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.
Mots clés : porous media, diffusion operator, anisotropy, non conforming meshes
@article{M2AN_2012__46_2_265_0,
author = {Eymard, Robert and Guichard, Cindy and Herbin, Rapha\`ele},
title = {Small-stencil {3D} schemes for diffusive flows in porous media},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {265--290},
publisher = {EDP-Sciences},
volume = {46},
number = {2},
year = {2012},
doi = {10.1051/m2an/2011040},
mrnumber = {2855643},
zbl = {1271.76324},
language = {en},
url = {http://www.numdam.org/articles/10.1051/m2an/2011040/}
}
TY - JOUR AU - Eymard, Robert AU - Guichard, Cindy AU - Herbin, Raphaèle TI - Small-stencil 3D schemes for diffusive flows in porous media JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2012 SP - 265 EP - 290 VL - 46 IS - 2 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an/2011040/ DO - 10.1051/m2an/2011040 LA - en ID - M2AN_2012__46_2_265_0 ER -
%0 Journal Article %A Eymard, Robert %A Guichard, Cindy %A Herbin, Raphaèle %T Small-stencil 3D schemes for diffusive flows in porous media %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2012 %P 265-290 %V 46 %N 2 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an/2011040/ %R 10.1051/m2an/2011040 %G en %F M2AN_2012__46_2_265_0
Eymard, Robert; Guichard, Cindy; Herbin, Raphaèle. Small-stencil 3D schemes for diffusive flows in porous media. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 46 (2012) no. 2, pp. 265-290. doi : 10.1051/m2an/2011040. http://www.numdam.org/articles/10.1051/m2an/2011040/
[1] , An introduction to multipoint flux approximations for quadrilateral grids. Comput. Geosci. 6 (2002) 405-432. Locally conservative numerical methods for flow in porous media. | MR | Zbl
[2] and , Well index in reservoir simulation for slanted and slightly curved wells in 3D grids. SPE J. 8 (2003) 41-48.
[3] , , and , Discretization on non-orthogonal, quadrilateral grids for inhomogeneous, anisotropic media. J. Comput. Phys. 127 (1996) 2-14. | Zbl
[4] , , , , and , A new finite-volume approach to efficient discretization on challenging grids. SPE J. 15 (2010) 658-669.
[5] , and , 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. | MR
[6] , and , A nine-point finite volume scheme for the simulation of diffusion in heterogeneous media. C. R. Math. Acad. Sci. Paris 347 (2009) 673-676. | MR | Zbl
[7] , and , A gradient reconstruction formula for finite-volume schemes and discrete duality, in Finite volumes for complex applications V. ISTE, London (2008) 161-168. | MR
[8] , , and , Convergence of discrete duality finite volume schemes for the cardiac bidomain model. Netw. Heterog. Media 6 (2011) 195-240. | MR
[9] and , Finite volume method for 2D linear and nonlinear elliptic problems with discontinuities. SIAM J. Numer. Anal. 46 (2008) 3032-3070. | MR | Zbl
[10] , and , Mimetic finite differences for elliptic problems. ESAIM: M2AN 43 (2008) 277-295. | Numdam | MR | Zbl
[11] and , A 3D discrete duality finite volume method for nonlinear elliptic equations. SIAM J. Sci. Comput. 33 (2011) 1739. | MR | Zbl
[12] , and , Convergence rate of a finite volume scheme for a two-dimensional convection-diffusion problem. ESAIM: M2AN 33 (1999) 493-516. | Numdam | MR | Zbl
[13] , , and , 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. | MR
[14] , , and , A 2D/3D discrete duality finite volume scheme. Application to ECG simulation. Int. J. Finite 6 (2009) 24. | MR
[15] and , A finite volume method for the Laplace equation on almost arbitrary two-dimensional grids. ESAIM: M2AN 39 (2005) 1203-1249. | Numdam | MR | Zbl
[16] , , and , A unified approach to mimetic finite difference, hybrid finite volume and mixed finite volume methods. Math. Models Methods Appl. Sci. 20 (2010) 265-295. | MR | Zbl
[17] and , Theory and practice of finite elements, Applied Mathematical Sciences 159. Springer-Verlag, New York (2004). | MR | Zbl
[18] , and , Hybrid finite element techniques for oil recovery simulation. Comput. Methods Appl. Mech. Eng. 74 (1989) 83-98. | MR | Zbl
[19] , and , Finite volume methods, in Handbook of numerical analysis, Handb. Numer. Anal. VII. North-Holland, Amsterdam (2000) 713-1020. | MR | Zbl
[20] , and , A new finite volume scheme for anisotropic diffusion problems on general grids: convergence analysis. C. R. Math. Acad. Sci. Paris 344 (2007) 403-406. | MR | Zbl
[21] , and , 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/. | MR | Zbl
[22] , , and , 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. | MR
[23] , , , , and , 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.
[24] , A control volume method to solve an elliptic equation on a two-dimensional irregular mesh. Comput. Methods Appl. Mech. Eng. 100 (1992) 275-290. | MR | Zbl
[25] and , 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. | MR | Zbl
[26] , Approximation of diffusion operators with discontinuous tensor coefficients on distorted meshes. Comput. Methods Appl. Mech. Eng. 192 (2003) 1939-1959. | MR | Zbl
[27] , 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. | MR | Zbl
[28] , A finite volume method for approximating 3D diffusion operators on general meshes. J. Comput. Phys. 228 (2009) 5763-5786. | MR | Zbl
[29] , 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. | MR | Zbl
Cité par Sources :
