Applications of Mathematics, Vol. 62, No. 6, pp. 561-577, 2017


Algebraic preconditioning for Biot-Barenblatt poroelastic systems

Radim Blaheta, Tomáš Luber

Received July 4, 2017.   First published December 7, 2017.

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
Classification MSC:  65F08
DOI:  10.21136/AM.2017.0179-17


References:
[1] D. N. Arnold, R. S. Falk, R. Winther: Preconditioning in $H (div)$ and applications. Math. Comput. 66 (1997), 957-984. DOI 10.1090/S0025-5718-97-00826-0 | MR 1401938 | Zbl 0870.65112
[2] O. Axelsson, R. Blaheta: Preconditioning of matrices partitioned in $2\times 2$ block form: eigenvalue estimates and Schwarz DD for mixed FEM. Numer. Linear Algebra Appl. 17 (2010), 787-810. DOI 10.1002/nla.728 | MR 2722647 | Zbl 1240.65090
[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. DOI 10.1007/s00791-013-0209-0 | MR 3148142
[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), 3-14. DOI 10.1007/978-3-319-26520-9 1 | MR 3480807
[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. DOI 10.1029/92wr02746
[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. DOI 10.1016/0021-8928(60)90107-6 | Zbl 0104.21702
[7] M. Benzi, G. H. Golub, J. Liesen: Numerical solution of saddle point problems. Acta Numerica 14 (2005), 1-137. DOI 10.1017/S0962492904000212 | MR 2168342 | Zbl 1115.65034
[8] D. Boffi, F. Brezzi, M. Fortin: Mixed Finite Element Methods and Applications. Springer Series in Computational Mathematics 44, Springer, Berlin (2013). DOI 10.1007/978-3-642-36519-5 | MR 3097958 | Zbl 1277.65092
[9] Decovalex 2019 project, Task G: EDZ evolution in sparsely fractured competent rock. http://decovalex.org/task-g.html.
[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). DOI 10.1093/acprof:oso/9780199678792.001.0001 | MR 3235759 | Zbl 1304.76002
[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. DOI 10.1029/92wr02339
[12] P. R. Halmos: Finite-Dimensional Vector Spaces. The University Series in Undergraduate Mathematics, D. van Nostrand Company, Princeton (1958). DOI 10.1007/978-1-4612-6387-6 | MR 0089819 | Zbl 0107.01404
[13] V. E. Henson, U. M. Yang: BoomerAMG: a parallel algebraic multigrid solver and preconditioner. Appl. Numer. Math. 41 (2002), 155-177. DOI 10.1016/S0168-9274(01)00115-5 | MR 1908755 | Zbl 0995.65128
[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.
[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.
[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. DOI 10.1515/rnam-2017-0009 | MR 3641710 | Zbl 06722604
[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. DOI 10.1002/nla.1959 | MR 3426324 | Zbl 06604518
[18] J. Kraus, S. Margenov: Robust Algebraic Multilevel Methods and Algorithms. Radon Series on Computational and Applied Mathematics 5, Walter de Gruyter, Berlin (2009). DOI 10.1515/9783110214833 | MR 2574100 | Zbl 1184.65113
[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. DOI 10.2478/cmam-2010-0017 | MR 2770296 | Zbl 1283.65100
[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.
[21] J. E. Warren, P. J. Root: The behavior of naturally fractured reservoirs. SPE J. 3 (1963), 245-255. DOI 10.2118/426-PA

Affiliations:   Radim Blaheta, Tomáš Luber, Institute of Geonics, Czech Academy of Sciences, Studenstká 1768, 708 00 Ostrava, Czech Republic, e-mail: radim.blaheta@ugn.cas.cz, tomas.luber@ugn.cas.cz

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