Nonlocal and localized analyses of conditional mean transient flow in bounded, randomly heterogeneous porous media

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1 ATR RSOURCS RSARCH, VOL. 40, 05104, doi: /2003r002099, 2004 Nonloal and loalized analyses of onditional mean transient flow in bounded, randomly heterogeneous porous media Ming Ye 1 and Shlomo P. Neuman epartment of Hydrology and ater Resoures, University of Arizona, Tuson, Arizona, USA Alberto Guadagnini ipartimento di Ingegneria Idraulia Ambientale e del Rilevamento, Politenio di Milano, Milan, Italy aniel M. Tartakovsky Theoretial ivision, Los Alamos National Laboratory, Los Alamos, New Mexio, USA Reeived 25 February 2003; revised 10 February 2004; aepted 17 February 2004; published 12 May [1] e onsider the numerial predition of transient flow in bounded, randomly heterogeneous porous media driven by random soures, initial heads, and boundary onditions without resorting to Monte Carlo simulation. After applying the Laplae transform to the governing stohasti flow equations, we derive exat nonloal (integrodifferential) equations for the mean and variane-ovariane of transformed head and flux, onditioned on measured values of log ondutivity Y = ln K. Approximating these onditional moment equations reursively to seond order in the standard deviation s Y of Y, we solve them by finite elements for superimposed mean-uniform and onvergent flows in a two-dimensional domain. An alternative onditional mean solution is obtained through loalization of the exat moment expressions. The nonloal and loalized solutions are obtained using a highly effiient parallel algorithm and inverted numerially bak into the time domain. A omparison with Monte Carlo simulations demonstrates that the moment solutions are remarkably aurate for strongly heterogeneous media with s Y 2 as large as 4. The nonloal solution is only slightly more aurate than the muh simpler loalized solution, but the latter does not yield information about preditive unertainty. The auray of eah solution improves markedly with onditioning. A preliminary omparison of omputational effiieny suggests that both the nonloal and loalized solutions for mean head and its variane require signifiantly less omputer time than is required for Monte Carlo statistis to stabilize when the same diret matrix solver is used for all three (we do not presently know how using iterative solvers would have affeted this onlusion). This is true whether the Laplae inversion and Monte Carlo simulations are onduted sequentially or in parallel on multiple proessors and regardless of problem size. The underlying exat and reursive moment equations, as well as the proposed omputational algorithm, are valid in both two and three dimensions; only the numerial implementation of our algorithm is two-dimensional. INX TRMS: 1829 Hydrology: Groundwater hydrology; 1869 Hydrology: Stohasti proesses; 3210 Mathematial Geophysis: Modeling; 3230 Mathematial Geophysis: Numerial solutions; KYORS: transient flow, unertainty, heterogeneity, spatial variability, loalization, parallel omputing Citation: Ye, M., S. P. Neuman, A. Guadagnini, and. M. Tartakovsky (2004), Nonloal and loalized analyses of onditional mean transient flow in bounded, randomly heterogeneous porous media, ater Resour. Res., 40, 05104, doi: /2003r Introdution [2] Hydrauli parameters vary randomly in spae and are therefore often modeled as spatially orrelated random fields. This, together with unertainty in foring terms (initial onditions, boundary onditions and soures), 1 Now at Paifi Northwest National Laboratory, Rihland, ashington, USA. Copyright 2004 by the Amerian Geophysial Union /04/2003R002099$ renders the groundwater flow equations stohasti. The solution of suh equations onsists of the joint, multivariate probability distribution of its dependent variables or, equivalently, the orresponding ensemble moments. It is advantageous to ondition the solution on measured values of input variables (parameters and foring terms) at disrete point in spae-time (onditioning on measured values of the dependent variables requires an inverse solution and is not disussed here). The equations an then be solved by onditional Monte Carlo simulation or by approximation. A typial solution inludes the first two 1of19

2 05104 Y T AL.: ANALYSIS OF CONITIONAL MAN TRANSINT FLO onditional moments (mean and variane-ovariane) of head and flux. The first moments onstitute optimum unbiased preditors of these random quantities, and the seond moments are measures of the assoiated predition errors. [3] The Monte Carlo approah is oneptually straightforward but requires knowing or assuming the joint multivariate distribution of all random inputs parameters and foring terms; is omputationally demanding; and laks theoretial onvergene riteria for moments higher than the mean (for whih suh riteria exist, e.g., Morgan and Henrion [1990] and Shapiro and Homem-de-mello [2000]). Its omputational burden stems from the need to (1) generate random realizations of the input parameters (e.g., log-ondutivity) on a grid that is muh finer than their spatial orrelation (integral) sale, so as to preserve their geostatistial properties; (2) solve the flow problem numerially on this fine grid; (3) repeat steps 1 and 2 many times (typially thousands) for a given set of random foring funtions, so as to minimize sampling error; and (4) repeat steps 1 3 for eah set of random foring funtions representing different senarios. [4] e fous on an alternative that allows omputing leading onditional ensemble moments of head and flux diretly by solving a reursive system of moment equations. The moment equations are distribution free and thus obviate the need to know or assume the multivariate distributions of random input parameters or foring terms. Tartakovsky and Neuman [1998a] developed exat integro-differential equations for the first two onditional moments of head and flux in a bounded, randomly heterogeneous domain. Their equations are nonloal and non-arian in that the mean flux depends on mean head gradients at more than one point in spae-time. Tartakovsky and Neuman [1998b] explored ways to loalize the exat onditional mean equations in real, Laplae- and/or infinite Fourier-transformed domains so as to render them arian. Suh loalization entails an approximation whih does not extend to moments higher than one. To render the nonloal moment equations workable, Tartakovsky and Neuman [1998a] approximated them reursively through expansion in powers of s Y, a measure of the onditional standard deviation of log ondutivity Y = ln K. Their reursive approximations are nominally limited either to mildly heterogeneous s Y 1 or to well-onditioned strongly heterogeneous media s Y > 1. The authors have not evaluated their expressions numerially. [5] In this paper we develop analogous reursive approximations in Laplae spae and solve them by a finite element algorithm patterned after that developed for steady state flow by Guadagnini and Neuman [1999a, 1999b]. The moment equations inlude Green s funtions that need to be omputed one for a given boundary onfiguration and are then appliable to a variety of foring senarios. For omparison, we ompute onditional mean head and flux by finite elements in Laplae spae using (1) a loalized version of the exat moment equations and (2) a standard Monte Carlo approah. e do so for superimposed mean-uniform and onvergent flows in a two-dimensional domain. All three sets of solutions are inverted numerially bak into the time domain by using a parallelized version of an algorithm due to Crump [1976] and e Hoog et al. [1982]. Previously, sequential or parallel versions of this algorithm were used to solve deterministi flow and transport problems by Sudiky [1989], Xu and Brusseau [1995], Gambolati et al. [1997], Pini and Putti [1997], and Farrell et al. [1998], and to ondut Monte Carlo simulations of transport by Naff et al. [1998]. e assess the relative auraies of the reursive nonloal and loalized solutions through omparison with the Monte Carlo results. e ompare the omputational effiienies of unonditional reursive nonloal and Monte Carlo solutions as funtions of grid size for the speial ase where a single diret matrix solver is applied to both. A more omprehensive omparison of omputational effiienies, in whih reursive nonloal moment and Monte Carlo solutions are obtained in manners that are optimal for eah, is outside the sope of this study. [6] Our approah differs from standard perturbative solutions [e.g., agan, 1982; Indelman, 1996, 2000, 2002; hang, 1999, 2002] in several ways. On a fundamental level, our approah originates in a set of onditional moment equations that are exat, ompat, formally inorporate boundary effets, provide a unique insight into the nature of the problem (as explained by Neuman [1997, 2002]), and lead to unique loalized moment equations that look like standard deterministi flow (and transport) equations, allowing one to interpret the latter within a onditional stohasti framework [Neuman and Guadagnini, 2000; Guadagnini and Neuman, 2001]. Both the nonloal and loalized equations desribe the spatial-time evolution of moments representing random funtions rendered statistially inhomogeneous (in spae) and nonstationary (in time) due to the ombined effets of soures, boundaries, and onditioning. To approximate the exat moment equations reursively, we use a valid expansion in terms of (deterministi) moments rather than a theoretially invalid expansion in terms of random quantities [e.g., agan, 1989; hang, 2002], whih may (but is not guaranteed) to yield valid results after subsequent averaging. [7] On a pratial level, our approah leads to loalized moment equations that are almost as easy to solve as standard deterministi flow equations, and to reursive nonloal moment equations written in terms of Green s funtions, whih are independent of internal soures and the magnitudes of boundary terms. One these funtions have been omputed for a given boundary onfiguration, they an be used repeatedly to obtain solutions for a wide range of internal soures and boundary terms (senarios). The same is not true for standard perturbation expressions suh as those developed for transient flow in bounded domains by hang [1999]. [8] The underlying exat and reursive moment equations, as well as the proposed omputational algorithm, are valid in both two and three dimensions, though we implement them numerially in two dimensions. 2. Statement of Problem [9] e onsider transient flow in a domain governed by ary s law qx; ð tþ ¼ KðxÞrhðx; tþ x 2 ð1þ 2of19

