United States
Environmental Protection
Agency
Simulating Radionuclide
Fate and Transport in the
Unsaturated Zone:
Evaluation and Sensitivity
Analyses of Select
Computer Models
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EPA/600/R-02/082
July 2002
Simulating Radionuclide Fate and
Transport in the Unsaturated Zone:
Evaluation and Sensitivity Analyses of
Select Computer Models
by
Jin-Song Chen, Ronald L. Drake and Zhixun Lin
Dynamac Corporation
3601 Oakridge Boulevard
Ada, OK 74820
David G. Jewett
Subsurface Protection and Remediation Division
National Risk Management Research Laboratory
Ada, OK 74820
Contract Number
68-C-99-256
Project Officer
David S. Burden
Subsurface Protection and Remediation Division
National Risk Management Research Laboratory
Ada, OK 74820
National Risk Management Research Laboratory
Office of Research and Development
U.S. Environmental Protection Agency
Cincinnati, OH 45268
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NOTICE
The U.S. Environmental Protection Agency, through its Office of Research and Development, partially funded and collaborated
in the research described here under Contract Number 68-C-99-256 to Dynamac Corporation. It has been subjected to the Agency's
peer and administrative review and has been approved for publication as an EPA document. Mention of modeling codes does not
constitute endorsement or recommendation for use.
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FOREWORD
The U.S. Environmental Protection Agency is charged by Congress with protecting the Nation's land, air, and water resources.
Under a mandate of national environmental laws, the Agency strives to formulate and implement actions leading to a compatible
balance between human activities and the ability of natural systems to support and nurture life. To meet these mandates, EPA's
research program is providing data and technical support for solving environmental problems today and building a science knowledge
base necessary to manage our ecological resources wisely, understand how pollutants affect our health, and prevent or reduce
environmental risks in the future.
The National Risk Management Research Laboratory is the Agency' s center for investigation of technological and management
approaches for reducing risks from threats to human health and the environment. The focus of the Laboratory's research program is
on methods for the prevention and control of pollution to air, land, water, and subsurface resources; protection of water quality in
public water systems; remediation of contaminated sites and ground water; and prevention and control of indoor air pollution. The
goal of this research effort is to catalyze development and implementation of innovative, cost-effective environmental technologies;
develop scientific and engineering information needed by EPA to support regulatory and policy decisions; and provide technical
support and information transfer to ensure effective implementation of environmental regulations and strategies.
Mathematical models are useful tools for determining soil screening levels of radionuclides in the unsaturated zone. However,
models require users to specify various parameters characteristic of the site and chemical of interest. These parameters are not
known without error. Many parameters vary over time and space in manners which are unknown. This is especially true when
models are used to predict future events. This uncertainty in input parameters is associated with an uncertainty in model output
which should be recognized by the model user. This report analyzes several transport models for unsaturated soils and quantifies the
sensitivity of model outputs to changes in input parameters. This information will help users understand the importance of different
parameters, identify parameters which must be determined at the site, interpret model results and apply their findings to specific
problems.
Stephen G. Schmelling, Acting Director
Subsurface Protection and Remediation Division
National Risk Management Research Laboratory
111
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ABSTRACT
Numerical, mathematical models of water and chemical movement in soils are used as decision aids for determining soil
screening levels (SSLs) of radionuclides in the unsaturated zone. Numerous transport and fate modeling codes exist for
predicting movement and degradation of these hazardous chemicals through soils. Many of these codes require extensive input
parameters which include uncertainty due to soil variability and unknown future meteorological conditions. The impacts of
uncertain model parameters upon pertinent model outputs are required for sound modeling applications. Model users need an
understanding of these impacts so they can collect the appropriate parameters for a given site and incorporate the uncertainties in
the model predictions into the decision making process. This report primarily summarizes the findings which address the
uncertainties and sensitivities of model outputs due to uncertain input parameters. However, the report also addresses the
sensitivity of simulated results to conceptual model selection, and the comparison of sensitivity results between models,
illuminating numerical differences and errors.
The objective of the parameter sensitivity studies was to determine the sensitivities and uncertainties of peak
contaminant concentrations and time to peak concentrations at the water table, as well as those for the time to exceed the
contaminant's MCL at a representative receptor well. The five models selected for these analyses were CHAIN, MULTIMED-DP
1.0, FECTUZ, CHAIN 2D, and HYDRUS. All of these are designed to estimate movement and fate of radionuclides through
unsaturated soils. The models span a range in detail and intended use. This report presents information on the sensitivity of
these codes to model conceptualization of radionuclide transport in the vadose zone, to numerical differences and errors, and to
changes and uncertainties in input parameters, as well as presenting information concerning the analysis and interpretation of
certain modeling components. The report does not intend to assess the appropriateness of any model for a particular use nor the
uncertainty due to the model chosen, but it does indicate the problems and limits of using certain modeling components for certain
physical applications.
Model parameters investigated include soil properties such as soil structure and texture, bulk density, water content, and
hydraulic conductivity. Chemical properties examined include distribution coefficient, degradation half-life, dispersion
coefficient, and molecular diffusion. Other site and soil characteristics such as equilibrium/nonequilibrium sorption sites, rooting
depth, recharge rate, hysteretic effects, and precipitation/evapotranspiration were examined. Model parameter sensitivity was
quantified in the form of sensitivity and relative sensitivity coefficients. The sensitivity coefficient is useful when calculating the
absolute change in an output due to a known change in a single parameter. Relative sensitivity is useful in determining the relative
change in an output corresponding to a specific relative change in one input parameter. Relative sensitivities are also used to
compare the sensitivities of different parameters. These results are presented in graphical and tabular forms.
This study identified the limitations and advantages of using the selected codes for assessing the transport and fate of
radionuclides in the unsaturated zone. This study also found the degree of uncertainty that exists in various model output
parameters due to the combination of sensitivities of input parameters, high parameter variabilities, model type with its particular
set of components, and the specific properties of the radionuclides. In addition, the study found that predicted movement of
radionuclides was greater when the natural variability of daily rainfall was incorporated into the model than when only an annual
flux was used. This is because major precipitation events (their daily averages in this case) result in larger fluxes of water and
higher leaching rates that are essentially smoothed over when annual averaged fluxes are used. The study reaffirms that
uncertainty is pervasive in natural systems and that results of modeling efforts presented in a deterministic fashion may be
misleading, unless the results of modeling studies are presented in terms of probabilities of various outcomes. Further, this report
evaluates model parameter sensitivity for a specific scenario, that is, radionuclide transport and fate through a 6 m homogeneous
soil column at the Las Cruses Trench Site in New Mexico. For other scenarios, the general model user should take great care in the
use of the results of the current study. Other sensitivity and uncertainty estimates may be required for the specific conditions and
parameters of interest.
IV
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Contents
Number
Page
Notice ii
Foreword iii
Abstract iv
List of Figures vii
List of Tables x
Section 1 Introduction 1
1.1. The Radionuclide SSL Effort 1
1.2 The Available Modeling Techniques for the Unsaturated Zone 2
1.3 Computer Model Uncertainty and Sensitivity 3
1.4. Report Organization 5
Section 2 Overview of Numerical Models for Simulating Radionuclide Transport 6
2.1 Model Selection for the Radionuclide SSL Effort 6
2.2 Model Description of the Selected Codes 7
2.2.1 The Variably Saturated Water Flow 8
2.2.2. Solute Transport Systems 14
2.2.3. Heat Transport Equation 16
Sections Sensitivity of Simulated Results to Conceptual Model Selection 18
3.1 Time and Space Scales 18
3.2 Domain Selection, Boundary and Initial Conditions 19
3.3 Density and Thermal Gradients 22
3.4 Facilitated Transport and Preferential Pathways 23
3.5 Scale Dependency in Heterogeneous Media 24
3.6 Chemical Adsorption, Chemical Reactions, and Decay Processes 25
3.7 Summary 28
Section 4 Parameter Sensitivity Analysis: Basic Elements
4.1. Computing Sensitivity Coefficients 30
4.2. An Application of Equations (4-1) and (4-2) to a Simple Model 31
Sections Parameter Sensitivity Analysis: Hypothetical Modeling Scenario 37
5.1 Site Selection Process 37
5.2 Selection of the Candidate Site 38
5.3 Characteristics of the Las Graces Trench Site in New Mexico 40
5.4. Development of a Conceptual Model 41
5.5 Base Parameter Selection 44
Section 6 Parameter Sensitivity Analysis: Implementation and Results 48
6.1 General Procedures for Parameter Sensitivity Analysis 48
6.2 Input Parameters for Constant Recharge Rate and Constant Water Content 48
6.3 Input Parameters for Constant Recharge Rate, but Variable Water Content 49
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6.4 Output Parameters Evaluated 51
6.5 Sensitivity Results for the HYDRUS Model 53
Section? Comparison of Sensitivity Results Between Models: Illuminating Numerical Differences 64
7.1 The Modeling Codes and Their Differences 64
7.2 The Parameter Sensitivity Results 66
7.3 Other Results Illuminating Numerical Differences/Errors Between the Models 93
7.3.1 Correction of the MULTIMED-DP 1.0 Code 93
7.3.2 Comparison of the Stehfest and DeHoog Inversion Algorithms 94
7.3.3. Increasing the Base Value of Dispersivity in the FECTUZ Code 94
Sections Summary and Conclusions 97
Section 9 References 102
APPENDICES
Appendix A Empirical Models of the Unsaturated Soil Hydraulic Properties Which Are Used
in the Various Models A-l
References A-5
AppendixB A Discussion on the Scaling of Field Soil-Water Behavior B-l
B.I Symmetry in Nature B-l
B.2 Similitude, Transformation Groups, Inspectional Analysis, Serf-Similarity .... B-2
B.3 Scale Dependence and Scale Invariance in Hydrology B-4
References B-10
Appendix C An Explanation of the Hysteretic Characteristics of Soil-Water Properties C-l
C.I The Origins and Applications of Hysteretic Phenomena C-2
C.2 Hysteresis Loops, Operators and Models C-2
C.3 Hysteresis in Soil-Moisture Parameters C-6
References C-ll
Appendix D The First-Order Decay Chains Used in the Various Models D-l
References D-4
Appendix E The Impact of Using a Nonuniform Moisture Distribution versus a
Uniform Distribution E-l
Appendix F The Impacts of Using Daily Precipitation Rates and Daily Evapotranspiration Rates
versus an Annual Average Recharge Rate F-l
References F-7
Appendix G The Impact of Considering a Layered Soil Column versus a Homogeneous
Soil Column G-l
References G-2
Appendix H A Detailed Analysis of Nonequilibrium Sorption of Pollutants, Mainly for the
Radionuclide 90Sr H-l
H.I A Simplified Version of the Transport Equations H-l
H.2 The Transport of "Tc with a Kd = 0.007 ml/g H-2
H.3 The Transport of "Tc with a Kd= 1.0 ml/g H-3
H.4 The Transport of 90Sr with a Kd= 1.0 ml/g H-4
H.5 The Transport of 90Sr with a Kd= 1.0 ml/g, f= 0, and Varying
Sorption Rates H-5
References H-8
Appendix I Results from the Transport and Fate of Other Radionuclides Not Considered in
the Main Text 1-1
I.I The CHAIN Governing Equations 1-2
1.2 Breakthrough Curves for 99Tc and It's Daughter, 99Ru 1-3
1.3. The Sensitivity of the Five Parent Radionuclides to Recharge Rate 1-5
References 1-8
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Figures
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Figure 1-1. Conceptual risk management spectrum for contaminated soil, where SSL is the soil screening
level, RL is the response level, and SSCG is a hypothetical, site-specific cleanup goal/level
(from U.S. EPA, 2000a) 1
Figure 2-1. Schematic of an unsaturated soil column, soil-water retention curve, and hydraulic
conductivity function, where subscript "s" indicates saturated conditions 11
Figure 2-2. An example of soil structure scaling for scale factors (ah,ae, ak) = (3/2, 2, 7/4). The ratios
W = hj/hj* = 3/2, (9S- 9r) - (9S*- 9r*) = (9r 9r) - (9j*- 9r*) = 2, and K/K^ K/K * = 7/4 13
Figure 3-1. Schematic of modeling applications for simulating three-dimensional field mapping units
using one-dimensional codes (a), and two-dimensional codes (b) 21
Figure 4-1. The graph of the relative sensitivity F in terms of the parameter p for a model defined by
Y = F(p) 30
Figure 4-2. Sensitivities and relative sensitivities of F with respect to a and b, with reference to the base
case in Equation (4-10) 34
Figure 4-3. Sensitivities and relative sensitivities of F with respect to c and d, with reference to the base
case in Equation (4-10) 35
Figure 4-4. Sensitivities and relative sensitivities of F with respect to e and f, with reference to the base
case in Equation (4-10) 36
Figure 5-1. The major segments of the decay chains for the five elements/isotopes considered to be
parents in these analyses (U.S. EPA, 2000b) 38
Figure 5-2. Daily precipitation and potential evapotranspiration (PET) at Las Graces Site, MM. PET is
calculated from daily climate data using Penman's equation (Jensen et al., 1990) 43
Figure 6-1. Sensitivity of "Tc breakthrough (through the 6m layer) to the system parameters using
the HYDRUS Model: (a) distribution coefficient, (b) recharge rate, (c) water content,
(d) bulk density, (e) dispersivity, (f) diffusion coefficient in water, (g) saturated conductivity,
(h) saturated water content, (i) residual water content, (j) van Genuchten retention parameter a,
(k) van Genuchten retention parameter P 54
Figure 6-2. Sensitivity and relative sensitivity of peak concentrations at the depth of 6m to the system
parameters using the HYDRUS Model: (a) distribution coefficient, (b) recharge rate, (c) water
content, (d) bulk density 55
(Cont.) Sensitivity and relative sensitivity of peak concentrations at the depth of 6m to the system
parameters using the HYDRUS Model: (e) dispersivity, (f) diffusion coefficient in
water, (g) saturated conductivity, (h) saturated water content 56
(Cont.) Sensitivity and relative sensitivity of peak concentrations at the depth of 6m to the system
parameters using the HYDRUS Model: (i) residual water content, (j) van Genuchten
retention parameter a, (k) van Genuchten retention parameter (3 57
Figure 6-3. Sensitivity and relative sensitivity of time to reach peak concentrations at the depth of 6m to the
system parameters using the HYDRUS Model: (a) distribution coefficient, (b) recharge rate, (c)
water content, (d) bulk density 58
(Cont.) Sensitivity and relative sensitivity of time to reach peak concentrations at the depth of
6m to the system parameters using the HYDRUS Model: (e) dispersivity, (f) diffusion
coefficient in water, (g) saturated conductivity, (h) saturated water content 59
(Cont.) Sensitivity and relative sensitivity of time to reach peak concentrations at the depth
of 6m to the system parameters using the HYDRUS Model: (i) residual water content, (j) van
Genuchten retention parameter a, (k) van Genuchten retention parameter (3 60
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Figure 6-4. Sensitivity and relative sensitivity of time to exceed MCL to the system parameters using the
HYDRUS Model: (a) distribution coefficient, (b) recharge rate, (c) water content, (d) bulk
density 61
(Cont.) Sensitivity and relative sensitivity of time to exceed MCL to the system parameters
using the HYDRUS Model: (e) dispersivity, (f) diffusion coefficient in water, (g) saturated
conductivity, (h) saturated water content 62
(Cont.) Sensitivity and relative sensitivity of time to exceed MCL to the system parameters
using the HYDRUS Model: (i) residual water content, (]) van Genuchten retention parameter
a, (k) van Genuchten retention parameter p 63
Figure 7-1. Sensitivity of "Tc breakthrough (through the 6m layer) to the distribution coefficient using
the CHAIN, MULTIMED-DP 1.0, FECTUZ, CHAIN 2D, HYDRUS Models, where Kd =
0.019 ml/g, 0.007 ml/g, and 0.001 ml/g 67
Figure 7-2. Sensitivity of "Tc breakthrough (through the 6m layer) to the recharge rate using the CHAIN,
MULTIMED-DP 1.0, FECTUZ, CHAIN 2D, and HYDRUS Models, where q = 0.030 cm/d,
0.024 cm/d, and 0.016 cm/d 68
Figure 7-3. Sensitivity of "Tc breakthrough (through the 6m layer) to the water content using the CHAIN,
MULTIMED-DP 1.0, FECTUZ, CHAIN 2D, and HYDRUS Models, where 9 = 0.22 cmVcm3,
0.16cm3/cm3, and 0.10 cmVcm 69
Figure 7-4. Sensitivity of "Tc breakthrough (through the 6m layer) to the bulk density using the CHAIN,
MULTIMED-DP 1.0, FECTUZ, CHAIN 2D, and HYDRUS Models, where p = 1.78 g/cm3,
1.70 g/cm3, and 1.62 g/cm3 70
Figure 7-5. Sensitivity of "Tc breakthrough (through the 6m layer) to the dispersion coefficient using
the CHAIN Model, where D = 2.20 cmVd, 1.00 cmVd, and 0.40 cmVd 71
Figure 7-6. Sensitivity of "Tc breakthrough (through the 6m layer) to the dispersivity using the
MULTIMED-DP 1.0, FECTUZ, CHAIN 2D, and HYDRUS Models, where DL = 5.33 cm,
4.53 cm, and 3.73 cm 72
Figure 7-7. Sensitivity of "Tc breakthrough (through the 6m layer) to the diffusion coefficient in water
using the CHAIN 2D and HYDRUS Models, where Dw = 2.53 cm2/d, 1.73 cm2/d, and 0.93
cm2/d 73
Figure 7-8. Sensitivity of "Tc breakthrough (through the 6m layer) to the saturated conductivity using the
MULTIMED-DP 1.0, FECTUZ, CHAIN 2D and HYDRUS Models, where Ks = 365 cm/d,
270 cm/d, and 175 cm/d 74
Figure 7-9. Sensitivity of "Tc breakthrough (through the 6m layer) to the saturated water content using the
MULTIMED-DP 1.0, FECTUZ, CHAIN 2D and HYDRUS Models, where 9S = 0.35 cmVcm3,
0.32 cmVcm3, and 0.29 cmVcm3 75
Figure 7-10. Sensitivity of "Tc breakthrough (through the 6m layer) to the residual water content using the
MULTIMED-DP 1.0, FECTUZ, CHAIN 2D and HYDRUS Models, where 9r = 0.103 cmVcm3,
0.083 cmVcm3, and 0.063 cmVcm3 76
Figure 7-11. Sensitivity of "Tc breakthrough (through the 6m layer) to the van Genuchten retention
parameter a using the MULTIMED-DP 1.0, FECTUZ, CHAIN 2D and HYDRUS Models,
where a = 0.059cm-1, 0.055cm-1, and 0.051cm-1 77
Figure 7-12. Sensitivity of "Tc breakthrough (through the 6m layer) to the van Genuchten retention
parameter (3 using the MULTIMED-DP 1.0, FECTUZ, CHAIN 2D and HYDRUS Models,
where (3 = 1.59, 1.51, and 1.43 78
Figure 7-13. Sensitivity and relative sensitivity of (a) peak concentration at the depth of 6m, (b) time to
reach peak concentration at the depth of 6m, and (c) time to exceed MCL at the receptor
well to the distribution coefficient using the CHAIN, MULTIMED-DP 1.0, FECTUZ,
CHAIN 2D and HYDRUS Models 80
Figure 7-14. Sensitivity and relative sensitivity of (a) peak concentration at the depth of 6m, (b) time to
reach peak concentration at the depth of 6m, and (c) time to exceed MCL at the receptor
well to the recharge rate using the CHAIN, MULTIMED-DP 1.0, FECTUZ, CHAIN 2D and
HYDRUS Models 81
Figure 7-15. Sensitivity and relative sensitivity of (a) peak concentration at the depth of 6m, (b) time to
reach peak concentration at the depth of 6m, and (c) time to exceed MCL at the receptor well
to the water content using the CHAIN, MULTIMED-DP 1.0, FECTUZ, CHAIN 2D and
HYDRUS Models 82
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Figure 7-16. Sensitivity and relative sensitivity of (a) peak concentration at the depth of 6m, (b) time to
reach peak concentration at the depth of 6m, and (c) time to exceed MCL at the receptor well
to the bulk density using the CHAIN, MULTIMED-DP 1.0, FECTUZ, CHAIN 2D and
HYDRUS Models 83
Figure 7-17. Sensitivity and relative sensitivity of (a) peak concentration at the depth of 6m, (b) time to
reach peak concentration at the depth of 6m, and (c) time to exceed MCL at the receptor
well to the dispersion coefficiient using the CHAIN Model 84
Figure 7-18. Sensitivity and relative sensitivity of (a) peak concentration at the depth of 6m, (b) time to
reach peak concentration at the depth of 6m, and (c) time to exceed MCL at the receptor
well to the dispersivity using the MULTIMED-DP 1.0, FECTUZ, CHAIN 2D, and
HYDRUS Models 85
Figure 7-19. Sensitivity and relative sensitivity of (a) peak concentration at the depth of 6m, (b) time to
reach peak concentration at the depth of 6m, and (c) time to exceed MCL at the receptor well
to the diffusion coefficient in water using the CHAIN 2D and HYDRUS Models 86
Figure 7-20. Sensitivity and relative sensitivity of (a) peak concentration at the depth of 6m, (b) time to
reach peak concentration at the depth of 6m, and (c) time to exceed MCL at the receptor well
to the saturated conductivity using the MULTIMED-DP 1.0, FECTUZ, CHAIN 2D, and
HYDRUS Models 87
Figure 7-21. Sensitivity and relative sensitivity of (a) peak concentration at the depth of 6m, (b) time to
reach peak concentration at the depth of 6m, and (c) time to exceed MCL at the receptor well
to the saturated water content using the MULTIMED-DP 1.0, FECTUZ, CHAIN 2D, and
HYDRUS Models 88
Figure 7-22. Sensitivity and relative sensitivity of (a) peak concentration at the depth of 6m, (b) time to
reach peak concentration at the depth of 6m, and (c) time to exceed MCL at the receptor
well to the residual water content using the MULTIMED-DP 1.0, FECTUZ, CHAIN 2D,
and HYDRUS Models 89
Figure 7-23. Sensitivity and relative sensitivity of (a) peak concentration at the depth of 6m, (b) time to
reach peak concentration at the depth of 6m, and (c) time to exceed MCL at the receptor well
to the van Genuchten retention parameter a using the MULTIMED-DP 1.0, FECTUZ,
CHAIN 2D, and HYDRUS Models 90
Figure 7-24. Sensitivity and relative sensitivity of (a) peak concentration at the depth of 6m, (b) time to
reach peak concentration at the depth of 6m, and (c) time to exceed MCL at the receptor
well to the van Genuchten retention parameter (3 using the MULTIMED-DP 1.0,
FECTUZ, CHAIN 2D, and HYDRUS Models 91
Figure 7-25. Water content distributions predicted by the HYDRUS, CHAIN 2D, FECTUZ, and
MULTIMED-DP 1.0 models. Note that the water contents (•) obtained from the originally
distributed MULTIMED-DP 1.0 code are in error. The corrected code gives a consistent water
content distribution (A.) with the other three models 93
Figure 7-26. Sensitivity of "Tc BTCs at 6m depth to water content, where (a) uses Stehfest Algorithm of
MULTIMED-DP 1.0, (b) uses DeHoog Algorithm of MULTIMED-DP 1.0, and (c) uses
DeHoog Algorithm of FECTUZ Code 95
Figure 7-27. Comparison of the "Tc BTCs for the HYDRUS, CHAIN 2D and CHAIN Models for the
base values of the input parameters with the ETC for the FECTUZ Model with the base
value of DL = 4.53 cm replaced by the value DL = 6.53 cm 96
APPENDICES
Figure A-l. Schematics of the soil-water retention curve (a) and the hydraulic conductivity function
(b) for the VC-Model (from Simuneketal., 1998) A-3
Figure B-l. Geometrically similar figures, where (a) is the reference figure with characteristic length L*,
(b) is a similar figure with characteristic length Lb L* = otjLj, with scale factor o^ = 2, and
(c) is a similar figure with a2L2 = L*, a2= l/i (fromGuymon, 1994) B-3
Figure B-2. The depiction of a set of vertical soil profiles ppp2,... distributed over a field mapping unit,
where z represents the local variable within a soil profile and R (i = 1,2,...) represent the
horizontal vectors in the xy-plane giving the global positioning of the vertical profiles,
P,,P2, B-5
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Figure B-3. (a) Unsealed observations of Seo(h), (b) Scaled observations of Seo(h*), showing the
reference relationship as a solid curve, (c) unsealed observations of K(Seo), and (d) scaled
observations of K*(Seo) (from Warrick, et al. 1977) B-6
Figure B-4. The application of linear scaling to a set of soil-moisture observations, resulting in a set of
m similar soil classes. The mth class of similar soils accounts for the soil structures
sml, sm2,..., smq within the soil textural class m. The number of similar soil classes
corresponds to the number of soil textural classes B-8
Figure C-l. A continuous hysteresis loop for a system whose state is given by the couple (u,v), where u
is the input and v is the output (after Visintin, 1994) C-2
Figure C-2. A defining sketch of Madelung's Rules for the memory attributes of ferromagnetic
hysteresis (after Brokate and Sprekels, 1996) C-3
Figure C-3a. A relay with hysteresis, or a delayed relay, defined by the parameters (a, b, uc, vc)
with repect to the system defined by states (u,v), after Visintin (1994) C-5
Figure C-3b. An approximation to a continuous hysteresis loop by a linear combination of a finite family of
delayed relays. The quantity Rf is the region inside the discontinuous loop formed by the finite
family of relays, after Visintin (1994) C-5
Figure C-4. A cross section of a soil pore and the solid soil particles that make up its walls, showing areas
drained by the pull of gravity, areas where water is held by capillary forces, and areas where
water is held by surface forces (e.g., van der Waals forces), after Miller and Donahue (1995) C-8
Figure C-5. The "ink bottle" effect demonstrating that draining/drying soils under the influence of
capillary forces retain more water at a given soil-water pressure than a wetting soil at the
same water pressure, after Guymon( 1994) C-9
Figure C-6. A hypothetical soil-moisture hysteresis loop which is discontinuous for pressure heads near
zero, showing the main drying and main wetting curves, and example primary and secondary
scanning curves, after Knox et al. (1993) C-10
Figure E-l. Water content distributions predicted by the HYDRUS, CHAIN 2D, FECTUZ, and
MULTIMED-DP 1.0 Models. Note that the water contents (•) obtained from the originally
distributed MULTIMED-DP 1.0 Code are in error. The corrected code gives a consistent
water content distribution (*) with the other three models E-2
Figure E-2. Comparison of the breakthrough curves predicted by the CHAIN, HYDRUS, CHAIN 2D,
FECTUZ, and MULTIIMED-DP 1.0 Models for the base case given in Section 6. The top
curves are for a nonuniform water content and the bottom curves are for 9 = 0.16
throughout the soil column. There are no CHAIN results in the top graph because 9 can
only be constant in this model E-3
Figure E-3. Sensitivity of "Tc breakthrough (through the 6 m layer) to the distribution coefficient using
the CHAIN 2D Model, for a nonuniform water content (top) and for a uniform water content
(bottom) E-4
Figure E-4. Sensitivity of "Tc breakthrough (through the 6 m layer) to the dispersivity using the CHAIN
2D Model, for a nonuniform water content (top) and for a uniform water content (bottom) E-4
Figure F-l. The annual precipitation amounts and the monthly average amounts in centimeters for the
Las Graces, NM Site, corresponding to the daily record in Figure 5.2 F-3
Figure F-2. The water stress response function for the Feddes Module of the HYDRUS Code
(Simunek, et al., 1998) F-4
Figure F-3. Cumulative amounts in centimeters of precipitation, actual evapotranspiration (ET), and net
recharge (precipitation minus actual ET) during a HYDRUS Model simulation using daily
variable precipitation and potential ET rates at the surface. Cumulative net recharge and
ET vary between the two figures because of differences in the root water uptake scenarios,
(hb h2, h3). The precipitation/PET segment from "a to b" is repeated from "b to c." F-4
Figure F-4. Comparison of predicted "Tc breakthrough curves (through the 6 m layer) using the variable
precipitation/actual ET versus uniform recharge rate in the HYDRUS Model. Average recharge
rate is calculated as the mean net amount of precipitation and actual ET from 0 to 12,000 days.
The net recharge varies between the two sets of curves due to the root-uptake scenario,
(hi,li2 ,h3) F-6
Figure G-l. Sensitivity of "Tc breakthrough (through the unsaturated zone with a water table at a depth
of 6 m) to the distribution coefficients in a layered soil and in a uniform soil using the
HYDRUS Model G-2
Figure H-l. The c- and s distribution of Equations (H-l) and (H-2) for 90Sr for (co,f) = (0,1),
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Number
(0.032 d-1, 0.47), (0.032d-!, 0). Distributions were derived by the HYDRUS Code,
(a) gives the concentration in solution and (b) gives the concentration on the soil matrix ............... H-4
Figure H-2. (a) Breakthrough curves for the liquid phase concentration at the 6 m level, (b)
Concentration curves for the nonequilibrium solid phase at the 6 m depth. Curves for co =
6.5 x 10-2d-! and 6.5 x IQ-M'1 are basically the same for both (a) and (b) ...................................... H-7
Figure H-3. Liquid phase concentration curves (a) and solid phase concentration curves (b) for 90Sr,
for various times and for co = 6.5 x lO^d'1, where zero depth is the surface and -600 cm
is the hypothetical water table [[[ H-8
Figure H-4. Liquid phase concentration curves (a) and solid phase concentration curves (b) for 90Sr,
for various times and for co = 6.5 x lO^d'1, where zero depth is the surface and -600 cm
is the hypothetical water table [[[ H-8
Figure H-5. Liquid phase concentration curves (a) and solid phase concentration curves (b) for 90Sr,
for various times and for co = 6.5 x lO^d'1, where zero depth is the surface and -600 cm
is the hypothetical water table [[[ H-9
Figure H-6. Liquid phase concentration curves (a) and solid phase concentration curves (b) for 90Sr, for
various times and for co = 6.5 x lO^d'1, where zero depth is the surface and -600 cm is the
hypothetical water table [[[ H-9
Figure 1-1. The normalized concentration of a radionuclide at the bottom of a soil column (the source
being at the top of the column) versus a hypothetical decay -mobility scale (DMS), where
I represents highly mobile, long-lived species, III represents highly immobile, short-lived
species, and II represents species with intermediate mobilities and half -lives ................................... I- 1
Figure 1-2. (a) Breakthrough curves of the daughter product, "Ru, from the "Tc decay using the
CHAIN and FECTUZ Models for base case simulation, and (b) the breakthrough curve
for"Tc. The text will explain a, b, c, d, e andf. [[[ 1-4
Figure 1-3. Sensitivity of radionuclide transport through the unsaturated zone to recharge rate (q)
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Tables
Number Page
Table 2-1. Comparison of the Model Components in Each of the Five Codes Being
Analyzed in this Report 9-10
Table 4-1. Sensitivities and Relative Sensitivities of Output F with Respect to the Six Input
Parameters, for Arbitrary Values of the Inputs 32
Table 4-2. Sensitivity and Relative Sensitivity of Output F to Individual Input Parameters with
Reference to the Base Case Given in Equation (4-10) 33
Table 5-1. Partial List of Radionuclide Contaminated and Disposal Sites in the U.S.
(U.S. EPA'sVISITT Database) 39
Table 5-2. Soil Hydraulic Properties at the Las Cruces Trench Site for SSG Model
Evaluation Study (from Wierenga, etal., 1991) 42
Table 5-3. Characteristics of the Las Cruces Trench Site for SSG Model Evaluation Study (from
Gee, etal., 1994) 42
Table 5-4. Base Values of Input Parameters for Unsaturated Zone Radionuclide Models
(from Wierenga, et al., 1991; Gee, et al., 1994; U.S. EPA, 2000ab; and U.S. EPA
VISITT Database) 45
Table 6-1. The Sensitivity Analysis Performed (•) for the Five Models Under the Assumption of
Constant Recharge Rate and Constant Water Content 50
Table 6-2. The Sensitivity Analysis Performed (•) for Four of the Five Models Under the
Assumption of Constant Recharge Rate and Variable Water Content, where "Base"
Represents the Base Parameter Values Given in Table 5-4, and the Water Content
Profile, 9(z), Varies with the Changing van Genuchten Parameters (Ks,9s,9r, a, B) 52
Table 6-3. Input Parameters for Each Model and the Range of Each Variable Parameter, Along
with the Base Value of the Parameter 52
Table 6-4. Relative Sensitivities for Cpeak, Tpeak and TMCL with Respect to the Input Parameters
forHYDRUS, Measured at the Base Values of the Input Parameters 53
Table 7-1. The Sensitivity Analyses Performed (•) for the Five Models Under the Assumption
of Constant Recharge Rate and Constant Water Content, and the Sensitivity Analyses
Performed (O) for Four of the Five Models Under the Assumption of Constant
Recharge Rate and Variable Water Content 65
Table 7-2. The Values of the Dispersion Factor, 9D, as Derived from Equation (6-4) and the
Base Values of D, DL, Dw, 9 and q and the Ranges of D, DL and Dw as given in Table 6-3 66
Table 7-3. Summary of Relative Sensitivities for the Outputs Obtained from the "Tc Breakthrough
Curves with Respect to All the Pertinent Input Parameters for All Models, Referenced
to the Base Values of the Input Parameters 92
Table 8-1. The Possible Numerical Differences and Errors (•) Separating the Five Models Tested
for Water Flow and Radionuclide Transport through the 6m, Vertical, Homogeneous
Soil Column 100
Table 8-2. The Figures in Section 7 Comparing "Tc BTCs and Output Sensitivities with Respect
to Model Input Parameters and Modeling Codes 101
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Number Page
APPENDICES
Table E-l Comparison of Results Derived from Figures E-3 and E-4 for the Distribution Coefficient
Kd and the Dispersivity DL, respectively. The Values of C k, Tpeak and TMCL are
Given for the Base Values of Kd and DL, and the Relative Sensitivities of These Output
Quantities to Kd and DL are Given, These Values Also Being Taken at the Base Values
ofKdandDL E-3
Table H-l. Comparison of Nonequilibrium and Equilibrium Results for "Tc BTCs for Kd Values of
0.007 and 1.0 ml/g H-3
Table H-2. Liquid Phase and Solid Phase Peak Concentrations at the Hypothetical Water Table
for a Sequence of Sorption Rates, along with the Corresponding Times to Arrive
at Those Peaks H-7
Table 1-1. Coefficients of the Advection, Diffusion, and Sink Terms in Equation (I-1) 1-3
Table 1-2. C k Normalized by Source Concentration C0 and Tpeak for Each Radionuclide and for
Three Recharge Rates 1-6
Table 1-3. The Relative Diffusion Factors and the Relative Decay Factors for the Five
Parent Radionuclides 1-7
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XlVl
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Section 1
Introduction
This report is ccncemed with the evaluation and sensitivity analysis of select ccnputer nails for sinulating
radicnuclide fate and transport in the unsaturated zone. The work reported here was perfomBd in support of the
U.S. Environmental Protection Agency' s (EPA's) project on Soil Screening Guidance (SSG) for Radicnjclides (U.S.
EPA, 200Cab). The models reviewed and evaluated in the current report form a grail subset of the models available
to the public, and those which are considered have not met U.S. EPA approval for the exclusive use in the Soil
Screening level (SSL) process. Other models also may be applicable to the SSL effort, depending on pollutant-and
site-specific cdrcurstances.
1.1. The Radionuclide SSL Effort
The U.S. EPA developed the Soil Screening Guidance for Radionuclides (U.S. EPA, 200Qab) as a tool to help
standardize and accelerate the evaluation and cleanup of soils contaminated with radioactive materials at sites on the
National Priorities List (NPL) with future residential land use. This guidance is intended for the appropriate
environmental professionals to be able to calculate risk-based, site-specific, SSts for radionuclictes in soil, t±us
allowing than to identify areas needing further investigation at NPL sites. However, these SSLs alone do not trigger
the need for response actions or define "unacceptable" levels of radionuclides in soil. In these guidance documents,
"screening" refers to the process of identifying and defining areas, radionuclides, and physical conditions at a
particular site that do> not ragjire further Federal action. Generally speaking, at sites where radionuclide
concentrations fall belcw SSLs, no further action or study is warranted under the Qonprehensive Environmental
Response, Compensation and Liability Act (CERCLA). Where radionuclide concentraticns equal or exceed SSLs,
further study or investigation, but not necessarily cleanup, is generally warranted.
In identifying and managing risks at sites, EPA considers a spectrum of radionuclide ooncentrations (Figure 1-1).
The level of concern associated with the ooncentrations in the figure depends en the likelihood of exposure to
radioactive soil ccntaminaticn at levels of potential concern to hnan health. As stated above, if the soil
concentrations fall between zero and SSL, no further study is warranted under CERCLA. If the soil concentrations
fall between SSL and RL (the response level), then further study and investigation are warranted, but response action
nay not be warranted. For sites requiring cleanup, the goal or level of cleanup, S3CG, nust fall belcw RL and be
suoh that no further action is regjirad.
No Further Study
Warranted
Under CERCLA
Site-Specific
Cleanup
Goal/Level
Response Action
Clearly
Warranted
-Be-
sses
SSL
RL
Increasing
Concentrations
Figure 1-1. Conceptual risk management spectrum for contaminated soil, where SSL is the soil screening level, RL is
the response level, and S3CG is a hypothetical, site-specific cleanup goal/level (from U.S. EPA, 200Qa).
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The soil screening process is a seven-step approach (U.S. EPA, 2000a) consisting of the following steps:
Developing a conceptual site model (CSM),
Comparing the CSM to the SSL scenario,
Defining data collection needs,
Sampling and analyzing soils at the site,
Calculating site-specific SSLs,
Comparing site soil radionuclide concentrations to calculated SSLs,
Determining which areas of the site require further study.
In following this process, several exposure pathways are addressed by the SSLs for radionuclides, such as produce
ingestion, soil ingestion, external radiation from soil and dust, and the consumption of drinking water. The models
considered in this report are concerned with the pathway from a surface source of radionuclides, down through the
unsaturated zone and finally leading to a ground-water aquifer which is a source of drinking water.
1.2 The Available Modeling Techniques for the Unsaturated Zone
Over the last ten to fifteen years, due in part to large increases in computer speed and memory, there has been an
explosion in the number of modeling techniques and numerical computer codes applicable to the investigation and
simulation of flow and transport through the unsaturated zone. The general discussion of subsurface contaminant
hydrogeology is covered in many recent texts, among which are those of Zheng and Bennett (1995), Fetter (1999),
Bedient et al. (1999) and Charbeneau (2000). In addition, there are several excellent books and reports concerned
with individual processes (e.g., chemical reactions, sorption processes, and media interface problems) and modeling
tools and their properties. These publications include texts on physicochemical hydrodynamics by Probstein (1994),
environmental chemodynamics by Thibodeaux (1996), and a text on multicomponent fluids by Drew and Passman
(1999). Two texts on modeling tools and their mathematical properties are those of Logan (2001) and
Nirmalakhandan (2002). A compilation of simple mathematical models for the estimation of infiltration rates of
water into the vadose zone is given in Ravi and Williams (1998) and Williams et al. (1998), while discussions of
porous medium thermodynamics are given in Bear and Nitao (1995) and Spanos (2002). The dynamics of fluids in
fractured rock, under both saturated and unsaturated conditions, have been covered in many references, such as Bear
(1993), Clemo and Smith (1997), Stockman (1997), Pruess et al. (1999), and Faybishenko et al. (2000). Because of
uncertainties in, lack of knowledge about, and the inherent randomness of subsurface hydrogeology, some
investigators have forsaken the "deterministic world" of the subsurface and have concentrated on the "random"
nature of subsurface processes and their stochastic simulation. Important references in this regard are those of Dagan
(1989), Chiles and de Marsily (1993), and Gelhar (1993). To better understand flow and transport processes on the
microscale, and how these processes affect macroscale flow and transport, investigators have applied cellular
automata and fractal techniques in their analyses, techniques that are very adaptable to the current upsurge in parallel
computational hardware and software. The use of fractal geometry in geology, geophysics and porous media is
discussed in Turcotte (1992), Adler and Thovert (1993), Flury and Fruhler (1995), and Gouyet (1996). Percolation
theories, homogenization, pore network models, and lattice gas automata have been used to analyze flow and
transport processes in homogeneous and heterogeneous porous media by authors such as Gao and Sharma (1994),
Hornung (1997), Kaiser (1997), Vollmayr et al. (1997), and Yortsos and Shing (1999). Another tool for bridging the
gap between molecular and macroscopic simulation of flows in porous media, multiphase flows, and colloidal
suspensions is that called "dissipative particle dynamics." This technique is described in Coveney and Novik (1996),
Groot and Warren (1997), and Boek and van der School (1998).
In the first half of the 1990s, three comprehensive documents were published which were concerned with
simulation codes for flow and the transport of solvents and heat in both saturated and unsaturated zones of the
subsurface. In 1993, van der Heijde and Elnawawy authored a report on the compilation of ground-water models for
saturated and unsaturated conditions. This compilation included analytical and numerical codes for the following
areas: saturated flow, unsaturated flow, solute transport, heat transport, gas flow and vapor transport, flow and
transport in fractured rock, hydrogeochemical models, optimization models for ground-water management, and
multiphase flow models. In 1994, van der Heijde authored a report on the compilation of unsaturated /vadose zone
models. This report considered about 90 modeling codes covering flow processes, solute transport and heat
transport. The final report of this series, van der Heijde and Kanzer (1995), considered ground-water model testing:
a systematic evaluation and testing of code functionality, performance, and applicability to practical problems. The
procedures were exemplified by applying them to a specific three-dimensional flow and solution transport code.
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Two other recent books on the flow and transport of pollutants in the subsurface environment are of interest.
Corwin et al. (1999) edited a report on the assessment of non-point source pollution in the vadose zone. Papers in
this report considered the application of geographic information systems (GIS), transfer functions, fuzzy logic,
fractals/ scaling, hierarchical theory and uncertainty analysis to flow and solute transport through heterogeneous soil
layers. Mackay (2001) detailed the intermedia transport of reactive organics through the environment using a series
of multiple-compartment, mass-balance modules. The modules included those of air, water, soil, sediments, and
biological entities.
1.3 Computer Model Uncertainty and Sensitivity
Model predictions are uncertain and erroneous because of uncertainties and errors that can occur at various
points in the modeling process and in the model's structure. These points can be illuminated by considering the
modeling protocol described in Bedient et al. (1999); namely:
1. Identify the purpose of the model and the site(s) to which it will be applied.
2. Develop a conceptual model of the system based on the model purpose and site application.
3. Select the governing conservation and constitutive equations describing the physical system, as well as the
computer codes required to solve the system. Verify that the governing equations describe the physical,
chemical and biological processes occurring in the physical system. Verify that the computer codes
accomplish stated objectives by comparing model results to analytical solutions of know problems.
4. Design the model for the site(s) at hand, including the selection of initial and boundary conditions, grid
design and grid size, time increments and parameters, and estimates of model parameters.
5. Calibrate the designed model by determining a set of model input parameters that approximate field-
measured pressure heads, flows and/or concentrations, thus establishing that the model can reproduce field-
measured values of the unknown variables.
6. Determine the effects of uncertainty on model results through the use of uncertainty and sensitivity analyses.
7. Verify the designed, calibrated model by testing the model's ability to reproduce another set of field
measurements using the model parameter set developed in the calibration process.
8. Predict site results for desired scenarios using the designed, calibrated, verified model.
9. Determine the effects of uncertainty on model predictions.
10. Present modeling design and results for the site(s) and scenarios in question.
11. Post audit and redesign model as necessary. Modify and refine the site model if the comparison of model
predictions against new sets of field data warrant such action.
The lists of modeling errors given by Guymon (1993), van der Heijde and Kanzer (1995), and Mulla and
Addiscott (1999) are related to the various steps given in this 11-step modeling protocol. Errors in
conceptualization or model structure occur when the processes and the assumptions represented by the model fail to
represent reality. An example is a model, which simulates solute transport using the convective-dispersive equation
over an area in which two-region, or macropore transport is significant. The failure of the model in this case may be
compensated for by using unrealistic large values for saturated hydraulic conductivity and solute dispersivity, but this
approach is usually not satisfactory. Errors in experimental measurement are also common due to biased sample
collection, sample handling and storage errors, or sample analysis errors. Quality control and quality assurance
procedures can be used to minimize such errors. Mulla and Addiscott (1999) state that a model can be considered
properly validated if the goodness-of-fit criterion for validation is less than the experimental error among
experimental replicates.
Since most mathematical systems require numerical, computer codes to solve them, the discretization in space
and time introduces error. Further, there are roundoff errors, which represent the differences between the "true"
representation of the dependent variables in the discretized formulas and the machine representation of those
variables in the computer (van der Heijde and Kanzer, 1995). Truncation errors are the difference between the true,
discretized representation of the variables and the exact value of the variables. Truncation errors are algorithm
errors and often occur because the distribution of the unknown variables is represented by truncated polynomial
expansions. As the number of polynomial elements increases in a particular expansion, the truncation errors tend to
decrease. As one might expect, the numerical solution technique used for a given mathematical system may have a
significant impact on the model results. For example, Mehl and Hill (2001) considered five common numerical
techniques for solving the advection-dispersion equation (i.e., finite difference, predictor-corrector, total variation
diminishing, method of characteristics, and modified method of characteristics). The results of their investigation
indicate that, depending on the solution technique employed and the choice of solute model dispersivity, model
calibration can produce significantly different estimated values of hydraulic conductivity that result in different
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simulated flow fields and potentially very different solute concentration predictions. As Bedient et al. (1999) state,
model calibration is very subjective, and in many cases, does not yield a unique set of parameters that reproduce field
conditions.
Guymon (1994) states that boundary conditions and initial conditions are always a problem and a source of
errors for unsaturated flow processes. For example, the land surface boundary condition for fluid flow is extremely
difficult to specify in practical field problems. About the only thing that one can do to evaluate errors associated
with boundary conditions is to conduct uncertainty analyses and sensitivity tests. The situation with respect to initial
conditions is no better since these conditions are rarely known with confidence. The deterministic nonlinear
equations that are used to describe fluid flux in the unsaturated zone may be sensitive to initial conditions; however,
the dissipative nature of the soil system is such that solutions often appear to be insensitive to initial conditions,
especially if sufficient time has elapsed since the start of the simulation.
Errors in model parameterization can result from a variety of sources and causes. One of the most important is
the uncertainty due to spatial and temporal variability (Mulla and Addiscott, 1999). Within the calibration site, the
relatively invariant soil properties of bulk density and soil pH can have coefficients of variation of 10% or so, while
the relatively variable soil properties such as soil hydraulic conductivity and solute diffusivity can have coefficients
of variation of 300% or greater. Solute velocities due to surface ponding and solute concentrations may have
coefficients of variation from 60 to 200%.
Uncertainties in model prediction can also result from errors in spatial scale transitions (Mulla and Addiscott,
1999). One type of error occurs when the source and sink terms or the transport and fate processes operating at the
scale of the calibration study (e.g., the field scale) are different from those operating at the prediction scale (e.g., the
watershed or aquifer scale). In ground-water modeling, this type of problem may be evidenced as changes in
dispersive or transformation processes, geological influences on transmissivity and flow direction, and recharge
patterns. A second type of scale-transition error is due to errors in extrapolation caused by spatial and temporal
averaging of model parameters, or due to bias caused when the calibration is not representative of the region over
which predictions are to occur. Thus, careful selection of the size and location of the calibration site and the time
scales for calibration are important. The essence of these spatial scale transition errors can be illustrated by the
folio wing two statements:
1. Processes that are significant at small scales may not be relevant at large scales.
2. New processes (e.g., dispersion) emerge in response to an increase in scale and the way to represent them
may depend on the original assumed model structure.
Investigators have employed various methods of uncertainty analysis and sensitivity testing to quantify the
uncertainties in model results due to the errors discussed in the above paragraphs. For example, Nofziger et al.
(1994) used Monte Carlo techniques in their uncertainty analysis of four unsaturated zone models for Superfund site
application. The sensitivity analysis invoked by these authors was the "informal approach" described by Helton
(1993). The crux of this approach involves varying one parameter or member of a set of assumptions, one at a time,
and observing the deviation in the resultant model prediction from a base-case prediction. In Helton's analysis in his
1993 paper, four "formal" uncertainty/sensitivity approaches were considered: differential analysis, Monte Carlo
analysis, response surface methodology, and Fourier amplitude sensitivity test (FAST). These approaches are based
on Taylor series, random sampling, response surface construction, and Fourier series, respectively. Although the
implementation of the formal approaches is more complex than the informal sensitivity approach, Helton found that
the formal approaches often yield more information concerning result uncertainty with less computational effort.
Further, he felt that Monte Carlo analysis is the most widely applicable approach for use in performance assessment
(e.g., for use in the performance assessment of radioactive waste disposal sites). In this regard, EPA has recently
published a summary report on a workshop on Monte Carlo analysis, U.S. EPA Risk Assessment Forum (1996).
Further, to aid in the implementation of the Monte Carlo uncertainty analyses of exposure estimates through the soil/
ground-water pathway, Meyer et al. (1997) have published generic probability distributions for unsaturated and
saturated zone soil hydraulic parameters. In addition, procedures are given for Bayesian updating of the generic
distributions when site-specific field data are available. Other uncertainty analysis techniques have been employed;
for example, reliability-based analysis given in Jang et al. (1994) and Hamed and Bedient (1999), optimization by
data fusion described in Eeckhout (1997), and fuzzy-random analysis and neural network/learning techniques
discussed in Ayyub (1998).
