The STANMOD Computer Software for Evaluating Solute
Transport in Porous Media Using Analytical Solutions ofConvection-Dispersion Equation

Versions 1.0 and 2.0

November 1999

U. S. SALINITY LABORATORYAGRICULTURAL RESEARCH SERVICE
U. S. DEPARTMENT OF AGRICULTURE
RIVERSIDE, CALIFORNIAThe STANMOD Computer Software for Evaluating Solute
Transport in Porous Media Using Analytical Solutions ofConvection-Dispersion Equation

Versions 1.0 and 2.0

byJiri Simunek, M. Th. van Genuchten, M. Sejna, N. Toride, and F. J. Leij

November 1999

U. S. SALINITY LABORATORYAGRICULTURAL RESEARCH SERVICE
U. S. DEPARTMENT OF AGRICULTURE
RIVERSIDE, CALIFORNIA

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DISCLAIMER
This report documents versions 1.0 and 2.0 of STANMOD, a software package for
evaluating solute transport in porous media using analytical solutions of the convection-dispersion
equation. The software has been verified against a large number of test cases. However, no warranty
is given that the program is completely error-free. If you do encounter problems with the code, finderrors, or have suggestions for improvement, please contact one of the authors at
U. S. Salinity Laboratory
USDA, ARS
450 West Big Springs Road
Riverside, CA 92507
Tel. 909-369-4865 (J. Simunek)
Tel. 909-369-4846 (M. Th. van Genuchten)
Fax. 909-342-4964
E-mail jsimunek@ussl.ars.usda.gov
rvang@ussl.ars.usda.gov

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ABSTRACT
Simunek, J., M. Th. van Genuchten, M. Sejna, N. Toride, F. J. Leij, 1999. The STANMOD
Computer Software for Evaluating Solute Transport in Porous Media Using Analytical Solutions ofConvection-Dispersion Equation, Versions 1.0 and 2.0, U.S. Salinity Laboratory, USDA, ARS,
Riverside, California.

This report documents versions 1.0 and 2.0 of STANMOD (STudio of ANalytical
MODels), a Windows based computer software package for evaluating solute transport in porous
media using analytical solutions of the convection-dispersion solute transport equation. Version 1.0
of STANMOD includes at present the following models: CXTFIT 2.0 [Toride et al., 1995], CFITM[van Genuchten, 1980], CFITIM [van Genuchten, 1981], and CHAIN [van Genuchten, 1985].
Version 2.0 of STANMOD, to be released in the spring of 2000, will also include the models
3DADE [Leij and Bradford, 1994] and N3DADE [Leij and Toride, 1997]. The original manuals of
all models included in STANMOD accompany this report.
The software package includes a modified and updated version of the CXTFIT code of
Toride et al. [1995] for estimating solute transport parameters using a nonlinear least-squares
parameter optimization method. This code may be used to solve the inverse problem by fitting avariety of mathematical solutions of theoretical transport models, based upon the one-dimensional
convection-dispersion (or advection-dispersion) equation (CDE), to experimental results. The
program may also be used to solve the direct or forward problem to determine concentrations as afunction of time and/or position. Three different one-dimensional transport models are considered:(i) the conventional CDE; (ii) the chemical and physical nonequilibrium CDEs; and (iii) a
stochastic stream tube model based upon the local-scale equilibrium or nonequilibrium CDE.
STANMOD also includes an updated version of the CFITM code of van Genuchten [1980]
for analyzing observed column effluent data using analytical solutions of the one-dimensional
equilibrium convective-dispersive transport equations. The code considers analytical solutions forboth semi-finite and finite columns. The model provides an easy to use, efficient and accuratemeans of determining various transport parameters by optimizing observed column effluent data.
CFITM represents a simple alternative to the much more comprehensive, but also more complex,
CXTFIT model.

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STANMOD also contains an updated version of the CFITIM code of van Genuchten[1981] for analyzing observed column effluent data using analytical solutions of the onedimensional equilibrium and nonequilibrium convective-dispersive transport equations. The code
includes analytical solutions for semi-finite columns. The nonequilibrium solutions for bicontinuum
model, the two-region flow model for physical nonequilibrium and the two-site sorption model forchemical nonequilibrium. The model provides an easy to use, efficient and accurate means ofdetermining various transport parameters by optimizing column effluent data.
In addition, STANMOD 1.0 includes the modified and updated CHAIN code of van
Genuchten [1985] for analyzing the convective-dispersive transport of solutes involved in sequential
first-order decay reactions . Examples are the migration of radionuclides, in which the chain
members form first-order decay reactions, and the simultaneous movement of various interacting
nitrogen or organic species.
STANMOD 2.0 includes the 3DADE code of Leij and Bradford [1994] for evaluating
analytical solutions for three-dimensional equilibrium solute transport in the subsurface. The
analytical solutions pertain to selected cases of three-dimensional solute transport during steadyunidirectional water flow in porous media having uniform flow and transport properties. The
transport equation contains terms accounting for solute movement by convection and dispersion, as
well as for solute retardation, first-order decay, and zero-order production. The 3DADE code can beused to solve the direct problem, i.e., the concentration is calculated as a function of time and spacefor specified model parameters, and the indirect (inverse) problem in which the program estimates
selected parameters by fitting one of the analytical solutions to specified experimental data.
Finally, STANMOD 2.0 incorporates the N3DADE code of Leij and Toride [1997] for evaluating
analytical solutions for a three-dimensional nonequilibrium solute transport in porous media. The
analytical solutions pertain to three-dimensional solute transport during steady unidirectional waterflow in porous media in systems of semi-infinite length in the longitudinal direction, and of infinite
length in the transverse direction. The solutions can be applied also to one- and two-dimensional
problems. The flow and transport properties of the medium are again assumed to be
macroscopically uniform. Nonequilibrium solute transfer can occur between two domains in either
the liquid or the absorbed phase. The transport equation contains terms accounting for solute
movement by advection and dispersion, as well as for solute retardation, first-order decay, and zeroorder production.