3 05104 Y T AL.: ANALYSIS OF CONITIONAL MAN TRANSINT FLO and the ontinuity equation S ¼ rqx; ð t Þþf ð x; t Þ x 2 ð2þ subjet to initial and boundary onditions hðx; 0Þ ¼ H 0 ðxþ x 2 ð3þ hðx; tþ ¼ Hðx; tþ x 2 G ð4þ qðx; tþnx ð Þ ¼ Qðx; tþ x 2 G N ð5þ where q is ary flux, x is a vetor of spae oordinates, t is time, K is a salar hydrauli ondutivity forming a orrelated random field, h is hydrauli head, S s is a deterministi speifi storage term, f is a random soure term, H 0 (x) is random initial head, H is randomly presribed head on irihlet boundaries G, Q is random flux into the flow domain aross Neumann boundaries G N, and n is a unit vetor normal to the boundary G = G [ G N pointing out of the domain. The foring terms f, H 0, H, and Q are taken to be unorrelated with eah other or with K. [10] All quantities in equations (1) (5) are defined on a onsistent nonzero support volume w, entered around x, whih is small in omparison to but suffiiently large for ary s law to be loally valid [Neuman and Orr, 1993]. This operational definition of w does not generally onform to a representative elementary volume (RV) in the traditional sense [Bear, 1972]. Conditioning and the ation of foring terms render the solution of equations (1) (5) nonstationary in spae-time. [11] The Laplae transform of a funtion g(t) is defined as [Carslaw and Jaeger, 1959] L½gt ðþš ¼ gðlþ ¼ 1 0 gt ðþe lt dt where l is a omplex Laplae parameter. The Laplae transform of a derivative is L dg ¼ lgðlþ gð0þ dt Applying equations (6) and (7) to (1) (5) yields transformed flow equations ð6þ ð7þ qx; ð lþ ¼ KðxÞrh x 2 ð8þ rqx; ð lþþs s ðxþlh ¼ f þs s ðxþh 0 ðxþ x 2 ð9þ h ¼ H x 2 G ð10þ where H, q, H, Q, and f are Laplae transforms of h, q, H, Q, and f, respetively. 3. xat Conditional Moment quations [12] Let hk(x)i be the ensemble mean of K(x) onditioned on w-sale measurements at a set of disrete points in. As suh, hk(x)i forms a relatively smooth optimum unbiased estimate of the flutuating random funtion K(x). The unknown hydrauli ondutivity K(x) differs from its known estimate hk(x)i by a zero mean random estimation error K 0 (x) suh that Likewise we write KðxÞ ¼ hkðxþi þ K 0 ðxþ hðx; l qx; ð l hk 0 ðxþi ¼ 0 ð12þ Þ ¼ h þ h0 h 0 ¼ 0 Þ ¼ hqx; ð lþi þ q 0 q 0 ¼ 0 ð13þ ð14þ 3.1. xat Conditional Mean quations [13] Substituting equations (12) (14) into (8) (11) and taking onditional ensemble mean yields the following exat onditional mean equations for the Laplae transform of head: hqx; ð lþi ¼ hkðxþi r h þ r r ¼ K 0 ðxþrh 0 x 2 ð15þ rhqx; ð lþi þ S s ðxþl h ¼ f þ Ss ðxþhh 0 ðxþi x 2 ð16þ h ¼ Hðx; lþ hqx; ð lþi nx ð Þ ¼ Q x 2 G x 2 G N ð17þ ð18þ where hf i, hh 0 i, hhi, and hqi are unonditional ensemble mean foring terms and r is a transformed residual flux. As shown in Appendix A, the latter is given impliitly by r ¼ with kernels a ðy; x; l Þr y hðy; lþ dy þ d ðy; x; l a ¼ K 0 ðxþk 0 ðyþr x r T y G ð y; x; l Þ Þr ðy; lþdy ð19þ ð20þ qnx ð Þ ¼ Q x 2 G N ð11þ 3of19 d ¼ K 0 ðxþr x r T y G ð y; x; l Þ ð21þ

4 05104 Y T AL.: ANALYSIS OF CONITIONAL MAN TRANSINT FLO Here G(y, x, l) is a transformed random Green s funtion defined in Appendix A. The kernels are similar to those obtained for steady state [Guadagnini and Neuman, 1999a] exept that they inlude the omplex Laplae parameter l. From equations (15) and (19) it is evident that the transformed residual and mean flux are nonloal in spae (they depend on mean head gradient at points other than x) and non-arian (there is no effetive or equivalent hydrauli ondutivity valid for arbitrary diretions of onditional mean flow in Laplae spae) xat Conditional Seond Moment quations [14] Let C h (x, y, t, s) =hh 0 (x, t)h 0 (y, s)i be the onditional ovariane of head orresponding to two points (x, t) and (y, s) in spae-time. Applying Laplae transform to C h (x, y, t, s) with respet to t yields a transformed onditional ovariane C h (x, y, l, s) =hh 0 (x, l)h 0 (y, s)i between the transformed head h(x, l) and the original head h(y, s). Similarly, C q (x, y, l, s) =hq 0 q T0 (y, s)i is a transformed onditional ross-ovariane tensor between the transformed flux q(x, l) and the original flux q(y, s), the supersript T denoting transpose. [15] e show in Appendix B that C h (x, y, l, s) satisfies exatly the equation h r x hkðxþi r x C h ðx; y; l; sþþp ðx; y; l; sþ i þ u ðx; y; sþr x h þ S s ðxþlc h ðx; y; l; sþ ¼ S s ðxþ H0 0 ðx Þh0 ðy; sþ f 0 h 0 ðy; sþ þ subjet to boundary onditions C h ðx; y; l; sþ ¼ H 0 h 0 ðy; sþ x 2 G x 2 h hkðxþi r x C h ðx; y; l; sþþp ðx; y; l; sþþu ðx; y; sþ i r x h nx ð Þ ¼ Q 0 h 0 ðy; sþ x 2 G N ð22þ ð23þ ð24þ where P (x, y, l, s) =hk 0 (x)r x h0 (x, l)h 0 (y, s)i is a third moment given expliitly by P ðx; y; l; sþ ¼ s r z K 0 ðxþr xh 0 r T z G ð z; y; s t Þ 0 r ðz; tþdzdt s K 0 ðxþk 0 ðþr z xh 0 r T z G ð z; y; s t Þ 0 r z hhðz; tþi dzdt þ S s ðþ z H0 0 ðþg z ð z; y; s ÞK0 ðxþr xh 0 dz s þ f 0 ðz; tþgðz; y; s tþk 0 ðxþr xh 0 dzdt 0 s hk 0 ðxþkðþ z H 0 ðz; tþr xh 0 r T z G ð z; y; s t Þi 0 G nz ðþdzdt s þ Q 0 ðz; tþk 0 ðxþ:gðz; y; s tþr xh 0 dzdt 0 G N ð25þ The onditional ross ovariane u (x, y,s)=hk 0 (x)h 0 (y, s)i between head and ondutivity is given by u ðx; y; sþ ¼ s r ðz; tþr z hgðz; y; s tþk 0 ðxþi dzdt 0 s r z hhðz; tþi hk 0 ðþr z z Gðz; y; s tþk 0 ðxþi dzdt ð26þ 0 as in equation (53) of Tartakovsky and Neuman [1998a]. xpliit expressions for hh 0 0h 0 i, h f 0 h 0 i, h H 0 h 0 i, and h Q 0 h 0 i are given in Appendix B. The onditional head variane C h (x, x, t, t) an be obtained through inverse transformation of C h (x, x, l, t) at time t. The latter is given expliitly by (Appendix B) C h ðx; x; l; tþ ¼ h 0 h 0 ðx; tþ ¼ K 0 ðyþrh 0 ðy; lþ r ygh 0 ðx; tþ dy r hðy; lþ K0 ðyþr y Gh 0 ðx; tþi dy þ f 0 ðy; lþgh 0 ðx; tþ dy þ S s ðyþ H0 0 ðyþ Gh 0 ðx; tþ dy H 0ðy; lþkðyþr y Gh 0 ðx; tþ ny ð Þdy G þ Q 0 ðy; lþgh 0 ðx; tþ dy G N ð27þ [16] The onditional ross-ovariane tensor C q (x, y, l, s) =hq 0 (x, l)q 0T (y, s)i of the flux is given expliitly by (Appendix B) C q ðx; y; l; sþ ¼ r r T ðy; sþ þ hkðxþi r x r T y C hðx; y; l; sþhkðyþ þ hkðxþi r x u r T y hhðy; sþi þ hkðxþi K 0 ðyþr x h 0 r T y h0 ðy; sþ þr x h rt y u ðx; y; sþhkðyþ þ K 0 ðxþk 0 h ðyþi r x h rt y hhðy; sþ þr x h h K0 ðxþk 0 ðyþr T y h0 ðy; sþ þ K 0 ðxþr xh 0 r T y h0 ðy; sþ hkðyþi þ K 0 ðxþk 0 ðyþr xh 0 rt y hhðy; sþi þ K 0 ðxþk 0 ðyþr xh 0 r T y h0 ðy; sþ i i ð28þ where u (y, x, l) =hk 0 (y) h 0 (x, l)i is given expliitly by (Appendix B) u ðy; x; l Þ ¼ r ðz; l r T z Þr z Gðz; x; lþk 0 ðyþ dz hðz; lþ K0 z ðþr z Gðz; y; lþk 0 ðyþ dz ð29þ 4of19

