Analysis of variance method for the equivalent conductivity of rectangular blocks

Transcription

1 WATER RESOURCES RESEARCH, VOL. 37, NO. 2, PAGES , DECEMBER 200 Analysis of variance method for the equivalent conductivity of rectangular blocks Daniele Veneziano and Ali Tabaei Department of Civil and Environmental Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts, USA Abstract. We use an analysis of variance (ANOVA) decomposition of the log hydraulic conductivity F(x) = In K(x) as the sum of an average value F plus main effects and interactions of various orders to evaluate the equivalent conductivity of rectangular blocks in D-dimensional space. Some of the ANOVA components are dealt with exactly. The effect of the other components is approximated through calibration to the effective conductivity Kef f under ergodic conditions. Our analysis applies to both isotropic and anisotropic lognormal K fields. We evaluate the theoretical findings through analytical and numerical comparisons with exact results and previously proposed approximations. We find that the ANOVA formula is nearly unbiased, although its performance is generally inferior to numerical renormalization. Advantages of the ANOVA formula are that it is less computationally demanding and explicitly shows the dependence of the block conductivity K b on the various components of fluctuation of F inside the block. Such dependence is more complex than previously assumed. Considering the ANOVA formula to be unbiased, we derive the bias of the geometric mean of the Cardwell-Parsons bounds for three-dimensional rectangular blocks.. Introduction The problem of determining the equivalent conductivity of finite blocks is fundamental to the efficient numerical analysis of flow through heterogeneous porous media. A number of methods have been proposed to upscale the point conductivity field K(x) to the block level. These methods range from heu- ristic combinations of theoretical bounds and numerical renor- Paper number 2000WR /0/2000WR00005 $09.00 simulation on the plane to compare our approximation to the exact block conductivity and to other existing approximations in a variety of cases. In most of the cases considered the ANOVA-based formula is nearly unbiased. An advantage of our approximation over existing formulas is that it explicitly shows how Kb depends on the various components of fluctuation of F inside the block. Such dependence is more complex than previously thought, especially in spaces of dimension D > 2. Assuming that the ANOVA-based formula is unbiased, we obtain the bias of the geometric average of the Cardwell-Parsons bounds as an estimator of the conductivity of three-dimensional (3-D) blocks. In numerical flow analysis, blocks typically have a square geometry. The extension to rectangular blocks is of interest for several reasons. One is that rectangular blocks are more appropriate in the case of heavily anisotropic media. Another is that if interest is in the equivalent conductivity of an aquifer with approximately rectangular shape, then the formula for blocks can be applied directly at the aquifer level without the need for numerical flow analysis. We start in section 2 with a brief review of previous work on malization techniques when the point conductivity K(x) is considered known (e.g., after numerical simulation from a stochastic model) to approximate probabilistic analyses of the flow when K(x) is modeled as a random field; see Renard and de Marsily [997] for a recent review. Here we develop an analytical approach (as opposed to heuristic arguments or numerical schemes) to approximate the directional equivalent conductivity Kb of rectangular blocks in D-dimensional space. Explicit results are obtained when the point conductivity K(x) is a homogeneous isotropic or anisotropic lognormal random field. Our approach is based on an analysis of variance (ANOVA) decomposition of the log conductivity F(x) = In K(x) inside the block. Specifically, F(x) is written as the spatial block average P plus main effects and the equivalent conductivity problem, followed in section 3 by interactions of various orders in the spatial coordinates the derivation of our main result for the case of isotropic Xl,..., xr. The exact value of K b under P plus all the main lognormal point conductivity. Section 4 extends the ANOVA effects and some of the interactions is derived analytically. This analysis to the anisotropic case. Section 5 makes analytical part of the analysis does not depend on the stochastic model of comparisons of the ANOVA results with exact block conduc- K, which, for example, needs not be lognormal or statistically tivities and other approximations. Numerical comparisons are homogeneous. The effect of the remaining interactions is obmade in section 6, using Monte Carlo simulation in the plane tained in approximation through calibration to the value Kef f with both isotropic and anisotropic point conductivity models and square or rectangular blocks. We conclude with a summary under ergodic conditions. This effect depends on the model of of the main results and an outlook on possible future work. K and should be modified if K(x) is not lognormal. We use theoretical analysis in D-dimensional space and Monte Carlo Copyright 200 by the American Geophysical Union. 2. Previous Results for Isotropic K 299 The vast literature on upscaling conductivity in parallel flow has been recently reviewed by Sanchez-Vila et al. [995], Wen

