Article
MSC:
65M08,
65N30,
76M10,
76M12,
76R99,
76S05 | MR 4218605 | Zbl 07332692 | DOI: 10.21136/AM.2020.0122-19
Full entry |
PDF
(0.9 MB)
Feedback
PDF
(0.9 MB)
Feedback
Keywords:
porous medium; nonconforming grid; finite volume scheme; a priori estimate; miscible fluid flow
porous medium; nonconforming grid; finite volume scheme; a priori estimate; miscible fluid flow
Summary:
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.
References:
[1] Bartels, S., Jensen, M., Müller, R.: 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] Becker, J., Grün, G., Lenz, M., Rumpf, M.: 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] Bradji, A., Fuhrmann, J.: 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] Brenner, K., Hilhorst, D., Vu-Do, H.-C.: 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] Chainais-Hillairet, C., Droniou, J.: 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] Chainais-Hillairet, C., Krell, S., Mouton, A.: 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] Chainais-Hillairet, C., Krell, S., Mouton, A.: 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] Ewing, R. E., Russell, T. F., Wheeler, M. F.: 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 | MR 0770577
[14] Ewing, R. E., Russell, T. F., Wheeler, M. F.: 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] Eymard, R., Gallouët, T., Herbin, R.: Finite volume methods. Handbook of Numerical Analysis, Volume 7 P. Ciarlet et al. North-Holland/ Elsevier, Amsterdam (2000), 713-1020. DOI 10.1016/S1570-8659(00)07005-8 | MR 1804748 | Zbl 0981.65095
[16] Eymard, R., Gallouët, T., Herbin, R.: 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] Eymard, R., Handlovičová, A., Herbin, R., Mikula, K., Stašová, O.: 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] Feng, X.: 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] Jaffre, J., Roberts, J. E.: 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] Russell, T. F.: 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] Sammon, P. H.: 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] Wang, H., Liang, D., Ewing, R. E., Lyons, S. L., Qin, G.: 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] Wang, H., Liang, D., Ewing, R. E., Lyons, S. L., Qin, G.: 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
Search
Partner of
