Recent Progress in Modelling Water Flow and Chemical Transport …hydrologie.org/redbooks/a204/iahs_204_0169.pdf · classical Richards' equations for unsaturated flow and the convection-dispersion

ffydrological Interactions Between Atmosphere, Soil and Vernation (Proceedings of the Vienna Symposium, August 1991). IAHS Publ. no. 204, 1991.

Recent Progress in Modelling Water Flow and Chemical Transport in the Unsaturated Zone

M. TH. VAN GENUCHTEN U.S. Salinity Laboratory, USDA, ARS, 4500 Glenwood Drive, Riverside, CA, 92501, USA

ABSTRACT This introductory paper reviews alternative approaches for modelling water and solute movement in the unsaturated (vadose) zone of field soils. A large number of analytical and numerical models are now available to predict unsaturated water flow and solute transport. The most popular models remain the classical Richards' equations for unsaturated flow and the convection-dispersion equation for solute transport. While deterministic solutions of these equations are important tools in research and management, their utility for predicting actual field-scale processes is increasingly being questioned. Problems caused by soil heterogeneity at a variety of spatial scales, and a lack of progress in improving our field measurement technology, have contributed to some disappointment with the classical models. Alternative deterministic and stochastic approaches have been developed to better deal with field-scale heterogeneity. This paper briefly reviews these models, and outlines several areas of research in need of further investigation.

INTRODUCTION

The importance of the unsaturated zone as an integral part of the hydrological cycle has long been understood. The zone plays an inextricable role in many aspects of hydrology, including infiltration, soil moisture storage, evaporation, plant water uptake, groundwater recharge, runoff, and soil erosion. Early studies of the unsaturated (vadose) zone focused primarily on water supply studies, inspired in part by attempts to optimally manage the root zone of agricultural soils for maximum crop production. Recently, studies of the unsaturated zone are increasingly motivated by concerns about soil and groundwater pollution from agricultural, industrial and municipal sources. Federal, state and local action and planning agencies, as well as the public at large, are now scrutinizing the intentional and unintentional release of surface-applied and soil-incorporated chemicals into the environment. Fertilizers and pesticides applied to agricultural lands inevitably move below the root zone, and may contaminate groundwaters. Chemicals migrating from municipal and industrial disposal sites, as well as

169

M. Th. van Genuchten 170

radionuclides from nuclear energy waste disposal facilities, represent similar environmental hazards.

Considerable progress has been obtained in the conceptual understanding and mathematical description of vadose zone flow and transport processes. A variety of analytical and numerical models are now available to predict water flow and solute transport between the land surface and the groundwater table. The most popular models continue to be the Richards' equation for unsaturated flow and the Fickian-based convection-dispersion equation for solute transport. Deterministic approaches based on these equations have and will continue to provide convenient tools for analyzing specific experiments on water and solute movement, and for extrapolating information from a limited number of field studies to different soil, crop and climatic conditions, as well as to different tillage and water management schemes. At the same time, the usefulness of the classical models for predicting actual field-scale water and solute processes is increasingly being questioned. Problems caused by preferential flow through soil macropores, spatial and temporal variability in the soil hydraulic properties, and various nonequilibrium processes affecting chemical transport, have raised serious questions about their accuracy for field-scale predictions. A number of alternative deterministic and stochastic models have been proposed to better deal with field-scale heterogeneities. These models have greatly increased our quantitative understanding of field-scale flow and transport processes, and in some cases have also resulted in better practical tools for management purposes. In this paper we shall briefly review several of these models, and outline a number of areas in need of further research and development. The review borrows extensively from a set of papers presented at a special IAHS workshop "Modelling of Water Movement and Chemicals in the Soil" at the 10th General Assembly of the International Union of Geodesy and Geophysics (IUGG) in Vienna, Austria, August 1991.

