Identifying Effective Parameters for Solute Transport Models in Heterogeneous Environments

Stefan Mayer and Timothy R. Ellsworth Department of Natural Resources and Environmental Science, University of Illinois, Urbana, IL

Dennis L. Corwin

USDA-ARS, U.S. Salinity Laboratory, Riverside, CA

Keith Loague Department of Geological and Environmental Sciences, Stanford University, Stanford, CA

Reliable simulation models are an essential prerequisite to accurate assessment of present and future consequences of human activities on ecosystem health and functioning. In particular, for non-point source pollution assessments in the vadose zone, chemical fate and transport models are required at field and watershed scales. However, developing such models that characterize adequately the processes occurring during an arbitrary period of time within an arbitrary spatial region in unsaturated soil is a challenging task due in part to the complexity of the processes and heterogeneity of the properties involved. This task requires developing, calibrating, and evaluating an appropriate process de- scription for the region of interest from relatively sparse spatial and temporal information. Here we discuss and point to some of the relevant literature addressing the various "hurdles" that need to be overcome in order to develop non-point source pollution transport models. Of central concern are spatial heterogeneities of physical properties over a range of scales, how these can be inferred from spatial data, how heterogeneities are effectively represented by macroscale parameters and how (or if) processes can (or should) be related over a range of effective scales. Upscaling from observed microscale processes could yield useful results.

INTRODUCTION

Transport in the vadose zone is governed by multiphase flow and transient effects due to changing water saturation levels [Sposito, 1995]. Transport models in porous media take a unique position among physical models. All difficulties in developing models are ultimately related to

Assessment of Non-Point Source Pollution in the Vadose Zone

Geophysical Monograph 108 Copyright 1999 by the American Geophysical Union

the complexity of the flow domain and to the related boundary condition at the solid-fluid interface, which has a dominant influence on the transport properties in porous media. The problem is one of identifying effective trans- port parameters, which implicitly include the influence of this spatially distributed boundary condition.

Modeling non-point source (NPS) pollutants in the vadose zone is a spatial problem complicated by the spatial variability of soil physical, chemical, and biological prop- erties that influence the transport process. The notion of modeling the transport of a NPS pollutant at every point within a spatial domain is technologically impractical at present because all the necessary model input parameters

120 IDENTIFYING EFFECTIVE PARAMETERS FOR SOLUTE TRANSPORT MODELS

and variables, both spatially and temporally, are not avail- able. Recently, the issue of spatial variability associated with modeling NPS pollution in the vadose zone has been addressed by coupling a one-dimensional model to a geographic information system [Cotwin et al., 1997]. The basis of this approach is the delineation of three- dimensional spatial domains that are assumed similar in those soil properties that influence the transport of the NPS pollutant of concern. Identifying effective parameters is a crucial aspect of this approach, where non-invasive or re- mote sensing techniques, and geostatistics are deemed essential for the measurement and estimation of effective

parameters.

"The main aim of the theory of flow through porous media... is to derive the laws governing the macroscopic variables" [Dagan, 1989] and the question of scales is central to the discussion [Wagenet and Hutson, 1995]. For instance, how are physical properties distributed over a range of scales? How can these distributions be estimated from given measurement support scales and sampling net- work scales? Can empirical models defined at different scales be related to each other? Can model predictions be extrapolated from transport properties at smaller and larger scales?

Models are useful for the description and understanding they offer of a specific physical process of interest, as well as for their ability to predict the behavior of the natural system studied. According to Narasirnhan [1995], models "help to recognize patterns in the structure of nature and the patterns so recognized are used to extrapolate and to predict."

In general, physical models are derived empirically from experience and observation that typically deal with phe- nomenological theories valid at a range of observable scales. Such phenomenological theories require an objec- tive relation between what can be measured and what the

model predicts. These theories are never proven, only corroborated or shown to be false by experimental studies.

Our ability to solve practical problems is intimately associated with our ability to measure field quantities, to infer the spatial and temporal distribution of such quan- tities and to relate them to empirical model parameters. In general, the current modeling ability combined with prac- tical data limitations is not reliable enough to resolve the detail needed for NPS pollutant transport prediction [Sahirni, 1993; Narasirnhan, 1998; Jury, 1998]. However, experimental and simulation methods have significantly improved over the past decade. Koltermann and Gorelick [1996] reviewed the various approaches that have been used to obtain maps of hydraulic property fields at scales ranging from multiple pores, to stratigraphic features, to the basin scale. These approaches can be complemented with detailed information of pore-scale processes [van

Genabeek and Rothman, 1996]. While such microscale data can only be obtained over a minute fraction of the transport domain, it nevertheless holds potential in im- proving predictions of macroscale transport parameters [Soll et al., 1997].

In this brief review, the various "difficulties" that a modeler is confronted with when developing a NPS trans- port model are discussed. While many models have been proposed and used with some amount of success, each one has limitations. The main difficulty is knowing the range of scales and the heterogeneous distribution of transport properties that need to be incorporated into a model. Re- lated hurdles are how to acquire and interpret data; how to infer effective transport parameters; and how to relate pa- rameters and models obtained for different scales.

APPLICATIONS AND LIMITATIONS OF NPS

TRANSPORT MODELS

A direct consequence of the solid - fluid interface is the connectivity of the fluid phase. This connectivity can be considerably more complex in multiphase flow, e.g. in the vadose zone, as it is also influenced by the distribution of a fluid - fluid interface. Many properties of the macroscopic system are determined by this connectivity [Berkowitz and Baalberg, 1993]. It is the modeler's task to determine what type of model is best suited to describe macroscopic transport given a microscopic fluid phase distribution, or rather, given the general lack of information on such a distribution.

Early models of flow in porous media were primarily motivated by such practical applications as resource re- covery, e.g. by estimating attainable well production at a given location, or the impact of sources and sinks on the water table. Since it was not necessary to provide a de- tailed description of the flow field, nor to consider mixing due to spatially variable flow rates [Narasimhan, 1998], the application of simple models, e.g. of Darcy's law, pro- vided satisfying answers.

The assessment of available water quantity has been combined with the need to assess water quality. This is linked to the transport of possible contaminants from their source to groundwater. Contaminated water must be considered distinct from drinking water and some estimate of the boundary between the two is needed. This boundary evolves as contaminants are transported through soils and in the groundwater. As a result, significantly more detailed transport models and data distributions are needed than for resource recovery estimates. Studies have shown that in many situations, coarse models that provide a satisfying description for average water flow are not useful to predict the evolution of contaminated water within the overall

water flow domain [Jury and Fliihler, 1992; Beven et

MAYER ET AL. 121

1993]. Significant efforts have been committed to and are still needed to improve on this situation.

