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.