CLASSICAL DESCRIPTIONS OF WATER AND SOLUTE MOVEMENT IN SOILS

Deterministic descriptions of water and solute movement in the unsaturated zone are generally based on the classical Richards' equation for unsaturated water flow and the convection-dispersion equation for solute transport. For one-dimensional vertical transfer, these equations are respectively

C ( h ) | £ = J - [ K ( h ) | * - K ( h ) ] + S (1) at dz dz

3(ps) + j ( e c ) = J . ( e D . g £ - q c W

dt dt dz dz (2)

where C is the soil water capacity, being the slope of the soil water retention curve, 0(h), 8 is the volumetric water content, h is the soil water pressure head (being negative for unsaturated conditions), t is time, z is distance from the soil surface downward, K is the hydraulic conductivity as a function of h or 8, s is the solute

171 Modelling water flow and chemical transport in the unsaturated zone

concentration associated with the solid phase of the soil, c is the solute concentration of the fluid phase, p is the soil bulk density, D is the solute dispersion coefficient, S and 0 are sources and sinks for water and solutes, and q is the volumetric fluid flux density given by Darcy's law as

q = - K ( h ) - ^ + K ( h ) (3) dz

A close scrutiny of the theoretical basis of equations (1), (2) and (3) reveals several assumptions which may be invalid for most field situations. For example, the equations assume that (a) the air phase plays a minor role in the partially saturated flow process so that a single equation can be used to describe saturated-unsaturated flow, (b) Darcy's equation is valid for near no-flow conditions as well as for flow through structured (aggregated) soils, (c) the osmotic and electrochemical components of the soil water potential are negligible, (d) the fluid density is independent of the solute concentration, and (e) matrix and fluid compressibilities may be neglected. Ignoring for now the problem of soil heterogeneity, mathematical solutions of the above water flow and solute transport equations are further complicated, and rendered less precise, by (a) the hysteretic nature of the soil hydraulic properties 8(h) and K(h), (b) the often significant effects of temperature and solute concentration on the hydraulic properties, (c) the extreme nonlinear dependency of the hydraulic conductivity on the pressure head, and (d) the lack of reliable and economic methods for measuring the unsaturated hydraulic and solute transport properties of undisturbed field soils.

The source/sink term S in equation (1) accounts primarily for water uptake by plant roots. Widely different approaches exist for modeling water uptake (e.g., Taylor et al., 1983). In a previous review, Molz (1981) listed 13 different models for the root water extraction term, several of which were based on approximations of Darcy's law for water flow between soil and roots. Typical examples of root water extraction terms are given by Hansen et ai. (1991) and Kuchment & Startseva (1991). Most expressions currently used for modelling water uptake as a function of water stress and other limiting factors (including soil salinity) are essentially empirical and contain parameters that depend on specific crop, soil and environmental conditions.

The source/sink term <p in equation (2) accounts for nutrient uptake and a variety of chemical and biological reactions and transformations insofar these processes are not included in the sorption/exchange term 3ps/<5t of equation (2). Degradation processes are usually approximated with zero-order and/or first-order rate terms. For microbially induced organic and inorganic transformations, the degradation process should also include provisions for the growth and maintenance metabolism of soil microbes. Proper description of the sink term <p for such situations remains a major problem. McLaren (1970) gave a helpful early review of the temporal and vectorial reactions of soil nitrogen. Frissel & van Veen (1980) and Iskandar (1981) provided additional overviews illustrating the level of complexity that can be obtained with elaborate nitrogen fate and transport models. Typical examples of nitrogen simulation models are given by Hansen et al. (1991) and Feher et al. (1991). Among the processes considered in these papers are


nitrification, denitrification, mineralization, nitrogen uptake by plants, and nitrogen leaching from the root zone. These and other studies indicate that microbial growth and maintenance are time- and space-dependent variables in the soil system, and that our current understanding of physical and chemical properties of soils far exceeds our understanding of microbially induced transformations during water flow in soils. In spite of this lack of understanding, nitrogen simulation models of the type discussed by Hansen êl aj. (1991) and Feher ej âl- (1991) are indispensable tools in both research and management. For example, they are extremely helpful for evaluating the comparative effects of alternative soil and water management practices and chemical application technologies on crop production and groundwater pollution.