Under a typical scenario, the main requirements for a transport model of NPS pollution are the ability to assess the state of water pollution and to predict the fate of con- taminants. A useful model will thus not only enable the preservation of water resources, but also provide guidance for economically optimal management. Specifically, avoid- ing problems is cheaper then fixing them. From a techni- cal point of view, measures to avoid potential contam- ination are easier to implement than remediation efforts.

There are obvious challenges to developing such an appropriate predictive model [Loague et al., 1998]. The prior history of the site, say of agricultural management in a field, may not be known. The physics of NPS pollutant transport in soils are also not known in sufficient detail to guide model development. The scale of the domain con- sidered and the structure of the included spatial hetero- geneities raise the practical questions of how much detail is needed for prediction and of how this information can be measured to support model development [Wagenet and Hutson, 1995]. Available data are generally extremely sparse and usually obtained only over a fraction of a percent of the overall region considered [Ellsworth, 1996]. Furthermore, transport parameters are not measured directly but inferred indirectly within the context of a transport model or a pedo-transfer function. Finally, model predictions carry over and potentially magnify all uncer- tainties that entered in their development.

Despite this long list of "difficulties" in modeling transport through the vadose zone, many models have been developed which describe at least a partial aspect of the transport in selected media. Berkowitz and Baalberg [1993] concede that "many transport processes can be suc- cessfully understood by considering an idealized transport of an abstract fluid through an abstract medium." Con- tinuum models based on the advection-dispersion equation [Bear, 1972], stochastic continuum models [Dagan, 1989; Gelhar, 1993], pore-scale network models [Saffman, 1959; Ioannidis et al., 1993], effective domain network models [Ewing and Gupta, 1993], percolation theory [Larson et al., 1981; Berkowitz and Baalberg, 1993], stochastic stream-tube models [Destouni and Cvetkovik, 1991], and transfer function models [Jury and Roth, 1990] are some examples of approaches used to describe transport in the vadose zone.

However, in an extensive review of transport models in fractured rocks, Sahimi [1993] concludes that most flow and transport processes in heterogeneous media are not yet understood. Jury [1998] confirms this conclusion specifi- cally for the transport modeling of the unsaturated zone. NPS pollution occurs through such media, of which frac- tured rock formations and most unsaturated soils are ex-

amples.

At the core of the problem is establishing the relation between the heterogeneous distribution of field properties and an appropriate distribution of effective transport pa- rameters. This can be attempted in the context of either deterministic or stochastic models that are either discrete or

continuous. A first requirement is to conduct careful data acquisition, from which a distribution of physical prop- erties in the field is inferred based on a primary model of the heterogeneity structure. Koltermann and Gorelick [1996] reviewed the many possible approaches yielding a map of field hydraulic properties. Then, secondary trans- port models and model parameters are derived and cali- brated, and model predictions are compared to data.

Matheron [1989] expressed concern that models devel- oped for a specific purpose or within a limited context are often used and abused beyond their range of validity. "Once operational concepts and the physical laws which support them have been brought together within the frame- work of a mathematical model, there is a great temptation to forget these limits and, blindly relying on the mathe- matical formalism, draw from the model conclusions which are well outside the domain of... validity .... (thus) overstepping the threshold of objectivity."

SCALES, HETEROGENEITIES, AND THEIR IMPLICATIONS FOR MODELING

The practical applications of continuum transport models were originally limited to a few simple analytical solutions. These evolved into increasingly detailed and sophisticated numerical simulations [van Genuchten and Sudicky, 1998]. As a result, the limiting factor for accurate transport pre- diction has shifted from model solution to parameter esti- mation and to related experimental methods. The dif- ficulties associated with this tend to increase with the

complexity of the spatial distribution of properties, with the heterogeneities and with the range of scales that need to be considered.

The issue of scales is implicitly or explicitly associated with modeling efforts [Wagenet and Hutson, 1995]. There may be naturally occurring scales consistent with physical phenomena or arbitrary scales imposed by our measure- ments and models, or by political and regulatory needs. The degree of difficulty in relating observations and predictions at different scales depends on the distributions of field properties and on how these properties are measured and modeled. Attempts to bridge scales through experimental and modeling efforts are influenced by the heterogeneity structure of the field. In practice, some as- sumption about this structure is always required to allow us to "extend" limited information to the unobserved. Usual-

ly, it is assumed that the studied phenomenon behaves at unobserved locations in a manner typical to what is ob- served at the measurement

122 IDENTIFYING EFFECTIVE PARAMETERS FOR SOLUTE TRANSPORT MODELS

There are some basic constructions concerning scales that all models have in common. Regardless of what scale or level of resolution one considers the studied system, a smaller scale can always be envisioned [Truesdell and Toupin, 1960]. The processes at all scales below resolution are not known or considered in detail. Rather, it is assumed that some form of local spatial averaging or the use of an expected value is acceptable for model parameters. At the other end of the spectrum, the amount of objective detail offered by the model depends on the relation between model parameter scale, sampling network scale and do- main scale [Beckie, 1996]. The domain scale of transport models developed for fields or watersheds is much larger than the scale of resolution of the transport parameters. Thus, only a minute fraction of the transport domain is sampled.

It is important to consider the influence of possible heterogeneities at smaller scales on the effective distribu- tion at the scale of locally averaged parameters. While some models (e.g. the advection-dispersion equation) neglect any such influence beyond the first moment (the spatially averaged parameter value), others include this variability. For example, de Marsily [1986] suggests that instead of using an empirical value for the dispersion coefficient based on averaged water velocity, it may be advantageous to retrieve this parameter as part of the solution of stochastic transport equations, where the small- scale variability of the water velocity is described as a stochastic process.

Among the natural or physical scales of transport in porous media are the molecular, pore and domain scales. They provide natural boundaries to the scales entering a model. Their magnitude may vary from case to case. The transport domain may be related to a lab, pedon, plot, field, formation, region or watershed scale. The pore size may vary over several orders of magnitude. It may be related to a typical pore diameter, or it may be defined as the integral scale describing the correlation length of a pore distribu- tion [Dagan, 1989].

These physical scales are conceptually related to models and model parameters using theoretical scales such as the microscopic, macroscopic, REV, local spatial average, continuum, correlation or integral scales. Definitions for these scales are not unique and their use may imply different meanings if used by different authors. For example, the microscopic scale has been associated with the fluid continuum scale [Bear, 1972], with the pore scale [Raats and Klute, 1968], and with a REV scale which is hypothesized to provide a quasi-homogeneous average of pore-scale velocity fluctuations [Bhattacharya and Gupta, 1983].