2 2920 VENEZIANO AND TABAEI: ANOVA EFFECTIVE CONDUCTIVITY and Gomez-Hernandez [996], and Renard and de Marsily [997]. Here we mention only a few results that are of significance for our analysis or will be used in subsequent sections to compare with our proposed method. Consider a D-dimensional rectangular region 2 with side lengths L,..., L z>, and assume that the log hydraulic conductivity inside 2 is a given deterministic function F(x) = In K(x). Let Q be the total flow when 2 is subjected to a largescale hydraulic gradient of amplitude J in the x direction and no flow is allowed through the rest of the boundary. The equivalent conductivity of the region in the direction of flow, Kb, is the spatially uniform hydraulic conductivity that produces the same flow Q; hence g b = Q(J IIiD 2 Li) -. In the case when K(x) is a random field, both the flow Q and the block conductivity Kb are, in general, random variables. However, for unbounded regions (L i ---> c ) and ergodic K(x), Q does not depend on the realization, and Kb becomes a deterministic quantity Kef f. We call this the ergodic case. First we mention a classic result for Kef f under isotropic K, which is exact up to second-order terms in the variance of F and is generally accurate for lognormal K fields. Through direct adaptation of the ergodic formula, we obtain a simple approximation to Kb for finite cubic regions. Then we review combinations of bounds and other heuristic approximations to Kb for finite rectangular regions. Although heuristic, these approximations are generally more accurate than the analytically derived formulas, especially for nonnormal or high-variance log conductivity fields. 2.. Ergodic Case At least for o-2 < and K isotropic lognormal, a formula first conjectured by Matheron [967] is known to be accurate. This formula gives the effective conductivity of an infinite D-dimensional region as the deterministic constant Like (), (2) is expected to be accurate only for small fluctuations f(x). A noticeable feature of (2) is that randomness of KbE R is exclusively due to the variability of the block average F. The random fluctuations of F around ' inside the block are assumed to have a deterministic (ergodic) effect given by the factor exp {(/2 - /D)(o-2 - o- )}. In reality, this effect is also random. For example, in some realizations the log conductivity F may be nearly constant inside the block whereas in other realizations, F may be highly variable. Improvements over (2) will be developed in section 3. For rectangular blocks, practical formulas for K b have been suggested on heuristic grounds, in most cases as interpolations between theoretical upper and lower bounds [Renard and de Marsily, 997]. The most useful bounds in this context are those of Cardwell and Parsons [945]. The lower Cardwell-Parsons bound, K, is the arithmetic average of the harmonic averages of K along lines parallel to the mean direction of flow, whereas the upper bound K - is the harmonic average of the arithmetic average of K on slices normal to the mean direction of flow. Thus, by denoting /Xtand/x[, as the arithmetic and harmonic averaging operators in direction t, the bounds for D = 2 and D = 3 may be written as K : Idl, x2 a Idl, X (K h, K -- x2 x3 x, T,., _ Xl X2 ) K -- Idl, h Idl, a (K) D: 2 ]'/'h/'l'a/'l'a I,/k} D = 3. /./,a/./,a/./,h [/ }, K - x. x3. x2, \ (3) Renard and de Marsily [997] attribute to Guerillot et al. [990] the suggestion to estimate the equivalent conductivity of rectangular blocks as the geometric average between K and K -, i.e., = (4) They also attribute to Lemouzy [99] the following modification of (4) for 3-D regions: where K = K, K 2 = K - K 3 t.,a..x2..x..x3 rcx t.,h t.,a and K 4 = Equation is exact for D = and D = 2 and is consistent with tx x3..x..x2 trx a t- 'h t- 'a Both (4) and (5) are based on heuristic argufirst-order and second-order perturbation analyses in the fluctuation f(x) = F(x) - me [e.g., Gutjahr et al., 978; Gelhar and Axness, 983; Dagan, 993] as well as with numerical simulations [Dykaar and Kitanidis, 992a, 992b; Neuman and Orr, 993; Sanchez-Vila et al., 995]. However, results from other analytical expansions are in contrast with () [Noetinger, 994; Abramovich and Indelman, 995; de Wit, 995]. It therefore appears that (), although generally accurate for isotropic lognormal K fields, is not exact when D > 2. ments. In section 5 we shall provide supporto (5) and evaluate the bias of (4) in three dimensions. Rather than combining the local conductivities first through arithmetic averaging and then through harmonic averaging or vice versa, as one does in the Cardwell-Parsons bounds, one can alternate between the two operations in progressively coarsening the discretization grid. This numerical renormalization is implemented by repeatedly replacing two contiguous discretization elements with a single element. If the contiguous elements are in parallel relative to the direction of flow, their 2.2. Finite Cubic Regions conductivities are combined through arithmetic averaging; oth- For the case when L i = L <, a simple heuristic modification of () is erwise, they are combined through harmonic averaging. If the first step uses harmonic averaging, then the resulting approximation K(b tends to be a lower bound, whereas if one starts with arithmetic averaging, the resulting approximation K(b gbe R -' exp P + - (rr2 - rr ), (2) tends to be an upper bound. Renard and LeLoc'h [996] have proposed to calculate K(b -) and K(b +) through these local renorwhere/' is the average value of F inside the D-dimensional malization procedures and then estimate the block conductivsquare region and o-} is the variance of P. The subscript ER ity as the geometric average (for ergodic) indicates that (2) is obtained from direct adaptation of Matheron's effective conductivity formula. Equation 2 has a form similar to expressions obtained by Rubin and Ku =,/K ¾ u (-)K u (+). (6) Gomez-Hernandez [990] and Sanchez-Vila et al. [995] for the ensemble average of the equivalent conductivity of cubic blocks. This formula applies under some conditions. First-order and second-order expansions and heuristic methods have also been used to deal with the case of anisotropic hydrauli conductivity.

