Modeling two-phase flow of immiscible fluids in porous media: Buckley-Leverett theory with explicit coupling terms

Continuum models that describe two-phase flow of immiscible fluids in porous media often treat momentum exchange between the two phases by simply generalizing the single-phase Darcy law and introducing saturation-dependent permeabilities. Here we study models of creeping flows that include an explicit coupling between both phases via the addition of cross terms in the generalized Darcy law. Using an extension of the Buckley-Leverett theory, we analyze the impact of these cross terms on saturation profiles and pressure drops for different couples of fluids and closure relations of the effective parameters. We show that these cross terms in the macroscale models may significantly impact the flow compared to results obtained with the generalized Darcy laws without cross terms. Analytical solutions, validated against experimental data, suggest that the effect of this coupling on the dynamics of saturation fronts and the steady-state profiles is very sensitive to gravitational effects, the ratio of viscosity between the two phases, and the permeability. Our results indicate that the effects of momentum exchange on two-phase flow may increase with the permeability of the porous medium when the influence of the fluid-fluid interfaces become similar to that of the solid-fluid interfaces.