport.

5.      There is no  way to directly relate observation of an asymptote to  an environmental benefit
       and little potential to ensure consistency in decision making. Vapor concentration of an
       asymptote at a highly contaminated site may be much higher than at a lesser contaminated
       site leading perhaps to less stringent remediation at the highly contaminated site.

6.      Observation of an effluent asymptote may be related to venting design (e.g., well
       screening and spacing) or gas flow patterns separate or in addition to rate-limited vapor
       transport. Since both design considerations and rate-limited vapor transport can  cause
       extensive effluent tailing and ineffectual removal of accessible contaminant mass, it is
       critical to first rigorously assess venting design, implementation, and associated
       monitoring  prior to concluding  that rate-limited vapor transport significantly constrains
       mass removal.

7.      An asymptote can be observed in gas extraction wells while much of the contaminant
       mass remains in soils. Observation of low asymptotic vapor concentrations in effluent
       gas is not a  sufficient condition to demonstrate progress in mass removal from
       contaminated soils.

8.      Different mass retention and rate coefficients can produce very similar effluent curves
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       suggesting that effluent and vapor probe concentration data can not be used to estimate
       mass transport coefficients and VOC concentration in the aqueous and solids phase.
       Thus, effluent and vapor probe concentration data is insufficient to quantitatively assess
       the progress of soil venting remediation.

9.      Some indicators of rate-limited transport in the field may be decreased vapor
       concentration with increased pore-gas velocity (flow variation) and extensive vapor
       concentration tailing accompanied by vapor rebound after cessation of operation (flow
       interruption). In both NAPL and non-NAPL contaminated soils, flow variation provides
       a simple yet powerful test of the validity of local equilibrium. For NAPL contaminated
       soils however, results of flow interruption results can be confusing if the propagation of
       evaporation-condensation fronts are not considered.

10.    One potential use  of flow interruption or rebound data is to use vapor concentrations and
       partition coefficients to estimate total soil concentration. However, there is presently so
       much uncertainty  and error inherent in estimation that calculations can at best only
       provide qualitative insight into soil concentration levels.
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13.    ASSESSMENT OF RATE-LIMITED VAPOR TRANSPORT WITH DIFFUSION
       MODELING

       After venting has been applied at a site for some time and VOCs in gas extraction wells
and vapor probes approach low levels, soil samples are often collected to evaluate actual
concentration reduction in soils. Lithologic examination of cores and sample analysis often
reveals much higher levels of VOCs remaining in lenses of fine-grained material compared to
surrounding coarser-grained material. This observation likely indicates that mass removal  from
fine-grained material is diffusion limited. It stands to reason then that the material properties of
these lenses (e.g., diffusion path length,  moisture content, organic carbon content) will eventually
dictate venting performance.  Thus, information collected during soil core collection should be
used in some quantitative manner.

       From previous discussions, it appears that well conceived field-scale tests and fairly
sophisticated data analysis will be necessary to elucidate and quantify rate-limited mass exchange
processes during venting. These tests however require an investment in research and time for
development. One obvious method of data analysis of soil cores is to conduct diffusion
modeling to assess time for mass removal under current operating conditions. While soil core
collection will likely not reveal the lateral extent or geometry of the fine-grained lense, other
properties required for modeling such as the thickness, moisture content, and organic carbon
content of the lense are readily available from field observation and sample analysis.  The
purpose then of this section is demonstrate this concept at an actual site.

13.1   Model Formulation

       Since lenses of fine-grained material often exhibit a plate-like structure, it would appear
inappropriate to conduct diffusion modeling in spherical coordinates.  Since the lateral extent and
actual geometric shape of a lense is typically unknown, an assumption will be made that the lense
extends infinitely thereby enabling the use of one-dimensional modeling.  The governing
equation describing one-dimensional diffusion is given as:
where CT = total soil concentration [ML"3] and k = effective diffusion coefficient [L2T4].  The
initial condition describing soil concentration as a function of depth in the lense is given by

CT(x,0) = f(x)                                                                   (13.2)

where f(x) is the initial total soil concentration distribution [ML"3]. Upper and lower boundary
conditions must now be specified. Since only vapor concentration will be known outside the
modeled domain, it would appear that first-type, time-dependent boundary conditions are
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appropriate for the upper and lower domains. The assumption is made here that these
concentrations are equivalent. Upper and lower boundary conditions then are expressed by
CT(L,t) =
                                                                                 (13.3)
                                                          (13.4)
where: L = diffusive path length [L], and g(t) = soil concentration at boundaries as a function of
time [ML"3]. The effective diffusion coefficient k can be defined by:
                                                                                 (13.5)
    H0g+0w+Kdpb

From equation (13.5), is assumed that NAPL is absent.
       A solution to equation (13.1) with initial and boundary conditions (13.2) through (13.4) is
found in Carslaw and Jaeger (1959), p. 104
             n=l
                                                                                 (13.6)
The expression for f(x') can be integrated in j partitions where:
           f
 J/(*')sin
(.  L
 L
nn
                                2=1
                                                                      (13.7)
An average matrix concentration is useful for comparison as a function of time. Integration of
(13.6) with respect to x from 0 to L, followed by dividing by L leads to:
                n=l
                        n
                     V
                                                                                 (13.8)
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A FORTRAN program written to implement equation (13.8) is given in Appendix G.

13.2   Site Description

       Soil cores collected and analyzed by a remedial contractor (Earth Tech, 1997) near
Building 763 of Norton AFB in California were used for diffusion modeling.  Building 763 was
used for aircraft maintenance for over 50 years. Information for diffusion modeling was
extracted from borehole CB-1 where a silty clay unit was found between 38.0 and 40.5 feet bgs.
Gravelly, silty sandy soils above and clayey, sandy, silty soils below this unit contained
trichloroethylene (TCE) concentrations near the detection limit (2 ug/kg) while as illustrated in
Figure 13.1, the silty clay unit contained concentrations ranging from 1300 to 6100 ug/kg TCE.
Figure 13.1 provides the initial condition for modeling purposes.  This suggested that vapor
transport was limited by diffusion from the silty clay lense to surrounding soils of higher
permeability. The current cleanup standard for TCE contaminated soil prescribed in the Record
of Decision (ROD) was 5 ug/1 TCE concentration in leachate as determined using the Toxic
Characteristics Leaching Procedure (TCLP) by EPA Method 1311/8240. Using soil property
information  collected during soil collection (95% water saturation, porosity = 38%, organic
carbon content = 0.68%) and literature values for chemical properties (dimensionless Henry's
law constant = 0.38, organic carbon - water partition coefficient = 126 cnrVg), this is equivalent
to approximately 10 ug/kg which was the goal set for modeling.
 o
£
 CD
                        0

                       10

                       20

                       30

                       40

                       50

                       60

                       70
                          0        2000       4000       6000
                                   TCE concentration (ug'kg)

Figure 13.1 Concentration (dry wt.) profile in clay lense.

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                                                    8000

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       The boundary conditions require knowledge of vapor concentration at the top and bottom
of the clay lense as a function of time. As illustrated in Figure 13.2, the decrease in vapor
concentration at the extraction well during the period of operation can be adequately described by
a simple analytical expression.  This expression can then be used to estimate future effluent
vapor concentrations at the well if venting were to continue indefinitely.  Vapor concentrations at
the well were then used to estimate vapor concentrations at the top and bottom of the clay lense
although it was preferable to have actual vapor concentration data as a function of time near the
soils being simulated. Unfortunately, this information was not available.  Table 13.1  summarizes
site-specific input data used for simulation.  For soil-air and soil-water tortuosity estimates, we
use a theoretically based model derived by Millington and Quirk (1961).

13.3   Results and Discussion

       Figure 13.3 illustrates the average concentration of TCE in the clay lense as a function of
time and four moisture saturations. At the moisture saturation at the time of sampling (95%), in
excess of 80 years would be required to reach a concentration of 10 ug/kg because the high
moisture content results in a very low effective diffusion coefficient. This level is reached in just
15 years when saturation  is reduced to 86% because of the nonlinear effect of the diffusion
coefficient on soil concentration. However, a water saturation of 95% is  expected to  remain
constant in the clay lense. Water saturation was varied in the model  simulations to illustrate  the
sensitivity of moisture content.

                                            y=a/(l+abx)

                                     a=367.96451    b=0.0023179721
                 1000
               g
               a
               a
               o
              O
               o
               a
              W
              O
                  100
                      0               200               400
                                       time of operation (days)
Figure 13.2 Best fit line of TCE vapor concentration profile as a function of time at the
extraction well.
gas
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Table 13.1  Input for diffusion modeling
                    Description
                                   Value
                                     Units
 pb
 H
volumetric air content

volumetric water content

porosity

bulk density

Henry's Law Constant for TCE

soil-water partition coeff.

free air diffusion coeff. for TCE
                    varies   cnr\ir/cm3soi]

                    varies   cm3water/cm3soU

                    0.38     cm3pore/cm3soil

                    1-68     gsoiidS/cm3soil

                    0.38     (ug/cm3air)/(ug/cm3water)

                    0.857*   (ug/gsolids)/(ug/cm3water)

                    6366.8   cm2/d
 a
free water diffusion coeff. for TCE   0.804   cnr/d
*Estimated by Kd = Koc*foc where foc= 0.0068, Koc= 126 cm3/g
10      20      30      40      50      60

          Time after initial operation (years)
                                                                       70
                                                                     80
Figure 13.3. Simulated average TCE concentration in clay lense as a function of time and
moisture content.
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       Vadose zone VOC transport simulations conducted by the remedial contractor indicated
that mass flux of remaining TCE in soils to ground water would be insignificant. For a well
characterized, designed, and monitored SVE system where contaminant mass flux to ground-
water will be insignificant, it may be difficult to justify prolonged venting operation to meet a 10
ug/kg TCE concentration level. Thus, simulations conducted here support venting closure at this
site.

13.4   Conclusions

       Diffusion modeling provides a simple yet powerful way of assessing rate-limited vapor
transport from discrete lenses of low permeability. Diffusion modeling could be extended to
two- or three-dimensions for more sophisticated analysis if assumptions on the geometric  shape
of lenses are made.
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14.    USE OF RESPIRATION TESTING TO MONITOR SUBSURFACE AEROBIC
       ACTIVITY

       During its initial use, venting was applied primarily to enhance volatilization of organic
compounds from unsaturated consolidated and unconsolidated geologic media. However, large-
scale tank experiments conducted by Texas Research Institute (TRI) (1980, 1984) demonstrated
that biodegradation could be a major mass removal mechanism for compounds which biodegrade
under aerobic conditions. In their studies, TRI (1980,  1984) indicated that biodegradation may
have accounted for as much as 38% of the mass removal of gasoline from vented soils. Field-
scale studies suggest that subsurface air circulation can indeed enhance aerobic biodegradation of
fuel oils and other aerobically biodegradable compound mixtures in oxygen deficient soils.
Subsurface advection of air allows rapid exchange of oxygen from the atmosphere as opposed to
gaseous diffusion which dominates in non-vented soils.

       The use of subsurface air circulation to enhance aerobic biodegradation is commonly
referred to as "bioventing".  Operationally, bioventing is supposed to be distinguished from
"conventional" venting in that lower pore-gas velocities or pore-volume exchange rates are used
for the former to minimize volatilization. For instance, Hinchee et al. (1991) state that
approximately one pore volume exchange of air per day is sufficient to support biodegradation.
The U.S. Army Corps of Engineers (USACE)(1995) state that recent field experience with full-
scale systems suggests that 0.25 to 0.5 pore volumes exchanges per day may be optimal to
maximize biodegradation while minimizing volatilization.  However as previously  discussed,
rate-limited vapor transport may also require low pore-gas velocities to optimize mass removal
efficiency through volatilization.  Thus, bioventing is an ill-defined term when one considers that
enhanced aerobic biodegradation occurs during venting whether intended or not. Bioventing
should possess a definition which clearly distinguishes the technology from conventional
venting. For example, the addition of methane to an air injection well to enable or enhance
cometabolism of trichloroethylene represents a significant departure from conventional venting
application and would logically fall within the realm of bioventing.

       For removal of VOCs there is a trade-off between maximizing the rate of total
hydrocarbon removal (volatilization plus biodegradation) and hydrocarbon removal  primarily
through biodegradation.  Because volatilization is a more rapid removal mechanism than
biodegradation, the application of high pore-gas velocities results in more rapid remediation.
However,  energy and vapor treatment costs are higher resulting in a trade-off between cost and
remediation time.  To properly assess this trade-off, quantification of factors affecting
volatilization and biodegradation are necessary.  Increased understanding of these factors may
eventually lead to successful prediction of remediation time under a variety of operating
conditions.

14.1    Discussion of Favorable Conditions for In-Situ Biodegradation

       Enhanced aerobic biodegradation through venting application has potential in soils
having: (1) organic compounds readily biodegradable under aerobic conditions; (2)

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environmental conditions (e.g., nutrients, moisture, temperatuare, pH, Eh, salinity, toxicity)
suitable for microbial growth; and (3) sufficient gas permeability to allow adequate air (oxygen)
circulation.  The rate of microbial growth or metabolism will be limited by the factor present at
its minimum level.

       Determination of limiting factors and practical alteration in the soil environment can
enhance biodegradation during venting application. Information on the "intrinsic" aerobic
biodegradability of many organic compounds and physiological tolerance limits for isolated
microbial populations is available from the literature.  However,  since microbial interactions in a
contaminated soil environment are complex,  laboratory based treatability studies are sometimes
conducted to evaluate the effect of various environmental variables. When treatability studies are
conducted, singular or multivariable evaluations of temperature, nutrients, and moisture are
common. For instance, Hinchee and Arthur (1991) evaluated the influence of moisture and
nutrient addition on JP-4 contaminated soils from Hill AFB.

       Laboratory microcosms possess distinct advantages over field testing.  Since microcosms
are replicable and their external environment is controllable, much greater statistical rigor can be
achieved than is possible in the field. Ease of manipulation allows relatively straightforward
evaluation relative to field studies of physical and geochemical factors affecting respiration rates
and correlation to actual biodegradation.  The question remains however whether these results
can be extrapolated to the field.  Physical and geochemical  conditions in the field will differ from
the laboratory environment.  Also, determination of microbial activity in disturbed, displaced
environmental samples incubated in the laboratory may be quantitatively and even qualitatively
different from the same determination in situ.

       Most field studies conducted to evaluate enhanced biodegradation during venting
application have been conducted in jet fuel and gasoline contaminated soils.  Most petroleum
related hydrocarbons are natural substrates for microorganisms which have evolved mechanisms
for their degradation.  They are anthropogenic or xenobiotic only in the sense that they occur at
waste sites at higher than accepted levels. For enhanced biodegradation to be feasible, an organic
chemical must be inherently biodegradable under aerobic conditions. n-Alkanes constitute a
significant proportion jet fuels (e.g.,  JP-4, JP-5, diesel fuel and gasoline. There are many
references on microbial attack and the diversity of microorganisms involved in alkane
metabolism. A wide range of n-alkanes are susceptible to biodegradation, however some
microorganisms preferentially utilize only short chain molecules while others metabolize only
longer chain structures.  The initial oxidation of n-alkanes involves the incorporation of
molecular oxygen, a common occurrence with all petroleum hydrocarbons. It is somewhat more
difficult for microorganisms to utilize cycloalkanes as a sole source of carbon and energy,
however, it is clear that individual or mixtures of microorganisms can metabolize cycloalkanes.
As with n-alkanes, molecular oxygen is incorporated, but in this case in two separate steps.
Branched alkanes and cycloalkanes in the CIO to C22 range are less biodegradable than n-
alkanes of equivalent size. Cycloalkane biodegradation requires the presence of two or more
species for complete metabolism (Dragun, 1988). Branched aliphatic hydrocarbons are generally
more resistant to biodegradation than unsubstituted hydrocarbons.  In general, as the number of

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substitutions increase, the greater the resistance to biodegradation. Oxygen plays a critical role in
the metabolism of aromatic hydrocarbons.  All of the major pathways require oxygen as a
coreactant. In addition, organisms catalyzing these reactions use oxygen as a terminal electron
acceptor.  Therefore, hydrocarbon metabolism puts a large demand on oxygen resources.
Aromatic hydrocarbons can generally be used as primary substrates under aerobic conditions. In
the absence of oxygen, degradation rates are slow.  There is a large body of information
describing the biodegradation of polyaromatic hydrocarbons (PAHs) under aerobic conditions.
Most studies on PAH biodegradation indicate that low solubility of PAHs is the primary factor
limiting biodegradation.

       In general, biodegradability increases with water solubility. VOCs with functional groups
containing nitrogen, sulfate, or oxygen typically are fairly water soluble and therefore are rapidly
degraded.  Low molecular weight aliphatic alcohols and ketones such as ethanol  and methyl ethyl
ketone (MEK) can biodegrade very rapidly in the presence of sufficient oxygen.  Aromatic
compounds such as toluene, ethylbenzene, and xylene biodegrade under aerobic conditions at
moderate rates. Benzene, however, has been found to biodegrade at relatively slow rates.

       Prior to venting, oxygen is typically depleted in soils containing high concentrations of
organic compounds which biodegrade under aerobic conditions. Loss of oxygen as a metabolic
electron acceptor induces a change in the activity and composition of the soil microbial
population. Soil venting provides air to the vadose zone and thus carries oxygen that can be used
as the terminal electron acceptor by soil microorganisms to biodegrade chemicals aerobically.
Air has a much greater potential than water for delivering oxygen to soil on a weight-to-weight
basis and volume-to-volume basis.

       The feasibility of enhanced biodegradation at a site may largely be dependent on the
ability to adequately circulate air through contaminated soils. Clayey soils pose a challenge in
that gas permeability can be very low due to low intrinsic permeability and high moisture
content. In these types of soils, diffusion may control VOC and oxygen transport. However,
high oxygen concentrations in more permeable soils provide a large driving force for diffusion.

       Soil water serves as the transport medium through which many nutrients and organic
chemicals diffuse to the microbial cell, and through which metabolic waste products are removed
(Sorensen and Sims,  1992). Atlas and Bartha (1993) state that optimal conditions for activity of
aerobic microorganisms occur between 50% to 70% of "water holding capacity", a term which is
not rigorously defined. Soil moisture should be maintained in the range of 40 to 70% of field
capacity. If soils are dried excessively, microbial activity can be seriously inhibited or stopped.
Because individual species are seldom eliminated in extremely wet or dry soil, the rewetting of a
dry soil should reestablish microorganism activity.  However, between extreme conditions (dry
vs. saturated), soil moisture content should have an undramatic effect on the microbiological
degradation of organic chemicals (Brown and Donnelly, 1983).

       Microbial metabolism and growth is dependent upon adequate supplies of essential
macro- and micronutrients.  At a minimum, the following nutrients must be available in the

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proper form and amount for microbial proliferation: nitrogen, phosphorus, potassium, sodium,
sulfur, calcium, magnesium, iron, manganese, zinc, copper, cobalt, and molybenum (Alexander,
1977). When an organic chemical enters the soil environment, the soil's supply of essential
elements is frequently inadequate to support desirable biodegradation rates (Dragun,  1988). If
soils are high in carbonaceous materials and low in nitrogen and phosphorus, the soils may
become depleted of available nitrogen and phosphorus (Sorensen and Sims,  1992). Microbial
cells contain 5 to 15 parts of carbon to 1 part nitrogen but 10:1  is a reasonable average for
aerobic microorganisms (Alexander, 1977). A carbon, nitrogen, phosphorus (C:N:P) ratio of
250:10:3  is considered optimum for biodegradation in soil but  100:10:2 has  been used in some
applications (Staps, 1989). In general, one unit of nitrogen assimilated into cell material is
accompanied by 10 units of carbon volatilized as CO2 (Alexander, 1977).  The 100:10:2 ratio
assumes that all carbon is assimilated into cell mass while the 250:10:3 ratio considers CO2
volatilization but does not consider recycling  of organic nitrogen or phosphorus. If the ratio of
C:N:P is wider than approximately 300:15:1 (weight basis), supplemental nitrogen or phosphorus
should be added (Alexander, 1977). Mineral  nutrients are typically supplied as soluble salts.
Minor nutrients are usually present in soils in sufficient amounts. The availability of nutrients to
microorganisms is  strongly influenced by pH. The easiest method to determine if adequate
nutrients  are present in soils at proper pH levels is to collect soil samples for analysis.

       One method of determining the effect of nutrient addition in the field on biodegradation is
monitoring increased oxygen depletion or carbon  dioxide production. Hinchee and Arthur  (1991)
conducted bench scale studies using soils from Hill AFB and found that both moisture and
nutrients  became limiting after aerobic conditions were achieved. This led to the addition  of
moisture  and nutrients in the field.  Moisture addition increased oxygen deficit in extraction well
offgas but nutrient  addition did not. Hinchee and  Ong (1992) postulate that  the failure to observe
increased oxygen demand after nutrient addition could be explained by a number of factors
including: (1) the failure of nutrients to move vertically through the soil column (pore-water and
soils were not sampled to determine if nutrients had percolated in soils to depths of interest); (2)
remediation of the  site was entering its final phase, and nutrient addition may have been too late
to result in an observed change; and (3) nutrients simply may not have been  limiting.  At Tyndall
AFB, Miller (1990) observed no changes in oxygen depletion or carbon dioxide production as a
result of moisture and nutrient addition. However pore-water and soils were not sampled to
determine if nutrients had percolated in soils to  depths of interest.

       While most microorganisms will tolerate a pH range of about 4 to 9,  the pH range within
which bioremediation processes are considered  to operate most efficiently is 6 to 8. Degradation
of hydrocarbons generates carbon dioxide, which  may lower pH.  Degradation of chlorine, sulfur,
and nitrogen containing compounds generates strong acids which require buffering capacity to
maintain  favorable pH conditions.  Soils  should be analyzed for pH prior to and at the
completion of venting ensure that adverse pH levels are not present.

       Microbiological reactions often increase as temperature increases.  The upper temperature
limit is thought to be about 50°C (122°F) because essential microbial enzymes are denatured
above this temperature.  Soils typically exhibit diurnal and seasonal temperature variation.

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Diurnal variation is less pronounced in wet soils because of the high specific heat of water.
Variation decreases with distance below the surface.  During field testing at Tyndall AFB, Miller
(1990) observed that O2 depletion rates were approximately twice as high at 25°C than at 18°C.
He used the van Hoff-Arrhenius equation to model of temperature effects.

       In many cases, oxygen consumption and degradation rates will be lower upon start-up
than following a period of acclimation. After the introduction of oxygen, microorganisms will
adapt to the composition and concentrations of biodegradable compounds.  Acclimation times
can vary from a few days to a few months.  Cessation of operation for more than a few days may
result in detrimental changes to microbial populations and necessitate another period of
acclimation before optimum performance is achieved.

14.2   Respiration Testing

       Proof of in-situ biodegradation requires demonstration that contaminant mass or
concentration reduction has decreased as a direct result of microbial activity. Proof however is
elusive because of difficulties in: (1) identification and enumeration of microorganisms
proliferating as a result of contaminants present in soils, (2) extreme spatial variability in
contaminant concentration, and (3) differentiation of concentration reduction due to abiotic and
biotic processes. Since measurement of biodegradation rates on a field scale is difficult and
expensive, indicators of aerobic microbial activity such as oxygen depletion and carbon dioxide
production  have been used to demonstrate microbial activity. At operating  sites, oxygen
depletion or carbon dioxide production rates are determined in vapor probes during well shut-in
periods. Zero or first-order kinetic constants are subsequently determined for each probe and
adjusted for a background oxygen depletion or carbon dioxide production rate from probes lying
in an uncontaminated area.  Hinchee and Ong (1992) and van Eyk and Vreeken (1989a,b) have
observed that O2 consumption is often a more sensitive measure of aerobic metabolism than CO2
production  because of CO2 sinks in soil (e.g., carbonate precipitation).  One exception was at
Tyndall AFB where soils had low alkalinity and pH (Miller, 1990). At Fallen NAS and Eielson
AFB where soils had higher alkalinity and pH, little carbon dioxide production was measured
(Hinchee and Ong, 1992).

       It has become common practice to use zero-order oxygen consumption rates to directly
estimate hydrocarbon degradation rates. A  reference compound such as benzene or hexane is
typically chosen for estimation purposes. The stoichiometry for mineralization of benzene is

C6H6 + 7.5  O2 - 6 CO2 + 3 H2O                                                     (14.1)
(3.1g02/gC6H6)

while for hexane it is

C6H14+9.5O2-6CO2 + 7H2O                                                    (14.2)
(3.5g02/gC6H14).
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Stoichiometric relationships allow derivation of an equivalency equation to estimated a
referenced hydrocarbon degradation rate to an oxygen consumption rate
   eg                    >
           PbK02/CIHy

where Kdeg is the reference hydrocarbon degradation rate (mg Kg'May"1), p02 = pure oxygen
density (gem"3), Ro2/CxHy = oxygen/hydrocarbon mass ratio (g g"1), and K02 = oxygen
consumption rate (% O2 day"1).

       One obvious limitation of this method is that the reference hydrocarbon used for the
stochiometric mass ratio between hydrocarbon degradation and oxygen consumption may not be
representative of many compounds present at a spill or hazardous waste site.  Another limitation
is that field data may show first-order degradation rates while equation (14.3) requires a zero-
order rate. Also, respiration tests provide estimates of total hydrocarbon degradation rates but
regulatory programs, specifically Superfund and RCRA, mandate specific contaminant
concentration reduction of which these tests provide little information. The greatest concern
however is the error involved in using oxygen depletion rates to directly calculate hydrocarbon
degradation rates. Equations (14. 1) and (14.2) represent the minimum oxygen requirement to
completely mineralize benzene and hexane respectively. They are based on the assumption that
either there is no cell synthesis or endogenous respiration, or that cell synthesis and endogenous
respiration are occurring at the same rate with no net accumulation of biomass.

       Oxygen serves as a terminal electron acceptor not only in the degradation of organic
matter but also in the oxidation of reduced inorganic compounds by microorganisms which
obtain energy through chemical oxidation. For example, different microbial populations carry
out the two steps of nitrification.  In the first reaction, molecular oxygen is incorporated into the
nitrite molecule. In the second step,  oxygen is derived from water and molecular oxygen serves
as an electron acceptor. In the presence of molecular oxygen, reduced sulfur compounds can
support microbial metabolism.  For instance H2S can be converted to elemental sulfur which can
subsequently be converted to H2SO4  or sulfate.  Under neutral and alkaline conditions, ferrous
iron is rapidly oxidized to ferric iron. The greenish-grey color of contaminated soils is often
associated with reduced iron. Reduced Mn+2 is stable under aerobic conditions at pH values
below 5.5 and at higher pH values under anaerobic conditions.  Various soil microorganisms
which catalyze manganese oxidation have been identified.

       Thus, use of oxygen consumption tests to estimate biodegradation rates may result in
overestimation of mass loss through biodegradation.  Perhaps the best use of respiration testing is
as an indicator of soil amendment (moisture, temperature, nutrients, etc.) effectiveness and of
microbial activity in general. Respiration testing could be used to indicate inhibition of
degradation. For instance, if oxygen and carbon dioxide monitoring  reveal low oxygen
consumption and carbon dioxide generation in soils containing high concentrations of readily
biodegradable compounds, further studies could be conducted to determine if biodegradation is

                                          215

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being limited by insufficient moisture content, toxicity (e.g. metals), or nutrients.  Persistent, low
oxygen levels may indicate an insufficient supply of oxygenated air at a location suggesting the
need for air injection, higher extraction well vacuum, additional extraction wells, or additional
soils characterization which may indicate high moisture content or the presence of immiscible
fluids impeding the flow of air.  Thus, respiration testing can be very useful in monitoring
venting operation.  Taken alone however, they do not provide proof of biodegradation and
quantitative estimation of biodegradation rates. When used with other indicators of increased
microbial activity or biodegradation however, respiration tests can provide one of several
convergent lines of independent evidence to document biodegradation.

14.3   Site Description

       Respiration testing was conducted at a field site, U.S. Coast Guard Base, Elizabeth City,
North Carolina, to evaluate the use of respiration testing as a general indicator of bioremediation.
In February, 1988, the U.S. Coast Guard detected a JP-4 fuel leak in a 10 cm (4 inch) diameter
pipeline at its support center in Elizabeth City, North Carolina.  Soils and ground water were
subsequently contaminated by JP-4 constituents over an area of approximately 1200 square
meters.

       Surficial soils at the site consisted of 1.5 m of a dark brown silty clay underlain by a
medium-grained sand. Ground water, which fluctuated ± 60 cm seasonally, was located about 2
m below the soil surface in the sand. During the spring, sandy soils were completely saturated
while during summer and fall, ground-water levels dropped to the point that 1.0 to 1.5 m of
sandy soils were available for venting. Ground-water temperature varied seasonally from 12 to
14°C.  Air injection tests were conducted to assess gas permeability and leakance. The program
AIR2D (Joss and Baehr, 1997) was used for parameter estimation. Average gas permeability
values were 8.22 x 10"8 cm2 assuming isotropic conditions and leakance was estimated at 2.06 x
10-10cm2.

       A soil venting system consisting of 42 air injection/extraction wells and 34
piezometer/vapor  probe clusters was installed at the site in December, 1992. Figure 14.1
provides a plan view of 18 piezometer/vapor probe clusters  (70A through 70X) used for VOC
and respiration monitoring and combined benzene, toluene,  ethylbenzene, and xylene (BTEX)
ground-water concentrations prior to commencement of venting in January, 1993. Vapor probes
were constructed from 1.3 cm diameter, 316 stainless  steel pipe having a length of 7.6 cm. The
pipe was manually slotted with a bandsaw and wrapped in a fine stainless steel mesh to exclude
particles. Brass compression fittings were used to connect the probes to 0.64 cm o.d. flexible
copper tubing which was connected at the surface to brass quick-connects.  The probes were
installed in coarse sand 0.33 meters below the base of the clay layer (1.3 meters below surface).
The annular space above the probes was sealed with granular bentonite to 0.3 meters below
surface and cement to the surface.  Differential pressure was measured with magnehelic gauges.
Vapor concentration was measured using a GasTech Trace-Techtor hydrocarbon analyzer
calibrated to hexane.  The  Trace-Techor utilizes a catalytic combustion sensor to oxidize
hydrocarbons and a change in electrical resistance to quantify vapor concentration.  The vapor

                                          216

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                                 70A      • 70J
                                      70U
                                     70H        • 701
                              70E
Figure 14.1  Dissolved combined benzene, toluene, ethylbenzene, and xylene (BTEX)
concentrations before venting at Elizabeth City USCG Base, January, 1993.

analyzer has three settings: 0 to 100 ppmv, 0 to 1,000 ppmv, and 0 to 10,000 ppmv.

       Soil sampling during installation of piezometer/vapor probe clusters was conducted to
delineate the vertical extent of initial contamination. In August, 1992, 320 samples were
obtained from 16 locations.  Samples were collected during hollow stem augering in stainless-
steel core barrels having an outside diameter and length of 5 and 120 cm respectively.  Samples
were preserved in 40 ml vials containing 5 ml methylene chloride and 5 ml distilled water and
                                         217

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analyzed for benzene, toluene, ethylbenzene, and xylene (BTEX), and total petroleum
hydrocarbons (TPH). In September, 1994, after 18 months of venting operation, soils were
sampled close (within 30 cm) of original sampling locations using a direct-push (Geoprobe)
technique.  A thin wall core barrel with a detachable piston was used to collect cores into plastic
liners which were cut into 15 cm section for sampling and preservation as before.  A total of 71
500     1000     1500   2000   2500    3000
            Concentration (mg/kg)
                                                                         3500
Figure 14.2 Total (wet) Petroleum Hydrocarbon (TPH) profile in TOE
   a   1.5
                    2000     4000      6000     8000     10000     12000
                               Concentration (mg/kg)
Figure 14.3 Total (wet) Petroleum Hydrocarbon (TPH) profile in TOD

                                      218

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samples from 8 locations were collected. Figures 14.2 and 14.3 illustrate initial vertical
distribution of TPH in boreholes TOD and TOE.  One meter MSL corresponds to the interface
between the sand and clay.  Thus, contaminated soils were generally restricted vertically between
the interface of the sand and clay loam and approximately 2 m below this interface. After T
months of operation, the first in-situ respiration or oxygen consumption test was conducted.
Percent oxygen concentration was measured with a Gastech model 3252OX meter which utilizes
an electrochemical cell for detection.  A purge time of 5 minutes was calculated based the flow
rate of the meters and volume of the probe and surrounding annulus. However, purge time was
reduced to one minute since O2 levels consistently stabilized within this time  frame. Data from
each oxygen depletion test was fitted to zero- and first-order kinetic relationships.  For zero-order
oxygen consumption kinetics:
which after integration and assumption that O2 = O2o at time zero equals

                                                                                  (14.5)
where O2o = O2 concentration at time zero.  For first-order kinetics:
          o
                                                                                  (14.6)
which after integration equals

                                                                                  (14.7)
or


O2 = O2oe~Kl°i'                                                                    (14.8)

where K102 = a first-order oxygen consumption rate [T"1]. Oxygen depletion rates were
determined by linear regression to (14.5) and (14. T) and reported with associated correlation
coefficients.  The significance of distinguishing zero- and first-order kinetics is reflected in the
formulation of (14.4) and (14.6) where oxygen depletion as a function to time is constant for the
former and a  function of oxygen concentration for the latter. If the oxygen depletion rate is a
function of oxygen concentration, then oxygen may become a limiting factor below some
concentration level above zero.  Thus, observation  of first-order kinetics has direct design
implications.  That is, oxygen levels would have to be maintained above some level to maximize
aerobic microbial activity or indirectly biodegradation.

                                           219

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14.4   Results and Discussion

       Figures (14.4) through (14.24) present the results of respiration tests while Table 14.1
provides a summary of zero- and first-order rates at each location. Rates of oxygen depletion
observed at the Elizabeth City facility were very rapid compared to rates observed at Hill and
Tyndall Air Force Bases which were also contaminated by JP-4 fuel oil. For comparison,

Table 14.1 Summary of zero- and first-order oxygen depletion rates at Elizabeth City USCG
Base
Vapor Probe
70A
70B
70C
70D
70E
70F
70G
70H
701
70J
70K
70M
70N
70P
70Q
70R
70S
70T
70U
70V
70X
zero-order rate (%O2/hr)
-0.1239
- 0.0434
-2.0257
-5.9280
-1.6179
-0.8259
-0.3432
- 1.7500
-3.0650
-1.5621
- 1.8664
-3.6147
-4.1243
-2.6200
-3.3093
- 1.7044
- 0.2977
-0.0128
-0.6036
- 0.9202
-0.5767
first-order rate (hr1)
- 0.0072
- 0.0023
-0.1780
	
-0.1080
- 0.0522
-0.0291
-0.8570
- 0.6990
-0.1716
-0.1176
-0.6456
	
-0.2151
-0.6216
-0.1389
-0.0343
	
-0.0547
-0.0578
-0.1307
                                           220

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consider first-order rates for oxygen consumption observed at a JP-4 spill location at Hill Air
Force Base in Utah ranging from -0.000366 to -0.06 hr"1 where at Elizabeth City first-order rates
varied from -0.00230 to -0.8570 hr"1, an increase of at least one order of magnitude.  At Tyndall
Air Force Base, an average zero-order oxygen consumption rate of-0.24 %O2/hr was observed
whereas at Elizabeth City zero-order rates varied from -0.013 to -5.93 %O2/hr.

       A number of observations can be drawn from these tests.  At 70A, 70B, 70G, 70S, and
70T, zero-order- and first-order oxygen depletion kinetics provide nearly identical fits to data
over the tested period making it impossible to identify the governing relationship. Longer-term
testing on the order of 50 to 100 hours at several of these locations would be necessary to discern
trends. At locations 70C, 701, 70J, 70M, 70P, and 70Q, oxygen depletion kinetics could follow
either zero- or first-order kinetics depending on the time at which time zero concentration was
first achieved.  The zero concentration data point was not used for linear regression in these plots
because zero concentration may have been maintained for  some time prior to measurement.
Thus, again the governing  relationship was indeterminate.  At locations 70E, 70F, 70H, 70K,
70R, 70U, and 70V it appears likely that oxygen depletion kinetics did not follow first-order
kinetics.  Since the time to zero concentration may have occurred near the x-axis intercept, it
appears possible that oxygen depletion kinetics followed zero-order kinetics.  At locations  70D
and 70N, it appears that oxygen depletion did follow zero-order kinetics. Lack of data at
intermediate times precluded determination of oxygen depletion kinetics at 70X.

