Three-Phase Fractional-Flow Theory of Foam-Oil Displacement in Porous Media with Multiple Steady States

Understanding the interplay of foam and non-aqueous phases in porous media is key to improving the design of foam for enhanced oil recovery and remediation of aquifers and soils. A widely used implicit-texture foam model predicts phenomena analogous to cusp catastrophe theory: the surface describing foam apparent viscosity as a function of fractional flows folds backwards on itself. Thus there are multiple steady states fitting the same injection condition J defined by the injected fractional flows. Numerical simulations suggest the stable injection state among multiple possible states but do not explain the reason.
We address the issue of multiple steady states from the perspective of wave propagation, using three-phase fractional-flow theory. The wave-curve method is applied to solve the two conservation equations for composition paths and wave speeds in 1D foam-oil flow. There is a composition path from each possible injection state J to the initial state I satisfying the conservation equations. The stable displacement is the one with wave speeds (characteristic velocities) all positive along the path from J to I. In all cases presented, two of the paths feature negative wave velocity at J; such a solution does not correspond to the physical injection conditions. A stable displacement is achieved by either the upper, strong-foam state or lower, collapsed-foam state, but never the intermediate, unstable state. Which state makes the displacement depends on the initial state of a reservoir. The dependence of the choice of the displacing state on initial state is captured by a boundary curve.