5 05104 Y T AL.: ANALYSIS OF CONITIONAL MAN TRANSINT FLO The onditional variane tensor C q (x, x, t, t) of flux is obtained through inverse transformation of C q (x, x, l, t) at time t. The latter is given expliitly by (Appendix B) C q ðx; x; l; tþ ¼ r r T ðx; tþ þ hkðxþi rh 0 r T h 0 ðx; tþ hkðxþ hkðxþi r r T hhðx; tþ þ hkðxþi hk 0 ðxþrh 0 r T h 0 ðx; tþ r h rt ðx; tþhkðxþi þr h h K0 ðxþk 0 ðxþi r T hhðx; tþ þr h K0 ðxþk 0 ðxþr T h 0 ðx; tþ i þ K 0 ðxþrh 0 r T h 0 ðx; tþ hkðxþi þ K 0 ðxþk 0 ðxþrh 0 rt hhðx; tþi þ K 0 ðxþk 0 ðxþrh 0 r T h 0 ðx; tþ ð30þ 4. Reursive Nonloal Conditional Moment Approximations [17] To render the above onditional moment equations workable, it is neessary to employ a suitable losure approximation. Tartakovsky and Neuman [1998a, 1999] developed reursive nonloal approximations in spae-time to leading orders of s Y. lsewhere [Ye, 2002], we develop reursive nonloal approximations to seond order in s Y in the Laplae domain. As the latter are similar in priniple to the former, we present them here without development Reursive Nonloal Conditional Mean Approximations in the Laplae omain [18] The reursive nonloal onditional mean flow equations are given to zero order in s Y by q ð0þ ¼ K GðxÞr h ð0þ x 2 ð31þ r q ð0þ þ S sðxþl h ð0þ ¼ f ðx; lþ þ S s ðxþhh 0 ðxþi x 2 ð32þ i i h ð0þ ¼ Hðx; lþ q ð0þ nx ð Þ ¼ Qðx; lþ and to seond order by q ð2þ ¼ K GðxÞ r h ð2þ ð Þ þ s2 Y x 2 G x 2 G N ð33þ ð34þ ðxþ r h ð0þ 2 þ r 2 x 2 ð35þ r q ð2þ þ S sðxþl h ð2þ ¼ 0 x 2 ð36þ h ð2þ ¼ 0 x 2 G ð37þ q nx x 2 G N ð38þ where the supersript designates orders of approximation in s Y, K G (x) = exphy(x)i is the onditional geometri mean of K(x), s 2 Y (x)=hy 02 (x)i is the onditional variane of Y(x), and r ð 2Þ ¼ a ð 2Þ r y h ð0þ ðy; lþ dy ¼ K G ðxþk G ðyþhy 0 ðxþy 0 ðyþi r x r T y G ð0þ r y h ð0þ ðy; lþ dy ð39þ where hg (0) (y, x, l)i is the zero-order approximation of the random Green s funtion and hy 0 (x)y 0 (y)i is the onditional ovariane of Y between points x and y. The first-order approximation is governed by a homogeneous equation subjet to homogeneous boundary onditions and so it vanishes identially, hh (1) (x, l)i 0. e approximate the mean head and flux by their leading terms up to seond order as h h ð0þ þ h ð2þ hqx; ð lþi q ð0þ þ q ð2þ ð40þ 4.2. Reursive Nonloal Conditional Seond Moment Approximations in the Laplae omain (0) [19] To seond order in s Y, C h (1) = C h = 0, and the seond-order approximation of the onditional head ovariane C h (x, y, l, s) is governed by h i r x K G ðxþr x C h ðx; y; l; sþ þ u 2Þ ðx; y; sþr x h þ S s ðxþlc h ðx; y; l; sþ ¼ S s ðxþ S s ðþc z H0 ðx; zþ G ðz; y; sþ s þ C f ðx; z; l; tþ G ðz; y; s tþ 0 dzdt ð41þ s C h ðx; y; l; sþ ¼ C H ðx; z; l; tþk G ðþ z 0 G r z G ðz; y; s tþ ð42þ h i K G ðxþr x C h ðx; y; l; sþþu 2Þ ðx; y; sþr x h nx ð Þ s ¼ 0 C Q ðx; z; l; tþ G ðz; y; s tþ G N dzdt x 2 G N ð43þ where C H0, C f, C H, and C Q are transformed ovarianes of the foring terms H 0, f, H, and Q, respetively: C H0 ðx; zþ ¼ H0 0 ðxþh 0 0 ðþ z C f ðx; z; l; tþ ¼ f 0 f 0 ðz; tþ ð44þ C H ðx; z; l; tþ ¼ H 0 H 0 ðz; tþ C Q ðx; z; l; tþ ¼ Q 0 Q 0 ðz; tþ For onsisteny, we assume that all random flutuations in foring terms are of order s Y. Perturbation expansion of 5of19

6 05104 Y T AL.: ANALYSIS OF CONITIONAL MAN TRANSINT FLO P (x, y, l, s) in equation (25) and u (x, y, s) in equation (26) yields the leading terms u ð 0Þ u ð 2Þ P ð0þ ðx; y; l; sþ ¼ P ð1þ ðx; y; l; sþ ¼ P ð2þ ðx; y; l; sþ 0 ð45þ ðx; y; sþ ¼ u ð 1Þ ðx; y; sþ 0 s ðx; y; sþ ¼ K G ðxþ K G ðþy z h 0 ðþy z 0 ðxþi 0 r z h ð0þ ðz;tþ r z G ð0þ ðz; y; s tþ dzdt ð46þ Perturbation of the onditional ovariane C h (x, x, l, t) yields leading terms C ð0þ h C ð2þ h ðx; x; l; tþ ¼ C ð1þ ðx; x; l; tþ 0 h ðx; x; l; tþ ¼ u ð 2Þ ðy; x; tþr y h ð0þ ðy; lþ r y G ð0þ dy t þ C f ðy; z; l; tþ G ð0þ ðz; x; t tþ G ð0þ dzdydt 0 þ S s ðyþs s ðþc z H0 ðy; zþ G ð0þ ðz; x; tþ G ð0þ dzdy t þ C H ðy; z; l; tþk G ðþr z z G ð0þ ðz; x; t tþ nz ðþ 0 G G K G ðyþr y G ð0þ ny ð Þdzdydt t þ C Q ðy; z; l; tþ G ð0þ ðz; x; t tþ G ð0þ 0 G N G N dzdydt ð47þ [20] Perturbation of the onditional ross-ovariane tensor of flux C q (x, y, l, s) yields leading terms C ð0þ q C ð2þ q ðx; y; l; sþ ¼ C ð1þ ðx; y; l; sþ 0 q ðx; y; l; sþ ¼ h K G ðxþk G ðyþ r x r T y C ð2þ h ðx; y; l; sþ i þ hy 0 ðxþy 0 ðyþi r x h ð0þ rt y h ð0þ ðy; sþ þ K G ðxþr x u ð 2Þ r T y h ð0þ ðy; sþ þ K G ðyþr x h ð0þ ð Þ ðx; y; sþ ð48þ rt y u 2 The leading terms of u (y, x, l) are u ð 2Þ ¼ K G ðyþ K G ðþy z h 0 ðþy z 0 ðyþ r T z h ð0þ ðz; lþ r z G ð0þ ðz; x; lþ dz i ð49þ Perturbation of C q (x, x, l, t) yields C ð0þ q C ð2þ q ðx; x; l; tþ ¼ C ð1þ q ðx; x; l; tþ 0 h i ðx; x; l; tþ ¼ K G ðxþk G ðxþ rh 0 ð2þ r T h 0 ðx; tþ þ s 2 Y ðxþr h ð0þ rt h ð0þ ðx; tþ K G ðxþr ð 2Þ r T h ð0þ ðx; tþ K G ðxþr h ð0þ ð ÞT ðx; tþ ð50þ r 2 where the ovariane tensor hrh 0 (x, l)r T h 0 (x, t) (2) i of head gradient is given by equation (B8) as h i ð2þ rh 0 r T h 0 ðx; tþ ¼ r x r T y G ð0þ r y h ð0þ ðy; lþ rt x u ð 2Þ ðy; x; tþdy t þ C f ðy; z; l; tþr x G ð0þ 0 r T x G ð0þ ðz; x; t tþ dzdydt þ S s ðyþs s ðþc z H0 ðy; zþr x G ð0þ r T x G ð0þ ðz; x; tþ dzdy t þ C H ðy; z; l; tþ 0 G G h i r x K G ðyþr y G ð0þ ny ð Þ h i r T x K GðÞr z z G ð0þ ðz; x; t tþ nz ðþdzdydt t þ C Q ðy; z; l; tþr x G ð0þ 0 G N G N r T x G ð0þ ðz; x; t tþ dzdydt ð51þ 5. Loalization of Conditional Mean Flow quations [21] As the transformed residual flux r (x, l) and mean flux hq(x, l)i are non-arian, the notion of effetive hydrauli ondutivity loses meaning in the ontext of flow predition based on ensemble mean quantities. In some speial ases, loalization of the mean equation is possible so that the flow beomes approximately arian in the mean. Loalization is valid when (1) rhh(x, l)i varies slowly in spae (not neessarily in time) wherever a (y, x, l) is nonzero and (2) r (x, l) does likewise wherever d (y, x, l) is nonzero. Then equation (19) an be approximated via r a dyr h þ d ðy; x; l Þdy r ð52þ 6of19

