Macroscopic Equations Derived from Space Averaging for Immiscible Two-Phase Flow in Porous Media

Abstract

Models of instabilities in porous media usually assume that the capillary pressure (the difference of pressure between the nonwetting and the wetting phase) depends on the radii of the macroscopic curvature of the two-phase front. However, this definition is not taken into account for modeling stable immiscible displacements in porous media whenever the heterogeneity of the porous medium may lead to high macroscopic curvature of the front. Before trying to solve flow equations in porous media under unstable conditions, a more accurate and complete set of equations for immiscible two-phase flow in porous media is required. Space averaging of microscopic equations valid at the pore level is used to define variables and equations that link these variables at the macroscopic scale. The thermodynamics of irreversible processes completes the set of equations. If some coupling between the flow of both phases is introduced, the relative permeability equation is proved to be valid, even with moving interfaces. Capillary pressure appears to be twofold; i. e. a static capillary pressure taking into account 1) the amount of interface (Gibbs-Duhem like equation) and 2) the area swept by the three-phase lines (Laplace like equation), as well as a dynamic capillary pressure related to fluid inertia. This is the first time that capillary pressure in porous media can be proved to be composed of three terms, each having an evident physical meaning. Under the assumptions of the present paper, the capillary pressure does not depend on macroscopic curvature, therefore.