       Given the previous discussion, it is evident that one of the goals of this investigation,
discernment of zero- and first-order oxygen depletion kinetics, was not achieved due to too few
data points at critical times. Unfortunately, this situation could not be addressed during
subsequent site visits as site accessibility became limited.  Oxygen depletion tests did however
demonstrate fairly intense  aerobic activity in highly contaminated areas compared to areas with
lesser contamination. For  instance, at locations such as 70D  and 701, zero-order oxygen
depletion rates of-5.93 and -3.07 %O2/hr were observed respectively while at 70B and 70T,
zero-order oxygen depletion rates of only -0.04 and -0.01 %O2/hr were observed respectively.  In
the portion of the site encompassed by high oxygen depletion rates (70D, 70N, 70M, 70P, 70R,
and 70Q), free LNAPL was observed in piezometers. Thus, as demonstrated here, respiration
tests can provide an inexpensive and rapid means of assessing in-situ aerobic metabolic activity.
From a soil venting monitoring perspective, respiration testing should be conducted periodically
to assess change in oxygen depletion rates. As remediation progresses, oxygen depletion rates
would be expected to decrease over time, eventually in some cases approaching background
conditions.

       Finally,  figures  14.25 and 14.26 illustrate the vertical distribution of benzene and TPH
respectively collected during installation of 70D and within 30 cm of 70D after 18 months  of
venting.  Sample comparison reveals that volatile BTEX compounds had been removed to  very
low concentrations but higher molecular weight and lower volatility compounds remained at
fairly high concentrations.  This situation is common at venting sites, where lower volatility and
more biologically recalcitrant compounds remain or only slowly dissipate in concentration.  The
fundamental question here is whether the remaining compounds pose an environmental risk.

                                           221

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              20 -
              15 -
              10 -
               5 -
                                                                            70A
                    O2 =-0.12391+ 18.564
                     R2 = 0.95 (zero-order)
                            O2 = 18.586e
                            R2 = 0.9589 (first-order)
                                           10           15
                                            Time (hours)
                                                20
                               25
Figure 14.4  Oxygen depletion test at vapor probe 70A. Solid and dashed lines denote linear
regression for zero- and first-order kinetics respectively.
              20 -
              15 -
              10
               5 -
                                                                            70B
 O2 = -0.0434t+19.329
R" = 0.4376 (zero-order)
O2=19.325e~l
R2 = 0.4507 (first-order)
                                          10           15
                                            Time (hours)
                                               20
                              25
Figure 14.5  Oxygen depletion test at vapor probe 70B.  Solid and dashed lines denote linear
regression for zero- and first-order kinetics respectively.
                                            222

-------
            20 -
            15 -
            10
             5 -
                                 O2 = -2.0257t + 16.489
                                     R2 = 0.9224
                                         10           15
                                           Time (hours)
                                                                            70C
                                                                        -0.1781
                                   O2= 17.315e
                                      R2 = 0.9768
                                                     25
Figure 14.6 Oxygen depletion test at vapor probe 70C.  Solid and dashed lines denote linear
regression for zero- and first-order kinetics respectively not using zero concentration data point.
            20 -
            15 -
            10 -
             5 -
O2 = -5.928t + 13.274
     R2=1.00
                                                                            70D
                                         10           15
                                           Time (hours)
                                        20
25
Figure 14.7 Oxygen depletion test at vapor probe 70D. Solid line denotes linear regression for
zero-order kinetics not using last data point at zero concentration.
                                           223

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            20 -
             15 -
             10 -
             5 -
             0
                                                                             70E
                                                            O2=18.421e
                                                              R2 = 0.9046
                O3 = -1.6179t+18.298
                    R2 = 0.8776
                0
    20
25
                             5           10           15
                                           Time (hours)
Figure 14.8 Oxygen depletion test at vapor probe 70E. Solid and dashed lines denote linear
regression for zero- and first-order kinetics respectively excluding last data point at zero
concentration.
              20 -
              15 -
              10 -
                                                                             70F
O2= 18.318 e
   R2 = 0.7624
                                                                          -0.05221
                   O2 = -0.8259t+18.163
                       R2 = 0.7538
                                          10           15
                                            Time (hours)
     20
25
Figure 14.9 Oxygen depletion test at vapor probe 70F.  Solid and dashed lines denote linear
regression for zero- and first-order kinetics respectively excluding last data point at zero
concentration.
                                            224

-------
             20 -
              15  -
              10 -
               5  -
                                                                             70G
                                                  O2= 16.296e
                                                  R2 = 0.9859 (first-order)
                      O2 = -0.3432t+ 15.972
                      R2 = 0.969 (zero-order)
                                          10           15
                                            Time (hours)
                                            20
                                   25
Figure 14.10  Oxygen depletion test at vapor probe 70G.  Solid and dashed lines denote linear
regression for zero- and first-order kinetics respectively.
             20 -
              15  -
          d  10
                                                                             70H
O2 = -1.75t+7.00
   R2 = 0.6447
                                                        -0.8571
O2 =11.562 e
   R2 = 0.7077
                                          10           15
                                            Time (hours)
                                            20
                                   25
Figure 14.11  Oxygen depletion test at vapor probe 70H.  Solid and dashed lines denote linear
regression for zero- and first-order kinetics respectively excluding last data point at zero
concentration.
                                            225

-------
            20 -
            15 -
            10 -
             5 -
                                                                              701
O2 =-3.065t+13.386
    R2 = 0.9420
                                                             O2 = 21.086e
                                                               R2 = 0.9934
                                                                        -0.6991
                                         10            15
                                           Time (hours)
                                                 20
25
Figure 14.12  Oxygen depletion test at vapor probe 701.  Solid and dashed lines denote linear
regression for zero- and first-order kinetics respectively excluding last data point at zero
concentration.
            20 -
             15  -
             10 -
              5  -
              0
                                                                              70J
                               O2 = -1.56211+12.657
                                   R2 = 0.7978
                                      O2= 13.147e
                                        R2 = 0.8414
                                                                    -o.iviet
                0
                                                 20
25
                             5            10           15
                                            Time (hours)
Figure 14.13  Oxygen depletion test at vapor probe 70J. Solid and dashed lines denote linear
regression for zero- and first-order kinetics respectively excluding last data point at zero
concentration.
                                            226

-------
             20 -
             15 -
             10 -
              5 -
                                                                            70K
                                                             O2 = 21.574e
                                                                R2 = 0.9890
                 O2 = -1.8664t + 20.893
                      R2 = 0.9979
                                          10           15
                                           Time (hours)
             20
             25
Figure 14.14  Oxygen depletion test at vapor probe 70K.  Solid and dashed lines denote linear
regression for zero- and first-order kinetics respectively excluding last data point at zero
concentration.
            20 -
                                                                            70M
                      O2 =-3.61471+16.987
                          R2 = 0.9786
                                         10
15
                                                               O2 = 27.445 e
                                                                   R2 = 0.939
20
                                                                           -0.64561
25
                                           Time (hours)
Figure 14.15  Oxygen depletion test at vapor probe 70M.  Solid and dashed lines denote linear
regression for zero- and first-order kinetics respectively excluding last data point at zero
concentration.
                                            227

-------
            20 -
            15 -
             10 -
             5 -
             0
                                                                            70N
O2 =-4.12431+ 10.414
    R2 = 0.8302
                0
                                             20
                           25
                             5           10           15
                                           Time (hours)
Figure 14.16. Oxygen depletion test at vapor probe TON.  Solid and dashed lines denote linear
regression using and not using last data point at zero concentration respectively.
            20 -
            15 -
            10
             5 -
             0 -
                             02 = -2
                                 R:
             621+18.456
             = 0.9475
                                                                -0.21511
O2=19.418e
    R2 = 0.9932
                                         10           15
                                           Time (hours)
                                             20
                                                                             70P
                           25
Figure 14.17. Oxygen depletion test at vapor probe TOP.  Solid and dashed lines denote linear
regression for zero- and first-order kinetics respectively excluding last data point at zero
concentration.
                                           228

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           20 -
            15 -
                                                                             70Q
                • '•
                        O2 =-3.30931+15.676
                             R2 = 0.9626
                                                               -0.62161
O2 = 24.70e
  R2 = 0.9465
                                         10           15
                                           Time (hours)
              20
25
Figure 14.18. Oxygen depletion test at vapor probe 70Q. Solid and dashed lines denote linear
regression for zero- and first-order kinetics respectively excluding last data point at zero
concentration.
             20  -
              15  -
              10  -
               5  -
                                                                             70R
                                O2 = -1.7044t+16.519
                                     R2 = 0.9661
             -0.13S91
    O2=17.00e
      R2 = 0.9917
                                          10           15

                                            Time (hours)
              20
25
Figure 14.19. Oxygen depletion test at vapor probe TOR.  Solid and dashed lines denote linear
regression for zero- and first-order kinetics respectively excluding last data point at zero
concentration.
                                            229

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              16

              14 -

              12 -

              10 -

               8 -

               6 -

               4

               2 -

               0
R = 0.6480
                                                 70S
O2 = -0.2977t+ 11.006
    R2 = 0.4984
                 0
                 10           15
                  Time (hours)
      20
25
Figure 14.20. Oxygen depletion test at vapor probe 70S. Solid and dashed lines denote linear
regression for zero- and first-order kinetics respectively.
           20 -
           15 -
           .10 -
            5 -
                                                  70T
                                             O2 = -0.01281+16.207
                                                 R2 = 0.0859
                                        10           15
                                          Time (hours)
                                         20
                  25
Figure 14.21. Oxygen depletion test at vapor probe 70T. Solid line denotes linear regression for
zero-order kinetics.
                                           230

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             20 -
             15 -
             10 -
              5 -
                                                                             70U
                                                              O2 = 12.6038 e
                                                                 R2 = 0.9557
                                                                          -0.05471
                    O2 =-0.603 61+12.525
                        R2 = 0.9465
                                          10           15
                                            Time (hours)
20
25
Figure 14.22. Oxygen depletion test at vapor probe 70U. Solid and dashed lines denote linear
regression for zero- and first-order kinetics respectively excluding last data point at zero
concentration.
            20 -
             15 -
             10  -
              5  -
                                                                             70V
                                                     O2= 18.3641 e
                                                         R2 = 0.9936
                                                                 -0.05781
                  O2 = -0.9202t+ 18.237
                       R2 = 0.9896
                                          10            15
                                           Time (hours)
20
25
Figure 14.23. Oxygen depletion test at vapor probe 70V. Solid and dashed lines denote linear
regression for zero- and first-order kinetics respectively excluding last data point at zero
concentration.
                                            231

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           20 -
           15 -
           lO
            5 -
          70X
                                                           O2 =-0.57671+ 13.114
                                                               R2 = 0.9823
                O2= 16.5138 e
                   R2 = 0.9978
                                        10           15
                                         Time (hours)
20
25
Figure 14.24.  Oxygen depletion test at vapor probe 70X. Solid and dashed lines denote linear
regression for zero- and first-order kinetics respectively excluding last data point at zero
concentration.

14.5   Conclusions

1.      Respiration tests provide an inexpensive and rapid means of assessing in-situ aerobic
       metabolic activity.  From a soil venting monitoring perspective, respiration testing should
       be conducted periodically to assess change in oxygen depletion rates. As remediation
       progresses,  oxygen depletion rates would be expected to decrease over time, eventually in
       some cases  approaching background conditions.

2.      It has become common practice to use zero-order oxygen consumption rates to directly
       estimate hydrocarbon degradation rates.  Limitations of this method include: (1) reference
       hydrocarbons used for the stochiometric mass ratio between hydrocarbon degradation and
       oxygen consumption may not be representative of compounds present at a spill or
       hazardous waste site; (2) field data may show first-order degradation rates; (3)  regulatory
       programs, specifically Superfund and RCRA, mandate specific contaminant
       concentration reduction; (4) the hydrocarbon-oxygen equivalency equation represents the
       minimum oxygen requirement to completely mineralize a reference compound based on
       the assumption that either there is no cell synthesis or endogenous respiration, and that
       oxygen does not serves a terminal electron acceptor in the oxidation of reduced inorganic
       compounds. Thus, it is possible that the hydrocarbon-oxygen equivalency equation will
       overestimate actual hydrocarbon degradation rates.
                                          232

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                               20
                                    Benzene  (mg/kg  soil)
                                          40         60         80
                                                                         100
                                                       Sep-94,  After
                                                       Remediation
Aug-92, Before
Remdiation
                320
Figure 14.25. Vertical distribution of benzene at TOD before and after venting

3.      The best use of respiration testing appears to be as a general indicator of soil amendment
       (moisture, temperature, nutrients, etc.) effectiveness and microbial activity. Respiration
       testing could be used to indicate inhibition of degradation.  For instance, if respiration
       testing reveals low oxygen consumption and carbon dioxide generation in soils containing
       high concentrations of readily biodegradable compounds, further studies could be
       conducted to determine if biodegradation is being limited by insufficient moisture
       content, toxicity, or nutrients. Persistent, low oxygen levels may indicate an insufficient
       supply of oxygenated air at a location suggesting the need for air injection, higher
       extraction well vacuum, additional extraction wells, or additional soils characterization
       which may indicate the presence of immiscible fluids impeding the flow of air.
                                           233

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                           2000
                                     TPH (mg/kg soil)
                                   4000     6000     8000
                                                            10000
                                                                     12000
                               Aug-92,  Before
                               Remediation
Sep-94,  After
Remediation
               320
Figure 14.26 Vertical distribution of total petroleum hydrocarbons (TPH) at TOD before and
after venting

4.      The significance of distinguishing zero- and first-order kinetics for oxygen consumption
       is that oxygen depletion is constant for the former and a function of oxygen concentration
       for the latter. If the oxygen depletion rate is a function of oxygen concentration, then
       oxygen may become a limiting factor below some concentration level above zero.  In the
       site study, discernment of zero- versus first-order oxygen depletion kinetics, was achieved
       at some but not all monitored locations due to too few data points at critical times. In
       areas of high contamination, oxygen depletion followed zero-order kinetics.

5.      Soil sampling at the test site after 18 months of venting revealed removal of BTEX
                                          234

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compounds to very low concentrations but poor removal of higher molecular weight and
lower volatility compounds.
                                 235

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15.    EVALUATION OF MASS EXCHANGE TO THE ATMOSPHERE AND GROUND
       WATER

       The inability to meet soil and soil-water concentration-based standards has led to the
development of alternative methods to assess venting performance and closure. In section 2, a
strategy for assessment of venting performance and closure was proposed based on regulatory
evaluation of four components: (1)  site characterization, (2) design, (3) monitoring, and (4) mass
flux to and from ground water. These four components form "converging lines of reasoning" or a
"preponderance of evidence";  an approach common in complex scientific analysis and decision
making.  Such an approach increases the likelihood of correctly assessing the performance of a
venting system and its suitability for closure.  Considering that venting is often applied to remove
contaminant mass from variably saturated media for ground-water protection, it would seem that
only an analysis of mass flux to and from ground water is necessary for assessment of soil
venting operation and closure. There is, however, so much uncertainty in vadose zone solute
transport modeling and difficulty in field validation that sole use of vadose zone mathematical
modeling for decision making appears unwise.  Nevertheless, mass flux evaluation is still a
crucial component of the performance assessment and closure strategy.

15.1   Modeling Approach

       Mathematical models which simulate soil-water movement to ground water vary in
complexity from simple and conservative algebraic water-balance equations that neglect
degradation and volatilization to more process descriptive finite-difference and finite-element
numerical codes incorporating Richards' equation for soil-water flow and mass transport kinetics.
Unsaturated zone modeling should  ideally: (1) be rigorous enough to incorporate major fate and
transport processes, (2) have input parameters that can be readily collected, and (3) be
sufficiently user-friendly to  enable use by consultants and regulators. Unfortunately,
simultaneous attainment of all of these goals is unrealistic. Lack of data to support spatial
discretization of soil properties (e.g., capillary  pressure parameters, hydraulic conductivity,
porosity, bulk density, moisture content, total organic carbon content) and contaminant
distribution, commonly limits the use of sophisticated mathematical models. One approach to
this problem is to use fictitious but  "reasonable" input for numerical two- or three-dimensional
modeling. This approach is applicable when attempting to simulate contaminant transport in a
layered or high heterogeneous soil environment, when complex boundary conditions exist, or
when consideration of mass transport kinetics is desirable. Another approach,  is to start with
commonly collected data and use fairly simplistic, one-dimensional, analytical screening models
to gain insight into the potential magnitude of solute transport to and from ground water.  This
approach is similar to screening models developed by Jury et al. (1984a-c).  More sophisticated
numerical modeling can then be utilized if additional supporting data becomes available or a
more  detailed analysis of the problem is desired. Regardless of sophistication, all modeling is a
simplification of reality containing  considerable uncertainty in  simulated results.

       When choosing and  interpreting output from a mathematical model, it is important to
                                           236

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keep in mind the intended objective of modeling. During regulatory decision making in regard to
unsaturated zone remediation, the purpose of mathematical modeling is often simply to
determine whether or not (yes or no) a long term potential risk to ground water exists.  If there is
a perceived  immediate or long-term risk, then unsaturated zone remediation is initiated or
continued. If not, then model output is combined with other factors such as the adequacy of site
characterization, to forgo remediation or commence closure.  In sharp contrast to fluid flow
(ground-water and gas) modeling efforts, vadose zone soil-water movement and contaminant
transport simulations are rarely calibrated or validated because of the time scale of modeling
(years), extensive site characterization needs, and related cost. Thus, for regulatory decision
making, one-dimensional analytical or numerical modeling may be appropriate under some
conditions. However, even at this level of modeling,  a sensitivity analysis of how model input
and selected boundary conditions affect model output and decision making is critical. The
objective here then is to conduct one-dimensional VOC transport simulations to assess the effect
of water saturation, NAPL saturation, degradation-half-life, and selection of boundary conditions
at the vadose zone - ground-water interface on model output and regulatory decision making in
regard to vadose zone remediation, specifically venting initiation or closure.  A one-dimensional
analytical, public domain code, termed VFLUX (Vertical Flux), was developed to support these
simulations.  Table 15.1 provides a summary of the model's characteristics and capabilities

       VFLUX was written to replace VLEACH (Ravi and Johnson, 1997), a one-dimensional,
finite-difference code, which is by far the most commonly used model to assess initiation or
closure of vadose zone remediation.  VFLUX has all the capabilities of VLEACH in addition to
allowing a time-dependent, first-type lower boundary condition, and consideration of the effect
of degradation and NAPL saturation on VOC removal and partitioning. The time-dependent
boundary condition is the centerpiece of the mass flux assessment because it dynamically links
performance of ground-water remediation to venting closure. VFLUX is a significant
improvement over VLEACH because of these increased capabilities and exact  method of
solution.  The governing equation with associated initial and boundary conditions of VLEACH
were solved  numerically, even though it could have easily been solved analytically. The
determination of mass flux requires calculation of a spatial derivative at the water table interface,
which is prone to large error when spatial discretization is coarse and a sharp concentration
gradient exists between grid blocks.  The analytical solutions of VFLUX provides exact values of
mass flux at any location. Also, in order to calculate  concentration profiles at a specified time,
the transport equation has to be solved numerically at each time step starting from the initial
distribution.  Analytical methods allow direct solution at the time of interest. In addition,
analytical solutions are not subject to convergence and stability problems arising from the use of
coarse  spatial grids and large time steps.  Thus, there  is less chance that an inexperienced user
will generate inaccurate output due to poor specification of numerical simulation criteria (e.g.,
grid size).
                                          237

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Table 15.1 Summary of VFLUX Model Capabilities
 Capability
               Description
 availability


 dimensionality

 method of solution

 upper boundary condition

 lower boundary condition

 initial condition

 mass transport kinetics

 degradation

 water saturation

 NAPL saturation


 infiltration rate

 computation soil-gas and soil-
 water tortuosity coefficients for
 diffusion

 hydrodynamic dispersion

 model output
               public domain (to be available without cost on EPA internet
               site)

               one-D

               analytical

               first-type (zero concentration)

               choice of first-type, time-dependent or finite, zero-gradient

               variable (input soil concentration as a function of depth)

               none - local equilibrium assumption

               first-order (input as degradation half-life)

               constant in time and space

               constant in time and space (NAPL assumed nonvolatile and
               nondegradable)

               constant

               Millington-Quirk relationship



               function of infiltration rate

               - parameter file summarizing input
               - flux file showing average concentration and flux to ground
                water at the atmosphere as a function of time
               - mass balance file showing remaining mass in soil profile
               and mass lost to ground water, atmosphere, and decay
15.2   Model Formulation

       Analysis of VOC transport in unsaturated media starts with a mass balance or continuity
equation for each organic compound in soil.  The continuity equation states that change in
storage equals the divergence of flux. This can be represented mathematically by
•Pbcs+encn}-r=--
                                                                                 (15.1)
                                          238

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where JT = mass flux across a vertical plane [ML"2!"1] and r = a reaction term [ML"3!"1]. This
equation can be represented more compactly as
where

CT=RCw=-^Cg                                                                (15.3)
_R_
H
and


R = H0+dw+Kocfocpb+	^	.                                    (15.4)
                                7=1
If it is assumed that NAPL composition does not change with time, R may be approximated by
                                                                                 05.5)
       Volatile organic compound transport in the vadose zone is caused by soil-water
advection, mechanical dispersion, and diffusion (aqueous and gaseous).  In derivation of the
advective-dispersive equation, mechanical dispersion and diffusion are combined to form an
overall dispersion coefficient. Transport by other mechanisms such as gaseous advection due to
barometric pressure variation, gas density variation, and infiltration fronts is normally considered
negligible.  Advective soil-water flux, Jw is expressed as

JW=VWCW                                                                      (15.6)

where qw = infiltration flux of water [LT1]. Flux due to molecular diffusion in the liquid and
gaseous phases (Jdw and Jdg respectively) is described by Pick's First Law


^,W=~DW^-                                                                 (15.7)
             dz

and
                                          239

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The negative sign enables values of flux to be positive since movement is from areas of greater
concentration to areas of lesser concentration resulting in a negative slope or gradient.  Pick's
First Law is a subset of a more general group of equations called "gradient flux laws" where flux
is a function of a some property (in this case diffusion coefficients) and a gradient. Other
examples are Darcy's Law (fluid flow), Fourier's Law (heat conduction), and Ohm's Law
(electrical conductivity). Diffusion will occur as long as a concentration gradient exists.

       In porous media, diffusion can not proceed as fast as it can in free water or gas because
solute molecules collide with solids and must travel tortuous paths around mineral grains. Thus,
diffusion coefficients for water and gas must be modified to account for the distance of the
flowpath followed by solute molecules. Modification is accomplished by use a tortuosity factor.
Flowpaths in a well-sorted soil would be expected to be shorter than those in a poorly sorted soil
in which smaller grains fill in the pore spaces of larger grains (Fetter, 1992). To incorporate the
effect of tortuosity, Picks First Law can be modified by


Jd,w=~TwDw                                                                     (15.9)
and
              ac
       A number of expressions have been developed to estimate tortuosity factors for gas and
aqueous diffusion. It stands to reason that the presence of soil-water will interfere with gas
diffusion and increase the flowpath for vapor molecules.  The theoretically based model derived
by Millington and Quirk (1961) relationship
     010/3
                                                                                  (15.11)
can be used to estimate Tg .  The equation illustrates the importance of volumetric gas content in
estimation of a gas tortuosity. This model has been shown to be in good agreement with data
over a wide range of water saturations (Sallam et al., 1984).  Soil-gas will interfere with aqueous
diffusion by forcing aqueous diffusion across pendular rings instead of directly across a pore.
Because liquid tortuosity modifies liquid diffusion in the same way that gas tortuosity modifies
gas diffusion, the Millington-Quirk (1961) tortuosity relation may also be used to estimate TW by
(Juryetal.  1991)


                                           240

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                                                                                  (15.12)
       Total mass flux across any plane in the vadose zone can now be represented by
                                                                                  (15.13)
where Dm is a mechanical dispersion coefficient [L].  Equation (15.13) is equivalent to:


j = ^LC  -~CT
      R
              R  dz
                                                                                  (15.14)
where
                                                                                  (15.15)
It is important to note from equation (15.15) that mechanical dispersion coefficient has been
combined with the aqueous diffusion coefficient to represent flux as a function of a concentration
gradient. This in essence describes mechanical dispersion as a Fickian process or flux that is a
function of a concentration gradient. In reality, it is known that mechanical dispersion is not a
Fickian process but the end result is the same, namely contaminant spreading.  Considering
mechanical dispersion as a Fickian process simplifies solution of the advective-dispersive
equation and has become the accepted method of handling overall dispersion which is considered
a combination of diffusion and mechanical dispersion.

       Because of the presence of a capillary fringe and variation in moisture content with
depth, the dispersion coefficient will not be constant in a homogeneous media. One method of
accounting for this variation is by specifying an effective dispersion coefficient (Deff) defined as
Deff=Dmvw+L
                 L
                                      1-1
                           dz
                                                                                  (15.16)
The integral expression in equation (15.16) can be approximated by
L
                         -i
L
 i\
I
                                                                1-1
                                                  10/3
                                                                                  (15.17)
                                           241

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where water saturation in each discrete layer (Swi) of length L; is represented.  In this way, the
reduction in dispersion due to the presence of a capillary fringe and other layers of media having
high water saturation can be explicitly accounted for. Equation (15.17) can then be used to
determine an effective water saturation for model input such that
          -,10/3
           w
                                                                        1-1
                                       I
                                                         °/3
                                                                   10/3
(15.18)
Johnson and Ettinger (1991) used a similar approach to calculate and effective diffusion
coefficient in unsaturated media to assess the impact of upward VOC migration into buildings.

       The only remaining factor in deriving the advective-dispersive equation is how to handle
the sink or degradation term.  Since degradation rates published in the literature are not
referenced to specific phase (e.g., aqueous), the reaction term, r, is expressed as a function of
total soil concentration and a first-order degradation constant (K)  [T"1]
                                                                                    (15.19)
Degradation half-life (t1/2) [T] equals:
      0.693
        K
                                                                                    (15.20)
Finally, the governing equation for one-dimensional transport of a volatile organic compound can
be expressed as:
              dz
- + D-
                           — KRC.
                                  T •
(15.21)
15.3   Representation of the Initial Condition

       Use of the advective-dispersive equation consists of choosing a set of initial and boundary
conditions which may be accepted as adequate to the real situation then finding an analytical or
numerical solution. Often selection of boundary conditions to the advective-dispersive equation
is neither straightforward nor intuitive. In simulations to follow, initial conditions describe total
(solids, NAPL, water, and air) soil concentrations as a function of depth at time zero. In
application, total soil concentrations would presumably be determined directly from soil core
analysis (solvent extraction). The initial condition can be represented by analytical expression or
by a piecewise continuous function. The latter approach was utilized in VFLUX because any
initial total soil concentration profile can be specified at any desired resolution without
                                            242

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modification of the code. Mathematically, the initial condition is represented by:

Cr(z,0) = /(z)                                                                    (15.22)

where f(z) [ML"3] is a piecewise continuous function.

15.4   Selection of the Upper Boundary Condition

       At the upper end of the modeled domain (z = 0), a first-, second-, or third-type boundary
condition could conceivably be specified for solute transport modeling.  Use of a first-type
boundary condition simply states that concentration is varied as a function of time or held
constant by some means at a boundary.  For instance, if it is assumed that clean air and water is
present at the surface, a first-type, constant (zero) concentration boundary condition could be
specified by

Cr (0,0 = 0.                                                                      (15.23)

A boundary condition of the second-type represents a fixed gradient at a boundary surface
expressed as a time-dependent function, a constant value, or zero.  For instance, the finite zero-

gradient condition represented by


                                                                                  (15.24)
could be used to simulate no dispersive flux at soil-atmosphere boundary. However, since
diffusive transfer from soil to the atmosphere is a desired output of modeling, this boundary
condition is inappropriate. A boundary condition of the third-type represents mass flux into the
modeled domain proportional to the difference in concentration at the boundary and surrounding
medium. The inlet third-type boundary condition for this application can be expressed by
                        = ^LCsurf(t)                                              (15.25)
                   z=o+   R  '

where CsurfT (t) is a "flux" concentration at the soil surface as a function of time. This boundary
condition though fixes flux into the domain when diffusive flux out of the domain is a desired
model output.  Jury et al. (1984 a,b) and Yates et al. (2000) formulated a third-type boundary
condition at the surface by assuming that VOCs had to pass through a boundary layer by vapor
diffusion before entering a well-mixed region of the atmosphere. This boundary layer could be a
surface crust, compacted zone, or a resistive material such as polyethylene. For this application,
an upper boundary expressed by
                                           243

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D dCT
             = h(c?rf(t)-CT(0,t))                                                 (15.26)


could be utilized where h is a mass transfer coefficient [LT"1] which could be approximated by h
= Dg/b where b is the thickness of air boundary [L].

       Selection of an appropriate boundary condition at the surface is complicated by mass
balance and concentration continuity considerations.  In a flowing fluid, the concentration of a
solute can be defined in two ways - "resident" and "flux" detection.  Kreft and Zuber (1978)
define resident detection as the mass of solute per unit volume of fluid contained in an
elementary volume at a given instant. They define flux detection as the mass of solute per unit
volume of fluid passing through a given cross section at an elementary time interval or the ratio
of solute flux to volumetric fluid flux.  Both concentrations though are in fact volume averaged
(Kreft and Zuber, 1986). Kreft and Zuber (1978) state that in practice, flux detection is realized
by measuring the mean concentration in an outflowing fluid.  Referring to miscible displacement
studies, Parker and van Genuchten (1986) state that in the case of a laboratory column where
flow from the entire cross-section is collected, a definite measure of flux concentration is
obtained.  Measurement of flux detection inside a column though is usually impossible without
disturbing flow lines. Parker and van Genuchten (1986) state that the question arises over the
magnitude of local perturbation in the flow field caused by fluid withdrawal and the installation
of sampling devices.  Uncertainty is compounded by problems with time averaging.
Measurement of resident concentration requires that the time to obtain the sample is small in
comparison to the time in which fluxes occur (Parker and van Genuchten, 1986).

       Kreft and Zuber (1978) also discuss differences between flux and resident injection. For
resident injection, a first-type boundary condition is utilized and concentration continuity across
the interface is maintained.  For flux injection, concentration continuity is lost across the
interface to allow flux across the interface as is apparent from equation (22). Kreft and Zuber
(1978) state that flux injection is achieved in practice by introducing fluid with a solute through
the whole entrance cross-section with no back mixing, van Genuchten and Parker (1984) state
that for miscible displacement experiments, a third-type boundary condition or flux injection
results in conservation of mass whereas resident injection does not maintain solute flux
continuity across the inlet position. In one-dimensional miscible displacement studies, it is now
generally agreed that the flux injection boundary condition is the correct inlet boundary condition
(van Genuchten and Parker, 1984; Parker and van Genuchten, 1984, Barry and Sposito, 1988;
and Novakowski, 1992 a,b). However, this is not necessarily true for gas displacement or
diffusive vapor transport. Gimmi and Fliihler, (1996) demonstrated that the selection of a
resident or flux injection boundary condition for gas transport is directly dependent on the ratio
of the cross-sectional areas of the tubing leading to a column and the column itself. Low gas
velocity and a large tubing to column ratio (in field-scale unsaturated solute transport, this ratio is
essentially unity) favor specification of a resident injection or type-one boundary condition.
They state that for flux injection, the influence of diffusion must be negligible compared to the
influence of mechanical dispersion. This typically occurs for miscible displacement experiments

                                          244

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but occurs for gas flow experiments only under conditions of high velocity and low tubing to
column cross-sectional area ratios. Gimmi and Fliihler (1996) state that for gaseous advective-
dispersive transport, the exchange of solutes at a boundary will resemble a resident injection
unless diffusive flux is restricted at the boundary.  They state that in most cases it should be
possible to model vapor exchange directly at the soil surface using a first-type boundary
condition unless mixing with the atmosphere is strongly restricted by an upper boundary.

15.5   Selection of Lower Boundary Condition

       Use of a third-type boundary condition similar to equation (15.25) would appear
unreasonable at the domain outlet because it again fixes flux out of the domain when in fact the
one objective of modeling is to calculate flux into and out of ground water. Flux from an outlet
can not be known a priori but must be measured or simulated.  Intuitively, a diffusive flux
boundary however could conceivably be used similar to equation (15.26). This boundary
condition though negates advection from the vadose zone to ground water. A first-type time-
dependent boundary condition represented by

CT(L,t) = g(t)                                                                    (15.27)

where g(t) = time-dependent soil concentration [ML"3], though could be used to link ground-
water and unsaturated zone aqueous concentrations at the base of the modeled domain. The
time-dependent term could be described by an analytical expression as is common for
exponential decay of a source term or by pulse loading of constant concentrations for fixed
periods of time. The latter approach takes advantage of the principle of superposition and allows
development of a general  analytical solution for any pulse loading function whereas use of
former requires either separate development of analytical solutions for each analytical loading
expression or modification of computer  code and subsequent numerical integration for time
dependent function. A first-type lower boundary condition though is not appropriate if one
wishes to use soil-water concentration at the outlet with a ground-water dilution attenuation
factor to assess regulatory compliance since outlet concentrations are set by the boundary
condition itself.

       At the base of modeled domain, a finite or semi-infinite (z goes to infinity) zero-gradient
boundary condition could also be utilized. Operationally, the primary difference between a finite
and semi-infinite domain zero-gradient lower boundary condition is that the former allows only
advective mass transfer to ground water while the latter allows diffusive and advective mass
transfer to ground water since the interface is not distinguishable from any other plane  over
which mass transfer occurs. Use of a semi-infinite zero-gradient boundary condition requires the
assumption that the exit boundary does not influence the system. In miscible displacement
laboratory column studies, this condition is ensured by the use of small diameter tubing at the
column effluent to prevent backward diffusion (Gimmi and Fliihler, 1996). This laboratory
configuration though is irrelevant to the  field.
                                          245

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       Neither a finite or semi-infinite zero-gradient boundary condition allows diffusive mass
transfer from ground water to the unsaturated zone. Also, the use of a zero-gradient boundary
condition is questionable when ground water containing elevated levels of VOCs underlies soils
containing low levels of VOCs.  In this situation, the primary direction of mass flux could be
from ground water to the vadose zone as a result of diffusive exchange. As Kreft and Zuber
(1986) state, it is difficult to imagine a boundary permeable to convective flow but impermeable
to dispersive flow. Kreft and Zuber (1986) and Parker and van Genuchten (1986) point out that
considerable ambiguity remains regarding the physical conditions under which various outlet
boundary conditions are valid. In VFLUX, the user was given the option of selecting a first-type,
time-dependent or a finite, zero-gradient lower boundary condition.

15.6   Derivation of Analytical Solutions for First- Type. Time-Dependent Lower Boundary
       Condition

       Equation (15.21) can be expressed in dimensionless units:


                         KCr                                                    (15.28)
 ar        az   az

where:
    RL2
                                                                                 (15.29)
Z=—                                                                           (15.30)
     L
                                                                                 (15.31)
     2D
                                                                                 (15.32)
      D

with initial and boundary conditions:

Cr(0,r) = 0                                                                      (15.33)

CT(\,T) = g(T)                                                                   (15.34)

Cr(Z,0) = /(Z)                                                                  (15.35)


                                          246

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The dimensionless parameter, A, is equivalent to a Peclet number if it is divided by the
volumetric moisture content and multiplied by two. In this model, A has been termed the half-
Peclet number.

