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

A quantitative understanding of three-phase flow in porous media is required to address many diverse processes in the subsurface, e.g., improved oil recovery, CO ₂ sequestration, and aquifer clean-up. In turn, all predictive models of three-phase flow originate from interpretations of one-dimensional laboratory experiments; when these interpretations are flawed, so are the models. In this paper we revisit the foundations of displacement theory in three-phase flow and provide the most general conditions for any relative permeability model to be physical anywhere in the saturation triangle. In doing so, we put to rest a controversy that has persisted in petroleum literature for the better part of the last six decades. When capillarity is ignored, the system of conservation laws describing incompressible immiscible flow should be strictly hyperbolic. This natural property of the system fails for most relative permeability models used today. 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 the most general analytical solution to the Riemann problem (constant initial and injected states) for three-phase flow, and describe the characteristic waves that may arise, concluding that only 9 combinations of rarefactions, shocks and rarefaction-shocks are possible. Some of these wave combinations have been overlooked by many because of the associated conceptual and mathematical difficulties. 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 streamtube simulators.