In the current report, the technique of uncertainty analysis will be the informal approach as defined by Helton
(1993) and invoked by Nofziger et al. (1994). In the U.S. EPA Risk Assessment Form (1996) report, this technique
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is called the simplest direct response method to explore changes in model output for a discrete or unit change in each
of the inputs, one at a time. For each of the varied inputs, all other inputs to the model are held at their nominal or
baseline values when the sensitivity is computed. These nominal values generally correspond to the means or
medians in a subsequent probabilistic analysis, if one was to be applied (e.g., a Monte Carlo uncertainty analysis).
Sensitivity analyses in this report will address process parameter sensitivity in general, and to some extent conceptual
model choice and numerical error.
1.4. Report Organization
This report is organized into seven sections, along with nine appendices. Section 2 is an overview of numerical
models for simulating radionuclide transport in the vadose zone. The section discusses model selection and includes
model description for the radionuclide SSL effort. Section 3 addresses the sensitivity of simulated results to
conceptual model selection. Model conceptualization and the effect of simplifying assumptions on model outcome
are discussed. Section 4 describes the basic elements and definitions of a sensitivity analysis. A simplified system is
provided to demonstrate the intricacies of an analysis. Section 5 provides the physical setting for the hypothetical
modeling scenario (i.e., the Las Graces Trench Site, NM) along with the development of the conceptual model for
the site, and the baseline parameter selection for the various models to be tested. Section 6 considers the
implementation of the parameter sensitivity analysis and reports the results obtained from the analysis. The general
impacts of the model's input parameters on the output quantities are specified and the relative sensitivity of each
model output to each input parameter is described. Section 7 compares the sensitivity results between the tested
model codes and presents a discussion on various numerical modeling errors. Section 8 gives the summary and
conclusions of this study, while Section 9 lists the cited references in alphabetical order by author.
The nine explanatory appendices cover features of the five tested model codes which are only briefly considered
in the main text, or which require a more extensive analysis than that given in the main text. The features considered
are as follows:
• Appendix A -
• Appendix B -
• Appendix C -
• Appendix D -
• Appendix E -
• Appendix F -
• Appendix G -
• Appendix H -
Empirical models of the unsaturated soil hydraulic properties which are used in the
various models.
A discussion on the scaling of field soil-water behavior.
An explanation of the hysteretic characteristics of soil-water properties.
The first-order decay chains used in the various models.
The impact of using a nonuniform moisture distribution versus a uniform distribution.
The impact of using daily precipitation and evapotranspiration rates versus an annual
average recharge rate.
The impact of considering a layered soil column versus a homogeneous soil column.
A detailed analysis of nonequilibrium sorption of pollutants, mainly for the
radionuclide 90Sr.
Appendix I - Results from the transport and fate of other radionuclides not considered in the
main text.
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Section 2
Overview of Numerical Models for Simulating Radionuclide Transport
The transport mechanisms and loss pathways for chemicals in the soil consist of many elements (Jury and
Valentine, 1986), such as:
Static Soil Properties - porosity, bulk density, organic carbon content, pH, soil temperature.
Flow and Transport Variables and Properties - saturated hydraulic conductivity, saturated water
content, matrix head-water content function, hydraulic conductivity function, solute dispersivity.
• Basic Chemical Properties - molecular weight, vapor pressure, water solubility, Henry' s constant,
liquid diffusion coefficient in water, partition coefficient, decay rate of compound.
Contaminant Source Characteristic - solute concentration of source, solute flux of source, solute
source decay rate.
• Time Dependent Parameters - water content and flux, infiltration or evaporation rate, solute
concentration and flux, solute velocity, volatilization flux, air entry pressure head.
• Soil Adsorption Parameters - distribution coefficient, isotherm parameters, organic carbon partition
coefficient.
• Tortuosity Functions - vapor and liquid diffusion tortuosities.
These variables and properties are related to one another by conservation equations (e.g., conservation of mass,
momentum and energy) and constitutive equations (e.g., state equations, sorption isotherms, dispersion relationships,
unsaturated soil hydraulic properties). Further, these equations are supplemented by sets of auxiliary equations (e.g.,
boundary and initial conditions). In general, for specific site/process contaminant scenarios, these mathematical
systems are sufficiently complex so that numerical solutions are required. Techniques and procedures as briefly
outlined in Section 1.2 are employed. However, for some pedagogical and screening scenarios, simple systems
employing analytical or semi-analytical solution techniques may be applicable. Such solutions can also be used to
verify numerical computer codes developed for the more complex scenarios.
In Section 1.1, the aims and objectives of the radionuclide SSL effort were briefly outlined. Based upon these
aims and objectives, the breadth of the modeling tools and codes outlined in Section 1.2 can be greatly reduced for
current applications. For example, the SSL effort will probably not be concerned with stochastic modeling and
stochastic codes, modeling and codes based on fractal geometry and percolation theory (i.e., discrete networks and
cellular automata), and fractured rock flow modeling. Further, for the radionuclides of concern in this report, no
vapor transport models will be explicitly required; however, there may be vapor modules in some of the candidate
codes finally chosen. In general, the final candidate list of codes was chosen in part from the codes listed in
publications such as van der Heijde and Elnawawy (1993), Nofziger et al. (1994), van der Heijde (1994) and Bedient
etal. (1999).
2.1 Model Selection for the Radionuclide SSL Effort
The evaluation of models' applicability to the SSL process for radionuclides was based on the following
considerations:
Can the model be used to simulate the transport and fate of five selected
radionuclides - tritium 3H, technetium "Tc, uranium 238U, strontium 90Sr, and plutonium 238Pu?
Can the model simulate the transport and fate of the selected radionuclides for a specific site test case?
Is the model appropriate for use in the SSL process?
What are the limitations of the model and how do they impact its intended use?
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Keeping these considerations in mind, a search was undertaken to locate and evaluate vadose zone flow and
transport models to determine their suitability for inclusion in the SSG document for radionuclides. The search
criteria sought to identify models which included at least first-order decay and radionuclide chain decay. Models
were also selected based on evidence of current revisions and updates to eliminate the evaluation of obsolete codes.
Whenever possible, the history of the models was also examined to determine if the codes had undergone name
changes or inclusions into other codes in order to eliminate redundancy. Using these criteria and the above
considerations, one-, two- and three-dimensional models in the public domain and the private sector were located
through library resources, internet searches, and personal communications.
In general, the search focused on the identification and preliminary evaluation of relatively simple one-
dimensional models which are most suited to the SSL process. The search produced four one-dimensional and two
two-dimensional models in the public domain. All six of these models have decay-chain transport modules in the
vadose zone, and all can calculate the leachate contaminant concentration entering the ground water. All four one-
dimensional models were chosen for analysis and evaluation, and only one of the two-dimensional models was
chosen. These five models are identified by the following acronyms:
• One-dimensional CHAIN Code (van Genuchten, 1985),
• One-dimensional MULTIMED-DP 1.0 Code (Liu et al., 1995; Sharp-Hansen et al., 1995; Salhotra et
al., 1995)
• One-dimensional FECTUZ Code (U. S. EPA, 1995ab),
• One-dimensional HYDRUS Code (Simunek et al., 1998),
Two-dimensional CHAIN 2D Code (Simunek and van Genuchten, 1994).
2.2 Model Description of the Selected Codes
The CHAIN Code is an analytical model which requires the assumption of uniform flow conditions. The
MULTIMED-DP 1.0 and FECTUZ Codes are semi-analytic models which approximate complex analytical solutions
using numerical methods such as numerical inverse transform modules. Transient and steady-state conditions can be
accommodated in both the analytical and semi-analytical models, while layered media are usually only
accommodated by the semi-analytical codes. The HYDRUS and CHAIN 2D Codes solve the pertinent sets of partial
differential equations using finite-difference or finite-element methods. The resolution in space and time of these
numerical models depends on the physical characteristics of the site in question, the computational resources
available to the modeler, and the purposes of the simulation. Numerical codes are used when simulating time-
dependent scenarios under spatially varying soil conditions and unsteady flow fields.
HYDRUS, CHAIN 2D, and CHAIN are stand-alone vadose zone models of the fate and transport of solutes in
the liquid, solid and gaseous phases. FECTUZ is the unsaturated module of EPA's composite model for leachate
migration with transformation products (EPACMTP), see USEPA 1995a and 1995b. This code is based on
enhancements to the VADOFT Code (Huyakorn and Buckley, 1987). FECTUZ simulates migration of contaminants
from a landfill or a surface impoundment through the unsaturated zone to an unconfined aquifer with a water table
fixed some L units below the surface. MULTIMED-DP was initially developed as a multimedia fate and transport
model to simulate contaminant migration from a waste disposal unit (e.g., from a landfill) through different pathways
which include air, surface water, soil and ground water. In Version 1.0, it simulates the transport and fate of first-
and second-generation transformation products through the saturated and unsaturated zones. This version has the
option to allow it to be used for unsaturated zone migration alone. It is this version, MULTIMED - DP 1.0, and the
unsaturated option that is analyzed in this report, see Liu, et al. (1995), Salhotra, et al. (1995), and Sharp-Hansen, et
al. (1995).
Except for dimensionality, the most comprehensive code is HYDRUS, which contains the greatest number of
physical processes. HYDRUS considers only the vertical dimension (z), while CHAIN 2D considers a vertical slab
in two dimensions (x, z). For vadose zone modeling, it is not always clear that a two-dimensional, vertical slab
model will give superior results to those obtained from a one-dimensional model for the real three-dimensional area
being simulated. If the region to be simulated is reasonably uniform in the horizontal planes of the surface and
subsurface layers, then a one-dimensional vadose zone model may suffice over that of the two dimensional model. If
the region to be simulated is rather uniform in one horizontal direction, while being heterogeneous in the orthogonal
horizontal direction (e.g., furrow irrigation of a field of row crops), then a two-dimensional, vertical slab model
oriented along the heterogeneous direction (e.g., perpendicular to the crop rows) would be superior to a one-
dimensional model. However, if the region to be simulated is rather heterogeneous and nonsymmetrical in the
-------
horizontal, then the choice between a one-dimensional model and a two-dimensional, vertical slab model may tilt in
the direction of the one-dimensional model being applied at several discreet, representative points within the region
of interest. Then through the use of superposition, scaling and geostatistics, the results of the several one-
dimensional solutions can be used to gain knowledge of the real three-dimensional system. Thus, one dimensional
vadose zone models need not take an inferior role to a two-dimensional, vertical slab model. Since HYDRUS is the
most comprehensive of the five models, the systems of equations discussed in this subsection with support from
Appendices A to D will be those associated with HYDRUS (Simunek et al., 1998). Table 2-1 indicates the model
components that will be discussed and the status of these components in each of the selected codes.
2.2.1 The Variably Saturated Water Flow
The variably saturated water flow in a vertical soil column (Figure 2-1) is governed by Darcy's Law:
(2-1)
and by the continuity of mass equation
36 3q
— + — = -S , (2-2)
dt dz
where h = h (z,t) is the water pressure head in units of length (L), q = q (z,t) is the Darcian fluid flux density in units
of length over time (LT-1), K = K (z,t) is the unsaturated hydraulic conductivity (LT-1), 9 = 9(z,t) is the volumetric
moisture content (L3L~3), and S = S (z,t) is the volume of water removed from a unit volume of soil per unit time due
to plant water uptake (L3L-3 T-1). As shown in Figure 2-1, z = 0 defines the bottom of the soil column which could
be taken as the water table of an unconfined aquifer. The value z = L is the soil surface and z = L - LR defines the
bottom of the root zone.
Combining Equations (2-1) and (2-2) leads to a modified Richards Equation (Richards, 1931):
ae a
dt
-S
(2-3)
For heterogeneous layers with time-independent properties, the soil hydraulic functions and K can be written as
follows (Simunek et. al. 1998):
), (2-4a)
K = K(z,t) = K(h,z) = Ks (z) Kr (h,z) , (2-4b)
where Ks(z) is the saturated hydraulic conductivity (LT-1), and K,.(h,z) is the relative hydraulic conductivity
(dimensionless). For homogeneous layers with time-independent properties, the parameters become:
e = e(z,t) = e(h), K = K(z,t) = K(h) . (2-5)
The moisture sink due to root water uptake is expressed as follows (Simunek et al., 1998):
S(z,t) = a(h,h,)Sp(z,t) , (2-6)
where a^h^) is the water stress response function (dimensionless), Sp(z,t) is the potential water uptake rate (T-1),
and hfy is the osmotic head (L) which is a linear function of the concentrations of all solutes in the soil moisture. Two
-------
Table 2-1. Comparison of the Model Components in Each of the Five Codes Being Analyzed in this Report.
MODEL COMPONENT
Variably Saturated Water
Flow
Root Water Uptake
Unsaturated Hydraulic
Properties
Scaling of Soil Hydraulic
Functions
Scaling for Temperature
Dependence of Soil
Hydraulic Functions
Hysteresis in Soil
Hydraulic Functions
Initial and Boundary
Conditions for the Water
Flow System
HYDRUS
One-Dimensional
Time Dependent Richards
Equation with Sink Term
Sink in Richards Equation
Which Depends on
Osmotic Head, Water
Stress, Root Distribution
See Appendix A
Brooks-Corey, van
Genuchten, and Vogel-
Cislerova Models
See Appendix B
Linear Scaling of Pressure
Head, Soil Moisture, and
Hydraulic Conductivity
Scaling Due to Parameter
Variations with
Temperature Applied to
Pressure Head and
Hydraulic Conductivity
See Appendix C
Applied to van Genuchten
and Vogel-Cislerova
Models for Moisture
Content and Hydraulic
Conductivity
Initial Condition on
Pressure Head; Head and
Flux Boundary Conditions;
Ponding and Seepage
CHAIN 2D
Two-Dimensional
Time Dependent Richards
Equation with Sink Term
Sink in Richards Equation
Which Depends on
Osmotic Head, Water
Stress, Root Distribution
See Appendix A
Vogel-Cislerova and van
Genuchten Models
See Appendix B
Linear Scaling of Pressure
Head, Soil Moisture, and
Hydraulic Conductivity
Scaling Due to Parameter
Variations with
Temperature Applied to
Pressure Head and
Hydraulic Conductivity
Not Considered in Model
Initial Condition on
Pressure Head; Head and
Flux Boundary Conditions;
Ponding and Seepage
FECTUZ
One-Dimensional
Steady State
Constant Infiltration Rate
Darcy's Law
Not Considered in Model
See Appendix A
van Genuchten Model
Not Considered in Model
Not Considered in Model
Not Considered in Model
Steady State, Thus no
Initial Condition; Pressure
Head Zero at Water Table,
Pressure Head Consistent
with Infiltration Rate at
Surface
MULTIMED_DP 1. 0
One-Dimensional
Steady State
Constant Infiltration Rate
Darcy's Law
To Some Extent,
Considered in the Landfill
Module, Not in the
Unsaturated Flow Zone
Module
See Appendix A
Brooks-Corey and
van Genuchten Models
Not Considered in Model
Not Considered in Model
Not Considered in Model
Steady State, Thus no
Initial Condition; Pressure
Head Zero at Water Table,
Pressure Head Consistent
with Infiltration Rate at
Surface
CHAIN
One-Dimensional
Steady State
Constant Flux and
Water Content
Throughout Soil Column
Not Considered in Model
Not Considered in Model
Not Considered in Model
Not Considered in Model
Not Considered in Model
Not Required Since Soil
Moisture and Flux Are
Specified Constants
-------
Table 2-1. Comparison of the Model Components in Each of the Five Codes Being Analyzed in this Report.
MODEL COMPONENT
Solute Transport Equations
Initial and Boundary
Conditions for the Solute
Transport Equation
Effective Dispersion
Coefficients
Temperature Dependence of
Transport and Reaction
Coefficients
Heat Transport Equations
HYDRUS
See Appendix D
One-Dimensional, Time-
Varying; Gas, Liquid and
Solid Phases; Oth Order
Production, 1st Order Decay;
Advection/ Dispersion;
Equilibrium/ Nonequilibrium
Sorption, Nonlinear Sorption
Initial Conditions for Liquid
and Solid Phases; Dirichlet,
Cauchy and Neumann
Boundary Conditions; Volatile
Solutes at Surface
Solute Dispersion Coefficient;
Gas Diffusion Coefficient;
Tortuosity Factors
Modified Arrhenius Equation
for Temperature Scaling of
Production, Degradation, and
Adsorption Coefficients
One-Dimensional Time-
Varying, Convection,
Dispersion Equation; Energy
Uptake by Roots; Thermal
Conductivity Coefficient; Heat
Capacities
CHAIN 2D
See Appendix D
Two-Dimensional, Time-
Varying; Gas, Liquid and
Solid Phases; Oth Order
Production, 1st Order Decay;
Advection/ Dispersion;
Equilibrium/ Nonequilibrium
Sorption, Nonlinear Sorption
Initial Conditions for Liquid
and Solid Phases; Dirichlet,
Cauchy and Neumann
Boundary Conditions; Volatile
Solutes at Surface
Solute Dispersion Matrix in
Two-Dimensions; Gas
Diffusion Coefficient;
Tortuosity Factors
Modified Arrhenius Equation
for Temperature Scaling of
Production, Degradation, and
Adsorption Coefficients
Two-Dimensional Time-
Varying, Convection, Dis-
persion Equation; Energy
Uptake by Roots; Thermal
Conductivity Matrix in Two-
Dimensions; Heat Capacities
FECTUZ
See Appendix D
One-Dimensional, Time-
Varying; Liquid and Solid
Phases; 1st Order Decay;
Advection/Dispersion;
Linear/Nonlinear Equilibrium
Sorption
Initial Concentration for
Liquid Phase; Dirichlet and
Cauchy Boundary Conditions;
Batemann Decaying Source
Terms
Solute Dispersion Coefficient;
Effective Molecular Diffusion
Coefficient
Not Considered in Model
Not Considered in Model
MULTIMED_DP 1. 0
See Appendix D
One-Dimensional, Time-
Varying; Gas, Liquid and
Solid Phases; 1st Order Decay;
Advection/ Dispersion; Linear
Equilibrium Sorption
Initial Concentration for
Liquid, Gas and Solid Phases;
Dirichlet and Cauchy, Separate
and Mixed, Boundary
Conditions
Solute Dispersion Coefficient;
Gas Diffusion Coefficient;
Tortuosity Factor
Hydrolysis Rates Temperature
Corrected Using Arrhenius
Equation
Not Considered in Model
CHAIN
See Appendix D
One-Dimensional, Time-
Varying; Liquid and Solid
Phases; 1st Order Decay;
Advection/Dispersion; Linear
Equilibrium Sorption
Initial Concentration for
Liquid Phase; Dirichlet and
Cauchy Boundary Conditions;
General and Batemann
Decaying Source Terms
Solute Dispersion Coefficient
Not Considered in Model
Not Considered in Model
-------
L
L-LR+
Surface
UHU
Water v Table
(Zone
Root Zone
Intermediate
Zone
Figure 2-1. Schematic of an unsaturated soil column, soil-water retention curve, and hydraulic conductivity
function, where subscript "s" indicates saturated conditions.
formulas for aCh,^) are given in terms of experimentally determined exponents and values of h and h^ at which the
water extraction is reduced by 50% during conditions of negligible stress. The quantity Sp(z,t) is the product of the
normalized water uptake distribution function b(z,t) in L-1 units and the potential transpiration rate Tp(t) in LT1 units.
The quantity b(z,t) is related to a root distribution function and the rooting depth LR(t). In turn, the quantity LR(t)
varies with a logistic -type growth function. Finally, the potential transpiration rate Tp(t) is related to the actual
transpiration rate obtained from field measurements.
In HYDRUS, three models of the unsaturated hydraulic properties, 6(h) and K(h), are given: they are the
Brooks and Corey (BC) Model (1964), the van Genuchten (VG) Model (1980), and the Vogel and Cislerova (VC)
Model (1988). The equations for these models are given in Appendix A. In addition to giving the analytical forms
of these models, Appendix A discusses two papers (Morel-Seytoux, et al. 1996; Nachabe, 1996) concerned with the
interchangeability of the BC- and VG-Models for infiltration calculations. Further, it was suggested by Morel-
Seytoux, et al., though not proved, that this interchange of models would be valid for drainage and evapotranspira-
tion calculations, as well. The VC-Model, as shown in Appendix A, is a modification of the VG-Model to add
flexibility in the description of the hydraulic properties near saturation. Also discussed in Appendix A is a new
model for the soil-water retention curve, 9(h), which the authors, Assouline, et al. (1998), claim to be more flexible
than the VG- or VC-Models.
The scaling features for soil -water behavior found in the HYDRUS and CHAIN 2D Models are part of the
fundamental process of searching for symmetry in nature. The application of these features is to simplify
calculations of soil-water behavior for vertically heterogeneous systems consisting of different homogeneous soil
layers, and to simplify and make more consistent the superposition of discrete one-dimensional (vertical dimension)
results distributed over a horizontally heterogeneous field. In Appendix B, the general topic of symmetry in nature is
briefly discussed, along with the analytical concepts of similitude, transformation groups, inspectional analysis and
serf-similarity. Finally, Appendix B discusses the scale dependence and scale invariance in the field of hydrology.
An excellent recent survey book on these topics is that edited by Sposito (1998).
One of the first papers on the similarity of soil-water behavior in unsaturated porous media is that of Miller and
Miller (1956). This work is based on geometric similarity and requires that similar porous media will have the same
porosities, a concept more fully satisfied in the laboratory than in the field. To account for the variation of porosity
in the field, Warrick, et al. (1977) replaced the use of 9 in the Miller and Miller Model by ratio 9/9s, where 9S is the
saturated moisture content. In essence, 9S becomes a new scale factor which accounts for the different internal
11
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geometries in field soils. Based on the results of these papers, as well as the principles of similitude and invariance,
Vogel, et al. (1991) introduced the transformations that form the foundation of the scaling relationships in the
HYDRUS and CHAIN 2D Codes. These transformations account for the linear variability in the soil moisture
parameters and are defined by the following:
h = ahh* , (2-7)
e(h)-er=ae[e*(h*)-e;], (2-8)
= akK*(h*), (2-9)
where 9r is the residual soil moisture, the asterisks denote reference quantities, and the three scale factors,
(ah,ae,ak), are assumed to be independent of one another. These linear transformations will explain the variability
due to soil structure within a given soil textural class, but will not account for the nonlinear phenomena expressed by
different soil textural classes. Figure 2-2 illustrates an example of soil structure scaling for a set of scaling factors
given by (ah,ae,ak) = (3/2, 2, 7/4). Thus, the above transformations become
h = 3h*/2, (2-7a)
6(h) - 6r= 6(3 h*/2) - 6r =2 [ 6* (h*) - 6r *] , (2-8a)
K(h) = K(3h*/2) = 7/4 K*(h*). (2-9a)
The values of the scaling factors, (a h, ae, a k), are determined from field sampling; and if representative of an
area in question, allows one to coalesce data sets (see Figure B-3 of Appendix B) and allows one to transfer
measured soil relationships to unsampled areas. For example, Rockhold et al. (1996) applied the transformations of
equations (2-7) to (2-9) in the simulation of water flow and the transport of tritium measured at an initially
unsampled domain of the Las Graces Trench Site, the site which forms the physical setting of our analyses. These
transformations allowed for successful transfer of soil-water properties from originally sampled areas at the site to
the domain used by Rockhold, et al. This study is a good example of the power of the scaling features in the
HYDRUS and CHAIN 2D Codes; however, in the present sensitivity analyses, these features will not be considered
since we use a single homogenous soil layer.
In capillary theory the dependence of the pressure head, h, on temperature variations comes from the variation of
surface tension with temperature, while those for the hydraulic conductivity, K, arise from the temperature
dependence of the dynamic viscosity, ^ (ML-lT~l), and the density, p(ML-3), of the soil water. The temperature
scaling factors are thus defined by:
a
(2-10)
K(9,T.) = K*(0) = akK*(0), (2.n)
where a(Ta) is the surface tension at the air-water interface (MT-2), Ta in the functional expressions is the temperature
(°K), ah is the temperature scaling factor for pressure head, and Otk is the temperature scaling factor for hydraulic
conductivity. For the current sensitivity analyses, the systems are taken to be isothermal and sensitivity to changes in
h and K are obtained by variations in other parameters rather than from temperature variations.
The soil-water retention curve, and to some extent the hydraulic conductivity curve, for a draining (drying) soil
is not the same as that for an imbibing (wetting) soil. Reasons for such a phenomenon in the unsaturated zone are the
entrapment of air in the soil pores and the surface tension effects occurring in the pore throats. This phenomenon is
called hysteresis and is a highly nonlinear characteristic which occurs in many mechanical, electrical, chemical,
biological, and geophysical systems. The formulation and analysis of hysteretic systems has a history from the early
1900's to the present. Appendix C is concerned with the origins and applications of hysteretic phenomena; hysteresis
loops, operators and models; and hysteresis in soil-moisture parameters. The study of hysteretic operators and
12
-------
3
-------
models lays the groundwork for the existence and uniqueness of solutions for hysteretic systems in soil physics, as
well as in other disciplines. The principles and techniques developed in the other disciplines are useful in soil
physics, and vice versa.
Appendix C discusses the background and theory for the hysteretic features presented in HYDRUS (Simunek et
al., 1998). These features are based on the VG- and VC-Models for the soil-moisture parameters and the
predominant feature that defines the difference between the main drying curve and the main wetting curve is the van
Genuchten parameter a, the inverse of the air-entry value or the bubbling pressure. The a for drying, ad, is less than
that for wetting, aw. A reasonable approximation is to let aw = 2ad. The drying scanning curves, as defined in
Appendix C, are linearly scaled from the main drying curve, while the wetting scanning curves are scaled from the
main wetting curve. In both cases, the scale factor a h in Equation (2-7) is taken to be unity. In the current analyses,
where a constant infiltration rate is postulated, the hysteretic features of the HYDRUS Model will not be invoked.
The intial and boundary conditions for the water flow system as given in HYDRUS (Simunek et al., 1998)
consist of a prescribed pressure head distribution at time zero, hj(z), 0 < z < L, and boundary conditions at the
surface z = L and the bottom of the soil profile z = 0. The type of boundary conditions possible are as follows:
Dirichlet Conditions
h(z,T) = h0(t), t>0, z = 0,L, (2-12)
Cauchv Conditions
-K — +1 =qo(t), t>0, z = 0,L, (2.13)
Neumann Conditions
— = 0, t > 0, only at z = 0 , (2-14)
dz
where h0 is a specified head and q0 is a prescribed soil water flux. The boundary conditions in Equations (2-12) to
(2-14) are assumed to be system-independent conditions. HYDRUS also considers the soil-air interface at z = L
under "current" atmospheric conditions of evaporation, precipitation, ponding, and infiltration. In addition, the code
considers the case when seepage occurs at the bottom of the soil column, at z = 0.
2.2.2. Solute Transport Systems
All five models under consideration in this report have time-dependent solute transport systems involving first-
order decay chains which are straight, branched and/or a combination of both, as well as systems where a set of
species can evolve independent of one another. The various decay chains which each model addresses are described
in Appendix D. The general form of the governing equations for the systems described in HYDRUS are as follows
(Simunek et al., 1998):
dt 3z 3z
flR r* -\- nr* = I flD r* I -I- T*1 fc ^r* -\- (~r (c Q Q ^ /o ^ ^\
"2^2,1 "^2,z |_" 2 2,zjz r2V^2/^l ^V^l^j 1^2' ' (2-16)
^Si _ TT/ *\ '\ * ' — 10
dt
where the "1" index is for the parent species, "2" index is for the daughter species, Rj is the retardation factor
14
-------
accounting for gaseous, equilibrium solid, and liquid phases of the solute, c; is the concentration of the liquid phase,
Dj is the dispersion coefficient for the 1th species, s* represents the concentration of the solute on the nonequilibrium
sites of the soil matrix, ^ is the degradation rate of s* and represents material lost to the system, ^' is the rate of
transfer of material from parent to daughter, and y; is the production of s;*. The function F^CJ) represents the
degradation of the solute (losses of material from the system) from gaseous, liquid and solid phases, transfer losses
from the parent passed on to the daughter for the three phases, losses due to plant uptake, and changes due to
temperature variations in the system. The function GJ(SJ*) accounts for the production of the solute in all three
phases, losses due to plant uptake, and the amount of solid phase at the nonequilibrium sorption sites. The function
G2(Cj ,Sj*, s2*) describes the amount of daughter material transferred from the parent in gaseous, liquid, equilibrium
solid and nonequilibrium solid phases, accounts for the production of the daughter species in all phases, and accounts
for root uptake losses. Finally, the function H(c1,s1*) in Equation (2-17) is a first order sorption rate of s* produced
by the difference between equilibrium and nonequilibrium concentrations.
The initial conditions corresponding to the system in Equation (2-15) to (2-17) require specifications of the
concentrations of all liquid phase solutes in the decay chain, as well as the species at the nonequilibrium sites in the
soil matrix; that is, we specify
Ci(z,0) = c10(z), Si*(z,0) = sl0*(z), 00, z = 0,L, (2-19)
Cauchy Conditions
dc
-0D; —- + qc, = q c,, t>0, z = 0,L , (2-20)
dz
Neumann Conditions
3c
V" = 0> t>0> Z = 0'L> (2-21)
dz
where q is the specified concentration at the boundaries or the specified concentration of the incoming/outgoing
fluid in Equation (2-20), and q is the downward fluid flux at the boundary. In addition to the conditions in Equation
(2-19) to (2-21), HYDRUS allows for vaporization losses of volatile solutes at the soil surface z = L.
The effective dispersion coefficients D1(L2T~1) in Equation (2-15) and (2-16) are defined by the following
(Simunek et al., 1998):
6D; = DiL q + 6Diwiw + avDigkigxg , (2-22)
where DlL is the longitudinal dispersivity (L) of the ith species, D1W is the molecular diffusion coefficient (L2^1) of
the 1th species in free water, TW is the dimensionless tortuosity factor in the liquid phase, a^ is the dimensionless air
content, Dlg is the molecular gas diffusion coefficient (L2T-i) of the ith species, klg is the dimensionless constant of
proportionality linearly linking c; to the gaseous solute concentration g;, and Tg is the dimensionless tortuosity factor
in the gas phase. The sum of the air content, a^ and the soil moisture content, 9, is equal to the saturated water
content, 9S. The empirical formulas for the tortuosity factors are given by:
15
-------
=e7/3e-2,
W S
=a
7/3-2
(2-23)
HYDRUS assumes that several of the parameters in the governing equations for the transport and fate of solutes
are temperature dependent, such as: the molecular diffusion coefficients, the zero-order production terms for all
phases, the first-order degradation rates (both for system losses and transfers from parent to daughter), and the
adsorption coefficients relating one phase to another phase. This dependency is given by a modified Arrhenius
equation defined by (Simunek et al., 1998):
A(Ta) = exp
E(Ta -Ta)
(2-24)
where Ta is the absolute temperature under consideration, Ta* is the absolute reference temperature, E is the
activation energy of the particular reaction or process being modeled in units of (ML2T-2M-1), and R is the universal
gas constant in units of (ML2 + T2M°K). If a (Ta) is the coefficient or parameter varying with temperature and a *
(Ta*) is the corresponding reference quantity dependent on the reference temperature, Ta*, then the relationship
between a and a* is given by:
a(Ta) = a*(Ta*) A(Ta).
(2-25)
However, for the current sensitivity analyses, the variations of the above parameters will be for isothermal conditions
and will be due to causes other than temperature changes.
2.2.3. Heat Transport Equation
The last model component listed in Table 2-1 is the heat transport equation. For the HYDRUS Code, the
governing equation for heat transfer is one-dimensional, time-varying, and with terms accounting for convection,
dispersion, and energy uptake by plant root water uptake. It is assumed that no heat is transferred by water vapor
diffusion and there is no latent heat transfer by vapor movement. The convection-dispersion equation takes the
following form (Simunek etal., 1998):
a[qTa]
-cwSTa ,
(2-26)
where cp(9) is the volumetric heat capacity of the porous medium (M -^ LT2°K), cw is the volumetric heat capacity
of the liquid phase (M -^ LT2°K), and ^(9) is the coefficient of the apparent thermal conductivity of the soil
(ML -H T3 °K). To a reasonable degree of accuracy, it can be shown that
dcp(6)^
d0 ~Cw'
Thus, combining Equations (2-2), (2-26) and (2-27) leads to the heat equation programmed in HYDRUS:
(2-27)
c(
-------
conditions require that the temperature be specified throughout the soil column at time zero. The boundary
conditions considered by HYDRUS are as follows:
Dirichlet Conditions
Ta(z,t) = T0(t), t>0, z = 0,L, (2-30)
Cauchy Conditions
+ Tacwq = T0(t)cwq0(t), t > 0, z = 0,L , (2-31)
3z
Neumann Conditions
9Ta
T2- = 0, t > 0, z = 0 , (2-32)
dz
Atmospheric Boundary Condition
1 > 0, z = 0 , (2.33)
5 *" 5
where T0 in Equation (2-31) is the temperature of the incoming fluid, (Ta) in Equation (2-33) is the average
temperature at the soil surface during period p, and A is its amplitude. In fact, in Equation (2-33), (Ta) is the
average daily temperature at the soil surface, p is one day, and A is the daily temperature amplitude. It is assumed
that time zero is at midnight and that the maximum temperature occurs at 1:00 pm.
As with the other temperature dependent components in HYDRUS, and some of the other codes, the heat
transfer equations are not considered in the current sensitivity analyses. It is believed that for radionuclide soil
screening purposes the computer codes need not incorporate the additional effects due to temperature variations.
Thus, all such corrections are dropped in the analyses which follow.
17
-------
Section 3
Sensitivity of Simulated Results to Conceptual Model Selection
As introduced in Section 1.3, the conceptual model is a simplification of the "real world" into a set of
mathematical equations (i.e., conservation equations, constitutive relationships, and auxiliary conditions) that
represent the most relevant physico-chemical processes thought to govern the problem at hand, in this case, the fate
and transport of radionuclides in the unsaturated zone. The reader should be aware that the "real world" itself is only
a conceptual model, or a composite of many conceptual models, that investigators embrace based on their field,
laboratory, theoretical and simulation studies. These more comprehensive models, which are called the "real world",
evolve and change in time as new and more extensive knowledge of the physico-chemical processes is obtained. One
should not think of the comprehensive models as one super computer code containing many interacting modules,
allowing for highly nonlinear processes, and time and space variations on all scales of interest. Such super computer
codes do not currently exist: and if they did, the interpretation of their results may be as complex as those seen in the
real world. These comprehensive conceptual models of the "real world" usually consist of equations and/or
inequalities relating physico-chemical variables and parameters for specific processes under various external
influences (e.g., flow of water through porous media, multicomponent/multiphase chemical processes, biological
processes, thermodynamic processes). When such a collection of comprehensive models is linked together into
single computer codes for situations such as the fate and transport of radionuclides in the unsaturated zone, many
simplifing assumptions are invoked in the individual comprehensive models or modules, and in the manner in which
these modules are linked together. The subject of this section is the sensitivity of the simulated results of
radionuclide fate and transport through the vadose zone to the simplifing assumptions of the individual modules as
they may occur in the computer codes selected to generate the results.
3.1 Time and Space Scales
Subsurface water and pollutant fate analyses are difficult because of the variabilities in time and space. One can
not easily see underground, and extensive measurements and observation are both difficult and expensive. The
medium through which the subsurface vapors, gases, liquids, and solids move is usually extremely hetergeneous, and
analyses often involve several spatial and temporal scales. Aquifers and their overlying soil layers many be on the
scale of 104 meters or more, and large-scale hetergeneities within an aquifer and the overlying soil may range from 1
to 102 meters. Surface source areas of pollutants may also have scales of 1 to 102 meters. The soil and rock pores
may have scales on the order of 1O2 to 1O4 meters or less, while cracks, fractures, worm holes and root channels
may be 100 times these values. Colloids which may freely pass through many of these pores have scales less than
10-5 meters. If adsorption and chemical processes are of interest, then analyses may require looking at scales of the
order of 1Q-7 meters, which is the order of the adhesive layers on the soil matrix. Added to these length scale
variations are a variety of time scales (Logan, 2001).
For example, precipitation/runoff/infiltration events may occur over severe storm periods lasting several minutes
to an hour or two, or may occur over moderate rainfall periods of a day or two. Water flow in soil and aquifers may
be a few meters per day to as low as a few millimeters per year. Significant variations in water flow can occur over
periods of several minutes to several hours. Certain chemical reactions may occur very rapidly, while radioactive
decay may require days to thousands of days to be consequential. Mineral deposition and dissolution may take
centuries or longer to have a substantial impact on subsurface processes. The mix of all these lengths and time
scales, some of which appear to be over scale continua, makes the selection of a problem's time/space domain
difficult, as well as making the problem's mathematical formulation and its solution difficult. In fact, the entire
conceptual model selection process is strongly tied to these time and space considerations and/or strongly influenced
by them.
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These variability and heterogeneity effects are evidenced in many ways, such as:
Affect gas and liquid flow in the subsurface.
Determine the validity and practicality of the selection of the problem's time/space domain.
Affect the choice of conditions on the boundaries of the space domain and affect the choice of the
initial conditions within the domain.
Dictate the presence or absence of the impacts of hy steretic effects and those of density and thermal
gradients.
Influence whether or not preferential flowpaths and facilitated transport (e.g., pollutant transport by
colloids) should be considered.
Affect the form of the dispersion processes for pollutant mass transport.
Enhance or suppress the impacts of chemical sorption, reaction and decay processes.
Influence multicomponent and multiphase systems.
Each of these items or concerns is important to the selection process of conceptual models, and they are
discussed in the following subsections with the respect to fate and transport of radionuclides in the unsaturated zone.
3.2 Domain Selection, Boundary and Initial Conditions
The fluid and water quality dynamics within the unsaturated zone are governed by the migration processes
within the zone and the interactions between this soil-water zone and the atmospheric/soil-surface zone, the surface-
water zone, and the ground-water zone (Luckner and Schestakow, 1991). The atmospheric/soil-surface zone affects
the soil-water dynamics through the processes of precipitation, infiltration, evaporation, transpiration, heat transfer,
vapor diffusion, pollutant transfer, exfiltration and fluid transfer at seepage surfaces. The surface-water zone
interacts with the soil-water zone by the process of interflow which transfers fluid, pollutants and heat between the
two zones. The soil-water and ground-water zones interact with one another through the processes of percolation
and capillary rise, and in so doing transfer fluids, pollutants and heat. Thus, the physical domain, D, for a given
modeling scenario in the unsaturated zone is a finite, three-dimensional, heterogeneous volume of a solid soil matrix
whose particles may have sorbed pollutants attached to them, and a soil void space containing air, vapors, and liquids
with dissolved pollutants. In addition, domain D may possess density gradients that affect the fluid and water quality
dynamics, the density gradients being produced by thermal effects, fresh/saline water interactions, and areas of free
pollutant products with densities higher or lower than pure water. Since domain D is finite in the real world, it must
possess a boundary, 3D. Parts of the boundary, 3Dj, may be fixed over the time domain, T, of interest in a given
modeling scenario, boundaries such as the soil surface, streams or river channels, lake or reservoir shores. Other
boundaries, 3D2, may be free, such as fresh/saline interfaces, water-table surfaces, and seepage zones. These free
boundaries, 3D2, are determined as part of the solution process of the governing equations in a modeling scenario,
thus highly complicating that solution process and often changing a linear system to a highly nonlinear system.
There may also be internal boundaries, 3D3, within domain D if the domain consists of distinct soil layers possessing
discontinuous properties as one moves from layer to layer, or if moisture and/or pollution fronts move through the
domain. The head, moisture, temperature, and pollutant conditions specified on the boundaries, 3Dj and 3D2
couple the soil-water zone to the atmospheric/soil-surface, surface-water, and ground-water zones. These real world
boundary conditions are time-varying, spatially-heterogeneous and often are highly uncertain. In addition, the fixed
vertical boundaries of the domain D may not be determined by natural interfaces such as rivers and lakes or by
anthropogenic interfaces such as walls and other structures, but may be chosen arbitrarily at some distance from the
area of immediate interest in a modeling scenario. The conditions specified on these boundaries may be specified in
terms of symmetry, periodicity or no-transport conditions, all of which may be contradictory to real world conditions.
Since modeling scenarios for the subsurface zone usually involve time variations and thus some starting point in the
time simulation, initial distributions of moisture content, hydraulic head, pollutant concentrations in solution, air and
gaseous vapors, sorbed pollutants, and temperature within domain D are required. Thus, a comprehensive
formulation of the fluid and pollutant dynamics in the unsaturated zone is often highly complex and highly uncertain.
This complexity and uncertainty force unsaturated zone scientists to make several assumptions and simplifications in
their investigations of specific physical systems.
Two trains of thought concerning choices for dimensionality and scale in the study of subsurface solute transport
and fate are exemplified by Carrera (1993) and Logan (2001). Carrera states that the anomalies that can not be
accounted for by the classical solute transport equations (Equations 2-15 to 2-17) are directly or indirectly caused by
heterogeneity in the subsurface. He states that current approaches for dealing with heterogeneity can be divided into
deterministic and stochastic procedures. Stochastic methods have been successful in explaining qualitatively some
anomalies of solute transport and fate; but because of their restrictive assumptions (e.g., statistical homogeneity and
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stationarity), they appear to be far from reaching a stage at which they can be used routinely for solving realistic field
problems. On the other hand, Carrera states that deterministic methods, when applied with care, have been
successively used in actual problems. Yet, he says it can be argued that these methods fail to account for small-scale
variability of concentration so that they would become ineffective when dealing with nonlinear processes, such as
chemical reactions.
Logan (2001) states the obvious that "models do not include all the details of the physical reality". He says that
in the best of all possible worlds, the model should give a reasonable description of some part of reality. This is why
one often separates out specific mechanisms and studies equations with only diffusion, with only advection, or with
only reaction-dispersion. Understanding the behavior of these simple models can then give one clues into the
behavior of more general problems. For example, if one can show that some simple, nonlinear reaction-advection-
dispersion equation possesses solutions that blow up (go to infinity, or have their derivatives go to infinity) in finite
time, then one has succeeded in creating a healthy skepticism about such systems in the real world. Many of the
systems in Logan (2001) are one-dimensional with semi-infinite or infinite domains, involving nonlinear equilibrium
adsorption processes, non-instantaneous and multi-site adsorption kinetics, nonlinear advection and reaction rates, in
situ bioremediation processes, or colloidal transport and porosity-reducing filtration processes. The similarity
solutions (see Appendix B) and traveling-wave solutions, the solution blow ups, and the colloidal clogging of
filtration media studied in these rather simplistic and ideal models may well be representative of such happenings in
the real world. Thus, in choosing the dimensionality of the domain D, one must keep in mind the objectives of the
study and the resolution of the heterogeneities that are to be considered.
One-dimensional models cannot simulate the curvatures and refractions of streamlines as water flows through
layered soil with spatially varying hydraulic properties, and cannot simulate water flow in sloping soil layers
(Miyazaki et al., 1993). These models cannot account for two- and three-dimensional soil heterogeneities and
anisotropies, cannot account for the nonsymmetric location of surface-water bodies, cannot simulate the
heterogeneous area! distribution of rainfall and vegetation cover, and cannot account for a heterogeneous source of
radionuclides. For example, McCord et al (1997) presented a numerical modeling approach for assessing the
impacts of geologic heterogeneity on groundwater recharge estimates derived from environmental tracers. They
stated that common to many of the environmental tracer methods used to infer recharge in arid environments is an
assumption of one-dimensional, vertical downward flow of water and solutes. However, it is known that fluid flux
rates through geologic materials can be spatially variable in more than one dimension owing to heterogeneities in
porous media properties. Local flow directions within the medium may not be vertical even when application of
water at the surface is spatially uniform and hydraulic gradients are vertical. Consequently, environmental tracer
movements are also spatially variable and the one-dimensional assumption involved to interpret unsaturated
environmental tracer concentration profiles may be unrealistic. The results of the McCord et al (1997) simulations
show that recharge inferred using environmental tracer methods is also highly spatially variable and the recharge
estimates obtained by tracer profiles (using the one-dimensional model) tend to overestimate recharge and may only
be accurate to within an order of magnitude, particularly in situations with significant media heterogeneity in two or
three dimensions.
However, one- and two-dimensional models possessing scaling modules as given in Equations (2-7) to (2-9) can
often efficiently treat linearly variable soil structures over field mapping units and thus present reasonable three-
dimensional pictures of flow and solute transport (Rockhold et al, 1996; and Rockhold, 1999). For example, Figure
3-la shows 13 locations in a field mapping unit where it is assumed that each location has a different soil-water
behavior, but each behavior is linearly related to each other through similar soil structures (Figure 2-2).
Thus, a one-dimensional code, such as HYDRUS, through the use of its scaling component, can be efficiently
applied at each location, and the results of the simulation can form a reasonable composite picture of radionuclide
transport and fate over the entire field mapping unit, the composites being formed through the use of scaling,
superposition, and geostatistical techniques. Similarly, a two-dimensional code, such as CHAIN 2D, can be used to
form a good composite for the transport and fate of pollutants through the vadose zone underlying the field of row
crops shown in Figure 3-lb. The vertical slab simulation along the four transects shown in the figure can be
efficiently carried out through the use of the scaling module in the code. From these four transects, a composite for
the field can be constructed as in the one-dimensional case.
Once the dimensionality of domain D is established, along with its dimensions in all directions, the next step is
to determine realistic boundary conditions on all segments of the boundary, 3D. These conditions may be constant
in time, time-varying, or cyclic; and the type of conditions may be Dirichlet, Neumann, or Cauchy (Bear and
Verruijt, 1987; van der Heijde, 1994). Conditions are required for fluid flow variables, solute transport variables,
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12
13
10
(a) Plan View of a
Field Mapping Unit
Furrow Irrigation
Row Crop
4
3
2
1
(b) Plan View of a
Field of Row Crops
Figure 3-1 Schematic of modeling applications for simulating three-dimensional field mapping units using one-
dimensional codes (a), and two-dimensional codes (b).
and heat transport variables. The boundary conditions dictate how the domain (i.e., the unsaturated zone) interacts
with its environment (i.e., atmospheric/soil-surface zone, surface-water zone, ground-water zone). If the boundary
conditions are time varying, then the corresponding environment is time varying; if the conditions are spatially
heterogeneous, then there exist spatially heterogeneous environmental conditions; and if the boundary conditions are
stochastically distributed, then there exists randomly variable environmental conditions. If the governing equations
of a given model contains an internal (i.e., internal to domain D) source on sink term, such as the root water uptake
term in Equation (2-3), then what happens in the external environment to domain D may influence the formation of
the source or sink term, such as given in Equation (2-6). Thus, specific solutions of the governing equations of a
given model are highly influenced by the specific conditions specified on the boundary, 3D, of domain D. However,
the influence of conditions on one part of the boundary may die off faster or slower, both spatially and temporarily,
than the influence of conditions specified on another part of the boundary. Which part of the boundary and which
boundary conditions are the most influential in a given model depends on the size, shape and dimensionality of
domain D, and on the set of governing equations and constitutive relationships. The determination of these
influences is part of the sensitivity analysis of conceptual model selection.
The initial conditions for a given model are the starting distributions of the flow variables and the heat and solute
transport variables within the domain D, along with the starting distributions of any adsorbed pollutant species on the
soil matrix. Usually for the pollutant species (e.g., radionuclides), a time-varying source of pollution is specified at
the soil surface. If this is the case, it is convenient and often reasonably realistic to set the initial concentrations of
the dissolved and adsorbed species at zero. The difficult initial distributions to determine are those for the flow
variables. The pollutant species, except for background concentrations in the soil water and on the soil matrix, may
well have a well-defined starting point; but the moisture content and hydraulic head within most unsaturated zone
domains are constantly varying due to the constantly varying conditions in the environmental zones that influence
these domains. What further complicates the specifications of the initial flow conditions is that these conditions
possess a "history" component. The history component accounts for the hysteretic characteristics of the soil-water
properties (see Visintin, 1994; Brokate and Sprekels, 1996; and Appendix C). For a given soil, the soil-water
retention curve and the hydraulic conductivity function of Figure 2-1 take on different forms depending on whether
the soil is drying or wetting. In addition, Fetter (1999) found from numerical simulation that the importance of the
hysteretic mode appears to be greater for the pressure head and water content than for pore velocity and the solute
front movement. Thus, the forms of the distribution of the flow variables at zero simulation time depend on the
history of soil conditions prior to zero time.
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In spite of the above difficulties, initial flow distributions must be specified. One common mode of specification
is to assume a constant, steady flow through domain D. In this way, history components and hysteretic effects are
eliminated, as well as time-varying infiltration rates at the soil surface. Miyazaki et al. (1993) assert that the steady
state of water flow is the state where water is moving continuously without storage or consumption in the soil. They
further state that saturated flows in groundwater and unsaturated flows in the vadose zone when suction is less that
the air entry value can be considered steady flow provided that the flow boundary conditions do not fluctuate. In the
field, vertical-down water flows under ponded water at the soil surface and lateral flows of groundwater are typical
steady flows. However, a completely steady flow in unsaturated soil can only be generated in the laboratory where
flow boundary conditions can be fixed. In nature, as stated above, flows of water in unsaturated soils are almost
always in unsteady states due to changes in the flow boundary conditions (e.g., variable precipitation and evaporation
rates, see Appendix F), changes in soil pore water storage, impacts of soil hysteresis, and the consumption of soil
water by plant roots.
