Multiscale two-stage solver for Biot’s poroelasticity equations in subsurface media

We propose a two-stage preconditioner for accelerating the iterative solution by a Krylov subspace method of Biot’s poroelasticity equations based on a displacement-pressure formulation. The spatial discretization combines a finite element method for mechanics and a finite volume approach for flow. The fully implicit backward Euler scheme is used for time integration. The result is a 2 × 2 block linear system for each timestep. The preconditioning operator is obtained by applying a two-stage scheme. The first stage is a global preconditioner that employs multiscale basis functions to construct coarse-scale coupled systems using a Galerkin projection. This global stage is effective at damping low-frequency error modes associated with long-range coupling of the unknowns. The second stage is a local block-triangular smoothing preconditioner, which is aimed at high-frequency error modes associated with short-range coupling of the variables. Various numerical experiments are used to demonstrate the robustness of the proposed solver.

[1]  Maria Vasilyeva,et al.  A Generalized Multiscale Finite Element Method for poroelasticity problems I: Linear problems , 2015, J. Comput. Appl. Math..

[2]  M. Benzi.  Preconditioning techniques for large linear systems: a survey , 2002 .

[3]  Poromechanics , 2020 .

[4]  Jeonghun J. Lee,et al.  Parameter-Robust Discretization and Preconditioning of Biot's Consolidation Model , 2015, SIAM J. Sci. Comput..

[5]  Hans Petter Langtangen,et al.  A parallel block preconditioner for large-scale poroelasticity with highly heterogeneous material parameters , 2012, Computational Geosciences.

[6]  Abimael F. D. Loula,et al.  On stability and convergence of finite element approximations of biot's consolidation problem , 1994 .

[7]  A. Kozlova,et al.  A Real-Field Multiscale Black-Oil Reservoir Simulator , 2015, ANSS 2015.

[8]  Cornelis W. Oosterlee,et al.  On an Uzawa smoother in multigrid for poroelasticity equations , 2017, Numer. Linear Algebra Appl..

[9]  Hamdi A. Tchelepi,et al.  Parallel Scalable Unstructured CPR-Type Linear Solver for Reservoir Simulation , 2005 .

[10]  Son-Young Yi.  Convergence analysis of a new mixed finite element method for Biot's consolidation model , 2014 .

[11]  Rainer Helmig,et al.  EfficientModeling of Flow and Transport in Porous Media Using Multiphysics andMultiscale Approaches , 2010 .

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

[13]  Ivan Lunati,et al.  An iterative multiscale finite volume algorithm converging to the exact solution , 2011, J. Comput. Phys..

[14]  Roland Masson,et al.  Hybrid Finite Volume Discretization of Linear Elasticity Models on General Meshes , 2011 .

[15]  Ruben Juanes,et al.  A locally conservative finite element framework for the simulation of coupled flow and reservoir geomechanics , 2007 .

[16]  JAN MARTIN NORDBOTTEN,et al.  Stable Cell-Centered Finite Volume Discretization for Biot Equations , 2015, SIAM J. Numer. Anal..

[17]  Jan M. Nordbotten,et al.  Robust fixed stress splitting for Biot's equations in heterogeneous media , 2017, Appl. Math. Lett..

[18]  J. Nordbotten.  Cell‐centered finite volume discretizations for deformable porous media , 2014 .

[19]  Rainer Helmig,et al.  Efficient Modeling of Flow and Transport in Porous Media Using Multi-physics and Multi-scale Approaches , 2015 .

[20]  Oleg Iliev,et al.  Multiscale Finite Elements for Linear Elasticity: Oscillatory Boundary Conditions , 2014 .

[21]  H. Tchelepi,et al.  Stability and convergence of sequential methods for coupled flow and geomechanics: Fixed-stress and fixed-strain splits , 2011 .

[22]  Cornelis W. Oosterlee,et al.  A systematic comparison of coupled and distributive smoothing in multigrid for the poroelasticity system , 2004, Numer. Linear Algebra Appl..

[23]  Kundan Kumar,et al.  Anderson accelerated fixed-stress splitting schemes for consolidation of unsaturated porous media , 2018, Comput. Math. Appl..

[24]  Jerome Droniou,et al.  FINITE VOLUME SCHEMES FOR DIFFUSION EQUATIONS: INTRODUCTION TO AND REVIEW OF MODERN METHODS , 2014, 1407.1567.

[25]  Hamdi A. Tchelepi,et al.  Monotone nonlinear finite-volume method for challenging grids , 2018, Computational Geosciences.

