Skip to content
Licensed Unlicensed Requires Authentication Published by De Gruyter June 19, 2019

A Hybrid High-Order Discretization Method for Nonlinear Poroelasticity

Michele Botti ORCID logo EMAIL logo , Daniele A. Di Pietro ORCID logo and Pierre Sochala


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.

MSC 2010: 65N08; 65N30; 76S05

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

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

Funding statement: 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).


[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. 10.1016/j.matpur.2006.04.004Search in Google 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. 10.1007/s002110100337Search in Google 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. 10.2516/ogst:2001043Search in Google Scholar

[4] M. A. Biot, General theory of threedimensional consolidation, J. Appl. Phys. 12 (1941), no. 2, 155–164. 10.1063/1.1712886Search in Google Scholar

[5] M. A. Biot, Nonlinear and semilinear rheology of porous solids, J. Geoph. Res. 78 (1973), no. 23, 4924–4937. 10.1029/JB078i023p04924Search in Google 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. 10.1137/15M1025505Search in Google Scholar

[7] D. Boffi, F. Brezzi And M. Fortin, Mixed Finite Element Methods and Applications, Springer Ser. Comput. Math. 44, Springer, Heidelberg, 2013. 10.1007/978-3-642-36519-5Search in 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. 10.1016/j.cma.2018.07.004Search in Google 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. 10.1137/16M1105943Search in Google 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. 10.1007/978-3-319-57394-6_56Search in 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. 10.1515/cmam-2018-0012Search in 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. 10.1016/j.cma.2010.04.005Search in Google Scholar

[13] O. Coussy, Poromechanics, J. Wiley & Sons, New York, 2004. 10.1002/0470092718Search in Google Scholar

[14] K. Deimling, Nonlinear Functional Analysis, Springer, Berlin, 1985. 10.1007/978-3-662-00547-7Search in 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. 10.1090/mcom/3180Search in 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. 10.1137/140993971Search in Google 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. 10.1016/ in Google 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. 10.1090/S0025-5718-10-02333-1Search in Google Scholar

[19] D. A. Di Pietro and A. Ern, Mathematical Aspects of Discontinuous Galerkin Methods, Math. Appl. (Berlin) 69, Springer, Heidelberg, 2012. 10.1007/978-3-642-22980-0Search in 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. 10.1016/j.cma.2014.09.009Search in Google 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. 10.1137/060676106Search in Google 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. 10.1007/978-3-319-94676-4_4Search in 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. 10.1007/978-3-319-79042-8Search in 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. 10.1007/s00211-014-0636-ySearch in Google 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. 10.1002/num.1046Search in Google Scholar

[26] P. Grisvard, Elliptic Problems in Nonsmooth Domains, Monogr. Stud. Math. 24, Pitman, Boston, 1985. Search in 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. 10.1002/nag.1062Search in Google 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. 10.1016/j.petrol.2013.04.005Search in Google 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. 10.1002/2017JB014892Search in Google Scholar

[30] V. Maz’ya and J. Rossmann, Elliptic Equations in Polyhedral Domains, Math. Surveys Monogr. 162, American Mathematical Society, Providence, 2010. 10.1090/surv/162Search in 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. 10.1016/S0920-4105(03)00021-4Search in Google 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. 10.1002/nme.1620370407Search in Google Scholar

[33] J. Nec̆as, Introduction to the theory of nonlinear elliptic equations, John Wiley & Sons, Chichester, 1986. Search in 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. 10.1007/s10596-008-9114-xSearch in Google 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. 10.1016/j.cma.2015.09.019Search in Google 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. 10.1137/S0036142999352394Search in Google Scholar

[37] R. E. Showalter, Diffusion in poro-elastic media, J. Math. Anal. Appl. 251 (2000), no. 1, 310–340. 10.1006/jmaa.2000.7048Search in Google 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. 10.1093/imanum/drw050Search in Google Scholar

[39] K. Terzaghi, Theoretical Soil Mechanics, Wiley, New York, 1943. 10.1002/9780470172766Search in Google Scholar

[40] A. Ženíšek, The existence and uniqueness theorem in Biot’s consolidation theory, Apl. Mat. 29 (1984), no. 3, 194–211. 10.21136/AM.1984.104085Search in Google Scholar

Received: 2018-05-07
Revised: 2019-05-24
Accepted: 2019-05-31
Published Online: 2019-06-19
Published in Print: 2020-04-01

© 2020 Walter de Gruyter GmbH, Berlin/Boston

Downloaded on 2.12.2022 from
Scroll Up Arrow