The water content distribution in a homogenous, unsaturated soil column under the influence of a steady
downward flow (i.e., a constant recharge rate, q) may be taken as a constant value, as in the CHAIN Model, or it may
be obtained from the steady-state solution of the Richards Equation (Equation 2-3, without the sink term S). This
steady-state solution gives a nonuniform moisture distribution in the homogenous soil column and can be obtained
from codes such as HYDRUS and CHAIN 2D. The differences of these two approaches on the fate and transport of
radionuclides are discussed in Appendix E. For multi-layered, unsaturated soil columns under the influence of a
steady downward flow, the water content distribution, as determined from the steady-state solution of the Richards
Equation, can be highly variable, depending upon the variability of the hydraulic properties as one passes from layer
to layer. These piecewise-continuous moisture distributions are shown for layered soils, with significant differences
in hydraulic properties from layer-to-layer, in Miyazaki et al. (1993) and Rockhold et al. (1997). The affects that a
layered soil under constant recharge versus that of a corresponding homogeneous layer on the fate and transport of
radionuclides are shown in Appendix G. However, the layered soil in this case does not possess large differences in
hydraulic properties as one passes from layer to layer. For example, the range of saturated hydraulic conductivities
over the nine layered column considered in Appendix G is from 172 cm/d to 539 cm/d, with a harmonic mean of 245
cm/d. Thus, differences in pertinent output parameters for radionuclide transport through the homogenous column
versus that of the nine-layered column vary from only 2 to 10 percent.
Appendix F compares the impacts on radionuclide fate and transport through a homogeneous soil column under
the influence of a steady-state recharge rate, q, against the time-varying conditions of daily precipitation rates and
daily evapotranspiration rates specified at the soil surface. As shown in Figure F-4, significant differences can occur
when steady-state flow conditions replace the more realistic daily, time-varying conditions. In arid and semi-arid
areas, the impacts are even more drastic when daily averages are replaced by storm event situations. For example,
Litaor et al (1998), through the use of large-scale rain simulations, found that the highly immobile radionuclide
species of plutonium and americium at the Colorado Rocky Flats Facility could be significantly remobilized under
severe storm conditions. Thus, it is assumed, but not proved, that the differences in radionuclide movement between
steady-state and time-varying flow conditions would even be greater than shown in Appendix F if daily averages
were replaced by sequences of realistic storm events.
3.3 Density and Thermal Gradients
For the objectives and purposes of the current study (i.e., radionuclide transport in the unsaturated zone), density
gradients produced by free pollutant products with densities greater than or less than pure water, and gradients
produced by fresh/saline water interfaces are assumed to be of lower order importance and will not be further
considered. Discussion of the importance of such gradients can be found in Domenico and Schwartz (1990) and
Bedient et al (1999). However, density gradients due to thermal effects may be of importance.
The temperature gradients in the soil form a driving force for movement of both liquid water and water vapor.
Therefore, a daily change in the temperature due to surface changes in wind velocity, net radiation from the
atmosphere, relative humidity, and air temperature near the ground may influence water movement in the soil. Such
temperature changes produce changes in soil water viscosity and surface tension which in turn change the hydraulic
conductivity and pressure head (Equation 2-10 and 2-11). In addition, these temperature gradients affect several of
the parameters (e.g., molecular diffusion coefficients, production coefficients, and decay rates) in the solute fate and
transport equations, see Equations (2-24) and (2-25). Temperature changes in the subsurface may also occur due to
the generation of heat during chemical reactions, the generation of heat by underground nuclear waste storage
repositories, and the heat produced in deep geological formations by hydrothermal systems. Whenever heat is
generated and temperature gradients produce significant changes in the flow of underground vapors and liquids, and
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produce significant changes in solute fate and transport, the energy balance must be added to the governing system of
equations, such as the balance given in Equation (2-26) to (2-28). Accounting for temperature variations in the upper
soil layers is definitely important in geographical areas experiencing annual fluctuations in air temperatures from
below freezing to above freezing. These fluctuations are, in turn, important for the fate and transport of
radionuclides from surface sources. Even in the arid areas around the Mojave Desert, Nevada, Andraski (1997)
found that both isothermal liquid and vapor flow, and nonisothermal vapor flow needed to be considered in the
conceptualization of flow under natural vegetated sites, as well as under sites where the vegetation had been
removed. Thus, the assumption of isothermal conditions (i.e., no heat flow considerations) for all fluid and solute
flow may lead to uncertainties that could be eliminated if the effects of thermal gradients, and their induced density
gradients are considered.
3.4 Facilitated Transport and Preferential Pathways
Facilitated transport of pollutants in the unsaturated zone, as well as in the ground-water zone, is the transport
of adsorbed contaminants on suspended particles or colloidal particles that are able to pass through the connected
pores of a porous medium. In Corapcioglu and Choi (1996), the simulation of colloid transport in the vadose zone
consists of four phases: an aqueous phase, the stationary solid matrix phase, a carrier (i.e., colloidal particles) phase,
and a stagnant air phase. The fact that colloidal particles can act as carriers enhances the transport of contaminants
in porous media by effectively reducing retardation, the R: value in Equations (2-15) and (2-16). McCarthy and
Zachara (1989) indicated that the highly immobile species of plutonium and americium were transported over a mile
from a liquid waste outfall at New Mexico's Los Alamos facility through preferential pathways by colloidal carriers.
These results are supported by the work of Litaor et al (1998) at Colorado's Rocky Flats facility. In this study, the
remobilization of large fluxes of americium and plutonium, initiated by severe storm events, was accomplished by
carrier suspended and colloidal particles. The dynamics of suspended and colloidal particle systems are fully
discussed in references such as van de Ven (1989) and Probstein (1994).
Colloids, other suspended particles, and microbial particles play another important role in the fate and
transported of radionuclides in the unsaturated zone, namely the change of the medium's porosity with time. Logan
(2001) calls this filtration. He says filtration occurs in a porous medium when the transported particles are actually
sieved by the solid porous matrix, thereby decreasing the porosity and possibly clogging the medium. He states that
this clearly can, and will, occur when the average size of the pores is smaller than the particles in solution, as is the
case for large molecules, bacteria and colloids migrating through clayey soil. But filtration can also occur when
smaller particles form sediments in domains with large porosities. In fact, Kretzschmar et al (1997) have evaluated
four different experimental systems for determining colloid deposition rates and collision efficiencies in natural
porous media. They found that colloid deposition generally followed a first-order kinetic rate law and they were able
to calculate collision efficiencies for the colloid deposition. Logan (2001) studied the filtration process through the
use of a one-dimensional, time-varying, deterministic system of equations in terms of the volume concentration of the
transported colloid, the volume concentration of the retained or immobile colloid, and the changing porosity of the
medium. Kaiser (1997) studied the clogging in a two-dimensional porous medium with small inhomogeneities
through the use of directed percolation techniques, Monte Carlo simulations, and fractal pore clogging (i.e., the
deposited colloids formed fractal clusters in the percolation network). The proposed models for study in the current
report do not allow for either colloidal transport of radionuclides or for colloidal reduction of porosity. As indicated
above, both phenomena under the right conditions could be very important.
The unsaturated zone is certainly not a homogeneous, porous medium. In the root zone there are numerous large
pores and cracks formed by such agents as plant roots, shrinkage cracks, and animal burrows. These macropores
can form preferential pathways for the movement of water and solute, both in the vertical and horizontal directions
through the root zone. This situation can lead to what is called "short-circuiting" or "bypass flow" of the infiltrating
water as it moves through the macropores at a rate much greater than would be expected from the hydraulic
conductivity of the soil matrix (Fetter, 1999). This bypass flow only takes places when either the macropores and
cracks are open to the atmosphere or the water pressure within the macropores and cracks is positive. On the other
hand, when ponded water infiltrates a soil matrix within which macropores and cracks are buried and when the soil
matrix is under suction, water will not infiltrate the buried macropores but will infiltrate only the soil matrix. In this
case, macropores are not preferential pathways for flow but are obstacles of the water flow (Miyazaki et al, 1993).
A second type of preferential flow is fingering which occurs when a uniformly infiltrating solute front, or
wetting front, is split into downward-reaching "fingers" due to instabilities in the wetting front. An instability often
occurs, due to pore-scale permeability variations, when an advancing wetting front reaches a boundary where a finer
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sediment overlies a coarser sediment. Under some conditions, fingering flows can also be generated in uniform soils.
Yao and Hendrickx (1996) state that instabilities in wetting fronts occur under the following conditions:
Infiltration of ponded water with compression of air ahead of the wetting front;
Surface desaturation and redistribution of water in the soil profile;
Water-repellent soils due to the plant litter and residues, organic fertilizers, and pesticides;
A water content increasing with depth; and
Continuous nonponding infiltration.
In several of these mechanisms which produce wetting front instabilities, flow hysteresis (Appendix C) plays a
major role.
There have been several recent studies concerned with these various conditions producing wetting front
instabilities. Wang et al (1997, 1998) considered the variations of water infiltration into the unsaturated zone under
the influence of air compression, air counterflow, and flow hysteresis. The first paper deals with theoretical
considerations and the second paper gives the results of testing this theory in the laboratory. Wang, Fey en and Elrick
(1998) derived generalized linear instability criteria for finger formation applicable for field scale assessment.
Wang, Feyen and Ritsema (1998) experimentally evaluated these criteria in the laboratory under the combined
effects of air entrapment, surface desaturation, soil layering, and water repellency. Further studies on the effects of
water repellency on infiltration, wetting front instability and vadose zone flow fingering were carried out by Wang et
al (2000), and Wang, Wu and Wu (2000).
A third type of preferential flow is funneling (Fetter, 1999; and Miyazaki et al, 1993). Funneling occurs in the
vadose zone below the root zone and is associated with stratified soil profiles. Sloping coarse-sand layers embedded
in fine-sand layers can impede the downward flow of water. The sloping layer will collect the water like the sides of
a funnel and direct the flow to the end of the layer. When the pressure of the funneled flow exceeds a critical value,
fingering flow will be generated in the coarse layer and the water again percolates vertically.
Miyazaki et al (1993) state that preferential flows in soils are generally regarded as a type of saturated flow, and
that evidence of unsaturated preferential flows is scarce. In addition, they state that the velocities and local quantities
of preferential flows are large compared with other saturated flows in soils. In the review by Gee at al (1991), it is
stated that some very heterogeneous soils exhibit little or no preferential flow when partially saturated. They
considered a test at Las Graces, New Mexico, where extremely heterogeneous soils were wetted at rates well below
saturation and no preferential flow was observed. In general, the visual wetting front for these tests was relatively
smooth and surprisingly uniform. However, Shurbaji and Campbell (1997) studied the phenomena of evaporation
and recharge in desert soil at three sites in southeastern New Mexico and found that possibly 85% of the recharge
occurs via movement of water through preferential pathways in the root zone. Preferential flow was evident at all
three sampling sites, and the long-term recharge rates at these sites were calculated to be 0.5, 0.8, and 2.4 mm/yr.
Yao and Hendrickx (1996) assert that worldwide it is thought that a major mechanism for ground-water
contamination is the passage of pollutants through preferential flow pathways. For modeling purposes, it should be
kept in mind that this is an event type phenomenon (e.g., storm events of tens of minutes to an hour or two) which is
currently not well parameterized for large time-step simulation.
3.5 Scale Dependency in Heterogeneous Media
Most subsurface flow and transport models are not sufficiently detailed to allow an explicit representation of all
dynamical scales in heterogeneous media. That is, most models are constructed at such a coarse scale of resolution
that unresolved subgrid variablity often exists. It is therefore important to understand the interaction between
unresolved dynamics and explicitly resolved dynamics. For example, Beckie et al (1994) examined when it is
possible to construct an accurate model for the explicitly resolved large scales, without an explicit description of the
subgrid scale dynamics; and they demonstrated how unresolved subgrid scale dynamics interact with resolved scale
dynamics. They showed that a universally valid resolved scale model can be constructed if the resolved dynamics
are sufficiently independent of the details of the subgrid scale dynamics; that is to say, there is a spectual gap
between the subgrid scales and the resolving scales. In the event of a spectral gap, a resolved model is composed of
a universal structure and accompanying model parameters, the parameters representing the effect of unresolved
dynamics upon resolved dynamics. Beckie et al (1994) showed theoretically, and numerically, that a local Darcy
law (Equation 2-1) for groundwater flow is a universally valid resolved scale model if the resolved and subgrid
scales of the hydraulic conductivity are separated in scale by a spectral gap. However, if the hydraulic conductivity
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possesses many scales of variability (which Neuman and DiFederico, 1998, assert is true of all real geologic media),
then a more general nonlocal Darcy law is a more appropriate model structure. Using numerical experiments,
Beckie et al (1994) found that when the nonlocal Darcy law is more appropriate, the errors in using the local Darcy
law with effective parameters are most significant at the smallest resolving scale of the nonlocal model, and are
minimal at scales between 8 and 16 times the resolving scale.
Carrera (1993) considered the applicability of the advection-dispersion-reaction equations in their standard form
(Equations 2-15 to 2-17) for simulating solute transport in heterogeneous media. He identified several solute
transport features observed in the field that are poorly simulated with such equations. The most notable of these is
scale dependency of dispersivity (dispersivity tends to increase with the scale of measurement). Other features are
long tails in breakthrough curves, skewness in spatial distribution of solutes, directional effects on arrival times,
porosity, and dispersivity, and the inability to account for small-scale variability of concentrations which affects
nonlinear processes such as chemical reactions. Carrera asserts that all of these effects are more or less caused by
the spatially correlated variability of hydraulic conductivity. He states that spatial variability can be simulated using
a stochastic approach and that this may be the correct framework for formulating solute transport. Even though
stochastic hydrogeology has been successful in explaining some facts observed in the field, he says that its
applicability to real problems is hampered by theoretical and practical difficulties, such as the stochastic stationarity
of hydraulic conductivity. Carrera says this conflicts with observations. For example, the scale effects of hydraulic
conductivity, small-scale variability of concentrations, and the possible unbounded growth of dispersivity suggest
that the logorithm of the hydraulic conductivity is usually non-stationary. The alternative to stochastic formulations
is a deterministic approach to solute transport, an approach based on the belief that dominant discrete
heterogeneities can be identified and their effects can be explicitly built into the model. Hence, the main advantage
of deterministic methods is their ability to incorporate qualitative information into their formulations, while the main
disadvantages arise from the lack of procedures for representing such qualitative information in a given model.
Thus, much effort is being expended to identify effective parameters for flow and solute transport models in
heterogeneous environments.
The flow and transport in heterogeneous porous media is highly dependent on the spatial variability of soil and
aquifer properties. However, it is impractical when modeling a real situation to obtain aquifer properties at the scale
of each "grid block" of a given model. Hence, tools are required to move information between scales (e.g., upscaling
from laboratory measurements to grid block scale), to incorporate the scale and location of test data when
interpolating to a fine grid, and to fill in missing information around sparse data areas. Some tools that have been
used, and are being used, for these purposes are given in the following books and papers:
Transport and divergence theorems developed in the context of generalized functions for changing
spatial scales in the analysis of subsurface flow and transport, Gray et al (1993).
A multi-scale, hydraulic conductivity distribution, reconstruction method based on forward and inverse
wavelet transforms in conjunction with a pseudo-fractal distribution for filling in missing information
around sparse data areas, Brewer and Wheatcraft (1994).
Comparison of two fast algorithms, one based on random walks and the other based on real-space
renormalization group methods, for upscaling permeability data from cores to reservoir grid blocks in
heterogeneous media, McCarthy (1995).
A review of the various upscaling techniques used to calculate the equivalent permeability of a
heterogeneous porous medium, and indicating in what circumstances they can be most suitably applied,
Renard and de Marsily (1997).
The use of the theory of homogenization as an upscaling procedure for modeling porous media on
micro-, meso- and macro-scales; percolation, two-phase flow and miscible displacement are also
considered, Hornung (1997).
The application of fractals in the unsaturated zone for characterizing heterogeneity and for modeling
soil moisture, biodegradation and solute transport, Crawford et al (1999).
A current assessment of parameter upscaling procedures and techniques for addressing vadose zone
heterogeneities, leading to an optimistic outlook on improving our ability to model pollutant transport
in this complicated subsurface zone, Mayer et al (1999).
3.6 Chemical Adsorption, Chemical Reactions, and Decay Processes
According to some investigators, ideal solute transport is transport through a homogenous porous media under
the influence of linear, instantaneous adsorption/desorption processes. Conversely, nonideal solute transport
includes a host of complicating processes leading to solute transport complexity, processes such as (Brusseau and
25
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Rao, 1989; Weber et al, 1991): nonlinear equilibrium sorption; partial, nonreversable sorption; linear and nonlinear,
nonequilibrium sorption; sorption capacity variability and rate-limited sorption; facilitated transport; hydraulic
conductivity variability due to heterogeneous media; vapor-phase processes; immiscible liquid phase processes; and
transformation reactions and decay rates. Brusseau and Rao (1989) list several induced and natural gradient tracer
experiments showing extended-tail breakthrough curves which were attributed to rate-limited sorption, hydraulic
conductivity heterogeneity, sorption capacity limits, and/or rate-limited desorption. Thus, the subject of this
subsection is the sensitivity of conceptual models based on varying chemical sorption, chemical transformation, and
chemical decay processes.
Kuox et al (1993) and Thibodeaux (1996) describe the various processes involved in solid-liquid soil reactions,
including those characterized as adsorption/desorption. Depending upon soil structure and texture, contaminant
characteristics, and the flow characteristics, the characteristic time scales for simple physical sorption are known to
vary from microseconds to months. In addition, experimental evidence has shown that often the solid-phase fraction
of a contaminant on the desorption cycle behaves much different than when it is on the adsorption cycle. This leads
to nonequilibrium states and hysteretic type adsorption/desorption cycles. For example, Knox et al (1993) states that
such hysteretic cycles affect the amount of water required in pump-and-treat remediation of subsurface aquifers.
Thibodeaux (1996) states that the validity of the ideal local equilibrium assumption (LEA) for solid-liquid reactions
depends on the degree of interaction between the macroscopic transport processes of water and hydrodynamic
dispersion, and the microscopic processes of molecular diffusion and sorbed-solute distribution in conjunction with
soil aggregate size, shape and composition. When the rate of change of solute mass during the microscopic sorption
processes is fast relative to the mean velocity of the bulk flow, the solid-liquid interactions are nearly instantaneous
with reference to bulk-flow time scales, and they conform to the LEA. Deviations from the LEA occur as the
interactions of the solute with the soil matrix become increasingly time dependent with respect to bulk-flow time
scales. This divergence also occurs as soil aggregates increase in size and complexity and pore-class heterogeneity
increases.
Several investigators have studied the effects of the deviations from LEA and linear adsorption/desorption,
deviations based on nonlinear adsorption and those based on nonequilibrium solute transport. Common nonlinear,
equilibrium isotherms are those of Freundlich,
s = acn , (3-1)
and Langmuir,
s= ac-(l+lk) (3-2)
where c is the concentration of the liquid phase in mg/L, s is the concentration of the solid phase on soil equilibrium
sites in mg/kg, and a, B, and n are constants with consistent physical units. The linear, equilibrium isotherm,
s = Kdc, (3-3)
arises from the Freundlich isotherm when n = 1 and a = Kd, the distribution parameter in ml/g. Logan (2001) and
Fetter (1999) list several other nonlinear equilibrium isotherms, such as the Langmuir two-surface sorption isotherm,
the generalized Langmuir isotherm and the quadratic and exponential isotherms.
Using a one-dimensional, deterministic, advection-dispersion-sorption model, Fetter (1999) found that the
Freundlich sorption isotherm with n>l produced a spreading pollutant front, while for 02. These differences were shown
to occur for both one- and two-dimensional domains. In addition, Logan (2001) pointed out that the Langmuir
isotherm limits the amount of material that can be adsorbed (i.e., s< o/B), but the Freundlich isotherm does not.
Thus, the use of the Freundlich isotherm should be done with extreme caution for realistic simulations. In other
studies, Bai et al (1997) constructed a triple-porosity, one-dimensional model of contaminant migration with linear
sorption in strongly heterogeneous media. They showed that extensive tailing in the breakthrough response occurred
over that due to single-porosity approaches and that multiple breakthrough fronts with reverse diffusion also
developed. Both behaviors were due to the strong heterogeneities. Berglund and Cvetkovic (1996) considered
contaminant displacement in aquifers under the coupled effects of flow heterogeneity and nonlinear sorption, using
five different isotherm equations, including those of Freundlich and Langmuir. They found that the effects of the
26
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choice of sorption isotherm is very significant and strongly dependent on the heterogeneity of the particular data set
considered.
The solute transport system of the HYDRUS Code, Equation (2-15) to (2-17), accounts for a two-site sorption
structure (i.e., equilibrium and nonequilibrium sites). This model was developed by van Genuchten and his
colleagues (van Genuchten and Wagenet, 1989; Toride et al, 1993). A two-dimensional version of this system also
occurs in the CHAIN 2D Code. In Appendix H, the impact of nonequilibrium sorption on the fate and transport of
radionuclides is considered. This impact can be great depending upon the mean recharge rate in the vadose zone,
and the relative relationships between the first-order rate for nonequilibrium sorption, the radioactivity decay rate,
and the linear distribution parameter Kd. In other studies, Simon etal (1997) and Logan (2001) determined the
properties of exact and approximate traveling wave solutions for solute transport with nonlinear and nonequilibrium
sorption. Often nonlinear sorption leads to the existence of moving concentration fronts of substances transported in
porous media that do not change shape (i.e., traveling wave solutions). The cause for the existence of such solutions
is a balance that develops between the self-sharpening effect of nonlinear sorption and the spreading effects of
dispersion and sorption kinetics. Traveling wave solutions also occur for the combined processes of linear
equilibrium and nonlinear nonequilibrium sorption. The nonequilbrium processes considered in Appendix H are
linear and do not possess traveling wave solutions. Finally, Chen and Wagenet (1995) compared the two-site,
equilibrium/nonequilibrium model with a continuously-distributed, sorption site heterogeneity. They found that the
deviations between the two-site model and the continuously-distributed model tended to be more severe as time
increases. Thus, as sorption-site heterogeneity increases, the applicability of the two-site model tends to decrease.
Murray (1989) and Logan (2001) have shown that diffusion-dispersion-reaction systems can have a variety of
solution forms when the reaction terms are nonlinear. For example, solutions can blow up in finite time; traveling
wave solutions can occur; solitons can be formed, which are special wave pulses which interact with one another so
as to keep their basic identity and take on a "particle-like" character; deterministic solutions can become chaotic as
time evolves and act as "random" quantities; and various stable and unstable spatial patterns can form. Even though
Fetter (1999) demonstrates that the chemistry of dissolved uranium is somewhat complex, the possible chemical
reactions of the five radionuclides listed in Section 2.1 will not be considered in this report. The only
transformations that will be considered are those related to adsorption/desorption and radioactive decay.
In Appendix I, the effects on the pollutant breakthrough curves of the five radionuclides (Section 2.1) due to
changing distribution coefficients (Kd of linear, equilibrium sorption), changing decay rates (|J,), and varying recharge
rates (q) were investigated using the CHAIN Model. The values of Kd and [i vary with the parent radionuclide under
consideration. Also given in Appendix I is an example of the differences between the one-dimensional transport and
fate of a parent and that of a daughter radionuclide. Oldenburg and Pruess (1996) indicate some daughter
characteristics in their two-dimensional simulation of mixing with first-order decay in variable-velocity porous media
flow that can not happen in the one-dimensional simulation given in Appendix I. These authors found that the
mixing due to advective dilution, hydrodynamic dispersion, and molecular diffusion is stronger in regions of higher
velocity and weaker in slower-moving regions. Further, they found that concentrations profiles normal to the flow
direction are displaced toward regions of slower flow in fields possessing velocity gradients. Hence, they found for
species undergoing first-order decay, that the effect of parent accumulation in regions of low velocity is enhanced for
the daughter species because of the following factors:
The rate of daughter production is proportional to the local concentration of the parent;
Mixing is proportional to the local velocity; and the lower the velocity, the less mixing and longer time
periods for decay.
Using a one-dimensional system, Logan (1996) studied the transport of a decaying tracer in a heterogeneous
porous medium subjected to rate-limited adsorption with a linear equilibrium isotherm. The medium was semi-
infinite and periodic boundary conditions of the Dirichlet and flux-type were examined. The heterogeneity was
simulated by using a scale-dependent dispersion coefficient that increased exponentially with distance, up to some
limiting value. The one-dimensional model was designed to give information about how adsorption and decay
effects can interact with heterogeneities in the medium. For example, when both adsorption and decay are present,
the amplitude of a pollutant wave in a homogeneous medium can exceed that in a heterogeneous medium, even
though the later has smaller dispersivity. Somewhat contradictory evidence was given by Miralles-Wilhelm and
Gelhar (1996), who performed a stochastic analysis of the transport and first-order decay of a solute plume in a
three-dimensional heterogeneous aquifer. They found that the characteristic timescale of the transient development
of all field-scale coefficients is reduced by the presence of a heterogeneous decay rate, and that all these trends are
27
-------
accentuated with increasing decay rate variability. Increase peak concentrations, earlier arrival times, and decreased
plume spreading are practical consequences of these derived results.
3.7 Summary
The selected models, identified in Section 2.1, are all deterministic and thus are incapable of stochastic
simulation, except for Monte Carlo simulations for certain parameter or boundary condition distributions. In
addition, four of the models (i.e., CHAIN, MULTIMED-DP 1.0, FECTUZ, and HYDRUS) are one-dimensional and
only CHAIN 2D is two-dimensional. Thus, three-dimensional features cannot be easily simulated by these models,
except for features such as those presented in Figure 3-1. Many two-dimensional features cannot be easily simulated,
as well, even with the use of CHAIN 2D. Density gradient problems cannot be addressed by these models, as well as
facilitated transport and preferential pathways. The scale dependency of hydraulic conductivity and dispersivity due
to soil heterogeneities are not considered by these models. Other processes and features that are not addressed by
these models are nonlinear reactions, nonlinear nonequilibrium sorption, nonequilibrium site heterogeneity, and
variable decay processes. To address precipitation/evapotranspiration/infiltration processes on a storm event basis
for the several-year periods required for some radionuclides will require extensive computational resources for the
models in this group which are capable of performing such simulations. Table 2-1 summarizes the five models'
capabilities and limitations with respect to the other features discussed in this section on conceptual model selection.
28
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Section 4
Parameter Sensitivity Analysis: Basic Elements
The process parameter sensitivity analysis of a model is a measure of the change in a selected model output
resulting from a specified change in an input parameter. Mathematically, the sensitivity coefficient, Sir of a model's
output, y;, to a model's input parameter, xp is given by
-
3Xj • (4-1)
For a model with n output quantities and m input parameters, the total number of sensitivity coefficients that may be
required in the analysis is given by the product, mn.
In the current analyses, the unsaturated zone model evaluation for the transport and fate of radionuclides, the
model input parameters include such quantities as the hydraulic conductivity, K, the maximum depth of plant roots,
Lm , the pore-size distribution index, n, the net infiltration rate, q, the first order rate constant, [i, and the dispersion
coefficient, D. The model outputs of interest include such quantities as the peak concentration, Cpeak, of a given
radionuclide, the time, Tpeak, at which peak concentrations reach the water table, and the time, TMCL, at which a
given radionuclide concentration will exceed a given quantity at a specified point. In this study the specified point is
the water table, which is the point where the radionuclide in question is leached into the ground water. The given
quantity is the product of the maximum critical level (MCL) of the contaminant in the ground water times a dilution
attentuation factor (DAF) where DAF > 1. The DAF is used for back-calculation of the soil screening levels (SSLs).
When the specified radionuclide's concentration exceeds the value (MCL)(DAF) at the water table, the contaminant
at the downgradient receptor is considered to pose a risk to human health and environment.
The units of Sy are those of y; divided by those of Xj. Since Sy in general, is unit dependent, it may be difficult
and confusing to compare sensitivities for different input parameters. These problems are overcome by introducing
a normalized form of Sy, called the relative sensitivity coefficient, defined by:
x,
<4'2)
By definition, Srij is dimensionless and can be considered an absolute sensitivity coefficient in the sense that it
reduces the difficulties and confusion inherent in the use of just Sy for sensitivity analyses.
Suppose we have a simple model given by the following arbitrary equation:
y = F(p), (4-3)
where p is the input parameter and y is the output of the model.
The sensitivity coefficient for this model is given by the ordinary derivative:
dy _ dF
FP = T~ " T~ ' (4-4)
dp dp
29
-------
The relative sensitivity coefficient F is defined by:
_ P
dF
Lrp
F(p) dp
(4-5)
Suppose that the graph of F^ versus p (Equation 4-5) is given by Figure 4-1. To aid in the interpretation of this
figure, rewrite Equation (4-5) as
F(p)
dp
—
p
(4-6)
Equation (4-6) states that a small percentage change in the input, 100 dp/p, in the neighborhood of p, produces a
percentage change in the output, 100 dF/F(p), in the neighborhood of p equal to 100 dp/p times Frp(p). Thus, looking
at Figure 4-1, one concludes the following:
1. A 10% increase in the input parameter p in the neighborhood of pj, based on the value of pj, produces
a (10a)% decrease in the value of y 1; or the quantity F(pj).
2. A 10% increase in the input parameter p in the neighborhood of p2, based on the value of p2, produces
a (10b)% increase in F(p2).
3. A 10% increase in the input parameter p in the neighborhood of p3, based on the value of p3, produces
a (10c)% increase in F(p3).
4.1. Computing Sensitivity Coefficients
Often models are too complicated for one to obtain sensitivity coefficients by the direct analytical approach
indicated in Equations (4-1) and (4-2). When this occurs, the sensitivity coefficients can be approximated by the
following difference equations:
s ~
=
and
(4-7)
a —/
Figure 4-1. The graph of the relative sensitivity F^ in terms of the parameter p for a model defined by y
F(p).
30
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Srij = "7 • > (4-8)
A -y- ~\T ~\T V ^ '
where Ay; is the change in y; due to a small change Ax j inxj. The above interpretation of the graph of F^ versus p
in Figure 4-1 follows directly from the last term in Equation (4-8); namely, the relative sensitivity can be expressed
as the ratio of the relative change of the output parameter Ay;/y; to the relative change of the input parameter
AXj / Xj . Forexample, if the relative sensitivities are -1.0 for the time to reach peak concentration to recharge
rate and +0.05 for the time to reach peak concentration to distribution coefficient, then it can be said that the time to
reach peak concentration is much more sensitive to recharge rate than to the distribution coefficient.
4.2. An Application of Equations (4-1) and (4-2) to a Simple Model
Because of the complexity of many models, the meaning of the variations in the sensitivity S and the relative
sensitivity Sr for a given output versus a given input parameter is often obscure. Thus, the choice of a simple model
with a variety of nonsimilar components will aid the investigator in understanding variations in S and Sr. The simple
model that we chose to present is one with a single output y and six input parameters (a,b,c,d,e,f). The defining
equation for this model is arbitrarily chosen for pedagogical reasons and is not representative of any physical
phenomenon. It is given by:
O j— —i
y = F(a,b,c,d,e,f) = — b + cd + eln(c2f) . (4-9)
eL J
For parameter a, the sensitivity of y with respect to a is denoted by Fa and the relative sensitivity is denoted by Fra.
This notation is used for the other 5 input parameters, as well. Table 4-1 lists the formulas for S and Sr
corresponding to the expression in Equation (4-9).
For a numerical example corresponding to the arbitrary formulas in Table 4-1, we require a base case and a domain
of definition for each of the input parameters. These numerical quantities are chosen as follows:
Base Case = F(a,b,c,d,e,f) = F(2,l,-l,3,4,5), (4-10)
-l
-------
Table 4-1. Sensitivities and Relative Sensitivities of Output F with Respect to the Six Input
Parameters, for Arbitrary Values of the Inputs.
Parameter
Sensitivity, S
Relative Sensitivity, Sr
a
e
b + cd + eln(c2f)
dc + 2e
dc + b + eln(c2f)
ac
e
cd
cd + b + eln(c2f)
b + cd
e£n(c2f) + b + cd
a
f
e£n(c2f) + b + cd
which is outside the domain of d. From these three input parameters (a,b,d), we can see the effects of linearity,
homogeneity and sign of the slope, as well as the domains of definition of the input parameters. Also in Figure 4-3 is
theplotof F(2,l,c,3,4,5) versus c. Because of the particular domain of c and the presence of ln(c4) inF, there is a
vertical asymptote at c = 0 in the graph of F versus c. In addition the linear-logarithmic variation of c in F results in
two zeros, one at a and one at p. The vertical asymptote shows up as a vertical asymptote in Fc but a zero in Frc.
The zeros (a, p) in F have no particular effect on Fc but become vertical asymptotes in Frc. The y zero in Fc carries
over into a zero of Frc. These zero/asymptotic relationships will exist between F, Fa, and Fra as long as we do not
run into indeterminant forms, 0/0 or °°/°°. For these cases, the asymptote and zeros in Fa, and Fra may disappear.
Finally, in Figure 4-4, we note that for the domain of definition in F(2,l,-l,3,e,5) there is a zero at a which produces
a vertical asymptote inFre. The quantity F(2,l,-1,3,4,1) has no zeros because of the restricted domain of definition
for f. One can also note the different characteristics in Fre and Frf at the right side of the domains of definition of e
and f, respectively. This is because the quantity F(2,l,-l,3,e,5) approaches a horizontal asymptote (F=3.219) for
sufficiently large e, while F(2,l,-l,3,4,f) approaches +°° in a logarithmic manner as f becomes large. Thus, by
studying these simple forms, one can better interpret the sensitivity and relative sensitivity results for the five models
being analyzed in this report.
32
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Table 4-2. Sensitivity and Relative Sensitivity of Output F to Individual Input Parameters with Reference to
the Base Case Given in Equation (4-10).
Parameter
a
b
c
d
e
f
Output F
Formula
Ui(5)- - a
L 2\
b 3
+ £n(25)
2 2
4 3c 1
ln(c ) + + ln(25) +
2 2
d 1
1 h fn(25)
1 1
4
- - + #j(25)
e
2 te(f ) - 1
Characterization
Linear and
Homogeneous in a.
Linear and Non-
homogeneous in b.
Linear-Logarithmic
and Nonhomo-
geneous in c.
Linear and Nonhomo-
geneous in d.
Nonhomogeneous in e,
Inverse Variation with e.
Logarithmic and
Nonhomogenous in f.
Sensitivity
s
1
9
1
2
4 3
p 9
\
2
A
e
2
—
f
Relative Sensitivity
sr
1
b
b - 3 + £n(625)
8
c + -
3
8/3 1 4
3 3
d
d - 1 - ln(625)
2
ln(5) e - 2
1
tn(f] -
2
33
-------
1.2
1.0
.8
F .6
1 ra
.4
.2
0
.4
-1012345
a
1.2
1.0
.8
"a .6
.4
.2
0
-1012345
0
F -2 -
rrb
.1 -
0 1/2 1 3/2 2
0 1/2 1 3/2 2
F
-02345
F
3 -
2 -
1 -
Figure 4-2.
0 1/2 1 3/2 2
Sensitivities and relative sensitivities of F with respect to a and b, with reference
to the base case in Equation (4-10).
-------
1 -
Frd
-2-
-3.
12345
-4 -3-2-10 12
-1.0
12345
d
3 -
F 2-
1 -
12345
d
Figure 4-3. Sensitivities and relative sensitivities of F with respect to c and d, with reference
to the base case in Equation (4-10).
35
-------
10
6 -
4 -
2 -
're o
-2 -
ia
a= llln (5)
1234567
e
'e 2-
1234567
1 234567
234567
f
Figure 4-4. Sensitivities and relative sensitivities of F with respect to e and f, with reference
to the base case in Equation (4-10).
36
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Section 5
Parameter Sensitivity Analysis: Hypothetical Modeling Scenario
The characteristics of a representative physical site are required to realistically study the sensitivity of the five
models proposed in this report for simulating the transport and fate of radionuclides in the unsaturated zone. Such a
site would be used to set the standards for base parameter choices in the sensitivity tests and for parameter range
determination. The ideal candidate site should have well characterized soil properties, and should have sufficient
chemical transport and fate data and climatological information for running the pertinent simulations. Once a site is
chosen, a conceptual model of that site is required so that the site data, the five proposed modeling codes, and the
intended application of the models are all consistent with one another. In what follows, we consider the site selection
process, selection of the candidate site, characteristics of the Las Graces Trench Site in New Mexico, development of
a conceptual model, and the base parameter selection.
5.1 Site Selection Process
The sensitivity studies reported in this document are part of an overall evaluation of the general applicability of
each model (i.e., HYDRUS, CHAIN 2D, FECTUZ, MULTIMED-DP 1.0, and CHAIN) for simulating the fate and
transport of radionuclides. The radionuclides of interest are the following five elements/isotopes (U.S. EPA, 2000b):
Element/Isotope Kd Default Values in ml/g
Plutonium (238Pu) 5
Strontium (90Sr) 1
Technetium ("Tc) 0.007
Tritium (3H) 0
Uranium (238U) 0.4 ,
where Kd is the soil/water partition coefficient for the linear Freundlich isotherm. These five elements/isotopes are
taken to be the parent species in the decay chains and the major segments of the chains are shown in Figure 5-1 along
with the corresponding half-lives of the species.
The quantity, Kd, in all five models being studied, is a lumped parameter representing known and unknown
phenomena. It tends to lose some of its meaning in the modeling world, while retaining its full meaning in the
laboratory. In the laboratory Kd is determined under carefully controlled conditions, but the real world cannot be
completely controlled or measured. Thus, the conditions surrounding the sorption phenomenon must be estimated,
and these estimates will only represent localized conditions in the vicinity of the sampling. Consequently, the large
heterogeneities of most, if not all, subsurface systems make it difficult to identify a single Kd value for the system.
Further, unless the model has a geochemical component, Kd will also represent everything one does not know about
the site's geochemistry. The relationships between Kd values, the chemical species, and a site's geochemistry are
discussed in U.S. EPA (1999ab, 2000b). However, in the current report, the default Kd values, given above, are
taken as the base values for the sensitivity analyses combined with an interval domain about these values, to partially
represent the heterogeneity found in the vadose zone.
Based on the above radionuclides, the site selection process consisted of identifying three features: site listing,
data availability, and existing field and modeling studies. By reviewing existing literature and databases, a list of
candidate sites was developed, see Table 5-1. These candidate sites, in general, were either contaminated with
radionuclides, or were current or future radioactive waste disposal sites. For example, Superfund sites with soil
contaminated by one or more of the above five radionuclides were identified in the U.S. EPA's VISITT database (i.e.,
the Vendor Information System for Innovative Treatment Technologies, Version 6.0). Many of these radionuclide
37
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PLUTONIUM,238Pu
88 yr. half-life
~v
URANIUM, 234U
2.4x 105yr. half-life
STRONTIUM, 9°Sr
29 yr. half-life
>.
YTTRIUM, 90y
64 hr. half-life
>.
ZIRCONIUM, 9°Zr
Stable
TECHNETIUM,"Tc
2.1 x 105yr half-life
V
RUTHENIUM, "Ru
Stable
TRITIUM, 3H
12 yr. half-life
•v
HELIUM-3,3He
Stable
URANIUM, 238U
4.5x 109yr. half-life
^
THORIUM, 234Th
24 day half-life
99.8%
PROTOACTINIUM,234m Pa
1 min half-life
URANIUM,234 U
2.4x 105yr. half-life
0.2%
PROTOACTINIUM, 234Pa
7 hr half-life
Figure 5-1. The major segments of the decay chains for the five elements/isotopes considered
to be parents in these analyses (U.S. EPA, 2000b).
contaminated sites were a result of past nuclear production and related activities, and are currently under the
environmental restoration program operated by the U. S. Department of Energy. With reference to low-level
radioactive waste storage sites, the most attractive areas were those having low annual precipitation, high
evapotranspiration, and thick unsaturated soils/sediments. An example of such a site is the Mojave Desert Waste-
Burial Site near Beatty, NV. In addition, the Nevada Test Site is a radionuclide contaminated site from nuclear
testing as well as an on-site/off-site low-level waste disposal facility. The only non-nuclear contaminated site and
non-nuclear storage site considered in this evaluation was the Las Graces Trench Research Site in New Mexico. The
field studies at this site have been used to provide data to test deterministic and stochastic models for water flow and
solute transport, Wierenga, et al. (1991), Hills, et al. (1991, 1994), and Rockhold, et al. (1996). The Las Graces
Trench Site was included in the list of candidate sites because it fit the above physical characteristics for waste
disposal facilities and because many detailed site characterization studies and field experiments have been conducted
at this location.
The availability of soil data and climatological records are essential for the proper simulation of water flow and
radionuclide transport in unsaturated soils. From the twenty-some sites in the original candidate list, four sites that
had the most available data were selected for further evaluation. These were the Las Graces Trench Site, the Beatty
Waste Storage Site, Idaho Falls National Laboratory and the Hanford Washington Site. In addition, the Perdido
Alabama Site was placed on the list; this was the site that was analyzed in the earlier sensitivity study of Nofziger, et
al. (1994). Actually, the Perdido Site, which is a benzene contaminated site in an area of high annual precipitation,
was assigned soil properties obtained from another site having the same soil series. Thus, no site-specific soil
properties were available for the Perdido site. This led to the elimination of the Perdido site from further
consideration. The existing field/modeling studies for the four remaining sites were reviewed and summarized. The
studies considered were water balance, recharge and infiltration experiments; tracer and radionuclide migration
studies; laboratory and field measurements of bulk density, Ks, and soil water retention curve data; and model
construction, testing, and simulation runs.
5.2 Selection of the Candidate Site
For the final four sites (Las Graces, Beatty, Idaho Falls, Hanford), additional factors and options were
considered in the choice of the best candidate site. Two of the important factors were the availability of required soil
data for the five pre-selected transport models in this study and the attainability of transport parameters from the
38
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Table 5-1. Partial List of Radionuclide Contaminated and Disposal Sites in the U.S. (U.S. EPA's VISITT Database).
SITE NAME
Location,
City
State
Contaminants
Site Characteristics
1 Rocky Flats Environ- Golden
mental Technology Site
Colorado
2 Idaho National
Engineering &
Environmental
Laboratory
3 Ethyl Corporation
4 DOE FUSRAP
St. Louis Site
Lreatability Study
5 Nevada Lest Site
Idaho Falls Idaho
Baton Rouge Louisiana
St. Louis Missouri
Mercury
Nevada
Plutonium, Uranium
Radioactive Materials,
Plutonium, Lritium
Uranium
Uranium
Plutonium, Lritium
6 Mojave Desert
Waste Burial Site
7 NL Industries, Inc.
8 Los Alamos Natl.
Laboratory
9 Las Cruces Lrench
Site
10 Sandia Natl. Lab.
11 West Valley Nuclear
12 Mound Demonstration
Project
13 Portsmouth Gaseous
Diffusion Plant (DOE)
14 EPASILE
Demonstration
15 Femald Feed Materials
Production Center
16 Apollo Fuel
Conversion Plant
17 Savannah River Site
18 INEL Pit 9 Pilot Project
19 K-25 Site
20 Hanford Site
21 DOE Morgantown
Energy Lech. Center
Beatty
Nevada
Various Radionuclides
Superfund Site. On-site waste
disposal
Radionuclide Contaminated Site:
Radioactive Waste Management
Complex was used for disposal
site for low-level and transuranic
radioactive waste
Superfund Site
Superfund Site
Hazardous Waste Site: Radio-
nuclide contaminated site from
nuclear testing: Low-level waste
disposal facility for both onsite
and off-site generated defense
low-level waste
Low-level radioactive waste and
hazardous chemical waste disposal
Pedricktown
Los Alamos
Las Cruces
Albuquerque
West Valley
Miamisburg
Piketon
Alliance
Fernald
Apollo
Aiken
Clem son
Oak Ridge
Richland
New Jersey
New Mexico
New Mexico
New Mexico
New York
Ohio
Ohio
Ohio
Ohio
Pennsylvania
South Carolina
South Carolina
Lennessee
Washington
Strontium
Plutonium, Uranium
None
Uranium, Plutonium
Strontium
Plutonium
Uranium, Lechnetium
Uranium
Uranium, Lechnetium
Uranium
Uranium
Uranium
Lechnetium, Uranium
Uranium, Strontium
Superfund Site
Superfund Site
Experimental Site
Superfund Site
Superfund Site
Superfund Site
Superfund Site
Superfund Site
Superfund Site
Superfund Site
Superfund Site
Superfund Site
Superfund Site
Superfund Site. C
Morgantown West Virginia Various Radionuclides
disposal.
Superfund Site
39
-------
existing site modeling studies. Selection of the final candidate site was also dependent upon decision-making
options. For example, when verification of modeling results is an important issue (i.e., comparison of modeling
results with field and laboratory experimental data) in addition to data availability, the Las Cruces Trench Site was
the best candidate. However, when reliable model input data is the only issue and model verification is less
important (i.e., comparison among the SSG screening models is more important than model field verification or
comparison of the screening models with more comprehensive modeling of the area), then the Beatty Waste-Burial
Site is a good alternative candidate.
Four of the five proposed SSG models, HYDRUS, CHAIN 2D, MULTIMED-DP 1.0, and FECTUZ, require van
Genuchten soil-water retention parameters (Table 2-1) These parameters plus dispersivities were available at the Las
Cruces Site. For the Beatty Site, the van Genuchten parameters were not available and would have to be derived
from the raw soil-water content/pressure head data (Andraski, 1996). Furthermore, a great amount of model testing at
the Las Cruces Site has been conducted for the transport and fate of tritium, bromide and chromium in the
unsaturated zone. Considering the location and climatological features of the site, and the abundance of existing
field data in conjunction with significant modeling studies, the Las Cruces Trench Site was chosen as the candidate
site for the model evaluation in the SSG for radionuclides.
5.3 Characteristics of the Las Cruces Trench Site in New Mexico
In the soil screening level (SSL) process, generic SSLs for radionuclides can be calculated based on a number of
default assumptions chosen to be protective of human health for most site conditions. However, these are expected
to be more conservative than calculated site-specific SSLs. When site-specific SSLs are of interest, a simulation
using an appropriate model and site-specific data is required. Thus, in the current study, a typical physical site is
required for the evaluation of the HYDRUS, CHAIN 2D, FECTUZ, MULTIMED-DP 1.0, and CHAIN Codes in the
SSL process.
The Las Cruces Trench Research Site in New Mexico was chosen as the typical physical site because:
1. The site characteristics met the criteria of radioactive waste disposal areas (low annual precipitation,
high annual evapotranspiration, and a deep water table);
2. The site had been subjected to extensive testing of its soil physical and chemical properties and soil-
moisture distribution and movement in the unsaturated zone;
3. Results of tracer tests (chloride, bromide and tritium) and metal movement (chromium) were available.
This site lies in the Chihuahuan Desert Province of southern New Mexico (Bailey, 1980). This province is
mostly desert and the Rio Grande and the Pecos River and a few of their larger tributaries are the only perennial
streams. The area has undulating plains with elevations near 1200 m from which somewhat isolated mountains rise
600 m to 1500 m. There are washes which are dry most of the year that fill with water following a rain. Basins that
have no outlets drain into shallow playa lakes that dry up during rainless periods. Extensive dunes of silica sand
cover parts of the province, and in places, there are dunes of gypsum sand, the most notable being the White Sands
National Monument near Alamogordo (a town 100 km northeast of Las Cruces).
The climate of the Chihuahuan Desert is distinctly arid and the spring and early summer are extremely dry.
During July the summer rains usually begin, and they continue through October (Bailey, 1980). In general, these
summer rains are local torrential storms. Average annual temperature in the province ranges from 10°C to 18°C.
Summers are hot and long, and winters are short but may include brief periods when temperatures fall below
freezing. The characteristic vegetation of the Chihuahuan Desert is a number of thorny shrubs. These shrubs
frequently grow in open stands, but sometimes form low closed thickets. Short grass often grows in association with
the shrubs. On deep soils, mesquite is usually the dominant plant; creosote bush covers great areas in its
characteristic open stand and is especially common on gravel fans. Royo (2000) says the creosote bush is a desert
plant "par excellence," a true xerophyte. It is a drought-tolerant shrub with small dark green leaves and has an
extensive double root system - both radial and deep - to accumulate water from both surface and ground water.
These plants can tolerate up to two years without precipitation. The leaves are coated with a varnish-like resin which
reduces water loss by evaporation. The original creosote bush can live to about 100 years old, but it can produce
clones of the parent as the bush ages. These clones are produced in a circular pattern of genetically-identical plants,
expanding outward at the rate of about one meter every 500 years. The "King Clone" family on BLM land near
Victorville, California, is estimated to be 11,700 years old.
40
-------
The actual experimental site is located on the New Mexico State University college ranch some 20 or more
kilometers due north of the city of Las Graces, New Mexico. The site is on a basin slope of Mount Summerford at
the north end of the Dona Ana Mountains (Wierenga, et al, 1991; Defense Mapping Agency, 1987). Geologically,
these mountains are a domal uplift complex composed of younger rhyolitic and the older andesitic volcanics which
were intruded by monzonite. The covered trench that provides horizontal access to the experimental plots and is
used to provide soil samples for these plots is an evacuated earthen box with dimensions 26.4 m long, 4.8m wide,
and 6.0 m deep.
The published soil hydraulic properties for this site, given by Wierenga, etal. (1991), are listed in Table 5-2.
