xvEPA
           United States
           Environmental Protection
           Agency
Office of Solid Waste
and Emergency Response
Washington DC 20460
July 1986
           Solid Watte
           Criteria for Identifying
           Areas of Vulnerable
           Hydrogeology Under the
           Resource Conservation
           and Recovery Act

           Appendix C

           Technical Methods for
           Calculating
           Time of Travel in the
           Unsaturated  Zone
           Interim Final

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                                       OSWER POLICY DIRECTIVE NO.

                                      9472-00-2A
     GUIDANCE CRITERIA FOR IDENTIFYING
      AREAS OF VULNERABLE HYDROGEOLOGY
                 APPENDIX C
 TECHNICAL GUIDANCE MANUAL FOR CALCULATING
TIME OF TRAVEL (TOT) IN THE UNSATURATED ZONE
           Office of Solid Waste
          Waste Management Division
    U.S. Environmental Protection Agency
             401 M Street, S.W.
          Washington, D.C.  20460
                Prepared by

    Batelle Project Management Division
    Office of Hazardous Waste Management
           601 Williams Boulevard
            Richland, WA   99352

            Under Subcontract to

              GCA CORPORATION
          GCA/TECHNOLOGY DIVISION
       Bedford, Massachusetts   01730
                 July 1986

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                                                            OSWER POLICY DIRECTIVE NO.


                                                          9472•00-2A  *
                                   DISCLAIMER
     This Final Report was furnished to the Environmental  Protection Agency
by the GCA Corporation, GCA/Technology Division,  Bedford,  Massachusetts 01730,
in fulfillment of Contract No.  68-01-6871,  Work Assignment No.  20.  The
opinions, findings, and conclusions  expressed are those of the  authors and not
necessarily those of the Environmental  Protection Agency or the cooperating
agencies.  Mention of company or product names is not to be considered as an
endorsement by the Environmental Protection Agency.
                                       ii

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                                                            USWtK KULiuY Ul
                                                          9472- 00"  2A
                                  EXECUTIVE SUMNBMQT
     This appendix to the Guidance Criteria for Identifying Areas of Vulnerable
Hydrogeology describes methods for calculating ground-water time of travel (TOT)
in the unsaturated zone.  The methods described in this appendix are intended for
use by hazardous waste facility permit applicants and writers in evaluating the
vulnerability of ground water to contamination.
     The appendix presents a review of the general theory of ground-water flow in
the unsaturated zone and describes the processes that control flow.  Equations are
presented which describe these processes and illustrate the relationships between
important parameters.
     Two general approaches are presented for calculating unsaturated zone TOT.  The
first approach involves the use of analytical solutions.  These solutions are simp-
lified approaches, appropriate for simple systems, and allow the analyst to directly
                        y
solve for TOT.  Two solutions are described, both for determination of steady state
TOT.  The first solution assumes a constant moisture content in the soil profile and
is appropriate for conditions dominated by gravity drainage.  The second solution
allows for variable moisture contents and is appropriate for conditions where
factors other than gravity drainage (e.g., capillary forces) are important.   The use
of these solutions is described and data requirements and sources of data are
identified.
     The second approach involves the use of unsaturated flow models.  Two general
classes of models are described,  numerical models and water balance models.   The
relative complexity of each type of model is described, as are data requirements,
output, and limitations.  Methods are presented for determining TOT from model
output for those models where TOT is not directly calculated by the model.

                                        iii

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     me approaches to  calculating  TOT are summarized  and  a decision tree is
presented to aid  in  selection of the  most appropriate  approach  for specific
applications.   Three  case  histories  are  presented using  data  from  actual
hazardous waste facilities to illustrate calculation of TOT.
                                      IV

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                                   CONTENTS

Executive Summary   .    .    .    .    .    .     .     •     •     •     •  iii
Figures   .    .    .    .    •    •    .    .     •     •     •     •     •   vi
Tables    ............. viii
Acknowledgements    ...........   ix
     1.   Introduction	1.1
     2.   Technical Background on Unsaturated Flow      .     .     .     .2.1
     3.   Technical Approaches for Determining TOT      ....  3.1
               Analytical Solution of TOT	3.1
               Modeling Solutions of TOT	3.3
               Selection of the Appropriate Method to
                 Determine TOT	3.11
     4.   Analytical Solutions of TOT	4.1
               Data Requirements and Sources .     .     .     .     ..     .4.1
               Description of Analytical Solutions      .     .     .     .  4.4
     5.   Unsaturated^'low Models	5.1
               Introduction   .........  5.1
               Example Numerical Code - UNSAT1D    	  5.7
               Example Water Balance Code  - HELP   .     .     .     .     .5.14
     6.   Examples of TOT Determination .......  6.1
               Example 1 ..........  6.1
               Example 2 ..........  6.5
               Example 3 ..........  6.11
     7.   References     .    .    .    .    .     .     .     .     .     .7.1
Glossary  .............  G.I

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                                   FIGURES

Number                                                                   Page

2.0-1     Graph of Moisture Content Versus Suction Head      .     .     .2.4

2.0-2     Graph of Hydraulic Conductivity Versus
            Moisture Content  	  2.5

3.2-1     Cross-Section View of the Node and Element
            Discretization of an Unsaturated Flow System
            Beneath a Waste Disposal. Facility	3.5

3.2-2     Cross-Section View of the Layer Representation of
            an Unsaturated Flow system for Use in a Typical
            Water Balance Model	3.6

3.2-3     Example Model Results Showing Migration
            of Wetting Front with Time  .    .    .     .     .     .     .  3.9

3.3-1     Summary of Selection of Approach for TOT Determination  .     .  3.12

4.2-1     Schematic of Example Single-Layered System    ....  4.7

4.2-2     Schematic of Example Multi-Layered System     .                 4.10
                       v
4.2.2-1   Discretization Between Grid Points 	  4.13

5.2-la    Moisture Profile for a Three Layer
            Unsaturated Flow System     	  5.12

5.2-lb    Cumulative Water Flux Versus Time Past Several
            Elevations in an Unsaturated Flow System    .     .     .     .5.12

6.2-1     Plot of Suction Head Versus Moisture
            Content for Case Study D    .    .    .     .     .     .     .6.7

6.2.2-1   Plot of Hydraulic Conductivity Versus Suction
            Head for Case Study D	6.9

6.2.2-2   Steady State Moisture Profile from Iterative
            Analytical Solution for Case Study D  .     .     .     .     .  6.10

6.3.1-1   Soil Water Characteristic Curve for Soil
            at Proposed Hanford Hazardous Waste Site         .     .     .6.13
                                      vi

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                              FIGURES (Cont'd.)               7^'OQ

Number                                                 .                  Page

6.3.1-2   Hydraulic Conductivity Versus Water Content
            Curve for Soil at Proposed Hanford
            Hazardous Waste Site	6.14

6.3.1-3   Simulated Pressure Head Versus Depth with
            Time at Proposed Hanford Hazardous Waste Site    .     .     .  6.15

6.3.2-1   Simulated Advance of Wetting Front at Proposed
            Hanford Hazardous Waste Site	.6.16

6.3.2-2   Simulated Rate of Leachate Discharge at Proposed
            Hanford Hazardous Waste Site     	  6.18

6.3.2-3   Simulated Cumulative Leachate Discharge at
            Proposed Hanford Hazardous Waste Site .....  6.19
                                     vn

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                                    TABLES

Number                                                                   Page

3.0-1     Relative Characteristics of Analytical
            Solutions and Unsaturated Flow Modeling    .    .    .    .3.1

4.1-1     Representative Values for Saturated  _
            Hydraulic Conductivity .    .    .    .....  4.3

4.1-2     Representative Values for Porosity ......  4.4

5.1-1     Summary of Unsaturated Zone Codes  ......  5.2

5.1-2     Sumnary of Unsaturated Code Documentation
            and Availability  .    .    .    .    .    .    .            5.6

5.2-1     Important Characteristics and Capabilities
            of UNSAT1D   ..........  5.8

5.2-2     Summary of UNSAT1D Input Data and Sources    .    .    .  •  .  5.10

5.3-1     Summary of Characteristics and
            Capabilities of HELP   ........  5.15
5.3.1-1   Summary of Irvut Data and Sources for HELP   ....  5.17

5.3.1-2   Summary of Output Data from HELP    ......  5.19

6.1.1-1   Typical Values for Slope of Soil Moisture
            Retention Curve   .........  6.4
                                     viii

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                                   ACKNOWLEDGEMENTS








      This appendix to the Guidance Criteria for Identifying Areas of Vulnerable



 Hydrogeology was prepared by Batelle's Project Management Division,  Office of



 Hazardous Waste Management under subcontract to GCA Corporation/  GCA/Technology



 Division.



      The appendix was written by Frederick W.  Bond,  James M. Doesburg, C.  Joseph



 English,  and Deborah J.  Stallings.  The authors wish to thank Alfred Leonard,



 Charles Young,  and Bob Farrell of GCA,  and Glen Galen of EPA's Office of Solid



 Waste,  for their assistance in the preparation of this appendix.   Special thanks



 are also extended to Peggy Monter, Nancy Painter,  Beth Eddy, and Dick Parkhurst,  of



'Battelle, for their typing and graphics assistance.
                                          IX

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                                 SECTION 1.0
                                 INTRODUCTION

     Subtitle C of the  Resource  Conservation  and Recovery Act  of 1976 (RCRA)
addresses  the  management  and  disposal   of  hazardous  waste.    Regulations
developed by  the  Environmental Protection Agency (EPA) from  RCRA legislation
(Regulations  for  Owners   and   Operators   of   Permitted   Hazardous   Waste
Facilities -  40 CFR  264 and Regulations for  Federally  Administered  Hazardous
Waste  Facilities  -  40 CFR  270)  require  owners and  operators  of  hazardous
waste  land   treatment,  storage,  and  disposal   (TSD)  facilities  to  provide
information  concerning  the  design,  construction, operation,'  and maintenance
of these facilities.   This information  is  provided in the  form of a RCRA Part
B  permit application.    Permit writers   in  the EPA  Regions  and  authorized
states  responsible   for  writing  permits  must   review  this   information  to
determine  if  the facility  will  meet  the  environmental  protection  goals
established in the RCRA regulations.
                      v
     A  major emphasis  of the above  environmental  protection  goals  is  the
protection of ground-water  resources that may be vulnerable  to contamination
originating  at  TSD  facilities  located   in  certain hydrogeologic  settings.
Therefore, much  of  the  permit application and  review process  addresses  the
adequacy  of  facility   design,   construction,   operation,  maintenance,  and
location with  respect  to ground-water  protection.   A considerable  amount of
guidance  has been developed  to  aid permit  writers in evaluating  potential
threats  to  ground water  posed  by  TSD  facilities.    This appendix  presents
methods  available for  predicting  ground-water  time of travel (TOT)  in  the
unsaturated zone.
     The intent of RCRA guidance and regulations is  to ensure that facilities
are  designed,  operated,   and  located  such  that  there   will   be  negligible
migration of  contamination  beyond  the barriers  of the  facility.  Assuming no
release,  the velocity  of  contaminant migration  to  the ground water beneath

                                     1.1

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the  site  is  of  little  or  no  importance.    The  regulations,  however,  also
recognize the  possiblity of  contaminant release  due to  failure  of facility
barriers.    The  contaminant   time  of  travel  to  the  ground  water  and  the
thickness of  the unsaturated  zone  then become  important  issues in assessing
the potential consequences of  site failure.
     Consideration of  unsaturated TOT  may  be important for  several reasons.
With  the  exception  of certain  hydrogeologic settings,  all  components  of a
facility permitted by  RCRA will  be  located  above the water table.   Therefore,
any  release  from a facility will migrate through  the unsaturated zone before
reaching the  water table.   The  TOT  through the unsatuated zone is important
in determining  how rapidly,  and to  what extent,  ground-water  resources will
be impacted  by  contaminant  release.   Consideration  of  unsaturated  TOT may be
necessary  to   determine   appropriate  monitoring   strategies  for  detecting
failures as well  as for developing appropriate corrective  actions.
     In some  locations, particularly those  characterized by arid climates  and
deep unsaturated  zones, flow  through  the unsaturated zone  may be minimal.  In
such cases,  the unsaturated zone may effectively  form  a buffer zone to  delay
contaminants  when  migrating  to  the  water  table.    Recognition of   such
conditions is important in assessing the adequacy of  a facility design.
     Calculation  of  TOT  in  the  unsaturated zone   is  not  a   trivial   task.
                                                                  »*
Simple  graphical  methods  useful for  estimating  TOT  in  the  saturated  zone
(e.g.,  construction  of flow  nets)  are not applicable  due  to nonlinearities
associated with unsaturated flow.   Solution of unsaturated  TOT must instead
rely  on  analytical   and   numerical  methods.    Available  techniques   vary
significantly with respect  to mathematical  complexity  and data requirements.
Selection of an  appropriate  technique must  consider  the objectives  of  the
application and  the availability of time, resources,  and data.
     The  purpose of  this   appendix  is  to  acquaint  permit  writers   and
applicants  with  the  techniques  available  for  determination of  TOT  in  the
unsaturated  zone.  The appendix  is  divided into six sections.   This section
provides an  introduction  to  the need  for  determination of TOT.   Section  2.0
presents technical background  material  regarding unsaturated  flow.   A general
discussion of  the two technical  approaches for  TOT determination,  analytical
methods,  and   unsaturated   flow  models   is  provided   in   Section    3.0.

