ANALYTICAL AND NUMERICAL MODELING OF PHYSICAL AND
CHEMICAL PROCESSES IN THE VADOSE ZONESHORT TITLE: MODELS FOR SOIL POLLUTION RISK ASSESSMENT

JIRKA ŠIMŮNEK
Department of Environmental Sciences, University of California
Riverside, Riverside, CA 92521, USA

Abstract. A large number of models for simulating water flow and solutetransport in the unsaturated zone are now used for a wide range of applications inresearch, management, and risk assessment of subsurface systems. Manymodels of varying degree of complexity and dimensionality have beendeveloped during the past several decades to quantify the basic physical andchemical processes affecting water flow and contaminant transport in theunsaturated zone. Modeling approaches range from relatively simple analyticaland semi-analytical solutions, to complex numerical codes. In this paper a brief
overview of more widely used analytical and numerical models is given. Sometypical problems in which the numerical codes have been applied are also
identified.

Keywords: Vadose zone; analytical models; numerical models; STANMOD; HYDRUS

1. Introduction
A large number of models for simulating water flow and solute transport in theunsaturated zone are now used for a wide range of applications in research,
management, and risk assessment of subsurface systems. Many models ofvarying degree of complexity and dimensionality have been developed during
the past several decades to quantify the basic physical and chemical processes
affecting water flow and contaminant transport in the unsaturated zone.
Modeling approaches range from relatively simple analytical and semianalytical solutions, to complex numerical codes. While analytical and semianalytical solutions are still popular for some applications, the ever-increasing
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Journal Information© 2005 Springer. Printed in the Netherlands.

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power of personal computers and the development of more accurate andnumerically stable solution techniques have motivated the much wider use ofnumerical codes in recent decades. The wide use of numerical models is alsosignificantly enhanced by their availability in both the public and commercial
domains, and by the development of sophisticated graphics-based interfacesthat can tremendously simplify their use.
2. Analytical Models
While under certain conditions (i. e., for linear sorption, a concentration
independent sink term, and a steady-state flow field) solute and heat transport
equations are linear equations, water flow equations are generally a highly
nonlinear because of the nonlinearity of the soil hydraulic properties.
Consequently, many analytical solutions have been derived in the past forsolute transport equations and these analytical solutions are now widely used
for analyzing solute transport under steady-state conditions. Many of theseanalytical solutions were compiled in STANMOD (STudio of ANalytical
MODels, www.hydrus2d.com) (Šimůnek et al., 1999b), a public domain
Windows based computer software package for evaluating solute transport inporous media using analytical solutions of the convection-dispersion solutetransport equation. It includes at present the following one-dimensional models:
CFITM (van Genuchten, 1980), CFITIM (van Genuchten, 1981), CHAIN (vanGenuchten, 1985), CXTFIT (Toride et al., 1995), and SCREEN (Jury et al.,1983), and two- and three-dimensional models: 3DADE (Leij and Bradford,
1994) and N3DADE (Leij and Toride, 1997).

Figure 1. The “Program Selection” dialog window in the STANMOD software package.

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2.1. ONE-DIMENSIONAL ANALYTICAL MODELS

An updated version of the CFITM code of van Genuchten (1980) that isincluded into STANMOD is intended for analyzing observed column effluent
data using analytical solutions of the one-dimensional equilibrium convectivedispersive transport equations. The code considers analytical solutions for bothsemi-finite and finite columns. The model provides an easy to use, efficient andaccurate means of determining various transport parameters by optimizing
observed column effluent data.
An updated version of the CFITIM code of van Genuchten (1981) can beused for analyzing observed column effluent data using analytical solutions of
the one-dimensional equilibrium and nonequilibrium convective-dispersive
transport equations. The code includes analytical solutions for semi-finite
columns. The nonequilibrium solutions consider the two-region dual-porosity(bi-continuum) flow model for physical nonequilibrium and the one-site or twosite sorption models for chemical nonequilibrium. CFITM and CFITIM
represent simple alternatives to the much more comprehensive, but also more
complex, CXTFIT model.
A modified and updated CHAIN code of van Genuchten (1985) analyzes
the convective-dispersive transport of solutes involved in sequential first-orderdecay reactions. Examples are the migration of radionuclides in which thechain members form first-order decay reactions, and the simultaneous
movement of various interacting nitrogen or organic species.
A modified and updated version of the CXTFIT code of Toride et al.
(1995) that is included in STANMOD is intended for estimating solute transportparameters using a nonlinear least-squares parameter optimization method. This
code may be used to solve the inverse problem by fitting a variety ofmathematical solutions of theoretical transport models, based upon the onedimensional convection-dispersion (or advection-dispersion) equation (CDE),
to experimental results. The program may also be used to solve the direct orforward problem to determine concentrations as a function 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-scaleequilibrium or nonequilibrium CDE.
Finally, the screening model of Jury et al. (1983), SCREEN, for behavior
assessment of trace organics in soils is included. This model is intended toclassify and screen organic chemicals for their relative susceptibility to different
loss pathways (volatilization, leaching, and degradation in the soil and air). The
model considers the following processes: gaseous diffusion, liquid dispersion,liquid convection, volatilization, sorption, and first-order degradation processes.