Some pertinent site characteristics obtained by Gee et al (1994) are given on Table 5-3. Table 5-2 shows that the
estimates for the hydraulic parameters were obtained for a uniform soil model and for a nine-layered soil model. The
layers in the layered soil model correspond to the nine soil layers identified at the site. The saturated hydraulic
conductivity for each soil layer was estimated by taking the geometric mean of the 50 laboratory-measured saturated
hydraulic conductivities obtained from each soil layer. Likewise, the water retention data from all 50 samples from a
given layer were used to estimate a,p,9r, and 9S for a single water retention curve for that layer. For the uniform soil
model, the geometric mean of 450 laboratory measured saturated hydraulic conductivities (nine layers with 50 per
layer) was used to estimate a uniform soil saturated hydraulic conductivity value. Likewise, the water retention data
for all 450 sample locations were simultaneously used to estimate single values for each of the parameters a,p,9r, and
9S in a least squares sense (Wierenga, et al., 1991).
In Porro and Wierenga (1993), the solute transport dispersivity (cm) was determined for six layers ranging over
0 < z < 500 cm. The values varied from 2.20 cm to 7.80 cm with an arithmetic average of 4.53 cm for the combined
layer of 500 cm. Minor adaptations of some of these data have been made in the conceptual model developed for the
sensitivity analyses. The details are given in the following subsection. Finally, Figure 5-2 shows the daily
precipitation and potential evapotranspiration (PET) at the Las Graces Site. The PET is calculated from daily
climatic data using Penman's general equation for a well-watered grass reference crop (Jensen, et al., 1990):
XEto = T • (Rn - G) + 6.43 (1 - T) W(e0 - e) , (5-1)
where the terms in Equation (5-1) are defined as:
'k = latent heat of vaporization in mega-joules per kilogram,
Eto = evapotranspiration rate (E^ from a well-watered grass reference crop, in kilograms per
meter squared per day,
^Eto = latent heat flux density in mega-joules per meter squared per day,
F = dimensionless parameter dependent upon surface elevation and air temperature, Table 6-1 of
Jensen, etal., 1990,
Pvj, = net radiation at the surface in mega-joules per meter squared per day,
G = heat flux density to the ground in mega-joules per meter squared per day,
W = a + b u2 = wind function in meters per second,
a,b = positive constants,
u2 = wind speed at 2m above surface in meters per second,
e0 = saturated vapor pressure of air at some height z in kPa,
e = water vapor pressure in air at height z in kPa,
PET = potential evapotranspiration rate in mm/d, given by Eto -^water density in kilograms per meter
squared per millimeter.
For the eighteen year period (1983-2000) shown in Figure 5-2, the annual precipitation ranged over the values from
11.5 cm/yrto 30.8 cm/yr, with an annual average of 22.5 cm/yr. For this same period, 28% of the precipitation came
in the January to June period, and 72% came in the July to December period. This is consistent with the
climatological description given by Bailey (1980).
5.4. Development of a Conceptual Model
The conceptual model for SSLs using detailed site-specific data is developed in a manner which is theoretically
and operationally consistent with the simplified methodology described in Section 2 of the Soil Screening Guidance
41
-------
Table 5-2. Soil Hydraulic Properties at the Las Graces Trench Site for SSG Model Evaluation Study
(reprinted from Water Resources Research, 1991, by PJ. Wierenga, R.G. Hills, and D.B. Hudson
with the permission of the American Geophysical Union, Washington, DC).
Layers Depth
(cm)
all 0 - 600
Saturated
Water
Content
(cnf/cnf)
0.321
Residual
Water
Content
(cnf/cnf)
Uniform Soil
0.083
van Genuchten
Alpha
Coefficient, a,
(cm1)
Model
0.055
van Genuchten
Beta
Coefficient, ft,
(—)
1.509
Saturated
Hydraulic
Conductivity, Ks,
(cm/d)
270.1
Layered Soil Model
1 0-15
2 15-140
3 140 - 205
4 205 - 250
5 250 - 305
6 305 - 370
7 370 - 460
8 460 - 540
9 540 - 600
0.348
0.343
0.336
0.313
0.302
0.294
0.310
0.325
0.306
0.095
0.091
0.085
0.071
0.072
0.090
0.073
0.083
0.078
0.042
0.062
0.060
0.068
0.040
0.070
0.027
0.041
0.047
1.903
1.528
1.574
1.537
1.550
1.711
1.418
1.383
1.432
539
250
267
300
250
334
221
172
226
Table 5-3. Characteristics of the Las Graces Trench Site for SSG Model Evaluation Study
(reprinted from Soil Sci. Soc. Am. J., 1994, by G.W. Gee, PJ. Wierenca, B.J. Andraski, M.H.
Young, MJ. Payer, and M.L. Rockhold with the permission of Soil Sci. Soc. Am. J., Madison,
WI).
Annual
Precipi-
tation
(cm/yr)
23
Annual
Potential
(Pan)
Evapora-
tion (cm/yr)
239
Annual
Potential
Recharge
(cm/yr)
8.7
Average
Daily Max.
Air Temper-
ature (°C)
28
Average
Daily Min
Air Temper-
ature (aC)
13
Elevation
(m)
1357
Depth to
Water
Table
(m)
60
Geology
Alluvial
Typical
Soil
Type
Berino fine
loamy sand
Typical
Vegeta-
tion
Creosote
bush
42
-------
"£•100
>§, 80
o 60
4°
o
£
a.
20
0
,1 UkiLiil
Las Cruces, NM
lllkJ I
Li
1983 1986 1989 1992
Calendar Year
1995
1998
2001
Figure 5-2. Daily precipitation and potential evapotranspiration (PET) at Las Cruces Site, NM. PET is
calculated from daily climate data using Penman's equation (Jensen et al., 1990).
for Radionuclides: Technical Background Document (U.S. EPA, 2000b). In so doing, it is assumed that the Las
Cruces Trench Site in New Mexico has been used as a waste disposal/storage facility where radionuclides from tank
leaks or improper waste disposal were released to the soil surface for a period of time with a specified total amount
of release set for each radionuclide (e.g., 10 mg/cm2 of 238U was released). Thus, a finite radionuclide contaminated
source is assumed. The driving force to send this material on a downward migration to the water table is the
infiltrated rainfall which produces a net annual recharge rate of 87 mm/yr. (Table 5-3). The site-specific soil
hydraulic properties given in the "all-layer" row of Table 5-2 and the mean layer dispersivity of 4.53 cm, obtained
for tritium transport through Berino fine loamy sand, are also used in the current analyses.
Further assumptions are:
The unsaturated zone is homogeneous although HYDRUS, CHAIN 2D, FECTUZ and MULTIMED-
DP 1.0 are capable of simulating layered soils and the hydraulic properties for layered soils are
available at the site.
There is no significant vapor pressure for most radionuclides (except for radon, which is not considered
in this study) and the dimensionless Henry's Law constant is assumed to be zero.
Only vertical flow and transport are considered; horizontal flow and transport are ignored, even though
CHAIN 2D is capable of simulating two-dimensional flow and transport.
There is no chemical or biological degradation in the unsaturated zone.
There is radioactive decay (Figure 5-1) in the unsaturated zone; however, since the published data on
decay rates (or half-lives) for radionuclides are reasonably accurate and precise, no sensitivity analysis
will be made on decay rates.
Complexation, oxidation-reduction, dissolution and precipitation, and ion-exchange are not considered
because these processes are not implemented in the five models being evaluated.
There is no facilitated transport (e.g., colloidal transport, preferential flow in fractures or root channels,
fingering pathways) of the radionuclides in the unsaturated zone.
The aquifer lying below the unsaturated zone is unconsolidated and unconfmed. However, flow and
transport of the radionuclides in the saturated zone are not considered and only leachate contributions
to the ground water at the center of the disposal area are of interest.
Initial concentrations of radionuclides in the soil are zero.
-------
In order to specify a total amount of radionuclide and a reasonable level of a radionuclide concentration release
from the waste source, a literature survey of radionuclide contamination in soils was conducted. Based on the survey
information, the radionuclide concentration released from the hypothesized waste source and the duration of
radionuclide release were determined. Thus, the total amount of radionuclide release can be obtained from the
product of the recharge rate, source concentration, and the duration time of waste release. The depth of radioactive
contaminated soil at the termination point of waste release from the source can be determined from the product of the
pore velocity times the duration time of waste release.
The decay series of the radionuclides in the sensitivity analyses in this report are based on the chain segments
given in Figure 5-1. The chain segments for plutonium-238 and uranium-238 are not complete but are sufficient for
current purposes because of the long half-life of uranium-234. Further, due to the time discretization in the five
models and the relatively short half-lives of the intermediate species in the strontium-90 and uranium-238 chains, the
following parent-daughter chains are only of importance in this report:
5.5 Base Parameter Selection
Table 5-4 provides the base values of selected input parameters to be used in the sensitivity analyses given in
later sections. These values are those that are basically used in the analyses of HYDRUS and CHAIN 2D. Subsets
of these values will be used in the analyses of FECTUZ, MULTIMED-DP 1.0 and CHAIN. Justification and
rationale for the use of these specific base parameters are presented in the following paragraphs.
The area of the disposal facility is arbitrarily taken as 400 m2, whose length and width are both 20 m. The one-
dimensional vertical models are assumed to be located at the center of the square, thus eliminating edge effects in the
unsaturated zone simulation. The total duration of the release of the radionuclide mass was arbitrarily chosen to be
1000 days; however, this value was reasonably consistent with the survey information mentioned in the previous
subsection.
The potential evapotranspiration (PET) from the surface of well-watered short grass was calculated from Las
Graces climatic data using Penman's equation (Jensen, et al. 1990). The weather/climatic data required for this
equation are temperature, relative humidity, wind speed, and solar radiation, along with the estimated albedo
coefficient of 0.21 and the 1357 m elevation of the site. Using the daily climatic data for the years 1983 to 2000, the
estimated value of PET is 204 cm/yr and the average precipitation at the site is 22.5 cm/yr. These values are
consistent with those reported by Gee, et al. (1994), see Table 5-3; thus, we use the annual potential recharge of 8.7
cm/yr given by Gee, et al. In Table 5-2, the all-layer residual moisture content is given as 0.083 and the all-layer
saturated moisture content is given as 0.321. It is expected that the "uniform" annual recharge of 8.7 cm/yr will
produce a soil moisture somewhere between 9r and 9S. Thus, we arbitrarily chose an initial water content, 9, equal
to the geometric mean of 9r and 9S, (9r9s )1/2. This value to two significant figures is 9 = 0.16.
As stated in the previous subsection, the total mass of the individual radionuclides released from the hypothetical
Las Graces disposal/storage site was chosen to be consistent with radionuclide releases from real sites throughout the
country. The relationship between the mass released and the concentration of the radionuclide in the recharge
water from the waste source is given by:
2 Duration (d) 1
Mass (mg/cm ) = Recharge (cm/yr) • • • Concentration (mg/L),
365 (d/yr) 1000 (cm'/L)
= (8.7) (fl)^) Concentration (mg/L), (5-2)
= 0.024 (Concentration, mg/L),
44
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Table 5-4. Base Values of Input Parameters for Unsaturated Zone Radionuclide Models (from Wierenga, et
al, 1991; Gee, et al., 1994; U.S. EPA, 2000ab; and U.S. EPA VISITT Database).
Parameters
Values
Source-Specific Parameters
Area of disposal facility (m2)
Width of disposal facility (m)
Length of disposal facility (m)
Mass release of Radionuclide 238U (mg/cm2)
Concentration of 238U in recharge water from waste source (mg/L)
Mass Release of Radionuclide "Tc (mg/cm2)
Concentration of "Tc in Recharge Water from Waste Source (mg/L)
Mass Release of Radionuclide 90Sr (mg/cm2)
Concentration of 90Sr in Recharge Water from Waste Source (mg/L)
Mass Release of Radionuclide 238Pu (mg/cm2)
Concentration of 238Pu in Recharge Water from Waste Source (mg/L)
Mass Release of Radionuclide 3H (mg/cm2)
Concentration of 3H in Recharge Water from Waste Source (mg/L)
Duration of Waste Source Being Completely Released (days)
Potential Recharge Rate (mm/yr)
Initial Water Content (cm3/cm3)
Soil Properties in Unsaturated Zone
Saturated Hydraulic Conductivity, Ks, (cm/d)
Porosity (-)
Residual Water Content (cm3/cm3)
Saturated Water Content (cm3/cm3)
Bulk Density (g/cm3)
van Genuchten Alpha Coefficient, a, (cm-1)
van Genuchten Beta Coefficient, p, (-)
Depth to Water Table (m)
Solute Transport Parameters
Decay Coefficient for Parent Product "Tc (1/d)
Decay Coefficient for Daughter Product "Ru (1/d)
Distribution Coefficient for "Tc (ml/g)
Distribution Coefficient for "Ru (ml/g)
Dispersivity (cm)
Diffusion Coefficient in Free Water (cm2/d)
Apparent Molecular Dispersion Coefficient (cm2/d)
Dispersion Coefficient (cm2/d)
400
20
20
10
417
3 x 10-4
1.25 x 10-2
4.8 x 10-3
0.2
2.4 x 10-9
1.0 x 10-7
2.6 x 10-9
l.lxlO-7
1000
87
0.16
270.1
0.358
0.083
0.321
1.70
0.055
1.509
6
9 x 10-9
Stable
0.007
5.0
4.53
1.73
0.33
1.01
or
Concentration (mg/L) = 41.7 (Mass, mg/cm2).
Using Equation (5-3) and the mass releases of 238U, "Tc, 90Sr, 238Pu and 3H in Table 5-4 results in the
corresponding concentrations of these species given in the table.
(5-3)
In order to keep the initial concentration and the total amount of the radionuclide entering the soil fixed for
varying recharge rates, q, in the sensitivity analyses, the duration of the source emissions in Equation (5.2) was
45
-------
allowed to vary. That is, the following product in Equation (5-2) was held fixed at its base value as the time duration
of the source varied with the recharge rate, q:
(Duration) x (Recharge Rate) = 8700 cm-d/yr. (5-4)
For the range of q used in the sensitivity analyses, 5.11 cm/yr < q < 10.95 cm/yr, the source duration ranges over the
interval 795 d < duration < 1703 d. However, for the species considered in this report, the differences in source time
duration have a negligible effect on the output values of concern in this report. That is, since the total mass of a
radionuclide and its initial concentration are held fixed, the differences produced by the total release times (i.e., the
pulse width of release) have sufficient time to smooth out in the soil column before the major parts of the
breakthrough curves (BTCs) are seen at the bottom, the 6 m depth, of the soil columns.
Table 5-3 lists the depth to water table at the Las Graces Test Site as 60 m. However, the detailed soil moisture
data are given only for the first 6 m, as listed in Table 5-2. Thus, for the hypothetical modeling scenario of this
report, we chose the 6 m depth to be the top of the water table in our sensitivity analyses. It is felt that this
assumption will meet the project's objectives. As stated in the previous subsection, this 6 m layer is taken to be
homogeneous with the soil properties listed in the all-layer row of Table 5-2, namely: 9S = 0.321, 9r = 0.083, VG-a =
0.055 cm-1, VG-(3 = 1.509, andKs = 270.1 cm/d.
Wierenga, et al. (1991) found that the bulk densities of the nine soil layers in Table 5-2 range in values from
1.66 to 1.74 g/cm3, thus giving a geometric mean of the end points of 1.70 g/cm3. In Table 5-3, Gee et al. (1994)
stated that the typical soil type at the Las Graces Trench Site is a Berino fine loamy sand. With these two pieces of
information and two "rales of thumb" used by Eagleson (1970), we can roughly determine the porosity (n) and the
effective porosity (rig) at the site. The following "rales of thumb" were used by Eagleson to analyze the properties of
a Touchet silt loam (ps = 2.60 g/cm3), a Columbia sandy loam (ps = 2.67 g/cm3), and an unconsolidated sand (ps =
2.71 g/cm3), where ps is the density of the solid matrix and p is the bulk density:
n=l-p/ps, (5-5)
ne = n-9r. (5-6)
Thus, using a density, ps, of 2.65 g/cm3 for Berino fine loamy sand and a bulk density of 1.70 g/cm3 gives a porosity
of 0.358 and an effective porosity of 0.275 for the all-layer soil column in Table 5-2.
The radionuclide used in the sensitivity analyses reported in Section 6 and 7 is technetium ("Tc). This species
possesses a rather long half-life and is highly mobile in the soil column, as seen by its default value for Kd (0.007
ml/g, as given in U.S. EPA 2000b). The decay coefficient for "Tc in Table 5-4 is derived from the radionuclide half-
life given in Figure 5-1 The daughter product of "Tc (i.e., ruthenium, "Ru) is stable, and its decay coefficient is
zero. The default values of Kd for "Ru is 5 ml/g, which indicates that this radionuclide is not very mobile in the soil
column. Because of "Tc's high mobility and long half-life, it possesses many of the characteristics of a conservative
species as it moves through the soil column. Conversely, a species such as plutonium (238Pu), with a shorter half-life
and a low mobility, may decay before reaching a receptor if the soil column is sufficiently long and facilitated
transport does not exist.
The dispersion coefficient, D, in the unsaturated zone is given by Hills, et al. (1991) as (also see Equations 2-22
and 2-23):
D=TwDw+DLq/0, (5-7)
where 9 is taken as the initial water content of 0.16 and |q| is the infiltration rate of 8.7 cm/yr. Taking the
dispersivity of 4.53 cm given for tritium transport at the Las Graces Site by Porro and Wierenga (1993), and the
above values of 9 and |q|, gives a value of 0.68 cm2/d for DL |q| 0"1 . The diffusion coefficient in free water is
assumed to be 1.73 cm2/d and the tortuosity factor TW is taken as 0.19 (Tomasko, etal., 1989). Thus, the value of
TWDW in Equation (5-7) is given as 0.33 cm2/d. Consequently, the sum of the two terms in Equation (5-7) is given as
1.01 cm2/d, the value of the dispersion coefficient in Table 5-4.
The HYDRUS and CHAIN 2D Codes have the capacity for accounting for water uptake by plant roots,
while the other three models do not have this feature (see Table 2-1). Therefore, root water uptake was only
considered in Appendix F to show its impact on radionuclide movement using the HYDRUS Code. As we have
46
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previously stated, the creosote bush (Larrea tridentata) is the dominant plant at the Las Graces Site. Gile, et al.
(1998) indicated that the root depth system of this plant varies with the soil environment and the slope of the terrain;
the depth of roots can extend 5 m, or so. Jenkins, et al. (1988) reported that the creosote bush roots at the Las Graces
Site have a vertical distribution over a range of 0.5 mto 3 m. Thus, for demonstration purposes, a root distribution
of 0.5 m to 2.5 m was used in the current study (see Appendix F).
47
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Section 6
Parameter Sensitivity Analysis: Implementation and Results
The basic elements of the parameter sensitivity analysis for the five models considered in this report are
illustrated in Section 4 through the use of a simple model, y = F(a,b,c,d,e,f). In the current section, these procedures
will be implemented for the HYDRUS Code as applied to the hypothetical modeling scenario described in Section 5.
In what follows, we consider the general procedures for parameter sensitivity analysis as applied to the five models
under investigation, the input parameters for constant recharge rate and constant water content, the input parameters
for constant recharge rate and variable water content, the output variables of interest to SSL application, and the
sensitivity analysis results for the HYDRUS Code.
6.1 General Procedures for Parameter Sensitivity Analysis
The procedures for conducting the parameter sensitivity analysis reported in this section and in Section 7 are
analogous to those given in Section 4.2 (Equations 4-9 to 4-13) for the simple model, y = F(a,b,c,d,e,f). For the
current case, the input/output relationships are represented by expressions of the following form:
Qij=Fij(Pbl>Pb2>...,PpJ...>Pbll)> (6-1)
where Qy represents the j* output for the ith model, Pbk is the base value of the kth parameter, Pp is the variable p*
parameter, and Fy- (-,-, ...,•) is the functional relationship between the n-1 base parameters and the one variable
parameter determined by the 1th model for the j* output. This functional relationship represents empirical physical
laws (e.g., Darcy's Law), conservation laws (e.g., the continuity equations) and constitutive equations (e.g., van
Genuchten soil parameter model).
The specific procedures for conducting a parameter sensitivity analysis for the current SSL application are as
follows:
1. Select a radionuclide from the list: 3H, "Tc, 238U, 90Sr, 238Pu;
2. Select a model for analysis from the list: CHAIN (i = 1), MULTIMED-DP 1.0 (i = 2), FECTUZ
(i = 3), CHAIN 2D (i = 4), HYDRUS (i = 5);
3. Choose the set of input parameters for the specific model selected in Step 2;
4. Select the model outputs of interest, j = 1,2,. ..,k;
5. Select a particular variable input parameter Pp and select its domain of variation,
Ap
-------
6D
at e + pKd
-JAC,
(6-3)
where p is the bulk density of the soil (g/cm3), c is the concentration of the liquid phase (mg/L), Kd is the distribution
coefficient (ml/g), D is the dispersion coefficient (cm2/d), and (J, is the decay rate (1/d) which is assumed fixed for a
given radionuclide. That is, for a given radionuclide, the decay rate, (J,, is the same for the sorbed phase and the
dissolved phase, and |A is sufficiently well known that no sensitivity analysis will be run on it. As in Equation (5-7),
9D can be written as:
= 0TwDw+DL|q
(6-4)
where Tw is a dimensionless tortuosity factor taken as 0.19, Dw is the molecular diffusion coefficient in free water
(cm2/d) and DL is the longitudinal dispersivity of the radionuclide (cm). Using the base values given in Table 5-4,
9iw Dw is about one-half of the value of DL |q|. The reason that the diffusion component of D is so important is
because of the low annual average recharge rate represented by |q|.
The denominators in the second and third terms in Equation (6-3) represent the product of the soil moisture and
the retardation factor R (unitless):
OR = 6 + pKd .
For the default values of Kd listed in Section 5 and the base values of 9 and p given in Table 5-4, we get the
following ratios, pKd + 9R, for the five radionuclides:
(6-5)
Radionuclides
Tritium (3H)
Technetium ("Tc)
Uranium (238U)
Strontium (90Sr)
Plutonium (238Pu)
0.000
0.012
0.680
1.700
8.500
OR
0.160
0.172
0.840
1.860
8.660
TpKd •*• 6R1 100
0.00%
6.98%
80.95%
91.40%
98.15%
These ratios show the relative importance of the distribution coefficient Kd to the product 9R as Kd increases. For
example, p and Kd are relatively unimportant in the transport and fate of technetium (99Tc), but very important for
that of plutonium (238Pu).
Equations (6-3) to (6-5) indicate how the seven input parameters (Kd,q,9,p,D,DL,Dw) influence the transport and
fate of radionuclides in the unsaturated zone when q and 9 are assumed to be constant input parameters. For
comparison purposes, all five models were analyzed under these constant q and 9 conditions for sensitivity to
variations of these seven parameters. The boundary conditions for the liquid phase concentration, c, were the source
term at the top of the 6m column, as described in Section 5.5, and a zero gradient at the bottom of the column.
However, the models do not all possess dispersion coefficients of the type shown in Equation (6-4). In the CHAIN
Code, only a constant D is specified, while MULTDVIED-DP 1.0 and FECTUZ only specify DL. CHAIN 2D and
HYDRUS specify both DLand Dw. The sensitivity analyses conducted for these five models for the seven input
parameters under the constant q and 9 assumptions are summarized in Table 6-1.
6.3 Input Parameters for Constant Recharge Rate, but Variable Water Content
For a constant recharge rate, q, and a variable water content, 9, in a vertical soil column, the governing equations
are Darcy's Law,
q = -K
(6-6)
and the equation of continuity,
49
-------
Table 6-1. The Sensitivity Analyses Performed (•) for the Five Models Under the Assumption of Constant
Recharge Rate and Constant Water Content.
Model
Input
Parameter
Sensitivity Analyses Performed
MULTIMED-
CHAIN DP 1.0 FECTUZ CHAIN 2D
HYDRUS
q
e
p
D
DL
D,,,
(6-7)
where q is in units of (cm/d), K is the unsaturated hydraulic conductivity (cm/d), h is the soil water pressure head
(cm), the z-coordinate is positive upward with the origin at the bottom of the layer, and S is a sink term due to plant
water uptake (cm3/cm3d). Combining Equations (6-6) and (6-7) leads to the modified Richards Equation:
-S
The van Genuchten Model for the parameters K(h) and 9(h) is as follows (see Appendix A):
K(h) = KsKr(h)
(6-8)
(6-9)
0(h) =
e, -e
^r, h<0
es, h>o,
(6-10)
where the effective water content Se is given by
S. =
e - e '
(6-11)
and where Ks is the saturated hydraulic conductivity (cm/d), Kr is the unitless relative hydraulic conductivity, 9S is
the saturated water content, 9r is the residual water content, a is the inverse of the air-entry value or bubbling
pressure head (cm-1), (3 is the unitless pore-size distribution index, and m is defined by
m = 1 - 1/P .
(6-12)
50
-------
For h < 0, Equation (6-10) can be rearranged to give h as a function of the effective water content Se:
For the parameter sensitivity results in this section and Section 7, it is assumed that the sink term, S, due to plant
water uptake is identically zero. Thus, from the continuity equation and from the assumption of constant recharge
rate, q, the water content, 9, is found to be independent of time. When 9 is independent of time, the modified
Richards Equation reduces to Darcy 's Law in Equation (6-6), and the water content becomes a space-varying
quantity, 9(z), through Equation (6-10).
As indicated in Table 2-1, the flow modules of MULTIMED-DP 1.0 and FECTUZ are based on Darcy's Law,
while those for HYDRUS and CHAIN 2D are based on Richards Equation. As stated above, the parameter
sensitivity analysis for the van Genuchten parameters (Ks,9s,9r,a,B ) are based on a constant recharge rate, q, and a
variable water content, 9(z). The determination of 9(z) for MULTIMED-DP 1.0 and FECTUZ follows from the
solution of Darcy 's Law, which is a nonhomogeneous, nonlinear, first-order ordinary differential equations in the
water pressure head, h(z). To solve this system, the head has to be specified at the bottom of the soil column (z = 0)
and at the top of the column (z = L). These boundary conditions for h are determined from the specifications of 9 at
the two boundaries, namely:
(1) 9= 9S at z = 0, giving h=0 at z = 0;
(2) 9 = Base value in Table 5-4 at z = L, giving a value of h at z = L derived from Equations (6-11) and (6-13).
Once h(z) is known, 9(z) can be determined from Equation (6-10). Given 9(z), the influence on the liquid phase
concentration, c, is experienced through Equation (2-15), of which Equation (6-3) is a simplified version. As the five
input parameters (Ks,9s,9r,a,B) are varied, one at a time, for the sensitivity analysis, different water content profiles,
9(z), are generated leading to different responses for the breakthrough curves (BTCs) of the liquid phase
concentrations.
The determination of 9(z) for HYDRUS and the one-dimensional version of CHAIN 2D is realized from the
steady -state solution of the Richards Equation. Using the expressions for K and 9 in Equations (6-9) and (6-10),
respectively, Richards Equation is a homogenous, nonlinear, partial differential equation in h, requiring both
boundary conditions and an initial condition. The boundary conditions for h are obtained in the same way as those
for MULTIMED-DP 1.0 and FECTUZ. The initial condition is specified as that value of h corresponding to the base
value of 9 given in Table 5-4, as calculated from Equation (6-13). Thus, the water content profile, 9(z), is given as
the steady-state solution of this initial, boundary value system for h(z,t), or 9(z,t), as given by equation (6-10).
Again, as the van Genuchten parameters (Ks,9s,9r,a,B) are varied, one at a time, for the sensitivity analysis, different
profiles, 9(z), are generated and the resultant BTCs for c are varied. Table 6-2 summarizes the sensitivity analyses
performed for the four models (CHAIN is not included) under the assumption of constant recharge rates, q, and
variable water content, 9(z).
Table 6-3 lists the base values of the twelve input parameters (Kd,q,9,p,D,DL,Dw,Ks,9s,9r,a,B), along with the
range of variation for each of the twelve parameters. The ranges of variation of the twelve parameters were chosen
to be reasonably consistent with the variations shown over the nine-layer soil column at the Las Graces Trench Site
given in Table 5-2, the hydrologic records shown in Figures 5-2 and F-l, and the physical properties of the Berino
fine loamy sand at the site. These base values and ranges were used in the parameter sensitivity analyses for all five
models being analyzed in this report.
6.4 Output Parameters Evaluated
The time evolution of the radionuclide concentration in the leachate at the bottom of the unsaturated zone (i.e.,
at the entry point to the ground-water zone) is of interest in the Soil Screening Level (SSL) process. The curve of
concentration versus time at this entry point is called the breakthrough curve (ETC). For sensitivity analysis, three
characteristics of these curves are of importance:
1 . The peak concentration of the radionuclide, Cpeak,
2. The time to reach peak concentration, Tpeak, and
3 . The time when the concentration of radionuclide is high enough so that the resulting concentration
at a receptor well will exceed the MCL (i.e., the time to reach MCL, denoted by TMCL).
51
-------
Table 6-2. The Sensitivity Analyses Performed (•) for Four of the Five Models Under the Assumption of Constant
Recharge Rate and Variable Water Content, where "Base" Represents the Base Parameter Values
Given in Table 5-4, and the Water Content Profile, 9(z), Varies with the Changing van Genuchten
Parameters (Ks,9s,9r, a, B).
Sensitivity Analyses Performed
Model
Input MULTIMED-
Parameter CHAIN DP 1.0
Kd Base
q Base
9 9(z)
p Base
D
DL Base
DW
9°
a
B
Table 6-3. Input Parameters for Each Model and the
Value of the Parameter.
Parameter, Pp
p Modela Symbol
1 1,2,3,4,5 Kd
2 1,2,3,4,5 q
3 1,2,3,4,5 9
4 1,2,3,4,5 p
5 1 D
6 2,3,4,5 DL
7 4,5 Dw
8 2,3,4,5 Ks
9 2,3,4,5 9S
10 2,3,4,5 9r
11 2,3,4,5 a
12 2,3,4,5 P
a 1 = CHAIN, 2 = MULTIMED-DP 1.0, 3
b B2 = 0.032 for CHAIN;
c B3 = 0.26 for CHAIN;
d A, = 1 60 fnr CHAIN-
FECTUZ CHAIN
Base
Base
9(z)
Base
Base
Base
Base
9(z)
Base
Base
Base
I
•
•
2D HYDRUS
Base
Base
9(z)
Base
Base
Base
I
•
•
Range of Each Variable Parameter, Along with the Base
Units
ml/g
cm/d
cm3/cm3
g/cm3
cm2/d
cm
cm2/d
cm/d
cm3/cm3
cm3/cm3
I/cm
1/1
= FECTUZ, 4 =
AP R
p rbp "p
0.001
0.016
0.10
1.62d
0.40
3.73
0.93
175
0.29
0.06
0.051
1.43e
= CHAIN 2D, 5 =
0.007 0.019
0.024 0.030b
0.16 0.22C
1.70 1.78
1.00 2.20
4.53 5.33
1.73 2.53
270 365
0.32 0.35
0.08 0.10
0.055 0.059
1.51 1.59e
HYDRUS;
A12 = 1.45 and B12 = 1.57 for FECTUZ
52
-------
In the SSL process, the concentration at a receptor well is assumed to exceed the MCL whenever the
concentration at the ground water entry point exceeds the MCL times a dilution attenuation factor (DAF). A value of
20 for DAF is proposed in U.S. EPA(2000ab), and this is the value used in this report. Thus, the three output
quantities that will be considered in the sensitivity analyses of this section and Section 7 are Cpeak, Tpeak and TMCL.
6.5 Sensitivity Results for the HYDRUS Model
As indicated by Tables 6-1 and 6-2, parameter sensitivity analyses for the HYDRUS Model were carried out for
the eleven input parameters (Kd,q,9,p,DL,Dw,Ks,9s,9r,a,B). For the first six parameters (Kd,q,9,p,DL,Dw) the
analyses were conducted under the constraints of constant recharge rates, q, and constant moisture content, 9. For
the van Genuchten parameters (Ks,9s,9r,a,B), the analyses were executed under the constraints of constant recharge
rate, q, and variable water content, 9(z). The BTCs for "Tc concentrations at the bottom of the homogeneous, 6
meter soil column were recorded, as given in Figure 6-la to 6-lk. Three BTCs for each parameter were obtained,
one for the base value of the parameter in question, Pp, one for the upper limit of the range in Table 6-3, Bp, and one
for the lower limit, Ap. Data from these BTCs were used to calculate the sensitivities and relative sensitivities of the
peak concentraitons at the depth of 6 meters to the eleven input parameters, see Figures 6-2a to 6-2k; the sensitivities
and relative sensitivities of the times to reach peak concentration at the depth of 6 meters to the eleven input
parameters, see Figures 6-3a to 6-3k; and the sensitivities and relative sensitivities of the time to exceed MCL to the
eleven input parameters, Figures 6-4a to 6-4k. The relative sensitivity curves show the rates of increase (+) or
decrease (-) of the outputs, Cpeak, Tpeak, and TMCL, to increases in the eleven input parameters.
Table 6-4 summarizes the relative sensitivities of Cpeak, Tpeak, and TMCL with respect to the eleven basic input
parameters for HYDRUS, the relative sensitivities being measured at the base values of the input parameters. The
dominant input parameters for all outputs are those which have the greatest influence on the advection term in
Equation (6-3), followed by those which have the greatest influence on the diffusion term. The parameters, p and
Kd, have a very small influence on the transport and fate of "Tc since 9 » pKd in the coefficients of Equation (6-
3). As one would expect from the statements about Equation (6-4), the outputs for DL and Dw are nearly the same,
as are their sensitivities. Thus, the ratios of the relative sensitivities of the outputs to DL and Dw should be roughly
in the same proportions as the ratio of their base values, 4.53 + 1.73 = 2.62 , which is the case in Table 6-4 if one
allows for round-off error. For the first six input parameters, Cpeak is most influenced by 9 and the sum of DL and
Dw, while Tpeak, and TMCL are most influenced by q and 9. The direction of these influences (increasing/
decreasing) is seen by the algebraic signs in Table 6-4 and is easily justified by referring to Equation (6-3).
The last five input parameters listed in Table 6-4 are concerned with the van Genuchten Model for 9(h) and
K(h), as given in Equations (6-9) to (6-13). These five parameters influence the distribution of soil moisture in the
soil column, and thus influence both the advection and diffusion terms in Equation (6-3). Because of the low annual
recharge rate of 87 mm/yr, the least influential parameter is Ks, followed by the soil structure parameter a. The most
influential input parameter for all the outputs is the soil texture coefficient (3. The next most influential parameters
are the upper and lower moisture bounds of the van Genuchten Model, respectively 9S and 9r.
Table 6-4. Relative Sensitivities for Cpeak, Tpeak and TMCL with Respect to the Input
Parameters for HYDRUS, Measured at the Base Values of the Input Parameters.
Relative Sensitivity (a), Base Values of the Input Parameters
Parameters Cpeak Tpeak TMCL
Kd
q
9
p
DL
Dw
Ks
es
9r
a
P
-0.06
+0.12
-1.10
-0.05
-0.28
-0.10
+0.05
-0.66
-0.29
+0.13
+1.38
+0.06
-1.10
+0.80
+0.06
-0.02
-0.01
-0.04
+0.57
+0.22
-0.05
-1.00
+0.07
-0.98
+0.86
+0.07
-0.07
-0.03
-0.04
+0.64
+0.24
-0.05
-1.03
53
-------
(a) Sensitivity of "To Breakthrough to Kd
(b) Sensitivity of "To Breakthrough to q
(o) Sensitivity of "To Breakthrough to 8
|
Distribution Coefficient (ml/g) HYDRUS
0.001
0.007
0.019
Reoharae Rate (om/d)
0.016
0.024
0.030
2000 4000 6000 8000 10000
Time (days)
HYDRUS Water Content (onf/orf)
0.010
0.016
0.022
2000 4000 6000 8000 10000
Time (days)
2000 4000 6000 8000 10000
Time (days)
(d) Sensitivity of "Tc Breakthrough to p (e) Sensitivity of "To Breakthrough to DL (f) Sensitivity of "To Breakthrough to D.
0.015| 1 1 1 1—- 1 0.015| 1 1 1 1 1 0.015
0.010
f
HYDRUS Bulk Dimity (a/on?)
1.62
1.70
1.78
0.010
HYDRUS DliporjMty (am)
3.73
4.53
5.33
_ _
2000 4000 6000 8000 10000
Time date
_ _
2000 4000 6000 8000 10000
Time «ay»))
Diffusion Coefficient In Water (orf f")
HYDRUS 0.93
1.73
2.53
2000 4000 6000 8000 10000
Time (days)
(g) Sensitivity of "To Breakthrough to K,
(h) Sensitivity of "To Breakthrough to 6S
(0 Sensitivity of MTo Breakthrough to 8r
|
Saturated Conductivity (cm O HYDRUS
175
270
365
HYDRUS Saturated Water Content (orf/onr)
0.29
0.32
0.35
2000 4000 6000 8000 10000
Time (days)
HYDRUS Residual Water Content (orf/onr)
0.06
0.08
0.10
2000 4000 6000 8000 10000
Time (days)
2000 4000 6000 8000 10000
Time (days)
(I) Sensitivity of "To Breakthrough to a
(k) Sensitivity of "Tc Breakthrough to
van Genuohten Retention Parameter a (onT1)
HYORUS 0.051
0.055
0.059
2000 4000 6000 8000 10000
Time (days)
van Ganuchten Retantlon Parameter 0 (—)
HYDRUS 1.43
1.51
1.59
2000 4000 6000 8000 10000
Time (days)
Figure 6-1. Sensitivity of "Tc breakthrough (through the 6m layer) to the system parameters using the HYDRUS
Model: (a) distribution coefficient, (b) recharge rate, (c) water content, (d) bulk density, (e) dispersi-
vity, (f) diffusion coefficient in water, (g) saturated conductivity, (h) saturated water content, (i) residual
water content, (]) van Genuchten retention parameter a, (k) van Genuchten retention parameter (3.
54
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Figure 6-3. Sensitivity and relative sensitivity of peak concentrations at the depth of 6m to the system parameters
using the HYDRUS Model: (e) dispersivity, (f) diffusion coefficient in water, (g) saturated
conductivity, (h) saturated water content.
56
-------
0.000
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-0.200
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van Genuchten Retention Parameter J3 (—)
0.050 0.052 0.054 0.056 0.058 0.060
van Genuchten Retention Parameter a (cm"1)
Figure 6-2. Sensitivity and relative sensitivity of peak concentrations at the depth of 6m to the system parameters
using the HYDRUS Model: (i) residual water content, (j) van Genuchten retention parameter a, (k)
van Genuchten retention parameter (3.
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°- 2000
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. 0.80
1 0.60
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20000
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-fc-
p 10000
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. 6000
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^ 4000
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o 2000
1 0
(g)
-
-
~s^
j^°^
HYDRUS "Tc
1 1 1 1
) 100 200 300 400 50
Saturated Conductivity (cm d"1)
(h)
-— —
-
1 1 1
HYDRUS "Tc
; __— — ;
_
i i i
.1 0.5 1.0 1.5 2.0 2.5 3.0
"- Diffusion Coefficient in Water (crrrd"1)
0.280 0.300 0.320 0.340 0.360
Saturated Water Content (cm3 cm"3)
Figure 6-3. Sensitivity and relative sensitivity of time to reach peak concentrations at the depth of 6m to the system
parameters using the HYDRUS Model: (e) dispersivity, (f) diffusion coefficient in water, (g) saturated
conductivity, (h) saturated water content.
59
-------
£ 0.30
'in
c 0.20
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50000
40000
t" 30000
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£" -0.50
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£ 5000
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HYDRUS "Tc
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1 1 1 1
Q OUUU
•a
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^ 2000
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HYDRUS "Tc
_
_
-
1 1 1
0 0.01 0.02 0.03 0.04 0.05 .E 1.60 1.65 1.70 1.75 1.80
.*- Po^hcii-.-iaPci + a/'j^m H~1 ^ ' Rnllynarioiti/^r-i / r* rrf5"!
Figure 6-4. Sensitivity and relative sensitivity of time to exceed MCL to the system parameters using the
HYDRUS Model: (a) distribution coefficient, (b) recharge rate, (c) water content, (d) bulk density.
61
-------
^ -0.10
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1 -0.20
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o
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to
^ 2000
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1 1 1 1
- _^—^ -
HYDRUS "Tc
iii
_>. -0.02
1 -0.04
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en
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CU
^ 2000
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o) 0
(g)
-!—---• '
-
HYDRUS "Tc
1 1 1 1
) 100 200 300 400 50
Saturated Conductivity (cm d"1)
(h)
1 1 1
-
HYDRUS "Tc
1 1 1
0.5 1.0 1.5 2.0 2.5 3.0
Diffusion Coefficient in Water (cm2 dH)
0.280 0.300 0.320 0.340 0.360
Saturated Water Content (cm3 cm"3)
Figure 6-4. Sensitivity and relative sensitivity of time to exceed MCL to the system parameters using the
HYDRUS Model: (e) dispersivity, (f) diffusion coefficient in water, (g) saturated conductivity, (h)
saturated water content.
62
-------
±r o.3o
1 0.20
CD
15 °-10
C£
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15000
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-i 6000
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^ 6000
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2 4000
_c
u
J 2000
0
- 0
(i)
^^^
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1 1 1 I I
^ .
3 *
1 1 1 1 1
HYDRUS "Tc
1 1 1 1 1
t" -0.50
1 -1.00
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CO
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0
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' (k)
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- ___— - — -
_
- _— — — — -
_
-
HYDRUS 99Tc
— — '
i i
0500.060 0.070 0.080 0.090 0.100 0.110 "> 1.400 1.450 1.500 1.550 1.600
Residual Water Content (cm3cm~3) ~ van Genuchten Retention Parameter jS ( — )
- ^—-——®-
1 1 I I
_
- __^---^^ -
-
1 1 1 1
HYDRUS 99Tc
-
-
iii
<" 0.050 0.052 0.054 0.056 0.058 0.060
~ van Genuchten Retention Parameter a (cm~1)
Figure 6-4. Sensitivity and relative sensitivity of time to exceed MCL to the system parameters using the HYDRUS
Model: (i) residual water content, (j) van Genuchten retention parameter a, (k) van Genuchten retention
parameter (3.
63
-------
Section 7
Comparison of Sensitivity Results Between Models:
Illuminating Numerical Differences
This section is concerned with the comparison of the parameter sensitivity results derived for the five models
being studied in this report. The input parameters being considered form subsets of the twelve parameters
(Kd,q,9,p,D,DL,Dw,Ks,9s,9r,a,p) described in Sections 6.2 and 6.3 The output parameters (Cpeak,Tpeak,TMCL) are
those introduced in Section 6.4. In what follows, we consider the modeling codes and their differences, the
parameter sensitivity results, and other results illuminating numerical differences and errors between the models.
7.1 The Modeling Codes and Their Differences
Table 2-1 lists the flow and transport modeling components, and the type of boundary conditions available to
each of the five codes being tested in this report. From this table, the depth and breadth of application of each code
can be determined. However, for the comparison of parameter sensitivity given in this section, we consider a
constant flow (i.e., recharge rate) through a 6m, vertical homogeneous soil column, with a constant or spatially
varying water content. The boundary conditions for the transport of the radionuclide, "Tc, consist of a stepwise
constant source at the top of the soil column and a zero concentration gradient at the bottom of the column.
As we have previously stated, the CHAIN Code is an analytical transport model which assumes uniform flow
conditions in the homogeneous soil column; that is, the soil moisture and recharge rate are both constant. Even
though CHAIN possesses an analytical solution for the pollutant breakthrough curves (BTCs) at the 6m level of the
soil column, numerical methods are required to generate the graphs of the BTCs from the closed-form analytical
expressions. Thus, the CHAIN Code can exhibit numerical error, with reference to the exact analytical expressions,
due to truncation of the series of functions representing the analytical solution in the numerical approximation
method, and due to computer round-off error.
The MULTIMED-DP 1.0 and FECTUZ Codes can handle constant recharge rates with constant or spatially
varying water contents. With respect to the BTCs for pollutant transport, these codes are semi-analytical models
which approximate the inversion of complex analytical solutions in transform space using numerical inverse
transform modules. Thus, the numerical errors most prevalent in these codes are those due to the numerical solution
of Darcy's Law for the pressure head (leading to a variable water content in the soil column) and those due to the
numerical inversion (i.e., a numerical Laplace inversion) of the analytical solutions for the pollutant concentration in
the Laplace transform space. If these two codes use the same differencing scheme or one of comparable accuracy for
solving Darcy's Law and the same or comparable numerical inversion schemes for the pollutant BTCs, then one
would expect that the BTCs for a given boundary-initial value problem would be nearly the same for both codes.
The CHAIN 2D (the one-dimensional, vertical version) and the HYDRUS Codes handle constant recharge rates
with constant, spatially varying, or temporally and spatially varying water contents. With respect to pollutant
transport, these codes solve the advection-dispersion-decay partial differential equations using finite-difference or
finite-element methods. Thus, the numerical errors most prevalent in these two codes are those due to the numerical
schemes used to solve the Richards Equation for the temporally and spatially varying pressure head (leading to a
temporally and spatially varying water content), those due to deriving a spatially varying water content from the
numerically obtained steady-state solution of the Richards Equation, and those due to the numerical schemes used to
obtain the BTCs for pollutant transport from the advection-dispersion-decay partial differential equations. If CHAIN
2D and HYDRUS use the same numerical solution schemes or ones of comparable accuracy for the Richards
Equation and the advection-dispersion-decay equations, then one would expect (as with MULTIMED-DP 1.0 and
FECTUZ) that the BTCs for a given boundary-initial value problem would be nearly the same for both codes.
64
-------
From the above discussions, one would be tempted to place the derived BTCs into three categories for the
current parameter sensitivity modeling scenario (the scenario as described in Sections 5, and 6.1 to 6.3). These three
categories, based on the potential for numerical error, would be those BTCs derived by the CHAIN Code, those
BTCs derived by the MULTDVIED-DP 1.0 and FECTUZ Codes, and those BTCs derived by the CHAIN 2D and
HYDRUS Codes. As pointed out in Sections 6.2 and 6.3 (see Tables 6-1 and 6-2), there are two other discriminating
factors between the codes for the current modeling scenario; namely: (1) the specification of a constant or spatially
varying water content in the homogeneous, 6m soil column, and (2) the construction of the dispersion factor, 9 D, in
Equation (6-4). Table 7-1 is a composite of Tables 6-1 and 6-2, and points out the impact that these two
discriminating factors have on the current parameter sensitivity analyses.
For the first seven parameters (Kd,q,9,p,D,DL,Dw) in Table 7-1, the five models were run under the constraints
of a constant recharge rate, q, and a constant water content, 9, using the base input values and input ranges given in
Table 6-3. As previously stated and as indicated in Table 7-1, CHAIN specifies a constant value for D and does not
resolve 9D into components as given in Equation (6-4). MULTIMED-DP 1.0 and FECTUZ only specify constant
values for DL in Equation 6-4, while CHAIN 2D (the one-dimensional, vertical version) and HYDRUS specify
constant values for both DL and Dw. Since the recharge rate, q, in the current sensitivity modeling scenario is so
small, the quantity 9iwDw in Equation (6-4) is relatively important as compared to the dispersion term DL|q|. Thus,
for the base values listed in Table 6-3, one would expect less total dispersion effects (i.e., effects due to 9D) on the
BTCs derived by MULTIMED-DP 1.0 and FECTUZ than for those derived by CHAIN 2D and HYDRUS. The
fourth row of Table 7-2 gives the numerical justification of this statement. As this row of values indicates, the base
value of D in the CHAIN Code gives a value for 9D which is nearly the same as those for CHAIN 2D and
HYDRUS, while the value of 9D for MULTIMED-DP 1.0 and FECTUZ is about 2/3 of the values for the values for
Table 7-1. The Sensitivity Analyses Performed (•) for the Five Models Under the Assumption of Constant
Recharge Rate and Constant Water Content, and the Sensitivity Analyses Performed (O) for Four of
the Five Models Under the Assumption of Constant Recharge Rate and Variable Water Content.
Model Input
Parameter
Sensitivity Analyses Performed
z
i
OH
Q
Q
W
S)
_
<
u
q
e
p
D
DL
Ks
es
er
a
P
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
65
-------
Table 7-2. The Values of the Dispersion Factor, 9D, as Derived from Equation (6-4) and the Base Values of
D, DL, Dw, 9 and q and the Ranges of D, DL and Dw as given in Table 6-3.
D
cm2/d
0.40
1.00
1.00
1.00
1.00
1.00
2.20
DL
cm
4.53
3.73
4.53
4.53
5.33
4.53
4.53
Dw
cm2/d
1.73
1.73
0.93
1.73
1.73
2.53
1.73
Djql
cm2/d
0.109
0.090
0.109
0.109
0.128
0.109
0.109
exwDw
cm2/d
0.053
0.053
0.028
0.053
0.053
0.077
0.053
9D in cm2/d
S;
«)
0.064
0.160
0.160
0.160
0.160
0.160
0.352
1
1-
1:
g"
§!
0.109
0.090
0.109
0.109
0.128
0.109
0.109
M
£*4
[^
0.109
0.090
0.109
0.109
0.128
0.109
0.109
§
§
1
0.162
0.143
0.137
0.162
0.181
0.186
0.162
g
1
0.162
0.143
0.137
0.162
0.181
0.186
0.162
the other three models. Thus, one would expect the BTCs for the base values of the input parameters to be nearly the
same for the CHAIN, CHAIN 2D and HYDRUS Codes, and nearly the same for the MULTIMED-DP 1.0 and
FECTUZ Codes, where the peak concentrations for the latter two models should be higher than those for the former
three models. This is, in fact, the case as we will see in Section 7.2. The numerical values for 9D in the other six
rows of Table 7-2 can be used to interpret the various sensitivity results given in the next subsections.
