Global Riemann solver and front tracking approximation of three-component gas floods
Authors:
Saeid Khorsandi, Wen Shen and Russell T. Johns
Journal:
Quart. Appl. Math. 74 (2016), 607-632
MSC (2010):
Primary 35F55
DOI:
https://doi.org/10.1090/qam/1444
Published electronically:
July 12, 2016
MathSciNet review:
3539024
Full-text PDF Free Access
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Abstract: We study a $2\times 2$ system of non-strictly hyperbolic conservation laws arising in three-component gas flooding for enhanced oil recovery. The system is not strictly hyperbolic. In fact, along a curve in the domain one family is linearly degenerate, and along two other curves the system is parabolic degenerate. We construct global solutions for the Riemann problem, utilizing the splitting property of thermodynamics from the hydrodynamics. Front tracking simulations are presented, using the global Riemann solver.
References
- K. Ahmadi, R. T. Johns, K. Mogensen, and R. Noman, Limitations of current method-of-characteristics (MOC) methods using shock-jump approximations to predict MMPs for complex gas/oil displacements, SPEJ 16 (2011), 743–750.
- Stefano Bianchini and Alberto Bressan, Vanishing viscosity solutions of nonlinear hyperbolic systems, Ann. of Math. (2) 161 (2005), no. 1, 223–342. MR 2150387, DOI 10.4007/annals.2005.161.223
- Alberto Bressan, Hyperbolic systems of conservation laws, Oxford Lecture Series in Mathematics and its Applications, vol. 20, Oxford University Press, Oxford, 2000. The one-dimensional Cauchy problem. MR 1816648
- S. E. Buckley and M. Leverett, Mechanism of fluid displacement in sands, Transactions of the AIME 146 (1942), 107–116.
- Olav Dahl, Thormod Johansen, Aslak Tveito, and Ragnar Winther, Multicomponent chromatography in a two phase environment, SIAM J. Appl. Math. 52 (1992), no. 1, 65–104. MR 1148319, DOI 10.1137/0152005
- Birol Dindoruk, Analytical theory of multiphase, multicomponent displacement in porous media, ProQuest LLC, Ann Arbor, MI, 1992. Thesis (Ph.D.)–Stanford University. MR 2687942
- A. M. Egwuenu, R. T. Johns, and Y. Li, Improved fluid characterization for miscible gas floods, SPERE 11 (2008), 655–665.
- V. M. Entov and A. Zazovsky, Nonlinear waves in physicochemical hydrodynamics of enhanced oil recovery, Multicomponent flows, International Conference on Porous Media: Physics, Models, Simulation, Moscow, 1997.
- Tore Gimse and Nils Henrik Risebro, Riemann problems with a discontinuous flux function, Third International Conference on Hyperbolic Problems, Vol. I, II (Uppsala, 1990) Studentlitteratur, Lund, 1991, pp. 488–502. MR 1109304
- Tore Gimse and Nils Henrik Risebro, Solution of the Cauchy problem for a conservation law with a discontinuous flux function, SIAM J. Math. Anal. 23 (1992), no. 3, 635–648. MR 1158825, DOI 10.1137/0523032
- F. G. Helfferich, Theory of multicomponent, multiphase displacement in porous media, SPEJ 21 (1980), 51–62.
- G. J. Hirasaki, Application of the theory of multicomponent, multiphase displacement to three-component, two-phase surfactant flooding, SPEJ 21 (1981), 191–204.
- Eli L. Isaacson and J. Blake Temple, Analysis of a singular hyperbolic system of conservation laws, J. Differential Equations 65 (1986), no. 2, 250–268. MR 861520, DOI 10.1016/0022-0396(86)90037-9
- Thormod Johansen, Aslak Tveito, and Ragnar Winther, A Riemann solver for a two-phase multicomponent process, SIAM J. Sci. Statist. Comput. 10 (1989), no. 5, 846–879. MR 1009544, DOI 10.1137/0910050
- Thormod Johansen and Ragnar Winther, The solution of the Riemann problem for a hyperbolic system of conservation laws modeling polymer flooding, SIAM J. Math. Anal. 19 (1988), no. 3, 541–566. MR 937469, DOI 10.1137/0519039
- T. Johansen and R. Winther, Mathematical and numerical analysis of a hyperbolic system modeling solvent flooding. ECMOR II – 2nd European Conference on the Mathematics of Oil Recovery, 1990.
