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The Solution by the Wave Curve Method of Three-Phase Flow in Virgin Reservoirs

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Abstract

There are two goals of this study. The first is to provide an introduction to the wave curve method for finding the analytic solution of a porous medium injection problem. Similar to fractional and chromatographic flow theory, the wave curve method is based on the method of characteristics, but it is applicable to an expanded range of physical processes in porous medium flow. The second goal is to solve injection problems for immiscible three-phase flow, as described by Corey’s model, in which a mixture of gas and water is injected into a porous medium containing oil and irreducible water. In particular we determine, for any choice of the phase viscosities, the proportion of the injected fluids that maximizes recovery around breakthrough time. Numerical simulations are performed to compare our solutions for Corey’s model with those of other models. For the injection problems we consider, solutions for Corey’s model are very similar to those for Stone’s model, despite the presence of an elliptic region in the latter; and they are very different from those for the Juanes-Patzek model, which preserves strict hyperbolicity. A nice feature of our analytical method is that it facilitates explaining both differences and similarities among the solutions for the three models considered.

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References

  • Azevedo A., Marchesin D.: Multiple viscous profile Riemann solutions in mixed elliptic-hyperbolic models for flow in porous media. In: Keyfitz, B., Shearer, M. (eds) Hyperbolic Equations that Change Type, IMA Volumes in Mathematics and its Applications, vol 27., pp. 1–17. Springer-Verlag, New York–Heidelberg–Berlin (1990)

    Google Scholar 

  • Azevedo A., Marchesin D.: Multiple viscous solutions for systems of conservation laws. Trans Amer Math Soc 347, 3061–3078 (1995)

    Article  Google Scholar 

  • Azevedo A., Marchesin D., Plohr B., Zumbrun K.: Nonuniqueness of nonclassical solutions of Riemann problems. Z Angew Math Phys 47, 977–998 (1996)

    Article  Google Scholar 

  • Azevedo A., Marchesin D., Plohr B., Zumbrun K.: Bifurcation of nonclassical viscous shock profiles from the constant state. Commun Math Phys 202, 267–290 (1999)

    Article  Google Scholar 

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

    Article  Google Scholar 

  • Azevedo, A., de Souza, A., Furtado, F., Marchesin, D.: Three-phase flow in a porous medium. In preparation (2009)

  • Aziz K., Settari A.: Petroleum Reservoir Simulation. Elsevier Applied Science, New York–London (1990)

    Google Scholar 

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

    Article  Google Scholar 

  • Bruining J., van Duijn C. J.: Uniqueness conditions in a hyperbolic model for oil recovery by steamdrive. Comput Geosciences 4, 65–98 (2000)

    Article  Google Scholar 

  • Bruining, J., Marchesin, D.: Maximal oil recovery by simultaneous condensation of alkane and steam. Phys Rev E 75, 036,312–1–16 (2007)

  • Buckley S., Leverett M.: Mechanisms of fluid displacement in sands. Trans AIME 146, 187–196 (1942)

    Google Scholar 

  • Čanić S.: Nonexistence of Riemann solutions for a model arising in petroleum engineering. Nonlinear Anal 4, 373–408 (2003)

    Article  Google Scholar 

  • Corey A., Rathjens C., Henderson J., Wyllie M.: Three-phase relative permeability. Trans AIME 207, 349–351 (1956)

    Google Scholar 

  • Courant R., Friedrichs K.: Supersonic Flow and Shock Waves. Springer-Verlag, New York (1976)

    Google Scholar 

  • Dafermos C.: Hyperbolic Conservation Laws in Continuum Physics, Grundlehren der mathematischen Wissenschaften ,vol 325. Springer-Verlag, New York (2000)

    Google Scholar 

  • Delshad M., Pope G., Lake L.: Two- and three-phase relative permeabilities of micellar fluids. SPE J 2, 327–337 (1987)

    Google Scholar 

  • de Souza A.: Stability of singular fundamental solutions under perturbations for flow in porous media. Mat Aplic Comput 11(2), 73–115 (1992)

    Google Scholar 

  • Falls A., Schulte W.: Theory of three-component, three-phase displacement in porous media. SPE Reservoir Engrg 7, 377–384 (1992)