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TABLE OF CONTENTSDISCLAIMER.................................................................................................................................iii
ABSTRACT .....................................................................................................................................v
TABLE OF CONTENTS................................................................................................................vii
LIST OF VARIABLES ...................................................................................................................ix
1. INTRODUCTION......................................................................................................................1
2. OVERVIEW OF INCLUDED MODELS ...................................................................................3
2.1. CFITM .................................................................................................................................3
2.2. CFITIM................................................................................................................................3
2.3. CHAIN.................................................................................................................................4
2.4. CXTFIT................................................................................................................................4
2.5. 3DADE.................................................................................................................................5
2.6. N3DADE..............................................................................................................................5
3. EXAMPLES ...............................................................................................................................7
3.1. Examples of the CFITM model............................................................................................7
3.2. Examples of the CFITIM model...........................................................................................7
3.3. Examples of the CHAIN model............................................................................................8
3.4. Examples of the CXTFIT model...........................................................................................8
3.4.1. Direct problems ..........................................................................................................8
3.4.1.1. Examples taken from Toride et al. [1995]......................................................93.4.1.2. Examples adopted from van Genuchten and Cleary [1979] .........................10
3.4.2. Stochastic problems ..................................................................................................11
3.4.3. Inverse problems ......................................................................................................11
3.5. Examples of the 3DADE model..........................................................................................123.6. Examples of the N3DADE model.......................................................................................13
4. USER MANUAL......................................................................................................................17
5. REFERENCES .........................................................................................................................19

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LIST OF VARIABLES
cconcentrationcf

flux concentrationcr

resident concentrationD

dispersion coefficientDxdispersion coefficient in the x-direction

Dydispersion coefficient in the y-directionffraction of sorption sites that equilibrate with the mobile region

Kd

distribution coefficientP

Peclet numberR

retardation factorrradial coordinate

t

time

t0

pulse timeTdimensionless time

T0

dimensionless pulse timev

pore-water velocity

x, y, z spatial coordinates
αmass transfer coefficientβdimensionless variable for partitioning in nonequilibrium transport modelsµ

first-order rate coefficient for decay

λzero-order rate coefficient for productionθimimmobile water content

θmmobile water content

σD

standard deviation of the logtransform of dispersion coefficientσv

standard deviation of the logtransform of pore-water velocity

ρvKd

correlation between the pore-water velocity, v, and the distribution coefficient, Kdωdimensionless mass transfer coefficient<..>

ensemble average, e.g., <v> is mean pore-water velocity

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1. INTRODUTION
The fate and transport of dissolved substances in soils and groundwater is generating
considerable interest because of concerns for the quality of the subsurface environment. The
behavior of solutes over relatively long spatial and temporal scales must generally be assessed withthe help of theoretical models since it is usually not feasible to carry out experimental studies oversufficiently long distances and/or time periods. Mathematical models are often also used to predict
solute concentrations before management strategies are implemented. Advances in software andhardware now permit the simulation of subsurface transport using sophisticated mathematicalmodels. Unfortunately, it is generally difficult to obtain reliable values for transport parameters suchas the pore-water velocity, the retardation factor, the dispersion coefficient, and/or degradation or
production parameters.
A large number of computer programs now exists for evaluating solute transport in porous
media using analytical solutions of the convection-dispersion equation. The purpose of this project
was to integrate the most widely used models into a one software package. We developed for thispurpose a suite of analytical models STANMOD (Versions 1.0) that includes, at present, thefollowing models for one-dimensional transport: CFITM [van Genuchten, 1980], CFITIM [vanGenuchten, 1981], CHAIN [van Genuchten, 1985], and CXTFIT (2.0) [Toride et al., 1995]. The
models 3DADE [Leij and Bradford, 1994] and N3DADE [Leij and Toride, 1997] for multidimensional transport problems will be available in version 2.0 of STANMOD to be released in thespring of 2000. In this report we briefly describe the different analytical models included in
STANMOD and review some typical examples that were included with the software. Detailed
descriptions of all models are given in the original manuals, which are included as PDF files on the
STANMOD CD. The manuals provide the governing transport equations, boundary and initialconditions, the derived analytical solutions, as well as description of several illustrative examples.
The graphics-based user-interface of STANMOD is for the MS Windows environment andis largely based on libraries developed for the HYDRUS-1D and HYDRUS-2D software packages[Simunek et al., 1998, 1999]. All computational programs were written in FORTRAN, and thegraphic interface in MS Visual C++.

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2. OVERVIEW OF INCLUDED MODELS
2.1. CFITM

The CFITM code of van Genuchten [1980] can be used for analyzing observed column
effluent data using analytical solutions of the one-dimensional equilibrium convection-dispersion
transport equations. The code includes analytical solutions for both infinite and finite columns. The
model provides a convenient, efficient and accurate means of determining transport parameters byfitting analytical solutions to observed column effluent data. The unknown parameters include thecolumn Peclet number, P, the retardation factor, R, and the dimensionless pulse time, T0 . CFITM
represents an easy to use, simple alternative to the more complex CXTFIT model reviewed in
section 2.4.

2.2. CFITIM

The CFITIM code of van Genuchten [1981] can be used for analyzing observed column
effluent data using analytical solutions for one-dimensional equilibrium and nonequilibrium
convective-dispersive transport. The code includes analytical solutions for infinite columns. The
nonequilibrium models are based on the assumption that, either for physical or chemical reasons,adsorption does not proceed at an equal rate in all parts of the soil medium. The model provides aneasy to use, efficient and accurate means of fitting various transport parameters to observed column
effluent data. Depending upon the exact form of the transport model, the program allows up to fivedifferent parameters to be estimated simultaneously. The unknown parameters include the column
Peclet number, P, the retardation factor, R, the dimensionless pulse time, T0 , and the dimensionlessparameters $ (a nonequilibrium partitioning coefficient), and T (a transfer coefficient).

