Upscaling of permeability of anisotropic heterogeneous formations: 2. General structure and small perturbation analysis

The general methodology developed in part 1 (Indelman and Dagan, this issue) of this study is applied to the detailed analysis of upscaled permeability. First, the structure of the dissipation function for the general conductivity, a tensor of stationary random components, is examined with the aid of dimensional analysis. It is shown that for arbitrary shapes of the numerical elements, the upscaled permeability has also this general structure, and the numbers of unknown parameters and equations match. This result suggests that the upscaling problem has an unique solution. In the particular case of scalar permeability of isotropic covariance of the actual formation, it is shown that a similar upscaled permeability is possible only for spherical (circular) numerical elements. Otherwise, the upscaled permeability has to be a tensor of anisotropic covariance. If the actual formation has a scalar permeability of axisymmetric covariance, upscaling preserves the last property only for axisymmetric partition elements, i.e., for a sphere, cylinder, and ellipsoid. Explicit expressions for the first moments of the upscaled permeability (mean, covariance) are derived at first order in the log permeability variance. The detailed computations for a scalar permeability of axisymmetric covariance and for axisymmetric numerical elements lead to simple results. The upscaled permeability expected values are components of an axisymmetric tensor, whereas the fluctuations are determined by a scalar random space function of anisotropic covariance. The general case of upscaling by a first‐order approximation is examined in Appendix B. These results will be applied to a few particular cases in part 3 (Indelman, this issue).