Numerical solutions to statistical moment

WATER RESOURCES RESEARCH, VOL. 34, NO. 3, PAGES 529 –538, MARCH 1998

Numerical solutions to statistical moment equations of groundwater flow in nonstationary, bounded, heterogeneous media Dongxiao Zhang Geoanalysis Group, Los Alamos National Laboratory, Los Alamos, New Mexico

Abstract. In this paper we investigate the combined effect of nonstationarity in log hydraulic conductivity and the presence of boundaries on flow in heterogeneous aquifers. We derive general equations governing the statistical moments of hydraulic head for steady state flow by perturbation expansions. Due to their mathematical complexity, we solve the moment equations by the numerical technique of finite differences. The numerical approach has flexibility in handling (moderately) irregular geometry, different boundary conditions, various trends in the mean log hydraulic conductivity, spatial variabilities in the magnitude and direction of mean flow, and different covariance functions, all of which are important factors to consider for real-world applications. The effect of boundaries on the first two statistical moments involving head is strong and persistent. For example, in the case of stationary log hydraulic conductivity the head variance is always finite in a bounded domain while the head variance may be infinite in an unbounded domain. As in many other stochastic models, the statistical moment equations are derived under the assumption that the variance of log hydraulic conductivity is small. Accounting for a spatially varying, large-scale trend in the log hydraulic conductivity field reduces the variance of log hydraulic conductivity. Although this makes the conductivity field nonstationary and significantly increases the mathematical complexity in the problem, it justifies the small-variance assumption for many aquifers.

1.

Introduction

It is well known that aquifer material properties such as hydraulic conductivity K and, to a lesser extent, porosity vary spatially. Since these spatial variabilities are difficult to describe deterministically, it has become quite common to approach subsurface flow and solute transport problems stochastically. During the last two decades, many stochastic theories have been developed to relate the statistical moments of hydraulic head, velocity, and other flow quantities to the statistical moments of hydraulic conductivity and the mean flow characteristics [e.g., Bakr et al., 1978; Dagan, 1979, 1982; Mizell et al., 1982; Gelhar and Axness, 1983; Rubin and Dagan, 1988, 1989; Rubin, 1990; Rubin and Dagan, 1992; Zhang and Neuman, 1992, 1996; Neuman and Orr, 1993]. However, applications of stochastic theories to real-world flow problems are rare. This may be largely due to the fact that the majority of previous works involve the assumptions that the log hydraulic conductivity field Y 5 ln K is second-order stationary and the flow domain is unbounded. However, in reality the hydraulic conductivity field may be nonstationary and flow domains are obviously bounded. Second-order (wide-sense) stationarity requires the mean log hydraulic conductivity ^Y& to be constant spatially and the two-point covariance C Y (x, x) to be dependent only on the relative distance between x and x. In many geostatistical applications, second-order stationarity is adopted as a working hypothesis, which works well when there is no apparent global This paper is not subject to U.S. Copyright. Published in 1998 by the American Geophysical Union. Paper number 97WR03607.

drift (trend). If a trend in ^Y& is observed but is intentionally ignored to achieve the (forced) stationarity as required in most previous stochastic theories, the resultant spatial variability in Y will be (artificially) large compared to accounting for the trend explicitly. Attempts have been made to address the issue of trending in the hydraulic conductivity field. Neuman and Jacobson [1984] and Rajaram and McLaughlin [1990] represented hydrologic variability as a stationary random field superimposed on a large-scale, deterministic trend. Rehfeldt et al. [1992] found that the spatial variability in the log hydraulic conductivity at the Columbus Air Force Base in Mississippi could be better explained when a spatial trend in the mean log conductivity is included explicitly. Loaiciga et al. [1993] cited an example of trending in hydraulic conductivity observed in the Eye-Dashwa Lakes Pluton near Atitokan, Ontario. A few stochastic theories have been developed to investigate the effects of linear trends in log hydraulic conductivity on flow in unbounded domains [Gelhar, 1993; Loaiciga et al., 1993; Rubin and Seong, 1994; Li and McLaughlin, 1995; Indelman and Rubin, 1995]. On the other hand, several researchers have investigated the effects of boundaries on flow in stationary media [e.g., Naff and Vecchia, 1986; Rubin and Dagan, 1988, 1989; Osnes, 1995]. To our knowledge, no attempt has been made to investigate the combined effect of boundaries and nonstationarity in log hydraulic conductivity on the statistical behaviors of flow, which is the primary purpose of this study. We do so through solving equations governing the statistical moments of hydraulic head for steady state flow in bounded aquifers with nonstationary log hydraulic conductivities. These equations are derived by first-order (in the variance of log hydraulic conductivity) perturbation expansions. Due to their mathemat-

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530

ZHANG: TECHNICAL NOTE

ical complexity we solve the moment equations by the numerical technique of finite differences. The numerical approach has flexibility in handling (moderately) irregular geometry, different boundary conditions, various trends in the log hydraulic conductivity, spatial variabilities in the magnitude and direction of mean flow, and different covariance functions.

