In this work, we present a macroscopic model for the flow of two immiscible and incompressible fluids in inhomogeneous porous medium. At the pore scale, the flow is governed by the fully Navier-Stokes equations while the evolution of the phase interface is captured by the Cahn-Hilliard equation. Using the volume averaging method, the upscaled equations describing the averaged behavior of two fluids at the Darcy scale are obtained, with unclosed terms related to spatial deviations. Then, spatial derivations are carefully modeled up to some undetermined coefficients, which could be evaluated by solving simplified closure problems in each representative volume element. In particular, the wetting behavior is incorporated into the averaged chemical potential. The differences between the proposed equations and the empirical two-phase Darcy-type models are discussed. Finally, a phase-field-based lattice Boltzmann model for the averaged equations is presented, and numerical results demonstrate the abilities of the proposed model.
Comment: 33 pages, 7 figures