3 ... VENEZIANO AND TABAEI: ANOVA EFFECTIVE CONDUCTIVITY 292 These results will be reviewed in section 4, where we deal with anisotropic K fields. 3. ANOVA Approach for Isotropic K Our method for determining the equivalent conductivity of rectangular blocks starts by making an ANOVA decomposition of F inside the block, F(X) -- P q- E ei(xi) q- E eij(xi, Xj) q-... q- el. '.D(X), i i<j where is the block average, e i is the first-order (main) effect of the coordinate xi, eo is the order-2 interaction between x and xs, and so on. Each % term with a list L of indices is obtained by first calculating the spatial average of F over the variables not in L, P, and then subtracting the spatial average P and all the terms el, with L' a subset of L. Hence ei(xi) -'- Pi(Xi) -- P, eij(xi, Xj) -'- Pij(Xi, Xj) -- P- ei(xi) -- ej(xj), eijk(xi, Xj, Xk) -'- Fijk(Xi, Xj, Xk) -- F - ei(xi) -- ej(xj) (8) -- ek(xk)- eij(xi, Xj)- eik(xi, Xk) and so on. As (8) indicates, the main effect ei(xi) is the fluctuation of F in the i th direction, after F has been averaged over all directionsj --/: i. Similarly, eo(x i, xi) is the fluctuation on the (x i, xi) plane after averaging over all directions other than i and j and after subtracting the main effects e i and e s. For background information on the ANOVA decomposition, see, for example, Shefig' [959]. Important properties of the ANOVA decomposition that will be used later in this section are that the e terms are orthogonal (pairwise uncorrelated) and have zero mean value. For the analysis that follows, it is conveniento group some of the terms on the right-hand side of (7) by setting F (xx) = F + e (xx) = average value of F(x) at xx, eint (x ) = sum of all interaction terms in which appears in the index list, eint :l(x) -'- sum of the remaining interaction terms. For example, for D = 2, and for D = 3, eintl(x) -- e2(xl ' X2) eint,q(x) = 0, eint,(x) : 82(X, X2) q- e3(x, X3) q- e23(x, X2, X3) eint,,(x) -- e23(x2 ' X3). With this grouping of terms, (7) becomes F(x) = 'i(x) q- Z ei(xi) q- eint,(x) q- eint,,(x) ß i> (7) (9) (lo) Next we find the exact value of K b for the case when eintl(x) = 0, i.e., for F(x) = F, (x) = F (x ) + 5;i> ei(xi) + eint :l(x) and finally approximate the effect of eintl(x). This way of proceeding is at the core of our ANOVA approach. 3.. Kb for F(x) = F,l(X) Since/' (x ) is the only term that depends on x, the log conductivity along any line parallel to the x x axis equals P (x x) plus some constant. Consequently, the flow lines are parallel to the x x axis and the flow is -D. Therefore, Kb can be obtained in two equivalent ways: () as the harmonic average over x of the arithmetic average of i> K,l(x) = exp { Pl(Xl) + E ei(xi) + eint, (x) } Kb* L2' ' ' LD over X2,..., XD,... K,l(x) dx2''' dxd dxl, (a) which is the Cardwell-Parsons upper bound, or (2) as the arithmetic average over x2,..., XD of the harmonic average of i> K :I(X) =exp [Pl(X) q- E ei(xi) q- eint,(x)] over x, = L2'''LD L2 LD L, K l(x) dxl Kb l which is the Cardwell-Parsons dx2''' dxd (b) lower bound. Since in this case the flow is l-d, the Cardwell-Parsons bounds coincide. Substitution of K (x) in either (a) or (b) with the expression immediately before (a) gives (xp [ = exp (exp ' (2) where angle brackets denote spatial averaging within the block. For example, the term in the denominator of (2) is (exp {-e )) = /L e - ( ) dx. For any given function F (x), (2) is exact. When F is the realization of a normal field, which is the case considered here, the e terms have normal distribution with zero mean and an accurate approximation to (exp{ e.}) is exp is th empirical variance of e.; for example, 2 ='i/l L2 2 [e 2(x, x2)] 2 dx dx2. When using this appro mation, (2) simplifies to K =exp [ + ( i) { 2} D=i exp P- 5,, - exp {_ F+ (O2 =- D=2, {_ exp F +5 ( 2 e2 + e3 2 + e23 _ a l) O = 3 ' (3)

4 = 2922 VENEZIANO AND TABAEI: ANOVA EFFECTIVE CONDUCTIVITY I I I I I I I I 0.85 ro=0.5 I n=3ø I I I I I I I I I I to= n= I I I I I I I I I I Figure. Bias factors B i and +_ one-sigma confidence intervals B i +_ MSEi/¾/- of the equivalent conductivity estimators in Table for square blocks of unit size. The conductivity field is isotropic with unit mean value, unit log variance, and Gaussian log correlation function. The correlation distance of In K, to, and the sample size n are indicated in each panel. In obtaining (3) we have used the fact that the 8 terms are independent; hence the expected value of their exponentiated sum is the product of the individual expected values. In many of our simulations with normal F fields we have found that (3) is indeed... an accurate approximaticm (ff(9. x, -] -- tee Figures and 2 and related discussion in section Effect of œint, (X) We now modify (3) to account for 8intl(X), i.e., for all the interactions that involve the spatial coordinate x. Equation 2 is similarly modified. For D =, there is no such interaction, and no correction is needed. For D = 2 we estimate the effect exp P - o- : '`2} D, Kb^N = exp p+ (6_ 2 _ o- ) '`2 D = 2, tf+ f_ (O 2+ O' :3,'2+ O' :23 -- O' :) Jff O'... } of 8intl = 82 through calibration to the ergodic, infinite-block case, for which (3) gives Keff, = exp {me}. Since this is also K ^ = exp R + o%,, (5) the