Introduction
Fractional flow theory has been applied by various authors to waterflooding, polymer flooding, carbonated waterflooding, alcohol flooding, miscible flooding, steamflooding, and various types of surfactant flooding. Many of the assumptions made by these authors are the same and are necessary for obtaining simple analytical or graphical solutions to the continuity equations. Typically, the major assumptions, which are sometimes not stated explicitly, are:one dimensional flow in a homogeneous, isotropic, isothermal porous medium,at most, two phases are flowing,at most, three components are flowing,local equilibrium exists,the fluids are incompressible,for sorbing components, the adsorption isotherm depends only on one component and has negative curvature,dispersion is negligible,gravity and capillarity are negligible,no fingering occurs,Darcy's law applies,the initial distribution of fluids is uniform, anda continuous injection of constant composition is injected, starting at time zero.
Several of these assumptions are relaxed easily. One of the most useful to relax is Assumption 12, continuous injection. The principles of chromatography can be applied to analyze the more interesting case of injecting one or more slugs. Most of these processes require slug injection of chemical or solvent to be economical. In fact, a lower bound on the slug size necessary to prevent slug breakdown can be obtained from a simple extension of fractional flow theory. In this and other extensions the common new feature is the need to evaluate more than one characteristic velocity. A second example of this is the extension of fractional flow theory from simultaneous immiscible two-phase flow (the classical Buckley-Leverett waterflood problem) to simultaneous immiscible three-phase flow (the classical oil/water/gas flow problem). A third example is the extension to nonisothermal cases. Here we need to consider the energy balance, mass balance, and velocity of a front of constant temperature. A fourth example is when one or more components are partitioning between phases. In all cases, mathematically, the extension is analogous to the generalization from the one-component adsorption problems (or two-component ion exchange problems with a stoichiometric constraint) to multicomponent sorption problems. The latter theory has been worked out in a very general way for many component systems using the concept of coherence. Pope et al. recently have applied this theory to reservoir engineering involving sorption problems.
SPEJ
P. 191＾
Three‐Phase Fractional‐Flow Theory of Foam‐Oil Displacement in Porous Media With Multiple Steady States