                                     1.2

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Sections 4.0  and  5.0  discuss  these  approaches  in  more detail,  and example
determinations of TOT are presented in Section 6.0.
                                      1.3

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                                  SECTION 2
                   TECHNICAL  BACKGROUND  ON  UNSATURATED FLOW

     The unsaturated zone  is  the  transition  region  between the atmosphere and-
the  saturated  ground-water  system.   Passage  of water  through this  zone  is
very dynamic  and depends on  detailed  variations in  the  hydraulic properties
of the water in the soil.
     Rainfall, irrigation, and  ponded  water are the  primary  sources  of water
to the  unsaturated  zone.   Redistribution  or downward movement of this water
through the soil  occurs  under the influence of  gravity  as long as there is a
sufficient quantity  present  to overcome the restraining  forces  of capillary
hydraulic potential.
     Water  is  removed  from the  surface  of  the   unsaturated  zone   by  the
processes of  evaporation  and/or transpiration.   The rates of  both processes
depend  directly  on  available  solar  energy and surface  winds.    Water also
moves out  the  bottom  «/  the  unsaturated zone as  drainage if  the soil-water
holding capacity is exceeded.    Drained  water  may possibly enter  the water
table depending on its depth.
     Water can also move  within  and  be  stored  in the unsaturated  zone.  Water
storage  is characterized by a  water  content  distribution.     Water  moves
through and  within  a  soil  via  two physical  mechanisms:   capillary Darcian
flow  (liquid   phase)   and  vapor  diffusion.    Darcian  flow  is described  by
hydraulic  conductivity and  matric  potential  gradients,  both  of which  are
highly  dependent  on  moisture   content.     Vapor   diffusion  controls   actual
surface evaporation and results from thermal gradients.
     A  soil  is  saturated  when  all  void  space  (space  not occupied  by solid
particles)  is  filled  with water.   An  unsaturated soil  contains air-filled
void space as well as water.   A measure  of  the quantity of water contained by
a  soil  is called  the   water  content  which  can  be  defined  either on  a
volumetric basis  (volume  of  water/total  volume of  soil,  water,  and voids) or
a mass basis (mass of water/mass of soil solids).
                                     2.1

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     Movement of water  in the unsaturated  zone  is always directed from  areas
of  higher   to  those   of  lower   potential   energy   (assuming   isothermal
conditions).   Total  soil  water potential  ($)  is  expressed as  (Feedes et  al..
1978):
where
    ^  =  the pneumatic potential arising from changes  in external pressure
    ^s =  osmotic  potential  arising from the attraction forces  of water to a
          higher solute concentration
    'J'  =  the matric  potential  arising  from capillary and adsorptive  forces
          of the soil matrix
    ^  =  the  gravitational  potential  expressing  the  potential  energy   of
          changes  in relative elevation changes.
     The  negative  of  the gradient  of total  potential is  the  force  causing
water movement in  a soil.
     The  gravitation potential  is  an important  component of the driving force
of  water downward through  the  unsaturated  zone below a  TSD  facility.  The
gravitational  potential   of  soil  water  at  each point is  determined  by  the
elevation  of the  poi'v-  relative  to some  arbitrary  reference  level.   The
matric  (or capillary)  potential is  a negative pressure  potential  resulting
from the adsorptive  forces of  the  soil  matrix.   The matric  potential  can  be
an  important  factor,  particularly  in   dry soils.    The   influence   of  the
pneumatic  and osmotic  potential is  almost always  quite small and, therefore,
they can  be disregarded.
     The  relation  between matric  potential  and soil  wetness (water content)
is  not  generally  a  unique  one  due to  a  phenomena  known  as  hysteresis.
Hysteresis  is  the  phenomenon where the  water content  of  a soil with  a given
matric potential  can be  different  depending on  whether the  soil is  wetting
(sorption)  or drying  (desorption) .   The  equilibrium water content at  a given
suction  is greater in  desorption than in sorption.  The hysteresis effect can
be   attributed   to  several   causes  which   include:      1)  the  "geometric
nonuniformity  of  individual  pores;   2)  entrapped  air;  and  3)  swelling   or

                                      2.2

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shrinking  phenomena.    Typically,  the  hysteresis  effect  is  small  and  is,
therefore, disregarded in the determination of TOT (Hillel, 1971).
     Movement of  water through  a porous medium  is  proportional  to  both the
hydraulic  conductivity of  the  medium  and  the  hydraulic  potential  gradient
across the medium.   The hydraulic conductivity,  K,  represents the ability of
a  soil  to transmit  water  from  locations  of  high  hydraulic  potential  to
locations  of  low  hydraulic  potential.    In  cases  of   unsaturated  flows,
hydraulic  conductivity  is  a function  of moisture  content (e), such that K can
be  represented  as K(e).   The functional relationship of  K(e) will  vary from
soil  to   soil.   Typically,  K(e)  will decrease rapidly  by several  orders  of
magnitude  from its maximum  saturated  value as water content decreases.
     As  stated  above,  the  moisture  content  can  vary within  the unsaturated
flow system.  As  the moisture content  changes, the  matric potential (suction
head or  negative  pressure  head)  and  the hydraulic  conductivity also change.
In  order  to  simulate  water  movement   in   the   unsaturated  zone,  these
relationships  between  moisture  content  and matric  potential,  and  moisture
content  (or matric  potential)  and hydraulic conductivity (soil  characteristic
curves)  must  be  known.   The  only exception to  this  is  for  cases  where the
moisture  content,  and therefore  the  matric head and  hydraulic  conductivity,
remain constant throuc^out  the  unsaturated  soil profile (unit gradient case).
Example  graphs  of  these two  relationships are  shown  in  Figures  2.0-1  and
2.0-2.
     Most  often   the  moisture  content,  matric  potential,   and  hydraulic
conductivity  relationships  are  not available for the  soils  of  interest  at  a
particular  site.     If  these  data  are not  available,  the  best  means  of
obtaining  site-specific  values  is  through  laboratory  measurements.    Field
measurements  can  be made,  but unlike  those  for  saturated  soils,  they are
typically  not as  accurate  and reliable as laboratory tests.
      The moisture  content  versus pressure  head  relationship can be  measured
 relatively easily,  whereas  methods  for  direct determination  of  hydraulic
 conductivity (K)   as  a function  of  moisture content  (9)  or  matric  potential
 (*m)  over the unsaturated  range of interest  are  experimentally difficult.   As
 a  result, the  K  versus  e  or  «/m  relationship  is  often   calculated  using
 analytical methods such as those presented by  Mualem  (1976a), Burdine  (1953),

                                      2.3

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       0.4000
       0.3000
 c
 OJ
4-1
 c
 o
CJ
0.2000
      0.1000
      0.0000
                                                   io5    io6    io7
                                 Suction  (cm)
     Figure 2.0-1.  Graph of Moisture Content Versus Suction
                                                       Head
                            2.4

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C

'i


u


>.
•4-»

>

4->
U
3
T>
C
O


(J
              10°
              io:2
              io:3
              io:4
              io:5
              io:6
              io:7
              io:8
              io:10
              io:11
              io:12
              io:13
              io:14
              io:15
              io:17-
               0.0000
                      0.1000
0.2000
0.3000
0.4000
                                     Moisture  Content
Figure 2.0-2.  Graph of Hydraulic Conductivity  Versus  Moisture  Content

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and Millington and Quirk  (1961).   The details of these analytical methods  are
discussed in Mualem (1976a).
     If  site-specific  data are  not available  and  cannot be  measured in  the
field  or laboratory,  a  last  resort  is  to  obtain  values  for representative
soils  from  the literature.  One  excellent reference  is  Maul em (19765) where
example  soil characteristic curves  for 45  soils for which actual conductivity
measurements  as  well  as  moisture content  as a function  of matric potential
were made.
                                     2.6

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                                 SECTION 3.0

                  TECHNICAL  APPROACHES  FOR  DETERMINING  TOT

     There  are  two  basic technical  approaches  for determining  TOT  in  the
unsaturated zone, analytical solutions  of  TOT,  and unsaturated flow modeling.
Both approaches  are  based on  the  same fundamental  equations, but  differ in
the number of simplifying assumptions made  in  order to  solve these equations.
As  a   result  of   the   simplifying   assumptions,   the   approaches   differ
significantly in  the time necessary to obtain  a solution,  in computational
difficulty, and  in  data  requirements.   Relative  characteristics of  the  two
approaches are summarized in  Table 3.0-1.
            Table  3.0-1.   Relative  Characteristics  of  Analytical
                           Solutions and Unsaturated Flow Modeling •
       Computational Time
           -• ^
       Data Requirements

       Complexity of Solution

       Time Dependency
 Analytical
  Solutions
Short
Low to Medium
Simple
Steady State
                                                       Unsaturated
                                                      Flow Modeling
Medium to Long
Medium to High
Complex
Steady State or Transient
3.1.  ANALYTICAL SOLUTION OF TOT
     The  analytical  solution of  unsaturated flow  TOT  is based  upon Darcy's
equation for one-dimensional flow
                                     3.1

-------
where
         q = flux in the vertical direction
        ^ = matric potential (suction head or negative pressure head)
     K(^) = hydraulic conductivity as a function of matric potential
     3i|/3z = hydraulic gradient  in the vertical direction
The  above  relation  is  identical.to  the relation  for saturated  flow except
that the hydraulic conductivity  is not constant.
     In  unsaturated flow,  both hydraulic  conductivity and  moisture content
are  nonlinear   functions  of   pressure  head.     Pressure  head,  hydraulic
conductivity, and  moisture content need  not be constant  throughout  the soil
column.   When  these variables are not  constant,  a direct analytical solution
of Darcy's equation  is not  possible for  unsaturated flow.   In order to obtain
an  approximate  solution,  simplifying  assumptions  must   be made.    Common
assumptions are:
•    one-dimensional flow in the vertical direction;
•    water flow  is steady state;
•    water table conditions exist at the lower boundary;
t    the upper boundary condition is constant flux;
•    soil  characteristics  (moisture  content  versus  matric  potential  and
     hydraulic conductivity versus matric  potential)  are constant with depth;
     and
                       y
•    the  hydraulic  gradient is  vertically down and equals unity (drainage  is
                                 3(k_
     due strictly to gravity, or —- = o).
                                  3^
     For  nonhomogeneous  soils, the  constant  property  assumption  can   be
approximated by  dividing  the  soil profile  into  a  series of layers, each layer
comprised of soils having approximately the  same characteristics.
     The  unit gradient  assumption  greatly  simplifies  the  analysis.   This
assumption  means that the  matric potential  and,  therefore,  moisture content
and  hydraulic  conductivity are  constant  with depth.   Using  this assumption,
it  is  possible  to  directly solve for moisture  content in terms  of  the flux
through  the  system and  saturated  soil  properties.    Knowing  the  moisture
content  and flux it is possible to calculate the pore water  velocity and TOT.
The  unit  gradient  assumption   is  generally  valid  if gravitational  forces
dominate other forces  (e.g., capillary forces).

                                     3.2

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     If the unit  gradient  assumption is not made,  the analytical solution to
unsaturated flow  becomes  more  complex.   In  this  case,  it is  necessary to
employ  an  iterative solution  for  pressure  head  and moisture  content.   This
iterative solution  is  time-consuming,  but  can be simplified through the use
of a computer.
     The  above  solutions  for  TOT  are  one-dimensional   solutions.    When
applying these  solutions  to specific sites,  it is  important to consider the
horizontal variability of  soil  characteristics.  If soil characteristics vary
spatially, the  solution  should be  applied  to  the soil  profile  having the
highest  hydraulic conductivity.   The  solution will  then  yield  the highest
velocity and shortest TOT  (e.g., worst case) for the unsaturated flow system.
     In  summary,  analytical solutions  provide a means  of  quickly estimating
TOT.    Several  assumptions  are required  to  perform  the  solutions. .   These
assumptions and,  therefore, the methods  themselves, are not  appropriate for
all  applications.   A detailed discussion  of two  analytical   approaches  of
calculating TOT are presented in Section 4.0.
3.2  MODELING SOLUTIONS OF TOT
3.2.1  Description of Model Types
     Unsaturated  flow models provide  another means  for determining TOT in the
unsaturated zone.   ThVs  section discusses  two general  types  of unsaturated
flow models and how they can be applied to solve for TOT.
     Two general  types of  unsaturated  flow  models  are  available.   These are
numerical  models  and  water   balance  models.     Numerical   models   solve
differential  equations describing water movement  within the unsaturated  zone.
These  equations  are  derived  by combining  mathematical  statements  for the
conservation of mass  and  energy with  equations which   have  been developed to
relate   these   statements   to   measurable  quantities  such   as  pressure,
temperature,  and  moisture  content.   Numerical  models  are able  to account for
the  nonlinearities  in soil  properties  and  the  variations of  properties  in
space.   Water balance models,  on  the  other  hand,  simulate unsaturated flow
systems  such  that the flow  into the system  is equal   to the flow out of the
system  plus or  minus storage within  the system for a  specific  area and for a
specific  period  of time.    These  models  generally  require  the  direct  or

                                     3.3

-------
indirect measurement of  soil moisture  and other properties which affect water
movement  within the  unsaturated zone.    Simplifying  assumptions  similar  to
those  used  with  analytical   solutions   are  often  used  with  water balance
models.
     The  two  types of models  represent  very  distinct  levels of complexity  in
their  methods of  solution  and their  required  input data.   Numerical models
typically  require  more data than  water balance models.   Numerical  solutions
use  either   a  finite  difference,   finite  element,   or  integrated  finite
difference  technique,  all of  which  require that  the  unsaturated  flow system
be  represented as  a  series of  nodes  and elements (Figure 3.2-1).   A complex
system may be  represented  in  the models  with  several  hundred nodes, and the
solution  complex  numerical  problems  may require  several  hours of  computer
time.
     Water  balance models,  on  the other  hand,  represent the unsaturated flow
system as  a   series  of  layers  of  geologic  materials   (Figure  3.2-2).     In
typical  applications,  such models  seldom  have  greater  than  ten  layers,  and
the  model input data  defines  the  properties of  each  layer.   Water balance
models use  a  "book-keeping"  approach  to keep track  of water  entering  and
exiting  the system, as well as  water  entering and  exiting  each layer within
the  system.   Water balance models  can  be solved  quite  rapidly by  computers
with solution times usu?  ly on the order  of minutes.
3.2.2   Determination of  TOT from Model Results
     The  equations used  to  develop the solutions used in the unsaturated flow
codes  do  not  have  velocity as  a variable,  nor  is  velocity a  primary output  of
the  models.   Therefore, additional  analysis  of model  results is necessary  to
derive TOT.   The purpose of the following  discussion  is  to present  methods  of
determining TOT from unsaturated model results.
     Four methods  of determining TOT  from model  results are presented below.
One  method deals  with  determining  the  travel  time for a  particle along a
travel path  in  the unsaturated  zone.    Another method deals with determining
the  time  required for  an   instantaneous  loading  to  migrate through  the
unsaturated  zone to the water table.   The other two methods use steady state
solutions  for moisture conditions  or contaminant concentrations.
                                      3.4

-------
                    Ground Surface
00
•

en
                                                  Waste
                                             Disposal Facility
                                                                                              Node

                                                                                            •Element
                                              Water Table
              Figure  3.2-1.
Cross-Section View of the Node and Element Discretization of  an
Unsaturated Flow System Beneath a Waste Disposal Facility

-------
             Ground Surface
CA>
                Water Table
                                                 Waste
                                                  Clay
                                            "V"
                                              Silty Sand
                                                Sand
              Figure 3.2-2.
Cross-Section View of the Layer Representation of an Unsaturated
Flow System for Use in a Typical Water Balance Model

