Mathematical Properties of Three-Phase Flow Equations
- Rafael E. Guzman | F. John Fayers
- Document ID
- Society of Petroleum Engineers
- SPE Journal
- Publication Date
- September 1997
- Document Type
- Journal Paper
- 291 - 300
- 1997. Society of Petroleum Engineers
- 5.4.7 Chemical Flooding Methods (e.g., Polymer, Solvent, Nitrogen, Immiscible CO2, Surfactant, Vapex), 5.3.2 Multiphase Flow, 5.3.1 Flow in Porous Media, 5.4.2 Gas Injection Methods, 5.2.1 Phase Behavior and PVT Measurements, 5.3.4 Reduction of Residual Oil Saturation, 4.5.7 Controls and Umbilicals, 5.5 Reservoir Simulation
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The equations for nondiffusive three-phase flow with gravity are examined according to the relative permeability model used. Three different classes of equations are distinguished (1) hyperbolic with a single umbilic point, (2) hyperbolic with multiple umbilic points, and (3) mixed hyperbolic and elliptic. The new equations with multiple isolated umbilic points are obtained when using Corey relative permeabilities that allow inflection points (i.e., extended Corey models). An example in which gravity and extended Corey relative permeabilities appear to give strictly hyperbolic equations is also included. The influences of viscosity, gravity, and relative permeabilities on the location of these umbilic points are shown. The relationship between the umbilic point(s) and the elliptic regions is discussed. The effect of gravity on the size of the elliptic regions is analyzed. It is shown that the size of the elliptic regions can be significantly increased or decreased by changing the gravity to viscous ratio, but without following an obvious systematic pattern.
Three-phase flow in porous media is a fundamental constituent of many oil recovery processes (e.g., gas injection, gas gravity drainage, surfacant flooding, and thermal recovery), and also occurs in the remediation of polluted aquifers. Despite its common occurrence and relevance, our mathematical and physical understanding of three-phase flow. The problem considered is the extension of the classical two-phase Buckley-Leverett5 theory to the flow of three phases. Two-phase Buckley-Leverett theory has been one of the most important tools in reservoir engineering, and its extension to three phases is long overdue. This paper presents an analysis of the mathematical properties of the equations of three-phase flow. Special interest is devoted to the study of singular points of the equations in the three-phase domain. This study is an essential first step in the construction of solutions to the three-phase Buckley-Leverett problem, as shown in a following paper.13 In fact, the presence of singular points dictates the way the solutions are constructed.
The model for three-phase flow is obtained by considering material balance of three incompressible and immiscible fluids, flowing at constant temperature inside a one dimensional homogeneous porous media. Diffusive forces are neglected. The flux terms of the material balance are calculated using Darcy's law extended to multiphase flow through the use of relative permeabilities.
The main feature of the three-phase flow equations is the presence of isolated points or regions where strict hyperbolicity fails. Failure of strict hyperbolicity occurs when the eigenvalues of the Jacobian matrix of the flux vector are real but become coincident, or they are complex conjugate pairs. A point where the eigenvalues are real but coincident is called an umbilic point. The system of equations is still hyperbolic at an umbilic point, whereas, it is elliptic when the eigenvalues are complex.
Two models for relative permeabilities have been used by many authors to show that the equations for three-phase flow are not strictly hyperbolic. Failure of strictly hyperbolic at a unique isolated umbilic point has been shown14,17,20 for Corey-type models. We refer to such equations as essentially hyperbolic. Mixed hyperbolic and elliptic behavior has been shown4,10,14,22,26 for problems using Stone-type models. The main assumptions in the Corey-type models are that the relative permeability of each phase is a unique function of its own saturation, and that the first and second derivatives of the relative permeabilities are positive whenever the phases are mobile. Stone-type models assume that the relative permeability for the wetting and nonwetting phases are unique functions of their own saturation, and that the intermediate wetting phase relative permeability is a function of two saturations.
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