United States 

Environmental Protection 

Agency 

Simulating Radionuclide 

Fate and Transport in the 

Unsaturated Zone: 

Evaluation and Sensitivity 

Analyses of Select 

Computer Models 
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
 
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. 

ii 
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 

iii 
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 
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
 

Section 3 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
 

Section 5 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 Cruces 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
 

v 
Number Page 

6.4 Output Parameters Evaluated .......................................................................................................... 51
 

6.5 Sensitivity Results for the HYDRUS Model .................................................................................. 53
 

Section 7 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
 

Section 8 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-1
 

References ...................................................................................................................... A-5
 

Appendix B	 A Discussion on the Scaling of Field Soil-Water Behavior............................................ B-1
 

B.1 Symmetry in Nature ......................................................................................... B-1
 

B.2 Similitude, Transformation Groups, Inspectional Analysis, Self-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-1
 

C.1 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-11
 

Appendix D The First-Order Decay Chains Used in the Various Models .......................................... D-1
 

References ...................................................................................................................... D-4
 

Appendix E The Impact of Using a Nonuniform Moisture Distribution versus a
 

Uniform Distribution .......................................................................................................E-1
 

Appendix F	 The Impacts of Using Daily Precipitation Rates and Daily Evapotranspiration Rates
 

versus an Annual Average Recharge Rate ....................................................................... F-1
 

References ....................................................................................................................... F-7
 

Appendix G	 The Impact of Considering a Layered Soil Column versus a Homogeneous
 

Soil Column .................................................................................................................... G-1
 

References ...................................................................................................................... G-2
 

Appendix H	 A Detailed Analysis of Nonequilibrium Sorption of Pollutants, Mainly for the
 

Radionuclide 90Sr ........................................................................................................... H-1
 

H.1 A Simplified Version of the Transport Equations ............................................ H-1
 

H.2 The Transport of 99Tc with a Kd = 0.007 ml/g ................................................. H-2
 

H.3 The Transport of 99Tc 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 ................................................................................................................... I-1
 

I.1 The CHAIN Governing Equations ..................................................................... I-2
 

I.2 Breakthrough Curves for 99Tc and It’s Daughter, 99Ru ..................................... I-3
 

I.3. The Sensitivity of the Five Parent Radionuclides to Recharge Rate .................. I-5
 

References ........................................................................................................................ I-8
 

vi 
Figures 

Number Page 

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 (αh,αθ, αk) = (3/2, 2, 7/4). The ratios
 

hs/hs

* = h1/h1* = 3/2, (θs- θr

) ÷ (θs

*- θr*) = (θ1- θr

) ÷ (θ1

*- θr*) = 2, and K1

/K1

* = Ks

/Ks

*

 = 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 Frp 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 Cruces Site, NM. PET is
 

calculated from daily climate data using Penman’s equation (Jensen et al., 1990) ............................ 43
 

Figure 6-1. Sensitivity of 99Tc 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 α,
 

(k) van Genuchten retention parameter β ........................................................................................... 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 α, (k) van Genuchten retention parameter β....................................................... 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 α, (k) van Genuchten retention parameter β .................................... 60
 

vii 
Number Page 

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, (j) van Genuchten retention parameter 

α, (k) van Genuchten retention parameter β....................................................................................... 63 

Figure 7-1.
	 Sensitivity of 99Tc 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 99Tc 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 99Tc breakthrough (through the 6m layer) to the water content using the CHAIN, 

MULTIMED-DP 1.0, FECTUZ, CHAIN 2D, and HYDRUS Models, where θ = 0.22 cm3/cm3, 

0.16 cm3/cm3, and 0.10 cm3/cm .......................................................................................................... 69 

Figure 7-4.
	 Sensitivity of 99Tc breakthrough (through the 6m layer) to the bulk density using the CHAIN, 

MULTIMED-DP 1.0, FECTUZ, CHAIN 2D, and HYDRUS Models, where ρ = 1.78 g/cm3, 

1.70 g/cm3, and 1.62 g/cm3 ....................................................................................................................................................................................................70 

Figure 7-5.
	 Sensitivity of 99Tc breakthrough (through the 6m layer) to the dispersion coefficient using 

the CHAIN Model, where D = 2.20 cm2/d, 1.00 cm2/d, and 0.40 cm2/d ............................................ 71 

Figure 7-6.
	 Sensitivity of 99Tc 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 99Tc 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 99Tc 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 99Tc breakthrough (through the 6m layer) to the saturated water content using the 

MULTIMED-DP 1.0, FECTUZ, CHAIN 2D and HYDRUS Models, where θs = 0.35 cm3

/cm3, 

0.32 cm3/cm3, and 0.29 cm3/cm3 .......................................................................................................................................................................................75 

Figure 7-10.
	 Sensitivity of 99Tc breakthrough (through the 6m layer) to the residual water content using the 

MULTIMED-DP 1.0, FECTUZ, CHAIN 2D and HYDRUS Models, where θr = 0.103 cm3

/cm3, 

0.083 cm3/cm3, and 0.063 cm3/cm3 ................................................................................................................................................................................76 

Figure 7-11.
	 Sensitivity of 99Tc breakthrough (through the 6m layer) to the van Genuchten retention 

parameter α using the MULTIMED-DP 1.0, FECTUZ, CHAIN 2D and HYDRUS Models, 

where α = 0.059cm-1, 0.055 cm-1, and 0.051 cm-1 ............................................................................. 77 

Figure 7-12.
	 Sensitivity of 99Tc breakthrough (through the 6m layer) to the van Genuchten retention 

parameter β using the MULTIMED-DP 1.0, FECTUZ, CHAIN 2D and HYDRUS Models, 

where β = 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 

viii 
� �

Number Page 

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 α 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 β 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 (�) with the other three models .......................................................................... 93 

Figure 7-26. Sensitivity of 99Tc 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 99Tc BTCs for the HYDRUS, CHAIN 2D and CHAIN Models for the 

base values of the input parameters with the BTC 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-1. Schematics of the soil-water retention curve (a) and the hydraulic conductivity function 

(b) for the VC-Model (from Simunek et al., 1998) .......................................................................... A-3 

Figure B-1. Geometrically similar figures, where (a) is the reference figure with characteristic length L*, 

(b) is a similar figure with characteristic length L1, L* = α1L1, with scale factor α1 = 2, and 

(c) is a similar figure with α2L2 = L*, α2 = 1⁄2 (from Guymon, 1994)............................................. B-3 

Figure B-2. The depiction of a set of vertical soil profiles p1,p2, ... distributed over a field mapping unit, 

where z represents the local variable within a soil profile and Ri (i = 1,2, ...) represent the 

horizontal vectors in the xy-plane giving the global positioning of the vertical profiles, 

p1,p2, ... ............................................................................................................................................. B-5 

ix 
Number Page 

Figure B-3. (a) Unscaled observations of Seo(h), (b) Scaled observations of Seo(h*), showing the 

reference relationship as a solid curve, (c) unscaled 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 

sm1, 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-1. 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-1. 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 θ = 0.16 

throughout the soil column. There are no CHAIN results in the top graph because θ can 

only be constant in this model ...........................................................................................................E-3 

Figure E-3. Sensitivity of 99Tc 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 99Tc 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-1. The annual precipitation amounts and the monthly average amounts in centimeters for the 

Las Cruces, 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 

(Šimúnek, 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, 

(h1, h2, h3). The precipitation/PET segment from “a to b” is repeated from “b to c.” ..................... F-4 

Figure F-4. Comparison of predicted 99Tc 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, 

(h1,h2 ,h3 ) .......................................................................................................................................... F-6 

Figure G-1. Sensitivity of 99Tc 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-1. The c- and s distribution of Equations (H-1) and (H-2) for 90Sr for (ω,f) = (0,1), 

x 
Number Page 

(0.032 d-1, 0.47), (0.032d-1, 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 ω = 

6.5 x 10-2d-1 and 6.5 x 10-1d-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 ω = 6.5 x 10-5d-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 ω = 6.5 x 10-4d-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 ω = 6.5 x 10-3d-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 ω = 6.5 x 10-2d-1, where zero depth is the surface and -600 cm is the 

hypothetical water table. ................................................................................................................... H-9 

Figure I-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 I-2. (a) Breakthrough curves of the daughter product, 99Ru, from the 99Tc decay using the 

CHAIN and FECTUZ Models for base case simulation, and (b) the breakthrough curve 

for 99Tc. The text will explain a, b, c, d, e and f. ............................................................................... I-4 

Figure I-3. Sensitivity of radionuclide transport through the unsaturated zone to recharge rate (q) 

using the CHAIN Model: (a) Tritium 3H; (b) Technetium 99Tc; (c) Uranium 238U; 

(d) Strontium 90Sr; and (e) Plutonium 238Pu. ..................................................................................... I-5 

Figure I-4. Cpeak ÷ Co 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. ............................................................................... I-7 

xi 
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’s VISITT Database) ......................................................................................................... 39
 

Table 5-2. Soil Hydraulic Properties at the Las Cruces Trench Site for SSG Model
 

Evaluation Study (from Wierenga, et al., 1991) ................................................................................. 42
 

Table 5-3. Characteristics of the Las Cruces Trench Site for SSG Model Evaluation Study (from
 

Gee, et al., 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, θ(z), Varies with the Changing van Genuchten Parameters (Ks,θs,θr, α, ß) .......................... 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
 

for HYDRUS, 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 (�) 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, θD, as Derived from Equation (6-4) and the
 

Base Values of D, DL, Dw, θ 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 99Tc 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 99Tc BTCs and Output Sensitivities with Respect
 

to Model Input Parameters and Modeling Codes ............................................................................. 101
 

xii 
Number Page 

APPENDICES 

Table E-1	 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. .....................................................................................................................................E-3
 

Table H-1.	 Comparison of Nonequilibrium and Equilibrium Results for 99Tc 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 I-1. Coefficients of the Advection, Diffusion, and Sink Terms in Equation (I-1). .................................... I-3
 

Table I-2. Cpeak Normalized by Source Concentration C0 and Tpeak for Each Radionuclide and for
 

Three Recharge Rates. ........................................................................................................................ I-6
 

Table I-3. The Relative Diffusion Factors and the Relative Decay Factors for the Five
 

Parent Radionuclides. ......................................................................................................................... I-7
 

xiii 
xiv� 
Section 1 

Introduction 

This report is concerned with the evaluation and sensitivity analysis of 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 models reviewed and evaluated in the current report form a small 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-specificcircumstances.


1.1. The Radionuclide SSL Effort 

The U.S. EPA developed the Soil Screening Guidance for Radionuclides (U.S. EPA, 2000ab) 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, SSLs for radionuclides in soil, thus


allowing them 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 require further Federal action. Generally speaking, at sites where radionuclide


concentrations fall below SSLs, no further action or study is warranted under the Comprehensive Environmental


Response, Compensation and Liability Act (CERCLA). Where radionuclide concentrations 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 concentrations (Figure 1-1).


The level of concern associated with the concentrations in the figure depends on the likelihood of exposure to


radioactive soil contamination at levels of potential concern to human 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


may not be warranted. For sites requiring cleanup, the goal or level of cleanup, SSCG, must fall below RL and be


such that no further action is required.


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

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), Chilès 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 Flühler (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 Schoot (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. 

2
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 fieldmeasured pressure heads, flows and/or concentrations, thus establishing that the model can reproduce fieldmeasured 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 

3
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 

following 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 

4
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 Cruces 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 – Empirical models of the unsaturated soil hydraulic properties which are used in the 

various models. 

• Appendix B – A discussion on the scaling of field soil-water behavior. 

• Appendix C – An explanation of the hysteretic characteristics of soil-water properties. 

• Appendix D – The first-order decay chains used in the various models. 

• Appendix E – The impact of using a nonuniform moisture distribution versus a uniform distribution. 

• Appendix F – The impact of using daily precipitation and evapotranspiration rates versus an annual 

average recharge rate. 

• Appendix G – The impact of considering a layered soil column versus a homogeneous soil column. 

• Appendix H – 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. 

5
 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 

et al. (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 99Tc, 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? 

6
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 onedimensional 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 onedimensional 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 (Šimúnek et al., 1998), 

• Two-dimensional CHAIN 2D Code (Šimúnek 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 timedependent 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 firstand 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 onedimensional model. However, if the region to be simulated is rather heterogeneous and nonsymmetrical in the 

7
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 onedimensional 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 (Šimúnek 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: 

∂h  

q = -K 

 ∂z

 + 1 

 , (2-1) 

and by the continuity of mass equation 

∂θ 

+ 

∂q

 = -S , (2-2) ∂t ∂z 

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), θ = θ(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): 

∂θ ∂   ∂h  

∂t

 = 

∂z 

K 

 ∂z 

+ 1 





 - S . 

For heterogeneous layers with time-independent properties, the soil hydraulic functions 

follows (Šimúnek et. al. 1998): 

�������������������, 

K = K(z,t) = K(h,z) = Ks (z) Kr (h,z) , 

(2-3) 

and K can be written as 

(2-4a) 

(2-4b) 

where Ks(z) is the saturated hydraulic conductivity (LT-1), and Kr(h,z) is the relative hydraulic conductivity 

(dimensionless). For homogeneous layers with time-independent properties, the parameters become: 

������������ ��������������������� ���� . (2-5) 

The moisture sink due to root water uptake is expressed as follows (Šimúnek et al., 1998): 

S(z,t) = �����φ ���p �������� (2-6) 

where α(h,h�) is the water stress response function (dimensionless), Sp(z,t) is the potential water uptake rate (T-1), 

and h� is the osmotic head (L) which is a linear function of the concentrations of all solutes in the soil moisture. Two 

8
Table 2-1. Comparison of the Model Components in Each of the Five Codes Being Analyzed in this Report. 

9

MODEL COMPONENT HYDRUS CHAIN 2D FECTUZ MULTIMED

_

DP 1.0 

Variably Saturated Water 

Flow 

One-Dimensional Time Dependent Richards 

Equation with Sink Term 

Two-Dimensional Time Dependent Richards 

Equation with Sink Term 

One-Dimensional Steady State Constant Infiltration Rate 

Darcy’s Law 

One-Dimensional Steady State 

Constant Infiltration Rate 

Darcy’s Law 

One-DimeSteady Sta

Constant FWater ConThrougho

Root Water Uptake Sink in Richards Equation 

Which Depends on Osmotic Head, Water 

Stress, Root Distribution 

Sink in Richards Equation Which Depends on Osmotic Head, Water 

Stress, Root Distribution 

Not Considered in Model To Some Extent, 

Considered in the Landfill 

Module, Not in the Unsaturated Flow Zone 

Module 

Not Consi

Unsaturated Hydraulic Properties 

See Appendix A Brooks-Corey, van 

Genuchten, and VogelCislerová Models 

See Appendix A 

Vogel-Cislerová and van 

Genuchten Models 

See Appendix A 

van Genuchten Model 

See Appendix A 

Brooks-Corey and van Genuchten Models 

Not Consi

Scaling of Soil Hydraulic 

Functions 

See Appendix B Linear Scaling of Pressure 

Head, Soil Moisture, and 

Hydraulic Conductivity 

See Appendix B 

Linear Scaling of Pressure 

Head, Soil Moisture, and 

Hydraulic Conductivity 

Not Considered in Model Not Considered in Model Not Consi

Scaling for Temperature 

Dependence of Soil Hydraulic Functions 

Scaling Due to Parameter 

Variations with Temperature Applied to Pressure Head and 

Hydraulic Conductivity 

Scaling Due to Parameter 

Variations with Temperature Applied to Pressure Head and 

Hydraulic Conductivity 

Not Considered in Model Not Considered in Model Not Consi

Hysteresis in Soil Hydraulic Functions 

See Appendix C Applied to van Genuchten and Vogel-Cislerová 

Models for Moisture 

Content and Hydraulic Conductivity 

Not Considered in Model Not Considered in Model Not Considered in Model Not Consi

Initial and Boundary 

Conditions for the Water 

Flow System 

Initial Condition on Pressure Head; Head and Flux Boundary Conditions; 

Ponding and Seepage 

Initial Condition on Pressure Head; Head and Flux Boundary Conditions; 

Ponding and Seepage 

Steady State, Thus no 

Initial Condition; Pressure 

Head Zero at Water Table, 

Pressure Head Consistent 

with Infiltration Rate at 

Surface 

Steady State, Thus no 

Initial Condition; Pressure 

Head Zero at Water Table, 

Pressure Head Consistent 

with Infiltration Rate at 

Surface 

Not Requ

Moisture 

Specified 
Table 2-1. Comparison of the Model Components in Each of the Five Codes Being Analyzed in this Report. 

