Theory for the Two-Phase Darcy’s Law Interface

Darcy’s law states that the velocity field is determined by the pressure gradient, the fluid viscosity, and the permeability of the porous medium. According to Darcy’s law, the velocity field is given by

(8-33)

In this equation:

•

u (SI unit: m/s) is the Darcy velocity vector

•

κ (SI unit: m2) is the permeability of the porous medium

•

μ (SI unit: Pa·s) is the fluid’s dynamic viscosity, and

•

p (SI unit: Pa) is the fluid’s pressure

The permeability, κ, represents the resistance to flow over a representative volume consisting of solid grains and pores.

The Two-Phase Darcy’s Law interface combines Darcy’s law (Equation 8-33) with the continuity equation for the average density ρ

(8-34)

here, εp is the porosity, defined as the fraction of the control volume that is occupied by pores. Inserting Darcy’s law (Equation 8-33) into the continuity equation (Equation 8-34) produces the governing equation

(8-35)

In the presence of two miscible fluid phases, the average density ρ and average viscosity μ depend on the composition of the mixture (Ref. 2). In the Two-Phase Darcy’s Law interface these dependencies are given by

(8-36)

(8-37)

(8-38)

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).