United States
                    Environmental Protection
                    Agency
Robert S. Kerr Environmental
Research Laboratory
Ada OK 74820	
                     Research and Development
EPA/600/SR-94/028   May 1994
& EPA         Project Su m mary
                    Identification and  Compilation of
                    Unsaturated/Vadose  Zone Models
                    Paul K.M. van der Heijde
                      Many ground-water contamination
                    problems are derived from sources at or
                    near the soil surface. Consequently, the
                    physical and (bio-)chemical behavior of
                    contaminants in the shallow subsurface is
                    of critical importance to the development
                    of protection and remediation strategies.
                    Mathematical models,  representing our
                    understanding of such  behavior, provide
                    tools useful in assessing the extent of
                    pollution problems and evaluating means
                    to prevent and remediate them.
                      In classifying models generally applied
                    to soil- and  ground-water pollution
                    problems, a distinction can  be made
                    between the transport of the contaminants
                    from the point of their introduction into the
                    subsurface (i.e., contaminant source) to
                    the location of concern (i.e.,  point of
                    exposure), and the  (bio-)chemical
                    transformations that may occur in the
                    subsurface. Models specifically simulating
                    fluid flow are referred to as flow models.
                    Models describing the movement of
                    dissolved chemicals and their interaction
                    with the soil or rock matrix in  terms of
                    concentrations and mass fluxes are often
                    referred to as contaminant transport
                    models or solute  transport  models.
                    Furthermore, somevadose modelsfocus
                    on the resulting fate of contaminants, in
                    particular, simulating the (bic-)chemical
                    changes and transformations that occur in
                    the subsurface. Increasingly, combinations
                    of these three model types are employed to
                    adequately simulate site-specific pollution
                    problems and their remediation.
                      To identify existing models foir simulation
                    of flow and contaminant transport in the
                    unsaturated subsurface, a database search
                    and literatu re reviewwere conducted. From
                    the review a catalogue was developed
                    consisting of approximately 100 flow and
transport models that may be used for the
simulation of flow and transportprocesses
in the unsaturated zone to determine the
effectiveness of soil remediation
schemes among other uses. The models
considered range from simple mass
balance calculations to sophisticated,
multi-dimensional  numerical simulators.
There are six categories of models listed,
including models for single-fluid flow,
coupled and uncoupled flow and solute
and/or heat transport, and solute transport
for given pressure head distribution.
Finally, models are listed that provide soil
parameters from column experiments on
soil samples.
  In the full report each model is described
in an uniform way by a set of annotations
describing its purpose, majorhydrological,
mathematicalandoperational characteristics,
input requirements, simulative capabilities,
level of documentation, availability, and
applicability. In some cases, the model
description includes comments made by
the model author and the investigator
concerning development, testing, quality
assurance and use, as well as references
of studies using the model and references
that are part of the documentation or
considered pertinent to the model.
  This Project Summary was developed
by EPA's Robert S. Kerr Environmental
Research Laboratory, Ada,  OK,  to
announce key findings of the research
project that is fully documented in  a
separate report of the same title (see
Project Report ordering information at
back).

