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Computational Methods in Applied Mathematics

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A Hybrid High-Order Discretization Method for Nonlinear Poroelasticity

Michele BottiORCID iD: https://orcid.org/0000-0003-1171-2874 / Daniele A. Di PietroORCID iD: https://orcid.org/0000-0003-0959-8830 / Pierre Sochala
Published Online: 2019-06-19 | DOI: https://doi.org/10.1515/cmam-2018-0142

Abstract

In this work, we construct and analyze a nonconforming high-order discretization method for the quasi-static single-phase nonlinear poroelasticity problem describing Darcean flow in a deformable porous medium saturated by a slightly compressible fluid. The nonlinear elasticity operator is discretized using a Hybrid High-Order method, while the Darcy operator relies on a Symmetric Weighted Interior Penalty discontinuous Galerkin scheme. The method is valid in two and three space dimensions, delivers an inf-sup stable discretization on general meshes including polyhedral elements and nonmatching interfaces, supports arbitrary approximation orders, and has a reduced cost thanks to the possibility of statically condensing a large subset of the unknowns for linearized versions of the problem. Moreover, the proposed construction can handle both nonzero and vanishing specific storage coefficients.

Keywords: Nonlinear Poroelasticity; Nonlinear Biot Problem; Korn’s Inequality; Hybrid High-Order Methods; Discontinuous Galerkin Methods; Polyhedral Meshes

MSC 2010: 65N08; 65N30; 76S05

References

  • [1]

    C. Amrouche, P. G. Ciarlet, L. Gratie and S. Kesavan, On the characterizations of matrix fields as linearized strain tensor fields, J. Math. Pures Appl. (9) 86 (2006), no. 2, 116–132. CrossrefGoogle Scholar

  • [2]

    M. A. Barrientos, G. N. Gatica and E. P. Stephan, A mixed finite element method for nonlinear elasticity: Two-fold saddle point approach and a-posteriori error estimate, Numer. Math. 91 (2002), no. 2, 197–222. CrossrefGoogle Scholar

  • [3]

    E. Bemer, M. Boutéca, O. Vincké, N. Hoteit and O. Ozanam, Poromechanics: From linear to nonlinear poroelasticity and poroviscoelasticity, Oil & Gas Sci. Technol. Rev. IFP 56 (2001), no. 6, 531–544. CrossrefGoogle Scholar

  • [4]

    M. A. Biot, General theory of threedimensional consolidation, J. Appl. Phys. 12 (1941), no. 2, 155–164. CrossrefGoogle Scholar

  • [5]

    M. A. Biot, Nonlinear and semilinear rheology of porous solids, J. Geoph. Res. 78 (1973), no. 23, 4924–4937. CrossrefGoogle Scholar

  • [6]

    D. Boffi, M. Botti and D. A. Di Pietro, A nonconforming high-order method for the Biot problem on general meshes, SIAM J. Sci. Comput. 38 (2016), no. 3, A1508–A1537. CrossrefGoogle Scholar

  • [7]

    D. Boffi, F. Brezzi And M. Fortin, Mixed Finite Element Methods and Applications, Springer Ser. Comput. Math. 44, Springer, Heidelberg, 2013. Google Scholar

  • [8]

    L. Botti, D. A. Di Pietro and J. Droniou, A Hybrid High-Order discretisation of the Brinkman problem robust in the Darcy and Stokes limits, Comput. Methods Appl. Mech. Engrg. 341 (2018), 278–310. CrossrefGoogle Scholar

  • [9]

    M. Botti, D. A. Di Pietro and P. Sochala, A Hybrid High-Order method for nonlinear elasticity, SIAM J. Numer. Anal. 55 (2017), no. 6, 2687–2717. CrossrefWeb of ScienceGoogle Scholar

  • [10]

    M. Botti, D. A. Di Pietro and P. Sochala, A nonconforming high-order method for nonlinear poroelasticity, Finite Volumes for Complex Applications VIII—Hyperbolic, Elliptic and Parabolic Problems, Springer Proc. Math. Stat. 200, Springer, Cham (2017), 537–545. Google Scholar

