TRANSIENT EFFECTIVE HYDRAULIC CONDUCTIVITIES UNDER SLOWLY AND RAPIDLYVARYING MEAN GRADIENTS IN BOUNDED 3-DIMENSIONAL RANDOM-MEDIA

We have shown elsewhere [Tartakovsky and Neumnn, this issue (a)] that in randomly heterogeneous media, the ensemble mean transient flux is g enerally nonlocal in space-time and therefore non-Darcian. We have als o shown [Tartakovsky and Neuman, this issue (b)] that there are specia l situations in which this flux can be localized so as to render it Da rcian in real or transformed domains. Each such situation gives rise t o an effective hydraulic conductivity which relates mean gradient to m ean flux at any point in real or transformed space-time. In this paper we develop first-order analytical expressions for effective hydraulic conductivity under three-dimensional transient flow through a box-sha ped domain due to a mean hydraulic gradient that varies slowly in spac e and time. When the mean gradient varies rapidly in time, the Laplace transform of the mean flux is local but its real-time equivalent incl udes a temporal convolution integral; we develop analytical expression s for the real-time kernel of this convolution integral. The box is em bedded within a statistically homogeneous natural log hydraulic conduc tivity field that is Gaussian and exhibits an anisotropic exponential spatial correlation structure. By the effective hydraulic conductivity of a finite box in such a field we imply the ensemble mean (expected value) of all random equivalent conductivities that one could associat e with the box under these conditions. We explore the influence of dom ain size, time, and statistical anisotropy on effective conductivity a nd include a simple new formula for its variation with statistical ani sotropy ratio in an infinite domain under steady state.