Introduction
  Many contamination  problems  find their
cause at or nearthe soil surface. Consequently,
the physical and (bio-)chemical. behavior of
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 these contaminants in the shallow subsurface
 is of critical importance to the development of
 protection and remediation  strategies.
 Mathematical models, representing  our
 understanding of such behavior, provide tools
 useful in assessing the extent of pollution
 problems and evaluating means to prevent
 and remediate them.
   Increasingly, detailed understanding and
 subsequent modeling of the near-surface
 zone are  crucial  in designing  effective
 remediation approaches. At many sites, this
 near-surface zone is only partially saturated
 with water, requiring  specially designed
 mathematical models. Thefull reportfocuses
 on models that might prove useful in
 simulating contaminant  levels in such
 partially saturated systems.
   Modeling contaminant behavior in  the
 unsaturated zone is generally aimed to
 address such Issues as [NRC,  1990]:
   • Determining the  arrival time of a
    contaminant at a  certain  depth;  this
    requires a prediction of the travel time
    forthecontaminant. Examples of depths
    of interest are the bottom of the root
    zone, the bottom of the treatment zone
    of a hazardous waste land treatment
    system facility, or the water table.
   • Predicting the amount of the surface-
    applied (or  spilled) contaminant that
    might  arrive at the depth of interest
    within  a certain time (or mass flux
    passing this  depth); this requires
    assessment   of   the   transport,
    (temporary)  retention, transformation
    and  degradation  (fate)  of  the
    contaminant.
   • Predicting the concentration distribution
    and/or the contaminant mass flux in the
    unsaturated zone (in both the aqueous
    and solid phases) at a particulartime or
    their changes over time.
  The latter purpose is of specific interest to
 this study  as it  relates to predicting  the
 amountof hazardous constituents remaining
 in the soil following a soil remediation or due
 to natural processes.
  Contaminating chemicals may leave the
 soil zone by leaching  downwards  to  the
 water table, by volatilization and escape to
 the  atmosphere,  by  (bio-)chemical
 transformation or degradation, and by plant
 uptake [Jury and Valentine,  1986]. Leaching
 constitutes mass  flow  of a chemical
 constituent and is the product of water flux
 and dissolved chemical concentration. Mass
 flow is dependent on the amount of applied
water, the water application intensity, the
saturated hydraulic conductivity of the soil,
the chemical concentration, the adsorption
sitedensity, and, indirectly, temperature [Jury
and Valentine, 1986]. Soils provide a strong
capacity for adsorbing chemicals and thus a
factor in determining the amount of chemical
 available for mass flux. This is due to the
 presence  of electrically  charged  clay
 minerals and organic matter and the large
 surface  areas of minerals and humus.
 Volatilization of chemical to the atmosphere
 takes place in the vapor phase of the soil
 and is controlled by chemical,  soil, and
 atmospheric conditions.  Volatilization is
 dependent on Henry's constant, chemical
 concentration, adsorption site density,
 temperature, water content, wind  speed,
 and water evaporation. Other potentially
 importanttransport processes include vapor
 and liquid  diffusion. Transformation and
 degradation processes determine the "fate"
 of the chemical of concern in the soil. The
 most important processes include chemical
 hydrolysis, (bio-)chemical transformations,
 and oxidation-reduction.
   In classifying models generally applied to
 soil-and ground-water pollution problems, a
 distinction  can be  made between  the
 transport of the contaminants from the point
 of their introduction into the subsurface (i.e.,
 contaminantsource) to the location of concern
 (i.e., pointof exposure), and the (bio-) chemical
 transformations that may occur  in  the
 subsurface.  A major transport mechanism
 results from the hydrodynamic behavior of
 contaminant carrying fluids or fluid phases in
 porousorfractured media. Models specifically
 simulating fluid flow are referred to as flow
 models. Models describing  the movement
 of dissolved chemicals and their interaction
 with the  soil or rock matrix in  terms of
 concentrations and mass fluxes are often
 referred to as contaminant transport models
 or solute transport  models. Furthermore,
 some vadose models focus on the resulting
 fate of contaminants, in particular, simulating
 the   (bio-)chemical   changes    and
 transformations that occur in the subsurface.
 The latter type of models may be based on
 a simple mass balance approach for the
 chemical of concern,  lumping spatial
 variations in a single valueforthe parameters
 of interest (e.g., SUMMERS model;  U.S.
 EPA, 1989, pp. 28-29), or it may constitute a
 set of complex equations describing the
 (bio-)chemical reactions of interest including
 a reaction constant database. Increasingly,
 combinations of these three model types are
 employed to adequately simulate site-specific
 pollution problems and their remediation (e.g.,
 Yehetal., 1993).
  The success of a given model depends
 on the accuracy and  efficiency with which
the physical  and (bio-)chemical processes
controlling   the behavior of  water  and
 introduced  nonaqueous liquids  and the
chemical and  biological species they
transport are simulated. The accuracy and
efficiency of the simulation, in turn, depend
heavily  on  the  applicability of  the
assumptions and  simplifications  adopted
 in the model, the availability and accuracy
 of  process  information   and  site
 characterization  data,  and  subjective
 judgments made by the modeler and
 management.
   As stated,  flow models simulate the
 movement of one or more fluids in porous or
 fractured rock. One such fluid is water; the
 others, if present, can be air or vapors such
 as  methane  (in soil)   or  immiscible
 nonaqueous phase liquids (NAPLs, in both
 fully and partially saturated systems) such
 as certain solvents, which may have a density
 distinct from water (LNAPLs, DNAPLs). In
 the context of the full report, only the flow of
 water (under unsaturated conditions) is
 considered. Most flow models are based on
 a mathematical formulation that considers
 the  hydraulic system  parameters as
 independent field information and hydraulic
 head, fluid pressure  or water content and
 fluid flux as dependent variables. They are
 used to calculate the steady-state spatial
 distribution, the changes in time in the spatial
 distribution, or the temporal distribution at a
 particular location of such variables as:
   •  hydraulic head, pressure head (or matric
     head), and suction head;
   •  saturation or moisture content;
   •  magnitude and direction of flow in terms
     of flow velocities or water mass fluxes;
   •  flowlines and travel times; and
   •  position of infiltration fronts.
   Inverse flow  models simulate the flow
 field to calculate the spatial distribution of
 unknown system parameters using field or
 experimental observations on  the state
 variables  such as hydraulic head,  fluid
 pressure, water content and fluid flux. Due
 to the complexity of the relationships between
 pressure head,  saturation and hydraulic
 conductivity, there are no truly  inverse
 models availableforflowin partially saturated
 porous media.
  The dominant parameter affecting flow
 and contaminant transport in the unsaturated
 zone is  hydraulic conductivity.  Since
 hydraulic  conductivity varies with water
 content, accurate  measurements of this
 parameter  are difficult to  make and very
 time-consuming. Therefore, theoretical
 methods have been developed to calculate
 the hydraulic conductivity from more easily
 measured soil water  retention data based
 on statistical pore-size distribution  models
 [van Genuchten etal., 1991]. The resulting
 functional relationship between pressure
 head and volumetric water content (i.e., soil-
 water retention function)  is presented  in
tabular form or as closed-form  analytical
 solutions that contain functional parameters
that are fitted to observed data. With the soil
water retention function known, the
 unsaturated hydraulic conductivity can be
calculated using the model of Mualem [1976].