32.3. CHAIN

The CHAIN code of van Genuchten [1985] can be used to analyze the convectivedispersive transport of solutes involved in sequential first-order decay reactions. The problem ofsolute transport involving sequential first-order decay reactions frequently occurs in soil systems.
Examples are the migration of radionuclides in which the chain members form first-order decay
reactions, and the simultaneous movement of various interacting nitrogen [Cho, 1972] or organic(e.g., pesticide) species. CHAIN is based on analytical solutions of the CDE that describe thesimultaneous convective-dispersive transport of up to four species involved in such consecutive
chain reactions. This program only considers the forward problem.

2.4. CXTFITThe program CXTFIT 2.1 may be used to estimate parameters in several analytical models
for solute transport during steady one-dimensional flow by fitting analytical solutions for the models
to observed laboratory or field solute transport data. The inverse problem is solved by minimizing
an objective function that consists of the sum of the squared differences between observed and fitted
concentrations. The objective function is minimized using a nonlinear least-squares inversion
method according to Marquardt [1963]. In addition, CXTFIT 2.1 may be used for the direct
problem to predict solute distributions versus time and/or space for specified model parameters.
CXTFIT 2.1 is an extension and update of earlier versions published by van Genuchten[1979, 1981], Parker and van Genuchten [1984] and Toride et al. [1995]. CXTFIT 2.1, as before,
uses the convection-dispersion equation, but with a larger number of analytical solutions pertaining
to various initial, boundary, and production value problems. The nonequilibrium transport models
also contain terms for zero-order production and first-order decay. Considerably more attention is
being paid to the use of stream tube models for simulating transport in heterogeneous fields, thus
reflecting the growing popularity of stochastic approaches for modeling field-scale solute transport.
A bivariate lognormal probability density function is used to quantify stochastic pore-water velocity

4and either stochastic dispersion, adsorption, or nonequilibrium solute transfer. Solute concentrations
across the field can be either in the resident mode or in two different types of flux-averaged modes.

2.5. 3DADEThe 3DADE code of Leij and Bradford [1994] is intended for evaluating analyticalsolutions for three-dimensional equilibrium solute transport in the subsurface. The analyticalsolutions pertain to selected cases of three-dimensional solute transport during steady unidirectional
water flow in porous media having uniform flow and transport properties. The transport equation
contains terms to account for solute movement by convection and dispersion, as well as for solute
retardation, first-order decay, and zero-order production. The 3DADE code can be used to solve thedirect problem, i.e., the concentration is calculated as a function of time and space for specified
model parameters, and the indirect (inverse) problem in which the program estimates selectedparameters by fitting one of the analytical solutions to specified experimental data. Transient
analytical solutions are evaluated for five different transport scenarios (three boundary value
problems and two initial value problems) in either a Cartesian or cylindrical coordinate system.
Simple steady-state solutions are also provided for three initial value problems.

2.6. N3DADEThe N3DADE code of Leij and Toride [1997] can be used to evaluate analytical solutions
for three-dimensional nonequilibrium solute transport in porous media. The analytical solutions
hold for three-dimensional solute transport during steady unidirectional water flow in porous mediaof semi-infinite length in the longitudinal direction, and of infinite length in the transverse
directions. The solutions can be also applied to one and two-dimensional problems. The flow andtransport properties of the medium are assumed to be macroscopically uniform. Nonequilibrium
solute transfer can occur between two domains in either the liquid or the absorbed phase. The
transport equation contains terms accounting for solute movement by advection and dispersion, as
well as for solute retardation, first-order decay, and zero-order production. Unlike 3DADE, the5

N3DADE code can be used only to solve the direct problem, i.e., concentrations are calculated as afunction of time and space for specified model parameters. A comprehensive set of specificsolutions is presented using Dirac, Heaviside and exponential functions to describe a variety of
initial, boundary and production profiles. A rectangular or circular inflow area is specified for theboundary value problem, while for the initial and production value problems the respective initialand production profiles are defined for parallelepipedal, cylindrical, or spherical regions of the soil.
Solutions are given for volume-averaged or resident concentrations, as well as for flux-averaged or
flowing concentrations.

63. EXAMPLESSTANMOD is installed with numerous examples that are divided into eight groups(workspaces): CFITM, CFITIM, CHAIN, Direct, Inverse, Stochast, 3DADE, and N3DADE. The
first three workspaces CFITM, CFITIM, and CHAIN contain examples of the corresponding
models. The next three workspaces Direct, Inverse, and Stochast contain examples of the direct,
inverse, and stochastic problems solved with the CXTFIT model. The last two workspaces 3DADE
and N3DADE contain examples of the 3DADE and N3DADE models. Users are advised to selectan example closest to their particular problem, copy this example and then simply modify the input
data. A list of all examples, together with brief descriptions, is given in Table 1 at the end of thischapter.

3.1. Examples of the CFITM model

The CFITM workspace contains two examples for the CFITM model. Both examples
pertain to the transport of chromium through a sand [Wierenga, 1980, unpublished data; van
Genuchten, 1980]. Observed effluent data from a 5-cm long soil column are fitted using ananalytical solution of the convection-dispersion solute transport equation. Example1 uses theanalytical solution for a semi-infinite system, while Example2 holds for a finite system. Peclet
numbers, P, and retardation factors, R, were fitted in both examples.

