Hostname: page-component-78c5997874-ndw9j Total loading time: 0 Render date: 2024-11-10T17:19:07.391Z Has data issue: false hasContentIssue false
  • English
  • Français

Mathematical study of a petroleum-engineering scheme

Published online by Cambridge University Press:  15 November 2003

Robert Eymard ,
Raphaèle Herbin  and
Anthony Michel
Robert Eymard
Affiliation:
Université de Marne-la-Vallée, 5 Bld Descartes, Bat. Lavoisier, 77454 Marne-la-Vallée, France. eymard@math.univ-mlv.fr.
Raphaèle Herbin
Affiliation:
Université de Aix-Marseille 1, 39 rue Joliot Curie, 13453 Marseille, France. herbin@cmi.univ-mrs.fr.
Anthony Michel
Affiliation:
Institut Français du Pétrole, 1 et 4 avenue Bois Préau, 92000 Rueil-Malmaison, France. anthony.michel@ifp.fr.
Get access

Abstract

Models of two phase flows in porous media, used in petroleumengineering, lead to a system of two coupled equations with ellipticand parabolic degenerate terms, and two unknowns,the saturation and the pressure.For the purpose of their approximation, a coupled scheme, consisting ina finite volume method together witha phase-by-phase upstream weighting scheme, is used in the industrial setting.This paper presents a mathematical analysis of this coupled scheme, first showingthat it satisfies some a priori estimates:the saturation is shown to remain in a fixed interval, anda discrete L2(0,T;H1 (Ω)) estimate is proved for both the pressureand a function of the saturation. Thanks to these properties,a subsequence of the sequence of approximate solutions is shown toconverge to a weak solutionof the continuous equationsas the size of the discretization tends to zero.

Keywords

Multiphase flowDarcy's lawporous mediafinite volume scheme.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 37 , Issue 6 , November 2003 , pp. 937 - 972
Copyright
© EDP Sciences, SMAI, 2003

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Alt, H.W. and DiBenedetto, E., Flow of oil and water through porous media. Astérisque 118 (1984) 89–108. Variational methods for equilibrium problems of fluids, Trento (1983).
Alt, H.W. and Luckhaus, S., Quasilinear elliptic-parabolic differential equations. Math. Z. 183 (1983) 311–341.
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.
Arbogast, T., Wheeler, M.F. and Zhang, N.-Y., 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. CrossRef
K. Aziz and A. Settari, Petroleum reservoir simulation. Applied Science Publishers, London (1979).
J. Bear, Dynamic of flow in porous media. Dover (1967).
J. Bear, Modeling transport phenomena in porous media, in Environmental studies (Minneapolis, MN, 1992). Springer, New York (1996) 27–63.
Brenier, Y. and Jaffré, J., Upstream differencing for multiphase flow in reservoir simulation. SIAM J. Numer. Anal. 28 (1991) 685–696. CrossRef
Carrillo, J., Entropy solutions for nonlinear degenerate problems. Arch. Rational. Mech. Anal. 147 (1999) 269–361. CrossRef
G. Chavent and J. Jaffré, Mathematical models and finite elements for reservoir simulation. Elsevier (1986).
Chen, Z., Degenerate two-phase incompressible flow. I. Existence, uniqueness and regularity of a weak solution. J. Differential Equations 171 (2001) 203–232. CrossRef
Chen, Z., Degenerate two-phase incompressible flow. II. Regularity, stability and stabilization. J. Differential Equations 186 (2002) 345–376. CrossRef
Chen, Z. and Ewing, R., Mathematical analysis for reservoir models. SIAM J. Math. Anal. 30 (1999) 431–453. CrossRef
Chen, Z. and Ewing, R.E., Degenerate two-phase incompressible flow. III. Sharp error estimates. Numer. Math. 90 (2001) 215–240. CrossRef
K. Deimling, Nonlinear functional analysis. Springer-Verlag, Berlin (1985).
Droniou, J., A density result in sobolev spaces. J. Math. Pures Appl. 81 (2002) 697–714. CrossRef
Enchéry, G., Eymard, R., Masson, R. and Wolf, S., 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. CrossRef
Ewing, R.E. and Heinemann, R.F., Mixed finite element approximation of phase velocities in compositional reservoir simulation. R.E. Ewing Ed., Comput. Meth. Appl. Mech. Engrg. 47 (1984) 161–176. CrossRef
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.
Eymard, R. and Gallouët, T., 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. CrossRef
Eymard, R., Gallouët, T., Ghilani, M. and Herbin, R., Error estimates for the approximate solutions of a nonlinear hyperbolic equation given by finite volume schemes. IMA J. Numer. Anal. 18 (1998) 563–594. CrossRef
Eymard, R., Gallouët, T., Herbin, R. and Michel, A., Convergence. Numer. Math. 92 (2002) 41–82. CrossRef
Eymard, R., Gallouët, T., Hilhorst, D. and Naït Slimane, Y., Finite volumes and nonlinear diffusion equations. RAIRO Modél. Math. Anal. Numér. 32 (1998) 747–761. CrossRef
R. Eymard, T. Gallouët and R. Herbin, Finite volume methods, in Handbook of numerical analysis, Vol. VII. North-Holland, Amsterdam (2000) 713–1020.
Eymard, R., Gallouët, T. and Herbin, R., Error estimate for approximate solutions of a nonlinear convection-diffusion problem. Adv. Differential Equations 7 (2002) 419–440.
Fabrie, P. and Gallouët, T., Modeling wells in porous media flow. Math. Models Methods Appl. Sci. 10 (2000) 673–709.
Feng, X., On existence and uniqueness results for a coupled system modeling miscible displacement in porous media. J. Math. Anal. Appl. 194 (1995) 883–910. CrossRef
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.
Forsyth, P.A., A control volume finite element approach to NAPL groundwater contamination. SIAM J. Sci. Statist. Comput. 12 (1991) 1029–1057. CrossRef
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.
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).
Kroener, D. and Luckhaus, S., Flow of oil and water in a porous medium. J. Differential Equations 55 (1984) 276–288. CrossRef
Kružkov, S.N. and Sukorjanskiĭ, S.M., 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.
Michel, A., A finite volume scheme for the simulation of two-phase incompressible flow in porous media. SIAM J. Numer. Anal. 41 (2003) 1301–1317. CrossRef
A. Michel, Convergence de schémas volumes finis pour des problèmes de convection diffusion non linéaires. Ph.D. thesis, Université de Provence, France (2001).
D.W. Peaceman, Fundamentals of Numerical Reservoir Simulation. Elsevier Scientific Publishing Co (1977).
A. Pfertzel, Sur quelques schémas numériques pour la résolution des écoulements multiphasiques en milieu poreux. Ph.D. thesis, Universités Paris 6, France (1987).
Vignal, M.H., Convergence of a finite volume scheme for an elliptic-hyperbolic system. RAIRO Modél. Math. Anal. Numér. 30 (1996) 841–872. CrossRef
Wang, H., Ewing, R.E. and Russell, T.F., Eulerian-Lagrangian localized adjoint methods for convection-diffusion equations and their convergence analysis. IMA J. Numer. Anal. 15 (1995) 405–459. CrossRef