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Models  have  been developed  to fit
mathematical functions to water retention
with known hydraulic conductivity or to water
retention and hydraulic  conductivity
simultaneously [van Genuchten etal., 1991].
These models may also be used to predict
hydraulic conductivity forgiven soil retention
data.
  Solute transport models are used to predict
movement or displacement, concentrations,
and mass balance components of water-
soluble  constituents and  to calculate
concentrations  or radiological  doses of
soluble radionuclides [van derHeijde etal.,
1988]. To do so,  solute transport models
incorporate various relevant physical and
chemical processes. Flow is represented in
the governing convective(-dispersive)
equation by thefiow velocity in theadvective
transport term. The velocities are also used
for the  calculation of the  spreading by
dispersion. If the velocity field is stationary,
it may be either calculated once using an
external flow program  or read into the
program as observed or interpreted data. If
the velocity field (i.e., spatial distribution of
velocities in terms of direction  and
magnitude)  is dependent on time and/or
concentration, then calculation of velocities
at each time step is required, either through
an internal flow simulation  module or an
external flow model linked by means of input
and output files. If a dissolved contaminant
is present in relative high concentrations,
changes in  its distribution during the
simulations might affect the flow behavior
through changes in the fluid density. In that
case, coupling of thefiow and solute transport
equations occurs through  an equation of
state, resulting in a system of equations that
needs to be solved simultaneously  (i.e.,
iteratively-sequentially [Huyakorn  and
Finder,  1983]).
   Generally, modeling the transformation
and  fate of chemical constituents  is
conducted in one of three possible ways
[van derHeijde etal., 1988]: (1) incorporating
simplified transformation orfateformulations
in the equation describing solute transport;
(2) formulating  a  mass-balance approach
to (bio-)chemical  transformation and  fate;
and  (3) by  coupling separate  equations
describing the (bio-)chemical processes with
the advective-dispersive transport equation.
Including transformation processes in solute
transport models  results  in  so-called
nonconservative (i.e., with respect to mass
in solution) transport and fate models. The
more complex  of these nonconservative
transport models may include advective
and  dispersive  transport,  molecular
diffusion, adsorption  (equilibrium  and
kinetics based), ion-exchange, radioactive
decay, and (bio-)chemical decay.