exact effective conductivity of the infinite block, we conclude that for D - 2, no correction for 82 is needed. where, is the total variance off due to components that do For D = 3, 8intl : 82 q- 83 q- 823, and we separately not contain in their index list and O'e 2 I,d is the sum of the assess the effects on K b of these three interaction terms. The variances 6-2 where L is a list of indices d digits long in which effect of 823 is estimated through calibration to the ergodic the first digit is. Equation 5 says that all the 8 components case when 2 = 3 = 0 and 0'e23: 2 0'. For this case, (3) that do not include in their index list make a contribution of gives Kerr, ' = exp {me}, whereas a good approximation to gef the type exp { } to the block conductivity, whereas compois exp {me + tr]:} (see ()). Therefore we estimate th effect nents with in their index list make contributions of the type of 823 to be a factor exp {g 23} on gb, '. To estimate the exp {(/2 - l/d) r 2}, where d is the length of the list. In the effect of 8]2, we consider the case when F = case of (2), components 8 with d > also contribute a factor 8]2, i.e., when K is decomposable as the product of a function of (xz, x2) and a function of x3. Now the flow lines are not exp {(/2 - i/d) o- } 2 or, more appropriately, a factor (exp {%/- 2/d. 8}) to K. straight, but they are parallel to the (xz, x2) plane and are independent of x3. Therefore the 3-D flow problem is a stack of 2-D problems in which, from the previous 2-D analysis, the 4. The ANOVA Approach for Anisotropic K effect of 82 on the equivalent conductivity is approximately The most obvious limitation of the model of K used in nil. A similar argument holds for section 3 is the assumption of isotropy. In stratified geologic We conclude that for D =, 2, and 3, the equivalent media, anisotropy may be very pronounced, with the correlaconductivity of a rectangular block with isotropic lognormal tion distance of In K acros strata often smaller by order of conductivity should be accurately estimated by magnitude or more than that on bedding planes. To evaluate exp '`2 '`2 '`2 /:3. (4) The subscript AN in gb indicates the ANOVA origin of the approximation. By induction we find that for rectangularegions in spaces of any dimension D,,' 2 q-

5 VENEZIANO AND TABAEI: ANOVA EFFECTIVE CONDUCTIVITY n=30 I I L2=0.25 I t I L2=0.5 I n=30 I I I I I I I I I I I I I I I I I I i I i i i i i i i i Figure 2. Same as Figure for rectangular blocks (simulation set 2 in Table ). K b whenk is anisotropic, we again multiply Kb, in (2) or (3) where r i is the correlation distance of the F field in the direcby a factor that accounts for the interactions with x. As we tion of xi and r h is the harmonic average of the correlation have done for the isotropic case, we approximate the effect of distances in the principal directions of anisotropy. Tartakovsky 8intl through calibration to ergodic results. and Neuman [998] have derived a first-order expression for The ergodic problem for anisotropic lognormal K has been the ensemble mean block conductivity under homogeneous studied by several authors (see review given by Renard and de anisotropic conditions. Their formula produces a result for- Marsily [997]). Gelhat and Axness [983] used a first-order mally similar to (8). perturbation method to obtain While any of the above results could be used to approximate the effect of 8intl in our approach, Ababou's formula is the most convenient one. Following the analysis steps of the iso- Keff,i = exp { mei l + ( l - go,i) rr ] }, (6) tropic case and assuming that the coordinate axes coincide with the principal directions of anisotropy, one obtains the followwhere Keff, i is the effective conductivity in the i th coordinate direction and #o,i is a quantity that depends on the space ing correction factors for Kb, : dimension D and the spectral density S e of F, as exp 2 r q- r (20) gd,i = o - Se(K) dk. (7) The first-orderesult in (6) is consistent with the exponential expression eff,,= exp go,,) }, (8) The completexpression for KbA N is obtained by multiplying which is a generalization of () and is the form suggested by Kb in (2) or (3) with the factors in (20). In the case of (2) Gelbar and ness [983] when is large. Indelman and the correction factors in (20) may be replaced with averaged Abramovich [994] have extended (6) to include second-order exponential terms, in analogy with what was done in section 3 terms in the variance of F. Their second-order e ansion is not for the isotropic case. consistent with (8). It is also worthwhile to consider a simpler approximation, Formulas of the Wpe given in (8), where go, is approxi- which stems from the assumption that the equivalent conducmated in different ways, have been proposed by Ababou [995] tivity does not change under the linear scaling x i -- x = and Neuman [994]. In particular, Ababou has suggested xi(ri/ri). This transformation maps the region 2 = {x: 0 -< X i L i} onto the region ll' = {x: 0 -< xi -< Li(r!/ri)} and Fh transforms F to F', where F' is isotropic. For the transformed isotropic problem, one can use the approximation in (4) or Keff, =exp mr+ 2 D ' (9) r2.2 + exp 2 r + r 2 ø' 2 2 r + r rir3 + r2r3 ) rir2 0' 23 D=3.

