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Relative Permeabilities for Strictly Hyperbolic Models of Three-Phase Flow in Porous Media

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Abstract

Traditional mathematical models of multiphase flow in porous media use a straightforward extension of Darcy’s equation. The key element of these models is the appropriate formulation of the relative permeability functions. It is well known that for one-dimensional flow of three immiscible incompressible fluids, when capillarity is neglected, most relative permeability models used today give rise to regions in the saturation space with elliptic behavior (the so-called elliptic regions). We believe that this behavior is not physical, but rather the result of an incomplete mathematical model. In this paper we identify necessary conditions that must be satisfied by the relative permeability functions, so that the system of equations describing three-phase flow is strictly hyperbolic everywhere in the saturation triangle. These conditions seem to be in good agreement with pore-scale physics and experimental data.

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References

  • Avraam, D. G. and Payatakes, A. C.: 1995, Flow regimes and relative permeabilities during steadystate 2-phase flow in porous-media, J. Fluid Mech. 293, 207–236.

    Google Scholar 

  • Azevedo, A. V. and Marchesin, D.: 1995, Multiple viscous solutions for systems of conservation laws, Trans. Am. Math. Soc. 347(8), 3061–3077.

    Google Scholar 

  • Azevedo, A. V., Marchesin, D., Plohr, B. and Zumbrun, K.: 2002, Capillary instability in models for three-phase flow, Z. Angew. Math. Phys. 53, 713–746.

    Article  Google Scholar 

  • Barenblatt, G. I., Entov, V. M. and Ryzhik, V. M.: 1990, Theory of Fluid Flows through Natural Rocks, Kluwer, Dordrecht.

    Google Scholar 

  • Bell, J. B., Trangenstein, J. A. and Shubin, G. R.: 1986, Conservation laws of mixed type describing three-phase flow in porous media, SIAM J. Appl. Math. 46(6), 1000–1017.

    Google Scholar 

  • Charny, I. A.: 1963, Subterranean Hydro-Gas Dynamics, Gostoptekhizdat, Moscow (in Russian).

    Google Scholar 

  • Coddington, E. A. and Levinson, N.: 1955, Theory of Ordinary Differential Equations, McGraw Hill, New York.

    Google Scholar 

  • Falls, A. H. and Schulte, W. M.: 1992, Theory of three component, three phase displacement in porous media, SPE Reserv. Eng. 7(3), 377–384.

    Google Scholar 

  • Fayers, F. J.: 1987, Extension of Stone's method I and conditions for real characteristics in three-phase flow, in: SPE Annual Technical Conference and Exhibition, Dallas, TX (SPE 16965).

  • Fayers, F. J. and Matthews, J. D.: 1984, Evaluation of normalized Stone's methods for estimating three-phase relative permeabilities, Soc. Pet. Eng. J. 24(2), 224–232; Petrol. Trans. AIME, 277.

    Google Scholar 

  • Geffens, T. M., Owens, W. W., Parrish, D. R. and Morse, R. A.: 1951, Experimental investigation of factors affecting laboratory relative permeability measurements, Petrol. Trans. AIME, 192, 99–110.

    Google Scholar 

  • Grader, A. S. and O'Meara Jr., D. J.: 1988, Dynamic displacement measurements of three-phase relative permeabilities using three immiscible fluids, in: SPE Annual Technical Conference and Exhibition, Houston, TX (SPE 18293).

  • Gray, W. G. and Hassanizadeh, S. M.: 1998, Macroscale continuum mechanics for multiphase porous-media flow including phases, interfaces, common lines and common points, Adv. Water Resour. 21, 261–281.

    Article  Google Scholar 

  • Guzmán, R. E. and Fayers, F. J.: 1997, Mathematical properties of three-phase flow equations, Soc. Pet. Eng. J. 2(3), 291–300.

    Google Scholar 

  • Hassanizadeh, S. M.: 1986, Derivation of basic equations of mass transport in porous media, Part 2. Generalized Darcy's and Fick's laws, Adv. Water Resour. 9, 207–222.

    Article  Google Scholar 

  • Hassanizadeh, S. M. and Gray, W. G.: 1993, Toward an improved description of the physics of twophase flow, Adv. Water Resour. 16(1), 53–67.

    Article  Google Scholar 

  • Hicks Jr., P. J. and Grader, A. S.: 1996, Simulation of three-phase displacement experiments, Transp. Porous Media 24, 221–245.

    Google Scholar 

  • Holden, L.: 1990, On the strict hyperbolicity of the Buckley-Leverett equations for three-phase flow in a porous medium, SIAM J. Appl. Math. 50(3), 667–682.

    Google Scholar 

  • Holden, H., Holden, L. and Risebro, N. H.: 1990, Some qualitative properties of 2×2 systems of conservation laws of mixed type, in: B. L. Keyfitz and M. Shearer (eds), Nonlinear Evolution Equations that Change Type, Springer-Verlag, New York, pp. 67–78.

    Google Scholar 

  • Jackson, M. D. and Blunt, M. J.: 2002, Elliptic regions and stable solutions for three-phase flow in porous media, Transp. Porous Media 48, 249–269.

    Article  Google Scholar 

  • Johnson, E. F., Bossler, D. P. and Naumann, V. O.: 1959, Calculation of relative permeability from displacement experiments, Petrol. Trans. AIME 216, 370–372.

    Google Scholar 

  • Juanes, R.: 2003, Displacement theory and multiscale numerical modeling of three-phase flow in porous media, PhD Dissertation, University of California at Berkeley.

