Nonlocal and localized finite element solution of conditional mean flow in randomly heterogeneous media

Abstract

This report considers the effect of measuring randomly varying local hydraulic conductivities K(x) on one's ability to predict deterministically, without upscaling, steady state flow in bounded domains driven by random source and boundary terms. Our aim is to allow optimum unbiased prediction of hydraulic heads h(x) and Darcy fluxes q(x) by means of their ensemble moments, , and c, conditioned on measurements of K(x). It has been shown earlier that these predictors satisfy a deterministic flow equation which contains an integro-differential "residual flux" term. This term renders c nonlocal and non-Darcian so that the concept of effective hydraulic conductivity looses meaning in all but a few special cases. Instead, the residual flux contains kernels which constitute nonlocal parameters that are conditional on hydraulic conductivity data and therefore nonunique. The kernels include symmetric and nonsymmetric second -rank tensors as well as vectors. We derive exact integro-differential equations for second conditional moments of head and flux which constitute measures of predictive uncertainty. We then develop recursive closure approximations for the moment equations through expansion in powers of a small parameter ay which represents the standard estimation error of In K(x). Finally, we solve these nonlocal equations to first order in a by finite elements on a rectangular grid in two dimensions. We also solve the original stochastic flow equations by conditional Monte Carlo simulation using finite elements on the same grid. Upon comparing our nonlocal finite element and conditional Monte Carlo results we find that the former are highly accurate, under either mean uniform or convergent flows, for both mildly and strongly heterogeneous media with a as large as 4 - 5 and spatial correlation scales as large as the length of the domain. Since conditional mean quantities are smooth relative to their random counterparts our method allows, in principle, resolving them on relatively coarse grids without upscaling. We also examine the quc on under what conditions can the residual flux be localized so as to render it approximately Darcian. One way to achieve such localization is to treat ' "draulic conductivity as if it was locally homogeneous and mean flow as if it was locally uniform. This renders the flux predictor Darcian according to c _ - Kc(x) \7c where Kc(x) is a conditional hydraulic conductivity tensor which depends on measurements of K(x) and is therefore a nonunique function of space. This function can be estimated by means of either stochastically- derived analytical formulae or standard inverse methods (in which case localization coincides with common groundwater modeling practice). We use the first approach and solve the corresponding localized conditional mean equation by finite elements on the same grid as before. Here the conditional hydraulic conductivity is given by the geometric mean KG(x). Upon comparing our localized finite element solution with a nonlocal finite element solution and conditional Monte Carlo results, we find that the first is generally less accurate than the second. The accuracy of the localized solution deteriorates rel tive to that of the nonlocal solution as one approaches points of conditioning and singularity, or as the variance and correla': ^n scale of the log hydraulic conductivity increase. Contrary to the nonlocal solution, locàlzation does not yield information about predictive uncertainty.