Upscaling of hydraulic conductivity in partially saturated heterogeneous porous formation

Abstract

This paper describes a procedure to upscale unsaturated hydraulic conductivity from the scale at which it is measured to a scale more amenable to actual flow simulations in partially saturated heterogeneous porous formations. In the approach adopted here the block conductivity is defined as the ratio of volume‐averaged flux to the volume‐averaged head gradient. The “point” values of the underlying formation properties (characterized by the saturated conductivity Ks and the pore size distribution parameter α) are viewed as realizations of independent random spatial functions. The general approach is applied to a partially saturated, heterogeneous, three‐dimensional statistically anisotropic formation, with statistical isotropy in the horizontal plane and arbitrarily oriented stratification characterized by a finite aspect ratio, for the case where the mean gradient exists only in the vertical direction. The procedure is illustrated by providing a second‐order approximation of the mean (in terms of the components of the block conductivity tensor) and a first‐order approximation of the covariance function of the block conductivity, which depend on the mean capillary pressure head H, the distribution of the “point” values of the underlying log unsaturated conductivity, and the block size. The results of this study allow assessment of the minimal domain size for which the concept of effective properties is appropriate and assessment of the maximal size of conductivity cells which preserve the heterogeneous structure of the porous formation with respect to simulation of flow and transport in the vadose zone. The application of the results to unconditional or conditional simulation of unsaturated flow and transport is straightforward, since knowledge of the mean and the covariance of the block unsaturated conductivity may lead to a relatively simple algorithm for generating its values on a grid at any level of aggregation desired.