-------
Determination of Particle Travel Time--
     This approach  to  TOT is appropriate for  determining  TOT associated with
any type of  loading  to the  system (steady state  or  transient).   The approach
is  based  upon   Darcy's  Equation  for  unsaturated  flow  (Equation  3-1)  and
requires that the  code be capable of  determining pressure  head  at each node
or layer boundary  for  each  time  step.   As  described by Darcy's Equation, the
velocity  between  two   nodes   (or  through  a  layer)  will   be equal  to  the
hydraulic  conductivity  between  the  two  nodes  times  the  hyraulic gradient
between  the  two  nodes.  These quantities  can  be  directly  determined from
matric  potential.   The  average velocity for the  element  can be converted to
pore  water  velocity  for a  time  step,  by  dividing  by   moisture  content.
Knowing  the  pore  water velocity  it  is  possible  to  determine  how  far  a
particle travels during  that time step.
     This  approach  to  TOT determination  involves tracking  the  position of a
particle present  at the surface  (or water  source) at the  start  of the  first
time  step.    Using  the  average velocity of  the  particle's  position for the
first  time  step,  the   displacement  of  the  particle  for  that  time  step  is
calculated.   After the  second  time  step, the  velocity  at the particle's new
position  is  calculated  and  the particle displaced  a  distance for 'that time
step.   This  process is  repeated until the  particle reaches the water  table.
The time required to reach the  water table is the  unsaturated  TOT.
     It  should  also be  noted that some  unsaturated  flow codes have  ancillary
programs that  will  extract  appropriate  data and  perform  the above  analysis.
If  desired,   it  is  possible  to modify  any  numerical code to  perform the
analysis.
Solution of  Steady-State Travel  Time--
     An  approach similar to  the previous  approach  can be used  to  determine
steady  state travel times through the  unsaturated zone.  A  model  is used to
solve  for  steady state  values  of hydraulic potential,  moisture  content, and
hydrauilc  conductivity  at  each  node.    These   nodal   values   are  used   to
determine  the  steady  state pure  water  velocity between  each pair of  nodes
(i.e.,  across each  element  in  the model).  Elemental velocities are  then used
to  determine travel times  for  each  element.   The  sum of all  the  elemental
travel  times along  a flow path  is the unsaturated TOT.
                                     3.7

-------
     The  above  approach  would be  appropriate for  determining  TOT  in  cases
where  steady state  conditions will  be  encountered (e.g.,  constant  surface
flux conditions).  A limitation  of the approach is  that the  TOT  is valid only
for steady state conditions and  does  not  yield the TOT for the period  leading
up to  steady state conditions.  For  example,  if a  column  of soil at  one  set
of  steady state  moisture conditions  suddenly receives  a new  or additional
constant  flux of  water  at  the  surface,  the  above approach can  be used  to
determine the TOT after new steady state conditions have  been  reached.    The
method will  not  determine the TOT  associated  with water which passed  through
the  soil  column  during the time when  new steady  state  conditions were  being
established.  Therefore,  the method will  not identify the  TOT associated with
the first particle of water to reach the  water table.
TOT Associated With  Instantaneous Loadings--
     The  advantage of  the above  approach is the  ability to determine  TOT  for
any type  of  loading.  The disadvantage is that the  solution  algorithm  may  not
be  included  in  the  code  being used.    The  following  approach  is  appropriate
for  virtually any  code,  but  is  limited to cases  involving large  transient
fluxes  at  the  surface.    Such  fluxes  represent  extreme  events;  including
natural  events  such  as extreme  precipitation,  or  artificial  events  such  as
sudden failure of a  landfill or  impoundment  liner.
     Introduction  of a^iarge  amount  of  infiltration  to  the unsaturated  zone
will  cause  the  propagation  of  a  wetting  front  downward through  the   soil
column.   Transient unsaturated flow models  may  be used to track  the progress
of  this  wetting front  through the unsaturated  zone.   The  time  required  for
the wetting  front to reach the  saturated zone may be taken as the TOT.   This
approach  is  illustrated in  Figure  3.2-3.  This  figure  shows the  variation  of
moisture  content  with  depth  and time.   The wetting front  appears as  an  area
where  there  is a  rapid  change  in moisture content with depth.  In  the  example
shown  in  Figure  3.2-3, the time for  the  wetting front to  reach  a  water  table
located  at  40 m  depth  would be approximately 90 hours.   If graphical  model
outputs  are  not  available,  the  moisture content  of each node (layer) should
be  examined  at  the end  of each  time step.   A  large  increase in moisture
content over  a time  step  signifies  the passing of  the wetting front.
                                     3.8

-------
      10
—     20
 Q.

 O)
O
      30
                     1  =  90 min   1.5 hr)
                     1 = 900 min (15 hr)
                  I   1 = 1,800 min  (30 hr)
                  I                       —
                       1  =  3,600 min  (60  hr)
                     1 - 5,400 min  (90  hr)
      40
          0      .05      .10      .15      .20      .25      .30


                                  Moisture Content,  0


       Figure 3.2-3.  Example Model Results Showing Migration
                      of Wetting  Front with Time


                                3.9
.35

-------
     An important consideration  in  the  above approach is determination of the
input source.   The  input must be  large enough to cause  the  wetting front to
migrate entirely through the  soil  column to the saturated zone.  For example
in areas with  deep  unsaturated zones having  low  moisture contents, even very
large  inputs  may never  reach the  saturated  zone regardless of  how long the
model  is  run.   Unfortunately,  there is  no general  guidance for  determining
how large an input is required.
     It  is  very important  to note that  use  of this approach  does not  imply
that variations of moisture  content  with time are  necessary for  unsaturated
flow  to  occur.    This  approach  is   applicable  only  for  transient  flow
conditions  and  only  when  the  water   input  is  large  enough  to  cause  a
significant perturbation to existing conditions.
Determination  of Contaminant  Travel Time  —
     Some unsaturated  codes have the capability  of  modeling  the transport of
contaminants   through   the  unsaturated  zone.   A   known   concentration  of
contaminants can  be input to  the  top of the soil column and the code used to
determine  the concentration  of   contaminants   leaving  the  bottom  of the
unsaturated soil column.   If  the simulation is run for a long period of  time,
steady   state   conditions  will   be   reached  where   the   concentration  of
contaminants  leaving  t^e soil column  is equal to  the  concentration entering
the  soil  column.   IfHhere  were  no dispersion  of  contaminants  (e.g.,  plug
flow),  the  time required  to  reach steady  state  conditions  would be equal to
the time  of  travel  for the contaminant.  Because of dispersion, however, the
average contaminant  TOT will  be  somewhat less than the time to reach steady-
state  conditions.   An  average  contaminant  TOT can  be  estimated as  some
fraction  of the  time to  reach  steady state conditions  (e.g.,  the  time to
reach  output  concentrations equal  to one-half the steady  state  value).   The
ground-water   TOT  can  be  related to  the   contaminant  TOT  by  use   of  a
retardation  factor.   The  retardation   factor is  equal  to  the ground-water
velocity  divided   by  the   contaminant velocity.     Dividing  the  average
contaminant  TOT by the retardation factor  will  yield the unsaturated ground-
water TOT.
                                     3.10

-------
3.3  SELECTION OF THE APPROPRIATE METHOD TO DETERMINE TOT
     A decision  tree  for selection of the  appropriate  method for determining
TOT  is  shown  in  Figure 3.3-1.   The  decisions  to  be  made in  this figure
consider  both  the characteristics  of the  site and  the availability  of the
data.   The  representation  of  the site  should be  as  realistic  as possible
depending on  the availability of the data.   For example,  if soil  properties
at  a  site exhibit complex  spatial  variability but data  are not available to
describe  this variability,  the assumption  of  simple  variability  should be
made.   Under these circumstances,  one  has to  realize  that the assumption(s)
made can  significantly impact the results.
     The  rule  of thumb for selecting the  appropriate procedures is to choose
the  simplest one  which  can  be applied  to your  specific  problem.    If the
unsaturated  flow system  can be represented as  a one-dimensional steady  state
problem with a  single material  type, and  the  flow  is  controlled  by gravity
drainage  (i.e.,  unit  gradient), the  simplest  analytical  approach  can be  used
to  obtain a  direct solution.   If, however, the unit  gradient assumption  does
not  apply but  the other   assumptions  are applicable,  an  analytical  method
using an  iterative solution scheme can  be applied.  All  transient problems,
and  problems    in  more  than  one  dimension  require  the  use  of  simulation
models.    Water  balvice  models  are   appropriate  for  simple   quasi-two-
dimensional  problems  where  flow is  controlled  by gravity drainage.  Numerical
models  should  be  used  for  all  higher  dimension  problems having  complex
geometry  and boundary conditions.
     The  direct  analytical  solution  technique  is  quick  and easy  to apply;
however,  the  assumptions  of  the  method  limit  its  applicability.    The
iterative analytical  solution  offers  more flexibility,  but limitations on the
dimensionality and complexity  of  the  problem restrict its applicability.  Due
to  the  limitations of the  analytical approaches,  simulation  modeling can be
the only means of obtaining reliable  results for complex problems.
     Application of modeling  is  typically limited by the availability of data
and the availability of  time.   The more  complex a model, the greater its  data
requirements  (both  in  time  and  space).    Acquisition  of these  data  often
requires  that field   and   laboratory  programs  accompany  model  development.
                                     3.11

-------
                    Unsaturated
                    Flow Problem
                   One-Dimensional
                       Problem
                           Yes
                    Steady State
                           Yes
                    Water Table
                     Boundary
                           Yes
                      Spatial
                   Variability
Iterative
Analytical
Solution
              No
                           Simple
Unit Gradient
                           Yes
                       Direct
                     Analytical
                      Solution
                        Spatial
                      Variability
                                               Simple
                        Gravity
                        Drainage
                       Controlled
No
                     Water Balance
                         Model
 Figure  3.3-1.   Summary of  Selection of Approach for TOT Determination
                                3.12

-------
While  complex  models  may  be  set  up  and  run  with   limited  data   using
simplifying assumptions, such  an  approach  does  not take full advantage of  the
capabilities of these models and is, therefore, of limited value.
     Development  and  application  of  models can  also be  very  time  consuming
(weeks  to  months).     Calibration  and  verification  of  models   may  take
considerably more time.   The time  required is  usually  proportional  to  the
complexity of the model.
     The   above   limitations  relate  to   modeling   in  general.    Specific
limitations related to  the  use of models to support permit writing activities
are:
1)   Time   requirements,   especially  for   development   and  calibration  of
     numerical  models,  may exceed  the  time  available  for preparation  and
     review of Part B applications.
2)   Data  supplied  with Part B applications may  be  inadequate  to develop  and
     calibrated  models, and  will   almost  certainly  be  inadequate  to  verify
     models.
     The  above  limitations do  not imply  that   the  use  of  unsaturated flow
modeling   is   inappropriate  to  support   permit  writing.     Rather,    these
limitations are  presented  to aid  the reader in determining the suitability of
modeling for Part B application.
                                     3.13

-------
                                 SECTION 4.0
                         ANALYTICAL SOLUTIONS OF TOT

     This section  provides  a more  detailed  discussion of  the  two analytical
approaches  for  calculating   TOT  presented  in  Section   3.0.     The  data
requirements  and the  sources  of  the  data  are  presented,  the   methods  are
explained,  and  example  calculations  are  provided.   It  will  be  shown  that
these methods provide a means of calculating TOT  that  is  easier, and less time
consuming than  using unsaturated  flow models.   However,  the  application  of
these methods is limited due to the assumptions used in their development.
4.1  DATA REQUIREMENTS AND SOURCES
     The  data  required by  the  analtyic.al solutions  for  calculating  TOT are
listed below:
•    stratigraphy of the site;
•    thickness of geol
-------
conductivity (K) and matric potential.  These charteristics   of  the  soil(s)
are required  for  any determination  of flow or  TOT in  the  unsaturated zone:
either analytically or through the use of models.
     Ideally, these  relationships  should be measured  in the laboratory using
soil  samples obtained  from  the  site.   If  laboratory measurements  are  not
possible, the following  simple  analytical  relationships between pressure head
and water content,  and  between conductivity and  matric potential  (Campbell,
1974) can be used:
      K *
where
     ^g = air entry matric potential;
     e$ = saturated water content;
      e = field water content;

      sat - saturated h-Jraulic conductivity;
     b = negative one times the slope of the log-log plot of  t  versus e; and
     n = 2 + 3/b.

     Using the  above  relationships it  is  necessary to know only the slope of
the  log-log plot of 4>m  versus e,  the  saturated hydraulic  conductivity,  and
the  saturated moisture  content.   The saturated  hydraulic conductivity can be
determined  in  the  field  or  measured  in  the  laboratory.  Field methods
are  preferred to laboratory methods, and are detailed  in  Appendix A, Section
1.0.  Default values, such as those listed in  Table A.1.1,  should be used only
as screening factors in choosing a proper  field  method.since they may
underestimate hydraulic conductivity by several  orders of magnitude.
     The saturated moisture content  (e$)  can also be obtained  from laboratory
measurements.   If  measurements  are  not  possible,  e  can  be  assumed  to  be

                                     4.2

-------
Table 4.1-1.   Representative Values for Saturated H>draulic  Conductivity
              (Source:   Freeze and Cherry,  1979)





1












ROCKS






Unconsolifloted k k K K K
deposits (dorcy) (crn2, (Cm/s)(m/s) (QQi/aav/f,2)




















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5
- 1
3 —1





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;
3


























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-10s rtO'3 rl02

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•10
• 1



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-1C'2
-10"3


-io-4

- IO"5
-to'6

-to-7
-to'8

-ID'4

-io-s
-io-6
-to-7
-tO'8



-ID'9



-10

- 1
-to"
-ID'2
-to'3



-io-4


-tO'10 - 1C'5
-to"1


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- to"3
IO"4

.o"5
10"6
-ID'6


-io-7

-ID'8
-ID'9

- 10"°
-IO"1
r- 1
.1C6
-IO"

-!0'2
-to-3
•to-4
-1C'5



•ID'6


-to'7
-,o-8


1C'9

10"°
10"'

10"2
IO"3


•IO5
1 W
•104
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-IO2


- tn
1 W

t

•10-'

ir-2
lu

10'3
to'4

10'5
io-6
io-7
                                  4.3

-------
equal  to  the  total   (actual)   porosity.     Representative   values  of  total
porosity are given in Table 4.1-2.