2.

2C h~x, x! ^Y~x!& C h~x, x! C Yh~x, x! 1 5 J i~x! xi xi xi x 2i

n i~x!

We consider steady state flow in saturated media satisfying the following continuity equation and Darcy’s law (1)

h~x! q i~x! 5 2K~x! xi

(2)

subject to boundary conditions h~x! 5 H~x! q~x! z n~x! 5 Q~x!

x [ GD x [ GN

(3) (4)

where q is specific discharge (flux), h(x) is hydraulic head, K(x) is hydraulic conductivity (assumed to be isotropic locally) and is treated as a random space function, i 5 1, z z z , k (where k is the number of space dimensions), H(x) is prescribed head on Dirichlet boundary segments G D , Q is prescribed flux across Neumann boundary segments G N , and n(x) 5 (n 1 , z z z , n k ) T is a unit vector outward normal to the boundary. In this study, H(x) is assumed to be deterministic (i.e., known with certainty), and Q is assumed to be zero such that G N is a no-flow (impervious) boundary. More general stochastic boundary conditions can be accommodated in the same manner as by Neuman and Orr [1993]. Substituting (2) into (1) and utilizing Y(x) 5 ln K(x) yields 2h~x! Y~x! h~x! 1 50 xi xi x 2i

(5)

Summation for repeated indices is implied. In this study, Y is treated as a random variable and is thus decomposed as Y(x) 5 ^Y(x)& 1 Y9(x) where ^Y(x)& is the mean log hydraulic conductivity and Y9(x) is the zero-mean fluctuation. In turn, h is also random. Since the randomness (spatial variability) of h depends on that of Y, one may expand h(x) as h (0) 1 h (1) 1 h (2) 1 z z z where h (n) 5 O( s nY ) and s Y is the standard deviation of Y [e.g., Dagan 1989]. By substituting these into (5) and collecting terms at separate order, and after some mathematical manipulations, one can obtain the following equations governing the first two moments involving head,

h ~0!~x! 5 H~x!

x [ GD

h ~0!~x! 50 xi

x [ GN

n i~x!

(6)

2^h ~2!~x!& ^Y~x!& ^h ~2!~x!& 1 52 @C Yh~x, x!# x5x xi xi xi xi x 2i

n i~x!

^h ~2!~x!& 5 0

x [ GD

^h ~2!~x!& 50 xi

x [ GN

n i~x!

(7)

C h~x, x! 5 0, xi

x [ GN

(8)

C Yh~x, x! 5 0

x [ GD

C Yh~x, x! 5 0, xi

x [ GN

(9)

In the above, h (0) is the zeroth-order mean head, ^h (2) & is the second-order mean head correction term, J i 5 2h (0) / x i is the negative of the (zeroth-order) mean hydraulic head gradient, C Yh 5 ^Y9(x)h (1) (x)& is the cross covariance between log hydraulic conductivity and head, and C h 5 ^h (1) (x)h (1) (x)& is the head covariance. Since ^h (1) & (the mean of first-order head correction term) is zero, ^h& 5 h (0) to zeroth or first order in s Y , ^h& 5 h (0) 1 ^h (2) & to second order. In the above, the covariances are of second order in s Y (or first order in s 2Y ). For uniform mean flow in unbounded domains, the secondorder correction term is identically zero when without any trend or with a linear trend either parallel or normal to the mean head gradient [Rubin and Seong, 1994; Li and McLaughlin, 1995; Indelman and Rubin, 1995]. However, as will be shown later, it is generally nonzero in bounded domains. In most stochastic analyses the log hydraulic conductivity Y is assumed to be second-order stationary such that its mean ^Y& is constant and the two-point covariance C Y (x, x) 5 ^Y9(x)Y9(x)& depends only on the relative distance r 5 x 2 x of the two points x and x. Hence the second terms in (6)–(9) disappear. In addition, the mean head gradient is assumed to be constant and unidirectional such that J i 5 d i1 J. In the presence of a linear trend in the mean log hydraulic conductivity, the mean head gradient is no longer constant spatially even if the trend is assumed to be aligned with the mean hydraulic gradient [Rubin and Seong, 1994; Li and McLaughlin, 1995; Indelman and Rubin, 1995]. In this study we consider a more general case of flow in a bounded domain of nonstationary log hydraulic conductivity. In turn, the flow field becomes nonstationary. Furthermore, in such a flow field both the magnitude and direction of the mean head gradient may vary spatially.

