Relative Permeabilities in Co-Current Three-Phase Displacements with Gravity

We study the character of the equations in the traditional formulation of one-dimensional immiscible three-phase flow with gravity, in the limit of negligible capillarity. We extend our previous analysis, presented in SPE 77539, to incorporate the effects of gravity. We restrict our analysis to co-current flow required for a displacement process; in cases of mixed co-current and counter-current flow, capillarity effects cannot be dropped from the formulation. The model makes use of the classical multiphase extension of Darcy's equation. It is well known that, if relative permeabilities are taken as fixed functions of saturations, the model yields regions in the saturation space where the system of equations is locally elliptic. We regard elliptic behavior as a nonphysical artifact of an incomplete formulation. We derive conditions on the relative permeabilities so that the system of governing equations is strictly hyperbolic. The key point is to acknowledge that a Darcy-type formulation is insufficient to capture all the physics of three-phase flow and that, consequently, the relative permeabilities are functional that depend on the fluid viscosity ratio and the gravity number. The derived conditions are consistent with the type of displacements that take place in porous media. By means of an illustrative example, we show how elliptic behavior can be removed, even when using simplistic relative permeability models.