7 05104 Y T AL.: ANALYSIS OF CONITIONAL MAN TRANSINT FLO and if [I R d (y, x, l)dy] 1 exists where I is the identity tensor, equation (52) an be rewritten in loal form as r k r h ð53þ where k ¼ I d dy 1 a dy ð54þ Substituting equation (53) into (15) yields the onditional mean arian expression hqx; ð lþi K ;app r h ð55þ where K,app (x, l) is a spatially varying onditional apparent hydrauli ondutivity tensor in the Laplae domain, given by K ;app ðx; l Þ ¼ hkðxþi I k ð56þ quations (52) (56) are idential to those obtained previously by Tartakovsky and Neuman [1998b] upon first developing moment expressions in spae-time and then applying the Laplae transform. There is no known way to loalize seond-moment equations. Hene loalization yields no information about preditive unertainty. [22] To render loalized mean equations workable, one needs to evaluate the apparent hydrauli ondutivity either by inverse method against measured values of head and flux or by reursive approximation. The latter approah yields to seond-order ;app ðx; lþ ¼ K GðxÞ 1 þ s2 Y ðxþ I 2 K G ðyþhy 0 ðxþy 0 ðyþi r x r T y G ð0þ dy K ½Š 2 ð57þ whih requires omputing the zero-order Green s funtion (not needed for inverse solution). 6. Numerial Approah [23] e solve the reursive nonloal moment equations and loalized mean flow equations in the Laplae domain using a Galerkin finite element sheme similar to that developed for steady state flow by Guadagnini and Neuman [1999b]; details beyond those given here are given by Ye [2002]. For purposes of Monte Carlo simulation, we use a standard Galerkin finite element sheme on the same grid to solve the Laplae-transformed stohasti flow equations (8) (11) for random realizations of hydrauli ondutivity. Independent Monte Carlo runs are onduted in parallel on multiple proessors. e then invert the results numerially from the Laplae bak into the time domain using a parallelized version of an algorithm devised by Crump [1976] and e Hoog et al. [1982]. The latter uses trapezoidal rule to disretize the Laplae parameter into k = 0, 1, 2,...,2M p + 1 values l k = l 0 + ikp/t where l 0 = ln()/(2t), i = ffiffiffiffiffiffi 1, T is half a Fourier period, and is a dimensionless disretization error. Following Crump [1976], Sudiky [1989], and Gambolati et al. [1997], we set T =0.8t max, where t max is the simulation period, = 10 6, and M = 23 to yield stable results. 7of19 Figure 1. Computational grid, boundary onditions, onditioning points (solid squares) and pumping well (solid irle). [24] Sudiky [1989] and Amore et al. [1999] have shown that sine the algorithm relies on a omplex Laplae parameter, it is more aurate than algorithms using realvalued Laplae parameters, whih tend to be unstable. Nevertheless, the Gibbs phenomenon may ause the algorithm to yield unstable solutions for t < 0.1t max (whih did not happen in our ase). To avoid suh instabilities, Gambolati et al. [1997] suggested varying t max so as to ensure that t always remains in the stable range. Another way to eliminate osillations at relatively small time is to inrease M [Pini and Putti, 1997]. [25] As finite element moment equations assoiated with various l k values are mutually independent, we solve them in parallel on multiple proessors, eah of whih yields a solution for one or more (depending on the number of proessors) l k values. Inverse head at any node j in the finite element grid is alulated aording to " h j ðþ¼ t 1 T exp ð l 0tÞ 1 2Mþ1 2 h j;0 þ X # Re h j;k expðikpt=tþ ð58þ k¼1 where h j,k is the orresponding transformed head at l k. Though it is possible to perform these expliit alulations in parallel, our use of a fast quotient-differene algorithm renders it unneessary. [26] To ompare the auray of the three solution methods (reursive nonloal and loalized moment; Monte Carlo) we base all of them on Laplae transformation (aording to Sudiky [1989] and Ye [2002], the latter is more aurate than traditional time marhing when applied to a deterministi or single Monte Carlo run). As Monte Carlo simulations are mutually independent we automatially assign eah to one proessor, whih performs the Laplae inversion sequentially for all l k values (rather than distributing the inversion among different proessors as we do in the ase of moment equations). Beause of ommuniation between proessors [Mendes and Pereira, 2003] this results in less than 100% effiieny. 7. Two-imensional xamples [27] e illustrate our approah on onditional and unonditional examples of superimposed mean uniform and onvergent flows in a two-dimensional retangular domain. The domain is subdivided into 800 square elements (20 rows and 40 olumns) of uniform size x 1 = x 2 = 0.2 measured in arbitrary onsistent length units (Figure 1). The length of the domain is L 1 = = 8 and its width is L 2 =20

8 05104 Y T AL.: ANALYSIS OF CONITIONAL MAN TRANSINT FLO = 4. A uniform deterministi head H L = 8 (in similar length units) is presribed on the left boundary (x 1 = 0) and a onstant head H R = 4 on the right boundary (x 1 = 8). The bottom (x 2 = 0) and top (x 2 = 4) boundaries are impermeable. Constant deterministi initial head H 0 = 4 is assigned to all but the presribed head boundary nodes of the grid. A pumping well (point sink) of deterministi strength Q p (in arbitrary onsistent units of length per time) is plaed at the enter node of the domain (x 1 =4,x 2 = 2). Transient flow is simulated over a period t sim = 20 of similar time units. Solutions of the finite element equations are obtained at disrete time steps t = 0.1, 0.3, 0.5, 0.7, 0.9, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 15, 18, and 20. e employ the same omputational grid and time steps for all moment and Monte Carlo solutions to render them diretly omparable. [28] The log hydrauli ondutivity Y(x) is taken to be statistially homogeneous and isotropi with the widely used [agan, 1989; hang, 2002] exponential ovariane funtion Cr ðþ¼s 2 Y exp r l ð59þ where r is separation distane (lag) between two points in the domain and l is the integral sale of Y. This hoie of ovariane funtion is purely for illustration purposes, our solution algorithms being equally apable of admitting other valid forms of this funtion. hereas an exponential ovariane auses steady state head variane to grow without limit as the distane between presribed head boundaries inreases, this is not a problem in a finite domain where boundaries render the head statistially nonhomogeneous [agan, 1989]. [29] For purposes of Monte Carlo simulation we assume that Y is multivariate Gaussian (no suh distributional assumption is required for the moment solutions). Random Y fields are generated using the sequential Gaussian simulation ode SGSIM [eutsh and Journel, 1998]. ah element is assigned a onstant ondutivity orresponding to the value generated at its enter. e generate 2500 unonditional realizations of Y with mean hy i = 0, variane s Y 2 = 4, and orrelation sale l = 1. The grid thus inludes a minimum of five elements per orrelation sale as reommended by, among others, Ababou et al. [1989]. [30] For purposes of onditional simulation, we measure Y (without error) aross an unonditional random field at 12 evenly distributed onditioning points shown by solid squares in Figure 1. e then generate 2500 orresponding onditional realizations of Y. Figure 2 depits images of one suh onditional realization and the onditional sample mean m Y and variane S Y 2 of all 2500 realizations. As expeted, the onditional Y fields are statistially nonhomogeneous in that their mean and variane vary with loation. [31] To render our omparison of nonloal and loalized moment solutions with Monte Carlo results meaningful, the same input statistis (mean, variane, and ovariane of Y ) are used for all three. e note that for omparative purposes it is not neessary that the Monte Carlo simulations fully stabilize, only that all three sets of input statistis be idential (this was originally pointed out by Guadagnini and Neuman [1999a, 1999b]). In pratie, nonloal and loalized solutions do not require generating random fields, Figure 2. Images of (a) a onditional realization of Y, onditional sample (b) mean m Y and () variane S Y 2 of NMC = 2500 realizations with unonditional s Y 2 =4,l =1. See olor version of this figure at bak of this issue. and one would typially infer the orresponding input statistis from measurements, using geostatistial methods. Upon plotting the sample mean and variane of head and flux at seleted points in spae-time versus the number of realizations (Ye [2002], not shown here), we find that whereas the onditional sample means beome effetively stable after about 1000 realizations, the onditional sample varianes ontinue to vary slowly even as the number of realizations approahes our maximum of Conditional Mean Hydrauli Head [32] Visual examination of onditional mean head ontours (Ye [2002], not shown here) obtained through reursive nonloal, loalized, and Monte Carlo solutions indiates that the three sets of solutions agree remarkably well. Figure 3 depits profiles of mean head and seondorder mean head orretions at time t = 5 along two setions. The loalized moment solution is seen to be as aurate (in omparison to the Monte Carlo results) as the nonloal reursive solution exept near the pumping well where it beomes relatively poor due to a steep inrease in mean head gradient. The orresponding nonloal solution is surprisingly aurate onsidering the strong heterogeneity of the underlying Y field (unonditional s 2 Y = 4). This is 8of19