3.

2h ~0!~x! ^Y~x!& h ~0!~x! 1 50 xi xi x 2i

x [ GD

2C Yh~x, x! ^Y~x!& C Yh~x, x! C Y~x, x! 1 5 J i~x! xi xi xi x 2i

Statistical Moment Equations

¹ z q~x! 5 0

C h~x, x! 5 0

Numerical Solution

Due to its mathematical complexity it is essentially impossible to obtain analytical solutions for the problem. In this study we solve the statistical moment equations by the numerical technique of finite differences in one and two dimensions. The solution is facilitated by recognizing that h (0) , ^h (2) &, C Yh , and C h are governed by the same type of equations but with different forcing terms. The spatial derivatives are discretized via the central difference approximations, P~x, x! < @P~ x 1 1 Dx 1, x 2; x 1, x 2! x1 2 P~ x 1 2 Dx 1, x 2; x 1, x 2!#~2Dx 1! 21

(10)


2P~x, x! < @P~ x 1 1 Dx 1, x 2; x 1, x 2! 2 2P~ x 1, x 2; x 1, x 2! x 21 2 21 1

1 P~ x 1 2 Dx 1, x 2; x 1, x 2!#~Dx !

(11)

where P stands for h (0) , ^h (2) &, C Yh , or C h . With this, the left-hand side (LHS) of the equations (6)–(9) becomes LHS 5 1 1 1 1

F F F F

F

1 DY1~x! 1 2Dx 1 Dx 21

2

2 2 2 P~ x 1, x 2; x 1, x 2! Dx 21 Dx 22

G

G

P~ x 1 1 Dx 1, x 2; x 1, x 2!

G G G

1 DY1~x! 2 P~ x 1 2 Dx 1, x 2; x 1, x 2! 2Dx 1 Dx 21 1 DY2~x! 1 P~ x 1, x 2 1 Dx 2; x 1, x 2! 2Dx 2 Dx 22 1 DY2~x! P~ x 1, x 2 2 Dx 2; x 1, x 2! 22 2Dx 2 Dx 2

(12)

where DY1(x) 5 ^Y(x)&/ x 1 and DY2(x) 5 ^Y(x)&/ x 2 . The coefficients in LHS are functions of known quantities. For h (0) and ^h (2) &, the dependency on x vanishes. For a onedimensional domain, the terms associated with x 2 disappear in the above. For the boundaries, P~x, x! 5 P 0~x!

x [ GD

n 1~x! @P~ x 1 1 Dx 1, x 2; x 1, x 2! 2 P~ x 1 2 Dx 1, x 2; x 1, x 2!# Dx 1 1

(13)

n 2~x! @P~ x 1, x 2 1 Dx 2; x 1, x 2! Dx 2

2 P~ x 1, x 2 2 Dx 2; x 1, x 2!# 5 0

Illustrative Examples and Results

For ease of illustration, we consider rectangular domains of size L 1 by L 2 . The left and right sides are specified as constant head boundaries: h( x 1 5 0) 5 H 0 and h( x 1 5 L 1 ) 5 H L . The lower ( x 2 5 0) and upper ( x 2 5 L 2 ) sides are no-flow boundaries. For the one-dimensional cases discussed later, L 2 5 0. Up to now, the nonstationarity in the log hydraulic conductivity field manifests itself in two ways: The mean ^Y& may vary spatially, and the two-point covariance C Y (x, x) may depend on the actual locations of x and x rather than only on their separation distance. The latter may be a result of conditioning. In the following examples we consider the case that the log hydraulic conductivity consists of a deterministic trend and a stationary fluctuation. That is to say, the mean varies spatially and the two-point covariance depends only on the relative distance. Although the numerical algorithm admits any covariance function, we consider only the exponential and Gaussian covariances in this study, C Y~x 2 x! 5 s 2Y exp

