Advertisement

Numerical Investigation on the Fixed-Stress Splitting Scheme for Biot’s Equations: Optimality of the Tuning Parameter

  • Jakub W. Both
  • Uwe Köcher
Conference paper
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 126)

Abstract

We study the numerical solution of the quasi-static linear Biot equations solved iteratively by the fixed-stress splitting scheme. In each iteration the mechanical and flow problems are decoupled, where the flow problem is solved by keeping an artificial mean stress fixed. This introduces a numerical tuning parameter which can be optimized. We investigate numerically the optimality of the parameter and compare our results with physically and mathematically motivated values from the literature, which commonly only depend on mechanical material parameters. We demonstrate, that the optimal value of the tuning parameter is also affected by the boundary conditions and material parameters associated to the fluid flow problem suggesting the need for the integration of those in further mathematical analyses optimizing the tuning parameter.

Notes

Acknowledgements

The research contribution of the second author was partially supported by E.ON Stipendienfonds (Germany) under the grant T0087 29890 17 while visiting University of Bergen.

References

  1. 1.
    M. Bause, Iterative coupling of mixed and discontinuous Galerkin methods for poroelasticity, in Numerical Mathematics and Advanced Applications – ENUMATH 2017 (Springer, Cham, 2018) pp. ??-??Google Scholar
  2. 2.
    M. Bause, F. Radu, U. Köcher, Space-time finite element approximation of the Biot poroelasticity system with iterative coupling. Comput. Methods Appl. Mech. Eng. 320, 745–768 (2017)MathSciNetCrossRefGoogle Scholar
  3. 3.
    M.A. Biot, General theory of three-dimensional consolidation. J. Appl. Phys. 12, 155–164 (1941)CrossRefGoogle Scholar
  4. 4.
    J.W. Both, M. Borregales, J.M. Nordbotten, K. Kumar, F.A. Radu. Robust fixed stress splitting for Biot’s equations in heterogeneous media. Appl. Math. Lett. 68, 101–108 (2017)MathSciNetCrossRefGoogle Scholar
  5. 5.
    J. Kim, H.A. Tchelepi, R. Juanes, Stability and convergence of sequential methods for coupled flow and geomechanics: fixed-stress and fixed-strain splits. Comput. Methods Appl. Mech. Eng. 200, 1591–1606 (2011)MathSciNetCrossRefGoogle Scholar
  6. 6.
    F. List, F.A. Radu, A study on iterative methods for solving Richards’ equation. Comput. Geosci. 20, 341–353 (2016)MathSciNetCrossRefGoogle Scholar
  7. 7.
    A. Mikelić, M.F. Wheeler, Convergence of iterative coupling for coupled flow and geomechanics. Comput. Geosci. 17, 451–461 (2013)MathSciNetCrossRefGoogle Scholar
  8. 8.
    A. Settari, F.M. Mourits, A coupled reservoir and geomechanical simulation system. Soc. Pet. Eng. J. 3, 219–226 (1998)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.University of BergenBergenNorway
  2. 2.Helmut-Schmidt-UniversityUniversity of the Federal Armed Forces HamburgHamburgGermany

Personalised recommendations