               Table 4.1-2.   Representative Values for Porosity
Material
Coarse Gravel
Medium Gravel
Fine Gravel
Coarse Sand
Medium Sand
Fine Sand
Silt
Clay
Porosity
28%
32%
34%
39%
39%
43%
46%
42%
     Analytical solutions of  travel  time assume.steady state flow of moisture
through the unsaturated  zone.   A simple approximation of steady state flux  is
to  assume  that  it  is  equal  to  the  net  infiltration  at  the  site.    Net
infiltration    is   equal    to   the   net    precipitation    minus   actual
evapotranspiration.    If such  information  is not  included  in  the Part  B
application,  it  should  'e  obtainable  from weather stations  or agricultural
research stations.
4.2  DESCRIPTION  OF ANALYTICAL  SOLUTIONS
     Two analytical  solutions to unsaturated travel  time are presented below.
Both solutions  require the  steady state assumptions  described in Section 3.0.
The  first  solution assumes that the hydraulic gradient  is equal to  1.   The
second  solution  allows  for  variable  moisture  content  in  the  soil  column
(i.e., hydraulic  gradient may not be equal  to  1).
4.2.1  Solution for Unit Hydraulic Gradient
     The  following solution  assumes  steady state flow and  a  unit hydraulic
gradient and  employs  the analytical  soil moisture, pressure, and conductivity
relationships described  earlier (Campbell, 1974).  Utilizing Darcy's equation
and  the soil  characteristic relationships described  by Campbell (1974), it  is

                                     4.4

-------
possible  to  derive  the  following  expression   for   moisture   content   as  a
function of steady state flux (Heller, Gee, and Myers, 1985)
     9 =
          \at
where
     q = steady state flux;
     i/    = saturated hydraulic conductivity;
     Nsat
     QS = saturated moisture content; and
     m = l/(2b + 3), where b is negative one times the slope of the  log-log
         plot of 
-------
                              Example Problem 1

     This first problem  provides  an example of  the  procedure for calculating
TOT  through  an unsaturated  zone  consisting  of  a  single  material  type.   A
schematic of this single layered system is shown in Figure 4.2-1.
     The following parameters  must be known (either measured in the field or
laboratory, or obtained from the  literature)  in  order  to apply this solution.
Example values of each parameter are provided.

                _ Parameter _    Example Value
                Flux (q)                       0.5 cm/yr
                Saturated water content (e )   0.31 m^/m^
                for soil profile
                The slope (-b) of a log-log    -3.162
                plot of *m versus e
                Saturated- hydraulic            5.4 x 10  cm/yr
                conductivity (Ksat) of the soil
                Length of unsaturated          3760 cm
                column (L)

Step 1;  Calculate m    v
     m
             1
         57374 + 3
       = 0.107

Step 2:  Calculate Steady-State Moisture Content e

     e =
       _ /   0.5 cm/yr   ,0.107  ,Q 31 m3
          5.4 x 104 cm/yr

                                     4.6

-------
Figure 4.2-1.   v-hematic of Example Single-Layered System
                             4.7

-------
       = 0.09 m3/m3
Step 3:  Calculate Travel Time (T)
         q

       = (3760 cm)(0.09 m3m3)
              0.5 cm/yr


       = 677 years
                                    4.8

-------
                              Example Problem 2

     This second problem  is  similar to  the  first except  that  it illustrates
the  procedure  for  calculating  TOT  through  a multi-layered  unsaturated flow
system.  A schematic of this multi-layered system is shown in Figure 4.?-?.
     The following  parameters must be known  in  order  to apply this solution.
Example values of each parameter are provided.
           Parameter
Flux (q)
                             Value
                              0.5 cm/yr
                                                       Notes
                                          Constant throughout section
    Saturated Water Content
      e
       si
       »s2
       Js3
    Slope of log-log plot
    of j, versus e(-b)
       l
                        0.31
                        0.40
                        0.42
                        -3.162
                        -3.475
                        -3.610
    Saturated hydraulic
    conductivity (K  .)
                   satj
K]
K;
                              5.4 x 10  cm/yr
                              1.0 x 10  cm/yr
                              0.5 x 104 cm/yr
    Length of unsaturated
    column (L)
       •1
                        1,000 cm
                        1,000 cm
                        1,760 cm
                                     4.9

-------
Figure 4.2-2.  Schematic of Example Multi-Layered System
                              4.10

-------
Step 1:  Calculate m for Each Layer
         Zb + 3
     ml = 0.107
     m2 = O-101
     m3 = 0.098

Step 2:  Calculate Steady-State Moisture Content for Each Layer

     e = (r^-f ee
          Nsat    s
     ej = 0.09
     e2 = 0.15
     93 = 0.17

The  total  travel  time  through the  section  in the  sum of  the  times  through
each of the layers.
Step 3;  Calculate Travel Time (T)
     T  = (1.000 cm)(0.09) = ,80
     '1      0.5 cm/yr       A   y
     T  9 (1,000 cm)(0.15) _ 300
     T2      0.5 cm/yr       JUU
     T  = (1,760 cm)(0.17) = 60Q
     T3      0.5 cm/yr       buu yr

     T = Tl * T2 + T3 = 1'080 yr
                                     4.11

-------
4.2.2  Solution for Variable Moisture Content
     Solution  of  the  variable  moisture  content  case   is  more  complex  and
requires  discretization  of  the  soil profile  into a  number of  node or  grid
points as  shown  in Figure 4.2.2-1.   The analytical solution for the case  is
(Jacobson, Freshley, and Dove, 1985)

     ^ =  4»M + A2i (q/K* - 1)

where

     ^ =  pressure head at the upper grid point;
     ^i-1  = Pressure head at the lower grid point;
      q =  flux through the soil column;
     Az-  - elevation difference between  grid points; and
     K* =  harmonic mean hydraulic conductivity between grid  points
                   AZ<
     K*
               i + 6ZM/KM
     Ki'Ki-l ~ hydraulic.,onductivity at the upper and  lower grid points,
     respectively.
     The  solution  begins with  the grid  point located  at  the lower boundary
(water table), where i^_j is known to be 0, and K^_j  is  known  from  \J>j_j     and
the soil  characteristic  curve.   The solution proceeds  iteratively  by assuming
a value of \b., determining K*, and then solving for ty-.     A   new   value   is
assumed for ty.  and  the  process  repeated  until   there  is  convergence  on  a
solution.  The calculated value of ^ is then  used as  \\>^_^  for the next pair
of grid points and the process is repeated.
     Once the  solution  has  determined the  pressure  head at every  grid  point,
the moisture  content and hydraulic  conductivity  at  every grid  point  can  be
obtained  from soil  characteristic curves.   Examples  of soil characteristic
curves, how  to use them, and  where to get them  is  provided  in Section 2.0.
Knowing the  moisture content and  hydraulic conductivity at  two grid points,
the travel time between  the grid points is  given  by
                                     4.12

-------
       Grid 1 + 1
       Grid i

K.  = Hydraulic
 1  •  Conductivity
     of Grid
       Grid i  - 1
       Grid i - 2
I


V

'i +1, i + 1
• ^^"
z ^
* » 1
•— —

Z1-l.
1 - 1
Zi - 2,
i-2
A
Node or
_— — Grid Point
J« ~
If 1
T AZ. A*.
h ^ 1
.-.L'LljL.

          Figure  4.2.2-1.  Discretization Between  Grid  Points
                                  4.13

-------
      11 =   K* dh.

The above equation  is  used  to  determine the travel time between every  pair  of
grid points.  These  travel  time segments are then  summed  to obtain the  total
travel time through the soil column.
     It  is  possible to perform the above  solutions  manually for very  simple
systems.   However,  as  with all  iterative  solutions,  the  process  can  be  very
time consuming.   Therefore,  the use of a computer  is recommended.  A computer
code  to perform the above  solutions for pressure  head and  travel  times has
been developed by Jacobson, Freshley, and Dove  (1985).
                                    4.14

-------
                                 SECTION 5.0
                           UNSATURATED FLOW MODELS

5.1  INTRODUCTION
     The two  types  of unsaturated  flow  models identified  in  Section  2.0 are
examined in  greater detail  in  this  section.    It  should  be  noted that  the
intent of this  appendix   is not to  recommend  specific  computer codes  for use
by  permit  writers  or  to  provide   detailed  instructions  in  the  use  of  any
particular codes.
     The purpose  is  to  demonstrate the use of two  codes which are considered
to be representative of codes in these two categories.
     A  large  number  of  unsaturated flow  codes  are  presently  available.
General characteristics of many of these codes are presented in Table 5.1-1.
     A partial  list  of available unsaturated  flow models is also contained in
EPA (1984).
     Selection  of  a cod; should be made by the  perspective user considering
such factors as:
1)   familiarity with the operation of an appropriate code;
2)   availability of data required by the code;
3)   applicability of the code  to  the specific problem (e.g.,  dimensionality,
     complexity of the system);
4)   acceptability and documentation of the code; and
5)   hardware availability.
     Familiarity with a  code  is  perhaps  the  most important consideration.  If
an  ana-lyst  is already familiar  with a particular  code,  that  code  should be
used provided  it  meets  the requirements of  the  application.  Availability of
data is  the  second most important  consideration.   The available  data  for an
application should  be  compared  with the data  requirements  of  the codes being
considered.   Applicability  of the code to the problem is  perhaps  equally as
                                     5.1

-------
                                       Table  5.1-1.   Summary of  Unsaturated  Zone Codes
                               Spatial Characteristics
in
ro
Code
Han
AMOCO
ALPURS

BETA- II
BRUTSAERT1
8RUTSAERT2
CMC
COOK
OELAAT

FEMHATER
FLUMP
GRANOAIF

GPS IN
MOMOLS
PORES
REEVES-DUGUIO
SHELL
SSC
STGtfT/MOGHT
SLH2
SUPERMOCK
TRACR30





TRIPH

TRUST
TSIE
UNFLOU

UNSAT1
UNSATIO
UNSAT?
VERGE

VS20
UAFE
Dimensions Discretization Method
Ho. 1
37
21

.40
33
39
38
29
48

24
5
30

31
52 X
49
22
42
41
27 X
46
47
SS





51

4
32
7

28 X
10 X
26
8

45
SO
2




X
X

X
X

X
X
X




X



X
X






X



X



X


X
X
3 FOM IFDM FEM Other Special Features
X
X Three- phase oil. water.
gas
X


X X
Two-phase oil £ gas
I Roots, evapotranspiration

X
X. V
X

X X
X
X X
X Radioactive decay
X X
X
X
X Roots, evapotranspiration
X Roots
X X 1- or 2 -phase flow with
tracer in either phase
(air or water)
Freundlich. Langmuir
sorption, radioactive
decay, capillary effects
X 3 -member decay chain

X I
X
X

X
X Roots, evapotranspiration
« Roots
X X

X Roots, evapotranspiration
X Coupled heat » 2-phase
Past Applications
Oil reservoir


Oil reservoir
Experimental
Experimental /Laboratory
Oil reservoir
Oil reservoir
Ground-water extraction
crop production


Underground nuclear
explosions

Experimental
Oil reservoir

Oil reservoir
Oil reservoir

Ground-water e>traction
Ground-water extraction
Tracer flow in unsat.
conditions, Radionu-
clude transport. Tracer
flow in fractured
system

Radioactive waste
disposal

Oil reservoir
Radioactive waste
disposal

Crop studies
Engineering design
Radioactive waste
storage

Confined underground
Principal Contact
AMOCO
Mobil Corp.

Intercomp.
Brutsaert
Brutsaert
CMC
Cook
Oe Laat

Yeh
Narasimhan
Morrison

Exxon
Rojstoc/er
UKAEA
Reeves
Shell Oil Co.
SSC
de Smedt
Oe Laat
Reed
Travis





Gureghian

Narasimhan
Tech. Soft. I Eng.
Pickens

van Genuchten
Bond
Neuman
Verge

Lappala
Travis
Comments
Proprietary Code
Proprietary Code

Proprietary Code


Proprietary Code
Proprietary Code
European Code





Proprietary Code

European Code

Proprietary Code
Proprietary Code
European Code
European Code

Can operate in
1. 2 or
3 dimensions






Proprietary Code








Can operate in
                                                          mass  transport (air  vapor
                                                          1 liquid) Accurate treat-
                                                          ment  of H^O Separate
                                                          velocity field phase
radioactive waste dis-
posal .  In-situ fossil
energy  recovery studies,
2-phase flow and tracer
studies
I  or
2  dimensions
        KEY:   FOM - finite difference method.
              IFDH • integrated finite difference method.
              FEM • finite element Method.

-------
                                                           Table  5.1-1.   Cont'd.
en

CO
           Tiame
  	Spatial Characteristics  _
Dimensions  Discretization Method
 1   7   I  FDll'TTBfnTM Oflier
BACHMAT
DUGUIO-REEVES
FECTRA
FEMUASTE
ML IRAN
MMT-DPRU
SCAT1D
SCAT20
TRNHDL
6
19
17
?5
64
44
35
36
2
X
X
X
X
X
X
X
X
X
                                         Otner       Special  features	Pas_t_*J)Plii:aJt'.ons_     ££.'1c_iPJ J Contact	Comments
      KEY:
 FOH  =  finite difference method.
IFPM  =  inteqrated finite difference method.
 FEM  =  finite element method.
                                                             Surface/ground water
                                                             Absorption & decay

                                                             li\J order decay, sorption
                                                                                       Ground-water  studies
                                                            Stochastic velocity field
                                                            Stochastic velocity field
                                                                                                   Backmat
                                                                                                   Ouguid

                                                                                                   Baca
                                                                                                   Yeh

                                                                                                   Reisenauer


                                                                                                   Simmons

                                                                                                   Oster
                                                                                                   Oster
                                                                                                   Av-Ron
                                                                                                          Middle-east  Code
                                                                                                          Compatible wi th
                                                                                                          Code No.  22

                                                                                                          Compatible with
                                                                                                          Code No.  22
                                                                                                          Compatible with
                                                                                                          Code No.  44

                                                                                                          Discrete  Parcel
                                                                                                          Random Walk
                                                                                                          Stochastic Code
                                                                                                          Stochastic Code
                                                                                                          Middle-east  Code

-------
                                                           Table  5.1-1.    Cont'd.
                                Spatial Characteristics
in
Code
Name
HANKS
MARINO
MCCANN
MOBIOIC
NMOOEL
SEGOL
SHAMTU
SUMATRA- 1
TARGET
TRANS
TRANSONE
TRANSTUO
UNFLW
MAT SOL
UHC
KEY: FOM -
1FON •
FEM •
Dimensions
No. I 2 3
1 X
S3 X
20 X
14 X
X
9 X
43 X
11 X
18 X
23 X
12 X
13 X
3 X
16 X
34 X
Discretization Method
FDM IFDM FEN Other
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
finite difference method.
integrated finite difference method.
finite element method.
                                                                  Special Features
                                                             Roots
                                                             Ist-order reactions
                                                             Heat transfer
0- & 1st-order decay
Heat transfer, elegant
numerical  solution.   0-  &
Ist-order  decay.   Vari-
able saturation.   Radio-
active decay products
                             Past Applications
                          Crop production studies
                                                                                        Water  loss  by
                                                                                        evaporation

                                                                                        Tailings  and chemical
                                                                                        waste  disposal, radio-
                                                                                        active waste disposal
                                                                                        Salinity  studies
 Principal Contact