6 2924 VENEZIANO AND TABAEI: ANOVA EFFECTIVE CONDUCTIVITY the corresponding approximation based on (2). It is interest- effects of x and x2. Since this factor is exact (except for the ing that the linear transformation leaves F and the components replacement of (2) with (3)), it is reasonable to view B = of variance. unchanged. Therefore this approximation con- /E [ exp { ( 6-2 2_ 6-2 ) }] as a bias factor for rue R. sists of using the isotropic formulas without modification. We close this section with a comparison of the Cardwell- Parsons bounds and the associated approximations K%u and KbL E in (3)-(5) with the ANOVA-based formula in (4). It is 5. Analytical Comparison of the ANOVA difficult to make a precise comparison because the bounds use Formula With Other Results the actual values of K inside the block whereas (4) uses the Before we evaluate the ANOVA approximation using nu- variance components of F. However, an interesting approximerical simulation (see section 6) we make a few analytical mate comparison results from stating the bounds in terms of comparisons. We start by examining (4) in a few special cases the variance components, as follows. Suppose that F = In K = for which the exact equivalent conductivity is known or other + el where ' is a constant and el is a zero-average analytical approximations are available. Then we compare the fluctuation in the space identified by the index list L. If i is in ANOVA formula for square regions in the plane with e, the index list, we approximate the averages lab'(k) and lab'(k) which is the value produced by the estimator in (2). Finally, we asexp{p L} and exp {P L}, respectively. If i is compare our results with the Cardwell-Parsons bounds and the not in the list, these averages are given exactly by lab'(k) = approximations in (4) and (5). Unless otherwise stated, K is /x '(K) = exp {P + el}. This approximate rule gives, after assumed to be isotropic lognormal. The special cases consid- some algebra, ered are () Case A, cubic blocks in 3-D: The variances 6-2 in (4) are random variables with identical distributions. The K = gb^ exp - & 22 same is true for the variances 6-2 Therefore, for D = 3 8j' E 5 (,,2 0'82 _3_ 0'83,,2-3- 0'823,,2_ 6-28 ) ' (0' ' '823 2 O'28)'3 I- - O [(o- o- -- 0'82 0'823 (o- - r-}). (2) ' D = 3. (22) Hence the approximation in (2) results from replacing the Hence, under the present approximation, Kbou in (4) and gbl E random variances in (4) with their mean values. Because of in (5) are related to Kb in (4) as the nonlinear dependence of gb on the variances this replacement does not produce the mean value of gba $. We conclude that (2) is biased. Moreover, KbA N in (4) and rue R in (2) depend in different ways on the components of the log con- Kb u = [KbA $ exp, ductivity variance. This shows that it is incorrecto express this KbL = guam, D - 3. dependence simply in terms of the total variance o-3 - o'3-, as is done in (2). Finally, (2) ignores the sample fluctuations of The interesting result of this analysis is that the ANOVA the variances o- 2, o- 2, and. Inclusion of these fluctuations, approximation is close to the geometric average of the Cardt tj 23 as is done in (4), should produce more accurate estimates of the block conductivity; (2) Case B, infinite region and ergodic well-parsons bounds for D = 2 and to the Lemouzy formula for D = 3. Equation 23 confirms that the geometric average of hydraulic conductivity: By construction, gbam in (4) reduces to the Cardwell-Parsons bounds is a biased estimator of the block the ergodic effective conductivity in (); (3) Case C, one- conductivity in three dimensions and gives its bias factor as dimensional flow (L i "--> O, i -> 2)' All first-order and inter- exp { }. action terms vanish except el. In particular, intl-- 0. There- In summary, we have found that (4) reduces to () under fore KbA m = Kb, P which is exact; (4) Case D, very thin region isotropic ergodic conditions (Case A) and improves on (2) for in the direction of flow (L--> 0)' In this case, = nt = 0 finite square regions (Case B). Equation 4 is exact for rectand Pl(Xl) = P; hence KbA m = Kb, = ff, which under the angular regions that are very short or very long in the direction present conditions is exact; and (5) Case E, only first-order of flow (Cases C and D) and remains accurate under certain effects; that is, F(x) in the block is the sum of independent anisotropiconditions (Case E). The reason for the accuracy of stationary Gaussian processes, each in one of the coordinates (4) for anisotropic