Solutions to the Three-Phase Buckley-Leverett Problem
- Rafael E. Guzman | F. John Fayers
- Document ID
- Society of Petroleum Engineers
- SPE Journal
- Publication Date
- September 1997
- Document Type
- Journal Paper
- 301 - 311
- 1997. Society of Petroleum Engineers
- 5.3.2 Multiphase Flow, 5.4 Enhanced Recovery, 4.1.2 Separation and Treating, 5.5 Reservoir Simulation, 5.3.1 Flow in Porous Media, 5.4.1 Waterflooding, 2.5.2 Fracturing Materials (Fluids, Proppant), 5.4.7 Chemical Flooding Methods (e.g., Polymer, Solvent, Nitrogen, Immiscible CO2, Surfactant, Vapex), 5.4.2 Gas Injection Methods, 5.2.1 Phase Behavior and PVT Measurements, 4.3.4 Scale, 5.3.4 Reduction of Residual Oil Saturation, 4.1.5 Processing Equipment, 4.5.7 Controls and Umbilicals
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This paper presents method of characteristics (MOC) solutions to the three-phase Buckley-Leverett problem with and without gravity, for the three classes of equations described by Guzmán and Fayers (1997).12 Thus, solutions for equations with single and multiple umbilic points, and mixed elliptic and hyperbolic equations are constructed. A close relation between the mathematical properties of the equations (in terms of location and quantity of umbilic points) and the physics of the displacement, the formation of oil and water banks, and the presence of a unique or infinite solution paths within the three-phase flow region for a class of solutions. Elliptic regions did not generate physically inadmissible effects and stable shocks with saturations on opposite sides of elliptic regions were constructed in all the cases examined.
Three-phase flow in porous media is a fundamental constituent of many oil recovery processes (e.g., gas injection, gas gravity drainage, surfactant 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 is not satisfactory. This paper is the second part of a study on the mathematical theory of three-phase flow. The problem considered is the extension of the classical two-phase Buckley-Leverett4 theory to the flow of three phases. This paper presents solutions to the three classes of three-phase flow equations described by Guzmán and Fayers (1997).12
Guzmán et al. (1994)13 presented three dimensional simulations of gas and water alternating gas displacements at typical reservoir conditions. These simulations suggest that a large volume of the reservoir experienced three-phase flow. Moreover, they found that uncertainties in three-phase relative permeabilities translated into significant uncertainties in oil recovery and production gas oil ratio. These results imply that additional studies in the theory of three-phase flow are well justified.
Despite the necessary simplifying assumptions, the Buckley-Leverett theory has been one of the most useful techniques in reservoir engineering. Although the theory may be more suitable for laboratory settings,10 the extension to three-phase flow can provide information to better interpret three dimensional simulations, new methods for numerical simulators, insight into the physics of displacements, and methods to interpret laboratory flow experiments.
Solutions for three-phase flow problems that assume Corey-type relative permeabilities and no gravity have been presented by many authors.1,5,6,18,27 Full solutions and a complete mathematical theory for Corey-type problems was presented by Isaacson et al. (1988).18 Bell et al. (1986)3 used finite differences to compute solutions with Stone-type relative permeabilities and to investigate the effect of the elliptic regions. Satisfactory solutions were found even when either the injection or initial conditions were inside an elliptic region. Example solutions were constructed using the method of characteristics by Gorell (1988)9 and Wingard and Orr (1994).31 Gorell (1988)9 ignored the imaginary part of the complex eigenvalues without a mathematical justification. Wingard and Orr (1994)31 included mass transfer and temperature effects, but the issues of three-phase flow were not presented in a systematic way.
This paper presents solutions for three-phase flow problems with and without gravity, using Corey, extended Corey, and Stone-type models. Practical conditions are simulated by using measured two-phase flow data and Stone's model I,28 and reasonable viscosity ratios and gravity numbers. It is shown that the structure of the new solutions is very different from the previous solutions to unique umbilic point problems1,5,6,18,27 (i.e., Corey-kr).
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