For the last five input parameters in Table 7-1 (Ks,9s,9r,a,p), only four models (CHAIN does not allow variable
9) are evaluated for constant recharge rate, q, and variable water content, 9(z). Based on the above discussions, one
would expect the BTCs for CHAIN 2D and HYRUS to be nearly the same, and those for MULTIMED-DP 1.0 and
FECTUZ to be nearly the same, where the peak concentrations for the latter two models should be higher than those
for the former two models. As we will see below, this is the case.
7.2 The Parameter Sensitivity Results
As indicated in Table 7-1, parameter sensitivity analyses are performed on all five models for the first four
parameters (Kd,q,9,p). CHAIN is the only model for which sensitivity analyses are performed for D. Sensitivity
analyses are performed for the six parameters (DL,Ks,9s,9r,a,p) on four of the five models (all except CHAIN), while
only CHAIN 2D and HYDRUS consider the parameter Dw. The "Tc BTCs at the bottom of the 6m, homogeneous
soil column are given in Figures 7-1 to 7-12, where the breakdown is as follows:
Figures 7-1 to 7-4 compare the BTCs for each of the five models for the first four parameters (Kd,q,9,p),
respectively;
Figure 1-5 gives the BTCs for the CHAIN Code for the parameter D; this figure is given for the purpose of
completion;
Figure 7-6 compares the BTCs for each of the four models (CHAIN being excluded) for the dispersivity,
DL;
Figure 7-7 compares the BTCs for CHAIN 2D and HYDRUS for the diffusion coefficient in water, Dw;
Figures 7-8 to 7-12 compare the BTCs for each of the four models (CHAIN being excluded) for the five van
Genuchten parameters (Ks,9s,9r,a,p), respectively.
These twelve figures of "Tc BTCs fulfill the expectations set forth in Section 7.1. These results, or
expectations, are summarized in the following list:
66
-------
0.015
0.010
0.005
Distribution Coefficient
HYDRUS
CHAIN 2D
FECTUZ
o>
E
c
o
c
-------
0.015
0.010
0.005
Recharge Rate = 0.030 cm/d
HYDRUS
CHAIN 2D
FECTUZ
MULTIMED-DP
CHAIN
0.015
0.010
"c 0.005
-------
0>
C
o
c
O)
0.015
0.010
0.005
OL
0
0.015
0.010
0.005
0.015
0.010
0.005
Water Content
HYDRUS
CHAIN 2D
FECTUZ
0.22 cnf/cnf
MULTIMED-DP
CHAIN
2000
4000
6000
8000
10000
Water Content = 0.16 cnf/cnf
HYDRUS MULTIMED-DP
CHAIN 2D CHAIN
FECTUZ
2000
4000
6000
8000
10000
Water Content
HYDRUS
CHAIN 2D
0.10 c n?/c rtf
MULTIMED-DP
CHAIN
FECTUZ
2000 4000 6000
Time (days)
8000
10000
Figure 7-3. Sensitivity of "Tc breakthrough (through the 6m layer) to the water content using the CHAIN,
MULTIMED-DP 1.0, FECTUZ, CHAIN 2D, and HYDRUS Models, where 9 = 0.22 cmVcm3, 0.16
cmVcm3, and 0.10 cmVcm3.
69
-------
U.UIO
0.010
0.005
0
C
On4 K.
.U1O
cr
n»
c Concentration (m<
o o
• •
o o
O -"•
o en o
s~ <
n rH «;
U.U1O
0.010
0.005
0
C
Bulk Density
HYDRUS
CHAIN 9D
_ FECTUZ
^*"v
/^
) 2000 4000
Bulk Density
HYDRUS
CHAIN 9D
_ FECTUZ
XN
<_ V
f\
) 2000 4000
Bulk Density
HYDRUS
CHAIN 9D
_ FECTUZ
XX
<•. v
/s
1 ^*^ 1
) 2000 4000
Time
= 1 .78 g/cn?
MULTIMED— DP
CHAIN
6000 8000 1 0000
= V.70 g/cn?
MULTIMFD DP
pUAIkl
6000 8000 1 0000
= V.62 g/cn?
MULTIMED— DP
CHAIN
l^feh^ l
6000 8000 1 0000
(days)
Figure 7-4. Sensitivity of "Tc breakthrough (through the 6m layer) to the bulk density using the CHAIN,
MULTIMED-DP 1.0, FECTUZ, CHAIN 2D, and HYDRUS Models, where p = 1.78 g/cm3, 1.70 g/
cm3, and 1.62 g/cm3.
70
-------
0.015
0.010
0.005
Dispersion Coefficient = 2.2 (cm2/d)
CHAIN
o>
E
c
o
c
o>
o
c
o
o
o
01
o>
0.015
0.010
0.005
2500
5000
7500
Dispersion Coefficient = 1.0 (cm2/d)
CHAIN
0.015
0.010
0.005
2500
5000
7500
Dispersion Coefficient = 0.4
CHAIN
2500 5000 7500
Time (days)
10000
10000
10000
Figure 7-5. Sensitivity of "Tc breakthrough (through the 6m layer) to the dispersion coefficient using the CHAIN
Model, where D = 2.20 cmVd, 1.00 cmVd, and 0.40 cmVd.
71
-------
0.015
0.010
0.005
o>
E
c
o
o>
0.015
0.010
0.005
0.015
0.010
0.005
HYDRUS
CHAIN 2D
FECTUZ
Dispersivity = 5.33 cm
MULTIMED-DP
J_
2000
4000
6000
8000
10000
HYDRUS
CHAIN 2D
FECTUZ
Dispersivity = 4.53 cm
MULTIMED-DP
2000
4000
6000
8000
10000
HYDRUS
CHAIN 2D
FECTUZ
Dispersivity = 3./3 cm
MULTIMED-DP
2000 4000 6000
Time (days)
8000
10000
Figure 7-6. Sensitivity of "Tc breakthrough (through the 6m layer) to the dispersivity using the MULTIMED-DP
1.0, FECTUZ, CHAIN 2D, and HYDRUS Models, where DL = 5.33 cm, 4.53 cm, and 3.73 cm.
72
-------
0.015
0.010
0.005
Diffusion Coefficient in Water
HYDRUS
o>
E
c
o
c
at
0.015
0.010
0.005
0.015
0.010
0.005
= 2.5I3 err? a"1
CHAIN 2D
2000
4000
6000
8000
10000
Diffusion Coefficient in Water
HYDRUS
CHAIN 2D
= 1 .3 en? a"1
2000
4000
6000
8000
10000
Diffusion Coefficient in Water
HYDRUS
CHAIN 2D
= 0.^3 err? of1
2000 4000 6000
Time (days)
8000
10000
Figure 7-7. Sensitivity of "Tc breakthrough (through the 6m layer) to the diffusion coefficient in water using the
CHAIN 2D and HYDRUS Models, where Dw = 2.53 cm2/d, 1.73 cm2/d, and 0.93 cm2/d.
73
-------
0.015
0.010
0.005
Saturated Conductivity
HYDRUS
CHAIN 2D
FECTUZ
o>
E
c
o
c
a>
o
c
o
o
o
o>
o>
0.015
0.010
0.005
0.015
0.010
0.005
T~Je5~
cm d"1
MULTIMED-DP
2000
4000
6000
8000
10000
Saturated Conductivity
HYDRUS
CHAIN 2D
FECTUZ
= 2*70
cm d"1
MULTIMED-DP
2000
4000
6000
8000 10000
Saturated Conductivity
HYDRUS
CHAIN 2D
FECTUZ
= i?5 cm d^
MULTIMED-DP
2000 4000 6000
Time (days)
8000
10000
Figure 7-8. Sensitivity of "Tc breakthrough (through the 6m layer) to the saturated conductivity using the
MULTIMED-DP 1.0, FECTUZ, CHAIN 2D and HYDRUS Models, where Ks = 365 cm/d, 270 cm/d,
and 175 cm/d.
74
-------
c
o
c
0
o
c
o
o
u
o>
o>
U.UIO
0.010
0.005
0
C
Ort 4 K.
.U1O
0.010
0.005
C
On* c
.Ul O
0.010
0.005
0
C
Saturated Water Content
HYDRUS
CHAIN 2D
_ FECTUZ
f
. J
) 2000 4000
Saturated Water Content
HYDRUS
CHAIN 9D
_ FECTUZ
£ \
r
) 2000 4000
Saturated Water Content
HYDRLJ^
CHAIN 2D
_ FECTUZ
A
f\
, J,
) 2000 4000
Time ((
= 0.35 cnr/cnf
MULTIMED DP
-
\
\
6000 8000 1 0000
= 0.32 cnr/crrf
-
V. '
6000 8000 1 0000
= 0.29 cnr/cnf
uiji TIMFD DP
-
6000 8000 1 0000
days)
Figure 7-9. Sensitivity of "Tc breakthrough (through the 6m layer) to the saturated water content using the
MULTIMED-DP 1.0, FECTUZ, CHAIN 2D and HYDRUS Models, where 9S = 0.35 cmVcm3, 0.32
cmVcm3, and 0.29 cmVcm3.
75
-------
U.UIO
0.010
0.005
0
C
0/"\4 C
.UlO
X
£ 0.010
c
o
D
"c 0.005
-------
0.015
0.010
0.005
o>
E
~c
o>
0.015
0.010
0.005
0.015
0.010
0.005
van Genuchten a = 0.059 ci
HYDRUS
CHAIN 2D
FECTUZ
MULTIMED-DP
2000
4000
6000
8000
10000
van Genuchten a = 0.055 cm"1
HYDRUS
CHAIN 2D
FECTUZ
MULTIMED-DP
2000
4000
6000
8000 10000
van Genuchten a = 0.051 ci
HYDRUS
CHAIN 2D
FECTUZ
MULTIMED-DP
2000 4000 6000
Time (days)
8000
10000
Figure 7-11. Sensitivity of "Tc breakthrough (through the 6m layer) to the van Genuchten retention parameter a
using the MULTIMED-DP 1.0, FECTUZ, CHAIN 2D and HYDRUS Models, where a = 0.059cm-1,
0.055 cm-1, and 0.051cm-1.
77
-------
U.UI3
0.010
0.005
0
C
0.015
a*
o>
^ 0.010
c
JO
2
"c 0.005
Q>
O
C
O
0
£ °
« C
w 0.015
0.010
0.005
0
van Genuchten 0 = 1.59
HYDRUS MULTIMED DP
CHAIN 2D
_ FECTUZ
/^
f V
) 2000 4000 6000 8000 10000
van Genuchten /? = 1.51
HYDRUS MLJLTIMED DP
CHAIN 2D
_ FECTUZ
/~\
£ V
) 2000 4000 6000 8000 10000
van Genuchten /? = 1.43
HYDRUS MULTIMED DP
CHAIN 2D
_ FECTUZ
£\
2000 4000 6000
Time (days)
8000
10000
Figure 7-12. Sensitivity of "Tc breakthrough (through the 6m layer) to the van Genuchten retention parameter (3
using the MULTIMED-DP 1.0, FECTUZ, CHAIN 2D and HYDRUS Models, where (3 = 1.59, 1.51,
and 1.43.
78
-------
The BTCs for the CHAIN 2D and HYDRUS Codes are nearly the same for all base parameter values and
upper and lower bounds of the parameter ranges, except for 9 = 0.10 in Figure 7-3; but even for this
exception, the peak concentration for CHAIN 2D is about 95% of that for the HYDRUS Code;
• For the first four parameters (Kd,q,9,p), the CHAIN BTCs are fairly close to those for CHAIN 2D and
HYDRUS, except for q = 0.016 cm/d in Figure 7-2; but even for this exception, the peak concentration for
CHAIN is about 87% of those for the CHAIN 2D and HYDRUS Codes;
• The BTCs for the MULTIMED-DP 1.0 and FECTUZ Codes are almost identical for all values of all
parameters considered.
The peak concentrations for the MULTIMED-DP 1.0 and FECTUZ BTCs range from 5% to 35% higher
than those for CHAIN 2D and HYDRUS, depending on the specific input parameter and its particular
value;
• For the base values of (Kd,q,9,p,DL), the concentration peaks of the MULTIMED-DP 1.0 and FECTUZ
BTCs are about 14% higher than those for CHAIN 2D and HYDRUS;
For the base values of the van Genuchten parameters (Ks,9s,9r,a,p), the concentration peaks of the
MULTIMED-DP1.0 and FECTUZ BTCs are about 21% higher than those of CHAIN 2D and HYDRUS.
The sensitivities and relative sensitivities of the output parameters (Cpeak, Tpeak, TMCL) to the twelve input
parameters (Kd,q,9,p,D,DL,Dw,Ks,9s,9r,a,p), corresponding to the BTCs given in Figures 7-1 to 7-12, are graphically
presented in Figures 7-13 to 7-24. The breakdown of these figures is as follows:
Figures 7-13 to 7-16 compare the output parameters and their sensitivities and relative sensitivities for each
of the five models for the first four input parameters (Kd,q,9,p), respectively;
Figure 7-17 gives the output parameters and their sensitivities and relative sensitivities for the CHAIN Code
for the input parameter, D; this figure is given for the purposes of completion;
Figure 7-18 compares the output parameters and their sensitivities and relative sensitivities for each of the
four models (CHAIN being excluded) for the dispersivity, DL;
Figure 7-19 compares the output parameters and their sensitivities and relative sensitivities for CHAIN 2D
and HYDRUS for the diffusion coefficient in water, Dw;
Figures 7-20 to 7-24 compare the output parameters and their sensitivities and relative sensitivities for each
of the four models (CHAIN being excluded) for the five van Genuchten parameters (Ks,9s,9r,a,p),
respectively.
As shown in Section 4 for the simple model, y = F(a,b,c,d,e,f), the relative sensitivity is a good measure for
comparing output parameter sensitivity to changing input parameters. For this reason, Table 7-3 was constructed
from the information contained in Figures 7-13 to 7-24. This table is a summary of the relative sensitivities for each
of the three output parameters for the "Tc BTCs with respect to all the pertinent input parameters for each of the
five models being tested. The relative sensitivities listed in the table are referenced to the base values of the input
parameters given in Table 6-3. The results evident from Table 7-3 are as follows:
The relative sensitivities are nearly the same for MULTIMED-DP 1.0 and FECTUZ for all output and input
parameters;
The relative sensitivities are nearly the same for CHAIN 2D and HYDRUS for all output and input
parameters;
The relative sensitivities are nearly the same for all models for the output parameters, Tpeak and TMCL, and
all input parameters;
79
-------
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o
-0.00
-0.02
£• -i
I
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0.020
0.015
0.010
0.005
0.04-
0.06- —
0.08 •
•no
HYDROS --
CHAIN 20
FECTUZ
MULTIMED-DP
CHAIN
0.20
£ 0.15
J °'
-5 0.05
0
50000
40000
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i 20000
I
1 10000
(b)
o 8000
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d 6000
" 4000
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5
0.005 0.010 0.015
Distribution Coefficient (ml/g)
0.020E
0.20
£ 0.15
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•5 0.05
0
50000
40000
I" 30000
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2
4000
2000
(c)
0.005 0.010 0.015 0.0201
Distribution Coefficient (ml/g) p
0 0.005 0.010 0.015 0.020
Distribution Coefficient (ml/g)
Figure 7-13. Sensitivity and relative sensitivity of (a) peak concentration at the depth of 6m, (b) time to reach peak concentration at the depth of 6m, and (c)
time to exceed MCL at the receptor well to the distribution coefficient using the CHAIN, MULTIMED-DP 1.0, FECTUZ, CHAIN 2D and
HYDRUS Models.
-------
(a)
(b)
(o)
u.ou
£. 0.40
| 0.30
0.20
1 0.10
o
09A
0.15
£•
£ 0.10
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X.
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1 — 1 L L _L-1_LJ '
1 1 1 1
— HYDRUS - - MULTIMED-DP
CHAIN 2D — CHAIN
-- FECTUZ
.
„, , j,»i — — •
•
i i i i
> 0.01 0.02 0.03 0.04 0.
U
£ -0.5
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^ -200000
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1 1 1 1
1 1 1 1
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^Kl^^
^^^^^
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> 0.01 0.02 0.03 0.04 0.
Figure 7-14. Sensitivity and relative sensitivity of (a) peak concentration at the depth of 6m, (b) time to reach peak concentration at the depth of 6m, and (c)
time to exceed MCL at the receptor well to the recharge rate using the CHAIN, MULTIMED-DP 1.0, FECTUZ, CHAIN 2D and HYDRUS
Models.
-------
(a)
(b)
(o)
oo
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-0.25
£
£ -0.50
g -0.75
w -1.00
£ -1.25
-1.50
_ft ftft,
-0.02
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CHAIN 2D — CHAIN
-- FECIUZ
*'** 0.10 0.20 0.
Wotor Content i o w o nT^)
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| 0.60
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Figure 7-15. Sensitivity and relative sensitivity of (a) peak concentration at the depth of 6m, (b) time to reach peak concentration at the depth of 6m, and (c)
time to exceed MCL at the receptor well to the water content using the CHAIN, MULTIMED-DP 1.0, FECTUZ, CHAIN 2D and HYDRUS
Models.
-------
(a)
(b)
(o)
oo
L.J
U.1U
f •
£ -0.10
c
<7> -0.20
1 -0.30
-0.40
0.0010
0.0006
£ 0.0002
| -0.0002
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-0.0010
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g> 0.015
g 0.010
S
Jt 0.005
D
°" 0
1
-
-
1 1 1
— HYDRUS - - MULTIMED-OP
CHAIN 20 — CHAIN
-- FECTUZ
-
-
60 1.65 1.70 1.75 1.
Bulk Density (g/onf)
U.2U
^ 0.15
c 0.10
•S 0.05
-------
(a)
(b)
(o)
u.u
t-0.2
-0.4
c
w -0.6
1 -0.8
-1.0
0,
-0.005
£•
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nfanAr«!nn flnmlltelmnt {eim*/A\
— U.UUU
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• -0.080
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-50
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§.
n annn
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CHAIN »»Tc
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) 0.5 1.0 1.5 2.0 2
nfanAntlnn Hnaff Infant ( nm* /A\
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£ 4000
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* 1 (
-^__ _
^^^
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CHAIN ^c
-
) 0.5 1.0 1.5 2.0 2
Hfaruinilnn Coafflnlant ^om*/H^
Figure 7-17. Sensitivity and relative sensitivity of (a) peak concentration at the depth of 6m, (b) time to reach peak concentration at the depth of 6m, and (c)
time to exceed MCL at the receptor well to the dispersion coefficient using the CHAIN Model.
-------
(a)
(b)
oo
(Jl
u
£ -0.10
c -0.20
•= -0.30
at
-0.40
-0.0002
^ -0.0004
g -0.0006
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-0.0010
5;
g 0.015
•^
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^^^^_
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— HYDRUS -- MULTIMED-DP
CHAIN 20 - - FECTUZ
•
.
.0 3.5 4.0 4.5 5.0 5.5 6
— u.uu
£. -0.02
i ~°-04
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-0.10
-20
t-40
c -60
--80
-------
(a)
(b)
(o)
oo
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£ -0.10
| -0.20
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Of
-0.40
0,
-0.0002
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-0.0010
^ 0.020
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-g 0.005
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.
— — — — ' "
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1 1 1 1
— HYDRUS
CHAIN 20
-
•
u
£ -0.01
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ae
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0,
-10
£
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n annn
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1 1 1 1
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-
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— u.uu
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1 -0.04
c
M -0.06
£ -0.08
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0,
-50
t
£ -100
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d 6000
2
-g 4000
D
06 2000
5
• 0
•
1 1 1 1
"To
-
•
0.5 1.0 1.5 2.0 2.5 3.0
Diffusion Coefficient In Water (on? d"1)
0.5 1.0 1.5 2.0 2.5 3.0
Diffusion Coefficient In Water (on? eT1)
0.5 1.0 1.5 2.0 2.5 3.0
Diffusion Coefficient In Water (oirf a"4)
Figure 7-19. Sensitivity and relative sensitivity of (a) peak concentration at the depth of 6m, (b) time to reach peak concentration at the depth of 6m, and (c)
time to exceed MCL at the receptor well to the diffusion coefficient in water using the CHAIN 2D and HYDRUS Models.
-------
(a)
(b)
(o)
. Senslttvlfy
» P F
; o 5 £
•5 —u.iu
Of
-0.20
t\ nnm
0.0005
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n noo
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g 0.010
o
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C
; _ ;
•
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— HYDRUS -- FECTUZ
CHAIN 20 -- MULTIMED-DP -
'
) 100 200 300 400 5C
Saturated Conductivity (cm d~*)
U.IU
1* 0.05
i °
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ae
-0.10
2,
1
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a 6000
g
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JC
°- 2000
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-
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• ^^ •
^ ^ ^
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) 100 200 300 400 5C
Saturated Conductivity (cm d"*)
U.IU
1* 0.05
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£ 4000
o
06 2000
• 0
K> E C
•
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. *~~- .
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) 100 200 300 400 5G
Saturated Conductivity (cm d"1)
Figure 7-20. Sensitivity and relative sensitivity of (a) peak concentration at the depth of 6m, (b) time to reach peak concentration at the depth of 6m, and (c)
time to exceed MCL at the receptor well to the saturated conductivity using the MULTIMED-DP 1.0, FECTUZ, CHAIN 2D, and HYDRUS
Models.
-------
(a)
(b)
(o)
oo
oo
u
1* -0.200
c -0.400
W
1 -0-600
-0.800
0,
-0.005
I -0.010
W -0.015
-0.020
n n9n
g* 0.015
g 0.010
o
o
-g 0.005
•
— ^
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. 22=====-"-^ .
— HYDRUS -- FECTUZ
CHAIN 20 -•-• MULTIMED-DP •
•
•
l.UU
£. 0.80
~ 0.60
C
V> 0.40
^ 0.20
0
9fWW>
15000
1 10000
OT 5000
n nnnn
o 6000
0
" 4000
°- 2000
&
• 0
•
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onnAn
15000
1 10000
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2
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06 2000
• 0
•
.
•
•
.
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0.280 0.300 0.320 0.340 0.360
Saturated Water Content (crn'onf*)
0.280 0.300 0.320 0.340
Saturated Water Content (cnr
0.360
0.280 0.300 0.320 0.340 0.360
Saturated Water Content (cm'cnf*)
Figure 7-21. Sensitivity and relative sensitivity of (a) peak concentration at the depth of 6m, (b) time to reach peak concentration at the depth of 6m, and (c)
time to exceed MCL at the receptor well to the saturated water content using the MULTIMED-DP 1.0, FECTUZ, CHAIN 2D, and HYDRUS
Models.
-------
(o)
(b)
(o)
u
^ -0.100
"5
| -0.200
V>
•= -0.300
Of
-0.400
-0.010
s| -0.020
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g* 0.015
g 0.010
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CHAIN 20 -- MULTIMED-DP -
-
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g °-20
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•x 0.10
o:
0
40000
30000
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d 6000
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°- 2000
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0.05 0.06 0.07 0.08 0.09 0.10 0.11
Residual Water Content (cm3 citf)
0.050 0.060 0.070 0.080 0.090 0.100 0.110
Residual Water Content (cm3 citf5)
0.050 0.060 0.070 0.080 0.090 0.100 0.110
Residual Water Content (om'onT3)
Figure 7-22. Sensitivity and relative sensitivity of (a) peak concentration at the depth of 6m, (b) time to reach peak concentration at the depth of 6m, and (c)
time to exceed MCL at the receptor well to the residual water content using the MULTIMED-DP 1.0, FECTUZ, CHAIN 2D, and HYDRUS
Models.
-------
(a)
(o)
u.zuu
£ 0.150
|
c 0.100
-5 0.050
0£
0
0.020
0.015
I 0.010
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c
OT 0.005
0
^ 0.020
g 0.015
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-
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050 0.052 0.054 0.056 0.058 O.C
Genuohten Retention Parameter a ( a
u
^ -0.05
i -°-10
•s -0.15
et
-0.20
0
-2000
£ -4000
| -6000
OT -8000
^10000
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2
£ 4000
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nf1) p van
• _—^-^ •
• ^^^-rrrr. '
o^""**
•
•
•
050 0.052 0.054 0.056 0.058 0.060
Genuohten Retention Parameter a (citT1)
Figure 7-23. Sensitivity and relative sensitivity of (a) peak concentration at the depth of 6m, (b) time to reach peak concentration at the depth of 6m, and (c)
time to exceed MCL at the receptor well to the van Genuchten retention parameter a using the MULTIMED-DP 1.0, FECTUZ, CHAIN 2D, and
HYDRUS Models.
-------
(o)
(b)
(o)
z.uu
1" 1.50
c 1.00
OT
•= 0.50
at
0
0.008
^ n nne
_>
g 0.004
* 0.002
0
fi n9n
^> o.ozu
g> 0.015
N^
g 0.010
o
jc 0.005
8
"• o
; ^^^;
•
"To
— HYDRUS -- FECTUZ
CHAIN 20 -- MULTIMED-DP -
•
u
^ -0.50
" -1.00
VI
•x -1.50
at
-2.00
-2000
^ —Af\nn
g -6000
5 -8000
*£r 10000
n ftnnn
S
d 6000
" 4000
g
°- 2000
j>
• 0
.,,:•/
• ______--—-
•
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•
u
^ -0^0
c -1.00
(ft
•s -1.50
at
-2.00
-2000
^ _4nnn
^
« -6000
5 -8000
^r 10000
>* fiAAA
Z
-J 6000
2
-c 4000
o
g
« 2000
_o
• 0
' „«=»———
•
"To
-_- . _ . _
•
1.400 1.450 1.500 1.550 1.600 E
van Genuohten Retention Parameter (I (—) *~
1.400 1.450 1.500 1.550 1.600 E
van Genuohten Retention Parameter /8 (—) *~
1.400 1.450 1.500 1.550 1.600
van Genuohten Retention Parameter /3 (—)
Figure 7-24. Sensitivity and relative sensitivity of (a) peak concentration at the depth of 6m, (b) time to reach peak concentration at the depth of 6m, and (c)
time to exceed MCL at the receptor well to the van Genuchten retention parameter (3 using the MULTIMED-DP 1.0, FECTUZ, CHAIN 2D, and
HYDRUS Models.
-------
Table 7-3. Summary of Relative Sensitivities for the Outputs Obtained from the "Tc Breakthrough Curves with Respect to All the Pertinent Input
Parameters for All Models, Referenced to the Base Values of the Input Parameters.
Model Input
Parameter
peak
Q
So
rt
5
FECTUZ
o
HYDRUS
peak
Q
So
rt
5
FECTUZ
U
HYDRUS
5
FECTUZ
<
ffi
0
a
8
Kd
q
e
p
D
DL
Dw
es
er
a
P
-0.06 -0.05 -0.05 -0.06 -0.06
+0.40 +0.00 +0.00 +0.12 +0.12
-1.10 -0.62 -0.62 -0.90 -1.10
-0.17 -0.05 -0.05 -0.05 -0.05
-0.45 ~~ ~~ ~~ ~~
~~ -0.36 -0.36 -0.29 -0.28
~~ ~~ ~~ -0.10 -0.10
~~ +0.04 +0.04 +0.05 +0.05
~~ -0.51 -0.51 -0.66 -0.66
~~ -0.20 -0.20 -0.29 -0.29
~~ +0.05 +0.06 +0.13 +0.13
~~ +0.86 +0.86 +1.30 +1.38
+0.06 +0.06 +0.06 +0.06 +0.06
-1.10 -1.10 -1.10 -1.10 -1.10
+0.80 +0.80 +0.81 +0.78 +0.80
+0.06 +0.06 +0.07 +0.06 +0.06
-0.02 ~~ ~~ ~~ ~~
~~ -0.02 -0.02 -0.02 -0.02
~~ ~~ ~~ -0.01 -0.01
~~ -0.05 -0.04 -0.04 -0.04
~~ +0.60 +0.60 +0.57 +0.57
~~ +0.22 +0.22 +0.22 +0.22
~~ -0.05 -0.08 -0.05 -0.05
~~ -1.00 -1.00 -1.00 -1.00
+0.07 +0.07 +0.07 +0.07 +0.07
-0.98 -1.00 -1.00 -0.98 -0.98
+0.85 +0.92 +0.92 +0.84 +0.86
+0.08 +0.07 +0.07 +0.07 +0.07
-0.10 ~~ ~~ ~~ ~~
~~ -0.09 -0.09 -0.07 -0.07
~~ ~~ ~~ -0.03 -0.03
~~ -0.05 -0.04 -0.04 -0.04
~~ +0.66 +0.66 +0.64 +0.64
~~ +0.26 +0.26 +0.24 +0.24
~~ -0.06 -0.09 -0.05 -0.05
~~ -1.13 -1.15 -1.03 -1.03
-------
For the output C k, there are some differences in the relative sensitivities between the CHAIN Model, the
MULTIMED-DP 1.0 and FECTUZ Models, and the CHAIN 2D and HYDRUS Models;
All output parameters for all models are rather insensitive to Kd and p, as one would expect for the "Tc
BTCs as discussed in Section 6-2 (i.e., 9 » p Kd);
Conversely, all output parameters for all models are rather sensitive to 9;
The output parameters, Tpeak and TMCL, for all models are also rather sensitive to q, while output Cpeak is
much less sensitive to q, the sensitivity being model-dependent;
For the van Genuchten parameters, the model output parameters are most sensitive to p\ followed by 9S,
while the output parameters are least sensitive to Ks and a.
7.3 Other Results Illuminating Numerical Differences/Errors Between the Models
For the sensitivity modeling scenario employed in this report, the major differences between the models arose
due to an error in coding, the use of different numerical inversion schemes (inversion from Laplace space to physical
space), and the treatment of the dispersion factor, 9D, in Equation (6-4).
7.3.7 Correction of the MULTIMED-DP 1.0 Code
When the van Genuchten Model for the water retention parameters (Ks,9s,9r,a,p) was considered in the
sensitivity analysis depicted in Table 7-1, an error in the originally distributed MULTIMED-DP 1.0 Code was
detected. This error arose from the solution of Darcy's Law for the pressure head, which gives the water content in
the soil column through the use of the van Genuchten Model. Figure 7-25 shows the water content profiles in the 6m
soil column derived from Darcy's Law for the MULTIMED-DP 1.0 and FECTUZ Codes, and from the steady-state
solution of Richards Equation for the CHAIN 2D and HYDRUS Codes. The water contents obtained through the
use of the originally distributed MULTIMED-DP 1.0 Code are inconsistent with respect to those obtained from the
other three codes. The reason for the inconsistency was found to be the incorrect use of the residual water content,
9r, in the van Genuchten Module. When this error was corrected, the new water content results became consistent
with those of the other three models, as shown in Figure 7-25. Figure 7-25 also shows that the steady-state solution
of the Richards Equation gives the same results as the solution of Darcy's Law, as we know it should from our
discussion in Section 6-3.
u
1 0
|2.0
£ 3.0
Q.
- ,. ,
0.20 0.30 0.40 0.
Water Content (cnr?/cir?)
Figure 7-25. Water content distributions predicted by the HYDRUS, CHAIN 2D, FECTUZ, and MULTIMED-
DP 1.0 models. Note that the water contents (•) obtained from the originally distributed
MULTIMED-DP 1.0 code are in error. The corrected code gives a consistent water content
distribution (A.) with the other three models.
93
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7.3.2 Comparison of the Stehfest andDeHoog Inversion Algorithms
All the major modules of the MULTIMED-DP 1.0 and FECTUZ Codes are the same (or nearly the same),
except the MULTIMED-DP 1.0 Code has four alternate inverse transform modules (analytical, convolution, Stehfest,
DeHoog), while the FECTUZ only uses the DeHoog Module. For the current uses of these codes, the analytical and
convolution modules are not applicable. Thus, one may ask: Are there any differences between the Stehfest and
DeHoog Algorithms? We found that the answer is YES to this question for the generation of "Tc BTCs using the
MULTIMED-DP 1.0 Code.
The Stehfest Algorithm (Stehfest, 1970) produced non-realistic oscillations in the "Tc BTCs for most of the
parameters studied in this report. The DeHoog Algorithm (DeHoog et al., 1982) is an improved inversion scheme
which was developed to eliminate errors in the use of the analytical and convolution algorithms and to reduce the
oscillatory conditions often produced by the Stehfest Algorithm. We compared the "Tc BTCs produced by the
Stehfest and DeHoog Algorithms in the MULTIMED-DP 1.0 Code for several different input parameters. The
following results were evident from this study:
For all input parameters tested, the Stehfest Algorithm produced oscillatory behavior and peak
concentrations lower than those produced by the DeHoog Algorithm;
For all input parameters tested, the DeHoog Algorithm produced "Tc BTCs almost identical to those
produced by the FECTUZ Code;
As an example of this comparison, Figure 7-26 shows the "Tc BTCs for 9 = 0.08, 0.16, 0.22, where (a)
illustrates the oscillatory behavior of the Stehfest Algorithm in MULTIMED-DP 1.0, (b) illustrates the
corrective behavior of the DeHoog Algorithm in MULTIMED-DP 1.0, and (c) indicates the near perfect
comparison between the FECTUZ and MULTIMED-DP 1.0 Codes, when the DeHoog Algorithm is used.
7.3.3. Increasing the Base Value of Dispersivity in the FECTUZ Code
As we have previously discussed, CHAIN only specifies the value for D in the dispersion factor, 9D, in Equation
(6-4), while MULTIMED-DP 1.0 and FECTUZ only specify the value for DL. CHAIN 2D and HYDRUS specify
both DL and Dw. Since the recharge rate, q, is so small in our sensitivity modeling scenario, there is a significant
difference between the base value of 9D for MULTIMED-DP 1.0 and FECTUZ, and that for CHAIN 2D and
HYDRUS (see the fourth row of Table 7-2). This effect was evident in the "Tc ETC curves shown in Figures 7-1 to
7-12, where the peak concentrations from MULTIMED-DP 1.0 and FECTUZ were higher than those of CHAIN 2D
and HYDRUS. To see if this difference was primarily due to 9D, we increased the base value of DL in FECTUZ to
6.53 cm. This gives a value of 9D for FECTUZ of 0.157 cm2/d, versus that of 0.160 cm2/d for CHAIN and 0.162
cm2/d for CHAIN 2D and HYDRUS. Figure 7-27 compares the "Tc BTCs for the base parameters for the
HYDRUS, CHAIN 2D and CHAIN Models versus the "Tc ETC for the FECTUZ Model with the base value of DL
= 4.53 cm replaced by the value DL = 6.53 cm. This comparison proves that the major differences between the
model BTCs in Figures 7-1 to 7-12 are due to the formulation of the dispersion factor, 9D, and not to other factors.
94
-------
(a)
(b)
(c)
U.UI3
CT
^^
o>
E
^ 0.010
c
JO
"5
L.
c
0>
o
§ 0.005
O
u
i-
m
Q
I 1 1 I 1
MULTIMED-DP 1.0 Water Content (crf/crf)
_, ,_ , A AQ
Stehfest u>uo
n 1 R
0.22
_
s~\
/ \
' / \
: y V'"^
• / A '\
: / :- 1 \ \
•: \ ~'~ 1 \ "^
/ / '•/ \ \
iL—J i A i v i N'^, i
0 2000N-/>OefO 6000 "§000 TOOOO 12000
Time (days)
Oni c
.UlO
cr
o>
E
^ 0.010
c
,0
|
"E
v-' 0.010
c
£
|
n^
0,22
-
; \ A
: / \ -^\
: / \ /
': / \/ \
7 \ ^-
^ / A \
-^ / '• '• \ \
•: / '• •'" V ^
i.-' _/ i X i > N i
0 2000 4000 6000 8000 10000 12000
Time (days)
Figure 7-26. Sensitivity of "Tc BTCs at 6m depth to water content, where (a) uses Stehfest Algorithm of
MULTIMED-DP 1.0, (b) uses DeHoog Algorithm of MULTIMED-DP 1.0, and (c) uses DeHooj
Algorithm of FECTUZ Code.
95
-------
o
0>
o
c
o
o
0.015
0.010
0.005
Uniform Water Content Distribution (Base Case)
HYDRUS
CHAIN 2D
FECTUZ (DL=6.53 cm)
CHAIN
2500 5000 7500
Time (days)
10000
Figure 7-27. Comparison of the "Tc BTCs for the HYDRUS, CHAIN 2D and CHAIN Models for the base values
of the input parameters with the ETC for the FECTUZ Model with the base value of DL = 4.53 cm
replaced by the value DL = 6.53 cm.
96
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Section 8
Summary and Conclusions
The purpose of this report was to evaluate and perform sensitivity analyses on select computer models for
simulating radionuclide fate and transport in the unsaturated zone. The work reported here was performed in support
of the U.S. Environmental Protection Agency's (EPA's) project on Soil Screening Guidance (SSG) for Radionuclides
(U.S. EPA, 2000ab). The U.S. EPA developed the SSG as a tool to help standardize and accelerate the evaluation
and cleanup of soil contaminated with radioactive materials at sites on the National Priority List (NPL) with future
residential land use. This guidance is intended for the appropriate environmental professionals to be able to calculate
risk-based, site-specific, soil screening levels (SSLs) for radionuclides in soil, thus allowing them to identify areas
needing further investigation and possible remediation at NPL sites. The models reviewed and evaluated in the
current report form a small subset of the models available to the public, and those which are considered here have not
met U.S. EPA approval for exclusive use in the SSL process. Other models may be applicable to the SSL effort,
depending on pollutant- and site-specific circumstances.
The model selection process for the radionuclide SSL effort is briefly described in Section 2.1, and resulted in
the following five models being chosen for evaluation and sensitivity analysis:
One-dimensional CHAIN Code (van Genuchten, 1985),
• One-dimensional MULTIMED-DP 1.0 Code (Liu et al., 1995; Sharp-Hansen et al., 1995; Salhotra et al.,
1995)
• One-dimensional FECTUZ Code (U. S. EPA, 1995ab),
• One-dimensional HYDRUS Code (Simunek et al., 1998),
Two-dimensional CHAIN 2D Code (Simunek and van Genuchten, 1994).
The CHAIN Code is an analytical model which requires the assumption of uniform flow conditions. The
MULTIMED-DP 1.0 and FECTUZ Codes are semi-analytic models which approximate complex analytical solutions
using numerical methods such as numerical inverse transform modules. Transient and steady-state conditions can be
accommodated in both the analytical and semi-analytical models, while layered media are usually only
accommodated by the semi-analytical codes. The HYDRUS and CHAIN 2D Codes solve the pertinent sets of partial
differential equations using finite-difference or finite-element methods. The resolution in space and time of these
numerical models depends on the physical characteristics of the site in question, the computational resources
available to the modeler, and the purposes of the simulation. Numerical codes are used when simulating time-
dependent scenarios under spatially varying soil conditions and unsteady flow fields. Table 2-1 lists, in a
comparative manner, the flow and transport model components of each of the five models. Further explanation of
the model components is given in Appendices A to D.
Computer model predictions are uncertain and erroneous because of uncertainties and errors that can occur at
various points in the modeling process and in the model's structure. Errors can occur in conceptualization or model
structure when processes and the assumptions represented by the model fail to represent the intended reality of the
physical problem at hand. Errors in experimental measurements of model input parameters and model output
parameters used for calibration and testing can also lead to uncertainties. For numerical computer codes, the
discretization in space and time introduces error; the computer, itself, introduces roundoff error; and truncation errors
are produced whenever analytical expressions are replaced by truncated series of elementary functions. Other errors
and uncertainties result from the imprecise specification of boundary and initial conditions, poor model
parameterization, and incorrect spatial scale transitions (see Section 1.3). These uncertainties and errors lead to a
three-tier sensitivity analysis:
97
-------
Sensitivity of simulated results to conceptual model selection (Section 3),
Model output parameter sensitivity to varying model input parameters (Sections 4 to 6),
Comparison of sensitivity results due to numerical differences and errors (Section 7).
The breadth and depth of the variability and heterogeneity of the phenomena that can occur in the fate and
transport of radionuclides in the unsaturated zone are briefly considered in Section 3 of this report. Phenomena
considered include the following: the selection of a physical problem's time/space domain, the choice of boundary
and initial conditions for the flow and transport variables, the potential impacts of hy steretic effects and density and
thermal gradients, the importance of preferential pathways and facilitated transport, the form of the transport
dispersion processes, the impacts of non-ideal sorption and reaction processes, the importance of decay processes
and the evolution of decay products, the influence of multicomponent and multiphase systems, and the effects of
highly heterogeneous media. Against this backdrop, the five selected models were compared (see Table 2-1). All of
the models are deterministic, and thus, are incapable of stochastic simulation, except for Monte Carlo simulations for
certain input parameter or boundary condition distributions. Four of the five models are one-dimensional and only
CHAIN 2D is two-dimensional. Thus, truly three dimensional features cannot be easily, or even correctly, simulated,
except for idealistic features such as those presented in Figure 3 -1. Density gradient problems cannot be addressed
by these models, nor can facilitated transport and preferential pathways. The scale dependency of hydraulic
conductivity and dispersivity due to soil heterogeneities is not considered by these models. Other processes and
features that are not addressed by these models are nonlinear reactions, nonlinear nonequilibrium sorption,
nonequilibrium site heterogeneity, and variable decay processes. To address precipitation/evapotranspiration/
infiltration processes on a storm event basis for the several-year periods required for some radionuclides will require
extensive computational resources for the models in this group which are capable of performing such simulations.
Appendices E to I present the sensitivity results of the various models to the following conceptualizations, given
respectively as: nonuniform moisture distribution versus uniform moisture distribution, daily variation of
precipitation/evaporation/infiltration versus annual recharge rate, layered soil column versus homogeneous soil
column, nonequilibrium sorption of pollutants versus equilibrium sorption of pollutants, and variations in fate and
transport of radionuclides with different distribution coefficients and half-lives.
The basic elements of parameter sensitivity analysis are defined and illustrated in Section 4. To carry out this
analysis for radionuclides in the unsaturated zone realistically requires the characteristics and properties from a
representative physical site. The site selection process and the choice of the candidate site are given in Sections 5.1
and 5.2, respectively. The candidate site, located in the semi-arid West, is the Las Graces Trench Research Site in
New Mexico. Some of the soil hydraulic properties at this site and other site characteristics are given in Tables 5-2
and 5-3 and Figure 5-2. The hypothetical modeling scenario for this site assumed a step-wise constant source of
radionuclides at the top of a 6m, vertical, homogeneous soil column with a zero concentration gradient at the bottom
of the column. The input parameters (the base values given in Table 5-4) and the output parameters used in the
sensitivity analyses are as follows:
Input Parameters
Kd = Distribution Coefficient Dw = Diffusion Coefficient in Water
q = Recharge Rate Ks = Saturated Hydraulic Conductivity
9 = Soil Water Content 9S = Saturated Water Content
p = Bulk Density 9r = Residual Water Content
D = Dispersion Coefficient a = van Genuchten Retention Parameter
DL = Dispersivity P = van Genuchten Retention Parameter
Output Parameters
Cpeak = P63^ Concentration of Radionuclide ETC at 6m.
Tpeak = Time to Reach Peak Concentration at 6m.
TMCL = Time When Radionuclide Concentration at 6m is High Enough to Exceed MCL at Receptor Well
Section 6 presents the parameter sensitivity analysis results for the "Tc breakthrough curves (BTCs) at the
bottom of the 6m, vertical, homogeneous soil column, where the BTCs were generated by the HYDRUS Code. The
BTCs for the eleven input parameters (Kd,q,9,p,DL,Dw,Ks,9s,9r,a,P) are given in Figure 6-1, where the BTCs for the
first six input parameters (Kd,q,9,p,DL,Dw) were obtained under the constraints of constant recharge rate, q, and
98
-------
constant moisture content, 9, while those for the last five parameters (Ks,9s,9r,a,p) were obtained under the
constraints of constant recharge rate, q, and variable soil water content, 9(z), see Sections 6.2 and 6.3, respectively.
The sensitivity and relative sensitivity results for these BTCs are given in the following figures:
Figures 6-2a to 6-2k; for Cpeak;
Figures 6-3a to 6-3k; for Tpeak;
Figures 6-4a to 6-4k; for TMCL.
Table 6-4 summarizes the relative sensitivities of (Cpeak, Tpeak, TMCL) to the eleven input parameters for the
HYDRUS Code, the sensitivities being measured at the base values of the input parameters. The dominant input
parameters for all three output parameters are those which have the greatest influence on the advection term in the
transport equations, followed by those which have the greatest influence on the dispersion term. The parameters, p
and Kd, have a very small influence on the transport and fate of "Tc since 9 » pKd in the coefficients of the
transport equations. For the first six input parameters (Kd,q,9,p,DL,Dw), Cpeak is most influenced by 9, while Tpeak
and TMCL are most influenced by both 9 and q. For the last five input parameters (from the van Genuchten Model,
Ks,9s,9r,a,p), the most influential input parameter for all three output parameters is p, followed by 9S, then followed
by9r, a,andKs.
Section 7 compares the sensitivity results between the five models, illuminating numerical differences and
errors. The numerical differences and errors separating the five models are summarized in Table 8-1. Each set of
sensitivity results are derived for the 6m, vertical, homogeneous soil column; and the first seven input parameters
(Kd,q,9,p,D,DL,Dw) are compared under the constraints of constant recharge rate, q, and constant water content, 9.
The last five input parameters (Ks,9s,9r,a,p) are compared under the constraints of constant recharge rate, q, and
variable water content, 9(z). Figures 7-1 to 7-12 compare the "Tc BTCs at the bottom of the 6m soil column for the
various models and input parameters, while Figures 7-13 to 7-24 compare the sensitivities and relative sensitivities of
(Cpeak' Tpeak, TMCL) to the twelve input parameters for the various models. The breakdown of these various figures
with reference to input parameters and models is given in Table 8-2. The results derived from these figures and
calculations are as follows:
The BTCs for the CHAIN 2D and HYDRUS Codes are nearly the same for all input parameter values
considered.
• For the input parameters (K d,q,9,p), the CHAIN BTCs are fairly close to those for CHAIN 2D and
HYDRUS.
• The BTCs for the MULTIMED-DP 1.0 and FECTUZ Codes are almost identical for all values of the input
parameters considered.
• For the base values of (K d,q,9,p,DL), the concentration peaks of the MULTIMED-DP 1.0 and FECTUZ
BTCs are about 14% higher than those for CHAIN 2D and HYDRUS, because of the differences in the
dispersion factor, 9D, as given in Table 8-1.
• For the base values of the van Genuchten parameters (K s,9s,9r,a,p), the concentration peaks of the
MULTIMED-DP 1.0 and FECTUZ BTCs are about 21% higher than those of CHAIN 2D and HYDRUS,
for the reason given above.
When the base value of D L is increased to compensate for the absence of 9iwDw in 9D (see Table 8-1),
FECTUZ and MULTIMED-DP 1.0 peak concentrations of the BTCs closely approach those of CHAIN 2D
and HYDRUS (see Figure 7-27).
• The relative sensitivities are nearly the same for MULTIMED-DP 1.0 and FECTUZ for all output
parameters and all input parameters.
The relative sensitivities are nearly the same for CHAIN 2D and HYDRUS for all output parameters and all
input parameters.
The relative sensitivities are nearly the same for outputs (T peak, TMCL) for all models and all input
parameters.
• For the output (C peak), there are some differences in the relative sensitivities for the three groups of models:
CHAIN, MULTIMED-DP 1.0 and FECTUZ, CHAIN 2D and HYDRUS.
All output parameters for all models are rather insensitive to K d and p, as one would expect for the "Tc
BTCs since 9 » pKd in the coefficients of the transport equations. This will not be the case for higher Kd
radionuclides.
Conversely, all output parameters for "Tc BTCs are rather sensitive to 9 for all models.
The output parameters (T k, TMCL) for all models are also rather sensitive to q, while Cpeak is much less
sensitive to q, the sensitivity being model-dependent.
99
-------
Table 8-1. The Possible Numerical Differences and Errors (•) Separating the Five Models Tested for Water Flow
and Radionuclide Transport through the 6m, Vertical, Homogeneous Soil Column.
Numerical Differences
and Errors
Models Tested in Report
MULTIMED-
CHAIN DP 1.0 FECTUZ CHAIN 2D HYDRUS
Computer Roundoff Errors
Analytical Solution Truncation
Errors for Transport Equations
Inverse Transform Scheme Errors
for Laplace Space Analytical
Solutions of Transport Equations
Finite Difference/Finite Element
Solution Errors for Transport
Equations
Constant Recharge Rate, q, and
Constant Water Content, 9
Constant Recharge Rate, q, and
Variable Water Content, 9(z)
Numerical Integration Errors of
Darcy'sLaw
Numerical Integration Errors of
Richards Equation
Numerical Errors in Deriving the
Steady State Solution of
Richards Equation
Specification of
9D = 9lwDw + DL|q|
in the Transport Equations
Possible Numerical Diffusion
in the Transport Equations
Specifies Specifies Specifies Specifies Specifies
only only only both both
D DL DL DLandDw DLandDw
For the van Genuchten parameters (K s,9s,9r,a,p), the model output parameters are most sensitive to p\
followed by 9S, while the output parameters are least sensitive to Ks and a.
Numerical diffusion for the current modeling scenario is rather insignificant (see Figure 7-27) since the D L-
adjusted FECTUZ BTCs (a model with no numerical diffusion) correspond closely to those for CHAIN 2D
and HYDRUS, models which may have numerical diffusion.
The strongest discriminating differences (see Table 8-1) between the five models are those due to the form
of 9D, and those due to the uniform/nonuniform water content. Other numerical differences/errors appear
to be of lower order effect for the current modeling scenario.