9 05104 Y T AL.: ANALYSIS OF CONITIONAL MAN TRANSINT FLO Figure 3. (a and b) Conditional mean head and ( and d) seond-order head orretions at time t =5 along longitudinal setion x 2 = 2 and transverse setion x 1 = 4 for s Y 2 =4,l =1. due to the seond-order orretion, whih is seen to be signifiant near the well. Results at times other than t =5 (not shown) exhibit similar patterns of behavior. [33] Figure 4 ompares temporal variations in onditional mean head at the pumping well (4.0, 2.0) and at an upstream point (2.0, 2.0) as omputed by the three methods of solution (note that the initial head aross the domain is 4, as is the head on the left boundary). All three solutions evolve at similar rates at both points. hereas the nonloal and loalized solutions agree losely with Monte Carlo results at the upstream point, the loalized solution seriously overestimates these results at the pumping well. Indeed, the seond-order nonloal orretion at the latter is signifiant at all times. It initially dereases with time and later slowly inreases Conditional Variane of Hydrauli Head [34] Figure 5 ompares profiles of onditional head variane at t = 5 obtained by the nonloal (dashed) and Monte Carlo (solid) solution methods. Although our nonloal results represent the lowest possible order of approximating head variane, they ompare remarkably well with MC results exept near the pumping well, where they underestimate preditive unertainty. lsewhere along the two setions the nonloal solution overestimates head unertainty by a small amount. Conditional head variane is zero at the upstream (left) and downstream (right) deterministi irihlet boundaries, inreasing toward the enter of the domain with a sharp rise near the pumping well. [35] Figure 6 depits profiles of onditional head variane along setion x 2 = 2 at times t = 0.5, 1, and 10. Profiles after t = 10 do not hange visibly with time and are therefore not shown. There is good qualitative agreement between the nonloal and Monte Carlo solutions, whih improves quantitatively with time. The manner in whih onditional head variane evolves with time at upstream point (2.0, 2.0) and downstream point (6.0, 2.0) is illustrated in Figure 7. Unertainty is seen to inrease and then derease monotonially with time upstream and inrease downstream of the well. There is no qualitative differene between the nonloal and Monte Carlo results, though they differ quantitatively upstream of the well at early time Conditional Covariane of Hydrauli Head [36] The ovariane is spatially nonhomogeneous due to onditioning and foring, with aeptable agreement between nonloal and Monte Carlo results (Ye [2002], not shown here). Figure 8 illustrates how the temporal onditional head ovariane C h (x 1 = y 1 =2,x 2 = y 2 =2,t, s) varies with time t relative to three referene times s = 0.5, 1, and 10. C h (x = y, t, s) is seen to be nonstationary in time due to the transient nature of the problem. For example, C h (x = y, t =5, s = 10) differs from C h (x = y, t = 15, s = 10) even though both are assoiated with the same time lag, jt sj =5. Agreement between the nonloal and Monte Carlo solutions improves as time and referene time inrease Conditional Mean Hydrauli Flux [37] Figure 9 depits profiles of longitudinal mean flux, seond-order mean flux orretion and residual flux at t = 5 along two setions. Analogous results for transverse mean flux are shown in Figure 10. Both the nonloal and loalized solutions ompare favorably with Monte Carlo simulations, the former more losely than the latter. Seond-order omponents of the longitudinal 9of19

10 05104 Y T AL.: ANALYSIS OF CONITIONAL MAN TRANSINT FLO Figure 4. Conditional mean head versus time at points (a) x 1 =4,x 2 = 2 (pumping well) and (b) x 1 = 2.0, x 2 =2.0 and seond-order head orretions at these points for s 2 Y =4, l =1. Figure 6. Profiles of onditional head variane along ross setions x 2 = 3 at times (a) t =0.5,(b)t = 1, and () t =10 obtained with NMC = 2500 Monte Carlo (solid urves) and nonloal moment (dashed urves) solutions for s 2 Y =4,l =1. mean flux are seen to be strongly affeted by the loation of onditioning points (open irles in Figures 9e and 9f ). Figure 9d indiates that longitudinal mean flux exhibits a maximum at onditioning point (3.1, 0.7) (x 1 /L 1 = ) where mean log ondutivity is the largest (Figure 2b), ausing mean longitudinal flux to onverge. Figure 11 illustrates how onditional longitudinal and Figure 5. Conditional head variane along setions (a) x 2 = 2 and (b) x 2 = 3 at time t = 5 obtained with NMC = 2500 Monte Carlo (solid urves) and nonloal moment (dashed urves) solutions for s Y 2 =4,l =1. 10 of 19 Figure 7. Conditional head variane versus time at points x 1 =2,x 2 = 2 (upstream) and x 1 =6,x 2 = 2 (downstream) with NMC = 2500 Monte Carlo (solid urves) and nonloal moment (dashed urves) solutions for s 2 Y =4,l =1.

11 05104 Y T AL.: ANALYSIS OF CONITIONAL MAN TRANSINT FLO Figure 8. Conditional head ovariane for various referene time s = 0.5, 1.0, and 10 at points x 1 = y 1 = 2.0, x 2 = y 2 = 2.0 obtained with NMC = 2500 Monte Carlo (solid urves) and nonloal moment (dashed urves) solutions for s 2 Y =4,l =1. transverse mean fluxes vary with time at an upstream point Conditional Cross-Covariane Tensor of Flux at ero Lag [38] The omponents of the onditional ross-ovariane tensor of flux at zero lag are designated by C q11 (x, x, t, t) (longitudinal flux variane), C q22 (x, x, t, t) (transverse flux variane), and C q12 (x, x, t, t) = C q21 (x, x, t, t) (ross-ovariane between longitudinal and transverse fluxes). Figures depit profiles of these omponents at time t = 5 as omputed by the nonloal and Monte Carlo methods. e note one again that although our nonloal solution represents the lowest possible order of approximating seond moments, these ompare remarkably well with the Monte Carlo results, even near the well. Flux varianes exhibit maxima in the viinity of the pumping well. The profile of C q11 (x, x, t, t) along setion x 2 = 1.9 exhibits loal minima at onditioning points (open irles in Figure 12a). In Figures 13 and 14, C q22 (x, x, t, t) and C q12 (x, x, t, t) are lose to zero near deterministi boundaries where transverse flux vanishes due to uniform deterministi boundary Figure 9. Profiles of onditional longitudinal mean flux, seond-order longitudinal mean flux orretion and residual flux at time t = 5 along setions (a ) x 2 = 1.9 and (d f) x 2 = 0.7 for s 2 Y =4,l =1. Open irles designate onditioning points. 11 of 19

