Mathematical study of a petroleum-engineering scheme
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 37 (2003) no. 6, p. 937-972

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.

DOI : https://doi.org/10.1051/m2an:2003062
Classification:  35K65,  76S05,  65M12
Keywords: multiphase flow, Darcy's law, porous media, finite volume scheme
@article{M2AN_2003__37_6_937_0,
     author = {Eymard, Robert and Herbin, Rapha\`ele and Michel, Anthony},
     title = {Mathematical study of a petroleum-engineering scheme},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     publisher = {EDP-Sciences},
     volume = {37},
     number = {6},
     year = {2003},
     pages = {937-972},
     doi = {10.1051/m2an:2003062},
     zbl = {1118.76355},
     mrnumber = {2026403},
     language = {en},
     url = {http://www.numdam.org/item/M2AN_2003__37_6_937_0}
}
Eymard, Robert; Herbin, Raphaèle; Michel, Anthony. Mathematical study of a petroleum-engineering scheme. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 37 (2003) no. 6, pp. 937-972. doi : 10.1051/m2an:2003062. http://www.numdam.org/item/M2AN_2003__37_6_937_0/

[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). | Zbl 0588.76166

[2] H.W. Alt and S. Luckhaus, Quasilinear elliptic-parabolic differential equations. Math. Z. 183 (1983) 311-341. | Zbl 0497.35049

[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. | MR 1035212 | Zbl 0696.76001

[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. | Zbl 0856.76033

[5] K. Aziz and A. Settari, Petroleum reservoir simulation. Applied Science Publishers, London (1979).

[6] J. Bear, Dynamic of flow in porous media. Dover (1967).

[7] J. Bear, Modeling transport phenomena in porous media, in Environmental studies (Minneapolis, MN, 1992). Springer, New York (1996) 27-63. | Zbl 0880.76083

[8] Y. Brenier and J. Jaffré, Upstream differencing for multiphase flow in reservoir simulation. SIAM J. Numer. Anal. 28 (1991) 685-696. | Zbl 0735.76071

[9] J. Carrillo, Entropy solutions for nonlinear degenerate problems. Arch. Rational. Mech. Anal. 147 (1999) 269-361. | Zbl 0935.35056

[10] G. Chavent and J. Jaffré, Mathematical models and finite elements for reservoir simulation. Elsevier (1986). | Zbl 0603.76101

[11] Z. Chen, Degenerate two-phase incompressible flow. I. Existence, uniqueness and regularity of a weak solution. J. Differential Equations 171 (2001) 203-232. | Zbl 0991.35047

[12] Z. Chen, Degenerate two-phase incompressible flow. II. Regularity, stability and stabilization. J. Differential Equations 186 (2002) 345-376. | Zbl 1073.35129

[13] Z. Chen and R. Ewing, Mathematical analysis for reservoir models. SIAM J. Math. Anal. 30 (1999) 431-453. | Zbl 0922.35074

[14] Z. Chen and R.E. Ewing, Degenerate two-phase incompressible flow. III. Sharp error estimates. Numer. Math. 90 (2001) 215-240. | Zbl 1097.76064

[15] K. Deimling, Nonlinear functional analysis. Springer-Verlag, Berlin (1985). | MR 787404 | Zbl 0559.47040

[16] J. Droniou, A density result in sobolev spaces. J. Math. Pures Appl. 81 (2002) 697-714. | Zbl 1033.46029

[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. | Zbl 1098.76625

[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. | Zbl 0545.76127

[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. | Zbl 0551.76079

[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. | Numdam | Zbl 0792.65073

[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. | Zbl 0973.65078

[22] R. Eymard, T. Gallouët, R. Herbin and A. Michel, Convergence. Numer. Math. 92 (2002) 41-82. | Zbl 1005.65099

[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. | Numdam | Zbl 0914.65101

[24] R. Eymard, T. Gallouët and R. Herbin, Finite volume methods, in Handbook of numerical analysis, Vol. VII. North-Holland, Amsterdam (2000) 713-1020. | Zbl 0981.65095

[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. | Zbl pre01700827

[26] P. Fabrie and T. Gallouët, Modeling wells in porous media flow. Math. Models Methods Appl. Sci. 10 (2000) 673-709. | Zbl 1018.76044

[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. | Zbl 0856.35030

[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] P.A. Forsyth, A control volume finite element approach to NAPL groundwater contamination. SIAM J. Sci. Statist. Comput. 12 (1991) 1029-1057. | Zbl 0725.76087

[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. | Zbl 0842.35126

[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] D. Kroener and S. Luckhaus, Flow of oil and water in a porous medium. J. Differential Equations 55 (1984) 276-288. | Zbl 0509.35048

[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. | Zbl 0372.35017

[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. | Zbl 1049.35018

[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] D.W. Peaceman, Fundamentals of Numerical Reservoir Simulation. Elsevier Scientific Publishing Co (1977).

[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] M.H. Vignal, Convergence of a finite volume scheme for an elliptic-hyperbolic system. RAIRO Modél. Math. Anal. Numér. 30 (1996) 841-872. | Numdam | Zbl 0861.65084

[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. | Zbl 0830.65095