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Small-stencil 3D schemes for diffusive flows in porous media

Published online by Cambridge University Press:  12 October 2011

Robert Eymard ,
Cindy Guichard  and
Raphaèle Herbin
Robert Eymard
Affiliation:
LAMA-CNRS UMR 8050, Université Paris-Est Marne-la-Vallée, 5 boulevard Descartes, Champs-sur-Marne, 77454 Marne-la-Vallée cedex 2, France. [email protected],
Cindy Guichard
Affiliation:
IFP Énergies nouvelles, 1 & 4, avenue de Bois-Préau, 92852 Rueil-Malmaison cedex, France. [email protected]
Raphaèle Herbin
Affiliation:
LATP-CNRS UMR 6632, Aix-Marseille Université, 39 rue Joliot Curie, 13453 Marseille, France. [email protected],
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Abstract

In this paper, we study some discretization schemes for diffusive flows in heterogeneous anisotropic porous media. We first introduce the notion of gradient scheme, and show that several existing schemes fall into this framework. Then, we construct two new gradient schemes which have the advantage of a small stencil. Numerical results obtained for real reservoir meshes show the efficiency of the new schemes, compared to existing ones.

Keywords

Porous mediadiffusion operatoranisotropynon conforming meshes
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 46 , Issue 2 , March 2012 , pp. 265 - 290
Copyright
© EDP Sciences, SMAI, 2011

