spatial variability in subsurface flow and transport: a review

Reliability Engineering and System Safety 42 (1993) 293-316

Spatial variability in subsurface flow and transport: a review

Allan L. Gutjahr Department of Mathematics, New Mexico Institute of Mining and Technology, Socorro, New Mexico 87801, USA

&

Rafael L. Bras Ralph M. Parsons Laboratory, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA

Stochastic models of spatial variation as they apply to both saturated and unsaturated flow and transport problems are examined in this paper. Both modeling and data interpretive geostatistical approaches are reviewed and an integrated discussion combining the two approaches given. The probabilistic content is of special interest for reliability and risk calculations for waste management and groundwater pollution studies.

INTRODUCTION

Massman & Freeze ~'2 used reliability theory in conjunction with models of spatial variability of the soil media in a methodology for risk-based engineering design of waste management sites. Their results indicate that failure probabilities, risks, associated costs and the monitoring network design are all sensitive to the degree and nature of the heterogeneity of the soil media where the waste management system resides. ~.2 It is then apparent that spatial variability of soil properties is an important phenomenon that must be taken into account when carrying out risk assessments, Some consequences of such variation are illustrated by results from the Borden Site experiment. 3"4 For example, Dagan 4 notes that the depth-averaged concentration has

1. a very irregular distribution in space, and 2. a much larger spread than that predicted by pore

scale dispersion.

Along similar lines, Gelhar 5 (see Fig. 1, adapted from Ref. 5) illustrates the result observed by many other researchers that longitudinal dispersivity estimates appear to increase with the scale of the

Reliability Engineering and System Safety 0951-8320/93/$06.00 © 1993 Elsevier Science Publishers Ltd, England.

experiment. Stochastic modelling of the heterogeneity of the geologic media can give an explanation of these and related observations.

Stochastic models of spatial variation as they apply to both saturated and unsaturated flow and transport problems are examined in the remainder of this paper. Both modelling and data interpretive geostatistical approaches will be reviewed and an integrated discussion combining the two approaches will be given.

Saturated flow models

Modelling efforts typically start from flow and transport equations for the solute. For steady state two-dimensional flow models, the single phase saturated flow equation used is

a a~ + _~__~

where T(x) is the transmissivity, q0(x) is the head or potential function and x = (x~, x2) is a location in two-dimensional space. Note that eqn (1) already includes a homogeneity assumption with regard to T(x); a more general model would take T(x) to be a tensor.

Classical hydrologic models are generally deterministic with T(x) assumed constant or at least

293

294 Allan L. Gutjahr, Rafael L. Bras

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data (from GelharS).

piecewise constant and known. Equation (1) is then solved for qg(x) using these known values and appropriate boundary conditions. Any uncertainty that arises is associated with the estimation of the presumed constant T values that have to be estimated via some statistical or inverse method. In reality T(x) does not have such a simple structure nor is it completely known or deterministic.

Parameter variation and discrete z o n e s

One attempt to model the uncertainty in transmissivity is to take it to be a simple random variable with some mean/~r and variance aZT. The 'field' T(x) could have means and variances that differ between zones or regions. For example, suppose that there are N regions with Tj representing the transmissivity in region/" for/" = 1 . . . . . N. For a given fixed set of T values, eqn (1) could be solved (possibly using numerical methods) to obtain a function, qg[T~, T2 . . . . . TN]. To account for the uncertainties associated with the transmissivities, the ~ values can be modelled as independent random variables with specified distributions or statistical properties, and the uncertainty (again in terms of a distribution or the moments) of ~p can be derived by standard probabilistic methods involving Jacobians or direct calculation. 6

In a more general framework, the Tj values need not be transmissivities and q0 need not be head. Conceptually there is a connective relationship

between some input parameters, T/, j = 1 . . . . . N, and another variable expressed as q~[T~ . . . . . T~] which may not be one-to-one or even analytically defined by a formula. The quantity q~ would be a solution to a differential equation or some other constitutive relationship between the variable q, and the parameters 7, J = 1 . . . . . N.

The parameters ~ could be

(a) transmissivities in different zones as discussed above; or

(b) conductivities and porosities in different regions; or

(c) velocities in different legs of a flow path. 7

The output functions could include

(a) head values; or (b) values of velocity; or (c) arrival time distributions at a point.

Dettinger & Wilson ~ and Sagar & Kiesel 9 applied this procedure to cases where the function ~ was specified and where there were a limited number of parameters.

Generally, q9 is hard to find explicitly and instead a Monte Carlo simulation approach could be used wherein

1. Independent realizations T a, j = 1 . . . . . N, i = 1 . . . . . 1 are generated.

2. q0[T~] . . . . . T~N] is obtained for each i, possibly by solving a differential equation or other connective relationships.

3. The statistics of cp (mean, variance, or distribution) are estimated from the results.

The resulting sensitivity analysis indicates the influence of the input parameters and their associated uncertainties on the output q0. The first step requires an assumption about the distributions as well as one of independence.

A direct simulation process may not be a very efficient way to carry out the sensitivity analysis when there are many parameters. One procedure that is useful for studying the various combinations of input parameters is Latin Hypercube Sampling (LHS). '° ' ' ' The assumption of independence in step (1) can be relaxed so that correlated observations are used. LHS can also be applied to cases that have correlation, though in this case the correlation structure is for the ranked values rather than the original observations.

A description of correlated LHS from lman & Conover '~ follows.

Presume as above, we have N parameters and 1 samples are desired. LHS for rank correlated data proceeds as follows:

(1) Find a decomposition of the rank correlation matrix, C = LL' where L is lower diagonal.

Spatial variability in subsurface f low and transport: a review 295

(2) Consider a set of scores: For example, the normal scores

a(i) = qg-'[i/(l + 1)l i = 1 . . . . . 1,

where q9 is the standard normal distribution. (3) Let R be an 1 x N matrix where each column is

a different random permutation of the scores a(i), i = 1 . . . . . 1.

(4) Form R* = RL' (5) Form a new 1 x N matrix Y where each row is

an independent observation from the desired distribution.

(6) Re-order the entries in column j of matrix Y so the elements have the same rank as the elements in R*.

The net result is that Y has the rank correlation C. Iman & Conover ~t also discuss a procedure for adjusting the output so that the rank covariances are close to the input covariances.

LHS is often applied to cases where many mean values could vary. It is a form of stratified sampling allowing better representation of the extremes.

LHS, as applied to spatial variability problems, has some limitations. First the input correlation is for the ranks and not the values. While the method implies any distribution could be generated for a parameter, connection between the rank correlation and regular correlation really requires knowledge of the joint distribution and is not immediate. Second, in spatial variability problems, the number o f values to be generated is very large--it is not unusual to generate as many as (128) 2 values in a two-dimensional case. The LHS method requires matrix decomposition of a (128) 2 by (128) 2 matrix which can be prohibitively expensive and also inaccurate.

Sensitivity analysis, as described above, was developed to study the effects of parameter variation; namely the problem where the estimates for the T values were modelled with uncertainty arising from statistical estimation or other sources. However, the formulation is general enough so that it can be applied to problems of spatial variability if there are certain zones within which the parameters remain relatively constant. An example of such a situation would be a stratified or layered medium.

The method outlined can also be used as a starting point for the perturbation or first-order, second- moment analyses discussed in greater detail below. For example, consider a Taylor series expansion for q0:

(r ,-

+ higher-order terms (2)

The linearized form (excluding the higher-order terms) can then be used to calculate variances and

covariances for the output. Typically one takes the #i to be the means, #j = E(~) .

The first-order, second-moment approach is illustrated in Dettinger & Wilson 8 and Townley & Wilson. ~2 If the results are obtained from a numerical model, the partial derivatives required can be calculated via an adjoint method.'3 In principle the method can also treat models that include correlation between the T values.

The main limitation of this discrete-parameter approach for modelling spatial heterogeneity is that only restricted kinds of spatial variability (constant over the zones or regions resulting from the necessary discretization) are treated. Inclusion of correlation between values can introduce considerable computational complexity. Interpretations of results as just representing measurement variability in the parameters may also obscure the inter-relationships that occur due to spatial heterogeneity.

