Multiscale finite-volume method for parabolic problems arising from compressible multiphase flow in porous media

The multiscale finite-volume (MSFV) method was originally developed for the solution of heterogeneous elliptic problems with reduced computational cost. Recently, some extensions of this method for parabolic problems have been proposed. These extensions proved effective for many cases, however, they are neither general nor completely satisfactory. For instance, they are not suitable for correctly capturing the transient behavior described by the parabolic pressure equation. In this paper, we present a general multiscale finite-volume method for parabolic problems arising, for example, from compressible multiphase flow in porous media. Opposed to previous methods, here, the basis and correction functions are solutions of full parabolic governing equations in localized domains. At the same time, to enhance the computational efficiency of the scheme, the basis functions are kept pressure independent and do not have to be recalculated as pressure evolves. This general approach requires no additional assumptions and its good efficiency and high accuracy is demonstrated for various challenging test cases. Finally, to improve the quality of the results and also to extend the scheme for highly anisotropic heterogeneous problems, it is combined with the iterative MSFV (i-MSFV) method for parabolic problems. As one iterates, the i-MSFV solutions of compressible multiphase problems (parabolic problems) converge to the corresponding fine-scale reference solutions in the same way as demonstrated recently for incompressible cases (elliptic problems). Therefore, the proposed MSFV method can also be regarded as an efficient linear solver for parabolic problems and studies of its efficiency are presented for many test cases.

[1]  H. Tchelepi,et al.  Adaptive Multiscale Finite-Volume Framework for Reservoir Simulation , 2007 .

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

[3]  T. Hughes,et al.  The variational multiscale method—a paradigm for computational mechanics , 1998 .

[4]  Patrick Jenny,et al.  Iterative multiscale finite-volume method , 2008, J. Comput. Phys..

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

[6]  Hamdi A. Tchelepi,et al.  Well Modeling in the Multiscale Finite Volume Method for Subsurface Flow Simulation , 2006, Multiscale Model. Simul..

[7]  Victor Ginting,et al.  Analysis of two-scale finite volume element method for elliptic problem , 2004, J. Num. Math..

[8]  T. Arbogast Implementation of a Locally Conservative Numerical Subgrid Upscaling Scheme for Two-Phase Darcy Flow , 2002 .

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

[10]  Yalchin Efendiev,et al.  An adaptive local–global multiscale finite volume element method for two-phase flow simulations , 2007 .

[11]  Patrick Jenny,et al.  Multiscale finite-volume method for compressible multiphase flow in porous media , 2006, J. Comput. Phys..

[12]  Hui Zhou,et al.  Operator-Based Multiscale Method for Compressible Flow , 2006 .

[13]  T. Hou,et al.  Multiscale Finite Element Methods for Nonlinear Problems and Their Applications , 2004 .

[14]  Knut-Andreas Lie,et al.  Mixed multiscale finite elements and streamline methods for reservoir simulation of large geomodels , 2005 .

[15]  L. Durlofsky,et al.  A coupled local-global upscaling approach for simulating flow in highly heterogeneous formations , 2003 .

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

[17]  Patrick Jenny,et al.  Treating Highly Anisotropic Subsurface Flow with the Multiscale Finite-Volume Method , 2007, Multiscale Model. Simul..

[18]  Louis J. Durlofsky,et al.  Efficient Incorporation of Global Effects in Upscaled Models of Two-Phase Flow and Transport in Heterogeneous Formations , 2006, Multiscale Model. Simul..

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

[20]  P. Jenny,et al.  Multiscale finite-volume method for density-driven flow in porous media , 2008 .

[21]  Thomas Y. Hou,et al.  Convergence of a multiscale finite element method for elliptic problems with rapidly oscillating coefficients , 1999, Math. Comput..

[22]  Patrick Jenny,et al.  Modeling complex wells with the multi-scale finite-volume method , 2009, J. Comput. Phys..

[23]  Patrick Jenny,et al.  Adaptive fully implicit multi-scale finite-volume method for multi-phase flow and transport in heterogeneous porous media , 2006, J. Comput. Phys..

[24]  Jørg E. Aarnes,et al.  On the Use of a Mixed Multiscale Finite Element Method for GreaterFlexibility and Increased Speed or Improved Accuracy in Reservoir Simulation , 2004, Multiscale Model. Simul..

[25]  Todd Arbogast,et al.  Numerical Subgrid Upscaling of Two-Phase Flow in Porous Media , 2000 .

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

[27]  Patrick Jenny,et al.  An efficient multi-scale Poisson solver for the incompressible Navier-Stokes equations with immersed boundaries , 2009, J. Comput. Phys..

[28]  Zhiming Chen,et al.  A mixed multiscale finite element method for elliptic problems with oscillating coefficients , 2003, Math. Comput..

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

[30]  Ruben Juanes,et al.  A locally conservative variational multiscale method for the simulation of porous media flow with multiscale source terms , 2008 .

[31]  Michael Andrew Christie,et al.  Tenth SPE Comparative Solution Project: a comparison of upscaling techniques , 2001 .

[32]  Patrick Jenny,et al.  A multiscale finite-volume method for three-phase flow influenced by gravity , 2006 .

[33]  H. Tchelepi,et al.  Multiscale finite-volume formulation for multiphase flow in porous media: black oil formulation of compressible, three-phase flow with gravity , 2008 .

[34]  Patrick Jenny,et al.  Well modeling in the multi‐scale finite‐volume context , 2007 .

[35]  Thomas Y. Hou,et al.  Convergence of a Nonconforming Multiscale Finite Element Method , 2000, SIAM J. Numer. Anal..