Abstract
We extend and generalize recently proposed finite volume methods using the framework of mixed finite element methods. Proposed discretizations are defined for tensor permeabilities and naturally produce a generalization of harmonic averaging, and are therefore well suited for heterogeneous and anisotropic media. They are locally mass conservative and work on extremely flexible distorted meshes. Flux variables can be excluded locally and the resulting discretizations for the pressures has the same stencils for Voronoi/PEBI grids as 2-point finite volume discretizations currently used in many simulators.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
I. Aavatsmark, T. Barkve, O. Be and T. Mannseth, Discretization on unstructured grids for inhomoge-neous, anisotropic media. Part I: Derivation of the methods, SIAM J. Sci. Statist. Comput. 19 (1998) 1700-1716.
T. Arbogast, M.F. Wheeler and I. Yotov, Mixed finite elements for elliptic problems with tensor coef-ficients as cell-centered finite differences, SIAM J. Numer. Anal. 34(2) (1997) 828-853.
K. Aziz and A. Settari, Petroleum Reservoir Simulation (Applied Science, New York, 1979).
F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods (Springer, New York, 1991).
Z. Cai, J. Jones, S.F. McCormick and T. Russell, Control volume mixed finite element methods, Comput. Geosci. 1 (1997) 289-315.
M.G. Edwards and C.F. Roger, A flux continuous scheme for the full tensor pressure equation, in: Proc. of 4th European Conf. on the Mathematics of Oil Recovery, eds. Christie et al., Norway.
M.G. Edwards and C.F. Roger, Finite volume discretization with imposed flux continuity for the general tensor pressure equation, Comput. Geosci. 2 (1999) 259-290.
Z.E. Heinemann, C. Brand, M. Munke and Y. Chen, Modeling reservoir geometry with irregular grids, SPE Reservoir Engrg. (1991) 225-232.
B. Heinrich, Finite Difference Methods on Irregular Networks (Akademie-Verlag, Berlin, 1987).
R. Herbin, An error estimate for a finite volume scheme for a diffusion-convection problem on a triangular mesh, Numer. Methods Partial Differential Equations 11(2) (1995) 165-174.
S.H. Lee, H.A. Tchelepi and L.F. DeChant, Implementation of a flux-continuous finite difference method for stratigraphic, hexahedron grids, SPE 51901, in: Proc. of the 15th SPE Reservoir Simulation Symposium, Houston, TX, USA (1999) pp. 231-241.
I.D. Mishev, Finite volume methods on Voronoi meshes, Numer. Methods Partial Differential Equations 14 (1998) 193-212.
C.L. Palagi and K. Aziz, Modeling vertical and horizontal wells with Voronoi grids, SPE Reservoir Engrg. (1994) 15-21.
D.W. Peaceman, Fundamentals of Numerical Reservoir Simulation (Elsevier, Amsterdam, 1997).
J.E. Roberts and J.M. Thomas, Mixed and hybrid methods, in: Handbook of Numerical Analysis, Vol. 2, eds. P.G. Ciarlet and J.L. Lions (Elsevier Science, Amsterdam, 1991) pp. 524-639.
M. Shashkov, Conservative Finite-Difference Methods on General Grids (CRC Press, New York, 1996).
J.M. Thomas and D. Trujillo, Analysis of finite volume methods, Technical Report 19, Université de Pau et des Pays de L'adour, Pau, France (1995).
J.M. Thomas and D. Trujillo, Convergence of finite volume methods, Technical Report 20, Université de Pau et des Pays de L'adour, Pau, France (1995).
S.K. Verma, Flexible grids for reservoir simulations, Ph.D. thesis, Stanford University, California (1996).
S.K. Verma and K. Aziz, Two-and three-dimensional flexible grids for reservoir simulation, in: Proc. of the 5th European Conf. on the Mathematics of Oil Recovery, Leoben, Austria (1996).
I. Yotov, Mixed finite element methods for flow in porous media, Ph.D. thesis, Rice University, TX (1996).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Mishev, I.D. Nonconforming Finite Volume Methods. Computational Geosciences 6, 253–268 (2002). https://doi.org/10.1023/A:1021214424953
Issue Date:
DOI: https://doi.org/10.1023/A:1021214424953