Avoiding Inconsistencies in the Relative Permeability Models of Three-Phase Flow in Porous Media

Traditional mathematical models of multiphase flow in porous media use a straightforward extension of Darcy's equation, and the key element of these models is the appropriate formulation of the relative permeability functions. It is well known that for one-dimensional flow of three immiscible incompressible fluids, when capillarity is neglected, most relative permeability models used today give rise to regions in the saturation space with elliptic behavior (the so-called elliptic regions). We argue that this behavior is not physical, but rather the result of an incorrect mathematical model. Inappropriateness of the formulation may have several sources but, in the context of Darcy-type models, the element that first needs to be revisited is the formulation of the relative permeabilities. Here we identify necessary conditions that must be satisfied by the relative permeability functions, so that the behavior of the system of equations describing three-phase flow is physically plausible. We show that these conditions are supported by pore-scale physics, and also in good agreement with experimental data.