Advertisement

Poroelasticity with Deformation Dependent Permeability

  • Sílvia BarbeiroEmail author
Conference paper
Part of the Mathematics in Industry book series (MATHINDUSTRY, volume 30)

Abstract

The poroelasticity theory that was originally developed in the context of geophysical applications has been successfully used to model the mechanical behavior of fluid-saturated living bone tissue. In this paper we focus on the numerical solution of the coupled fluid flow and mechanics in Biot’s consolidation model of poroelasticity. The method combines mixed finite elements for Darcy flow and Galerkin finite elements for elasticity. The permeability tensor in the model is allowed to be a nonlinear function on the deformation, since this influence has relevance in the case of biological tissues like bone. We deal with the nonlinear term by considering a semi-implicit in time scheme. We provide the a priori error estimates for the numerical solution of the fully discretized model. For efficiency, we also explore an operator splitting strategy where the flow problem is solved before the mechanical problem, in an iterative process.

Notes

Acknowledgements

This work was partially supported by the Centre for Mathematics of the University of Coimbra—UID/MAT/00324/2019, funded by the Portuguese Government through FCT/MEC and co-funded by the European Regional Development Fund through the Partnership Agreement PT2020.

References

  1. 1.
    Barbeiro, S., Wheeler, M.F.: A priori error estimates for the numerical solution of a coupled geomechanics and reservoir flow model with stress-dependent permeability. Comput. Geosci. 14, 755–768 (2010)CrossRefGoogle Scholar
  2. 2.
    Berger, L., Bordas, R., Kay, D., Tavener, S.: A stabilized finite element method for finite-strain three-field poroelasticity. Comput. Mech. 60(1), 51–68 (2017)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Biot, M.A.: General theory of three-dimensional consolidation. J. Appl. Phys. 12, 155–164 (1941)CrossRefGoogle Scholar
  4. 4.
    Brezzi, F., Fortin, M.: Mixed and Hybrid Finite Element Methods. Springer Series in Computational Mathematics, vol. 15, p. 350. Springer, New York (1991)Google Scholar
  5. 5.
    Cardoso, l., Fritton, S.P., Gailani, G., Benalla, M., Cowin, S.C.: A review of recent advances in the assessment of bone porosity, permeability, and interstitial fluid flow. J. Biomech. 46(2), 253–265 (2013)Google Scholar
  6. 6.
    Cowin, S.C.: Bone poroelasticity. J. Biomech. 32, 217–238 (1999)CrossRefGoogle Scholar
  7. 7.
    Cowin, S.C., Sadegh, A.M.: Non-interacting modes for stress, strain and energy in anisotropic hard tissue. J. Biomech. 24(9), 859–67 (1991)CrossRefGoogle Scholar
  8. 8.
    Dana, S., Wheeler, M.F.: Convergence analysis of two-grid fixed stress split iterative scheme for coupled flow and deformation in heterogeneous poroelastic media. Comput. Methods Appl. Mech. Eng. 341, 788–806 (2018)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Goulet, G.C., Cooper, D.M.L., Coombe, D., Zernicke, R.F.: Validation and application of iterative coupling to poroelastic problems in bone fluid flow. Bull. App. Mech. 5(17), 6–17 (2009)Google Scholar
  10. 10.
    Kim, J., Tchelepi, H., Juanes, R.: Stability, Accuracy and efficiency of sequential methods for coupled flow and geomechanics. SPE J. 16(02), (2009)Google Scholar
  11. 11.
    Kowalczyk, P., Kleiber, M.: Modelling and numerical analysis of stresses and strains in the human lung including tissue-gas interaction. Eur. J. Mech. A. Solids 13(3), 367–393 (1994)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Lai, W., Mow, V.: Drag-induced compression of articular cartilage during a permeation experiment. Biorheology 17(1–2), 111 (1980)CrossRefGoogle Scholar
  13. 13.
    Nedelec, J.C.: Mixed finite elements in \(\mathbb {R}^3\). Numer. Math. 35(3), 315–341 (1980)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Phillips, P.J., Wheeler, M.F.: A coupling of mixed and continuous Galerkin finite element methods for poroelasticity II: The discrete-in-time case. Comput. Geosci. 11(4), 145–158 (2007)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Raghavan, R., Chin, L.Y.: Productivity changes in reservoirs with stress-dependent permeability. SPE Reserv. Eval. Eng. 7(4), 308–315 (2004)CrossRefGoogle Scholar
  16. 16.
    Raviart, P.A., Thomas, J.M.: A mixed finite element method for second order elliptic problems. Lect. Notes Math. Springer 606, 292–315 (1977)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Ryser, M.D., Komarova, S.V., Nigam, N.: The cellular dynamics of bone remodeling: a mathematical model. SIAM J. Appl. Math. 70(6), 1899–1921 (2010)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Wheeler, M.F., Gai, X.: Iteratively coupled mixed and Galerkin finite element methods for poro-elasticity. Numer. Methods Partial Differ. Eq. 23, 785–797 (2007)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.CMUC, Department of MathematicsUniversity of CoimbraCoimbraPortugal

Personalised recommendations