  • [11]

    M. Botti and R. Riedlbeck, Equilibrated stress tensor reconstruction and a posteriori error estimation for nonlinear elasticity, Comput. Methods Appl. Math. (2018), 10.1515/cmam-2018-0012. Google Scholar

  • [12]

    M. Cervera, M. Chiumenti and R. Codina, Mixed stabilized finite element methods in nonlinear solid mechanics Part II: Strain localization, Comput. Methods Appl. Mech. Engrg. 199 (2010), no. 37–40, 2571–2589. CrossrefGoogle Scholar

  • [13]

    O. Coussy, Poromechanics, J. Wiley & Sons, New York, 2004. Google Scholar

  • [14]

    K. Deimling, Nonlinear Functional Analysis, Springer, Berlin, 1985. Google Scholar

  • [15]

    D. A. Di Pietro and J. Droniou, A Hybrid High-Order method for Leray–Lions elliptic equations on general meshes, Math. Comp. 86 (2017), no. 307, 2159–2191. Google Scholar

  • [16]

    D. A. Di Pietro, J. Droniou and A. Ern, A discontinuous-skeletal method for advection-diffusion-reaction on general meshes, SIAM J. Numer. Anal. 53 (2015), no. 5, 2135–2157. CrossrefWeb of ScienceGoogle Scholar

  • [17]

    D. A. Di Pietro, J. Droniou and G. Manzini, Discontinuous skeletal gradient discretisation methods on polytopal meshes, J. Comput. Phys. 355 (2018), 397–425. CrossrefGoogle Scholar

  • [18]

    D. A. Di Pietro and A. Ern, Discrete functional analysis tools for discontinuous Galerkin methods with application to the incompressible Navier–Stokes equations, Math. Comp. 79 (2010), no. 271, 1303–1330. CrossrefGoogle Scholar

  • [19]

    D. A. Di Pietro and A. Ern, Mathematical Aspects of Discontinuous Galerkin Methods, Math. Appl. (Berlin) 69, Springer, Heidelberg, 2012. Google Scholar

  • [20]

    D. A. Di Pietro and A. Ern, A Hybrid High-Order locking-free method for linear elasticity on general meshes, Comput. Methods Appl. Mech. Engrg. 283 (2015), 1–21. CrossrefGoogle Scholar

  • [21]

    D. A. Di Pietro, A. Ern and J.-L. Guermond, Discontinuous Galerkin methods for anisotropic semidefinite diffusion with advection, SIAM J. Numer. Anal. 46 (2008), no. 2, 805–831. Web of ScienceCrossrefGoogle Scholar

  • [22]

    D. A. Di Pietro and R. Tittarelli, An introduction to Hybrid High-Order methods, Numerical Methods for PDEs, SEMA SIMAI Springer Ser. 15, Springer, Cham (2018), 75–128. Google Scholar

  • [23]

    J. Droniou, R. Eymard, T. Gallouët, C. Guichard and R. Herbin, The Gradient Discretisation Method, Math. Appl. (Berlin) 82, Springer, Cham, 2018. Google Scholar

  • [24]

    J. Droniou and B. P. Lamichhane, Gradient schemes for linear and non-linear elasticity equations, Numer. Math. 129 (2015), no. 2, 251–277. Web of ScienceCrossrefGoogle Scholar

  • [25]

    G. N. Gatica and E. P. Stephan, A mixed-FEM formulation for nonlinear incompressible elasticity in the plane, Numer. Methods Partial Differential Equations 18 (2002), no. 1, 105–128. CrossrefGoogle Scholar

  • [26]

    P. Grisvard, Elliptic Problems in Nonsmooth Domains, Monogr. Stud. Math. 24, Pitman, Boston, 1985. Google Scholar

  • [27]

    J. B. Haga, H. Osnes and H. P. Langtangen, On the causes of pressure oscillations in low-permeable and low-compressible porous media, Int. J. Numer. Anal. Methods Geomech. 36 (2012), no. 12, 1507–1522. CrossrefWeb of ScienceGoogle Scholar

