A macroscopic model for immiscible two-phase flow in porous media

This work provides the derivation of a closed macroscopic model for immiscible two-phase, incompressible, Newtonian and isothermal creeping steady flow in a rigid and homogeneous porous medium without considering three-phase contact. The mass and momentum upscaled equations are obtained from the pore-scale Stokes equations, adopting a two-domain approach where the two fluid phases are separated by an interface. The average mass equations result from using the classical volume averaging method. A Green's formula and the adjoint Green's function velocity pair problems are used to obtain the pore-scale velocity solutions that are averaged to obtain the upscaled momentum balance equations. The macroscopic model is based on the assumptions of scale separation and the existence of a periodic representative elementary volume allowing a local description as usually postulated for upscaling. The macroscopic momentum equation in each phase includes the generalized Darcy-like dominant and viscous coupling terms and, importantly, an additional compensation term that accounts for surface tension effects to momentum transfer that is, otherwise, incompletely captured by the Darcy terms. This interfacial term, as well as the dominant and viscous coupling permeability tensors, can be predicted from the solutions of two associated closure problems that coincide with those reported in the literature. The relevance of the compensation term and the upscaled model validity are assessed by comparisons with direct numerical simulations in a model two-dimensional periodic structure. Upscaled model predictions are found to be in excellent agreement with direct numerical simulations.