  • Juanes, R. and Patzek, T. W.: 2003, Relative permeabilities in co-current three-phase displacements with gravity, in: SPE Western Regional/AAPG Pacific Section Joint Meeting, Long Beach, CA (SPE 83445).

  • Juanes, R. and Patzek, T. W.: 2004, Analytical solution to the Riemann problem of three-phase flow in porous media, Transp. Porous Media 55(1), 47–70.

    Article  Google Scholar 

  • Keyfitz, B. L.: 1990, Shocks near the sonic line: a comparison between steady and unsteady models for change of type, in: B. L. Keyfitz and M. Shearer (eds), Nonlinear Evolution Equations that Change Type, Springer-Verlag, New York, pp. 89–106.

    Google Scholar 

  • Lenhard, R. J. and Parker, J. C.: 1987, A model for hysteretic constitutive relations governing multiphase flow, 2. Permeability-saturation relations, Water Resour. Res. 23(2), 2197–2206.

    Google Scholar 

  • Lenormand, R., Touboul, E. and Zarcone, C.: 1988, Numerical models and experiments on immiscible displacements in porous media, J. Fluid Mech. 189, 165–187.

    Google Scholar 

  • Leverett, M. C. and Lewis, W. B.: 1941, Steady flow of gas-oil-water mixtures through unconsolidated sands, Petrol. Trans. AIME 142, 107–116.

    Google Scholar 

  • Marchesin, D. and Medeiros, H. B.: 1989, A note on the stability of eigenvalue degeneracy in nonlinear conservation laws of multiphase flow, in: W. B. Lindquist (ed.), Current Progress in Hyperbolic Systems: Riemann Problems and Computations, American Mathematical Society, Providence, RI, pp. 215–224.

    Google Scholar 

  • Marchesin, D. and Plohr, B. J.: 2001, Wave structure in WAG recovery, Soc. Pet. Eng. J. 6(2), 209–219.

    Google Scholar 

  • Muskat, M.: 1949, Physical Principles of Oil Production, McGraw-Hill, New York.

    Google Scholar 

  • Oak, M. J.: 1990, Three-phase relative permeability of water-wet Berea, in: SPE/DOE Seventh Symposium on Enhanced Oil Recovery, Tulsa, OK (SPE/DOE 20183).

  • Oak, M. J., Baker, L. E. and Thomas, D. C.: 1990, Three-phase relative permeability of Berea sandstone, J. Pet. Technol. 42(8), 1054–1061.

    Google Scholar 

  • Odeh, A. S.: 1959, Effect of viscosity ratio on relative permeability, Petrol. Trans. AIME 216, 346–353.

    Google Scholar 

  • Osoba, J. S., Richardson, J. G., Kerver, J. K., Hafford, J. A. and Blair, P. M.: 1951, Laboratory measurements of relative permeability, Petrol. Trans. AIME 192, 47–56.

    Google Scholar 

  • Sahni, A., Guzmán, R. and Blunt, M.: 1996, Theoretical analysis of three phase flow experiments in porous media, in: SPE Annual Technical Conference and Exhibition, Denver, CO (SPE 36664).

  • Shearer, M.: 1988, Loss of strict hyperbolicity of the Buckley-Leverett equations for three phase flow in a porous medium, in: M. F. Wheeler (ed.), Numerical Simulation in Oil Recovery, Springer-Verlag, New York, pp. 263–283.

    Google Scholar 

  • Shearer, M. and Trangenstein, J. A.: 1989, Loss of real characteristics for models of three-phase flow in a porous medium, Transp. Porous Media 4, 499–525.

    Article  Google Scholar 

  • Silin, D. and Patzek, T.: 2004, On Barenblatt's model of spontaneous countercurrent imbibition, Transp. Porous Media 54(3), 297–322.

    Article  Google Scholar 

  • Stklyanin, Y. I.: 1960, The motion of a mixture of three liquids in a porous medium (in Russian), Izv. Akad. Nauk SSSR, Otd. Tekh. Nauk, Mekhanika i Mashinostroenie 2(5).

  • Stone, H. L.: 1970, Probability model for estimating three-phase relative permeability, J. Pet. Technol. 23(2), 214–218; Petrol. Trans. AIME249.

    Google Scholar 

  • Trangenstein, J. A.: 1989, Three-phase flow with gravity, in: W. B. Lindquist (ed.), Current Progress in Hyperbolic Systems: Riemann Problems and Computations, American Mathematical Society, Providence, RI, pp. 147–159.

    Google Scholar 

  • Truesdell, C. and Noll, W.: 1965, The Non-Linear Field Theories of Mechanics (Second edition, 1992), Springer-Verlag, Berlin.

    Google Scholar 

  • Wyckoff, R. D. and Botset, H. G.: 1936, The flow of gas-liquid mixtures through unconsolidated sands, Physics 7, 325–345.

    Google Scholar 

  • Yortsos, Y. C., Xu, B. and Salin, D.: 1997, Phase diagram of fully-developed drainage in porous media, Phys. Rev. Lett. 79(23), 4581–4584.

    Article  Google Scholar 

  • Zauderer, E.: 1983, Partial Differential Equations of Applied Mathematics, John Wiley & Sons, New York.

    Google Scholar 

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Authors
  1. Ruben Juanes
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  2. Tadeusz W. Patzek
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Juanes, R., Patzek, T.W. Relative Permeabilities for Strictly Hyperbolic Models of Three-Phase Flow in Porous Media. Transport in Porous Media 57, 125–152 (2004). https://doi.org/10.1023/B:TIPM.0000038251.10002.5e

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