Adaptive numerical homogenization for upscaling single phase flow and transport

Abstract We propose an adaptive multiscale method to improve the efficiency and the accuracy of numerical computations by combining numerical homogenization and domain decomposition for modeling flow and transport. Our approach focuses on minimizing the use of fine scale properties associated with advection and diffusion/dispersion. Here a fine scale flow and transport problem is solved in subdomains defined by a transient region where spatial changes in transported species concentrations are large while a coarse scale problem is solved in the remaining subdomains. Away from the transient region, effective macroscopic properties are obtained using local numerical homogenization. An Enhanced Velocity Mixed Finite Element Method (EVMFEM) as a domain decomposition scheme is used to couple these coarse and fine subdomains [1] . Specifically, homogenization is employed here only when coarse and fine scale problems can be decoupled to extract temporal invariants in the form of effective parameters. In this paper, a number of numerical tests are presented for demonstrating the capabilities of this adaptive numerical homogenization approach in upscaling flow and transport in heterogeneous porous medium.

[1]  Mary F. Wheeler,et al.  Enhanced velocity mixed finite element methods for modeling coupled flow and transport on non-matching multiblock grids , 2011 .

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

[3]  Yalchin Efendiev,et al.  Generalized Multiscale Finite Element Methods for Wave Propagation in Heterogeneous Media , 2013, Multiscale Model. Simul..

[4]  I. Babuska,et al.  Generalized Finite Element Methods: Their Performance and Their Relation to Mixed Methods , 1983 .

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

[6]  Yalchin Efendiev,et al.  Adaptive multiscale model reduction with Generalized Multiscale Finite Element Methods , 2016, J. Comput. Phys..

[7]  Grigorios A. Pavliotis,et al.  Multiscale Methods: Averaging and Homogenization , 2008 .

[8]  Alain Bourgeat,et al.  Effective Macrodiffusion in Solute Transport through Heterogeneous Porous Media , 2005, Multiscale Model. Simul..

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

[10]  A. Bensoussan,et al.  Asymptotic analysis for periodic structures , 1979 .

[11]  Ivan Yotov,et al.  A multipoint stress mixed finite element method for elasticity on quadrilateral grids , 2018, Numerical Methods for Partial Differential Equations.

[12]  Martin Vohralík,et al.  Robust a Posteriori Error Control and Adaptivity for Multiscale, Multinumerics, and Mortar Coupling , 2013, SIAM J. Numer. Anal..

[13]  Grégoire Allaire,et al.  Homogenization of the stokes flow in a connected porous medium , 1989 .

[14]  A. Mikelić,et al.  Homogenization of nonstationary Navier-Stokes equations in a domain with a grained boundary , 1991 .

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

[16]  Ivan Yotov,et al.  Implementation of a mortar mixed finite element method using a Multiscale Flux Basis , 2009 .

[17]  Hamdi A. Tchelepi,et al.  Adaptive multiscale finite-volume method for nonlinear multiphase transport in heterogeneous formations , 2009, J. Comput. Phys..

[18]  G. Allaire.  Homogenization and two-scale convergence , 1992 .

[19]  S. Whitaker.  Flow in porous media I: A theoretical derivation of Darcy's law , 1986 .

[20]  V. Zhikov,et al.  Homogenization of Differential Operators and Integral Functionals , 1994 .

[21]  Giancarlo Sangalli,et al.  Variational Multiscale Analysis: the Fine-scale Green's Function, Projection, Optimization, Localization, and Stabilized Methods , 2007, SIAM J. Numer. Anal..

[22]  Todd Arbogast,et al.  Operator-based approach to upscaling the pressure equation , 1998 .

[23]  Andro Mikelic,et al.  Rigorous Upscaling of the Reactive Flow through a Pore, under Dominant Peclet and Damkohler Numbers , 2006, SIAM J. Math. Anal..

[24]  Todd Arbogast,et al.  Multiscale mortar mixed methods for heterogeneous elliptic problems , 2012 .

[25]  D. W. Peaceman.  Interpretation of well-block pressures in numerical reservoir simulation with nonsquare grid blocks and anisotropic permeability , 1983 .

[26]  David A. Barajas-Solano,et al.  Hybrid Multiscale Finite Volume Method for Advection-Diffusion Equations Subject to Heterogeneous Reactive Boundary Conditions , 2015, Multiscale Model. Simul..

[27]  Mary F. Wheeler,et al.  Compositional flow modeling using a multi-point flux mixed finite element method , 2014, Computational Geosciences.

[28]  E Weinan,et al.  The heterogeneous multiscale method* , 2012, Acta Numerica.

[29]  I. Babuska,et al.  Special finite element methods for a class of second order elliptic problems with rough coefficients , 1994 .

[30]  Mary F. Wheeler,et al.  Enhanced Velocity Mixed Finite Element Methods for Flow in Multiblock Domains , 2002 .

[31]  Mary F. Wheeler,et al.  Mortar Upscaling for Multiphase Flow in Porous Media , 2002 .

[32]  T. Arbogast.  Mixed Multiscale Methods for Heterogeneous Elliptic Problems , 2012 .

[33]  A. Weiser,et al.  On convergence of block-centered finite differences for elliptic-problems , 1988 .

[34]  B. D. Veubeke.  Displacement and equilibrium models in the finite element method , 1965 .

[35]  M. Wheeler,et al.  Coupling Different Numerical Algorithms for Two Phase Fluid Flow , 2000 .

[36]  G. Taylor.  Dispersion of soluble matter in solvent flowing slowly through a tube , 1953, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[37]  C. L. Farmer.  Upscaling: a review , 2002 .

[38]  Gergina Pencheva,et al.  A Global Jacobian Method for Mortar Discretizations of a Fully Implicit Two-Phase Flow Model , 2014, Multiscale Model. Simul..

[39]  Louis J. Durlofsky,et al.  Upscaling of geocellular models for reservoir ow simulation: A review of recent progress , 2003 .

[40]  Vivette Girault,et al.  Mortar multiscale finite element methods for Stokes–Darcy flows , 2014, Numerische Mathematik.

[41]  Todd Arbogast,et al.  A Nonlinear Mixed Finite Eelement Method for a Degenerate Parabolic Equation Arising in Flow in Porous Media , 1996 .

[42]  Yalchin Efendiev,et al.  Multiscale Finite Element Methods: Theory and Applications , 2009 .

[43]  Mary F. Wheeler,et al.  Mortar coupling and upscaling of pore-scale models , 2008 .

[44]  Yalchin Efendiev,et al.  Generalized multiscale finite element methods (GMsFEM) , 2013, J. Comput. Phys..

[45]  T. Hou,et al.  Analysis of upscaling absolute permeability , 2002 .

[46]  Susan D. Hovorka,et al.  Monitoring CO 2 Storage in Brine Formations: Lessons Learned from the Frio Field Test One Year Post Injection , 2006 .

[47]  Todd Arbogast,et al.  Numerical Subgrid Upscaling for Waterflood Simulations , 2001 .

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

[49]  Jan M. Nordbotten,et al.  A Multipoint Stress Mixed Finite Element Method for Elasticity on Simplicial Grids , 2018, SIAM J. Numer. Anal..

[50]  E Weinan,et al.  Heterogeneous multiscale methods: A review , 2007 .

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

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

[53]  Yalchin Efendiev,et al.  An Adaptive Multiscale Method for Simulation of Fluid Flow in Heterogeneous Porous Media , 2006, Multiscale Model. Simul..