skip to main content
DOE PAGES title logo U.S. Department of Energy
Office of Scientific and Technical Information

Title: Scalable algorithms for three-field mixed finite element coupled poromechanics

Abstract

We introduce a class of block preconditioners for accelerating the iterative solution of coupled poromechanics equations based on a three-field formulation. The use of a displacement/velocity/pressure mixed finite-element method combined with a first order backward difference formula for the approximation of time derivatives produces a sequence of linear systems with a 3 x 3 unsymmetric and indefinite block matrix. The preconditioners are obtained by approximating the two-level Schur complement with the aid of physically-based arguments that can be also generalized in a purely algebraic approach. A theoretical and experimental analysis is presented that provides evidence of the robustness, efficiency and scalability of the proposed algorithm. In conclusion, the performance is also assessed for a real-world challenging consolidation experiment of a shallow formation.

Authors:
 [1];  [2]; ORCiD logo [3]
  1. Stanford Univ., CA (United States). Energy Resources Engineering
  2. Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States). Atmospheric, Earth and Energy Division
  3. Univ. of Padova (Italy). Dept. of Civil, Environmental and Architectural Engineering
Publication Date:
Research Org.:
Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)
Sponsoring Org.:
USDOE National Nuclear Security Administration (NNSA)
OSTI Identifier:
1463016
Alternate Identifier(s):
OSTI ID: 1397767
Report Number(s):
LLNL-JRNL-737320
Journal ID: ISSN 0021-9991; 889967; TRN: US1902250
Grant/Contract Number:  
AC52-07NA27344; AC52-07-NA27344
Resource Type:
Accepted Manuscript
Journal Name:
Journal of Computational Physics
Additional Journal Information:
Journal Volume: 327; Journal Issue: C; Journal ID: ISSN 0021-9991
Publisher:
Elsevier
Country of Publication:
United States
Language:
English
Subject:
58 GEOSCIENCES; 97 MATHEMATICS AND COMPUTING; 42 ENGINEERING; Poromechanics; Preconditioners; Iterative methods; Mixed formulation; Algebraic multigrid

Citation Formats

Castelletto, Nicola, White, Joshua A., and Ferronato, Massimiliano. Scalable algorithms for three-field mixed finite element coupled poromechanics. United States: N. p., 2016. Web. https://doi.org/10.1016/j.jcp.2016.09.063.
Castelletto, Nicola, White, Joshua A., & Ferronato, Massimiliano. Scalable algorithms for three-field mixed finite element coupled poromechanics. United States. https://doi.org/10.1016/j.jcp.2016.09.063
Castelletto, Nicola, White, Joshua A., and Ferronato, Massimiliano. Mon . "Scalable algorithms for three-field mixed finite element coupled poromechanics". United States. https://doi.org/10.1016/j.jcp.2016.09.063. https://www.osti.gov/servlets/purl/1463016.
@article{osti_1463016,
title = {Scalable algorithms for three-field mixed finite element coupled poromechanics},
author = {Castelletto, Nicola and White, Joshua A. and Ferronato, Massimiliano},
abstractNote = {We introduce a class of block preconditioners for accelerating the iterative solution of coupled poromechanics equations based on a three-field formulation. The use of a displacement/velocity/pressure mixed finite-element method combined with a first order backward difference formula for the approximation of time derivatives produces a sequence of linear systems with a 3 x 3 unsymmetric and indefinite block matrix. The preconditioners are obtained by approximating the two-level Schur complement with the aid of physically-based arguments that can be also generalized in a purely algebraic approach. A theoretical and experimental analysis is presented that provides evidence of the robustness, efficiency and scalability of the proposed algorithm. In conclusion, the performance is also assessed for a real-world challenging consolidation experiment of a shallow formation.},
doi = {10.1016/j.jcp.2016.09.063},
journal = {Journal of Computational Physics},
number = C,
volume = 327,
place = {United States},
year = {2016},
month = {10}
}

Journal Article:

Citation Metrics:
Cited by: 17 works
Citation information provided by
Web of Science

Save / Share:

Works referenced in this record:

Production-Induced Compaction of a Sandstone Reservoir: The Strong Influence of Stress Path
journal, August 2000

  • Hettema, M. H. H.; Schutjens, P. M. T. M.; Verboom, B. J. M.
  • SPE Reservoir Evaluation & Engineering, Vol. 3, Issue 04
  • DOI: 10.2118/65410-PA

Discrete fracture model for coupled flow and geomechanics
journal, January 2016

  • Garipov, T. T.; Karimi-Fard, M.; Tchelepi, H. A.
  • Computational Geosciences, Vol. 20, Issue 1
  • DOI: 10.1007/s10596-015-9554-z

Numerical simulation and prediction of regional land subsidence caused by groundwater exploitation in the southwest plain of Tehran, Iran
journal, February 2016