       Equation (15.28) subject to equations (15.33) through (15.35), is linear and hence can be
solved by the principle of superposition. The solution can be written as:

CT(Z,T} = C^(Z,T) + CITI (Z,T]                                                  (15.36)

where C\ (Z,T) and CnT (Z,T) are  subject to different initial and boundary conditions. The first
solution, C!T (Z,T) is due to a nonuniform initial condition, and the second solution, CnT (Z,T) is
due to a nonhomogeneous lower boundary condition.  The conditions are as follows:
C/(Z,0) = /(Z)                                                                (15.37)


C/(0,r) = 0                                                                    (15.38)


C/(l,r) = 0                                                                    (15.39)

and

C/7(Z,0) = 0                                                                   (15.40)


C/7(0,r) = 0                                                                   (15.41)


C»(\J) = g(T)                                                                 (15.42)

A solution to C!T (Z,T) can be obtained by the separation of variables technique

            00
C^(Z,r) = ^4, exp(AZ-
-------
To avoid numerical integration of complex analytical functions, f(Z) can be given by as a
piecewise-constant function
               Zk
-------
The residues at p = - (A2+n2n2+K) = - con2 (for n=l,2,...) are
exp(A(Z-]
                                                                                 (15.53)
Therefore,
 F(Z,r) = exp(A(Z-l))
sinh(^Z)
 sinh(£)
                                        (-1)"
                                                    (15.54)
Now, the solution for any arbitrary function g(T) can be approximated as closely as desired by a
sum of j discrete step functions (superposition with time) as follows:
 g(T) =
                                                                                (15.55)
                       7=1
where the asterisk denotes soil concentration at the vadose zone - ground-water interface and
U(T-Tj) is the Heaviside unit-step function defined such that it is equal to one when T>Tj, and
equal to zero when Tl(T-T.) .
                                              en
                                         n=l    n
                                              U(T-Tj]
                                                                                 (15.57)
Average total soil concentration of a compound as a function of dimensionless time, T, can be
calculated by:
                                          249

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                                                                              (15.58)
or
           w=i
                         A
                               22
             00    2 ( -i\n -coiT
                 n (-1) e
                               i+(-i
                                                      U(T-TJ
                                                             U(T-T,
                                                l + (-l)w+V




                                                                              (15.59)



Total mass per unit area of a compound at any given dimensionless time, T, can be calculated by
       = LCT(avg}=L^CT(Z,T}dZ.
                                                                              (15.60)
Total (advective and diffusive) mass flux of a compound at the ground-water - unsaturated zone

interface (Z=l) at any given dimensionless time, T, can be expressed by
           R
                       R  az
                                                                              (15.61)
                              z=i
or
                                        250

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 /
J
                     D
                R   RL
                          °° n2e~c
       A + £coth(£) + 2;r2^	
                                          n=l
            ~C
                   qw    D
                   R   RL
                                    ;S
                                     w=l
                                                  n2e
    D
   ~RL
                                              U(T-TJ]    (15.62)
Since the orientation of the z-axis is positive downwards, positive values of mass flux denote
migration to ground water while negative values of mass flux denote migration from ground
water to the vadose zone.  Cumulative total (advective/diffusive) mass transfer to and from
ground water at any selected dimensionless time, T, can be expressed by
MT(l,T} =
RLL
 D
JT(\,T)dT
                                                                           (15.63)
or
MT(\J} =
                   D
i^lT_T\_
 D  \     J>
               (A +
                  A
                             ••£-
                                               «=i
                                    K)
                                                 1    "V!
                                                 1-e
                                          W=l
                                 K)
                                              U(T-T,)
                                                                            (15.64)

Total mass flux (in this case only diffusive transfer) of a compound, at any dimensionless time,
T, to the atmosphere (Z=0) can be calculated by
            D
                                                                            (15.65)
                   z=o
or
                                        251

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                  \T)=-°-
                     '    RL
                                n=l
                                                         n=l
D_
RL
                    E
              sinh(^)
                  v  '
+
2*2£-
                                                         w=l
                                                                                              (15.66)
                                                  U(T-Tt]
              Since the orientation of the z-axis is positive downwards and a first-type boundary condition of
              zero constant concentration at the surface has been set, mass flux to land surface will always
              contain negative values. Cumulative diffusive mass transfer to the atmosphere at any selected
              dimensionless time, T, can be calculated by
                         KT2  rT

                                JT(Q,T}dT
                                                                 (15.67)
             or
              MT(0,T) = -
Z
*z-
  W=l
                                     -A
                                         sinh(^)
                                                             w=l
                                                              z=l
                                                                                          U(T-T,)
                                                                                              (15.68)
             When mass flux at the unsaturated zone - capillary fringe is unidirectional to the capillary fringe,
             mass lost to decay can now be calculated by
                             (T) -MT (1, T) -MT (0, T).
                                                                 (15.69)
              15.7   Derivation of Analytical Solutions for Zero-Gradient Lower Boundary Condition

                    Consider equation (15.28) subject to
                      = 0
                                                                 (15.70)
                   Z=l
             instead of equation (15.34). Separation of variables where
                                                       252

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CT(Z,T) = F(Z)G(T)                                                             (15.71)




can be used to reduce the governing partial differential equation into two ordinary differential

equations
 d2F   _ . dF
     -2A —+(
-------
      fe~AZ /(Z)sin(4,Z)
-------
jT(\,T) = ^CT(\,T) = ^y 4/A-^jsm(/lB).                                   (15.87)
          R           R ^
                         n=l

Cumulative advective mass transfer to ground water at any selected dimensionless time, T, can
be expressed by

           RT2  rT
MT(\,T} =	  \JT(\,T}dT                                                     (15.88)
or
                                                                                (15.89)
Flux at the surface is

                           D
            RL  dZ  z=0
                                                                                (15'90)
Cumulative diffusive mass transfer to the atmosphere at any selected dimensionless time, T, can
be calculated by
           KT2
                  JT(0,T)dT                                                    (15.91)
or

                      \\-e
                                                                                (15.92)
              n=\         ^n

As before, mass lost to decay is calculated as before.

       Appendix H provides sample input and output files and the source code for VFLUX.
Model input is in a free FORTRAN format.  This format provides easy and rapid model.  Text
output files can easily be open as a spreadsheet file and manipulated for graphing.

15.8   Example Simulations

       Table 15.2 summarizes input used for VFLUX simulations. Degradation half-life, water
saturation, and NAPL saturation were left as variable as these input parameters likely represent

                                         255

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the greatest sources of uncertainty and sensitivity to model output. Use of equation (15.18)
assuming a vadose zone thickness (unsaturated media + capillary fringe) of 520 cm, a capillary
fringe thickness of 10 cm, and saturation values above the capillary fringe varying from 0.1 to 0.9
resulted in  effective water saturations varying from 0.703 to 0.902.  Thus, effective water
saturations of 0.7, 0.8, and 0.9 were used for model simulation to bracket and provide for a wide
range of model output possibilities.  With the exception of simulations conducted to assess the
impact of infiltration rate on mass flux to ground water, illustrated in Figure 15.4, a relatively
low infiltration rate (0.035 cm/d or 5.04 inches/year) was selected for remaining simulations to
represent venting application in arid regions of the United States and in more humid regions
where an asphalt cap or synthetic liner is placed on the soil surface to minimize infiltration.
Trichloroethene (TCE) was chosen for simulation because of its frequent detection at Superfund
sites.

Table 15.2 Input for modeling

sw
sn
11
Pb
Pn
/-i sat
^ w
H
Kd
M0
M;
Dg
DW

qw
Dm
K
Description
water saturation
NAPL saturation
porosity
bulk density
density of NAPL
water solubility for TCE
Henry's Law Constant for TCE
soil-water partition coefficient
average molecular weight of NAPL
molecular weight of TCE
free air diffusion coefficient for TCE
free water diffusion coefficient for
TCE
infiltration flux of water
longitudinal mechanical dispersivity
first-order degradation rate
Value
variable
variable
0.35
2.15
1.462
1.1E-03
0.38
0.126
131.5
131.5
6366.8
0.804

0.035
30.0
variable
Units
Cm water/ Cm pore space
cm NAPL/cm pore space
Cm pore space' Cm soil
§solids'Cm soil
§NAPL'Cm NAPL
g/cm3
(ug/cm3air)/(ug/cm3water)
(ug/gsoiids)/(ug/cm3water)
g/mole
g/mole
cm2/d
cm2/d

cm/d
cm
1/d
                                           256

-------
       A hypothetical Gaussian-shaped soil concentration profile, illustrated in Figure 15.1, was
used to represent the initial condition.  Soil concentrations were input in 10 cm increments for
model simulation.  A hypothetical ground-water concentration versus time profile, typical of
many systems undergoing pump and treat aquifer remediation and illustrated in Figure 15.2, was
used to represent the lower first-type, time-dependent boundary condition. Ground-water
concentrations were input at one year increments for model simulation.  Mass flux to ground
water as a function of three infiltration rates (0.000, 0.035, and 0.050 cm/d) and two water
saturations (0.7 and 0.9) is illustrated in Figure 15.3. As would be expected, mass flux to ground
water increased significantly with increased infiltration rate (advective transport) and decreased
water saturation (diffusive transport). Of particular interest however, is the duration and
potentially environmentally significant magnitude of diffusive mass flux to ground water at high
water saturation and zero infiltration rate illustrated in the embedded image of Figure  15.3.  This
figure reveals why isolating highly contaminated soil (even low permeability soil having a high
moisture content) in a capped landfill decreases but does not eliminate mass flux to ground
water. Isolation of highly contaminated soil or waste may not be environmentally protective in
many circumstances and thus may simply delay but not preclude the use of a source control
technology such as venting.

       Mass flux to ground water as a function of time and water saturation for first-type, time-
dependent and zero-gradient lower boundary conditions are illustrated in Figure 15.4.  Since the
orientation of the z-axis is positive downwards, positive mass flux for the  first-type lower
            1800 T
                  0
100
                                         200         300
                                           Depth (cm)

Figure 15.1 Hypothetical initial soil concentration profile.
400
500
                                           257

-------
     0
20
                                         40           60
                                           Time (years)
Figure 15.2 Hypothetical ground-water concentration profile
     80
                                                                              100
   120 -i
   100 -
&  80 -
    60 -
 o
5
 o
•^-»
I
u.
 01
            	qw = 0.000 cm/d, Sw = 0.7
            	qw = 0.035 cm/d, Sw = 0.7
            	qw = 0.050 cm/d, Sw = 0.7
                          	qw = 0.000 cm/d, Sw = 0.9
                          	qw = 0.035 cm/d, Sw = 0.9
                          	qw = 0.050 cm/d, Sw =0.9
     0
       0
  10
                                    15
35
                                           20     25    30
                                            Time (years)
Figure 15.3 Mass flux to ground water as a function of time, infiltration rate, and water
saturation for a first-type, time-dependent (TD) lower boundary condition. NAPL absent, no
degradation.
                                 258

-------
90 -r
                                              	Sw = 0.7, TD
                                              	Sw = 0.7,ZG
                                              - - -  Sw = 0.8, TD
                                              	Sw = 0.8,ZG
                                              	Sw = 0.9, TD
                                              	Sw = 0.9,ZG
   0
10
15
35
40    45
50
                                          20    25    30
                                            Time (years)
Figure 15.4 Mass flux to ground water as a function of time and water saturation for first-type,
time-dependent (TD) and zero-gradient (ZG) lower boundary conditions.  NAPL absent, no
degradation, qw = 0.035 cm/d.

boundary condition denotes flux from the vadose zone to ground water while negative mass flux
numbers denotes mass flux by diffusion from ground water to the vadose zone.  As previously
mentioned, for the finite, zero-gradient lower boundary condition, mass transfer from ground
water to the vadose zone can not be simulated by virtue of the specified boundary condition.
Thus, mass flux values for the finite, zero-gradient lower boundary condition are always positive.
For soil concentration vertical profile illustrated in Figure 15.1, mass flux to ground water is
always positive for the first-type boundary condition indicating a need to initiate or continue
venting application.  For the first-type, time-dependent lower boundary condition, increasing
water saturation results in a smaller magnitude but longer duration pulse of TCE to ground water
with time to reach the maximum mass flux rate shifted to  increasing time. For the finite, zero-
gradient lower boundary condition, there is also a shift in increasing time to reach the maximum
mass flux rate to ground water but the magnitude of the peak mass flux rate remains about the
same. Since the infiltration rate was held constant at 0.035 cm/d, the shift in peak mass flux rate
to increasing time is due to an increase in the retardation factor from increased water saturation.

       As illustrated in Figure  15.5, increased water saturation results in decreased diffusive loss
to the atmosphere (1- mass fraction to ground water since degradation loss = 0) and hence a
greater mass fraction of initial mass going to the ground water for both types of boundary
conditions. However, the mass fraction of initial mass lost to ground water is  greater for the
first-type, time-dependent lower boundary condition compared to the finite, zero-gradient
                           259

-------
                                   •Sw = 0.7, TD
                                    Sw = 0.8, TD
                                    Sw = 0.9, TD
               •Sw = 0.7, ZG
                Sw = 0.8, ZG
            ---- Sw = 0.9, ZG
         -a
         o
         o

                             10
20
30
40
50
                                           Time (years)
Figure 15.5 Mass fraction of initial mass to lost to ground water as a function of time and water
saturation for first-type, time-dependent (TD) and zero-gradient lower boundary conditions.
NAPL absent, no degradation, qw = 0.035 cm/d.

boundary condition for any given water saturation. This is likely due to the fact that the first-
type, time-dependent lower boundary condition allows advective and diffusive transfer across the
lower boundary while the finite, zero-gradient lower boundary conditions allows only advective
transfer.

       Figure 15.6 illustrates average soil concentration as a function of time and water
saturation for the two lower boundary conditions.  Average soil concentration computation
supplements mass balance calculations in gaining easy and rapid insight into patterns of mass
removal with time. Figure 15.6 shows that average soil concentration was reduced below 1
mg/kg after approximately 50 years for both lower boundary conditions. However, concentration
reduction was always more rapid with the use of the first-type, time-dependent boundary
condition, especially  at lower water saturations because of diffusive movement across the vadose
zone - ground-water interface.  Figures 15.3  through 15.6  clearly demonstrate that water
saturation and selection of a lower boundary condition has a profound affect on the rate of
concentration reduction and mass flux from the modeled domain and thus should be carefully
considered when simulating VOC transport in the unsaturated zone.

       Figures 15.7 and 15.8 illustrate differences in soil  concentration as a function of time and
depth for the two lower boundary conditions at a water saturation value of 0.9. Soil
concentration at the vadose zone - ground-water interface  is directly determined by the specified
                                           260

-------
             200
                                                                  — Sw = 0.7, TD
                                                                  — Sw = 0.7, ZG
                                                                  - - Sw = 0.8. TD
                                                                  •-Sw = 0.8,ZG
                                                                  -  - • Sw = 0.9, TD
                                                                  •-• Sw = 0.9, ZG
                              10    15
                       20    25    30
                        Time (years)
35
40
                                                                         45
50
Figure 15.6 Average total soil concentration as a function of time and water saturation for first-
type, time-dependent (TD) and zero-gradient (ZG) lower boundary conditions. NAPL absent, no
degradation, qw = 0.035 cm/d.
             700
             600 -
          ^500-
           o
           o
           o
          O
             400 -
             300 -
          - 200 -
             100 -
               0
	0.5 years, TD
	0.5 years, ZG
	1.0 years, TD
	1.0 years, ZG
- - - 2.0 years, TD
	2.0 years, ZG
— - -6.0 years, TD
	6.0 years, ZG
— - • 10.0 years, TD
	10.0 years, ZG
                                                        TD and ZG lines overlap
                  0
         100
  400
        500
                                        200        300
                                            Depth (cm)
Figure 15.7 Total soil concentration as a function of time (0.5 - 10.0 years) and depth for first-
type, time-dependent (TD) and zero-gradient (ZG) lower boundary conditions. Water saturation
= 0.9, NAPL absent, no degradation, qw = 0.035 cm/d.
                                           261

-------
          OB
         ^
          o
          o
         U
          o
         VI
160

140 -

120 -

100 -

 80 -

 60 -

 40 -

 20 -
               0
	15 years, TD
	15 years, ZG
	20 years, TD
	20 years, ZG
• • • 25 years, TD
	25 years, ZG
	30 years, TD
	30 years, ZG
— • '50 years, TD
	50 years, ZG
                 0
               100
                    200
300
400
500
                                           Depth (cm)
Figure 15.8 Total soil concentration as a function of time (15-50 years) and depth for first-
type, time-dependent (TD) and zero-gradient (ZG) lower boundary conditions. Water saturation
= 0.9, NAPL absent, no degradation, qw = 0.035 cm/d.

first-type, time-dependent, lower boundary condition whereas soil concentration for the finite,
zero-gradient lower boundary condition is allowed to vary independently of temporally varying
ground-water concentrations.  The effect of the lower boundary condition propagates up the soil
column (toward z = 0) with time. For early time (0.5, 1.0, and 2.0  years), soil concentrations
are nearly identical throughout the entire profile for the two boundary conditions.  At 6 and 10
years simulation time though, soil concentration profiles for the two boundary conditions are
identical  only at depths less than 300 and 200 cm respectively and show clear deviation with
approach to the water table. By 20 years simulation time, the effect of lower boundary condition
selection has propagated up the entire soil column. Differences in soil concentration at the
vadose zone - ground-water interface for the two boundary conditions increase with time up until
approximately 10 years after which they begin to decrease as the contaminant pulse moves
through the soil.

       Figures 15.9, and  15.10 illustrate the effect that  selection of degradation half-life has on
mass flux to ground water, cumulative mass lost to degradation, and average soil concentration
as a function of time.  As would be expected, selection  of a high degradation rate or low half-life
significantly decreases the magnitude and duration of mass flux to ground water, increases the
cumulative mass fraction lost to degradation, and decreases average soil concentration with time
for both boundary conditions. Model output results are very sensitive to selection of a
degradation half-life, an input parameter in which little  is known at the time of modeling.  Small
                                           262

-------
            o
            a

                        — - -half-life = 1 year, TD
                        	half-life = 1 year, ZG
                        	half-life = 5 years, TD
                        	half-life = 5 years, ZG
                        	half-life = 10 years, TD
                            half-life =10 years, ZG
                        — - -half-life = 50 years, TD
                        	half-life = 50 years, ZG
                            half-life = 100 years, TD
                            half-life = 100 years, ZG
                                 10    15     20     25    30
                                                Time (years)
                                  40     45
                    50
Figure 15.9 Mass flux to the capillary fringe as a function of time and degradation half-life for
first-type, time-dependent (TD) and zero-gradient (ZG) lower boundary conditions.  Water
saturation = 0.9, NAPL absent, qw = 0.035 cm/d.
                            • half-life = 1 year, TD
                           -half-life = 5 years, TD
                            half-life = 10 years, TD
                            half-life = 50 years, TD
                           -half-life = 100 years, TD
              	half-life = 1 year, ZG
              	half-life = 5 years, ZG
              	half-life=10years^ZG
              	half-life = 50 years, ZG
              	half-life = 100 years, ZG
                                10
15
20     25     30
  Time (years)
35
40
45
50
Figure 15.10  Mass fraction of initial mass lost to degradation as a function of time and
degradation half-life for first-type, time-dependent (TD) and zero-gradient (ZG) lower boundary
conditions. Water saturation = 0.9, NAPL absent, qw = 0.035 cm/d.
                                               263

-------
changes in low half-lives (e.g., 1, 5 years) substantially affect model output whereas large
changes in large half-lives (e.g., 50 to 100 years) have minimal effect.  Notice however from
Figure 15.10 that use of a finite, zero-gradient lower boundary condition consistently results in
more cumulative mass loss due to degradation compared to a first-type, time-dependent lower
boundary condition especially at degradation half-lives of  5 and 10 years. This is likely due to
the to the observation that the zero-gradient boundary condition results in higher prolonged
concentrations in the soil column compared to the first-type boundary condition. In advection-
dispersion problems, delayed transport provides a greater opportunity for degradation.

       Figures 15.11, 15.12 and 15.13 illustrate the effect that of NAPL residual saturation on
mass flux to ground water, cumulative mass fraction lost to ground water, and average soil
concentration. Formulation of equation (15.5) required constant volumetric NAPL content and
that NAPL composition not change with time. Thus, NAPL is represented in VFLUX as a
nonvolatile, nondegradable oil matrix. A volatile, degradable NAPL would decrease in
volumetric content and change in composition (increasing mole fractions of compounds with
lower vapor pressure) with time. Because of significantly increased retardation, NAPL saturation
decreases the magnitude but dramatically increases the duration of mass flux to ground water for
both boundary conditions.  For instance, for the first-type boundary condition with no NAPL
saturation, mass flux to ground water ceases after 15 years but continues for over 50 years at a
NAPL saturation of 0.01. At higher NAPL saturations (0.05), as illustrated in the embedded
figure in Figure 15.11, smaller but still potentially environmentally significant mass flux occurs
to ground water in excess of the simulation period due to exceeding high retardation factors.  As
illustrated in figures 15.12 and 15.13, at a NAPL saturation of 0.05, little mass removal or
concentration reduction occurs over a 100 year period. The argument of allowing VOCs in the
vadose zone to migrate freely to ground water for subsequent removal during pump- and treat is
less tractable when NAPL is present in soils because of exceeding long duration of mass flux to
ground water.  From Figures 15.11, 15.12. and 15.13, it is apparent that model output is very
sensitive to specification of NAPL saturation and thus care must be exercised when inputting
NAPL saturation values into VOC transport codes.

       An important observation for all of the previous simulations is that at high initial soil
concentrations, both lower boundary conditions indicated significant mass flux to ground water
suggesting the need to initiate or continue venting operation.  It is useful now to consider the case
of much lower contaminant concentrations in the unsaturated zone, typical of prolonged venting
application, to see if both lower boundary conditions result in consistent decision making on
whether to initiate or continue venting application. Initial soil concentration levels were reduced
by three orders of magnitude.  Figure  15.14 illustrates mass flux to ground water as a function of
water saturation and boundary condition. Mass flux profiles for the finite, zero-gradient
boundary condition are as illustrated in Figure 15.4 but at a 1000 fold reduction.  Use of a first-
type, time-dependent lower boundary condition though having the same ground-water
concentration profile as before though results in entirely different mass flux profile compared to
Figure 15.4. At a water saturation of 0.7, the direction of mass flux is initially from the vadose
zone to ground water but then switches to mass flux from ground water to the vadose zone after
                                           264

-------
                                     So = 0.0, TD
                                     So = 0.001, TD
                                     So = 0.005. TD
                                     So = 0.01. tD
                                     So = 0.05; TD
                                            -So = 0.0,ZG
                                         	So = 0.001, ZG
                                         	So = 0.005.ZG
                                         	So = 0.01.ZG
T n -,

1.5 -
1.0 -
0.5 -
-0.5 -
i n



x'
S .... 	
r^ ' ' ' 1
)» 20 40 60 80 IfflO
•

10
15
20    25     30
  Time (years)
                                            35
                                                                    40
                                                             45
                                       50
Figure 15.11  Mass flux to ground water as a function of time and NAPL saturation for first-
type, time-dependent (TD) and zero-gradient (ZG) lower boundary conditions. Water saturation
= 0.7, no degradation, qw = 0.035 cm/d.
         fr,

          VI
0.7 -i

0.6 -

0.5 -

0.4 -




0.2 -

0.1 -

0.0 -
                                  •So = 0.0, TD
                                     -So = 0.00,ZG
                               	So = 0.001, TD	So = 0.001, ZG
                               	So = 0.005, TD 	So = 0.005, ZG
                               	So = 0.01,TD
                               — - 'So = 0.05. TD
                                  	So = 0.01,ZG
                                  	So = 0.05.ZG
0
                       10
                20    30
40    50     60

  Time (years)
                               70
                               80
                               90    100
Figure 15.12  Mass fraction of initial mass lost to ground water as a function of time and NAPL
saturation for first-type, time-dependent (TD) and zero-gradient (ZG) lower boundary conditions.
Water saturation = 0.7, no degradation, qw = 0.035 cm/d.
                                            265

-------
            'an 200
            1$ 180 -
	 So = 0.0, TD
	 So = 0.001,TD
— - So = 0.01,TD
	 So = 0.05,TD
	 So = 0.0, ZG
	 So = 0.001, ZG
	 So = 0.01,ZG
	 So = 0.05,ZG
                   o
               10
20    30   40    50    60    70   80    90    100
                                           Time (years)
Figure 15.13 Average total soil concentration as a function of time and NAPL saturation for
first-type, time-dependent (TD) and zero-gradient (ZG) lower boundary conditions. Water
saturation = 0.7, no degradation, qw = 0.035 cm/d.
          •B
          c<3
8
          C<3
          o
          CO
          CO
          C3
             I
                                                         Sw = 0.7, TD
                                                         Sw = 0.7, ZG
                                                         Sw = 0.8, TD
                                                               	Sw = 0.8,ZG
                                                               	Sw = 0.9, TD
                                                                  Sw = 0.9.ZG
                -0.02
                          10
                      20    30
             40    50   60
               Time (years)
70    80   90    100
Figure 15.14 Mass flux to ground water as a function of time and water saturation (0.7 - 0.9) for
first-type, time-dependent (TD) and zero-gradient (ZG) lower boundary conditions. NAPL
absent, no degradation, qw = 0.035 cm/d.
                                           266

-------
about 10 years.  For higher water saturations (0.8 and 0.9) the direction of mass flux is
consistently from the vadose zone to ground water as vapor diffusion to the atmosphere is
repressed. Figure 15.15 illustrates average soil concentration as a function of time.  The average
soil profile illustrates the "inhibitive" effect that the first-type, time-dependent lower boundary
condition can have on mass removal from the vadose zone especially at low water saturations.
At low initial soil concentration levels, the first-type, time-dependent lower boundary condition
consistently results in longer average soil concentration or mass reduction compared to the finite,
zero-gradient lower boundary condition.  This is opposite of what previously observed in Figure
15.4 at higher initial soil concentration levels.  For the first-type, time-dependent lower boundary
condition, Figure 15.15 indicates that residual  mass remains in soil for over 100 years compared
to complete removal after 40 or 50 years for the finite, zero-gradient lower boundary condition.
Also, notice that for the first-type, time-dependent lower boundary condition, lower water
saturation prolongs the amount of time for average concentration or mass reduction in soil  - again
opposite of what was previously observed at high  soil concentration  levels.

       Figures 15.16 and 15.17 provide further insight as to how selection of the lower boundary
condition affects simulation results when water saturation is held constant at 0.9.  As
illustrated in Figures 15.16 and 15.17, for the  finite, zero-gradient lower boundary condition, soil
concentration at the vadose zone - ground-water interface slowly increase with
               0.45
               0.00 4.
                   0
10    20   30    40    50    60
                   Time (years)
70    80    90   100
Figure 15.15 Average total soil concentration as a function of time for the low initial soil
concentration profile for first-type, time-dependent (TD) and zero-gradient (ZG) lower boundary
conditions. NAPL absent, no degradation, qw = 0.035 cm/d.
                                           267

-------
           0.5 years. TD
           0.5 years, ZG
       	1.0 years. TD
       	1.0 years, ZG
       	2.0 years. TD
            2.0 years, ZG
       	6.0 years. TD
       	6.0 years, ZG
       	10.0 years, TD
          -  10 years, ZG
0.0
    0
100
400
500
                                        200         300
                                            Depth (cm)
Figure 15.16  Total soil concentration as a function of time (0.5 - 10 years) and depth for low
initial soil concentration profile and first-type, time-dependent (TD) and zero-gradient (ZG)
lower boundary conditions. Water saturation = 0.9, NAPL absent, no degradation, qw = 0.035
cm/d.
0.35
             15 years, TD
             15 years, ZG
         	20 years, TD
         	20 years, ZG
         	25 years, TD
         	25 years, ZG
         	30 years, TD
         	30 years, ZG
         	50 years, TD
         	50 years, ZG
0.00
     0
 100
400
500
                                         200         300
                                             Depth (cm)
Figure 15.17  Total soil concentration as a function of time (15-50 years) and depth for low
initial soil concentration profile and first-type, time-dependent (TD) and zero-gradient (ZG)
lower boundary conditions.Water saturation = 0.9, NAPL absent, no degradation, qw=0.035 cm/d.
                              268

-------
time as the contaminant mass moves vertically downward peaking at a concentration of about 0.2
mg/kg at about 10 years time. From 10 to 50 years, the concentration at the interface slowly
recedes to levels which would be difficult to detect in the field. For the first-type, time-
dependent lower boundary condition however, the soil concentration at the interface immediately
peaks to  1.4 mg/kg at early time due to mass flux from ground water and then slowly recedes
over time. After 50 years simulation time though, concentration levels near the interface remain
elevated  due to vapor diffusion from ground-water.  Similar to the high concentration profile,  the
affect of selection of the lower boundary condition propagates up the soil column with time only
this time at a much faster rate.  At time 0.5 years, there is large disparity in concentration profiles
at about 400 cm. Concentration versus depth profiles at late time reflect what is frequently
observed in the field during soil gas surveys to track ground-water plumes in initially
uncontaminated soils.  At late time, soil contamination is caused by vapor diffusion from
contaminated ground water.

      Finally, Figure 15.18 illustrates differences in soil-water concentrations with depth as a
function  of time (15 to 50 years) and depth.  Since equilibrium conditions are presumed to
prevail, Figure 15.18 is the same as Figure 15.17 adjusted by the retardation factor. Often, soil-
water concentrations at MCLs or MCLs adjusted for a dilution attenuation factor (often 10) are
required  at the vadose zone - ground-water interface for vadose zone remediation closure,
especially in the  State of California.  Since the MCL for TCE is 5 ug/1, a soil-water concentration
                          15 years, TD
                          15 years, ZG
                     -- 20 years, TD
                     ---- 20 years, ZG
                     --- 25 years, TD
                     ...... 25 years, ZG
                     --- 30 years, TD
                     ----- 30 years, ZG
                     --- 50 years, TD
                     ----- 50 years, ZG
              0.0
                 0
100
400
500
                                       200         300
                                           Depth (cm)
Figure 15.18 Soil-water concentration as a function of time (15-50 years) and depth for low
initial soil concentration profile for first-type, time-dependent (TD) and zero-gradient (ZG) lower
boundary conditions. Water saturation = 0.9, NAPL absent, no degradation, qw = 0.035 cm/d.
                                           269

-------
between 5 and 50 ug/1 or 0.005 to 0.05 mg/1 would be required for closure. With the finite,
zero-gradient boundary condition, it would take in excess of 50 years to achieve this level
assuming a water saturation level of 0.9 and no degradation. Using a first-type, time-dependent
lower boundary condition however, these soil-water concentration levels would not be achievable
because of vapor diffusion from ground water. If this type of modeling reflects reality, then
venting closure would be unattainable.

       Given model output summarized in Figures 15.14 through 15.18 for soils contaminated at
low levels, regulatory decision making on whether to initiate, continue, or cease venting or any
other vadose zone remediation technology is directly contingent upon selection of the lower
boundary condition. It would appear that use of a finite, zero-gradient lower boundary condition
would require continued venting application, whereas, use of the first-type, time-dependent lower
boundary condition would most likely lead to a decision to cease venting application.  This
disagreement in decision making was not present for more highly contaminated soils since mass
flux to ground water overwhelmed mass flux from the ground water.

15.9   Conclusions

1.      Selection of lower boundary conditions and highly sensitive input parameters such as
       water saturation, NAPL saturation, and degradation half-life significantly affect model
       output and decision making whether to initiate or cease venting application, especially
       when soil concentrations are low. A first-type, time-dependent lower boundary condition
       allows mass transfer across the vadose zone - ground-water interface by infiltration and
       diffusion. A finite, zero-gradient lower boundary condition only allows mass transfer
       across the interface by infiltration with soil and soil-water concentrations independent of
       boundary conditions. For more highly contaminated soils, the first-type, time-dependent
       lower boundary condition results in a higher magnitude but shorter duration pulse of mass
       to ground water and a higher cumulative mass fraction to ground water compared to the
       finite, zero-gradient lower boundary condition. For more highly contaminated soils,
       regardless of which lower boundary condition is chosen, model simulations clearly
       indicate a need to initiate or continue venting application.  For soils at lower contaminant
       levels however, use of a first-type, time-dependent lower boundary condition may result
       in the primary direction of mass flux being from ground water to the vadose zone perhaps
       questioning the need to initiate or continue venting. The finite, zero-gradient lower
       boundary condition on the other hand results in sustained but low magnitude mass flux to
       ground water perhaps prompting a decision to initiate or continue venting.  Thus, the two
       boundary conditions result in contradictory decisions.

2.      As previously discussed, a coherent soil venting strategy should link progress in ground-
       water remediation to vadose zone remediation. A first-type, time-dependent lower
       boundary conditions accomplishes this with relative ease.  Regulators though typically
       use the model VLEACH and a finite, zero-gradient lower boundary condition when
       assessing venting closure. This approach is more conservative than use of a first-type,
                                          270

-------
time-dependent lower boundary condition. Differences in model output inherent in use of
these two boundary conditions have been explained and illustrated.  Thus, if a finite,
zero-gradient lower boundary condition is utilized in vadose zone modeling to assess
closure of a venting system, limitations associated with this approach can be understood
and taken into account during decision making.
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                                  APPENDIX A (MFROAINV)

Sample Input File for MFROAINV

               /* input file for test 3
298.0          /* temperature (K)
-56.90           /* mass pumping rate (g/sec)
518.16          /* thickness of formation (cm)
396.24          /* distance to top of sandpack from surface (cm)
426.72         /* distance to bottom of sandpack from surface (cm)
7.5              /* radius of sandpack (cm)
l.e-10           /* convergence criterion
5. le-07,5.4e-07   /* minimum and maximum radial permeability (cmA2)
1.09,1.20        /* minimum and maximum anisotropy ratio (K_r/K_z)
5000           /* number of random parameter guesses
25              /* number of "best" parameter sets for forward problem
15             /*number of observations
152.4   91.44   0.9985247       /* observations   r(cm)   z(cm)   P/P(atm)
152.4   274.32  0.9950823
152.4   426.72  0.9881975
304.8   91.44   0.9990165
304.8   274.32  0.9972953
304.8   426.72  0.9950823
457.2   91.44   0.9993853
457.2   274.32  0.9984017
457.2   426.72  0.9979100
609.6   91.44   0.9996312
609.6   274.32  0.9992623
609.6   426.72  0.9992623
914.4   91.44   0.9997049
914.4   274.32  0.9995082
914.4   426.72  0.9993361
                                                299

-------
Sample Output File for MFROAINV
   Steady-State Multi-Well Parameter Estimation
        Finite-Radius Model
      Top BC: Open to Atmosphere
Parameter Estimates Ordered from Best to Worst

 K r         K z        RMS-Error   IFLAG
0.5272E-06
0.5269E-06
0.5280E-06
0.5271E-06
0.5281E-06
0.5282E-06
0.5275E-06
0.5262E-06
0.5287E-06
0.5263E-06
0.4614E-06
0.4615E-06
0.4606E-06
0.4616E-06
0.4609E-06
0.4602E-06
0.4604E-06
0.4616E-06
0.4606E-06
0.4624E-06
0.5243E-03
0.5243E-03
0.5243E-03
0.5243E-03
0.5243E-03
0.5243E-03
0.5243E-03
0.5243E-03
0.5243E-03
0.5243E-03
0
0
0
0
0
0
0
0
0
0
(only first ten shown)
Observed Versus Modeled Fit (only first ranked fit shown)

Parameter set -  1

K_r = 0.5272E-06
K  z = 0.4614E-06
    R(cm)
Z(cm)
P observed    P model
0.1524E+03
0.1524E+03
0.1524E+03
0.3048E+03
0.3048E+03
0.3048E+03
0.4572E+03
0.4572E+03
0.4572E+03
0.6096E+03
0.6096E+03
0.6096E+03
0.9144E+03
0.9144E+03
0.9144E+03
0.9144E+02
0.2743E+03
0.4267E+03
0.9144E+02
0.2743E+03
0.4267E+03
0.9144E+02
0.2743E+03
0.4267E+03
0.9144E+02
0.2743E+03
0.4267E+03
0.9144E+02
0.2743E+03
0.4267E+03
0.9985E+00
0.995 1E+00
0.9882E+00
0.9990E+00
0.9973E+00
0.995 1E+00
0.9994E+00
0.9984E+00
0.9979E+00
0.9996E+00
0.9993E+00
0.9993E+00
0.9997E+00
0.9995E+00
0.9993E+00
0.9984E+00
0.9939E+00
0.9893E+00
0.9989E+00
0.9967E+00
0.9952E+00
0.9993E+00
0.998 1E+00
0.9975E+00
0.9996E+00
0.9989E+00
0.9986E+00
0.9999E+00
0.9996E+00
0.9995E+00
                                               300

-------
Source Code for MFROAINV (author: Ravi Varadhan)

(A FORTRAN program for estimation of radial and vertical gas permeability in a domain open to the atmosphere)

c Parameter estimation for the steady-state solution for finite well radius
c BC: Open to the atmosphere on top
c Multiple monitoring locations
c version 1.0  07/19/99
   implicit real*8 (a-h,o-z)
   dimension rr(100),zz(100),pobs(100),p_r(5000),p_z(5000),
   >errmet(5000),wksp(5000),iwksp(5000),iflag(5000)
   integer*4 istart,ifinish
   character* 15 prefix,fnamel,fname2
   character*4 suffixI,suffix2
   write(*,*)' Parameter Estimation for Steady-State Multi-Well Test'
   write(*,*)' Finite-Radius Solution - BC: Open to Atmosphere'
   write(*,*)
   write(*,*)'Enter the Prefix of the input file:  '
   write(*,*)'An output file is created w/ same prefix'
   write(*,*)' Input-file Prefix ?  '
   read(*,'(a)')prefix
   call getlen(prefix,' ',nO,nl)
   suffixl='.inp'
   suffix2='.dat'
   fnamel=prefix(nO:nl)//suffixl
   fname2=prefix(nO:nl)//suffix2
   open( 10,fname 1, status='unknown')
   open(20,fname2,status='unknown')
   open(30,'seed.inp',status='unknown')
   read(10,*)
   read(10,101)temp,xmpr,xb,xd,xl,rw,tol
   read(10,*)prmin,prmax
   read(10,*)rmin,rmax
   read(10,101)nguess
   read(10,101)nbest
   read(10,101)nsim
   do 2 i=l,nsim
   read(10,*)rr(i),zz(i),pobs(i)
2   continue
   patm= 1013200.
   visc=1.76e-04
   rgas=8.314e07
   xmw=28.8
   pi=4.0*datan(1.0)
c  patm = atmospheric pressure (g/cm/secA2)
c  vise = viscosity of gas (g/cm/sec)
c  rgas = universal gas constant
c  temp = temperature of gas (K)
c  xmw = molecular weight of gas (g/mole)
c  xmpr = mass withdrawal rate  (g/sec)
c  prad = radial permeability of the formation (cmA2)
c  pz = vertical permeability


