Equivalent Permeability Tensor of Heterogeneous Media: Upscaling Methods and Criteria (Review and Analyses)

Abstract

When conducting numerical upscaling, either for a fractured or a porous medium, it is important to account for anisotropy because in general, the resulting upscaled conductivity is anisotropic. Measurements made at different scales also demonstrate the existence of anisotropy of hydraulic conductivity. At the “microscopic” scale, the anisotropy results from the preferential flatness of grains, presence of shale, or variation of grain size in successive laminations. At a larger scale, the anisotropy results from preferential orientation of highly conductive geological features (channels, fracture families) or alternations of high and low conductive features (stratification, bedding, crossbedding). Previous surveys of homogenization techniques demonstrate that a wide variety of approaches exists to define and calculate the equivalent conductivity tensor. Consequently, the resulting equivalent conductivities obtained by these different methods are not necessarily equal, and they do not have the same mathematical properties (some are symmetric, others are not, for example). We present an overview of different techniques allowing a quantitative evaluation of the anisotropic equivalent conductivity for heterogeneous porous media, via numerical simulations and, in some cases, analytical approaches. New approaches to equivalent permeability are proposed for heterogeneous media, as well as discontinuous (composite) media, and also some extensions to 2D fractured networks. One of the main focuses of the paper is to explore the relations between these various definitions and the resulting properties of the anisotropic equivalent conductivity, such as tensorial or non-tensorial behavior of the anisotropic conductivity; symmetry and positiveness of the conductivity tensor (or not); dual conductivity/resistivity tensors; continuity and robustness of equivalent conductivity with respect to domain geometry and boundary conditions. In this paper, we emphasize some of the implications of the different approaches for the resulting equivalent permeabilities.