MODELS FOR SOIL POLLUTION RISK ASSESSMENT

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1.0

1.0

P1

P1
P2

0.6

P3

0.6

P4

0.4

0.4

P2

0.8

0.8

P3
P4
P5

P5

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Time [-]

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Figure 2. Examples of breakthrough curves at the end of the soil column (40 cm long) demonstrating
the effect of the retardation factor (R=1, 1.75, 2.5, 3.5, and 5) and the mass transfer coefficient(ω=0.02, 0.2, 0.5, 1.5, 7.5, and 1000) in the two region model. Calculations were carried out using
CXTFIT2 in STANMOD.

2.2. TWO- AND THREE-DIMENSIONAL ANALYTICAL MODELSSTANMOD also includes the 3DADE code of Leij and Bradford (1994) for
evaluating analytical solutions for three-dimensional equilibrium solutetransport in the subsurface. The analytical solutions pertain to selected cases ofthree-dimensional solute transport during steady unidirectional water flow inporous media having uniform flow and transport properties. The transport
equation contains terms accounting for solute movement by convection anddispersion, as well as for solute retardation, first-order decay, and zero-orderproduction. The 3DADE code can be used to solve the direct problem, i.e., theconcentration is calculated as a function of time and space for 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 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 beapplied 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 theliquid 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.

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Figure 3. The “Model Code” dialog window for the 3DADE model in the STANMOD software
package.

3. Numerical Models
Although a large number of analytical solutions also exist for water flow equation,
they can generally be applied only to drastically simplified problems. The majorityof applications for water flow in the vadose zone require a numerical solution of
the Richards equation.
Numerical methods are in general superior to analytical methods in terms of
being able to solve much more practical problems. They allow users to design
complicated geometries that reflect complex natural geologic and hydrologic
conditions, control parameters in space and time, prescribe realistic initial andboundary conditions, and implement nonlinear constitutive relationships.
Numerical methods usually subdivide the time and spatial coordinates into smaller
pieces, such as finite differences, finite elements, and/or finite volumes, andreformulate the continuous form of governing partial differential equations interms of a system of algebraic equations. In order to obtain solutions at certain
times, numerical methods generally require intermediate simulations (timestepping) between the initial condition and the points in time for which thesolution is needed.
There is a large number of numerical models that are available for evaluating

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variably-saturated water flow and solute transport in the subsurface. Some of thesemodels are in the public domain, such as MACRO (Jarvis, 1994), SWAP (vanDam et al., 1997), UNSATH (Fayer, 2000), VS2DI (Healy, 1990), and HYDRUS1D (Šimůnek et al., 1998; www.hydrus2d.com), while others are in thecommercial domain, such as HYDRUS-2D (Šimůnek et al., 1999a) and
MODFLOW-SURFACT (HydroGeoLogic, 1996). These models vary widely interms of their complexity, sophistication, and ease of use. Although some models
are still being run under the DOS operating system, with associated difficulties ofpreparing input files and interpreting tabulated outputs, many others, especiallythose in the commercial domain, are supported by sophisticated graphics-based
interfaces that tremendously simplify their use (e.g., Šimůnek et al., 1998; 1999a).
Models have recently become increasingly sophisticated in terms of thetype and complexity of solute transport processes that can be simulated.
Transport models are no longer being limited to solutes undergoing relatively
simple chemical reactions such as linear sorption and first-order decay, but now
consider also a variety of nonlinear sorption and exchange processes, physicaland chemical nonequilibrium transport, volatilization, gas diffusion, colloid
attachment/detachment, decay chain reactions, and many other processes. For
example, the general formulation of the transport equations in the HYDRUS
codes allows one to simulate not only non-adsorbing or linearly sorbing
chemicals, but also a variety of other contaminants, such a viruses (Schijven andŠimůnek, 2001), colloids (Bradford et al., 2002), cadmium (Seuntjens et al.,2001), and hormones (Casey et al., 2003, 2004), or chemicals involved in thesequential biodegradation of chlorinated aliphatic hydrocarbons (Schaerlaekens etal., 1999; Casey and Šimůnek, 2001).
A substantial amount of research has been recently devoted to developing
methods for describing preferential/nonequilibrium flow in structured media(macroporous soils). A variety of dual-porosity, dual-permeability, multiporosity, and/or multi-permeability models (e.g., Gerke and van Genuchten,
1993; Jarvis, 1994; Šimůnek et al., 2003) has been developed to describe
preferential flow. Dual-porosity and dual-permeability models both assume thatthe porous medium consists of two interacting regions, one associated with theinter-aggregate, macropore, or fracture system, and the other one comprising
micropores (or intra-aggregate pores) inside soil aggregates or the rock matrix.
While dual-porosity models assume that water in the matrix is stagnant, dualpermeability models allow for water flow in the matrix as well. For a recent
comprehensive review of various modeling approaches used to simulate
preferential flow see Šimůnek et al. (2003).
Several of the dual-porosity and dual-permeability features were recently
included in the HYDRUS software packages (Šimůnek et al., 2003, 2005).
Examples of their application to a range of laboratory and field data involving