Continuous stochastic models

The limitations of the discretized model, and the initial intent of that approach to address problems in parameter variation rather than spatial variability or heterogeneity issues, led to the introduction of continuous stochastic models in the mid- to late-1970s. In this formulation, media properties like hydraulic conductivity are represented as random spatial processes or random fields. As a result the solutions to the flow and transport equations similarly yield spatial stochastic processes.

One of the earliest papers that took this conceptual view was by Freeze. ~4 He investigated a one- dimensional flow model using a Monte Carlo approach. Actually his model yielded a simple explicit analytical result connecting head and the conductivities that were constant within blocks. Because he took the conductivities to be uncorrelated from block-to-block ~5 and considered just a one- dimensional medium, the results generated are not generally valid. However, the paper represents a very significant conceptual step in the study of spatial variability.

The correlation of a statistically homogeneous stochastic process can be expressed in terms of the spectral density, which is a Fourier transform of the covariance function.

1 ~ . . exp/ i ~] g g S(k)= T _" . j=,

x C(~) d~, . . . . . d~p (3)

where C(~) is the correlation function and S(k) is the spectral density, dependent on wavelengths k. A useful aspect of the spectral approach is that for appropriate statistically homogeneous processes the


process itself can be represented as a generalized Fourier-Stieltjes transform. ~6"~7 This result, referred to as the spectral representation theorem, simplifies many of the required calculations. Let V(x) be a statistically homogeneous random field with mean zero, covariance function C(~) continuous at the origin and spectral density S(k). Then there exists a unique (with probability one) complex random process Z(k) such that

V(x )= f ~ eik'x dZ(k) (4)

where (i) E[dZ(k)] = 0

0 if kl :# k2 (ii) E[dZ(k0 dZ(k2)] = S(k) dk if k, = k~ = k

The power of the spectral approach is that it takes correlated terms in the original space domain and replaces them by terms in the wave-number (k) domain that are uncorrelated but related to the original process. It is completely analogous to a change of basis or matrix transformation. The spectral density plays the role of the eigenvalues and, the dZ process the role of eigenvectors. In a linear system (often used to approximate the original nonlinear system) this means that the frequency or wave- number components are uncorrelated and do not interfere with each other. The representation is useful for both the analytical approach to spatial variability and for the Monte Carlo approach where it can be used to construct a simple algorithm for generating realizations or copies of random fields and where spectral approaches can be used to solve the equations, ts

Following is an outline of the stochastic approach to flow problems which uses the above concepts. For the sake of illustration we concentrate on eqn (1) in the two-dimensional case. The result of this analysis yields moment or distributional information about the output random field, in terms of the statistical properties of the input process, T(x).

In the perturbation approach, 5'''~-2~ eqn (1) is rewritten after dividing by T(x):

V[ln T(x)] - Vqo(x) + V2q~(x) = 0 (5)

[ 0 a_~2], is used ' where the gradient operator, V = Ox~ '

Variables are then perturbed around a nominal value; In T(x) and qv(x) are represented as sums of a non-random mean term and a term that is random but has mean 0. Thus,

In T(x) = Elln T(x)] + f ( x ) (6a)

qg(x) = E[qo(x)] + h(x) (6b)

where both f (x) and h(x) have mean zero. In addition

we assume that E[Vq0(x)] = - J . Substituting eqns (6a,b) into (5), taking expected values and subtracting expected values, we get

V2h(x) = J " W (7)

for the perturbation equation. Finally, if both f (x) and h(x) are statistically homogeneous (see Mizell et al. 2~ and Gutjahr & Gelhar 22 for necessary and sufficient conditions for the stationarity of h), then f (x) and h(x) will have unique representations of the form specified by the spectral representation theory (eqn (4)) where the dZ terms are related through

dZh(k) = - i ( J - k ) dZr(k)/Ikl 2 (8)

The relationship (8) contains the information needed for variance and covariance connections. In particular eqn (8) can be used to obtain spectral relationships between the f and h spectra and, by an extension of the spectral representation theorem, to get cross-spectra. The covariances and cross- covariances are then easily calculated from these spectral results. Figure 2, adapted from Bakr et al.t~) shows the correlation functions for f (x) (exponential) and the output head correlation for a three- dimensional model. The input is statistically isotropic, while the output correlation depends on the orientation of the separation vector to the flow direction.

The approach is simple, representing a linearization of the original system and assuming a constant uniform mean gradient. However, it does give some useful and important qualitative and quantitative information. For example, if f (x) has variance a} and

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C ) \ '~ \

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\ I \ \

\ \

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: ~ : ! F : ~ : r i I r

Fig. 2. Correlation functions for log-permeability (solid) and heads; 90 degrees from flow direction (short dashes); 45 degrees from flow direction (medium dashes) with flow

direction (long dashes).

q)

© C ) ~ r

O


correlation scale ~ (e.g. a value where the correlation function has an e-fold drop, p(3.)~ C1()O/o~=e-~) , then in two- and three-dimensions,

0 2 = Var[h(x)] = C)~20~ IJI 2 (9)

The constant C depends weakly on the covariance form as well as dimension.. Furthermore, the h(x) process has a simple statistically anisotropic structure with a correlation scale longer than that of f(x).

The linearized spectral analysis has been extended to include transport, 23-26 and it has been applied to unsaturated flow 27-28 and transport. 3~r-32

Linearization imposes restrictions on the allowable magnitude of the input variance 0}. Furthermore, the model assumes an infinite domain; though Naff & Vecchia 33 and Rubin & Dagan 34-36 have extended the results to finite domains. Rubin & Dagan, 34-36 in particular, show that for both unsteady and steady nonuniform flow in two-dimensions, the head covariance within three correlation lengths perpen- dicular to boundaries is influenced by the boundary conditions while beyond that point the boundary effects are essentially negligible. Dagan 37 showed that the first-order approximations are good so long as o~ is smaller than 1. If h(x) and f(x) are jointly Gaussian, then the spectral results obtained via the perturbation method are exact. 38

Graham & McLaughlin 39"4° and Bonano et al. 26 use the perturbation method to derive partial differential equations for moments in a transport problem. The latter equations were then solved either analytically 26 or numerically. Graham & McLaughlin 39 compared their results to a Monte-Carlo approach and observed good agreement. They also illustrate the fact that individual plumes can differ significantly from each other even though the underlying random field maintains the same statistical structure. An important point of their analysis is that conditioning (discussed later in this paper) can lead to substantial improvements in predictions 4~ (Fig. 3).

Gelhar et al. 23 and Gelhar & Axness 24 used a perturbation approach to model transport and the added dispersion due to the correlation between the random velocity and concentration. Gelhar & Axness 24 derived relationships between dispersivities and effective conductivity. Variances of conductivity 25"42"43 can be large and as noted in the work of Graham & McLaughlin 39,~ the differences in concentration plumes between realizations can be substantial.

In addition to variance results the spectral approach can be used to obtain effective values that apply to the averaged equations. 2° Two other approaches to this problem are Monte Carlo simulation 44,45 and averaging methods. 46

Monte Carlo procedures are appealing because boundary conditions and other special features can be

included. However, this method does require generation of correlated input fields. One method that has traditionally been used is the Turning Bands Method. 47-~9 Recent studies show that this method, for a finite number of lines, may lead to serious nonisotropic behavior in the input fields. 5c~'5~ Gutjahr 5" proposes an alternative procedure based on the Spectral Representation Theorem and the Fast Fourier Transform to overcome this problem. After the input fields are generated, the equations can be solved numerically to get the output head fields. The procedure can be time consuming since it needs to be repeated for every input field.

On the other hand, Monte Carlo simulations can give insight into problems not readily treated by analytical methods. For example, Rubin & Gomez- Hernandez 52 use a Monte Carlo approach to study upscaling, where values for blocks that are the order of a correlation length of the input field are desired. While effective values apply to domains large in size when compared to the correlation scale, the block conductivities apply to smaller regions and are useful for numerical modelling. Rubin & Gomez- Hernandez 52 used a Monte Carlo approach as well as an analytical approach to show that the block transmissivity depends on the block size and is generally larger than the geometric mean.