3.2. Examples of the CFITIM model

The CFITIM workspace includes five examples for the CFITIM model. Example1
demonstrates the used of CFITIM on a generated data set using the assumption of physical nonequilibrium solute transport (Figure 1 of van Genuchten [1980]). Four parameters (the Peclet
number, P, the retardation factor, R, and the dimensionless coefficients β (a nonequilibrium
partitioning coefficient) and T (transfer coefficient)) were fitted simultaneously using the generated
effluent curve for a 30-cm long soil column. Example2 and Example3 consider the movement oftritiated water through a Glendale clay loam in a 30-cm long column. The nonequilibrium transport

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model with five fitted parameters (P, R, β, T, and the dimensionless pulse time, T0 ) was considered
for Example2 (Figure 1 of van Genuchten [1980]), while the linear equilibrium adsorption modelwith three fitted parameters (P, R, and T0 ) was used for Example3. Examples Fig79a (Figure 7.9a of
Toride et al. [1995]) and Fig79b (Figure 5 of van Genuchten [1980] and Figure 7.9b in Toride et al.
[1995]) consider the transport of tritiated water and boron, respectively, through Glendale clay loam
in a 30-cm long column. In both examples, parameters of the non-equilibrium transport model are
optimized against the effluent curves.

3.3. Examples of the CHAIN modelTwo examples are presented here to illustrate typical results that can be obtained with theprogram. The first example (Nitrogen) applies to the three-species nitrification chain (NH4 + - NH2- NH3-) (Figure 1 of van Genuchten [1985]). The second example deals with the radionuclide decay
chain (238 Pu - 234 U - 230 Th - 226 Ra) (Figure 2 of van Genuchten [1985]), which serves as anillustration of modeling subsurface radionuclide transport.

3.4. Examples of the CXTFIT model

Examples of the CXTFIT model are divided into three workspaces. The first group (Direct)
contains direct problems, the second group (Inverse) involves inverse problems, and the third group(Stochast) contains stochastic examples. Most examples were taken from the original CXTFIT
manual [Toride et al., 1995], several other examples from a book chapter by van Genuchten and
Cleary [1979], and two examples from a book chapter by Leij and Toride [1998].

3.4.1. Direct problems

The first 2 examples (Fig51 and Fig52) were taken from a book chapter by Leij and Toride[1998], the next 9 examples (Fig71 through Fig78) from the original CXTFIT manual [Toride et al.,
1995], and the last 15 examples (Fig105 through Fig1012b) from a book chapter by van Genuchten
and Cleary [1979].

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Examples Fig51 and Fig52 demonstrate the use of the nonequilibrium solute transport
options in the CXTFIT code. Examples Fig51 (Figure 5.1 of Leij and Toride [1998]) and Fig52(Figure 5.2 of Leij and Toride [1998]) use the first-order physical nonequilibrium model to calculate
concentrations in the mobile and immobile phases as a function of time at a depth of 50 cm for a 2-dpulse input at the soil surface. Four different combinations of the mobile water content (β = θm /θ =0.25, 0.5, 0.75, and 0.99) and the transfer rate (ω = 0.2, 1, 5, 1000) are used in example Fig51, while
five different fractions of sorption sites (f = 0, 0.25, 0.5, 0.75, 1) that equilibrate with the mobile
region are used in example Fig52.

3.4.1.1. Examples taken from Toride et al. [1995]

The first two examples (Fig71 and Fig72) deal with solutions of the direct problem forequilibrium transport. Example Fig71 (Figure 7.1 of Toride et al. [1995]) illustrates the effect of
the first-order decay constant, µ (=0, 0.25, 0.5, 1 d-1) on solute distribution. Resident concentrations
at t =7.5 d were calculated for a single pulse input of duration 5 d starting at t = 0 to a solute-freesoil profile. Examples Fig72a and Fig72b (Figures 7.2a and 7.2b of Toride et al. [1995]) calculate
flux (cf) and resident (cr) concentrations for two values of the Peclet number, P (= 2, 10),respectively, as a function of relative distance when solute-free water is applied to a soil having a
stepwise initial resident distribution.
The next four examples (Fig75 and Fig78) deal with solutions of the direct problem fornonequilibrium transport. Examples Fig75 and Fig76 were included to demonstrate differences
between the one-site and two-site nonequilibrium sorption models. Example Fig75 (Figure 7.5 of
Toride et al. [1995]) show the effects of the first-order mass transfer rate coefficient, α (= 0.08, 0.2,
10, 1000 d-1), on breakthrough curves in terms of the flux-averaged concentration, as a result ofapplying a Dirac delta input function to an initially solute free soil. Example Fig76 (Figure 7.6 ofToride et al. [1995]) calculates breakthrough curves according to the two-site nonequilibrium CDE
for four values of the fraction of sorption sites (f = 0, 0.3, 0.7, and 0.999), and using the same initialand boundary conditions as for Fig75. Values of 0.08 and 0.2 d-1 for the first-order mass transferrate, α, were used in Fig76a and Fig76b, respectively. Examples Fig77a and Fig77b (Figures 7.7a
and 7.7b of Toride et al. [1995]) calculate breakthrough curves using different sets of R, α, and fvalues in the two-site nonequilibrium CDE for Dirac delta input and pulse input, respectively. The

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last example (Fig78) involving a direct problem concerns deterministic nonequilibrium transport as
described by an initial value problem. This case calculates equilibrium and nonequilibrium resident
concentrations and total concentration profiles at T = 1 for three values of the partitioning
coefficient β (= 0.1, 0.5, and 0.9).
3.4.1.2. Examples adopted from van Genuchten and Cleary [1979]