Of the five models tested in this report and for the given modeling scenario, there are basically three
groupings: CHAIN, MULTIMED-DP 1.0 and FECTUZ, CHAIN 2D and HYDRUS.
100
-------
Table 8-2. The Figures in Section 7 Comparing "Tc BTCs and Output Sensitivities with Respect to Model Input
Parameters and Modeling Codes.
Model
Input
Parameter
Kd
q
e
p
D
DL
Dw
Ks
es
er
a
3
CHAIN
7-1
7-2
7-3
7-4
7-5
Figures
MULTIMED-
DP 1.0
7-1
7-2
7-3
7-4
7-6
7-8
7-9
7-10
7-11
7-12
Comparing BTCs
FECTUZ
7-1
7-2
7-3
7-4
7-6
7-8
7-9
7-10
7-11
7-12
CHAIN 2D
7-1
7-2
7-3
7-4
7-6
7-7
7-8
7-9
7-10
7-11
7-12
HYDRUS
7-1
7-2
7-3
7-4
7-6
7-7
7-8
7-9
7-10
7-11
7-12
Figures Comparing Sensitivities
CHAIN
7-13
7-14
7-15
7-16
7-17
MULTIMED-
DP 1.0
7-13
7-14
7-15
7-16
7-18
7-20
7-21
7-22
7-23
7-24
FECTUZ
7-13
7-14
7-15
7-16
7-18
7-20
7-21
7-22
7-23
7-24
CHAIN 2D
7-13
7-14
7-15
7-16
7-18
7-19
7-20
7-21
7-22
7-23
7-24
HYDRUS
7-13
7-14
7-15
7-16
7-18
7-19
7-20
7-21
7-22
7-23
7-24
101
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Section 9
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Appendix A
Empirical Models of the Unsaturated Soil Hydraulic Properties
Which Are Used in the Various Models
One-dimensional soil-water movement in a partially saturated rigid porous medium is described in the HYDRUS
Code(Simunek etal, 1998) by a modified form of the Richards Equation:
(A-l)
where 9 is the soil-water volumetric content (L3L~3), h is the soil-water pressure head (L), a negative quantity for
unsaturated conditions, K is the unsaturated hydraulic conductivity function (UP1), t is time (T), z is the vertical
coordinate (L), positive upward with an origin at some finite depth in the soil, and S(h,z) is a sink term defined as the
volume of water removed from a unit volume of soil per unit time due to plant water uptake (L3L-3T-!). The
formulation of Equation (A-l) assumes that the air phase plays an insignificant role in the liquid flow process and
that water flow due to thermal gradients can be neglected. The general form of K is given by :
K = K(h,z) = Ks(z) K,.(h,z) , (A-2)
where Ks is the saturated hydraulic conductivity (LT-1) and K,. is the relative hydraulic conductivity (1).
The HYDRUS Code features three models for the unsaturated soil hydraulic properties, 9 and K. In
homogeneous media, these properties can be expressed as nonlinear functions of the soil-water pressure head, 9(h)
and K(h). The three models are analytical expressions denoted by BC, VG, and VC, respectively given by Brooks and
Corey (1964), van Genuchten(1980), and Vogel and Cislerova (1988).
The BC-Model is defined by the following equations:
S =•
ah" , h < -I/a
(A'3)
K = KsSfn + c + 2, (A-4)
where Se is the effective water content, 9S is the saturated water content, 9r is the residual water content, a is the
inverse of the air-entry value or bubbling pressure, n is a pore-size distribution index, and c is a pore-connectivity
parameter (c was assumed to be 2.0 in the original Brooks-Corey study). Thus, this model consists of six free
parameters (9r, 9S, Ks, n, c, a). The quantities (9r, 9S, Ks) can be measured in the laboratory or in the field, while the
parameters (n,c,a) are considered to be empirical coefficients affecting the shapes of the hydraulic functions.
The VG-Model is defined by the analytical expressions given below:
A-l
-------
0(h) =
-,h<0
h>0
(A-5)
K(h)=KsS:[l-(l-Sl/m)mJ,
where
m=l-l/n, n
(A-6)
(A-7)
In some of the codes, other than that of HYDRUS, the parameter m is denoted by y, and the parameter n is denoted by
P. The pore-connectivity parameter, c, has been estimated to be about 0.5 as an average for many soils (Mualem,
1976). As in the BC-Model, this model contains the six parameters (9r, 9S, Ks, n, c, a).
The third set of hydraulic equations implemented in HYDRUS are the defining equations of the VC-Model, which
are modifications of the VG-Equations to add flexibility in the description of the hydraulic properties near saturation.
These modified equations are as follows:
0(h) =
9m-9a
hh
(A-8)
K(h)=
KsKr(h) ,
Y ,(h-hk)(Ks
. S '
h hs _
(A-9)
where
K.
'F(9r)-F(9)f
.F(9r)-F(6k)J
(A-10)
F(0) =
1 -
(A-H)
(A-12)
This model allows for a nonzero minimum capillary height, h,, by replacing 9S in the VG-Model by an extrapolated
parameter, 9m, which is slightly more than 9S (Figure A-la). In general, this change from 9S to 9m has little to no
effect on the soil-water retention curve; but its effect on the shape and value of K may be considerable, especially for
fine-textured soils when n is small (i.e., 1.0 < n < 1.3). A further increase in the flexibility of the analytical expressions
of the VC-Model is experienced by replacing 9r in the equation for 9(h) by another extrapolated parameter 9a < 9r.
A-2
-------
o
.2
"c
o
U
0
I
(a)
1% 0
o
3
T3
O
U
.O
"5
D
T3
h. 0
Pressure Head, h Pressure Head, h
Figure A-l. Schematics of the soil-water retention curve (a) and the hydraulic conductivity
function (b) for the VC-Model (from Simuneketal, 1998).
This model retains the physical meaning of 9r and 9S as measurable quantities while adding two new fitting
parameters, 9a and 9m. Equation (A- 10) assumes that the predicted hydraulic conductivity function is matched to a
measured value of the function, Kk = K(9k), at some water content 9k satisfying
9k<9s, Kk 9m~ 9k ~ 9S= Kk-Ks
the VC-Model reduces to the VG-Model.
(A-14)
The question arises, "Which model should a user employ?" The claim is that the VC-Model is more versatile
than the VG-Model, but three more unknown parameters are required to use the VC-Model. Of the BC- and VG-
Models, which is the best to use? Morel-Seytoux et al. (1996) defined an equivalence which provides a simple way to
convert BC-parameters into VG-parameters, and vice versa, when one set of parameters is known and preservation of
the maximum value of a physical characteristic called the "effective capillary drive" is guaranteed. This quantity is
defined by:
r'-rwdhc, (A-15)
o
where k^, is the relative permeability (or conductivity) to water and hc is the capillary pressure head, a positive
quantity equal to minus h. This equivalence makes infiltration capacity calculations insensitive to the model used
(BC or VG) to represent the soil hydraulic properties. It is strictly a matter of convenience for the user which
expressions are used. As one would expect, this equivalence only applies for situations which lead to high water
contents in some part of the domain of interest. However, because of the equivalence criterion which preserves the
asymptotic behavior of capillary pressure at low water contents and thus preserves the hydraulic gradient, the
authors claim there are at least plausible grounds that the parameter equivalence will be reasonably satisfactory for
situations involving drainage and evapotranspiration. But, the application of this paper to these latter processes
should be further tested.
Further support for the equivalence relationship of Morel-Seytoux et al. (1996) is given by the work of Nachabe
(1996). The work of this author was based on the fact that infiltration tests in the field routinely involve
measurements of sorptivity and macroscopic capillary length, the latter being related to the effective capillary drive
defined in Equation A-15. In addition, these two factors are strongly related through a shape factor b, which is a
measure of the nonlinearity of the soil water hydraulic diffusivity D(9), (L2T-!):
A-3
-------
d0
The macroscopic capillary length Xs, which is equivalent to the wetting front suction head, is defined by
J D(0) d0 ,
0)-K(0n)
where 90 is the supply water content applied at the surface and 9n is the dry, antecedent water content in the soil,
ahead of the wetting front. The sorptivity, S ,is well approximated by
S2 = J(00+0-20n)D(0)d0, (A.18)
en
where S , (LT~1/2), is defined by the early stages of an infiltration test. In Nachabe's study, relationships were
developed between hs, S, and shape factor b, and the parameters of the BC- and VG-Models for K(9) and D(9).
Numerical simulations using a dimensionless form of Richards Equation showed that the predicted infiltration rate is
not very sensitive to small variations in the shape factor, b, provided ^s is the same in all the simulations. This result
is important because b is difficult to determine accurately in the field. Thus, this similarity of the dimensional
infiltration solutions under small variations inb shows that ^s is a scale-factor (see Appendix B) because:
(1) ^s renders predictions of infiltration rates fairly insensitive to the models used for K(9) and D(9); and
(2) ^s reduces the number of parameters needed to characterize infiltration.
Therefore, given the generalized infiltration solution from this scale analysis, an infiltration curve into a particular soil
at a particular location and time can be deduced by choosing the proper units in time and space (i.e., scaling) of this
generalized solution, thus minimizing the computations for obtaining infiltration curves over spatially varying
domains. Further, the small range of variation of the shape factor, b, and the insensitivity to this variation by \-
scaled infiltration rates based on the BC- and VG-Models used in the Richards Equation supports the use of linear
scaling in the HYDRUS Code, see Appendix B.
In spite of the results obtained by the two previous investigations, researchers are still looking for better soil-
water retention curve (WRC) models. Assouline et al. (1998) introduced a new conceptual model for WRC's. Before
introducing this new concept, the authors gave a brief, but comprehensive, survey of WRC construction from the
work of Brooks and Corey (1964) to recent studies based on fractal and multifractal soil structures. The proposed
concept and model for the WRC given by Assouline et al. (1998) are based on two assumptions:
(1) A soil structure resulting from a uniform random fragmentation process where the probability of
fragmentation of an aggregate of soil material is proportional to its size; and
(2) A power function describing a relationship between the volume of the aggregates and the corresponding
pore volumes.
This new model WRC was tested on 12 different soil types that presented a wide range of textures and
mechanical analysis. The model accurately reproduced the very different shapes of the respective WRC's, from the
step-function shape of a sand to the sigmoidal almost linear shape of a clay. Also, the new model accurately
reproduced measured data over the whole range of water contents, from saturation conditions to water contents at
the wilting point. Compared with the VG-Model and another popular WRC-Model, the new model of Assouline et al.
exhibited increased flexibility and reproduced the measured data in the high and low water content ranges better than
the two other models. Thus, the authors claim two major benefits for the new WRC-Model:
(1) For this new two fitting parameter model, there is a gain in accuracy and flexibility over competing two fitting
parameter models, such as the BC- and VG-Models.
A-4
-------
(2) The conceptual basis on which the WRC-Model is constructed forms a basis for further research toward a
more physically based relationship between soil structure and its corresponding hydraulic properties.
References
Assouline, S, D.Tessier and A. Bruand. 1998. "A conceptual model of the soil water retention curve." Water Resour.
Res.. 34 (2\ 223-231.
Brooks, R. H. andA.T. Corey. 1964. Hydraulic Properties of Porous Media. Hydrology Paper No. 3. Colorado State
University. Fort Collins, CO.
Morel-Seytoux, H.I, RD. Meyer, M. Nachabe, J. Touma, M. Th. van Genuchten and R. J. Lenhard. 1996. "Parameter
equivalence for Brooks-Corey and van Genuchten soil characteristics: Preserving the effective capillary drive."
Water Resour. Res.. 32 (5), 1251-1258.
Mualem, Y. 1976. " A new model for predicting the hydraulic conductivity of unsaturated porous media." Water
Resour. Res.. 12(3), 513-522.
Nachabe, M. H. 1996. "Macroscopic capillary length, sorptivity, and shape factor in modeling the infiltration rate."
Soil Sci. Soc. Am. J. 60,957-962.
Simunek, I, K.Huang and M.Th. van Genuchten. 1998. The HYDRUS Code for Simulating the One-Dimensional
Movement of Water. Heat. andMultiple Solutes in Variably-Saturated Media. Version6.0. Res. Rep. No. 144.
U.S. Salinity Laboratory, Ag. Res. Ser, USD A, Riverside, CA.
van Genuchten, M. Th. 1980. "A closed-form equation for predicting the hydraulic conductivity of unsaturated
soils." Soil Sci. Soc. Am. J.. 44,892-898.
Vogel, T andM. Cislerova. 1988. "On the reliability of unsaturated hydraulic conductivity calculated from the
moisture retention curve." Transport in Porous Media. 3,1-15.
A-5
-------
Appendix B
A Discussion on the Scaling of Field Soil-Water Behavior
The scaling features for soil-water behavior found in the HYDRUS and CHAIN 2D Codes are part of a bigger
picture in science which includes the concepts of symmetry, invariance, and conservation. Further, it is difficult to
meaningfully discuss scaling without involving some of these more fundamental concepts. For example, Skoglund
(1967) defines scaling as a technique for relating the variables of a "prototype" to those of a "model" at
corresponding points and times in the two systems. Further, he says that scaling often requires that the ratios of the
corresponding dimensional variables of the "prototype" and the "model" at corresponding points and times be
constant. These constants are called scaling factors and are commonly used in the design and interpretation of
physical and mathematical models. But Skoglund says that this often obscures the more fundamental process of
searching for symmetry in nature. In what follows, the roles between scaling, symmetry, invariance, and
conservation are explained.
B.I Symmetry in Nature
Rosen (1995) says a symmetric situation is one which possesses the possibility of change; and if such a change
occurs, some aspect of the situation remains unchanged. Thus, the situation is said to be symmetric under the
change, or transformation, with respect to that aspect. Consequently, there are two essential components of
symmetry in nature (Rosen, 1995):
1. Possibility of change - it must be possible to perform the change, although the change does not actually
have to be performed.
2. Immunity - some aspect of the natural situation remains unchanged if the change is actually performed.
If a change is possible but some aspect of the situation is not immune to it, asymmetry exists. If there is no
possibility of change, with reference to the physical reality of the situation, then the concepts of symmetry and
asymmetry are inapplicable.
In the geophysical sciences, as opposed to several situations in fundamental particle physics, pure mathematical
symmetry is not the rule; but approximate symmetry does exist. That is, in natural situations there are aspects which
are approximately immune to possible transformations. There is no approximation in the transformation or in its
possibility; it must indeed be possible to perform the transformation. The approximation is in the immunity; that is,
some aspect of the natural situation must change by only a "little" when some transformation is performed (Rosen,
1995). The concept of "little" is based on the variations of the aspect in question being much less in magnitude than
some characteristic length, time, mass, velocity, or what have you. Thus, the situation is said to be approximately
symmetric under the transformation with respect to that aspect. In spite of the lack of perfect symmetry in the
biological and geophysical sciences, the search for symmetry is a fruitful undertaking as it has been and is in the
more fundamental areas of physics (Rosen, 1995).
As a science matures its reproductive and predictive features lead to the formulation of fundamental laws which
are usually expressed in mathematical statements and equations given in some system of coordinates (i.e., a reference
frame). Hence, Rosen (1995) states that the searchfor symmetry in nature consists of transformations, immunity from
these changes, frames of reference, and non-immunity from the changes (or asymmetry). This search requires a
general formalism of symmetry. Rosen shows that this formalism is couched in the mathematical language of
transformations and symmetry, which is known as group theory. As geometry is the appropriate mathematics for
investigating space, and calculus is the appropriate mathematics for investigating motion, group theory is the
mathematics of symmetry. It is essential for the search of symmetry in all fields of science, including investigations in
soil science and subsurface geology.
B-l
-------
B.2 Similitude, Transformation Groups, Inspectional Analysis, Self-Similarity
The search for symmetry in physical systems invokes the use of concepts such as similitude, dimensional
analysis, similar physical systems, inspectional analysis, transformations, groups, and self similarity. Skoglund (1967)
says that similitude includes both the concepts of dimensional analysis and similar physical systems. Dimensional
analysis is the special mathematics of units. All scientific measurements are referred to four internationally
recognized fundamental or primary units of measurement: the kilogram, meter, second, and degree Kelvin. The
fundamental or primary concepts of physical things and systems are mass M, length L, time T, and temperature 0.
These units of measurement and the primary concepts, although somewhat arbitrarily chosen, are the minimum
number required for all macroscopic measurements of physical phenomena. From a microscopic point of view (i.e.,
individual molecules, atoms and subatomic particles), the fundamental units are the kilogram, meter and second.
Thus, both viewpoints, macroscopic and microscopic, can be treated as one with respect to similitude concerns.
Dimensions comprise a class of units for the same variable, where the primary dimensions are M, L, T and 0.
For example, the universal gas constant in units of cal/degree K has the physical dimensions of ML2/T2 0. The
fundamental principle of dimensional substitution (Skoglund, 1967) states that any "complete equation of physics"
is satisfied when the units or dimensions of the variables are substituted for the variables themselves. Thus, any
"complete equation" can be converted to a nondimensional form by dividing by a variable or group of variables with
the same units as each of the sides of the equation. This forms a corollary of the principle of dimensional
substitution, namely, that any "complete equation" can be converted into a nondimensional form. Therefore, any
"complete equation" is independent of units, or unit free. "Complete equations" are also said to be dimensionally
homogeneous. In essence, the above properties of a complete physical equation form a set of equivalent definitions
for such an entity.
Birkhoff (1955) states that it is often claimed that "only dimensional homogeneous equations can be regarded as
having fundamental physical significance." However, not all physically significant equations are true in all units! In
fact, there exist no known fundamental units with respect to which all known physical laws are unit free. Birkhoff
says no "Swift's Lilliput" can exist, for to do so would require changing some of the fundamental physical constants,
such as the speed of light and Planck's constant in quantum mechanics. Nevertheless, for the current applications
we are not limited in this regard. Birkhoff shows that for normal velocity distributions and temperature variations, the
Newtonian mechanics of fluids possess as independent, fundamental units the quantities of mass, length, time and
temperature. Thus, the above concepts of complete physical equations and dimensional substitution are valid for
our current applications.
Two physical systems are said to be similar to each other if their pertinent, corresponding variables (e.g.,
velocity, acceleration, mass distribution, force fields and temperature fields) are proportional at corresponding
locations and times. For example, we have previously introduced the terms prototype and model, ^prototype of a
given problem is the physical system itself. A model of the problem is any system that is similar to it.
Mathematically speaking, and this is the only way that similarity can be precisely stated, the first requirement for the
similarity of two systems is that their nondimensional governing equations be the same. Further, to guarantee that
two similar systems possess the same nondimensional solutions it is required that both systems be well-formulated
or well-posed (Flavin and Rionero, 1996) and that their nondimensional initial and boundary conditions be equivalent
at corresponding points and times in their respective geometric domains. Thus, the complete establishment of
similarity, its existence and its properties, dictates that these requirements be met. Anything less than this leads to a
similarity between systems that is less than whole.
The various levels of similarity of importance in continuum mechanics are described in Skoglund (1967). A
model is geometrically similar to a prototype if all the geometric features of the model can be related to those of the
prototype by a single scale factor, as illustrated in Figure B-1. Kinematic similarity is concerned with similar
"particles" (solid or fluid particles) possessing similar velocities and accelerations at corresponding points and times
in similar geometries. Dynamic similarity is similarly defined for systems involving masses, forces, velocities,
accelerations, geometries, and time. Finally, the thermodynamic similarity of a prototype continuum and its
continuum model is based on the requirements and results derived from the principles of the conservation of mass,
momentum and energy, along with pertinent initial and boundary conditions for the dependent variables.
Mathematically speaking (obviously, there is also a very strong experimental component of the search), the
search for symmetry in the fields of continuum mechanics, which includes subsurface processes, first consists of
writing down the governing equations and initial and boundary conditions on which the area of continuum
mechanics in question is based. Then this system is tested for its invariance (i.e., similarity) under groups of
B-2
-------
Figure B-l. Geometrically similar figures, where (a) is the reference figure with characteristic lengthL*, (b) is a
similar figure with characteristic length!^, L* = ocjLj, with scale factor ocj = 2, and (c) is a similar
figure with a2L2 = L*, a2 = 1A (reprinted from UnsaturatedZone Hydrology, 1994, by Gary L.
Guymon, with permission of Pearson Education, Inc., Upper Saddle River, NJ).
transformations of interest. This procedure is called inspectional analysis by Birkhoff (1955). Forexample, two fluid
motions denoted by O and O' are dynamically similar if they can be described by Newtonian coordinate systems (i.e.,
systems in which Newton's laws of motion are valid) which are related by the transformation of space, time, and mass
of the form:
x' = a x, y' = a y, z' = a z, t' = P t, m' = yin,
(B-l)
where (a, P, y) are constant scale factors, and x, y, z, t, m have their usual meanings of length, time and mass. If T,
T(O) = O', represents the transformation in Equation (B-l) with scale factors (a, P, y), then T'1, T'1 (O') = O,
represents its inverse transformation with scale factors (or1, P'1, y1). The successive application of T and T'1, or T'1
and T, leads back to the original fluid motion. Thus, the product T^T or TT'1 defines the identity transformation I
with scale factors (1,1,1). Suppose we have two transformations of the type given in Equation (B-l): T1 with scale
factors (ab pb YI) and T2 with scale factors (a2, P2, y2). The product of Tj and T2, apply Tl to O and T2 to Ob is
defined by: T2 (T10) = T2 Oj = O2, where the scale factors of T2Tj are given by (a2 a1; P2 P1; y2 YJ). Transformations
which satisfy the above properties form a group. Thus, as stated above, Newton's three laws of motion (for speeds
much less than the speed of light) are invariant under this group of transformations. And in general, if the
hypotheses of a given physical theory (e.g., governing equations and initial and boundary conditions) are invariant
under a given group G of transformations, then the conclusions (e.g., system's solutions) are invariant under G; and
these invariances are related to the system's symmetries.
Another important group of transformations in continuum mechanics is the ten parameter Galilei-Newton Group
defined by Birkhoff (1955):
1. The space translation subgroup,
x =x2 + c2,
(B-2)
B-3
-------
2. The time translation subgroup,
t' = t + c; (B-3)
3. The three parameter subgroup of rigid motions,
xi = aikxk , (B-4)
k=l
||aik || = most general 3x3 orthogonal matrix;
4. The three parameter subgroup of moving axes translated with constant velocity,
Xj=Xi-bit. (B-5)
Newton's three laws of motion, for speeds much less than the speed of light, are also invariant under the Galilei-
Newton Group as they are under the group in Equation (B-l). Under these transformations, the definitions of the
material constants (density, viscosity, etc.) are left unchanged for constant mass in the original and primed systems.
In Birkhoff 's classical work of 1955, he concluded that because there are no general uniqueness theorems for the
combined system of Navier-Stokes Equations, the continuity equation, the equation of state, and general initial and
boundary conditions, complete inspectional analysis of hydrodynamic modeling is not possible. Such analysis
awaits fundamentally new theorems about partial differential equations. Although much advancement has taken
place since 1955, this is still the case in many areas of hydrodynamics, especially in some areas of interest to
subsurface scientists, such as hysteresis (Visintin, 1994) and phase transitions (Brokate and Sprekels, 1996).
Self-similarity is a special condition of a single system (Skoglund, 1967). A system is serf-similar if there exist a
separable variable of the governing equation and the initial and boundary conditions of the system. This separable
variable is called the similarity variable, and it permits one to reduce the number of variables/parameters which need
to be considered in the system. Such variables are very valuable in the search for symmetric solutions of special
partial differential and integral equations which satisfy special initial and boundary conditions. Classical examples of
the application of self-similarity variables are the solutions of certain diffusion systems, the Prandll boundary -layer
equations of viscous fluids, and the coagulation equation describing interacting liquid and solid particles. Whether
or not a system is serf-similar is not obvious, and the discovery of similarity variables may be a tedious process.
However, the use of group-theoretic techniques and other mathematical tools reduces some of this tedium as
evidenced by the work of Rogers and Ames (1989), Hill (1992), and Sachdev (2000). Several examples of interest to
the subsurface scientist are detailed in these three books.
B.3 Scale Dependence and Scale Invariance in Hydrology
Sposito ( 1 998a) has edited a recent survey concerned with symmetry and asymmetry in the field of hydrology.
The symmetric features identified in the field are related to the concept of scale invariance, while the asymmetric
features are related to scale dependence. If a system's features are symmetric, then there are no intrinsic scales for
these features, be they length scales, time scales, velocity scales, or what have you. The symmetric features are
similar for "any positive numerical values" of the scale factors in question. Of course, the use of the word "any" is
only mathematical; based on realistic physical considerations, there are both upper and lower limits on the values of
the pertinent scale factors. If some feature is asymmetric or becomes asymmetric for some reason, we say that this
feature's symmetry is broken and the feature becomes scale dependent. An intrinsic scale exists for the feature.
Thus, as the pertinent scales vary, the feature of the system changes in a nonsimilar manner.
The review of Sposito (1998a) is concerned with a variety of topics such as: scale analyses for land-surface
hydrology, scaling of river networks, spatial variability and scale invariance in hy drologic regionalization, scaling of
field soil-water behavior, scaling invariance of Richards Equation and its application to watershed modeling, scale
issues of heterogeneity and stochastic modeling of scale-dependent macrodispersion in vadose-zone hydrology,
dilution of nonreactive solutes in heterogeneous porous media and scale effects in large-scale solute-transport
models, and scale effects in fluid flow through fractured geologic media and transport in multiscale permeability
fields. Many of these topics are of direct concern to investigators of soil-moisture distribution and solute transport
in the unsaturated, subsurface media. In what follows, we will briefly consider some of the topics in this review and
show how they are related to the scaling features found in the HYDRUS and CHAIN-2D Codes.
B4
-------
One of the first papers on the similarity of soil-water behavior in unsaturated porous media and a paper related to
the subject at hand is that of Miller and Miller (1956). Their work is based on the principle of geometric similitude in
the microscopic arrangement of pores and solid particles, as illustrated in Figure B-l. Such similar porous media will
have the same porosities and the same volumetric water content 9. Thus, these similarity assumptions resulted in the
following scaling relationships:
Soil-Water Property
Symbol and Dimensions
Scaling Relationships
Volumetric Water Content
Soil-Water Pressure Head
Hydraulic Conductivity
em
h[L]
K[LT-!]
9* = 9
h* = ah
K* = or2K
where the asterisk denotes the referenced quantities, or the scaled properties that are the same for all the Miller-Miller
similar porous media. The quantity a denotes the scale factor for the pores and solid particles combined (as shown
in Figure B-l). The Miller-Miller similitude has been successfully applied to laboratory studies of sandy soils but has
not been as successful in the analysis of field soils. The reason for this is that the saturated soil moisture content 9S
usually varies from place to place in a field such as shown in Figure B-2. In this figure, 9S will usually vary (and thus,
the 9-distribution) as one moves from vertical profile to vertical profile. That is to say, the soil profile p; has a
different internal geometry than that of soil profile PJ, i # j. Geometric similitude is not satisfied.
Warrick, et al. (1977) accounted for the fact that total porosity of a field soil is highly variable even within a given
soil mapping unit by replacing 9 in the Miller-Miller similitude by Seo = 9/9s, where the residual soil moisture content
9r in the effective water content Se is taken to be zero. In essence, these authors introduced another scale factor into
the Miller-Miller system, namely the saturated soil moisture content 9S which accounts for the different internal
geometries in field soils. The resulting scaling relationships for this new system are as follows:
Soil-Water Property
Symbol and Dimensions
Scalins Relationships
Relative Saturation
Soil-Water Pressure Head
Hydraulic Conductivity
h[L]
K[LT-i]
c * = c
•Jeo oeo
h* = ahh
K*=ak-2K
Soil Surface
TR"
Figure B-2. The depiction of a set of vertical soil profiles ppp2,... distributed over afield mapping unit, where z
represents the local variable within a soil profile and R. (i = 1,2,...) represent the horizontal vectors
in the xy-plane giving the global positioning of the vertical profiles, p,,p2,....
B-5
-------
where the asterisk denotes the referenced quantities which may be fieldwide averages, whereas the un-asterisked
quantities denote field-plot-specific profiles of the quantities in question. The scale factors ah and ak are, in general,
not equal as in the Miller-Miller similitude case.
Warrick, et al. (1977) avoided a search for a microscopic physical length, as required by geometric similarity, by
merely deriving values of a,, that minimized the sum of squares
N M
where N represents the number of macroscopic locations within a field soil and M and number of observations of h
as a function of Seo. Figures B-3abcd illustrate this procedure for observations taken over an agriculture field
consisting of panoche soil. Figure B-3a shows 840 measurements of (9,h) plotted as h vs Seo, where the soil samples
were taken at six soil depths at 20 sites within the field, giving an N = 120. Each of these 120 soil samples were
analyzed in the laboratory with seven values of h, giving various values of Seo for each h, thus M is equal to 7. The
result of minimizing S S in Equation (B -6) was the coalescing of the data in Figure B -3 a into the single curve h* (Seo)
shown in Figure B-3b, the h*-curve being a low order rational function in Seo. Similarly, the 2640 values of (K,9) in
Figure B-3c, plotted as Seo vs K, were coalesced and described by the regression curve indicated in Figure B-3d. This
regression relationship is a cubic polynomial in Seo for the logarithm to the base e of K*. Warrick, et al. found that the
scale factors for ah were not equal to those for scaling K,ak. Further, it should be noted that the values of h(9) scaled
in Figure B-3ab were those measured in the laboratory on soil cores removed from the field, and the values of K(9)
scaled in Figures B-3cd relied on the laboratory results for h(9) to obtain estimates of 9(t) based on field tensiometric
measurements.
1
D
l.U
0.8
0.6
0.4
IT>
1
*
1
i
i i
a
"
OBSERVATIONS
» • FROM 6 SOIL
. • * t DEpTHS
{ill
0 -200 -400
Pressure Head, h in cm
1.0
0.8
0.6
0.4
0.2
SCALED
OBSERVATIONS
I
0 -200 -400
Reference Head, h* in cm
0.2 0.4 0.6 0.8 1.0
Saturalion,
0.2 0.4 0.6 0.8 1.0
Saturation, Sso
Figure B-3. (a) Unsealed observations of S eo(h), (b) Scaled observations of Seo(h*), showing the reference
relationship as a solid curve, (c) unsealed observations of K(Seo), and (d) scaled observations of
K*(Seo) (reprinted from Water Resources Research, 1977, by A.W. Warrick, G.J. Mullen andD.R.
Nielsen with the permission of the American Geophysical Union, Washington, DC).
B-6
-------
Based on the early work of Miller and Miller (1956) and Warrick, et al. (1977), as well as the principles of
similitude and invariance, Vogel, et al. (1991) introduced and studied the system of transformations that form the
foundation of the scaling relationships in the HYDRUS and CHAIN 2D Codes. These linear transformations
account for the linear variability in the soil moisture parameters and are defined by the following:
h=ahh*, (B-7)
e(h)-er=ae[e*(h*)-e;], (&*>
K(h) = akK*(h*), (B-9)
where once again the asterisk denotes reference quantities and the three scale factors (ah,ae,ak) are assumed to be
independent of one another.
Vogel, et al. (1991) claim that these linear transformations will explain the variability due to soil structure within a
given soil textural class, but will not account for the nonlinear phenomena expressed by different soil textural classes.
For example, if the soil-water behavior is represented by one of the models considered in Appendix A, say the VG-
Model, then the a fitting parameter in the model relates to the linear scaling factors and is dominated by the soil
structure. The shape factors in the quantities 9(h) and K(h) of the VG-Model are expressed by the n fitting parameter
which is highly correlated with soil texture and represents phenomena not accounted for in Equations (B-7) to (B-9).
Thus, the scaling laws in Equations (B-7) to (B-9), with their scale factors (ah ,(Xe ,(Xk ), reference hydraulic
characteristics (h*, 9*, K*), and the parameters of the reference quantities derived from analytical expressions (e.g.,
the BC-, VG-, or VC-Models), identify a set of similar soil classes where each member of the set corresponds to a
given textural class. Hence, this linear transformation allows an investigator to resolve a set of soil measurements
(from one field mapping unit, or a set of field mapping units) into a set of similar soil classes where the similar class
distinction is given by the various soil textural classes found in the data. Within a given similarity class, the soils are
linearly related by the transformations in Equations (B-7) to (B-9), and this relationship is dominated by varying soil
structure within the given textural class. These variations in soil structure may be due to certain linearly changing,
heterogeneous layers in a vertical profile and/or due to global changes in soil structure as one changes horizontal
location within field mapping units (Figure B-2). Figure B-4 is a schematic which summarizes the similarity
classification process.
Other features of the Vogel, etal. (1991) similarity system include the resolution of the scaling factors
(ah ,ae ,ak ) into products of two subfactors. One subfactor accounts for the local variation within a given profile
(the z-variation of profiles p l, p2,... in Figure B -2) and the other subfactor accounts for the global variation between
profiles (the R1; R2,... locations inFigure B-2). However, the reference quantities (h*, 9*, K*) are constructed to be
independent of global variations, while being locally dependent on some type of mean-field profile. Further, the
authors consider the invariance of Richards Equation (Appendix A and the main text) for a set of soil profiles with
respect to the transformations in Equations (B-7) to (B-9), supplemented by linear transformations in the variable, z,
time, t, infiltration rate, v, cumulative infiltration, I, and cumulative outflow, Q. Richards Equation was shown to be
invariant under these transformations, thus giving invariant solutions to the system, provided certain restrictions
were met. These restrictions were placed on the local and global subfactors of the scaling factors, and on the initial
and boundary conditions of the system. The results of the derived invariances led to simple relationships between
the soil-moisture scaling factors (ah ,ae ,ak ) and the dynamic scaling factors (% az, o^, a:, aQ). Because of the
restrictions placed on the system's auxiliary conditions, these invariances were shown to hold only for the following
situations:
1. Constant head infiltration into a stratified soil profile.
2. Water redistribution within a soil-water system with zero flux boundaries.
3. Drainage flow with constant suction head boundary conditions, such as in laboratory outflow experiments.
Finally, Vogel, et al. (1991) outline the procedures for identifying the (cth ,(Xe ,(Xk ) scaling factors and reference
quantities (h*, 9*, K*) from measured soil hydraulic data, as well as the indirect identification procedures of these
quantities from measured dynamic characteristics of soil-water systems, such as those obtained from field infiltration
tests and laboratory outflow studies. The authors conclude that these linear transformation procedures and their
resultant dynamical similarities produce the following advantages in the analyses of soil-moisture processes:
B-7
-------
Nonlinear Variability in Soils
Linear Variability in Soils
Shape Parameters Correlate with Soil Texture
Scaling Parameters Dominated by
Soil Structure
Figure B-4. The application of linear scaling to a set of soil-moisture observations, resulting in a set of m similar
soil classes. The m* class of similar soils accounts for the soil structures sml, sm2,..., smq within
the soil textural class m. The number of similar soil classes corresponds to the number of soil
textural classes.
1. There exists savings in the experimental and computing efforts required in the analyses of heterogeneous
soils;
2. There exists simplifications in the analyses of certain phenomena, such as the
derivation of 9 and K distributions from outflow experiments and infiltration tests; and
3. There exists efficient and viable methods of relating laboratory scale results to field scale results.
Rockhold, et al. (1996) applied the basic principles found in the previous three studies to simulate the water flow
and the transport of tritium measured at an initially unsampled domain of the Las Graces Trench Site. The hydraulic
parameters in this simulation were obtained in the following manner. From water-retention data for core samples
collected at the site (but not from the domain in question), the scale factors ah in h* = ah h were determined from a
scale-mean BC-Model (Appendix A) and the measured moisture data. Parameters for the soil-water retention were
then used in the Burdine (1953) relative-permeability model to estimate K(9). Saturated values forthe hydraulic
conductivity Ks were measured in the field at nearly 600 locations. The Ks data were scaled according to
B-8
-------
K(0)=a^K*(0). The probability distribution for ah and ak was found to be lognormal. The geostatistical horizontal
variograms for the log-transformed scale factors (ah, ak) show similar spatial structures to lags of 4 to 6 meters, which
is an indication of a Miller-Miller similitude structure. Thus, Rockhold, et al. (1996) found that it was not necessary to
invoke three independent factors (ah,ae,ak)to condition the hydraulic properties of the field to run their simulations.
Rather they used a constant value for 9S, constant values for the slopes of Seo(h) and K(Seo), and the same
distribution for ah and ak to condition the hydraulic properties of the field in their simulations. Simulations of water
flow adequately agreed with those measured without any further "calibration" of the model. Hence, the authors
showed the power of scaling analyses in the simulation of unsaturated flow and transport processes.
In the Sposito (1998a) review, which was previously mentioned, there are six sections that are of particular
interest to the subject at hand. These are Sections 5 to 10. In Section 5, Nielsen, et al. (1998) gives a detailed survey
of the scaling of field soil-water behavior, from the work of Miller and Miller, to the use of fractal and multifractal
concepts for representing soil structures. In addition, they review the work which combines linear-variability scaling
with the inverse solution of the Richards Equation to estimate in situ hydraulic properties of the soil. Although many
of the techniques surveyed have been successfully applied in various problems, the authors conclude that there
exists no generalized theory at this time for the comprehensive scaling of the behavior of field soil-water regimes.
In Section 6, Sposito (1998b) considers the relationship between scaling invariance and Richards Equation (the
Richards Equation which is free of sources and sinks). Using group analysis (Lie Groups in particular), Sposito
shows that the invariance of the pore size distribution under scaling leads to the result that similar media will show
mathematical identical forms of Se (ah) but not for 9 (ah). This invariance is compatible with a broad range of particle
sizes and the nonuniform porosity of field soils. Further, the scale invariance is a more general concept than fractal
serf-similarity. The Lie Group analysis of the Richards Equation, using the transformation
(e',t',z')=(ne,5t,az), (B-10)
where (|J,,8,a) are independent scale factors, requires that the parameters D(9), K(9), and h(9) mustbe/www or
exponential functions if the Richards Equation is scale invariant. For the derivation of invariant solutions of the
transformed Richards Equation, compatible (invariant) initial and boundary conditions are also required. The scale
invariance of the Richards Equation implies the scale invariance of Se, implies that the BC-Function can be a model for
h(9), and also implies that h(9) can be represented by a generic fractal model Perrier, et al. (1996). In addition, Sposito
(1998b) considers the effects of broken symmetries, for example the effects of time being unsealed, and the effects of
length being unsealed.
In Section 7, Haverkamp, et al. (1998) consider the scaling of Richards Equation and its application to watershed
modeling. They make a strong plea for the use of dimensional analysis and inspectional analysis (i.e., dynamic
similarity) as opposed to just the scaling of a soil's static hydraulic characteristics. This latter procedure fails to scale
vadose-zone behavior in a general way. With this in mind, this section is concerned with cumulative-infiltration
curves I(t) analyzed for different surface boundary conditions and for subsurface water movement governed by the
Richards Equation. They found that for general field soils, there is no unique dynamic similarity for the behavior of
soil-water movement in the vadose zone (see Figure B -4). Only for two soils (Green and Ampt, 1911; and Gardner,
1958) does unique dynamic similarity exist. However, these two soils define the limits of the envelope for all possible
similarity classes that exist for general field soils. For the description of general infiltration, three scaling parameters
are required, while only two are required forthe two limiting cases. Further, because of these two limiting cases, in
practical applications such as watershed modeling, one can assume that there exists a unique dynamic similarity, thus
simplifying certain hydrologic analyses.
In Section 8, Yeh (1998) investigates the key scale issues of heterogeneity in vadose-zone hydrology. He
defines a field scale representative elementary volume (FSREV) and considers the effects on specifying the hydraulic
parameters when our domain of interest is greater than FSREV and when it is less. He considers the use of stochastic
methods to give better estimates of these parameters, but dogmatically states that only large amounts of data can
lessen the uncertainties in vadose-zone parameters and can make the stochastic results statistically meaningful.
In Section 9, Russo (1998) considers the problem of solute transport through partially saturated heterogeneous
porous formations, while inSection 10, Kapoor and Kitanidis (1998) are concerned with the dilution of nonreactive
solutes in heterogeneous porous media. The key results of these papers show that solute plumes may not spread as
much in the longitudinal direction as expected in homogeneous cases, that the travel distances required for the
longitudinal component of the macrodispersion tensor to approach its asymptotic value (i.e., Fickian behavior) can be
B-9
-------
exceedingly large, and that current standard approaches of estimating plume concentrations can severely
overestimate dilution in heterogeneous media.
References
Birkhoff, G. 1955. Hydrodynamics: A Study in Logic. Fact and Similitude. Dover Publications, Inc. New York, NY.
186pp.
Brokate, M. and J. Sprekels. 1996. Hysteresis and Phase Transitions. Springer-Verlag, Inc. New York, NY, 357 pp.
Burdine, N.T 1953. "Relative permeability calculations from size distribution data." Am. Inst. Min. Metal. Pet. Eng.
198,71-77.
Flavin, J.N. and S. Rionero. 1996. Qualitative Estimates for Partial Differential Equations: An Introduction. CRC
Press. BocaRaton,FL. 368pp.
Gardner, W.R. 1958. "Some steady-state solutions of the unsaturated moisture flow equation with application to
evaporation from a water table." SoilSci. 85. 228-232.
Green, W.H. and G. A. Ampt. 1911. "Studies in soil physics: I. The flow of air and water through soils." J. Agric. Sci.
4,1-24.
Guymon,G.L. 1994. Unsaturated Zone Hydrology. PTR Prentice Hall. Englewood Cliffs, NJ. 210pp.
Haverkamp, R., J.-Y Parlange, R. Cuenca, P. J. Ross and T S. Steenhuis. 1998. "Scaling of the Richards equation and
its application to watershed modeling." In Scale Dependence and Scale Invariance in Hydrology. Edited by G.
Sposito. Cambridge University Press. New York, NY. 190-223.
Hill, J.M. 1992. Differential Equations and Group Methods for Scientists and Engineers. CRC Press. BocaRaton, FL
201pp.
Kapoor, V and P. Kitanidis. 1998. "Dilution of nonreactive solutes in heterogeneous porous media." In Scale
Dependence and Scale Invariance in Hydrology. Edited by G. Sposito. Cambridge University Press. New York,
NY, 291-313.
Miller, E.E. and R.D. Miller. 1956. "Physical theory for capillary flow phenomena." J. Appl. Phys. 27. 324-332.
Nielson, D.R., J.W. Hopmans andK. Reichardt. 1998. " An emerging technology for scaling field soil-water behavior."
In Scale Dependence and Scale Invariance in Hydrology. Edited by G. Sposito. Cambridge University Press.
New York, NY. 136-166.
Perrier, E., M. Rieu, G. Sposito and G. de Marsily. 1996. "Models of the water retention curve for soils witha fractal
pore size distribution." Water Resour. Res. 32(10). 3025-3031.
Rockhold, M.L., R.E. Rossi and R.G. Hills. 1996. "Application of similar media scaling and conditional simulation for
modeling water flow and tritium transport at the Las Cruses Trench Site." Water Resour. Res. 32(3). 595-609.
Rogers, C. and W.F. Ames. 1989. Nonlinear Boundary Value Problems in Science and Engineering. Academic Press,
Inc. New York, NY. 416pp.
Rosen, J. 1995. Symmetry in Science: An Introduction to the General Theory. Springer-Verlag. New York, NY. 213
pp.
Russo, D. 1998. "Stochastic modeling of scale-dependent macrodispersion in the vadose zone." In Scale
Dependence and Scale Invariance in Hydrology. Edited by G. Sposito. Cambridge University Press. New York,
NY 266-290.
Sachdev, PL. 2000. Serf-Similarity and Beyond: Exact Solutions of Nonlinear Problems. Chapman &Hall/CRC. Boca
Raton, FL. 319pp.
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Skoglund, V.J. 1967. Similitude: Theory and Applications. International Textbook Company. Scranton, PA 320pp.
Sposito, G. 1998a. Editor of Scale Dependence and Scale Invariance in Hydrology. Cambridge University Press. New
York, NY 423 pp.
Sposito, G. 1998b. "Scaling invariance and the Richards equation." In Scale Dependence and Scale Invariance in
Hydrology. Edited by G. Sposito. Cambridge University Press. New York, NY. 167-189.
Visintin,A. 1994. Differential Models of Hysteresis. Springer-Verlag. New York, NY. 407pp.
Vogel, T.,M. Cislerova and J. W. Hopmans. 1991. "Porous media with linearly variable hydraulic properties." Water
Resour.Res. 27(10). 2735-2741.
Warwick, A.W, G.W. Mullen and D.R. Nielsen. 1977. "Scaling field-measured soil hydraulic properties using a similar
media concept." WaterResour. Res. 13. 355-362.
Yeh, T. -C.J. 1998. "Scale issues of heterogeneity in vadose-zone hydrology: In Scale Dependence and Scale
Invariance in Hydrology. Edited by G. Sposito. Cambridge University Press. New York, NY. 224-265.
B-ll
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Appendix C
An Explanation of the Hysteretic
Characteristics of Soil-Water Properties
The HYDRUS Code is the only code of the five considered in this report to introduce the phenomenological
concept of hysteresis in the formulation of the soil-water properties, 9(h) and K(h). This phenomenon is certainly not
unique to the field of soil physics, for it has a long history in many engineering and scientific systems such as
mechanical, electrical, chemical, biological, physical, and geophysical systems, and even in psychological systems.
The presence of hysteretic components in a system tends to make the system highly nonlinear, may produce
instabilities and discontinuities, and certainly introduces memory dependent processes. Thus, hysteresis will make
easy problems complex, and complex problems even more complicated. But hysteresis is a natural phenomenon that
does exist and must be dealt with in many areas of soil physics, as well as in other scientific areas.
There are two sets of inquiries which one is concerned with when dealing with such a phenomenon. One set
consists of the following questions: In what physical systems does it occur? Under what conditions is the
phenomenon important? And, how does one formulate the phenomenological component in mathematical terms and
incorporate it into the governing equations of the system in question? The other set of inquiries deals with the well-
posedness of the formulated systems containing the phenomenon. The questions in this set of inquiries include:
Does a solution exist for the formulated system? Is the solution obtained unique for the given governing equation
and initial and boundary conditions? And, is the solution continuous with respect to the system parameters, the
form of the phenomenological component, and the initial and boundary conditions?
With reference to the first set of inquiries for the phenomenon of hysteresis, one runs across such concepts as
hysteresis loops, hysteresis operators, hysteretic models, partial differential equations with memory, and
discontinuous hysteresis, as well as ideas from catastrophe theory. Because of the complexity introduced by
hysteresis operators and the memory components, it is necessary to analyze the resultant systems with respect to the
concepts of well-posedness, our second set of inquiries. The tools used to answer this set of inquiries are highly
mathematical and involve such fields as functional analysis, semigroups, variational inequalities, and differential
inclusions. In the following subsections we will only briefly cover some of the key points in the first set of inquiries
and will not discuss any of the mathematical tools used in the second set of inquiries. The subsections that follow
are designated as:
(1) The Origins and Application of Hysteretic Phenomena,
(2) Hysteresis Loops, Operators and Models,
(3) Hysteresis in Soil-Moisture Parameters.
Although this appendix does not cover the mathematical aspects of hysteretic phenomena, we list several
pertinent references for the mathematically-inclined reader:
(1) For the origins and applications of hysteresis, the formulation of hysteresis operators and models, and the
investigations of the well-posedness of the systems see Visintin (1994), and Brokate and Sprekels (1996).
(2) For the theory and application of functional analysis see Rudin (1991) and Edwards (1995).
(3) For the formulation and applications of catastrophe theory see Gilmore (1981), and Castrigiano and Hayes
(1993).
(4) For the application of variational inequalities in porous media and other mechanical systems see Chipot
(1984), andHlavaceketal. (1988).
(5) For the theory and applications of semigroups in linear and nonlinear systems see Goldstein (1985) and
Miyadera(1992).
(6) For the definition, use, and solution of differential inclusions see Aubin and Cellina (1984) and Visintin
(1994).
C-l
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C. 1 The Origins and Applications of Hysteretic Phenomena
Hysteresis is a phenomenon that occurs in rather different situations. It can be a byproduct of fundamental
physical mechanisms, such as phase transitions; or a consequence of a degradation or imperfection, like the play in a
mechanical system; or a deliberate construct of a system in order to monitor the system's behavior, as in the case of
heat control via thermostats (Brokate and Sprekels, 1996). In physics, we encounter hysteresis in plasticity, friction,
ferromagnetism, ferroelectricity, superconductivity, adsorption and desorption, and other processes. Recently, shape
memory effects have been observed and exploited in some new materials. Hysteresis also occurs in the engineering
disciplines, such as in porous media filtration, granular motion, semiconductors, spin glasses, and mechanical
damage and fatigue. In addition, the phenomenon appears in chemical, economical, psychological, and biological
processes (Visintin, 1994).
As indicated above, hysteresis effects are often caused by phase transitions which are accompanied by abrupt
changes of some of the involved physical quantities, as well as the absorption or release of energy in the form of
latent heat. The area of the hysteresis loop (Figure C-l) gives a measure of the amount of energy that has been
dissipated or absorbed during the phase transformation. For example, phase transitions possess hysteresis effects
when there is undercooling prior to nucleation and superheating prior to vaporization. Other phase transitions
possessing hysteresis effects are as follows (Visintin, 1994):
Phase Transition
ferromagnetic
ferroelectric
liquid-vapor
martinsite
phase separation
C.2 Hysteresis Loops, Operators and Models
Order Parameter
magnetization
polarization
reduced density
strain
concentration
External Field
magnetic field
electric field
pressure
stress
chemical potential differences
Figure C-1 shows a typical continuous, closed hysteresis loop for a system whose state is defined by the two
scalar variables (u, v). The variable u represents the input to the system (i.e., the independent variable) and v
represents the output (i.e., the dependent variable). Both u and v depend on time t, and evolve as time increases in a
CONTINUOUS HYSTERESIS LOOP
Figure C-l. A continuous hysteresis loop for a system whose state is given by the couple (u,v), where u
is the input and v is the output (reprinted from Differential Models of Hysteresis, 1994, by Augusto
Visintin, with permission of Springer-Verlag, GmbH & Co. KG, Heidelberg, Germany).
