For the first time, we apply three-phase fractional-flow theory combined with the wave-curve method to better understand the mechanisms of foam displacements with oil in porous media, employing a widely used foam model. Fractional-flow theory demonstrates that oil saturation in foam-created oil banks never exceeds the upper limit for stable foam, fmoil (i.e. an oil saturation above which foam is killed); see (Tang et al., 2019c) and section 3.4 below. This constraint suggests a criterion for creating significant oil banks: for the surfactant formulation, fmoil must be far above the initial oil saturation. We identify key factors controlling foam and oil-bank propagation: fmoil, foam quality, the regime in which foam is injected and foam strength at both injection and initial states. The mechanisms of these factors are revealed through a material balance on gas: any factor increasing gas volume injected while maintaining adequate foam strength, or reducing gas saturation in the foam region, accelerates foam propagation, and vice versa. Also, an optimal foam injection strategy is identified: inject foam in the low-quality regime near the transition foam quality (Tang et al., 2019a, 2019b), at which mobility reduction is at its maximum. This rule's universality needs to be further verified. Fractional-flow solutions, free of numerical artifacts, can be used to benchmark numerical simulators and machine-learning approaches for foam processes.

Foam is remarkably effective in the mobility control of gas injection for enhanced oil recovery (EOR) processes and CO₂ sequestration. Our goal is to better understand immiscible three-phase foam displacement with oil in porous media. In particular, we investigate (i) the displacement as a function of initial (I) and injection (J) conditions and (ii) the effect of improved foam tolerance to oil on the displacement and propagation of foam and oil banks. We apply three-phase fractional-flow theory combined with the wave-curve method (WCM) to find the analytical solutions for foam-oil displacements. An n-dimensional Riemann problem solver is used to solve analytically for the composition path for any combination of J and I on the ternary phase diagram and for velocities of the saturations along the path. We then translate the saturations and associated velocities along a displacement path to saturation distributions as a function of time and space. Physical insights are derived from the analytical solutions on two key aspects: the dependence of the displacement on combinations of J and I and the effects of improved oil-tolerance of the surfactant formulation on composition paths, foam-bank propagation and oil displacement. The foam-oil displacement paths are determined for four scenarios, with representative combinations of J and I that each sustains or kills foam. Only an injection condition J that provides stable foam in the presence of oil yields a desirable displacement path, featuring low-mobility fluids upstream displacing high-mobility fluids downstream. Enhancing foam tolerance to oil, e.g. by improving surfactant formulations, accelerates foam-bank propagation and oil production, and also increases oil recovery. Also, we find a contradiction between analytical and numerical solutions. In analytical solutions, oil saturation (S_o) in the oil bank is never greater than the upper-limiting oil saturation for stable foam (fmoil in our model). Nevertheless, in numerical simulations, S_o may exceed the oil saturation that kills foam in the oil bank ahead of the foam region, reflecting a numerical artifact. This contradiction between the two may arise from the calculation of pressure and pressure gradient using neighboring grid blocks in a numerical simulation. The analytical solutions we present can be a valuable reference for laboratory investigation and field design of foam for gas mobility control in the presence of oil. More significantly, the analytical solutions, which are free of numerical artifacts, can be used as a benchmark to calibrate numerical simulators for simulating foam EOR and CO2 storage processes.

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.