Hanks
Marino
McCann
Couchat
Selim

Segol
Vauclin
van Genuchten
Dames 4 Moore
                                                   Walker

                                                   van Genuchten
                                                   Shapiro
                                                   Kapuler
                                                   Gaudet
                                                   Crooks
    Comments
                                                                                                                                    European Code
                                                                       European  Code
Proprietary Code
                                                                                                                                    European Code
                                                                                                                                    Integrated Com-
                                                                                                                                    partment Method

-------
                                               TABLE 5.1.1.  Cont'd.
     Code
         Spatial Characteristics	
    Dimensions  Discretization Method
  Name
No.  1  2  3
SESOIL
 (Seasonal
 Soil Model)
PRZM
 (Pesticide
 Root Zone
 Model)
FDM  IFDM  FEM  Other Special Features

                      Single constituent
                      migration through
                      unsaturated zone;
                      user-friend ly

                      Cal^~lates soil
                      moisture charac-
                      teristics, crop
                      root growth,
                      pesticide
                      application and
                      soil transport
     Past
 Applications

Hydrologic,
Sediment and
pollutant fate
simulation

Pesticide
migration
through
unsaturated
root zone
  Principal
   Contact

Bona Zountas, M.
A.D. Little
(617) 864-5770
                                                                           Carsel,  R.F.
                                                                           U.S.  EPA
                                                                           Environmental
                                                                            Research Lab
                                                                           Athens,  GA
                                                                           EPA Pub. No.
                                                                           600/3-84-109
Comments
                                                                                            Analytical
                                                                                              Model
                 Numerical
                   Model

-------
important.     Applicability  involves   consideration  of   such   factors  as
dimensionality  (e.g.,  one-dimensional flow  versus two-dimensional  flow) and
complexity  (e.g.,  number of layers,  degree  of  inhomogeneity).   Acceptability
may  also be  an  important consideration,  particularly  within  a   regulatory
framework.  In  all  cases,  an  effort should be made to select codes  which have
been  fully documented   and  verified  against  standard  solutions.    Lastly,
hardware  requirements  may be important.   Codes which  require  large computer
systems  are inappropriate  if such systems are not  available.
     The  reader interested  in  evaluating  and  selecting  an  unsaturated flow
code for a particular  application  is referred  to the  review  of unsaturated
codes prepared  by Oster  (1982).
     Once  a code has  been selected,  detailed  instructions on  operation and
use should  be  obtained  from the user's manual for  the  code.   Availability of
code  documentation  is  summarized   in  Table  5.1-2.    References   to   user's
manuals  for unsaturated  flow codes  are provided in  Oster  (1982).
     Models are developed and applied to understand  and predict the behavior
of  complex physical systems  and processes.   Physical  systems,  such  as the
unsaturated  zone,  display characteristic  behavior  in  response  to physical
laws.   This behavior is described  (either exactly or approximately) in  terms
of  mathematical expressions  (e.g.,  differential   equation  describing  flow).
Many of these  expresses are not  amenable to  analytical solution  and are
transformed into approximate  solutions in the form of  computer codes.   These
codes  form  the  framework upon  which  models  are  developed.    Models  are
developed  through  assignment  of representative   data  to  the  computer  code.
These data  are  assigned to represent and describe the  physical properties of
the   system   being  modeled   (e.g.,   spatial   distribution   of   hydraulic
conductivities).
     A  model,   therefore,  consists  of two  components,  the  computer code and
the input data  for  the  code.  The accuracy  of a  model is dependent on both of
these components.   Codes must adequately describe  the processes of  importance
for the particular  system being modeled.   Input data must be  provided  which
are representative  of the  properties  of the  system.
     Model  results  are  nothing more than  solutions to  complex mathematical
expressions.    The  mere ability of a model  to  produce  results  says nothing

                                      5.5

-------
Table  5.1-2.   Summary of Unsaturated Code  Documentation  and Availability
                (Source:  Oster, 1982)
                          Documentation

No.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
IS
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
4B
49
50
51
52
S3
54
55
Code
Name
HANKS
TRNHDL
UNFLU
TRUST
FLUW
BACHMAT
UNFLOH
VERGE
SEGOL
UNSAT1D
SUMATRA- i
TRANSONE
TRANS TWO
MOB1DIC
NHOOEL
WATSOl
FECTRA
TARGET
OUGUIO-REEVES
HCCANN
ALPURS
REEVES-OUGUID
TRANS
FEMVATER
FENMSTE
UNSAT2
STGMT/MOGyT
UNSAT1
COOK
GANOALF
GPSIM
TSAE
BRUTSAERT1
UHC
SCAT10
SCAT2D
AMOCO
CMC
BRUTSAERT2
BETA II
ssc
SHELL
SHAMTU
MMT-OPRW
VS20
SUM- 2
SUPERMOCK
OELAAT
PORES
UAFE
TRIPM
NOMOLS
MARINO
MLTRAN
TRACR30
Model
Description
X
X
X
X
X

X
X
X
X
X


X

X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
V
X
X
X
X
X






X
X
X

X



X
X
X
X

Published
Applications
X


X
X

X
X
X
X
X




X


X




X
X
X

X














X


X

X


X


X

User's
Manual



X
X


X
X
X
X





X




X

X
X
X
X
X





X








X

X

X






X

Applicat ions
                                        User's  Laboratory  Field ___^	
                                                 Data     Data Available  Proprieta~
               Code Availabi1ity_
                                      5.6

-------
                                                           947  2 • 00-2A
with respect  to  the accuracy  or  validity of those results.   The accuracy  of
model  results and,  hence,  the  accuracy of  the model  itself  are  typically
assessed  through  the process  of  calibration.    During  calibration,  the model
input  data  are  adjusted until the model  accurately predicts conditions which
are known to  exist.   Calibration  itself, however, is still not proof of model
accuracy.     Accuracy  can   be  further   tested  through  the   process   of
verification.   During verification,  the  calibrated model is  used to predict
known  conditions  at  a different  time than  that  tested  in  the calibration
process.
     The  processes  of  calibration  and verification  are  often time-consuming
and  expensive.    Therefore,  many times  they  are  not  performed.   Lack   of
calibration does  not necessarily  mean that the model  is inaccurate, only that
the  accuracy  of the  model has  not been established.
     Because  of   the   large  number  of  unsaturated  flow  codes  currently
available,  and  for  simplicity of discussion,  only  one  code  representative  of
each type of  model  will be  examined:  UNSAT1D  (UNSATurated ^Dimensional),  an
example  of a numerical  code; and  HELP  Ojydrologic  Evaluation  of Landfill
Performance),  an  example  of  a  water balance  code.    Characteristics  of the
codes,  as well as  the required  input data,  input  data  sources,  and  output
data,  will  be  discussed  to  provide permit  writers  and applicants  with  a
description of  the  appvjach  to  unsaturated  flow modeling.    The limitations
and  applicability of  each code  for  the  determination of  TOT will  also  be
discussed.
5.2  EXAMPLE  NUMERICAL  CODE  -  UNSAT1D
•5.2.1  General Characteristics
     The  UNSAT1D code was  originally developed  to  describe  water movement
under  typical agricultural conditions  (Gupta,  et al,  1978).    The  code and
its   auxilliary  programs  were  later   revised  and  incorporated  into   an
unsaturated flow  modeling system  (Bond,  Cole,  and Gutknecht, 1982).
     UNSAT1D  is  a  one-dimensional,   finite  difference  code which solves the
differential  equation  for ground-water  flow under saturated  and unsaturated
conditions.    It  simulates  infiltration,  vertical  seepage,  and  plant  root
uptake as a  function  of  the hydraulic  properties  of a  soil,  soil  layering,

                                      5.7

-------
root growth  characteristics,  evapotranspiration  rates, and  frequency,  rate,
and  amount  of  precipitation  and/or  irrigation.      UNSAT1D  can  be  used  to
estimate  ground-water  recharge,   irrigation  and  consumptive  use  of water,
irrigation return  flows,  and other processes associated with unsaturated and
saturated soils  which can be  represented  as  one-dimensional  (Bond, Freshley,
and Gee, 1982).
     The UNSAT1D modeling system  consists  of one computer  code which solves
the  flow  equation  and  several   supporting  codes  which   are  used  for  the
preparation  of  input  data  and for  evaluation  and display  of model  results.
Use  of  UNSAT1D  requires  site-specific  input  data  and  an  understanding  of
ground-water  flow  theory.    Input data  requirements  depend on  the  problem
being  solved,  but in most cases  include  the  soil profile  description of the
site;  the hydraulic properites of each layer of  the profile; characteristics
of  vegetation  at the site;  the means  by  which  water is applied to the site;
and  climatic  data  for  the  site.    The  model   output   includes   soil-water
potential  (suction), moisture  content,  and  water flux  at  each  node (depth
increment) for each  time  interval  considered  by the model.
     Table  5.2-1  provides  a  summary of . the  important   characteristics  and
capabilities of  UNSAT1D.

     TABLE 5.2-1.   Important Characteristics  and Capabilities  of UNSAT1D

1)   Simulates partially-saturated ground-water flow.
2)   Simulates infiltration, vertical seepage, and plant root  uptake.
3)   Derives solution  using  a  finite difference, fully  implicit method.
4)   Describes one-dimensional flow  in a vertical or horizontal  direction.
5)   Accomodates homogeneous,  heterogeneous,  or layered soil  profiles.
6)   Simulates up  to  ten  soil  layers.
7)   Simulates   rain,  sprinkler  or  flood   irrigation,   or  constant  head
     condition for the  upper boundary.
8)   Simulates   lower  boundary  conditions as  water  table,  dynamic, quasi-
     dynamic,  or unit  gradient.
                                     5.8

-------
                                                            OSWER POLICY DIRECTIVE NO.

5.2.2  General Approach to Application                    Q 4 T Q  •  0 0 " 2A  ^
     The first  step  in application  of  a numerical model  is  development of a
conceptual   model  of  the  site  being  considered.    A  conceptual   model  must
identify the  important features  and  characteristics of the site and describe,
in a  qualitative way,  the relationships  between  the  various  components  and
processes.     The  conceptual  model  of   a  site  must   be  developed  before  an
analyst  can  develop  a  mathematical   approximation  of  the  system.     For
unsaturated  flow,  important  data  required  to  develop  a  conceptual  model
include stratigraphic  data describing soil layering  at  the site and climatic
data  describing net  precipitation  and  evaporation.    It should  be apparent
that   substantial  knowledge   of  geohydrology  is   required  to   develop  a
conceptual model.
     Once  a conceptual model  has  been  developed,   the  analyst  must translate
the  conceptual  model  to  a  mathematical  model by  supplying appropriate input
data to the  computer  code.  The first step in developing a numerical model is
development  of  a  finite difference  or  finite element grid  network.   UNSAT1D
is  a one  dimensional  finite  difference code,  so grid  development  involves
specifying   a   vertical  or  horizontal  array  of  nodes   (depending  on   the
application).
     Once  the grid has teen  established,  input data  must  be supplied to  the
model.  Data describing soil  properties and  characteristics  must  be supplied
at each node in the grid.  Input  data may be obtained in the field, measured
in the  laboratory,  or obtained  theoretically.   The  necessary  input data  for
UNSAT1D  and  their  sources  or  methods  of  estimation  are  summarized  in
Table 5.2-2.
     As indicated  in  Table 5.2-2  a considerable  amount  of data are required
to  operate   the  UNSAT1D   Model   Sequence.     One  of  the  difficulties  of
unsaturated  flow  modeling  is that  data  requirements  often  exceed  the amount
of  available measured data.   Various  theoretical and  .laboratory techniques
may be  used  to  estimate  some  of these data, and some  data may be generated or
estimated  using the  supporting  programs contained  within the  UNSAT1D Model
Sequence.   These  programs must be  run before UNSAT1D can be used to simulate
a particular  unsaturated flow problem.
                                     5.9

-------
        Table 5.2-2.   Surmrary of UNSAT1D  Input Data  and Sources
        Input Parameter
• Depth of soil layers and lower
  boundary condition

• Soil hydraulic properties
  1) soil-water retention relation-
     ship, saturated volumetric mois-
     ture content  (es), and saturated
     hydraulic conductivity.
  2) hydraulic conductivity vs.
     water content.

  3) initial moisture content

  4) field density

• Precipitation and irrigation
  with hourly distribution

• Potential evapotranspiration with
  diurnal variation

• Plant growth behavioV
  1) leaf-area index
  2) root growth and density

  3) growing season
	Data Source or Estimation3	

• Must be known or measured via field
  dri1 ling
  1) laboratory measurements of moisture
     contents at various suction heads.
     fls may be assumed to be porosity.

  2) calculated from soil-water reten-
     tion relationship and saturated
     hydraulic conductivity.
  3) measured from samples or estimated
     from water balance history.
  4) measured from samples.

 • Obtain from nearest weather station
  and agricultural sources.

 • Obtain from weather/experimental
  station or calculate with detailed
  climatic data.

  1) published for some plants.
  2) published for some plants.  May
    be assumed over growing season.
  3) available from weather service or
    agricultural organizations
aThe degree of estimation acceptable depends on accuracy  required .in model.
                                     5.10

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     There  are  four  data  preparation  programs  within  the  model  sequence.
Brief descriptions of these programs and their functions are given below.
HYDRAK  -  This  program  estimates  the  hydraulic  conductivity  versus  water
content or matric potential relationships for each soil.
EXTEND  -  This  program  extrapolates  additional  data  points  from  the  high
suction head/low water content end of the soil moisture characteristic curve.
POLYFIT  -  This  program enters the soil  moisture  characteristic data into the
UNSAT1D  code  in the form  of polynomial  expressions,  the  preferred  form for
these data.
FAOPET  - This program estimates  daily  potential  evapotranspiration (PET) for
the  site, as  required  by  the  model  when  these data  are  not  specifically
available.
     Output   from   UNSAT1D   includes  the  soil   water  potential  (suction),
moisture  content,  and  soil  water flux  rates at each node  for'each time step.
Examples  of  graphical  output  from  UNSAT1D  showing moisture  content  versus
depth  and cumulative  drainage  or  flux versus  time past  certain elevations
within an  unsaturated soil  profile are shown in Figures 5.2-la  and 5.2-lb.
5.2.3  Determination of  TOT from Model Results
     The  output of  UNSAT1D does  not  include TOT  for the unsaturated zone.
Therefore,  the  model  results  must be  analyzed   using  one of  the techniques
described  in  Section  372.  to determine  TOT.   For  transient  simulations, TOT
can be estimated by  following  the migration of the  wetting front.  For steady
state  simulations,  the  steady   state  model  solutions  for  nodal  values  of
matric potential and moisture  content  can be used to calculate velocities and
travel times  across each model element.
5.2.4  Limitations
     UNSAT1D  was  developed to  predict  the amount and  rate of water entering
and moving through  a partially  saturated flow system.   The code accomplishes
this  task • by simulating one-dimensional  flow  through  the system  using the
differential  equation

     C(e)  4*  -  '  (K (e) I?} - S
                                     5.11

-------
Q.
Ol
a
       10  -
                  Volumetric Moisture Content


        Figure 5.2-la.  Moisture Profile for a  Three  Layer
                        Unsaturated Flow System
  o>
 o
  OJ
Figure 5.2-lb.
          Time  (days)

Cumulative Water Flux Versus Time Past  Several
Elevations in an Unsaturated Fl'ow System

               5.12

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where
     C(e) = soil water differential capacity;
        e = volumetric water content;
        4» = pressure head;
        t = time;
        z = vertical coordinate;
     K(e) = hydraulic conductivity;
     H = hydraulic head; and
     S = source/sink term.