Examples Fig105 and Fig105a (Figure 10.5 of van Genuchten and Cleary [1979])
demonstrate the effect of the first-order decay coefficient µ (= 0, 0.1, 0.2, 0.35) on Picloram
movement through Norge loam using either a pulse or step input, respectively, and assuming
applicability of the equilibrium CDE. Example Fig108 (Figure 10.8 of van Genuchten and Cleary[1979]) demonstrates the effect of the dimensionless mass transfer coefficient ω (= 0.001, 0.28, 0.7,
1.7, 2.8, 7.0, 1000000) on calculated effluent curves for 2,4,5-T movement through Glendale clay
loam using the two-region physical nonequilibrium CDE model). Example Fig109 (Figures 10.9a
and 10.9b of van Genuchten and Cleary [1979]) demonstrates the effect of the dimensionlesspartitioning coefficient β (= 0.2, 0.35, 0.5, 0.65, 0.80, 0.99) on calculated effluent curves from, andspatial concentration distributions in, an aggregated sorbing medium, respectively, assuming tworegion physical nonequilibrium transport. Example Fig1010 (Figures 10.10a and 10.10b of van
Genuchten and Cleary [1979]) demonstrates the effect of the retardation factor R (= 1.0, 1.75, 2.5,
3.5, and 5.0) on calculated effluent curves from, and spatial concentration distributions in, anaggregated sorbing medium, respectively, again assuming two-region physical nonequilibriumtransport. Example Fig1011 (Figures 10.11a and 10.11b of van Genuchten and Cleary [1979])
calculates solute concentration versus time and distance for an aggregated sorbing medium,
respectively, assuming two-region physical nonequilibrium model, as affected by the dimensionlessmass transfer coefficient ω (= 0.02, 0.2, 0.5, 1.5, 7.5, 1000). Finally, Example Fig1012 (Figures
10.12a and 10.12b of van Genuchten and Cleary [1979]) demonstrates the effect of the Peclet
number P (= 5, 15, 40, 100, 10000) on calculated effluent curves from, and spatial concentration
distribution in, an aggregated sorbing medium, respectively, assuming two-region physicalnonequilibrium transport.

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3.4.2. Stochastic problems

Examples presented in this group demonstrate the use of the stochastic stream tube model in
the CXTFIT code. The first two example (Fig45a and Fig45b) calculates field-scale resident
concentrations, cr, versus depth resulting from the instantaneous application of a solute to thesurface as a BVP (variable mass) and an IVP (constant mass) (Figure 4.5 of Toride et al. [1995]).
The next two examples (Fig47a and Fig47b) calculate field-scale resident concentrations versus
depth as a result of a pulse-type solute application of constant duration (Figure 4.7 of Toride et al.
[1995]).
The next example (Fig710) demonstrates the effect of variability in the pore-water velocity,v, on the field-scale resident concentration profile and the distribution of the variance for cr in thehorizontal plane (Figure 7.10 of Toride et al. [1995]). This example calculates the mean resident
concentration and its variance as a function of depth at t = 3 d for three values of σv (=0.1, 0.3, 0.5)
as a result of a 2-d solute application to a solute-free soil. Examples Fig711a, Fig711b, and Fig711c
calculate the breakthrough curves for three types of field-scale concentration modes (Figure 7.11 inToride et al. [1995]).
Example Fig713 demonstrates the effect of correlation (ρvKd = -1, 0, +1) between the porewater velocity, v, and the distribution coefficient, Kd , on calculated field-scale resident
concentration, cr, profiles (Figure 7.13 of Toride et al. [1995]). Field-scale concentrations at t = 5 dresulting from a Dirac delta input at t = 0 are calculated versus depth for either perfect or nocorrelation between v and Kd . Finally, examples Fig714a and Fig714b calculate field-scale resident
and total concentrations, respectively, assuming stochastic nonequilibrium solute transport. Bothexamples assume a negatively correlated v and Kd (ρvKd = -1) using three values of the mass transfer
coefficient ω (Figure 7.14 of Toride et al. [1995]).
3.4.3. Inverse problems

The parameter estimation option of the CXTFIT model is demonstrated with sevenexamples. The first three examples (Fig73a, Fig73b, and Fig74) deal with equilibrium solutetransport, and the next two examples (Fig79a and Fig79b) with nonequilibrium solute transport.
Two additional examples (Fig712 and Fig715) consider field-scale stochastic solute transport.

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In the first two examples (Fig73a and Fig73b), the pore-water velocity, v, and dispersion
coefficient, D, are estimated from breakthrough curves measured at three different depths (11, 17,
and 23 cm) with four-electrode electric conductivity sensors. Breakthrough curves were a result of(a) continuous application of a 0.001 M NaCl solution to an initially solute-free saturated sand, and(b) leaching with solute free water during unsaturated conditions, respectively. In the third inverse
example (Fig74), the pore-water velocity, v, dispersion coefficient, D, and the duration of the KClapplication for a pulse input, t 0 , to an undisturbed sandy soil are estimated from breakthrough curves
measured at a depth of 10 cm with a TDR probe.
The two examples of nonequilibrium solute transport (Fig79a corresponding to Figure 7.9aof Toride et al. [1995], and Fig79b corresponding to Figure 5 of van Genuchten [1980] and Figure
7.9b of Toride et al. [1995]) consider transport of tritiated water and boron, respectively, throughGlendale clay loam in a 30-cm long column. In both examples parameters of the non-equilibriumtransport model were optimized against effluent curves. The same examples are also consideredusing the CFITIM model.
The stochastic option of CXTFIT, together with parameter estimation, is demonstrated with
examples Fig712 and Fig715. The first example (Figure 7.12 of Toride et al. [1995]) pertains toresident concentrations in a 0.64-ha field to which a bromide pulse was applied for 1.69 d followedby leaching with solute-free water [Jury et al., 1982]. The stream tube model was used to estimate
the mean pore-water velocity <v>, the mean dispersion coefficient <D>, and their standard
deviations σv and σD, respectively. The second example (Figure 7.15 of Toride et al. [1995])
demonstrates the estimation of parameters in the stream tube model for reactive transport using ahypothetical data set. The standard deviation, σKd , and the coefficient of correlation between v, andKd, i.e., ρvKd , were fitted to the hypothetical data, while keeping <v>, σv, and <Kd > constant.
3.5. Examples of the 3DADE model

