Upscaling of permeability of anisotropic heterogeneous formations: 3. Applications

Abstract

The general results of Indelman and Dagan (this issue (a), (b)) are applied to some cases of interest in applications. The exact solution of upscaling of stratified formations is used in order to assess the range of applicability of the first‐order analysis. The explicit expressions of the two first moments of the upscaled permeability are derived for media of two‐dimensional isotropic and three‐dimensional axisymmetric actual log permeability of Gaussian covariance and for parallelepipedic (rectangular for two‐dimensional) averaging elements. The effects of the actual log permeability covariance and of the anisotropy of partition elements on the upscaled permeability statistics are analyzed. The asymptotic behavior of the upscaled permeability for large and small blocks is discussed for general heterogeneity. The strong dependence of the small element limit on the properties of actual heterogeneity and on the shape of the elements is illustrated by typical examples. In contrast, the homogenization of the upscaled media by large blocks depends primarily on the volume (area for two dimensions) of the partition element relative to the one based on the correlation scale of the log permeability.