TRANSIENT FLOW IN BOUNDED RANDOMLY HETEROGENEOUS DOMAINS-2 - LOCALIZATION OF CONDITIONAL MEAN EQUATIONS AND TEMPORAL NONLOCALITY EFFECTS

In randomly heterogeneous porous media one cannot predict flow behavio r with certainty. One can, however, render optimum unbiased prediction s of such behavior by means of conditional ensemble mean hydraulic hea ds and fluxes. We have shown in paper 1 [Tartakovsky and Neuman, this issue] that under transient flow, these optimum predictors are governe d by nonlocal equations, In particular, the conditional mean flux is g enerally nonlocal in space-time and therefore non-Darcian. As such, it cannot be associated with an effective hydraulic conductivity except in special cases, Here we explore analytically situations under which localization is possible so that Darcy's law applies in real, Laplace, and/or infinite Fourier transformed spaces, approximately or exactly, with or without conditioning. We show that the corresponding conditio nal effective hydraulic conductivity tensor is generally nonsymmetric. An alternative to Darcy's law in each case, valid under mean no-flow conditions along Neumann boundaries, is a quasi-Darcian form that incl udes only a symmetric tensor which, however, does not constitute a bon a fide effective hydraulic conductivity. Both lack of symmetry and dif ferences between Darcian and quasi-Darcian forms disappear to first (b ut not necessarily higher) order of approximation in the (conditional) variance of natural log hydraulic conductivity. We adopt such an appr oximation to investigate analytically the effect of temporal nonlocali ty on one-and three-dimensional mean flows in infinite, statistically homogeneous media, Our results show that temporal nonlocality may mani fest itself under either monotonic or oscillatory time variations in t he mean hydraulic gradient. The effect of temporal nonlocality increas es with the variance of log hydraulic conductivity and is more pronoun ced in one dimension than in three.