Mathematical study of a petroleum-engineering scheme

Robert Eymard; Raphaèle Herbin; Anthony Michel

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique (2003)

  • Volume: 37, Issue: 6, page 937-972
  • ISSN: 0764-583X

Abstract

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Models of two phase flows in porous media, used in petroleum engineering, lead to a system of two coupled equations with elliptic and parabolic degenerate terms, and two unknowns, the saturation and the pressure. For the purpose of their approximation, a coupled scheme, consisting in a finite volume method together with a phase-by-phase upstream weighting scheme, is used in the industrial setting. This paper presents a mathematical analysis of this coupled scheme, first showing that it satisfies some a priori estimates: the saturation is shown to remain in a fixed interval, and a discrete L 2 ( 0 , T ; H 1 ( Ø ) ) estimate is proved for both the pressure and a function of the saturation. Thanks to these properties, a subsequence of the sequence of approximate solutions is shown to converge to a weak solution of the continuous equations as the size of the discretization tends to zero.

How to cite

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Eymard, Robert, Herbin, Raphaèle, and Michel, Anthony. "Mathematical study of a petroleum-engineering scheme." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 37.6 (2003): 937-972. <http://eudml.org/doc/245709>.

@article{Eymard2003,
abstract = {Models of two phase flows in porous media, used in petroleum engineering, lead to a system of two coupled equations with elliptic and parabolic degenerate terms, and two unknowns, the saturation and the pressure. For the purpose of their approximation, a coupled scheme, consisting in a finite volume method together with a phase-by-phase upstream weighting scheme, is used in the industrial setting. This paper presents a mathematical analysis of this coupled scheme, first showing that it satisfies some a priori estimates: the saturation is shown to remain in a fixed interval, and a discrete $L^2(0,T;H^1(Ø))$ estimate is proved for both the pressure and a function of the saturation. Thanks to these properties, a subsequence of the sequence of approximate solutions is shown to converge to a weak solution of the continuous equations as the size of the discretization tends to zero.},
author = {Eymard, Robert, Herbin, Raphaèle, Michel, Anthony},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},
keywords = {multiphase flow; Darcy’s law; porous media; finite volume scheme; Multiphase flow; Darcy's law},
language = {eng},
number = {6},
pages = {937-972},
publisher = {EDP-Sciences},
title = {Mathematical study of a petroleum-engineering scheme},
url = {http://eudml.org/doc/245709},
volume = {37},
year = {2003},
}

TY - JOUR
AU - Eymard, Robert
AU - Herbin, Raphaèle
AU - Michel, Anthony
TI - Mathematical study of a petroleum-engineering scheme
JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
PY - 2003
PB - EDP-Sciences
VL - 37
IS - 6
SP - 937
EP - 972
AB - Models of two phase flows in porous media, used in petroleum engineering, lead to a system of two coupled equations with elliptic and parabolic degenerate terms, and two unknowns, the saturation and the pressure. For the purpose of their approximation, a coupled scheme, consisting in a finite volume method together with a phase-by-phase upstream weighting scheme, is used in the industrial setting. This paper presents a mathematical analysis of this coupled scheme, first showing that it satisfies some a priori estimates: the saturation is shown to remain in a fixed interval, and a discrete $L^2(0,T;H^1(Ø))$ estimate is proved for both the pressure and a function of the saturation. Thanks to these properties, a subsequence of the sequence of approximate solutions is shown to converge to a weak solution of the continuous equations as the size of the discretization tends to zero.
LA - eng
KW - multiphase flow; Darcy’s law; porous media; finite volume scheme; Multiphase flow; Darcy's law
UR - http://eudml.org/doc/245709
ER -

