Applications of Mathematics

, Volume 62, Issue 6, pp 561–577 | Cite as

Algebraic preconditioning for Biot-Barenblatt poroelastic systems

Article

Abstract

Poroelastic systems describe fluid flow through porous medium coupled with deformation of the porous matrix. In this paper, the deformation is described by linear elasticity, the fluid flow is modelled as Darcy flow. The main focus is on the Biot-Barenblatt model with double porosity/double permeability flow, which distinguishes flow in two regions considered as continua. The main goal is in proposing block diagonal preconditionings to systems arising from the discretization of the Biot-Barenblatt model by a mixed finite element method in space and implicit Euler method in time and estimating the condition number for such preconditioning. The investigation of preconditioning includes its dependence on material coefficients and parameters of discretization.

Keywords

poroelasticity double permeability preconditioning Schur complement 

MSC 2010

65F08 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    D. N. Arnold, R. S. Falk, R. Winther: Preconditioning in H(div) and applications. Math. Comput. 66 (1997), 957–984.MathSciNetCrossRefMATHGoogle Scholar
  2. [2]
    O. Axelsson, R. Blaheta: Preconditioning of matrices partitioned in 2 × 2 block form: eigenvalue estimates and Schwarz DD for mixed FEM. Numer. Linear Algebra Appl. 17 (2010), 787–810.CrossRefMATHGoogle Scholar
  3. [3]
    O. Axelsson, R. Blaheta, P. Byczanski: Stable discretization of poroelasticity problems and efficient preconditioners for arising saddle point type matrices. Comput. Visual Sci. 15 (2012), 191–207.MathSciNetCrossRefMATHGoogle Scholar
  4. [4]
    O. Axelsson, R. Blaheta, T. Luber: Preconditioners for mixed FEM solution of stationary and nonstationary porous media flow problems. Large-Scale Scientific Computing. Int. Conf. Lecture Notes in Comput. Sci. 9374, Springer, Cham, 2015, pp. 3–14.1Google Scholar
  5. [5]
    M. Bai, D. Elsworth, J.-C. Roegiers: Multiporosity/multipermeability approach to the simulation of naturally fractured reservoirs. Water Resources Research 29 (1993), 1621–1633.CrossRefGoogle Scholar
  6. [6]
    G. I. Barenblatt, I. P. Zheltov, I. N. Kochina: Basic concepts in the theory of seepage of homogeneous liquids in fissured rocks (strata). PMM, J. Appl. Math. Mech. 24 (1961), 1286–1303 (In English. Russian original.); translation from Prikl. Mat. Mekh. 24 (1960), 852–864.CrossRefMATHGoogle Scholar
  7. [7]
    M. Benzi, G. H. Golub, J. Liesen: Numerical solution of saddle point problems. Acta Numerica 14 (2005), 1–137.MathSciNetCrossRefMATHGoogle Scholar
  8. [8]
    D. Boffi, F. Brezzi, M. Fortin: Mixed Finite Element Methods and Applications. Springer Series in Computational Mathematics 44, Springer, Berlin, 2013.Google Scholar
  9. [9]
    Decovalex 2019 project, Task G: EDZ evolution in sparsely fractured competent rock. http://decovalex.org/task-g.html.Google Scholar
  10. [10]
    H. C. Elman, D. J. Silvester, A. J. Wathen: Finite Elements and Fast Iterative Solvers: with Applications in Incompressible Fluid Dynamics. Numerical Mathematics and Scientific Computation, Oxford University Press, Oxford, 2014.Google Scholar
  11. [11]
    H. H. Gerke, M. T. Van Genuchten: A dual-porosity model for simulating the preferential movement of water and solutes in structured porous media. Water Resources Research 29 (1993), 305–319.CrossRefGoogle Scholar
  12. [12]
    P. R. Halmos: Finite-Dimensional Vector Spaces, The University Series in Undergraduate Mathematics, D. van Nostrand Company, Princeton, 1958.MATHGoogle Scholar
  13. [13]
    V. E. Henson, U. M. Yang: BoomerAMG: a parallel algebraic multigrid solver and preconditioner. Appl. Numer. Math. 41 (2002), 155–177.MathSciNetCrossRefMATHGoogle Scholar
  14. [14]
    Q. Hong, J. Kraus: Parameter-robust stability of classical three-field formulation of Biot’s consolidation model. Available at arXiv:1706.00724 (2017), 20 pages.Google Scholar
  15. [15]
    S. H. S. Joodat, K. B. Nakshatrala, R. Ballarini: Modeling flow in porous media with double porosity/permeability: A stabilized mixed formulation, error analysis, and numerical solutions. Available at arXiv:1705.08883 (2017), 49 pages.Google Scholar
  16. [16]
    A. E. Kolesov, P. N. Vabishchevich: Splitting schemes with respect to physical processes for double-porosity poroelasticity problems. Russ. J. Numer. Anal. Math. Model. 32 (2017), 99–113.MathSciNetCrossRefMATHGoogle Scholar
  17. [17]
    J. Kraus, M. Lymbery, S. Margenov: Auxiliary space multigrid method based on additive Schur complement approximation. Numer. Linear Algebra Appl. 22 (2015), 965–986.MathSciNetCrossRefMATHGoogle Scholar
  18. [18]
    J. Kraus, S. Margenov: Robust Algebraic Multilevel Methods and Algorithms. Radon Series on Computational and Applied Mathematics 5, Walter de Gruyter, Berlin, 2009.Google Scholar
  19. [19]
    J. M. Nordbotten, T. Rahman, S. I. Repin, J. Valdman: A Posteriori error estimates for approximate solutions of the Barenblatt-Biot poroelastic model. Comput. Methods Appl. Math. 10 (2010), 302–314.MathSciNetCrossRefMATHGoogle Scholar
  20. [20]
    C. Rodrigo, X. Hu, P. Ohm, J. H. Adler, F. J. Gaspar, L. Zikatanov: New stabilized discretizations for poroelasticity and the Stokes’ equations. Available at arXiv:1706.05169 (2017), 20 pages.Google Scholar
  21. [21]
    J. E. Warren, P. J. Root: The behavior of naturally fractured reservoirs. SPE J. 3 (1963), 245–255.CrossRefGoogle Scholar

Copyright information

© Institute of Mathematics of the Academy of Sciences of the Czech Republic, Praha, Czech Republic 2017

Authors and Affiliations

  1. 1.Institute of GeonicsCzech Academy of SciencesOstravaCzech Republic

Personalised recommendations