TRANSIENT FLOW IN BOUNDED RANDOMLY HETEROGENEOUS DOMAINS-1 - EXACT CONDITIONAL MOMENT EQUATIONS AND RECURSIVE APPROXIMATIONS

We consider the effect of measuring randomly varying local hydraulic c onductivities K(x) on one's ability to predict transient flow within b ounded domains, driven by random sources, initial head, and boundary c onditions. Our aim is to allow optimum unbiased prediction of local hy draulic heads h(x, t) and Darcy fluxes q(x, t) by means of their ensem ble moments, [h(x, t)](c) and [q(x, t)](c), conditioned on measurement s of K(x). We show that these predictors satisfy a compact determinist ic flow equation which contains a space-time integrodifferential ''res idual flux'' term. This term renders [q(g, t)](c) nonlocal and non-Dar cian so that the concept of effective hydraulic conductivity looses me aning in all but a few special cases, Instead, the residual flux conta ins kernels that constitute nonlocal parameters in space-time that are additionally conditional on hydraulic conductivity data and thus nonu nique. The kernels include symmetric and nonsymmetric second-rank tens ors as well as vectors, We also develop nonlocal equations for second conditional moments of head and flux which constitute measures of pred ictive uncertainty. The nonlocal expressions cannot be evaluated direc tly without either a closure approximation or high-resolution conditio nal Monte Carlo simulation. To render our theory workable, we develop recursive closure approximations for the moment equations through expa nsion in powers of a small parameter which represents the standard est imation error of natural log K(x). These approximations are valid to a rbitrary order for either mildly heterogeneous or well-conditioned str ongly heterogeneous media. They allow, in principle, evaluating the co nditional moments numerically on relatively coarse grids, without upsc aling, by standard methods such as finite elements.