MODEL COMPONENT HYDRUS CHAIN 2D FECTUZ MULTIMED_DP 1.0 

Solute Transport Equations See Appendix D 

One-Dimensional, TimeVarying; Gas, Liquid and 

Solid Phases; 0th Order 

Production, 1st Order Decay; 

Advection/ Dispersion; 

Equilibrium/ Nonequilibrium 

Sorption, Nonlinear Sorption 

See Appendix D Two-Dimensional, TimeVarying; Gas, Liquid and 

Solid Phases; 0th Order 

Production, 1st Order Decay; 

Advection/ Dispersion; 

Equilibrium/ Nonequilibrium Sorption, Nonlinear Sorption 

See Appendix D One-Dimensional, TimeVarying; Liquid and Solid 

Phases; 1st Order Decay; 

Advection/Dispersion; 

Linear/Nonlinear Equilibrium 

Sorption 

See Appendix D One-Dimensional, TimeVarying; Gas, Liquid and 

Solid Phases; 1st Order Decay; Advection/ Dispersion; Linear 

Equilibrium Sorption 

See AppeOne-Dim

Varying; 

Phases; 1

Advectio

Equilibri

Initial and Boundary 

Conditions for the Solute 

Transport Equation 

Initial Conditions for Liquid 

and Solid Phases; Dirichlet, 

Cauchy and Neumann Boundary Conditions; Volatile 

Solutes at Surface 

Initial Conditions for Liquid 

and Solid Phases; Dirichlet, 

Cauchy and Neumann Boundary Conditions; Volatile 

Solutes at Surface 

Initial Concentration for 

Liquid Phase; Dirichlet and 

Cauchy Boundary Conditions; 

Batemann Decaying Source Terms 

Initial Concentration for 

Liquid, Gas and Solid Phases; 

Dirichlet and Cauchy, Separate 

and Mixed, Boundary 

Conditions 

Initial Co

Liquid P

Cauchy B

General a

Decaying 

Effective Dispersion 

Coefficients 

Solute Dispersion Coefficient; 

Gas Diffusion Coefficient; 

Tortuosity Factors 

Solute Dispersion Matrix in 

Two-Dimensions; Gas 

Diffusion Coefficient; 

Tortuosity Factors 

Solute Dispersion Coefficient; 

Effective Molecular Diffusion 

Coefficient 

Solute Dispersion Coefficient; 

Gas Diffusion Coefficient; 

Tortuosity Factor 

Solute D

Temperature Dependence of 

Transport and Reaction 

Coefficients 

Modified Arrhenius Equation 

for Temperature Scaling of 

Production, Degradation, and 

Adsorption Coefficients 

Modified Arrhenius Equation 

for Temperature Scaling of 

Production, Degradation, and 

Adsorption Coefficients 

Not Considered in Model Hydrolysis Rates Temperature 

Corrected Using Arrhenius 

Equation 

Not Con

Heat Transport Equations One-Dimensional TimeVarying, Convection, 

Dispersion Equation; Energy 

Uptake by Roots; Thermal 

Conductivity Coefficient; Heat 

Capacities 

Two-Dimensional TimeVarying, Convection, Dispersion Equation; Energy 

Uptake by Roots; Thermal 

Conductivity Matrix in TwoDimensions; Heat Capacities 

Not Considered in Model Not Considered in Model Not Cons

10
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 α(h,h�) 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 LT-1 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, �(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 Cislerová (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 MorelSeytoux, et al., though not proved, that this interchange of models would be valid for drainage and evapotranspiration 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, θ(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 

self-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 θ in the Miller and Miller Model by ratio θ/θs, where θs is the 

saturated moisture content. In essence, θs becomes a new scale factor which accounts for the different internal 

11
�

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 = � � � (2-7) h

* * * ��������r ����� 

� �� ������

 � (2-8) 

* * K(h) = � � �� � , (2-9) k

where θr is the residual soil moisture, the asterisks denote reference quantities, and the three scale factors, 

(αh,αθ,αk), 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 (αh,αθ,αk) = (3/2, 2, 7/4). Thus, the above transformations become 

h = 3h*/2, (2-7a) 

θ(h) - θr = θ(3h*/2) - θr =2 [ θ* (h*) - θr 

*] , (2-8a) 

K(h) = K(3h*/2) = 7/4 K*(h*). (2-9a) 

The values of the scaling factors, (α h, αθ, α 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 Cruces 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, μ (ML-1T -1), and the density, ρ(ML-3), of the soil water. The temperature 

scaling factors are thus defined by: 

��� � a * * h(Ta ) = * h = � � � h (2-10)

* 

� ��� � a * * K(���a ��� * � �������k � ���� (2-11) ��� � � a 

where �(Ta) is the surface tension at the air-water interface (MT-2), Ta in the functional expressions is the temperature 

(0K), �h is the temperature scaling factor for pressure head, and �k 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
Figure 2-2. An example of soil structure scaling for scale factors (αh,αθ, αk) = (3/2, 2, 7/4). The ratios hs/hs

* = 

h1/h1

* = 3/2, (θs- θr

) ÷ (θs

*- θr*) = (θ1- θr) ÷ (θ1*- θr*) = 2, and K1

/K1

* = Ks

/Ks

*

 = 7/4. 

13
 

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 (Šimúnek 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 α, the inverse of the air-entry value or the bubbling pressure. The α for drying, αd, is less than 

that for wetting, αw. A reasonable approximation is to let αw = 2αd. 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 α 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 (Šimúnek et al., 1998) 

consist of a prescribed pressure head distribution at time zero, hi

(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) = ho(t), t > 0, z = 0,L , (2-12) 

Cauchy Conditions 

∂h  -K 

 ∂z 

+ 1 = qo (t), t > 0, z = 0, L , (2-13)  

Neumann Conditions 

∂h = 0, t > 0, only at z = 0 , (2-14) ∂z 

where ho is a specified head and qo 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 firstorder 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 

(Šimúnek et al., 1998): 

* ��1 

∂

∂ 

c

t

1 ���

∂

∂ 

c

z

1 � ∂

∂ 

z 





 ��1 

∂

∂ 

c

z

1 



 + F1 ��1 ��1 � �1 �� 1 

��� (2-15) 

* ��2 �2,t ����2,z � ��2 �2,z ,z + F2 ��2 ��1 ����2 ��1 ,�′ 

1 

��2 ��� (2-16) 

* ∂si = H(ci ,s

*

i ) - (�i ��′ i ��

*

i ���i �������������� (2-17) ∂t 

where the “1” index is for the parent species, “2” index is for the daughter species, Ri is the retardation factor 

14 
accounting for gaseous, equilibrium solid, and liquid phases of the solute, ci

 is the concentration of the liquid phase, 

Di is the dispersion coefficient for the ith species, si

* represents the concentration of the solute on the nonequilibrium 

sites of the soil matrix, μi is the degradation rate of si

* and represents material lost to the system, μi

� is the rate of 

transfer of material from parent to daughter, and γi is the production of si

*. The function Fi

(ci) 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 G1(s1

*) 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(c1,si

*, 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(ci,si

*) in Equation (2-17) is a first order sorption rate of si* 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) = cio(z), si

*(z,0) = si0*(z), 0<z<L, (2-18) 

where cio and si0* are prescribed functions of z. The standard boundary conditions in the HYDRUS model are as 

follows: 

Dirichlet Conditions 

ci (z,t) = ci (z,t), t > 0, z = 0,L , (2-19) 

Cauchy Conditions 

∂c -��i ∂z

i �����i ��� �i ���������������������� (2-20) 

Neumann Conditions 

∂ci

∂z 

= 0 , t > 0, z = 0, L , (2-21) 

where ci 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 Di(L2T-1) in Equation (2-15) and (2-16) are defined by the following 

(Šimúnek et al., 1998): 

��i ����iL � ����iw �w ��� v �ig �ig �g � (2-22) 

where DiL is the longitudinal dispersivity (L) of the ith species, Diw is the molecular diffusion coefficient (L2T-1) of 

the ith species in free water, τw is the dimensionless tortuosity factor in the liquid phase, av is the dimensionless air 

content, Dig is the molecular gas diffusion coefficient (L2T-1) of the ith species, kig is the dimensionless constant of 

proportionality linearly linking ci

 to the gaseous solute concentration gi, and τg is the dimensionless tortuosity factor 

in the gas phase. The sum of the air content, av, and the soil moisture content, θ, is equal to the saturated water 

content, θs. The empirical formulas for the tortuosity factors are given by: 

15
� � �7/3 �-2 

������� �����

7/3 �-2

� (2-23) w s g v s 

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 (Šimúnek et al., 1998): 

* 

A(Ta ) = exp 



 E(Ta - T

*

a ) 

 , (2-24)  RT aa T  

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 ÷ T2MoK). 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 (Šimúnek et al., 1998): 

∂ c (���a 

  p ∂[qTa ] ∂  ∂Ta 

∂t 

+ cw ∂z = 

∂z 

����

∂z 

 ����w��a �� (2-26) 

where cp(θ) 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 λ(θ) is the coefficient of the apparent thermal conductivity of the soil

(ML ÷ T3

°K .) To a reasonable degree of accuracy, it can be shown that

) 

dcp (��

≅ c . w (2-27) d�

Thus, combining Equations (2-2), (2-26) and (2-27) leads to the heat equation programmed in HYDRUS: 

∂T ∂T ∂  ∂Ta  c (�� ∂t

a ����w � ∂za � ∂z 

 ���� ∂z  � p (2-28)

The apparent thermal conductivity coefficient λ(θ) is given by the empirical formula 

�������� ��� � �� �

1/2 ����� � � (2-29) 1 2 3 w 

where (b1,b2,b3) are empirical constants with units of (ML ÷ T3�

K ) and � is the thermal dispersivity (L). The initial 

16 
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 

∂Ta -� � � �

∂z aw � ���0 ����w �0 �����������������������, (2-31) 

Neumann Conditions 

∂Ta

∂z 

= 0, t > 0, z = 0 , (2-32) 

Atmospheric Boundary Condition 

 T0 (t) = a 

2�� �� T +A sin - p 2 

 

  1  , t > 0, z = 0 , (2-33)  

where T0 in Equation (2-31) is the temperature of the incoming fluid, T a 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), T a 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 10-2 to 10-4 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 10-7 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. 

18
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 hysteretic 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 surfacewater 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, �D. Parts of the boundary, �D1, 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, �D2, may be free, such as fresh/saline interfaces, water-table surfaces, and seepage zones. These free 

boundaries, �D2, 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, �D3, 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, �D1 and �D2 

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 

19
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-advectiondispersion 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 areal 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 threedimensional pictures of flow and solute transport (Rockhold et al, 1996; and Rockhold, 1999). For example, Figure 

3-1a 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 3-1b. 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, �D. 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, 

20
Furrow Irrigation 

Row Crop 

1 

8 

7 

6 

12 

13 

9 

10 

2 

3 

4 

11 

5 

4 

3 

2 

1 

(a) Plan View of a (b) Plan View of a 

Field Mapping Unit Field of Row Crops 

Figure 3-1 Schematic of modeling applications for simulating three-dimensional field mapping units using onedimensional 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, �D, 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. 

21
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 

22
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 Ri

 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 

23
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, Feyen 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 Cruces, 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 

24
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
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 = αcn , (3-1) 

and Langmuir, 

s = αc ÷ (1 + ßc) (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 α, ß, 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 α = 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>1 produced a spreading pollutant front, while for 0<n<1, the front was selfsharpening. For n=1 (i.e., the linear isotherm) the classical “Gaussian” breakthrough curve was produced. These 

results are supported by Logan (2001) and Sachdev (2000). Sachdev (2000) found that the asymptotic behavior of 

the solute plume differed for the following three ranges of n: 0<n<1, 1<n<2, and n>2. 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< α/ß), 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
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 et al (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 (μ), and varying recharge 

rates (q) were investigated using the CHAIN Model. The values of Kd and μ 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 semiinfinite 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
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, Sij, of a model’s 

output, yi, to a model’s input parameter, xj, is given by 

S= ∂yi 

ij ∂x . (4-1) 

j 

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, μ, 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 Sij are those of yi

 divided by those of xj. Since Sij, 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 Sij, called the relative sensitivity coefficient, defined by: 

S =

xj ∂yi 

rij yi ∂x . (4-2) 

j 

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 Sij 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: 

F = dy ≡ 

dF 

p dp dp . (4-4) 

29
�

�

�

�

�

The relative sensitivity coefficient Frp is defined by: 

F = p � dF . rp (4-5) F(p) dp 

Suppose that the graph of Frp versus p (Equation 4-5) is given by Figure 4-1. To aid in the interpretation of this 

figure, rewrite Equation (4-5) as 

dF 

= Frp (p) dp . (4-6) F(p) p 

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 p1 , based on the value of p1 , produces 

a (10a)% decrease in the value of y1, or the quantity F(p1). 

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: 

Sij ≅ ��i , (4-7) ��j 

and 

��

�� ��

�������

Figure 4-1. The graph of the relative sensitivity Frp in terms of the parameter p for a model defined by y = 

F(p). 

30
�� �

Srij ≅ 

��i � j

= 

��i ÷ 

�� j , (4-8) 

ji �i � j 

where ��i is the change in yi due to a small change �� j in xj. The above interpretation of the graph of Frp 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 ��i ��i to the relative change of the input parameter 

�� j ����j �� For example, 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: 

a 

y = F(a,b,c,d,e,f) = 

b + cd + e�n(c2

f)

 . (4-9) 

e 

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(2,1,-1,3,4,5) , (4-10) 

-1 ≤≤ a 5, 0 ≤ b ≤ 2, -4 ≤ c ≤ 2, (4-11) 

1 ≤≤ d 5, 1 ≤ e ≤ 7, 2 ≤ f ≤ 8 . (4-12) 

The functional value, F, and the sensitivity and relative sensitivity of y with respect to a given input parameter are 

determined by letting the given input parameter be arbitrary and all other input parameters take on the values of the 

base case. Thus, if the given input parameter is c, the functional value and the sensitivities are 

F(2,1,c,3,4,5), Fc(2,1,c,3,4,5), Frc(2,1,c,3,4,5), (4-13) 

similarly for the other five parameters. Table 4-2 lists the forms of F, Fα and Frα, where α represents the parameters 

a,b,c,d,e,f. The characteristics of the relationship between the output y and the input α dictate the forms of the 

sensitivities Fα and Frα over the domain of the input α. Figures 4-2 to 4-4 show the variations of F, Fα and Frα over 

the domain of the input α. 

From Figure 4-2, we note that F(a,1,-1,3,4,5) is homogeneous in a and linearly varies with a, thus both Fa and 

Fra are constants over the domain of a. In fact, Fra must be equal to unity. Also from this figure, we see that F(2,b,-

1,3,4,5) is linear in b, not homogeneous, and possesses a positive slope. Linearity guarantees that Fb is a constant 

over the domain of b, but Frb is a strictly monotonically increasing function of b which would approach a horizontal 

asymptote (Frb=1) if the domain of b was great enough. In Figure 4-3, we note the F(2,1,-1,d,4,5) is linear in d, not 

homogeneous, and possesses a negative slope. The linearity guarantees that Fd is a constant, actually negative, while 

Fb was positive. Further, Frd is now a strictly monotonically decreasing function of d and approaches a vertical 

asymptote at 

d =+ 1 �n(625) ≈ 7.438 , (4-14) 

31
Table 4-1. Sensitivities and Relative Sensitivities of Output F with Respect to the Six Input 

Parameters, for Arbitrary Values of the Inputs. 

a 

f 

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 

the plot of F(2,1,c,3,4,5) versus c. Because of the particular domain of c and the presence of �nc ( ) in F, there is a 4 

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 α and one at �. The vertical asymptote shows up as a vertical asymptote in Fc but a zero in Frc. 

The zeros (α, �) in F have no particular effect on Fc but become vertical asymptotes in Frc. The � zero in Fc carries 

over into a zero of Frc. These zero/asymptotic relationships will exist between F, Fα, and Frα as long as we do not 

run into indeterminant forms, 0/0 or ���. For these cases, the asymptote and zeros in Fα, and Frα may disappear. 

Finally, in Figure 4-4, we note that for the domain of definition in F(2,1,-1,3,e,5) there is a zero at α which produces 

a vertical asymptote in Fre. The quantity F(2,1,-1,3,4,f) 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(2,1,-1,3,e,5) approaches a horizontal asymptote (F�3.219) for 

sufficiently large e, while F(2,1,-1,3,4,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. 

Parameter Sensitivity, S Relative Sensitivity, Sr 

a ( ) 1 2 b + cd + e c f 

e 

 n    � 1 

b 

a 

e 

2 

b 

b + cd + e �n(c f)

c 

a 2e d + 

e c 

      2 

dc + 2e 

dc + b + e �n(c f)

d 

ac 

e 

2 

cd 

cd + b + e �n(c f)

e [ ] 2 

a - b + cd 

e 

2 

b + cd 

e (c f) + b + cd - �n

f 2 

e 

e ( �n c f) + b + cd

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). 