12 05104 Y T AL.: ANALYSIS OF CONITIONAL MAN TRANSINT FLO Figure 10. Profiles of onditional transverse mean flux, seond-order transverse mean flux orretion and residual flux at time t = 5 along setions (a ) x 2 = 1.9 and (d f ) x 2 = 0.7 for s 2 Y =4,l =1. onditions. The profile of C q12 (x, x, t, t) is more or less antisymmetri about the pumping well along setion x 2 = 1.9 (Figure 14a) and uniform along setion x 2 =2.5 (Figure 14b). [39] Figures illustrate profiles of C q11 (x, x, t, t) and C q22 (x, x, t, t) along setion x 2 = 1.9 at times t = 0.5, 1, and 10. Variation with time after t = 10 is negligible. Similar profiles of C q12 (x, x, t, t) hange slightly over time. The nonloal solution agrees reasonably well with Monte Carlo results, somewhat better at late than at early time. Both C q11 (x, x, t, t) and C q22 (x, x, t, t) are generally larger upstream than downstream of the pumping well. Figure 17 shows how omponents of the ross-ovariane tensor of flux evolve with time upstream and downstream of the well Comparison of Conditional and Unonditional Moment Solutions [40] For the sake of brevity, we do not illustrate unonditional solutions orresponding to the above flow problem. Instead, we list in Table 1 the maximum normalized absolute (MNA) and root mean square (MNRMS) deviations of onditional and unonditional nonloal moment solutions from orresponding Monte Carlo results, and in Table 2 the maximum and maximum averaged (MA) varianes of head and flux as obtained from the onditional and unonditional moment solutions. MNA and MNRMS are auray measures defined as MNA ¼ max i;j MNRMS ¼ max j P i;m t j Pi;MC t j P i;mc t j X N 2 P i;m t j Pi;MC t j N i¼1 6 1 X N P i;mc t j N i¼1 1=2 ð60þ where P i,m (t j ) is a moment (mean or variane of head or flux) omputed at node i and time t j using nonloal moment equations, and P i,mc (t j ) is the same moment omputed by Monte Carlo simulation, N being the total number of nodes. The preditive variane measures in Table 2 are maximum = max i; j 1 P i,m (t j ) and MA = max j N P N i¼1 P i,m (t j ) where P now stands for the variane of head or flux. [41] Table 1 demonstrates that onditioning enhanes the overall auray of the nonloal moment solution, as ompared to Monte Carlo results. Table 2 shows that onditioning brings about a signifiant redution in the maximum and overall preditive variane of head and flux. The latter is seen, for example, upon omparing profiles of 12 of 19

13 05104 Y T AL.: ANALYSIS OF CONITIONAL MAN TRANSINT FLO Figure 11. Conditional (a) longitudinal and (b) transverse mean flux versus time at point x 1 = 1.9, x 2 = 1.9 (upstream) for s 2 Y =4,l =1. Figure 13. Profiles of onditional variane of transverse flux along setions (a) x 2 = 1.9 and (b) x 2 = 2.5 at time t =5 obtained with NMC = 2500 Monte Carlo (solid urves) and nonloal moment (dashed urves) solutions for s 2 Y =4,l =1. Open irles designate onditioning points. Figure 12. Profiles of onditional variane of longitudinal flux along setions (a) x 2 = 1.9 and (b) x 2 = 2.5 at time t =5 obtained by NMC = 2500 Monte Carlo (solid urves) and nonloal moment (dashed urves) solutions for s 2 Y =4,l =1 (onditioning points are marked by irles). 13 of 19 Figure 14. Profiles of onditional ross ovariane between longitudinal and transverse flux at zero lag along setions (a) x 2 = 1.9 and (b) x 2 = 2.5 at time t = 5 obtained with NMC = 2500 Monte Carlo (solid urves) and nonloal moment (dashed urves) solutions for s 2 Y =4,l = 1. Open irles designate onditioning points.

14 05104 Y T AL.: ANALYSIS OF CONITIONAL MAN TRANSINT FLO of eah method optimal; as exploring all these possibilities would be outside the sope of this paper, we onsider our omparison of omputational effiienies to be of a preliminary nature. [43] Table 3 ompares Monte Carlo (MC) and moment (M) runtimes required to ompute onditional mean and variane of head and flux in the numerial example disussed earlier using one, four, and eight proessors on a University of Arizona SGI 2000 superomputer (reently replaed by a muh more powerful HP TRU64 superomputer). CPU run time is measured by a portable bath system (PBS), whih manages jobs submitted to an isolated queue. Our nonloal moment method onsistently outperforms the Monte Carlo method by a signifiant margin. This is true whether we use one or multiple proessors. For example, using one proessor to ompute onditional mean head takes about 4 times as long with 2500 Monte Carlo runs as with the nonloal approah; omputing onditional head variane takes twie as long. These ratios of run time inrease onsiderably with an inrease in the number of proessors. The same holds true for the unonditional ase. [44] In addition to grid size, numerial sheme, matrix solver, and method of parallelization, omputational effiieny is also influened by the number of random soure terms, number and loation of onditioning points, and the number of Monte Carlo runs. In this preliminary omparison we vary the grid size while keeping all other fators Figure 15. Profiles of onditional variane of longitudinal flux along setion x 2 = 1.9 at times (a) t =0.5,(b)t = 1, and () t = 10 for s 2 Y = 4, l = 1. Open irles designate onditioning points. unonditional variane of head (Figure 18) and longitudinal flux (Figure 19) at time t = 5 with orresponding profiles of onditional variane (Figures 5 and 12, respetively). A omparison of Figures 19 and 12 illustrates the improvement in auray ahieved through onditioning. 8. Preliminary Comparison of Computational ffiienies [42] e ompare the omputational effiienies of reursive nonloal moment and Monte Carlo methods in terms of run times, rather than memory, as (1) optimizing one usually degrades the other and (2) memory is not a major limitation on modern superomputers, whih we use for our examples. In general, the reursive nonloal method is assoiated with a larger number of variables (e.g., ovariane of log hydrauli ondutivity, zero- and seond-order head and flux) and therefore requires more memory than does the Monte Carlo method. To ensure that run times are omparable, we use the same omputational grid, finite element sheme, and matrix solver for both methods. Parallel omputing is done in FORTRAN using message passing interfae (MPI), whih is portable aross a variety of omputing platforms, oupled with single program multiple data (SPM) programming, whih allows the same ode to run on multiple proessors. e reognize that different grid sizes, numerial shemes, matrix solvers, and methods of parallelization may be required to render the run time Figure 16. Profiles of onditional variane of transverse flux along setion x 2 = 1.9 at times (a) t =0.5,(b)t = 1, and () t = 10 for s 2 Y = 4, l = 1. Open irles designate onditioning points. 14 of 19

15 05104 Y T AL.: ANALYSIS OF CONITIONAL MAN TRANSINT FLO Table 2. Maximum and Maximum Averaged (MA) Conditional and Unonditional Moment Solutions Conditional Unonditional Quantities Maximum MA Maximum MA Head variane Flux q 1 variane Flux q 2 variane Figure 17. Conditional (a) variane of longitudinal flux, (b) variane of transverse flux, and () ross ovariane between longitudinal and transverse flux at a zero lag versus time at points x 1 = 1.9, x 2 = 1.9 (upstream) and x 1 = 5.9, x 2 = 1.9 (downstream) with NMC = 2500 Monte Carlo (solid urves) and nonloal moment (dashed urves) solutions for s 2 Y =4,l =1. fixed, without onditioning. In partiular, we employ a diret LU fatorization method (using the LSACB, LFTCB, and LFSCB routines of the IMSL library on and 2000 Monte Carlo runs. As full stabilization of all orresponding sample statistis may require many more suh runs (e.g., 20,000 in the ase of S. Li et al. [2003] and 9000 and in the ase of L. Li et al. [2003]), our omparison is biased in this sense in favor of the Monte Carlo method. It is biased in favor of the nonloal reursive method in that the latter would entail extra terms for seond moments of random soures. [45] Figure 20a depits ratios between runtimes required for Monte Carlo and reursive nonloal omputations of mean head and head variane on the University of Arizona HP TRU64 superomputer using one and 16 proessors. Our onditional example runs about 3 times faster on the HP than on the SGI mahine. Following optimization of our odes on the HP mahine, we find that as grid size inreases eightfold from 800 to 6400 finite elements, the run time ratio for mean head dereases by a fator of 7.5 from to 9.18 and that for head variane by a fator of 11 from to 1.36 with one proessor. hen 16 proessors are used, these ratios drop slightly in a manner that does not appear to have pratial signifiane. The drop is due to a somewhat better parallel performane of the Monte Carlo method in omparison to the nonloal reursive algorithm, as is seen in Figure 20b; the latter ompares the speedups of the Monte Carlo, nonloal mean head, and nonloal head Table 1. Maximum Normalized Absolute (MNA) and Root- Mean-Square (MNRMS) eviations of Conditional and Unonditional Moment Solutions From Corresponding Monte Carlo Results Conditional Unonditional Quantities MNA, % MNRMS, % MNA, % MNRMS, % Mean head Mean flux q Mean flux q 2 NA NA Head variane Flux q 1 variane Flux q 2 variane Figure 18. Unonditional head variane along setions (a) x 2 = 2 and (b) x 2 = 3 at time t = 5 obtained with NMC = 2500 Monte Carlo (solid urves) and nonloal moment (dashed urves) solutions for s 2 Y =4,l =1. 15 of 19