Nine examples (Example1 through Example9) are presented here to illustrate the use of the
3DADE model. The first five examples (Example1 through Example5) solve direct problems, while
the last four examples (Example6 through Example9) inverse parameter estimation problems.
Example1 calculates steady-state concentration profiles for a diffuse solute source in one
quadrant of the soil surface (Figure 2a of Leij et al. [1991]). Example2 and Example 3 calculate12transient concentration profiles for transport from a rectangular solute source at the surface using
either a first- or third-type boundary condition (Figure 4 of Leij et al. [1991]), respectively.
Example4 calculates transient concentration profiles for transport from a parallelepipedal initial
distribution (Figure 3 of Leij et al. [1991]). Finally, Example5 calculates transient concentration
profiles for transport from a circular solute source at the surface using a third-type boundary
condition (Figure 7 of Leij et al. [1991]).
Inverse problems are demonstrated using four examples (Example6 through Example9). The
first example (Example6) considers (similarly to Example1) solute application in one quadrant of
the soil surface. The parameters R, Dx, and Dy (retardation factor, and dispersion coefficients in thex- and y-directions, respectively) are fitted using a breakthrough curve at a specified position and thesteady-state profile at a selected transect. The second inverse example (Example7) involves theestimation of the parameters R, µ, λ, Dx, Dy, and Dz (retardation factor, first-order rate coefficient for
decay, zero-order rate coefficient for production, and dispersion coefficients in the x-, y- and zdirections, respectively) for solute transport from a parallelepipedal initial distribution.
Breakthrough curves at ten positions along the x coordinate and two transverse profiles were used
for the problem. This optimization example was repeated (Example8) for a data set in which errorsof "10 % were imposed on the transverse concentration profiles (Figure 2 of Leij et al. [1994]). The
final example (Example9) concerns the application of a solute pulse from a circular area at the soil
surface. Parameters t 0 , Dx, and Dr (pulse time, and dispersion coefficients in the x- and r-directions,
respectively) were estimated using concentrations at several spatial locations at a specific time.

3.6. Examples of the N3DADE model

Five examples are presented here to illustrate the use of the N3DADE model for calculating
concentration profiles using the nonequilibrium solute transport CDE. The first two examples
demonstrate the use of N3DADE to solve boundary value problems (BVP), the next two examples
initial value problems (IVP), and the last example a production value problem (PVP).
The first example (Figures 6, 7a, and 7b of Leij and Toride [1997]) calculates breakthroughcurves at a depth of 50 cm (Exampl1a) and the flux-averaged spatial concentration distribution(Exampl1b) for instantaneous solute application from a disk having radius of 2.5 cm at the soil
surface. The problem involves a circular geometry. The second example (Exampl2a and Exampl2b,

13

Figures 8 and 9 of Leij and Toride [1997]) pertain to flux-averaged concentration profiles resulting
from the continuous application of solute to a rectangular surface area (-2.5 < y < 2.5, -2.5 < z <2.5). Exampl2a calculates equilibrium, nonequilibrium and total concentrations versus longitudinal
distance at three different times, while Exampl2b calculates concentrations in the transverse plane at
two longitudinal positions.
The third example (Example3, Figure 10 of Leij and Toride [1997]) considers an initialvalue problem (rectangular) with solute initially located in the regions 5 < x < 15 (c = 1) and 25 < x< 35 (c = 0.5) for 15 < y < 25 and -100 < z < 100. The fourth example (Example4, Figure 11 of Leij
and Toride [1997]), which also pertains to an initial value problem (in this case spherical), assumes
that the solute has initially a maximum value at the point given by x=5, y=0, and z=5, and that thesolute concentration decreases exponentially from the maximum.
The last example (Example5, Figure 12 of Leij and Toride [1997]) involves solute
production in a cylindrical region of the soil (0 < x < 10 and 0 < r < 2.5). The problem is modeled asa production value problem (PVP) with a heaviside function for the longitudinal and transversal
directions, with production in the equilibrium phase equal to 0.5, and in the nonequilibrium phaseequal to 1.

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Table 1. Summary of examples included in STANMOD.

Model/Workspace

Name

Brief Description

CFITM/CFITM

Example1
Example2

Example 2A: Chromium (Column Number 4) - Semi-Finite Sysem
Example 2B: Chromium (Column Number 4) - Finite Sysem

CFITIM/CFITIM

Example1
Example2
Example3
Fig79aFig79bGenerated data, Nonequilibrium model
Example 2D: Tritiated water (EXP. 5-2); Nonequilibrim model
Example 2H: Tritiated water (EXP. 5-2); Linear adsorption model
Tritium effluent curve from Glendale clay loam; Nonequilibrim model
Boron effluent [exp. 3-1, van Genuchten, 1974]; Nonequilibrim model