Mathematical Solutions in
Unsaturated/Vadose Models
  Most mathematical  models for the
simulation of flow and solute transport in the
unsaturated zone are distributecl-parameter
models, either deterministic or stochastic
[van  der Heijde  et al.,  1988].  Their
mathematical framework consists of one or
more partial differential equations describing
the flow and/or transport and fate processes,
as well as initial and boundary conditions
and solution  algorithms. Some of these
models assume that the processes active in
the system are stochastic in nature  or, at
least, that the process variables may be
described by probability distributions. In such
stochastic models, system responses are
characterized  by statistical distributions
estimated  by  solving  a deterministic
governing equation.
  The governing  equations for flow and
transport in the unsaturated zone are usually
solved either analytically or numerically.
Analytical models contain a closed-form or
analytical solution  of the field  equations
subject to  specified initial and boundary
conditions. To obtain  these analytical
solutions, simplifying assumptions have to
be made regarding  the nature  of the soil-
water-solute system, geometry, and external
stresses, often limiting their  application
potential. Because of the complex nature of
single-  and multi-phase flow  in the
unsaturated  zone and  the  resulting
nonlinearity of the governing equation(s),
very few analytical flow solutions have been
published [Bear,  1979]. With  respect to
transport and fate the situation is somewhat
different. Many one-, two-,  and  three-
dimensional analytical  solutions  for the
classical convection-dispersion equation
exist, often requiring a uniform flow field.
Some of these  solutions, specifically one-
dimensional solutions, can be used  in the
unsaturated  zone  assuming  a uniform
vertical soil water flux.
  In  semi-analytical models, complex
analytical solutions are approximated, often
using numerical techniques. In  the case of
unsaturated flow, semi-analytical solutions
may  be derived  by using  analytical
expressions for the relationships between
the dependent variables and the hydraulic
parameters  and   involving  numerical
integration [Bear, 1979]. Models based on a
closed-form solution for either the space or
time domain, and which contain additional
numerical approximations  for the other
domain, are also considered semi-analytical
models. Various quasi-analytical techniques
and approximate (analytical) equations have
been developed for simulating infiltration of
water  in soils [El-Kadi,  1983]. These
techniques are also  used for  the one-
dimensional  transport of  solutes [van
Genuchten and Alves, 1982].
  In numerical models, a discrete solution is
obtained in both the space and time domains
by using  numerical approximations of the
governing partial differential equation. As a
result of these approximations,  the
conservation of mass and accuracy in the
prediction variable are not always assured
(becauseof truncation and round-off errors),
and thus, these need to be verified for each
application. Spatial and temporal resolution
in applying such models is user-defined. If
the governing equations are nonlinear, as is
thecase in simulating flow in the unsaturated
zone, linearization often precedes the matrix
solution [Remson et al., 1971;  Huyakorn
and Finder,  1983].  Usually, solution of
nonlinear equations is achieved employing
nonlinear matrix methods such as the Picard,
Newton-Raphson,  and   Chord-Slope
methods  [Huyakorn and Finder, 1983].
  The numerical solution techniques used
for approximating the spatial components of
the governing  flow equations in the
unsaturated  zone are primarily the finite-
difference methods (FD), the integral finite-
differencemethods(IFDM),andtheGalerkin
finite-element method (FE). In most cases,
time is approximated by finite-difference
techniques resulting in an explicit, (weighted)
implicit or fully implicit solution scheme. A
finite-difference solution is obtained  by
approximating the  derivatives of the
governing equation.  In  the finite-element
approach an integral equation is formulated
first, followed by the numerical evaluation of
the integrals over the discretized flow or
transport domain. The formulation of  the
solution in each approach results in a set of
algebraic equations that are then solved
using  direct or iterative matrix methods.
Specific schemes may be required for  the
constitutive relationships, specifically in the
presence of hysteresis.
  There are many numerical considerations
in selecting  a model for simulation of a
particular  soil-water-solute   system.
Simulating flow in relatively wet soils (e.g.,
nearly saturated conditions and  ponding)
requires expression of the Richard's equation
in terms  of hydraulic head, matric head or
suction head, especially when parts of the
modeled soil system becomefully saturated.