16 05104 Y T AL.: ANALYSIS OF CONITIONAL MAN TRANSINT FLO Figure 19. Unonditional variane of longitudinal flux along setions (a) x 2 = 1.9 and (b) x 2 = 2.5 at time t =5 obtained by NMC = 2500 Monte Carlo (solid urves) and nonloal moment (dashed urves) solutions for s 2 Y =4, l =1. variane solutions with number of proessors for the largest grid of 6400 elements, a speedup of 1:1 being ideal. It is ommon for speedup to derease with the number of proessors due to an inrease in overhead aused by fators suh as system idling and ommuniation between proessors. The speedup of nonloal mean head omputation is lower than that of nonloal variane omputation beause the former requires solving four oupled moment equations (zero-order head, zero-order Green s funtion, seond-order residual flux, and seond-order head) and the latter only three (zero-order head, zero-order Green s funtion, and seond-order head variane). 9. Conlusions [46] Our work leads to the following major onlusions: [47] 1. It is possible and omputationally feasible to render optimum unbiased preditions of transient groundwater flow, and to assess the orresponding predition unertainty in bounded, randomly heterogeneous porous media onditional on measurements without resorting to Monte Carlo simulation. e have done so by solving Laplae-transformed reursive onditional nonloal moment equations using finite elements and numerial inversion of the results bak into the time domain (use of the Laplae transform is limited to linear problems suh as ours). Although our theoretial approah allows aounting formally for unertainty in initial, boundary, and soure terms, we have not yet explored this feature of our solution method numerially. [48] 2. The moment equations we use are distribution-free and thus obviate the need to know or assume the multivariate distributions of random input parameters or foring terms, as is required for the Monte Carlo approah. [49] 3. Our approah differs fundamentally from standard perturbative solutions in that it originates in a set of onditional moment equations whih are exat, ompat, formally inorporate boundary effets, provide a unique insight into the nature of the problem, and lead to unique loalized moment equations that look like standard deterministi flow (and transport) equations, allowing one to interpret the latter within a onditional stohasti frame- Table 3. Run Times, in Minutes, of Monte Carlo (MC) and Nonloal Moment (M) Solutions Required to Compute Conditional Mean and Variane of Head and Flux Using P = One, Four, and ight Proessors on the University of Arizona SGI 2000 Superomputer Variables hhi hqi s h 2 s q 2 P = 1 MC M P = 4 MC M P = 8 MC M Figure 20. (a) Run time ratio between 2000 Monte Carlo runs and mean head and head variane solutions for different problem sizes using one and 16 proessors. (b) Speedup of these solutions for problem of 6400 elements using multiple proessors. 16 of 19

17 05104 Y T AL.: ANALYSIS OF CONITIONAL MAN TRANSINT FLO work. Both the nonloal and loalized equations desribe the spae-time evolution of moments representing random funtions rendered statistially inhomogeneous (in spae) and nonstationary (in time) due to the ombined effets of soures, boundaries, and onditioning. To approximate the exat moment equations reursively, we use a valid expansion in terms of (deterministi) moments rather than a theoretially invalid expansion in terms of random quantities, whih may (but is not guaranteed to) yield valid results after subsequent averaging. Our approah leads to loalized approximations whih do not arise from standard perturbation shemes. [50] 4. Our approah differs omputationally from standard perturbative solutions in that it leads to loalized moment equations that are almost as easy to solve as standard deterministi flow equations, and to reursive nonloal moment equations written in terms of Green s funtions, whih are independent of internal soures and the magnitudes of boundary terms. One these funtions have been omputed for a given boundary onfiguration, they an be used repeatedly to obtain solutions for a wide range of internal soures and boundary terms (senarios). [51] 5. Though our loalized algorithm is mathematially muh simpler and omputationally more effiient than the nonloal algorithm, it has the disadvantage of being less aurate and unable to provide information about preditive unertainty. [52] 6. Our nonloal reursive moment solution annot be guaranteed to onverge for strongly heterogeneous media with log hydrauli ondutivity standard deviation, s Y,of order 1 or larger. Yet upon submitting it to a severe test by onsidering superimposed mean uniform and onvergent flows in a strongly heterogeneous medium with s Y =4,the solution proved to be remarkably aurate upon onditioning it on 12 measured log ondutivity values. Removing the onditioning points aused auray to diminish but the solution remained aeptable. Hene the method appears to be appliable to omplex flows in strongly heterogeneous media with or without onditioning. [53] 7. Conditioning was shown to bring about a signifiant redution in the preditive unertainty of head and flux. [54] 8. A preliminary omparison of superomputer runtimes for mean head and its variane suggests that our nonloal moment algorithm onsistently outperforms the Monte Carlo method by a signifiant margin when the same diret matrix solver is used for all three (we do not presently know how using iterative solvers would have affeted this onlusion). The ratio between Monte Carlo and nonloal reursive run times diminishes toward an apparent asymptote as problem size inreases, regardless of number of proessors. [55] 9. Our nonloal algorithm is potentially well suited for groundwater optimization problems and for the investigation of various flow senarios in randomly heterogeneous aquifers. This is so beause the orresponding finite element matries are independent of internal soure terms or the magnitudes of initial and boundary terms. Hene they an be fatored one and then used repeatedly to ompute the effet of varying the foring terms on predited head and flux and on the assoiated predition errors. This is in ontrast to the Monte Carlo method, whih requires repeating all simulations whenever there is a hange in foring terms. [56] 10. The underlying exat and reursive moment equations, as well as the proposed omputational algorithm, are valid in both two and three dimensions, though we have implemented them here in two dimensions. Appendix A [57] The random Green s funtion G(y, x, l), assoiated with equations (8) (11), satisfies r y KðyÞr y G þ Ss ðyþlg ¼ dðy xþ y 2 ða1þ subjet to homogeneous boundary onditions G ¼ 0 y 2 G ða2þ r y Gny ð Þ ¼ 0 y 2 G N ða3þ where d(y x) is the ira delta funtion. G(y, x, l) is symmetri in the Laplae domain. Substituting equations (12) (14) into (8) (11), taking onditional ensemble mean, and subtrating the latter from the former yields h i h i r KðxÞrh þr K 0 ðxþr h 0 r K 0 ðxþrh 0 S sðxþlh ¼ S s ðxþh0 0 x 2 ða4þ subjet to h 0 ¼ H 0 x 2 G ða5þ h KðxÞrh 0 þk 0 i ðxþr h K0 ðxþrh 0 nx ð Þ ¼ Q 0 x 2 G N ða6þ xpressing equations (A4) (A6) in terms of y, multiplying by G(y, x, l), integrating over, and applying Green s first identity twie gives the desired expression: h 0 ¼ hk 0 ðyþr y h 0 ðy; lþi r y Gdy K 0 ðyþr y hhðy; lþi r y Gdy þ f 0 ðy; lþgdy þ S s ðyþh0 0 ð y ÞG ð y; x; l Þdy H 0 ðy; lþkðyþr y Gny ð Þdy G þ Q 0 ðy; lþgdy ða7þ G N Applying the operator K 0 (x)r to equation (A7), taking onditional ensemble mean, and reognizing that driving 17 of 19