Parameter Output F Sensitivity Relative Sensitivity 

Formula Characterization S r 

a 

1 

(5) - a 

2 

n  

  � Linear and 

1 

(5) -

2 

�n 1 

Homogeneous in a. 

b 

b - + (25) 

2 

�n Linear and Non1 

2 

b 

b - 3 + �n(625) homogeneous in b. 

c 

4 + 

3c 

(c ) + + (25) 

2 

1 

2 

�n Linear-Logarithmic 

4 

c 

+ 8/3 

8 

c + 

3 

1 (c ) + + (5) 3 c + � n and Nonhomogeneous in c. 

d 

d 

- + + (25) 

2 

�n Linear and Nonhomo1 

2 

− d 

d - 1 - �n(625) geneous in d. 

e 

4 - + (25) 

e 

�n Nonhomogeneous in e, 2 

4 

e 

2 

�n(5) e - 2 Inverse Variation with e. 

f 2(f �n ) − 1 Logarithmic and 

2 

f 

1 

2 

1 

�n(f) 

Nonhomogenous in f. 

S

 



3 

2 

�n 3 

2 

4

3 n �

1 

2 

-

33
� �

�

�

�

�

1.2 .4 

1.0 

.3 

.8 

���

.6 ���

.2 

.4 

.1 

.2 

0 0 

-1 0 1 2 3 4 5 0 1 / 2 1 3 / 2 

1.2 1 

1.0 

.8 

�� .6 �� 1⁄2 

.4 

.2 

0 

-1 0 1 2 3 4 5 

6 

5 

4 

3 

� 2 �

1 

0 

-1 

-2 

-1 0 1 2 3 4 5 

0 

0 1 / 2 1 3 / 2 2 

4 

3 

2 

1 

0 

0 1 / 2 1 3 / 2 2 

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

2
1.2 

γ α β 

0 

1.0 

.8 

.6 1 

.4 

Frc .2 Frd 

0 -2 

-.2 

-.4 

-.6 -3 

-.8 1 2 3 4 5 

-4 -3 -2 -1 0 1 2 

c d 

10 

0 8 

6 

γ = −8/3 

γ 

-.2 

4 -.4 

2 Fd -.6 

FC 0 -.8 

-2 

-1 .0 

-4 1 2 3 4 5 

-6 

-8 d 

-4 -3 -2 -1 0 1 2 

c 4 

10 

α�-0 .4 7 1 

α β 

β + 0 .3465 

�

�

3 

8 

6 F 2 

4 

F 2 

1 0 

-2 

-4 0 

-4 -3 -2 -1 0 1 2 1 2 3 4 5 

c 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 

α= 2 /ln (5 ) 

α 

6 

8 

6 5 

4 

2 4 

Fre 0 

-2 Frf 3 

-4 

2 

-6 

-8 

1 -10 

1 2 3 4 5 6 7 

e 0 

2 3 4 5 6 7 8 

4 f 

1.0 

.9 

3 .8 

.7 

.6 Fe 2 

Ff .5 

.4 

1 .3 

.2 

.1 

0 0 

1 2 3 4 5 6 7 2 3 4 5 6 7 8 

e f 

3 

2

α 

4 

3 

F 1 F 

2 

0 

1 

-1 0 

1 2 3 4 5 6 7 2 3 4 5 6 7 8 

e 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
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 Cruces 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 (99Tc) 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

PLUT O N IU M ,

23 8Pu 

88 yr. half-life 

URA N IUM , U 

2 .4 x 105y r. h a lf-life 

2 34 

ST R O NT IU M , 90 Sr 

2 9 y r. h a lf-life 

YTT R IU M , 90 Y 

6 4 h r. h a lf-life 

Z IR C O N IU M , 90 Zr 

S ta b le 

T E CHNE T IUM, 99 Tc 

2.1 x 1 05 

y r. h a lf-life 

RUT HE N IUM ,99 Ru 

S ta b le 

T R IT IU M ,3 H 

12 yr. half-life 

HE L IUM -3 ,

3 He 

S ta b le 

URA N IU M , U 

4.5 x 10 9 yr. half-life 

23 8 TH O R IU M , 234 Th 

2 4 day ha lf-life 

P R OT OA C T IN IU M , 234 m Pa 

1 m in h a lf-life 

URA N IUM , U 

2.4 x 10 5 y r. h a lf-life 

P R O T O A C T IN IU M , 234 Pa 

7 h r h a lf-life 

99 .8 % 23 4 

0 .2% 

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 WasteBurial 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 Cruces 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 Cruces 

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 Cruces 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 Cruces, 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

Table 5-1. Partial List of Radionuclide Contaminated and Disposal Sites in the U.S. (U.S. EPA’s VISITT Database). 

Location, 

SITE NAME City State Contaminants Site Characteristics 

1	 Rocky Flats Environmental Technology Site 

2	 Idaho National 

Engineering & 

Environmental 

Laboratory 

3 Ethyl Corporation 

4	 DOE FUSRAP 

St. Louis Site 

Treatability Study 

5 Nevada Test Site 

6	 Mojave Desert 

Waste Burial Site 

7 NL Industries, Inc. 

8	 Los Alamos Natl. 

Laboratory 

9	 Las Cruces Trench 

Site 

10 Sandia Natl. Lab. 

11 West Valley Nuclear 

12	 Mound Demonstration 

Project 

13	 Portsmouth Gaseous 

Diffusion Plant (DOE) 

14	 EPA SITE 

Demonstration 

15	 Fernald 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 Tech. Center 

Golden Colorado 

Idaho Falls Idaho 

Baton Rouge Louisiana 

St. Louis Missouri 

Mercury Nevada 

Beatty Nevada 

Pedricktown New Jersey 

Los Alamos New Mexico 

Las Cruces New Mexico 

Albuquerque New Mexico 

West Valley New York 

Miamisburg Ohio 

Piketon Ohio 

Alliance Ohio 

Fernald Ohio 

Apollo Pennsylvania 

Aiken South Carolina 

Clemson South Carolina 

Oak Ridge Tennessee 

Richland Washington 

Morgantown West Virginia 

Plutonium, Uranium 

Radioactive Materials, 

Plutonium, Tritium 

Uranium


Uranium


Plutonium, Tritium


Various Radionuclides


Strontium


Plutonium, Uranium


None


Uranium, Plutonium


Strontium


Plutonium


Uranium, Technetium


Uranium


Uranium, Technetium


Uranium


Uranium


Uranium


Technetium, Uranium


Uranium, Strontium


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: Radionuclide 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 

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. On-site waste 

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 soilmoisture 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 Cruces, 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, 4.8 m wide, 

and 6.0 m deep. 

The published soil hydraulic properties for this site, given by Wierenga, et al. (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 α,β,θr, and θs 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 α,β,θr, and 

θs 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 Cruces 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): 

λEt0 = Γ � (Rn - G) + 6.43 (1 - Γ) W(e0 − e) , (5-1) 

where the terms in Equation (5-1) are defined as: 

λ = latent heat of vaporization in mega-joules per kilogram, 

Et0 = evapotranspiration rate (Et

) from a well-watered grass reference crop, in kilograms per 

meter squared per day, 

λEt0 = latent heat flux density in mega-joules per meter squared per day, 

Γ = dimensionless parameter dependent upon surface elevation and air temperature, Table 6-1 of 

Jensen, et al., 1990, 

Rn = 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 Et0 ÷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/yr to 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 Cruces Trench Site for SSG Model Evaluation Study 

(reprinted from Water Resources Research, 1991, by P.J. Wierenga, R.G. Hills, and D.B. Hudson 

with the permission of the American Geophysical Union, Washington, DC). 

Saturated Residual van Genuchten van Genuchten Saturated 

Layers Depth Water Water Alpha Beta Hydraulic 

(cm) Content Content Coefficient, α, Coefficient, β, Conductivity, Ks, 

(cm3

/cm3) (cm3/cm3) (cm-1) (—) (cm/d) 

Uniform Soil Model 

all 0 - 600 0.321 0.083 0.055 1.509 270.1 

Layered Soil Model 

1 0 - 15 0.348 0.095 0.042 1.903 539 

2 15 - 140 0.343 0.091 0.062 1.528 250 

3 140 - 205 0.336 0.085 0.060 1.574 267 

4 205 - 250 0.313 0.071 0.068 1.537 300 

5 250 - 305 0.302 0.072 0.040 1.550 250 

6 305 - 370 0.294 0.090 0.070 1.711 334 

7 370 - 460 0.310 0.073 0.027 1.418 221 

8 460 - 540 0.325 0.083 0.041 1.383 172 

9 540 - 600 0.306 0.078 0.047 1.432 226 

Table 5-3.
 Characteristics of the Las Cruces Trench Site for SSG Model Evaluation Study 

(reprinted from Soil Sci. Soc. Am. J., 1994, by G.W. Gee, P.J. Wierenca, B.J. Andraski, M.H. 

Young, M.J. Fayer, and M.L. Rockhold with the permission of Soil Sci. Soc. Am. J., Madison, 

WI). 

Annual 

Precipitation 

(cm/yr) 

Annual 

Potential 

(Pan) 

Evaporation (cm/yr) 

Annual 

Potential 

Recharge 

(cm/yr) 

Average 

Daily Max. 

Air Temperature (o

C) 

Average 

Daily Min 

Air Temperature (o

C) 

Elevation 

(m) 

Depth to 

Water 

Table 

(m) Geology 

Typical 

Soil 

Type 

Typical 

Vegetation 

23 239 8.7 28 13 1357 60 Alluvial 

Berino fine 

loamy sand 

Creosote 

bush 

42

43

for Radionuclides: echnical Background Document (U.S. EPA, 2000b). 

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). 

source is assumed. 

infiltrated rainfall which produces a net annual recharge rate of 87 mm/yr. able 5-3). 

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 MULTIMEDDP 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 unconfined. , 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.

Figure 5-2. Daily precipitation and potential evapotranspiration (PET) at Las Cruces Site, NM. 

calculated from daily climate data using Penman’s equation (Jensen et al., 1990).

/d)

T In so doing, it is assumed that the Las

Thus, a finite radionuclide contaminated

The driving force to send this material on a downward migration to the water table is the

(T The site-specific soil

However

PET is
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 onedimensional 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 

Cruces 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 θr and θs. Thus, we arbitrarily chose an initial water content, θ, equal 

to the geometric mean of θr and θs, (θrθs )1/2. This value to two significant figures is θ = 0.16. 

As stated in the previous subsection, the total mass of the individual radionuclides released from the hypothetical 

Las Cruces 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) , 3

365 (d/yr) 1000 (cm /L) 

1 = (8.7) ( 1000 )(1000 )Concentration (mg/L) , (5-2) 365 

= 0.024 (Concentration, mg/L) , 

44

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 99Tc (mg/cm2)


Concentration of 99Tc 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, α, (cm-1)


van Genuchten Beta Coefficient, β, (–)


Depth to Water Table (m)


Solute Transport Parameters 

Decay Coefficient for Parent Product 99Tc (1/d)


Decay Coefficient for Daughter Product 99Ru (1/d)


Distribution Coefficient for 99Tc (ml/g)


Distribution Coefficient for 99Ru (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


1.1 x 10-7


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). (5-3) 

Using Equation (5-3) and the mass releases of 238U, 99Tc, 90Sr, 238Pu and 3H in Table 5-4 results in the 

corresponding concentrations of these species given in the table. 

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 Cruces 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: θs = 0.321, θr = 0.083, VG-α = 

0.055 cm-1, VG-β = 1.509, and Ks = 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 Cruces Trench Site is a Berino fine loamy sand. With these two pieces of 

information and two “rules of thumb” used by Eagleson (1970), we can roughly determine the porosity (n) and the 

effective porosity (ne) at the site. The following “rules of thumb” were used by Eagleson to analyze the properties of 

a Touchet silt loam (ρs = 2.60 g/cm3), a Columbia sandy loam (ρs = 2.67 g/cm3), and an unconsolidated sand (ρs = 

2.71 g/cm3), where ρs is the density of the solid matrix and ρ is the bulk density: 

n = 1 - ρ/ρs , (5-5) 

ne = n - θr . (5-6) 

Thus, using a density, ρs, 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 (99Tc). 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 99Tc in Table 5-4 is derived from the radionuclide halflife given in Figure 5-1 The daughter product of 99Tc (i.e., ruthenium, 99Ru) is stable, and its decay coefficient is 

zero. The default values of Kd for 99Ru is 5 ml/g, which indicates that this radionuclide is not very mobile in the soil 

column. Because of 99Tc’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 = τ wDw + DL q / ��� (5-7) 

where θ 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 Cruces Site by Porro and Wierenga (1993), and the -1 above values of θ and q , gives a value of 0.68 cm2/d for Dq � ��The diffusion coefficient in free water is L 

assumed to be 1.73 cm2/d and the tortuosity factor τw is taken as 0.19 (Tomasko, et al., 1989). Thus, the value of 

τwDw 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

previously stated, the creosote bush (Larrea tridentata) is the dominant plant at the Las Cruces 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 Cruces 

Site have a vertical distribution over a range of 0.5 m to 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

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,...,Pp, ..., Pbn), (6-1) 

where Qij represents the jth output for the ith model, Pbk is the base value of the kth parameter, Pp is the variable pth 

parameter, and Fij (·,·,...,·) is the functional relationship between the n-1 base parameters and the one variable 

parameter determined by the ith model for the jth 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, 99Tc, 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 < Pp < Bp ; (6-2) 

6. Run model simulations using the n - 1 base input parameter values and the one variable input 

parameter Pp, thus giving the functional relationships given in Equation (6-1); 

7. Calculate the sensitivity S and the relative sensitivity Sr of the model outputs to the various input 

parameters following the approach described in Equations (4-1) to (4-8); 

8. Repeat the procedure given in Steps 1 to 7 for each pertinent input parameter and for each model 

and radionuclide. 

6.2 Input Parameters for Constant Recharge Rate and Constant Water Content 

There is no flow module in the CHAIN Code, and the recharge rate, q, and the water content, θ, are specified, 

constant, input parameters for given homogeneous layers. Thus, the governing differential equation for the parent 

species is defined as follows: 

48
ρ θ ρ θ

∂c q ∂c �� ∂2

�

∂t

 + 

��� ��d ∂�

= 

������d ∂�

2 - ���� (6-3) 

where ρ 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 μ is the decay rate (1/d) which is assumed fixed for a 

given radionuclide. That is, for a given radionuclide, the decay rate, μ , is the same for the sorbed phase and the 

dissolved phase, and μ is sufficiently well known that no sensitivity analysis will be run on it. As in Equation (5-7), 

θD can be written as: 

�������w �w ����L � � (6-4) 

where τw 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, 

θτw Dw is about one-half of the value of Dq . The reason that the diffusion component of D is so important is L 

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): 

�����������d � (6-5) 

For the default values of Kd listed in Section 5 and the base values of θ and ρ given in Table 5-4, we get the 

following ratios, ρKd ÷ θR, for the five radionuclides: 

Radionuclides ρKd θR [ρKd ÷ θR] 100 

Tritium (3H) 0.000 0.160 0.00% 

Technetium (99Tc) 0.012 0.172 6.98% 

Uranium (238U) 0.680 0.840 80.95% 

Strontium (90Sr) 1.700 1.860 91.40% 

Plutonium (238Pu) 8.500 8.660 98.15% 

These ratios show the relative importance of the distribution coefficient Kd to the product θR as Kd increases. For 

example, ρ 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,θ,ρ,D,DL,Dw) influence the transport and 

fate of radionuclides in the unsaturated zone when q and θ are assumed to be constant input parameters. For 

comparison purposes, all five models were analyzed under these constant q and θ 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 MULTIMED-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 θ 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, θ, in a vertical soil column, the governing equations 

are Darcy’s Law, 

∂h  

q = -K 

 ∂z 

+ 1 

 , (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. 

Sensitivity Analyses Performed 

Model 

Input MULTIMEDParameter CHAIN DP 1.0 FECTUZ CHAIN 2D HYDRUS 

Kd � � � � �

q � � � � �

θ � � � � �

ρ � � � � �

D �

DL � � � �

Dw � �

∂� ∂�

∂t

 + 

∂z 

= -S , (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: 

∂� ∂   ∂h  

∂t

 = 

∂z 

K 

 ∂z 

+ 1 





 - S . (6-8) 

The van Genuchten Model for the parameters K(h) and θ(h) is as follows (see Appendix A): 

K(h) = KsKr (h) 

2 m 

= K S 1/2 



1 - (1 - Se

1/m ) 

 , (6-9) se 

 � � s � �r �r + m � ����������

������  



1 + �� 

 (6-10)  

�s �������� ≥ ����

where the effective water content Se is given by 

��� � S = e

r , (6-11) � �� � s r 

and where Ks is the saturated hydraulic conductivity (cm/d), Kr is the unitless relative hydraulic conductivity, θs is 

the saturated water content, θr is the residual water content, α is the inverse of the air-entry value or bubbling 

pressure head (cm-1), β is the unitless pore-size distribution index, and m is defined by 

m = 1 - 1/β . (6-12) 

50 
For h < 0, Equation (6-10) can be rearranged to give h as a function of the effective water content Se: 

h = h(Se ) = -

1 S-1

e 

/m - 1

 

1/� 

. (6-13)

�  

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, θ, is found to be independent of time. When θ 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, θ(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,θs,θr

,α,ß ) are based on a constant recharge rate, q, and a 

variable water content, θ(z). The determination of θ(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 θ at 

the two boundaries, namely: 

(1) θ = θs at z = 0, giving h = 0 at z = 0;


(2) θ = 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, θ(z) can be determined from Equation (6-10). Given θ(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,θs,θr

,α,ß) are varied, one at a time, for the sensitivity analysis, different water content profiles, 

θ(z), are generated leading to different responses for the breakthrough curves (BTCs) of the liquid phase 

concentrations. 