[26]  Ronaldo I. Borja,et al.  Stabilized low-order finite elements for coupled solid-deformation/fluid-diffusion and their application to fault zone transients , 2008 .

[27]  Olav Møyner,et al.  Use of Multiple Multiscale Operators To Accelerate Simulation of Complex Geomodels , 2017 .

[28]  Yalchin Efendiev,et al.  Upscaling of Mixed Finite Element Discretization Problems by the Spectral AMGe Method , 2016, SIAM J. Sci. Comput..

[29]  Yalchin Efendiev,et al.  Multiscale model reduction for shale gas transport in poroelastic fractured media , 2018, J. Comput. Phys..

[30]  O. Iliev,et al.  Multiscale finite element coarse spaces for the application to linear elasticity , 2013 .

[31]  Joshua A. White,et al.  Block-partitioned solvers for coupled poromechanics: A unified framework , 2016 .

[32]  Hadi Hajibeygi,et al.  Algebraic dynamic multilevel method for compositional flow in heterogeneous porous media , 2018, J. Comput. Phys..

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

[34]  Y. Saad,et al.  GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems , 1986 .

[35]  Mary F. Wheeler,et al.  A coupling of mixed and continuous Galerkin finite element methods for poroelasticity II: the discrete-in-time case , 2007 .

[36]  Mary F. Wheeler,et al.  A multiscale fixed stress split iterative scheme for coupled flow and poromechanics in deep subsurface reservoirs , 2018, J. Comput. Phys..

[37]  Hamdi A. Tchelepi,et al.  Block-preconditioned Krylov Methods for Coupled Multiphase Reservoir Flow and Geomechanics , 2016 .

[38]  Hamdi A. Tchelepi,et al.  Multiscale finite-element method for linear elastic geomechanics , 2017, J. Comput. Phys..

[39]  Peter Arbenz,et al.  Large scale micro finite element analysis of 3D bone poroelasticity , 2014, Parallel Comput..

[40]  L. Zikatanov,et al.  New stabilized discretizations for poroelasticity and the Stokes’ equations , 2017, Computer Methods in Applied Mechanics and Engineering.

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

[42]  Nicola Castelletto,et al.  Thermo‐hydro‐mechanical modeling of fluid geological storage by Godunov‐mixed methods , 2012 .

[43]  Ronaldo I. Borja,et al.  Block-preconditioned Newton–Krylov solvers for fully coupled flow and geomechanics , 2011 .

[44]  Jean H. Prevost.  Two‐way coupling in reservoir–geomechanical models: vertex‐centered Galerkin geomechanical model cell‐centered and vertex‐centered finite volume reservoir models , 2014 .

[45]  Ruben Juanes,et al.  Stability, Accuracy, and Efficiency of Sequential Methods for Coupled Flow and Geomechanics , 2011 .

[46]  R. Eymard,et al.  Finite Volume Methods , 2019, Computational Methods for Fluid Dynamics.

[47]  Antonin Settari,et al.  A Coupled Reservoir and Geomechanical Simulation System , 1998 .

[48]  Nicola Castelletto,et al.  A fully coupled 3-D mixed finite element model of Biot consolidation , 2010, J. Comput. Phys..

[49]  Andro Mikelić,et al.  Convergence of iterative coupling for coupled flow and geomechanics , 2013, Computational Geosciences.

[50]  F. Radu,et al.  Space–time finite element approximation of the Biot poroelasticity system with iterative coupling , 2016, 1611.06335.

[51]  Carmen Rodrigo,et al.  On the fixed-stress split scheme as smoother in multigrid methods for coupling flow and geomechanics , 2017 .

[52]  J. Pasciak,et al.  A preconditioning technique for indefinite systems resulting from mixed approximations of elliptic problems , 1988 .

[53]  Damiano Pasetto,et al.  A reduced order model‐based preconditioner for the efficient solution of transient diffusion equations , 2017 .

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

[55]  Nicola Castelletto,et al.  Scalable algorithms for three-field mixed finite element coupled poromechanics , 2016, J. Comput. Phys..

[56]  Ruben Juanes,et al.  Coupled multiphase flow and poromechanics: A computational model of pore pressure effects on fault slip and earthquake triggering , 2014 .

[57]  Dennis Eichmann,et al.  Theory Of Linear Poroelasticity , 2016 .

[58]  L. Zikatanov,et al.  Robust Block Preconditioners for Biot's Model , 2017, CSE 2017.