HF F

C Y~x 2 x! 5 s 2Y exp

2

2

~ x i 2 x i! 2 l 2i

GJ G 1/ 2

p ~ x i 2 x i! 2 4 l 2i

(14)

where A is the coefficient matrix, P is the solution vector for one of the four moments h (0) , ^h (2) &, C Yh , and C h , and R is a vector containing information about the right-hand side of each equation and the boundary conditions. While the vector R is different for these four moments, the matrix A is exactly the same for all of them. In this study, we solve the linear algebraic equations by lower-upper (LU) decomposition with forward and back substitution [Press et al., 1986]. It is obvious that the matrix A only has to be decomposed once for each problem setup, though the substitution has to be performed as many times as the number of different right-hand side vectors. For a specific grid the equation governing h (0) or ^h (2) & only has to be solved once, and that governing C h (x, x) needs to be solved for each selected reference point x. However, since the equation for C Yh (x, x) is written with respect to x, it has to be solved as many times as the number of nodes on the grid for x in order to obtain the derivatives C Yh (x, x)/ x i (required for solving C h and ^h (2) &). Although the grid for x can be different from that for x, we use the same for both x and x.

(15) (16)

where s 2Y is the variance and l i is the correlation scale of Y along the x i axis. For l1 5 l2 5 l, the covariance functions are isotropic. Unless stated otherwise, in the examples the covariance is assumed to take the exponential function of (15). The mean ^Y(x)& can take any functional form as well. To illustrate the effect of a deterministic trend in ^Y&, we consider two special cases. In one case, ^Y& varies linearly (if a 3 5 a 4 5 a 5 5 0) or quadratically according to ^Y~x!& 5 a 0 1 a 1x 1 1 a 2x 2 1 a 3x 1x 2 1 a 4x 21 1 a 5x 22

x [ GN

For P 5 h (0) , P 0 5 H(x); otherwise, P 0 [ 0. The right-hand side of (6) for h (0) is zero; hence it can be solved readily. The equation governing C Yh (equation (9)) should be solved before doing so for ^h (2) & and C h because the latter depend on the solution of C Yh . With these discretizations, the moment equations become AP 5 R

4.

531

(17)

where a i are known coefficients and, in particular, a 0 5 ^Y(0)&. Since ^Y& can vary in both the x 1 and x 2 directions, its direction may change spatially and may not be aligned with the mean hydraulic gradient. In the second case, ^Y(x)& varies periodically along x 1 according to ^Y~x!& 5 a 0 1 a 1 sin ~a 2x 1!

(18)

Numerical solutions were tested against the analytical solution of head variance in a bounded domain by Osnes [1995]. In the latter the mean ^Y& is constant, and the covariance C Y must take the separated exponential form which is slightly different from the exponential covariance of (15). By specifying the separation exponential covariance in our numerical code, an excellent agreement was obtained with Osnes [1995] which, in turn, compared well with Monte Carlo simulations. Test runs indicate that using the discretization of Dx i 5 0.5 l gives approximately the same results as using Dx i 5 0.25 l . Based on the discussion in the last section, we see that solving for the cross covariance CYh and its derivatives consumes a significant portion of the total CPU time. Furthermore, the CPU time increases rapidly with the size of the domain. For example, it takes about 54 s, 6.9 min and 55.8 min on a Sun workstation (Ultra 1 Model 140) for a rectangular domain of 5l by 5l, 10l by 5l, and 20l by 5l (all with Dx i 5 0.25 l ), respectively. Below, we illustrate the results for the first two moments involving head for flow in one- and two-dimensional bounded domains in the presence of various trends in the mean log hydraulic conductivity. In the examples, s 2Y 5 1 and l 5 1.

532


Figure 1. (a) Mean head ^h& and (b) horizontal negative mean head gradient J 1 along the horizontal centerline ( x 2 5 L 2 / 2), (c) normalized head variance s 2h along the horizontal centerline and (d) normalized head variance s 2h along the transverse ( x 1 5 L 1 / 2) centerline in a domain of size 10l by 10l with linear trends parallel or normal to the mean flow.

4.1.

