Enriched multi-point flux approximation for general grids

It is well known that the two-point flux approximation, a numerical scheme used in most commercial reservoir simulators, has O(1) error when grids are not K-orthogonal. In the last decade, the multi-point flux approximations have been developed as a remedy. However, non-physical oscillations can appear when the anisotropy is really strong. We found out the oscillations are closely related to the poor approximation of pressure gradient in the flux computation.In this paper, we propose the control volume enriched multi-point flux approximation (EMPFA) for general diffusion problems on polygonal and polyhedral meshes. Non-physical oscillations are not observed for realistic and strongly anisotropic heterogeneous material properties described by a full tensor. Exact linear solutions are recovered for grids with non-planar interfaces, and a first and second order convergence are achieved for the flux and scalar unknowns, respectively.

[1]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[2]  T. Schmidt,et al.  Box schemes on quadrilateral meshes , 1993, Computing.

[3]  Jim E. Jones,et al.  Control‐volume mixed finite element methods , 1996 .

[4]  G. T. Eigestad,et al.  Improved streamlines and time-of-flight for streamline simulation on irregular grids , 2007 .

[5]  Philippe G. Ciarlet,et al.  The finite element method for elliptic problems , 2002, Classics in applied mathematics.

[6]  Khalid Aziz,et al.  A Control Volume Scheme for Flexible Grids in Reservoir Simulation , 1997 .

[7]  Ivar Aavatsmark,et al.  Monotonicity conditions for control volume methods on uniform parallelogram grids in homogeneous media , 2005 .

[8]  Ragnar Winther,et al.  Convergence of multipoint flux approximations on quadrilateral grids , 2006 .

[9]  Mary F. Wheeler,et al.  Superconvergence of the Velocity in Mimetic Finite Difference Methods on Quadrilaterals , 2005, SIAM J. Numer. Anal..

[10]  I. M. Nguena,et al.  A finite volume approximation for second order elliptic problems with a full matrix on quadrilateral grids: derivation of the scheme and a theoretical analysis , 2006 .

[11]  Ivar Aavatsmark,et al.  Monotonicity of control volume methods , 2007, Numerische Mathematik.

[12]  G. T. Eigestad,et al.  On the convergence of the multi-point flux approximation O-method: Numerical experiments for discontinuous permeability , 2005 .

[13]  F. Brezzi,et al.  A FAMILY OF MIMETIC FINITE DIFFERENCE METHODS ON POLYGONAL AND POLYHEDRAL MESHES , 2005 .

[14]  B. Rivière,et al.  Part II. Discontinuous Galerkin method applied to a single phase flow in porous media , 2000 .

[15]  I. Aavatsmark,et al.  Discretization on Non-Orthogonal, Curvilinear Grids for Multi-Phase Flow , 1994 .

[16]  L. Durlofsky.  Accuracy of mixed and control volume finite element approximations to Darcy velocity and related quantities , 1994 .

[17]  Xiao-Hui Wu,et al.  Effect of Grid Deviation on Flow Solutions , 2005 .

[18]  J. D. Wilson,et al.  Shape Functions for Velocity Interpolation in General Hexahedral Cells , 2002 .

[19]  I. Aavatsmark,et al.  An Introduction to Multipoint Flux Approximations for Quadrilateral Grids , 2002 .

[20]  Zhiqiang Cai,et al.  The finite volume element method for diffusion equations on general triangulations , 1991 .

[21]  Konstantin Lipnikov,et al.  Convergence of the Mimetic Finite Difference Method for Diffusion Problems on Polyhedral Meshes , 2005, SIAM J. Numer. Anal..

[22]  Michael G. Edwards,et al.  A Flux Continuous Scheme for the Full Tensor Pressure Equation , 1994 .

[23]  Ruben Juanes,et al.  Robust streamline tracing for the simulation of porous media flow on general triangular and quadrilateral grids , 2006, J. Comput. Phys..

[24]  Ivar Aavatsmark,et al.  A Compact MPFA Method with Improved Robustness , 2006 .

[25]  D. Rose,et al.  Some errors estimates for the box method , 1987 .

[26]  Y. Kuznetsov,et al.  New mixed finite element method on polygonal and polyhedral meshes , 2005 .

[27]  Do Y. Kwak,et al.  A General Framework for Constructing and Analyzing Mixed Finite Volume Methods on Quadrilateral Grids: The Overlapping Covolume Case , 2001, SIAM J. Numer. Anal..

[28]  Mikhail Shashkov,et al.  Solving Diffusion Equations with Rough Coefficients in Rough Grids , 1996 .

[29]  Y. Kuznetsov,et al.  Mixed Finite Element Method on Polygonal and Polyhedral Meshes , 2003 .

[30]  Jan M. Nordbotten,et al.  Discretization on quadrilateral grids with improved monotonicity properties , 2005 .

[31]  Kellen Petersen August.  Real Analysis , 2009 .

[32]  J. David Moulton,et al.  Convergence of mimetic finite difference discretizations of the diffusion equation , 2001, J. Num. Math..

[33]  Hamdi A. Tchelepi,et al.  Implementation of a Flux-Continuous Finite Difference Method for Stratigraphic, Hexahedron Grids , 1999 .

[34]  T. F. Russell,et al.  Relationships among some locally conservative discretization methods which handle discontinuous coefficients , 2004 .