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However, application of this form of the
Richard's  equation  causes  significant
convergence problems when simulating an
infiltration front in extremely dry soil
conditions; in the latter case, formulation of
Richard's equation should be based on
saturation  or  mixed  pressure-saturation
[Huyakorn  and Finder,  1983; Celia et al.,
1990], An advantage of the mixed form is
that it allows the transition from unsaturated
to saturated conditions while  maintaining
numerical mass conservation [Celia et al.,
1990].  Also,  significant mass balance
problems- might occur when site-specific
conditions result in highly nonlinear model
relationships [Celia et al., 1990].
  Other issues that should be addressed in
selecting a model for simulating flow in the
unsaturated zone are the possible needs for
double  precision versus single precision
variables,  the time-stepping approach
incorporated, the definition used for intercell
conductance (e.g., harmonic mean versus
geometric mean), and, if present, the way
steady-state simulation  is achieved (most
models do not provide steady-state flow
solutions). Some problems encountered with
specific models (or modeling techniques)
Include code limitations on gridding flexibility,
numerical  problems in zones  with high-
contrast soil  or  rock properties, and
Inaccuracy and instability inareas where the
flow field changes significantly in magnitude
and direction. In  some cases, avoiding
inaccuracy and instability problems require
very small spatial and temporal increments,
making multi-dimensional simulations
expensive or even unfeasible. Sometimes,
an adaptive  time-stepping  scheme is
implemented in the computer program to
optimize time-step requirements.
  Typical numerical techniques encountered
In solving the convective-dispersive solute
transport equation in the unsaturated zone
are comparable to those employed in
simulating solute transport in the saturated
zone  and include various finite-difference
methods,  the integral finite-difference
method, various Galerkin finite-element
formulations, and variants of the method of
characteristics [Yeh et al.,  1993]. As with
flow, time is generally approximated by finite-
difference techniques resulting in an explicit,
(weighted) implicit or fully implicit solution
scheme.
  Typical  problems found  in applying
traditional finite-difference and finite-element
techniques to simulatecontaminanttransport
in both the saturated and unsaturated zones
Include  numerical   dispersion  and
oscillations. Numerical dispersion is referred
to when the  actual  physical  dispersion
mechanism of the contaminant transport
cannot be distinguished from the front-
smearing  effects of the  computational
scheme [Huyakorn and Finder, 1983]. For
the finite-difference method, this problem
can  be  reduced  by using  the  central
difference   approximation.   Spatial
concentration oscillations (and  related
overshoot and undershoot) may occur near
a sharp concentration front in an advection-
dominated transport system. Remedies for
these problems are found to some extent in
the reduction of grid increments ortime-step
size, or by  using upstream weighting for
spatial derivatives. The use  of weighted
differences (combined upstream and central
differences) ortheselection of other methods
(e.g., the method of characteristics, and the
Laplace transform Galerkin method)
significantly reduces the occurrence of these
numerical problems.