Numerical modeling of injection, stress and permeability enhancement during shear stimulation at the Desert Peak Enhanced Geothermal System
journal, September 2015

  • Dempsey, David; Kelkar, Sharad; Davatzes, Nicholas
  • International Journal of Rock Mechanics and Mining Sciences, Vol. 78
  • DOI: 10.1016/j.ijrmms.2015.06.003

Fully coupled wellbore-reservoir modeling of geothermal heat extraction using CO 2 as the working fluid
journal, January 2015


Geomechanical issues of anthropogenic CO2 sequestration in exploited gas fields
journal, October 2010


3D geomechanical modeling for CO2 geological storage in faulted formations. A case study in an offshore northern Adriatic reservoir, Italy
journal, March 2014

  • Teatini, Pietro; Castelletto, Nicola; Gambolati, Giuseppe
  • International Journal of Greenhouse Gas Control, Vol. 22
  • DOI: 10.1016/j.ijggc.2013.12.021

Geomechanical behavior of the reservoir and caprock system at the In Salah CO2 storage project
journal, May 2014

  • White, J. A.; Chiaramonte, L.; Ezzedine, S.
  • Proceedings of the National Academy of Sciences, Vol. 111, Issue 24
  • DOI: 10.1073/pnas.1316465111

Bone poroelasticity
journal, March 1999


General Theory of Three‐Dimensional Consolidation
journal, February 1941

  • Biot, Maurice A.
  • Journal of Applied Physics, Vol. 12, Issue 2
  • DOI: 10.1063/1.1712886

On stability and convergence of finite element approximations of Biot's consolidation problem
journal, February 1994

  • Murad, Márcio A.; Loula, Abimael F. D.
  • International Journal for Numerical Methods in Engineering, Vol. 37, Issue 4
  • DOI: 10.1002/nme.1620370407

Stabilized low-order finite elements for coupled solid-deformation/fluid-diffusion and their application to fault zone transients
journal, September 2008

  • White, Joshua A.; Borja, Ronaldo I.
  • Computer Methods in Applied Mechanics and Engineering, Vol. 197, Issue 49-50
  • DOI: 10.1016/j.cma.2008.05.015

Stability and monotonicity for some discretizations of the Biot’s consolidation model
journal, January 2016

  • Rodrigo, C.; Gaspar, F. J.; Hu, X.
  • Computer Methods in Applied Mechanics and Engineering, Vol. 298
  • DOI: 10.1016/j.cma.2015.09.019

A locally conservative finite element framework for the simulation of coupled flow and reservoir geomechanics
journal, July 2007


A coupling of mixed and continuous Galerkin finite element methods for poroelasticity I: the continuous in time case
journal, March 2007


A coupling of mixed and continuous Galerkin finite element methods for poroelasticity II: the discrete-in-time case
journal, March 2007


A fully coupled 3-D mixed finite element model of Biot consolidation
journal, June 2010

  • Ferronato, Massimiliano; Castelletto, Nicola; Gambolati, Giuseppe
  • Journal of Computational Physics, Vol. 229, Issue 12
  • DOI: 10.1016/j.jcp.2010.03.018

On the causes of pressure oscillations in low-permeable and low-compressible porous media: PRESSURE OSCILLATIONS IN LOW-PERMEABLE POROUS MEDIA
journal, July 2011

  • Haga, Joachim Berdal; Osnes, Harald; Langtangen, Hans Petter
  • International Journal for Numerical and Analytical Methods in Geomechanics, Vol. 36, Issue 12
  • DOI: 10.1002/nag.1062

Large scale micro finite element analysis of 3D bone poroelasticity
journal, July 2014


Convergence analysis of a new mixed finite element method for Biot's consolidation model
journal, February 2014

  • Yi, Son-Young
  • Numerical Methods for Partial Differential Equations, Vol. 30, Issue 4
  • DOI: 10.1002/num.21865

The role of preconditioning in the solution to FE coupled consolidation equations by Krylov subspace methods
journal, February 2009

  • Ferronato, Massimiliano; Pini, Giorgio; Gambolati, Giuseppe
  • International Journal for Numerical and Analytical Methods in Geomechanics, Vol. 33, Issue 3
  • DOI: 10.1002/nag.729

Block-preconditioned Newton–Krylov solvers for fully coupled flow and geomechanics
journal, April 2011


RMCP: Relaxed Mixed Constraint Preconditioners for saddle point linear systems arising in geomechanics
journal, May 2012

  • Bergamaschi, Luca; Martínez, Ángeles
  • Computer Methods in Applied Mechanics and Engineering, Vol. 221-222
  • DOI: 10.1016/j.cma.2012.02.004

Stability, Accuracy, and Efficiency of Sequential Methods for Coupled Flow and Geomechanics
journal, June 2011

