ABSTRACT
Many natural rock systems contain small patches of different permeability which affect the flow of fluids through them. As these heterogeneities become smaller and more numerous, they become harder to model numerically. We consider two dimensional flows through such systems, where each patch is many radii away from another. For finite numbers of heterogeneities, they can be replaced by single dipoles to leading order, and this approximation predicts a number of features of the flow. In some circumstances, the entire system can be described by assigning an average permeability tensor to it. Several infinite systems including vertical and horizontal bands are also considered, leading to quite simple expressions for the flow rate and average permeability. The theory is extended to the case where there is a small change in capillary pressure within each heterogeneity as well as a nearly-unit mobility ratio at the displacement front. First-order corrections for these effects are found and applied to the model.
