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 behavior with certainty. One can, however, render optimum unbiased predictions of such behavior by means of conditional ensemble mean hydraulic heads and fluxes. We have shown in paper 1 [Tartakovsky and Neuman, this issue] that under transient flow, these optimum predictors are governed by nonlocal equations. In particular, the conditional mean flux is generally 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 conditional 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 includes only a symmetric tensor which, however, does not constitute a bona fide effective hydraulic conductivity. Both lack of symmetry and differences between Darcian and quasi-Darcian forms disappear to first (but not necessarily higher) order of approximation in the (conditional) variance of natural log hydraulic conductivity. We adopt such an approximation to investigate analytically the effect of temporal nonlocality on one- and three-dimensional mean flows in infinite, statistically homogeneous media. Our results show that temporal nonlocality may manifest itself under either monotonic or oscillatory time variations in the mean hydraulic gradient. The effect of temporal nonlocality increases with the variance of log hydraulic conductivity and is more pronounced in one dimension than in three.