Effective hydraulic conductivity of bounded, strongly heterogeneous porous media

We develop analytical expressions for the effective hydraulic conductivity K(e) of a three-dimensional, heterogeneous porous medium in the presence of randomly prescribed head and flux boundaries. The log hydraulic conductivity Y forms a Gaussian, statistically homogeneous and anisotropic random field with an exponential autocovariance. By effective hydraulic conductivity of a finite volume in such a field, we mean the ensemble mean (expected value) of all random equivalent conductivities that one could associate with a similar volume under uniform mean flow. We start by deriving a first-order approximation of an exact expression developed in 1993 by Neuman and Orr. We then generalize this to strongly heterogeneous media by invoking the Landau-Lifshitz conjecture. Upon evaluating our expressions, we find that K(e) decreases rapidly from the arithmetic mean K(A) toward an asymptotic value as distance between the prescribed head boundaries increases from zero to about eight integral scales of Y. The more heterogeneous is the medium, the larger is K(e) relative to its asymptote at any given separation distance. Our theory compares well with published results of spatially power-averaged expressions and with a first- order expression developed intuitively by Kitanidis in 1990.