K is that one effect of anisotropy is to x,..., xd' In this case the F field is homogeneous Gaussian modify the first-order variances 6-2 and the interactions that but not isotropic. Since all the interactions are zero, intl-- 0 and KbA m --- Kb,l, which is exact. are treated exactly by (4). Finally, (4) is consistent with Lemouzy's estimate in (5) for rectangular blocks in three di- Next we compare KbA m in (4) with rue R in (2) for square mensions. blocks on the plane. For 2-D square blocks, rue R reduces to exp 6. {P}, the geometric mean of K inside the block, whereas (4) ß - I,,2 2 i,, Numerical Evaluation gives Kb^ N = exp {F j(o' )}. The factor exp { ( - )} in the latter expression is the influence of the main K = Kb^ exp & 22 g I = Kj = gb^ N exp - 5 ( ) K2 = KJ: KbA N exp 5 ( ) K3 = guam exp K4 = guam exp D = 3 (23) In this section we make a more detailed numerical assess- ment of the ANOVA formulas for various 2-D cases. We

7 VENEZIANO AND TABAEI: ANOVA EFFECTIVE CONDUCTIVITY 2925 Table. Numerical Simulation: Approximations and Bounds for the Block Conductivity Evaluated in Each Simulation Case Number i Approximation or Bound Text Reference gba N = exp {P}((exp{ 2})/(exp{-- })) equation (2) 2 KbA N = exp {P + (o' 2- o' )} equation (4) 3 K, lower Cardwell-Parsons bound equation (3) 4 K -, upper Cardwell-Parsons bound equation (3) 5 Kbc u = V'K K - equation (4) 6 K(b -), from local renormalization section 2 7 K(b +), from local renormalization section 2 8 KbRE = V'K(-)K b b (+) equation (6) 9 KbE R = exp {/'} equation (2) Set a Set 2 b same as in Set square subblocks and approximation in set, i - 2 Set 3 c KbA N = exp {P + (( - b/2)( + b)) 22}(exp{82})/(ex p {-8 })) KbA N = exp {P}(exp{82}>/<ex p {-8 }> Same as in Set Same as in Set 2, after rescaling to isotropy same as in Set equation (4) for each subblock equations (2) and (20) equation 2 (linear rescaling; see end of section 3) Same as in Set Same as in Set 2 ablock geometry: unit square blocks (L = L 2 = ); log conductivity and its correlation: isotropic with r 0 = 0.25, 0.5,, 2. bblock geometry: rectangular blocks (L = ; L 2 = 0.25, 0.5, 2, 4); log conductivity and its correlation: isotropic with r 0 =. CBlock geometry: unit square blocks (L = L 2 = ); log conductivity and its correlation: anisotropic with (r 0, b) = (0.5, 0.25), (, 0.25), (2, 0.25), or (0.5, 4). assume that the log conductivity F is a normal field with unit imation in (4) to each square block and combined the block variance and correlation function of the double-exponential conductivities through arithmetic or harmonic averaging detype pending on whether the blocks are arranged in parallel or in series. This method has been used also for anisotropic K after (24) rescaling the region to make K isotropic (estimator 0 for set Pr(r,r2)=exp{-(ro2 b- ro2j}, 3). where r and r 2 are distances in the two coordinate directions, For each set and parameter combination in Table, we have simulated the K field n times. For each simulation we have ro is the correlation distance in the mean direction of flow, and bro is the correlation distance in the orthogonal direction. In evaluated the exact block conductivity Kb through numerical the isotropic case, b = and (24) reduces to p(r) = exp flow analysis and calculated the approximations Kb, listed in {-r2/r( }, with r as the separating distance. The main reason Table. To evaluate the approximations, we have calculated we have chosen this correlation function is to minimize the the following statistics of the ratios % = gb,/gb: () the mean errors in the numerical evaluation of the flow field by finite bias factor, B/- E[%]; (2) the normalized mean square error, differences and therefore in the numerical estimation of the MSEi = V'Var [%]; and (3) the correlation matrix [Po] of the "exact" block conductivities. These errors are small if the de- ratios (%, /i)' Results are presented in Figures -3 as plots of rivatives of the F field have smooth local fluctuations, which is the bias factor B i and the +_ one-sigma confidence intervals on the case for the correlation function in (24). B i, B +_ MSEdX/-, for the various estimators. The sample To ensure sufficient numerical accuracy, we have made pre- size n is given in each figure panel. Notice that the vertical liminary runs in which the discretization grid was progressively scale varies from panel to panel and that in some cases the refined until the numerical estimates of the total flow through one-sigma interval on B i is very narrow. Consider first the the block became