A Darcy's Law Extension and a New Capillary Pressure Equation for Two-Phase Flow in Porous Media
Authors: Pavone D. R. 
Published: December 2007

ABSTRACT
Using a technique introduced by Marie6, including distribution, averaging with weight functions and thermodynamics of irreversible processes, we propose a new derivation of the standard equations for two-phase flow in porous media.
We start from mass, momentum and energy balance equations valid at the pore level where interfacial discontinuities are described by distributions. Averaged mass, momentum and energy balance equations are derived. Unfortunately, the momentum balance equation falls to produce the generalized Darcy equation. Hence, thermodynamics of irreversible processes is used to generate two phenomenological laws.
One law is a generalized Daxcy-like equation composed of standard terms plus viscous and temperature couplings. The other one is a new capillary pressure equation. It is not standard because it is the sum of three terms:
P c = γ a b ∂ A a s ∂ ( Φ S a ) cos ( θ ) + γ a b ∂ A a b ∂ ( Φ S a ) + π ∂ ( ϕ S a ) ∂ t
where ∂Aas∂(ΦSa) and ∂Aab∂(ΦSa) the derivatives of liquida-solids and liquida-liquidb( interfacial areas with respect to the saturation (5a), 8 is the wettability angle and ϕ is the porosity. The first term comes from the three-phase-linea_b_s motion at the pore level, the second comes from interfacea_b, extension and the third is dynamic in nature. When computed in a conical capillary, this new capillary pressure equation is reduced to the standard Lapace law.
Morrow and Mungan15-16 measured static capillary pressure with different wettability angles. We show that their results are exactly matched by an affine function of cos(θ) (Pc = α cos(θ)+β), instead of the expected and usual simple linear function (Pc = αcos(θ)). Under static conditions (third term canceled), this is in agreement with our theoretical results. Therefore, we can now extract from their data the values of α=∂Aas∂(ΦSa) and β=∂Aab∂(ΦSa) turn out to be of the same order of magnitude (~ 1 μm2/μm3) Moreover, we can derive fluida/fluidband fluida/solids interfacial areas as a function of the saturation.
Results can be used to estimate capillary pressure in reservoirs from capillary pressure measurements in the lab. They are a tool for measuring interfacial areas in porous media.

BACKGROUND
Oil recovery processes are based on the displacement of one fluid by another in porous media. The problem is to model fluid displacements without solving them in each pore.
The simplest idealized mathematical structure is bundles of nonintersecting tubes. Dullien1 investigated more elaborate structures in which each tube radius may have axial variations. Since fluid retention is not explained by such models, Chatzis and Dullien2 investigated two-dimensional networks.
Statistics do not require complete porous media description. Hence, the percolation theory was applied to capillary displacement (Larson et al.3). However, the percolation theory applied to porous media deals only with the minimum amount of porosity or fluid required for flow.
In fractal geometry, boundaries are not located exactly, but they are assumed to be self-similar. Jacquin and Adler4 found that some geological structures are fractal. Adler5 attempted to calculate the properties of such a medium from a continuous point of view. The longitudinal flow of a Newtonian fluid in the Stokes approximation has been computed, and single-phase permeability has been derived.
Space averaging was introduced to justify the generalized Darcy equation for two-phase flow. Space averaging does not require the location or shape of pore boundaries but fails to predict constitutive parameters which must be evaluated empirically.
Attempts made to justify these laws can be divided into two groups. The first uses volume averaging over a "representative elementary volume" (Marie6, Slattery7, Whitaker8, Kalaydjian9, Pavone13). The second assumes that porous media are periodic, an'd space averaging is done over a period (Auriault and Sanchez-Palencia10).
Our work is related to the first group. In 1982, Marie6-11 treated the two-phase case and formulated the basis of a volume averaging technique. He derived a generalized Darcy equation with couplings between phases and temperature and a capillary pressure equation that was the sum of a static and a dynamic term. The static part of the capillary pressure is proportional to the mean curvature of the interfaces.