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

Abstract

We start from mass and momentum balance equations which are valid at the pore level. Then, space averaging is used to define macroscopic variables and to derive equations that link these variables. That is the way a macroscopic mass balance equation and a macroscopic momentum balance equation are derived. But, the mometum balance equation fails to produce the generalised Darcy equation. Hence, thermodynamics of irreversible processes is used to generate two phenomenological laws. One is a generalised Darcy-like equation, the other is a capillary pressure equation. The genera.lised Darcy-like equation is standard except that it includes viscous and temperature couplings. The derived capillary pressure equation takes macroscopic fluid/fluid and fluid/solid interfacial areas into account plus a dynamic term. Finally, an explicit calculation of the derived capillary pressure equation in a conical capillary is given. It validates the derived equation and indicates a new way to compute capillary pressure in porous media.