Prediction of transient flow in random porous media by conditional moments

Abstract

This dissertation considers the effect of measuring randomly varying local hydraulic conductivity K(x) on one's ability to predict transient flow within bounded domains, driven by random sources, initial head distribution, and boundary functions. The first part of this work extends the steady state nonlocal formalism by Neuman and Orr [1992] in order to obtain the prediction of local hydraulic head h(x, t) and Darcy flux q(x, t) by means of their ensemble moments (c) and (c)conditioned on measurements of K(x). These predictors satisfy a deterministic flow equation which contains a nonlocal in space and time term called a "residual flux". As a result, (c) is nonlocal and non-Darcian so that an effective hydraulic conductivity K(c) does not generally exist. It is shown analytically that, with the exception of several specific cases, the well known requirement of "slow time-space variation" in uniform mean hydraulic gradient is essential for the existence of K(c). In a subsequent chapter, under this assumption, we develop analytical expressions for the effective hydraulic conductivity for flow in a three dimensional, mildly heterogeneous, statistically anisotropic porous medium of both infinite extent and in the presence of randomly prescribed Dirichlet and Neumann boundaries. Of a particular interest is the transient behavior of K(c) and its sensitivity to degree of statistical anisotropy and domain size. In a bounded domain, K(c) (t) decreases rapidly from the arithmetic mean K(A) at t = 0 toward the effective hydraulic conductivity corresponding to steady state flow, K(sr), K(c), exhibits similar behavior as a function of the dimensionless separation distance ρ between boundaries. At ρ = 0, K(c) = K(A) and rapidly decreases towards an asymptotic value obtained earlier for an infinite domain by G. Dagan. Our transient nonlocal formalism in the Laplace space allows us to analyze the impact of other than slow time-variations on the prediction of (c),. Analyzing several functional dependencies of mean hydraulic gradient, we find that this assumption is heavily dependent on the (relaxation) time-scale of the particular problem. Finally, we formally extend our results to strongly heterogeneous porous media by invoking the Landau-Lifshitz conjecture.