The determination of θ(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 θ 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 θ given in Table 5-4, as calculated from Equation (6-13). Thus, the water content profile, θ(z), is given as 

the steady-state solution of this initial, boundary value system for h(z,t), or θ(z,t), as given by equation (6-10). 

Again, as the van Genuchten parameters (Ks,θs,θr,α,ß) are varied, one at a time, for the sensitivity analysis, different 

profiles, θ(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, θ(z). 

Table 6-3 lists the base values of the twelve input parameters (Kd,q,θ,ρ,D,DL,Dw,Ks,θs,θr,α,ß), 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 Cruces Trench Site 

given in Table 5-2, the hydrologic records shown in Figures 5-2 and F-1, 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 (BTC). 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, θ(z), Varies with the Changing van Genuchten 

Parameters (Ks,θs,θr, α, ß). 

Sensitivity Analyses Performed 

Model 

Input MULTIMEDParameter CHAIN DP 1.0 FECTUZ CHAIN 2D HYDRUS 

Kd Base Base Base Base 

q Base Base Base Base 

θ θ(z) θ(z) θ(z) θ(z) 

ρ Base Base Base Base 

D 

DL Base Base Base Base 

Dw Base Base 

Ks � � � � 

θs � � � � 

θr � � � � 

α � � � � 

ß � � � � 

Table 6-3.
 Input Parameters for Each Model and the Range of Each Variable Parameter, Along with the Base 

Value of the Parameter. 

Parameter, Pp 

p Modela Symbol Units Ap Pbp Bp 

1 1,2,3,4,5 Kd ml/g 0.001 0.007 0.019 

2 1,2,3,4,5 q cm/d 0.016 0.024 0.030b 

3 1,2,3,4,5 θ cm3/cm3 0.10 0.16 0.22c 

4 1,2,3,4,5 ρ g/cm3 1.62d 1.70 1.78 

5 1 D cm2/d 0.40 1.00 2.20 

6 2,3,4,5 DL cm 3.73 4.53 5.33 

7 4,5 Dw cm2/d 0.93 1.73 2.53 

8 2,3,4,5 Ks cm/d 175 270 365 

9 2,3,4,5 θs cm3/cm3 0.29 0.32 0.35 

10 2,3,4,5 θr cm3/cm3 0.06 0.08 0.10 

11 2,3,4,5 α 1/cm 0.051 0.055 0.059 

12 2,3,4,5 β 1/1 1.43e 1.51 1.59e 

a 1 = CHAIN, 2 = MULTIMED-DP 1.0, 3 = FECTUZ, 4 = CHAIN 2D, 5 = HYDRUS;


b B2 = 0.032 for CHAIN;
 c B3 = 0.26 for CHAIN;


d A4 = 1.60 for CHAIN;
 e 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,θ,ρ,DL,Dw,Ks,θs,θr,α,ß). For the first six parameters (Kd,q,θ,ρ,DL,Dw) the 

analyses were conducted under the constraints of constant recharge rates, q, and constant moisture content, θ. For 

the van Genuchten parameters (Ks,θs,θr

,α,ß), the analyses were executed under the constraints of constant recharge 

rate, q, and variable water content, θ(z). The BTCs for 99Tc concentrations at the bottom of the homogeneous, 6 

meter soil column were recorded, as given in Figure 6-1a to 6-1k. 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, ρ and 

Kd, have a very small influence on the transport and fate of 99Tc since θ >> ρKd 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 θ and the sum of DL and 

DW, while Tpeak , and TMCL are most influenced by q and θ. 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 θ(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 α. The most 

influential input parameter for all the outputs is the soil texture coefficient β. The next most influential parameters 

are the upper and lower moisture bounds of the van Genuchten Model, respectively θs and θr . 

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 @ Base Values of the Input Parameters 

Parameters Cpeak Tpeak TMCL 

Kd -0.06 +0.06 +0.07 

q +0.12 -1.10 -0.98 

θ -1.10 +0.80 +0.86 

ρ -0.05 +0.06 +0.07 

DL -0.28 -0.02 -0.07 

Dw -0.10 -0.01 -0.03 

Ks +0.05 -0.04 -0.04 

θs -0.66 +0.57 +0.64 

θr -0.29 +0.22 +0.24 

α +0.13 -0.05 -0.05 

β +1.38 -1.00 -1.03 

53
54

Figure 6-1. Sensitivity of 99Tc breakthrough (through the 6m layer) to the system parameters using the HYDRUS

Model: ficient, (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 α, (k) van Genuchten retention parameter β.

(a) distribution coef
55

Figure 6-2. Sensitivity and relative sensitivity of peak concentrations at the depth of 6m to the system parameters

using the HYDRUS Model: ficient, (b) recharge rate, (c) water content, (d) bulk

density.

(a) distribution coef
56

Figure 6-3. Sensitivity and relative sensitivity of peak concentrations at the depth of 6m to the system parameters

using the HYDRUS Model: , (f) diffusion coefficient in water, (g) saturated

conductivity, (h) saturated water content.

(e) dispersivity
57

Figure 6-2. Sensitivity and relative sensitivity of peak concentrations at the depth of 6m to the system parameters

using the HYDRUS Model: α, (k)

van Genuchten retention parameter β.

(i) residual water content, (j) van Genuchten retention parameter 
58

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: ficient, (b) recharge rate, (c) water content,

(d) bulk density.

(a) distribution coef
59

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: , (f) diffusion coefficient in water, (g) saturated

conductivity, (h) saturated water content.

(e) dispersivity
60

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: (i) residual water content, (j) van Genuchten retention parameter

α, (k) van Genuchten retention parameter β.
61

Figure 6-4. Sensitivity and relative sensitivity of time to exceed MCL to the system parameters using the

HYDRUS Model: ficient, (b) recharge rate, (c) water content, (d) bulk density.

(d)

(a) distribution coef
62

Figure 6-4. Sensitivity and relative sensitivity of time to exceed MCL to the system parameters using the

HYDRUS Model: , (f) diffusion coefficient in water, (g) saturated conductivity, (h)

saturated water content.

(e) dispersivity
63

Figure 6-4. Sensitivity and relative sensitivity of time to exceed MCL to the system parameters using the HYDRUS

Model: α, (k) van Genuchten retention

parameter β.

(i) residual water content, (j) van Genuchten retention parameter 
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,θ,ρ,D,DL,Dw,Ks,θs,θr,α,β) 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, 99Tc, 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 MULTIMED-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, θ 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,θ,ρ,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, θ, 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 θD 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 θτwDw 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 θD) 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 θD which is nearly the same as those for CHAIN 2D and 

HYDRUS, while the value of θD 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 (�) for Four of 

the Five Models Under the Assumption of Constant Recharge Rate and Variable Water Content. CHAIN MULTIMED-DP 1.0 FECTUZ CHAIN 2D 

HYDRUS 

Sensitivity Analyses Performed 

Model Input 

Parameter 

Kd � � � � �

q � � � � �

θ � � � � �

ρ � � � � �

D �

DL � � � �

Dw � �

Ks � � � �

θs � � � �

θr � � � �

α � � � �

β � � � �

65
Table 7-2. The Values of the Dispersion Factor, θD, as Derived from Equation (6-4) and the Base Values of 

L and Dw D, D as given in Table 6-3. L, Dw, θ and q and the Ranges of D, D CHAIN MULTIMED­ DP 1.0 

FECTUZ 

CHAIN 2D 

HYDRUS 

0.40 4.53 1.73 0.109 0.053 0.064 0.109 0.109 0.162 0.162 

1.00 3.73 1.73 0.090 0.053 0.160 0.090 0.090 0.143 0.143 

1.00 4.53 0.93 0.109 0.028 0.160 0.109 0.109 0.137 0.137 

1.00 4.53 1.73 0.109 0.053 0.160 0.109 0.109 0.162 0.162 

1.00 5.33 1.73 0.128 0.053 0.160 0.128 0.128 0.181 0.181 

1.00 4.53 2.53 0.109 0.077 0.160 0.109 0.109 0.186 0.186 

2.20 4.53 1.73 0.109 0.053 0.352 0.109 0.109 0.162 0.162 

θD in cm 2

/d 

D 

cm 2

/d 

DL 

cm 

Dw 

cm 2/d 

DL�q� ���

cm 2

/d 

θτwDw 

cm 2/d 

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 θD 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,θs,θr,α,β), only four models (CHAIN does not allow variable 

θ) are evaluated for constant recharge rate, q, and variable water content, θ(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,θ,ρ). CHAIN is the only model for which sensitivity analyses are performed for D. Sensitivity 

analyses are performed for the six parameters (DL,Ks,θs,θr

,α,β) on four of the five models (all except CHAIN), while 

only CHAIN 2D and HYDRUS consider the parameter Dw. The 99Tc 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,θ,ρ), 

respectively; 

• Figure 7-5 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,θs,θr,α,β), respectively. 

These twelve figures of 99Tc BTCs fulfill the expectations set forth in Section 7.1. These results, or 

expectations, are summarized in the following list: 

66

67

Figure 7-1. Sensitivity of 99Tc 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.
68

Figure 7-2. Sensitivity of 99Tc 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, 

d, and 

0.024 cm/

0.016 cm/d.
69

Figure 7-3. Sensitivity of 99Tc breakthrough (through the 6m layer) to the water content using the CHAIN,

MULTIMED-DP 1.0, FECTUZ, CHAIN 2D, and HYDRUS Models, where θ = 0.22 cm3/cm3, 0.16

cm3/cm3, and 0.10 cm3/cm3.
70

Figure 7-4. Sensitivity of 99Tc breakthrough (through the 6m layer) to the bulk density using the CHAIN,

MULTIMED-DP 1.0, FECTUZ, CHAIN 2D, and HYDRUS Models, where ρ = 1.78 g/cm3, 1.70 g/

cm3, and 1.62 g/cm3.
71

Figure 7-5. Sensitivity of 99Tc breakthrough (through the 6m layer) to the dispersion coefficient using the CHAIN

Model, where D = 2.20 cm2

/d, 1.00 cm2/d, and 0.40 cm2/d.
72

Figure 7-6. Sensitivity of 99Tc 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.
73

Figure 7-7. Sensitivity of 99Tc 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.
74

Figure 7-8. Sensitivity of 99Tc 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.
75

Figure 7-9. Sensitivity of 99Tc breakthrough (through the 6m layer) to the saturated water content using the

MULTIMED-DP 1.0, FECTUZ, CHAIN 2D and HYDRUS Models, where θs = cm3/cm3, 0.32

cm3/cm3, and 0.29 cm3/cm3.

0.35 
76

Figure 7-10. Sensitivity of 99Tc breakthrough (through the 6m layer) to the residual water content using the

MULTIMED-DP 1.0, FECTUZ, CHAIN 2D and HYDRUS Models, where θr = 0.103 cm3/cm3,

0.083 cm3/cm3, and 0.063 cm3/cm3.
77

Figure 7-11. Sensitivity of 99Tc breakthrough (through the 6m layer) to the van Genuchten retention parameter α

using the MULTIMED-DP 1.0, FECTUZ, CHAIN 2D and HYDRUS Models, where α = 0.059cm-1,

0.055 cm-1, and 0.051 cm-1.
78

Figure 7-12. Sensitivity of 99Tc breakthrough (through the 6m layer) to the van Genuchten retention parameter β

using the MULTIMED-DP 1.0, FECTUZ, CHAIN 2D and HYDRUS Models, where β = 1.59, 1.51,

and 1.43.
• 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 θ = 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,θ,ρ), 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,θ,ρ,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,θs,θr,α,β), 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,θ,ρ,D,DL,Dw,Ks,θs,θr

,α,β), 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,θ,ρ), 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,θs,θr,α,β), 

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 99Tc 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
80

Figure 7-13. Sensitivity and relative sensitivity of (a) peak concentration at the depth of 6m, (b) time to reach peak concentration at th

time to exceed MCL at the receptor well to the distribution coefficient using the CHAIN, MULTIMED-DP 1.0, FECTUHYDRUS Models. 
81

Figure 7-14. Sensitivity and relative sensitivity of (a) peak concentration at the depth of 6m, (b) time to reach peak concentration at th

time to exceed MCL at the receptor well to the recharge rate using the CHAIN, MULTIMED-DP 1.0, FECTUZ, CHAIN Models. 
82

Figure 7-15. Sensitivity and relative sensitivity of (a) peak concentration at the depth of 6m, (b) time to reach peak concentration at th

time to exceed MCL at the receptor well to the water content using the CHAIN, MULTIMED-DP 1.0, FECTUZ, CHAIN Models. 
83

Figure 7-16. Sensitivity and relative sensitivity of (a) peak concentration at the depth of 6m, (b) time to reach peak concentration at th

time to exceed MCL at the receptor well to the bulk density using the CHAIN, MULTIMED-DP 1.0, FECTUZ, CHAIN Models. 
84

Figure 7-17. Sensitivity and relative sensitivity of (a) peak concentration at the depth of 6m, (b) time to reach peak concentration at th

time to exceed MCL at the receptor well to the dispersion coefficient using the CHAIN Model. 
85

Figure 7-18. Sensitivity and relative sensitivity of (a) peak concentration at the depth of 6m, (b) time to reach peak concentration at th

time to exceed MCL at the receptor well to the dispersivity using the MULTIMED-DP 1.0, FECTUZ, CHAIN 2D, and H
86

Figure 7-19. Sensitivity and relative sensitivity of (a) peak concentration at the depth of 6m, (b) time to reach peak concentration at th

time to exceed MCL at the receptor well to the diffusion coefficient in water using the CHAIN 2D and HYDRUS Model
87

Figure 7-20. Sensitivity and relative sensitivity of (a) peak concentration at the depth of 6m, (b) time to reach peak concentration at th

time to exceed MCL at the receptor well to the saturated conductivity using the MULTIMED-DP 1.0, FECTUZ, CHAIN Models. 
88

Figure 7-21. Sensitivity and relative sensitivity of (a) peak concentration at the depth of 6m, (b) time to reach peak concentration at th

time to exceed MCL at the receptor well to the saturated water content using the MULTIMED-DP 1.0, FECTUZ, CHAIModels. 
89

Figure 7-22. Sensitivity and relative sensitivity of (a) peak concentration at the depth of 6m, (b) time to reach peak concentration at the 

time to exceed MCL at the receptor well to the residual water content using the MULTIMED-DP 1.0, FECTUZ, CHAIN Models. 
90

Figure 7-23. Sensitivity and relative sensitivity of (a) peak concentration at the depth of 6m, (b) time to reach peak concentration at th

time to exceed MCL at the receptor well to the van Genuchten retention parameter α using the MULTIMED-DP 1.0, FE

HYDRUS Models. 
91

Figure 7-24. Sensitivity and relative sensitivity of (a) peak concentration at the depth of 6m, (b) time to reach peak concentration at the 

time to exceed MCL at the receptor well to the van Genuchten retention parameter β using the MULTIMED-DP 1.0, FEHYDRUS Models. 
Table 7-3. Summary of Relative Sensitivities for the Outputs Obtained from the 99Tc Breakthrough Curves with Respect to All th

Parameters for All Models, Referenced to the Base Values of the Input Parameters. 

92

CHAIN 

MULTIMED­

DP 1.0 

FECTUZ 

CHAIN 2D 

HYDRUS 

CHAIN 

MULTIMED­

DP 1.0 

FECTUZ 

CHAIN 2D 

HYDRUS 

CHAIN 

MULTIMED­

DP 1.0 

FECTUZ 

Kd -0.06 -0.05 -0.05 -0.06 -0.06 +0.06 +0.06 +0.06 +0.06 +0.06 +0.07 +0.07 +0.07 

q +0.40 +0.00 +0.00 +0.12 +0.12 -1.10 -1.10 -1.10 -1.10 -1.10 -0.98 -1.00 -1.00 

θ -1.10 -0.62 -0.62 -0.90 -1.10 +0.80 +0.80 +0.81 +0.78 +0.80 +0.85 +0.92 +0.92 ρ -0.17 -0.05 -0.05 -0.05 -0.05 +0.06 +0.06 +0.07 +0.06 +0.06 +0.08 +0.07 +0.07 

D -0.45 __ __ __ __ -0.02 __ __ __ __ -0.10 __ __ 

DL __ -0.36 -0.36 -0.29 -0.28 __ -0.02 -0.02 -0.02 -0.02 __ -0.09 -0.09 

Dw __ __ __ -0.10 -0.10 __ __ __ -0.01 -0.01 __ __ __ 

Ks __ +0.04 +0.04 +0.05 +0.05 __ -0.05 -0.04 -0.04 -0.04 __ -0.05 -0.04 

θs __ -0.51 -0.51 -0.66 -0.66 __ +0.60 +0.60 +0.57 +0.57 __ +0.66 +0.66 

θr __ -0.20 -0.20 -0.29 -0.29 __ +0.22 +0.22 +0.22 +0.22 __ +0.26 +0.26 

α 

__ 

+0.05 +0.06 +0.13 +0.13 

__ 

-0.05 -0.08 -0.05 -0.05 

__ 

-0.06 -0.09 

β __ +0.86 +0.86 +1.30 +1.38 __ -1.00 -1.00 -1.00 -1.00 __ -1.13 -1.15 

C peak 

Model Input 

Parameter 

T peak T MCL 
93

• For the output Cpeak, 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 ρ, as one would expect for the 99Tc

BTCs as discussed in Section 6-2 (i.e., θ >> ρ Kd);

• Conversely, all output parameters for all models are rather sensitive to θ;

• 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 β, followed by θs,

while the output parameters are least sensitive to Ks and α.