C-2
-------
manner dictated by the system in question. When u increases from Uj to u2 , then the state of the system (u, v)
evolves along the lower curve of the loop, ABC. If u increases beyond u2, then the state (u, v) follows the curve from
C to E. If u now decreases from its value at point E, then the state (u, v) moves along the curve from E to C. As u
decreases from u2 to ub the state (u, v) follows the upper curve of the loop CD A. If u decreases beyond ub then (u,
v) follows the curve from A to F. Now, if u begins to increase from the u at point F, the cycle begins to repeat itself.
If u is between Uj and u2 and the evolution of the state (u, v) is inverted, (u, v) will move into the interior of the
region bounded by ABCDA, (i.e., region R). In standard examples, the couple (u, v) can attain any interior point of R
by a suitable choice of the input function u. These last two properties follow from the memory aspect of hysteresis.
As early as (1905), Madelung attempted to formalize these memory attributes into a set of rules governing the
experimentally observed branchings and loopings of ferromagnetic hysteresis. The rules which were developed are
as follows (see Figure C-2):
(1) As the state of the system (u, v) moves down curve F from point C, it arrives at turning point A, where the
state (u, v) now moves up curve Fj. Rule 1 states that Tl is uniquely determined by the coordinates of A.
(2) Point B is now taken as any point on curve Tl where the motion of the state (u, v) is inverted, point B
becomes a new turning point. Rule 2 says that the evolution of the state is now along curve F2 and that
curve starts at B and terminates at A.
(3) If the input value u decreases beyond its value for point A, Rule 3 says the state (u, v) follows along curve
F from A to D.
These rules of Madelung can be applied at any point on the loop ABCDA in Figure C-l, and at any point in its
interior R if previous state evolutions have reached that interior point.
We assume that the evolution of v is uniquely determined by that of u, and such a result is made precise by
formulating the concept of a hysteresis operator, W: u —> v. However, we know from above, that whenever
MADELUNG'S RULES
v
0
Figure C-2. A defining sketch of Madelung's Rules for the memory attributes of ferromagnetic
hysteresis (reprinted from Hysteresis and Phase Transitions, 1996 by Martin Brokate, Jtirgen
Sprekels, with permission of Springer-Verlag, GmbH & Co. KG, Heidelberg, Germany).
C-3
-------
U!
-------
V '
b
(uc,vc;
-a
Figure C-3a. A relay with hysteresis, or a delayed relay, defined by the parameters (a, b, u c, vc) with repect to the
system defined by states (u,v) (reprinted from Differential Models of Hysteresis, 1994, by Augusto
Visintin, with permission of Springer-Verlag, GmbH & Co. KG, Heidelberg, Germany).
Figure C-3b. An approximation to a continuous hysteresis loop by a linear combination of a finite family of
delayed relays. The quantity Rf is the region inside the discontinuous loop formed by the finite
family of relays (reprinted from Different! al Models of Hysteresis, 1994, by Augusto Visintin, with
permission of Springer-Verlag, GmbH & Co. KG, Heidelberg, Germany).
C-5
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complemented with pertinent initial conditions. Optimal control problems for ordinary differential equations with
hysteresis require specific attention since, even though hysteresis operators are in some sense continuous, they are
usually not differentiable. Understanding such systems cannot be accomplished by merely setting up a "numerical
solution" scheme, programming it, and generating computer output. Mathematical analyses are required for these
complex problems.
Constitutive laws in continuum mechanics formulated in terms of hysteresis operators lead in a natural way to
the partial differential equations of the conservation of mass, momentum and interval energy being coupled with
hysteresis operators. Mathematically speaking, the most interesting and complicated situations are those where the
hysteresis operators appear in the principal part of the partial differential equations. For these systems, basic
existence and uniqueness results are strongly linked in non-obvious ways to the properties of the hysteresis
diagrams and of the memory structure. For such situations, Visintin (1994) considers two model problems: the heat
equation (comparable to the concentration equation) with hysteresis,
du , d r -, 32u 32u
— + — P(u) - — - — = f(x,z,t) , (Ola)
dt dt dx dz
and the wave equation with hysteresis,
a2u a
dt2 dx
(C-4b)
where P(«) is the Preisach Model of hysteresis or some other hysteresis model. Although not studied in detail,
Visintin (1994) introduced two systems of prime interest to the subsurface investigator. One of these systems is the
partial differential equation model describing the growth of bacteria in the presence of nutrients. This phenomenon
exhibits a pattern of growing rings in response to nutrient diffusion. The constitutive relation between the
concentrations of nutrients and the activity of the bacteria leads to a bistability region in the "nutrient space", which
in turn leads to the occurrence of hysteresis in the evolution of the concentration of the bacteria. The other system
considered by Visintin is of direct interest to the problem at hand. This problem is the study of unsaturated water
flow through porous media whose constitutive laws, 9(h) and K(h), account for compressibility, and the history of
drainage/wetting, thus leading to hysteretic effects. In a simplified setting, the "Richards Equation" is written as:
f- [e + P(6)]-|^-f [P(6)]=-S, (C-5)
dt dz dz
where P(9) is the hysteretic constitutive equation. Suitable initial and boundary conditions must be specified for
Equation (C-5), and S and the starting point of P(9) must be given.
C.3 Hysteresis in Soil-Moisture Parameters
In Volume 2 of the Annual Review of Fluid Mechanics, Philip (1970) says that the usual state of the unsaturated
zone is that wetting and drying are simultaneously occurring in different regions of the zone, and that this leads to
the occurrence and the importance of the phenomenon of hysteresis. This phenomenon is most pronounced for
media and moisture ranges where capillarity is dominant. Philip says that the study of this problem started in the
late 1800 's with the investigation of interfacial stability. However, it was not until the late 1920 's that these ideas were
systematically applied to subsurface phenomenon. During this period, it was recognized that many of the possible
equilibrium interface configurations in capillary-porous media are unstable and that the configuration changes are
often spontaneous and uncontrollable. Such changes involve irreversibility and produce hysteretic effects in the
soil-moisture parameters. In addition, Philip says that there is another source of the discontinuities seen in stable
interface configurations, that source is due to the noncontinuous lines of solid-liquid-gas contact which exist due to
the geometrical impossibility of contact lines on some parts of the solid surface. The significance of this latter effect
was illustrated by Philip by making interface configuration calculations in a tube with a radius varying periodically
with the axial coordinate. Following up on the interfacial stability discussions of Philip are discussions given by
Lucknerand Schestakow (1991) in their book on subsurface processes. These discussions are concerned with
C-6
-------
interfacial tension, contact angles between gaseous-liquid interfaces and surface of the solid phase (i.e., a measure of
wettability), and the differential between contact angles at the advancing front of the wetting liquid phase and that at
the receding front (i.e., there exists a hysteresis of contact angles).
In some respects the above comments of Philip can be thought of as the more theoretical aspects of hysteresis in
soil-moisture parameters, while the more phenomenological aspects of the problem are framed in the language of soil
water potential and pressure head. This latter approach is more in line with the hysteresis components considered in
the HYDRUS Code. Water potential, symbolized by \|/, is the free energy of water, or its capacity to do work (Miller
and Donahue, 1995; Mauseth, 1998). When \|/ > 0, the water is capable of doing positive work, while \|/ < 0 means that
work must be done on the water to remove it from its given location and move it to a reference pool (e.g., moving
adsorbed water on soil particles to a free pool of water). The free energy of water can be increased by heating it,
putting it under pressure, and by elevating it. Conversely, the energy of water is reduced by cooling it, reducing its
pressure, and lowering it with respect to gravity. Water's capacity to do work can also be decreased by its adherence
to a solid substance due to hydrogen bonding between the water molecules and the material.
Soil water potential is a combination of the following effects (Miller and Donahue, 1995):
1. The surface area of soil particles and small soil pores that adsorb water - matrix potential \|/m;
2. The effects of dissolved substances - solute or osmotic potential \|/s;
3. The atmospheric or gas pressure effects - pressure potential \|fp;
4. The position of the water with respect to a reference state, a state usually taken as zero - gravitational
potential \|f
In summary, the water potential for soils independent of a reference state is given by
Vw=Vm+Vs+Vp, (06)
and the total water potential is defined by
Vt=Vw+Vg- (C-7)
Water moves whenever there is a difference of water potential within a specific mass of water; this is true in
subsurface soils, within plants, and between soils and living cells. However, if the water potentials of two regions are
equal, the regions are in equilibrium and there is no net movement of water. Temperature differences are usually not a
factor in the subsurface because the solutions being compared are assumed to be at nearly the same temperatures.
Even the temperature differences between leaves and roots in a plant are usually not of significance for water
movement.
With reference to the components in Equations (C-6) and (C-7), the quantity \|/s is zero for pure water and is
always negative for solutions. The quantity \|/p can either be positive or negative and the matrix potential \|/m is a
measure of water's adhesion to nondissolved structures such as soil particles, cell walls and membranes. Since
adhesion can only decrease water's free energy, \|/m is always negative. In soils \|/m is very important because so
much soil water is tightly bound to soil particles; while in living cells, \|/m is usually much less important than \|/s and
\|/p. Miller and Donahue (1995) state that most productive soils do not have a depth of water standing on them (i.e.,
|\|/p is small) and most soils have few salts (i.e., |\|/s is small), thus
Vt=Vw=Vm- (08)
So for most situations, \|/m is the dominant component of \|/t and represents about 95% or more of \|/t. For living cells,
the cell potential \|/c is fairly well approximated by
Vc=ys+yP^o. (c-9)
The description of hysteresis given by Guymon (1994) starts with how water is distributed in a pore and how \|/m
varies through the cross section of that pore. An example of such a pore is given in Figure C-4. In this figure are
shown areas which are drained by the pull of gravity, areas where water is held by capillary forces, and areas where
C-7
-------
Figure C-4. A cross section of a soil pore and the solid soil particles that make up its walls, showing areas
drained by the pull of gravity, areas where water is held by capillary forces, and areas where water
is held by surface forces (e.g., van der Waals forces) (reprinted from Soils in Our Environment,
1995,7th Ed., by R. W. Miller and R.L. Donahue, with permission of Pearson Education, Inc., Upper
Saddle River, NJ).
water is held by surface forces. Point a in the figure represents a point where air begins to enter the pore; for fine
grained soils, the air entry pore water pressure head is about -0.1 atm. Point b is located in the area of water being
held under the influence of capillary forces and may be represented by a water pressure head on the order of -3 to -5
atm. The concept of field capacity is defined as the pore water pressure at which gravity drainage is negligible and
for agriculture soils it occurs at about -2 to -3 atm. Capillary forces dominate surface forces over the range of
pressures from the air entry pressure to -1 atm and beyond in the negative direction down to a pressure as low as
possibly -5 atm. The permanent wilting point is the point at which plants can no longer extract water from the soil;
this may occur at from -5 to -15 atm and is indicated by Point c in Figure C-4. Point d, within the area where the water
film is held by surface forces, may represent a location where the pressure head is as low as -60 atm. Within a few
water molecules of the surface of the soil particles, Point e in Figure C-4, the water pressure may be as low as -8000
atm. With reference to hysteresis, Guymon (1994) states that the soil-water hysteretic memory of soils is
predominantly a capillary effect (as was also stated by Philip, 1970). In the ranges of soil water pressures where
chemical and other surface forces predominate, hysteresis is not evident. In addition, soil memory also ceases when
and where the pressure head is zero or greater. Thus, Guymon says that hysteretic memory effects commonly occur
in the pore pressure range from -5 atm to 0. Of course, the lower limit is dependent upon the soil structure and
texture.
The main cause of hysteresis advanced by Guymon is what is called the "ink bottle" effect as illustrated in
Figure C-5. This figure (Figure C-5a) illustrates that during a drying cycle starting from saturated conditions (h,, = 0),
water is drained from the soil. However, because of the presence of narrow necks, the capillary forces at a given
negative pressure head, hj < h0 = 0, will tend to keep some of the lower large pores filled or partially filled with water
as shown in Figure C-5a. Prior to the initiation of a wetting cycle, it is assumed in Figure C-5b that the soil has
reached the residual moisture content with a soil pressure head, h2 « hj < ho = 0. When the wetting cycle begins,
the soil moisture increases and h2 —> hj. Since the larger pores are starting from the "empty position" with respect to
capillary forces, the water retained in them is due to the lower narrow necks as h2 —> \ and is not due to the upper
narrow necks. Therefore, at the given soil-water pressure head hb the water content during drying is higher than that
during wetting. A secondary cause of hysteresis identified by Guymon is that due to wettability, or the so-called
"rain drop" effect. This effect is described as follows: a wetted soil while drying retains more sorbed water than a dry
soil while being wetted. Thus, the tendency for a higher water content during drying than during wetting is
enhanced.
C-8
-------
THE INK BOTTLE EFFECT
Air, Vapors
and
Gases
(a) A Draining Pore
(b) A Wetting Pore
Figure C-5. The "ink bottle" effect demonstrating that draining/drying soils under the influence of capillary
forces retain more water at a given soil-water pressure than a wetting soil at the same water
pressure (reprinted from Unsaturated Zone Hydrology, 1994, by Gary L. Guymon, with permission
of Pearson Education, Inc., Upper Saddle River, NJ).
Guymon (1994), as well as Luckner and Schestakow (1991), claims that significant effects of hysteresis have been
observed in the soil retention curves, 9(h), while only minor hysteresis effects have been observed in the
unsaturated hydraulic conductivity function, K(h). These authors believe that for most practical problems the
hysteretic effects experienced by K(h) may be ignored, especially since there is such a large error associated with
determining K(h). In spite of these conclusions, the HYDRUS Code does consider hysteresis in both 9(h) and K(h).
A hypothetical soil-moisture hysteresis loop with a discontinuity as the pressure head approaches zero is
illustrated in Figure C-6. The discontinuity is displayed by the positive difference between the saturated moisture
content for the main drying curve and that for the main wetting curve:
e? - er > o.
(C-10)
The reason for this difference is entrapped air in some of the soil pores, a condition that may be removed with time as
the moisture redistributes. Mathematically speaking, the varying discontinuity violates the time invariance of the
hysteresis loop and presents a time-varying condition that is most difficult to verify and document in the field, and to
simulate by modeling.
Also shown in Figure C-6 are sets of primary and secondary scanning curves for wetting and drying cycles.
These scanning curves follow the rules of Madelung depicted in Figure C-2. Knowing the time history of the wetting
and drying cycles throughout a soil column allows an investigator to know at each time and space location where he/
she is at with respect to the moisture state (9,h) on the main drying and wetting curves or the moisture state (9,h) on
scanning curves within the main loop's interior. However, this is much easier to say than to precisely formulate (as
indicated by the previous subsection), and much easier to say than to verify and document in the field. As indicated
by the HYDRUS Code, one can set up numerical procedures which seem to parrot the soil-moisture memory effects in
a soil column; but one should be cautious of the simulated outputs with respect to their verification, or lack thereof,
and their field documentation.
The main drying curve shown in Figure C-6 may be experimentally determined, but is more often expressed in
terms of one of the analytical forms found in Appendix A. The parameters occurring in the analytical model are
evaluated from field or laboratory data. Given the main drying curve, the main wetting curve is obtained by changing
(according to experimental results) one or more of the parameters in the main drying curve. The primary and
secondary scanning curves are usually obtained by a scaling procedure applied to the main drying and wetting
C-9
-------
„
O ^
CD jjj
I (1)
-1- o:
Q) §
3 O
<" £?
e/j ®
0) z
Q_ |
CD
is
^^ °?
=0=1
CO
-ATM
Primary Scanning Curves
Secondary Scanning Curves
o e
Volumetric Moisture Content
Figure C-6. A hypothetical soil-moisture hysteresis loop which is discontinuous for pressure heads near zero,
showing the main drying and main wetting curves, and example primary and secondary scanning
curves (reprinted from Subsurface Transport and Fate Processes, 1993, by R.C. Knox, D.A.
Sabatini, andLW. Canter, with permission of CRC Press, Inc., Boca Raton, FL).
curves (Lucknerand Schestakow, 1991). In the case of the HYDRUS Code, the analytical VG- or VC-Model of
Appendix A is used to give the main drying curve with experimentally derived parameters. Allowance is made in the
code for 9d being equal to 0™, or 9d being greater than 0™. Usually, the following relationships for the main
curves are assumed:
6rd = 67 = 6r, nd = nw = n, ad < aw. (C-ll)
Thus, the main wetting curve is primarily differentiated from the main drying curve by the choice of ad < aw, and
possibly by the saturated moisture contents if 0™ < 9d. If 0™ = 9d, the loop of the main curves closes as the
pressure head approaches zero.
In the HYDRUS Code, the drying scanning curves are scaled from the main drying curve, while those for wetting
are scaled from the main wetting curve. The scale factor for the pressure head in this process is unity, ah = 1. As
previously mentioned, this code does introduce hysteretic effects into the relationship K versus h. As with soil
moisture, allowance is made for differences between dry and wet saturated hydraulic conductivities:
Ksw < Ksd.
The construction of the hysteresis loop for K(h) is similar to that of the moisture hysteresis loop.
(C-12)
C-10
-------
References
Aubin, J.-P. and A. Cellina. 1984. Differential Inclusions: Set-Valued Maps and Viability Theory. Springer-Verlag.
New York, NY. 342pp.
Brokate, M. and J. Sprekels. 1996. Hysteresis and Phase Transitions. Springer-Verlag. New York, NY. 357pp.
Castrigiano, D.P.L. and S.A. Hayes. 1993. Catastrophe Theory. Addison-Wesley Publ. Co., Reading, MA. 250pp.
Chipot, M. 1984. Variational Inequalities and Flow in Porous Media. Springer-Verlag. New York, NY. 118pp.
Edwards, R.E. 1995. Functional Analysis: Theory and Applications. Dover Publ., Inc., New York, NY. 783pp.
Gilmore, R. 1981. Catastrophe Theory for Scientists and Engineers. John Wiley & Sons. New York, NY. 666pp.
Goldstein, J. A. 1985. Semigroups of Linear Operators and Applications. Oxford University Press. New York, NY. 241
pp.
Guymon,G.L. 1994. Unsaturated Zone Hydrology. PTR Prentice Hall. Englewood Cliffs, NJ. 210pp.
Hlavacek, I., J. Haslinger, J. Necas and J. Lovisek. 1988. Solution of Variational Inequalities in Mechanics. Springer-
Verlag. New York, NY. 275pp.
Knox, R.C.,D. A. Sabatini andL.W. Canter. 1993. Subsurface Transport and Fate Processes. Lewis Publ., Boca
Raton, FL. 430pp.
Luckner, L. and WM. Schestakow. 1991. Migration Processes in the Soil and Groundwater Zone. Lewis Publ.
Chelsea, MI. 485 pp.
Madelung, E. 1905. " Uber Magnetisierung durch schnell verlaufende Strome and die Wirkungsweise des
Rutherford-Marconi-schenMagnetdetektors." Ann. Phys. 17. 861-890
Mauseth, J.D. 1998. Botany: An Introduction to Plant Biology. 2/e. Multimedia Enhanced Edition. Jones and
BartlettPubl. Boston, MA. 837pp.
Miller, R. Wand R.L.Donahue. 1995. Soils in Our Environment. 7th Edition. Prentice Hall, Englewood Cliffs, NJ. 649
pp.
Miyadera, I. 1992. Nonlinear Semigroups. Translations of Mathematical Monographs. Vol. 109. Amer. Math Soc.
Providence, RI. 230pp.
Philip, J.R. 1970. "Flow in porous media." In: Annual Review of Fluid Mechanics. Vol.2. Annual Reviews, Inc. Palo
Alto,CA. 177-204.
Rubin. W 1991. Functional Analysis. 2ndEdition. McGraw-Hill, Inc. New York, NY. 424pp.
Visintin,A. 1994. Differential Models of Hysteresis. Springer-Verlag. New York, NY. 407pp.
C-ll
-------
Appendix D
The First-Order Decay Chains Used in the Various Models
The one-dimensional version of the transport equations for solutes involved in sequential
first-order decay reactions consists of the following:
i
dt dz dz
6R2 ^ + q^ = eD2 + F2(c2)c2 +G2(Cl,s,s2), (D-2)
dt dz dz |_ oz
3s*
— J-=H(ci,s* )-(m +n[)s*+Yi, i=U, (D-3)
dt
where 9 is the soil moisture content, "1" is the index for the parent species, "2" is the index for the daughter species,
R is the retardation accounting for gaseous and liquid solute phases and the equilibrium solid phase, c is the
concentration of the liquid phase, q is the Darcian fluid flux density, D is the dispersion coefficient, s* represents the
nonequilibrium solid phase of the solute, /j, is the degradation rate of s*, \i' is the rate of transfer of material from
parent to daughter, and y is the production of s* . The function F(c) represents degradation of the solute from
gaseous, liquid and solid phases, transfer losses from parent passed on to daughter for the three phases, losses due
to plant uptake, and changes due to the temperature variations in the system. The function Gl ( s, ) accounts for
production of the solute in all three phases, losses due to plant uptake, and the amount of solid phase at the
nonequilibrium sorption sites. The function G2 (q ,s* ,s*2 ) accounts for the daughter material transferred from the
parent in the gaseous, liquid, equilibrium solid and nonequilibrium solid phases, accounts for the production of the
daughter species in all phases, and accounts for root uptake losses. Finally, the function H(ci5s*) could be thought
of as the first-order sorption rate of s* due to the difference between equilibrium concentrations and nonequilibrium
concentrations.
The system given in Equations (D-l) to (D-3) is basically that which is presented in the HYDRUS Model. The
CHAIN 2D version of these equations replaces the vertical coordinate (z) by the vertical slab coordinates (x, z), the
quantity q by components in the x and z directions, and the dispersion coefficient D by a 2 x 2 matrix of dispersion
coefficients. The FECTUZ Code considers the one-dimensional version of the above system, but without
nonequilibrium sorption, zero-order production teams, root uptake and accounting for temperature variations. The
MULTIMED-DP 1 .0 Code is similar to FECTUZ but it only accounts for linear equilibrium sorption; it can also change
rate constants to account for temperature. Finally, the CHAIN Code is similar to that of MULTIMED-DP 1.0 but no
account is made of temperature variations and the first-order decay of the parent becomes the zero-order production
of the daughter.
In the following paragraphs, we list the type of reaction chains that are considered in the various codes. Specific
species and products are not identified in this discussion, only the types of chains that can be formulated in the
system in question. Both HYDRUS and CHAIN 2D will allow the user to specify six solutes, either coupled in an
unidirectional chain; or totally uncoupled, where each species is independent of the other. Examples that have been
run have had the following forms:
D-l
-------
Radionuclides
>•
Nitrogen Compounds
Gas
. .: .. .. .
Products
Uninterrupted Chain —
One Reaction Path for Pesticides
Gas
'! 'I 'I '
Product Product Product
Products
Interrupted Chain-
Two Independent Reaction Paths for Pesticides
Gas
Gas
Product
Product Product
Product
The FECTUZ Code can handle up to seven different chemical species, either in straight, unidirectional chains, or
in branched chains. Systems for which solutions have been obtained include the following:
Straight Chains
N<7
Simple Splitting Chains
D = Number of Daughters < 6
r|ij = Splitting Factors
D-2
-------
Simple Converging Chains
N<7
Seven-Member Branching Chain
•H12+ 1113 = T124+ T125 =
= 1
The MULTIMED-DP 1.0 Code only considers parents, daughters and granddaughters; and within each
subgroup, the code only considers 3, 4, and 4 variables (species), respectively. Pathways that have been considered
in hydrolysis transformations include those given below:
Hydrolysis Pathways
2
<
The CHAIN Code, as reported in van Genuchten (1985), considers four species in a straight chain. The
radioactive decay chain example problem given in this paper consists of the following radionuclides:
238
234
230
226
Pu-
Th-
Ra
The author states that the most critical species is the last one in the chain, 226Ra. This species is a high biological
hazard and is highly mobile, the retardation factor R4 is rather low.
D-3
-------
References
van Genuchten, M.Th. 1985. "Convective-dispersive transport of solutes involved in sequential first-order decay
reactions." Computers and Geosciences. 11(2): 129-147.
D4
-------
Appendix E
The Impact of Using a Nonuniform Moisture Distribution
versus a Uniform Distribution
Constant values of q, 9, and h were assumed for all models in the sensitivity analyses for the input parameters
Kd, q, 9, p, D, DL and Dw. On the other hand, a nonuniform water content distribution was employed by MULTIMED-
DP 1.0, FECTUZ, CHAIN 2D and HYDRUS for the analyses involving Ks, 9r, 9S, a, and (3. One should assume that
the nonuniform distribution is the "standard" and the uniform distribution is the "approximation." However, when
the van Genuchten Model for the water retention parameters (Ks 9S, 9r, a, (3) was considered in the sensitivity
analyses reported in Sections 6 and 7, an error in the originally distributed MULTIMED-DP 1.0 Code was detected.
The error arose from the solution of the governing equations for the pressure head, h, and the moisture content, 9.
When corrected, all systems were defined for a constant recharge rate, q, and boundary conditions resulting in the
following conditions for 9: 9 = base value = 0.16, at the surface; and at 6 m, 9 = saturated value = 0.32. Figure E-l
shows the results of solving Darcy's Law for MULTIMED-DP 1.0 and FECTUZ, and solving the Richards Equation
for CHAIN 2D and HYDRUS.
The water contents obtained from using the originally distributed MULTIMED-DP 1.0 Code are inconsistent with
respect to those obtained from the other three codes. The reason for the inconsistency was found to be the incorrect
use of the residual water content in the van Genuchten Module. When this error was corrected, the new water
content results became consistent with the results of the FECTUZ Code, as one would expect, and also became
consistent with the steady state (remember q and the boundary conditions are fixed) solutions for HYDRUS and
CHAIN 2D. For the simulations conducted in this report (i.e., time durations up to 10,000 days, and times to peak
concentrations of from 3000 to 8000 days), the approach to steady state moisture distributions for both HYDRUS and
CHAIN 2D is very fast. For the purposes of this report, the variable moisture content obtained from the van
Genuchten Model and Richards Equation is a steady state distribution as shown in Figure E-l. Further, one would
expect that all four models should give the same distribution (as the figure indicates) if the MULTIMED-DP 1.0 Code
is properly given. The revised code for MULTIMED-DP 1.0 is the one that was used in all of the sensitivity analyses
and any other subsequent calculations using this code.
Before comparing the breakthrough curves (BTCs) of "Tc for a uniform moisture content (9 = 0.16) with those
for a nonuniform distribution (Figure E-l), one should note the degree of nonlinearity in the distribution shown in
Figure E-l. Because of the homogeneity of the 6m layer, one notes that the steady state distribution is nearly
uniform(9 ~ 0.16) for the first 4m of depth. At about 4m, the distribution begins to break away from this uniform
value. However, even at a depth of 5.5m, the distribution has only deviated about 33% of the total difference (0.32 -
0.16). Thus, about 67% of the total deviation of the distribution takes place in the last 0.5m of the layer.
Consequently, one would assume that changes from going from a uniform water content to a nonuniform water
content (of the type in Figure E-l) would not be too great.
This comparison between uniform and nonuniform distributions of water content is illustrated in Figure E-2
using the breakthrough curves (BTCs) of "Tc predicted by the four models: MULTIMED-DP 1.0, FECTUZ, CHAIN
2D and HYDRUS. The bottom graph in this figure gives the BTCs predicted by all five models, including the CHAIN
Model, for the base values of the parameters given in Section 6. In the top graph, all input parameters are kept at
their base values except for 9 which is replaced by the distribution given in Figure E-1. No CHAIN results are given
in the top graph since 9 can only be a constant in this model.
In comparing the BTCs for uniform water content versus the nonuniform values, one sees that the nonuniform 9
reduces the Cpeak and increases the Tpeak and TMCL for each model. This is reasonable if we remember that the
nonuniform 9-distribution increases the overall moisture in the soil column versus that for 9 = 0.16 throughout the
column. Increasing moisture reduces the advection term in the transport equation and increases the diffusion term.
E-l
-------
u
1 0
I 2'°
£ 3.0
0.
- ,
0.10 0.20 0.30 0.40 0.50
Water Content (cnf/cnf)
Figure E-l. Water content distributions predicted by the HYDRUS, CHAIN 2D, FECTUZ, and MULTIMED-DP
1.0 Models. Note that the water contents (•) obtained from the originally distributed MULTIMED-
DP 1.0 Code are in error. The corrected code gives a consistent water content distribution (*) with
the other three models.
This results in lower Cpeak values for the nonuniform BTCs, produces delays in the Tpeak values and TMCL values
with respect to those for the uniform case, and produces an increase in diffusive spread of the BTCs for the
nonuniform case over that of the uniform case, although this spread is only about 1.08 times that for the uniform
case.
To get a better quantitative effect of the impact of using the uniform approximation for the sensitivity analyses of
the first seven input parameters (Kd, q, 9, p, D, DL ,DW), we used the CHAIN 2D Model to derive the BTCs and
sensitivities for the parameters Kd and DL for both the uniform and nonuniform water contents. These comparative
BTCs are given in Figure E-3 (Kd comparison) and Figure E-4 (DL comparison). As in Figure E-2, the Cpeak values for
the nonuniform water content BTCs have been uniformly reduced over the ranges of the input parameters compared
to those for a constant 9, while the T k values and TMCL values have been increased uniformly over the input
ranges. Table E-l lists the output vames for the uniform and nonuniform water content cases, values which are
measured at the base values of the input parameters Kd and DL. In addition, the table gives the relative sensitivities
of these three outputs to the inputs Kd and DL, again referenced to the base values of Kd and DL. As one can see,
the relative sensitivities for the uniform and nonuniform cases are basically the same. The percent differences of the
output values, based on the nonuniform case as the standard, are as follows:
Percent Differences
Output
*~peak
T
1peak
TMCL
15. 5% Decrease
6.7% Increase
6.4% Increase
12.1% Decrease
5. 9% Increase
6.1% Increase
Based upon these results and the BTCs in Figures E-2 to E-4, we believe that the use of the uniform approximation for
9 for the analysis of the sensitivity of the three outputs to the input parameters Kd, q, 9, p, D, DL and Dw is both valid
and representative of the sensitivities that would be obtained from the nonuniform water content distribution.
Further, using this uniform approximation allows us to directly compare the results of the CHAIN Model with the
other four models, as illustrated in the bottom graphs shown in Figure E-2. However, it should be emphasized that
these results are for a homogeneous layer under steady state conditions.
E-2
-------
0.015
0.010
0.005
o>
E
c
o
c
o
0.015
o
o
o
r
0.010
0.005
Nonuniform Water Content Distribution
HYDRUS
CHAIN 2D
FECTUZ
MULTIMED-DP 1.0
Uniform Water Content Distribution (Base Case)
A CHAIN HYDRUS
CHAIN 2D
FECTUZ
MULTIMED-DP 1.0
2500 5000 7500
Time (days)
10000
Figure E-2. Comparison of the breakthrough curves predicted by the CHAIN, HYDRUS, CHAIN 2D, FECTUZ,
and MULTIIMED-DP 1.0 Models for the base case given in Section 6. The top curves are for a
nonuniform water content and the bottom curves are for 9 = 0.16 throughout the soil column.
There are no CHAIN results in the top graph because 9 can only be constant in this model.
Table E-l Comparison of Results Derived from Figures E-3 and E-4 for the Distribution Coefficient Kd and the
Dispersivity DL, respectively. The Values of Cpeak, Tpeak and TMCL are Given for the Base Values of
Kd and DL, and the Relative Sensitivities of These Output Quantities to Kd and DL are Given, These
Values Also Being Taken at the Base Values of Kd and DL.
Output
c-
Tpeak
TMCL
Property
Base Value (mg/1)
Relative Sensitivity
Base Value (d)
RelativeSensitivity
Base Value (d)
Relative Sensitivity
i
Uniform
Water Content
0.00732
-0.06
4766
+0.06
3554
+0.07
din ml/g
Nonuniform
Water Content
0.00634
-0.05
5108
+0.06
3799
+0.07
DL in cm
Uniform
WaterContent
0.00725
-0.28
4737
-0.02
3552
-0.08
Nonuniform
WaterContent
0.00647
-0.25
5036
-0.02
3783
-0.07
E-3
-------
0.0101
"5; 0.008
o 0.006
§ 0.004
I
o 0.002
Distribution Coefficient (ml/g)
0.001 CHAIN 2D
0.007
0.019
(nonunifotm water content)
0 2500 5000 7500 10000
Time (days)
0.010
0.008
o 0.006
0.004
0.002
Distribution Coefficient (ml/g) CHAIN 2D
0.001
0.007
0.019
(Uniform water content)
2500 5000 7500
Time (days)
10000
Figure E-3. Sensitivity of "Tc breakthrough (through the 6 m layer) to the distribution coefficient using the
CHAIN 2D Model, for a nonuniform water content (top) and for a uniform water content (bottom).
0.015
0.010
o.oos
CHAIN 2D (nonuniform water content)
Dlsperalvlty (am)
3.73
4.53
5.33
4000 6000 8000 10000
Time (days)
0.015
0.010
o
o
t—
s
CHAIN 2D DtspersMty (cm)
(uniform water content) 3.73
5.33
2000 4000 6000 8000 10000
Time (days)
Figure E-4. Sensitivity of "Tc breakthrough (through the 6 m layer) to the dispersivity using the CHAIN 2D
Model, for a nonuniform water content (top) and for a uniform water content (bottom).
E4
-------
Appendix F
The Impacts of Using Daily Precipitation Rates and Daily
Evapotranspiration Rates versus an Annual Average Recharge Rate
In the sensitivity analyses of Sections 6 and 7, the impact of the "natural cycles" of precipitation/runoff/
infiltration/evapotranspiration was assumed to be a constant mean annual recharge rate, q, which is applied year-
after-year for a total simulation time of 25 to 40 years. It is the constant rate, q, which carries the radionuclides down
through the homogeneous 6 m layer to the hypothetical water table at its lower boundary. The result of this ideal
hydrologic mechanism is the classical "bell-shaped" breakthrough curve (ETC) shown throughout this report. These
"bell-shaped" BTCs are probably sufficient to demonstrate the sensitivities of the three output quantities to the
twelve input parameters given in Section 6. However, this "ideal mechanism" is at most a very crude approximation
of the real mechanisms which produce ground-water contamination from radionuclides passing through the
unsaturated zone.
The attributes of radionuclide movement through the vadose zone depend on the details of the time and space
variations of the "cycles" of precipitation/runoff/infiltration/ evapotranspiration. Closely linked to these cycles are
the mechanisms of flow hysteresis and flow through preferential pathways. In fact, Yao and Hendrickx (1996) state
that worldwide it is thought that a major mechanism for ground-water contamination is the passage of pollutants
through preferential flow paths, which is an event type phenomenon not currently well parameterized for large time-
step simulations. These preferential flow paths can be produced by macropores such as soil cracks, old root channels
or animal paths; by spatial variability of hydraulic conductivity; or by unstable wetting fronts. The occurrence of
unstable wetting fronts is especially important since this mechanism can produce preferential pathways in
homogeneous layers which do not contain macropores and significant changes in hydraulic conductivity, such as the
6 m layer used in the current report. In several of the mechanisms which produce wetting front instabilities,./?*™;
hysteresis (Appendix C) plays a major role, see Section 3.4.
An example of the impact of flow hysteresis on the results of single storm events is given in Eagleson (1970).
He reported the results of a numerical experiment on the redistribution of moisture in a soil profile after the cessation
of infiltration. In this simulation it was found that percolation with hysteresis was much slower than percolation
without hysteresis. This simulation occurred over a two hour period and the percolation was through sandy soil.
Such a result applied to a soil with vegetation would allow a longer time for evapotranspiration to act under
hysteretic flow than under nonhysteretic flow, thus producing a greater loss of subsurface moisture under flow
hysteresis.
Modeling codes required to simulate the above phenomena which lead to flow fingering and preferential
pathways for moisture and pollutant transport through the vadose zone to the ground water should be two-
dimensional in character and preferably three-dimensional, have the ability to handle event precipitation/runoff/
infiltration/evapotranspiration, have the ability to simulate air compressibility and flow hysteresis, and the ability to
vary soil properties in a significant manner. In addition, such simulations would require time steps on the order of an
hour or less, and space increments fine enough to simulate a two-dimensional pattern of flow fingers in the horizontal
field of interest. Such simulations are beyond the objectives of the current report and beyond the capabilities of the
five models under consideration, although the one-dimensional HYDRUS Code considers most of the pertinent
hydrologic processes required for such simulations. Thus, even though the above preferential pathway phenomena
are potentially very important for the transport and fate of radionuclides through the vadose zone, they will not and
cannot be considered further in this appendix. Further, the time and space increments required for an analyses of such
phenomena would be prohibitive for the current applications, where our simulations run from 10,000 days to 15,000
days.
F-l
-------
To compromise, the simulations considered in this appendix are based on daily precipitation amounts without
allowance for runoff, and on daily evapotranspiration rates based on two different root water uptake stress response
distributions (Equations 2-3 and 2-6). Considering the 18 years of daily precipitation amounts shown in Figure 5-2,
one notes that there is only one peak above 80 mm/d, two peaks above 40 mm/d, and about 30 peaks above 20 mm/d.
These three categories, respectively, convert to 0.33 cm/hr, 0.17 cm/hr, and 0.08 cm/hr. However, one should keep in
mind that these hourly rates are lower, and probably much lower, than storm event rates. Further the frequency of
occurrence of the higher rates is greater when one uses storm events as opposed to using the daily precipitation rates.
In the work of Yao and Hendrickx (1996), for New Mexico type soils and hydrologic conditions, it was found
that wetting front instabilities and fingering occurred for infiltration rates between 0.3 and 12 cm/hr, incomplete
wetting without distinct development of fingering occurred for infiltration rates between 0.12 and 0.3 cm/hr, and
stable wetting front migration occurred for infiltration rates below 0.12 cm/hr. Assuming these laboratory results are
convertible to the homogeneous 6 m layer used in the current simulations, one would expect wetting front instability
to occur on many days throughout the total record of 6570 days. Thus, the complexities of wetting front instability
for storm events may be inconsistent with the assessment of the impacts of daily precipitation/evapotranspiration on
the transport of radionuclides in the unsaturated zone. However, such complexities, as stated above, require finer
time and space scale simulations than those used in the current study.
The precipitation record used in the current study, as mentioned above, is the one shown in Figure 5-2. Figure F-
1 shows this same record in terms of annual precipitation amounts and monthly average amounts for the 18 years of
record at the Las Cruces Site. As one can see from Figure F-la, nine of the first ten years of record are above the
annual average of 22.5 cm; the last eight years of the record are equal to or less than the mean annual average. For
the monthly averages, each of the first five months has a monthly value less than 1 cm. The monthly averages begin
to rise sharply in June; and from July to October, the monthly averages are greater than 2 cm per month. November's
average drops below 2 cm, while December returns to just over 2 cm.
As given in Equations (2-3) and (2-6), the sink term S in the modified Richards Equation for the soil moisture
distribution in the soil profile is expressed as the product of a water stress response function a(h) times a potential
water uptake rate Sp(z, t). In turn, the quantity Sp(z, t) is a product of the normalized water uptake distribution
function b(z, t) and the potential transpiration rate Tp(t). The quantity b (z, t) is related to the root distribution
function and in the current study was taken as uniform over an interval from 50 cm below the surface to 250 cm
below the surface. This type of root distribution is probably reasonable for mesquite tree and creosote bush
vegetative cover over semi-dry regions, such as the Las Cruces Site (see Royo, 2000; and Miller and Donahue,
1995). The potential transpiration rate was calculated from the site's climate data using Penman's Equation (Jensen
et al, 1990). The water stress response function a(h) used in the current simulation was the Feddes Module of the
HYDRUS Code (Simunek, et al, 1998). This function is the simple trapezoid shown in Figure F-2. The quantity hj
is the value of the pressure head below which roots start to extract water from the soil (a value of -20 to -50 cm); h2
is the pressure head below which roots start to extract water at the maximum possible rate, a(h2) =1; h3 is the
pressure head below which roots can no longer extract water at the maximum rate; and finally h4 is the pressure head
below which root water uptake ceases, this being usually equal to the wilting point. Water uptake by roots can be
easily varied by changing one or more of the values of the triplet (h1; h2 h3). In the current simulations, two sets of
this triplet were used to give two different scenarios of root water uptake. Two modifications that were not
considered in these simulations were root growth and decline, and the characteristics that desert plants use to reduce
water uptake and evapotranspiration rates, such as leaf coatings and root shrinkage which produces root-soil air gaps
that are highly resistant to moisture transfer (Nobel, 1994). Such modifications are probably important in the first
five months of the year (Figure F-lb) and during the dry years (Figure F-la). Thus, the results that follow may be
giving higher actual evapotranspiration (ET) rates than occur in nature, and lower recharge rates.
Figure F-3 summarizes the results of the two root water uptake scenarios run on the daily precipitation and PET
data for the Las Cruces Site. These two sets of curves give the cumulative amounts of precipitation, actual ET and
net recharge (precipitation minus actual ET, no allowance for runoff) in centimeters. The HYDRUS Model was used
to calculate the actual ET values, and thus the recharge rates. The chief characteristics of these two sets of curves are
given in the following notes:
The records for both sets of curves consist of three pieces, 0 to a, a to b, and b to c, covering a total of
14,140 days.
F-2
-------
(A
**
O
0)
p
1
o
aBejeAv A|muo|/\|
I
S2
re
0)
p
|Bnuuv
60
2
g aa
a a
aE
CH
' '^
" ^
it
.&
'o
.8
H
60
'H -3
S 8
F-3
-------
0)
w
^ 1.0--
.
at
81
0) *•
" e
**
a
0.5--
Q
£
LU
Figure F-2.
Soil Water Pressure Head, h
The water stress response function for the Feddes Module of the HYDRUS Code (Simunek, et al..
1998).
1000
800
600
400
- 200
^
0
1000
800
600
400
200
Cumul. Net Recharge (Avg. 67 mm/yr)
Cumul. Actual ET
Cumul. Precipitation
b
Cumul. Net Recharge (Avg. 54 mm/yr)
Cumul. Actual ET
Cumul. Precipitation
b
3000
6000 9000
Time (days)
12000 15000
Figure F-3. Cumulative amounts in centimeters of precipitation, actual evapotranspiration (ET), and net
recharge (precipitation minus actual ET) during a HYDRUS Model simulation using daily variable
precipitation and potential ET rates at the surface. Cumulative net recharge and ET vary between
the two figures because of differences in the root water uptake scenarios, (hj, h2, h3). The
precipitation/PET segment from "a to b" is repeated from "b to c."
F-4
-------
The records for the interval 0 to a cover 1000 days and represent a constant recharge rate and precipitation
rate (no ET taking place) of 0.024 cm/hr. This period was used to let the radionuclides distribute
throughout the "upper layers" of the 6 mvadose zone under the action of the base values given in Section 6.
Once the source of radionuclides quit emitting material at 1000 days, the daily precipitation and PET
records in Figure 5-2 were applied to the system for the transport and fate of the radionuclides contained in
the "upper layers" of the 6 m zone. Thus, the second piece of the records in Figure F-3 runs from a = lOOOd
to b = 7570d.
When the simulation reached 7570 days, the daily precipitation and PET records in Figure F-2 were
repeated, giving the third piece of the records in Figure F-3, which runs from b = 7570d to c = 14,140d.
The average recharge rates, 67 mm/yr for the top set of curves in Figure F-3 and 54 mm/yr for the bottom
set, were obtained from the graphs by taking the cumulative recharge values at 12,000 days and converting
these values to mm/yr.
Considering the wettest part of the 18 year record, the first ten years, the mean annual recharge rate for the
top set of curves in Figure F-3 over the first 4650 days (10 years plus 1000 days) is about 84 mm/yr and that
for the bottom set is 59 mm/yr.
The high frequency "oscillations" in all the cumulative curves shown in Figure F-3 are due to daily and
weekly variations in the precipitation and PET records.
Ignoring these oscillations in the cumulative precipitation curves (which, of course, are the same in both sets
of curves), one sees a curve made up of five line segments:
1. 0 to lOOOd, 87 mm/yr recharge rate for initial spreading of pollutants,
2. lOOOd to 5015d, first eleven years of record, the wettest period,
3. 5015d to 7570d, last seven years of record, the driest period,
4. 7570d to ll,585d, repeat of the wettest period,
5. ll,585d to 14,140d, repeat of driest period.
Ignoring the daily and weekly "oscillations" in the bottom set of curves in Figure F-3, one sees that the root
water uptake scenario is such that the divergence between the cumulative precipitation and cumulative
actual ET in each of the four segments from lOOOd to 14,140d is roughly constant from segment to segment.
This result produces a net recharge cumulative curve which is roughly a straight line, which means that after
the first 1000 days the pollutants are being transported by a constant recharge rate of about 52 mm/yr.
Once again ignoring the daily and weekly "oscillations" in the top set of curves in Figure F-3, one sees that
the root water uptake scenario is such that the divergence between the cumulative precipitation and
cumulative actual ET in the segment from lOOOd to 5015d is greater than that for the bottom set of curves,
giving a greater recharge rate of about 83 mm/yr. In the next segment, 5015d to 7570d, the cumulative
precipitation curve is approximately parallel to the cumulative actual ET, giving a recharge rate which is
relatively small at 20 mm/yr. The last two segments, 7570d to ll,585d and ll,585d to 14,140d, are a repeat
of the previous two segments.
The impacts that the variable recharge rates in Figure F-3 have on the transport and fate of radionuclides through
the 6 m of the vadose zone are shown by the "Tc breakthrough curves (BTCs) given in Figure F-4. The solid BTCs
in this figure corresponds to the daily computed recharge rates, while the dashed BTCs correspond to calculations
based on constant recharge rates (67 mm/yr and 54 mm/yr) for the entire time duration of the simulations. For both
sets of BTCs, the moisture distribution throughout the soil column is nonuniform (i.e., the modified Richards
Equation was solved). Observations that one can make about these results are as follows:
The solid curve in the bottom set of curves is basically the classical "bell-shaped" curve as one might expect
by the nearly constant recharge rate of 52 mm/yr after the first 1000 days of a rate of 87 mm/yr. The dashed
curve is the ETC for a constant recharge rate of 54 mm/yr from zero to the end of the simulation.
The solid curve in the bottom set of curves varies from that of the dashed curve because of the first 1000
days of pollutant distribution under a recharge rate of 87 mm/yr and because of daily variations of
F-5
-------
o
f
0.012
0.010
0.008
0.006
0.004
0.002
0
0.012
0.010
0.008
0.006
0.004
0.002
Variable Precip/Actural ET HYDRUS
Uniform Net Recharge
Average Net Recharge: 67 mm/yr)
Variable Precip/Actual ET HYDRUS
Uniform Net Recharge
Average Net Recharge: 54 mm/yr)
5000 10000
Time (days)
15000
Figure F-4. Comparison of predicted "Tc breakthrough curves (through the 6 m layer) using the variable
precipitation/actual ET versus uniform recharge rate in the HYDRUS Model. Average recharge
rate is calculated as the mean net amount of precipitation and actual ET from 0 to 12,000 days.
The net recharge varies between the two sets of curves due to the root-uptake scenario, (hbh2 ,h3).
precipitation and PET. The Cpeak of the solid curve is 1.16 times that of the dashed curve, and the Tpeak of
the solid curve is 1.01 times that of the dashed curve. At the scale shown in Figure F-4, the daily and
weekly "oscillations" are not apparent, but their cumulative effects are present.
The relationship between the dashed curve in the bottom set, which reflects a nonuniform moisture
distribution, and the corresponding uniform moisture content result is such that the Cpeak decreases by 14%
and the Tpeak increases 5.5% over those for the uniform case, based on the standard being the nonuniform
case. These percentages are consistent with those found for the parameters Kd and DL in Appendix E.
The solid curve in the top set of curves in Figure F-4 does not possess the classic "bell-shape". This is
primarily due to the nonlinear cumulative recharge shown in the top set of curves in Figure F-3. The
average recharge rate is 87 mm/yr for the first 1000 days, 83 mm/yr for the next 4015 days, followed by 20
mm/yr for the next 2555 days, then 83 mm/yr for 4015 days, and finally 20 mm/yr for the last 2550 days.
The first part of the solid curve in the top set is somewhat like that of the base case for HYDRUS given in
Section 6, except the variable precipitation Cpeak is 1.08 times higher and its T k 0.88 times smaller. After
5015 days, the average recharge rate drops sharply to 20 mm/yr, thus giving a long "heavy tail" to the
distribution. At 7570 days the larger recharge rate kicks in again, producing a cutoff of the tail at about
12,500 days, or so.
The dashed curve in the top set of Figure F-4 is the nonuniform moisture distribution ETC for a constant
recharge rate of 67 mm/yr. Comparing this curve to results for the uniform moisture case, one sees that the
Cpeak decreases by 13.6% and the Tpeak increases by 2.9% over those for the uniform case, based on the
standard being the nonuniform case. These results are consistent with those of Appendix E.
F-6
-------
References
Eagleson, P.S. 1970. Dynamic Hydrology. McGraw-Hill Book Co. New York, NY. 462pp.
