Deterministic solution of stochastic groundwater flow equations by nonlocalfiniteelements A. Guadagnini^ & S.P. Neuman^

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1 Deterministic solution of stochastic groundwater flow equations by nonlocalfiniteelements A. Guadagnini^ & S.P. Neuman^ Milano, Italy; ^Department of. Hydrology & Water Resources, The University ofarizona, Tucson, AZ 85721, U.S.A. Abstract We consider the effect of measuring randomly varying hydraulic conductivities K(x) on the prediction of groundwater flow in a bounded porous domain under uncertainty. Hydraulic head is governed by a stochastic Poissbn equation subject to random source and boundary terms. We present a system of exact nonlocal deterministic equations for optimum unbiased predictors of these quantities and for measures of corresponding prediction errors. We then develop recursive approximations for these equations and solve them to leading order in the variance of In K(x) by nonlocal Galerkm finite elements. Our results compare well with Monte Carlo simulations of mean uniform and convergent flows in media with large variance and arbitrary correlation. 1 Introduction We consider steady state groundwater flow governed by ) jceo (1) /,(*)=#(*) %et^ (2) -9(x)-n(jc) = g(.x) jcel^ (3) where Q isflowdomain, q isflux,k is random conductivityfield,h is head,/ is random source, His random head on Dirichlet boundary 1% Q is random flux across Neumann boundary T, n(jc) is unit outward normal to boundary F = r^ul^r, and the forcing terms/ //, Q are prescribed in statistically independent

2 348 Computer Methods in Water Resources XII manners. Let (A% be the ensemble mean (optimum unbiased estimate) of K conditioned on measurements, with random estimation error K W = f (a) - (f(x)), (f We ^0. (4) We "define corresponding conditional ensemble moments (optimum unbiased flux and head predictors) (q)^ and (h), so that 9 %*) = #)-(#)), (?'W>^0 (5) A '(a) = A (jc) - (A(jc)>, (A W) ^0 (6) where q ' and h ' are associated random prediction errors. Taking the conditional ensemble mean of (1)- (3) yields where the nonlocal, non-darcian "residual flux" r<, is give )> *el^ (9) () Here a^ is a second-rank positive semidefinite symmetric tensor and 6. and c^ are vectors independent of forcing terms, while h,, is the solution of (7) - (9) for r, = 0. Eqs. (7) - () are exact. We also derived^, but do not present here, exact equations for second conditional moments of h and q. Expanding in powers ofoy, the unconditional standard deviation of Y' = Y- (Tic where Y= In /T yields', to second-order in Oy, (h),» W% + W% and (q)c ~ (q ^%+ (q }c where the zero- and second-order terms,(h^)^ and (h^)^ satisfy respectively the recursive equations V - [K<,(x)V(h (x».] + </(*)> = 0 (11) xet» (12) }} x e r^ (13) -V -<f">(*)>, =0 </i^(jc)x=0 ^er^ (15) -<^(jc)>,-n(jc) = 0 xet» (16) Here K^ is the conditional geometric mean of AT, ^(G^(y,x)\ (18) c is the zero-order Green's function associated with (11) - (13), and is the conditional auto-covariance of 7 between points x and y.

3 Computer Methods in Water Resources XII 349 The second-order flux approximation is clearly nonlocal and non-darcian. The conditional covariance of head, C^(x, y} = (h'(x) h'(y)), is approximated to 2^-order by C^ - Cj*\ The latter satisfies^ ^ (20) = L(Q' 000 (z)>,<g""(z,:v)>, dz * e r, where we take/ \H\ Q ' to be of order 7'. The conditional cross-covariance ChKcfcj) = (K\x)h '(y))c is approximated to 2"^-order as ^(^^)=-^(^)L^XA^Xz)XvxG^\z^))^,a (22) A 2 -order approximation of the conditional flux covariance tensor, ),, is given by (23) 2 Nonlocal Finite Element Equations We solve (11) - (16) for forcing terms of zero variance by a Galerkin finite element scheme in a rectangular domain with square elements using bilinear shape functions. The 0^-order part is standard and thus not discussed. The finite element equations corresponding to (14) -(16) are^ P+S n = l2...n- (24) where U is the set of all N* nodes not on F^, h^ is conditional mean head at node m, and the coefficients are defined by W^ (25) (26) K ^ M^, JV^, ^ S = e=\ i=l e'=\ (27)