Kitanidis 46 considered effective hydraulic conductivity for gradually varying flows where the correlation length of the hydraulic head is considerably larger than for the conductivity. He used a general method of volume averaging and spatial moments to derive effective conductivity tensors and applied the result to several known cases, including the stochastic model of Gutjahr et al. 2° This same averaging method can also be applied to the transport or dispersion problem (Kitanidis, P. K., 1990, pers. comm). The generality of the averaging method is a desirable and useful feature and its application to known cases has been successful.

The predictions for spreading and transport are especially important for risk assessment calculations. Many of the results cited above are for ensemble results rather than specific sites. For specific sites, various methodologies have been proposed. 53"54 The approach of Nelson et al. 54 summarized below indicates how a flow model can be used to carry out risk evaluations. The steps for carrying out a risk evaluation include:

1. generating joint realizations for head and transmissivities using estimated covariances and flow equation connections;

2. computing velocities in the discretized model within each block; and

3. carrying out particle tracking or streamline calculations to obtain estimated probability


~ , , , , , i ,,, ?

(v.c / / ~ I ~=ity

260 Days

Fig. 3. Comparison of prior and posterior conditional mean concentration and velocity fields at 260 days after injection of tracer at Borden site (from Graham & McLaughlin4~).

distributions at given locations given input of contamination at another location.

While this approach is promising, problems that remain to be resolved are:

1. estimation of covariance parameters recognizing the interconnections implied by the flow equations;

2. proper joint conditioning using all the data available; and

3. estimating parameters based on sparse, error- prone or imprecise data.

Optimal estimation: kriging

The first step in kriging-geostatistical studies is to estimate the second-moment properties of the random field in question. This may be the covariance for stat ionary/homogeneous fields or the generalized

covariance. The latter represents the second-moment properties of generalized differences of the field. These differences are designed to filter out the unknown drifts or mean and hence avoid estimation of first-order moments. For example, the use of the semivariogram, or variance of first-order, simple, differences, eliminates unknown constant mean structures.

Variogram or covariance estimation based on observations V(xi), i = 1 . . . . . N, is almost always based on simple moment estimates. If a parametric form for the variogram or covariance is postulated or if distributional information is assumed for the data, minimum quadratic mean or restricted maximum likelihood estimators can be used. 55'56

Estimating the variogram is usually not straightforward. Removal of trends can bias the estimates, outliers can severely influence the results, and generally there are no simple confidence bounds for

Spatial variability in subsurface flow and transport: a review 299

the final estimates. If trends are present, higher-order differences can be used to remove the trends though controversy exists concerning such a procedure. 57 Recently Isaacs and Srivastava 5~'59 proposed a non-ergodic general covariance (and variogram) estimation method that is promising for data with trends.

Formal fitting procedures like least squares estimation have to consider the implications of independence and homoscedastic assumptions. Max- imum likelihood approaches generally assume normality or log normality of residuals. Commonly the raw variogram or covariance estimates are fitted with a functional form on an informal basis.

More specifically the variogram estimates at different lags are correlated. In addition even under restrictive assumptions (e.g. joint normality) the variances of the estimators are hard to obtain. The situation is analogous to covariance estimation in conventional times series. ~

In any case, it should be emphasized that variogram estimates (in the absence of other information) do not have an associated well developed statistical theory. Even asymptotic results are not readily available. If the data are assumed to have a particular distribution (e.g. multivariate normal) and the variogram has a specified parametric form (e.g. exponential), then one can use conventional statistical techniques like maximum likelihood to do the estimation.

The next step in most geostatistical studies is to estimate or predict values at non-observed values using the variogram information. Standard kriging uses unbiased minimum mean squared estimations for the unobserved values. Namely if one has observation V(xi), i = 1 . . . . . N, an estimator of the form ~"(Xo) = )-~JT=l )~iV(xi) is used where the Zi values are chosen to minimize E[V(xo) - l?(xo)] 2.

Kriging estimators have several attractive properties: 6j

1. The estimators are exact interpolators. 2. The kriging weights and kriging variances

depend on the locations and covariance functions, but not on the actual V(xi) values.

3. Kriged values are correlated, smoother than the actual field, but more variable than the mean value.

4. The kriging estimators l?(Xo) can be viewed as an approximation to the conditional mean and the associated kriging variance o[(Xo) can be viewed as an average of the conditional variances.

5. ~'(Xo)-t-20rk(Xo) is an approximate 95% toler- ance interval in the case where V(x) is a Gaussian (jointly normal) field.

Property (5) must be interpreted with care. Only in the multivariate normal case can this interval be used

to assess exact probability values. The kriging variance is primarily a measure of data configuration and the role of different configurations on predictions. 62

Details of kriging and its various extensions can be found in sources like: Matheron; 48"63 Delfiner; 64 Bras & Rodriguez-Iturbe; 65 and Journel & Huijbregts. 49

Several applications of kriging to groundwater and risk assessment problems appear in recent literature. Zirscky et al.66 and Zirscky & Harris 67 used kriging in a study of a dioxin spill. Zirscky & Harris 67 imply kriging may be useful for delineating high contaminant concentration areas. Pucci & Murashige 68 used kriging in conjunction with trend surface analysis to get hydraulic conductivity values and water surface levels and determine areas where added measurements would be useful.

Cooper & Istok 69"7° and Istok & Cooper 71 apply kriging to a groundwater pollution problem. Cooper & Istok TM used moving neighborhoods to account for changes in the means and then used kriging to select areas with high concentrations. They also used block kriging 7~ to estimate the total contaminant present and locations of subdomains with high concentrations. The results can be used for risk evaluation and design of a sampling program.

Bras & Vomvoris 72 applied kriging and higher-order intrinsic random field theory to estimate concentrations remaining after chemical clean-up. They note that a large amount of the work involved was simply assessing the data and verifying the data base. They used two different types of data (inter-related by a regression model) and, assuming normality, did probability calculations (Fig. 4).

Indicator and soft kriging are extensions that appear to be promising in groundwater transport. This procedure, along with indicator conditional simulation, has been espoused by Journel. 75-77 The objective here is to predict the conditional pi'obability that V(xo) is less than some value, Vo, given the data:

P[V(x,,) <-Vo I V(x,) . . . . , V(xn)] (10)

This problem can be put into a kriging framework by encoding the data into indicator variables.

For a set of cutoff values, Vk, the indicator functions

I[V k: V(x/)] = 1, if V(xj) < Vk

= 0, if V(Xj) > Vk (11)

are introduced. Namely, I[Vk:V(x]) ] for fixed Vk simply indicates when the data value observed is less than Vk; hence a sequence of 0s and ls replaces the original data. As Vk changes, different sequences result though some order relations obviously have to be satisfied; e.g. l[vl:V(xj)] -<l[v2: V(xi) ], if vl -< v2. The expected value of I[vo : V(x)] is just P[V(x) <- Vo].

By treating the values of the indicators as the field


CONTOURS I 0.1 2 0 . 2 3 0 . 4 4 0 . 6 5 0 . 8

Fig. 4. Probability of finding

3.62

- 5 0

depth-averaged concentrations greater than 100 ppm (15.2 m x 15-2 m x 1.5 m) (from Bras & Vomvoris72).

5

over blocks 5 0 f t × 5 0 f t × 5 f t

of interest, covariances and variograms can again be obtained. Note,

E { I[vl : V (x~)ll[v2 : V(xo)]}

= P[V(x,)--< Vl, V(xb) --< v2] (12)

Namely, the covariance function for the indicators is related to the joint probability function for two values and two locations. While it is possible to examine cross-covariances based on these joint distributions, in many cases it is sufficient to examine the covariances involving only l[vo : V(xj)] and l[vo : V(xo)].

After the covariances or variograms for the indicators are available, kriging can be used to estimate the desired conditional probabilities. The kriging equations will be similar to those previously seen:

P(V(xo) <- vo [ l[v,, : V(x,)] . . . . . l[v,,: V(x,)])

= ~ Zff[v,,: V(xj)] j - i

For K cutoff values vl . . . . . VK, there will be K values at each prediction location, and these are used to estimate the conditional distribution of V(x) at xo.