                                                   301

-------
c  pleak = permeability of the leaky confining unit
c  xb = thickness of the formation (cm)
c  xbp = thickness of the leaky unit
c  xl = distance from the bottom of leaky unit to the bottom of well screen
c  xd = distance from the bottom of leaky unit to the top of well screen
c  rw = nominal well radius (cm)
c  aa = radial cordinate (cm)
c  bb = vertical cordinate
c  cc = elapsed time (sec)
c  nsim = number of times solution will be generated
c  pobs = measured pneumatic pressure responses
   write(20,*)'    Steady-State Multi-Well Parameter Estimation'
   write(20,*)'         Finite-Radius Model  '
   write(20,*)'       Top BC: Open to Atmosphere '
   write(20,*)
   xll=xl/xb
   xdd=xd/xb
   r_w=rw/xb
   call timer(istart)
   read(30,*)iseed
   if(iseed.gt.0)iseed=-iseed
   do  lj=l,nguess
   iflag(j)=0
   errmet(j)=0.
   sqerr=0.
   guess 1 =ranl (iseed)
   guess2=ranl (iseed)
   p_r(j)=prmin*(prmax/prmin)**guessl
   ratio=rmin+(rmax-rmin) *guess2
   p_z(])=p_r(j)/ratio
   prad=p_r(j)
   pz=p_z(j)
   a=dsqrt(prad/pz)
   xm=xmpr*visc*rgas*temp/(pi*xmw*prad*patm*patm*xb)
c
   do  5 i=l,nsim
   r=rr(i)/xb
   z=zz(i)/xb
   n=0
   sum=0.0
10  continue
   n=n+l
   rm=n-0.5
   if(n.gt.lOOO)then
   iflag(j)=l
   goto 15
   endif
   vn=rm*pi
   wn=vn/a
                                                  302

-------
   fn=besskO(wn*r)/(wn*r_w*besskl(wn*r_w))
   tsin=dsin(vn*z)
   tcos=dcos(vn*xdd)-dcos(vn*xll)
   trig=tcos*tsin
   fact=l./(vn*(xll-xdd))
   add=trig*fact*fn
   sum=sum+add
   if(dabs(sum).le.l.e-14)goto 15
   if(dabs(add/sum).gt.tol)goto 10
15  continue
   psqrd=1.0+2.0*xm*sum
   if(psqrd.lt.0.)then
   iflag(j)=2
   enmet(j)=l.elO
   goto 1
   endif
   popatm=dsqrt(psqrd)
   sqerr=sqerr+(popatm-pobs(i)) * *2
5   continue
   errmet(j )=dsqrt(sqerr/nsim)
   write(*,*)j,prad,pz,errmet(j),iflag(j)
1   continue
   call sorts (nguess,errmet,p_r,p_z,iflag,wksp,iwksp)
   write(20,107)
   write(20,102)
   do 20j=l,nguess
   write(20,103)p_r(j),p_zCJ),errmetCJ),iflagCJ)
20  continue

   do30j=l,nbest
   prad=p_r(j)
   pz=p_z(j)
   write(20,104)j,prad,pz
   write(20,105)
   a=dsqrt(prad/pz)
   xm=xmpr*visc*rgas*temp/(pi*xmw*prad*patm*patm*xb)
c
   do 35 i=l,nsim
   r=rr(i)/xb
   z=zz(i)/xb
   n=0
   sum=0.0
40  continue
   n=n+l
   rm=n-0.5
   if(n.gt.lOOO)goto45
   vn=rm*pi
   wn=vn/a
   fn=besskO(wn*r)/(wn*r_w*besskl(wn*r_w))
   tsin=dsin(vn*z)
   tcos=dcos(vn*xdd)-dcos(vn*xll)
   trig=tcos*tsin
                                                  303

-------
   fact=l./(vn*(xll-xdd))
   add=trig*fact*fn
   sum=sum+add
   if(dabs(sum).le.l.e-14)goto 45
   if(dabs(add/sum).gt.tol)goto 40
45  continue
   psqrd=1.0+2.0*xm*sum
   if(psqrd.ge.0.)then
   popatm=dsqrt(psqrd)
   else
   popatm=0.
   endif
   write(20,106)rr(i),zz(i),pobs(i),popatm
35  continue
30  continue
   call timer(ifinish)
   write(*,*)Time of execution (sec)  =',(ifinish-istart)/100.
101  format(gl5.0)
102  format(/5x,'K_r', 1 lx,'K_z',9x,'RMS-Error',5x,'IFLAG'/)
103  format(2x,el0.4,4x,el0.4,5x,el0.4,5x,i2)
104  format(//20x, 'Parameter set - ',i4//,3x,'K_r = ',e!0.4/
   >3x,'K_z = ',e!0.4//)
105  format(10x,'R (cm)',8x,'Z (cm)',9x,'P_observed',8x,'P_model'/)
106  format(7x,el0.4,5x,el0.4,8x,el0.4,6x,el0.4)
107  format(//10x, 'Parameter Estimates Ordered from Best to Worst'//)
   rewind(30)
   write(30,*)-iseed
   stop
   end

   Double Precision Function BesskO(x)
   implicit real*8 (a-h,o-z)
   datapl,p2,p3,p4,p5,p6,p7/-0.57721566,0.4227842,0.23069756,
   >0.348859d-l,0.262698d-2,0.10750d-3,0.74d-5/
   dataql,q2,q3,q4,q5,q6,q7/1.25331414,-0.7832358d-l,0.2189568d-l,
   >-0.1062446d-l,0.587872d-2,-0.25154d-2,0.53208d-3/
   if(x.le.2.0)then
   y=x*x/4.0
   besskO=(-dlog(x/2.0)*bessiO(x))+(pl+y*(p2+y*(p3+y*(p4+y*(p5+y*(p6+
   >Y*p7))))))
   else
   y=2.0/x
   besskO=(dexp(-x)/dsqrt(x))*(ql+y*(q2+y*(q3+y*(q4+y*(q5+y*(q6+y*q7)
   endif
   return
   end

   DOUBLE PRECISION FUNCTION BESSIO(W)
   IMPLICIT real*8 (A-H,O-Z)
   T=W/3.75
   IF(DABS(T).LE.1.0) THEN
                                                  304

-------
   Y=T*T
   BESSIO=1.0+Y*(3.5156229+Y*(3.0899424+Y*(1.2067492+Y*(0.2659732+
   >Y*(0.0360768+0.0045813*Y)))))
   ELSE
   AW=DABS(W)
   Y=3.75/AW
   TEMP=039894228+Y*(0.01328592+Y*(0.00225319+Y*(-0.00157565+Y*(
   >0.00916281+Y*(-0.02057706+Y*(0.02635537+Y*(-0.01647633+
   >0.00392377*Y)))))))
   BESSIO=DEXP(AW)*TEMP/DSQRT(AW)
   END IF
   RETURN
   END

   double precision FUNCTION besskl(x)
CU  USES bessil
   implicit real*8 (a-h,o-z)
   DOUBLE PRECISION pl,p2,p3,p4,p5,p6,p7,ql,q2,q3,q4,q5,q6,q7,y
   SAVE p 1 ,p2,p3 ,p4,p5,p6,p7,ql ,q2,q3 ,q4,q5,q6,q7
   DATA pl,p2,p3,p4,p5,p6,p7/1.0dO,0. 15443 144dO,-0.67278579dO,
   *-0.18156897dO,-0.1919402d-l,-0.110404d-2,-0.4686d-4/
   DATAql,q2,q3,q4,q5,q6,q7/1.25331414dO,0.23498619dO,-0.3655620d-l,
   *0.1504268d-l,-0.780353d-2,0.325614d-2,-0.68245d-3/
   if (x.le.2.0)then
    y=x*x/4.0
    besskl=(dlog(x/2.0)*bessil(x))+(1.0/x)*(pl+y*(p2+y*(p3+y*(p4+y*
   *(p5+y*(p6+y*p7))))))
   else
    y=2.0/x
    besskl=(dexp(-x)/dsqrt(x))*(ql+y*(q2+y*(q3+y*(q4+y*(q5+y*(q6+y*
   endif
   return
   END

   double precision FUNCTION bessil(x)
   implicit real*8 (a-h,o-z)
   DOUBLE PRECISION pl,p2,p3,p4,p5,p6,p7,ql,q2,q3,q4,q5,q6,q7,q8,q9,y
   SAVE p 1 ,p2,p3 ,p4,p5,p6,p7,ql ,q2,q3 ,q4,q5,q6,q7,q8,q9
   DATApl,p2,p3,p4,p5,p6,p7/0.5dO,0.87890594dO,0.51498869dO,
   *0.15084934dO,0.2658733d-l,0.301532d-2,0.32411d-3/
   DATAql,q2,q3,q4,q5,q6,q7,q8,q9/0.39894228dO,-0.3988024d-l,
   *-0.362018d-2,0.163801d-2,-0.1031555d-l,0.2282967d-l,-0.2895312d-l,
   *0.1787654d-l,-0.420059d-2/
   if(abs(x).lt.3.75)then
    y=(x/3.75)**2
    bessil=x*(pl+y*(p2+y*(p3+y*(p4+y*(p5+y*(p6+y*p7))))))
   else
    ax=abs(x)
    y=3.75/ax
    bessil=(dexp(ax)/dsqrt(ax))*(ql+y*(q2+y*(q3+y*(q4+y*(q5+y*(q6+y*
   *(q7+y*(q8+y*q9))))))))
                                                305

-------
    if(x.lt.0.)bessil=-bessil
   endif
   return
   END

   double precision FUNCTION ranl(idum)
   implicit real*8 (a-h,o-z)
   INTEGER idum,IA,IM,IQ,IR,NTAB,NDIV
   PARAMETER (IA=16807,IM=2147483647,AM=1./IM,IQ=127773,IR=2836,
   *NTAB=32,NDIV= 1+(IM-1)/NTAB,EPS=1.2e-7,RNMX= 1 .-EPS)
   INTEGER j ,k,iv(NTAB),iy
   SAVE iv,iy
   DATA iv /NTAB*0/, iy /O/
   if (idurale.O.or.iy.eq.O) then
    idum=max(-idum, 1)
    dollj=NTAB+8,l,-l
     k=idum/IQ
     idum=IA*(idum-k*IQ)-IR*k
     if (idum.lt. 0) idum=idum+IM
     if (j.le.NTAB) iv(j)=idum
11   continue
    iy=iv(l)
   endif
   k=idum/IQ
   idum=IA*(idum-k*IQ)-IR*k
   if (idum.lt. 0) idum=idum+IM
   j=l+iy/NDIV
   iy=iv(j)
   iv(j)=idum
   ranl=dminl(AM*iy,RNMX)
   return
   END

   SUBROUTINE sorts (n,ra,rb,rc,id,wksp,iwksp)
   implicit real*8 (a-h,o-z)
   INTEGER n,iwksp(n),id(n)
   dimension ra(n),rb(n),rc(n),wksp(n)
CU  USES indexx
   INTEGER j
   call indexx(n,ra,iwksp)
   do llj=l,n
    wksp(j)=ra(j)
11  continue
   do 12j=l,n
    ra(j )=wksp(iwksp(j))
12  continue
   do 13j=l,n
    wksp(j)=rb(j)
13  continue
   do 14j=l,n
    rb(j )=wksp(iwksp(j))
14  continue
                                              306

-------
   do 15j=l,n
    wksp(j)=rc(j)
15  continue
   do 16j=l,n
    rc(j)=wksp(iwksp(j))
16  continue
   do 17j=l,n
    wksp(j)=id(j)
17  continue
   do 18j=l,n
    id(j )=wksp(iwksp(j))
18  continue
   return
   END

   SUBROUTINE indexx(n,arr,indx)
   implicit real*8 (a-h,o-z)
   INTEGER n,indx(n),M,NSTACK
   dimension arr(n)
   PARAMETER (M=7,NSTACK=50)
   INTEGER i,indxt,ir,itempj j stack,k,l,istack(NSTACK)
   do llj=l,n
    indx(j)=j
11  continue
   jstack=0
   1=1
   ir=n
1   if(ir-l.lt.M)then
    do!3j=l+l,ir
     indxt=indx(j)
     a=arr(indxt)
     do!2i=j-l,l,-l
      if(arr(indx(i)).le.a)goto 2
      indx(i+l)=indx(i)
12    continue
     i=0
2     indx(i+l)=indxt
13   continue
    if(j stack, eq. 0)return
    ir=istack(j stack)
    l=istack(j stack-1)
    jstack=jstack-2
   else
    k=(l+ir)/2
    itemp=indx(k)
    indx(k)=indx(l+l)
    indx(l+l)=itemp
    if(arr(indx(l+1)). gt.arr(indx(ir)))then
     itemp=indx(l+l)
     indx(l+l)=indx(ir)
     indx(ir)=itemp
    endif
                                                  307

-------
     if(arr(indx(l)).gt.arr(indx(ir)))then
      itemp=indx(l)
      indx(l)=indx(ir)
      indx(ir)=itemp
     endif
     if(arr(indx(l+1)) .gt. arr(indx(l)))then
      itemp=indx(l+l)
      indx(l+l)=indx(l)
      indx(l)=itemp
     endif

     j=ir
     indxt=indx(l)
     a=arr(indxt)
3    continue

     if(arr(indx(i)).lt.a)goto 3
4    continue

     if(arr(indx(j)).gt.a)goto 4
     if(j.lt.i)goto 5
     itemp=indx(i)
     indx(i)=indx(j)
     indx(j)=itemp
     goto 3
5    indx(l)=indx(j)
     indx(j)=indxt
     jstack=jstack+2
     if(j stack. gt.NSTACK)pause 'NSTACK too small in indexx'
     if(ir-i+1. ge j -l)then
      istack(jstack)=ir
      istack(j stack-l)=i

     else
      istack(jstack)=j-l
      istack(j stack-1)=1

     endif
    endif
    goto  1
    END
c
    subroutine getlen(string,chr,nO,nl)
c
c   to determine the length of a string excluding any blank padding & heading
c   and return string(nO:nl) nO... position of first non-blank letter
c                  nl... position of last non-blank letter
c   else return the original string
c   ***Use  to keep blank space at the begining or the end of the
c    ***  string
c
    character*(*) string
                                                     308

-------
    character* 160 stringl
    character*! chr
    stringl=''
    inO=0
    inl=0
    do 10 i=len(string),l,-l
    if(string(i:i).ne.chr) then
     if(inl.eq.O) then
     if(string(i:i).eq.'|') then
     nl=i-l
     else
     nl=i
     endif
     inl=l
     endif
     goto 11
    endif
10   continue
11   do20i=l,len(string)
    if(string(i:i).ne.chr) then
    if(in0.eq.O) then
    if(string(i:i).eq.'|') then
    nO=i+l
    else
    nO=i
    endif
    inO=l
    nk=nO-l
c    write(6,*) '|'//string(nO:nl)//'|', nO,nl
    do 30 k=nO,nl
    if(string(k:k).ne.'') then
     nk=nk+l
     if(k.eq.nO) then
       string 1=string(k:k)
     else
       stringl=stringl(l :nk-nO)//string(k:k)
     endif
    endif
c    write(6,*) '|'//stringl(l:nk-nO)//'|'
30   continue
    string=stringl
    nl=nl-nO+l
    nO=l
    endif
    return
    endif
20   continue
    nO=-l
    nl=-l
    return
    end
                                                      309

-------
                                    APPENDIX B (SAIRFLOW)
Sample Input File for SAIRFLOW
283.             /* temperature (K)
-75.11          /* mass pumping rate (g/sec)
6.1 Oe-07        /* radial permeability (cmA2)
4.89e-07         /* vertical permeability (cmA2)
0.2             /* air-filled porosity
518.16           /* thickness of formation  (cm)
396.24          /* distance to top of sandpack (cm)
426.72          /* distance to bottom of sandpack (cm)
7.5              /* well radius of sandpack (cm)
l.Oe-06          /* convergence criterion
74,50           /*number of r and z points
10.0,20.0,30.0,40.0,50.0,60.0,70.0,80.0,90.0,100.0,
110.0,120.0,130.0,140.0,150.0,160.0,170.0,180.0,190.0,200.0,
210.0,220.0,230.0,240.0,250.0,260.0,270.0,280.0,290.0,300.0,
310.0,320.0,330.0,340.0,350.0,360.0,370.0,380.0,390.0,400.0,
410.0,420.0,430.0,440.0,450.0,460.0,470.0,480.0,490.0,500.0,
520.0,540.0,560.0,580.0,600.0,620.0,640.0,680.0,700.0,720.0,
740.0,760.0,780.0,800.0,820.0,840.0,860.0,880.0,900.0,920.0,
940.0,960.0,980.0,1000.0 /* radial distances (cm)
20.0,40.0,50.0,60.0,70.0,80.0,90.0,100.0,
110.0,120.0,130.0,140.0,150.0,160.0,170.0,180.0,190.0,200.0,
210.0,220.0,230.0,240.0,250.0,260.0,270.0,280.0,290.0,300.0,
310.0,320.0,330.0,340.0,350.0,360.0,370.0,380.0,390.0,400.0,
410.0,420.0,430.0,440.0,450.0,460.0,470.0,480.0,490.0,500.0,
510.0,518.0     /* vertical distance (cm)
n               /* yes/no flag for particle tracking
1.0             /* maximum step size (cm)
                                                   310

-------
Sample Output File for SAIRFLOW
  Results of Single-Well Airflow Simulation

  Steady-state Finite-Radius Solution
  Open to the Atmosphere
  R start
Z  start
Potential
Streamline
                                                      v r
                                                                   v z
v (cm/s)
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
100E+02
200E+02
300E+02
400E+02
500E+02
600E+02
700E+02
800E+02
900E+02
100E+03
(Only first 10
R_start
0
0
0
0
0
0
0
0
0
0
.100E+02
.200E+02
.300E+02
.400E+02
.500E+02
.600E+02
.700E+02
.800E+02
.900E+02
.100E+03
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
200E+02
200E+02
200E+02
200E+02
200E+02
200E+02
200E+02
200E+02
200E+02
200E+02
.999142E+00
.999143E+00
.999145E+00
.999149E+00
.999154E+00
.999159E+00
.999166E+00
.999174E+00
.999182E+00
.999191E+00
.613049E+01
.612470E+01
.611509E+01
.610167E+01
.608450E+01
.606365E+01
.603918E+01
.601120E+01
.597980E+01
.594508E+01
-0.
-0.
-0.
-0.
-0.
-0.
-0.
-0.
-0.
-0.
4157E-03
1630E-02
2655E-02
3618E-02
4540E-02
5424E-02
6269E-02
7073E-02
7833E-02
8546E-02
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
.3034E+00
.3030E+00
.3021E+00
.3008E+00
.2992E+00
.2971E+00
.2948E+00
.2920E+00
.2890E+00
.2857E+00
0
0
0
0
0
0
0
0
0
0
.3034E+00
.3030E+00
.3021E+00
.3008E+00
.2992E+00
.2972E+00
.2948E+00
.2921E+00
.2891E+00
.2858E+00
simulations shown)
Z
0
0
0
0
0
0
0
0
0
0
start
.200E+02
.200E+02
.200E+02
.200E+02
.200E+02
.200E+02
.200E+02
.200E+02
.200E+02
.200E+02
Potential
.999142E+00
.999143E+00
.999145E+00
.999149E+00
.999154E+00
.999159E+00
.999166E+00
.999174E+00
.999182E+00
.999191E+00
Norm_Streamline
.100000E+01
.999057E+00
.997488E+00
.995299E+00
.992499E+00
.989097E+00
.985107E+00
.980542E+00
.975419E+00
.969757E+00




























































(Only first 10 simulations shown)
                                                 311

-------
Source Code for SAIRFLOW (author Ravi Varadhan)

(A FORTRAN program for computation of pore-gas velocity, streamlines, and particle tracking for one well in a
domain open to the atmosphere)
C  Particle tracking algorithm for Travel time calculation
C  Airflow due to one pumping well
C  Finite well radius - Steady state flow field
C
C  Runge-Kutta Second Order scheme is used in Particle-Tracking
c
   implicit real*8 (a-h,o-z)
   dimension r(50),z(50),strm(50,50),pot(50,50)
   common/cseries/xk,zb,zd,zl,prad,pz,a,visc,theta,rw,tol,patm
   character* 15 prefix,fnamel,fname2
   character*4 suffixI,suffix2
   character*! ptrack
   write(*,*)' Steady-State Airflow Simulation for a Single Well'
   write(*,*)' Finite-Radius Solution with Borehole Storage '
   write(*,*)' Top BC: Open to Atmosphere'
   write(*,*)
   write(*,*)'Enter the Prefix of the input file:  '
   write(*,*)'An output file is created w/ same prefix'
   write(*,*)' Input-file Prefix ? '
   read(*,'(a)')prefix
   call getlen(prefix,' ',nO,nl)
   suffixl='.inp'
   suffix2='.dat'
   fname 1 =prefix(nO: n 1 )//suffix 1
   fname2=prefix(nO:nl)//suffix2
   open(20,fname 1, status='unknown')
   open(30,fname2,status='unknown')
   read(20,10 l)temp,xmpr,prad,pz,theta,zb,zd,zl,rw,tol
   read(20,*)nr,nz
   read(20,*)(r(i),i=l,nr)
   read(20,*)(z(i),i=l,nz)
   read(20,'(a)')ptrack
   read(20,*)delx
c
   write(*,*)'    Steady-state, Finite Radius-Open to Atmosphere'
   write(*,*)
   call timer(istart)
   patm= 1013200.
   visc=1.76e-04
   rgas=8.314e07
   xmw=28.8
   pi=4.0*datan(1.0)
   a=dsqrt(prad/pz)
   qp=visc*rgas*temp/xmw*xmpr
   xk=2.0*qp/(pi*pi*prad*(zl-zd))
   write(30,106)
                                                   312

-------
    if(ptrack.ne.'y' .and. ptrack.ne.'Y')then
    write(30,102)
    else
    write(30,104)
    endif
    strmax=-l.elO
    do 5 j=l,nz
    zO=z(j)
    if(z0.eq.zb)zO=0.999*zb
    do 5 i=l,nr
    iO=r(i)
    ttime=0.0
    rnew=rO
    znew=zO
    call series(rnew,znew,vr,vz,phiO,psiO)
    vresO=dsqrt(vr*vr+vz*vz)
    pot(ij)=phiO
    strm(ij)=psiO
    if(psi0.gt.strmax)strmax=psiO
    if(ptrack.ne.'y' .and. ptrack.ne.'Y')then
    write(30,103)rO,zO,phiO,psiO,vr,vz,vresO
    goto 5
    endif
    write(*,*)'r(0),z(0),psi =  ',rnew,znew,psiO
C  loop for particle tracking
1   continue
    rold=rnew
    zold=znew
    call series(rold,zold,vr,vz,phi,psi)
    vres=dsqrt(vr*vr+vz*vz)
    delt=delx/vres
    delrl=vr*delt
    delzl=vz*delt
    call series(rold+delrl ,zold+delz 1 ,vr,vz,phi,psi)
    delr2=vr*delt
    delz2=vz*delt
    rnew=rold+(delr 1 +delr2)/2.
    znew=zold+(delz 1 +delz2)/2.
    ttime=ttime+delt
    dist 1=dsqrt((rnew-rw) * *2+(znew-zd) * *2)
    dist2=rnew-rw
    dist3=dsqrt((rnew-rw) * *2+(znew-zl) * *2)
    if(znew.lt.zd)dist=distl
    if(znew.ge.zd .and.  znew.lt.zl)dist=dist2
    if(znew.ge.zl)dist=dist3
    if(dist.ge.delx)goto 1
C  Particle tracking loop is complete
    if(rnew. le .rw)rnew=rw
2   continue
    call series(rnew,znew,vr,vz,phi,psi)
    vres=dsqrt(vr*vr+vz*vz)
    write(*,*)'r(t),z(t),t,psi =  ',rnew,znew,ttime,psi
                                                    313

-------
   write(30,105)rO,zO,phiO,psiO,vr,vz,vres,ttime
5   continue
   write(30,107)
   do 6j=l,nz
   do 6 i=l,nr
   write(30,108)r(i),z(j),pot(ij),strm(ij)/stnnax
6   continue
   call timer(ifinish)
   write(*,*)'Executiontime (sec): ',(ifinish-istart)/100.
101  format(g!5.0)
102  fonnat(4x,'R_start',7x,'Z_start',7x,'Potential',7x;Streaniline',
   >8x,V_r',13x,V_z',10x,V(cm/s)'/)
103  format(2x,el0.3,4x,el0.3,5x,ell.4,5x,ell.4,4x,ell.4,5x,ell.4,4x,
104  fonnat(3x,'R_start',6x,'Z_start',6x,'Potential',6x;Streaniline',
   >9x,V_r',12x,V_z',llx,V (cm/s)',7x, Travel-Time (s)'/)
105  format(lx,el0.3,3x,el0.3,3x,ell.4,5x,ell.4,5x,ell.4,5x,ell.4,5x,
   >ell.4,5x,ell.4)
106  format(//4x,65('*')//16x,'Results of Single-Well Airflow ',
   >'Simulation'//10x,'Steady-state Finite-Radius Solution'/,
   >10x,'Opento the Atmosphere'//4x,65('*')/)
107  format(///4x,'R_start',7x,'Z_start',7x,'Potential',7x,
   >'Norm_Streamline'/)
108  format(3x,el0.3,4x,el0.3,5x,ell.4,5x,ell.4)
   stop
   end

   subroutine series(r,z,vr,vz,phi,psi)
   implicit real*8 (a-h,o-z)
   common/cseries/xk,zb,zd,zl,prad,pz,a,visc,theta,rw,tol,patm
   pi=4.0*datan(1.0)
   n=0
   sumphi=0.0
   sumpsi=0.0
   sumvr=0.0
   sumvz=0.0
10   continue
   n=n+l
   if(n.gt. 1000) goto 15
   rm=n-0.5
   xmn=rm*pi/zb
   sterm=dsin(xmn*z)
   cterm=dcos(xmn*z)
   am=(dcos(xmn*zd)-dcos(xmn*zl))/(xmn*rw/a*besskl(xmn*rw/a))
   if(r.le.0.0)then
   write(*,*)'Step size for particle tracking is too large'
   return
   endif
   addphi=sterm*am/rm*besskO(xmn*r/a)
   addpsi=cterm*am/rm*bessk 1 (xmn*r/a)
   addvr=sterm*am*xmn/rm*bessk 1 (xmn*r/a)
   addvz=cterm*am*xmn/rm*besskO(xmn*r/a)
                                                   314

-------
   sumphi=sumphi+addphi
   sumpsi=sumpsi+addpsi
   sumvr=sumvr+addvr
   sumvz=sumvz+addvz
   if (dabs(sumvr).le.l.e-14 .or. dabs(sumvz).le.l.e-14)goto 15
   if(dabs(addvr/sumvr).gt.tol .or. dabs(addvz/sumvz).gt.tol)goto 10
15  phi=xk*sumphi+patm*patm
   psi=-xk*r*sumpsi/(patm*patm)
   vr=-xk/a*sumvr*(-prad/(2.*visc*dsqrt(phi)))/theta
   vz=xk*sumvz*(-pz/(2.*visc*dsqrt(phi)))/theta
   phi=phi/(patm*patm)
   return
   end

   DOUBLE PRECISION FUNCTION BESSIO(W)
   IMPLICIT REAL*8 (A-H,O-Z)
   T=W/3.75
   IF(DABS(T).LE.1.0) THEN
   Y=T*T
   BESSIO=1.0+Y*(3.5156229+Y*(3.0899424+Y*(1.2067492+Y*(0.2659732+
   >Y*(0.0360768+0.0045813*Y)))))
   ELSE
   AW=DABS(W)
   Y=3.75/AW
   TEMP=039894228+Y*(0.01328592+Y*(0.00225319+Y*(-0.00157565+Y*(
   >0.00916281+Y*(-0.02057706+Y*(0.02635537+Y*(-0.01647633+
   >0.00392377*Y)))))))
   BESSIO=DEXP(AW)*TEMP/DSQRT(AW)
   END IF
   RETURN
   END

   Double Precision Function BesskO(x)
   implicit real*8 (a-h,o-z)
   datapl,p2,p3,p4,p5,p6,p7/-0.57721566,0.4227842,0.23069756,
   >0.348859d-l,0.262698d-2,0.10750d-3,0.74d-5/
   dataql,q2,q3,q4,q5,q6,q7/1.25331414,-0.7832358d-l,0.2189568d-l,
   >-0.1062446d-l,0.587872d-2,-0.25154d-2,0.53208d-3/
   if(x.le.2.0)then
   y=x*x/4.0
   besskO=(-dlog(x/2.0)*bessiO(x))+(pl+y*(p2+y*(p3+y*(p4+y*(p5+y*(p6+
   >y*p7))))))
   else
   y=2.0/x
   besskO=(dexp(-x)/dsqrt(x))*(ql+y*(q2+y*(q3+y*(q4+y*(q5+y*(q6+y*q7)
   endif
   return
   end

   DOUBLE PRECISION FUNCTION BESSIl(W)
   IMPLICIT REAL*8 (A-H,O-Z)
                                               315

-------
   T=W/3.75
   IF(DABS(T).LE.1.0) THEN
   Y=T*T
   BESSI1=W*(0.5+Y*(0.87890594+Y*(0.51498869+Y*(0.15084934+Y*
   >(0.02658733+Y*(0.00301532+0.00032411*Y))))))
   ELSE
   AW=DABS(W)
   Y=3.75/AW
   TEMP=0.39894228+Y*(-0.03988024+Y*(-0.00362018+Y*(0.00163801+Y*
   >(-0.01031555+Y*(0.02282967+Y*(-0.02895312+Y*(0.01787654-
   >0.00420059*Y)))))))
   BESSI1=DEXP(AW)*TEMP/DSQRT(AW)
   IF(W.LT.O.)BESSI1=-BESSI1
   END IF
   RETURN
   END

   Double Precision Function Besskl(x)
   implicit real*8 (a-h,o-z)
   datapl,p2,p3,p4,p5,p6,p7/l.dO,0.15443144dO,-0.67278579dO,
   >-0.18156897dO,-0.1919402d-l,-0.110404d-2,-0.4686d-4/
   dataql,q2,q3,q4,q5,q6,q7/1.25331414,0.23498619dO,-0.3655620d-l,
   >0.1504268d-l,-0.780353d-2,0.325614d-2,-0.68245d-3/
   if(x.le.2.0)then
   y=x*x/4.0
   besskl=(dlog(x/2.0)*bessil(x))+(l./x)*(pl+y*(p2+y*(p3+y*(p4+y*(p5+
   >y*(p6+y*p7))))))
   else
   y=2.0/x
   besskl=(dexp(-x)/dsqrt(x))*(ql+y*(q2+y*(q3+y*(q4+y*(q5+y*(q6+y*q7)
   endif
   return
   end
c
   subroutine getlen(string,chr,nO,nl)
c
c  to determine the length of a string excluding any blank padding & heading
c  and return string(nO:nl) nO... position of first non-blank letter
c                 nl . . . position of last non-blank letter
c  else return the original string
c  ***Use | to keep blank space at the begining or the end of the
c   *** string
c
   character*(*) string
   character* 160 stringl
   character*! chr
   stringl=' '
   inO=0
   inl=0
   do 10 i=len(string),l,-l
   if(string(i:i).ne.chr) then
                                                 316

-------
     if(inl.eq.O) then
     if(string(i:i).eq.'|') then
     nl=i-l
     else
     nl=i
     endif
     inl=l
     endif
     goto 11
   endif
10   continue
11   do20i=l,len(string)
   if(string(i:i).ne.chr) then
    if(in0.eq.O) then
    if(string(i:i).eq.'') then
    nO=i+l
    else
    nO=i
    endif
    inO=l
    nk=nO-l
c    write(6,*) T//string(nO:nl)//T, nO,nl
    do 30 k=nO,nl
    if(string(k:k).ne.'|') then
     nk=nk+l
     if(k.eq.nO) then
      string 1=string(k:k)
     else
      stringl=stringl(l :nk-nO)//string(k:k)
     endif
    endif
c    write(6,*) T//stringl(l:nk-nO)//T
30   continue
    string=stringl
    nl=nl-nO+l
    nO=l
    endif
    return
   endif
20   continue
   nO=-l
   nl=-l
   return
   end
                                                    317

-------
                                  APPENDIX C (MFRLKINV)
Sample Input File for MFRLKINV
/* Title - This is a sample
293.0
-11.38
451.5
190.5
451.
12.7
l.e-10
1.74e-07,1.85e-07
2.9,9.3
2.1e-10,1.0e-9
5000
1
6
457.0,229.0,0.997049
1097.0,229.0,0.999238
1237.0,229.0,0.999189
1478.0,229.0,0.999336
1582.0,269.0,0.999607
1584.0,229.0,0.999754
1932.0,286.0,0.999705
input file for the MFRLKINV code
/* temperature (K) (TEMP)
/* mass pumping rate (g/sec)  (XMPR)
/* thickness of formation (XB)
/* distance to top of well screen (XD)
 /* distance to bottom of well screen (XL)
/* radius of the borehole
/* convergence criterion
/* minimum and maximum radial permeability (cmA2)
 /* minimum and maximum anisotropy ratio (K_r/K_z)
/* minimum and maximum leakance (cm)
/* number of random parameter guesses
 /* number of "best" parameter sets for forward problem
                                                318

-------
Sample Output File for MFRLKINV
   Steady-State Multi-Well Parameter Estimation
        Finite-Radius Model
      Top BC: Semi-Confining Layer
Parameter Estimates Ordered from Best to Worst

   K r        K z         Leakance      RMS-Error  IFLAG
0.1752E-06
0.1777E-06
0.1780E-06
0.1751E-06
0.1768E-06
0.1764E-06
0.1782E-06
0.1800E-06
0.1759E-06
0.1821E-06
0.5618E-07
0.5791E-07
0.5497E-07
0.5626E-07
0.5480E-07
0.5135E-07
0.4436E-07
0.4191E-07
0.4902E-07
0.4174E-07
0.2257E-09
0.2181E-09
0.2264E-09
0.2282E-09
0.2254E-09
0.2519E-09
0.3000E-09
0.3259E-09
0.2680E-09
0.3319E-09
0.1216E-03
0.1216E-03
0.1216E-03
0.1216E-03
0.1218E-03
0.1218E-03
0.1219E-03
0.1219E-03
0.1220E-03
0.1220E-03
0
0
0
0
0
0
0
0
0
0
(Only top ten listed)

Observed Versus Modeled Results

Parameter set -  1

  K_r=0.1752E-06
  K_z = 0.5618E-07
  KTb(leaky) = 0.2257E-09
    R(cm)
Z(cm)
P observed   P model
0.4570E+03
0.1097E+04
0.1237E+04
0.1478E+04
0.1582E+04
0.1584E+04
0.2290E+03
0.2290E+03
0.2290E+03
0.2290E+03
0.2690E+03
0.2290E+03
0.9970E+00
0.9992E+00
0.9992E+00
0.9993E+00
0.9996E+00
0.9998E+00
0.9970E+00
0.999 1E+00
0.9993E+00
0.9995E+00
0.9996E+00
0.9996E+00
                                               319

-------
Source Code for MFRLKINV (author: Ravi Varadhan)

(A FORTRAN program for estimation of radial gas permeability, vertical gas permeability, and leakance in a semi-
confined domain)

c Parameter estimation for the steady-state solution for finite well radius
c
c BC: Semi-confining unit on top
c Multiple monitoring locations
c
c version 1.0  07/21/99
c
   implicit real*8 (a-h,o-z)
   common/ceigen/xlk
   dimension pobs(100),p_r(5000),p_z(5000),beta(5000),errmet(5000),
   >rr(100),zz(100),wksp(5000),iwksp(5000),iflag(5000),bn(1000)
   integer*4 istart,ifmish
   character* 15 prefix,fnamel,fname2
   character*4 suffixI,suffix2
   write(*,*)' Parameter Estimation for Steady-State Multi-Well Test'
   write(*,*)' Finite-Radius Solution - BC: Leaky-Layer on Top'
   write(*,*)
   write(*,*)'Enter the Prefix of the input file:  '
   write(*,*)'An output file is created w/ same prefix'
   write(*,*)' Input-file Prefix ?  '
   read(*,'(a)')prefix
   call getlen(prefix,' ',nO,nl)
   suffixl='.inp'
   suffix2='.dat'
   fname 1 =prefix(nO: n 1 )//suffix 1
   fname2=prefix(nO:nl)//suffix2
   open( 10,fname 1, status='unknown')
   open(20,fname2,status='unknown')
   open(30,'seed.inp',status='unknown')
   read(10,*)
   read(10,101)temp,xmpr,xb,xd,xl,rw,tol
   read(10,*)prmin,prmax
   read(10,*)rmin,rmax
   read(10,*)xlkmin,xlkmax
   read(10,101)nguess
   read(10,101)nbest
   read(10,101)nsim
   do 2 i=l,nsim
   read(10,*)rr(i),zz(i),pobs(i)
2   continue
   patm= 1013200.
   visc=1.76e-04
   rgas=8.314e07
   xmw=28.8
   pi=4.0*datan(1.0)
c  patm = atmospheric pressure (g/cm/secA2)
c  vise = viscosity of gas (g/cm/sec)