stable. In the isotropic case with the above simulations in Figures and 2, for which K is isotropic. In all correlation function, this happens for grid spacings A <_ ro/64. these cases the geometric average of K? ) and K?), estimator Hence, in all the numerical analyses we have chosen A to satisfy A _< min {ro, bro}/64. We have made three sets of runs with square and rectangular blocks and isotropic or anisotropi conductivity, using the 8, is unbiased and has a very small error variance. Next in performance, with a slightly higher bias and error variance, are the ANOVA formula (estimator ) and the geometric average of the Cardwell-Parsons bounds (estimator 5). The simplified parameters in Table. Estimators -9 for sets and 2 already ANOVA estimator 2 is slightly less accurate than estimator. have been discussed in previou sections. For set 2 (rectangular One case in which estimator 2 does not perform well is when blocks), one additional estimator (estimator 0) has been con- the blocks are rectangular and are narrow in the direction of sidered. Since the ANOVA methods have been observed to flow; see case L 2 = 4 in Figure 2. Estimator 9, KbER = exp perform better for square than for rectangular blocks (see below), we have partitioned the rectangular regions into square blocks, connected in series or in parallel relative to the direction of flow. We have then applied the ANOVA approx- {P}, has been suggested only for square regions and is included in Figure 2 only for the purpose of comparison. As expected, this estimator is highly biased for rectangular blocks, the sign of the bias depending on the direction of elongation of

8 2926 VENEZIANO AND TABAEI: ANOVA EFFECTIVE CONDUCTIVITY.2. E 0.9 (.' ]; ro=0.5 I ro=o.5 I t 0.6 I I I I I I I I I I [] [] ß 0.95 I ro=l I t I I n=3o I Figure 3. Same as Figure for anisotropiconductivity (simulation set 3 in Table ). the block. For square blocks, estimator 9 is essentially unbiased Figures -3 compare estimators of the block conductivity Kb but has a large error variance relative to the other estimators. in terms of their bias and error variance. That comparison does This is especially true when the length of the block side is close not reflect any probabilistic dependence that might exist to the correlation distance of the In K field; see cases with ro = among the estimators. Information on dependence is displayed 0.5,, and 2 in Figure. For rectangular blocks, estimator 0 in Table 2 through the correlation matrix of the ratios 7i = performs very well, at a level comparable with the best numer- Kt,,/Kt, for the methods i in Table. The correlation matrix is ical renormalization method (estimator 8). for the case of square blocks and isotropic log conductivity, In the anisotropi case, Figure 3, estimator 8 continues to with r o = 0.25 (top left of Figure ). Table 2 shows that the outperform the other methods, whereas the ANOVA estima- ANOVA estimators are very highly correlated. The reason is tors and 2 have an uneven performance. Estimator 2, which that (2) and (3) produce very similar values of Ku, and the rescales the support to make K isotropic and then uses esti- errors of Ku and Kb2 are entirely (estimator ) or predomimator 2 of set, performs closely to the geometric average of nantly (estimator 2) due to the way the effect of the interaction the Cardwell-Parsons bounds (estimator 5). As one can see si,tl is approximated. The latter approximation, obtained from Table, estimator is obtained by multiplying estimator through calibration to ergodic results, is the same for the two 2 by the factor exp {(( - b/2) ( + b)) ) ' Therefore, estimators. One may further notice from Table 2 that while the which of () or (2) is larger depends on whether the anisotropy Cardwell-Parsons bounds K and K - have high negative corratio b is larger or smaller than. Contrary to the isotropic relation, the estimators K? ) and K? ) are uncorrelated. The case, estimator 0 produces biased estimates and is not rec- negative correlation between K and K - may be explained ommended when K is highly anisotropic. The estimator Kb., = using (22). That equation shows that in approximation the two exp {P}, whic has not been suggested for anisotropic hydrau- bounds equal the ANOVA estimator divided or multiplied by lic conductivity, performs poorly. exp { 2). Hence the negative correlation comes primarily Table 2. Correlation Matrix of the Block Conductivity Ratios )t i -- Kb/K b for Different Estimators i a Estimators aresults are for simulation set in Table, with r o = 0.25.