References

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  1. [1] H.W. Alt and E. DiBenedetto, Flow of oil and water through porous media. Astérisque 118 (1984) 89–108. Variational methods for equilibrium problems of fluids, Trento (1983). Zbl0588.76166
  2. [2] H.W. Alt and S. Luckhaus, Quasilinear elliptic-parabolic differential equations. Math. Z. 183 (1983) 311–341. Zbl0497.35049
  3. [3] S.N. Antontsev, A.V. Kazhikhov and V.N. Monakhov, Boundary value problems in mechanics of nonhomogeneous fluids. North-Holland Publishing Co., Amsterdam (1990). Translated from the Russian. Zbl0696.76001MR1035212
  4. [4] T. Arbogast, M.F. Wheeler and N.-Y. Zhang, A nonlinear mixed finite element method for a degenerate parabolic equation arising in flow in porous media. SIAM J. Numer. Anal. 33 (1996) 1669–1687. Zbl0856.76033
  5. [5] K. Aziz and A. Settari, Petroleum reservoir simulation. Applied Science Publishers, London (1979). 
  6. [6] J. Bear, Dynamic of flow in porous media. Dover (1967). 
  7. [7] J. Bear, Modeling transport phenomena in porous media, in Environmental studies (Minneapolis, MN, 1992). Springer, New York (1996) 27–63. Zbl0880.76083
  8. [8] Y. Brenier and J. Jaffré, Upstream differencing for multiphase flow in reservoir simulation. SIAM J. Numer. Anal. 28 (1991) 685–696. Zbl0735.76071
  9. [9] J. Carrillo, Entropy solutions for nonlinear degenerate problems. Arch. Rational. Mech. Anal. 147 (1999) 269–361. Zbl0935.35056
  10. [10] G. Chavent and J. Jaffré, Mathematical models and finite elements for reservoir simulation. Elsevier (1986). Zbl0603.76101
  11. [11] Z. Chen, Degenerate two-phase incompressible flow. I. Existence, uniqueness and regularity of a weak solution. J. Differential Equations 171 (2001) 203–232. Zbl0991.35047
  12. [12] Z. Chen, Degenerate two-phase incompressible flow. II. Regularity, stability and stabilization. J. Differential Equations 186 (2002) 345–376. Zbl1073.35129
  13. [13] Z. Chen and R. Ewing, Mathematical analysis for reservoir models. SIAM J. Math. Anal. 30 (1999) 431–453. Zbl0922.35074
  14. [14] Z. Chen and R.E. Ewing, Degenerate two-phase incompressible flow. III. Sharp error estimates. Numer. Math. 90 (2001) 215–240. Zbl1097.76064
  15. [15] K. Deimling, Nonlinear functional analysis. Springer-Verlag, Berlin (1985). Zbl0559.47040MR787404
  16. [16] J. Droniou, A density result in sobolev spaces. J. Math. Pures Appl. 81 (2002) 697–714. Zbl1033.46029
  17. [17] G. Enchéry, R. Eymard, R. Masson and S. Wolf, Mathematical and numerical study of an industrial scheme for two-phase flows in porous media under gravity. Comput. Methods Appl. Math. 2 (2002) 325–353. Zbl1098.76625
  18. [18] R.E. Ewing and R.F. Heinemann, Mixed finite element approximation of phase velocities in compositional reservoir simulation. R.E. Ewing Ed., Comput. Meth. Appl. Mech. Engrg. 47 (1984) 161–176. Zbl0545.76127
  19. [19] R.E. Ewing and M.F. Wheeler, Galerkin methods for miscible displacement problems with point sources and sinks — unit mobility ratio case, in Mathematical methods in energy research (Laramie, WY, 1982/1983). SIAM, Philadelphia, PA (1984) 40–58. Zbl0551.76079
  20. [20] R. Eymard and T. Gallouët, Convergence d’un schéma de type éléments finis–volumes finis pour un système formé d’une équation elliptique et d’une équation hyperbolique. RAIRO Modél. Math. Anal. Numér. 27 (1993) 843–861. Zbl0792.65073