Unsaturated/Vadose Model Data
Requirements
  The number and  type of parameters
required for modeling flow and transport
processes in soils depend on the type of
model chosen. These parameters can be
categorized  as  control  parameters
(controlling the operation of the computer
code),  discretization data (grid and  time
stepping), and material parameters. The
material parameters can be grouped in six
sets [Jury and Valentine, 1986]: static soil
properties, water transport and retention
functions, basic chemical properties, time-
dependent parameters, soil  adsorption
parameters, and tortuosity functions. Table
1 lists many of the relevant material model
parameters.

Selection of Models
  To identify existing models for simulation
of flow and contaminant transport in the
unsaturated subsurface, a database search
and literature review have been conducted.
The database search was focused on the
MARS model annotation database of the
IGWMC, which contains  about  650
descriptions  of  soil- and  ground-water
simulation models.  Information  for the
Table 1.   Selected Material Parameters for Flow and Transport Parameters in Soils
          (After Jury and Valentine, 1986)
  Static Soil Properties
      porosity
      bulk density
      particle size
      specific surface area
      organic carbon content
      cation exchange capacity
      pH
      soil temperature

  Flow and Transport Variables and Properties
      saturated hydraulic conductivity
      saturated water content
      matric head-water content function
      hydraulic conductivity function
      dispersion coefficient or dispersivity

  Basic Chemical Properties
      molecular weight
      vapor pressure
      water solubility
      Henry's constant
      vapor diffusion coefficient in air
      liquid diffusion coefficient in water
         octanol-water or oil-water partition
         coefficient
      half-life or decay rate of compound
      hydrolysis rate(s)

  Contaminant Source Characteristics
      solute concentration of source
      solute flux of source
      source decay rate
  Time Dependent Parameters
      water content
      water flux
      infiltration rate
      evaporation rate
      solute concentration
      solute flux
      solute velocity
      air entry pressure head
      volatilization flux

  Soil Adsorption Parameters
      distribution coefficient
      isotherm parameters
      organic carbon partition coefficient

  Tortuosity Functions
      vapor diffusion tortuosity
      liquid diffusion tortuosity

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literature review has been obtained from
various sources,  including  the  IGWMC
literature collection of more than 3000 titles
and about 20 serials, and through interlibrary
loan. Additional information was  received
from the U.S. EPA Center for Subsurface
Modeling Support (CSMoS) located at
RSKERL, Ada, Oklahoma. After reviewing
the model's  documentation and other
pertinent literature, additional information
was obtained from model authors and code
custodians when necessary.
  From  the  review a  catalogue was
developed consisting of approximately 100
flow and transport models that may be used
for the simulation of flow  and transport
processes in  the unsaturated  zone to
determine the effectiveness of  soil
remediation schemes among other uses.
The models considered range from simple
mass balance calculations to sophisticated,
multi-dimensional numerical simulators.
There  are six categories  of models listed,
including models for single-fluid flow, coupled
and uncoupled flow and solute and/or heat
transport, and solute transport for given
pressure head distribution. Finally, models
are listed that provide soil parameters from
column experiments on soil samples.
  In the full report each model is described in
a uniform way by a set of annotations defining
the  purpose  and  major  hydrologic,
mathematical and operational characteristics,
input requirements, simulative capabilities,
level of documentation,  availability, and
applicability.  In some cases, the model
description includes comments made by the
model  author and  IGWMC staff concerning
development, testing, quality assurance and
use, as well as references of studies using
the model and references that are part of the
documentation or considered pertinent to
the model.
  It should be noted that the full report does
not pretend to be complete in its listing of
appropriate models. Moreover, many codes
have been developed primarily for research
purposes and are not readily available. Also,
there are many simple models  based on
mass balance  evaluation or analytical
solution of  highly simplified systems not
presented in this catalogue. An  effort has
been made to select those 'simple' models
that  either are known for their  use in a
regulatory or enforcement mode or that are
considered representative for a certain type
of model. The report does not discuss multi-
fluid flow and  associated transport of
contaminants since a considerable amount
of research is  currently focussed  on
understanding   and  mathematically
describing the physics and chemistry of these
systems.

