Three-Phase Displacement Theory: An Improved Description of Relative Permeabilities

In this paper, we revisit the displacement theory of three-phase flow and provide conditions for a relative permeability model to be physical anywhere in the saturation triangle. When capillarity is ignored, most relative permeability functions used today yield regions in the saturation space where the system of equations is locally elliptic, instead of hyperbolic. We are of the opinion that this behavior is not physical, and we identify necessary conditions that relative permeabilities must obey to preserve strict hyperbolicity. These conditions are in agreement with experimental observations and pore-scale physics. We also present a general analytical solution to the Riemann problem (constant initial and injected states) for three-phase flow, when the system satisfies certain physical conditions that are natural extensions of the two-phase flow case. We describe the characteristic waves that may arise, concluding that only nine combinations of rarefactions, shocks, and rarefaction-shocks are possible. Some of these wave combinations may have been overlooked but can potentially be important in certain recovery processes. The analytical developments presented here will be useful in the planning and interpretation of three-phase displacement experiments, in the formulation of consistent relative permeability models, and in the implementation of streamline simulators.