Linear Trends

Figure 1 is for the case of linear trends parallel or normal to the mean flow in a square domain of size 10l by 10l. Constant heads 100 and 90 are specified for the respective left and right boundaries, and the two lateral sides are no-flow boundaries. The covariance function of Y takes the exponential form of (15). Figures 1a and 1b show the zeroth-order mean head ^h& and the horizontal negative mean head gradient J 1 along the horizontal centerline ( x 2 5 L 2 / 2). In the legend, a 1 and a 2 are defined in (17) and are the slopes of a linear trend along the x 1 and x 2 directions. The coordinates x i in the figures are normalized by the length of the domain L i . Due to the specific geometry and boundary conditions, the (zeroth-order) mean head is a straight line along the horizontal direction and the horizontal negative mean head gradient is constant in the absence of any trend (a i 5 0); the mean head is constant along the transverse direction and the transverse mean gradient is zero in the whole domain. In the case of a 1 5 20.1 and a 2 5 0, ^Y& decreases linearly with the distance from x 1 5 0 in the horizontal direction, ^h& is generally larger than that without a trend (Figure 1a), and J 1 increases exponentially with the dis-

tance from the left boundary (Figure 1b). Physically speaking, the (pressure) head builds up in the left portion of the domain because the right portion with low hydraulic conductivity acts like a barrier to flow and the (mean) flow must occur from the left to the right with the specific geometry and boundaries. Mathematically speaking, (6) can be rewritten as J 1 / x 1 1 a 1 J 1 5 0 in the case of a linear trend parallel to the mean flow, which has the solution J 1 5 J 1 (0) exp (2a 1 x 1 ). In the case of a 1 5 0 and a 2 5 0.1, ^Y& increases with x 2 in the direction normal to the mean flow. The presence of a linear trend normal to the mean flow does not affect the mean head and the mean head gradients under the specific boundary conditions. Figures 1c and 1d show the head variance along the horizontal ( x 2 5 L 2 / 2) and transverse ( x 1 5 L 1 / 2) centerlines. The head variance s 2h (x) 5 C h ( x 1 5 x 1 , x 2 5 x 2 ) is normalized with respect to l 2 s 2Y . In the absence of any trend in the mean log hydraulic conductivity ^Y&, along the horizontal centerline the head variance is zero at the left and right (constant head) boundaries due to the boundary constraint, increases toward the center of the domain, and reaches its peak there because the effect of constant head boundaries is mini-


533

Figure 2. Normalized head covariance C h ( x 1 , x 2 ; y 1 , y 2 ) in a domain of size 10l by 10l with a linear trend (a 1 5 0.1) for (a and b) exponential and (c and d) Gaussian covariance functions for log hydraulic conductivity. mum at the domain center. Along the transverse centerline the head variance is minimum at the center of the domain and increases toward the lateral (no-flow) boundaries (for this particular geometry). The effects of domain size and ratio between the horizontal and transverse length of a rectangular domain on the head variance have recently been investigated by Osnes [1995] for the case without any trend. For the case of a linear trend with a negative slope in the x 1 direction (a 1 5 20.1 and a 2 5 0), the head variance profile along the horizontal centerline is skewed toward the right portion of the domain. That is to say, the head variance is higher in the area where the negative mean head gradient is large. That the head variance is larger in the right portion of the domain is also consistent with the fact that the coefficient of variation of Y is larger there because s Y is the same and ^Y& is smaller. In this case, the head variances remain symmetric about the domain center along the transverse direction. In the presence of a linear trend normal to the mean flow (a 1 5 0 and a 2 5 0.1), the head variance along the horizontal direction is the same as that in the case free of any trend; the variance along the transverse centerline is, however, skewed toward the upper portion (0.5L 2 , x 2 , L 2 ) of the domain where the coefficient of variation of Y is lower. This is different from our

earlier finding for the case of the linear trend parallel to the mean flow that the head variance is large in the area where the coefficient of variation of Y is large. This contradiction may be explained by realizing that the flow domain can be viewed as a series of layers normal or parallel to the mean flow in the case of a linear trend parallel or normal to it, respectively. In the case of flow normal to bedding, the flow is mainly controlled by the blocks with small (log) hydraulic conductivities; but in the case of flow parallel to bedding, the flow converges into the high-conductivity layers. Figure 2 shows the normalized head covariance C h (x, y) along the horizontal and transverse centerlines in a square domain of size 10l by 10l with a linear trend in ^Y& parallel to the mean flow (a 1 5 0.1) for exponential (Figures 2a and 2b) and Gaussian (Figures 2c and 2d) covariance functions. Here y is the reference point like x in (8). The head covariance is normalized with respect to l 2 s 2Y . When y 5 x, the head covariance (solid curves) becomes the head variance illustrated earlier. In the presence of a linear trend with a positive slope, along the horizontal centerline (Figures 2a and 2c) the head variances are skewed toward the left boundary, exactly opposite to the case of a linear trend with a negative slope as shown in Figure 1c, and the head covariance is not symmetric even

