The Two-Phase Darcy’s Law interface combines Darcy’s law (Equation 4-1) with the continuity equation (Equation 4-2) for the average density ρ. In the presence of two miscible fluid phases, the average density ρ and average viscosity μ depend on the composition of the mixture (Ref. 1). In the Two-Phase Darcy’s Law interface these dependencies are given by

Here, s1 and s2 represent the saturation of each phase, ρ1 and ρ2 the densities, and κr1 and κr2 the relative permeabilities. When either of the fluids is compressible, its density can be related to the pressure (for instance using the ideal gas law).

Beside the continuity equation for the mixture, the Two-Phase Darcy’s Law interface also solves the transport equation for the fluid content of one of the phases, c1 = s1ρ1:

Here, Dc (SI unit: m2/s) is the capillary diffusion coefficient, which can be directly specified or derived from a capillary pressure expression. Normally, the fluid content c1 will be the concentration of the wetting phase.

When capillary pressure is selected as capillary model, the capillary diffusion coefficient is computed from the saturation of one of the phases and the capillary pressure:

The capillary pressure is defined as the pressure difference between the phases, and it can be defined as a function of saturation, pc(s1).