4 350 Computer Methods in Water Resources XII (28) Here fa(x) is a bilinear Lagrange interpolation function associated with node n;npis the number of nodes on 1^; M^ = My are the number of elements e and e ' in the x- and j-planes, respectively; N^=N^ = 4 (nodes per element in the jcand^-planes); x? is the midpoint off; Q^ is (G^ (y, jc)>, at node / off due to a unit source at nodey ofe% and /^' is /^ (y* ) at node k of element e\ For simplicity, we take AT to be uniform in each element and evaluate (Y\e)Y\e ')>^ between element midpoints. The finite element equations associated with (19) - (21) read where C is head covariance between nodes^ in j-plane and m in jc-plane, C,\ = i:\ Ql^, (*' )L,, v^,, (^i. v ^^ or; \ (32) (33) (34) where M, = M^ = M^ is the number of elements e " in the z-plane; Ijf = 4 (nodes per element in the z-plane); andrp^ (x} is cross-covariance of head hkcjy \ / at node jy in the j-plane and Katx, Finally, + _n I^Kcrn^ 1 +..«,«_ ' ^/I,H ~~ where n is a node in element e and m is a node in element e\ 3 Example We use a grid of length L\ = 18 and width L^ = 8 with 40 rows and 90 columns of size AXj = A = 0.2 as shown in Figure 1. Head is prescribed deterministically as H ^ = at x, = 0 on the upstream (left) and H^ = 0 at Xi = 18 on the downstream (right) boundaries. Flux is prescribed

5 Computer Methods in Water Resources XII 351 no-flux no-flux Figure 1: Computational grid; ( ) well and (*) conditioning points location (J =1 200 /' ^ g = 1.50 i nonlocal FE, ' - -. NMC = Figure 2: Contours of longitudinal mean flux as obtained for 0y = 1,?t = 1, 2 = by Monte Carlo simulations and nonlocal finite elements.

6 352 Computer Methods in Water Resources XII e = = 1 nonlocal FE NMC = looq/t t Figure 3: Contours of transverse mean flux as obtained for (jy = I, A =1, Q- by Monte Carlo simulations and nonlocal finite elements. ^ 4 - ṉ 3 - i - /; nonlocal FE /' NMr 1 ODD / ' i T 2 - ( ) reference point // f in y plane // 1 * 1-0^%^ / ^ 0 22_^^ ^ ^ ^/ - i\\ Sections: (b) A A (A- A) %2 = 4 ^A (B-B)%, = 9 \\\ ^^ B-B - - i - U ;...- t #. t. * i i Figure 4: (a) Contours and (b) sections of conditional head covariance C^(^; yj - 9, ^ = 4) forcry = 1, X = 1, Q - as computed by Monte Carlo simulations and nonlocal finite elements.

7 Computer Methods in Water Resources XII (b) j_ 20 (c) 20 c,,, (d) Figure 5: Components of the flux covariance tensor C^«/(JC, j)with reference to yi = 8.9, j2 = 3.9 as obtained for cry = 1, A = 1, g = by Monte Carlo simulations (solid) and nonlocal (dashed) finite elements along the longitudinal section (%? = 3.9).

8 354 Computer Methods in Water Resources XII deterministically to be zero across the lateral boundaries at ^ = 0 (bottom) and ^ = 8 (top). A well pumps at a deterministic rate Q from the center node fa = 9, ^ = 4). The natural log hydraulic conductivity field Y is taken to be statistically homogeneous and isotropic with exponential autocovariance C^(r)-a\e~"^ where r is separation distance and A is the autocorrelation (integral) scale. The field is conditioned on 16 measurement points (Figure 1) with values obtained from a randomly generated realization of Y. Figures 2-3 compare longitudinal and transverse mean fluxes obtained by nonlocalfiniteelements and Monte Carlo (MC) simulations by standard finite elements on the same grid with of=l,a=\ and Q=. The number of MC runs is indicated in the figures. Figure 4 shows the corresponding head autocovariance function C^(x,y) with reference to the well. Figure 5 depicts all four components of the flux covariance tensor C <,,,(.*,.y) for / = 1,2, with reference to a point y\ = 8.9, % =3.9 near the well, as functions of longitudinal and transverse separation distances ^ and ^. As expected, the covariance is nonsymmetric. Agreement between nonlocal and MC finite element results is seen to be very good in all cases. In general, we find that the agreement between nonlocal and MC finite element results improves as of decreases. Good agreement is obtained for of as large as 4 and A ranging between 1 (length scale of an element) and 18 (length of the domain). 4 References [1] S. P. Neuman, and S. Orr, Prediction of steady state flow in nonuniform geologic media by conditional moments: Exact nonlocal formalism, effective conductivities, and weak approximation, Water Re sour. Res., 29(2), , [2] S. P. Neuman, D. Tartakovsky, T.C. Wallstrom, and C.L. Winter, Correction to "Prediction of steady state flow in nonuniform geologic media by conditional moments: Exact nonlocal formalism, effective conductivities, and weak approximation", Water Resour. Res., 32(5), , [3] A. Guadagnini, and S. P. Neuman, Nonlocal and Localized Finite Element Solution of Conditional Mean Flow in Randomly Heterogeneous Media, Tech. Rep. HWR-97-0, Department of Hydrology and Water Resources, The University of Arizona, Tucson, 1997.