18 05104 Y T AL.: ANALYSIS OF CONITIONAL MAN TRANSINT FLO fores are statistially unorrelated with hydrauli ondutivity and hene the random Green s funtion, leads to K 0 ðxþrh ¼ K 0 ðxþr x r T y G ð y; x; l Þ K 0 ðyþr h 0 ðy; lþ dy hk 0 ðxþk 0 ðyþr x r T y G ð y; x; l Þ r y hðy; lþ dy ða8þ This oupled with r (x, l) = hk 0 (x)r h 0 (x, l)i = hk 0 (x)rh(x, l)i leads immediately to equations (19) (21). Appendix B [58] Multiplying equations (A4) (A6) by h 0 (y, s) and taking onditional ensemble mean gives equations (22) (24). xpressing equation (B5) of Tartakovsky and Neuman [1998a] in terms of y and s, s h 0 ðy; sþ ¼ hk 0 ðþr z z h 0 ðz; tþi r z Gðz; y; s tþdzdt 0 s K 0 ðþr z z hhðz; tþi r z Gðz; y; s tþdzdt 0 þ S s ðþh z 0 0 ðþg z ð z; y; s Þdz s þ f 0 ðz; tþgðz; y; s tþdzdt 0 s KðÞr z z Gðz; y; s tþnz ðþh 0 ðz; tþdzdt 0 G s þ Q 0 ðz; tþgðz; y; s tþdzdt ðb1þ G N 0 premultiplying by K 0 (x)r xh0 (x, l) and K 0 (x), and taking onditional ensemble mean leads to the mixed onditional moments (25) and (26), respetively. Multiplying equation (B1) by H 0 0, f 0, H 0, and Q 0, and taking onditional ensemble mean gives expliit expressions for the mixed onditional moments in equations (22) (24): H0 0 ðx Þh0 ðy; sþ ¼ S s ðþh z 0 0 ðxþh 0 0 ðþ z hgðz; y; sþi dz ðb2þ f 0 h 0 ðy; sþ s ¼ 0 s H 0 h 0 ðy; sþ ¼ 0 f 0 f 0 ðz; tþ hgðz; y; s tþi dzdt G H 0H 0 ðz; tþ hkðþ z r y Gðz; y; s tþ nz ðþdzdt ðb3þ ðb4þ Q 0 h 0 ðy; sþ ¼ s Q 0 Q 0 ðz; tþ hgðz; y; s tþi dzdt ðb5þ 0 G N Multiplying equation (A7) by h 0 (x, t) and taking onditional ensemble mean gives equation (27). Substituting equations (12) (14) into (8) yields q 0 ¼ r hkðxþi rh 0 K 0 ðxþr h K0 ðxþrh 0 ðb6þ xpressing equation (B13) of Tartakovsky and Neuman [1998a] in terms of y and s, q 0 ðy; s Þ ¼ r ðy; sþ hkðyþi rh 0 ðy; sþ K 0 ðyþrhhðy; sþ K 0 ðyþrh 0 ðy; sþ ðb7þ premultiplying the transpose of equation (B7) by (B6), and taking onditional ensemble mean gives equation (28). Multiplying equation (A7) by K 0 (y) and taking ensemble mean yields equation (29). Applying the operator r x to equation (A7), postmultiplying by r x T h 0 (x, s), and taking onditional ensemble mean leads to hr xh 0 r T x h0 ðx; tþi ¼ hr x r T y G ð y; x; l Þr ðy; lþr T x h0 ðx; tþi dy hk 0 ðyþr x r T y G ð y; x; l Þr yhhðy; lþi r T x h0 ðx; tþi dy þ hf 0 ðy; lþr x Gr T x h0 ðx; tþi dy þ S s ðyþhh0 0 ðyþr xgr T x h0 ðx; tþi dy hkðyþr x r T y G ð y; x; l Þny ð Þ H 0 ðy; lþr T x h0 ðx; tþi dy G þ h Q 0 ðy; lþr x Gr T x h0 ðx; tþi dy ðb8þ G N [59] Aknowledgments. This researh was supported by the U.S. Nulear Regulatory Commission under ontrat NRC and by NSF/ITR grant AR Referenes Ababou, R.,. MLaughlin, L.. Gelhar, and A. F. B. Tompson (1989), Numerial simulation of three-dimensional saturated flow in randomly heterogeneous porous media, Transp. Porous Media, 4, Bear, J. (1972), ynamis of Fluid in Porous Media, over, Mineola, N. Y. Carslaw, H. S., and J. C. Jaeger (1959), Condution of Heat in Solids, Oxford Univ. Press, New York. Crump, K. S. (1976), Numerial inverse of Laplae transform using a Fourier series approximation, J. Asso. Comput. Mah., 23(1), agan, G. (1982), Analysis of flow through heterogeneous random aquifers: 2. Unsteady flow in onfined formations, ater Resour. Res., 18(5), agan, G. (1989), Flow and Transport in Porous Formations, Springer- Verlag, New York. Amore, L., G. Laetti, and A. Murli (1999), An implementation of a Fourier series method for the numerial inversion of the Laplae transform, ACM Trans. Math. Software, 25(3), e Hoog, F. R., J. H. Knight, and A. N. Stokes (1982), An improved method for numerial inversion of Laplae transform, SIAM J. Si. Stat. Comput., 3(3), eutsh, C. V., and A. G. Journel (1998), GSLIB: Geostatistial Software Library and User s Guide, 2nd ed., Oxford Univ. Press, New York. Farrell,. A., A.. oodbury, and. A. Sudiky (1998), Numerial modeling of mass transport in hydrogeologi environments: Performane omparison of the Laplae transform Galerkin and Arnoldi model redution shemes, Adv. ater Resour., 21, Gambolati, G., C. Gallo, and C. Panioni (1997), Comment on A ombined Laplae transform and streamline upwind approah for nonideal transport of solute in porous media by Linlin Xu and Mark L. Brusseau, ater Resour. Res., 33(2), i 18 of 19

19 05104 Y T AL.: ANALYSIS OF CONITIONAL MAN TRANSINT FLO Guadagnini, A., and S. P. Neuman (1999a), Nonloal and loalized analyses of onditional mean steady state flow in bounded, randomly nonuniform domains: 1. Theory and omputational approah, ater Resour. Res., 35(10), Guadagnini, A., and S. P. Neuman (1999b), Nonloal and loalized analyses of onditional mean steady state flow in bounded randomly nonuniform domains: 2. Computational examples, ater Resour. Res., 35(10), Guadagnini, A., and S. P. Neuman (2001), Reursive onditional moment equations for advetive transport in randomly heterogeneous veloity fields, Transp. Porous Media, 42(1/2), Indelman, P. (1996), Average of unsteady flows in heterogeneous media of stationary ondutivity, J. Fluid Meh., 310, Indelman, P. (2000), Unsteady soure flow in weakly heterogeneous porous media, Comput. Geosi., 4, Indelman, P. (2002), On mathematial models of average flow in heterogeneous formations, Transp. Porous Media, 48, Li, L., H. A. Thelepi, and. hang (2003), Perturbation-based moment equation approah for flow in heterogeneous porous media: Appliability range and analysis of high-order terms, J. Comput. Phys., 188, Li, S. G.,. MLaughlin, and H. S. Liao (2003), A omputational pratial method for stohasti groundwater modeling, Adv. ater Resour., 26, Mendes, B., and A. Pereira (2003), Parallel Monte Carlo river (PMC) A software pakage for Monte Carlo simulations in parallel, Comput. Phys. Commun., 151(1), Morgan, M. G., and M. Henrion (1990), Unertainty: A Guide to ealing with Unertainty in Qualitative Risk and Poliy Analysis, Cambridge Univ. Press, New York. Naff, R. L.,. F. Haley, and. A. Sudiky (1998), High-resolution Monte Carlo simulation of flow and onservative transport in heterogeneous porous media: 1. Methodology and flow results, ater Resour. Res., 34, Neuman, S. P. (1997), Stohasti approah to subsurfae flow and transport: A view to the future, in Subsurfae Flow and Transport: A Stohasti Approah, editedbyg.aganands.p.neuman,pp , Cambridge Univ. Press, New York. Neuman, S. P. (2002), Foreword, in Stohasti Methods for Flow in Porous Media, pp. ix xi, Aademi, San iego, Calif. Neuman, S. P., and A. Guadagnini (2000), A new look at traditional deterministi flow models and their alibration in the ontext of randomly heterogeneous media, in Calibration and Reliability in Groundwater Modelling: Coping ith Unertainty, edited by F. Stauffer et al., IAHS Publ., 265, Neuman, S. P., and S. Orr (1993), Predition of steady state flow in nonuniform geologi media by onditional moments: xat nonloal formalism, effetive ondutivities, and weak approximation, ater Resour. Res., 29(2), Pini, G., and M. Putti (1997), Parallel finite element Laplae transform method for the nonequilibrium groundwater transport equation, Int. J. Numer. Methods ng., 40, Shapiro, A., and T. Homem-de-mello (2000), On the rate of onvergene of optimal solutions of Monte Carlo approximations of stohasti program, SIAM J. Control Optim., 11(1), Sudiky,. A. (1989), The Laplae transform Galerkin tehnique: A timeontinuous finite element theory and appliation to mass transport in groundwater, ater Resour. Res., 25(8), Tartakovsky,. M., and S. P. Neuman (1998a), Transient flow in bounded randomly heterogeneous domains: 1. xat onditional moment equations and reursive approximations, ater Resour. Res., 34(1), Tartakovsky,. M., and S. P. Neuman (1998b), Transient flow in bounded randomly heterogeneous domains: 2. Loalization of onditional mean equations and temporal nonloality effets, ater Resour. Res., 34(1), Tartakovsky,. M., and S. P. Neuman (1999), xtension of Transient flow in bounded randomly heterogeneous domains: 1. xat onditional moment equations and reursive approximations, ater Resour. Res., 35(6), Xu, L., and M. L. Brusseau (1995), A ombined Laplae transform and streamline upwind approah for nonideal transport of solute in porous media, ater Resour. Res., 31(10), Ye, M. (2002), Parallel finite element Laplae transform algorithm for transient flow in bounded randomly heterogeneous domains, Ph.. dissertation, Univ. of Ariz., Tuson. hang,. (1999), Quantifiation of unertainty for fluid flow in heterogeneous petroleum reservoirs, Physia, 133, hang,. (2002), Stohasti Methods for Flow in Porous Media, Aademi, San iego, Calif. A. Guadagnini, ipartimento di Ingegneria Idraulia Ambientale e del Rilevamento, Politenio di Milano, Piazza Leonardo da Vini 32, Milan, Italy. (alberto.guadagnini@polimi.it) S. P. Neuman, epartment of Hydrology and ater Resoures, University of Arizona, Tuson, A 85721, USA. (neuman@hwr.arizona. edu). M. Tartakovsky, Theoretial ivision, Group T-7, MS B256, Los Alamos National Laboratory, Los Alamos, NM 87545, USA. (dmt@lanl. gov) M. Ye, Paifi Northwest National Laboratory, Rihland, 620 S Fifth Avenue, Suite 810, Rihland, Portland, OR 97204, USA. 19 of 19

20 05104 Y T AL.: ANALYSIS OF CONITIONAL MAN TRANSINT FLO Figure 2. Images of (a) a onditional realization of Y, onditional sample (b) mean m Y and () variane S 2 Y of NMC = 2500 realizations with unonditional s 2 Y =4,l =1. 8of19