9 VENEZIANO AND TABAEI: ANOVA EFFECTIVE CONDUCTIVITY 2927 from the interaction term 2, which affects the bounds in opposite ways. 7. Conclusions Except for a few special cases, the analytical evaluation of the equivalent conductivity of square or rectangular blocks in heterogeneous media has remained an elusive goal. The best available procedures are numerical or heuristic and do not give much insight into the effect of various components of fluctuation of K or In K inside the block. We have developed semianalytical approximations to Kb based on an ANOVA decomposition of the log conductivity inside the block. The decomposition partitions the total fluctuation of In K into main effects and interactions of various orders among the spatial coordinates. Some of the components of fluctuation are dealt with exactly, while others are accounted for in approximation. The approximations we have used are based on calibration to ergodic cases (very large blocks), for which the effective conductivity Kef f is known either exactly or with good approximation. We have found through numerical simulation that () for square blocks and K isotropic the ANOVA approximations are essentially unbiased and (2) for rectangular blocks and K isotropic and for square blocks and K anisotropic the ANOVA approximations are somewhat biased. Different treatment of the components of fluctuation that are not dealt with exactly might produce improved estimates of Kb in the latter cases. The main value of the ANOVA approach is that it shows how K b is affected by the components of fluctuation of In K that are dealt with analytically. These are all the main effects and the interactions that do not involve the mean direction of flow. In the absence of the latter interactions, K b is given exactly by (2). This result holds irrespective of the probabilistic model of K(x), the elongation of the block, and the space dimension; in particular, it holds for isotropic or nonisotropic, homogeneous or nonhomogeneous K fields and for any rectangular block in D-dimensional space. For homogeneous lognormal K fields the simpler expression in (3) produces accurate approximations. Acknowledgments. This work was supported by the Italian CNR under a Cooperative Agreement for the Study of Climatic Changes and Hydrogeologic Risks and by the National Science Foundation under grant CMS References Ababou, R., Random porous media flow on large 3-D grids: Numerics, performance and application to homogenization, in Environmental Studies: Mathematical, Computational and Statistical Analysis, IMA Vol., vol. 79, edited by M. F. Wheeler, pp. -25, Springer-Verlag, New York, 995. Abramovich, B., and P. Indelman, Effective permeability of log-normal isotropic random media, J. Phys. A Math. Gen., 28, , 995. Cardwell, W. T., and R. L. Parsons, Average permeabilities of heterogeneous oil sands, Trans. Am. Inst. Min. Metall. Pet. Eng., 60, 34-42, 945. Dagan, G., High-order correction of effective permeability of heterogeneous isotropic formations of log-normal conductivity distribution, Trans. Porous Media, 2, , 993. De Wit, A., Correlation structure dependence of the effective permeability of heterogeneous porous media, Phys. Fluids, 7(), , 995. Dykaar, B. B., and P. K. Kitanidis, Determination of the effective hydraulic conductivity for heterogeneous porous media using a numerical spectral approach,,, Water Resour. Res., 28(4), 55-66, 992a. Dykaar, B. B., and P. K. Kitanidis, Determination of the effective hydraulic conductivity for heterogeneous porous media using a numerical spectral approach, 2, Results, Water Resour. Res., 28(4), 67-78, 992b. Gelhar, L. W., and C. L. Axness, Three-dimensional stochastic analysis of macrodispersion in aquifers, Water Resour. Res., 9(), 6-8, 983. Guerillot, D., J. L. Rudkiewitz, C. Ravenne, G. Renard, and A. Galli, An integrated model for computer-aided reservoir description: From outcrop study to fluid flow simulations, Rev. Inst. Fr. Pet., 45(), 990. Gutjahr, A. L., L. W. Gelhar, A. A. Bakr, and J. R. McMillan, Sto- chastic analysis of spatial variability in subsurface flows, 2, Evaluation and application, Water Resour. Res., 4(5), , 978. Indelman, P., and B. Abramovich, A higher-order approximation to effective conductivity in media of anisotropic random structure, Water Resour. Res., 30(6), , 994. Lemouzy, P., Calcul de la permeabilite absolute effective, RF , Inst. Fr. de Pet., Paris, 99. Matheron, G., Elements Pour une Theorie des Milieux Poreaux, 66 pp., Masson et Cie, Paris, 967. Neuman, S. P., Generalized scaling of permeabilities: Validation and effect of support scale, Geophys. Res. Lett., 2(5), , 994. Neuman, S. P., and S. Orr, Prediction of steady state flow in nonuniform geologic media by conditional moments: Exact nonlocal formalism, effective conductivities, and weak approximation, Water Resour. Res., 29(2), , 993. Noetinger, B., The effective permeability of a heterogeneous porous medium, Trans. Porous Media, 5, 99-27, 994. Renard, P., and G. de Marsily, Calculating equivalent permeability: A review, Adv. Water Resour., 20(5-6), , 997. Renard, P., and G. LeLoc'h, A new upscaling technique for the permeability of porous media: The simplified renormalization, Comptes Rendus de lmcad. Des Sci., Ser. II Fascicule, 323(0), , 996. Rubin, Y., and J. Gomez-Hernandez, A stochastic approach to the problem of upscaling of conductivity in disordered media: Theory and unconditional numerical simulations, Water Resour. Res., 26(4), 69-70, 990. Sanchez-Vila, X., J.P. Girardi, and J. Carrera, A synthesis of approaches to upscaling of hydraulic conductivities, Water Resour. Res., 3(4), , 995. Sheff6, H., The Analysis of Variance, John Wiley, New York, 959. Tartakovsky, D. M., and S. P. Neuman, Transient effective hydraulic conductivities under slowly and rapidly varying mean gradients in bounded three-dimensional random media, Water Resour. Res., 34(), 2-32, 998. Wen, X., and J. Gomez-Hernandez, Upscaling hydraulic conductivities in heterogeneous media: An overview, J. Hydrol., 83(/2), 9-32, 996. (Received October 26, 2000; revised February 28, 200; accepted June 23, 200.)