534


Figure 3. (a) Horizontal negative mean head gradient J 1 along the horizontal centerline, (b) transverse gradient J 2 along the transverse centerline, (c) normalized head variance s 2h along the horizontal centerline, and (d) normalized head variance s 2h along the transverse centerline in a domain of size 10l by 10l with different quadratic trends. when y 1 5 L 1 / 2. When the reference point y 1 5 0.25L 1 , the head covariance is zero at the left boundary, increases with distance from there, reaches its peak shortly after x 1 5 y 1 , and then decreases toward the right boundary; for y 1 5 0.75L 1 the head covariance reaches its peak shortly before x 1 5 y 1 . The reason for the peak head covariance’s not occurring at x 1 5 y 1 may stem from the fact that the head covariance at two points is an increasing function of the variances at these two points and a decreasing function of the distance between them. Along the transverse centerline (Figures 2b and 2d), the head covariances are not zero at the lateral boundaries no matter where the reference point y 2 is. When y 2 is at the center of the transverse line, the head covariance is symmetric and has its peak exactly at x 2 5 y 2 5 L 2 / 2; otherwise, the head covariance profile is not symmetric, and its peak appears at a point close to but not exactly at x 2 5 y 2 . Comparing Figures 2a and 2b with Figures 2c and 2d reveals that the general effects of boundaries and trending in log hydraulic conductivity are qualitatively similar for the exponential and Gaussian covariance functions. It is seen from Figures 1c and 1d that the peak head variances are higher in the presence of trends in the mean log

hydraulic conductivity. This is so because the head variances are normalized by l 2 s 2Y . As discussed in Section 1, the variance in log hydraulic conductivity may be reduced by accounting for the large-scale trend in it. Therefore the absolute head variance may be much smaller in this case than when the trend is ignored. It may be of interest to mention that the head variance is infinite in a two-dimensional unbounded domain with a log hydraulic conductivity field characterized by a constant mean and an exponential covariance function [e.g., Dagan, 1989], while the head variance is always finite and varies with distance from the boundaries in a bounded domain. That is to say, for a domain of finite size, the effect of the boundaries is always present in the head variance regardless of how many correlation scales may lie between a given point and the boundaries. However, the boundary impacts on the velocity field are limited to only within a few correlation scales, as found in previous studies. 4.2.

Quadratic Trends

Figure 3 shows the horizontal gradient J 1 along the horizontal centerline, transverse gradient J 2 along the transverse centerline, and the normalized head variance along the horizontal


535

Figure 4. (a) Mean head ^h& and (b) horizontal negative mean head gradient J 1 along the horizontal centerline, (c) normalized head variance s 2h along the horizontal centerline, and (d) normalized head variance s 2h along the transverse centerline in a domain of size 10l by 10l with different periodic trends. and transverse centerlines in a square domain of size 10l by 10l for different quadratic trends defined by (17). The boundary conditions are the same as in the previous case of Figure 1, and J 1 is 1 in the absence of any trend. For the quadratic trends the slopes depend on the coordinates x 1 and/or x 2 . As a result, the horizontal (negative) mean head gradients J 1 deviate significantly from 1 (Figure 3a) compared to those with the linear trends in Figure 2. For the case of a 3 5 0.05 and a i 5 0 (i 5 1, 2, 4, and 5), the mean ^Y& increases with both horizontal distance from x 1 5 0 and transverse distance from x 2 5 0. Due to this nonuniformity in the mean log hydraulic conductivity field, a transverse component of the (negative) mean head gradient is present and varies spatially (Figure 3b) even for the square domain with constant heads at two opposite sides and no-flow boundaries for the two other sides. In this case the horizontal (negative) mean head gradient decreases from the left to the right in the horizontal direction, in contrast to how the mean log hydraulic conductivity ^Y& varies; as a result the head variance is skewed (higher) toward the left portion of the domain (Figure 3c). The transverse gradient J 2 is positive along the transverse centerline (Figure 3b) and is skewed toward the lower portion of the domain (0 , x 2 /L 2 , 0.5), and correspondingly, the head variance is skewed toward