In the context of transport problems, one can imagine particles emplaced at an initial site which then travel through the domain. The number of particles at a particular location and time is proportional to the probability distribution at location x. Thus it would appear that indicator kriging is well suited to answer questions about concentrations at a location given data at other locations.

The question of trends in the values again has to be faced in both covariance estimation and prediction. In addition inconsistencies can occur in the distribution functions; namely, the estimated distribution may not be monotonic in the V-values. The latter can be overcome by fitting monotonic functions to the estimated values.

The formalism of indicator kriging can be used to include qualitative or 'soft' data of several types. At a particular location there may be no hard data, but an

assessment of the probability distribution given data in the surrounding area could be available. For example, in examining permeability at a location, relatively simple information about geology and soil type could yield some rough information about the distribution of the non-measured value at the given location. Translated this would give a prior distribution for V ( x i ) : P [ V ( x i ) ~ v] ~- G(v :xi). Then at the indicated location xi, the data value can be encoded but this time using l[vo:V(xi)] = G(vo:xi), the probability value (see Fig. 5).

Other kinds of soft data (e.g. interval constraints) can be included in this approach. Indicator kriging and use of soft data is very promising because it makes use of a wider range of data and directly predicts quantities of interest.

The geostatistical methods discussed above have estimation or prediction as their primary focus. The kriging procedure yields smoothed paths for the conditional means and conditional variance approximations. In many applications it is important to know the variability that can occur in a particular region given the observations. One could argue that, in fact, this is the main a im-- to predict what can happen in a specific instance rather than in general for related fields.

There are several procedures for doing this; in the standard case all involve some distributional assumptions. However, for indicator conditioning Journei 77 and Journel & Alabert TM have proposed a method that is essentially non-parametric.

Standard conditional simulations start by uncondi- tionally generating a realization using the proposed mean and covariance estimates from the variability characterization phase of the study. The unconditioned path is then conditioned to agree with the data while still satisfying the specified variogram behavior. If V,(x) denotes the unconditioned generated realization, 12k(X) the kriged estimator based on the data, and l~',k(X) a kriged estimator for V,(x) based on data taken at x~ . . . . . x, (the same corresponding location as in the actual field), then the conditioned


v(×2)=11; K lo :v )=o

V(x )=8; I(10:V)=1

x 3

- - - J /

~: . . . . . > ~ ,,

1 0 v

At x 3 , I ( lO :V) = 0 ( 1 0 : x 3 )

Fig. 5. Illustration for soft kriging framework. Indicator function for V = 10 and indicated distribution.

path is V¢~(x) = l?~(x) + [V,(x) - l?,k(x)]. This conditioned path has the following properties:

(i) Vc.~(x) agrees with the observations:

Vc~(x/) = V(xi) i = 1 . . . . . n

(ii) The mean of V~(x) is 17"k(x); the kriging estimator and the variance is the kriging variance.

(iii) The variogram of Vc~(x) is the initial variogram.

Figure 6 illustrates conditional paths along with the kriged path from a hypothetical field. Two hypothetical paths are shown (dashed lines) along with the smoothed kriged paths. Three observations are assumed and indicated by squares; as required by the kriging and conditional simulation method, all three paths agree at the observations. The conditional path will be less variable than the true path but more variable than the kriged path.

Model-based geostatistical studies

The applications previously discussed and many of the geostatistical applications in the literature are non-model based in the sense that the field V(x) is considered in isolation or only statistical relationships between two fields are considered. Yet many flow and dispersion phenomena are explained at least partially by physical and mathematical models. As discussed previously, if the input parameters (transmissivities, etc.) are considered to be random fields, then output quantities (head, velocities, concentration, etc.) will also be random fields. Therefore the statistical descriptors like covariances and means for the fields will be related to each other. Model-based geostatistical studies use physical relationships to obtain consistent covariance structures for the various variables and then use these results within the kriging formalism to estimate variables of interest.

Several model-based studies have been treated by

, . , . . , , , , . , . , .

>7

I

r l i i i r i i , i r i i i i , ~ i i i i i i i i ~ i i i i i i i i i I i i r i i i ~ i , ~ i r i I

0 2 4- 6 8 10 x

Fig. 6. Conditional simulations (dashed paths) and kriged path (solid). Correlation scale = 1, exponential covariance function.

302 A l l a n L . Gut jahr , Ra fae l L . Bras

Gutjahr; TM Gutjahr & W i l s o n y Kitanidis & Vomvoris; 8~ Hoeksema & Kitanidis; 82 Clifton & Neumann; 83 and Bonano et al. 84 More detailed discussion of these results and related studies is given in Gutjahr. 8s'86

To illustrate model-based geostatistical studies, we will use the results of Bonano et al. x4 that are completely based on the concepts of Kitanidis & Vomvoris s~ and Hoeksema & Kitanidis. ~2 The approach can be described in five primary steps.

First a covariance model is proposed for the log transmissivity field appearing in eqn (5). The proposed covariance model has several unknown parameters, normally one to three parameters.

Second, a perturbation analysis like that discussed in the previous section is used to develop a relationship between heads and log transmissivities (eqn (7)). This relationship, after discretization through finite differences, is then used to develop expressions relating the covariance (now matrices) of head measurements to the covariance of log transmissivity and covariance of head boundary conditions. Similar expressions are obtained for the cross-covariance between heads measurements and log transmissivities and heads and boundary conditions. Darcy's equation for seepage velocity

1 exp(ln T) 3q~ (13) v, = - nb ~x~

is linearized and used to relate the covariance of

velocities and the cross-covariance of velocities and measured heads and transmissivities to the covariance model of log-transmissivities.

In a third step measurements of transmissivities and/or heads are used to estimate the parameters of the log transmissivity covariance model which, from the second step, also parameterized all other necessary covariances and cross-covariances. This step involves a numerical procedure described by Kitanidis & Lane 5~ and assumes a specific covariance form and multivariate normality of the log transmissivities.

In a fourth step kriging and co-kriging techniques use the now available covariances to estimate conditional mean fields and conditional kriging covariances of head, transmissitivies and velocities in the x- and y-directions. All the analysis is done in the vertically integrated, two-dimensional, domain described by eqns (1) and (5). Co-kriging refers to linear estimation of one random field from observations of a related multivariate set. In this case the four estimated fields mentioned above are obtained using limited observations of transmissivities and/or heads. These measurements were the ones used in the third steps to obtain covariance parameters•

In a fifth step the observations and the available covariances are used in a conditional simulation (described in the previous section) to obtain possible realizations of any of the random fields of interest, particularly seepage velocities.

Bonano et al. ~4 used the above procedure to study

l . O

0 . 9

E

~ - - 0 . 7

t :~0. 5

b- b- ~[~0 .5

> ' 0 . 4 4_1 .,.-t

- ~ 0 . 3

0 - 0 . 2

o!

Plot B. t

I

- I

I I t

- - 3 . 0 mi. Co r r . Length i . . . . . ~ mi. Corr Length - - - - -510 mi. Corr Length - - - ~ 0 . 0 mi. Co r r . Length I

Log ( Time (yrs . ) )

Fig. 7. Effect of correlation length of log transmissivity on the cumulative density function of travel time al.~).

(from Bonano et


groundwater time at a hypothetical nuclear repository site. They used conditionally simulated velocity fields with a particle tracking model to: (a) generate a flow path from the edge of the repository to the accessible environment; (b) estimate the groundwater travel time along each flow path; and (c) construct a cumulative distribution function of the groundwater travel time. Figure 7 shows the sensitivity of characteristic travel time distributions to changes in the correlation length of log transmissivities.

It is important to finish this section by emphasizing the differences between straight applications of kriging or geostatistics and model-based geostatistical studies. The key difference is that in the latter, estimated covariances are restricted to be consistent with an interpretation of the basic equations of motion in porous media as relationships between random fields. Hence, covariances of head, transmissivities and boundary conditions are not independent and cannot be estimated in isolation.

