Upscaled equations for two-phase flow in highly heterogeneous porous media: Varying permeability and porosity

Abstract

In this article we consider a two-phase flow model in a highly heterogeneous porous column. The porous column consists of homogeneous blocks, where the porosity and permeability vary from one block to the other. The flow direction is perpendicular to the layering of the porous column, and hence can be approximated by one-dimensional model equations. The periodic change in porosity and absolute permeability enforce the fluid to be trapped at the interface between the blocks, leading to a highly varying saturation. In order to capture the effective behavior, upscaled equations for the average saturation are derived via homogenization. This technique relies on a notion of periodicity and allows averaging over any number of blocks that may have any internal distributions of the rock parameters. Moreover, the present article also derives effective equations for randomly distributed layers of different porosity and absolute permeability. Numerical experiments are performed which show good agreement between the averaged solutions of the original micro-scale equations and the solutions of the upscaled equations, also in the case of randomly distributed layers. In particular these numerical experiments show how the internal distribution of the permeability and porosity affect the effective behavior of the flow.