Adaptive numerical homogenization of non-linear diffusion problems

We propose an efficient numerical strategy for simulating fluid flow through porous media with highly oscillatory characteristics. Specifically, we consider non-linear diffusion models. This scheme is based on the classical homogenization theory and uses a locally mass-conservative formulation. In addition, we discuss some properties of the standard non-linear solvers and use an error estimator to perform a local mesh refinement. The main idea is to compute the effective parameters in such a way that the computational complexity is reduced without affecting the accuracy. We perform some numerical examples to illustrate the behaviour of the adaptive scheme and of the non-linear solvers. Finally, we discuss the advantages of the implementation of the numerical homogenization in a periodic media and the applicability of the same scheme in non-periodic test cases such as SPE10th project.

[1]  Peter Knabner,et al.  Error estimates for a mixed finite element discretization of some degenerate parabolic equations , 2008, Numerische Mathematik.

[2]  Olav Møyner,et al.  A multiscale restriction-smoothed basis method for high contrast porous media represented on unstructured grids , 2016, J. Comput. Phys..

[3]  F. Radu,et al.  Mixed finite elements for the Richards' equation: linearization procedure , 2004 .

[4]  Gurpreet Singh,et al.  Multiscale methods for model order reduction of non-linear multiphase flow problems , 2018, Computational Geosciences.

[5]  Hadi Hajibeygi,et al.  Multiscale gradient computation for flow in heterogeneous porous media , 2017, J. Comput. Phys..

[6]  F. Radu,et al.  A study on iterative methods for solving Richards’ equation , 2015, Computational Geosciences.

[7]  P. Donato,et al.  An introduction to homogenization , 2000 .

[8]  Peter Knabner,et al.  Order of Convergence Estimates for an Euler Implicit, Mixed Finite Element Discretization of Richards' Equation , 2004, SIAM J. Numer. Anal..

[9]  Iuliu Sorin Pop,et al.  A modified L-scheme to solve nonlinear diffusion problems , 2019, Comput. Math. Appl..

[10]  Carsten Carstensen,et al.  All first-order averaging techniques for a posteriori finite element error control on unstructured grids are efficient and reliable , 2003, Math. Comput..

[11]  A. K. Nandakumaran,et al.  Homogenization of a parabolic equation in perforated domain with Neumann boundary condition , 2002 .

[12]  P. Renard,et al.  Calculating equivalent permeability: a review , 1997 .

[13]  On the homogenization of degenerate parabolic equations , 2000 .

[14]  U. Hornung.  Homogenization and porous media , 1996 .

[15]  Ben Schweizer,et al.  An Adaptive Multiscale Finite Element Method , 2014, Multiscale Model. Simul..

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

[17]  Luc Tartar,et al.  The General Theory of Homogenization: A Personalized Introduction , 2009 .

[18]  Carsten Carstensen,et al.  Error reduction and convergence for an adaptive mixed finite element method , 2006, Math. Comput..

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

[20]  Luca Bergamaschi,et al.  MIXED FINITE ELEMENTS AND NEWTON-TYPE LINEARIZATIONS FOR THE SOLUTION OF RICHARDS' EQUATION , 1999 .

[21]  M. Celia,et al.  A General Mass-Conservative Numerical Solution for the Unsaturated Flow Equation , 1990 .

[22]  J. Bear,et al.  Introduction to Modeling of Transport Phenomena in Porous Media , 1990 .

[23]  Carsten Carstensen,et al.  A posteriori error control in low-order finite element discretisations of incompressible stationary flow problems , 2001, Math. Comput..

[24]  A. K. Nandakumaran,et al.  Homogenization of a parabolic equation in perforated domain with Dirichlet boundary condition , 2002 .

[25]  Carsten Carstensen,et al.  A unifying theory of a posteriori finite element error control , 2005, Numerische Mathematik.

[26]  E Weinan,et al.  The Heterogeneous Multiscale Method Based on the Discontinuous Galerkin Method for Hyperbolic and Parabolic Problems , 2005, Multiscale Model. Simul..

[27]  Knut-Andreas Lie,et al.  An Introduction to Reservoir Simulation Using MATLAB/GNU Octave , 2019 .

[28]  Jan M. Nordbotten,et al.  A robust linearization scheme for finite volume based discretizations for simulation of two-phase flow in porous media , 2015, J. Comput. Appl. Math..

[29]  Assyr Abdulle,et al.  Adaptive finite element heterogeneous multiscale method for homogenization problems , 2011 .

[30]  Ben Schweizer,et al.  Adaptive heterogeneous multiscale methods for immiscible two-phase flow in porous media , 2013, Computational Geosciences.

[31]  Assyr Abdulle,et al.  A short and versatile finite element multiscale code for homogenization problems , 2009 .

[33]  J. Nordbotten,et al.  Iterative Linearisation Schemes for Doubly Degenerate Parabolic Equations , 2017, Lecture Notes in Computational Science and Engineering.

[34]  Hadi Hajibeygi,et al.  Algebraic dynamic multilevel (ADM) method for fully implicit simulations of multiphase flow in porous media , 2016, J. Comput. Phys..

[35]  E Weinan,et al.  Heterogeneous multiscale method: A general methodology for multiscale modeling , 2003 .

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

[37]  F. Radu,et al.  Upscaling of Non-isothermal Reactive Porous Media Flow with Changing Porosity , 2016, Transport in Porous Media.

[38]  Gergina Pencheva,et al.  Adaptive mesh refinement with an enhanced velocity mixed finite element method on semi-structured grids using a fully coupled solver , 2018, Computational Geosciences.

[39]  Yerlan Amanbek,et al.  Adaptive numerical homogenization for upscaling single phase flow and transport , 2019, J. Comput. Phys..

[40]  R. Dutton,et al.  Delaunay triangulation and 3D adaptive mesh generation , 1997 .

[41]  Stephan Luckhaus,et al.  Quasilinear elliptic-parabolic differential equations , 1983 .

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

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

[44]  Marián Slodicka,et al.  A Robust and Efficient Linearization Scheme for Doubly Nonlinear and Degenerate Parabolic Problems Arising in Flow in Porous Media , 2001, SIAM J. Sci. Comput..

[45]  E Weinan,et al.  The Heterogeneous Multi-Scale Method , 2002 .

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

[47]  Jan M. Nordbotten,et al.  A robust, mass conservative scheme for two-phase flow in porous media including Hölder continuous nonlinearities , 2018 .

[48]  P. Henning,et al.  Multiscale mixed finite elements , 2015, 1501.05526.