UNSATURATED FLOW MODELS

Unsaturated flow in porous media at scales on the order of tens of centimeters or higher is generally presumed to obey a Darcy-type equation:

qx = -Kx(O) Ocp(O) (14a) Ox

aq (o) qy=-Ky(O) Oy (14b)

q~ = -Kz(O) Oq)(O) K~(O) (14c) Oz

where q is the moisture flux in the ith direction; q0(0) is the soil water tension or matrix potential, a function of soil moisture 0; and Ki(O) is the hydraulic conductivity in the ith direction, also a function of moisture 0.

Using the above equations in the continuity equation (conservation of mass) for an isothermal, single phase, system leads to

a--~-=~x rx(O) ax J ay Ky(O)

+--az Kz(o) + N Kz(o) (15a)

or alternatively, expressing everything in terms of matrix potential

C(qo) o , 7y

+ + N Kz( ) (15b)

where C(~p)= O0/Oq) is called the moisture capacity

and has units of inverse length. Equation (15b) is known as Richard's equation. Equations (15) have some significant differences from the saturated flow equation (1). These differences flavor the stochastic interpretations of unsaturated flow and make the analysis of heterogeneities in the soil much more difficult. First is the obvious nonlinearity of eqn (15). The 'parameters' of the equation, C(qg) and Ki(qg) cannot be interpreted as stochastic fields but are stochastic functions of the state. The nature of these functions may vary significantly in space as we will illustrate later.

Second, in contrast to eqn (1), the expression for flow in unsaturated porous media is purposely shown in its unsteady form. Interesting unsaturated flow usually occurs during transients. It is the change in moisture, the movement of a moisture front, that is normally of interest. The analysis of a time-varying nonlinear flow is much more difficult, even more so from a stochastic point of view since commonplace stationarity assumptions are then invalid.

In order to study eqn (15), it is first necessary to parameterize the relationships between hydraulic conductivity, matrix potential, and soil moisture. One difficulty is that these relationships are known to be hysteretic. For example, hydraulic conductivity can take two values for a given soil moisture, depending on whether you are in a wetting (infiltration) or drying (redistribution) event. This hysteresis is generally attributed to pore scale phenomena although it may also arise as a result of heterogeneities in the soil, as we will see later. Parameterizing hysteresis has eluded researchers and has been treated by only a f e w . 87-9° In most applications singled valued functions are used for the unsaturated flow parameters. Common functions are those suggested by Brooks and Corey; 9~ Gardner; 92 van Genuchten; 93 and Mualem. 94 They are;

Brooks & Corey: 9~

~(s) = ~(1)s -I/m

o r

Gardner: 92

K(s) = K(1)s"

K(qo) = K ( 1 ) ( ~ I ) ) '~

C(qo) = aqo d

K(~) = K(1)exp(-a~ I~l)

Van Genuchten 92 and Mualem:94

1 s( o) -

(1 + laqol") m

K(cp) = K(1)st[1 - (1 - slim)"] 2

where m = 1 - 1/n.

(16a)

(16b)

(16c)


2 1 0

,,c

~ 10.2 -

F- 0

C21 -3 Z 10 -- 0 0 0

~ 10 4

C~ >- I

5 10 I I I

50 1 O0 150

CAPILLARY TENSION HEAD, • (cm)

I

200 250

Fig. 8. Variability of the relationship between hydraulic conductivity and capillary tension at a site (from Mantoglou & Gelhar, 1987a).

In the above equations, qg(1) and K(1) stand for the values of matrix potential and hydraulic conductivity at saturation (s = 1), respectively.

Heterogeneity or spatial variability of the above functions is illustrated in Fig. 8 where typical field values are shown. It is clear that the spatial variability can be introduced by changes in the various parameters in any of the above parameterizations. Unfortunately the number of parameters is large and the choice of parameterization remains arbitrary, with little more significance than an empirical fit of a convenient function.

Given the difficulties outlined in the past few paragraphs, it is not surprising that studies of spatial variability and its effect on unsaturated flow are limited, relative to work in saturated media. Following are brief descriptions of some of the problems and approaches taken. First we discuss a particular type of heterogeneity in soils that is very relevant to disposal of waste in unsaturated porous media; that is, fractured versus matrix flow and flow in structured (macroporous) soils. Second, we treat Monte Carlo approaches to study the effect of soil heterogeneity in unsaturated flow. Third, we discuss the use of simplified analytical solutions of the flow equations to derive stochastic properties of soil moisture and flux distributions. Fourth, we discuss spectral approaches for solving stochastic interpretations of the unsaturated flow problem. These are mostly directed to deriving 'equivalent' large scale parameters resulting from local scale heterogeneity in analogy to the saturated flow problem. Finally, we treat perturbation solutions involving numerical integrations in the time domain.

Dual medium models of unsaturated flow in fractured rock and structured soils

Dual medium models of unsaturated flow are commonly applied to both fractured rock and to structured (or macroporous) soils. Fractures and macropores are difficult to locate and map, but can have a significant effect on the behavior of the bulk flow system. The nature of this effect depends greatly on whether the medium is saturated or unsaturated. Most early research on flow through fractured rock considered only saturated media. Recently, a number of saturated fractured flow models have been extended to deal with unsaturated conditions. A brief but informative summary of currently available fracture flow models is provided by Yeh et al. 95 All of these models identify two distinct flow systems: a matrix system and a fracture system. The matrix is treated as a continuum which behaves much like a classical unsaturated soil. The fracture system is typically modelled in one of two ways: (1) as a network of discrete flow channels distributed throughout the matrix; or (2) as a second continuum superimposed upon the matrix. Examples of the first (discrete fracture) approach include the saturated flow models of Huyakorn et al. , 96 Andersson & Thunvik, 97 and EIIsworth, 9~ and the unsaturated flow models of Wang & Narasimhan 99 and Rasmussen.""~ This approach requires that the location, orientation, and other geometrical properties of individual fractures be specified throughout the bulk medium. Given available methods for measuring fracture properties (primarily boreholes), discrete fracture models are impractical for most field-scale predictive applica-


tions. Nevertheless, they can be useful in research studies.

The usefulness of a hypothetical discrete fracture analysis is illustrated by the work of Wang & Narasimhan. 99 The model of unsaturated flow through a fractured medium postulates a regular network of fractures embedded in a rock matrix with known spatially uniform properties. The matrix and fracture flow equations are coupled by interaction terms which depend on the matrix-fracture flow area. A version of Mualem's 94 capillary bundle theory is used to derive the matrix and fracture saturation-pressure and conductivity-pressure functions as well as a function describing the relationship between matrix-fracture flow area and fracture saturation.

Wang & Narasimhan 99 use their laboratory scale deterministic model to analyze the properties of a hypothetical block of fractured rock having properties similar to the Topopah Spring tuff found near Yucca Mountain, Nevada. The Wang & Narasimhan analysis indicates that unsaturated flow occurs primarily in the tuff rock matrix. The fracture system appears to influence only the response to large flow transients, through its effect on the storage properties of the composite porous medium. The implication is that the effective properties of the bulk medium are the same as those of the rock matrix for a wide range of conditions.

For situations where field-scale predictions are required, the second (continuum) approach for modelling fracture flow is undoubtedly the most practical alternative. Examples of this approach include the saturated flow models of Kazemi l°~ and Moench "~2 and the unsaturated flow model of Klavetter & Peters. ~°3 Dual medium continuum models adopt separate flow equations for the fracture and matrix systems. In the case of unsaturated flow, these equations are most often versions of Richard's equation, each having its own saturation-pressure and conductivity-pressure functions. The matrix and fracture flow equations are coupled by a mass transfer term having the general form:

Qf,, = qg(q~f - % , ) (17)

where q9 I and tp,~ are the fracture and matrix heads, and to is a constitutive transfer coefficient. In principle, the coupled flow equations can be solved for qgi and q0,,, subject to appropriate boundary conditions. The bulk flow can then be obtained by superimposing the contributions from each flow system. In practice, it is advisable to simplify this computationally demanding two-component problem.