  • Kim, Jihoon; Tchelepi, Hamdi A.; Juanes, Ruben
  • SPE Journal, Vol. 16, Issue 02
  • DOI: 10.2118/119084-PA

Block-partitioned solvers for coupled poromechanics: A unified framework
journal, May 2016

  • White, Joshua A.; Castelletto, Nicola; Tchelepi, Hamdi A.
  • Computer Methods in Applied Mechanics and Engineering, Vol. 303
  • DOI: 10.1016/j.cma.2016.01.008

An investigation of the stability of numerical solutions of Biot's equations of consolidation
journal, July 1975


Stable discretization of poroelasticity problems and efficient preconditioners for arising saddle point type matrices
journal, August 2012

  • Axelsson, Owe; Blaheta, Radim; Byczanski, Petr
  • Computing and Visualization in Science, Vol. 15, Issue 4
  • DOI: 10.1007/s00791-013-0209-0

Bi-CGSTAB: A Fast and Smoothly Converging Variant of Bi-CG for the Solution of Nonsymmetric Linear Systems
journal, March 1992

  • van der Vorst, H. A.
  • SIAM Journal on Scientific and Statistical Computing, Vol. 13, Issue 2
  • DOI: 10.1137/0913035

A preconditioning technique for indefinite systems resulting from mixed approximations of elliptic problems
journal, January 1988


Efficient preconditioning of the linearized Navier–Stokes equations for incompressible flow
journal, March 2001

  • Silvester, David; Elman, Howard; Kay, David
  • Journal of Computational and Applied Mathematics, Vol. 128, Issue 1-2
  • DOI: 10.1016/S0377-0427(00)00515-X

Performance and analysis of saddle point preconditioners for the discrete steady-state Navier-Stokes equations
journal, February 2002

  • Elman, Howard C.; Silvester, David J.; Wathen, Andrew J.
  • Numerische Mathematik, Vol. 90, Issue 4
  • DOI: 10.1007/s002110100300

Block triangular preconditioners for symmetric saddle-point problems
journal, April 2004


Numerical solution of saddle point problems
journal, April 2005


GMRES: A Generalized Minimal Residual Algorithm for Solving Nonsymmetric Linear Systems
journal, July 1986

  • Saad, Youcef; Schultz, Martin H.
  • SIAM Journal on Scientific and Statistical Computing, Vol. 7, Issue 3
  • DOI: 10.1137/0907058

Any Nonincreasing Convergence Curve is Possible for GMRES
journal, July 1996

  • Greenbaum, Anne; Pták, Vlastimil; Strakoš, Zdenvk
  • SIAM Journal on Matrix Analysis and Applications, Vol. 17, Issue 3
  • DOI: 10.1137/S0895479894275030

Some basic stress diffusion solutions for fluid-saturated elastic porous media with compressible constituents
journal, January 1976


Importance of poroelastic coupling in dynamically active aquifers of the Po River Basin, Italy
journal, September 2000

  • Gambolati, Giuseppe; Teatini, Pietro; Baú, Domenico
  • Water Resources Research, Vol. 36, Issue 9
  • DOI: 10.1029/2000WR900127

Stability and convergence of sequential methods for coupled flow and geomechanics: Fixed-stress and fixed-strain splits
journal, March 2011

  • Kim, J.; Tchelepi, H. A.; Juanes, R.
  • Computer Methods in Applied Mechanics and Engineering, Vol. 200, Issue 13-16
  • DOI: 10.1016/j.cma.2010.12.022

Convergence of iterative coupling for coupled flow and geomechanics
journal, October 2012


Convergence of iterative coupling of geomechanics with flow in a fractured poroelastic medium
journal, June 2016

  • Girault, Vivette; Kumar, Kundan; Wheeler, Mary F.
  • Computational Geosciences, Vol. 20, Issue 5
  • DOI: 10.1007/s10596-016-9573-4

Convergence analysis of multirate fixed-stress split iterative schemes for coupling flow with geomechanics
journal, November 2016

  • Almani, T.; Kumar, K.; Dogru, A.
  • Computer Methods in Applied Mechanics and Engineering, Vol. 311
  • DOI: 10.1016/j.cma.2016.07.036

Flow of compressible fluid in porous elastic media
journal, January 1973

  • Ghaboussi, J.; Wilson, E. L.
  • International Journal for Numerical Methods in Engineering, Vol. 5, Issue 3
  • DOI: 10.1002/nme.1620050311

An accuracy condition for consolidation by finite elements
journal, January 1981

  • Vermeer, P. A.; Verruijt, A.
  • International Journal for Numerical and Analytical Methods in Geomechanics, Vol. 5, Issue 1
  • DOI: 10.1002/nag.1610050103

Ill-conditioning of finite element poroelasticity equations
journal, August 2001

  • Ferronato, Massimiliano; Gambolati, Giuseppe; Teatini, Pietro
  • International Journal of Solids and Structures, Vol. 38, Issue 34-35
  • DOI: 10.1016/S0020-7683(00)00352-8

