Theory of Three-Component, Three-Phase Displacement in Porous Media

Abstract

Summary A theory of multicomponent, multiphase displacement in porous media isfurther developed for three-component, three-phase porous media is furtherdeveloped for three-component, three-phase systems. In regions of atie-triangle in which one or two of the phases is (are) immobile, thedescription of solutions to the phases is (are) immobile, the description ofsolutions to the differential equations that govern flow is shown to beindependent of the way relative permeabilities depend on saturations. Whenthree phases flow simultaneously, key features of the mathematical solutionphases flow simultaneously, key features of the mathematical solution space canbe described analytically for an exponent-type relative permeability model.permeability model. Introduction The theory and calculational procedures set forth by Helfferich (calledcoherence theory) can reveal the salient features of fluid/fluid displacementin porous media. Heretofore, applications of this approach to equilibriumthree-phase problems have been restricted to constant-phase-velocity-ratiodisplacements (straight-line relative permeability curves), drainages in whichoil and water are replaced by a vapor phase of infinite mobility, and floodsthat produce a very limited range of three-phase compositions. (Other problemshave been described without being fully solved; see Ref. 4.) By applyingcoherence theory under less restrictive conditions, we uncovered additional anddifferent features of flow within threephase regions. In this paper, we developthe mathematics behind applying Helfferich's theory and methods to isothermal, three-component, three-phase displacements. We also detail numerical techniquesthat facilitate solving the nonlinear equations that arise in the analysis. Features of the mathematical space depicting solutions to the equations beforeinitial and boundary conditions are specified (the so-called composition pathgrid), illustrated, and discussed. Results that make known the nature ofthree-phase displacements and explain experimentally observed phenomena archighlighted in SPE 19678. Theory Assumptions. In coherence theory, which can be considered a generalizationof Buckley-Leverens and fractional-flow theories, equations governingconservation of mass, momentum, and energy during idealized displacements inporous media are formulated and solved. To formulate a manageable problem. thefollowing assumptions and limitations are imposed.The flow is 1D andperpendicular to the gravitational field.Properties of the porous mediumare uniform,There are no dispersive phenomena.All phases areincompressible.The linear momentum balance for each phase is governed by Darcy's law.The effect of pressure on phase and fractional-flow behavior isnegligible.The system is isothermal.Partial molar volumes of componentsare constant.The fluids consist of three nonadsorbing components thatalways partition into three phases.