                                                   320

-------
c  rgas = universal gas constant
c  temp = temperature of gas (K)
c  xmw = molecular weight of gas (g/mole)
c  xmpr = mass withdrawal rate (g/sec)
c  prad = radial permeability of the formation (cmA2)
c  pz = vertical permeability
c  xlk = leakance of the confining unit (cm): xlk = k'/b'
c  por = gas filled porosity
c  xb = thickness of the formation (cm)
c  xbp = thickness of the leaky unit
c  xl = distance from the bottom of leaky unit to the bottom of well screen
c  xd = distance from the bottom of leaky unit to the top of well screen
c  rw = nominal well radius (cm)
c  aa = radial cordinate (cm)
c  bb = vertical cordinate
c  cc = elapsed time (sec)
c  nsim = number of times solution will be generated
c  pobs = measured pneumatic pressure responses
c
   write(20,*)'	'
   write(20,*)'   Steady-State Multi-Well Parameter Estimation'
   write(20,*)'         Finite-Radius Model '
   write(20,*)'      Top BC: Semi-Confining Layer'
   write(20,*)'	'
   write(20,*)
   xll=xl/xb
   xdd=xd/xb
   r_w=rw/xb
   call timer(istart)
   read(30,*)iseed
   do lj=l,nguess
   iflag(j)=0
   errmet(j)=0.
   sqerr=0.
   guess 1=ranl (iseed)
   guess2=ranl (iseed)
   guess3=ranl (iseed)
   p_r(j)=prmin*(prrnax/prrnin)**guessl
   ratio=rmin+(rmax-rmin) *guess2
   p_z(j)=p_r(j)/ratio
   beta(j )=xlkmin*(xlkmax/xlkmin) * *guess3
   prad=p_r(j)
   pz=p_z(j)
   xlk=beta(j)/(pz/xb)
   a=dsqrt(prad/pz)
   xm=xmpr*visc*rgas*temp/(pi*xmw*prad*patm*patm*xb)
c
   call vnfunc(bn)
   do 5 i=l,nsim
   r=rr(i)/xb
   z=zz(i)/xb
   n=0
                                                  321

-------
   sum=0.0
10  continue
   n=n+l
   if(n.gt.lOOO)then
   iflag(j)=l
   goto 15
   endif
   vn=bn(n)
   tcos=dcos(vn*(l.-z))
   tsin=dsin(vn*( 1. -xdd))-dsin(vn*( 1 .-xll))
   trig=tcos*tsin
   fact=xlk/(vn*(xll-xdd)*(xlk+dsin(vn)**2))
   wn=vn/a
   fn=besskO(wn*r)/(wn*r_w*besskl(wn*r_w))
   add=trig*fact*fn
   sum=sum+add
   if(dabs(sum).le.l.e-14)goto 15
   if(dabs(add/sum).gt.tol)goto 10
15  continue
   psqrd=1.0+2.0*xm*sum
   if(psqrd.lt.0.)then
   iflag(j)=2
   enmet(j)=l.elO
   goto 1
   endif
   popatm=dsqrt(psqrd)
   sqerr=sqerr+(popatm-pobs(i)) * *2
5   continue
   errmet(j )=dsqrt(sqerr/nsim)
   write(*,*)j,prad,pz,beta(]),errmet(j),iflag(j)
1   continue
   call sorts (nguess,errmet,p_r,p_z,beta,iflag,wksp,iwksp)
   write(20,107)
   write(20,102)
   do 20j=l,nguess
   write(20,103)p_r(j),p_z(j),beta(j),errmet(j),iflag(j)
20  continue
c
   do 30 j=l,nbest
   prad=p_r(j)
   pz=p_z(j)
   xlk=beta(j)/(pz/xb)
   write(20,104)j,prad,pz,beta(j)
   write(20,105)
   a=dsqrt(prad/pz)
   xm=xmpr*visc*rgas*temp/(pi*xmw*prad*patm*patm*xb)
   call vnfunc(bn)
c
   do 35 i=l,nsim
   r=rr(i)/xb
   z=zz(i)/xb
   n=0
                                                   322

-------
   sum=0.0
40   continue
   n=n+l
   if(n.gt.lOOO)goto45
   vn=bn(n)
   tcos=dcos(vn*(l.-z))
   tsin=dsin(vn*( 1. -xdd))-dsin(vn*( 1 .-xll))
   trig=tcos*tsin
   fact=xlk/(vn*(xll-xdd)*(xlk+dsin(vn)**2))
   wn=vn/a
   fn=besskO(wn*r)/(wn*r_w*besskl(wn*r_w))
   add=trig*fact*fn
   sum=sum+add
   if(dabs(sum).le.l.e-14)goto 45
   if(dabs(add/sum).gt.tol)goto 40
45   continue
   psqrd=1.0+2.0*xm*sum
   if(psqrd.ge.0.)then
   popatm=dsqrt(psqrd)
   else
   popatm=0.
   endif
   write(20,106)rr(i),zz(i),pobs(i),dsqrt(psqrd)
35   continue
30   continue
   call timer(ifinish)
   write(*,*)Time of execution (sec)  =',(ifinish-istart)/100.
101  format(gl5.0)
102  format(/5x,'K_r', 1 lx,'K_z', 10x,'Leakance',9x,'RMS-Error',4x,
   >'IFLAG'/)
103  format(2x,el0.4,4x,el0.4,5x,el0.4,8x,el0.4,4x,i2)
104  format(//20x,'Parameter set - ',i4//,3x,'K_r = ',e!0.4/
   >3x,'K_z = ',el0.4/3x,'K/b(leaky) = ',e!0.4//)
105  format(7x,'R (cm)', 10x,'Z (cm)', 1 lx,'P_observed',7x,'P_model'/)
106  format(5x,el0.4,7x,el0.4,8x,el0.4,6x,el0.4)
107  format(//10x,'Parameter Estimates Ordered from Best to Worst'//)
   rewind(30)
   write(30,*)-iseed
   stop
   end
c
   subroutine vnfunc(x)
   implicit real*8 (a-h,o-z)
   dimension x(1000)
   external eigen
   common/ceigen/b
   pi=4.*datan(1.0)
   xl=l.e-06
   x2=pi
   kount= 1
1   continue
   zero=rtsafe(eigen,xl,x2,l.e-06)
                                                   323

-------
x(kount)=zero
xl=zero+0.5*pi
x2=zero+1.5*pi
kount=kount+l
if(kount.le.lOOO)goto 1
return
end

Double Precision FUNCTION rtsafe(funcd,xl,x2,xacc)
implicit real*8 (a-h,o-z)
EXTERNAL funcd
PARAMETER (MAXIT=100)
callfuncd(xl,fl,df)
call funcd(x2,fh,df)
if((fl.gt.0..and.fh.gt.0.).or.(fl.lt.0..and.fh.lt.0.))write(*,*)
*'root must be bracketed in rtsafe'
if(fl.eq.0.)then
  rtsafe=xl
  return
else if(fh.eq.0.)then
  rtsafe=x2
  return
else if(fl.lt.0.)then
  xl=xl
  xh=x2
else
  xh=xl
  xl=x2
endif
rtsafe=.5*(xl+x2)
dxold=dabs(x2-xl)
dx=dxold
call funcd(rtsafe,f,df)
do llj=l,MAXIT
  if(((rtsafe-xh)*df-f)*((rtsafe-xl)*df-f).ge.0..or. dabs(2.*
*f).gt.dabs(dxold*df)) then
   dxold=dx
   dx=0.5*(xh-xl)
   rtsafe=xl+dx
   if(xl. eq. rtsafe)return
  else
   dxold=dx
   dx=f/df
   temp=rtsafe
   rtsafe=rtsafe-dx
   if(temp.eq.rtsafe)return
  endif
  if(dabs(dx).lt.xacc) return
  call funcd(rtsafe,f,df)
  if(f.lt.O.) then
   xl=rtsafe
  else
                                                 324

-------
     xh=rtsafe
    endif
1 1  continue
   write(*,*)'rtsafe exceeded maximum iterations'
   return
   END

   subroutine eigen(x,f,df)
   implicit real*8 (a-h,o-z)
   common/ceigen/b
   f=b*dcos(x)-x*dsin(x)
   df=-b*dsin(x)-dsin(x)-x*dcos(x)
   return
   end

   Double Precision Function BesskO(x)
   implicit real*8 (a-h,o-z)
   datapl,p2,p3,p4,p5,p6,p7/-0.57721566,0.4227842,0.23069756,
  >0.348859d-l,0.262698d-2,0.10750d-3,0.74d-5/
   dataql,q2,q3,q4,q5,q6,q7/1.25331414,-0.7832358d-l,0.2189568d-l,
  >-0.1062446d-l,0.587872d-2,-0.25154d-2,0.53208d-3/
   if(x.le.2.0)then
   y=x*x/4.0
   besskO=(-dlog(x/2.0)*bessiO(x))+(pl+y*(p2+y*(p3+y*(p4+y*(p5+y*(p6+
  >y*p7))))))
   else
   y=2.0/x
   besskO=(dexp(-x)/dsqrt(x))*(ql+y*(q2+y*(q3+y*(q4+y*(q5+y*(q6+y*q7)
   endif
   return
   end

   DOUBLE PRECISION FUNCTION BESSIO(W)
   IMPLICIT real*8 (A-H,O-Z)
   T=W/3.75
   IF(DABS(T).LE.1.0) THEN
   Y=T*T
   BESSIO=1.0+Y*(3.5156229+Y*(3.0899424+Y*(1.2067492+Y*(0.2659732+
  >Y*(0.0360768+0.0045813*Y)))))
   ELSE
   AW=DABS(W)
   Y=3.75/AW
   TEMP=0.39894228+Y*(0.01328592+Y*(0.00225319+Y*(-0.00157565+Y*(
  >0.00916281+Y*(-0.02057706+Y*(0.02635537+Y*(-0.01647633+
  >0.00392377*Y)))))))
   BESSIO=DEXP(AW)*TEMP/DSQRT(AW)
   ENDIF
   RETURN
   END

   double precision FUNCTION besskl(x)


                                               325

-------
CU  USES bessil
   implicit real*8 (a-h,o-z)
   DOUBLE PRECISION pl,p2,p3,p4,p5,p6,p7,ql,q2,q3,q4,q5,q6,q7,y
   SAVEpl,p2,p3,p4,p5,p6,p7,ql,q2,q3,q4,q5,q6,q7
   DATA pl,p2,p3,p4,p5,p6,p7/1.0dO,0. 15443 144dO,-0.67278579dO,
   *-0.18156897dO,-0.1919402d-l,-0.110404d-2,-0.4686d-4/
   DATAql,q2,q3,q4,q5,q6,q7/1.25331414dO,0.23498619dO,-0.3655620d-l,
   *0.1504268d-l,-0.780353d-2,0.325614d-2,-0.68245d-3/
   if(x.le.2.0)then
    y=x*x/4.0
    besskl=(dlog(x/2.0)*bessil(x))+(1.0/x)*(pl+y*(p2+y*(p3+y*(p4+y*
   *(P5+y*(p6+y*p7))))))
   else
    y=2.0/x
    besskl=(dexp(-x)/dsqrt(x))*(ql+y*(q2+y*(q3+y*(q4+y*(q5+y*(q6+y*
   endif
   return
   END

   double precision FUNCTION bessil(x)
   implicit real*8 (a-h,o-z)
   DOUBLE PRECISION pl,p2,p3,p4,p5,p6,p7,ql,q2,q3,q4,q5,q6,q7,q8,q9,y
   SAVEpl,p2,p3,p4,p5,p6,p7,ql,q2,q3,q4,q5,q6,q7,q8,q9
   DATApl,p2,p3,p4,p5,p6,p7/0.5dO,0.87890594dO,0.51498869dO,
   *0.15084934dO,0.2658733d-l,0.301532d-2,0.32411d-3/
   DATAql,q2,q3,q4,q5,q6,q7,q8,q9/0.39894228dO,-0.3988024d-l,
   *-0.362018d-2,0.163801d-2,-0.1031555d-l,0.2282967d-l,-0.2895312d-l,
   *0.1787654d-l,-0.420059d-2/
   if(abs(x).lt.3.75)then
    y=(x/3.75)**2
    bessil=x*(pl+y*(p2+y*(p3+y*(p4+y*(p5+y*(p6+y*p7))))))
   else
    ax=abs(x)
    y=3.75/ax
    bessil=(dexp(ax)/dsqrt(ax))*(ql+y*(q2+y*(q3+y*(q4+y*(q5+y*(q6+y*
   *(q7+y*(q8+y*q9))))))))
    if(x.lt.0.)bessil=-bessil
   endif
   return
   END

   double precision FUNCTION ranl(idum)
   implicit real*8 (a-h,o-z)
   INTEGER idum,IA,IM,IQ,IR,NTAB,NDIV
   PARAMETER (IA=16807,IM=2147483647,AM=1./IM,IQ=127773,IR=2836,
   *NTAB=32,NDIV= 1+(IM-1)/NTAB,EPS=1 .2e-7,RNMX= 1 .-EPS)
   INTEGER j ,k,iv(NTAB),iy
   SAVE iv,iy
   DATA iv /NTAB*0/, iy /O/
   if (idum.le.0.or.iy.eq.O) then
    idum=max(-idum, 1 )
                                                326

-------
    do llj=NTAB+8,l,-l
     k=idum/IQ
     idum=IA*(idum-k*IQ)-IR*k
     if (idum.lt. 0) idum=idum+IM
     if (j.le.NTAB) iv(j)=idum
11   continue
    iy=iv(l)
   endif
   k=idum/IQ
   idum=IA*(idum-k*IQ)-IR*k
   if (idum.lt. 0) idum=idum+IM
   j=l+iy/NDIV
   iy=iv(j)
   iv(j)=idum
   ranl=dminl(AM*iy,RNMX)
   return
   END

   SUBROUTINE sort3(n,ra,rb,rc,rd,ie,wksp,iwksp)
   implicit real*8 (a-h,o-z)
   INTEGER n,iwksp(n),ie(n)
   dimension ra(n),rb(n),rc(n),rd(n),wksp(n)
CU  USES indexx
   INTEGER j
   call indexx(n,ra,iwksp)
   do llj=l,n
    wksp(j)=ra(j)
11  continue
   do 12j=l,n
    ra(j )=wksp(iwksp(j))
12  continue
   do 13j=l,n
    wksp(j)=rb(j)
13  continue
   do 14j=l,n
    rb(j )=wksp(iwksp(j))
14  continue
   do 15j=l,n
    wksp(j)=rc(j)
15  continue
   do 16j=l,n
    rc(j)=wksp(iwksp(j))
16  continue
   do 17j=l,n
    wksp(j)=rd(j)
17  continue
   do 18j=l,n
    rd(j)=wksp(iwksp(j))
18  continue
   do 19j=l,n
    wksp(j)=ie(j)
19  continue
                                                327

-------
   do20j=l,n
    ie(j )=wksp(iwksp(j))
20  continue
   return
   END

   SUBROUTINE indexx(n,arr,indx)
   implicit real*8 (a-h,o-z)
   INTEGER n,indx(n),M,NSTACK
   dimension arr(n)
   PARAMETER (M=7,NSTACK=50)
   INTEGER i,indxt,ir,itemp j j stack,k,l,istack(NSTACK)
   do llj=l,n
    indx(j)=j
11  continue
   jstack=0
   1=1
   ir=n
1   if(ir-l.lt.M)then
    do!3j=l+l,ir
     indxt=indx(j)
     a=arr(indxt)
     do!2i=j-l,l,-l
       if(arr(indx(i)).le.a)goto 2
       indx(i+l)=indx(i)
12     continue
     i=0
2     indx(i+l)=indxt
13   continue
    if(j stack, eq. 0)return
    ir=istack(j stack)
    l=istack(j stack-1)
    jstack=jstack-2
   else
    k=(l+ir)/2
    itemp=indx(k)
    indx(k)=indx(l+l)
    indx(l+l)=itemp
    if(arr(indx(l+1)). gt.arr(indx(ir)))then
     itemp=indx(l+l)
     indx(l+l)=indx(ir)
     indx(ir)=itemp
    endif
    if(arr(indx(l)).gt.arr(indx(ir)))then
     itemp=indx(l)
     indx(l)=indx(ir)
     indx(ir)=itemp
    endif
    if(arr(indx(l+l)).gt.arr(indx(l)))then
     itemp=indx(l+l)
     indx(l+l)=indx(l)
     indx(l)=itemp
                                                  328

-------
     endif

     j=ir
     indxt=indx(l)
     a=arr(indxt)
3    continue

     if(arr(indx(i)).lt.a)goto 3
4    continue

     if(arr(indx(j)).gt.a)goto 4
     if(j.lt.i)goto 5
     itemp=indx(i)
     indx(i)=indx(j)
     indx(j)=itemp
     goto 3
5    indx(l)=indx(j)
     indx(j)=indxt
     jstack=jstack+2
     if(jstack.gt.NSTACK)write(*,*) 'NSTACKtoo small in indexx'
     if(ir-i+1. ge j -l)then
      istack(jstack)=ir
      istack(j stack-l)=i

     else
      istack(jstack)=j-l
      istack(j stack-1)=1

     endif
    endif
    goto 1
    END
    subroutine getlen(string,chr,nO,nl)
c
c   to determine the length of a string excluding any blank padding & heading
c   and return string(nO:nl) nO... position of first non-blank letter
c                  nl...  position of last non-blank letter
c   else return the original string
c   ***Use | to keep blank space at the begining or the end of the
c    *** string
c
    character*(*) string
    character* 160 stringl
    character*! chr
    stringl=''
    inO=0
    inl=0
    do 10 i=len(string),l,-l
    if(string(i:i).ne.chr) then
     if(inl.eq.O) then
                                                     329

-------
     if(string(i:i).eq.'|') then
     nl=i-l
     else
     nl=i
     endif
     inl=l
     endif
     goto 11
   endif
10   continue
11   do20i=l,len(string)
   if(string(i:i).ne.chr) then
    if(in0.eq.O) then
    if(string(i:i).eq.'') then
    nO=i+l
    else
    nO=i
    endif
    inO=l
    nk=nO-l
c    write(6,*) T//string(nO:nl)//'|', nO,nl
    do 30 k=nO,nl
    if(string(k:k).ne.'|') then
     nk=nk+l
     if(k.eq.nO) then
      string 1=string(k:k)
     else
      stringl=stringl(l :nk-nO)//string(k:k)
     endif
    endif
c    write(6,*) '|'//stringl(l:nk-nO)//'|'
30   continue
    string=stringl
    nl=nl-nO+l
    nO=l
    endif
    return
   endif
20   continue
   nO=-l
   nl=-l
   return
   end
                                                    330

-------
                                        Appendix D (TFRLK)

Sample Input File for TFRLK code

/* Title - This is a sample input file for the TFRLK code
298.0           /* temperature (K) (TEMP)
1.0              /* mass pumping rate (g/sec)  (XMPR)
2000.0           /* time of pumping (sec)
600.0           /* thickness of formation  (XB)
410.            /* distance to top of well screen (XD)
470.             /* distance to bottom of well screen (XL)
10.0             /* radius of the well (Rw)
l.e-09           /* convergence criterion
1
0.10             /* air-porosity
1
15000.0         /* well-bore storage volume
1
1 .Oe-09          /* radial permeability (cmA2/s)
6
2.e-09,1.0E-09,3.33E-10,2.0E-10,1.40E-10,1.0E-10       /* vertical permeability (cmA2/s)
1
l.e-11          /* leakance (cm)
1
10.0            /* radial distances (cm)
1
440.0            /* vertical distance (cm)
100             /*times(s)
1.0,2.0,3.0,4.0,5.0,6.0,7.0,8.0,9.0,10.0,
15.0,20.0,25.0,30.0,35.0,40.0,45.0,50.0,55.0,60.0,
70.0,80.0,90.0,100.0,110.0,120.0,130.0,140.0,150.0,175.0,
200.0,225.0,250.0,275.0,300.0,325.0,350.0,375.0,400.0,450.0,
500.0,550.0,600.0,650.0,700.0,750.0,800.0,850.0,900.0,950.0,
1000.0,1100.0,1200.0,1300.0,1400.0,1500.0,1600.0,1700.0,1800.0,1900.0,
2000.0,2001.0,2002.0,2003.0,2004.0,2005.0,2006.0,2007.0,2008.0,2009.0,
2010.0,2015.0,2020.0,2025.0,2030.0,2035.0,2040.0,2045.0,2050.0,2100.0,
2200.0,2300.0,2400.0,2500.0,2600.0,2700.0,2800.0,2900.0,3000.0,3100.0,
3200.0,3300.0,3400.0,3400.0,3500.0,3600.0,3700.0,3800.0,3900.0,4000.0
                                                   331

-------
Sample Output File for TFRLK code
              Results of Sensitivity Analysis

     Transient Finite-Radius Solution with Well-Bore Storage (
     Semi-confining Layer on Top
           Parameter Set:  1

     Porosity =   0.1000
     Storage =0.1500E+05
     K_radial =0.1000E-08
     K_vertical = 0.2000E-08
     Leakance =0.1000E-10
                      t
P/P_atm     Flag
0.100E+02
0.100E+02
0.100E+02
0.100E+02
0.100E+02
0.100E+02
0.100E+02
0.100E+02
0.100E+02
0.100E+02
0.440E+03
0.440E+03
0.440E+03
0.440E+03
0.440E+03
0.440E+03
0.440E+03
0.440E+03
0.440E+03
0.440E+03
0.100E+01
0.200E+01
0.300E+01
0.400E+01
0.500E+01
0.600E+01
0.700E+01
0.800E+01
0.900E+01
0.100E+02
0.10468E+01
0.10849E+01
0.11179E+01
0.11464E+01
0.11711E+01
0.11933E+01
0.12131E+01
0.12307E+01
0.12470E+01
0.12608E+01
0
0
0
0
0
0
0
0
0
0
(first 10 simulations for parameter set)
                                                 332

-------
Source Code for TFRLK (author: Ravi Varadhan)

c Forward problem for the transient solution for finite well radius
c including well-bore storage
c Top BC: Semi-Confining Layer
c
c version 1.0  07/14/99
c
   implicit real*8 (a-h,o-z)
   dimension rr(100),zz(100),tt(100),poros(20),stor(20),p_r(20),p_z
   >(20),p_lk(20),bn(1000)
   common/cfunc/r,r_w,vn,a,gama
   common/ceigen/xlk
   character* 15 prefix,fnamel,fname2
   character*4 suffixI,suffix2
   write(*,*)' Forward Problem - Transient State'
   write(*,*)' Finite-Radius Solution with Borehole Storage '
   write(*,*)' Top BC: Semi-Confining Layer'
   write(*,*)
   write(*,*)'Enter the Prefix of the input file: '
   write(*,*)'An output file is created w/ same prefix'
   write(*,*)' Input-file Prefix ?  '
   read(*,'(a)')prefix
   call getlen(prefix,' ',nO,nl)
   suffixl='.inp'
   suffix2='.dat'
   fnamel=prefix(nO:nl)//suffixl
   fname2=prefix(nO:nl)//suffix2
   open( 10,fname 1, status='unknown')
   open(20,fname2,status='unknown')
   read(10,*)
   read(10,110)temp,xmpr,ttpump,xb,xd,xl,rw,tol
   patm=1013200.
   visc=1.76e-04
   rgas=8.314e07
   xmw=28.8
   pi=4.0*datan(1.0)
c  patm = atmospheric pressure (g/cm/secA2)
c  vise = viscosity of gas (g/cm/sec)
c  rgas = universal gas constant
c  temp = temperature of gas (K)
c  xmw = molecular weight of gas (g/mole)
c  xmpr = mass withdrawal rate  (g/sec)
c  por = air-filled porosity
c  prad = radial permeability of the formation (cmA2)
c  pz = vertical permeability
c  plk = permeability of the leaky confining unit
c  xb = thickness of the formation (cm)
c  xbp = thickness of the leaky unit
c  xl = distance from the bottom of leaky unit to the bottom of well screen
c  xd = distance from the bottom of leaky unit to the top of well screen
c  rw = nominal well radius (cm)
                                                   333

-------
c
    xll=xl/xb
    xdd=xd/xb
    r_w=rw/xb
c
c parameters to be varied are: porosity,wellbore-storage,k_r,k_z,leakance
c
    read(10,*)npor
    read(10,*)(poros(i),i=l,npor)
    read(10,*)nstor
    read( 10, *)(stor(i),i= 1 ,nstor)
    read(10,*)nprad
    read( 10, *)(p_r(i),i= 1 ,nprad)
    read(10,*)npz
    read( 10, *)(p_z(i),i= 1 ,npz)
    read(10,*)nlk
    read(10,*)(p_lk(i),i=l,nlk)
    nparm=npor+nstor+nprad+npz+nlk
    read(10,*)nr
    read(10,*)(rr(i),i=l,nr)
    read(10,*)nz
    read(10,*)(zz(i),i=l,nz)
    read(10,*)nt
    read(10,*)(tt(i),i=l,nt)
    write(20,101)
    mlinv=-l
    kount=0
    do 1 i=l,nparm
    por=poros(l)
    vstor=stor(l)
    prad=p_r(l)
    pz=p_z(l)
    plk=p_lk(l)
    if(i.eq.l)goto 2
    if((i-npor).eq.l)goto 1
    if((i-npor-nstor). eq. 1 )goto 1
    if((i-npor-nstor-nprad).eq. l)goto 1
    if((i-npor-nstor-nprad-npz).eq. l)goto 1
    if(i.le.npor)por=poros(i)
    if(i.gt.npor .and. i.le.(npor+nstor))vstor=stor(i-npor)
    if(i.gt.(npor+nstor) .and. i.le.(npor+nstor+nprad))prad=p_r
   >(i-npor-nstor)
    if(i.gt.(npor+nstor+nprad) .and. i.le.(npor+nstor+nprad+npz))
   >pz=p_z(i-npor-nstor-nprad)
    if(i.gt.(npor+nstor+nprad+npz))plk=p_lk(i-npor-nstor-nprad-npz)
c
2   continue
    kount=kount+l
    write(20,102)kount
    write(*,*)'porosity =',por
    write(*,*)'storage =',vstor
    write(*,*)'K_r =',prad
                                                     334

-------
   write(*,*)'K_z  =',pz
   write(*,*)'leakance  =',plk
   write(*,*)
   write(20,103)por,vstor,prad,pz,plk
   gama=vstor/(2. *pi*(xl-xd) *xb *xb *por)
   a=dsqrt(prad/pz)
   xm=xmpr*visc*rgas*temp/(pi*xmw*prad*patm*patm*xb)
   tpump=ttpump*prad*patm/(visc*por*xb*xb)
   xlk=plk/(pz/xb)
   if(i.eq.l .or. i.gt.(npor+nstor+nprad+npz))callvnfunc(bn)
c
   write(20,104)
   do 5 jl=l,nz
   z=zz(jl)/xb
   do6j2=l,nr
   r=rr(j2)/xb
   do7j3=l,nt
   t=tt(j 3) *prad*patm/(visc *por*xb *xb)
   iflag=0
   n=0
   sum=0.0
10   continue
   n=n+l
   if(n.gt.lOOO)goto  15
   vn=bn(n)
   call linv(mlinv,t,fnrt)
   call linv(mlinv,t-tpump,fnrtp)
   tcos=dcos(vn*(l.-z))
   tsin=dsin(vn*( 1. -xdd))-dsin(vn*( 1 .-xll))
   trig=tcos*tsin
   fact=xiy(vn*(xll-xdd)*(xlk+dsin(vn)**2))
   ans=fnrt-fnrtp
   add=trig*fact*ans
   sum=sum+add
   if(dabs(sum).le.l.e-14)goto 15
   if(dabs(add/sum).gt.tol)goto 10
15   continue
   psqrd=1.0+2.0*xm*sum
   if(psqrd.lt.0.)then
   popatm=0.
   iflag=999
   else
   popatm=dsqrt(psqrd)
   endif
   write(20,105)rr(j2),zz(j I),tt(j3),popatm,iflag
7   continue
   write(20,106)
6   continue
   write(20,106)
5   continue
c
1   continue
                                                   335

-------
101  format(//4x,65('*')//25x,'Results of Sensitivity Analysis'//10x,
   ^Transient Finite-Radius Solution with Well-Bore Storage & '/,
   >10x,'Semi-confining Layer on Top'//4x,65('*')/)
102  format(///20x,'Parameter Set: ',i2/)
103  format(10x,'Porosity = ',f!0.4/10x,'Storage = ',e!0.4/10x,
   >'K_radial  = ',el0.4/10x,'K_vertical = ',e!0.4/10x,'Leakance =',
   >e 10.47)
104  format(8x,'r',15x,'z',14x,'t',15x,'P/P_atm',8x,'Flag'/)
105  format(3x,el0.3,7x,el0.3,5x,el0.3,7x,el2.5,6x,i3)
106  format(/)
110  format(glO.O)
   stop
   end

   subroutine linv(m,t,f)
   implicit real*8 (a-h,o-z)
   common/cfunc/r,r_w,vn,a,gama
   dimension g(50),v(50),h(25)
   f=0.
   if(t.le.0.)return
   n=14
   if(m.ne.n)then
   call auxvec(n,g,v,h)
   m=n
   endif
   ss=dlog(2.dO)/t
   xp=0.dO
   visum=0.dO
   do  50 1=1,n
   s=l*ss
   wn=dsqrt(s+(vn/a) * *2)
   pnum=besskO(wn*r)
   pden=s*(wn*r_w*besskl(wn*r_w)+s*gama*besskO(wn*r_w))
   fbar=pnum/pden
   xp=xp+v(l)*fbar
   visum=v(l)/l+visum
50   continue
   f=ss*xp
   return
   end
c
   subroutine auxvec(n,g,v,h)
   implicit real*8 (a-h,o-z)
   dimension g(0:50),v(50),h(25)
   g(0)=l
   nh=n/2
   do  90 i=l,n
   g(i)=i*g(i-l)
90   continue
   h(l)=2.dO/g(nh-l)
   do  100 i=2,nh
   h(i)=i**nh*g(2*i)/(g(nh-i)*g(i)*g(i-l))
                                                   336

-------
100  continue
   sn=2*mod(nh,2)-l
   vsum=0.dO
   do 140 i=l,n
   v(i)=0
   k2=minO(i,nh)
   do 130 k=kl,k2
   v(i)=v(i)+h(k)/(g(i-k)*g(2*k-i))
130  continue
   v(i)=sn*v(i)
   vsum=vsum+v(i)
   sn=-sn
140  continue
   return
   end
c
   Double Precision Function BesskO(x)
   implicit real*8 (a-h,o-z)
   datapl,p2,p3,p4,p5,p6,p7/-0.57721566,0.4227842,0.23069756,
   >0.348859d-l,0.262698d-2,0.10750d-3,0.74d-5/
   dataql,q2,q3,q4,q5,q6,q7/1.25331414,-0.7832358d-l,0.2189568d-l,
   >-0.1062446d-l,0.587872d-2,-0.25154d-2,0.53208d-3/
   if(x.le.2.0)then
   y=x*x/4.0
   besskO=(-dlog(x/2.0)*bessiO(x))+(pl+y*(p2+y*(p3+y*(p4+y*(p5+y*(p6+
   >y*p7))))))
   else
   y=2.0/x
   besskO=(dexp(-x)/dsqrt(x))*(ql+y*(q2+y*(q3+y*(q4+y*(q5+y*(q6+y*q7)
   endif
   return
   end

   DOUBLE PRECISION FUNCTION BESSIO(W)
   IMPLICIT real*8 (A-H,O-Z)
   T=W/3.75
   IF(DABS(T).LE.1.0) THEN
   Y=T*T
   BESSIO=1.0+Y*(3.5156229+Y*(3.0899424+Y*(1.2067492+Y*(0.2659732+
   >Y*(0.0360768+0.0045813*Y)))))
   ELSE
   AW=DABS(W)
   Y=3.75/AW
   TEMP=0.39894228+Y*(0.01328592+Y*(0.00225319+Y*(-0.00157565+Y*(
   >0.00916281+Y*(-0.02057706+Y*(0.02635537+Y*(-0.01647633+
   >0.00392377*Y)))))))
   BESSIO=DEXP(AW)*TEMP/DSQRT(AW)
   ENDIF
   RETURN
   END
                                               337

-------
c
   double precision FUNCTION besskl(x)
CU  USES bessil
   implicit real*8 (a-h,o-z)
   DOUBLE PRECISION pl,p2,p3,p4,p5,p6,p7,ql,q2,q3,q4,q5,q6,q7,y
   SAVEpl,p2,p3,p4,p5,p6,p7,ql,q2,q3,q4,q5,q6,q7
   DATA pl,p2,p3,p4,p5,p6,p7/1.0dO,0. 15443 144dO,-0.67278579dO,
   *-0.18156897dO,-0.1919402d-l,-0.110404d-2,-0.4686d-4/
   DATAql,q2,q3,q4,q5,q6,q7/1.25331414dO,0.23498619dO,-0.3655620d-l,
   *0.1504268d-l,-0.780353d-2,0.325614d-2,-0.68245d-3/
   if(x.le.2.0)then
    y=x*x/4.0
    besskl=(dlog(x/2.0)*bessil(x))+(1.0/x)*(pl+y*(p2+y*(p3+y*(p4+y*
   *(p5+y*(p6+y*p7))))))
   else
    y=2.0/x
    besskl=(dexp(-x)/dsqrt(x))*(ql+y*(q2+y*(q3+y*(q4+y*(q5+y*(q6+y*
   endif
   return
   END
c
   double precision FUNCTION bessil(x)
   implicit real*8 (a-h,o-z)
   DOUBLE PRECISION Pl,p2,p3,p4,p5,p6,p7,ql,q2,q3,q4,q5,q6,q7,q8,q9,y
   SAVEpl,p2,p3,p4,p5,p6,p7,ql,q2,q3,q4,q5,q6,q7,q8,q9
   DATApl,p2,p3,p4,p5,p6,p7/0.5dO,0.87890594dO,0.51498869dO,
   *0.15084934dO,0.2658733d-l,0.301532d-2,0.32411d-3/
   DATAql,q2,q3,q4,q5,q6,q7,q8,q9/0.39894228dO,-0.3988024d-l,
   *-0.362018d-2,0.163801d-2,-0.1031555d-l,0.2282967d-l,-0.2895312d-l,
   *0.1787654d-l,-0.420059d-2/
   if(abs(x).lt.3.75)then
    y=(x/3.75)**2
    bessil=x*(pl+y*(p2+y*(p3+y*(p4+y*(p5+y*(p6+y*p7))))))
   else
    ax=abs(x)
    y=3.75/ax
    bessil=(dexp(ax)/dsqrt(ax))*(ql+y*(q2+y*(q3+y*(q4+y*(q5+y*(q6+y*
   *(q7+y*(q8+y*q9))))))))
    if(x.lt.0.)bessil=-bessil
   endif
   return
   END
c
   subroutine vnfunc(x)
   implicit real*8 (a-h,o-z)
   dimension x( 1000)
   external eigen
   common/ceigen/b
   pi=4.*datan(1.0)
   xl=l.e-06
   x2=pi
                                                  338

-------
   kount=l
1   continue
   zero=rtsafe(eigen,xl,x2,l.e-06)
   x(kount)=zero
   xl=zero+0.5*pi
   x2=zero+1.5*pi
   kount=kount+l
   if(kount.le.lOOO)goto 1
   return
   end
c
   Double Precision FUNCTION rtsafe(funcd,xl,x2,xacc)
   implicit real*8 (a-h,o-z)
   EXTERNAL tuned
   PARAMETER (MAXIT=100)
   callfuncd(xl,fl,df)
   call funcd(x2,fh,df)
   if((fl.gt.0..and.fh.gt.0.).or.(fl.lt.0..and.fh.lt.0.))write(*,*)
   *'root must be bracketed in rtsafe'
   if(fl.eq.0.)then
     rtsafe=xl
     return
   else if(fh.eq.0.)then
     rtsafe=x2
     return
   else if(fl.lt.0.)then
     xl=xl
     xh=x2
   else
     xh=xl
     xl=x2
   endif
   rtsafe=.5*(xl+x2)
   dxold=dabs(x2-xl)
   dx=dxold
   call funcd(rtsafe,f,df)
   do llj=l,MAXIT
     if(((rtsafe-xh)*df-f)*((rtsafe-xl)*df-f).ge.0..or. dabs(2.*
   *f).gt.dabs(dxold*df)) then
      dxold=dx
      dx=0.5*(xh-xl)
      rtsafe=xl+dx
      if(xl. eq. rtsafe)return
     else
      dxold=dx
      dx=f/df
      temp=rtsafe
      rtsafe=rtsafe-dx
      if(temp.eq.rtsafe)return
     endif
     if(dabs(dx).lt.xacc) return
     call funcd(rtsafe,f,df)
                                                    339