Equations (1) and (2) are formulated assuming isothermal soil conditions. In reality, most physical, chemical and microbial processes in the soil are strongly influenced by soil temperature. This also applies to water flow itself, including the indirect effects of temperature on the unsaturated soil hydraulic properties (e.g., Hopmans & Dane, 1986). Hence, a complete description of unsaturated zone transfer processes requires also consideration of heat flow and its nonlinear effect on most physical, chemical and biological processes taking place in the soil-plant system. Kuchment & Startsova (1991) give a comprehensive mathematical treatment of water and heat flow in the soil-plant-atmosphere system. Gusev (1991) similarly provides a set of models which illustrate the complex and interactive relationships between water and heat flow in partially frozen, snow-covered soils. This author used an integral balance method for predicting water and temperature distributions in frozen soils, including the closely related processes of soil freezing, snow melt, water infiltration, and surface runoff. An equally refined but isothermal analysis of infiltration and overland flow is given by Lee ej ai. (1991).

Equations (1) and/or (2) have been solved using a variety of analytical and numerical finite difference and finite element techniques, both for one- and multidimensional applications. Since this review focuses primarily on conceptual issues, no attempt is made to provide an exhaustive review of all available literature on vadose zone flow and transport modelling. We refer to recent publications by van der Heijde (1985), Gùven et al. (1990) and Sudicky & Huyakorn (1991) for helpful reviews of recent developments in analytical and numerical modelling.

NONEQUILIBRIUM SOLUTE TRANSPORT

For conditions of steady-state water flow in homogeneous soils (q and 8 are constant in time and space), neglecting the source term <p, and assuming linear sorption such that the adsorbed concentration (s) is linearly related to the solution concentration (c) through a distribution coefficient, k (i.e., s=kc), equation (2) reduces to the much simpler

REl=B^l-vEl (4) 3t dz2 dz


where v = q/8 is the average pore water velocity, and R = 1 + pk/8 is the solute retardation factor.

The use of a constant retardation factor R in equation (4) assumes the occurrence of equilibrium-type interactions between the liquid and solid phases of the soil. While the use of a linear isotherm can greatly simplify the mathematics of a transport problem (e.g., Valocchi, 1984), adsorption-desorption and exchange reactions are generally nonlinear and usually depend also on the presence of competing species in the soil solution. This, in turn, may require the consideration of solution chemistry and/or ion exchange principles. Widely different solution techniques have been proposed to deal with the resulting multi-component transport problem (e.g., Rubin & James, 1973; Miller & Benson, 1983); a recent review of specific models is given by Mangold & Tsang (1991).

The assumption of instantaneous (equilibrium) sorption is also being questioned. A number of chemical-kinetic and diffusion-controlled "physical" models have been used to describe nonequilibrium transport (Wagenet, 1983). Among these, the most popular and simplest one is the first-order linear rate equation

i i=a(kc-s) (5) at

where a is a first-order rate constant. Unfortunately, the kinetic adsorption parameters in equation (5) have frequently been found to vary as a function of the pore water velocity. More refined nonequilibrium transport models invoke the two-site sorption or two-region (double-porosity) assumptions. Two-site models assume that sorption sites in a soil can be divided into two fractions each exhibiting different equilibrium and kinetic adsorption properties (Selim et al, 1976; Parker & Jardine, 1986). Two-region or "mobile-immobile" type dual-porosity models, on the other hand, assume that the sorption rate is controlled by the rate at which solutes diffuse from relatively mobile (flowing) liquid regions to reaction sites in equilibrium with immobile (dead-end) water. Diffusion into and out of these immobile water pockets is generally modeled as an apparent first-order exchange process (Coats & Smith, 1964; De Smedt & Wierenga, 1984). An example application of the two-region model for transient unsaturated flow is given by Mermoud & Gaillard (1991).