One possible simplification is to follow the approach suggested by Klavetter & Peters. ~o3 They note that the unsaturated simulations performed by Wang & Narasimhan 99 indicated that matrix and fracture pressure heads are nearly the same in the Yucca

Mountain case study. Based on this observation, Klavetter & Peters ~°3 replace the two unknown pressure heads in their dual medium continuum model by a single unknown which characterizes the bulk medium. They then add the matrix and fracture flow equations and identify composite (or effective) storage and conductivity terms which depend on the properties of both the matrix and the fracture system. These effective parameters can be estimated with a constitutive approach based on capillary bundle theory (see Burdine ~°4) or with more traditional macroscopic field methods (see KluteH~-~). Although the Klavetter & Peters approach is admittedly an oversimplification of reality, it probably represents the state-of-the-art in applied modelling of unsaturated fractured media.

In addition to the fracture flow models discussed above, there are many models of unsaturated flow through structured soils. Such soils are less consolid- ated than fractured rock and generally lie close to the surface. They typically contain macropores which act as preferred pathways under saturated conditions and as obstacles to flow under unsaturated conditions. Edwards et al. ~vo present a classic simulation study of the effect of a single large macropore on the bulk flow properties of an unsaturated medium. Germann, 1°7 Germann & Beven, 1°~ and Beven & Clarke 1~9 have all proposed various types of continuum models which describe the bulk flow behavior of unsaturated structured soils. These models are primarily intended to provide better understanding of infiltration processes, particularly during intense rainfall events. In this case, they serve a somewhat specialized purpose which bridges the gap between laboratory and field scales.

Monte Carlo simulations

There are a number of ways to investigate the field-scale effects of spatial heterogeneity. One of the most conceptually straightforward is to simulate flow through a large region with synthetically generated heterogeneous soil properties. Ababou 'H~ used this approach to derive the pressure and moisture content distributions which result when the parameters of the unsaturated hydraulic conductivity function are allowed to vary continuously throughout a three- dimensional domain. This is probably the most ambitious Monte Carlo exercise to date. The correlation scales assigned to the random soil properties were approximately 1 meter while the domain size was about 25 by 10 by 7 meters. A computational grid of 125 000 nodes was required to obtain accurate solutions for this wide range of scales. Typical small-scale tension (matrix potential) distributions generated for a three-dimensional infiltration problem are shown in Fig. 9.'H~ Large-scale behavior


E

8 t~

"t=

o

(.9

_o nn

8

a

0-

1-

2-

3-.

4-

,5

p,o,,=, B '

Ababou M o d e l

...... at y = _ . ~2.0 m

0 -

1-

2-

3-

4

5-

I f

t Prohle B 2

Ababou Model a t y =

-4.8 m

1 -

2" 1

o ' - " "11

2-

3-

4

"6 -4 "2 0 Distance from Wet Strip Centerline (m)

Profile B 3 /

Aloaloou Model a t y = 9.8 m

i !

. . . . . . . . . ,L ~ [

Mean - r r r - - Model 4 6 Tei°n'~°rH(cm°fwater)- 8(~

W 80 - i/~',~

" 2 : - i 4 4[:, - ! J

Fig. 9. Matrix potential resulting from three-dimensional simulation with stochastic soil properties, comparison with results from a model with mean properties (from Ababou ~"').

can be studied by computing spatial averages of these small scale distributions.

Ababou's stochastic simulation approach provides a detailed description of the moisture content distribution obtained with a single hypothesized distribution of small-scale soil properties. Any given site may have a much different soil property distribution and, therefore, a much different small-scale moisture content distribution. The site-to-site variability of hydrologic properties can be investigated, at least in principle, with a Monte Carlo analysis which treats each possible site as an independent replicate drawn at random from a hypothetical population (or ensemble). Monte Carlo studies of unsaturated flow have generally been limited to very simple or very specialized problems, primarily because of the computational demands imposed by problems of realistic size and dimensionality. ~ m~ It is questionable to what extent the results of these simple simulation

experiments can be generalized to more realistic situations.

Others have used the Monte Carlo approach to solve particular problems of flow in unsaturated porous media. Warrick et al. ~2 assumed that water content is independent of depth and that hydraulic conductivity is an exponential function of soil moisture (i.e. Gardner's parameterization). They then suggested a simple equation for drainage as a function of saturated hydraulic conductivity and a parameter of the K - O relationship, a,. Both K(1) and c are assumed log-normally distributed and samples from this distribution are used to find the mean flux and variance. They conclude that the mean flux is larger than that resulting from using mean parameters in the flux equation. A similar exercise is that of Rao et al. ~

A fair amount of Monte Carlo exercises have been performed to study the phenomena of runoff production particularly on a hillsiope. Runoff is


naturally related to infiltration and soil moisture distribution. Examples are Freeze, 1~4'~'5 Sharma & Seely, 116 Sharma et al., tIT and Smith & Hebbert. 118 EI-Kadi ~9 assumed log normality and exponential spatial correlation of saturated hydraulic conductivity. He then numerically solved a one-dimensional Richard's equation, sampling from the stochastic parameter distribution.

Andersson & Shapiro ~1 used a Monte Carlo analysis to compare against analytical results (to be discussed later). They numerically solved the one- dimensional, steady-state Richard's equation, assuming uncertainty only in the saturated hydraulic conductivity, which was generated as a log-normally distributed variable along the vertical discretization of their soil column. There is correlation between adjacent blocks in their discretization. They studied the standard deviation and correlation for pressure head as a function of depth in their one-dimensional soil column.

R u s s o 12°'121 combined geostatistical approaches with simulation. In 1983 ~2° he used kriging techniques to estimate the saturated hydraulic conductivities and the parameter in a Brooks-Corey type parameterization of conductivity over space from a limited set of measurements. The resulting conditional mean fields are used in a linearized steady-state solution for infiltration from a shallow circular pond to find the matrix potential distribution. In 1986 TM conditional simulation of parameters were used to study soil salinity, soil moisture, and crop yield distributions.

Analytical , simplified solut ions

It is sometimes possible in studies of uniform soils to circumvent the computational burden of Monte Carlo methods by deriving the ensemble moments of local hydrologic variables directly from analytical solutions of the small-scale soil flow equation. This is the approach taken by Andersson & Shapiro, jl~ Dagan & Bresler, j22 and Bresler & Dagan. 123"124 For the most part these authors do not explicitly examine the statistical properties of spatially averaged variables. Instead, they assume that all small-scale soil properties and hydrologic variables are locally stationary and ergodic. In this case, spatial averages approach ensemble means (which vary slowly over space) as the averaging scale becomes sufficiently large.

The work of Dagan & Bresler ~22 and Bresler & Dagan123,124 illustrates how analytical methods may be used to derive field-scale models of unsaturated flow. Their approach assumes that flow moves only in the vertical direction through hypothetical stream tubes which extend from the ground surface to the water table. Each tube is assumed to have a different vertically homogeneous saturated hydraulic conduc-

tivity drawn at random from a specified probability distribution. Milly 125 calls this a 'parallel' model of heterogeneity, reflecting the assumption that flow moves through many parallel paths which are hydrologically uncoupled. Such a model ignores horizontal moisture movement by construction. This is a significant assumption which needs to be critically examined in applications (such as those involving stratified soils) where lateral spreading may be significant.

Using the parallel model, Dagan & Bresler ~22 derive expressions for the ensemble mean of the moisture content (as a function of depth) for the case of surface infiltration and subsequent redistribution. A Green & Ampt ~26 solution of Richard's equation is used to express the percent saturation, s, at any time and depth, xi, as a function f ( x t , t, Ks) of the saturated hydraulic conductivity. All other soil parameters are assumed to be known constants. The saturation ensemble mean is then derived from the following discrete approximation to the mathematical expecta- tion integral:

1 N g(x,, t) = ~l .~ f (x,, t, K~.) (18)

i = l

Here the hydraulic conductivity probability density has been divided into N intervals of equal area and K~. is the midpoint of interval i. Bresler & Dagan 123 assume that this probability density is log normal, with a known mean and variance.