  • [28]

    L. Hu, P. H. Winterfield, P. Fakcharoenphol and Y. S. Wu, A novel fully-coupled flow and geomechanics model in enhanced geothermal reservoirs, J. Pet. Sci. Eng. 107 (2013), 1–11. Web of ScienceCrossrefGoogle Scholar

  • [29]

    M. D. Jin and L. Zoback, Fully coupled nonlinear fluid flow and poroelasticity in arbitrarily fractured porous media: A hybrid-dimensional computational model, J. Geophys. Res. Solid Earth 122 (2017), 7626–7658. Web of ScienceCrossrefGoogle Scholar

  • [30]

    V. Maz’ya and J. Rossmann, Elliptic Equations in Polyhedral Domains, Math. Surveys Monogr. 162, American Mathematical Society, Providence, 2010. Google Scholar

  • [31]

    S. E. Minkoff, C. M. Stone, S. Bryant, M. Peszynsak and M. F. Wheeler, Coupled fluid flow and geomechanical deformation modeling, J. Pet. Sci. Eng. 38 (2003), 37–56. CrossrefGoogle Scholar

  • [32]

    M. A. Murad and A. F. D. Loula, On stability and convergence of finite element approximations of Biot’s consolidation problem, Internat. J. Numer. Methods Engrg. 37 (1994), no. 4, 645–667. CrossrefGoogle Scholar

  • [33]

    J. Nec̆as, Introduction to the theory of nonlinear elliptic equations, John Wiley & Sons, Chichester, 1986. Google Scholar

  • [34]

    P. J. Phillips and M. F. Wheeler, Overcoming the problem of locking in linear elasticity and poroelasticity: An heuristic approach, Comput. Geosci. 13 (2009), 5–12. CrossrefGoogle Scholar

  • [35]

    C. Rodrigo, F. J. Gaspar, X. Hu and L. T. Zikatanov, Stability and monotonicity for some discretizations of the Biot’s consolidation model, Comput. Methods Appl. Mech. Engrg. 298 (2016), 183–204. CrossrefGoogle Scholar

  • [36]

    D. Schötzau and C. Schwab, Time discretization of parabolic problems by the hp-version of the discontinuous Galerkin finite element method, SIAM J. Numer. Anal. 38 (2000), no. 3, 837–875. CrossrefGoogle Scholar

  • [37]

    R. E. Showalter, Diffusion in poro-elastic media, J. Math. Anal. Appl. 251 (2000), no. 1, 310–340. CrossrefGoogle Scholar

  • [38]

    I. Smears, Robust and efficient preconditioners for the discontinuous Galerkin time-stepping method, IMA J. Numer. Anal. 37 (2017), no. 4, 1961–1985. Web of ScienceGoogle Scholar

  • [39]

    K. Terzaghi, Theoretical Soil Mechanics, Wiley, New York, 1943. Google Scholar

  • [40]

    A. Ženíšek, The existence and uniqueness theorem in Biot’s consolidation theory, Apl. Mat. 29 (1984), no. 3, 194–211. Google Scholar

About the article

Received: 2018-05-07

Revised: 2019-05-24

Accepted: 2019-05-31

Published Online: 2019-06-19


Funding Source: Agence Nationale de la Recherche

Award identifier / Grant number: ANR-10-LABX-20

Award identifier / Grant number: ANR-15-CE40-0005

This work was partially funded by the Bureau de Recherches Géologiques et Minières. The work of M. Botti was additionally partially supported by Labex NUMEV (ANR-10-LABX-20) ref. 2014-2-006. The work of D. A. Di Pietro was additionally partially supported by project HHOMM (ANR-15-CE40-0005).


Citation Information: Computational Methods in Applied Mathematics, ISSN (Online) 1609-9389, ISSN (Print) 1609-4840, DOI: https://doi.org/10.1515/cmam-2018-0142.

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