- Thormod Johansen and Ragnar Winther, The Riemann problem for multicomponent polymer flooding, SIAM J. Math. Anal. 20 (1989), no. 4, 908–929. MR 1000728, DOI 10.1137/0520061
- Russell Taylor Johns, Analytical theory of multicomponent gas drives with two-phase mass transfer, ProQuest LLC, Ann Arbor, MI, 1992. Thesis (Ph.D.)–Stanford University. MR 2687944
- R. T. Johns, B. Dindoruk, and F. M. Orr, Analytical theory of combined condensing/vaporizing gas drives, SPE Adv. Tech. Series 1 (1993), 7–16.
- R. T. Johns and F. M. Orr, Miscible gas displacement of multicomponent oils, SPEJ 1 (1996), 39–50.
- Ruben Juanes and Knut-Andreas Lie, Numerical modeling of multiphase first-contact miscible flows. II. Front-tracking/streamline simulation, Transp. Porous Media 72 (2008), no. 1, 97–120. MR 2377328, DOI 10.1007/s11242-007-9139-y
- Barbara L. Keyfitz and Herbert C. Kranzer, A system of nonstrictly hyperbolic conservation laws arising in elasticity theory, Arch. Rational Mech. Anal. 72 (1979/80), no. 3, 219–241. MR 549642, DOI 10.1007/BF00281590
- S. Khorsandi, K. Ahmadi, and R. T. Johns, Analytical solutions for gas displacements with bifurcating phase behavior, SPEJ 19 (2014), 943–955.
- S. Kruzhkov, First-order quasilinear equations with several space variables, Math. USSR Sb. 10 (1970), 217–273.
- T. C. LaForce and R. T. Johns, Effect of quasi-piston-like flow on miscible gasflood recovery, SPE Western Regional Meeting (2005).
- K.-A. Lie and R. Juanes, A front-tracking method for the simulation of three-phase flow in porous media, Comput. Geosci. 9 (2005), no. 1, 29–59. MR 2158504, DOI 10.1007/s10596-005-5663-4
- P. D. Lax, Hyperbolic systems of conservation laws. II, Comm. Pure Appl. Math. 10 (1957), 537–566. MR 93653, DOI 10.1002/cpa.3160100406
- Tai Ping Liu, The entropy condition and the admissibility of shocks, J. Math. Anal. Appl. 53 (1976), no. 1, 78–88. MR 387830, DOI 10.1016/0022-247X(76)90146-3
- M. Michelsen, The isothermal flash problem. Part II. Phase-split calculation, Fluid Phase Equilibr. 9 (1982), 21–40.
- F. M. Orr, R. T. Johns, and B. Dindoruk, Development of miscibility in four-component CO2 floods, SPERE 8 (1993), 135–142.
- D. Y. Peng and D. B. Robinson, A new two-constant equation of state, Industrial & Engineering Chemistry Fundamentals 15 (1976), 59–64.
- A. P. Pires, P. G. Bedrikovetsky, and A. A. Shapiro, A splitting technique for analytical modelling of two-phase multicomponent flow in porous media, Journal of Petroleum Science and Engineering 51 (2006), 54–67.
- G. A. Pope, The application of fractional flow theory to enhanced oil recovery, SPEJ 20 (1980), 191–205.
- H-K. Rhee, R. Aris, and N. R. Amundson, On the theory of multicomponent chromatography, Philos. Trans. Roy. Soc., London Ser. A 267 (1970), 419–455.
- C. J. Seto and F. M. Orr, Analytical solutions for multicomponent, two-phase flow in porous media with double contact discontinuities, Transport in Porous Media 78 (2008), 161–183.
- J. A. Smoller, On the solution of the Riemann problem with general step data for an extended class of hyperbolic systems, Michigan Math. J. 16 (1969), 201–210. MR 247283
- Blake Temple, Systems of conservation laws with invariant submanifolds, Trans. Amer. Math. Soc. 280 (1983), no. 2, 781–795. MR 716850, DOI 10.1090/S0002-9947-1983-0716850-2
- David H. Wagner, Equivalence of the Euler and Lagrangian equations of gas dynamics for weak solutions, J. Differential Equations 68 (1987), no. 1, 118–136. MR 885816, DOI 10.1016/0022-0396(87)90188-4
- Y. Wang and F. M. Orr, Analytical calculation of minimum miscibility pressure, Fluid Phase Equilibria 139 (1997), 101–124.