Assuming steady-state water flow, Nkedi-Kizza et aL (1984) recently showed that the two-site and two-region models can be put into the same dimensionless form using model-specific parameters. They used this information to show that effluent curves from laboratory soil columns alone cannot be used to differentiate between chemical and physical phenomena that cause apparent non-equilibrium. This means that independent experiments (such as batch studies or displacement experiments with nonreactive tracers) are needed for verification of the two types of models. Similar problems related to nonlinear and nonequilibrium sorption also pertain to the transport of organic solutes (MacKay et al, 1985; Pinder & Abriola, 1986). Depending upon the type of organic involved, models to predict their transport in the unsaturated zone may also need to account for volatilization, microbial, chemical and photochemical transformations, and


possibly multiphase flow (McCarty eî al., 1984; Widdowson §1 al , 1988; Kaluarachchi & Parker, 1990).

While the above models based on equations (1) and (2) have been, and undoubtedly will remain, indispensable tools in the research and management of unsaturated zone transfer processes (Addiscott & Wagenet, 1985), growing evidence in the literature suggests that solutions of these classical models may not accurately describe transfer processes in most natural field soils (Sposito gt al , 1986a; Nielsen et âl-, 1986). Factors contributing to this failure to describe field-scale processes are (1) preferential movement of water and solutes through large, continuous macro-pores in the soil, and (2) spatial and temporal variability of field-scale flow and transport properties. These two problem areas, and alternative approaches to deal with them, are briefly discussed below.

TRANSPORT IN STRUCTURED SOILS

Valid questions have arisen in the literature about the usefulness of equations (1) and (2) for describing flow and transport processes in structured soils characterized by large continuous voids, such as natural interaggregate pores, interpedal voids, earthworm and gopher holes, decayed root channels, or drying cracks in desiccated fine-textured soils. The movement of water and solutes in such soils can be substantially different from that in relatively homogeneous materials (Beven & Germann, 1982; White, 1985). A helpful discussion of these issues is given by Villholth ejt al. (1991). Additional evidence of preferential flow is provided by Ohte Et al- (1991) for chloride transport in forest soils.

Attempts to describe water flow in unsaturated structured soils have generally centered on two-domain, two-region, or bicontinuum approaches. One domain consists of the soil matrix in which water flow is described with the conventional Darcian-based unsaturated water flow equation, while the other domain consists of either a single macropore, or of a statistical network of macropores, through which water flows primarily under the influence of gravity (Yeh & Luxmoore, 1982; Germann & Beven, 1985; Wang & Narasimhan, 1985; among others). Several of these formulations have been inspired by the closely related problem of water flow and solute transport through partially saturated fractured rock. A helpful comprehensive review of these models is given by Wang (1991).

While macropore flow itself has important implications in subsurface hydrology in general, and on infiltration and unsaturated water flow in particular, its main implications are in the accelerated movement of surface-applied fertilizers or pollutants into and through the unsaturated zone (McLay ej al., 1991; Villholth ÊÎ al-, 1991; Ohte fit al-, 1991). A large number of two-region type models have been developed over the years to describe this type of preferential solute transport. Like the quasi-empirical first-order mobile-immobile equations, these models assume that the chemical is transported through a single well-defined pore or crack of known geometry, or through the voids between well-defined uniformly-sized aggregates. Contrary to those previous models, however, Fickian-based diffusion equations are used to more rigorously describe the transfer of solutes from the


larger pores into micropores of the soil matrix. A variety of models, often focusing on transport in structured soil and groundwater systems, currently exist (Huyakorn St âl., 1983; Rasmuson, 1984; van Genuchten, 1985; Goltz & Roberts, 1986; Wang, 1991). While these models have been successfully tested in the laboratory, their field verification has started only recently (van Genuchten ÊÎ âl-, 1990; Gee et âl., 1991; Sudicky & Huyakorn, 1991).

Although two-region models are conceptually pleasing and have led to improved predictions, the question remains whether or not geometry-based solute transport models for structured soils are too complicated for routine use in research or management. They require a large number of parameters which are not easily measured independently, especially in the field. In contrast, the classical Fickian-based transport equation is much simpler to use and requires fewer parameters. Moreover the classical model may well be applicable to certain limiting situations predicated by the spatial scale of the transport problem. Several attempts have been made to define conditions for which the much simpler classical model (as well as the first-order mobile-immobile model) may be valid, in which case the effects of soil matrix (intra-aggregate) diffusion can be lumped into an effective dispersion coefficient, D (e.g., Parker & Valocchi, 1986).