    Google Scholar 

  • Fayers F. S.: Some theoretical results concerning the displacement of a viscous oil by a hot fluid in a porous medium. Fluid Mech 13, 65–76 (1962)

    Article  Google Scholar 

  • Frid H., Liu I. S.: Oscillation waves in Riemann problems for phase transitions. Quart Appl Math 56, 115–135 (1998)

    Google Scholar 

  • Gel’fand, I.M.: Some problems in theory of quasilinear equations. Amer Mat Soc Trans, ser 2 29:295–381, english transl (1963)

    Google Scholar 

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

    Google Scholar 

  • Guzmán R. E., Fayers F. J.: Solutions to the three-phase flow Buckley-Leverett problem. SPE J 2(3), 301–311 (1997)

    Google Scholar 

  • Helfferich F. G.: General theory of multicomponent, multiphase displacement in porous media. Soc Pet Engrg J 21, 51–62 (1981)

    Google Scholar 

  • Hirasaki G. J.: Application of the theory of multicomponent, multiphase displacement to three-component, two-phase surfactant flooding. Soc Pet Engrg J 21, 191–204 (1981)

    Google Scholar 

  • Holden H.: On the riemann problem for a prototype of a mixed type conservation law. Comm Pure Appl Math 40, 229–264 (1987)

    Article  Google Scholar 

  • Holden H., Holden L., Riesebro N. H.: Some qualitative properties of 2 × 2 systems of conservation laws of mixed type. In: Keyfitz, B., Shearer, M. (eds) Hyperbolic Equations that Change Type, IMA Volumes in Mathematics and its Applications vol 27., pp. 67–78. Springer-Verlag, New York–Heidelberg–Berlin (1990)

    Google Scholar 

  • Isaacson E., Marchesin D., Plohr B., Temple J. B.: Multiphase flow models with singular Riemann problems. Mat Apl Comput 11(2), 147–166 (1992)

    Google Scholar 

  • Juanes R., Patzek T.: Relative permeabilities for strictly hyperbolic models of three-phase flow in porous media. Transport in Porous Media 57, 125–152 (2004)

    Article  Google Scholar 

  • LaForce, T., Jessen, K.: Analytical and numerical investigation of multicomponent multiphase WAG displacements. In: Proceedings of the SPE Annual Technical Conference and Exhibition, November 2007, Anaheim, California (2007)

  • LaForce T., Jessen K., F M Orr J.: Four-component gas/water/oil displacements in one dimension: Part I, Structure of the conservation law. Transport in Porous Media 71, 199–216 (2007)

    Article  Google Scholar 

  • LaForce T., Jessen K., F M Orr J.: Four-component gas/water/oil displacements in one dimension: Part II, Example solutions. Transport in Porous Media 72, 1573–1634 (2008)

    Article  Google Scholar 

  • Lambert, W., Marchesin, D., Bruining, J.: The Riemann solution for the injection of steam and nitrogen in a porous medium. Transport in Porous Media to appear (2009)

  • Lax, P.: Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves, Regional Conference Series Lectures in Applied Mathematics, vol 11. Society for Industrial and Applied Mathematics. Philadelphia, Pennsylvania (1973)

  • Liu T. P.: The Riemann problem for general 2 × 2 conservation laws. Trans Amer Math Soc 199, 89–112 (1974)

    Article  Google Scholar 

  • Liu T. P.: The Riemann problem for general systems of conservation laws. J Differential Equations 18, 218–234 (1975)

    Article  Google Scholar 

  • Marchesin D., Plohr B.: Wave structure in WAG recovery. SPE J 6, 209–219 (2001)

    Google Scholar 

  • Medeiros H.: Stable hyperbolic singularities for three-phase flow models in oil reservoir simulation. Acta Applicandae Mathematica 28, 135–159 (1992)

    Article  Google Scholar 

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

  • Oleĭnik O.: On the uniqueness of the generalized solution of the Cauchy problem for a non-linear system of equations occuring in mechanics. Uspehi Mat Nauk 12, 169–176 (1957)

    Google Scholar 

  • Pope G.: The application of fractional flow theory to enhanced oil recovery. Soc Petr Engrg J 20, 191–205 (1980)