     The above  equation  is  a  general  solution for one-dimensional flow in the
unsaturated zone.   The  only  limitations  to  this solution  are that  it  is a
one-dimensional  solution  and  that it does not  account  for migration of water
in the vapor phase.
     The use  of a one-dimensional  solution  may  or  may not pose  limitations,
depending  on  the characteristics of  the site.   If  materials  present  at the
site are highly inhomogenous,  a two-dimensional model may be more appropriate
(assuming,  of   coursev  that   there   are   adequate  data   to   define  the
inhomogeneity).   A  two-dimensional  model may also  be more  appropriate for
cases where there may be  significant  lateral  flow.   For example,  leaks from a
surface  impoundment  located  above  dry  soil  would  be  expected to  undergo
significant lateral migration due to capillarity.
     Numerical  methods  of  solution  require  finer   resolution of  data  than
other methods.   This  increases the  amount of  data  that must  be obtained to
set  up  a   model.    In  addition,  the  numerical  solution  of  the  nonlinear
unsaturated flow problem is quite sensitive  to input data.   Values of  input
data must  be  reasonably  close to actual values in order to obtain a solution.
Therefore,  numerical  models are  difficult to use with  "default" data values
since such values may not yield a solution.
     The   numerical  method  of  solution  also  requires  a  great  deal  of
computational   time.    The  methods   are  most  appropriate  for   large  mini-

                                     5.13

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computers or  main-frame computers.   Even  with such  computers,  solutions may
require several hours of central processing unit (CPU) time.
     Lastly,  numerical  models  are very  complex  and  require  a good  deal of
understanding on the part  of  the analyst.  The  analyst  must have significant
experience  and  understanding  in  order to  develop  the  conceptual model  of  a
site and  set  up the finite difference  or  finite  element grid.   In addition,
several  runs  of  the model  are  often required  before  input data  sets are
adequately  adjusted to  produce a  solution.    The  analyst  must  have enough
understanding of the workings  of the  model to be able to adjust and calibrate
the model.
     With  the exception of  the limitation due  to one-dimensional  flow, the
above  limitations apply to all  numerical models.
5.3  EXAMPLE WATER  BALANCE CODE - HELP
     The  HELP  code was   developed   and  adapted  from  the  EPA's  Hydrologic
Simulation  Model  for  Estimating Percolation  at Solid  Waste  Disposal   Sites
(HSSWDS)  and  from  the  U.  S.  Department of Agriculture's Chemical Runoff and
Erosion  from  Agricultural  Management Systems (CREAMS)  code  (Walski  et  al.,
1983).   HELP  is  a quasi-two-dimensional  hydrologic  model  which  rapidly and
economically  estimates  the amount of runoff,  drainage,  and  leachate that may
be  expected  to  result   from  operation  of  landfills.    HELP' performs   a
sequential  daily analyses  of water inflow and outflow that  takes  into account
the effects of runoff, evapotranspiration, percolation, and lateral drainage
on the water  balance for  a  particular site.    The  code  was not  developed to
account for lateral inflow and  surface  runon.
     HELP  produces  daily,  monthly,  and annual  water budgets  which describe
both vertical  flow  through a  landfill profile and horizontal flow through its
drainage  layers.    The model  requires   climatological  data and  soil  and
landfill  design data.    Site-specific  data should  be used  for  the analysis;
however,  if these data  are not available  a substantial amount of climatic and
soil  data  are  maintained  within  the model,   as  well as  default  options for
vegetative  covers.   The model's  output includes summary data describing the
water  balance for  each layer  in  the  model.    These data  can be provided on  a
daily, monthly, or  yearly  basis  depending  on  the needs of the  user.
                                     5.14

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     Table  5.3-1 provides  a  summary  of  the  important  characteristics  and
capabilities of HELP.
5.3.1  General Approach to Application
     As  with  numerical models,  aoplication of water  balance models  requires
formulation  of a conceptual  model  of the  site  and  representation  of  this
conceptual  model  within the  framework of  the  mathematical  model.   The  HELP
model represents  the  system  as  a series of layers.  Four  types of  layers  are
allowed:
1)   those  which  only  allow vertical percolation;
?.)   those  which  inhibit vertical percolation (barrier layers);
3)   those  which  allow lateral drainage;  and
4)   waste  layers.
Application  of the  model requires  the analyst to  review  the stratigraphy of
the  site  and  translate  the stratigraphy  into  a  series  of  layers  of  the
appropriate type.

      Table 5.3-1.  Summary of Characteristics and Capabilities of HELP

1)   Simulates partially-saturated ground-water flow.
2)   Performs    a   sequential    daily   analysis    to    determine   runoff,
     evapotranspiratiorjf percolation,  and lateral drainage.
3)   Uses  a quasi-two-dimensional water budget approach.
4)   Describes  two-dimensional  flow  for  both  vertical  flow  through   the
     profile and  horizontal flow through  drainage  layers.
5)   Applies to  a wide variety of landfill  designs.
6)   Simulates up to nine layers.
7)   Maintains default climatic  and  soil  data,  as well as default options  for
     site  vegetation.
8)   Assumes  gravitational  forces  to  be the most  important  force  for  fluid
     movement  (capillary  Forces  are ignored) thereby  greatly simplifying  the
     unsaturated  flow  solution.
                                     5.15

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     HELP requires  input data  similar  to  those for  UNSAT1D, but  with less
spatial  resolution.   One advantage of  the  less stringent  data  requirements
and  less  complex numerics  of the  water balance models  is  that  it makes it
easier to use default values.  The  HELP  model  maintains an internal data base
of default data  values.   This data  base includes  five years of climatic data
for  102  cities  in the United States  and default characteristics  for 21 soil
types.    The model   also  makes  available  seven  default  options  for site
vegetation.   The necessary  input  data and  their  sources  and   methods  of
estimation are summarized in Table 5.3.1-1.
     HELP model  output consists  of a summary  of  all  default or user-provided
input  information (except daily precipitation) used for  the simulation and a
summary of the analysis  computed  by the  model.  The analysis summary includes
a  table of  annual  totals  for  each  year of  simulation;  a table  of average
monthly  totals   for  all  years  simulated by  the  model;   a  table  of average
annual  totals  for all years  of  simulation;  and a  table  of  peak  daily  values
for  all years of simulation.   A summary of  the  information  contained in each
table  is provided in  Table 5.3.1-2.    If the user is  interested  in monthly
output, HELP can produce tables  which report  monthly  totals for  all years of
simulation.  These tables include:
•    precipitation;
•    runoff;
•    evapotranspiration;
•    percolation  from base of landfill cover;
•    percolation  from base of landfill;
t    lateral drainage from base of  landfill  cover;  and
•    lateral drainage from base of  landfill.
     If  daily  output  is desired, the  model  provides  daily values  for each
Julian date of each  year of simulation.   In addition  to the above data, these
tables include:
•    head at base of  landfill cover;
•    head at base of  landfill; and
•    soil moisture content of the evaporative  zone.
                                     5.16

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          Table  5.3.1-1.   Summary of  Input  Data  and  Sources for HELP


Climatologic Data

     Daily precipitation values
          All can  be  obtained  from data  base  or  measured  for each  year of
          interest (2  -  20 year)  --  libraries,  universities, agricultural and
          climatologic research  facilities,  and the  National  Climatic Center
          are possible sources.

     Mean monthly temperature

          Measured  for  each  year or  single  set   of  data  for all years --
          libraries,  universities,  agricultural   and  climatologic  research
          facilities, and the National Climatic Center are possible sources.

     Mean monthly solar  radiation factors

          Measured  for  each  year or  single  set   of  data  for all years --
          agricultural  publications,  solar  heating hand  books,  and  general
          reference works are possible -sources.

     Winter cover factors

          Measured  for  each  year or  single  set   of  data  for all years --
          libraries,  universities,  agricultural   and  climatologic  research
          facilities, and the National Climatic Center are possible sources.

     Leaf area indices (LAI)

          Measured  for  e^ch  year or  single  set   of  data  for all years —
          various refere,Xes, including USDA's  publication,  "Climate  and Man,
          Year Book of Agriculture," are possible sources.


Vegetative Cover Data

     Root zones or evaporative zone depth

          Choose some  of seven  vegetative cover options  from data base, or
          must be known  or measured/observed on site.


Design and Soil  Data

     Landfill profile

          Modeled from data base or observed/measured on site.
                                     5.17

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                      Table 5.3.1-1.  Cont'd.
Soil data

     From   data   base    (21    default    soil    types'  "aVailabie";or
     observed/measured on site.   Observations/measurements  must include:
     porosity,  field  capacity,  wilting  point,   hydraulic  conductivity,
     and evaporation coeffecients for each soil  layer of profile.

Soil compaction

     From data  base or  use  soil data  representative or compacted soil
     (from observation of site).

Design  data  —  Number  of  layers  and  their  descriptions  —  type,
thickness,  slope,  and   maximum  lateral   distance   to  a  drain  (if
applicable).

     Observed/measured on site.

Design data  —  Whether or not  synthetic membranes used  in  the landfill
cover and/or liner

     Observed/measured on site.
                                5.18

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               Table  5.3.1-2.   Summary of Output Data from HELP


Analysis Summary Table No. 1

     Table  of  annual  totals  for  each  year  of operation  simulated  by the
model:
a)   precipitation;
b)   runoff;
c)   evapotranspiration  (total  of  surface  and  soil  evaporation  and   plant
     transpiration);
d)   percolation from base of cover;
e)   drainage from base of cover;
f)   soil water at beginning of year;
g)   soil water at end of year;
h)   snow water at beginning of year; and
i)   snow water at end of year.


Analysis Summary Table No. 2

     Table of  average  monthly totals for all  years  of operation simulated by
the model:
a)   precipitation;
b)   runoff;
c)   evapotranspiration;
d)   percolation from base of cover;  and
e)   drainage from base of cover.


Analysis Summary Table fl',* 3
                        ^' '

     Table of  average  annual  totals  for all years  of operation simulated by
the model:
a)   Precipitation;
b)   runoff;
c)   evapotranspiration;
d)   percolation from base of cover;  and
e)   drainage from base of cover.


Analysis Summary Table No. 4

     Table of  peak daily values  for all years  of  operation  simulated by the
model:
a)   precipitation;
b)   runoff;
c)   percolation from base of cover;
d)   drainage from base of cover;
e)   maximum head on base of cover;
f)   snow water;
g)   maximum soil moisture for vegetative layer; and
h)   minimum soil mositure for vegetative layer.
                                 -   5.19

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5.3.2  Determination of TOT from Model Results
     As discussed earlier,  travel  times can be  determined  as particle travel
timer.,  travel  times associated  with  instantaneous loadings,  or  steady state
conditions.   Application of  the first  approach requires  that  model  outputs
include the  moisture content within  a layer and  the  flux out  of  the layer.
Since water  balance models  assume average conditions  throughout  a  layer, the
instantaneous pore  water  velocity through a layer  can be approximated as the
flux out of the layer divided by the average moisture  content of the layer.
     Determination  of  the travel times  associated  with instantaneous (pulse)
loadings may  not  be appropriate  with  water balance models.   Because  of the
averaging  that  occurs  within  a  layer,  there  is   a  loss  of  resolution.
Therefore, it is difficult to detect the migration of  wetting fronts.
     Water  balance   models  can  be  used  to  solve  for steady  state  moisture
content  for  each   layer.    These  steady  state  conditions  can be   used  to
determine  steady  state  travel  times  through  the  landfill  soil column.   As
discussed  in  Section 3.2, however,  this  TOT  is  the steady state TOT  and not
the TOT of the first particle of water leaving the site.
     Water balance  models are not  suited' for contaminant  transport problems
so that TOT cannot  be determined from contaminant TOT.
5.3.3  Limitations
     The HELP code  was  designed  to  develop long-term water balance models for
landfills to predict generation  of  leachate  from landfills.  The soil profile
and  landfill  are divided into  layers and  water is  budgeted to  each  layer
based  on  a  mass  balance  between water  flowing  into each  layer  and  water
flowing out  of each  layer.   The code  allows  for lateral  drainage  from some
layers, giving it quasi-two-dimensional capabilities.
     Unlike  numerical   models,  HELP  is  not  based  on  a  general  solution  to
unsaturated  flow  and,  therefore, has several limitations  with respect to its
analytical  capabilities.    For example,  HELP  simplifies unsaturated  flow by
assuming that gravity  is  the driving force  for  all  fluid movement; capillary
forces  and the effects of vegetation are  ignored  by  the  model.   Therefore,
the  solution obtained  by HELP  is  no  more  rigorous  than  those  obtained  by
analytical methods.