Bresler & Dagan lz3 found that mean vertical flow through their simplified heterogeneous flow system could not be reproduced with a Richard's equation based on field-scale effective parameters, even though Richard's equation is assumed to apply within each small-scale soil column. This result led them to the conclusion that 'the traditional deterministic approach for solving the (unsaturated) flow equations cannot be justified for solving flow problems in spatially variable fields'. Although this unequivocal conclusion should be viewed skeptically, given the many assumptions and simplifications involved, the Dagan & Bresler work certainly suggests that traditional methods may not always be able to accurately predict spatially averaged hydrologic variables.

As previously mentioned Andersson & Shapiro ~ also analytically studied issues of heterogeneity in unsaturated porous media. Using a perturbation technique they found a simple solution to the one-dimensional steady-state Richard's equation. The solution is applicable for regions far from the surface of the soil. Their expression for matrix potential is a simple function of saturated hydraulic conductivity (a van Genuchten-type parameterization was used). Presuming that saturated hydraulic conductivity is stochastic with known first- and second-moments, they


derived the first- and second-moments of soil moisture and matrix potential. They found that their analytical results agreed well with Monte Carlo exercises, numerically solving the original steady-state Richard's equation. Andersson & Shapiro TM point out the possible problems of extrapolating one-dimensional results to three-dimensional reality. As in saturated media, the standard deviation of hydraulic head may be exaggerated by one-dimensional analysis where water is artificially constrained in the flow paths it may take.

Dagan & Bresler ~27 and Bresler & Dagan ~2~~2'~ studied the influence of heterogeneous soils on dispersive solute transport in unsaturated porous media. In Ref. 128, the authors used a simple solution to the one-dimensional convective-dispersive equation assuming a known constant specific discharge through the medium. They neglected the effect of pore-scale dispersion and attributed the spreading of solute to convection only. Soil variability is attributed only to horizontal heterogeneity of saturated hydraulic conductivity. The specific discharge (leaching at the surface) is assumed to follow a uniform distribution everywhere in space. In Ref. 129, Bresler & Dagan extend their analysis to account for pore-scale dispersion. Spatial heterogeneity was then introduced by a variable (stochastic) vertical specific discharge, v, which is a function of soil properties; a constant in time, but variable in space recharge, R; and a variable pore-scale dispersivity, Z. Again a simple analytical solution to the convective-dispersive equation function of the above stochastic variables is used. Their results are best illustrated in Fig. 10 which gives the probability function P(z, t, c). It represents the area of the field which at depth z and time t is at a concentration less than c. Their major conclusion is that 'the average concentration profile in the heterogeneous field, ((z, t), cannot be generally

1.0 2~o 3 9 f (

0.8 " ~ s z ~e

f PIC) 8

0.2 (b~

0 V ~ = = * ,IV ,J~ I ~J J = 0 0.2 0 4 0 .6 0 .8 I.O 0 .2 0 .4 0 .6 0 .8 1.0

D I M E N S I O N L E S S C O N C E N T R A T I O N C

Fig. 10. The probability function P(z, t; C) presented as a function of C for various fixed z and t. (a) Panoche soil (ov=1-16, )t=3cm, D,=O.O2cm2/hr. r = l , o,=O, sR=0, t=24h); (b) Bet Dagan soil (cry=0.4, &=3cm, D~,=0-02, r=0-1, 0 , = 0 , sR=0, t=5h) . The numbers labeling the curves are values of z in centimeters (Bresler

and Dagan ~:").

modelled as the solution of a convection-diffusion equation with constant coefficients. Thus the traditional approach to solute transport is not warranted even for this gross characterization of the transport phenomenon, and different tools have to be employed for large fields.

Spectral approaches

Many traditional models of field-scale unsaturated flow are usually based on the effective parameter concept. Among the many reasons for this choice, one of the most cogent is the constraint imposed by data limitations. Practical modelling studies must rely on soil properties estimated from a relatively small number of scattered soil samples, using laboratory techniques which are costly and time-consuming. In such cases effective parameters are typically estimated by aggregating or contouring the available samples. These alternatives adopt the tacit assumption that effective properties are simple weighted averages of local variables. This assumption can, in fact, be incorporated into a more quantitative 'regionalized variable' aggregation approach based on geostatistical concepts. 4'~'65 The use of sophisticated spatial inter- polation technique does not, however, change the fact that effective properties, if they exist at all, may not be simple averages. It is particularly important that the effects of larger scale structural features, such as soil layering or vertical gradation, be incorporated into the effective conductivity calculations. These can introduce large-scale anisotropy which may not be apparent from laboratory analyses of individual soil samples.

Recently, a number of researchers have attempted to develop more rigorous techniques for identifying and estimating effective unsaturated flow parameters. These methods typically adopt stochastic models of small-scale heterogeneity similar to those described in the previous subsection. Expressions for effective parameters are then derived rather than postulated. An example of this approach is provided by the theory of Yeh et al. 27 29 and Montaglou & Gelhar. 3~32 In this theory the saturated hydraulic conductivity and several unsaturated soil properties are assumed to be random functions which vary over all three dimensions. This 'series-parallel' description of spatial heterogeneity ~2~ is more realistic than the 'parallel' model adopted by Dagan & Bresler. ~22 The random soil properties are characterized by their first and second moments and the ensemble mean of the pressure head is derived from a perturbation analysis of Richard's equation. This ensemble mean is interpreted to be an estimate of the spatially averaged head over a scale which is large compared to local fluctuations. Ergodicity is assumed, as in the Dagan & Bresler 122 approach.


The perturbation method used by Yeh eta/. 2 7 - 2 9 and Mantoglou & Gelhar '~-32 yields a mean flow equation which resembles Richard's equation when the specific capacity is spatially invariant or when the system is in steady-state. In the former case, the equation for the mean pressure head t~ in the Cartesian system (X1, X2, X3) with x, vertically upward is:

c~ ,:gg((~) (19) c (¢ ) ~ - = v . [g (¢)v¢] + ax------~

where C(~) is the spatially invariant specific capacity function and /~(t~) is the effective hydraulic conductivity tensor. The general expression for this tensor, which is quite complex, depends on the sample statistics of the small-scale properties as well as the mean pressure head and its time and space derivatives. The effective hydraulic conductivity expression simplifies considerably for the special case of a rapidly wetting or drying layered soil in a region where the head gradients are relatively small and the unsaturated soil properties are uncorrelated with one another. If the mean flow direction is vertically downward, the components of the effective conductivity tensor reduce to:

Drying conditions

/~11((~ 9) = /~22((~ 9) ~--" K33(~) = Kc, exp[(&)(q~) - a2(p/&]

(20a) Wetting conditions

,(¢) = Kc. exp[(&)(q 0 R, k

1+ q~2°2°J °} a2q)/&] &A(2cOCT/ Ox, + 1)

(20b)

gzz(qg) = K33((o) -2 2 2 ] 1 + q0 a~/a? a2q)/&

= Kc, exp (&)((~) &J~(2a(p/Ox, + 1)

(20c)

where the K, are the three components of the conductivity tensor; Kc, =exp( f ) is the geometric mean of the local saturated hydraulic conductivity K~.; f, o}, and ~ are the mean, variance, and correlation

2 scale of log K,-; and & and cr~ are the mean and variance of the local soil parameter cr used in the Gardner model. Some typical plots of these functions, taken from Mantoglou & Gelhar, 32 are presented in Fig. 11.

It is important to note that the functions presented in eqns (20) are assumed to apply throughout the flow domain simulated with eqn (19). The specific values of unsaturated hydraulic conductivity obtained from these functions depend on the local mean pressure head (~, which generally varies over both time and space. The vertical and horizontal conductivities differ during wetting conditions but are the same during drying. Since the wetting anisotropy ratio depends on the mean pressure head, it also varies over time and space. The complexity of these derived conductivity- pressure functions suggests that simple weighted averages of local soil property measurements may be poor estimates of effective parameters, particularly in layered soils. Several investigators have recently tried to verify the above results, particularly state- dependent anisotropy and hysteresis, in the field and via numerical simulation (McCord & Stephens; ~2° Polmannn39).

