Upscaled conductivity in gravity‐dominated flow through variably saturated heterogeneous formations

Abstract

We investigated effects of a few characteristics of a heterogeneous, unsaturated formation on the vertical (Kv) and horizontal (Kh) components of the upscaled conductivity tensor, under conditions of steady state, gravity‐dominated, unsaturated flow, using Gardner's [1958] two‐parameter (Ks, α) model for the local unsaturated conductivity. Results suggest that relatively large averaging domains are required in order to obtain asymptotic, effective conductivities devoid of the domain size. This is particularly so in relatively dry, stratified, fine‐textured (with appreciable capillary forces) formations and when (1) the variability in the soil parameter α is not small compared with the variability in saturated conductivity, Ks (ζ ≫ 0); (2) the fluctuations of logα are negatively correlated with those of logKs (rfa < 0); and (3) the correlation length scale of logα is small compared to that of logKs (υ < 1). Larger averaging domains are required in order to ignore the uncertainty about the upscaled log conductivities, particularly in stratified, coarse‐textured formations, and when υ > 1. Findings of this study suggest that the asymptotic Kv diminishes in dry, stratified, coarse‐textured formations and when ζ ≫ 0; for the physically plausible situation in which rfa > 0, however, the diminishing of the asymptotic Kv is balanced by increasing rfa. On the other hand, the increase in the asymptotic Kh, and, concurrently, in the effective anisotropy ratio, due to increasing stratification and decreasing water saturation, is balanced by decreasing capillary forces, as well as by decreasing ζ and increasing (positive) rfa, and to a lesser extent, by increasing υ. Application of the results of this study for assessment of (1) the minimal domain size for which the concept of effective properties is appropriate; and (2) the maximal size of conductivity cells which preserve the heterogeneous structure of the underlying formation with respect to simulation of flow and transport in the vadose zone, are briefly discussed.