-------
     if(f.lt.O.) then
      xl=rtsafe
     else
      xh=rtsafe
     endif
11   continue
    write(*,*)'rtsafe exceeded maximum iterations'
    return
    END
c
    subroutine eigen(x,f,df)
    implicit real*8 (a-h,o-z)
    common/ceigen/b
    f=b*dcos(x)-x*dsin(x)
    df=-b*dsin(x)-dsin(x)-x*dcos(x)
    return
    end
c
    subroutine getlen(string,chr,nO,nl)
c
c   to determine the length of a string excluding any blank padding & heading
c   and return string(nO:nl) nO... position of first non-blank letter
c                  nl... position of last non-blank letter
c   else return the original string
c   ***Use | to keep blank space at the begining or the end of the
c    *** string
c
    character*(*) string
    character* 160 stringl
    character*! chr
    stringl=''
    inO=0
    inl=0
    do 10 i=len(string),l,-l
    if(string(i:i).ne.chr) then
     if(inl.eq.O) then
     if(string(i:i).eq.'|') then
     nl=i-l
     else
     nl=i
     endif
     inl=l
     endif
     goto 11
    endif
10   continue
11   do20i=l,len(string)
    if(string(i:i).ne.chr) then
    if(in0.eq.O) then
    if(string(i:i).eq.'') then
    nO=i+l
    else
                                                      340

-------
    nO=i
    endif
    inO=l
    nk=nO-l
c   write(6,*) T//string(nO:nl)//'|', nO,nl
    do 30 k=nO,nl
    if(string(k:k).ne.'|') then
     nk=nk+l
     if(k.eq.nO) then
      string 1=string(k:k)
     else
      stringl=stringl(l :nk-nO)//string(k:k)
     endif
    endif
c   write(6,*) '|'//stringl(l:nk-nO)//T
30   continue
    string=stringl
    nl=nl-nO+l
    nO=l
    endif
    return
   endif
20  continue
   nO=-l
   nl=-l
   return
   end
                                                   341

-------
                                    Appendix E (FRLKINV)

Sample Input File for FRLKINV

/*  Title - This is a sample input file for the FRLKINV code
298.0       /* temperature (K) (TEMP)
0.215       /* mass pumping rate (g/sec) (XMPR)
76.0        /* time of pumping (sec)
521.2       /* thickness of formation (XB)
198.1        /* distance to top of well screen (XD)
259.1        /* distance to bottom of well screen (XL)
10.48       /* radius of the well (Rw)
l.e-06       /* convergence criterion
0.01,0.1       /* minimum and maximum air-porosity
4.e-10,3.e-09   /* minimum and maximum radial permeability (cmA2)
1., 10.       /* minimum and maximum anisotropy ratio (K_r/K_z)
l.e-14,l.e-10    /* minimum and maximum leakance (K'/b1) (cm)
7500., 12500.    /* Minimum and Maximum Storage volume (cmA3)
3000         /* number of random parameter guesses
10          /* number of "best" parameter sets for forward problem
500         /* number of observations  (NSIM)
0       1.000885189
0.5      1.010032139
1       1.034817425
1.5      1.051636012
2       1.066684221
2.5      1.078191675
3       1.08881394
3.5      1.098551016
4       1.106222652
4.5      1.H4484414
(only first 10 observations shown)
                                               342

-------
Sample Output File for FRLKINV
   Transient Single-Well Parameter Estimation
   Finite-Radius Model with Well-Bore Storage
      Top BC: Semi-Confining Layer
Parameter Estimates Ordered from Best to Worst
K_r
0.1812E-08
0.1763E-08
0.1763E-08
0.1764E-08
0.1796E-08
0.1805E-08
0.1744E-08
0.1829E-08
0.1714E-08
0.1722E-08
K_z
0.2009E-09
0.2311E-09
0.2228E-09
0.2347E-09
0.2122E-09
0.1999E-09
0.2300E-09
0.1864E-09
0.2559E-09
0.2467E-09
Leakance
0.7196E-11
0.6141E-11
0.6459E-10
0.7404E-11
0.6801E-11
0.3735E-10
0.5049E-10
0.4269E-11
0.5804E-10
0.8233E-11
Porosity
0.8013E-02
0.8087E-02
0.8181E-02
0.8228E-02
0.8015E-02
0.8924E-02
0.8388E-02
0.8148E-02
0.974 1E-02
0.9450E-02
Storage
0.1179E+05
0.1186E+05
0.1186E+05
0.1164E+05
0.1151E+05
0.1148E+05
0.1167E+05
0.1156E+05
0.1171E+05
0.1188E+05
RMS-Error
0.3228E-02
0.3229E-02
0.3273E-02
0.3287E-02
0.3300E-02
0.3365E-02
0.3369E-02
0.3374E-02
0.3388E-02
0.3420E-02
IFLAG
0
0
0
0
0
0
0
0
0
0
(Only first 10 estimates shown)

Observed Versus Simulated Pressure

Parameter set -  1

  K_r=0.1812E-08
  K_z = 0.2009E-09
  K7b(leaky) = 0.7196E-ll
  Air-porosity = 0.8013E-02
  Storage = 0.1179E+05

    Time (sec)    P_observed    P_model
O.OOOOE+00
0.5000E+00
0.1000E+01
0.1500E+01
0.2000E+01
0.2500E+01
0.3000E+01
0.3500E+01
0.4000E+01
0.4500E+01
0.1001E+01
0.1014E+01
0.1032E+01
0.1048E+01
0.1062E+01
0.1074E+01
0.1086E+01
0.1096E+01
0.1106E+01
0.1115E+01
0.1000E+01
0.1021E+01
0.1039E+01
0.1055E+01
0.1070E+01
0.1084E+01
0.1096E+01
0.1107E+01
0.1117E+01
0.1127E+01
(Only first 10 observations shown)
                                                343

-------
Source Code for FRLKINV

c Parameter estimation for the transient solution for finite well radius
c including well-bore storage
c
c Single-well Test
c BCLeaky layer on top
c
c Author: VaradhanRavi
c version 1.0  07/21/99
c
   implicit real*8 (a-h,o-z)
   common/cfunc/r,r_w,vn,a,gama
   common/ceigen/xlk
   dimension tt(3600),pobs(3600),poros(5000),p_r(5000),p_z(5000),
   >beta(5000),errmet(5000),wksp(5000),iwksp(5000),iflag(5000),
   >stor(5000),bn(500)
   integer*4 istart,ifinish
   character* 15 prefix,fnamel,fname2
   character*4 suffixI,suffix2
   write(*,*)' Parameter Estimation for Transient Single-Well Test'
   write(*,*)' Finite-Radius Solution - BC: Leaky Layer on Top'
   write(*,*)
   write(*,*)'Enter the Prefix of the input file:  '
   write(*,*)'An output file is created w/ same prefix'
   write(*,*)' Input-file Prefix ? '
   read(*,'(a)')prefix
   call getlen(prefix,' ',nO,nl)
   suffixl='.inp'
   suffix2='.dat'
   fname 1 =prefix(nO: n 1 )//suffix 1
   fname2=prefix(nO:nl)//suffix2
   open( 10,fname 1, status='unknown')
   open(20,fname2,status='unknown')
   open(30,'seed.inp',status='unknown')
   read(10,*)
   read(10,101)temp,xmpr,ttpump,xb,xd,xl,rw,tol
   read( 10, *)pormin,pormax
   read(10,*)prmin,prmax
   read(10,*)rmin,rmax
   read(10,*)xlkmin,xlkmax
   read( 10, *)vstmin,vstmax
   read(10,101)nguess
   read(10,101)nbest
   read(10,101)nsim
   read(10,*)(tt(i),pobs(i),i=l,nsim)
   patm= 1013200.
   visc=1.76e-04
   rgas=8.314e07
   xmw=28.8
   pi=4.0*datan(1.0)
c  patm = atmospheric pressure  (g/cm/secA2)
                                                   344

-------
c  vise = viscosity of gas (g/cm/sec)
c  rgas = universal gas constant
c  temp = temperature of gas (K)
c  xmw = molecular weight of gas (g/mole)
c  xmpr = mass withdrawal rate (g/sec)
c  prad = radial permeability of the formation (cmA2)
c  pz = vertical permeability
c  xlk = leakance of the confining unit (cm):  xlk = k'/b'
c  por = gas filled porosity
c  xb = thickness of the formation (cm)
c  xbp = thickness of the leaky unit
c  xl = distance from the bottom of leaky unit to the bottom of well screen
c  xd = distance from the bottom of leaky unit to the top of well screen
c  rw = nominal well radius (cm)
c  nsim = number of times solution will be generated
c  pobs = measured pneumatic pressure responses
c
   write(20,*)'	'
   write(20,*)'   Transient Single-Well Parameter Estimation'
   write(20,*)'   Finite-Radius Model with Well-Bore Storage '
   write(20,*)'      Top BC: Semi-Confining Layer'
   write(20,*)'	'
   write(20,*)
   xll=xl/xb
   xdd=xd/xb
   r_w=rw/xb
   call timer(istart)
   read(30,*)iseed
   do  lj=l,nguess
   iflag(j)=0
   errmet(j)=0.
   sqerr=0.
   guess 1=ranl (iseed)
   guess2=ranl (iseed)
   guess3=ranl (iseed)
   guess4=ranl (iseed)
   guess5=ranl (iseed)
   poros(])=pormin+(pormax-pormin)*guessl
   p_r(j)=prrnin*(prmax/prmin)**guess2
   ratio=rmin+(rmax-rmin) *guess3
   p_z(])=p_r(j)/ratio
   beta(j )=xlkmin*(xlkmax/xlkmin) * *guess4
   stor(j )=vstmin+(vstmax-vstmin) *guess5
   prad=p_r(j)
   pz=p_z(j)
   por=poros(j)
   xlk=beta(j)/(pz/xb)
   vstor=stor(j)
   gama=vstor/(2. *pi*(xl-xd) *xb *xb *por)
   a=dsqrt(prad/pz)
   xm=xmpr*visc*rgas*temp/(pi*xmw*prad*patm*patm*xb)
   tpump=ttpump*prad*patm/(visc*por*xb*xb)
                                                  345

-------
   call vnfunc(bn)
   mlinv=-10
   do 5 i=l,nsim
   r=r_w
   z=(xdd+xll)/2.
   t=tt(i)*prad*patm/(visc*por*xb*xb)
   n=0
   sum=0.0
10   continue
   n=n+l
   if(n.gt.500)then
   iflag(j)=l
   goto 15
   endif
   vn=bn(n)
   call linv(mlinv,t,fnrt)
   call linv(mlinv,t-tpump,fnrtp)
   tcos=dcos(vn*(l.-z))
   tsin=dsin(vn*( 1 . -xdd))-dsin(vn*( 1 .-xll))
   trig=tcos*tsin
   fact=xlk/(vn*(xll-xdd)*(xlk+dsin(vn)**2))
   ans=fnrt-fnrtp
   add=trig*fact*ans
   sum=sum+add
   if(dabs(sum).le.l.e-14)goto 15
   if(dabs(add/sum).gt.tol)goto 10
15   continue
   psqrd=1.0+2.0*xm*sum
   if(psqrd.lt.0.)then
   iflag(j)=2
   enmet(j)=l.elO
   goto 1
   endif
   popatm=dsqrt(psqrd)
   sqerr=sqerr+(popatm-pobs(i)) * *2
5   continue
   errmet(j )=dsqrt(sqerr/nsim)
   write(*,*)j,prad,pz,beta(j),por,stor(j),ernnet(j),iflag(j)
1   continue
   call sorts (nguess,errmet,poros,p_r,p_z,beta,stor,iflag,wksp,iwksp)
   write(20,107)
   write(20,102)
   do 20j=l,nguess
   write(20,103)p_r(j),p_z(j),beta(j),poros(j),stor(j),errmet(j),
20   continue
c
   do 30 j=l,nbest
   prad=p_r(j)
   pz=p_z(j)
   por=poros(j)
                                                    346

-------
   xlk=beta(j)/(pz/xb)
   write(20,104)j,prad,pz,beta(j),por,stor(j)
   write(20,105)
   gama=vstor/(2 . *pi*(xl-xd) *xb *xb *por)
   a=dsqrt(prad/pz)
   xm=xmpr*visc*rgas*temp/(pi*xmw*prad*patm*patm*xb)
   tpump=ttpump*prad*patm/(visc*por*xb*xb)
   call vnfunc(bn)
c
   mlinv=-10
   do 35 i=l,nsim
   r=r_w
   z=(xdd+xll)/2.
   t=tt(i)*prad*patm/(visc*por*xb*xb)
   n=0
   sum=0.0
40   continue
   n=n+l
   if(n.gt.500)goto  45
   vn=bn(n)
   call linv(mlinv,t,fnrt)
   call linv(rnlinv,t-tpump,fnrtp)
   tcos=dcos(vn*(l.-z))
   tsin=dsin(vn*( 1 . -xdd))-dsin(vn*( 1 .-xll))
   trig=tcos*tsin
   fact=xlk/(vn*(xll-xdd)*(xlk+dsin(vn)**2))
   ans=fnrt-fnrtp
   add=trig*fact*ans
   sum=sum+add
   if(dabs(sum).le.l.e-14)goto 45
   if(dabs(add/sum).gt.tol)goto 40
45   continue
   psqrd=1.0+2.0*xm*sum
   if(psqrd.ge.0.)then
   popatm=dsqrt(psqrd)
   else
   popatm=0.
   endif
   write(20, 106)tt(i),pobs(i),popatm
35   continue
30   continue
   call timer(ifinish)
   write(*,*)Time of execution (sec)  =',(ifinish-istart)/100.
101  format(gl5.0)
102  format(/4x,'K_r', 1 lx,'K_z', 10x,'Leakance',6x,'Porosity',7x,
   >'Storage',7x,'RMS-Error',4x,'IFLAG'/)
103  format(lx,el0.4,4x,el0.4,5x,el0.4,4x,el0.4,5x,el0.4,5x,el0.4,4x,
104  format(//20x,'Parameter set - ',i4//,3x,'K_r = ',e!0.4/
   >3x,'K_z = ',e!0.4/3x,'K/b(leaky) = ',elO. 4/3 x, 'Air-porosity = ',
   >e!0.4/3x,'Storage = ',e!0.4//)
105  format(7x,Time (sec)',8x,'P_observed',8x,'P_model'/)
                                                   347

-------
106  format(7x,el0.4,8x,el0.4,7x,el0.4)
107  format(//10x,'Parameter Estimates Ordered from Best to Worst'//)
108  format(/3x,'K_r = ',e 10.4/3x,'K_r/K_z = ',e 10.4/3x,'K/b(leaky) = '
   > el0.4/3x,'Air-porosity = ',e 10.4/3 x,'Storage = ',e!0.4/)
   rewind(30)
   write(30,*)-iseed
   stop
   end

   subroutine linv(m,t,f)
   implicit real*8 (a-h,o-z)
   common/cfunc/r,r_w,vn,a,gama
   dimension g(50),v(50),h(25)
   f=0.
   if(t.le.0.)return
   n=14
   if(mne.n)then
   call auxvec(n,g,v,h)
   m=n
   endif
   ss=dlog(2.dO)/t
   xp=0.dO
   visum=0.dO
   do 50 1=1,n
   s=l*ss
   wn=dsqrt(s+(vn/a) * *2)
   pnum=besskO(wn*r)
   pden=s*(wn*r_w*besskl(wn*r_w)+s*gama*besskO(wn*r_w))
   fbar=pnum/pden
   xp=xp+v(l)*fbar
   visum=v(l)/l+visum
50   continue
   f=ss*xp
   return
   end

   subroutine auxvec(n,g,v,h)
   implicit real*8 (a-h,o-z)
   dimension g(0:50),v(50),h(25)
   g(0)=l
   nh=n/2
   do 90 i=l,n
   g(i)=i*g(i-l)
90   continue
   h(l)=2.dO/g(nh-l)
   do  100 i=2,nh
   h(i)=i**nh*g(2*i)/(g(nh-i)*g(i)*g(i-l))
100  continue
   sn=2*mod(nh,2)-l
   vsum=0.dO
   do  140 i=l,n
   v(i)=0
                                                   348

-------
   k2=minO(i,nh)
   do 130 k=kl,k2
   v(i)=v(i)+h(k)/(g(i-k)*g(2*k-i))
130  continue
   v(i)=sn*v(i)
   vsum=vsum+v(i)
   sn=-sn
140  continue
   return
   end

   subroutine vnfunc(x)
   implicit real*8 (a-h,o-z)
   dimension x(500)
   external eigen
   common/ceigen/b
   pi=4.*datan(1.0)
   xl=l.e-06
   x2=pi
   kount=l
1   continue
   zero=rtsafe(eigen,xl,x2,l.e-06)
   x(kount)=zero
   xl=zero+0.5*pi
   x2=zero+1.5*pi
   kount=kount+l
   if(kount.le.500)goto 1
   return
   end

   Double Precision FUNCTION rtsafe(funcd,xl,x2,xacc)
   implicit real*8 (a-h,o-z)
   EXTERNAL tuned
   PARAMETER (MAXIT=100)
   callfuncd(xl,fl,df)
   call funcd(x2,fh,df)
   if((fl.gt.0..and.fh.gt.0.).or.(fl.lt.0..and.fh.lt.0.))write(*,*)
   *'root must be bracketed in rtsafe'
   if(fl.eq.0.)then
    rtsafe=xl
    return
   else if(fh.eq.0.)then
    rtsafe=x2
    return
   else if(fl.lt.0.)then
    xl=xl
    xh=x2
   else
    xh=xl
    xl=x2
   endif
                                                   349

-------
   rtsafe=.5*(xl+x2)
   dxold=dabs(x2-xl)
   dx=dxold
   call funcd(rtsafe,f,df)
   do llj=l,MAXIT
    if(((rtsafe-xh)*df-f)*((rtsafe-xl)*df-f).ge.0..or. dabs(2.*
   *f).gt.dabs(dxold*df)) then
     dxold=dx
     dx=0.5*(xh-xl)
     rtsafe=xl+dx
     if(xl. eq. rtsafe)return
    else
     dxold=dx
     dx=f/df
     temp=rtsafe
     rtsafe=rtsafe-dx
     if(temp.eq.rtsafe)return
    endif
    if(dabs(dx).lt.xacc) return
    call funcd(rtsafe,f,df)
    if(f.lt.O.) then
     xl=rtsafe
    else
     xh=rtsafe
    endif
11  continue
   write(*,*)'rtsafe exceeded maximum iterations'
   return
   END

   subroutine eigen(x,f,df)
   implicit real*8 (a-h,o-z)
   common/ceigen/b
   f=b*dcos(x)-x*dsin(x)
   df=-b*dsin(x)-dsin(x)-x*dcos(x)
   return
   end

   Double Precision Function BesskO(x)
   implicit real*8 (a-h,o-z)
   datapl,p2,p3,p4,p5,p6,p7/-0.57721566,0.4227842,0.23069756,
   >0.348859d-l,0.262698d-2,0.10750d-3,0.74d-5/
   dataql,q2,q3,q4,q5,q6,q7/1.25331414,-0.7832358d-l,0.2189568d-l,
   >-0.1062446d-l,0.587872d-2,-0.25154d-2,0.53208d-3/
   if(x.le.2.0)then
   y=x*x/4.0
   besskO=(-dlog(x/2.0)*bessiO(x))+(pl+y*(p2+y*(p3+y*(p4+y*(p5+y*(p6+
   >y*p7))))))
   else
   y=2.0/x
   besskO=(dexp(-x)/dsqrt(x))*(ql+y*(q2+y*(q3+y*(q4+y*(q5+y*(q6+y*q7)
                                                   350

-------
   endif
   return
   end

   DOUBLE PRECISION FUNCTION BESSIO(W)
   IMPLICIT real*8 (A-H,O-Z)
   T=W/3.75
   IF(DABS(T).LE.1.0) THEN
   Y=T*T
   BESSIO=1.0+Y*(3.5156229+Y*(3.0899424+Y*(1.2067492+Y*(0.2659732+
   >Y*(0.0360768+0.0045813*Y)))))
   ELSE
   AW=DABS(W)
   Y=3.75/AW
   TEMP=039894228+Y*(0.01328592+Y*(0.00225319+Y*(-0.00157565+Y*(
   >0.00916281+Y*(-0.02057706+Y*(0.02635537+Y*(-0.01647633+
   >0.00392377*Y)))))))
   BESSIO=DEXP(AW)*TEMP/DSQRT(AW)
   ENDIF
   RETURN
   END

   double precision FUNCTION besskl(x)
CU  USES bessil
   implicit real*8 (a-h,o-z)
   DOUBLE PRECISION pl,p2,p3,p4,p5,p6,p7,ql,q2,q3,q4,q5,q6,q7,y
   SAVE p 1 ,p2,p3 ,p4,p5,p6,p7,ql ,q2,q3 ,q4,q5,q6,q7
   DATA pl,p2,p3,p4,p5,p6,p7/1.0dO,0. 15443 144dO,-0.67278579dO,
   *-0.18156897dO,-0.1919402d-l,-0.110404d-2,-0.4686d-4/
   DATAql,q2,q3,q4,q5,q6,q7/1.25331414dO,0.23498619dO,-0.3655620d-l,
   *0.1504268d-l,-0.780353d-2,0.325614d-2,-0.68245d-3/
   if (x.le.2.0)then
    y=x*x/4.0
    besskl=(dlog(x/2.0)*bessil(x))+(1.0/x)*(pl+y*(p2+y*(p3+y*(p4+y*
   *(p5+y*(p6+y*p7))))))
   else
    y=2.0/x
    besskl=(dexp(-x)/dsqrt(x))*(ql+y*(q2+y*(q3+y*(q4+y*(q5+y*(q6+y*
   endif
   return
   END

   double precision FUNCTION bessil(x)
   implicit real*8 (a-h,o-z)
   DOUBLE PRECISION pl,p2,p3,p4,p5,p6,p7,ql,q2,q3,q4,q5,q6,q7,q8,q9,y
   SAVE p 1 ,p2,p3 ,p4,p5,p6,p7,ql ,q2,q3 ,q4,q5,q6,q7,q8,q9
   DATApl,p2,p3,p4,p5,p6,p7/0.5dO,0.87890594dO,0.51498869dO,
   *0.15084934dO,0.2658733d-l,0.301532d-2,0.32411d-3/
   DATAql,q2,q3,q4,q5,q6,q7,q8,q9/0.39894228dO,-0.3988024d-l,
   *-0.362018d-2,0.163801d-2,-0.1031555d-l,0.2282967d-l,-0.2895312d-l,
   *0.1787654d-l,-0.420059d-2/
                                               351

-------
   if(abs(x).lt.3.75)then
    y=(x/3.75)**2
    bessil=x*(pl+y*(p2+y*(p3+y*(p4+y*(p5+y*(p6+y*p7))))))
   else
    ax=abs(x)
    y=3.75/ax
    bessil=(dexp(ax)/dsqrt(ax))*(ql+y*(q2+y*(q3+y*(q4+y*(q5+y*(q6+y*
   *(q7+y*(q8+y*q9))))))))
    if(x.lt.0.)bessil=-bessil
   endif
   return
   END

   double precision FUNCTION ranl(idum)
   implicit real*8 (a-h,o-z)
   INTEGER idum,IA,IM,IQ,IR,NTAB,NDIV
   PARAMETER (IA=16807,IM=2147483647,AM=1./IM,IQ=127773,IR=2836,
   *NTAB=32,NDIV= 1+(IM-1)/NTAB,EPS=1.2e-7,RNMX= 1 .-EPS)
   INTEGER j ,k,iv(NTAB),iy
   SAVE iv,iy
   DATA iv /NTAB*0/, iy /O/
   if (idum.le.0.or.iy.eq.O) then
    idum=max(-idum, 1)
    do llj=NTAB+8,l,-l
     k=idum/IQ
     idum=IA*(idum-k*IQ)-IR*k
     if (idum.lt. 0) idum=idum+IM
     if (j.le.NTAB) iv(j)=idum
11   continue
    iy=iv(l)
   endif
   k=idum/IQ
   idum=IA*(idum-k*IQ)-IR*k
   if (idum.lt. 0) idum=idum+IM
   j=l+iy/NDIV
   iy=iv(j)
   iv(j)=idum
   ranl=dminl(AM*iy,RNMX)
   return
   END

   SUBROUTINE sort3(n,ra,rb,rc,rd,re,rf,ig,wksp,iwksp)
   implicit real*8 (a-h,o-z)
   INTEGER n,iwksp(n),ig(n)
   dimension ra(n),rb(n),rc(n),rd(n),re(n),rf(n),wksp(n)
CU  USES indexx
   INTEGER j
   call indexx(n,ra,iwksp)
   do llj=l,n
    wksp(j)=ra(j)
11  continue
   do 12j=l,n
                                               352

-------
    ra(] )=wksp(iwksp(j))
12  continue
   do 13j=l,n
    wksp(j)=rb(j)
13  continue
   do 14j=l,n
    rb(j )=wksp(iwksp(j))
14  continue
   do 15j=l,n
    wksp(j)=rc(j)
15  continue
   do 16j=l,n
    rc(j)=wksp(iwksp(j))
16  continue
   do 17j=l,n
    wksp(j)=rd(j)
17  continue
   do 18j=l,n
    rd(j )=wksp(iwksp(j))
18  continue
   do 19j=l,n
    wksp(j)=re(j)
19  continue
   do20j=l,n
    re(j )=wksp(iwksp(j))
20  continue
   do21j=l,n
    wkspCJ)=rf(j)
21  continue
   do22j=l,n
    rf(j)=wksp(iwksp(j))
22  continue
   do23j=l,n
    wksp(j)=ig(j)
23  continue
   do24j=l,n
    ig(j)=wksp(iwksp(j))
24  continue
   return
   END

   SUBROUTINE indexx(n,arr,indx)
   implicit real*8 (a-h,o-z)
   INTEGER n,indx(n),M,NSTACK
   dimension arr(n)
   PARAMETER (M=7,NSTACK=50)
   INTEGER i,indxt,ir,itempj j stack,k,l,istack(NSTACK)
   do llj=l,n
    indx(j)=j
11  continue
   jstack=0
   1=1
                                                353

-------
    ir=n
1   if(ir-l.lt.M)then
     do!3j=l+l,ir
      indxt=indx(j)
      a=arr(indxt)
      dol2i=j-l,l,-l
       if(arr(indx(i)).le.a)goto 2
       indx(i+l)=indx(i)
12     continue
      i=0
2      indx(i+l)=indxt
13    continue
     if(j stack, eq. 0)return
     ir=istack(j stack)
     l=istack(j stack-1)
     jstack=jstack-2
    else
     k=(l+ir)/2
     itemp=indx(k)
     indx(k)=indx(l+l)
     indx(l+l)=itemp
     if(arr(indx(l+1)). gt.arr(indx(ir)))then
      itemp=indx(l+l)
      indx(l+l)=indx(ir)
      indx(ir)=itemp
     endif
     if(arr(indx(l)).gt.arr(indx(ir)))then
      itemp=indx(l)
      indx(l)=indx(ir)
      indx(ir)=itemp
     endif
     if(arr(indx(l+l)).gt.arr(indx(l)))then
      itemp=indx(l+l)
      indx(l+l)=indx(l)
      indx(l)=itemp
     endif
     i=l+l
     j=ir
     indxt=indx(l)
     a=arr(indxt)
3    continue
      i=i+l
     if(arr(indx(i)).lt.a)goto 3
4    continue
      j=j-l
     if(arr(indx(j)).gt.a)goto 4
     if(j.lt.i)goto 5
     itemp=indx(i)
     indx(i)=indx(j)
     indx(j)=itemp
     goto 3
5    indx(l)=indx(j)
                                                     354

-------
     indx(j)=indxt
     jstack=jstack+2
     if(jstack.gt.NSTACK)write(*,*) 'NSTACKtoo small in indexx'
     if(ir-i+1. ge j -l)then
      istack(jstack)=ir
      istack(j stack-l)=i
      ii=j-l
     else
      istack(jstack)=j-l
      istack(j stack-1)=1
      l=i
     endif
    endif
    goto 1
    END
    subroutine getlen(string,chr,nO,nl)
c
c   to determine the length of a string excluding any blank padding & heading
c   and return string(nO:nl) nO... position of first non-blank letter
c                  nl...  position of last non-blank letter
c   else return the original string
c   ***Use  to keep blank space at the begining or the end of the
c    *** string
c
    character*(*) string
    character* 160 stringl
    character*! chr
    stringl=''
    inO=0
    inl=0
    do 10 i=len(string),l,-l
    if(string(i:i).ne.chr) then
     if(inl.eq.O) then
     if(string(i:i).eq.'|') then
     nl=i-l
     else
     nl=i
     endif
     inl=l
     endif
     goto 11
    endif
10   continue
11   do20i=l,len(string)
    if(string(i:i).ne.chr) then
    if(in0.eq.O) then
    if(string(i:i).eq.'|') then
    nO=i+l
    else
    nO=i
                                                     355

-------
    endif
    inO=l
    nk=nO-l
c   write(6,*) T//string(nO:nl)//T, nO,nl
    do 30 k=nO,nl
    if(string(k:k).ne.'|') then
     nk=nk+l
     if(k.eq.nO) then
      string 1=string(k:k)
     else
      stringl=stringl(l :nk-nO)//string(k:k)
     endif
    endif
c   write(6,*) T//stringl(l:nk-nO)//T
30   continue
    string=stringl
    nl=nl-nO+l
    nO=l
    endif
    return
   endif
20  continue
   nO=-l
   nl=-l
   return
   end
                                                  356

-------
                                   APPENDIX F (MAIRFLOW)
Sample Input File for MAIRFLOW

293.            /* temperature (K)  (TEMP)
1.42e-07          /* radial permeability (cmA2)  (PRAD)
1.44e-07         /* vertical permeability (cmA2)  (PZ)
3.62e-l 1          /* leaky-layer permeability (cmA2) (PLEAK)
465.5           /* thickness of formation (cm) (XB)
1.0              /* thickness of leaky-layer (cm) (XBLK)
0.24             /* air-filled porosity (THETA)
6                /* number of wells
1707., 1250.       /* X,Y coordinates of TW-1 (cm)
208.,465.         /* screened interval of TW-1 (cm)
12.7             /* effective well radius of TW-1 (cm)
-11.0            /* pumping rate of TW-1 (std. cu.ft per min)(+/- injection/extraction)
0                /* integer flag for particle tracking of TW-1 (0/1 no/yes)
3109.,2743       /* X,Y coordinates of TW-3 (cm)
190.,465.        /* screened interval of TW-3 (cm)
12.7            /* effective well radius of TW-3 (cm)
-20.0           /* pumping rate of TW-3 (std. cu.ft per min)(+/- injection/extraction)
0               /* integer flag for particle tracking of TW-3 (0/1  no/yes)
4816.,3353.      /* X,Y coordinates of TW-4 (cm)
193.,465.        /* screened interval of TW-4 (cm)
12.7             /* effective well radius of TW-4 (cm)
-9.0             /* pumping rate of TW-4 (std. cu.ft per min)(+/- injection/extraction)
0                /* integer flag for particle tracking of TW-4 (0/1 no/yes)
3688.,1219.      /* X,Y coordinates of TW-5 (cm)
213.,465.         /* screened interval of TW-5 (cm)
12.7            /* effective well radius of TW-5 (cm)
-29.0           /* pumping rate of TW-5 (std. cu.ft per min)(+/- injection/extraction)
0               /* integer flag for particle tracking of TW-5 (0/1  no/yes)
1219.,3200.      /* X,Y coordinates of TW-6 (cm)
198.,465.         /* screened interval of TW-6 (cm)
12.7            /* effective well radius of TW-6 (cm)
-11.0           /* pumping rate of TW-6 (std. cu.ft per min)(+/- injection/extraction)
0               /* integer flag for particle tracking of TW-6 (0/1  no/yes)
2195.,4084.      /* X,Y coordinates of TW-7 (cm)
178.,465.        /* screened interval of TW-7 (cm)
12.7            /* effective well radius of TW-7 (cm)
-24.0           /* pumping rate of TW-7 (std. cu.ft per min)(+/- injection/extraction)
0                /* integer flag for particle tracking of TW-7 (0/1 no/yes)
O.,6000.         /* X-min and X-max for simulations
0. ,6000.         /* Y-min and Y-max for simulations
460.0,460.       /* Z-min and Z-max for simulations
50.,50., 10.       /* X,Y,Z grid spacing for simulation (cm)
2.4e-03         /* minimum darcy velocity for adequate airflow (cm/s)
0.0             /* step size for particle tracking (cm)
                                                  357

-------
Sample Output File for MAIRFLOW




Total pumping (g/s) =   58.7993904877061




  X(cm)    Y(cm)       Z (cm)     P/P_atm
qx
qy
qz
q (cm/s)
O.OOOE+00
0.500E+02
0.100E+03
0.150E+03
0.200E+03
0.250E+03
0.300E+03
0.350E+03
0.400E+03
0.450E+03
O.OOOE+00
O.OOOE+00
O.OOOE+00
O.OOOE+00
O.OOOE+00
O.OOOE+00
O.OOOE+00
O.OOOE+00
O.OOOE+00
O.OOOE+00
0
0
0
0
0
0
0
0
0
0
.460E+03
.460E+03
.460E+03
.460E+03
.460E+03
.460E+03
.460E+03
.460E+03
.460E+03
.460E+03
0
0
0
0
0
0
0
0
0
0
.99888E+00
.99884E+00
.99880E+00
.99876E+00
.99872E+00
.99868E+00
.99864E+00
.99860E+00
.99855E+00
.99851E+00
0.6013E-03
0.6186E-03
0.6360E-03
0.6537E-03
0.6714E-03
0.6893E-03
0.7071E-03
0.7249E-03
0.7425E-03
0.7599E-03
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
.4918E-03
.5144E-03
.5382E-03
.5631E-03
.5893E-03
.6166E-03
.6452E-03
.6752E-03
.7066E-03
.7394E-03
0
0
0
0
0
0
0
0
0
0
.2656E-05
.2744E-05
.2835E-05
.2929E-05
.3025E-05
.3124E-05
.3225E-05
.3329E-05
.3436E-05
.3545E-05
0.7768E-03
0.8045E-03
0.8332E-03
0.8628E-03
0.8933E-03
0.9248E-03
0.9573E-03
0.9906E-03
0.1025E-02
0.1060E-02
(Only first 10 simulations shown)
                                              358

-------
Source Code for MAIRFLOW (author Ravi Varadhan)

(A FORTRAN program written to compute a three-dimensional pressure and pore-gas velocity field from injection or
extraction of multiple wells)

c :
c MAIRFLOW- Airflow induced by Multiple Wells
c :
C Analytical solution for steady air flow in a well field.
C Top BC: Semi-Confining Layer
c Refer to Baehr and Huh (1995) - WRR - for the solution
c Analytical solution obtained using the superposition principle.
c Particle tracking using the Simple Runge-Kutta (second-order) method.
c
c This code could be used to space venting wells such that there is adequate
c airflow between wells. It could also be used in bioventing applications
c to identify whether there is any flow of contaminated air to the surface.
    implicit real*8 (a-h,o-z)
    common/ceigen/h
    common/cparam/bn,xb,a,prad,pz,visc,patm
    common/cwell/nwell,xw(200),yw(200),xl(200),xd(200),rw(200),xm(200)
    dimension bn(100),x(1000),y(1000),z(1000),xmpr(200),itrack(200)
    character*! reply
    character* 15 prefix,fname 1 ,fname2,fname3
    character*4 suffixI,suffix2,suffix3
    write(*,*)' Steady-State Airflow Simulation for Multiple-Wells'
    write(*,*)' Finite-Radius Solution with Borehole Storage '
    write(*,*)' Top BC: Semi-Confining Layer'
    write(*,*)
    write(*,*)'Enter the Prefix of the input file: '
    write(*,*)'An output file is created w/ same prefix'
    write(*,*)' Input-file Prefix ? '
    read(*,'(a)')prefix
    call getlen(prefix,' ',nO,nl)
    suffixl='.inp'
    suffix2='.dat'
    suffix3='.ptk'
    fname 1 =prefix(nO: n 1 )//suffix 1
    fname2=prefix(nO:nl)//suffix2
    f name 3 =prefix(nO: n 1 )//suffix3
    open( 10,fname 1, status='unknown')
    open(20,fname2,status='unknown')
    open(3 0,fname3, status='unknown')
    read(10,101)temp,prad,pz,pleak,xb,xblk,theta
    patm=1013200.
    visc=1.76e-04
    rgas=8.3143e07
    xmw=28.8
    a=dsqrt(prad/pz)
    h=pleak*xb/(pz*xblk)
    pi=4.0*datan(1.0)
    call vnfunc(bn)