the lower portion of the domain (Figure 3d). When a quadratic trend has only one component in the horizontal direction (a 4 5 20.03 and a i 5 0 where i 5 1, 2, 3, and 5), the transverse component of the mean head gradient is zero (Figure 3b) and the head variance is symmetric in the transverse direction (Figure 3d). The horizontal gradient J 1 increases from the left to the right boundary where the coefficient of variation of Y is highest, and the head variance increases from zero at the left boundary, reaches its peak near the right boundary, and then decreases to zero at the right boundary, reflecting the combined effect of boundaries and trending in log hydraulic conductivity. It is seen from Figures 3a and 3b that both the magnitude and direction of the (negative) mean head gradient may vary spatially in the presence of trends even for this regular geometry and under these simple boundary conditions. Under more realistic situations with irregular geometry and complex boundary conditions, the mean head gradient may vary spatially in both magnitude and direction even without any trend in the mean log hydraulic conductivity. Hence the assumption of uniform and/or unidirectional mean head gradient commonly made in most previous stochastic theories may be inadequate for real-world applications. However, it is essentially

536


Figure 5. Second-order mean head correction term ^h (2) & versus distance x 1 /L 1 in one-dimensional domains of various sizes in the presence of (a) linear, (b) quadratic, and (c) periodic trends, and (d) the maximum value of ^h (2) & as a function of domain size in the absence of any trend. impossible to analytically solve the stochastic problem with spatially varying mean head gradients in bounded domains due to its mathematical complexity. Therefore we believe that numerical techniques will play a significant role in applying stochastic approaches to real-world problems. 4.3.

Periodic Trends

Figure 4 shows the mean head and horizontal negative mean head gradient along the horizontal centerline, and the normalized head variance along the horizontal and transverse centerlines in a square domain of size 10l by 10l for different periodic trends in ^Y& along the x 1 direction. The boundary conditions are the same as in the previous cases. The periodic trend is a sinusoid given by (18) with a 1 and a 2 characterizing its magnitude and frequency. Since the trends are only in the horizontal direction, the transverse mean gradient J 2 is zero. When a 1 5 1.0 and a 2 5 0.5, the mean head decreases slowly in the left portion but quickly in the right portion of the domain (Figure 4a), resulting in lower negative mean head gradient J 1 in the left and higher J 1 in the right portion of the domain (Figure 4b). When a 2 is increased from 0.5 to 1.0 and a 1 remains the same, the mean head is flat near the left and right boundaries but it declines rapidly near the center of the

domain, generating lower J 1 near the boundaries and higher J 1 at the domain center. When a 1 is further increased from 1.0 to 2.0 while a 2 remains at 1.0, the mean head becomes even flatter near the boundaries and steeper at the domain center, and the gradient J 1 is thus much lower near the boundaries and higher at the domain center. A comparison between Figure 4b and Figure 4c reveals that the head variance in the horizontal centerline is large wherever J 1 is large, except at and near the constant head boundaries. This is consistent with our earlier observations for the cases of linear and quadratic trends. The head variances along the transverse centerlines are symmetric about the domain center (Figure 4d). 4.4.

Second-Order Mean Head Correction Term

In the previous figures, the mean head is of zeroth or first order in s Y . The second-order mean head ^h& 5 h (0) 1 ^h (2) &. In this subsection we look at the magnitude of the correction term ^h (2) & and the effects of trends in ^Y& and domain sizes on it. Figure 5 shows the second-order mean head correction term ^h (2) & versus distance x 1 /L 1 in one-dimensional domains of various sizes in the presence of different trends. Here, J 1 5 1, l 5 1 and s 2Y 5 1. For the case free of any trend (a 1 5 0 in Figure 5a), ^h (2) &


537

Figure 6. Second-order mean head correction term ^h (2) & along horizontal and transverse centerlines in a domain of size 5l by 5l with (a and b) linear and (c and d) quadratic trends.