Consideration of a macroscopic scale is subjective and can draw on past experience, geological information, or on

implicit theoretical requirements of the model. Dullien [1991] considers its choice to be an intuitive process related to the experimental data. It can also be guided by a hypothesis on spatial averaging, as is the case with the REV scale or with the integral scale. In general, it is assumed - although not often verified - that the effect of microscale fluctuations is captured in the macroscopic description. The theoretical REV scale is a typical exam- ple of a macroscopic scale based on theoretical hypotheses (see subsection The Theoretical REV Concept). The integral scale A (if it exists) is another possible macro- scopic scale, which is used within a stochastic model context. It is defined as the spatial scale at which the variance of measurements at scale A within an increasing region of size S decrease proportional to A/S, or as where n = S/A [Matheron, 1989]. In a model-based design, it is the minimum spatial scale required to obtain an independent sample of the stochastic process.

The choice of a macroscopic scale may also be guided by other practical considerations. It may be related to a possible resolution of the domain scale given finite re- sources, as well as to the resolution required for a useful prediction of spatial transport distribution. The model and model parameters must provide estimates of pollutant fate that are reliable and objective as far as the data allows. For instance, even if the concept of a theoretical REV does not correspond to reality, a related model may be useful if it provides reliable answers to our problems.

An operational connection between physical and param- eter or model scales can only be established by the measurement scale, which is defined by the interaction of an instrument with measured quantities. Model parameters are only objective to the extent that they can be related to measurements and thus to physical entities [Bear, 1972; Baveye and Sposito, 1984; Cushman, 1986].

Most geological media are heterogeneous at practical scales of measurement [Dagan, 1989; Berkowitz and Baalberg, 1993] and assumptions of spatial or statistical homogeneity are not supported. The physical scales or range of physical scales of those heterogeneities determine the transport properties. For example, observed hydraulic dispersivities increase by several orders of magnitude with the size of the study site [Neuman, 1990; Gelhar, 1993]. McCoy et al. [1994] list the distribution of macropores in soils and of fractures in rocks as extreme cases of spatial heterogeneities. In such materials it is difficult or impos- sible to model transport phenomena with classical theory.

Two hypothetical scenarios can be considered. Either, heterogeneities exist at distinct ranges of scales, with inter- mediate ranges being locally stationary [Bachmat and Bear, 1986]. Or, there are no intermediate ranges of quasi- stationarity, and heterogeneities are effective at all scales [Berkowitz and Braester, 1991

MAYERETAL. 123

If the variability structure is such that heterogeneities can be grouped in identifiable and separate ranges of scales, then the use of a quasi-stationarity assumption with- in discrete steps of scales seems appropriate. Bhattacharya and Gupta [1983] assumed the existence of such a hetero- geneity structure when they used two distinct REV scales. The smaller REV scale was used to derive transport prop- erties by averaging over "many" pores, and the larger, called Darcy scale REV by the authors, was used to characterize the dispersion in the field.

Neuman [1994] reported a discrete hierarchy of scales for geological media. This type of variability can be related to nested structures with intermediate sills in the cor-

responding variogram [Gelhar, 1993]. There is a well- developed methodology for modeling such a system. The macroscopic transport scale is defined in any one of the ranges of quasi-stationarity. Fluctuations at smaller scales are only included in an average sense in effective transport parameters. Fluctuations at larger scales are treated as a stochastic process.

In the case of heterogeneities distributed over all scales, no specific meaning can be attached to intermediate scales. In particular, the definition of an REV is not meaningful since no intermediate range of scales exist in which field properties are quasi-stationary [Berkowitz and Braester, 1991]. A possible modeling approach is to assume that measured data is part of a stationary distribution over a fraction (say, less then one tenth) of the domain scale, and part of a deterministic trend at larger scales. However, such a distinction is subjective, since one cannot devise a unique decomposition of heterogeneous structures into random and deterministic features [Cressie, 1991 ].

In practical applications either limited or no a priori knowledge exists of the pore-scale heterogeneity and of the corresponding variability of transport processes. Further- more, extensive measurement in the field is usually not a possibility. Given prior experiences with heterogeneous formations, and unless data analysis reveals clear spatial trends, it is necessary to infer a probability distribution and a spatial structure from sparse data. As a consequence, inferred transport parameters are often conceptualized as random variables or random functions [Gelhar, 1993].

MODELS AND MEASUREMENTS

For any modeling approach to be valid and useful in terms of calibration and prediction, it must be closely related to what can be determined experimentally [Wagenet and Hutson, 1995]. Ties between model param- eters and observations must be clear. Especially, measure- ment scale and model parameter scale must be appro- priately related. Obtaining and interpreting experimental

data within the context of a transport model is another of the "hurdles" that the modeler must overcome.

Since instruments typically measure a spatially averaged quantity, it seems useful to define a corresponding weighted-spatial average as the macroscopic scale [Baveye and Sposito, 1984]. Conversely, Cushman [1986] suggests that future instrumentation be developed to measure at the scale of theoretical interest.

Moltyaner [1989] used two types of instruments to measure radiotracer concentrations in a study aimed at esti- mating hydrodynamic dispersion at the local scale. Care was taken that the respective instrument responses were combined with the correct weighting functions to infer the field-variable values. It was concluded that these weighting functions could be approximated by "table"-type functions. In this study, the spatial extent of these "table" functions defined the local sample support scales, which in turn were assumed to define the local scale of a continuum repre- sentation.

Geostatistics offers a conceptual framework to quantify the ties between measurement and model scales and to

relate them to a statistical model of the distribution of field

properties. The variance between measured data depends on the size of the sample support and on the structure of spatial heterogeneities. Assume that a physical property is described by a random process that is second order station- ary. If the distribution of this property is characterized by a pure nugget effect, i.e. if there is no spatial correlation between data points, then the observed variance, also call- ed the dispersion variance at a finite sample support within the field, decreases rapidly as sample support (the measurement scale) increases. If measurements in the field are linearly correlated up to some limiting separation distance (the correlation range of the field), then the disper- sion variance also decreases with increasing sample sup- port, albeit more slowly then in the uncorrelated case. The larger the correlation scale, the slower the decrease in ob- served variance [Warrick et al., 1998; Mayer et al., 1997].