STOCHASTIC APPROACHES

Two-region models simulating preferential movement of water and solutes through soils represent attempts to deal with pore structure heterogeneity at spatial scales somewhere intermediate between laboratory-scale measurements and the larger field scale. As such they are useful for predicting the predominantly vertical transport in structured but areally homogeneous field soils. Unfortunately, few soils are areally homogeneous (Gee gt al-, 1991; Sudicky & Huyakorn, 1991). This is further illustrated in Figure 1 which shows measured bromide concentrations 399

BROMIDE Oi . 1 . 1 . 1

"E 0.5 -

3 0 0 2.5 5.0 7.5 10.0 12.5 15.0

DISTANCE ALONG TRANSECT (m)

FIG. 1 Observed gravimetric bromide concentrations (mg per kg dry soil) 399 days after application of a bromide tracer pulse to the soil surface (after Schulin et al., 1987).

M. Th. van Genuchten ne

days after application of an areally uniform instantaneous bromide pulse to the surface of a field soil in Switzerland. The unequal distribution of the tracer, especially horizontally along the transect, raises important questions about how to design effective field sampling programs, which field instrumental methods to use for this sampling, and how to simulate the heterogeneous field-scale transport process.

The enormous variability of the subsurface hydrological environment, and the imprecision with which parameters and processes can be measured, has led to the adoption of stochastic models and geostatistical procedures to assist in the prediction and monitoring of contaminant transport in the unsaturated zone. A large number of stochastic approaches are currently available (Dagan, 1989; Jury & Roth, 1990). For convenience, the stochastic approaches are grouped here into scaling theories, Monte Carlo methods, and stochastic-continuum models. A common assumption of all field-scale stochastic transport models is that parameters are treated as random variables with discrete values assigned according to a given probability distribution. In practice, the stochastic approach cannot be used without several simplifying assumptions, including (1) the stationarity hypothesis which assumes that a random transport parameter has the same probability density function (pdf) at every point in the field (the mean and variance are constant), and (2) the ergodicity hypothesis which states that ensemble averages can be replaced by spatial averages, and that spatial replicates can be used to construct the appropriate pdf s for the transport parameters.

Current scaling theories applied to field-scale flow and transport problems have evolved from the early work of Miller & Miller (1956) on microscopic geometric similitude. The approach considers different regions of a heterogeneous field soil to be similar if their microscopic geometric structures are scale magnifications of each other. Transport parameters at any point within a given field soil are related to the parameters at an arbitrary reference point (*) through length scale ratios, or scaling factors a (= X/X*, where X represents the microscopic length scale). Hydraulic conductivity and soil water retention parameters of a particular region in the field are then calculated from those of the reference soil by means of a set of prescribed equations (Miller, 1980; Nielsen et al., 1983; Sposito & Jury, 1985). Recent work by Jury ej al. (1987) suggests that two scaling factors may be needed for soils which are not strictly similar. A two-parameter water content scaling procedure was also used by Shouse et al- (1991) to estimate the hydraulic properties of a layered soil profile. Most previous applications of scaling theory to field problems assumed that a is a random variable characterized by a certain probability density function. The method has been a central part of the stochastic flow and transport models of Bresler & Dagan (1983) and Dagan & Bresler (1983).

Monte Carlo simulations assume that the flow and transport parameters are random variables with values assigned from a joint pdf. The water flow or solute transport models are repeatedly run with coefficient values from the assumed pdf until a large number of possible outcomes has been generated. These outcomes are then used to calculate sample means and variances of the underlying stochastic transfer process. Anderson & Shapiro (1983) used this type of simulation to study steady-state unsaturated water flow in a heterogeneous soil. Amoozegard-Fard ej


al. (1982) used the same method to show that of all factors influencing field-scale solute transport, the pore-water velocity had the greatest influence, while local dispersion phenomena were only of secondary importance. A more detailed study by Persaud et a i (1985) reached similar conclusions.