                                     5.20

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     The  water  balance  approach   also  lacks  the  resolution  possible  with
numerical  models.    In  the  water  balance  solution,   soil   properties   are
assigned  by  layer and  the solution  yields the  average  moisture content  for
the  entire  layer.    Using  such  an  approach,   subtle  effects  such  as   the
migration of a wetting front may not  be seen.
     The  HELP  code  was developed  to  simulate the migration  of  leachate  from
landfills.    Application  of  the  code  to   include  migration  through   the
unsaturated  zone beneath  a landfill  will   require  addition  of  several  soil
layers  beneath the  landfill.    There are  internal  limits  to the  number of
layers  that  can  be  simulated  and  internal  requirements  for  certain types of
layers.   Therefore,  it  is not  possible  to  simulate  a  large  number of  soil
layers  beneath the  landfill.   Because of  limited spatial resolution, the  code
is  probably best  applied  to  sites  in  humid  areas having  thin unsaturated
zones.  This limitation was  confirmed by  a  recent comparison between HELP  and
                                                  7
the UNSAT1D numerical code  (Thompson  and Tyler, 1983).
                                     5.21

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                                 SECTION 6.0
                        EXAMPLES OF TOT DETERMINATION

     This section  presents  examples of  determining  unsaturated  zone  TOT for
several  proposed  hazardous  waste  facilities.    Examples  1 and  2 demonstrate
the use of analytical  solutions  for determining steady  state  TOT at proposed
hazardous waste  disposal  facilities in  the  Gulf Coastal  Plain  and  Basin and
Range physiographic  regions.   Example 3  demonstrates  the use of the UNSAT1D
numerical model for  determining  TOT associated with an  accidental  spill at a
proposed  hazardous   waste   site   in   the   Columbia-Snake   River   plateau
physiographic region.
6.1  EXAMPLE 1 - Case Study G
     Case study G  is a land disposal facility located near the northern edge
of the  Gulf Coastal  Plain.    A  review  of  the  data  provided  in the  Part B
permit application for  this  facility is presented in  the Case Study Appendix
to the Phase II Location Guidance (see Appendix C).
     The case  study  G  facility  is  underlain  by approximately 21 to  34 m of
fine to  medium  grained  quartz sand, with  limited  occurrences of silt, clay,
and lignite beds.  This sand  layer  beneath  the facility forms the water-table
aquifer.  Depths to  ground water  at the facility range from 6 to 15  m.  This
shallow  aquifer is  recharged by  rainfall   which  averages  119   cm/yr  at the
facility.
6.1.1  Description of Method and Data
     Unsaturated TOT was calculated using  the one-dimensional  steady state
analytical   solution   described  by  Heller,  Gee,  and Myers  (1985).    This
solution assumes that the hydraulic  gradient in the  unsaturated zone is equal
to one.   The unit gradient  assumption   implies  that  flow  in  the unsaturated
zone  is  dominated  by  gravity  (i.e.,  capillary  forces  are  negligible).
                                     6.1

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Because  the site  is  located  in  a  humid  region  having  a moderately  high
rainfall  (119  cm/yr),  soils  in  the  unsaturated  zone  should be  fairly moist
(i.e.,  at or  above  field capacity).    Therefore,  the  unit gradient assumption
is probably valid for this site.
     As   discussed   in  Section   4.0,  this   analytical   solution  requires
relatively few input data compared to other methods.  These data are:
•    soil profile and depth;
•    soil characteristics  (saturated hydraulic  conductivity,  saturated water
     content, moisture content versus pressure head); and
•    steady state moisture flux.
The assumptions used to  develop  input  data and the limitations resulting from
these assumptions are described below.
Soil Profile and Depth—
     The  analytical  method may  be applied  to single- or  multi-layered  soil
profiles.   Geologic  cross sections  of the site  identify lenses  of  clay and
silt  within the  sand.    However,  these  lenses  are  not  continuous  over the
site.   Therefore, a  uniform  profile  of  sand was  assumed.    In   view  of the
higher  permeability of sand compared  to silt and clay, this is a  conservative
assumption  (i.e., yields lower TOT).
     The  thickness  of the  unsaturated zone  was  estimated from  the geologic
cross  sections and  repoted  ground-water  surface  .elevations.    The minimum
distance  from  the bottom of  the facility to  groundwater  was  estimated to be
6 m.   This  minimum distance  was selected for  the  analysis to yield a worst
case.
Soil Characteristics--
     Saturated  hydraulic  conductivities  for  soils  at  the  facility  were
determined  by  aquifer  tests.   The geometric mean conductivity from tests of
shallow  wells  was  0.079  cm/sec.  For  lack  of other  data, this  value was used
for saturated conductivity in TOT calculations.
     It  should be noted  that  laboratory permeameter  results are the  preferred
source  of saturated conductivity data.   Field measurements from aquifer tests
are easier  to obtain, however,  and are  expected to  be  the major source of
such data presented in Part B applications.   The following limitations to the
use of  these data should be recognized:
                                     6.2

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t    Aquifer test  results  indicate the hydraulic  conductivity of material  in
     the saturated  zone.   Unsaturated TOT calculations  require the saturated
     conductivity  of  material  in  the  unsaturated  zone.    In  this  example,
     materials  in  the two  zones are  very  similar  and  the use  of saturated
     conductivities is not expected to be a major  source of error.
•    Aquifer   test   results  represent   horizontal   saturated  conductivity.
     Unsaturated  TOT  calculations  require  vertical  hydraulic conductivity.
     There  can  be significant differences (e.g.,  order  of magnitude) between
     vertical   and   horizontal   conductivity  for   some   materials.     The
     relationship   between   vertical   and  horizontal  conductivity  for  the
     materials at the site  is not known.   Use of  aquifer  test results should
     be recognized  as a potential source of error.
     No  saturated  moisture  content  data  were  presented   in  the  Part  B
application.    As  described  in  Section  4.0,   total  porosity   is  a  good
approximation  to  saturated  moisture  content.   Because  no  porosity data were
presented  in  the  permit  application,  the default  values presented in Section
4.0 were used.   A  value of  0.41  was  used to represent  the average saturated
moisture content  based  on  the  default  porosities  for  fine  sand  and medium
sand.
     No  data describing  the moisture retention  characteristics  of  soils  at
the site were  provided _n the Part 3 permit application.    Typical values  of
the slope  of  the  moisture  retention  curves  ("b" values)  for different soil
textures are  presented  by  Hall  et al.   (1977).   These  values are  shown  in
Table 6.1.1-1.   A  value  of 4.0 was  selected as  representative  of the sandy
soil at the site.
Moisture Flux—
     No  information  was  presented  in   the  Part  3 application  describing
moisture flow  through the  unsaturated zone.   The  yearly average rainfall  for
the site was reported to be 119 cm.   A conservative assumption  would be  to
ignore   runoff,   evaporation,   and   transpiration    and   assume  that   all
precipitation  is  available   for  recharge.    This assumption  would   tend   to
maximize the unsaturated TOT.
                                     6.3

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          Table 5.1.1-1.  Typical Values for Slope of Soil Moisture
                           Retention Curve (b) Source:  Hall- et al., 1977
                               Soil  Texture
                             Clay              11.7
                             Silty Clay         9.9
                             Silty Clay Loam    7.5
                             Clay Loam          8.5
                             Sandy Clay Loam    7.5
                             Sandy Silt Loam    5.4
                             Silt,Loam          4.8
                             Sandy Loam         6.3
                             Loamy Sand         5.6
                             Sand               4.0
Summary —

     The  following  summarizes   the   input   parameters   for  the  analytical
solution:

•    depth to ground water, L = 6.0 m;

•    saturated hydraulic conductivity, K  .  = 0.079 cm/sec;

•    saturated moisture content, 9f_.  = 0.41;
                                  sac
•    negative one times  the  slope of  log-log  characteristic curve, b  = 4.0;

     and

•    moisture flux,  q =H19 cm/yr.

6.1.2  Solution of TOT

     The  following  solution  follows   the  same  steps  as  those  presented  in

Section 4.0.
Step 1:  Calculate m
                                 °'091
              J    2)(4.0) + 3

Step 2:  Calculate Moisture Content e
          T--   e
           sat    sat
                                     6.4

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                                       0.091
       	(119 cm/yr)  	
       (U.O/y cm/sec M-Jit Wo,UUU sec/yr)     (0.41)
     = 0.16

Step 3:  Calculate Travel Time (T)
              (6.0 m) (0.16)	 _ n pi u_ _ onn J,..-
         (119 cm/yr)(0.01 m/cm) ' °'81 yr ' 29° days
6.2  EXAMPLE 2 - Case Study D
     This Case Study  D  Facility is  a landfill located in the Amargosa Desert,
in the Basin and Range  physiographic  province.   A review of the data provided
in the  Part B permit application  for this facility  is  presented  in the Case
Study Appendix  and the Phase  II Location  Guidance Manual  (see  Appendix C).
This  review  .constitutes  the  only  source  of   site-specific  data  for  the
unsaturated TOT calculation presented below.
     The  site is  underlain  by  at  least  170 m  of  alluvial  and  valley-fill
deposits,  primarily  sands,  gravels,  and  cobbles of   local  origin.   These
alluvial  and  valley-fill  materials  form the  water-table aquifer  at the site.
The depth to ground-water at the site is approximately 90 m.
     The  climate  at the  site  is characterized by  very  low rainfall and high
evaporation.    The   average   rainfall   at  the   site   is  11.4  cm/yr,  with
evaporation  and  potential  evapotranspiration  estimated  at  254   cm/yr  and
91 cm/yr, respectively.
     Analytical methods of unsaturated  travel time are  based on  steady state
flow through the unsaturated zone.   If  the steady state  flux is not known,  it
can be  estimated as  the  net  recharge  at  the site.  A  value of  net recharge
for the  site  of  0.064 cm/yr  is reported in Appendix  C.   Because  of this very
low  flux,  the  assumption of  a  unit  hydraulic  gradient  may not  be  valid.
Therefore,  unsaturated  travel  time was   calculated using  both   analytical
solutions presented in  Section  4.0
                                     6.5

-------
6.2.1  Unit Gradient Analytical Solution
     The  data  requirements  for  this  analytical  method  were  described   in
Example 1.   Soil  characteristic data for  the  site (Appendix C)  were  used  to
construct  soil  characteristic curves for a  typical  soil  at the site.   A plot
of suction head versus moisture content is shown  in  Figure  6.2-1.   The slope
of  the  linear  portion  of  this  curve  was measured  to obtain  a  "b" value  of
3.3.  The  saturated moisture content and  saturated  hydraulic conductivity  of
this soil  are 0.40  and  265  cm/day,  respectively.  From the cross-section data
for the site, the  average depth from the bottom of the  landfill  to the water
table is 76 m.  The above data were used to calculate the travel time for the
steady state flux of 0.064  cm/yr.
Step 1:   Calculate m

     m = --    =             = °'10
Step 2:  Calculate Moisture Content 6


     9 = (^)m9sat
       -       0.064 cm/:>r       °-10 (0.40)
         (265 cm/day) (365 cm/yr)

       = 0.10

Step 3:  Calculate Travel Time (T)
               (76 m) (0.10)
         (0.064 cm/yr)(0.01 m/cm)

       = 12,000 yrs
                                     6.6

-------
         100-
Ld

Z
o

c5

oo
           10-
            1-
          0.1-
Figure 6.2-1.
             0.1                 1
                    THETA
         Plot of Suction Head Versus Moisture
         Content for Case Study D

                 6.7

-------
     The above  equation  can also be  used  to solve for  a  travel  time for the
first 100  ft  below the  facility.   This  100 ft  travel  time  is  4,800 years/
which is well  above the 100-yr location guidance criterion.
     It  should  be  noted  that   this  analytical   solution   is  not   strictly
applicable at  the  site.   The steady  state  moisture content  is so low that  it
does  not  fall  within the  linear,  central  portion  of the  curve  where the
solution  technique is  applicable.    At this  low  moisture  content  (moisture
contents  measured   at  the  site  were  all  at   or   below  the  wilting point),
capillary forces would be significant and the  unit gradient assumption is not
appropriate.
6.2.2   Iterative Analytical Solution
     Because  of   the  low  moisture  contents  at  the  site,  the   iterative
analytical  solution  described   by  Jacobson,  Freshley,  and  Dove  (1985)   is
probably  more  appropriate.    This  solution   allows  for  variable   moisture
contents within  the soil profile.  The  data requirement  for  this method,  in
addition  to  the  steady  state  flux,  are  soil   characteristics   curves for
moisture  content  and  hydraulic  conductivity.    The  curve  for  suction   head
versus  moisture content  was  shown previously  in  Figure 6.2.1-1.   A plot  of
hydraulic conductivity versus suction head  for a typical  soil  at  the site  is
shown  in  Figure 6.2.2-1.   The  data  from  these curves  were  used  to  generate
tables  of  suction  head cV!d moisture  content,  and  suction  head and  hydraulic
conductivity for use  in the solution.
     The above  data were  used with the iterative solution to calculate travel
time  for   a  steady state  flux  of  0.064  cm/yr.     To  employ  the  iterative
solution,  a  grid  system was constructed to  represent the  site.   The   grid
system  was constructed  to represent  the site.    The  grid consisted, of 251
nodes,  uniformly  spaced  at  1  ft,  for  a total depth of 250  ft.   A  boundary
condition  of  0  suction  head  (saturation)  was set  for  the  bottom  node  to
represent  the  water table.  The  iterative  solution was then applied  to  solve
for  the steady state moisture  profile in the  soil column.    This profile  is
shown  in  Figure  6.2.2-2.   Knowing  the  steady  state  moisture  content and
suction head  at each  node,  the  travel  time between each pair of nodes was
calculated, and  these  nodal  travel  times summed to give the total  travel  time
                                     6.8

-------
        0.01J
       0.001J
Q
g    O.OOOId
    0.00001J
   O.OOOOOlH—r-rrrr
         III)   I I I I Illlj  F  I rriTIT)   TTTTTTTTl
0.1       1        10     100    1000
      SUCTION  HEAD, FT.
       Figure 6.2.2-1.
       Plot of Hydraulic Conductivity Versus
       Suction Head for Case Study D

             6.9

-------
     250-i
     200-
tf
	•*
O
§
UJ
      50-
         0.1        0.2        0.3        0.4
              MOISTURE  CONTENT
Figure 6.2.2-2.
Steady State Moisture Profile from Iterative
Analytical Solution for Case Study D
                    6.10

-------
through the  soil  column.  The  steady state travel time  was  16,000 yrs.  The
travel times  for  the first  101  nodes were  summed  to give the  100 ft  travel
time.  This  travel  time  is  5,100 yrs, which is much  greater  than the  location
guidance criterion of 100 yrs.
     It should  be noted  that the  travel  times  obtained from this  iterative
solution  are greater than those  obtained from the  unit gradient  assumption.
The  reason  for this  is  the  nonuniform distribution  of  moisture contents  in
the  soil  column.   Because  of this moisture  distribution, the suction head  is
not  uniform  through the soil  column and the  hydraulic  gradient is  less  than
1, giving longer  travel  times.   Use  of  the  unit  hydraulic gradient  solution,
therefore, yields a conservative answer.
6.3  EXAMPLE  3 - Pulse Loading to the Unsaturated Zone
     The  following  example  uses  an  actual  case  study  to  illustrate how the
UNSAT1D  code  can  be  used  to  estimate  TOT  through  the   unsaturated  zone
associated with a  pulse  loading (i.e.,  transient  flow).  The  example  selected
was  performed for  the  Washington Department  of  Ecology (1979)  to  determine
the  environmental  impacts  of establishing a hazardous waste  disposal  facility
at a site in  south central Washington.
     The  study  addressed the potential  adverse impacts  that  might occur  in  a
number of different  areac,  to include:   earth, air,  water,  flora,'fauna, and
elements  of  the  human  Environment.   The portion  of the  study dealing  with
water  looked  at the  potential  for  migration from  a surface spill,  through the
unsaturated  zone,  into  the  saturated   zone,  and  eventually to  a  discharge
point; in this case  the  Columbia River.  The  purpose of the  unsaturated  flow
modeling  was specifically  to estimate  the  TOT from a hypothetical accidental
liquid spill  at  the surface vertically  downward  through the  unsaturated  zone
to the water  table.
6.3.1  Simulation Details
     The  scenario   simulated  with  the  UNSAT1D  code  was  a  hypothetical
accidental  liquid  spill  of 424,000  liters  spread over an  area  of 930  square
meters.   For a  one-dimensional  simulation,  this  volume of spill  is equivalent
to an  initial ponding of 37.5 cm.   Based on percolation  rates for the soil  at
the  proposed facility,  the  depth  of the ponding was linearly reduced  over  a