Jensen, M.E.,R.D. Burman, and R.G. Allen. 1990. "Evapotranspiration and irrigation water requirements." ASCE
Manuals and Reports on Engineering Practice. No. 70. American Society of Civil Engineers, NY. pp.332.
Miller, R.W. and R.L. Donahue. 1995. Soils in Our Environment. 7th Edition. Prentice Hall, Englewood Cliffs, NJ.
649 pp.
Nobel, P.S. 1994. "Root-soil responses to water pulses in dry environments." In: Exploitation of Environmental
Heterogeneity by Plants: Ecophysiological Processes Above- and Belowground. M.M. Caldwell and R.W.
Pearcy, Eds. Academic Press. New York, NY. pp. 285-304.
Royo, A..R. 2000. "Desert Plant Survival." DesertUSA Newsletter. http://www.desertusa.com/du_plantsurv.html.
5pp.
Simunek, I, K. Huang, andM.Th. van Genuchten. 1998. The HYDRUS Code for Simulating the One-Dimensional
Movement of Water. Heat, and Multiple Solutes in Variably-Saturated Media. Research Report No. 144. U.S.
Salinity Laboratory. USD A, ARS. Riverside, CA.
Yao, T-M. and J.M.H. Hendrickx. 1996. "Stability of wetting fronts in dry homogeneous soils under low infiltration
rates." Soil Sci. Soc. Am. J. 60, 20-28.
F-7
-------
Appendix G
The Impact of Considering a Layered Soil Column
versus a Homogeneous Soil Column
In this appendix we are concerned with the impact on the flow and transport, and the sensitivity analyses of
replacing the 6 m homogeneous layer (the "all" layer in Table 5-2) by the 9-layer soil column given in Table 5-2. It
is recognized that these impacts will be considerably less than the impacts due to layering reported in Miyazaki et al.
(1993) and Rockhold et al. (1997). However, the objective of the current impact analysis is to assess the differences
of using the 9-layered soil column for the Las Graces Trench Site versus the single all-layer soil column, and to
decide if the use of one layer is comparable to the nine layers at this site.
As stated, Table 5-2 gives the soil hydraulic properties (KS,9S, 9r, a, P) at the Las Graces Trench Site for the first
6m of the soil profile. The 6 m homogeneous layer used in the sensitivity analyses in Sections 6 and 7 is the layer
with the hydraulic properties listed in the"all layer" row of the table. However, as seen in this table, the 600 cm layer
is broken down into 9 sublayers, having different thicknesses and hydraulic properties. The biggest outlier of all the
9 layers is, as one would expect, the 15 cm thick surface layer which contains the most organic matter and is the most
highly disturbed, both by natural and anthropogenic processes. The pathway through this surface layer represents
only about 2.5 percent of the total pathway through the 600 cm layer; thus, its effect on the total passage of "Tc
through the 6m vadose zone to the hypothetical water table is very minor. For the remaining eight layers, the range of
values for the five soil properties about their base values (Section 6) and the corresponding maximum percentage
deviation (based on the base value) are given below:
172 cm/d < Ks = 270 < 334 cm/d, 36.3 %,
0.294 <9S = 0.32 < 0.343, 8.1%,
0.071<9r = 0.08 < 0.091, 13.8%,
0.027 cm-1 < a = 0.055 < 0.070 cm-1, 50.9 %,
1.383
-------
c
o
o
c
o
o
0.010
0.008
0.006
0.004
0.002
HYDRUS Distribution Coefficient (ml/g)
Layered ° Uniform 0.001
Layered • Uniform 0.007
Layered * Uniform 0.019
2500 5000 7500
Time (days)
10000
Figure G-l. Sensitivity of "Tc breakthrough (through the unsaturated zone with a water table at a depth of 6 m)
to the distribution coefficients in a layered soil and in a uniform soil using the HYDRUS Model.
Cpeak anc* decrease T k and TMCL. In the layer from 305 to 370 cm, 9S and (3 tend to increase C k and decrease
Tpeak ancl TMCL> while 9r has the opposite effect. In the final three layers (370 to 600 cm), (3 tends to decrease Cpeak
and increase T k and TMCL, while the effects due to 9S and 9r are fairly close to those of the uniform case. From
this collection of "facts," it is difficult to dogmatically say how the layered and uniform cases should compare. Thus,
relying on the computations, one arrives at the following results, where percentages are based on the layered results
as being the standard and decreases/increases are related to changes from the uniform case:
Kd(ml/g)
0.001
0.007
0.019
(-*
^peak
9. 9% decrease
9. 5% decrease
9.0% decrease
T
peak
3. 4% increase
2.7% increase
2.7% increase
TMCL
2.4% increase
2.2% increase
2.0% increase
As suggested above, the differences between the results for the uniform soil and the layered soil are not too
great. And because of this, it seems reasonable to expect the sensitivity studies given in this report for the 6m
homogeneous layer are also applicable for the layered soil described in Table 5-2. The differences in Cpeak are
assumed to be within 10% of one another and those for Tpeak and TMCL are assumed to be around 3 to 4%, or less, of
one another. However, as initially stated, these conclusions will not be valid if the layering is of the type considered
by Miyazaki et al. (1993) and Rockhold et al. (1997), layering with significant changes in hydraulic properties from
layer to layer in the soil column.
References
Miyasaki, T., S. Hasegawa and T. Kasubuchi. 1993. Water Flow in Soils. Marcel Dekker, Inc., New York, NY,
296 pp.
Rockhold, M.L., C.S. Simmons and M.F. Payer. 1997. "An analytical solution technique for one-dimensional,
steady vertical water flow in layered soils." Water Resour. Res. 33(4):897-902.
G-2
-------
Appendix H
A Detailed Analysis of Nonequilibrium Sorption of Pollutants,
Mainly for the Radionuclide 90Sr
Thibodeaux (1996) states that the validity of the local equilibrium assumption (LEA) in solid-liquid soil
reactions depends on the degree of interaction between the macroscopic transport processes of water flow and
hydrodynamic dispersion, and the microscopic processes of molecular diffusion and sorbed-solute distribution in
conjunction with soil aggregate size. When the rate of change of solute mass during microscopic sorption processes
is fast relative to bulk flow, the interaction is nearly instantaneous, comparatively speaking, and it conforms with the
LEA. Deviations from local equilibrium occur as the interactions of the solute with the porous media become
increasingly time dependent with respect to the time scales of the bulk flow. This divergence also occurs as soil
aggregates increase in size and complexity, and the pore-class heterogeneity increases. In the present case, the
transport of radionuclides through the unsaturated zone is taking place under the influence of a very low annual
recharge rate, which leads to a hydrodynamic dispersion of only two times that of molecular diffusion. Under these
conditions, sites that were in local equilibrium at higher recharge rates will still be in local equilibrium. Some new
sites where nonequilibrium conditions existed or no liquid-solid reactions occurred at higher recharge rates will now
become under the LEA, while other new sites will pass from no action sites to nonequilibrium sites, and some will be
nonequilibrium sites under both the higher and lower flows. Thus, the adsorption-desorption cycle of a given
pollutant through a given soil profile has a variety of characteristic time scales dependent upon the type of liquid-
solid reaction mechanisms at the various sites in the soil matrix. Therefore, it seems reasonable that for "all"
recharge flows one considers nonequilibrium sorption as well as equilibrium conditions.
H.1 A Simplified Version of the Transport Equations
Although there are other models (see Section 3.6), the system used in CHAIN 2D and HYDRUS is a two-site
(equilibrium-nonequilibrium) model developed by van Genuchten and his colleagues (van Genuchten and Wagenet,
1989; Toride et al., 1993). The one-dimensional version of the transport equations using this model (i.e., the
HYDRUS Code) is given in Section 2 and Appendix D. It is this version that is used to generate results for the
current appendix. However, for purposes of discussion and argument, the following simplified set of transport
equations for the liquid phase pollutant concentration, c, and the nonequilibrium solid phase concentration, s, is
considered:
ac 0rwD +D
wwL
D/P '
at 6 + pfKd az 6 + pfKd az2 6 + pfKd
3s
— = D/P - US , (H-2)
3t
where ^ is the radioactive decay rate for both s and c, and D/P is a decay/production, sorption transfer function
defined by
D/P = 0)[(l-f)Kd C-s]. (H-3)
H-l
-------
The quantity, f, is a parameter fixed by the soil structure. When f = 1, there are no nonequilibrium sorption sites in
the soil and when f = 0, there are no equilibrium sorption sites. When 0 < f < 1, there are both equilibrium and
nonequilibrium sites. The quantity, co, is the first-order rate constant for nonequilibrium sorption in inverse time
units. Thus, we have two new input parameters for the nonequilibrium sorption process, (co,f).
When D/P = 0 (i.e., the nonequilibrium sites act as equilibrium sites), these transport equations reduce to the
following:
3c , q 3c _ OTWDW + DL q 32C
- TT - HC , (H-4)
3t 0 + pfKd 3z 0 + pfKd 3z
s = (1 - f) Kdc . (H-5)
Equations (H-4) and (H-5) define the situation when there are no nonequilibrium sites, which occurs when f = 1; or if
0 < f < 1, the equations define the situation when co is sufficiently large so as to dominate the effect of the decay rate,
H.2 The Transport of "Tc with a Kj = 0.007 ml/g
As a first example of Equations (H-l) and (H-2), let us consider the transport of "Tc through the 6 m soil
column. The initial distributions of (c,s) in the soil profile are taken as zero, and a source of c is postulated at the
surface. For "Tc, the decay rate, (J,, has a very small effect on the breakthrough curves (BTCs) since lifetimes for
these curves in the given system are from 10,000 days to 12,000 days for the default Kd of 0.0007 ml/g, while the
half-life of "Tc is 2.1 x 105 years. Thus the important decay/production in these two equations is D/P and not [i. As
the process begins, t > 0, pollutant c begins to be spread throughout the water in the soil profile. This then produces
a source of s via the D/P term; thus, s begins to be spread throughout the soil matrix in the column. As long as s is
less than (1 - f)Kdc, D/P is a sink term for c and a source term for s. If s is greater than (1 - f)Kdc, then D/P is a
source term for c and a sink term for s. Since the only other term in the s-equation is a very small sink term (i.e., |J,s),
s will approximately be equal to (1 - f)Kdc for sufficiently large co. Thus, under sufficiently high sorption rates co, we
have the "equilibrium" condition:
S - (1 - f)KdC . (H-6)
Equation (H-6) shows that for low Kd-pollutants, the amount of material adsorbed at the nonequilibrium sites for any
one time is very small, while for high Kd-pollutants, the opposite is usually true. Consequently, even though the low-
q problems considered in this report may partially violate the LEA, the resultant effect of nonequilibrium sorption
can still be very small if the pollutant in question (i.e., "Tc) has a very small Kd.
Using the HYDRUS Code with the nonuniform distribution for 9 and the base values of Section 6 for the other
10 standard input parameters, the BTCs of "Tc for different values of (co, f) were derived. One set of BTCs were
obtained for f = 0.47 and co = 0.003, 0.032, and 0.320^. Another set of BTCs were obtained for co = 0.032^ and f
= 0.27, 0.47, 0.67. These five curves were compared with the equilibrium case, f = 1 and co = 0, given by the solid
curve in the top set of curves in Figure E-2. It should be noted that when one is comparing such sets of curves,
Equations (H-l) and (H-2) are not exactly the ones solved by the HYDRUS Code since 9 is not constant but is
variable with depth as shown in Figure E-l. However, for order-of-magnitude arguments, Equations (H-l) to (H-6)
are adequate.
Considering Equations (H-l) and (H-2), one sees that the parameters (co, f) affect three processes (terms):
advection, diffusion, the decay/production term D/P. When s reaches its equilibrium value of (l-f)Kdc, the D/P term
is turned off and the s-distribution will track the c-distribution if c and s do not again get out of balance. This
tracking is such that when s = (l-f)Kdc, the c-distribution is the distribution that occurs when there are no
nonequilibrium sites. That is, the existence of active nonequilibrium sites requires an s-distribution which is not
equal to the equilibrium sorption condition given by (l-f)Kdc.
H-2
-------
For "Tc, with its default Kd of 0.007 ml/g and its large half-life, the c- and s-distributions should not get out of
balance once they are in balance. Further, the transfer of material between solution and the soil matrix is relatively
small for the (co, f) given above:
[(l-f)Kdcl = 0.51%ofc(t).
(H-7)
These results are verified when the "nonequilibrium" "Tc BTCs for the five combinations of (co, f) mentioned above
are compared with one another and with the equilibrium case for (co, f) = (0, 1). All of these BTCs are within 1% or
2% of one another at any given time. Thus, for a soil profile of 6 m, the nonequilibrium formulation given in
Equation (H-l) and (H-2) reduces to an equilibrium formulation for a radionuclide with a Kd of 0.007 ml/g and a
half-life of 76,650,000 days, if co is sufficiently large.
H.3 The Transport of "Tc with a Kd = 1.0 ml/g
For illustrative purposes, we next considered "Tc with a Kd equal to unity and (co, f) equal to (0.032d~1, 0),
(0.032d~1, 0.47) and (0, 1). The results of these numerical experiments were basically the same as the Kd of 0.007
ml/g results (i.e., nonequilibrium conditions quickly approach equilibrium conditions), even though the maximum
amount of material transferred between solution and soil matrix is much greater than that given in Equation (H-7):
[(l-f)Kdc]_ = 100%ofc(t),
(H-8)
Table H-l summarizes both sets of results (i.e.,K = 0.007, 1.0 ml/g) for three selected times and concentration
pairs (c, s) for the "Tc BTCs. The equilibrium cases are given by the pair (co, f) = (0,1), while the "nonequilibrium"
cases are given by the pairs (co, f) = (0.032, 0.47) and (0.032, 0). Columns headed by c and s are rounded off
calculated values from the HYDRUS Code, while the column headed by (l-f)Kdc is a check to see if c and s are in
equilibrium sorption balance. To round-off error, for both Kd values of 0.007 and 1.0 ml/g, all times, and a half-life
of 76,650,000 days, the equilibrium and "nonequilibrium" BTCs are the same and the c- and s-distributions are in
Table H-l. Comparison of Nonequilibrium and Equilibrium Results for "Tc BTCs for Kd Values of 0.007 and
1.0 ml/g.
Radionuclide Kd
ml/g
(d-i)
en
(d-i)
f
Time
(d)
c
mg/L
mg/kg
s
mg/kg
Technetium
"Tc 0.007 9.043xlO-9 0
0.007 9.043 xlO-9 0.032
1 9.043 xlO-9 0
1 9.043 xlO-9 0.032
1 9.043 xlO-9 0.032
0.47
0.47
3750
5005
6250
3750
5005
6250
40,000
46,500
60,000
40,000
46,500
60,000
40,000
46,500
60,000
0.00105
0.00646
0.00227
0.00111
0.00644
0.00222
3.67 x lO-4
6.22 x lO-4
1.38x10-4
3.67x10-4
6.12x10-4
1.38x10-4
3.67x10-4
6.02 x lO-4
1.38x10-4
0
0
0
4.12xlO-6
2.39xlO-5
8.24 x 10-6
0
0
0
1.94 x lO-4
3.24 x lO-4
7.31xlO-5
3.67 x lO-4
6.02 x lO-4
1.38 x lO-4
0
0
0
4.12xlO-6
2.39x10-5
8.19x10-6
0
0
0
1.90 x lO-4
3.24 x lO-4
7.35 x lO-5
3.63 x lO-4
6.01 x lO-4
1.40 x lO-4
H-3
-------
equilibrium sorption balance through the formula (l-f)Kdc = s. In addition, one sees a slight drop in the Cpeak (at t =
46,500d) for the Kd = 1.0 ml/g case from 6.22 x 10'4 mg/L for the equilibrium case (co, f) = (0, 1), to 6.12 x 10'4 mg/
L for (co, f) = (0.032.0.47) to 6.02 x 10'4 mg/L for (co, f) = (0.032, 0).
H.4 The Transport of 90Sr with a Kj = 1.0 ml/g
The third case that we considered was the base case scenario for90Sr; this radionuclide has a default Kd of 1.0
ml/g and a half-life of 10,585 days. Thus, one would expect significant decay over the lifetime of the BTCs for this
radionuclide. Figure H-la shows the BTCs for the concentrations of 90Sr in solution for the parameter pairs (co, f) =
(0, 1), (0.032d-!, 0.47), and (0.032d-!, 0). The BTCs for these three parameter pairs are basically equal, which
means that the two "nonequilibrium" cases, (co, f), (0.032d~1, 0.47), and (0.032CT4, 0), are the same as the equilibrium
case (co, f) = (0, 1). Further, Figure H-lb shows that the D/P term is essentially zero for all three parameter pairs (this
is trivially so for co = 0 and f = 1):
D/P = eo[(l -f)Kd c-s] = 0, for all time t. (H-9)
The s-distributions for f = 0 and 0.47 track those of the c-distributions through the relationship s = (1 -f)Kdc =
(1 -f)c.
The results in Table H-l and those shown in Figure H-lab support the claim that nonequilibrium sorption has
very little effect for the low recharge rates considered (0.014 < q < 0.032 cm/d), for the sorption reaction rates which
range over 0.003 < co < 0.32d~1, and for the specific radionuclides considered in this report, with Kd values that range
from 0 to 5 ml/g. This claim is further supported by the magnitude of the coefficients in Equations (H-l) and (H-2).
For example, in the 90Sr case, the coefficients for (co,f) = (0.032d~1, 0.47) are as follows:
(a)
1.5x10-*
o>
1.0x10-*
o 5. 0x10"*
O
HYDRUS MSr
Kd=1.0 ml/g
o = 0.032 f (1=0)
Equilibrium Sorption ff=1)
' (f=0.47)
o = 0.032
20000 40000
Time (days)
60000
80000
(b)
1.0x10-*
8. 0x10-*
6.0x10-*
4.0x10"*
2.0x10"*
Kd=1.0 ml/g
HYDRUS MSr
a = 0.032 dH (f=0)
Equilibrium Sorption (f=1)
a = 0.032 d"1 (f=0. 47)
16000 32000 48000 64000 80000
Time (days)
Figure H-l. The c- and s distribution of Equations (H-l) and (H-2) for 90Srfor (co,f) = (0,1),
(0.032 d'1, 0.47), (0.032d-!, 0). Distributions were derived by the HYDRUS Code, (a) gives the
concentration in solution and (b) gives the concentration on the soil matrix.
H-4
-------
Nonhomogeneous
Equation Radioactive Decay Sorption Decay Sorption Term
Eq. (H-l) -6.55xlO-5c -3.01 x 10'2c +5.67 x !Q-2s
Eq. (H-2) -6.55xlO-5s -3.20xlQ-2s +1.70 x 10'2c
From these coefficients, we note that the radioactive decay term is negligible as compared to the transfer of 90Sr
between solution and soil matrix. Thus, even though the radioactive decay greatly reduces the Cpeak values of the
90 Sr BTCs over their lifetimes of 65,000 to 80,000 days, this decay has little influence on the much faster adsorption/
desorption processes. Further, as shown in Figures H-la and H-lb, the nonequilibrium sorption process is
sufficiently fast that the D/P term goes to zero and the nonequilibrium sites, in reality, become equilibrium sites. A
more detailed analysis is given in the following paragraphs.
H.5 The Transport of 90Sr with a K,, = 1.0 ml/g, f = 0, and Varying Sorption Rates
We have found that the nonequilibrium sorption sites for 99Tc, in essence, acted as equilibrium sites for the low
recharge rates considered in this report when the sorption rates co are taken to be equal to or greater than 0.003d'1. In
addition, the nonequilibrium sorption sites for 90Sr acted as equilibrium sites for a sorption rate co of 0.032d-L.
Therefore, the question that arises is the following: At what sorption rates co will these systems truly possess
nonequilibrium sites that are not in equilibrium balance with the liquid phase concentration? We answered this
question for 90Sr, whose Kd value is taken as 1.0 ml/g and whose decay rate (J, is given as 6.55 x lO^d'1. However,
the reader should be aware that the values of co for which 90Sr possesses "true" nonequilibrium sites in this Las
Graces soil may be physically unrealistic. The current exercise was conducted to both understand this module of the
HYDRUS Code and to indicate the range of values for co that are required to unbalance equilibrium sorption
conditions for the very low recharge rates considered in this report.
For the radionuclide, 90Sr, withaKd= 1.0 ml/g and anf =0 (i.e., no equilibrium sorption sites, only
nonequilibrium sites), Equations (H-l) and (H-2) become:
— + v — = D—- - (AC - — co(c-s), (H-10)
at az az 6
3s .
— + (co + |A)S = coc, (H-ll)
at
where the coefficients (v,D) are defined by
v = q/6, D = TWDW + DL q /0 . (H-12)
Equation (H-ll) is a is a first order, linear, nonhomogeneous equation for s where s(z,0) is taken to be zero, as stated
in Section H.2. Solving Equation (H-ll) results in the following:
t
s(z,t) = co J exp[-(co + n)(t - TI)]C(Z,TI) dr| .
Combining Equations (H-10) and (H-13) produces the following linear, integrodifferential equation for the liquid
phase concentration c:
— +v— =D— +
at az az2
+
t
)[exp[-(co + jj,)(t -r|)l c(z,r|) dr| - c
(H-14)
col .-.-...- - .. -
0
H-5
-------
The integral term in Equation (H-14) radically changes the system defined by Equation (H-4) by introducing a
"history" dependence in the evolution of the liquid phase concentration. This integral term shows that the value of
c(z,t) at any point z in the soil column not only depends on the current time, but also depends on the entire history of
the evolution of c(z,r|) from the initial time r| = 0 to the current time r| = t at the point z in question. As we have seen
in Section H.2, this integral term loses its historic dependence if co is sufficiently large. But what happens if co is
sufficiently small? To answer this question, we let co equal a sequence of values from 6.5 x lO^d'1 to 6.5 x KHd'1,
remembering that [i equals 6.55 x lO^d'1.
Before considering the results obtained from the HYDRUS calculations, two comments are worth emphasizing:
(1) Equations (H-10) to (H-14) represent a "stripped-down" version of the HYDRUS Code used to
simulate the transport of 90Sr with parameters (Kd,f) set equal to (1.0, 0). The HYDRUS Code uses a
variable soil moisture, 9(z), in the simulation, while the "stripped-down" version does not. Equations
(H-10) to (H-14) are only used for explanatory and argumentative purposes.
(2) The "stripped-down" version of the HYDRUS Code (although f need not be zero) has been recently
solved by Drake et al. (2002), though the use of Laplace Transformations, the theory of residues, and
the Bromwich Integral in the complex plane. The z-domain was defined by 0 < z < L, the source at z =
0 was a variable stepwise-continuous function, and the condition at z = L was a variable flux of
pollutants. The exact solution is in the form of a single definite integral involving elementary functions
of the independent variables (z,t) and the system parameters.
Figures H-2a and H-2b depict the c- and s-distributions at the bottom of the 6 m soil column for the sequence of
co values listed in the above paragraph. Figure H-2a gives the usual BTCs for the concentration of 90Sr in solution at
the hypothetical water table (i.e., the bottom of the 6 m column). These BTCs show the usual "bell-shape" for co >
6.5 x lO-M'1 (keep in mind that Figure H-2a is a semi-log plot), but are radically different from the "bell-shape" for
co = 6.5 x lO^d'1 and 6.5 x 10-5cH. For these lowest two values of co, the "history" term in Equation (H-14) has the
greatest effect. Figure H-2b gives the corresponding concentration curves (CCs) for the solid phase at the water table
level of 6 m. As with the BTCs, the CCs show a characteristic "bell-shape" for co > 6.5 x lO^d'1, and a non "bell-
shape" for the lowest two cos. In fact, as co increases in size one can see, by careful analysis of the curves in Figures
H-2a and H-2b, that the BTCs approach the CCs. That is, as co increases, the nonequilibrium sites "convert" to
equilibrium sites, the "history" term in Equation (H-14) becomes less effective, and Equations (H-l) and (H-2)
change over to Equations (H-4) and (H-5).
Table H-2 reinforces the above conclusion. In this table we have listed the peak values of the BTCs and CCs at
the 6 m water table, C k and S k, respectively, and their corresponding times to Tpeak. The values for the BTCs
and CCs are quite different for the lowest two co values, while from co equal to 6.5 x KHd'1 and higher, the values for
the BTCs begin to quickly approach those for the CCs. For co > 0.032d'1, the BTCs and CCs tend to give the same
results. This means that the nonequilibrium sites become equilibrium sites (or act like equilibrium sites) for the low
recharge rates used in this analysis whenever the sorption rate co > 0.032d~1 and the pollutant is 90Sr. For other
pollutants with different Kds and decay rates, \i, this limiting sorption rate will vary, but the general trend is valid.
The final reinforcement of the above conclusion was made by considering the time evolution of the c- and s-
distributions throughout the entire 6 m soil column. Figures H-3 to H-6 give the liquid phase concentrations (the "a"
figures) and the solid phase concentrations (the "b" figures) throughout the soil column (0 for the surface and -600
cm for the water table) for various fixed times and various values of the sorption rate co. Figure H-3 is for a rate co of
6.5 x lO^d'1. It is obvious that the c- and s-distributions are not in balance, and nonequilibrium sorption conditions
do exist. An interesting observation is that the earlier time distributions for c have the classic "bell-shape," while the
later time distributions tend to lose this shape (also see the dotted curve in Figure H-2a). The reason for the non-bell-
shapes is that the "history" term in Equation (H-14) has had more time to accumulate the effects from earlier
distributions and has thus changed the classic diffusion equation. Figure H-4 is for a rate co of 6.5 x lO'M'1. These
distributions also indicate the existence of nonequilibrium conditions for the earlier time curves; while for the later
time curves, the c-distributions are rapidly approaching those of the solid phase, which indicates the approach of
equilibrium sorption at all sites in the soil matrix. Figure H-5 is for a rate co of 6.5 x lO^d'1. As one can easily see,
the shapes of the c-distributions are nearly the same as those for the s-distributions; however, the peaks of the s-
distributions are slightly greater than those of the c-distributions at the earlier times. As time increases the peaks for
both sets of distributions approach one another and the sorption sites approach equilibrium. In addition, both sets of
curves exhibit the classic "bell-shape" which indicates the reduced roll of the "history" term in Equation (H-14). The
H-6
-------
(a)
10-"
10-*
10"*
I/I
(b)
10-*
1.2x10-*
.
-
. Kd=1.0 ml/g
HYDRUS "Sr
B = 6. 5x 1 0-« d-'
a = 6.5x10-* d-(
a = 6. 5x 1 0-* d-1
a =6.5x10-* d"1
B = 6. 5x 1 0-* d'1
B = 6. SxlO-' d"1
f =0
f =0
f =0
f =0 -
f =0
f =0
1.0x10-*
8.0x10-
o>
£
I/I
E
*c
.a
= 6.0x10-*
4.0x10-
2.0x10-
20000 40000
Time (days)
60000
80000
HYDRUS
•°Sr
Kd=1.0 ml/g
= 6. 5x10-* dH
= 6. 5x10-= dH
= 6.5x10-* d-*
= 6.5x10-* d-*
= 6.
f =0
f=0
f=0
f=0
f=0
f =0
20000 40000
Time (days)
60000
80000
Figure H-2. (a) Breakthrough curves for the liquid phase concentration at the 6 m level, (b) Concentration
curves for the nonequilibrium solid phase at the 6 m depth. Curves for co = 6.5 x lO^d'1 and 6.5 x
KHd'1 are basically the same for both (a) and (b).
Table H-2. Liquid Phase and Solid Phase Peak Concentrations at the Hypothetical Water Table for a Sequence
of Sorption Rates, along with the Corresponding Times to Arrive at Those Peaks.
Sorption
Rate, CD
(d-1)
6.5 x 10-6
*6.5 x 10-5
6.5 x 10-4
6.5 x 10-3
**3.2xlO-2
6.5 x lO'2
6.5 x lO'1
Solid Phase Concentration
Speak (mg/kg) Tpeak (d)
6.5 x 10-4
6.7 x 10-4
4.1xlO-4
5.6 x 10-4
6.2 x 10-4
6.2 x 10-4
6.2 x 10-4
6,300
9,500
37,100
43,100
43,500
43,500
43,500
Liquid Phase Concentration
Cpeak(mg/L)
6.3 x lO'2
6.2 x lO'3
4.6 x lO'4
5.7 x lO'4
6.2 xlO'4
6.2 xlO'4
6.2 xlO'4
T ,(d)
peak v '
4,900
4,700
35,200
43,200
43,500
43,500
43,500
Decay Rate p for 90Sr is 6.55 x
Data from Figure H-1.
H-7
-------
final sets of distributions are given in Figure H-6, which are for co = 6.5 x 10-2cH; the curves for co = 6.5 x KHcH
were found to be "graphically" equivalent to those for co = 6.5 x lO^d'1. As one can see, for all times greater than
or equal to 2000 days, the c- and s-distributions are the same, and there are no nonequilibrium sites within the soil
column.
References
Drake, R.L., J-S. Chen and D.G. Jewett (2002). "An exact solution for the assessment of nonequilibrium sorption of
radionuclides in the vadose zone." Waste Management 2002 Conference. February 24-28, 2002, Tucson, AZ.
Available from www.wmsym.org/wm02 (click on WM'02 Proceedings).
Thibodeaux, L.J. 1996. Environmental Chemodynamics: Movement of Chemicals in Air. Water, and Soil. 2nd
Edition. John Wiley & Sons, Inc. New York, NY. 593pp.
Toride, N., F.J. Leij, and M.T. van Genuchten. 1993. "A comprehensive set of analytical solutions for
nonequilibrium solute transport with first-order decay and zero-order production." Water Resour. Res. 29(7),
2167-2182.
van Genuchten, M. Th., and RJ. Wagenet. 1989. "Two-site/two-region models for pesticide transport and
degradation: theoretical development and analytical solutions." Soil Sci. Soc. Am. J. 53, 1303-1310.
(a)
(b)
X. 8.0x10-*
ot
&
g 6.0x10-"
1
§ 4.0x10"*
Q
C
o
o
m 2.0x10-"
1.6x10-"
•c 1.2x10-"
.Q
"5
S 8.0x10-"
o
z
o 4.0x10""
HYDRUS Kd=1.0 ml/g
"°Sr a = 6.5x10"* d~* (f^))
2000 days
3000 days
4000 days
5000 days
-100 -200 -300 -400 -500 -600
Depth (cm)
HYDRUS Kd=1.0 ml/g
'"Sr a = 6.5x10-" f (f=0)
2000 days
3000 days
4000 days
5000 days
10000 days
20000 days
•r'-.T^rsgg™
-100 -200 -300 -400 -500 -600
Depth (cm)
r
HYDRUS Kd=1.0 ml/g
•"Sr a = 6.5x10-* dM (f=0)
2000 days
4000 days
6000 days
8000 days
10000 days
20000 days
-100 -200 -300 -400 -500 -600
Depth (cm)
5. OxIO-2
4. OxIO-2
3. CxIO"2
2.0x10-"
1.0x10-"
HYDRUS
"Sr a = e.
Kd=1.0 ml/g
* d-1 (f=0)
2000 days
4000 days
6000 days
8000 days
10000 days
20000 days
0 -100 -200 -300 -400 -500 -600
Depth (cm)
Figure H-3. Liquid phase concentration curves (a)
and solid phase concentration curves
(b) for 90Sr, for various times and for
co = 6.5 x 10-5d'1, where zero depth
is the surface and -600 cm is the
hypothetical water table.
Figure H-4. Liquid phase concentration curves (a) and
solid phase concentration curves (b) for
90Sr, for various times and for co = 6.5 x
10-4d-!, where zero depth is the surface
and -600 cm is the hypothetical water
table.
H-8
-------
(b)
HYDRUS Kd=1.0 ml/g
ir a = 6.5x10-" d"1 (f=0)
2000 days
4000 days
6000 days
BOOO days
20000 days
40000 days
(a)
-100 -200 -300 -400 -500 -600
Depth (om)
8.0x10-*! 1 1 1 1
HYDRUS Kd=1.0 ml/g
•°Sr a = 6.5X10-1 f (f=0)
2000 days
4000 days
6000 days
c 11 \ 8000 days
£ I \ 20000 days
£ I 40000 days
= 4.0x10-4 '
0 -100 -200 -300 -400 -500 -600
Depth (cm)
Figure H-5. Liquid phase concentration curves (a)
and solid phase concentration curves
(b) for 90Sr, for various times and for co
= 6.5 x lO-M'1, where zero depth is
the surface and -600 cm is the
hypothetical water table.
(b)
HYDRUS Kd=1.0 ml/g
"Sr o = 6.5x10-* d-1 (f=0)
2000 days
4000 days
6000 days
8000 days
20000 days
40000 days
-100 -200 -300 -400 -500 -600
Depth (om)
v? 8. Ox10-*r
oi | HYDRUS
Kd=1.0 ml/g
= 6.5x10-* dH (f=0)
2000 days
4000 days
6000 days
BOOO days
20000 days
40000 days
-100 -200 -300 -400 -500 -600
Depth (om)
Figure H-6. Liquid phase concentration curves (a) and
solid phase concentration curves (b) for 90Sr,
for various times and for co = 6.5 x lO^d'1,
where zero depth is the surface and -600 cm
is the hypothetical water table.
H-9
-------
Appendix I
Results from the Transport and Fate of Other
Radionuclides Not Considered in the Main Text
The fate and transport of radionuclides in a finite soil column (e.g., the five parent radionuclides listed in Section
5.1 passing through the 6 m soil column considered in this report) are strongly influenced by the distribution
coefficient, Kd, the radioactive decay rate, u, and the recharge rate, q. One could imagine that there exists some
decay-mobility scale (DMS) that represents the basic characteristics of a radionuclide as it passes through the finite
column. For example, for the smaller values of the DMS, the radionuclide could be long-lived and highly mobile in
the soil column, and thus have characteristics somewhat similar to a conservative species, Region I of the curve in
Figure 1-1. In this figure, the ordinate is the concentration, C, at the bottom of the soil column, normalized by the
concentration, C0, of the source at the top of the column, the ordinate being denoted as [C/C0]B. The values of
[C/C0]B for Region I of the curve, which are less than one, represent diffusive dilution but very little dilution due to
radioactive decay. For the larger values of DMS, the radionuclides could be short-lived and highly immobile in the
soil column, thus producing small values of [C/C0]B as shown in Region III of the curve in Figure I-1. The
radionuclides with intermediate half-lives and values of Kd would be characterized by the intermediate values of
DMS and Region II of the curve. However the exact form of the DMS is unknown, but its qualitative properties are
as represented in Figure 1-1. One reason a DMS is difficult to quantify is the fact that Kd is a lumped parameter
which represents many unknown geochemical processes in natural soil columns. Thus, a Kd value for a given
radionuclide is not unique but is highly dependent on soil structure, texture, and geochemistry (U.S. EPA 1999a,b;
2000a,b). Consequently, a DMS value for a given radionuclide would not be expected to be unique either. But, the
qualitative trends stated above, and shown in Figure 1-1, are potentially correct, and have value in relating diffusive
dilution and radioactive decay to pollutant travel time.
Decay-Mobility Chart
(Arbitrary Scale)
DMS
Figure 1-1. The normalized concentration of a radionuclide at the bottom of a soil column (the source being at the
top of the column) versus a hypothetical decay-mobility scale (DMS), where I represents highly mobile,
long-lived species, III represents highly immobile, short-lived species, and II represents species with
intermediate mobilities and half-lives.
1-1
-------
The five parent radionuclides given in Section 5.1, along with their default Kd values and their half-lives listed in
Figure 5-1, can be located "on the curve" in Figure 1-1 for this report's 6 m, homogeneous soil column. These
proposed locations are as follows:
"Tc and 238U are represented by Region I;
Travel time through the 6 m column for 238U is about 5 times greater than that for "Tc, thus diffusive
dilution is much greater for 238U than for "Tc;
3H is represented by the left-hand side of Region II;
90 Sr is represented by the right-hand side of Region II; and
238pu is represented by Region III.
This appendix reports the results of two sets of simulations illustrating the validity of the placement of the five
parent radionuclides on the curve in Figure I-1. Using the CHAIN and FECTUS Codes, the differences between the
transport and fate of a parent and that of a daughter are displayed for "Tc and its daughter, ruthenium "Ru, for the
6 m, homogenous soil column. The other set of calculations used the CHAIN Code to compute the Cpeak - and Tpeak-
values for the breakthrough curves (BTC's) at the 6 m level, for each of the five radionuclides under the influence of
a range of recharge rates, thus verifying the proposed locations of the radionuclides "along the curve" in Figure I-1.
LI The CHAIN Governing Equations
The CHAIN Equations covering the transport and fate for both these comparative exercises are as follows:
at e + pKd
eo a2c
+ pK, 3z2
at
9D
pic; az2
- u c
,
+
e + PKd
- ^-Hr
6 + pK*d
(1-2)
where c is the concentration of the parent in mg/L and the asterisked (*) quantities represent those for the daughter.
Using the base values for (q, 9, p, D) given in Section 6, the default values of Kd for the radionuclides given in
Section 5, and the half-lives given in Figure 5-1, the coefficients for the advection term, the diffusion term, and the
sink term in Equation (I-1) can be evaluated. These coefficients for the five radionuclides are given in Table I-1.
The relationship between \a and the half-life, t1/2, is given by:
H= (Ioge2)-t
1/2
(1-3)
Table 1-1 shows that as Kd increases from 0 to 5 ml/g, the seepage velocity (advection term coefficient)
decreases by a factor of 54, as does the diffusion term coefficient. Even though the diffusive mechanism decreases
greatly, it may still be very important because the great decrease in seepage velocity allows the diffusion to act over a
longer time, as exemplified by the following:
Radionuclide
3H
"Tc
238U
90Sr
238pu
(600 cm) -=- (Advection
Term Coefficient)
4,000 d
4,300 d
21,000 d
46,500 d
216,600 d
1-2
-------
Table 1-1. Coefficients of the Advection, Diffusion, and Sink Terms in Equation (1-1).
Radionuclide
Tritium 3H
Technetium "Tc
Uranium 238U
Strontium 90Sr
Plutonium 238Pu
Initial
Concentration
In Recharge
Water Cg
(mg/L)
l.lxlO-7
1.25 X ID'2
417
2.0 x ID'1
1.0 x 10-7
Advection Term*
Default
Value
Kd
(ml/g)
0
0.007
0.4
1
5
Half-Life
ty
(d)
4380
76,650,000
1.6425xl012
10,585
32,120
First-Order *
Decay Rate
M
(a-1)
1.5825xlO-4
9.043xlO-9
4.2201xlO-13
6.5484xlO-5
2.1580xlO-5
Coefficient
q
e + PKd
(cm/d)
0.15000
0.13962
0.02857
0.01290
0.00277
Diffusion Term*
Coefficient
6D
e + PKd
(cm2/d)
1.00000
0.93077
0.19048
0.08602
0.01848
q, 0, p, and D are assigned the base values in Section 6;
Kd is assigned values given in Column Three;
|i is obtained from half-life t1/2 using Equation (1-3).
Comparing the times, (600 cm + advection term coefficient), with the half-lives listed in Table 1-1, one observes
that the first-order decay rate [i will probably be most important for the transport and fate of 238Pu through the 6 m
layer, followed by that of 90Sr, followed by that of 3H. Because of the very long half-lives for 99Tc and 238U, their
decay rates are not very important over the 6 m layer with respect to the BTCs of these parents. However, with
respect to the production of the daughter, 99Ru from 99Tc and 234U from 238U, these very small decay rates may be
important.
1.2 Breakthrough Curves for 99Tc and It's Daughter, 99Ru
For the decay chain, 99Tc -> 99Ru, 99Tc has a very low decay rate \i which has negligible effect on the ETC for
99Tc over the 6 m depth of the current simulation. However, this low decay rate does produce a small amount of the
stable radionuclide, 99Ru, while the 99Tc moves from its surface source (of a 1,000 days duration) down through the
6 m soil column. The lifetime of the moving source of 99Ru equals the lifetime of the 99Tc BTCs for the given
recharge rate q = 0.0024 cm/d. This lifetime is equal to or less than 10,000 days while the half-life of 99Tc is about
76,650,000 days. The travel time for 99Ru through the layer is much greater than that for 99Tc because the Kd for
99Ru is 5 ml/g versus that of 0.007 ml/g for 99Tc. However, the travel time for 99Ru through the 6 m layer should be
on the same order as that of 238Pu, namely about 200,000 days, since the Kd values of 99Ru and 238Pu are taken to be
equal. However, 238Pu has a significant decay rate (238Pu -> 234U), while 99Ru is stable; |J* in Equation (1-2) is zero
for 99Ru. Since FECTUZ has a different 9D-value than CHAIN (see Equation 1-2), these two codes were used to see
the effects of diffusion on the BTCs of 99Ru for the 6 m layer. The diffusion term coefficient for FECTUZ is about
2/3 that of CHAIN.
The relative magnitude of the concentration c* of 99Ru produced by the decay of 99Tc can be obtained from
Equation (1-2) if one ignores all terms except the source term and the time derivative of c*:
C* = 1.8 x 10'
J c(t)dt .
(1-4)
The integral term in Equation (1-4) can be approximated from the ETC for 99Tc under the influence of a 0.024
cm/d recharge rate, as given in Figure I-2b. We can roughly approximate the area under this curve as given by 0.001
mg/L times 10,000 d. Thus, the concentration of 99Ru at 10,000 d is equal to 1.8 x 10'9 mg/L by Equation (1-4). The
FECTUZ/CHAIN result given in Figure I-2a is very nearly equal to this value at t = 10,000 d. Thus, over the first
10,000 d of the evolution of 99Ru, the dominant term in Equation (1-2) is the source of 99Ru being produced by the
decaying 99Tc.
1-3
-------
(a)
o>
E
c
o
c
-------
parent at the top of the column. However, the evolutions of parent and daughter species can be quite complex in
two-dimensional and three-dimensional, heterogeneous soil columns, as pointed out in Section 3.6 (Oldenburg and
Pruess, 1996).
1.3. The Sensitivity of the Five Parent Radionuclides to Recharge Rate
Figures 1-3 (a to e) illustrate the sensitivity of the transport and fate of the five parent radionuclides through the 6
m, homogenous soil column to a variable recharge rate, q, using the CHAIN Code for the simulation. The five sets
of curves are given in order of increasing default Kd values. As indicated by Equation (1-1), these Kd values govern
the rates of pollutant passage through the 6 m layer. In fact, a good representation of characteristic times of passage
(a)
5. On 10-"
•.3.75K10-"
2.5tt10-°
R.ohorg. Rat* (om/d)
0.014
CHAIN
Tim* (day*)
(d)
2. OilO
"Sr
(b)
0.010
I
I
'; 0.004
i
0.002
"To
Rtoharg* Rot* (om/d)
0.014
0.024
0.032
CHAIN
Tim* (days)
(e)
(c)
100.000
?
I
| 40.000
Rcoharg* Rat* (om/d)
0.014
0.024
0.032
CHAIN
2.0»fO
Raoharge Roto (om/d)
0.014
0.024
0.032
CHAIN
Tlm« (days)
Rwhargc Roto (om/d)
= 8:811
0.032
CHAIN
Tim. (days)
Tim* (days)
Figure 1-3. Sensitivity of radionuclide transport through the unsaturated zone to recharge rate (q) using the CHAIN
Model: (a) Tritium 3H; (b) Technetium "Tc; (c) Uranium 238U; (d) Strontium 90Sr; and (e) Plutonium
238Pu.
1-5
-------
can be obtained from the quotient 600 cm + v, where v is the seepage velocity, or coefficient of the advection term in
Equation (I-1):
v =
q
PKd
(1-5)
These characteristic times are listed in Table 1-2 for the five radionuclides and the three recharge rates q. Also listed
in this table are the normalized peaks of the BTCs shown in Figures 1-3 (a to e). These peaks are normalized by the
respective source concentrations of the radionuclides given in Table 1-1. Corresponding to these normalized C k
values are the times to peak concentration, T k, given in the last column. One should note the rather close
correspondence between the "characteristic time" values and the values of T k given in Table 1-2. The final column
to note in this table is the "type of decay." The descriptors given in the table can be made more explicit from the log-
log plot of Cpeak ^ C0 versus Tpeak in Figure 1-4.
To further illuminate the results in Table 1-2 and Figure 1-4, we introduce a relative diffusion factor (DIP) and a
relative decay factor (DEC). The value of DIP is defined as the quotient of the species Tpeak for q = 0.024 cm/d
divided by the Tpeak for the highly mobile radionuclide, 3H, at a q = 0.024 cm/d. The value of DEC is defined as the
quotient of the species Tpeak for q = 0.0024 cm/d divided by the species decay half-life. Table 1-3 lists these factors
for the five parent radionuclides. As DIP increases, the amount of diffusive dilution increases, while an increase in
DEC produces an increase in the amount of radioactive decay. Using these two factors, the locations of the five
curve segments in Figure 1-4 can be easily explained:
The diffusive dilutions of 3H and "Tc are nearly equal since their DIP values are nearly equal, and the
time ranges of their curve segments are nearly the same.
The curve segment for 3H is below that for "Tc because 3H has significantly decayed over the time
interval, while the decay of "Tc has been insignificant, although not zero as seen in Figure I-2a.
The curve segment for 238U, as that of "Tc, has only experienced insignificant decay, but is lower than
the curve segments for "Tc and 3H, and further to the right, because its DIP is almost five times that of
"Tc and 3H. The 238U's normalized concentrations are lower only because of diffusive dilution.
Table 1-2. Cpeak Normalized by Source Concentration C0 and Tpeak for Each Radionuclide and
for Three Recharge Rates.
Radionuclide
Tritium 3H
Technetium "Tc
Uranium 238U
Strontium 90Sr
Plutonium 238Pu
*Type
of Decay
Significant
Very Little
Very Little
Very Significant
Very Significant
*Type of Decay with respect **
to the 6 m soil column.
Recharge * * Characteristic Time
Rate q (600cm) + v
(cm/d) (d)
0.014
0.024
0.032
0.014
0.024
0.032
0.014
0.024
0.032
0.014
0.024
0.032
0.014
0.024
0.032
v = seepage velocity
6860
4000
3000
7370
4300
3225
36,000
21,000
15,750
79,715
46,500
34,875
371,145
216,600
162,375
q
9 + PK,
* "Normalized
peak
C *+C
peak o
1.72x10-'
3.23 x lO'1
4.18x10-'
4.49 x 10-1
5.64 x 10-1
6.31x10-'
9.98 x 10-2
1.29x10-'
1.48x10-'
4.47 x 10-4
3.28 xlO'3
7.44 x 10-3
1.66xlO-5
2.20 x 10-4
6.48 x 10-4
T ,
peak
(d)
7300
4400
3300
8000
4750
3560
35,500
21,000
15,900
66,100
43,100
33,200
287,000
194,000
152,000
*** C = Concentration at Source,
in Table I- 1.
1-6
-------
10 10 10 10
Time to Peak Concentration (Tpeak In days)
Figure 1-4. Cpeak -^ C0 versus Tpeak in days for five radionuclides, showing the effects of various Kd values and
various decay rates, where the time intervals of the curve segments are determined by the range of
discharge rates for the 6 m soil column.
The curve segments for 90Sr and 238Pu are both much lower and to the right of those 3H, "Tc and 238U
because their DEC values are much larger and their DIP values are also much larger. This means that
their normalized concentrations are lower than the other three species because of both greater diffusion
and greater decay.
The curve segment for 238Pu is lower and further to the right of that of 90Sr since its DIP is four and
one-half times that of 90Sr and its DEC is about one and one-half times that of 90Sr. This means that
over comparable time segments (i.e., the Tpeak values for the range of q for the 6 m soil column) more
of the 238Pu has diffused and decayed than that of 90 Sr.
Table 1-3. The Relative Diffusion Factors and the Relative Decay Factors for the Five Parent Radionuclides.
For Recharge Rate, q = 0.024 cm/d
Radionuclide
Tritium 3H
Technetium "Tc
Uranium 238U
Strontium 90Sr
Plutonium 238Pu
*Type
of Decay
Significant
Very Little
Very Little
Very Significant
Very Significant
Relative Diffusion
Factor
1.000
1.080
4.773
9.795
44.091
Relative Decay
Factor
1.005
0.000
0.000
4.072
6.040
*Type of Decay with respect to the 6 m soil profile.
1-7
-------
References
Oldenburg, CM. and K. Pruess. 1996. "Mixing with first-order decay in variable-velocity porous media flow."
Transport in Porous Media. 22:161-180.
U.S. EPA. 1999a. Understanding Variations in Partition Coefficients. Kd. Values: Model. Methods of
Measurement, and Application of Chemical Reaction Codes. Volume I. EPA 402/R-99/004A. Office of
Radiation and Indoor Air & Office of Solid Waste and Emergency Response. U.S. EPA. Washington, DC.
U.S. EPA. 1999b. Understanding Variations in Partition Coefficients. Kd. Values: Review of Geochemistry and
Available Kd Values for Cadmium. Cesium. Chromium. Lead. Plutonium. Radon. Strontium. Thorium. Tritium.
(3HX and Uranium. Volume II. EPA402/R-99/004B. Office of Radiation and Indoor Air and Office of Solid
Waste and Emergency Response. U.S. EPA. Washington, DC.
U.S. EPA. 2000a. Soil Screening Guidance for Radionuclides: User's Guide. EPA 540/R-00/007. Office of
Radiation and Indoor Air and Office of Solid Waste and Emergency Response. U.S. EPA. Washington, DC.
U.S. EPA. 2000b. Soil Screening Guidance for Radionuclides: Technical Document. EPA 540/R-00/006. Office
of Radiation and Indoor Air and Office of Solid Waste and Emergency Response. U.S. EPA. Washington, DC.
1-8
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