Stochastic continuum models were initially used primarily in groundwater studies as illustrated by the studies of Gelhar gt al (1979) and Gelhar & Axness (1983). In these models all random variables are represented by the sum of their mean values plus random fluctuations which, when substituted into the convection-dispersion equation (Eq. 4), lead to a new mean transport model with additional terms. The modified model is evaluated by deriving first-order approximations for the fluctuations and solved by means of Fourier transforms. The approach, among other things, leads to a macro-scale dispersion coefficient whose value is reached asymptotically as distance and/or time increase. Spatial correlations of solute velocity variations characterized by its autocorrelation function, have been shown to play important roles in the derivation of the asymptotic convection-dispersion equation (Sposito et al., 1986a).

A different continuum approach was followed by Simmons (1982) who neglected the dispersion coefficient D in equation (4), and developed a formal theoretical approach using the pore-water velocity and the travel time as random variables. Jury (1982) initially also neglected D in his development of their transfer function model (TFM) of solute transport. The TFM leads to an estimation of the distribution of travel times from the soil surface down to some reference depth L. Solute transport is characterized by a travel time probability density function fL(t), which for many soil transport processes may be represented as lognormal. The flux concentration in the profile is represented with a convolution integral of fL(t) and the imposed flux concentration at the soil surface. Recent applications of the transfer function model are given by White et. al., (1986) and Dyson & White (1987), whereas its relationship to the classical one-region (Eq. 4) and first-order (mobile-immobile) two-region transport models is discussed by Sposito et al. (1986b). Transfer function models are expected to find increasingly wider applications in subsurface solute transport as its underlying theory is being strengthened by the incorporation of a variety of physical, chemical and biological processes (Jury, 1986; Jury & Roth, 1991), including transient water flow (Jury et al-, 1990).

Still other statistical approaches exist. For example, Knighton & Wagenet (1987) simulated solute transport using a continuous Markov process. Fractal-mathematical approaches (Wheatcraft & Cushman, 1991), random walk particle methods (Kinzelbach, 1988; Tompson & Gelhar, 1990), and a variety of procedures based on moment analysis (Barry & Sposito, 1990; Cvetkovic, 1991) are also providing new opportunities for studying solute transport in heterogeneous soil and aquifer systems. More work in these and related areas of research can be expected in the near future.

CONCLUDING REMARKS

This review shows that a large number of widely different approaches are available for modelling laboratory and field-scale solute transport. While deterministic


models have been, and will continue to be, invaluable tools in research and management, their usefulness for predicting actual field-scale distributions in time and space is increasingly being questioned. The stochastic approaches appear especially useful for estimating solute travel times in the vadose zone, as well as for predicting areally averaged solute transport loadings to underlying groundwater systems. Their evaluation in the field has thus far been limited to only a handful of data sets. Hence, a number of carefully executed field experiments are needed to determine their validity for describing field-scale processes, especially for relatively deep unsaturated profiles (Gee el aj., 1991; Sudicky & Huyakom, 1991).

The development and use of simulation models has significantly increased our quantitative understanding of the main physical, chemical and microbiological processes operative in the unsaturated zone. The proliferation of computer models in research and management will likely continue as computer costs keep decreasing and the need for more realistic predictions increases. Unfortunately, the simulation of field-scale processes requires considerable effort in quantifying spatially and temporally varying soil hydraulic and solute transport parameters. Thus, the completeness of experimental data, and the accuracy of the estimated model parameters, may eventually become the critical factors determining the usefulness of site-specific simulations (Ostrowski, 1991). Many of our current methods for measuring relevant unsaturated flow and transport parameters are largely those that were introduced several decades ago (Dane & Molz, 1991). Thus, new methods and technologies of measurement are critically needed to keep pace with our ability to simulate increasingly complex laboratory and field systems. A number of potentially powerful methods based on parameter estimation techniques have recently been introduced (Wagner & Gorelick, 1986; Kool & Parker, 1988; Mishra & Parker, 1989). One application to the measurement of the unsaturated hydraulic properties of heterogeneous (layered) soils is illustrated by Shouse ÊÎ al., 1991). It is important that work in these and related areas of research continues.

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