CHAIN/CHAIN

NitrogenRadionucNitrogen chain [Cho, 1972]
Radionuclide transport

CXTFIT/Direct

Fig51
Fig52
Fig71
Fig72aFig72b
Fig75
Fig76aFig76b
Fig77aFig77b
Fig78
Fig105
Fig105aFig108
Fig109
Fig109aFig109b
Fig1010
Fig1010aFig1010b
Fig1011
Fig1011aFig1011b
Fig1012
Fig1012aFig1012bFirst-order nonequilibrium model, effect of β and ωFirst-order nonequilibrium model, effect of β and f
Fig.7-1: The deterministic CDE (BVP+PVP)
Fig.7-2a: Flux vs. (resident) conc. for the IVP, Cf(Z) (a) P=2 (b) P=10
Fig.7-2b: (Flux) vs. resident conc. for the IVP, Cf(Z) (a) P=2 (b) P=10
Fig.7-5: Nonequilibrium one-site CDE (β=1/R, ω=0.08, 0.2, 1.0, 10)
Fig.7-6a: Two-site CDE (ω =0.08, f=0, 0.3, 0.7,0.99875)
Fig.7-6b: Two-site CDE (ω =0.2, f =0, 0.3, 0.7,0.99875)
Fig.7-7a: Two-site CDE for βR=0.22 - Dirac input
Fig.7-7b: Two-site CDE for βR=0.22 - pulse input
Fig.7-8: IVP for the nonequilibrium CDE
Equilibrium model, Effect of first-order decay, Step Input
Equilibrium model, Effect of first-order decay, Pulse Input
One-Site Model, Effect of mass-transfer coefficientTwo-site model, Effect of mobile/immobile water ratio, Cf (T)Two-site model, Effect of mobile/immobile water ratio, Cf (Z)Two-site model, Effect of mobile/immobile water ratio, Cr (Z)Two-site model, Effect of retardation factor, Cf (T)Two-site model, Effect of retardation factor, Cf (Z)Two-site model, Effect of retardation factor, Cr (Z)Two-site model, Effect of mass transfer coefficient, Cf (T)Two-site model, Effect of mass transfer coefficient, Cf (Z)Two-site model, Effect of mass transfer coefficient, Cr (Z)Two-site model, Effect of Peclet number, Cf (T)Two-site model, Effect of Peclet number, Cf (Z)Two-site model, Effect of Peclet number, Cr (Z)15

Table 1. (continued).

Model/Workspace

Name

Brief Description

CXTFIT/Inverse

Fig73aFig73b
Fig74
Fig79aFig79b
Fig712
Fig715

Fig.7-3a: Steady saturated flow in a sand column
Fig.7-3b: Steady saturated flow in a sand column
Fig.7-4: Estimation of duration time (MASS = 1 in Block B)
Fig.7-9a: Tritium effluent curve from Glendale clay loam
Fig.7-9b: Boron effluent curve (exp. 3-1, van Genuchten, 1974)
Fig.7-12: Field-scale bromide movement (after Jury at al., 1982)
Fig.7-15: Hypothetical field-scale reactive solute transport

CXTFIT/Stochast

Fig45aFig45b
Fig47aFig47b
Fig710
Fig711a
Fig711b
Fig711c
Fig713
Fig714a
Fig714b

Fig4-5: Stream tube model (STM) with random v, BVP vs. (IVP)
Fig4-5: Stream tube model (STM) with random v, (BVP) vs. IVP
Fig4-7: STM with random v, Constant and (variable) duration
Fig4-7: STM with random v, (Constant) and variable duration
Fig.7-10: STM with random v, Effect of σv
Fig.7-11: STM with random v, ensemble-averaged flux conc.
Fig.7-11: STM with random v, field-scale flux conc.
Fig.7-11: STM with random v, field-scale resident conc.
Fig.7-13: STM with random v and Kd , effect of ρvKd
Fig.7-14: Nonequilibrium field-scale transport , field-scale cr
Fig.7-14: Nonequilibrium field-scale transport , field-scale ct

3DADE/3DADE

Example1
Example2
Example3
Example4
Example5
Example6
Example7
Example8
Example9

Figure 2A: Diffuse source in semi-infinite region of surface, Steady-state.
Figure 4: Rectangular source at surface, First-type BC.
Figure 4: Rectangular source at surface, Third-type BC.
Figure 3B: Parallelepipedal initial distribution, Third-type BC.
Figure 7: Circular source at surface, Third-type BC.
Inverse: Diffuse source in semi-infinite region of surface, First-type BC.
Inverse: Parallelepipedal initial distribution, Third-type BC.
Inverse: Parallelepipedal initial distribution, Third-type BC.
Inverse: Circular source at surface, First-type BC.

N3DADE/N3DADE

Exampl1a
Exampl1b
Exampl2a
Exampl2b
Example3
Example4
Example5

BVP: Fig. 6: Instantaneous application from disc (cm,d)
BVP: Fig. 7: Instantaneous application from disc (cm,d)
BVP: Fig. 8: Heaviside application finite rectangle
BVP: Fig. 9: Heaviside application finite rectangle
IVP: Fig. 10: Heaviside initial, Finite rectangle
IVP: Fig. 11: Exponential distribution about (5,0,0), Spherical coordinate
PVP: Fig. 12: Heaviside production, Circular coordinate16

4. USERS MANUALThe computational modules of the STANMOD software package (i.e., CFITM, CFITIM,CHAIN, CXTFIT, 3DADE, N3DADE) were all written in FORTRAN. However, the interactive
graphics-based interface for the MS Windows environment was written in C++. In addition to
information given in this chapter and the original manuals of the different codes, extensive contextsensitive on-line help is made part of the interface. By pushing the F1 button, or clicking on theHelp button while working in any window, the user obtains information about the window content.
In addition, context-sensitive help is available using the "SHIFT+F1" help button. In this mode themouse cursor changes to a help cursor (a combination of arrow + question mark), and the user canproceed by clicking on the object for which help is needed (i.e., a menu item, toolbar button, orother feature). At this point a help file will be displayed, giving information about the item on which
the user clicked.
STANMOD is the main program unit defining the overall computational environment of thesystem. This module controls execution of the program and determines which other optional
modules are necessary for a particular application. The module contains a project manager and boththe pre-processing and post-processing units. The pre-processing unit includes specification of allnecessary parameters to successfully run the FORTRAN codes. The post-processing unit consists ofsimple x-y plots for graphical presentation of the results and a dialog window that displays an
ASCII output file.
The work for a new project should begin by opening the Project Manager, and giving a
name and brief description of the new project. Users must also decide where to save the project's
data (to which workspace). After clicking OK, the user has to select the specific program (e.g.,
CXTFIT or CFTM) for which the input is to be created. Then select the Type of Problem commandfrom the Input Menu. From this point on the program will navigate the user through the entire
process of entering input files. The user may either select particular commands from a menu, or
allow the interface to lead him/her through the process of entering input data by repeatedly selecting
the Next button. Alternatively, clicking the Previous button will return the user to the previous
window.
A Project Manager is used to manage data of existing projects, and to help locate, open,
copy, delete and/or rename desired projects or their input or output data. A "project" represents any