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, θD, in Equation (6-4).

7.3.1 Correction of the MULTIMED-DP 1.0 Code

When the van Genuchten Model for the water retention parameters (Ks,θs,θr

,α,β) was considered in the

sensitivity analysis depicted in Table 7-1, an error in the originally distributed MULTIMED-DP 1.0 Code was

detected. ’s Law for the pressure head, which gives the water content in

the soil column through the use of the van Genuchten Model. 

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. 

use of the originally distributed MULTIMED-DP 1.0 Code are inconsistent with respect to those obtained from the

other three codes. 

θr, in the van Genuchten Module. 

with those of the other three models, as shown in Figure 7-25. 

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.

Figure 7-25. Water content distributions predicted by the HYDRUS, CHAIN 2D, FECTUZ, and MULTIMEDDP 1.0 models. �) obtained from the originally distributed

MULTIMED-DP 1.0 code are in error. 

distribution (�) with the other three models.

����������� ��������������

������

��������

����������� �����������

������

This error arose from the solution of Darcy

Figure 7-25 shows the water content profiles in the 6m

The water contents obtained through the

The reason for the inconsistency was found to be the incorrect use of the residual water content,

When this error was corrected, the new water content results became consistent

Figure 7-25 also shows that the steady-state solution

Note that the water contents (

The corrected code gives a consistent water content
7.3.2 Comparison of the Stehfest and DeHoog 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 99Tc BTCs using the 

MULTIMED-DP 1.0 Code. 

The Stehfest Algorithm (Stehfest, 1970) produced non-realistic oscillations in the 99Tc 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 99Tc 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 99Tc BTCs almost identical to those 

produced by the FECTUZ Code; 

• As an example of this comparison, Figure 7-26 shows the 99Tc BTCs for θ = 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, θD, 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 θD 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 99Tc BTC 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 θD, we increased the base value of DL in FECTUZ to 

6.53 cm. This gives a value of θD 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 99Tc BTCs for the base parameters for the 

HYDRUS, CHAIN 2D and CHAIN Models versus the 99Tc BTC 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, θD, and not to other factors. 

94
95

Figure 7-26. Sensitivity of 99Tc 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.

����

����

����

����

����

����

����������

Stehfest

DeHoog

DeHoog

(a)

(b)

(c)
96

Figure 7-27. Comparison of the 99Tc BTCs for the HYDRUS, CHAIN 2D and CHAIN Models for the base values

of the input parameters with the BTC for the FECTUZ Model with the base value of DL = 4.53 cm

replaced by the value DL = 6.53 cm.

��
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 (Šimúnek et al., 1998), 

• Two-dimensional CHAIN 2D Code (Šimúnek 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 timedependent 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 hysteretic 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 Cruces 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 

θ = Soil Water Content θs = Saturated Water Content 

ρ = Bulk Density θr = Residual Water Content 

D = Dispersion Coefficient α = van Genuchten Retention Parameter 

DL = Dispersivity β = van Genuchten Retention Parameter 

Output Parameters 

Cpeak = Peak Concentration of Radionuclide BTC 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 99Tc 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,θ,ρ,DL,Dw,Ks,θs,θr,α,β) are given in Figure 6-1, where the BTCs for the 

first six input parameters (Kd,q,θ,ρ,DL,Dw) were obtained under the constraints of constant recharge rate, q, and 

98
constant moisture content, θ, while those for the last five parameters (Ks,θs,θr

,α,β) were obtained under the 

constraints of constant recharge rate, q, and variable soil water content, θ(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, ρ 

and Kd, have a very small influence on the transport and fate of 99Tc since θ >> ρKd in the coefficients of the 

transport equations. For the first six input parameters (Kd,q,θ,ρ,DL,Dw), Cpeak is most influenced by θ, while Tpeak 

and TMCL are most influenced by both θ and q. For the last five input parameters (from the van Genuchten Model, 

Ks,θs,θr

,α,β), the most influential input parameter for all three output parameters is β, followed by θs, then followed 

by θr, α, and Ks. 

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,θ,ρ,D,DL,Dw) are compared under the constraints of constant recharge rate, q, and constant water content, θ. 

The last five input parameters (Ks,θs,θr

,α,β) are compared under the constraints of constant recharge rate, q, and 

variable water content, θ(z). Figures 7-1 to 7-12 compare the 99Tc 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,θ,ρ), 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,θ,ρ,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, θD, as given in Table 8-1. 

• For the base values of the van Genuchten parameters (K s,θs,θr,α,β), 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 θτwDw in θD (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 ρ, as one would expect for the 99Tc 

BTCs since θ >> ρKd in the coefficients of the transport equations. This will not be the case for higher Kd 

radionuclides. 

• Conversely, all output parameters for 99Tc BTCs are rather sensitive to θ for all models. 

• The output parameters (T peak, 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. 

Models Tested in Report 

Numerical Differences MULTIMEDand Errors 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 ransport Equations 

Finite Difference/Finite Element �

Solution Errors for Transport 

Equations 

Constant Recharge Rate, q, and �

Constant Water Content, θ 

Constant Recharge Rate, q, and �

Variable Water Content, θ(z) 

Numerical Integration Errors of �

Darcy’s Law 

Numerical Integration Errors of �

Richards Equation 

Numerical Errors in Deriving the �

Steady State Solution of 

Richards Equation 

Specification of Specifies Specifies Specifies Specifies Specifies 

θD = θτwDw + DL�q� only only only both both 

in the Transport Equations D DL DL DL and Dw DL and Dw 

Possible Numerical Diffusion �

in the Transport Equations 

� � � �

�

T

�

� � � �

� � �

�

�

�

�

• For the van Genuchten parameters (K s,θs,θr,α,β), the model output parameters are most sensitive to β, 

followed by θs, while the output parameters are least sensitive to Ks and α. 

• Numerical diffusion for the current modeling scenario is rather insignificant (see Figure 7-27) since the D Ladjusted 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 θD, 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 99Tc BTCs and Output Sensitivities with Respect to Model Input 

Parameters and Modeling Codes. CHAIN MULTIMED­ DP 1.0 FECTUZ 

CHAIN 2D 

HYDRUS 

CHAIN 

MULTIMEDDP 1.0 

FECTUZ 

CHAIN 2D 

HYDRUS 

Kd 7-1 7-1 7-1 7-1 7-1 7-13 7-13 7-13 7-13 7-13 

q 7-2 7-2 7-2 7-2 7-2 7-14 7-14 7-14 7-14 7-14 

θ 7-3 7-3 7-3 7-3 7-3 7-15 7-15 7-15 7-15 7-15 

ρ 7-4 7-4 7-4 7-4 7-4 7-16 7-16 7-16 7-16 7-16 

D 7-5 7-17 

DL 7-6 7-6 7-6 7-6 7-18 7-18 7-18 7-18 

Dw 7-7 7-7 7-19 7-19 

Ks 7-8 7-8 7-8 7-8 7-20 7-20 7-20 7-20 

θs 7-9 7-9 7-9 7-9 7-21 7-21 7-21 7-21 

θr 7-10 7-10 7-10 7-10 7-22 7-22 7-22 7-22 

α 7-11 7-11 7-11 7-11 7-23 7-23 7-23 7-23 

β 7-12 7-12 7-12 7-12 7-24 7-24 7-24 7-24 

Figures Comparing BTCs 

Model 

Input 

Parameter 

Figures Comparing Sensitivities 

101
Section 9 

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109
 

 





� �

� � �

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 et al, 1998) by a modified form of the Richards Equation: 

∂� ∂   ∂�  

∂t

 = 

∂z 

K 

 ∂z 

+ 1 





 - S , 

(A-1) 

where θ 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 (LT-1), 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-1). The 

formulation of Equation (A-1) 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) Kr(h,z) , (A-2) 

where Ks is the saturated hydraulic conductivity (LT-1) and Kr is the relative hydraulic conductivity (1). 

The HYDRUS Code features three models for the unsaturated soil hydraulic properties, θ and K. In 

homogeneous media, these properties can be expressed as nonlinear functions of the soil-water pressure head, θ(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 Cislerová (1988). 

The BC-Model is defined by the following equations: 

-n ��� �r  �� ���������� Se = =   , (A-3) s r 1 , h ≥ -1/�  

K = K S2/n + c + 2

, (A-4) se 

where Se is the effective water content, θs is the saturated water content, θr is the residual water content, α 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 (θr, θs, Ks, n, c, α). The quantities (θr, θs, Ks) can be measured in the laboratory or in the field, while the 

parameters (n,c,α) 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-1
 

� � �

� � �

 � � � s r  �r + m � � < � (A-5) n �������  



1 + �� 

  �

  �s � ���������� ≥ �  

2 

s e

c 



1 - (1 - S

1

e

/m )

m 

 K(h)=K S , (A-6)

where 

m = 1 -1/n, n > 1. (A-7) 

In some of the codes, other than that of HYDRUS, the parameter m is denoted by γ, and the parameter n is denoted by 

β. 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 (θr, θs, Ks, n, c, α). 

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: 

 � �� � m a  �a � m ������������ n s  ������  



1 + �� 

  � (A-8)   �s ������������ ≥ �s  

K K (h) , h ≤ h s r k    

K(h)= 

 

Kk + (h - hk )(Ks - Kk ) , hk < h < hs 

 

, 

 h - hk  (A-9) s 


 Ks , h ≥ hs 


where 

c 2 

� �� �(�)  

K =

Kk  Se   F ( )r 

r � �� �(�k )



 , (A-10) Ks 



Sek 



 

 F ( )r 

  ��� � 

1/m 

m 

a F(��������    � (A-11) 

  � � �a  m 

S =	 ek

k r . (A-12) s r 

This model allows for a nonzero minimum capillary height, hs, by replacing θs in the VG-Model by an extrapolated 

parameter, θm, which is slightly more than θs (Figure A-1a). In general, this change from θs to θm 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 θr in the equation for θ(h) by another extrapolated parameter θa < θr. 

A-2 
� �

�

� � � �

�

���������������θ 

���

θ 

θ�

�������������������������

��� �

� �

���������������� ����������������

Figure A-1. Schematics of the soil-water retention curve (a) and the hydraulic conductivity 

function (b) for the VC-Model (from Simunek et al., 1998). 

This model retains the physical meaning of θr and θs as measurable quantities while adding two new fitting 

parameters, θa and θm. Equation (A-10) assumes that the predicted hydraulic conductivity function is matched to a 

measured value of the function, Kk = K(θk), at some water content θk satisfying 

θk < θs

 , Kk < Ks . (A-13) 

The VC-Model contains 10 unknown parameters: θr, θs, θa, θm, α, c, n, Ks, Kk and θk. However, as in the VGModel, c is often assumed to be 0.5. When 

θa = θr, θm = θk = θs, Kk = Ks (A-14) 

the VC-Model reduces to the VG-Model. 

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 VGModels, 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: 

∞ 

H =∫ krwdhc , cm (A-15) o 

where krw 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(θ), (L2T-1): 

A-3
dh D(��������� � (A-16) d�

The macroscopic capillary length λs, which is equivalent to the wetting front suction head, is defined by 

�o 1 �s �

K(�o �������n � �

∫ 

n 

�������� � (A-17) 

where θo is the supply water content applied at the surface and θn is the dry, antecedent water content in the soil, 

ahead of the wetting front. The sorptivity, S ,is well approximated by 

�o 2 

S= ∫ (�o ����� ��n ) ������� � (A-18) 

�n 

where S , (LT-1/2), is defined by the early stages of an infiltration test. In Nachabe’s study, relationships were 

developed between λs, S , and shape factor b, and the parameters of the BC- and VG-Models for K(θ) and D(θ). 

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 λs 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 in b shows that λs is a scale-factor (see Appendix B) because: 

(1) λs renders predictions of infiltration rates fairly insensitive to the models used for K(θ) and D(θ); and 

(2) λs 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 λsscaled 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 soilwater 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. and A.T. Corey. 1964. Hydraulic Properties of Porous Media. Hydrology Paper No.3. Colorado State 

University. Fort Collins, CO. 

Morel-Seytoux, H.J., P.D. 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. 

Šimùnek, J., K. Huang and M. Th. van Genuchten. 1998. The HYDRUS Code for Simulating the One-Dimensional 

Movement of Water, Heat, and Multiple Solutes in Variably-Saturated Media. Version 6.0. Res. Rep. No. 144. 

U.S. Salinity Laboratory, Ag. Res. Ser., USDA, 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. and M. Cislerová. 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.1 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 search for 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-1
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 Θ. 

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 Θ. 

For example, the universal gas constant in units of cal/degree K has the physical dimensions of ML2/T2 Θ. 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. A 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
(b ) 

L 1 

L* 

(a ) 

(c ) 

L2 

Figure B-1. Geometrically similar figures, where (a) is the reference figure with characteristic length L*, (b) is a 

similar figure with characteristic length L1, L* = α1L1, with scale factor α1 = 2, and (c) is a similar 

figure with α2L2 = L*, α2 = 1⁄2 (reprinted from Unsaturated Zone 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). For example, two fluid 

motions denoted by Φ and Φ’ 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’ = α x, y’ = α y, z’ = α z, t’ = β t, m’ = γ m, (B-1) 

where (α, β, γ) are constant scale factors, and x, y, z, t, m have their usual meanings of length, time and mass. If T, 

T(Φ) = Φ’, represents the transformation in Equation (B-1) with scale factors (α, β, γ), then T-1, T-1 (Φ’) = Φ, 

represents its inverse transformation with scale factors (α-1, β-1, γ-1). The successive application of T and T-1, or T-1 

and T, leads back to the original fluid motion. Thus, the product T-1T 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-1): T1 with scale 

factors (α1, β1, γ1) and T2 with scale factors (α2, β2, γ2). The product of T1 and T2, apply T1 to Φ and T2 to Φ1, is 

defined by: T2 (T1 Φ) = T2 Φ1 = Φ2, where the scale factors of T2T1 are given by (α2 α1, β2 β1, γ2 γ1). 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, 

x1 ′ = x1 + c1, x′ 2 = x2 + c2 , x′ 3 = x3 + c3 ; (B-2) 

B-3
2. The time translation subgroup, 

t′ = t + c ; (B-3) 

3. The three parameter subgroup of rigid motions, 3 

x′ i = ∑ aik xk , (B-4) 

k = 1 

aik = most general 3x3 orthogonal matrix; 

4. The three parameter subgroup of moving axes translated with constant velocity, 

x = ′ xi - bit . (B-5) i 

Newton’s three laws of motion, for speeds much less than the speed of light, are also invariant under the GalileiNewton Group as they are under the group in Equation (B-1). 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 self-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 self-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 (1998a) 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 hydrologic 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. 

B-4
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-1. Such similar porous media will 

have the same porosities and the same volumetric water content θ. Thus, these similarity assumptions resulted in the 

following scaling relationships: 

Soil-Water Property Symbol and Dimensions Scaling Relationships 

Volumetric Water Content θ[1] θ* = θ 

Soil-Water Pressure Head h[L] h* = α h 

Hydraulic Conductivity K[LT-1] K* = α -2K 

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 α denotes the scale factor for the pores and solid particles combined (as shown 

in Figure B-1). 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 θs 

usually varies from place to place in a field such as shown in Figure B-2. In this figure, θs will usually vary (and thus, 

the θ-distribution) as one moves from vertical profile to vertical profile. That is to say, the soil profile pi 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 θ in the Miller-Miller similitude by Seo = θ/θs, where the residual soil moisture content 

θr

 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 θs 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 Scaling Relationships 

Relative Saturation Seo[1] Seo* = Seo 

Soil-Water Pressure Head h[L] h* = αhh 

Hydraulic Conductivity K[LT-1] K* =αk

-2K 

z 

Soil Surface 

R4 

R5 

R2 

R3 

P3 

P2 

P5 

P4 

P1 

R1 

y 

0 x 

Figure B-2. The depiction of a set of vertical soil profiles p1

,p2

i

in the xy-plane giving the global positioning of the vertical profiles, p

, ... 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 

1

,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 αh and αk 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 αr that minimized the sum of squares 

NM 

2 SS = ∑∑ (h*-� � � � ri (B-6) r = 1 i = 1 

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 (θ,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 SS in Equation (B-6) was the coalescing of the data in Figure B-3a 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,θ) 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 αh were not equal to those for scaling K,αk. Further, it should be noted that the values of h(θ) scaled 

in Figure B-3ab were those measured in the laboratory on soil cores removed from the field, and the values of K(θ) 

scaled in Figures B-3cd relied on the laboratory results for h(θ) to obtain estimates of θ(t) based on field tensiometric 

measurements. 