                                     6.11

-------
10 day period  until  it had all  infiltrated.   After infiltration, a condition
of no surface evaporation was  assumed  in  the  model, which gave a conservative
drainage prediction.
     One-dimensional vertical  flow  beneath  the spill was assumed.  Neglecting
the  lateral  movement   of   the  infiltrate  due  to  capillary  forces  further
contributed to a conservative  estimate of TOT.
     The vertical distance  between  the spill  and the constant water table  was
52 m.   A  homogeneous  soil  profile was assumed.   The soil moisture retention
characteristics  (Figure  6.3.1-1) and the hydraulic  conductivity versus water
content  relationship   (Figure   6.3.1-2)  for  the   soil  were  obtained  from
laboratory  measurements of soil samples taken  from a  well  constructed near
the  proposed  site.   The  saturated hydraulic  conductivity  equaled  1.7 x 10
cm/sec throughout  the  entire  soil  profile.   For  use in the  model,  the soil
properties   (Figures  6.3.1-1   and   6.3.1-2)   were  fit   with   logarithmic
polynomials.
     The vertical column was  defined with 53 nodes  in the  UNSAT1D  model.   The
node spacing  was uniform at 1  m.   The  initial  pressure conditions (pressure
head)  in the  model  were  set  to  equilibrium (i.e.,  pressure head  equal   to
negative  one  times  the  elevation  above   the   water   table),   as  shown   in
Figure 6.3.1-3.
     During  the time  th; t  the  spill  was  infiltrating, changes  in  pressure
head near  the surface  were rapid and, therefore,  a small time  step  of 0.02
hours  was   used.    This time  step  was  used  for  the  first  20  days  of  the
simulation.   After  20  days the time  step  was doubled  after every iteration
until  a  maximum time  step  of 24  hours  was reached.  The simulation  was  run
fo-  a total time period of  300 years.
6.3.2  Model Predictions
     The UNSAT1D model  results,  in terms of pressure head  versus depth  in  the
soil profile  at various points  in time, are  shown in  Figure  6.3.1-3.   This
figure  illustrates  the advance  of  the  wetting front at  0 and 10 days,  and at
10,  100, and  300 years.
     Figure  6.3.2-1  illustrates the  advance  of the  wetting  front  with time.
This figure shows the  time  required  for  5 cm of  leachate to seep past a given
                                     6.12

-------
CO
            100,000
         !
        •o
        •o
        l-
        • 3
        t/>
        t-
        o.
             10,000
1,000
               100
                10


                     0.00
                            0.10
                                                               0.20

                                                   Moisture  Content
0.30
                     Figure 6.3.1-1.  Soil Water Characteristic Curve  for  Soil  at
                                      Proposed Hanford Hazardous Waste Site
                                      (Source:   Washington Department of Ecology, 1979)
                                                                                             J
                                                                                              j
                                                                                            J
                                                                                                    	j
0.40

-------

U
3
•o
c
o
CJ
     10
          _-,
     10
       -1
    10
      -2
    10
      -4
     10
-5
     10
       -7
     10
       -8
        0.00
         Figure 6.3.1-2.
                                            0.20

                                     Moisture Content
0.30
...  J


 0.40
                   Hydraulic Conductivity Versus Water Content Curve  for

                   Soil at Proposed Hanford Hazardous Uaste Site
                   (Source:  Washington Department of Ecology, 1979)

-------
 E
 u
0.
O)
o
           j

    6,000  •


          0
                                                                    '  0 Days
                                                                             r-t-r- T T -i r-
1,000
2,000         3,000          4,000


        Suction  Head  (cm)
.-.,. i.. ,

 5,000
             Figure 6.3.1-3.   Simulated Pressure Head Versus Depth with  Time
                               at  Proposed Hanford Hazardous Waste Site
                               (Source:  J^hington Department of Ecology,  1979)
.  J

 6,000

-------
     1,000
     2,000
E
o
O.
O
o
     3,000
     4,000
     5,000  -
                                 TT
     6,000  I............ .•_,


           0          10





           Figure  6.3.2-1
    . j.


    20
30
                                                               r-r-VI I I l-» I-T-J-T TfTTTn T l-| I r I T ri r I :
   40


Time (yr)
50
60
1..1-1 - . . I.


      70
Simulated  Advance of Uetting Front at

Proposed Hanford Hazardous Waste Site

(Source:  Washington  Department of  Ecology, 1979)
                                                                                                   i
                                                                                                   i
80

-------
depth  (depth  of  leachate  in  a  one-dimensional  case is  equal  to  volume of
leachate  in  a  three-dimensional   case).     Based  on   the  data  shown  in
Figure 6.3.2-1  the  model  predicted that  approximately  73  years would be
required for 5 cm of leachate from the spill to reach the water table.
     The seepage rate of  leachate  into  the water table (depth of  leachate per
10 year interval)  is  shown  in Figure 6.3.2-2.   Although  the first arrival of
leachate  occurs  in  40 years,  the  maximum rate  occurs  at  100 years.   The
                                          ?                             o
maximum leakage rate over the entire 930 m  area was approximately 3.5 m /yr.
     The  time  required  for percentages  of the total  leachate from the spill
to arrive  at  the water table  is  illustrated  in Figure 6.3.2-3.   The  results
indicate that after  300 years (the total simulation period),  about 80% of the
leachate has reached the water table.
6.3.3  Summary
     The  results  of  the  TOT estimates  with  the UNSAT1D model  show  that the
first  arrival of  infiltrate  from  the  spill  was approximately 40  years after
the  spill occurred,  the maximum  seepage rate  occurred approximately 100 years
after  the  spill,  and  more  than  300   years   are  required  for  all  of  the
infiltrate  to  reach  the water  table.    This case  study  provides an excellent
example of  how  a  numerical model  can be used  to estimate  the travel  time for
a pulse loading.
                                     6.17

-------
                  4.0
                 3.0
00
            >»

            o
            e
            o
            0>
            +-)
            to
2.0  l-
           (O
           -tr
           o
           
                 1.0  I
                  0
                                                 100
                                                                              200
                                                          Time (yr)
                      Figure  6.3.2-2.
                       Simulated Rate of Leachate Discharge  at
                       Proposed Hanford Hazardous Waste  Site

                       (Source:   Washington Department of Ecology,  1979)
                                                                                          j

                                                                                         300

-------
UD
             t.
             Q)
             •o
             c

             o

             CD
             
             (O
             x:
             u
             (0
             o>
01
D)
i-
KO
.C
u
             o-
             >
             •o
            O
                   90
                   80
                   70
      60
      50
                  40
                   30
                   20
                   10
                                              V
                                                  100
                                                                  200
.. . .  J

    300
                                                              Time (yr)
                  Figure 6-3.2-3.
                       Simulated Cumulative Leachate Discharge at
                       Proposed Hanford  Hazardous Waste Site
                       (Source:  uAingtori Department of Ecology,  1979)

-------
                                 SECTION 7.0

                                 REFERENCES


Bates, R.  I.,  and  J.  A.  Jackson.   1980.   Glossary  of  Geology.   American
Geological Institute, Falls Church, VA.

Bond, F.  W.,  C.  R.  Cole and P.  J.  Gutknecht.   1982.   Unsaturated Groundwater
Flow Model  (UNSAT1D)  Computer  Code Manual.  EPRI  CS-2434-CCM,  Electric Power
Research Institute,  Palo Alto,  CA.

Bond, F.  W.,  M.  D.  Freshley and G. W.  Gee.   1982.   Unsaturated Flow Modeling
on  a  Retorted   Oil  Shale  Pile.    PNL-4284,   Pacific  Northwest  Laboratory,
Richland, WA.

Burdine,  N.T.     1953.    "Relative   Permeability   Calculations  from  Size
Distribution Data."  Transactions AIME, Vol. 198, pp. 71-78.

Campbell,  G.  S.    1974.    "A  Simple Method  for  Determining  Unsaturated
Conductivity from Moisture Retention Data."  Soil  Science,  Vol. 117, pp. 311-
314.

EPA.   1984.   Procedures  for Modeling  Flow  Through  Clay Liners  to Determine
Required  Liner  ThicknessEPA/530-SW-84-001,  U.S.EnvironmentalProtection
Agency, Office of Solidv-Jaste,  Washington, DC.

Feedes,  R.  A.,  P.  J.  Kowalik  and H.  Zaradny.    1978.   Simulation  of Field
Water Use and Crop Yield.   John Wiley and Sons, New York, NY.

Freeze,  R.   A.,   and  J.  A.  Cherry.    1979.    Groundwater.   Prentice-Hall,
Englewood Cliffs, NJ.

Gupta, S. K.,  K. Tanju, D.  Nielsen,  J. Biggar, C.  Simmons  and J. Maclntyre.
1978.   Field Simulation of Soil-Water Movement  with  Crop-Water Extraction.
Water  Science  and Engineering  Paper  No,  4013,  Department   of  Land,  Air,  and
Water Resources, University of California, Davis, CA.

Hall, -D.  G.  M.,  A.  J. Reeve, A. J. Thomasson and  V.  F.  Wright.  1977.  Water
Retention,  Porosity, and  Density  of  Field  Soils.    Soil   Survey Technical
Monograph 9, Rothamsted Experimental Station, Harpenden, England.

Heller,  P.  R.,  G.  W.  Gee  and  D.  A.  Myers.    1985.   Moisture  and Texttiral
Variations  in Unsaturated  Soils/Sediments  Near  the  Hanford  Wye  Barricade.
PNL-5377,'Pacific Northwest Laboratory, Richland, WA.
                                     7.1

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Jacobsen, E.  A.,  M.  D.  Freshley  and  F. H.  Dove.   1985.   Investigations of
Sensitivity and Uncertainty in Some  Hydrologic  Models of  Yucca  Mountain and
Vicinity.    DRAFT.PNL-5306,SAND84-7212,  PacificNorthwest  Laboratory,
Rich land, WA.

Millington, R.  J.,  and J.  P. Quirk.   1961.   "Permeability of Porous Solids."
Transactions Faraday Society, Vol. 57, pp. 1200-1207.

Mualem, Y.  1976a.   "A New Model  for Predicting the Hydraulic Conductivity of
Unsaturated Porous Media."  Water Resources Research, Vol.  12, pp. 513-522.

Mualem,  Y.    1976b.   A Catalogue  of the Hydraulic  Properties  of Unsaturated
Soils.  Haifa,  Israel, Israel Institute  of Technology.

Oster,  C.  A.   1982.   Review of Ground-Water  Flow and Transport Models in the
Unsaturated  Zone.     NUREG/CR-2917,  PNL-4427,    U.  S.   Nuclear  Regulatory
Commission, Washington, DC.

Thompson, F.  L.,  and S. W. Tyler.   1984.   Comparison of Two Groundwater  Flow
Models  - UNSAT1D  and  HELP.   EPRI  CS-3695, Electric  Power Research Institute,
Palo Alto, CA.

Walski,  T.  M.,  J. M. Morgan, A.  C.  Gibson  and P. R.  Schroeder.   1983.   User
Guide  for the  Hydroloqic  Evaluation  of Landfill  Performance  (HELP)  Model,
DRAFT.    U.  S.  Environmental  Protection  Agency,  Office  of  Reasearch and
Development, Municipal Environmental Research Laboratory, Cincinnati, OH.
Washington  Department  of  Ecology.    1979.    Environmental   Impact  Statement,
Proposed  Hazardous  I
Ecology, Olympia, WA
Proposed  Hazardous Waste  Site,  Supplement  Final.    Washington  Department of
   _^_
                                      7.2

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                                  GLOSSARY*


Capillarity -- The  action by which  a fluid, such  as  water, is  drawn  up  (or
depressed) in small interstices or tubes as a result of surface tension.

Evapotranspiration -- Loss of water  from  a land area through transpiration of
plants and evaporation from the soil.

Flux  --  The specific  discharge of  water, volumetric flow  divide'd  by cross-
sectional area (Freeze and Cherry, 1979).

Heterogeneous —  Spatial  variation  of physical  properties (Freeze and Cherry,
1979).

Homogeneous  --  No  spatial   variation  of  physical   properties   (Freeze   and
Cherry, 1979).

Hydraulic-Conductivity Boundary —  The boundary  between  two materials having
different hydraulic conductivities (Freeze and Cherry, 1979).

Hydraulic Gradient  -- The rate  of  change of total head  per unit of distance
of flow at a given point.

Hydraulic Head  -- The sum of the  elevation  at a point  and  the pressure head
at the point (Freeze and .Vierry, 1979).

Infiltration —  The flow of  a  fluid into a solid  substance through pores or
small openings.

Inhomogeniety -- Heterogeniety.

Moisture Content  — The  amount  of  moisture in a given soil  mass, expressed as
weight of water divided by weight of oven-dried soil.

Percolation -- See infiltration.

Pore Water Velocity -- See seepage velocity.

Pressure  Head  --  The  portion  of  the  total  hydraulic  head  due  to  fluid
pressure (Freeze and Cherry, 1979).

Seepage  —  The act or process  involving the slow  movement  of  water or other
fluid through a porous material such as soil.
                                     G.I

-------
Seepage Velocity  --  The rate  at  which seepage water  is  discharged through a
porous medium  per  unit area of pore  space perpendicular to  the  direction of
flow.

Steady State  — Flow  where  the magnitude and direction  of  the flow velocity
at any point are constant with  time (Freeze and Cherry, 1979).

Transient  —  Flow where the  magnitude and direction  of  the  flow velocity at
any point varies with  time (Freeze and Cherry, 1979).

Unsaturated Zone  -- A subsurface zone  containing water under pressure less
than that of the atmosphere,  including water head  by capillarity.

Vadose Zone -- See unsaturated  zone.

Water Table -- The surface of  a  body of unconfined ground  water at which the
pressure is equal to that of  the  atmosphere.


*A11 definitions are after Bates  and  Jackson, 1980, unless otherwise noted.
                                     G.2

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