Applications of Mathematics, first online, pp. 1-29


The generalized finite volume SUSHI scheme for the discretization of the peaceman model

Mohamed Mandari, Mohamed Rhoudaf, Ouafa Soualhi

Received May 19, 2019.   Published online September 15, 2020.

Abstract:  We demonstrate some a priori estimates of a scheme using stabilization and hybrid interfaces applying to partial differential equations describing miscible displacement in porous media. This system is made of two coupled equations: an anisotropic diffusion equation on the pressure and a convection-diffusion-dispersion equation on the concentration of invading fluid. The anisotropic diffusion operators in both equations require special care while discretizing by a finite volume method SUSHI. Later, we present some numerical experiments.
Keywords:  porous medium; nonconforming grid; finite volume scheme; a priori estimate; miscible fluid flow
Classification MSC:  76M10, 76M12, 76S05, 76R99, 65M08, 65N30
DOI:  10.21136/AM.2020.0122-19

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References:
[1] S. Bartels, M. Jensen, R. Müller: Discontinuous Galerkin finite element convergence for incompressible miscible displacement problems of low regularity. SIAM J. Numer. Anal. 47 (2009), 3720-3743. DOI 10.1137/070712079 | MR 2576518 | Zbl 1410.76154
[2] J. Becker, G. Grün, M. Lenz, M. Rumpf: Numerical methods for fourth order nonlinear degenerate diffusion problems. Appl. Math., Praha 47 (2002), 517-543. DOI 10.1023/B:APOM.0000034537.55985.44 | MR 1948194 | Zbl 1090.35086
[3] A. Bradji, J. Fuhrmann: Some abstract error estimates of a finite volume scheme for a nonstationary heat equation on general nonconforming multidimensional spatial meshes. Appl. Math., Praha 58 (2013), 1-38. DOI 10.1007/s10492-013-0001-y | MR 3022767 | Zbl 1274.65251
[4] K. Brenner, D. Hilhorst, H.-C. Vu-Do: The generalized finite volume SUSHI scheme for the discretization of Richards equation. Vietnam J. Math. 44 (2016), 557-586. DOI 10.1007/s10013-015-0170-y | MR 3541150 | Zbl 1348.35093
[5] C. Chainais-Hillairet, J. Droniou: Convergence analysis of a mixed finite volume scheme for an elliptic-parabolic system modeling miscible fluid flows in porous media. SIAM J. Numer. Anal. 45 (2007), 2228-2258. DOI 10.1137/060657236 | MR 2346377 | Zbl 1146.76034
[6] C. Chainais-Hillairet, S. Krell, A. Mouton: Study of discrete duality finite volume schemes for the Peaceman model. SIAM J. Sci. Comput. 35 (2013), A2928-A2952. DOI 10.1137/130910555 | MR 3141755 | Zbl 1292.76044
[7] C. Chainais-Hillairet, S. Krell, A. Mouton: Convergence analysis of a DDFV scheme for a system describing miscible fluid flows in porous media. Numer. Methods Partial Differ. Equations 31 (2015), 723-760. DOI 10.1002/num.21913 | MR 3332290 | Zbl 1325.76128
[8] J. Douglas, Jr.: The numerical simulation of miscible displacement in porous media. Computational Methods in Nonlinear Mechanics. North-Holland, Amsterdam, 1980, 225-237. MR 0576907 | Zbl 0439.76087
[9] J. Douglas, Jr.: Numerical methods for the flow of miscible fluids in porous media. Numerical Methods in Coupled Systems. Wiley Series in Numerical Methods in Engineering. Wiley, Chichester, 1984, 405-439. Zbl 0585.76138
[10] J. Douglas, Jr., R. E. Ewing, M. F. Wheeler: A time-discretization procedure for a mixed finite element approximation of miscible displacement in porous media. RAIRO, Anal. Numér. 17 (1983), 249-265. DOI 10.1051/m2an/1983170302491 | MR 0702137 | Zbl 0526.76094