Figure B-3. (a) Unscaled observations of S eo(h), (b) Scaled observations of Seo(h*), showing the reference 

relationship as a solid curve, (c) unscaled observations of K(Seo), and (d) scaled observations of 

K*(Seo) (reprinted from Water Resources Research, 1977, by A.W. Warrick, G.J. Mullen and D.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 = αh h* , (B-7) 

* ��������r ����� 

�����������

 , (B-8) 

K(h) = � ������ � (B-9) k

where once again the asterisk denotes reference quantities and the three scale factors (� ��� ��� ) are assumed to be h 

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 VGModel, then the α fitting parameter in the model relates to the linear scaling factors and is dominated by the soil 

structure. The shape factors in the quantities θ(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 (� ��� ��� ) , reference hydraulic h 

characteristics (h*, θ*, 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, et al. (1991) similarity system include the resolution of the scaling factors

(� ��� ��� ) into products of two subfactors. One subfactor accounts for the local variation within a given profile h 

(the z-variation of profiles p1, p2, ... in Figure B-2) and the other subfactor accounts for the global variation between 

profiles (the R1, R2, ... locations in Figure B-2). However, the reference quantities (h*, θ*, 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 (� ��� ��� ) and the dynamic scaling factors (αt, αz, αv, αI, αQ). Because of the h 

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 (� ��� ��� ) scaling factors and reference h 

quantities (h*, θ*, 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 

Shape Parameters Correlate with Soil Texture 

Linear Variability in Soils 

Scaling Parameters Dominated by 

Soil Structure 

Soil 

Observations 

Texture 

1 

Texture 

2 

Texture 

m 

S11 

S12 

S1n 

S21 

S22 

S2p 

Sm1 

Sm2 

S mq 

Class 

1 

Class 

2 

Class 

m 

SOIL 

STRUCTURES 

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 sm1, 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 θ 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 Cruces 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 αh in h* = αh 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(θ). Saturated values for the hydraulic 

conductivity Ks were measured in the field at nearly 600 locations. The Ks data were scaled according to 

B-8
K(����k

2 ������ The probability distribution for αh and αk was found to be lognormal. The geostatistical horizontal 

variograms for the log-transformed scale factors (αh, αk) 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 (� ��� ��� ) to condition the hydraulic properties of the field to run their simulations. h 

Rather they used a constant value for θs, constant values for the slopes of Seo(h) and K(Seo), and the same 

distribution for αh and αk 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 (αh) but not for θ (αh). 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 

self-similarity. The Lie Group analysis of the Richards Equation, using the transformation 

(�′ ′ ��� ���′)�(����������)� (B-10) 

where (μ,δ,α) are independent scale factors, requires that the parameters D(θ), K(θ), and h(θ) must be power 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(θ), and also implies that h(θ) 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 unscaled, and the effects of 

length being unscaled. 

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 for the 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 in Section 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. 

186 pp. 

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. Boca Raton, FL. 368 pp. 

Gardner, W.R. 1958. “Some steady-state solutions of the unsaturated moisture flow equation with application to 

evaporation from a water table.” Soil Sci. 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. 210 pp. 

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. Boca Raton, FL 

201 pp. 

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 and K. 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 with a 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. 416 pp. 

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, P.L. 2000. Self-Similarity and Beyond: Exact Solutions of Nonlinear Problems. Chapman & Hall/CRC. Boca 

Raton, FL. 319 pp. 

B-10
Skoglund, V.J. 1967. Similitude: Theory and Applications. International Textbook Company. Scranton, PA 320 pp. 

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. 407 pp. 

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.” Water Resour. 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-11
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, θ(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 wellposedness 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), and Hlavá�ek et al. (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-1

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-1) 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 Order Parameter External Field 

ferromagnetic magnetization magnetic field 

ferroelectric polarization electric field 

liquid-vapor reduced density pressure 

martinsite strain stress 

phase separation concentration chemical potential differences 

C.2 Hysteresis Loops, Operators and Models 

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 

C 

E 

R 

v 

D 

A 

B 

F 

0 

u1 

u2 u 

Figure C-1. 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 u1 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 u1, the state (u, v) follows the upper curve of the loop CDA. If u decreases beyond u1, 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 u1 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 Γ from point C, it arrives at turning point A, where the 

state (u, v) now moves up curve Γ1. Rule 1 states that Γ1 is uniquely determined by the coordinates of A. 

(2) Point B is now taken as any point on curve Γ1 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 Γ2 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 

Γ from A to D. 

These rules of Madelung can be applied at any point on the loop ABCDA in Figure C-1, 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 

C 

B 

v 

D 

A 

0 u 

Γ 

Γ2 

Γ1 

Γ 

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, Jürgen 

Sprekels, with permission of Springer-Verlag, GmbH & Co. KG, Heidelberg, Germany). 

C-3 
� �

u1<u(t)<u2 , v(t) is not determined by the value of u(t) at the same instant, but also depends upon the previous 

evolution of u (the memory effect) and possibly on the initial state of the system. Therefore, the operator W should 

be written as: 

v(t) = W [ ] u ≡ W [u(t),H(u),u0 ,v0 ], (C-1) 

where H(u) represents the pertinent history of u and (u0, v0) is the initial state. 

For hysteresis, we also require that the path of the couple [u(t), v(t)] be invariant with respect to any increasing 

time homeomorphism (Visintin, 1994). This means that at any instant of time t, v(t) only depends on the range of the 

restriction on u (i.e., the values of u(t�) over the time interval o t ≤≤ ′ t) , and on the order in which the values are 

attained as t� increases from o to t. There is no dependence upon the time derivatives of u: du/dt, d2u/dt2,etc. In fact, 

these time derivatives (du/dt, d2u/dt2, ...) may not exist in general, or at least not exist at some points. This condition 

is essential for the existence of Figure C-1; for if it did not hold, the path of the state ( u, v ) would depend on (du/dt, 

dv/dt)and Figure C-1 would be invalid. This condition is commonly called rate independence, and means that the 

output is invariant with respect to changes of the time scale. Thus, hysteresis is characterized by two features: 

memory and rate independence. Rate independent memory is persistent and thus scale invariant, while rate 

dependent memory is typically fading and thus scale dependent (Visintin, 1994). 

The above restrictions on hysteretic phenomena excludes any viscous type memory such as those represented 

by time convolution. Typical hysteretic phenomena such as ferromagnetism, ferroelectricity and plasticity possess 

memory effects that are not purely rate independent since the hysteresis is coupled with viscous-type effects. But if 

the evolving state is not too fast, (du/dt, dv/dt) is relatively small in magnitude, and the rate independent component 

prevails. Thus, hysteresis is more evident in slow regimes, and rate dependent effects get larger as rates increase and 

vanish as (du/dt, dv/dt) approaches zero. Finally, hysteresis loops as shown in Figure C-1 need not appear in the 

physical systems listed in the above paragraphs. Some regions which are surrounded by hysteresis curves may be 

unbounded, some regions may be bounded but the hysteresis curves are discontinuous at various points, and some 

systems may possess rate dependent loops (e.g., electric losses in ferromagnetism due to eddy currents), see Visintin 

(1994). 

Visintin (1994) states that maybe the most powerful hysteresis model among those available for analysis is the 

Preisach Model. This model is the superposition of a continuous family of elementary discontinuous hysteresis 

models, called delayed relays. The Preisach Model was originally proposed for scalar ferromagnetism and offers a 

good qualitative description of several phenomena in that field. Since then, it has been regarded by engineers and 

physicists as a fundamental tool in several areas. In fact, it has been applied to hysteretic phenomena in the field of 

unsaturated flow through porous media. 

The underlying idea of the Preisach Model is quite simple and relies on the delayed relay shown in Figure C-3a. 

This relay with hysteresis can be defined by the four parameters shown in the figure: width a, height b, center point 

(uc, vc). Thus, we denote this relay by the expression R(a, b, uc, vc). The linear combination of a finite family of such 

delayed functions R(ai

, bi, uci, vci), i = 1, 2, ..., n, yields a hysteresis loop as shown in Figure C-3b, whose enclosed 

region is denoted by Rf. This construction, in the limit, allows one to approximate a fairly large class of continuous 

hysteresis laws (including those for soil-moisture parameters) and yields an operator which acts in the space of 

continuous time functions. With reference to Figure C-3b, this limiting procedure is such that the discontinuous 

hysteresis loop approaches the continuous loop which is the envelope of the finite combination of delayed relays. 

Further, the region Rf of the discontinuous loop approaches the region R of the continuous loop as shown in Figure 

C-1. Finally, the limiting operator, or Preisach Model, is denoted by 

v = P(u) . (C-2) 

The formal mathematical definition of the Preisach Model is given in Visintin (1994) and in Brokate and Sprekels 

(1996). 

Visintin (1994) considers the effects of incorporating the Preisach Model, as well as other hysteretic models, into 

systems governed by ordinary and partial differential equations. For example, the coupling of rate independent 

hysteretic nonlinearities with ordinary differential equations leads to interesting and complicated problems in the 

theory of nonlinear oscillations, such as the oscillator with a hysteretic restoring force: 

2 d u(t)

 + P(u) = f(t) , (C-3) dt2 

C-4 
v 

b 

a 

(u ,v ) c c 

0 u 

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). 

v 

u 0 

Rf 

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 Differential Models of Hysteresis, 1994, by Augusto Visintin, with 

permission of Springer-Verlag, GmbH & Co. KG, Heidelberg, Germany). 

C-5 
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, 

2 2 ∂ ∂ u ∂ u

∂ 

u

t 

+ 

∂

∂ 

t [P(u)] - ∂x

2 - ∂z2 = f(x,z,t) , (C-4a) 

and the wave equation with hysteresis, 

2 ∂ u ∂   ∂u  

∂t

2 - ∂x 

P 

 ∂x 





 

= 0 , (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, θ(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: 

∂2

�

∂

∂ 

t [��� ����]�� ∂z

2 � ∂

∂ 

z 

[����] ������� (C-5) 

where P(θ) 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(θ) 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 

Luckner and Schestakow (1991) in their book on subsurface processes. These discussions are concerned with 

C-6

ψ

ψ

ψ

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 ψ, is the free energy of water, or its capacity to do work (Miller 

and Donahue, 1995; Mauseth, 1998). When ψ > 0, the water is capable of doing positive work, while ψ < 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 ψm; 

2. The effects of dissolved substances – solute or osmotic potential ψs; 

3. The atmospheric or gas pressure effects – pressure potential ψp; 

4. The position of the water with respect to a reference state, a state usually taken as zero – gravitational 

potential ψg. 

In summary, the water potential for soils independent of a reference state is given by 

�w ���m ���s ����p , (C-6) 

and the total water potential is defined by 

� �� � � � � (C-7) t w g

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 ψs is zero for pure water and is 

always negative for solutions. The quantity ψp can either be positive or negative and the matrix potential ψm 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, ψm is always negative. In soils ψm is very important because so 

much soil water is tightly bound to soil particles; while in living cells, ψm is usually much less important than ψs and 

ψp. Miller and Donahue (1995) state that most productive soils do not have a depth of water standing on them (i.e., 

�ψp� is small) and most soils have few salts (i.e., �ψs� is small), thus 

�t ≅ �w ≅ �m � (C-8) 

So for most situations, ψm is the dominant component of ψt and represents about 95% or more of ψt. For living cells, 

the cell potential ψc is fairly well approximated by 

�c ≅ �s + �p ≤ �� (C-9) 

The description of hysteresis given by Guymon (1994) starts with how water is distributed in a pore and how ψm 

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
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��

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����

����

��

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��

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�

�

�

�

�

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 (h0 = 0), 

water is drained from the soil. However, because of the presence of narrow necks, the capillary forces at a given 

negative pressure head, h1 < 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 << h1 < h0 = 0. When the wetting cycle begins, 

the soil moisture increases and h2 � h1. 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 � h1 and is not due to the upper 

narrow necks. Therefore, at the given soil-water pressure head h1, 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
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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, θ(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 θ(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: 

d w � �� � � �� (C-10) s s

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 (θ,h) on the main drying and wetting curves or the moisture state (θ,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 
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, and LW. Canter, with permission of CRC Press, Inc., Boca Raton, FL). 

curves (Luckner and 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 w w code for �d being equal to � � or �d being greater than � � Usually, the following relationships for the main s s s s 

curves are assumed: 

�d ���w ��� �����d ����w ������������d ����w � (C-11) r r r

Thus, the main wetting curve is primarily differentiated from the main drying curve by the choice of αd < αw, and

d possibly by the saturated moisture contents if �sw < �sd 

� If �s

w = �s � 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, αh = 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: 

w K ≤ Kd

. (C-12) s s

The construction of the hysteresis loop for K(h) is similar to that of the moisture hysteresis loop. 

C-10 
References 

Aubin, J.-P. and A. Cellina. 1984. Differential Inclusions: Set-Valued Maps and Viability Theory. Springer-Verlag. 

New York, NY. 342 pp. 

Brokate, M. and J. Sprekels. 1996. Hysteresis and Phase Transitions. Springer-Verlag. New York, NY. 357 pp. 

Castrigiano, D.P.L. and S.A. Hayes. 1993. Catastrophe Theory. Addison-Wesley Publ. Co., Reading, MA. 250 pp. 

Chipot, M. 1984. Variational Inequalities and Flow in Porous Media. Springer-Verlag. New York, NY. 118 pp. 

Edwards, R.E. 1995. Functional Analysis: Theory and Applications. Dover Publ., Inc., New York, NY. 783 pp. 

Gilmore, R. 1981. Catastrophe Theory for Scientists and Engineers. John Wiley & Sons. New York, NY. 666 pp. 

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. 210 pp. 

Hlava�ek, I., J. Haslinger, J. Ne�as and J. Lovišek. 1988. Solution of Variational Inequalities in Mechanics. SpringerVerlag. New York, NY. 275 pp. 

Knox, R.C., D.A. Sabatini and L.W. Canter. 1993. Subsurface Transport and Fate Processes. Lewis Publ., Boca 

Raton, FL. 430 pp. 

Luckner, L. and W.M. Schestakow. 1991. Migration Processes in the Soil and Groundwater Zone. Lewis Publ. 

Chelsea, MI. 485 pp. 

Madelung, E. 1905. “ �ber Magnetisierung durch schnell verlaufende Str�me and die Wirkungsweise des 

Rutherford-Marconi-schen Magnetdetektors.” Ann. Phys. 17. 861-890 

Mauseth, J.D. 1998. Botany: An Introduction to Plant Biology, 2/e. Multimedia Enhanced Edition. Jones and 

Bartlett Publ. Boston, MA. 837 pp. 

Miller, R.W. and 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. 230 pp. 

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. 2nd Edition. McGraw-Hill, Inc. New York, NY. 424 pp. 

Visintin, A. 1994. Differential Models of Hysteresis. Springer-Verlag. New York, NY. 407 pp. 

C-11

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: 

* ��1 

∂

∂ 

c

t

1 � �

∂

∂ 

c

z

1 � ∂

∂ 

z 





 ��1 

∂

∂ 

c

z

1 



 + F1 ��1 ��1 ���1 �� 1 

��� (D-1) 

��2


∂c2 � �∂c2 � ∂ 



 ��2 

∂c2 

 + F2 ��2 ��2 ���2 ��1,�1

*

,�

*

2 ��� (D-2) ∂t ∂z ∂z  ∂z  

* ∂si = H(ci ,s*

i ) - (�i ��′ i ��

*

i ����i �������������� (D-3) ∂t 

where θ 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, � is the degradation rate of s*, μ’ is the rate of transfer of material from 

parent to daughter, and γ 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 G1 (s1 ) 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 G (c1,s1 ,s2 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(ci ,si ) could be thought * of as the first-order sorption rate of s1 due to the difference between equilibrium concentrations and nonequilibrium 

concentrations. 

The system given in Equations (D-1) 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-1
� � � �

���

� � �

Radionuclides 

Nitrogen Compounds 

��������

Uninterrupted Chain --

One Reaction Path for Pesticides 

�

� �

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Interrupted Chain-­

Two Independent Reaction Paths for Pesticides 

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 

�

η 

η 

η 

D = Number of Daughters < 6 

ηlj = Splitting Factors 

D 

∑ � �� � 1j j = 1 

D-2
�

� � � �

Simple Converging Chains 

� � ���

N < 7 

Seven-Member Branching Chain 

�

η η 

�

η η 

�

η η 

η12 + η13 = η24 + η25 = η36 + η37 = 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 

� �

� �

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: 

��� ��� ��� ���

�� � �� �����

The author states that the most critical species is the last one in the chain, 226 Ra. 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. 