- H. Yuan and R. T. Johns, Simplified method for calculation of minimum miscibility pressure or enrichment, SPEJ 10 (2005), 416–425.
References
- K. Ahmadi, R. T. Johns, K. Mogensen, and R. Noman, Limitations of current method-of-characteristics (MOC) methods using shock-jump approximations to predict MMPs for complex gas/oil displacements, SPEJ 16 (2011), 743–750.
- Stefano Bianchini and Alberto Bressan, Vanishing viscosity solutions of nonlinear hyperbolic systems, Ann. of Math. (2) 161 (2005), no. 1, 223–342. MR 2150387, DOI 10.4007/annals.2005.161.223
- Alberto Bressan, Hyperbolic systems of conservation laws, The one-dimensional Cauchy problem, Oxford Lecture Series in Mathematics and its Applications, vol. 20, Oxford University Press, Oxford, 2000. MR 1816648
- S. E. Buckley and M. Leverett, Mechanism of fluid displacement in sands, Transactions of the AIME 146 (1942), 107–116.
- Olav Dahl, Thormod Johansen, Aslak Tveito, and Ragnar Winther, Multicomponent chromatography in a two phase environment, SIAM J. Appl. Math. 52 (1992), no. 1, 65–104. MR 1148319, DOI 10.1137/0152005
- Birol Dindoruk, Analytical theory of multiphase, multicomponent displacement in porous media, ProQuest LLC, Ann Arbor, MI. Thesis (Ph.D.)–Stanford University, 1992. MR 2687942
- A. M. Egwuenu, R. T. Johns, and Y. Li, Improved fluid characterization for miscible gas floods, SPERE 11 (2008), 655–665.
- V. M. Entov and A. Zazovsky, Nonlinear waves in physicochemical hydrodynamics of enhanced oil recovery, Multicomponent flows, International Conference on Porous Media: Physics, Models, Simulation, Moscow, 1997.
- Tore Gimse and Nils Henrik Risebro, Riemann problems with a discontinuous flux function, Third International Conference on Hyperbolic Problems, Vol. I, II (Uppsala, 1990) Studentlitteratur, Lund, 1991, pp. 488–502. MR 1109304
- Tore Gimse and Nils Henrik Risebro, Solution of the Cauchy problem for a conservation law with a discontinuous flux function, SIAM J. Math. Anal. 23 (1992), no. 3, 635–648. MR 1158825, DOI 10.1137/0523032
- F. G. Helfferich, Theory of multicomponent, multiphase displacement in porous media, SPEJ 21 (1980), 51–62.
- G. J. Hirasaki, Application of the theory of multicomponent, multiphase displacement to three-component, two-phase surfactant flooding, SPEJ 21 (1981), 191–204.
- Eli L. Isaacson and J. Blake Temple, Analysis of a singular hyperbolic system of conservation laws, J. Differential Equations 65 (1986), no. 2, 250–268. MR 861520, DOI 10.1016/0022-0396(86)90037-9
- Thormod Johansen, Aslak Tveito, and Ragnar Winther, A Riemann solver for a two-phase multicomponent process, SIAM J. Sci. Statist. Comput. 10 (1989), no. 5, 846–879. MR 1009544, DOI 10.1137/0910050
- Thormod Johansen and Ragnar Winther, The solution of the Riemann problem for a hyperbolic system of conservation laws modeling polymer flooding, SIAM J. Math. Anal. 19 (1988), no. 3, 541–566. MR 937469, DOI 10.1137/0519039
- T. Johansen and R. Winther, Mathematical and numerical analysis of a hyperbolic system modeling solvent flooding. ECMOR II – 2nd European Conference on the Mathematics of Oil Recovery, 1990.
- Thormod Johansen and Ragnar Winther, The Riemann problem for multicomponent polymer flooding, SIAM J. Math. Anal. 20 (1989), no. 4, 908–929. MR 1000728, DOI 10.1137/0520061
- Russell Taylor Johns, Analytical theory of multicomponent gas drives with two-phase mass transfer, ProQuest LLC, Ann Arbor, MI. Thesis (Ph.D.)–Stanford University, 1992. MR 2687944
- R. T. Johns, B. Dindoruk, and F. M. Orr, Analytical theory of combined condensing/vaporizing gas drives, SPE Adv. Tech. Series 1 (1993), 7–16.
- R. T. Johns and F. M. Orr, Miscible gas displacement of multicomponent oils, SPEJ 1 (1996), 39–50.
