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 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.
Mots clés : 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: Mod\'elisation math\'ematique et analyse num\'erique}, pages = {937--972}, publisher = {EDP-Sciences}, volume = {37}, number = {6}, year = {2003}, doi = {10.1051/m2an:2003062}, mrnumber = {2026403}, zbl = {1118.76355}, language = {en}, url = {http://www.numdam.org/articles/10.1051/m2an:2003062/} }
TY - JOUR AU - Eymard, Robert AU - Herbin, Raphaèle AU - Michel, Anthony TI - Mathematical study of a petroleum-engineering scheme JO - ESAIM: Modélisation mathématique et analyse numérique PY - 2003 SP - 937 EP - 972 VL - 37 IS - 6 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an:2003062/ DO - 10.1051/m2an:2003062 LA - en ID - M2AN_2003__37_6_937_0 ER -
%0 Journal Article %A Eymard, Robert %A Herbin, Raphaèle %A Michel, Anthony %T Mathematical study of a petroleum-engineering scheme %J ESAIM: Modélisation mathématique et analyse numérique %D 2003 %P 937-972 %V 37 %N 6 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an:2003062/ %R 10.1051/m2an:2003062 %G en %F M2AN_2003__37_6_937_0
Eymard, Robert; Herbin, Raphaèle; Michel, Anthony. Mathematical study of a petroleum-engineering scheme. ESAIM: Modélisation mathématique et analyse numérique, Tome 37 (2003) no. 6, pp. 937-972. doi : 10.1051/m2an:2003062. http://www.numdam.org/articles/10.1051/m2an:2003062/
[1] Flow of oil and water through porous media. Astérisque 118 (1984) 89-108. Variational methods for equilibrium problems of fluids, Trento (1983). | Numdam | Zbl
and ,[2] Quasilinear elliptic-parabolic differential equations. Math. Z. 183 (1983) 311-341. | Zbl
and ,[3] Boundary value problems in mechanics of nonhomogeneous fluids. North-Holland Publishing Co., Amsterdam (1990). Translated from the Russian. | MR | Zbl
, and ,[4] 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
, and ,[5] Petroleum reservoir simulation. Applied Science Publishers, London (1979).
and ,[6] Dynamic of flow in porous media. Dover (1967).
,[7] Modeling transport phenomena in porous media, in Environmental studies (Minneapolis, MN, 1992). Springer, New York (1996) 27-63. | Zbl
,[8] Upstream differencing for multiphase flow in reservoir simulation. SIAM J. Numer. Anal. 28 (1991) 685-696. | Zbl
and ,[9] Entropy solutions for nonlinear degenerate problems. Arch. Rational. Mech. Anal. 147 (1999) 269-361. | Zbl
,[10] Mathematical models and finite elements for reservoir simulation. Elsevier (1986). | Zbl
and ,[11] Degenerate two-phase incompressible flow. I. Existence, uniqueness and regularity of a weak solution. J. Differential Equations 171 (2001) 203-232. | Zbl
,[12] Degenerate two-phase incompressible flow. II. Regularity, stability and stabilization. J. Differential Equations 186 (2002) 345-376. | Zbl
,[13] Mathematical analysis for reservoir models. SIAM J. Math. Anal. 30 (1999) 431-453. | Zbl
and ,[14] Degenerate two-phase incompressible flow. III. Sharp error estimates. Numer. Math. 90 (2001) 215-240. | Zbl
and ,[15] Nonlinear functional analysis. Springer-Verlag, Berlin (1985). | MR | Zbl
,[16] A density result in sobolev spaces. J. Math. Pures Appl. 81 (2002) 697-714. | Zbl
,[17] 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
, , and ,[18] 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
and ,[19] 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
and ,[20] 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
and ,[21] 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
, , and ,[22] Convergence. Numer. Math. 92 (2002) 41-82. | Zbl
, , and ,[23] Finite volumes and nonlinear diffusion equations. RAIRO Modél. Math. Anal. Numér. 32 (1998) 747-761. | Numdam | Zbl
, , and ,[24] Finite volume methods, in Handbook of numerical analysis, Vol. VII. North-Holland, Amsterdam (2000) 713-1020. | Zbl
, and ,[25] Error estimate for approximate solutions of a nonlinear convection-diffusion problem. Adv. Differential Equations 7 (2002) 419-440.
, and ,[26] Modeling wells in porous media flow. Math. Models Methods Appl. Sci. 10 (2000) 673-709. | Zbl
and ,[27] On existence and uniqueness results for a coupled system modeling miscible displacement in porous media. J. Math. Anal. Appl. 194 (1995) 883-910. | Zbl
,[28] 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] A control volume finite element approach to NAPL groundwater contamination. SIAM J. Sci. Statist. Comput. 12 (1991) 1029-1057. | Zbl
,[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
[31] Multiphase Flow and Transport Processes in the Subsurface: A Contribution to the Modeling of Hydrosystems. Springer-Verlag Berlin Heidelberg (1997). P. Schuls (Translator).
,[32] Flow of oil and water in a porous medium. J. Differential Equations 55 (1984) 276-288. | Zbl
and ,[33] 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
and ,[34] A finite volume scheme for the simulation of two-phase incompressible flow in porous media. SIAM J. Numer. Anal. 41 (2003) 1301-1317. | Zbl
,[35] Convergence de schémas volumes finis pour des problèmes de convection diffusion non linéaires. Ph.D. thesis, Université de Provence, France (2001).
,[36] Fundamentals of Numerical Reservoir Simulation. Elsevier Scientific Publishing Co (1977).
,[37] Sur quelques schémas numériques pour la résolution des écoulements multiphasiques en milieu poreux. Ph.D. thesis, Universités Paris 6, France (1987).
,[38] Convergence of a finite volume scheme for an elliptic-hyperbolic system. RAIRO Modél. Math. Anal. Numér. 30 (1996) 841-872. | Numdam | Zbl
,[39] Eulerian-Lagrangian localized adjoint methods for convection-diffusion equations and their convergence analysis. IMA J. Numer. Anal. 15 (1995) 405-459. | Zbl
, and ,Cité par Sources :