  21. [21] R. Eymard, T. Gallouët, M. Ghilani and R. Herbin, Error estimates for the approximate solutions of a nonlinear hyperbolic equation given by finite volume schemes. IMA J. Numer. Anal. 18 (1998) 563–594. Zbl0973.65078
  22. [22] R. Eymard, T. Gallouët, R. Herbin and A. Michel, Convergence. Numer. Math. 92 (2002) 41–82. Zbl1005.65099
  23. [23] R. Eymard, T. Gallouët, D. Hilhorst and Y. Naït Slimane, Finite volumes and nonlinear diffusion equations. RAIRO Modél. Math. Anal. Numér. 32 (1998) 747–761. Zbl0914.65101
  24. [24] R. Eymard, T. Gallouët and R. Herbin, Finite volume methods, in Handbook of numerical analysis, Vol. VII. North-Holland, Amsterdam (2000) 713–1020. Zbl0981.65095
  25. [25] R. Eymard, T. Gallouët and R. Herbin, Error estimate for approximate solutions of a nonlinear convection-diffusion problem. Adv. Differential Equations 7 (2002) 419–440. Zbl1173.35563
  26. [26] P. Fabrie and T. Gallouët, Modeling wells in porous media flow. Math. Models Methods Appl. Sci. 10 (2000) 673–709. Zbl1018.76044
  27. [27] X. Feng, On existence and uniqueness results for a coupled system modeling miscible displacement in porous media. J. Math. Anal. Appl. 194 (1995) 883–910. Zbl0856.35030
  28. [28] P.A. Forsyth, A control volume finite element method for local mesh refinements, in SPE Symposium on Reservoir Simulation. number SPE 18415, Texas: Society of Petroleum Engineers Richardson Ed., Houston, Texas (February 1989) 85–96. 
  29. [29] P.A. Forsyth, A control volume finite element approach to NAPL groundwater contamination. SIAM J. Sci. Statist. Comput. 12 (1991) 1029–1057. Zbl0725.76087
  30. [30] Gérard Gagneux and Monique Madaune-Tort, Analyse mathématique de modèles non linéaires de l’ingénierie pétrolière. Springer-Verlag, Berlin (1996). With a preface by Charles-Michel Marle. Zbl0842.35126
  31. [31] R. Helmig, Multiphase Flow and Transport Processes in the Subsurface: A Contribution to the Modeling of Hydrosystems. Springer-Verlag Berlin Heidelberg (1997). P. Schuls (Translator). 
  32. [32] D. Kroener and S. Luckhaus, Flow of oil and water in a porous medium. J. Differential Equations 55 (1984) 276–288. Zbl0509.35048
  33. [33] S.N. Kružkov and S.M. Sukorjanskiĭ, Boundary value problems for systems of equations of two-phase filtration type; formulation of problems, questions of solvability, justification of approximate methods. Mat. Sb. (N.S.) 104 (1977) 69–88, 175–176. Zbl0372.35017
  34. [34] A. Michel, A finite volume scheme for the simulation of two-phase incompressible flow in porous media. SIAM J. Numer. Anal. 41 (2003) 1301–1317. Zbl1049.35018
  35. [35] A. Michel, Convergence de schémas volumes finis pour des problèmes de convection diffusion non linéaires. Ph.D. thesis, Université de Provence, France (2001). 
  36. [36] D.W. Peaceman, Fundamentals of Numerical Reservoir Simulation. Elsevier Scientific Publishing Co (1977). 
  37. [37] A. Pfertzel, Sur quelques schémas numériques pour la résolution des écoulements multiphasiques en milieu poreux. Ph.D. thesis, Universités Paris 6, France (1987). 
  38. [38] M.H. Vignal, Convergence of a finite volume scheme for an elliptic-hyperbolic system. RAIRO Modél. Math. Anal. Numér. 30 (1996) 841–872. Zbl0861.65084
  39. [39] H. Wang, R.E. Ewing and T.F. Russell, Eulerian-Lagrangian localized adjoint methods for convection-diffusion equations and their convergence analysis. IMA J. Numer. Anal. 15 (1995) 405–459. Zbl0830.65095

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