References
Bear, J. 1979. Hydraulics of Groundwater.
  McGraw-Hill, New York, NY.
Celia, M.A., E.T. Bouloutas, and R.L Zarba.
  1990.  A Genera!  Mass-Conservative
  Numerical Solution for the  Unsaturated
  F'low Equation. Water Resources Res.,
  Vol. 26(7), pp. 1483-1496.
El-Kadi, A.I. 1983. Modeling Infiltration for
  Water   Systems.  GWMI   83-09,
  International  Ground Water Modeling
  Center, Holcomb  Research  Institute,
  Indianapolis, IN.
Huyakorn,  P.S., and G.F. Pinder.  1983.
  Computational  Methods in Subsurface
  Flow. Academic Press, New York, NY.
Jury, W.A., and R.L. Valentine. 1986.
  Transport Mechanisms  and  Loss
  Pathways for Chemicals in Soil. In: S.C.
  Hern and S.M. Melancon (eds.), Vadose
  Zone Modeling of Organic Pollutants,
  Lewis Publishers, Inc., Chelsea, Ml, pp.
  37-60.
Mualem, Y. 1976. A New Model for Predicting
  the Hydraulic Conductivity of Unsaturated
  Porous Media.  Water Resources Res.,
  Vol. 12(3), pp. 513-522.
National Research Council (NRC). 1990.
  Ground  Water Models—Scientific  and
  Regulatory Applications.  National
  Academy Press, Washington, DC.
Remson, I., G.M. Hornberger, and F.J. Molz.
  1981. Numerical Methods in Subsurface
  Hydrology. Wiley Interscience, New York,
  NY.
U.S. Environmental Protection  Agency.
  1989. Determining Soil Response Action
  Levels Based on Potential Contaminant
  Migration  to  Ground   Water:   A
  Compendium of Examples. EPA/540/2-
  89/057,  Office  of Emergency  and
  Remedial Response, Washington, DC.
van der Heijde, P.K.M., A.I. El-Kadi, and
  S.A. Williams.  1988. Groundwater
  Modeling: An Overview and Status Report.
  EPA/600/2-89/028,  U.S.  Environmental
  Protection   Agency,   R.S.   Kerr
  Environmental Research Lab., Ada, OK.
van Genuchten, M.Th., and W.J. Alves. 1982.
  Analytical Solutions  of the One-
  Dimensional Convective-Dispersive
  Solute Transport Equation. Techn. Bull.
  1661, U.S. Dept. of Agriculture, Riverside,
  CA.
van Genuchten, M.Th., F.J. Leij,  and S.R.
  Yates.  1991. The RETC Code  for
  Quantifying the Hydraulic Functions of
  Unsaturated Soils. EPA/600/2-91/065,
  U.S. Environmental Protection Agency,
  R.S. Kerr Environmental Research Lab.,
  Ada, OK.
Yeh, T.-C., R. Srivastava, A. Guzman, and
  T. Harter. 1993. A Numerical Model for
  Water Flow and Chemical Transport in
  Variably Saturated Porous Media. Ground
  Water, Vol. 31 (4), pp. 634-644.

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  Paul K.M. van der Heijde is with the international Ground Water Modeling Center,
    Institute for Ground-Water Research and Education, Colorado School of Mines,
    Golden, CO 80401-1887.
  Joseph Williams is the EPA Project Officer (see below).
  The complete report, entitled/'Identification and Compilation of UnsaturatedA/adose
    Zone Models," (Order No. PB 94-157773; Cost: $27.00, subject to change) will be
    available only from
         National Technical Information Service
         5285 Port Royal Road
         Springfield, VA 22161
         Telephone: 703-487-4650
  The EPA Project Officer can be contacted at:
         Robert S. Kerr Environmental Research Laboratory
         U.S. Environmental Protection Agency
         Ada, OK 74820
United States
Environmental Protection Agency
Center for Environmental Research Information
Cincinnati, OH 45268

Official Business
Penalty for Private Use
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         EPA
   PERMIT No. G-35
EPA/600/SR-94/028

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