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References

Aavatsmark, I., An introduction to multipoint flux approximations for quadrilateral grids. Comput. Geosci. 6 (2002) 405–432. Locally conservative numerical methods for flow in porous media. CrossRef
Aavatsmark, I. and Klausen, R., Well index in reservoir simulation for slanted and slightly curved wells in 3D grids. SPE J. 8 (2003) 41–48. CrossRef
Aavatsmark, I., Barkve, T., Boe, O. and Mannseth, T., Discretization on non-orthogonal, quadrilateral grids for inhomogeneous, anisotropic media. J. Comput. Phys. 127 (1996) 2–14. CrossRef
Aavatsmark, I., Eigestad, G., Heimsund, B., Mallison, B., Nordbotten, J. and Oian, E., A new finite-volume approach to efficient discretization on challenging grids. SPE J. 15 (2010) 658–669. CrossRef
L. Agelas, D.A. Di Pietro and R. Masson, A symmetric and coercive finite volume scheme for multiphase porous media flow problems with applications in the oil industry, in Finite volumes for complex applications V. ISTE, London (2008) 35–51.
Agelas, L., Eymard, R. and Herbin, R., A nine-point finite volume scheme for the simulation of diffusion in heterogeneous media. C. R. Math. Acad. Sci. Paris 347 (2009) 673–676. CrossRef
B. Andreianov, M. Bendahmane and K. Karlsen, A gradient reconstruction formula for finite-volume schemes and discrete duality, in Finite volumes for complex applications V. ISTE, London (2008) 161–168.
Andreianov, B., Bendahmane, M., Karlsen, K.H. and Pierre, C., Convergence of discrete duality finite volume schemes for the cardiac bidomain model. Netw. Heterog. Media 6 (2011) 195–240.
Boyer, F. and Hubert, F., Finite volume method for 2D linear and nonlinear elliptic problems with discontinuities. SIAM J. Numer. Anal. 46 (2008) 3032–3070. CrossRef
Brezzi, F., Buffa, A. and Lipnikov, K., Mimetic finite differences for elliptic problems. ESAIM: M2AN 43 (2008) 277–295. CrossRef
Coudière, Y. and Hubert, F., A 3D discrete duality finite volume method for nonlinear elliptic equations. SIAM J. Sci. Comput. 33 (2011) 1739. CrossRef
Coudière, Y., Vila, J.-P. and Villedieu, P., Convergence rate of a finite volume scheme for a two-dimensional convection-diffusion problem. ESAIM: M2AN 33 (1999) 493–516. CrossRef
Y. Coudière, C. Pierre, O. Rousseau and R. Turpault, 2D/3D discrete duality finite volume scheme (DDFV) applied to ECG simulation. A DDFV scheme for anisotropic and heterogeneous elliptic equations, application to a bio-mathematics problem: electrocardiogram simulation, in Finite volumes for complex applications V. ISTE, London (2008) 313–320.
Coudière, Y., Pierre, C., Rousseau, O. and Turpault, R., A 2D/3D discrete duality finite volume scheme. Application to ECG simulation. Int. J. Finite 6 (2009) 24.
Domelevo, K. and Omnes, P., A finite volume method for the Laplace equation on almost arbitrary two-dimensional grids. ESAIM: M2AN 39 (2005) 1203–1249. CrossRef
Droniou, J., Eymard, R., Gallouët, T. and Herbin, R., A unified approach to mimetic finite difference, hybrid finite volume and mixed finite volume methods. Math. Models Methods Appl. Sci. 20 (2010) 265–295. CrossRef
A. Ern and J.-L. Guermond, Theory and practice of finite elements, Applied Mathematical Sciences 159. Springer-Verlag, New York (2004).
Eymard, R., Gallouët, T. and Joly, P., Hybrid finite element techniques for oil recovery simulation. Comput. Methods Appl. Mech. Eng. 74 (1989) 83–98. CrossRef
R. Eymard, T. Gallouët and R. Herbin, Finite volume methods, in Handbook of numerical analysis, Handb. Numer. Anal. VII. North-Holland, Amsterdam (2000) 713–1020.
Eymard, R., Gallouët, T. and Herbin, R., A new finite volume scheme for anisotropic diffusion problems on general grids: convergence analysis. C. R. Math. Acad. Sci. Paris 344 (2007) 403–406. CrossRef
Eymard, R., Gallouët, T. and Herbin, R., Discretisation of heterogeneous and anisotropic diffusion problems on general non-conforming meshes, SUSHI: a scheme using stabilisation and hybrid interfaces. IMA J. Numer. Anal. 30 (2010) 1009–1043. see also http://hal.archives-ouvertes.fr/. CrossRef
R. Eymard, C. Guichard, R. Herbin and R. Masson, Multiphase flow in porous media using the VAG scheme, in Finite Volumes for Complex Applications VI – Problems and Persepectives, edited by J. Fort, J. Furst, J. Halama, R. Herbin and F. Hubert. Springer Proceedings in Mathematics (2011) 409–417.
R. Eymard, G. Henry, R. Herbin, F. Hubert, R. Kloefkorn and G. Manzini, 3D benchmark on discretization schemes for anisotropic diffusion problem on general grids, in Finite Volumes for Complex Applications VI – Problems and Persepectives, edited by J. Fort, J. Furst, J. Halama, R. Herbin and F. Hubert. Springer Proceedings in Mathematics (2011) 95–130.
Faille, I., A control volume method to solve an elliptic equation on a two-dimensional irregular mesh. Comput. Methods Appl. Mech. Eng. 100 (1992) 275–290. CrossRef
R. Herbin and F. Hubert, Benchmark on discretization schemes for anisotropic diffusion problems on general grids for anisotropic heterogeneous diffusion problems, in Finite Volumes for Complex Applications V, edited by R. Eymard and J.-M. Hérard. Wiley (2008) 659–692.
Hermeline, F., Approximation of diffusion operators with discontinuous tensor coefficients on distorted meshes. Comput. Methods Appl. Mech. Eng. 192 (2003) 1939–1959. CrossRef
Hermeline, F., Approximation of 2-D and 3-D diffusion operators with variable full tensor coefficients on arbitrary meshes. Comput. Methods Appl. Mech. Eng. 196 (2007) 2497–2526. CrossRef
Hermeline, F., A finite volume method for approximating 3D diffusion operators on general meshes. J. Comput. Phys. 228 (2009) 5763–5786. CrossRef
G. Strang, Variational crimes in the finite element method, in The mathematical foundations of the finite element method with applications to partial differential equations (Proc. Sympos., Univ. Maryland, Baltimore, Md. 1972). Academic Press, New York (1972) 689–710.