Numerical calculation of equivalent grid block permeability tensors for heterogeneous porous media

Abstract

A numerical procedure for the determination of equivalent grid block permeability tensors for heterogeneous porous media is presented. The method entails solution of the fine scale pressure equation subject to periodic boundary conditions to yield, upon appropriate averaging of the fine scale velocity field, the coarse scale or equivalent grid block permeability. When the region over which this coarse scale permeability is computed constitutes a representative elementary volume (REV), the resulting equivalent permeability may be interpreted as the effective permeability of the region. Solution of the pressure equation on the fine scale is accomplished through the application of an accurate triangle‐based finite element numerical procedure, which allows for the modeling of geometrically complex features. The specification of periodic boundary conditions is shown to yield symmetric, positive definite equivalent permeability tensors in all cases. The method is verified through application to a periodic model problem and is then applied to the scale up of areal and cross sections with fractally generated permeability fields. The applicability and limitations of the method for these more general heterogeneity fields are discussed.