[11] J. Douglas, Jr., R. E. Ewing, M. F. Wheeler: The approximation of the pressure by a mixed method in the simulation of miscible displacement. RAIRO, Anal. Numér. 17 (1983), 17-33. DOI 10.1051/m2an/1983170100171 | MR 0695450 | Zbl 0516.76094
[12] J. Douglas, Jr., J. E. Roberts: Numerical methods for a model for compressible miscible displacement in porous media. Math. Comput. 41 (1983), 441-459. DOI 10.2307/2007685 | MR 0717695 | Zbl 0537.76062
[13] R. E. Ewing, T. F. Russell, M. F. Wheeler: Simulation of miscible displacement using mixed methods and a modified method of characteristics. SPE Reservoir Simulation Symposium. Society of Petroleum Engineers, San Francisco, 1983, ID SPE-12241-MS. DOI 10.2118/12241-MS
[14] R. E. Ewing, T. F. Russell, M. F. Wheeler: Convergence analysis of an approximation of miscible displacement in porous media by mixed finite elements and a modified method of characteristics. Comput. Methods Appl. Mech. Eng. 47 (1984), 73-92 (1984). DOI 10.1016/0045-7825(84)90048-3 | MR 0777394 | Zbl 0545.76131
[15] R. Eymard, T. Gallouët, R. Herbin: Finite volume methods. Handbook of Numerical Analysis, Volume 7 (P. Ciarlet et al., eds.). North-Holland/ Elsevier, Amsterdam, 2000, 713-1020. DOI 10.1016/S1570-8659(00)07005-8 | MR 1804748 | Zbl 0981.65095
[16] R. Eymard, T. Gallouët, R. Herbin: Discretization of heterogeneous and anisotropic diffusion problems on general nonconforming meshes SUSHI: A scheme using stabilization and hybrid interfaces. IMA J. Numer. Anal. 30 (2010), 1009-1043. DOI 10.1093/imanum/drn084 | MR 2727814 | Zbl 1202.65144
[17] R. Eymard, A. Handlovičová, R. Herbin, K. Mikula, O. Stašová: Applications of approximate gradient schemes for nonlinear parabolic equations. Appl. Math., Praha 60 (2015), 135-156. DOI 10.1007/s10492-015-0088-4 | MR 3320342 | Zbl 1340.65187
[18] X. Feng: On existence and uniqueness results for a coupled systems modeling miscible displacement in porous media. J. Math. Anal. Appl. 194 (1995), 883-910. DOI 10.1006/jmaa.1995.1334 | MR 1350201 | Zbl 0856.35030
[19] J. Jaffre, J. E. Roberts: Upstream weighting and mixed finite elements in the simulation of miscible displacements. RAIRO, Modélisation Math. Anal. Numér. 19 (1985), 443-460. DOI 10.1051/m2an/1985190304431 | MR 0807326 | Zbl 0568.76096
[20] T. F. Russell: Finite elements with characteristics for two-component incompressible miscible displacement. SPE Reservoir Simulation Symposium. Society of Petroleum Engineers, New Orleans, 1982, ID SPE-10500-MS. DOI 10.2118/10500-MS
[21] P. H. Sammon: Numerical approximations for a miscible displacement process in porous media. SIAM J. Numer. Anal. 23 (1986), 508-542. DOI 10.1137/0723034 | MR 0842642 | Zbl 0608.76084
[22] H. Wang, D. Liang, R. E. Ewing, S. L. Lyons, G. Qin: An approximation to miscible fluid flows in porous media with point sources and sinks by an Eulerian-Lagrangian localized adjoint method and mixed finite element methods. SIAM J. Sci. Comput. 22 (2000), 561-581. DOI 10.1137/S1064827598349215 | MR 1780614 | Zbl 0988.76054
[23] H. Wang, D. Liang, R. E. Ewing, S. L. Lyons, G. Qin: An improved numerical simulator for different types of flows in porous media. Numer. Methods Partial Differ. Equations 19 (2003), 343-362. DOI 10.1002/num.10045 | MR 1969198 | Zbl 1079.76044

Affiliations:   Mohamed Mandari (corresponding author), Mohamed Rhoudaf, Ouafa Soualhi, Faculty of Sciences, Moulay Ismail University, B.P. 11201 Zitoune, Meknès, Morocco, e-mail: mandariart@gmail.com, rhoudafmohamed@gmail.com, ouafasoualhi@gmail.com


 
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