- 03 10

-04 _ 10

lO°S - >

10 .o6

,5 '"2- 10-°7 _

-r 10°8 _

',,= 10 -o9 _

-lO 10

KG= 0.00116 cm/,'

o 2 = 0.37

~ , _ ^ A = 0.0506 crrf'

~ K22 o 2 = 0,00013 crn 2

- . . . . . . . . .

Steady-state; c3H/~ = 0.0 ~ , ~ I

'1( Wetting; bH/~t =-0.0001 cm/s

> Drying; ~H/3t = +0.0001 cm/s i i i i i

50 1 O0 150 200 250

Mean Tension, H (cm of water)

Fig. 11. Elements of hydraulic conductivity tensor as a function of tension and under wetting and drying conditions (from Mantoglou & Gelhar, 32 figure courtesy of Dennis McLaughlin).


Propagation of uncertainty: numerical-perturbation approaches

Monte Carlo approaches, like the ones previously mentioned, are extremely numerically burdensome. Analytical approaches generally require significant simplifications, including the assumption of infinite domain, unrealistic boundary conditions, and steady- state or near steady-state conditions. Treating unsaturated flow and transport jointly is practically impossible under an analytical framework. Like in the case of saturated porous media, the other alternative is to perform numerical integrations of piecewise linearized systemsY

Protopapas ~3~ and Protopapas & B r a s 133")34 have recently applied the numerical-perturbation approach to the unsteady, unsaturated, flow and transport problem in one vertical dimension. Their motivation was to study the influence of spatially varying soil properties and climate fluctuations on crop yields; hence the work also treats a phenomenological model of crop growth.

The approach involved discretizing the flow equation (15) and a transport equation,

~t (0c)+3--~-9z (qc, = -~zO [D*(O'q)~zz] (21,

in one dimension. In the above 0 is soil moisture; c is concentration; q is Darcy's flux resulting from the flow model; and D* is a dispersion term accounting for molecular and macro dispersivity as a function of moisture 0 and q.

A Taylor series expansion, up to linear terms, around a nominal solution and a discretization of space and time leads to a state-space matrix equation, where the states are the perturbed matrix potential, q~', and concentration, c', and where the input is a vector of soil parameters, ~', (perturbations) that control the functions K(q0) and c(qg):

[~]+,l =[AT'B ;. 0 ][qo~] [m[ l Ck+lJ AT'C, AFIB, JL c'k J + AF l

(22) where subscripts f and t indicate matrices resulting from discretizations and linearization of the flow and transport equations, respectively. The dimensions of the matrices are: A I, A,, B I, B,, C,(N x N), H r, Ht(N × ~N) where N is the number of discretization nodes and /~ is the number of soil parameters in the study. The entries of B I, H I, C,, and/4, depend on the nominal values of the state and the soil parameters.

Given a mean vector and covariance matrix of the soil parameters, it is possible to propagate their uncertainty to the states and obtain corresponding first- and second-moments of their matrix potential and concentrations.

Protopapas & Bras TM compared perturbation results to Monte Carlo simulations. Figure 12, left-hand panel, gives the first-order linearization profiles (mean and variance) of matrix potential (top), solute concentration (middle), and saturation (bottom) for a Bet Dagan soil with variance of saturated hydraulic conductivity of (100cm/d) 2. Results are given at 6 (crosses) and 18 (points) hours of infiltration with a constant flux and concentration at the top boundary. The right-hand panels gives the corresponding coefficient of variation. Some of their more interesting observations are:

1. The standard deviation profile of q~, s, or c are not statistically homogeneous (constant in depth), although the variance of the saturated conductivity or the capillarity index is homogeneous. The transient characteristics of the flow are reflected clearly in the variance profiles.

2. At the moisture front the standard deviation profiles of q0 and s show a maximum value, which increases with time. A minimum value of variance is observed in the transition from the steady-state region to the peak variance. The behavior of the coefficient of variation is qualitatively similar.

3. In the region behind the front a steady-state matric potential value of variance is reached. This value is predictable.

4. In all studied cases the coefficient of variation of soil moisture is less than that of the matrix potential.

5. The standard deviation profile from concentration shows a Gaussian pattern with a maximum value at the center of the front, which also increases with time.

Quoting from their conclusions: 'State-space methods possess a number of ad-

vantages compared to the spectral methods: (1) they directly deal with boundary conditions and finite domain problems; (2) they explicitly study time- varying problems; in fact, under steady-state flow conditions, these methods become attractively simple applications; (3) they do not require statistical homogeneity in the covariance structure of the input soil properties; mean and covariance of the soil properties can vary for different regions of the medium (4) the same algorithms that solve the deterministic system of equations can be used in the propagation of uncertainty; and (5) state-space formulations also lend themselves to deal with problems of parameter estimation and simulation.'

'In contrast, some nondesirable features of the state-space methods are as follows: (1) they involve matrices of high dimensionality as a direct conse- quence of the discretization of the partial differential equations on a nodal grid; and (2) like the spectral


Me.'lo Potential Iota) C.V. M , , t~ I~tentlel 0.0 tOO.O 400.0 100.0 100.0 1000.0 0.0 0.,I 0.4 0.41 0.II 1.0

Solute Concentration C.V. Solute Conoe~trlUon o.o oa o, o., o., ,.o o.o oa oa.,, o.e o., ,.o

il 0.0

i .

OS- ~ °

Degree of Sotumtion 0.2 0 .4 0 . 0

I I I

St. D e v i ~ I

t i o

0 . | 0.11 1.0 I I

C.V. I : ) ,~o, of SotumtJon 1.0 0 . 0 0 .2 0 . 4 D.II

~'1~i 6 hr °'[

Fig. 12. First-order linearization profiles of matrix potential (top), solute transport (middle), and saturation (bottom) for a Bet Dagan soil with variance of hydraulic conductivity of (100 cm/d):. Mean, standard deviations and coefficients of variation are

shown 6 and 18 hours into an infiltrating event (from Protopapas and Bras~34).


methods, they are first-order second-moment methods, since only first-order terms are kept in the linearization and only up to second-moment statistical analysis is possible without making assumptions on the joint distribution of the inputs.

CONCLUSIONS

This review has emphasized recent developments in the areas of groundwater flow and dispersion. This research field is an active one with very direct relationships to risk assessment.

Many of the models have a probabilistic basis that emphasizes both the uncertainty in the measurements and in the geologic media. This formulation leads quite naturally to predictions of outflows, arrival times, dispersion, etc., that are again probabilistic.

The more simplified theoretical models give qualitative results indicating which parameters control aspects of flow and dispersion. For example, in the saturated flow models it is apparent that the correlation scale of the permeability is a significant parameter for the associated predictions.

The complex models can be used in specific cases to make associated predictions, often in conjunction with a Monte Carlo approach.

Models with specified parameter values can be important for the study of theoretical properties and for sensitivity analysis. However, the added conditioning imposed by the observations lead to significant improvements in predictions for particular regions.

Flow and transport in unsaturated regions have added complications since the equations are, among other things, highly nonlinear. Adding other features like fractures and interaction between fractures and the porous media leads to complex but more realistic models.

Effective parameters for all cases (saturated, unsaturated, and fractured systems) can be very useful. However they have generally been derived for large domains using rather restrictive assumptions. Effective parameters that could be applied to grid blocks used in Monte Carlo simulations would be especially useful and work is progressing on several fronts for these cases.

The nonlinearities present in the equations present challenges for both modelling studies and data incorporation. As more efficient computer codes become available for solving highly heterogeneous systems, some of the Monte Carlo analyses needed will be more useful. However because of the complexity of the equations, a coupling of both a Monte Carlo and a theoretical approach would appear to show greater promise.

ACKNOWLEDGEMENTS

The section on Dual Medium Models of Unsaturated Flow in Fractured Rock and Structured Soils as well as parts of Monte Carlo Simulations, Analtyical, Simplified Solutions and Spectral Approaches were taken from a Review of Parameter Estimation and Uncertainty Analysis Techniques for Models of Unsaturated Flow which is a report of Rafael L. Bras, Consulting Hydrologist to Sandia National Labora- tories, and was largely written by Professor Dennis McLaughlin of M.1.T. The cooperation of Dr McLaughlin is appreciated.

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