Consolidation Des Sols (Étude Mathématique)
journal, September 1953


Mandel's problem revisited
journal, June 1996


Accuracy and convergence properties of the fixed-stress iterative solution of two-way coupled poromechanics: Accuracy and convergence properties of the fixed-stress iterative solution of two-way coupled poromechanics
journal, June 2015

  • Castelletto, N.; White, J. A.; Tchelepi, H. A.
  • International Journal for Numerical and Analytical Methods in Geomechanics, Vol. 39, Issue 14
  • DOI: 10.1002/nag.2400

A Block FSAI-ILU Parallel Preconditioner for Symmetric Positive Definite Linear Systems
journal, January 2010

  • Janna, Carlo; Ferronato, Massimilano; Gambolati, Giuseppe
  • SIAM Journal on Scientific Computing, Vol. 32, Issue 5
  • DOI: 10.1137/090779760

Enhanced Block FSAI Preconditioning Using Domain Decomposition Techniques
journal, January 2013

  • Janna, Carlo; Ferronato, Massimilano; Gambolati, Giuseppe
  • SIAM Journal on Scientific Computing, Vol. 35, Issue 5
  • DOI: 10.1137/120880860

FSAIPACK: A Software Package for High-Performance Factored Sparse Approximate Inverse Preconditioning
journal, February 2015

  • Janna, Carlo; Ferronato, Massimiliano; Sartoretto, Flavio
  • ACM Transactions on Mathematical Software, Vol. 41, Issue 2
  • DOI: 10.1145/2629475

Incomplete Cholesky Factorizations with Limited Memory
journal, January 1999


Numerical performance of projection methods in finite element consolidation models
journal, January 2001

  • Gambolati, Giuseppe; Pini, Giorgio; Ferronato, Massimiliano
  • International Journal for Numerical and Analytical Methods in Geomechanics, Vol. 25, Issue 14
  • DOI: 10.1002/nag.190

Iterative methods for the solution of the navier equations of elasticity
journal, August 1978


Displacement decomposition—incomplete factorization preconditioning techniques for linear elasticity problems
journal, March 1994


On parallel solution of linear elasticity problems: Part I: theory
journal, March 1998


deal.II---A general-purpose object-oriented finite element library
journal, August 2007

  • Bangerth, W.; Hartmann, R.; Kanschat, G.
  • ACM Transactions on Mathematical Software, Vol. 33, Issue 4
  • DOI: 10.1145/1268776.1268779

An overview of the Trilinos project
journal, September 2005

  • Heroux, Michael A.; Phipps, Eric T.; Salinger, Andrew G.
  • ACM Transactions on Mathematical Software, Vol. 31, Issue 3
  • DOI: 10.1145/1089014.1089021

A coupled MFE poromechanical model of a large-scale load experiment at the coastland of Venice
journal, October 2014

  • Castelletto, Nicola; Gambolati, Giuseppe; Teatini, Pietro
  • Computational Geosciences, Vol. 19, Issue 1
  • DOI: 10.1007/s10596-014-9450-y

    Works referencing / citing this record:

    Aspects of Solvers for Large-Scale Coupled Problems in Porous Media
    journal, September 2019

    • Nägel, Arne; Logashenko, Dmitry; Schroder, Jacob B.
    • Transport in Porous Media, Vol. 130, Issue 1
    • DOI: 10.1007/s11242-019-01323-w

    Conservative discontinuous finite volume and mixed schemes for a new four-field formulation in poroelasticity
    journal, January 2020

    • Kumar, Sarvesh; Oyarzúa, Ricardo; Ruiz-Baier, Ricardo
    • ESAIM: Mathematical Modelling and Numerical Analysis, Vol. 54, Issue 1
    • DOI: 10.1051/m2an/2019063

    Robust iterative schemes for non-linear poromechanics
    journal, April 2018

    • Borregales, Manuel; Radu, Florin A.; Kumar, Kundan
    • Computational Geosciences, Vol. 22, Issue 4
    • DOI: 10.1007/s10596-018-9736-6

    Low‐order stabilized finite element for the full Biot formulation in soil mechanics at finite strain
    journal, February 2019

    • Monforte, Lluís; Navas, Pedro; Carbonell, Josep Maria
    • International Journal for Numerical and Analytical Methods in Geomechanics, Vol. 43, Issue 7
    • DOI: 10.1002/nag.2923

    Multiscale two-stage solver for Biot’s poroelasticity equations in subsurface media
    journal, November 2018

    • Castelletto, Nicola; Klevtsov, Sergey; Hajibeygi, Hadi
    • Computational Geosciences, Vol. 23, Issue 2
    • DOI: 10.1007/s10596-018-9791-z