Analytical solutions for the distribution of pressure in permeable formations with ellipsoidal inclusions

摘要

The reservoir up-scaling problem has been receiving increased attention in recent years. Over the past decade or so, there has been increasing interest in development of computationally efficient methods to determine effective properties or permeability. Those properties were traditionally computed from detailed numerical solutions of the actual reservoir realization. This is an indirect approach requiring substantial computer resources particularly in 3D problems in which the number of grid-blocks often become impractically large. A contrasting strategy is the direct approach in which the effective properties are computed directly from the statistical description of the medium without the aid of an actual reservoir realization. This method will be particularly important for multiphase problems. Among the direct methods, a particularly promising one which motivated this study, is the self-consistent approximation for determining the electric conductivity of heterogeneous media and multiphase materials. In reservoir engineering context, the self-consistent approximation has been recently applied to determine effective permeabilities. This approximation needs analytical solutions for the fluctuation of pressure created in an otherwise homogeneous matrix of infinite dimensions by the submersion of inclusions. The existing solutions are based on models which have limitations on the orientation of permeability tensors and perhaps largely in the geometry of the inclusions. Mathematical models have been developed which strongly generalize the existing inclusion models serving as a basis for the self-consistent approximation. 21 refs., 9 figs., 2 tabs.