Multiscale Finite-Volume CVD-MPFA Formulations on Structured and Unstructured Grids

This paper presents the development of finite-volume multiscale methods for quadrilateral and triangular unstructured grids. Families of Darcy-flux approximations have been developed for consistent approximation of the general tensor pressure equation arising from Darcy's law together with mass conservation. The schemes are control-volume distributed (CVD) with flow variables and rock properties sharing the same control-volume location and are comprised of a multipoint flux family formulation (CVD-MPFA). The schemes are used to develop a CVD-MPFA based multiscale finite-volume (MSFV) formulation applicable to both structured and unstructured grids in two dimensions. The basis functions are a key component of the MSFV method, and are a set of local solutions, usually defined subject to Dirichlet boundary conditions. A generalization of the Cartesian grid Dirichlet basis functions described in [P. Jenny, S. H. Lee, and H. A. Tchelepi, J. Comput. Phys., 187 (2003), pp. 47--67] is presented here for unstructured grids. Whilst the transition from a Cartesian grid to an unstructured grid is largely successful, use of Dirichlet basis functions can still lead to pressure fields that exhibit spurious oscillations in areas of strong heterogeneity. New basis functions are proposed in an attempt to improve the pressure field solutions where Neumann boundary conditions are imposed almost everywhere, except corners which remain specified by Dirichlet values.

[1]  Ivar Aavatsmark,et al.  Discretization on Unstructured Grids For Inhomogeneous, Anisotropic Media. Part II: Discussion And Numerical Results , 1998, SIAM J. Sci. Comput..

[2]  Olav Møyner,et al.  The Multiscale Finite-Volume Method on Stratigraphic Grids , 2014 .

[3]  Eirik Keilegavlen,et al.  Physics‐based preconditioners for flow in fractured porous media , 2014 .

[4]  Ivar Aavatsmark,et al.  Discretization on Unstructured Grids for Inhomogeneous, Anisotropic Media. Part I: Derivation of the Methods , 1998, SIAM J. Sci. Comput..

[5]  Ivar Aavatsmark,et al.  Multiscale mass conservative domain decomposition preconditioners for elliptic problems on irregular grids , 2011 .

[6]  M. Shashkov,et al.  The mimetic finite difference discretization of diffusion problem on unstructured polyhedral meshes , 2006 .

[7]  Thierry Gallouët,et al.  A cell-centred finite-volume approximation for anisotropic diffusion operators on unstructured meshes in any space dimension , 2006 .

[8]  Knut-Andreas Lie,et al.  A comparison of multiscale methods for elliptic problems in porous media flow , 2008 .

[9]  W. Marsden.  I and J , 2012 .

[10]  H. Tchelepi,et al.  Multi-scale finite-volume method for elliptic problems in subsurface flow simulation , 2003 .

[11]  F. Hermeline,et al.  A Finite Volume Method for the Approximation of Diffusion Operators on Distorted Meshes , 2000 .

[12]  Johannes Mykkeltveit,et al.  Symmetric Positive Definite Flux-Continuous Full-Tensor Finite-Volume Schemes on Unstructured Cell-Centered Triangular Grids , 2008, SIAM J. Sci. Comput..

[13]  Yalchin Efendiev,et al.  Accurate multiscale finite element methods for two-phase flow simulations , 2006, J. Comput. Phys..

[14]  M. Shashkov,et al.  The Numerical Solution of Diffusion Problems in Strongly Heterogeneous Non-isotropic Materials , 1997 .

[15]  Michael G. Edwards,et al.  Finite volume discretization with imposed flux continuity for the general tensor pressure equation , 1998 .

[16]  Michel Fortin,et al.  Mixed and Hybrid Finite Element Methods , 2011, Springer Series in Computational Mathematics.

[17]  Patrick Jenny,et al.  A hierarchical fracture model for the iterative multiscale finite volume method , 2011, J. Comput. Phys..

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

[19]  Michael G. Edwards,et al.  A family of MPFA finite-volume schemes with full pressure support for the general tensor pressure equation on cell-centered triangular grids , 2011, J. Comput. Phys..

[20]  Mayur Pal,et al.  CVD-MPFA Based Multiscale Formulation on Structured and Unstructured Grids , 2012 .

[21]  L. Durlofsky.  A triangle based mixed finite element–finite volume technique for modeling two phase flow through porous media , 1993 .

[22]  J. Nordbotten,et al.  On the relationship between the multiscale finite-volume method and domain decomposition preconditioners , 2008 .

[23]  Michael G. Edwards,et al.  Unstructured, Control-Volume Distributed, Full-Tensor Finite-Volume Schemes with Flow Based Grids , 2002 .

[24]  Todd Arbogast,et al.  A Multiscale Mortar Mixed Finite Element Method , 2007, Multiscale Model. Simul..

[25]  Louis J. Durlofsky,et al.  Finite Difference Simulation of Geologically Complex Reservoirs With Tensor Permeabilities , 1998 .

[26]  K. Aziz,et al.  Petroleum Reservoir Simulation , 1979 .

[27]  Barry Koren,et al.  Finite-volume scheme for anisotropic diffusion , 2016, J. Comput. Phys..

[28]  Bradley T. Mallison,et al.  Compact Multiscale Finite Volume Method for Heterogeneous Anisotropic Elliptic Equations , 2008, Multiscale Model. Simul..

[29]  C. L. Farmer,et al.  A Global Optimization Approach to Grid Generation , 1991 .

[30]  Stein Krogstad,et al.  Multiscale mixed/mimetic methods on corner-point grids , 2008 .

[31]  Mayur Pal,et al.  Multiscale Formulations with CVD-MPFA Schemes on Structured and Unstructured Grids , 2013, ANSS 2013.

[32]  Mary F. Wheeler,et al.  A Multipoint Flux Mixed Finite Element Method , 2006, SIAM J. Numer. Anal..

[33]  Thomas Y. Hou,et al.  A Multiscale Finite Element Method for Elliptic Problems in Composite Materials and Porous Media , 1997 .

[34]  Patrick Jenny,et al.  Adaptive Multiscale Finite-Volume Method for Multiphase Flow and Transport in Porous Media , 2005, Multiscale Model. Simul..