Thus, some of the variance of a physical property is contained within the finite sample support. The remaining variance can be estimated from the data as the dispersion variance, or variance at given sample support within the field. Parker and Albrecht [1987] showed how the sample support influences the estimate of transport model param- eter variability. As sample size was increased, the ob- served variance of conductivity between samples was re- duced, suggesting that more of the variance was contained within each sample. This is consistent with Krige's rela- tion, which relates variances of a parameter within and be- tween sample supports [Journel and Huijbregts, 1978]. Similarly, the estimated variance between measurements of the average pore water velocity observed in tracer experi- ments decreased with increasing sample support, while

124 IDENTIFYING EFFECTIVE PARAMETERS FOR SOLUTE TRANSPORT MODELS

estimated average effective dispersion coefficient in- creased.

It is widely accepted that the effective dispersivity increases with the size of the domain considered [Gelhar, 1993]. For example, Killey and Moltyaner [1988] obtained experimental data at the Twin-lake tracer test and con- cluded that transport was mainly influenced by hetero- geneities of the hydraulic conductivity field at two separate scales. Six main regions in the aquifer having distinct hydrological properties defined the larger scale. They observed that within each of those main regions, the effec- tive longitudinal dispersivity was comparable in magnitude to results obtained from laboratory measurements. Specif- ically, intermediate scale strata present within each region did not significantly influence the effective longitudinal dispersivity. The whole-aquifer dispersion on the other hand was found to be significantly influenced by the large- scale heterogeneities between those regions.

However, the reliability of such dispersivity estimates tends to decrease with increasing field scales. Gelhar et al. [1992] reviewed 59 different field sites and concluded, "the largest scale for high reliability data was only 250 m."

Bouma [1990] pointed out that soil physical measure- ments and the methods to infer physical parameters from data often erroneously imply that soils are isotropic and homogeneous. To improve sampling, he suggested that sample volumes should be a function of soil structure, e.g. by taking into account possible heterogeneities such as obtained from a spatial mix of sandy and clayey soils. He further suggested that possible interactions between the instrument scale and soil heterogeneities could be esti- mated by using similar instruments having different sample support.

Baveye and Sposito [1985] derived the theoretical implications that space- and time-varying instrument sup- port scales have on macroscale transport equations. The example of a neutron probe response to varying soil water content was given.

Estimating the variance of physical transport properties in the field is an important factor in deriving effective transport parameters. To improve upon such an estimation, a method was developed to infer the variance contained within the sample support [Ellsworth and Boast, 1996; Restrepo et al., 1997]. A prerequisite is the use of multiple sample supports and that the interaction between instru- ment and measurement is linear, and the assumption that a simple analytical model describes the variogram [Warrick et al., 1998]. This method thus holds promise to free the modeler from some of the constraints imposed by the measurement scale.

For most practical NPS pollution considerations, trans- port predictions at field, regional, or watershed scales are of interest. Ideally, direct measurement at such large scales

would allow the development of an empirical model valid at those scales. However, in practice this may not be feasi- ble and the domain scale process must be inferred from measurements distributed within. In general, modelers of NPS pollutants are faced with economic (cost of sampling and sample analysis) and physical (disturbance by instru- mentation of the "real" physics) constraints when obtaining such a distribution of data. Ever-smaller fractions are

measured with increasing domain size. Also, local resolu- tion decreases with increasing sample support. It is neces- sary to deal with the resolution of spatial variability and the uncertainty of its estimate as limited by the density of the sampling grid. A potential improvement to this situation may lie in the advent of improved remote sensing methods.

Beckie [1996] reviewed how model parameters are related to measurement scale and network sampling scale, where the latter is defined by the spacing between measurement locations. He pointed out that the distribution of model parameters at subnetwork scale (i.e., at a scale smaller than the network sampling scale) is not resolved. The need to infer such a distribution from limited data is

called the closure problem by Beckie [1996]. It is error prone and depends on the combination of network sam- pling scale and heterogeneity scale. A method is offered to quantify the average error of parameter estimation as a result of the closure problem.

MACROSCALE PARAMETERS

Much, if not all of soil water physics is based on the conceptual picture introduced by Buckingham [1907]. He defined the capillary potential (the soil water matric, potential) and related its gradient to water flux [Sposito, 1986]. In doing so, the existence of macroscopically measurable, continuous and differentiable quantities has been implied.

The use of the continuum approach and of macro- scopically averaged transport parameters for modeling the vadose zone thus has a long history. It is a classical engineering approach that has been successfully applied to solve a wide array of problems. Until recently, there was little choice in the matter since the precise description of transport at the pore scale could not be compared to experimental evidence. For some model applications, such detail may also be of little importance for model applica- tions.

Estimating effective macroscopic transport parameters is a key component in model development. Moltyaner and Killey [ 1988] reported on the Twin-lake tracer tests. These were conducted in part to estimate effective aquifer disper- sion coefficients to be used with a model based on the

advection-dispersion equation. Hess et al. [1992] reported on the Cape-Cod tracer tests. The resulting estimates

MAYER ET AL. 125

hydraulic conductivity distributions and of macrodisper- sivities were used in a stochastic transport model [Gelhar, 1993].

For theoretical considerations, and to understand the limits of application of transport models, some under- standing is needed on how measured data, estimated parameters, and the assumptions underlying a transport model are related. For example, Miller (1980) described the process of relating pore-scale information to effective transport parameters as "the usual tightrope act of defining macroscopic properties, of whatever nature, from discrete microscopic entities, whether pores or atoms."

The details of the implied upscaling process are often ignored and macroscopic parameters are estimated directly from regularized data. However, it may be instructive to consider such a process in greater detail. This may serve to better relate macroscale data and parameters given the natural heterogeneities in the field. Finally, with detailed pore-scale data now available and measurable, this may lead to a more comprehensive definition of transport parameters.

The Theoretical REV Concept

As discussed by Baveye and Boast [1998] there are two different kinds of REVs, one theoretical and the other experimental. Herein, only the frequently invoked theoret- ical REV is considered, which is a theoretical assumption underlying the existence of continuum models of porous media transport [Bear, 1972; Bachmat and Bear, 1986].

Introductory fluid mechanics books (see e.g. [Batchelor, 1967]) relate the existence and definition of an REV of a

fluid to the existence of several distinct scales. First, the REV is assigned to the smallest scale large enough to smoothly average out local and temporal fluctuations at molecular scales. Averaged quantities such as density or velocity are associated with the centroid of the averaging REV. The second assumption is that the scale of the REV is smaller then the typical variation scales of the parameters in the field. This theoretical concept is equiva- lent with the assumption that heterogeneities are manifest at clearly distinct scales and that the REV scale must be within a range of scales defined by local statistical homogeneity.