MODELS FOR SOIL POLLUTION RISK ASSESSMENT

7transient flow and solute transport are given by Šimůnek et al. (2001), Zhang etal. (2004), Köhne et al. (2004, 2005), Kodešová et al. (2005), Haws et al. (2005),
and Pot et al. (2005), among others.
Numerical models are, however, still lacking that would fully describe
colloid transport and colloid-facilitated solute transport in a complex
subsurface environment. This complex process requires knowledge of variably
saturated water flow, colloid transport, dissolved contaminant transport, andcolloid-facilitated contaminant transport. Colloids are inorganic and/or organic
constituents that are generally chemically reactive. Inorganic colloids areprimarily fine-sized mineral soil constituents, while organic colloids are organic
matter-based. Transport equations must be formulated for both colloids andcontaminants, in all its forms. Equations therefore must be written for the total
contaminant, for contaminant sorbed kinetically or instantaneously to the solid
phase, and for contaminant sorbed to mobile colloids, to colloids attached to thesoil solid phase, and to colloids accumulating at the air-water interface. Some ofthese processes were recently incorporated in the HYDRUS software package(van Genuchten and Šimůnek, 2004). A complete description of the colloidfacilitated transport can be, for example, found in Hornberger et al. (1992) andvan Genuchten and Šimůnek (2004).
4. Biogeochemical Models
Significant efforts have recently been carried out to couple physical flow andtransport models with geochemical models to simulate even more complex
reactions, such as surface complexation, precipitation/dissolution, cationexchange, and/or biological reactions. Geochemical models can be divided into
two major groups: those with specific chemistry and general models (Šimůnek
and Valocchi, 2002). Models with specific chemistry are limited in the number of
species they can handle and their application is quite restricted to problems
having a prescribed chemical system. They are, however, much easier to use andcan be more computationally efficient than general models. Typical examples ofmodels with specified chemistry are models simulating the transport of majorions, such as LEACHM (Wagenet and Hutson, 1987) and UNSATCHEM(Šimůnek and Suarez, 1994; Šimůnek et al., 1996), and various reclamation
models (Šimůnek and Valocchi, 2002). These models typically considertransport of major ions and their mutual reactions such as complexation, cationexchange, and precipitation/dissolution. Also models simulating carbon andnitrogen cycles are becoming a standard feature of many environmental models,such as COUP (Jansson and Karlberg, 2001) or HYDRUS-2D coupled withCW2D (Langergraber and Šimůnek, 2005). These models typically distributeorganic matter, carbon, and organic and mineral nitrogen over multiple

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MODELS FOR SOIL POLLUTION RISK ASSESSMENT

computational pools, while allowing organic matter to be decomposed bymultiple microbial biomass populations. They can account for most of the
major reaction pathways, such as inter-pool transfer of carbon and nitrogen,nitrification (ammonium to nitrate-N), denitrification (leading to the productionof N2 and N2O), volatilization losses of ammonia (NH3), and microbial biomass
growth and death. HYDRUS-2D coupled with CW2D (Langergraber andŠimůnek, 2005) was specifically developed to describe processes in constructedwetlands.
Models with generalized chemistry provide users with much more freedom indesigning their particular chemical systems; possible applications of these models
are also much wider. Users can either select species and reactions from large
geochemical databases, or are able to define their own species with particularchemical properties and reactions. Most codes with general geochemistry are
limited to fully saturated porous media with constant water contents andvelocities, and evaluate only solute transport and biogeochemical reactions(e.g., PHREEQC, Parkhurst and Appelo (1999)). Only few models allow thevelocity field to be calculated internally. Several codes for transient unsaturated
flow have also been coupled to general biogeochemistry models. These include,for example, 3DHYDROGEOCHEM (Yeh and Cheng, 1999) and HP1 (Jacquesand Šimůnek, 2005).
4.1. HP1