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

[60]  Cornelis W. Oosterlee,et al.  Multigrid method for nonlinear poroelasticity equations , 2015, Comput. Vis. Sci..

[61]  Luca Bergamaschi,et al.  Novel preconditioners for the iterative solution to FE-discretized coupled consolidation equations , 2007 .

[62]  Joachim Berdal Haga,et al.  On the causes of pressure oscillations in low‐permeable and low‐compressible porous media , 2012 .

[63]  Joshua A. White,et al.  Accuracy and convergence properties of the fixed‐stress iterative solution of two‐way coupled poromechanics , 2015 .

[64]  Hongwu Zhang,et al.  Coupling upscaling finite element method for consolidation analysis of heterogeneous saturated porous media , 2009 .

[65]  Ajh Arjan Frijns,et al.  A four-component mixture theory applied to cartilaginous tissues : numerical modelling and experiments , 2000 .

[66]  Hamdi A. Tchelepi,et al.  Hybrid Multiscale Formulation for Coupled Flow and Geomechanics , 2016 .

[67]  Hui Zhou,et al.  Two-Stage Algebraic Multiscale Linear Solver for Highly Heterogeneous Reservoir Models , 2012, Annual Simulation Symposium.

[68]  Hadi Hajibeygi,et al.  Algebraic multiscale method for flow in heterogeneous porous media with embedded discrete fractures (F-AMS) , 2016, J. Comput. Phys..

[69]  Susanne C. Brenner,et al.  Domain Decomposition Methods in Science and Engineering XXIV , 2018 .

[70]  B. Schrefler,et al.  The Finite Element Method in the Static and Dynamic Deformation and Consolidation of Porous Media , 1998 .

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

[72]  Ludmil T. Zikatanov,et al.  Stability and monotonicity for some discretizations of the Biot’s consolidation model , 2016 .

[73]  Gene H. Golub,et al.  Numerical solution of saddle point problems , 2005, Acta Numerica.

[74]  J. Nordbotten,et al.  Finite volume methods for elasticity with weak symmetry , 2015, 1512.01042.

[75]  Yixuan Wang,et al.  Adaptive algebraic multiscale solver for compressible flow in heterogeneous porous media , 2014, J. Comput. Phys..

[76]  Maria Vasilyeva,et al.  A generalized multiscale finite element method for poroelasticity problems II: Nonlinear coupling , 2015, J. Comput. Appl. Math..

[77]  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..

[78]  Luca Bergamaschi,et al.  RMCP: Relaxed Mixed Constraint Preconditioners for Saddle Point Linear Systems arising in Geomechanics , 2012 .

[79]  Yousef Saad,et al.  Iterative methods for sparse linear systems , 2003 .

[80]  Roland Masson,et al.  TP or not TP, that is the question , 2014, Computational Geosciences.

[81]  J. Meijerink,et al.  An iterative solution method for linear systems of which the coefficient matrix is a symmetric -matrix , 1977 .

[82]  Frédéric Nataf,et al.  Spillane, N. and Dolean Maini, Victorita and Hauret, P. and Nataf, F. and Pechstein, C. and Scheichl, R. (2013) Abstract robust coarse spaces for systems of PDEs via generalized eigenproblems in the overlaps , 2018 .

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

[84]  H. Klie,et al.  A Time-Stepping Scheme for Coupled Reservoir Flow and Geomechanics , 2005 .

[85]  Jostein R. Natvig,et al.  Constrained pressure residual multiscale (CPR-MS) method for fully implicit simulation of multiphase flow in porous media , 2015, J. Comput. Phys..

[86]  Mary F. Wheeler,et al.  A coupling of mixed and continuous Galerkin finite element methods for poroelasticity I: the continuous in time case , 2007 .

[87]  Hamdi A. Tchelepi,et al.  Monotone multiscale finite volume method , 2016, Computational Geosciences.

[88]  Martin J. Blunt,et al.  Tenth SPE Comparative Solution Project: a comparison of upscaling techniques , 2001 .

[89]  Hamdi A. Tchelepi,et al.  Algebraic multiscale solver for flow in heterogeneous porous media , 2014, J. Comput. Phys..

[90]  Bradley T. Mallison,et al.  Cell-centered nonlinear finite-volume methods for the heterogeneous anisotropic diffusion problem , 2017, J. Comput. Phys..

[91]  Ludmil T. Zikatanov,et al.  A nonconforming finite element method for the Biot's consolidation model in poroelasticity , 2016, J. Comput. Appl. Math..