D-4
Appendix E

The Impact of Using a Nonuniform Moisture Distribution

versus a Uniform Distribution

Constant values of q, θ, and h were assumed for all models in the sensitivity analyses for the input parameters 

Kd, q, θ, ρ, D, DL and Dw. On the other hand, a nonuniform water content distribution was employed by MULTIMEDDP 1.0, FECTUZ, CHAIN 2D and HYDRUS for the analyses involving Ks, θr

, θs, α, and β. 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, θs, θr, α, β) 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, θ. 

When corrected, all systems were defined for a constant recharge rate, q, and boundary conditions resulting in the 

following conditions for θ: θ = base value = 0.16, at the surface; and at 6 m, θ = saturated value = 0.32. Figure E-1 

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-1. 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 99Tc for a uniform moisture content (θ = 0.16) with those 

for a nonuniform distribution (Figure E-1), one should note the degree of nonlinearity in the distribution shown in 

Figure E-1. Because of the homogeneity of the 6m layer, one notes that the steady state distribution is nearly 

uniform (θ � 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-1) 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 99Tc 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 θ which is replaced by the distribution given in Figure E-1. No CHAIN results are given 

in the top graph since θ 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 θ 

reduces the Cpeak and increases the Tpeak and TMCL for each model. This is reasonable if we remember that the 

nonuniform θ-distribution increases the overall moisture in the soil column versus that for θ = 0.16 throughout the 

column. Increasing moisture reduces the advection term in the transport equation and increases the diffusion term. 

E-1
E-2

Figure E-1. Water content distributions predicted by the HYDRUS, CHAIN 2D, FECTUZ, and MULTIMED-DP

1.0 Models. �) obtained from the originally distributed MULTIMEDDP 1.0 Code are in error. �) with

the other three models.

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������

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, θ, ρ, 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. 

BTCs are given in Figure E-3 (Kd comparison) and Figure E-4 (DL comparison). peak values for

the nonuniform water content BTCs have been uniformly reduced over the ranges of the input parameters compared

to those for a constant θ, while the Tpeak values and TMCL values have been increased uniformly over the input

ranges. able E-1 lists the output values for the uniform and nonuniform water content cases, values which are

measured at the base values of the input parameters Kd and DL. 

of these three outputs to the inputs Kd and DL, again referenced to the base values of Kd and DL. 

the relative sensitivities for the uniform and nonuniform cases are basically the same. 

output values, based on the nonuniform case as the standard, are as follows:

 Percent Differences

Output Kd DL

Cpeak 15.5% Decrease 12.1% Decrease

Tpeak 6.7% Increase 5.9% Increase

TMCL 6.4% 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

θ for the analysis of the sensitivity of the three outputs to the input parameters Kd, q, θ, ρ, 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. , it should be emphasized that

these results are for a homogeneous layer under steady state conditions.

Note that the water contents (

The corrected code gives a consistent water content distribution (

These comparative

As in Figure E-2, the C

T

In addition, the table gives the relative sensitivities

As one can see,

The percent differences of the

However
E-3

Table E-1 Comparison of Results Derived from Figures E-3 and E-4 for the Distribution Coefficient Kd and the

Dispersivity DL, respectively. alues 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 Property Kd in ml/g DL in cm

Uniform Nonuniform Uniform Nonuniform

Water Content Water Content WaterContent WaterContent

Cpeak Base Value (mg/l) 0.00732 0.00634 0.00725 0.00647

Relative Sensitivity -0.06 -0.05 -0.28 -0.25

Tpeak Base Value (d) 4766 5108 4737 5036

RelativeSensitivity +0.06 +0.06 -0.02 -0.02

TMCL Base Value (d) 3554 3799 3552 3783

Relative Sensitivity +0.07 +0.07 -0.08 -0.07

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. 

nonuniform water content and the bottom curves are for θ = 0.16 throughout the soil column.

There are no CHAIN results in the top graph because θ can only be constant in this model.

The V

The top curves are for a
E-4

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Figure E-3. Sensitivity of 99Tc 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).

�������� ��������������

Figure E-4. Sensitivity of 99Tc 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).
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 yearafter-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 (BTC) 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 timestep 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, flow 

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 twodimensional 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-1
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 F1 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-1a, 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 α(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 α(h) used in the current simulation was the Feddes Module of the 

HYDRUS Code (Šimúnek, et al, 1998). This function is the simple trapezoid shown in Figure F-2. The quantity h1 

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, α(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-1b) and during the dry years (Figure F-1a). 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
F-3

Figure F-1. The annual precipitation amounts and the monthly average amounts in centimeters for the Las Cruces, NM Site,corresponding to the daily record in Figure 5.2

	


3 53 02 52 0

22.5

1 51 0

5

0

83 84 85 86 87 88 89 90 91 92 93

M ean Annual Precipitation

94 95 96 97 98 99 00


	

 

3.6

0

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec




	






3.33.02.72.42.11.81.51.20.90.60.3
F-4

Figure F-2. The water stress response function for the Feddes Module of the HYDRUS Code (Šimúnek, et al.,

1998).

	

	

		

1.0

0.5

0

h3 h2 h1 0 h4

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, (h1, h2, h3). The

precipitation/PET segment from “a to b” is repeated from “b to c.”













yr)

yr)
• 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 m vadose 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 = 1000d 

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 1000d, 87 mm/yr recharge rate for initial spreading of pollutants, 

2. 1000d 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 11,585d, repeat of the wettest period, 

5. 11,585d 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 1000d 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 1000d 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 11,585d and 11,585d 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 99Tc 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 BTC 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
F-6

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. 

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. d 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”. 

primarily due to the nonlinear cumulative recharge shown in the top set of curves in Figure F-3. 

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 Tpeak 0.88 times smaller. 

5015 days, the average recharge rate drops sharply to 20 mm/yr, thus giving a long “heavy tail” to the

distribution. ger 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 BTC for a constant

recharge rate of 67 mm/yr. 

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. 

Figure F-4. Comparison of predicted 99Tc breakthrough curves (through the 6 m layer) using the variable

precipitation/actual ET versus uniform recharge rate in the HYDRUS Model. verage 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, (h1,h2 ,h3 ).

�� yr)

yr)

At the scale shown in Figure F-4, the daily and

These percentages are consistent with those found for the parameters K

This is

The

After

At 7570 days the lar

Comparing this curve to results for the uniform moisture case, one sees that the

These results are consistent with those of Appendix E.

A
References 

Eagleson, P.S. 1970. Dynamic Hydrology. McGraw-Hill Book Co. New York, NY. 462 pp. 

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. 

5 pp. 

Šimúnek, J., K. Huang, and M.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. USDA, 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 Cruces 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,θs, θr, α, β) at the Las Cruces 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 99Tc 

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 < θs = 0.32 < 0.343, 8.1 %, 

0.071 < θr = 0.08 < 0.091, 13.8 %, 

0.027 cm-1 < α = 0.055 < 0.070 cm-1, 50.9 %, 

1.383 < β = 1.51 < 1.711 , 13.3 %. 

The sensitivity results derived for the HYDRUS Code in Section 6 , a code which can handle layered soil 

columns,show that of these five parameters, Ks is the least sensitive, followed closely by α; and β is the most 

sensitive, followed by θs and then by θr. The quantity β appears to be about twice as sensitive as θs, and θs about 

twice as sensitive as θr. Combining these relative sensitivity results with the above ranges and percent deviations, one 

sees that calculating the breakthrough curves (BTCs) for 99Tc using the layered profile should not depart too 

radically from the homogeneous layer calculations. 

Figure G-1 illustrates one set of differences between the homogeneous-layer calculations and the layered-soil 

calculations, using the HYDRUS Code. The comparative results are for three different values of the distribution 

coefficient Kd shown in the figure (0.001, 0.007, 0.019 ml/g). For both the layered and uniform soil profiles, the 

nonuniform soil moisture content was used, i.e., the Richards Equation was solved for constant recharge rate q. For 

the uniform layer case, the base values in Section 6 were used (i.e., the base values for q, ρ, DL, Dw, Ks, θs, θr, α, β). 

For the layered case, the various values of Ks, θs, θr

, α, and β in Table 5-2 were used, along with the base values of 

q, ρ, DL, and Dw given in Section 6. 

The results in this figure indicate that the Cpeak values for the layered soil are less than those for the uniform 

layer, while the values for Tpeak and TMCL are greater. Considering the parameters which are the most sensitive (β, 

θs, θr), the changes in the outputs due to θs and θr are in the opposite directions to those for β. Thus, in the first three 

layers (0 to 205 cm), θs and θr tend to reduce Cpeak and increase Tpeak and TMCL over those for the uniform layer 

case, while β has the opposite effect. In the next two layers (205 to 305 cm), all three parameters tend to increase 

G-1
G-2

Figure G-1. Sensitivity of 99Tc 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 and decrease Tpeak and TMCL. θs and β tend to increase Cpeak and decrease

Tpeak and TMCL, while θr has the opposite effect. β tends to decrease Cpeak

and increase Tpeak and TMCL, while the effects due to θs and θr are fairly close to those of the uniform case. 

this collection of “facts,” it is difficult to dogmatically say how the layered and uniform cases should compare. 

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) Cpeak Tpeak TMCL

0.001 9.9% decrease 3.4% increase 2.4% increase

0.007 9.5% decrease 2.7% increase 2.2% increase

0.019 9.0% decrease 2.7% increase 2.0% increase

As suggested above, the differences between the results for the uniform soil and the 

great. 

homogeneous layer are also applicable for the layered soil described in Table 5-2. ferences 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. , 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. Water Flow in Soils. Marcel Dekker, Inc., New York, NY,

296 pp.

Rockhold, M.L., C.S. Simmons and M.F. Fayer. “An analytical solution technique for one-dimensional,

steady vertical water flow in layered soils.” Water Resour. Res. 

In the layer from 305 to 370 cm, 

In the final three layers (370 to 600 cm), 

From

Thus,

layered soil are not too

And because of this, it seems reasonable to expect the sensitivity studies given in this report for the 6m

The dif

However

1993. 

1997. 

33(4):897-902.
 

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 liquidsolid 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: 

∂c q ∂c ��w �w ���L � ∂2

c � + = 

∂t ��� ���d ∂� �������d ∂�

2 - ����

�������d 

��� � (H-1) 

∂s

 = D/P - ���� (H-2) ∂t 

where μ is the radioactive decay rate for both s and c, and D/P is a decay/production, sorption transfer function 

defined by 

( )�d � � � D/P ≡ � ��� � (H-3)

H-1
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, ω, 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, (ω,f). 

When D/P = 0 (i.e., the nonequilibrium sites act as equilibrium sites), these transport equations reduce to the 

following: 

∂c q ∂c ��w �w ���L � ∂2

c 

∂t

 + 

�������d ∂�

= 

� �����d ∂z

2 - ���� (H-4) 

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 ω is sufficiently large so as to dominate the effect of the decay rate, 

μ. 

H.2 The Transport of 99Tc with a Kd = 0.007 ml/g 

As a first example of Equations (H-1) and (H-2), let us consider the transport of 99Tc 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 99Tc, the decay rate, μ, 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 99Tc is 2.1 x 105 years. Thus the important decay/production in these two equations is D/P and not μ. 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., μs), 

s will approximately be equal to (1 - f)Kdc for sufficiently large ω. Thus, under sufficiently high sorption rates ω, 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 lowq 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., 99Tc) has a very small Kd. 

Using the HYDRUS Code with the nonuniform distribution for θ and the base values of Section 6 for the other 

10 standard input parameters, the BTCs of 99Tc for different values of (ω, f) were derived. One set of BTCs were 

obtained for f = 0.47 and ω = 0.003, 0.032, and 0.320d-1. Another set of BTCs were obtained for ω = 0.032d-1 and f 

= 0.27, 0.47, 0.67. These five curves were compared with the equilibrium case, f = 1 and ω = 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-1) and (H-2) are not exactly the ones solved by the HYDRUS Code since θ is not constant but is 

variable with depth as shown in Figure E-1. However, for order-of-magnitude arguments, Equations (H-1) to (H-6) 

are adequate. 

Considering Equations (H-1) and (H-2), one sees that the parameters (ω, f) affect three processes (terms): 

advection, diffusion, the decay/production term D/P. When s reaches its equilibrium value of (1-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 = (1-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 (1-f)Kdc. 

H-2
μ ω

For 99Tc, 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 (ω, f) given above: 

[(1 - f) Kdc] ≅ 0.51% of c(t) . (H-7) max

These results are verified when the “nonequilibrium” 99Tc BTCs for the five combinations of (ω, f) mentioned above 

are compared with one another and with the equilibrium case for (ω, 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-1) 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 ω is sufficiently large. 

H.3 The Transport of 99Tc with a Kd = 1.0 ml/g 

For illustrative purposes, we next considered 99Tc with a Kd equal to unity and (ω, 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): 

[(1 - f) Kdc]max = 100% of c(t), (H-8) 

Table H-1 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 99Tc BTCs. The equilibrium cases are given by the pair (ω, f) = (0,1), while the “nonequilibrium” 

cases are given by the pairs (ω, 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 (1-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-1.
 Comparison of Nonequilibrium and Equilibrium Results for 99Tc BTCs for Kd Values of 0.007 and 

1.0 ml/g. 

Radionuclide Kd μ ω f Time c (1-f)Kdc s 

ml/g (d-1) (d-1) (d) mg/L mg/kg mg/kg 

Technetium 

99Tc 0.007 9.043 x10-9 0 1 3750 0.00105 0 0 

5005 0.00646 0 0 

6250 0.00227 0 0 

0.007 9.043 x10-9 0.032 0.47 3750 0.00111 4.12 x 10-6 4.12 x 10-6 

5005 0.00644 2.39 x 10-5 2.39 x 10-5 

6250 0.00222 8.24 x 10-6 8.19 x 10-6 

1 9.043 x10-9 0 1 40,000 3.67 x 10-4 0 0 

46,500 6.22 x 10-4 0 0 

60,000 1.38 x 10-4 0 0 

1 9.043 x10-9 0.032 0.47 40,000 3.67 x 10-4 1.94 x 10-4 1.90 x 10-4 

46,500 6.12 x 10-4 3.24 x 10-4 3.24 x 10-4 

60,000 1.38 x 10-4 7.31 x 10-5 7.35 x 10-5 

1 9.043 x10-9 0.032 0 40,000 3.67 x 10-4 3.67 x 10-4 3.63 x 10-4 

46,500 6.02 x 10-4 6.02 x 10-4 6.01 x 10-4 

60,000 1.38 x 10-4 1.38 x 10-4 1.40 x 10-4 

H-3
H-4

equilibrium sorption balance through the formula (1-f)Kdc = s. peak (at t =

46,500d) for the Kd = 1.0 ml/g case from 6.22 x 10-4 mg/L for the equilibrium case (ω, f) = (0, 1), to 6.12 x 10-4 mg/

L for (ω, f) = (0.032.0.47) -4 mg/L for (ω, f) = (0.032, 0).

H.4 The Transport of 90Sr with a Kd = 1.0 ml/g

The third case that we considered was the base case scenario for 90Sr; this radionuclide has a default Kd of 1.0

ml/g and a half-life of 10,585 days. 

radionuclide. 90Sr in solution for the parameter pairs (ω, f) =

(0, 1), (0.032d-1, 0.47), and (0.032d-1, 0). 

means that the two “nonequilibrium” cases, (ω, f), (0.032d-1, 0.47), and (0.032d-1, 0), are the same as the equilibrium

case (ω, f) = (0, 1). , Figure H-1b shows that the D/P term is essentially zero for all three parameter pairs (this

is trivially so for ω = 0 and f = 1):

D/P [ d = � �� ����� � � � ≅ �� 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-1 and those shown in Figure H-1ab 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 < ω < 0.32d-1, and for the specific radionuclides considered in this report, with Kd values that range

from 0 to 5 ml/g. ficients in Equations (H-1) and (H-2).

For example, in the 90Sr case, the coefficients for (ω,f) = (0.032d-1, 0.47) are as follows:

Figure H-1. The c- and s distribution of Equations (H-1) and (H-2) for 90Sr for (ω,f) = (0,1),

(0.032 d-1, 0.47), (0.032d-1, 0). 

concentration in solution and (b) gives the concentration on the soil matrix.

(a)

(b)

In addition, one sees a slight drop in the C

to 6.02 x 10

Thus, one would expect significant decay over the lifetime of the BTCs for this

Figure H-1a shows the BTCs for the concentrations of 

The BTCs for these three parameter pairs are basically equal, which

Further

]

This claim is further supported by the magnitude of the coef

Distributions were derived by the HYDRUS Code, (a) gives the
 

 





Nonhomogeneous 

Equation Radioactive Decay Sorption Decay Sorption Term 

Eq. (H-1) -6.55 x 10-5c -3.01 x 10-2c 

-6.55 x 10-5s -3.20 x 10-2s 

+5.67 x 10-2s 

Eq. (H-2) +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 

90Sr 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-1a and H-1b, 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 Kd = 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 ω 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 ω of 0.032d-1.. 