Following a similar line of arguments (for a detailed discussion, see e.g. [Bear, 1972]), a theoretical REV scale is defined for transport in a porous medium. It is assumed that there exists a natural range of scales that are both significantly larger than underlying small-scale fluctua- tions, and significantly smaller than the scale of macro- scopic fluctuations in heterogeneous media [Bachmat and Bear, 1986]. Local statistical homogeneity is thus implied over a range of intermediate scales. A simple example of

such a medium is given by the micromodel of Corapcioglu et al. [1997], in which theoretical results at the REV scale compared well to experimental data.

The practical application of continuum models can be limited due to scaling and existence issues associated with the REV. Baveye and Sposito [1984] give examples for such scaling issues, showing that if soil moisture content varies as a result of wetting or drying, the REV is not invariant. Furthermore, if pore-scale fluctuations and macroscopic heterogeneities are not separated by a range of scales, then theoretical assumptions underlying the existence of an REV are not valid. Neuman (1990) concluded that statistical homogeneity of hydraulic conductivity is at best a local phenomenon and that "one must question the utility of associating medium properties with REVs."

Some of these difficulties can be avoided if one uses a

modified approach. Baveye and Sposito (1984) suggested the use of a relativist concept, which integrated the microscopic properties of porous media with a weighted- averaging procedure over a finite volume defined by the instrument used to measure a given macroscopic property or parameter. They derived macroscopic balance equations under the assumptions that all measurements (i.e. hydraulic or parameter. They derived macroscopic balance equations under the assumptions that all measurements (i.e. hydraulic pressure, water flux, water content, etc.) can be described by the same weighting function.

Overall, the existence of meaningful local averages is closely tied to the distribution of heterogeneities in the field. If no intermediate scales of quasi-stationarity can be identified, then no meaningful REV can be defined below the domain scale [Berkowitz and Braester, 1991]. This point is of practical concern if assumptions implied by the use of continuum models are violated and if this causes the

model to fail to predict transport to sufficient accuracy.

Spatial Averages and Expected Values

There is no doubt that a macroscopic description includes transport parameters that represent some form of average and that predictions can only be made in some average sense. There are, however, different viewpoints on how to obtain such an average, what it represents, and mainly, how to tie it in with observable quantities.

There are several commonly used approaches relating macroscale parameters to microscale processes. The first is the spatial average of pore-scale variability as obtained in an REV, or in a weighted-spatial integration. This is analogous to the experimental approach, where the meas- ured data are regularized over a finite-sample support and thus encompass all variability at smaller scales [Baveye and Sposito, 1984; Bachmat and Bear, 1986]. The

126 IDENTIFYING EFFECTIVE PARAMETERS FOR SOLUTE TRANSPORT MODELS

approach uses expectations of a given value in a statistical sense. This is tied to the experimental picture of a distribu- tion of data points in the field, combined with an assumed model of stationarity or ergodicity, such that the spatial distribution of data can be used to estimate local expecta- tions and distributions [de Marsily, 1986; Dagan, 1989; Gelhar, 1993]. Other possible approaches are given by the theory of homogenization and by the use of renormaliz- ation group theory, which are subsequently discussed.

First developed to deal with partial differential equations having rapidly oscillating coefficients, the theory of homogenization has been applied to model transport in porous media (see e.g. [Hornung, 1997]). Instead of obtaining a spatial average of microscopic properties by defining a finite volume sufficiently large to smooth out rapid oscillations of the underlying parameters, homoge- nization consists of multiplying these parameters by a scaling factor which tends to a limit small enough to make the scaled parameter in a given volume appear smooth. For statistically homogeneous media, it has been claimed that this approach is the right way to do "volume averag- ing" [Badea and Bourgeat, 1997].

Renormalization group theory is based on an iterative approach, which obtains a distribution of locally averaged parameter values. For example, the microscale distribution of a permeability field is successively replaced by spatially averaged, coarser fields [Gautier and Noetinger, 1997]. In a slightly different approach, within the context of discrete pore-scale network models, pore-scale fluctuations are averaged using a defined-stepwise-scaling procedure that replaces several small pores by a few larger pores [Gavrilenko and Gueguen, 1998]. Reasonable agreement is obtained between coarse representations and full solutions. However, the upscaling procedure involves some arbitrary assumptions governing the local transport averaging within each iterative step.

Since model development for porous medium transport has been strongly influenced by classical field equations, it is instructive to take into account some special notions on the use of spatially averaged or expected parameter values in the context of continuum mechanics. Truesdell and

Toupin [1960] reject the concept of spatial averaging in favor of the viewpoint of statistical mechanics when relating discrete distributions to continuous fields. The field concept provides an exact theory in terms of expected values, rather than in terms of spatial averages. In fact, the authors suggest that "to speak of an element of volume in a gas as a region large enough to contain many molecules but small enough to be used as an element of integration is not only loose but also needless and bootless."

A few further considerations can be added. First, proc- esses at the molecular scale cannot be measured directly. Thus, the statistical distribution of molecular behavior must be inferred indirectly. Second, a simple field theory,

as well as thermodynamics, only emerges in approximation from statistical mechanics [Truesdell and Toupin, 1960]. Finally, the development of geostatistics [Journel and Huijbregts, 1978] and especially the ergodic hypothesis provides an operational tool to relate the expected value at a point to the spatial average over a finite region. Geo- statistics guides the careful use of spatial data and of the uncertainty associated with unknown variability resulting from finite sample supports as well as from data sparsity. In a sense, it provides a framework to improve on the "loose" approach bemoaned by Truesdell and Toupin [1960].

The question of how to obtain objective macroscopic parameters is tied to the upscaling process as well as to the parameter in question. For instance, the spatial distribu- tions and thus the appropriate averaging process for poros- ity and for water fluxes are different. Bear [1972] distinguished between a transport REV and a porosity REV. Baveye and Sposito [ 1984] also point out that such distinctions are important when considering averages. According to Bachmat and Bear [1986], a common scale for all REVs must be found before representing the domain by a continuum. Given the particular nature of transport in porous media, and especially in the vadose zone, meeting this latter condition may be the exception rather than the rule.

