Fractional-flow theory has proven useful for understanding the factors that control foam displacements and as a benchmark against which to test foam simulators. Most applications of fractional-methods to foam have excluded oil. Recently, Mayberry and Kam (SPE 100964) presented out fractional-flow solutions for foam injection with a constant effect of oil on the foam. We extend fractional-flow methods to foam displacements with oil, using the effects of oil and water saturations on foam as represented in the STARS simulator. There can be abrupt shifts in the composition paths at the limiting water and oil saturations for foam stability. In the immiscible three-phase SAG displacements examined, if foam collapses at the initial oil saturation in the reservoir, there is a very-small-velocity shock from the injected condition to complete foam collapse. The displacement is nearly as inefficient as if no foam were present at all. It does not matter in these cases whether foam is weakened by low water saturation. The displacement is efficient, however, if foam is unaffected by oil but weakened at low water saturation. These results may reflect our foam model, where foam is only partially destroyed at low water saturations but is completely destroyed by high oil saturation. Two idealized models for three-phase displacements can be represented at two-phase displacements with chemical or miscible shocks: A model for a first-contact miscible gas flood with foam suggests an optimal water fraction in foam that puts the gas front just slightly ahead of the foam (surfactant) front. An idealized model of a surfactant flood with foam for mobility control suggests it is important to inject a sufficiently high water fraction in the foam that the gas front is behind the surfactant front as the flood proceeds. We present simulations to verify the solutions obtained with fractional-flow methods.


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