The HP1 program (Jacques and Šimůnek, 2005) resulted from coupling
HYDRUS-1D (Šimůnek et al., 1998) with the PHREEQC geochemical code
(Parkhurst and Appelo, 1999). This new, very powerful code contains modules
simulating (1) transient water flow in variably-saturated media, (2) the transportof multiple components, (3) mixed equilibrium/kinetic biogeochemical
reactions, and (4) heat transport. HP1 is a significant expansion of theindividual HYDRUS-1D and PHREEQC programs by preserving andcombining most of their original features and capabilities. The code still uses
the Richards equation for simulating variably-saturated water flow andadvection-dispersion type equations for heat and solute transport. However, theHP1 program can now simulate also a broad range of instantaneous and kinetic
low-temperature biogeochemical reactions in water, the vadose zone and inground water systems, including interactions with minerals, gases, exchangers
and sorption surfaces based on thermodynamic equilibrium, kinetic, or mixed
equilibrium-kinetic reactions. HP1 uses the operator-splitting approach with noiterations during one time step (non-iterative sequential approach). The
accuracy of this operator-splitting approach for (a) a kinetic reaction network(i.e., sequential and parallel kinetic degradation reactions) by comparing HP1

MODELS FOR SOIL POLLUTION RISK ASSESSMENT

9with an analytical solution for TCE-degradation and (b) for mixed equilibriumand kinetic reactions for different flow conditions (steady-state and transient)has been evaluated by Jacques et al. (2005b). HP1 is available on request fromwww.sckcen.be/hp1.
Jacques et al. (2003, 2005), Jacques and Šimůnek (2005), and Šimůnek etal. (2006) demonstrated the versatility of HP1 on several examples such as a)
the transport of heavy metals (Zn2+, Pb2+, and Cd2+) subject to multiple cationexchange reactions, b) transport with mineral dissolution of amorphous SiO2
and gibbsite (Al(OH)3), c) heavy metal transport in a medium with a pHdependent cation exchange complex, d) infiltration of a hyperalkaline solution
in a clay sample (this example considers kinetic precipitation-dissolution of
kaolinite, illite, quartz, calcite, dolomite, gypsum, hydrotalcite, and sepiolite), e)long-term transient flow and transport of major cations (Na+, K+, Ca2+, andMg2+) and heavy metals (Cd2+, Zn2+, and Pb2+) in a soil profile, f) cadmium
leaching in acid sandy soils, g) radionuclide transport, h) long term uranium
migration in agricultural field soils following mineral P-fertilization, and i) fateand subsurface transport of explosives.
There is an increasing concern about the presence of explosives andenergetics in the subsurface environment. Such chemicals are the result of the
manufacture, distribution, testing and/or unsafe disposal of ammunition, but canalso be released into the environment during a terrorist attack. Šimůnek et al.
(2006) showed an application of HP1, in which they simulated the transport ofthe major explosive (the parent product TNT, 2,4,6-trinitrotoluene), and its
various metabolites (the daughter products) 2ADNT, 4ADNT, and TAT, as they
are being created sequentionally by degradation of the parent compounds. HP1
further allows the parent and daughter compounds to have different mobilities
in the subsurface as dictated by their specific dissolution, sorption and transport
properties. In its most general case, the model permits contaminants to reside in
all three phases, i.e., the liquid, solid (precipitated and sorbed), and gaseous
phases. This example indicated that ground water may be more vulnerable toleaching of TNT daughter products (notably TAT) than of the parent compound
itself, and that monitoring for the daughter products may provide an early
warning of possible TNT leaching.
These various examples demonstrate that the coupling of HYDRUS-1D and
PHREEQC leads to a potentially very powerful tool for simulating a broad
range of interacting physical, chemical and biological processes affecting
transport of contaminants in soils.
Acknowledgment: This paper is based on work supported by the Terrestrial
Sciences Program of the Army Research Office (Terrestrial Processes andLandscape Dynamics and Terrestrial System Modeling and Model Integration).

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