Homogenize coupled Stokes–Cahn–Hilliard system to Darcy’s law for two-phase fluid flow in porous medium by volume averaging

A technique of local volume averaging is applied to obtain general equations that depict mass and momentum transport of incompressible two-phase flow in porous media. Starting from coupled Stokes–Cahn–Hilliard equations for incompressible two-phase fluid flow, the averaging is performed without oversimplifying either the porous media or the fluid mechanical relations. The resulting equations are Darcy’s law for two-phase flow with medium parameters which could be evaluated by experiment. The Richards equation of the mixed form can be deduced from the resulting equations. The differences between the resulting equations and the empirical two-phase fluid flow model adopted in oil industry are discussed using several numerical examples.