                                                   359

-------
   read(10,*)nwell
   totpmp=0.
c
   do 1 i=l,nwell
   read( 10, *)xw(i),y w(i)
   read(10,*)xd(i),xl(i)
   read(10,*)rw(i)
   read(10,*)xmpr(i)
   read(10,*)itrack(i)
   xmpr(i)=xmpr(i)*471.947*28.8/(82.05*293.0)
   totpmp=totpmp+dabs(xmpr(i))
   qp=xmpr(i)*visc*rgas*temp/xmw
   xm(i)=2.*h*qp*a*xb/(pi*prad*(xl(i)-xd(i))*rw(i))
1   continue
c
   read(10,*)xmin,xmax
   read( 10, *)ymin,ymax
   read(10,*)zmin,zmax
   read(10,*)dx,dy,dz
   read(10,*)qmin
   read(10,*)delx
   nx=nint((xmax-xmin)/dx)+1
   ny=nint((ymax-ymin)/dy)+1
   nz=nint((zmax-zmin)/dz)+1
c
   write(*,108)
   write(20,*)
   write(*,*)Total pumping(g/s) =',totpmp
   write(20,*)Total pumping(g/s)  =',totpmp
   write(20,102)
c
   kount=0
   pmin=l.elO
   do 5 k=l,nz
   z(k)=zmin+(k-l)*dz
   do 5 j=l,ny
   y(j)=yniin+(j-l)*dy
   do 5 i=l,nx
   x(i)=xmin+(i-1) *dx
   call solution(x(i),y(j),z(k),popatm,qx,qy,qz)
   spdisch=dsqrt(qx*qx+qy*qy+qz*qz)
   write(20,103)x(i),y(j),z(k),popatm,qx,qy,qz,spdisch
   if(spdisch.le.qmin)kount=kount+l
   pmin=dminl (pmin,popatm)
5   continue
   pinadq= 100. *kount/(nx*ny *nz)
   pmin=( 1.-pmin) *407.
   write(*,*)'Maximum vaccum in the domain (in water)  ',pmin
   write(20,*)'Maximum vaccum in the domain (in water) ',pmin
   write(20,*)'% Volume receiving inadequate airflow =',pinadq
   write(*,*)'% Volume receiving inadequate airflow  =',pinadq
                                                  360

-------
c Particle tracking segment
c
   write(*,108)
   write(*,*)' Particle Tracking is going to commence '
   write(*,*)' Do you want to continue (y/n)?  '
   read(*,109)reply
   if (.not.(reply.eq.'y' .or. reply.eq.'Y')) stop
   distl=dsqrt((xmax-xmin)**2+(ymax-ymin)**2)/(nwell*100.0)
   dist2=(zmin+zmax)/(2.*100.)
   if(delx.le.0.)delx=dminl(distl,dist2)
   write(*,*)'Max. particle tracking step (cm):',delx
   do 20 i=l,nwell
   if(itrack(i).eq.0)goto 20
   sign=l
   if(xmpr(i).lt.0.)sign=-l
   write(30,108)
   write(30,106)xw(i),yw(i)
   if(sign.gt.0)then
   write(30,*)' Forward Tracking from Injection Well'
   kount=0
   else
   write(30,*)' Backward Tracking from Extraction Well'
   endif
   write(30,105)
   do 30 iz=l,5
   zz=xd(i)+dfloat(iz-l.)*(xl(i)-xd(i))/4.
   do30iang=l,12
   alpha=dfloat(iang-1.) *pi/6.
   xx=xw(i)+rw(i)*dcos(alpha)
   yy=y w(i)+rw(i) *dsin(alpha)
   xnew=xx
   ynew=yy
   znew=zz
   ttime=0.
   nsteps=0
25   continue
   nsteps=nsteps+l
   angold=datan(ynew/xnew)
   xold=xnew
   yold=ynew
   zold=znew
   call solution(xold,yold,zold,popatm,qx,qy,qz)
   qres=dsqrt(qx*qx+qy*qy+qz*qz)
   if(qres.lt.l.e-07)then
   write(30,110)arpha*180./pi,zz
   goto 30
   endif
   delt=derx*theta/qres
   xkl=delt*qx/theta
   ykl=delt*qy/theta
   zkl=delt*qz/theta
   call solution(xold+xk 1 ,yold+yk 1 ,zold+zk 1 ,popatm,qx,qy,qz)
                                                    361

-------
   xk2=delt*qx/theta
   yk2=delt*qy/theta
   zk2=delt*qz/theta
   xnew=xold+sign*(xkl+xk2)/2.
   ynew=yold+sign*(yk 1 +yk2)/2.
   znew=zold+sign*(zkl+zk2)/2.
   ttime=ttime+delt
   if(znew.le. l.e-07)then
   znew=0.
   if(sign.gt.0)kount=kount+l
   write(30,107)alpha*180./pi,zz,xnew,ynew,znew,ttime/86400.
   write(*,107)alpha*180./pi,zz,xnew,ynew,znew,ttime/86400.
   goto 30
   endif
   do40j=l,nwell
   if(j.eq.i)goto 40
   dist=dsqrt((xw(j )-xnew) * *2+(y w(j )-ynew) * *2)
   if(dist.le.rw(j))then
   write(30,107)alpha*180./pi,zz,xwCJ),yw(j),znew,ttime/86400.
   write(*,107)alpha*180./pi,zz,xwCJ),yw(j),znew,ttime/86400.
   goto 30
   endif
40   continue
   angnew=datan(ynew/xnew)
   if(nsteps.gt.lOOO .and. dabs(angold-angnew).lt.l.e-15)then
   write(30,11 l)alpha*180./pi,zz
   goto 30
   endif
   goto 25
30   continue
   if(sign.gt.0)then
   escape=100.*dfloat(kount)/60.
   write(30,108)
   write(30,*)'Percent airflow escaping to the surface from injection
   >well:  ',escape
   endif
20   continue
C
101  format(gl5.0)
102  format(/4x,'X (cm)',6x,'Y (cm)',6x,'Z (cm)',9x,'P/P_atm', 1 Ix,
   >'qx', 15x,'qy', 14x,'qz', 1 lx,'q (cm/s)'/)
103  format(2x,e9.3,3x,e9.3,4x,e9.3,5x,ell.5,5x,ell.4,6x,ell.4,5x,ell.4
   >4x,el0.4)
104  format(3x,'Particle-tracking from the Well @ ',2x,e 10.4,2x,
   >e 10.47)
105  format(/10x,'Origin',20x,'Destination',16x,Travel time(d)'//,
   >2x,'Angle(deg)',6x,'Z',12x,'X',9x,'Y',9x,'Z'/)
106  format(4x,'Particles released from the Well at:',2(2x,e9.2)/)
107  format(4x,f5.1,4x,e9.3,6x,e9.3,2x,e9.3,2x,e9.3,8x,e9.3)
108  format(//)
109  format(al)
110  format(4x,f5. l,4x,e9.3,8x,'Stagnation Point')
                                                    362

-------
Ill  format(4x,f5.1,4x,e9.3,8x,'Indeterminate')
   stop
   end

   subroutine solution(xx,yy,zz,popatm,qx,qy,qz)
   implicit real*8 (a-h,o-z)
   common/ceigen/h
   common/cparam/bn,xb,a,prad,pz,visc,patm
   common/cwell/nwell,xw(200),yw(200),xl(200),xd(200),rw(200),xm(200)
   dimension bn(100)
   sump=0.
   sumqx=0.
   sumqy=0.
   sumqz=0.
   do 20 nw=l,nwell
   rr=dsqrt((xx-xw(nw)) * *2+(yy-yw(nw)) * *2)
   if(rr.lt.rw(nw))rr=rw(nw)
   xdd=xd(nw)
   xll=xl(nw)
   rrw=rw(nw)
   n=0
   suml=0.0
   sum2=0.0
   sum3=0.0
10  continue
   n=n+l
   if(n.gt.lOO)goto 15
   vn=bn(n)
   tcos=dcos(vn*( 1.-zz/xb))
   if((xll-xdd)/xb .le. 0.99) then
   tsin=(dsin(vn*( 1. -xdd/xb))-dsin(vn*( 1 .-xll/xb)))/(vn*vn*(h+dsin
   >(vn)**2))
   else
   tsin=(xll-xdd)/(xb *(h+dsin(vn) * *2))
   endif
   trig=tcos*tsin
   bessO=besskO(vn*rr/a/xb)
   bess l=besskl (vn*rrw/a/xb)
   addl=bess0^essl*trig
   add2=trig*(-vn/a/xb)*besskl(vn*rr/a/xb)/bessl
   add3=tsin*bess0^essl*(vn/xb)*dsin(vn*(l.-zz/xb))
   suml=suml+addl
   sum2=sum2+add2
   sum3=sum3+add3
   if(suml.eq.O. .or. sum2.eq.O. .or.sum3.eq.0.)goto 15
   convg=dmaxl(dabs(addl/suml),dabs(add2/sum2),dabs(add3/sum3))
   if(convg.gt.l.e-07)goto 10
15  continue
   sump=sump+xm(nw)*suml
   sumqx=sumqx+xm(nw)*sum2*(xx-xw(nw))/rr
   sumqy=sumqy+xm(nw)*sum2*(yy-yw(nw))/rr
   sumqz=sumqz+xm(nw)*sum3
                                                 363

-------
20   continue
   phi=sump+patm*patm
   popatm=dsqrt(phi/patm/patm)
   qx=-prad/(2. *visc *dsqrt(phi)) *sumqx
   qy=-prad/(2. *visc*dsqrt(phi))*sumqy
   qz=-pz/(2. *visc *dsqrt(phi)) *sumqz
   return
   end

   subroutine vnfunc(x)
   implicit real*8 (a-h,o-z)
   dimension x(*)
   external eigen
   common/ceigen/h
   pi=4.*datan(1.0)
   xl=l.e-07
   x2=pi
   kount= 1
1   continue
   zero=rtsafe(eigen,xl,x2,l.e-06)
   x(kount)=zero
   xl=zero+0.5*pi
   x2=zero+1.5*pi
   kount=kount+l
   if(kount.le.lOO)goto 1
   return
   end

   Double Precision FUNCTION rtsafe(funcd,xl,x2,xacc)
   implicit real*8 (a-h,o-z)
   EXTERNAL tuned
   PARAMETER (MAXIT=100)
   callfuncd(xl,fl,df)
   call funcd(x2,fh,df)
   if((fl.gt.0..and.fh.gt.0.).or.(fl.lt.0..and.fh.lt.0.))pause
   *'root must be bracketed in rtsafe'
   if(fl.eq.0.)then
     rtsafe=xl
     return
   else if(fh.eq.0.)then
     rtsafe=x2
     return
   else if(fl.lt.0.)then
     xl=xl
     xh=x2
   else
     xh=xl
     xl=x2
   endif
   rtsafe=.5*(xl+x2)
   dxold=dabs(x2-xl)
   dx=dxold
                                                   364

-------
   call funcd(rtsafe,f,df)
   do llj=l,MAXIT
    if(((rtsafe-xh)*df-f)*((rtsafe-xl)*df-f).ge.0..or. dabs(2.*
   *f).gt.dabs(dxold*df) ) then
     dxold=dx
     dx=0.5*(xh-xl)
     rtsafe=xl+dx
     if (xl. eq. rtsafe)return
    else
     dxold=dx
     dx=f/df
     temp=rtsafe
     rtsafe=rtsafe-dx
     if(temp.eq.rtsafe)return
    endif
    if(dabs(dx).lt.xacc) return
    call funcd(rtsafe,f,df)
    if(f.lt.O.) then
     xl=rtsafe
    else
     xh=rtsafe
    endif
1 1  continue
   pause 'rtsafe exceeding maximum iterations'
   return
   END

   subroutine eigen(x,f,df)
   implicit real*8 (a-h,o-z)
   common/ceigen/h
   f=h*dcos(x)-x*dsin(x)
   df=-h*dsin(x)-dsin(x)-x*dcos(x)
   return
   end

   Double Precision Function BesskO(x)
   implicit real*8 (a-h,o-z)
   datapl,p2,p3,p4,p5,p6,p7/-0.57721566,0.4227842,0.23069756,
   X).348859d-l,0.262698d-2,0.10750d-3,0.74d-5/
   dataql,q2,q3,q4,q5,q6,q7/1.25331414,-0.7832358d-l,0.2189568d-l,
   >-0.1062446d-l,0.587872d-2,-0.25154d-2,0.53208d-3/
   if(x.le.2.0)then
   y=x*x/4.0
   besskO=(-dlog(x/2.0)*bessiO(x))+(pl+y*(p2+y*(p3+y*(p4+y*(p5+y*(p6+
   >y*p7))))))
   else
   y=2.0/x
   besskO=(dexp(-x)/dsqrt(x))*(ql+y*(q2+y*(q3+y*(q4+y*(q5+y*(q6+y*q7)
   endif
   return
   end
                                                    365

-------
   DOUBLE PRECISION FUNCTION BESSIO(W)
   IMPLICIT REAL*8 (A-H,O-Z)
   T=W/3.75
   IF(DABS(T).LE.1.0) THEN
   Y=T*T
   BESSIO=1.0+Y*(3.5156229+Y*(3.0899424+Y*(1.2067492+Y*(0.2659732+
   >Y*(0.0360768+0.0045813*Y)))))
   ELSE
   AW=DABS(W)
   Y=3.75/AW
   TEMP=039894228+Y*(0.01328592+Y*(0.00225319+Y*(-0.00157565+Y*(
   >0.00916281+Y*(-0.02057706+Y*(0.02635537+Y*(-0.01647633+
   >0.00392377*Y)))))))
   BESSIO=DEXP(AW)*TEMP/DSQRT(AW)
   END IF
   RETURN
   END

   double precision FUNCTION besskl(x)
CU  USES bessil
   implicit real*8 (a-h,o-z)
   DOUBLE PRECISION pl,p2,p3,p4,p5,p6,p7,ql,q2,q3,q4,q5,q6,q7,y
   SAVE p 1 ,p2,p3 ,p4,p5,p6,p7,ql ,q2,q3 ,q4,q5,q6,q7
   DATA pl,p2,p3,p4,p5,p6,p7/1.0dO,0. 15443 144dO,-0.67278579dO,
   *-0.18156897dO,-0.1919402d-l,-0.110404d-2,-0.4686d-4/
   DATAql,q2,q3,q4,q5,q6,q7/1.25331414dO,0.23498619dO,-0.3655620d-l,
   *0.1504268d-l,-0.780353d-2,0.325614d-2,-0.68245d-3/
   if (x.le.2.0)then
    y=x*x/4.0
    besskl=(dlog(x/2.0)*bessil(x))+(1.0/x)*(pl+y*(p2+y*(p3+y*(p4+y*
   *(p5+y*(p6+y*p7))))))
   else
    y=2.0/x
    besskl=(dexp(-x)/dsqrt(x))*(ql+y*(q2+y*(q3+y*(q4+y*(q5+y*(q6+y*
   endif
   return
   END

   double precision FUNCTION bessil(x)
   implicit real*8 (a-h,o-z)
   DOUBLE PRECISION pl,p2,p3,p4,p5,p6,p7,ql,q2,q3,q4,q5,q6,q7,q8,q9,y
   SAVE p 1 ,p2,p3 ,p4,p5,p6,p7,ql ,q2,q3 ,q4,q5,q6,q7,q8,q9
   DATApl,p2,p3,p4,p5,p6,p7/0.5dO,0.87890594dO,0.51498869dO,
   *0.15084934dO,0.2658733d-l,0.301532d-2,0.32411d-3/
   DATAql,q2,q3,q4,q5,q6,q7,q8,q9/0.39894228dO,-0.3988024d-l,
   *-0.362018d-2,0.163801d-2,-0.1031555d-l,0.2282967d-l,-0.2895312d-l,
   *0.1787654d-l,-0.420059d-2/
   if(abs(x).lt.3.75)then
    y=(x/3.75)**2
    bessil=x*(pl+y*(p2+y*(p3+y*(p4+y*(p5+y*(p6+y*p7))))))
   else
                                                366

-------
     ax=abs(x)
     y=3.75/ax
     bessil=(dexp(ax)/dsqrt(ax))*(ql+y*(q2+y*(q3+y*(q4+y*(q5+y*(q6+y*
   *(q7+y*(q8+y*q9))))))))
     if(x.lt.0.)bessil=-bessil
   endif
   return
   END
    subroutine getlen(string,chr,nO,nl)
c
c   to determine the length of a string excluding any blank padding & heading
c   and return string(nO:nl) nO... position of first non-blank letter
c                  nl...  position of last non-blank letter
c   else return the original string
c   ***Use | to keep blank space at the begining or the end of the
c    *** string
c
    character*(*) string
    character* 160 stringl
    character*! chr
    stringl=''
    inO=0
    inl=0
    do 10 i=len(string),l,-l
    if(string(i:i).ne.chr) then
     if(inl.eq.O) then
     if(string(i:i).eq.'|') then
     nl=i-l
     else
     nl=i
     endif
     inl=l
     endif
     goto 11
    endif
10   continue
11   do20i=l,len(string)
    if(string(i:i).ne.chr) then
    if(in0.eq.O) then
    if(string(i:i).eq.'') then
    nO=i+l
    else
    nO=i
    endif
    inO=l
    nk=nO-l
c    write(6,*) '|'//string(nO:nl)//'|', nO,nl
    do 30 k=nO,nl
    if(string(k:k).ne.'|') then
      nk=nk+l
                                                      367

-------
     if(k.eq.nO) then
      string 1=string(k:k)
     else
      stringl=stringl(l :nk-nO)//string(k:k)
     endif
    endif
c   write(6,*) '|'//stringl(l:nk-nO)//'|'
30   continue
    string=stringl
    nl=nl-nO+l
    nO=l
    endif
    return
   endif
20  continue
   nO=-l
   nl=-l
   return
   end
                                                   368

-------
                           APPENDIX G (Vapor Diffusion)

Source Code for Vapor Diffusion Problem (author: Dominic DiGiulio)
PROGRAM Vapor_Diff
IMPLICIT NONE
EXTERNAL Func
REAL*8,PARAMETER::Pi=3.141592654
REAL*8,DIMENSION(:),ALLOCATABLE: :arr_v !variables
REAL*8,DIMENSION(:),ALLOCATABLE::arr_z Idepth array
REAL*8,DIMENSION(:),ALLOCATABLE::arr_c !concentration array
REAL*8,DIMENSION(:),ALLOCATABLE::arr_t Itime array
REAL*8,DIMENSION(:),ALLOCATABLE::arr_w !water saturation array
REAL*8::func,a,b,s,t,tt
INTEGER: :M,J,Q,P
INTEGER: :N_conc ! number of discrete soil concentrations in simulation
INTEGER: :N_times ! number of times to simulate
INTEGER: :N_sat ! number of moisture saturations to simulate
! variables
REAL*8::theta_a ! volumetric air content
REAL*8::theta_w ! volumetric water content
REAL*8::theta_o ! volumetric NAPL content
REAL*8::rho_b  Ibulk density [g/cm**3]
REAL*8::rho_o  ! density of NAPL [g/cm**3]
REAL*8::H    IHenry's constant [dimensionless]
REAL* 8:: K_d   ! solids/water partition coefficient
REAL*8::S_w   Iwater solubility [g/cm**3]
REAL* 8:: M_o   ! average molecular weight of NAPL [g/mole]
REAL*8::M_i   Imolecular weight of compound [g/mole]
REAL* 8:: D_a   ! free air gas diffusion coefficient [cm* * 2/d]
REAL* 8: :D_w   ! free water diffusion coefficient [cm* * 2/d]
REAL* 8:: L     ! length of domain [cm]
REAL*8::R          Iretardation factor [dimensionless]
REAL*8::por   Iporosity
REAL* 8:: Si_a   Itortuosity factor for air phase [dimensionless]
REAL*8::Si_w  Itortuosity factor for water phase [dimensionless]
REAL*8::k           !diffusion coefficient [cm**2/d]
REAL*8::K_a   !soil-air partition coefficient [dimensionles]
REAL* 8: :NAPL_sat,water_sat
REAL* 8: :term 1 ,term2,term3,sum,sum_l ,B_n,C_n,C_T_avg

COMMON/cfunc/tt

OPEN (l,FILE='diff_var.in')
OPEN (2,FILE='conc.in')
OPEN (3,FILE='depth.in')
                                         369

-------
OPEN (4,FILE='time.in')
OPEN (5,FILE='avg_conc.ouf)
OPEN (6,FILE='water_sat.in')

ALLOCATE (arr_v(15)) !allocate 15 spaces for variables
READ (1,*) arr_v  Iread diff_var.in
N_conc=arr_v(13) Iread number of discrete soil cone.from diff_var.in (13th line)
N_times=arr_v(14) Iread number of sim.times from diff_var.in (14th line)
N_sat=arr_v( 15)  I read number of water saturations from diff_var.in (15th line)
ALLOCATE (arr_c(N_conc)) I allocate space in arr_c for N_conc
ALLOCATE (arr_z(N_conc-l)) I allocate space in arr_z for N_conc - 1 depths
              I do not include first and last depth z=0 and L
ALLOCATE (arr_t(N_times)) I allocate space forN_times
ALLOCATE (arr_w(N_sat))  I allocate space forN_sat
READ (2,*) arr_c Iread cone.in file
READ (3,*) arr_z Iread depth.in file
READ (4,*) arr_t Iread time.in file
READ (6,*) arr_w Iread water_sat.in

por=arr_v(l)
NAPL_sat=arr_v(2)
rho_b=arr_v(3)
rho_o=arr_v(4)
S_w=arr_v(5)
H=arr_v(6)
K_d=arr_v(7)
M_i=arr_v(8)
M_o=arr_v(9)
D_a=arr_v(10)
D_w=arr_v(ll)
L=arr_v(12)

WRITE(5,*)   "time (yrs)   avg. cone. (ug/Kg)"
WRITE(5,*)

DO P=l,N_sat
water_sat=arr_w(P)
WRITE(5,*)
WRITE(5,20) water_sat
20FORMAT(1X,F10.2)
theta_w=por*water_sat
theta_a=por* (1.0-water_sat-NAPL_sat)
theta_o=por*NAPL_sat
R=theta_a*H+theta_w+K_d*rho_b+(M_i*rho_o*theta_o)/(S_w*M_o)
K_a=R/H
Si_a=(theta_a* *3.333)/por* *2
Si_w=(theta_w* *3.333)/por* *2
                                            370

-------
k=(Si_a*D_a*H+Si_w*D_w)/R
a=0.0

DO Q=l,N_times
t=arr_t(Q)*365

 ! summation loop
 sum=0.0
 DO M= 1,10000,2

  tt=k*M**2*Pi**2*t/L**2
       b=tt
  Icalc B_n
  CALL qtrap(func,a,b,s)
  B_n=K_a*s !ug/cm**3 soil

  Icalc A_n
  sum_l=0.0
  DO J=l,N_conc-l
  term 1=(arr_c(J+1 )-arr_c(J))* dcos(arr_z( J) * M* Pi/L)
  sum_l=sum_l+terml
  END DO
  C_n=l./2.*(sum_l+arr_c(l)+(-l)**(M+l)*arr_c(5)) lug/kg
       C_n=0.001*rho_b*C_n !ug/cm**3

 terml=C_n*dexp(-tt)
 term2=B_n
 term3=l./(M**2*Pi**2)*(terml+term2)
 sum=sum+term3
 IF ((dABS(term3/SUM) <= l.OE-04) .or. (M > 1000)) EXIT
 PRINT 30, term3/SUM,M,t/365
 30 FORMAT(E10.5,I10,F10.2)
 END DO
 C_T_avg=8. * sum* (1000/rho_b) ! ug/kg
WRITE(5,10) t/365.0,C_T_avg
10FORMAT(1X,2F10.2)
END DO

END DO
CONTAINS

SUBROUTINE qtrap(func,a,b,s)
   INTEGER JMAX
   REAL* 8 a,b,func,s,EPS
   EXTERNAL func
   PARAMETER (EPS=l.e-5, JMAX=20)
!    USES trapzd
                                         371

-------
   INTEGER j
   REAL* 8 olds
   olds=-l.e30
   do llj=l,JMAX
    call trapzd(func,a,b,s,j)
    if (dabs(s-olds).lt.EPS*dabs(olds)) return
    if (s.eq.0..and.olds.eq.0..and.j.gt.6) return
    olds=s
11  continue
   ! pause 'too many steps in qtrap'
   END SUBROUTINE qtrap

SUBROUTINE trapzd(func,a,b,s,n)
   INTEGER n
   REAL* 8 a,b,s,func
   EXTERNAL func
   INTEGER itj
   REAL* 8 del,sum,tnm,x
   if (n.eq.l) then
    s=0.5 * (b-a) * (func(a)+func(b))
   else
    it=2**(n-2)
    tnm=it
    del=(b-a)/tnm
    x=a+0.5*del
    sum=0.
    do llj=l,it
     sum=sum+func(x)
     x=x+del
11   continue
    s=0.5 * (s+(b-a) * sum/tnm)
   endif
   return
   END SUBROUTINE trapzd

END PROGRAM Vapor_Diff

FUNCTION func(x)
REAL*8:: func,x,tt
COMMON/cfunc/tt
a=0.71761
b=0.0023179721
c=0.0054
fx=a*c/(1.0+a*b*x)
func=dexp(x-tt)*fx
END FUNCTION func
                                          372

-------
                                       APPENDIX H (VFLUX)

Example Input File for VFLUX

0.35       /* porosity
0.0        /* NAPL saturation (napl volume / pore volume)
0.70       /* water saturation (water volume / pore volume)
0.38       /* dimensionless Henry's law constant
131.5       /* average molecular weight of NAPL (g/mole)
131.5       /* molecular weight of compound (g/mole)
1100.0      /* aqueous solubility of pure-phase compound (mg/L)
126.       /* organic carbon partition coefficient (mL/g)
0.001       /* fraction organic carbon (unitless)
2.15       /* bulk density of soil (g/cmA3)
1.462       /* density of NAPL (g/cmA3)
0.035       /* average annual infiltration flux (cm/d)
6367.0      /* free-air diffusion coefficient of compound (cmA2/d)
0.804       /* free-water diffusion coefficient of compound (cmA2/d)
30.0       /* longitudinal dispersivity (cm)
3.65e6      /* degradation half life of compound (d)
11         /* number of depths for initial soil concentration
50      1006.635905   /* depth (cm) and the measured soil concentration (ug/kg)
100     4809.263006
150     35200.72357
200     266876.3314
250     1528360.581
300     210637.9515
350     28260.94547
400     3882.013805
450     913.4462901
500     656.0163567
11        /* number of times for groundwater concentration (mg/1)
1       4.999994228
5       1.998388846
10      1.445123207
15      1.165824718
20      0.984167078
30      0.74981552
40      0.598388867
50      0.489243244
60      0.405386745
80      0.282682286
74              /* number of times (years) for outputting results
0.5,1.0,1.5,2.0,2.5,3.0,3.5,4.0,4.5,5.0,5.5,6.0,6.5,7.0,7.5,
8.0,8.5,9.0,9.5,10.0,10.5,11.0,11.5,12.0,12.5,13.0,13.5,14.0,14.5,15.0,
15.5,16.0,16.5,17.0,17.5,18.0,18.5,19.0,19.5,20.0,20.5,21.0,21.5,22.0,22.5,
23.0,23.5,24.0,24.5,25.0,25.5,26.0,26.5,27.0,27.5,28.0,28.5,29.0,29.5,30.0,
35.0,40.0,45.0,50.0,55.0,60.0,65.0,70.0,75.0,80.0,85.0,90.0,95.0,100.0 /* time unit is  (year)
27              /* number of depths (cm) for outputting results
0.0,20.0,40.0,60.0,80.0,100.0,120.0,140.0,160.0,180.0,200.,220.0,240.0,260.0,280.0,
300.0,320.0,340.0,360.0,380.0,400.0,420.0,440.0,460.0,480.0,500.0,518.16  /*  depth unit is (cm)
y       /* flag for mass balance if y or Y
                                                  373

-------
Example Parameter File for VFLUX

***** Input Parameters prescribed by the user *****

Porosity =  0.350000000000000
NAPL saturation =  0.000000000000000
Water saturation =  0.700000000000000
Dimensionless Henrys constant =  0.380000000000000
Average molecular weight of NAPL =   131.500
Molecular weight of compound of interest =   131.500
Aqueous solubility of compound (mg/L)  =   1100.00000000000
Organic carbon partition coefficient (cc/g) =   126.000
Fraction organic carbon content = 0.100000000000000D-002
Bulk density of soil (g/cc) =  2.15000000000000
density of NAPL (g/cc) =  1.46200000000000
Average annual recharge rate (cm/d) = 0.350000E-01
Free-air diffusion coefficient of compound (cmA2/d) =  6367.00000000000
Free-water diffusion coefficient of compound (cmA2/d) = 0.804000000000000
Hydrodynamic dispersivity (cm) =  30.0000000000000
Degradation half-life of compound (d) =  3650000.00000000
Number of initial soil concentrations =     11
Initial soil concentration versus depth (cm)
  0.500E+02    0.101E+04
  0.100E+03    0.481E+04
  0.150E+03    0.352E+05
  0.200E+03    0.267E+06
  0.250E+03    0.153E+07
  0.300E+03    0.211E+06
  0.350E+03    0.283E+05
  0.400E+03    0.388E+04
  0.450E+03    0.913E+03
  0.500E+03    0.656E+03
  0.520E+03    0.645E+03
Number of groundwater concentrations =     11
Groundwater concentrations versus time (yrs)
  0.100E+01    0.500E+01
  0.500E+01    0.200E+01
  0.100E+02    0.145E+01
  0.200E+02    0.984E+00
  0.300E+02    0.750E+00
  0.400E+02    0.598E+00
  0.500E+02    0.489E+00
  0.600E+02    0.405E+00
  0.800E+02    0.283E+00
  0.100E+03    0.195E+00
Number of time values for simulation results =      75
Times (yr) at which results are produced
  0.000000000000000    0.500000000000000
  1.50000000000000     2.00000000000000
                       3.50000000000000
                       5.00000000000000
                       6.50000000000000
3.00000000000000
4.50000000000000
6.00000000000000
 1.00000000000000
2.50000000000000
4.00000000000000
5.50000000000000
7.00000000000000
                                               374

-------
 7.50000000000000
 9.00000000000000
 10.5000000000000
 12.0000000000000
 13.5000000000000
 15.0000000000000
 16.5000000000000
 18.0000000000000
 19.5000000000000
 21.0000000000000
 22.5000000000000
 24.0000000000000
 25.5000000000000
 27.0000000000000
 28.5000000000000
 30.0000000000000
 45.0000000000000
 60.0000000000000
 75.0000000000000
 90.0000000000000
8.00000000000000
9.50000000000000
11.0000000000000
12.5000000000000
14.0000000000000
15.5000000000000
17.0000000000000
18.5000000000000
20.0000000000000
21.5000000000000
23.0000000000000
24.5000000000000
26.0000000000000
27.5000000000000
29.0000000000000
35.0000000000000
50.0000000000000
65.0000000000000
80.0000000000000
95.0000000000000
0.000000000000000
60.0000000000000
120.000000000000
180.000000000000
240.000000000000
300.000000000000
360.000000000000
420.000000000000
480.000000000000
8.50000000000000
10.0000000000000
11.5000000000000
13.0000000000000
14.5000000000000
16.0000000000000
17.5000000000000
19.0000000000000
20.5000000000000
22.0000000000000
23.5000000000000
25.0000000000000
26.5000000000000
28.0000000000000
29.5000000000000
40.0000000000000
55.0000000000000
70.0000000000000
85.0000000000000
100.000000000000
  27
                     40.0000000000000
                     100.000000000000
                     160.000000000000
                     220.000000000000
                     280.000000000000
                     340.000000000000
                     400.000000000000
                     460.000000000000
                     518.160000000000
Number of depth values for simulation results =
Depths (cm) at which results are produced
                       20.0000000000000
                       80.0000000000000
                       140.000000000000
                       200.000000000000
                       260.000000000000
                       320.000000000000
                       380.000000000000
                       440.000000000000
                       500.000000000000
Length of domain (cm) =  520.000
****** Calculated Model Parameters *******
Volumetric NAPL saturation =  0.000000000000000
Volumetric water saturation =  0.245000000000000
Volumetric air saturation = 0.105000000000000
Lumped diffusion coefficient - D (cmA2/d) =  15.1325989906785
Half-Peclet number - a =  0.601350768916067
First-order decay constant (/d) =  0.189903E-06
Dimensionless decay constant  =  0.188601131871932D-002
Parameter Delta =  0.602916875360662
Aqueous solubility (g/cc) = 0.109999999722277D-002
Soil-water partition coefficient - Kd = 0.126000
Total Retardation coefficient - R =  0.555800004357100
                                             375

-------
Example Flux File for VFLUX

Average Soil Concentration as a Function of Time
   Time
   (yr)

 O.OOOOE+00
 0.5000E+00
 0.1000E+01
 0.1500E+01
 0.2000E+01
 0.2500E+01
 0.3000E+01
 0.3500E+01
 0.4000E+01
 0.4500E+01
(First 10 times shown)
Average Soil Concentration
(ug/kg)

0.1790E+06
0.1739E+06
0.1528E+06
0.1287E+06
0.1073E+06
0.8909E+05
0.7393E+05
0.6131E+05
0.5085E+05
0.4217E+05
Mass flux To/From Groundwater
Time Total Mass-flux Advective flux Dispersive flux
(yr)
O.OOOOE+00
0.5000E+00
0.1000E+01
0.1500E+01
0.2000E+01
0.2500E+01
0.3000E+01
0.3500E+01
0.4000E+01
0.4500E+01
(first 10 times
(ug/cmA2/d)
O.OOOE+00
0.419E+02
0.845E+02
0.854E+02
0.754E+02
0.642E+02
0.539E+02
0.450E+02
0.374E+02
0.311E+02
shown)

O.OOOE+00
0.175E+00
0.175E+00
0.105E+00
0.105E+00
0.874E-01
0.874E-01
0.771E-01
0.771E-01
0.699E-01


O.OOOE+00
0.417E+02
0.843E+02
0.853E+02
0.753E+02
0.641E+02
0.538E+02
0.449E+02
0.374E+02
0.310E+02

Mass Flux to and from Atmosphere
Time
(yr)
O.OOOOE+00
0.5000E+00
0.1000E+01
0.1500E+01
0.2000E+01
0.2500E+01
0.3000E+01
0.3500E+01
0.4000E+01
0.4500E+01
(first 10 times
Flux to Atmosphere
(ug/cmA2/d)
O.OOOE+00
-0.488E+02
-0.643E+02
-0.557E+02
-0.457E+02
-0.373E+02
-0.306E+02
-0.252E+02
-0.208E+02
-0.172E+02
shown)
                                              376