A number of reasons can be given for this, irrespective of whether an REV concept or a more general averaging procedure is adopted. In saturated flow, the pore-scale water flow rate is approximately proportional to the 4th power of the effective pore diameter. A distribution of effective pore diameters thus yields a distribution of pore water flow rates spanning 4 times as many orders of magnitude. Unlike the flux vector, the scalar porosity does not depend on the orientation of individual pores, or on their connectivity. Furthermore, transport is strongly influ- enced by water content, and the effective porosity available to transport may differ significantly from the total porosity. Finally, the physics at fluid-fluid and fluid-solid interfaces needs to be considered. For example, a further reduction of effective transport porosity may result from an adhesive hygroscopic layer. The local average thus reflects the combined properties of the geometry of the porous medi- um, of the water content, of the derived local transport paths, and of the specific physics of transport in each individual path.

EMPIRICAL MODELS AND "FIRST PRINCIPLES"

From a practical point of view, a description of transport at a macroscale is of particular interest to NPS pollutant modelers. Sposito [1986] suggests that solute transport scientists measure directly what is of interest, derive and evaluate corresponding empirical theories, and not

MAYER ET AL. 127

about how they are related to other, possibly smaller scale processes. Of necessity, most, if not all, of existing theory was developed following this suggestion. Empirical mod- els are developed from observations at the measurement scale combined with basic assumptions on soil water physics, such as Buckingham's [1907] hypothesis on the capillary potential and the macroscale relationship between potential gradient and water flux.

Since the measurement scale is usually much larger than the pore scale, empirical models have no direct "link" to the "hidden" processes at smaller scales. Attempts to bridge this gap, to verify, prove, or improve the empirical models, have led to numerous attempts to relate them to other models. The latter are typically valid at smaller scales. These are often referred to as "first principles", although at best this means that they have been verified to an extent that their validity is not in doubt.

Similarly, classical continuum mechanics was first developed as an empirical theory from observed data, followed by justifying arguments based on averaging molecular scale processes. In both cases, it seems that the main function of defining an REV scale was to provide an argument, which justified and lent credibility to the use of empirical models developed at the macroscopic scale [Baveye and Boast, 1998]. In both cases, such a "confirmation" of empirical theory by bridging scales was criticized [Truesdell and Toupin, 1960; Baveye and Sposito, 1984].

For practical applications, smaller scales are only of interest if they serve to better understand the large-scale process from some upscaling. This is not the case for common fluid mechanics applications. The Navier-Stokes equations are supported by a large body of experimental evidence, which gives a high level of confidence to the phenomenological theory. The equations are valid at all scales larger then the lower limit imposed by the continuity assumption. The gap between molecular scale processes and the continuum scale spans several orders of magnitude. The macroscale parameters (e.g., viscosity, density, pressure, velocity, etc.) can be measured or inferred from data to great accuracy. Note, however, that particular applications such as the Brownian motion of isolated small particles offer well-documented examples in which molec- ular scale processes are used to explain transport of a particle at the continuum scale.

A similar line of argument can be offered for some applications of porous media transport models. The linear relationship between pressure drop and water flux described by Darcy's law, which is an empirical model inferred from experimental observation, has been verified in many circumstances and thus is used with a high degree of confidence. Verification of this model based on an

upscaling exercise does not improve its application.

However, solute transport models in the vadose zone do not enjoy the same high level of confidence as the Navier- Stokes equations. On the contrary, the current tendency seems to be to invalidate existing models rather than to confirm their predictions with experimental data. We are thus at an impasse in our modeling efforts.

In general, if an empirical model is well supported by experience, then there is no practical need to "verify" that model by relating it to "first principles". On the contrary, such a "verification" approach can be misleading if it involves assumptions that are not well supported. In that case, rather than "validating" an empirical model that was already accepted at the outset, a reverse process can occur, i.e. incorrect assumptions may gain credibility because they lead to an accepted result. Clearly, if experience fails to support the use of an empirical model, then "verifying" this model from "first principles" casts serious doubts on the "verification" process.

Two scenarios can be envisioned in which relating a given empirical model valid at a certain scale to processes observed at different scales is useful. If only accepted theory is used to define both the "first principles" and the upscaling procedure, and if an accepted empirical model is recovered, then a scientifically beautiful relation between processes at different scales can be found. A practical result is obtained if a similar procedure helps to uncover as yet unknown models or model extensions.

Our ability to develop an empirical theory for transport in the vadose zone is limited by data requirements. What exactly is measured in attempts to estimate transport parameters is a topic of discussion [Cushman, 1986; van Genuchten and Leij, 1992]. The validity of currently used transport models appears to be limited to special experi- mental or field conditions. Even if a given model is consistent with a transport data set at a given scale, little if any reliable information can be extracted as to the transport at larger scales, at other locations, or given a different set of boundary conditions. Perhaps the appropriate empirical model for vadose zone transport has not yet been uncovered. Nor is there evidence that such a model, valid at different locations and over a range of scales, should exist. Therefore, any approach, including upscaling from pore-scale processes, should be used to improve on the current situation.

PORE-SCALE PROCESSES

Results obtained in heterogeneous and unsaturated media have time and again lead to questioning the validity of the continuum approach. Sahimi [1993] concluded that this is due to large-scale correlations and connectivity of fluid phases. In such a case the implicit upscaling used to define macroscopic transport parameters needs to

128 IDENTIFYING EFFECTIVE PARAMETERS FOR SOLUTE TRANSPORT MODELS

reconsidered and possibly based on a more detailed knowl- edge of the processes at the microscale [Soll et al., 1997].

Until recently, direct studies at the pore scale were beyond experimental abilities. Thus, a detailed knowledge of pore-scale processes could not contribute to studies of porous medium transport. How the physics at the pore level translate into transport properties at a hypothetical continuum scale was a question that could not be answered [Raats and Klute, 1968; Sposito, 1986]. As Bear [1972] put it, the unknown processes at the microscale had to be lumped into "parameters of uncertainty."

The current outlook on including detailed microscale studies can be viewed more optimistically. Especially, the traditional concept that detailed transport properties at the microscale cannot be directly observed has begun to erode over the past decade. Significant effort has been commit- ted to describing the microscopic detail of porous media and of related transport properties both experimentally and with numerical simulations [van Genabeek and Rothman, 1996]. Use of micromodels, fluorescent particle tracking, scanning electron microscopy (SEM), nuclear magnetic resonance (NMR) relaxation, synchrotron computed microtomography (CMT) or X-ray microtomography are some of the experimental techniques used to measure microscale processes. A recent article in Eos [Soll et al., 1997] summarizes a conference on using studies at the pore scale to improve understanding of the macroscopic transport.

The first question to answer is, what is an acceptable transport model at the pore scale? If we consider Newtonian fluids such as water or air in sufficiently large pores (e.g. having an effective diameter greater than 1 micron), the Navier-Stokes equations are accepted for momentum transport. They simplify to the linear Stokes equations, if inertia effects can be neglected.