    Google Scholar 

  • Pope G., Lake L., Helfferich F.: Cation exchange in chemical flooding: Part 1–basic theory without dispersion. Soc Petr Engrg J 18, 418–444 (1978)

    Google Scholar 

  • Riemann G. F. B.: Uber die fortpflanzung ebener luftwellen von endlicher schwingungsweite. Abhandlungen der Gesellschaft der Wissenschaften zu Göttingen 8, 43 (1860)

    Google Scholar 

  • Rossen W. R., van Duijn C. J.: Gravity segregation in steady-state horizontal flow in homogeneous reservoirs. J Pet Sci Eng 43, 99–111 (2004)

    Article  Google Scholar 

  • Sahni, A., Guzman, R., Blunt, M.: Theoretical analysis of three phase flow experiments in porous media. In: Proceedings of the SPE Annual Technical Conference and Exhibition, October 1996 (SPE 36664), Denver, Colorado (1996)

  • Schaeffer D., Shearer M.: The classification of 2 × 2 systems of non-strictly hyperbolic conservation laws, with application to oil recovery. Comm Pure Appl Math 40, 141–178 (1987)

    Article  Google Scholar 

  • Schecter S., Marchesin D., Plohr B.: Structurally stable Riemann solutions. J Differential Equations 126, 303–354 (1996)

    Article  Google Scholar 

  • Schulte W., Falls A.: Features of three-component, three-phase displacement in porous media. SPE Reservoir Engrg 7, 426–432 (1992)

    Google Scholar 

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

    Article  Google Scholar 

  • Smoller J.: Shock Waves and Reaction-Diffusion Equations, 2nd edn. Springer-Verlag, New York (1994)

    Google Scholar 

  • Stone H.: Probability model for estimating three-phase relative permeability. J Petr Tech 22, 214–218 (1970)

    Article  Google Scholar 

  • Temple J. B.: Systems of conservation laws with coinciding shock and rarefaction curves. Contemp Math 17, 143–151 (1983)

    Google Scholar 

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Authors and Affiliations

  1. Departamento de Matemática, Universidade de Brasília, 70910-900, Brasília, DF, Brazil

    Arthur V. Azevedo

  2. D. de Matemática e Estatística, U. Federal de Campina Grande, 58109-970, Campina Grande, PB, Brazil

    Aparecido J. de Souza

  3. Department of Mathematics, University of Wyoming, 82071, Laramie, WY, USA

    Frederico Furtado

  4. Instituto de Matemática Pura e Aplicada, E.D. Castorina 110, 22460, Rio de Janeiro, RJ, Brazil

    Dan Marchesin

  5. Physics of Condensed Matter and Complex Systems Group, Los Alamos National Laboratory, 87544, Los Alamos, NM, USA

    Bradley Plohr

Authors
  1. Arthur V. Azevedo
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  2. Aparecido J. de Souza
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  3. Frederico Furtado
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  4. Dan Marchesin
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  5. Bradley Plohr
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Corresponding author

Correspondence to Frederico Furtado.

Additional information

This study was supported in part by: FEMAT under Grant 04/10, UnB under Grant FUNPE 2005, CNPq under Grants 306609/2004-5, 620017/2004-0, 490696/2004-0, 491148/2005-4, 304168/2006-8, 472067/2006-0, 474121/2008-9, 620029/2004-8, 620025/2006-9, 478668/2007-4; 401961/2008-7, IMPA-PCI 680022/2006-6; Instituto do Milênio IM/AGIMB; FAPERJ under Grants E-26/152.525/2006, E-26/102.723/2008, E-26/112.220/2008, E-26/110.310/2007, E-26/112.112/2008, and E-26/110.414/2008; CAPES 014/2008; DE-FC26-08NT4; and the U. S. Department of Energy. The authors gratefully acknowledge the hospitality of IMPA, UFCG, UnB, and UW during this study.

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Azevedo, A.V., de Souza, A.J., Furtado, F. et al. The Solution by the Wave Curve Method of Three-Phase Flow in Virgin Reservoirs. Transp Porous Med 83, 99–125 (2010). https://doi.org/10.1007/s11242-009-9508-9

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