is antisymmetric about the domain center and its magnitude is small. In the presence of linear trends (Figure 5a), ^h (2) & is no longer antisymmetric and its peak value increases with the slope of the trend and the domain size. It seems that the important quantity is a 1 L 1 , the product of the slope and the domain size. For a 1 L 1 5 50, the correction term is zero except at near the left boundary. By comparing the curves of a 1 L 1 5 1, 10, and 50, we see that the portion of the domain where the correction term is zero increases with a 1 L 1 . This is consistent with previously reported results that the secondorder mean head correction term is zero in unbounded domains with linear trends [Rubin and Seong, 1994; Li and McLaughlin, 1995]. Figure 5b shows the cases of quadratic trends. As in the case of linear trends, the peak values for the correction term increase with the coefficient a 4 and the domain size L 1 , and the portion of the domain where the correction term is zero increases with the product a 4 L 1 . In the case of periodic trends (Figure 5c), the magnitude and frequency of the correction term increase with the magnitude and frequency of the periodic trend when the size of a domain is fixed. The magnitude of the correction term decreases as the domain size increases. For example, it is practically zero when L 1 5 50 l . Figure 5d shows the maximum value of the second-

order correction term as a function of ratio between the domain size L 1 and the correlation scale l for both the exponential and Gaussian covariance functions of Y in the absence of any trend in Y. It is seen that the second-order correction term decreases almost exponentially with the domain size L 1 / l . Figure 6 shows the second-order mean head correction term ^h (2) & along the horizontal and transverse centerlines in a square domain of size 5l by 5l in the presence of linear and quadratic trends. Constant heads 100 and 95 are specified for the respective left and right boundaries, and the two lateral boundaries are impervious. For the cases with linear trends which vary in both the horizontal and transverse directions (Figures 6a and 6b), the peak values of the correction term increase with the slopes a 1 and a 2 . However, as in the onedimensional case of linear trends (Figure 5a), the portion of the domain where the correction term is zero increases with the product of the slope and the domain size. This is also true for the cases of quadratic trends (Figures 6c and 6d).

5.

Summary and Conclusions

We investigated the combined effect of nonstationarity in log hydraulic conductivity and the presence of boundaries on

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flow in heterogeneous aquifers. We did so through solving equations governing the statistical moments of hydraulic head for steady state flow, which were derived by perturbation expansions. Due to their mathematical complexity, we solve the moment equations by the numerical technique of finite differences. The numerical approach has flexibility in handling (moderately) irregular geometry, different boundary conditions, various trends in the mean log hydraulic conductivity, spatial variabilities in the magnitude and direction of mean flow, and different covariance functions, all of which are important factors to consider for real-world applications. As in many other stochastic models, the statistical moment equations are derived under the assumption that the variance of log hydraulic conductivity is small. Accounting for a spatially varying, large-scale trend in the log hydraulic conductivity field reduces the variance of log hydraulic conductivity. This justifies the small-variance assumption for many aquifers, although it makes the conductivity field nonstationary and significantly increases the mathematical complexity in the problem. We discussed the effects of nonstationarity in the log hydraulic conductivity field and boundary conditions based on some oneand two-dimensional examples. In the two-dimensional examples, the domain is rectangular with two opposite constant head and two opposite no-flow boundaries. The log hydraulic conductivity consists of a known, deterministic trend and a stationary fluctuation. The trend may be linear, quadratic, or periodic in space, and the covariance is either exponential or Gaussian. This paper leads to the following major conclusions: 1. The effects of boundaries on the first two statistical moments involving head are strong and persistent. For example, in the case of stationary log hydraulic conductivity the head variance is always finite in a bounded domain while the head variance may be infinite in an unbounded domain. However, the boundary impacts on the velocity field may be limited to within a few correlation scales. 2. The presence of a trend in the mean log hydraulic conductivity has a strongly impact on the flow behavior in heterogeneous aquifers. It may render the mean head gradient spatially variable in both magnitude and direction under situations where the mean gradient is otherwise constant. Together with the influence of boundaries, it makes the covariances involving head highly nonstationary and nonsymmetric. 3. The second-order mean head correction term is generally nonzero in a bounded domain, although its magnitude is small compared to the zeroth-order (first-order) mean head. In the absence of any trend or in the presence of a periodic trend in the mean log hydraulic conductivity, the magnitude of the correction term decreases with the increase of domain size. In the presence of a linear or quadratic trend, although the peak value of the correction term increases with the slope of the trend and the domain size, the portion of the domain where the correction term is practically zero increases with the product of the slope and the domain size. Acknowledgments. This work was funded by LDRD project 97528 from Los Alamos National Laboratory. The author wishes to thank the three anonymous reviewers for their helpful comments which resulted in a much improved manuscript.

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(Received October 23, 1997; revised December 9, 1997; accepted December 10, 1997.)