Abstract
We consider a multirate iterative scheme for the quasi-static Biot equations modelling the coupled flow and geomechanics in a porous medium. The iterative scheme is based on undrained splitting where the flow and mechanics equations are decoupled with the mechanics solve followed by the pressure solve. The multirate scheme proposed here uses different time steps for the two equations, that is, uses q flow steps for each coarse mechanics step and may be interpreted as using a regularization parameter for the mechanics equation. We prove the convergence of the scheme and the proof reveals the appropriate regularization parameter and also the effect of the number of flow steps within coarse mechanics step on the convergence rate.
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References
M.A. Biot, General theory of three-dimensional consolidation. J. Appl. Phys. 12 (2), 155–164 (1941)
X. Gai, R.H. Dean, M.F. Wheeler, R. Liu, Coupled geomechanical and reservoir modeling on parallel computers, in The SPE Reservoir Simulation Symposium, Houston, 3–5 Feb 2003
X. Gai, S. Sun, M.F. Wheeler, H. Klie, A timestepping scheme for coupled reservoir flow and geomechanics on nonmatching grids, in SPE Annual Technical Conference and Exhibition, Dallas (2005). SPE97054
V. Girault, K. Kumar, M.F. Wheeler, Convergence of iterative coupling of geomechanics with flow in a fractured poroelastic medium. Ices report 15-05, Institute for Computational Engineering and Sciences, The University of Texas at Austin, Austin, 2015
J. Kim, H.A. Tchelepi, R. Juanes, Stability, accuracy, and efficiency of sequential methods for coupled flow and geomechanics, in The SPE Reservoir Simulation Symposium, Houston, 2–4 Feb 2009. SPE119084
A. Mikelić, M.F. Wheeler, Convergence of iterative coupling for coupled flow and geomechanics. Comput. Geosci. 17, 455–461 (2013)
A. Mikelić, B. Wang, M.F. Wheeler, Numerical convergence study of iterative coupling for coupled flow and geomechanics. Comput. Geosci. 18, 325–341 (2014)
P.J. Phillips, M.F. Wheeler, A coupling of mixed and continuous Galerkin finite element methods for poroelasticity. I. The continuous in time case. Comput. Geosci. 11 (2), 131–144 (2007)
I.S. Pop, F. Radu, P. Knabner, Mixed finite elements for the Richards’ equation: linearization procedure. J. Comput. Appl. Math. 168 (1–2), 365–373 (2004)
F.A. Radu, J.M. Nordbotten, I.S. Pop, K. Kumar, A robust linearization scheme for finite volume based discretizations for simulation of two-phase flow in porous media. J. Comput. Appl. Math. 289, 134–141 (2015)
A. Settari, F.M. Mourits, Coupling of geomechanics and reservoir simulation models. in Computer Methods and Advances in Geomechanics, ed. by H.J. Siriwardane, M.M. Zema (Balkema, Rotterdam, 1994), pp. 2151–2158
L. Shan, H. Zheng, W.J. Layton, A decoupling method with different subdomain time steps for the nonstationary Stokes-Darcy model. Numer. Methods Part. Differ. Equ. 29 (2), 549–583 (2013)
R.E. Showalter, Diffusion in poro-elastic media. J. Math. Anal. Appl. 251 (1), 310–340 (2000)
M.F. Wheeler, I. Yotov, A multipoint flux mixed finite element method. SIAM J. Numer. Anal. 44, 2082–2106 (2006)
Acknowledgements
K. Kumar acknowledges the support of Statoil-UiB Akademia grant that allowed him to travel to ENUMATH conference at Ankara and CSM, ICES, UT Austin for the hospitality where part of the work was completed.
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Kumar, K., Almani, T., Singh, G., Wheeler, M.F. (2016). Multirate Undrained Splitting for Coupled Flow and Geomechanics in Porous Media. In: Karasözen, B., Manguoğlu, M., Tezer-Sezgin, M., Göktepe, S., Uğur, Ö. (eds) Numerical Mathematics and Advanced Applications ENUMATH 2015. Lecture Notes in Computational Science and Engineering, vol 112. Springer, Cham. https://doi.org/10.1007/978-3-319-39929-4_41
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DOI: https://doi.org/10.1007/978-3-319-39929-4_41
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