Coarse scale models of two phase flow in heterogeneous reservoirs: volume averaged equations and their relationship to existing upscaling techniques

Abstract

Coarse scale equations applicable to two phase flow in heterogeneous reservoirs are developed via a volume average of the fine scale saturation equation. The resulting coarse scale description, in the form of an average saturation equation and a sequence of moment equations, is then used to assess several methods for the upscaling of detailed reservoir characterizations. The approaches considered are the use of pseudo‐relative permeabilities, the nonuniform coarsening of the fine grid geostatistical model, and the use of higher moments of the fine scale variables. Using the volume averaged equations as a framework, the underlying assumptions, strengths and limitations of each of these methods are discussed. Previous findings demonstrating the accuracy of coarse scale relative permeability functions based on higher moments are readily explained in terms of the volume average saturation and moment equations. Finally, the volume averaged equations are simplified for the unit mobility ratio case and applied to the direct solution of a model coarse scale problem.