17particular problem to be solved by STANMOD. The project name (8 letters maximum), as well as abrief description of the project, helps to locate a particular problem. Input and output data for eachproject are placed in a subdirectory with the same name as the project. Projects are represented by afile project_name.cxt and the project_name subdirectory.
The Project Manager gives users considerable freedom in terms of organizing his/her
projects. The projects are grouped into Workspaces that can be placed anywhere in accessible
memory, i.e., on local and/or network hard drives. The Workspace can be any existing accessible
subdirectory (folder). STANMOD is installed together with several default workspaces. For
example, the Direct, Inverse, and Stochast workspaces located in the STANMOD subdirectory
contain test examples for, respectively, direct, inverse, and stochastic problems that can be analyzed
with the CXTFIT program. We suggest that the user creates his/her own workspaces, e.g., theMyDirect and MyInverse workspaces, and keeps the provided examples intact for further reference.
Projects can be copied with the Project Manager only within a particular workspace. Users cancopy projects between workspaces using standard file managing software, e.g., Windows Explorer.
In that case users must copy both the subdirectory of a particular project and the project_name.cxt
file. Another way of copying a project between Workspaces is to first open the project and thenusing the command Save as to save this project to a new location.

18

5. REFERENCES
Cho, C. M., Convective transport of ammonium with nitrification in soil, Can. Jour. Soil Sci., 51(3),339-350, 1972.
Jury, W. A., L. H. Stolzy, and P. Shouse, A field test of a transfer function model for predicting
solute transport, Water Resour. Res., 18, 369-375, 1982.
Leij, F. J., T. H. Skaggs, and M. Th. van Genuchten, Analytical solutions for solute transport inthree-dimensional semi-infinite porous media, Water Resour. Res., 27(10), 2710-2733, 1991.
Leij, F. J., and S. A. Bradford, 3DADE: A computer program for evaluating three-dimensional
equilibrium solute transport in porous media, Research Report No. 134, U. S. Salinity
Laboratory, USDA, ARS, Riverside, CA, 1994.
Leij, F. J., and N. Toride, N3DADE: A computer program for evaluating nonequilibrium threedimensional equilibrium solute transport in porous media, Research Report No. 143, U. S.
Salinity Laboratory, USDA, ARS, Riverside, CA, 1997.
Leij, F. J., and N. Toride, Analytical solutions for nonequilibrium transport models, in Selim, H. M.,
and L. Ma (eds.) Physical Nonequilibrium in Soils, Modeling and Application, Ann Arbor
Press, Chelsea, Michigan, 1998.
Marquardt, D. W., An algorithm for least-squares estimation of nonlinear parameters, SIAM J. Appl.
Math. 11, 431-441, 1963.
Parker, J. C., and M. Th. van Genuchten, Determining transport parameters from laboratory andfield tracer experiments, Bull. 84-3, Va Agric. Exp. St., Blaacksburg, Va, 1984.
Simunek, J., K. Huang, M. Sejna, and M. Th. van Genuchten, The HYDRUS-1D software package
for simulating the one-dimensional movement of water, heat, and multiple solutes in variablysaturated media. Version 1.0, IGWMC - TPS - 70, International Ground Water ModelingCenter, Colorado School of Mines, Golden, Colorado, 186pp., 1998.
Simunek, J., M. Sejna, and M. Th. van Genuchten, The HYDRUS-2D software package forsimulating two-dimensional movement of water, heat, and multiple solutes in variably
saturated media. Version 2.0, IGWMC - TPS - 53, International Ground Water ModelingCenter, Colorado School of Mines, Golden, Colorado, 251pp., 1999.
Toride, N., F. J. Leij, and M. Th. van Genuchten, A comprehensive set of analytical solutions fornonequilibrium solute transport with first-order decay and zero-order production, Water
Resour. Res., 29, 2167-2182, 1993.
Toride, N., F. J. Leij, and M. Th. van Genuchten, The CXTFIT code for estimating transportparameters from laboratory or field tracer experiments. Version 2.0, Research Report No. 137,
U. S. Salinity Laboratory, USDA, ARS, Riverside, CA, 1995.
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van Genuchten, M. Th., Mass transfer studies in sorbing porous media, Ph. D. thesis, New Mexico
State Univ., Las Cruces, New Mexico, 1974.
van Genuchten, M. Th., Determining transport parameters from solute displacement experiments,
Research Report No. 118, U. S. Salinity Laboratory, USDA, ARS, Riverside, CA, 1980.
van Genuchten, M. Th., Non-equilibrium transport parameters from miscible displacement
experiments, Research Report No. 119, U. S. Salinity Laboratory, USDA, ARS, Riverside,CA, 1981.
van Genuchten, M. Th., Convective-dispersive transport of solutes involved in sequential first-orderdecay reactions, Computers & Geosciences, 11(2), 129-147, 1985.
van Genuchten, M. Th., and R. W. Cleary, Movement of solutes in soil: Computer-simulated andlaboratory results, in Bolt, G. H. (ed.) Soil Chemistry B. Physico-Chemical Models, Elsevier,
Amsterdam, 1979.

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