In very small pores, the fluid continuum approximations may not hold. Bear [1972] quotes that for a Knudsen number, i.e. the ratio of the molecular mean free path length over the pore scale, of less then 1%, the fluid con- tinuum model is acceptable. In a more recent review, Koplik and Banavar [1995] conclude, "continuum flow requires only 10 or so molecules per direction of space." Continuum parameters such as viscosity of the fluid or phenomena such as Poiseuille flow or Taylor-Aris disper- sion correctly described the locally averaged (continuum) behavior of molecular hydrodynamic simulations in very small flow regions. Note, however, that possible electro- chemical interactions between the fluid molecules and the

solid wall, i.e. the influence of a hygroscopic layer on transport, were not considered.

Since the heterogeneity of the porous medium influences the distribution of fluid phases and thus of pore-scale transport, a second key question for a better understanding

of macroscale transport is how flow and transport behave in a sample of interconnected pores [Berkowitz and Baalberg, 1993]. Dullien [ 1991] reviews the description of porous media at the pore level. He provides an overview on how 3-D pore-scale networks can be reconstructed from serial cross sections. He suggests that the derivation of macroscale parameters based on information on pore-scale morphology could improve our physical picture of porous medium transport. Three-dimensional computer recon- structions of the pore structure from serial sample sections were performed to provide a basis for transport modeling.

Wan et al. [1996] combined 2-D microetched glass model representations of slices through porous media with fluorescent particle image tracking. They visualized detailed flow phenomena such as at the transition between fractures (larger pores) and the matrix (characterized by small pore sizes). They also observed the change in film thickness at the fracture-matrix interface with changing applied potential. Gvirtzman et al. [ 1987] obtained pictures of the water phase distribution using cold stage SEM and verified the existence of mobile and immobile water.

These and similar studies may help to improve our under- standing of the distribution of water, and thus of effective flow paths in the vadose zone.

van Genabeek and Rothman [1996] reviewed the possibility of deriving phenomenological theories of macroscale transport from microscale observations using numerical simulations. X-ray microtomography was pre- sented as a non-invasive imaging technique. Pore-scale geometry could be measured to a resolution of 1 micron and used as a basis for numerical simulations. Instead of

simplified pore-scale network models, full numerical solu- tions of the Stokes equations were used.

These authors also suggest going beyond the use of first- order statistics when relating pore-scale processes to macroscale transport parameters. Rather, they suggest that "much progress could be made by attempting to correlate second-order statistics, i.e. correlation functions of fluxes, with geometric properties (also perhaps correlation func- tions) of the porous medium." Such estimates of the variance as well as of the expectation of transport param- eters require significantly more data, which is made avail- able through the detailed imaging studies and simulations at the pore scale.

BRIDGING SCALES

Physical processes are often effective over a broad range of scales. For example, oceanography can be concerned with currents near a beach, in a bay, or in the entire oceanic basin. Similarly, transport in porous media can consider flow in a single pore, in a column, or in larger domain sizes. The transport detail at smaller scales is usually

MAYER ET AL. 129

and of little interest at the domain scale. However, we

have no a priori knowledge of sub-measurement scale transport, the cumulative effect of which determines the macroscale transport. It is thus instructive and at times necessary to relate phenomena at different scales to each other, a process that can loosely be called bridging scales.

To a greater or lesser extent, a "leap of faith" is always needed when bridging scales and this may be the greatest "hurdle" that a modeler needs to overcome. For example, some assumption must be made to interpolate between sparse data. If a statistical model is chosen and if the ergodic hypothesis is accepted, then the distribution of data at unsampled locations can be estimated as being part of a distribution. This in turn allows relating the sampled and inferred distribution of physical properties to a distribution at a larger scale. The "leap of faith" is accepting the statis- tical model and the ergodic hypothesis. It can be trusted given previous experience or tested after the fact using experimental data. Dagan [1986] reviewed how statistical theory can be used to bridge from pore to regional scales. Gelhar [1986] provided an example to show how sto- chastic theory can be applied by the practitioner in field studies.

As previously stated, it is frequently assumed that micro- and macroscales are related in a similar manner in

continuum mechanics and in porous media transport. As a result, bridging scales from pore to macroscale transport often implies a similar "leap of faith" as used when bridging from molecular to continuum scale. However, there are a number of differences between the physics governing the distribution of the associated microscale processes. Such differences should be considered before making analogies and using similar upscaling arguments.

Molecular scale processes are dynamic and the associ- ated expectations or averages include both spatial and temporal fluctuations. Available instrumentation averages over a time that is large compared to the time scale of molecular fluctuations, as well as over a volume that is large compared to the molecular scale. Pore-scale transport processes on the other hand are static at the time scale of measurement. If the temporal fluctuations related to changes in water content are neglected, the conceptual random nature is defined by the geometry of the fluid-solid interface and is thus only related to its spatial distribution. In typical applications, there is neither a conceptual nor a measurable temporal average that might help to reduce the variance of the measured data, which is entirely contained in its spatial distribution.

Another significant difference lies in the conceptual picture of how molecular scale processes relate to fluid "particles" on one hand, and how pore-scale processes re- late to the macroscopic description on the other. To obtain spatial averages or expected values of either density,

viscosity or flow rate of a fluid "particle", a sufficiently large number of molecules either in a statistical or in a spatial sense must be sampled. These averages are a prop- erty of the physics of molecular interactions, and do not depend on further constraints such as complex boundary conditions. Similarly, to obtain a local porosity value, a sufficiently large number of pores must be included. In this case, the spatial average or expected value is a property of the geometry of the porous medium and is not affected by dynamic physical processes. Finally, to obtain a local filtration velocity, a sufficient number of actual local flow paths should be included. The distribution of these flow paths is defined by an interaction of the fluid phases (liquid and gas) and the solid matrix. The resulting flow paths are thus governed by a combination of geomet- rical constraints (a very complex boundary condition), and fluid-fluid and fluid-solid interfaces, which in turn can depend on the prior history of wetting and drying cycles.

While this complexity at the microscale does not neces- sarily imply a complex behavior at a macroscale, it certain- ly is an indication that relating microscale to macroscale processes is a formidable task. It is also an indication that bridging scales is not necessarily achieved in the same way as for classical field theory. Dooge [1986] relates most hydrological problems to a "law of medium numbers", suggesting that these problems are too complex to be de- scribed by simple mechanical models, and not sufficiently random to allow for a meaningful average description using statistical mechanics. At the end of a brief review, Sposito [1986] concluded that neither a rigorous, nor a comprehensive answer exists relating the microscopic water behavior to macroscopic transport properties.

