Two‐phase flow through porous media

In the context of a previously proposed new description for multiphase flow through porous media, we examine constitutive equations for gi, the force per unit volume which phase i exerts upon the pore walls and other adjacent fluids. The principle of material difference is used to determine the most general dependence of gi upon saturation gradient for a nonoriented (no porosity gradient) permeable structure. Three problems are solved and the results compared with analyses based upon our interpretation of Muskat's equations. The first concerns a semi‐infinite porous column in air, the bottom of which is saturated with a liquid. The same saturation distribution is found with our approach as with Muskat's equations. The analysis of the second problem indicates the relation between our relative permeability and that of Muskat. The jump balances for mass and for momentum at a surface of discontinuity (or shock wave) with respect to volume averaged variables are derived. The significance of solutions in which such surfaces of discontinuity appear is discussed. In the third problem, a plane surface of discontinuity with respect to local saturation moves through an infinite permeable structure. This is similar to the displacement of oil by an aqueous solution in a secondary recovery process. The results are comparable to those with Muskat's equations only in the limit where capillary pressure is neglected (the Buckley‐Leverett case). When capillary pressure differs from zero, Muskat's equations do not admit a shock wave solution.