Foam Displacements with Multiple Steady States
Authors: Rossen William Richard , Bruining Johannes 
Published: March 2007

Summary
A number of experimental and theoretical studies suggest that the fractional-flow function for foam can be either multivalued in water saturation or else can comprise distinct fractional-flow curves for two or more foam regimes, with jumps between them where each regime reaches its limiting condition. We construct fractional-flow solutions for these cases. When such a foam is employed in a surfactant-alternating-gas (SAG) process, the usual "tangency" condition is modified and the foam can be considerably weaker than the foam formed at what appears to be the point of tangency of the multivalued fractional-flow function. If the capillary-pressure function Pc (Sw) differs between foam regimes, that difference can substantially change the nature of the displacement. This alters the effective fractional-flow function and hence the global solution of the equations. It is therefore important to determine how capillary pressure varies in foam displacements, by direct measurement in situ if possible. Special care is needed in numerical simulation of processes using fixed grids if capillary pressure depends directly on foam regime. Using gridblocks that are too large can weaken the effect of capillary pressure that would enforce the correct shock on the small scale. Using an upscaled fractional-flow function appears to eliminate this problem, however.

Introduction
Foams are injected into geological formations for gas diversion in improved oil recovery (IOR) (Schramm 1994; Rossen 1996), acid diversion in matrix acid well stimulation (Gdanski 1993), and environmental remediation (Hirasaki et al. 2000). In IOR and environmental remediation, it is often useful to inject gas and surfactant solution in alternating slugs, a process called SAG injection. SAG injection holds several advantages over continuous coinjection of gas and liquid, as described elsewhere (Shi and Rossen 1998; Shan and Rossen 2004).
Method of Characteristics and Fractional-Flow Theory. Many problems involving conservation equations can be formulated in the so-called hyperbolic framework. The ensuing equations can then be solved using the method of characteristics (see, for example, Smoller (1980), after page 266). The solution consists of rarefactions or spreading waves, constant states, and shocks. Additional conditions are also required to obtain a unique solution. Numerical solutions of the equations can, in effect, pick out the wrong uniqueness conditions and give inaccurate results. A complete analysis requires the traveling-wave representation of a shock. Bruining and Van Duijn (2000, 2007) present an example in which the conditions on the traveling wave are essential to identifying the correct solution of the macroscopic equations. Bruining et al. (2002, 2004) give a regularization procedure for constructing such a traveling wave solution for an application of steam injection. The conditions on the traveling wave must be kept in mind when using a graphical procedure for finding the solution with the method of characteristics, a technique introduced by Buckley and Leverett (1941). It is widely used for petroleum engineering applications [see also Pope (1980), who generalized fractional-flow theory to deal with more complex problems]. In this formulation, the uniqueness condition is called the Welge tangent condition (1952).