The problem remains that transport models in the vadose zone do not usually offer accurate predictions of solute arrival times and distributions. Thus there is a need to look

to all avenues that may improve upon these models, including the potential of better understanding, new param- eter identification, or even upscaling based on pore-scale studies. For this to be useful, these approaches should yield some benefit beyond "retrieving" what was accepted ma- croscale theory at the outset.

Discrete models, as used by percolation theory [Larson et al., 1981; Berkowitz and Baalberg, 1993] or as defined by pore-scale networks [Dullien, 1991; Ioannidis et al., 1993], have been used extensively to relate pore-scale processes to macroscale transport properties. For example, the hydraulic conductivity of a given soil is known to vary by orders of magnitude with changing water content. The influence of pore-scale distributions and connectivity of effective flow paths are commonly invoked to explain this. They are naturally described with discrete models. Such models gain credibility to the extent that the distribution of pores can be related to experimental

130 IDENTIFYING EFFECTIVE PARAMETERS FOR SOLUTE TRANSPORT MODELS

van Genabeek and Rothman [1996] give a review of recent advances in using pore-scale observations to model transport in rocks. Significant contributions can be made if microscopic observations lead to a better understanding of transport at a macroscale. Furthermore, such pore-scale observations can suggest the definition of new, physically meaningful macroscale transport parameters. This would guide experimental verification at the macroscale of a hypothesis obtained from the upscaling approach, as op- posed to simply "verifying" previously established macro- scale models.

Clearly, such direct upscaling studies from the pore scale are limited. From an experimental point of view, only a minutely small fraction of the field can be sampled to such detail. From a modeling point of view, a full simulation over scales including macropores (soils) or fractures (rocks) as well as detail at the pore scale is well beyond computational reach.

Progress and potential for improving our understanding at the microscale should not deter from the fact that there is

a real need for better modeling abilities at very large scales. A range of heterogeneities in porous medium properties may influence transport at regional or watershed scales. It is possible that a gradual upscaling from the conservation of mass and momentum equations to models valid at larger hydrologic scales can not be established. In this case, Dooge [ 1986] suggested that "we may be forced to skip some scales and seek entirely new laws of hydrologic behavior." At the present, it cannot be decided whether a solute transport model at such large scales needs to be very complex or very simple [Baveye and Boast, 1998].

Ideally, direct measurement at such scales would allow inferring a corresponding empirical model. However, if data cannot be obtained at such large scales, upscaling appears to be the only other alternative. Possibly, the insights gained from careful upscaling studies bridging the gap between known phenomena (e.g. pore scale to lab scale or lab scale to field scale) can guide further upscaling efforts, The conceptual picture would be one of successive upscaling steps between locally homogeneous scales, where intermediate results are grouped in effective transport parameters. This is in part supported by Neuman [1994], who reported a discrete hierarchy of scales for geological media.

A possible alternative is a stepwise upscaling by grouping intermediate results in effective transport parameters. This would be an improvement over traditional renormalization group theory [Gautier and Noetinger, 1997; Gavrilenko and Gueguen, 1998], as the iterative scaling laws would be based on experimental observations and direct simulations, rather than on intuition. Never- theless, it is evident that the challenges to overcome in

developing practical and reliable non-point source pollu- tion transport models are daunting.

SUMMARY

Porous media models are distinct from most other

physical models by the extent to which they have to incorporate the influence of the boundary conditions at the fluid-solid interface on the flow and transport properties. Intuitively, appealing parallels with models used in different contexts should thus be looked at cautiously. The complex boundary condition is distributed throughout the transport domain and its influence is usually absorbed into effective properties of the transport equations. If the boundary condition is statistically homogeneous above a certain scale, then transport model derivations are usually met with success. If the distribution is heterogeneous over a range of scales, then modeling efforts are met with many difficulties and only partial success in predicting transport properties. In general, the distribution is both too complex to be represented by a simple deterministic model and not sufficiently random to be represented by a simple sto- chastic model based on effective average properties.

At the core of the problem is establishing the relation between the heterogeneous distribution of field properties and a distribution of transport parameters. The first step is to estimate a distribution of field properties based on sparse, regularized data. The possible influence of unre- solved scales within the sample support and between data locations on the sampling network needs to be included in the data analysis. Data acquisition should be guided by the need to identify and resolve the spatial structure of possible heterogeneities, and thus should allow estimation of spatial variability over a range of scales. Usually, data can only be obtained over a small fraction of the field. In the case

of heterogeneous data distributions, a statistical model is needed to infer a probable distribution of field properties in the domain.

Model development, the distribution of model param- eters, and estimates of parameter values must have a clear relation to the distribution of field properties. Such a relation can be established if data suggests that the distribution is quasi-stationary over an intermediate range of scales. In this case, effective transport parameters can be estimated in terms of spatial averages or statistical expectations from the distribution at smaller scales. Alternatively, since the distribution of field properties is more closely related to experimental observation, it is suggested that such spatial averages be estimated for the field properties first. Then, those estimates can be used to infer the related transport parameters at the given scale. For this to be consistent, it should be verified that both

MAYER ET AL. 131

field property and transport parameter distributions are quasi-stationary over the same range of scales.

Bridging scales from pore to macroscale processes has often been criticized since it did not seem to further our

understanding of transport in porous media, and since it was based on unverified assumptions about the pore-scale processes and on the existence of meaningful spatial aver- ages. Advances in experimental and simulation techniques may have allowed overcoming some such criticism. De- tailed imaging studies of pore-scale distributions and full solutions of the Stokes equations in the corresponding do- main have offered new insights on how pore-scale physics translates into macroscale transport. Combined with data obtained over a range of scales and with careful upscaling procedures taking into account the distribution of hetero- geneities, this offers an optimistic outlook' on improving our ability to model pollution transport in the vadose zone.

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Timothy R. Ellsworth, Dept. of Natural Resources & Environmental Science, 1102 South Goodwin Ave., University of Illinois, Urbana, IL 61801. Phone: 217- 333-2055, fax: 217-333-9817, e-mail:

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Keith Loague, Dept. of Geological and EnVironmental Stefan Mayer, Dept. of Natural Resources & Sciences, Stanford University, Stanford, CA 94305-2115. Environmental Science, S-208 Turner Hall, University of Phone: 650-723-3090, fax: 650-725-0979, e-mail: Illinois, Urbana, IL 61801. Phone: 217-333-2055, e- keith•pangea. stanford.edu.