-------
Example of Mass Balance File for VFLUX
  (Note: The mass units are - ug/cmA2)
  Time(yr)    M_remaining    M_Atm
M Gw
M_decay
O.OOOE+00
0.500E+00
0.100E+01
0.150E+01
0.200E+01
0.250E+01
0.300E+01
0.350E+01
0.400E+01
0.450E+01
0.500E+01
0.550E+01
0.600E+01
0.650E+01
0.700E+01
0.750E+01
0.800E+01
0.850E+01
0.900E+01
0.950E+01
0.100E+02
0.105E+02
0.110E+02
0.115E+02
0.120E+02
0.125E+02
0.130E+02
0.135E+02
0.140E+02
0.145E+02
0.150E+02
0.155E+02
0.160E+02
0.165E+02
0.170E+02
0.175E+02
0.180E+02
0.185E+02
0.190E+02
0.195E+02
0.200E+02
0.205E+02
0.210E+02
0.215E+02
0.2002E+06
0.1945E+06
0.1708E+06
0.1439E+06
0.1199E+06
0.9960E+05
0.8265E+05
0.6855E+05
0.5685E+05
0.4715E+05
0.3910E+05
0.3243E+05
0.269 1E+05
0.2232E+05
0.1853E+05
0.1538E+05
0.1278E+05
0.1061E+05
0.8822E+04
0.7336E+04
0.6107E+04
0.5085E+04
0.4240E+04
0.3538E+04
0.2957E+04
0.2474E+04
0.2075E+04
0.1742E+04
0.1467E+04
0.1238E+04
0.1049E+04
0.8902E+03
0.7598E+03
0.6503E+03
0.5602E+03
0.4843E+03
0.4220E+03
0.3691E+03
0.3260E+03
0.2890E+03
0.2589E+03
0.2329E+03
0.2119E+03
0.1934E+03
O.OOOOE+00
0.3513E+04
0.1462E+05
0.2567E+05
0.3491E+05
0.4245E+05
0.4863E+05
0.5370E+05
0.5789E+05
0.6135E+05
0.642 1E+05
0.6658E+05
0.6854E+05
0.7016E+05
0.7151E+05
0.7263E+05
0.7356E+05
0.7432E+05
0.7496E+05
0.7549E+05
0.7593E+05
0.7629E+05
0.7660E+05
0.7685E+05
0.7706E+05
0.7723E+05
0.7738E+05
0.7750E+05
0.7760E+05
0.7768E+05
0.7775E+05
0.778 1E+05
0.7786E+05
0.7790E+05
0.7793E+05
0.7796E+05
0.7799E+05
0.780 1E+05
0.7803E+05
0.7804E+05
0.7806E+05
0.7807E+05
0.7808E+05
0.7809E+05
O.OOOOE+00
0.2182E+04
0.1471E+05
0.3059E+05
0.4532E+05
0.5808E+05
0.6885E+05
0.7787E+05
0.8538E+05
0.9163E+05
0.968 1E+05
0.1011E+06
0.1047E+06
0.1076E+06
0.1101E+06
0.1121E+06
0.1138E+06
0.1152E+06
0.1163E+06
0.1173E+06
0.1181E+06
0.1187E+06
0.1193E+06
0.1197E+06
0.1201E+06
0.1204E+06
0.1207E+06
0.1209E+06
0.1210E+06
0.1212E+06
0.1213E+06
0.1214E+06
0.1215E+06
0.1216E+06
0.1216E+06
0.1217E+06
0.1217E+06
0.1217E+06
0.1218E+06
0.1218E+06
0.1218E+06
0.1218E+06
0.1218E+06
0.1218E+06
O.OOOOE+00
0.7146E+01
0.1396E+02
0.2010E+02
0.2498E+02
0.2880E+02
0.3196E+02
0.3459E+02
0.3675E+02
0.3856E+02
0.4006E+02
0.4130E+02
0.4233E+02
0.4319E+02
0.4389E+02
0.4448E+02
0.4497E+02
0.4538E+02
0.4572E+02
0.4600E+02
0.4623E+02
0.4643E+02
0.4659E+02
0.4673E+02
0.4684E+02
0.4694E+02
0.4702E+02
0.4709E+02
0.4714E+02
0.4719E+02
0.4723E+02
0.4727E+02
0.4730E+02
0.4732E+02
0.4734E+02
0.4736E+02
0.4738E+02
0.4739E+02
0.474 1E+02
0.4742E+02
0.4743E+02
0.4744E+02
0.4745E+02
0.4745E+02
                                            377

-------
Source Code for VFLUX (authors: Ravi Varadhan and Dominic DiGiulio)

c This code, VFLUX, calculates an exact solution to the one-simensional
c solute transport equation under time dependent boundary conditions.
c The code calculates soil concentration profile, average soil concentrations
c and mass flux to/from groundwater.
c Refer to DiGiulio et al. (1999) for details regarding the analytical model
c
c Date: March 1999
c
c Code modified 04/21/99 to account for zero-gradient BC at the bottom
c of the soil column and also for g(T)=0; with these features the code
c almost completely supersedes VLEACH - the only feature lacking is
c the ability to simulate multiple polygons
c
c Modified on 06/29/99 to generate a summary of mass balance at user-
c specified times. This option slows the execution of the code due to
c slower convergence of the resulting fourier series for cumulative mass
c flux.
c
c modified on October 12, 1999 - moved the  line, calculating
c the conversion of aqsol from (mg/L) to (g/cc), up before the determination
c of retardation factor
c
   implicit real*8 (a-h,o-z)
   dimension ctinit(500),zct(500),t(100),z(100),ctavg(100),
   >gwcon(500),tgw(500),eign(1000),an(1000),cmgw(100),cmatm(100)
   realmavg,mcomp,l,lambda,koc,kd,infil
   character* 10 prefix,fname 1 ,fname2,fname3 ,fname4,fname5
   character*4 suffixI,suffix2,suffix3,suffix4,suffix5
   character*! mbl
   common/cmain/a
   common/cflux/an,eign,biglamb,tgw,gwcon,r,d,infil,l,ngw
   external funcd
   write(*,*)'Enter the Prefix of the  input file: '
   write(*,*)'3 different output files are created w/ same prefix'
   write(*,*)' Input-file Prefix ? '
   read(*,'(a)')prefix
   call getlen(prefix,' ',nO,nl)
   suffixl='.inp'
   suffix2='.par'
   suffix3='.prf
   suffix4='.flx'
   suffix5='.mbl'
   fnamel=prefix(nO:nl)//suffixl
   fname2=prefix(nO:nl)//suffix2
   fname3=prefix(nO:nl)//suffix3
   fname4=prefix(nO:nl)//suffix4
   fname5=prefix(nO:nl)//suffix5
   open( 10,fname 1, status='unknown')
   open(20,fname2,status='unknown')
   open(3 0,fname3, status='unknown')
                                                   378

-------
   open(40,fname4,status='unknowri)
   open(50,fname5,status='unknowri)
c
c The following parameter definitions and units are used:
c
c por = void volume / total volume
c vnapl = NAPL volume / total volume
c vwater = water volume / total volume
c henry = dimensionless henry's law coefficient
c mavg = average molecular weight (g/mole)
c mcomp = molecular weight of component of interest
c aqsol = aqueous solubility (mg/L)
c koc = organic carbon partition coefficient (mL/g)
c foe = fraction organic carbon (unitless)
c rhob = bulk density of soil (g/cmA3)
c rhonapl = density of NAPL (g/cmA3)
c infil = average annual infiltration rate (cm/d)
c dair = free-air diffusion coefficient of compound (cmA2/d)
c dwater = free-water diffusion coefficient of compound (cmA2/d)
c disp = longitudinal dispersivity (cm)
c Wife = half-life of degradation (d)
c zct = depths at which initial soil concentrations are measured (cm)
c ctinit = initial soil concentrations (ug/kg)
c tgw = times at which groundwater concentrations are measured (yr)
c gwcon = groundwater concentrations (mg/L)
c t = times at which outputs are calculated (yr)
c z = depths at which soil concentration profile is evlauated (cm)
c
   read(10,101)por,vnapl,vwater,henry,mavg,mcomp,aqsol,koc,foc,rhob,
   >rhonapl
   read(10,101)infil,dair,dwater,disp,hlife
   read(10,101)nzct
   do 41 i=l,nzct
   read(10,*)zct(i),ctinit(i)
41   continue
   read(10,101)ngw
   if(ngw.le.0)goto 43
   do 42 i=l,ngw
   read(10,*)tgw(i),gwcon(i)
42   continue
43   continue
   read(10,101)nt
   read(10,*)(t(i),i=l,nt)
   if(t(l).ne.0.)then
   do 44 i=nt+l,2,-l
   t(i)=t(i-l)
44   continue
   t(l)=0.
   nt=nt+l
   endif
   read(10,101)nz
   read(10,*)(z(i),i=l,nz)
                                                   379

-------
   l=zct(nzct)
   read(10,'(al)')mbl
c
   write(20,*)'***** Input Parameters prescribed by the user *****'
   write(20,*)
   write(20,*)'Porosity =',por
   write(20,*)'NAPL saturation =', vnapl
   write(20,*)'Water saturation  =',vwater
   write(20,*)'Dimensionless Henrys constant =',henry
   write(20,*)'Average molecular weight of NAPL  =',mavg
   write(20,*)'Molecular weight of compound of interest =',mcomp
   write(20,*)'Aqueous solubility of compound (mg/L) =',aqsol
   write(20,*)'Organic carbon partition coefficient (cc/g) =',koc
   write(20,*)'Fraction organic carbon content =',foc
   write(20,*)'Bulk density of soil (g/cc) =',rhob
   write(20,*)'density of NAPL (g/cc) =',rhonapl
   write(20,*)'Average annual recharge rate (cm/d) =',infil
   write(20,*)'Free-air diffusion coefficient of compound (cmA2/d) =
   >',dair
   write(20,*)'Free-water diffusion coefficient of compound (cmA2/d)
   >=',dwater
   write(20,*)'Hydrodynamic dispersivity (cm) =',disp
   write(20,*)'Degradation half-life of compound (d)  =',hlife
   write(20,*)
   write(20,*)'Number of initial soil concentrations =',nzct
   write(20,*)
   write(20,*)'Initial soil concentration versus depth (cm)'
   do 120 i=l,nzct
   write(20,109)zct(i),ctinit(i)
120  continue
   write(20,*)
   write(20,*)'Number of groundwater concentrations =',ngw
   write(20,*)
   write(20,*)'Groundwater concentrations versus time (yrs)'
   if(ngw.gt.0)then
   do 130 i=l,ngw
   write(20,109)tgw(i),gwcon(i)
130  continue
   endif
   write(20,*)
   write(20,*)'Number of time values for simulation results =',nt
   write(20,*)
   write(20,*)'Times (yr) at which results are produced '
   write(20,*)(t(i),i=l,nt)
   write(20,*)
   write(20,*)'Number of depth values for simulation results =',nz
   write(20,*)'Depths (cm) at which results are produced '
   write(20,*)(z(i),i=l,nz)
   write(20,*)
   write(20,*)'Length of domain (cm)  =',1
c
c output times are converted from years to days
                                                    380

-------
c
   do 45 i=l,nt
   t(i)=t(i)*365.2
45   continue
c
c NAPL and water saturations are converted to bulk volumetric contents
c
   vnapl=vnapl*por
   vwater=vwater*por
c
c soil concentrations which are input as (ug/kg) are converted to (ug/cmA3)
c
   sum=0.
   do46k=l,nzct
   ctinit(k)=ctinit(k)*rhob/1000.
   sum=sum+ctinit(k)*(zct(k)-zct(k-l))
46   continue
   ctavg(l)=sum/l
c
c The conversion of aquqous solubility from mg/L to g/cc was moved to just
c below this line so that the Retardation factor could be properly
c calculated
c
   aqsol=aqsol* 1 .e-06
   vair=por-vnapl-vwater
   kd=koc*foc
   r=vair*henry+vwater+kd*rhob+mcomp*rhonapl*vnapl/(aqsol*mavg)
   psia=vair**(10./3.)/(por*por)
   psiw=vwater**(10./3.)/(por*por)
   d=dair*psia*henry+dwater*psiw+disp*infil/vwater
   a=infil*l/(2.*d)
   lambda=0.
   if(hlife.gt.0.)lambda=dlog(2.0)/hlife
   biglamb=lambda*r*l*l/d
   delta=dsqrt(a*a+biglamb)
   pi=4.*datan(1.0)
c
   write(20,*)
   write(20,*)'****** Calculated Model Parameters ******* '
   write(20,*)
   write(20,*)'Volumetric NAPL saturation =',vnapl
   write(20,*)'Vorumetric water saturation =',vwater
   write(20,*)'Volumetric air saturation  =',vair
   write(20,*)'Lumped diffusion coefficient - D (cmA2/d) =',d
   write(20,*)'Half-Peclet number - a  =',a
   write(20,*)'First-order decay constant (/d) =',lambda
   write(20,*)'Dimensionless decay constant  =',biglamb
   write(20,*)'Parameter Delta  =',delta
   write(20,*)'Aqueous solubility (g/cc)  =',aqsol
   write(20,*)'Soil-water partition coefficient - Kd =',kd
   write(20,*)'Total Retardation coefficient - R - ,r
                                                   381

-------
c groundwater concentrations (mg/L) are converted to interface soil
c concentrations (ug/cmA3)
c times are converted from (yrs) to (days)
c
   if(ngw.gt.0)then
   do 60 k=l,ngw
   tgw(k)=tgw(k)*365.2
   gwcon(k)=gwcon(k)*r
60   continue
   endif
c
c
c calculation of eigen values
c
   if(ngw.ge.0)then
   do lnl=l,1000
   eign(nl)=nl*pi
1   continue
   else
   xl=0.01
   x2=pi
   do 1001 kount=l,1000
   zero=rtsafe(funcd,xl,x2,1 .e-06)
   eign(kount)=zero
   xl=zero+0.5*pi
   x2=zero+1.5*pi
1001  continue
   endif
c
c calculation of fourier coeffiecients
c
   do 1003 1=1,1000
   suml=0.
   do 1002k=l,nzct
   zzctk=zct(k-l)/l
   zzctkl=zct(k)/l
   hzk=dexp(-a*zzctk)*(a*dsin(eign(i)*zzctk)+eign(i)*dcos(eign(i)*
   >zzctk))
   hzk 1 =dexp(-a*zzctkl) *(a*dsin(eign(i) *zzctkl )+eign(i) *dcos(eign(i) *
   >zzctkl))
   suml=suml+ctinit(k)*(hzk-hzkl)
1002  continue
   if(ngw.ge.O)xnorm=l./2.
   if(ngw.lt.0)xnorm=(a+a*a+eign(i)**2)/(2.*(a*a+eign(i)**2))
   an(i)=suml/(xnorm*(a*a+eign(i)**2))
1003  continue
c
c calculation of soil concentration profile
c
   do 3 1=1,nt
   tt=d*t(i)/(r*l*l)
   if(tt.eq.0.)goto 3
                                                   382

-------
c
c The profile is not calculated for t=0, since it is the
c initial distribution, which is known.
c
   write(30,103)t(i)/365.2
   write(30,108)
   do 4j=l,nz
   zz=z(j)/l
   sum2=0.
   n2=0
5   continue
   n2=n2+l
   if(n2.gt.lOOO)goto6
   wn2sq=biglamb+a*a+eign(n2) * *2
   term=an(n2)*dexp(-wn2sq*tt)
   sterm=dsin(eign(n2) *zz)
   add=term*sterm
   sum2=sum2+add
   if(abs(sum2).le.l.e-14)goto 6
   if(abs(term/sum2).gt.l.e-07)goto 5
6   continue
   ctl=dexp(a*zz)*sum2
   ct2=0.
   ct3=0.
   if(ngw.le.0)goto 71
   n3=0
   sum3=0.
   fact=l
7   continue
   n3=n3+l
   if(n3.gt.lOOO)goto8
   fact=-fact
   wn3 sq=biglamb+a*a+(eign(n3)) * *2
   term=n3 *fact/(wn3 sq) *dexp(-wn3 sq*tt)
   sum3=sum3 +term*dsin(eign(n3) *zz)
   if(abs(sum3).le.l.e-14) goto 8
   if(abs(term/sum3).gt.l.e-07)goto 7
8   continue
   ct2=gwcon(l)*dexp(a*(zz-l.))*(2.*pi*sum3+dsinh(delta*zz)/dsinh
   >(delta))
   ct3=0.
   do 9k=l,ngw
   ttgw=d*tgw(k)/(r*l*l)
   if(ttgw.ge.tt)goto 9
   n4=0
   sum4=0.
   fact=l
10   continue
   n4=n4+l
   if(n4.gt.lOOO)goto 11
   fact=-fact
   wn4sq=biglamb+a*a+(eign(n4)) * *2
                                                  383

-------
   term=n4*fact/(wn4sq)*dexp(-wn4sq*(tt-ttgw))
   sum4=sum4+term*dsin(eign(n4)*zz)
   if(abs(sum4).le.l.e-14) goto 11
   if(abs(term/sum4).gt.l.e-07)goto 10
11   continue
   Ct3=ct3+(gwcon(k+l)-gwcon(k))*dexp(a*(zz-l.))*(2.*pi*sum4+
   >dsinh(delta*zz)/dsinh(delta))
9   continue
71   continue
   ct=ctl+ct2+ct3
   ctmass=ct*1000./rhob
   cair=ct*henry/r
   csolid=ct*kd/r
   cwater=ct/r
   write(30,109)z(j),ctmass,csolid,cair,cwater
4   continue
3   continue
c
c Calculation of average soil concentration
c
   write(40,104)
   write(40,105)t(l)/365.2,ctavg(l)*1000./rhob
   do 30 i=l,nt
   tt=d*t(i)/(r*l*l)
   if(tt.eq.0.)goto 30
   sum5=0.
   n5=0
12   continue
   n5=n5+l
   if(n5.gt.lOOO)goto 13
   wn5 sq=biglamb+a*a+(eign(n5)) * *2
   tl=a*dsin(eign(n5))-eign(n5)*dcos(eign(n5))+dexp(-a)*eign(n5)
   term=an(n5)*dexp(-wn5sq*tt)*tl/(a*a+(eign(n5))**2)
   sum5=sum5 +term
   if(abs(sum5).le.l.e-14)goto 13
   if(abs(term/sum5).gt.l.e-07)goto 12
13   continue
   ctavg 1=dexp(a) *sum5
   ctavg2=0.
   ctavg3=0.
   if(tt.eq.O. .or. ngw.le.0)goto 72
   n6=0
   sum6=0.
   fact=l.
14   continue
   n6=n6+l
   if(n6.gt.lOOO)goto 15
   fact=-fact
   wn6sq=biglamb+a*a+(eign(n6))**2
   term=(eign(n6))**2/wn6sq/(a*a+(eign(n6))**2)*dexp(-wn6sq*tt)*
   >(fact*dexp(-a)-l.)
   sum6=sum6+term
                                                   384

-------
   if(abs(sum6).le.l.e-14)goto 15
   if(abs(term/sum6).gt.l.e-07)goto 14
15   continue
   if(abs(a-delta).ge. l.e-14)then
   ctavg2=gwcon(l)*((a*dsinh(delta)-delta*dcosh(delta)+delta*dexp(-a)
   >)/(dsinh(delta)*(a*a-delta*delta))+2.*sum6)
   else
   ctavg2=gwcon(l)*((lVdelta-dexp(-delta)/dsinh(delta))/2.+2.*sum6)
   endif
   do 16 k=l,ngw
   ttgw=d*tgw(k)/(r*l*l)
   if(ttgw.ge.tt)goto 16
   n7=0
   sum7=0.
   fact=l.
17   continue
   n7=n7+l
   if(n7.gt.lOOO)goto 18
   fact=-fact
   wn7sq=biglamb+a*a+(eign(n7))**2
   term=(eign(n7))**2/wn7sq/(a*a+(eign(n7))**2)*dexp(-wn7sq*(tt-ttgw)
   >)*(fact*dexp(-a)-l.)
   sum7=sum7+term
   if(abs(sum7).le.l.e-14)goto 18
   if(abs(term/sum7).gt.l.e-07)goto 17
18   continue
   if(abs(a-delta).ge. l.e-14)then
   ctavg3=ctavg3+(gwcon(k+l)-gwcon(k))*((a*dsinh(delta)-delta*dcosh
   >(delta)+delta*dexp(-a))/(dsinh(delta)*(a*a-delta*delta))+2.*sum7)
   else
   ctavg3=ctavg3+(gwcon(k+l)-gwcon(k))*((l./delta-dexp(-delta)/dsinh
   >(delta))/2.+2.*sum7)
   endif
16   continue
72   continue
   ctavg(i)=ctavgl+ctavg2+ctavg3
   write(40,105)t(i)/365.2,ctavg(i)*1000./rhob
30   continue
c
c Calculation of groundwater flux
c
   write(40,106)
   do40i=l,nt
   tt=d*t(i)/(r*l*l)
   call gwflux(tt,gwflx,advflx,dspflx)
   write(40,107)t(i)/365.2,gwflx,advflx,dspflx
40   continue
c
c Calculation of flux to the atmosphere
c
   write(40,110)
   do 50 i=l,nt
                                                    385

-------
   tt=d*t(i)/(r*l*l)
   call atmflux(tt,atmflx)
   write(40,l 1 l)t(i)/365.2,atmflx
50   continue
c
c Summary of mass balance
c
c
   if(mbl.eq.'y' .or. mbl.eq.'Y')then
c
   if(lambda.eq.0.)write(50,l 12)
   if(lambda.gt.0.)write(50,114)
   do 80 i=l,nt
   tt=d*t(i)/(r*l*l)
   call gwmass(tt,cgwm)
   call atmass(tt,catm)
   cmatm(i)=-catm*r*l*l/d
   cmgw(i)=cgwm*r*l*l/d
   cmdis=ctavg( 1) *l-(ctavg(i) *l+cmatm(i)+cmgw(i))
   write(50,113)t(i)/365.2,ctavg(i)*l,cmatm(i),cmgw(i),cmdis
80   continue
c
   endif
c
101  format(glO.O)
102  format(glO.O,glO.O)
103  format(75('*')/'Soil Concentration Profile @ time (yr)=',e!0.3/)
104  format(75('*')/5x, Time', 12x,'Average Soil Concentration '/5x,
   >'(yr)',21x,'(ug/kg)'/)
105  format(2x,el0.4,16x,el0.4)
106  format(75('*'),//10x,'Mass flux To/From Groundwater'//5x,Time',
   >9x, Total Mass-flux',4x,'Advective flux',4x,'Dispersive flux'/5x,
   >'(yr)', 12x,'(ug/cmA2/d)'/)
107  format(3x,el0.4,8x,el0.3,8x,el0.3,10x,el0.3)
108  format(/6x,'Depth',9x,'C_soil',9x,'C_solids',9x,'C_air',8x,
   >'C_water'/,7x,'(cm)',9x,'(ug/kg)',7x,'(ug/kg)',9x,'(ug/cmA3)',
   >5x,'(ug/cmA3)'/)
109  format(4x,e9.3,5x,el0.3,5x,el0.3,5x,el0.3,5x,el0.3)
110  forrmt(75('*'),/7x,Time', 10x,'Flux to Atmosphere'/,7x,'(yr)',
   >13x,'(ug/cmA2/d)'/)
111  format(3x,el0.4,12x,el0.3)
112  format(///5x,20('*'),3x,'Summary of Mass Balance',3x,20('*')/3x,
   >'(Note: The mass units are - ug/cmA2)'///5x,Time(yr)',5x,
   >'M_remaining',7x, 'M_Atm', 11 x, 'M_Gw', 13 x, 'M_error'/)
113  format(3x,e9.3,7x,el0.4,6x,el0.4,4x,el0.4,8x,el0.4)
114  format(///5x,20('*'),3x,'Summary of Mass Balance',3x,20('*')/3x,
   >'(Note: The mass units are - ug/cmA2)'///5x,Time(yr)',5x,
   >'M_remaining',7x,'M_Atm',llx,'M_Gw',13x,'M_decay'/)
   stop
   end
c
   double precision FUNCTION rtsafe(funcd,xl,x2,xacc)


                                                   386

-------
    implicit real*8 (a-h,o-z)
    EXTERNAL funcd
    PARAMETER (MAXIT=100)
    callfuncd(xl,fl,df)
    call funcd(x2,fh,df)
    if((fl.gt.0..and.fh.gt.0.).or.(fl.lt.0..and.fh.lt.0.))pause
   *'root must be bracketed in rtsafe'
    if(fl.eq.0.)then
     rtsafe=xl
     return
    else if(fh.eq.0.)then
     rtsafe=x2
     return
    else if(fl.lt.0.)then
     xl=xl
     xh=x2
    else
     xh=xl
     xl=x2
    endif
    rtsafe=.5*(xl+x2)
    dxold=dabs(x2-xl)
    dx=dxold
    call funcd(rtsafe,f,df)
    do llj=l,MAXIT
     if(((rtsafe-xh)*df-f)*((rtsafe-xl)*df-f).ge.O..or. dabs(2.
   *f).gt.dabs(dxold*df)) then
      dxold=dx
      dx=0.5*(xh-xl)
      rtsafe=xl+dx
      if(xl. eq. rtsafe)return
     else
      dxold=dx
      dx=f/df
      temp=rtsafe
      rtsafe=rtsafe-dx
      if(temp.eq.rtsafe)return
     endif
     if(dabs(dx).lt.xacc) return
     call funcd(rtsafe,f,df)
     if(f.lt.O.) then
      xl=rtsafe
     else
      xh=rtsafe
     endif
11   continue
    pause  'rtsafe exceeding maximum iterations'
    return
    END
c
    subroutine funcd(x,f,df)
    implicit real*8 (a-h,o-z)
                                                     387

-------
   common/cmain/a
   f=x*dcos(x)+a*dsin(x)
   df=dcos(x)-x*dsin(x)+a*dcos(x)
   return
   end
c
   subroutine gwflux(tt,gwflx,advflx,dspflx)
   implicit real*8 (a-h,o-z)
   dimension an(1000),eign(1000),tgw(500),gwcon(500)
   real l,infil
   common/cflux/an,eign,biglamb,tgw,gwcon,r,d,infil,l,ngw
   common/cmain/a
   advflx2=0.
   advflx3=0.
   dspflxl=0.
   dspflx2=0.
   dspflx3=0.
   advflx=0.
   dspflx=0.
   gwflx=0.
   delta=dsqrt(a*a+biglamb)
   if(tt.eq.0.)goto73
   sum8=0.
   n8=0
   fact=l.
19  continue
   n8=n8+l
   if(n8.gt.lOOO)goto20
   fact=-fact
   wn8sq=biglamb+a*a+(eign(n8)) * *2
   if(ngw. ge .0)term=an(n8) *dexp(-wn8sq*tt) *eign(n8) *fact
   if(ngw.lt.0)term=an(n8)*dexp(-wn8sq*tt)*dsin(eign(n8))
   sum8=sum8+term
   if(abs(sum8).le.l.e-14)goto 20
   if(abs(term/sum8).gt.l.e-07)goto 19
20  continue
   dspflxl=-sum8*dexp(a)*d/(r*l)
   if(ngw.lt.0)then
   advflx=infil*dexp(a)*sum8/r
   gwflx=advflx+dspflx
   elseif(ngw.eq.0)then
   dspflx=dspflxl
   gwflx=advflx+dspflx
   endif
   if(ngw.le.O) goto 73
   n9=0
   sum9=0.
21  continue
   n9=n9+l
   if(n9.gt.lOOO)goto22
   wn9sq=biglamb+a*a+(eign(n9)) * *2
   term=(eign(n9))**2/wn9sq*dexp(-wn9sq*tt)
                                                 388

-------
   sum9=sum9+term
   if(abs(sum9).le.l.e-14)goto 22
   if(abs(term/sum9).gt.l.e-07)goto 21
22  continue
   advflx2=gwcon( 1) *infil/r
   dspflx2=-gwcon(l)*d/(r*l)*(a+delta*dcosh(delta)/dsinh(delta)+2.*
   >sum9)
   do 23 k=l,ngw
   ttgw=d*tgw(k)/(r*l*l)
   if(ttgw.ge.tt)goto 23
   nlO=0
   sumlO=0.
24  continue
   nlO=nlO+l
   if(n!0.gt.lOOO)goto25
   wnl Osq=biglamb+a*a+(eign(nl 0)) * *2
   term=(eign(nlO))**2/wnlOsq*dexp(-wnlOsq*(tt-ttgw))
   suml 0=suml 0+term
   if(abs(sumlO).le.l.e-14)goto 25
   if(abs(term/sumlO).gt.l.e-07)goto 24
25  continue
   advflxS =advflx3 +(gwcon(k+1 )-gwcon(k)) *infil/r
   dspflx3=dspflx3-d/(r*l)*(gwcon(k+l)-gwcon(k))*(a+delta*dcosh(delta
   >)/dsinh(delta)+2.*sumlO)
23  continue
   advflx=advflx2+advflx3
   dspflx=dspflx 1 +dspflx2+dspflx3
   gwflx=advflx+dspflx
73  continue
   return
   end
c
   subroutine atmflux(tt,atmflx)
   implicit real*8 (a-h,o-z)
   dimension an(1000),eign(1000),tgw(500),gwcon(500)
   real l,infil
   common/cflux/an,eign,biglamb,tgw,gwcon,r,d,infil,l,ngw
   common/cmain/a
   atmflxl=0.
   atmflx2=0.
   atmflx3=0.
   if(tt.eq.0.)goto 74
   delta=dsqrt(a*a+biglamb)
   sum!2=0.
   n!2=0
51  continue
   nl2=n!2+l
   if(n!2.gt.lOOO)goto52
   wnl 2sq=biglamb+a*a+(eign(nl 2)) * *2
   term=an(nl2)*dexp(-wnl2sq*tt)*eign(nl2)
   suml 2=suml 2+term
   if(abs(suml2).le.l.e-14)goto 52
                                                 389

-------
   if(abs(term/suml2).gt.l.e-07)goto 51
52  continue
   atmflxl=-suml2*d/(r*l)
   if(ngw.le.0.)goto 74
   n!3=0
   sum!3=0.
   fact=l.
53  continue
   nl3=n!3+l
   fact=-fact
   if(n!3.gt.lOOO)goto54
   wnl 3 sq=biglamb+a*a+(eign(nl 3)) * *2
   term=fact*(eign(nl 3)) * *2/wnl 3 sq*dexp(-wnl 3 sq*tt)
   suml3=suml3+term
   if(abs(suml3).le.l.e-14)goto 54
   if(abs(term/suml3).gt.l.e-07)goto 53
54  continue
   atmflx2=-d/(r*l)*gwcon(l)*dexp(-a)*(delta/dsinh(delta)+2.*suml3)
   do 55 k=l,ngw
   ttgw=d*tgw(k)/(r*l*l)
   if(ttgw.ge.tt)goto 55
   n!4=0
   sum!4=0.
   fact=l
56  continue
   nl4=n!4+l
   fact=-fact
   if(n!4.gt.lOOO)goto57
   wnl 4sq=biglamb+a*a+(eign(nl 4)) * *2
   term=fact*(eign(nl4))**2/wnl4sq*dexp(-wnl4sq*(tt-ttgw))
   suml 4=suml 4+term
   if(abs(suml4).le.l.e-14)goto 57
   if(abs(term/suml4).gt.l.e-07)goto 56
57  continue
   atmflxS=atmflx3 -d/(r*l) *(gwcon(k+1 )-gwcon(k)) *dexp(-a) *(delta/
   >dsinh(delta)+2. *sum!4)
55  continue
74  continue
   atmflx=atmflx 1 +atmflx2+atmflx3
   return
   end
c
   subroutine gwmass(tt,cgwm)
   implicit real*8 (a-h,o-z)
   dimension an(1000),eign(1000),tgw(500),gwcon(500)
   real l,infil
   common/cflux/an,eign,biglamb,tgw,gwcon,r,d,infil,l,ngw
   common/cmain/a
   delta=dsqrt(a*a+biglamb)
   if(tt.eq.0.)goto73
   sum21=0.
   n8=0
                                                 390

-------
   fact=l.
19  continue
   n8=n8+l
   if(n8.gt.lOOO)goto20
   fact=-fact
   wn8sq=biglamb+a*a+(eign(n8)) * *2
   if(ngw.ge.0)cmtrm=an(n8)*eign(n8)*fact*(l.-dexp(-wn8sq*tt))/wn8sq
   if(ngw.lt.0)cmtrm=an(n8)*dsin(eign(n8))*(l.-dexp(-wn8sq*tt))/wn8sq
   sum21=sum21 +cmtrm
   if(abs(sum21).le.l.e-14)goto 20
   if(abs(cmtrm/sum21).gt.l.e-07)goto 19
20  continue
   if(ngw.lt.0)cgwm=infil*dexp(a)*sum21/r
   if(ngw.eq.0)cgwm=-sum21 *dexp(a)*d/(r*l)
   if(ngw.le.O) goto 73
   cmgwl=-dexp(a)*sum21 *d/(r*l)
   n9=0
   sum22=0.
21  continue
   n9=n9+l
   if(n9.gt.lOOO)goto22
   wn9sq=biglamb+a*a+(eign(n9)) * *2
   cmtrm=(eign(n9))**2*(l.-dexp(-wn9sq*tt))/wn9sq**2
   sum22=sum22+cmtrm
   if(abs(sum22).le.l.e-14)goto 22
   if(abs(cmtrm/sum22).gt.l.e-07)goto 21
22  continue
   cmgw2=gwcon(l)/r*(infil*tt-d/l*((a+delta*dcosh(delta)/dsinh(delta)
   >)*tt+2.*sum22))
   cmgw3=0.
   do 23 k=l,ngw
   ttgw=d*tgw(k)/(r*l*l)
   if(ttgw.ge.tt)goto 23
   nlO=0
   sum23=0.
24  continue
   nlO=nlO+l
   if(n!0.gt.lOOO)goto25
   wnl Osq=biglamb+a*a+(eign(nl 0)) * *2
   cmtrm=(eign(nlO))**2*(l.-dexp(-wnlOsq*(tt-ttgw)))/wnlOsq**2
   sum23=sum23+cmtrm
   if(abs(sum23).le.l.e-14)goto 25
   if(abs(cmtrm/sum23).gt.l.e-07)goto 24
25  continue
   cmgw3=cmgw3+(gwcon(k+l)-gwcon(k))/r*(infil*(tt-ttgw)-d/l*((a+delta
   >*dcosh(delta)/dsinh(delta))*(tt-ttgw)+2.*sum23))
23  continue
   cgwm=cmgw 1 +cmgw2+cmgw3
73  continue
   return
   end
                                                 391

-------
   subroutine atmass(tt,catm)
   implicit real*8 (a-h,o-z)
   dimension an(1000),eign(1000),tgw(500),gwcon(500)
   real l,infil
   common/cflux/an,eign,biglamb,tgw,gwcon,r,d,infil,l,ngw
   common/cmain/a
   if(tt.eq.0.)goto 74
   delta=dsqrt(a*a+biglamb)
   sum24=0.
   n!2=0
51  continue
   nl2=n!2+l
   if(n!2.gt.lOOO)goto52
   wnl 2sq=biglamb+a*a+(eign(nl 2)) * *2
   cmtrm=an(nl2)*eign(nl2)*(l.-dexp(-wnl2sq*tt))/wnl2sq
   sum24=sum24+cmtrm
   if(abs(sum24).le.l.e-14)goto 52
   if(abs(cmtrm/sum24).gt.l.e-07)goto 51
52  continue
   catm=-sum24*d/(r*l)
   if(ngw.le.0.)goto 74
   n!3=0
   sum25=0.
   fact=l.
53  continue
   nl3=n!3+l
   fact=-fact
   if(nl3.gt.lOOO)goto54
   wnl 3 sq=biglamb+a*a+(eign(nl 3)) * *2
   cmtrm=fact*(eign(nl3))**2*(l.-dexp(-wnl3sq*tt))/wnl3sq**2
   sum25=sum25+cmtrm
   if(abs(sum25).le.l.e-14)goto 54
   if(abs(cmtrm/sum25).gt.l.e-07)goto 53
54  continue
   cmatml=-d/(r*l)*gwcon(l)*dexp(-a)*(delta/dsinh(delta)*tt+2.*sum25)
   cmatm2=0.
   do 55 k=l,ngw
   ttgw=d*tgw(k)/(r*l*l)
   if(ttgw.ge.tt)goto 55
   n!4=0
   sum26=0.
   fact=l
56  continue
   nl4=n!4+l
   fact=-fact
   if(n!4.gt.lOOO)goto57
   wnl 4sq=biglamb+a*a+(eign(nl 4)) * *2
   cmtrm=fact*(eign(nl4))**2*(l.-dexp(-wnl4sq*(tt-ttgw)))/wnl4sq**2
   sum26=sum26+cmtrm
   if(abs(sum26).le.l.e-14)goto 57
   if(abs(cmtrm/sum26).gt.l.e-07)goto 56
57  continue
                                                 392

-------
   cmatm2=cmatm2-d/(r*l)*(gwcon(k+l)-gwcon(k))*dexp(-a)*(delta/dsinh
   >(delta)*(tt-ttgw)+2.*sum26)
55   continue
   catm=catm+cmatml+cmatm2
74   continue
   return
   end
    subroutine getlen(string,chr,nO,nl)
c
c   to determine the length of a string excluding any blank padding & heading
c   and return string(nO:nl) nO... position of first non-blank letter
c                  nl... position of last non-blank letter
c   else return the original string
c   ***Use | to keep blank space at the begining or the end of the
c    *** string
c
    character*(*) string
    character* 160 stringl
    character*! chr
    stringl=''
    inO=0
    inl=0
    do 10 i=len(string),l,-l
    if(string(i:i).ne.chr) then
     if(inl.eq.O) then
     if(string(i:i).eq.'|') then
     nl=i-l
     else
     nl=i
     endif
     inl=l
     endif
     goto 11
    endif
10   continue
11   do20i=l,len(string)
    if(string(i:i).ne.chr) then
    if(in0.eq.O) then
    if(string(i:i).eq.'') then
    nO=i+l
    else
    nO=i
    endif
    inO=l
    nk=nO-l
c    write(6,*) T//string(nO:nl)//T, nO,nl
    do 30 k=nO,nl
    if(string(k:k).ne.'|') then
     nk=nk+l
     if(k.eq.nO) then
                                                    393

-------
      string 1=string(k:k)
     else
      stringl=stringl(l :nk-nO)//string(k:k)
     endif
    endif
c   write(6,*) T//stringl(l:nk-nO)//T
30   continue
    string=stringl
    nl=nl-nO+l
    nO=l
    endif
    return
   endif
20  continue
   nO=-l
   nl=-l
   return
   end
                                                  394

-------