- Ruben Juanes and Knut-Andreas Lie, Numerical modeling of multiphase first-contact miscible flows. II. Front-tracking/streamline simulation, Transp. Porous Media 72 (2008), no. 1, 97–120. MR 2377328, DOI 10.1007/s11242-007-9139-y
- Barbara L. Keyfitz and Herbert C. Kranzer, A system of nonstrictly hyperbolic conservation laws arising in elasticity theory, Arch. Rational Mech. Anal. 72 (1979/80), no. 3, 219–241. MR 549642, DOI 10.1007/BF00281590
- S. Khorsandi, K. Ahmadi, and R. T. Johns, Analytical solutions for gas displacements with bifurcating phase behavior, SPEJ 19 (2014), 943–955.
- S. Kruzhkov, First-order quasilinear equations with several space variables, Math. USSR Sb. 10 (1970), 217–273.
- T. C. LaForce and R. T. Johns, Effect of quasi-piston-like flow on miscible gasflood recovery, SPE Western Regional Meeting (2005).
- K.-A. Lie and R. Juanes, A front-tracking method for the simulation of three-phase flow in porous media, Comput. Geosci. 9 (2005), no. 1, 29–59. MR 2158504, DOI 10.1007/s10596-005-5663-4
- P. D. Lax, Hyperbolic systems of conservation laws. II, Comm. Pure Appl. Math. 10 (1957), 537–566. MR 0093653
- Tai Ping Liu, The entropy condition and the admissibility of shocks, J. Math. Anal. Appl. 53 (1976), no. 1, 78–88. MR 0387830
- M. Michelsen, The isothermal flash problem. Part II. Phase-split calculation, Fluid Phase Equilibr. 9 (1982), 21–40.
- F. M. Orr, R. T. Johns, and B. Dindoruk, Development of miscibility in four-component CO2 floods, SPERE 8 (1993), 135–142.
- D. Y. Peng and D. B. Robinson, A new two-constant equation of state, Industrial & Engineering Chemistry Fundamentals 15 (1976), 59–64.
- A. P. Pires, P. G. Bedrikovetsky, and A. A. Shapiro, A splitting technique for analytical modelling of two-phase multicomponent flow in porous media, Journal of Petroleum Science and Engineering 51 (2006), 54–67.
- G. A. Pope, The application of fractional flow theory to enhanced oil recovery, SPEJ 20 (1980), 191–205.
- H-K. Rhee, R. Aris, and N. R. Amundson, On the theory of multicomponent chromatography, Philos. Trans. Roy. Soc., London Ser. A 267 (1970), 419–455.
- C. J. Seto and F. M. Orr, Analytical solutions for multicomponent, two-phase flow in porous media with double contact discontinuities, Transport in Porous Media 78 (2008), 161–183.
- J. A. Smoller, On the solution of the Riemann problem with general step data for an extended class of hyperbolic systems, Michigan Math. J. 16 (1969), 201–210. MR 0247283
- Blake Temple, Systems of conservation laws with invariant submanifolds, Trans. Amer. Math. Soc. 280 (1983), no. 2, 781–795. MR 716850, DOI 10.2307/1999646
- David H. Wagner, Equivalence of the Euler and Lagrangian equations of gas dynamics for weak solutions, J. Differential Equations 68 (1987), no. 1, 118–136. MR 885816, DOI 10.1016/0022-0396(87)90188-4
- Y. Wang and F. M. Orr, Analytical calculation of minimum miscibility pressure, Fluid Phase Equilibria 139 (1997), 101–124.
- H. Yuan and R. T. Johns, Simplified method for calculation of minimum miscibility pressure or enrichment, SPEJ 10 (2005), 416–425.
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Additional Information
Saeid Khorsandi
Affiliation:
Department of Energy and Mineral Engineering, 110 Hosler Building, Penn State University, University Park, Pennsylvania 16802-5000
Email:
sxk482@psu.edu
Wen Shen
Affiliation:
Department of Mathematics, Penn State University, University Park, State College, Pennsylvania 16802
MR Author ID:
613346
Email:
wxs27@psu.edu
Russell T. Johns
Affiliation:
Department of Energy and Mineral Engineering, 110 Hosler Building, Penn State University, University Park, Pennsylvania 16802-5000
MR Author ID:
952550
Email:
rjohns@psu.edu
Received by editor(s):
August 12, 2015
Published electronically:
July 12, 2016
Article copyright:
© Copyright 2016
Brown University
