We have shown elsewhere [Tartakovsky and Neuman, this issue (a)] that in randomly heterogeneous media, the ensemble mean transient flux is generally nonlocal in space-time and therefore non-Darcian. We have also shown [Tartakovsky and Neuman, this issue (b)] that there are special situations in which this flux can be localized so as to render it Darcian in real or transformed domains. Each such situation gives rise to an effective hydraulic conductivity which relates mean gradient to mean 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-shaped domain due to a mean hydraulic gradient that varies slowly in space and time. When the mean gradient varies rapidly in time, the Laplace transform of the mean flux is local but its real-time equivalent includes a temporal convolution integral; we develop analytical expressions for the real-time kernel of this convolution integral. The box is embedded within a statistically homogeneous natural log hydraulic conductivity 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 associate with the box under these conditions. We explore the influence of domain size, time, and statistical anisotropy on effective conductivity and include a simple new formula for its variation with statistical anisotropy ratio in an infinite domain under steady state.
ASJC Scopus subject areas
- Water Science and Technology