Therefore, the question that arises is the following: At what sorption rates ω 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 μ is given as 6.55 x 10-5d-1. However, 

the reader should be aware that the values of ω for which 90Sr possesses “true” nonequilibrium sites in this Las 

Cruces 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 ω that are required to unbalance equilibrium sorption 

conditions for the very low recharge rates considered in this report. 

For the radionuclide, 90Sr, with a Kd = 1.0 ml/g and an f = 0 (i.e., no equilibrium sorption sites, only 

nonequilibrium sites), Equations (H-1) and (H-2) become: 

2 ∂c ∂c ∂ c �

∂t

 + v 

∂z = D ∂z

2 - ���� �

�������� (H-10) 

∂s

 + (��������� �� , (H-11) ∂t 

where the coefficients (v,D) are defined by 

v = q/�����������w �w � �L � ����� (H-12) 

Equation (H-11) 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-11) results in the following: 

t 

s(z,t) = ��∫ ���� �(�����)(��� �) ����������� (H-13) 

0 

Combining Equations (H-10) and (H-13) produces the following linear, integrodifferential equation for the liquid 

phase concentration c: 

2 ∂c ∂c ∂ c 

+ v = D - ����

2 ∂t ∂z ∂z 

t 

+ �� 

 

�∫ ��� �(�����)(��� �) ��������� � �

 � (H-14) 

�  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,η) from the initial time η = 0 to the current time η = t at the point z in question. As we have seen 

in Section H.2, this integral term loses its historic dependence if ω is sufficiently large. But what happens if ω is 

sufficiently small? To answer this question, we let ω equal a sequence of values from 6.5 x 10-6d-1 to 6.5 x 10-1d-1, 

remembering that μ equals 6.55 x 10-5d-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, θ(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 

ω 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 ω > 

6.5 x 10-4d-1 (keep in mind that Figure H-2a is a semi-log plot), but are radically different from the “bell-shape” for 

ω = 6.5 x 10-6d-1 and 6.5 x 10-5d-1. For these lowest two values of ω, 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 ω > 6.5 x 10-4d-1, and a non “bellshape” for the lowest two ωs. In fact, as ω 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 ω increases, the nonequilibrium sites “convert” to 

equilibrium sites, the “history” term in Equation (H-14) becomes less effective, and Equations (H-1) 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, Cpeak and Speak, respectively, and their corresponding times to Tpeak. The values for the BTCs 

and CCs are quite different for the lowest two ω values, while from ω equal to 6.5 x 10-4d-1 and higher, the values for 

the BTCs begin to quickly approach those for the CCs. For ω > 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 ω > 0.032d-1 and the pollutant is 90Sr. For other 

pollutants with different Kds and decay rates, μ, 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 sdistributions 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 ω. Figure H-3 is for a rate ω of 

6.5 x 10-5d-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-bellshapes 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 ω of 6.5 x 10-4d-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 ω of 6.5 x 10-3d-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 sdistributions 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
H-7

Figure H-2. (a) Breakthrough curves for the liquid phase concentration at the 6 m level. 

curves for the nonequilibrium solid phase at the 6 m depth. ω = 6.5 x 10-2d-1 and 6.5 x

10-1d-1 are basically the same for both (a) and (b).

C

���

���

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 Solid Phase Concentration Liquid Phase Concentration

Rate, ω

(d-1) peak (mg/kg) Tpeak (d) Cpeak (mg/L) Tpeak (d)

6.5 x 10-6 6.5 x 10-4 6,300 6.3 x 10-2 4,900

*6.5 x 10-5 6.7 x 10-4 9,500 6.2 x 10-3 4,700

6.5 x 10-4 4.1 x 10-4 37,100 4.6 x 10-4 35,200

6.5 x 10-3 5.6 x 10-4 43,100 5.7 x 10-4 43,200

**3.2 x 10-2 6.2 x 10-4 43,500 6.2 x 10- 4 43,500

6.5 x 10-2 6.2 x 10-4 43,500 6.2 x 10- 4 43,500

6.5 x 10-1 6.2 x 10-4 43,500 6.2 x 10- 4 43,500

* Decay Rate μ for 90Sr is 6.55 x 10-5d-1.

** Data from Figure H-1.

(b) Concentration

Curves for 

S
H-8

final sets of distributions are given in Figure H-6, which are for ω = -2d-1; the curves for ω = -1d-1

were found to be “graphically” equivalent to those for ω = -2d-1. 

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. ucson, AZ.

Available from www.wmsym.org/wm02 (click on WM’02 Proceedings).

Thibodeaux, L.J. Environmental Chemodynamics: Movement of Chemicals in Air, Water, and Soil. nd

Edition. iley & Sons, Inc. ork, NY. 

Toride, N., F.J. Leij, and M.T. van Genuchten. “A comprehensive set of analytical solutions for

nonequilibrium solute transport with first-order decay and zero-order production.” Water Resour. Res. 

2167-2182.

van Genuchten, M. Th., and R.J. Wagenet. “Two-site/two-region models for pesticide transport and

degradation: theoretical development and analytical solutions.” Soil Sci. Soc. Am. J. 53, 1303-1310.

Figure H-3. Liquid phase concentration curves (a)

and solid phase concentration curves

(b) for 90Sr, for various times and for

ω = -5d-1, where zero depth

is the surface and -600 cm is the

hypothetical water table.

C

���

���

Figure H-4. Liquid phase concentration curves (a) and

solid phase concentration curves (b) for

90Sr, for various times and for ω = 

10-4d-1, where zero depth is the surface

and -600 cm is the hypothetical water

table.

C

���

���

6.5 x 10 6.5 x 10

6.5 x 10 As one can see, for all times greater than

February 24-28, 2002, T

1996. 2

John W New Y 593 pp.

1993. 

29(7),

1989. 

6.5 x 10

6.5 x
H-9

Figure H-5. Liquid phase concentration curves (a)

and solid phase concentration curves

(b) for 90Sr, for various times and for ω

= -3d-1, where zero depth is

the surface and -600 cm is the

hypothetical water table.

C

���

���

Figure H-6. Liquid phase concentration curves (a) and

solid phase concentration curves (b) for 90Sr,

for various times and for ω = 6.5 x 10-2d-1,

where zero depth is the surface and -600 cm

is the hypothetical water table.

C

���

���

6.5 x 10
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, μ, 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 I-1. In this figure, the ordinate is the concentration, C, at the bottom of the soil column, normalized by the 

concentration, Co, of the source at the top of the column, the ordinate being denoted as [C/Co]B. The values of 

[C/Co]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/Co]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 I-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 I-1, are potentially correct, and have value in relating diffusive 

dilution and radioactive decay to pollutant travel time. 

Figure I-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

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 I-1 for this report’s 6 m, homogeneous soil column. These 

proposed locations are as follows: 

• 99Tc and 238U are represented by Region I; 

• Travel time through the 6 m column for 238U is about 5 times greater than that for 99Tc, thus diffusive 

dilution is much greater for 238U than for 99Tc; 

• 3H is represented by the left-hand side of Region II; 

• 90Sr 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 99Tc and its daughter, ruthenium 99Ru, for the 

6 m, homogenous soil column. The other set of calculations used the CHAIN Code to compute the Cpeak - and Tpeakvalues 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. 

I.1 The CHAIN Governing Equations 

The CHAIN Equations covering the transport and fate for both these comparative exercises are as follows: 

∂c q ∂c �� ∂2

�

∂t

 + 

��� ��d ∂�

= 

������d ∂�

2 - ���� (I-1) 

* * ∂c q ∂c �� ∂2

�

* 

** ��� ��d 

∂t

 + 

��� ��*

d ∂�

= 

������*

d ∂�

2 - � � ���

� ����*

d 

������ (I-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, θ, ρ, 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 μ and the half-life, t1/2, is given by: 

��� (���e �) ÷ t. (I-3) 1/2 

Table I-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 (600 cm) ÷ (Advection 

Term Coefficient) 

3H 4,000 d 

99Tc 4,300 d 

238U 21,000 d 

90Sr 46,500 d 

238Pu 216,600 d 

I-2

μ

Table I-1. Coefficients of the Advection, Diffusion, and Sink Terms in Equation (I-1). 

Initial Advection Term* Diffusion Term* 

Concentration Default Half-Life First-Order* Coefficient Coefficient 

in Recharge Value t

1⁄2 Decay Rate q ��

Water C0 Kd μ ������d ������d 

Radionuclide (mg/L) (ml/g) (d) (d-1) (cm/d) (cm2/d) 

Tritium 3H 1.1 x 10-7 0 4380 1.5825x10-4 0.15000 1.00000 

Technetium 99Tc 1.25 x 10-2 0.007 76,650,000 9.043x10-9 0.13962 0.93077 

Uranium 238U 417 0.4 1.6425x1012 4.2201x10-13 0.02857 0.19048 

Strontium 90Sr 2.0 x 10-1 1 10,585 6.5484x10-5 0.01290 0.08602 

Plutonium 238Pu 1.0 x 10-7 5 32,120 2.1580x10-5 0.00277 0.01848 

*
 q, θ, ρ, and D are assigned the base values in Section 6; 

Kd is assigned values given in Column Three; 

μ is obtained from half-life t1/2 using Equation (I-3). 

Comparing the times, (600 cm ÷ advection term coefficient), with the half-lives listed in Table I-1, one observes 

that the first-order decay rate μ 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. 

I.2 Breakthrough Curves for 99Tc and It’s Daughter, 99Ru 

For the decay chain, 99Tc ��99Ru, 99Tc has a very low decay rate μ which has negligible effect on the BTC 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; μ* in Equation (I-2) is zero 

for 99Ru. Since FECTUZ has a different θD-value than CHAIN (see Equation I-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 (I-2) if one ignores all terms except the source term and the time derivative of c*: 

t 

c* ≅ 1.8 x 10-10 ∫ c(t)dt . (I-4) 

0 

The integral term in Equation (I-4) can be approximated from the BTC 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 (I-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 (I-2) is the source of 99Ru being produced by the 

decaying 99Tc. 

I-3
I-4

Figure I-2. (a) Breakthrough curves of the daughter product, 99Ru, from the 99Tc decay using the CHAIN and

FECTUZ Models for base case simulation, and (b) the breakthrough curve for 99Tc. 

explain a, b, c, d, e and f.

�

�

�

�

�

�

���

���

� �

�

Figures I-2ab, as stated above, show the BTC’s at the 6 m soil depth for the parent (99Tc in Figure I-2b) and for

the daughter (99Ru in Figure I-2a) using the CHAIN and FECTUZ Codes for the base case simulation given in

Section 6. 

BTC’s. Point a represents the beginning of the 99Tc BTC for either the CHAIN or FECTUZ simulation. Since 99Ru

is continually being generated by 99Tc, although very slowly, Point a also represents the initiation of the 99Ru BTC.

From Point a to Point b, the 99Ru BTC sharply rises as shown in Figure I-2a. Point b on the 99Tc BTC is close to the

inflection point of the “bell-shaped” curve. Point c on the 99Tc BTC, the remaining mass of the 99Tc released by

the surface source during its 1000 day lifetime has moved out of the 6 m thick soil layer. 

the 99Ru BTC first reaches its maximum value. 

Because of the large Kd of 5 ml/g for 99Ru, there is still a reservoir of this stable radionuclide in the 6 m soil column.

As seen by both the CHAIN and FECTUZ Codes, this reservoir produces a constant, or nearly constant, peak in the

BTC from Point c to Point d (a total time interval of more than 140,000 days). Point d to Point e in Figure I2a, one notes the classical “bell-shaped” curve as the 99Ru BTC goes from a maximum value at Point d to zero at

about Point e. 

Points d and e. Points d and f, the FECTUZ result is above that of CHAIN; the reverse is true from Point

f to e. 

difference (in the right direction) in the models from Point a to Point b. 

quantify graphically.

Overall, the CHAIN and FECTUZ Codes tell a reasonable and consistent story concerning the evolution of

parent and daughter radionuclides through a 6 m, homogenous soil column, with a uniform, 1000 d source of the

The text will

The salient points, a, b, c, d, f, e, in the two graphs represent key features in the evolution of the two

At 

Point c is also the time that

The times (a,b,c) are roughly equal to (2700, 5600, 7800) days.

From 

The only major difference between CHAIN and FECTUZ (and this is pretty small) occurs between

Between 

This is reasonable since FECTUZ has less diffusion than that of CHAIN. There is also a very slight

The magnitude of this difference is hard to
I-5

parent at the top of the column. 

two-dimensional and three-dimensional, heterogeneous soil columns, as pointed out in Section 3.6 (Oldenburg and

Pruess, 1996).

I.3. 

Figures I-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. 

of curves are given in order of increasing default Kd values. d values govern

the rates of pollutant passage through the 6 m layer. 

Figure I-3. Sensitivity of radionuclide transport through the unsaturated zone to recharge rate (q) using the CHAIN

Model: (a) Tritium 3H; (b) Technetium 99Tc; (c) Uranium 238U; (d) Strontium 90Sr; and (e) Plutonium

238Pu.

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���

���

���

����

����

CHAIN

CHAIN

CHAIN

238U

CHAIN

CHAIN

����

��

��

��

��

���

���

���

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However, the evolutions of parent and daughter species can be quite complex in

The Sensitivity of the Five Parent Radionuclides to Recharge Rate

The five sets

As indicated by Equation (I-1), these K

In fact, a good representation of characteristic times of passage
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): 

q v = 

��� ��d 

. (I-5) 

These characteristic times are listed in Table I-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 I-3 (a to e). These peaks are normalized by the 

respective source concentrations of the radionuclides given in Table I-1. Corresponding to these normalized Cpeak

values are the times to peak concentration, Tpeak, given in the last column. One should note the rather close 

correspondence between the “characteristic time” values and the values of Tpeak given in Table I-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 loglog plot of Cpeak ÷ C0 versus Tpeak in Figure I-4. 

To further illuminate the results in Table I-2 and Figure I-4, we introduce a relative diffusion factor (DIF) and a 

relative decay factor (DEC). The value of DIF 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 I-3 lists these factors 

for the five parent radionuclides. As DIF 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 I-4 can be easily explained: 

• The diffusive dilutions of 3H and 99Tc are nearly equal since their DIF 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 99Tc because 3H has significantly decayed over the time 

interval, while the decay of 99Tc has been insignificant, although not zero as seen in Figure I-2a. 

• The curve segment for 238U, as that of 99Tc, has only experienced insignificant decay, but is lower than 

the curve segments for 99Tc and 3H, and further to the right, because its DIF is almost five times that of 

99Tc and 3H. The 238U’s normalized concentrations are lower only because of diffusive dilution. 

Table I-2. Cpeak Normalized by Source Concentration C0

 and Tpeak

for Three Recharge Rates. 

for Each Radionuclide and 

Recharge **Characteristic Time ***Normalized 

*Type Rate q (600cm) ÷ v Cpeak Tpeak

Radionuclide of Decay (cm/d) (d) Cpeak ÷ Co (d) 

Tritium 3H Significant 

Technetium 99Tc Very Little 

Uranium 238U Very Little 

Strontium 90Sr Very Significant 

Plutonium 238Pu Very Significant 

0.014 6860 1.72 x 10-1 7300 

0.024 4000 3.23 x 10-1 4400 

0.032 3000 4.18 x 10-1 3300 

0.014 7370 4.49 x 10-1 8000 

0.024 4300 5.64 x 10-1 4750 

0.032 3225 6.31 x 10-1 3560 

0.014 36,000 9.98 x 10-2 35,500 

0.024 21,000 1.29 x 10-1 21,000 

0.032 15,750 1.48 x 10-1 15,900 

0.014 79,715 4.47 x 10-4 66,100 

0.024 46,500 3.28 x 10-3 43,100 

0.032 34,875 7.44 x 10-3 33,200 

0.014 371,145 1.66 x 10-5 287,000 

0.024 216,600 2.20 x 10-4 194,000 

0.032 162,375 6.48 x 10-4 152,000 

*Type of Decay with respect ** v = seepage velocity = q . *** Co = Concentration at Source, 

to the 6 m soil column. in Table I-1. ��� ��d 

I-6 
Figure I-4. Cpeak ÷ Co 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, 99Tc and 238U 

because their DEC values are much larger and their DIF 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 DIF 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 90Sr. 

Table I-3. The Relative Diffusion Factors and the Relative Decay Factors for the Five Parent Radionuclides. 

For Recharge Rate, q = 0.024 cm/d 

*Type Relative Diffusion Relative Decay 

Radionuclide of Decay Factor Factor 

Tritium 3H Significant 1.000 1.005 

Technetium 99Tc Very Little 1.080 0.000 

Uranium 238U Very Little 4.773 0.000 

Strontium 90Sr Very Significant 9.795 4.072 

Plutonium 238Pu Very Significant 44.091 6.040 

*Type of Decay with respect to the 6 m soil profile. 

I-7 
References 

Oldenburg, C.M. 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, 

(3H), and Uranium. Volume II. EPA 402/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. 

I-8
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