×
Borregales, Manuel; Kumar, Kundan; Radu, Florin Adrian; Rodrigo, Carmen; Gaspar, Francisco José

A partially parallel-in-time fixed-stress splitting method for Biot’s consolidation model. (English) Zbl 1442.65250

Comput. Math. Appl. 77, No. 6, 1466-1478 (2019).
Summary: In this work, we study the parallel-in-time iterative solution of coupled flow and geomechanics in porous media, modelled by a two-field formulation of Biot’s equations. In particular, we propose a new version of the fixed-stress splitting method, which has been widely used as solution method of these problems. This new approach forgets about the sequential nature of the temporal variable and considers the time direction as a further direction for parallelization. The method is partially parallel-in-time. We present a rigorous convergence analysis of the method and numerical experiments to demonstrate the robust behaviour of the algorithm.

Cited in 21 Documents

MSC:

65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
65Y05 Parallel numerical computation
76S05 Flows in porous media; filtration; seepage
86A05 Hydrology, hydrography, oceanography

Keywords:

iterative fixed-stress splitting scheme; parallel-in-time; Biot’s model; poromechanics; finite elements; convergence analysis

Software:

deal.ii
Cite Review PDF
Full Text: DOI arXiv Link

References:

[1] Terzaghi, K., Theoretical Soil Mechanics (1943), Wiley: Wiley New York
[2] Biot, M. A., General theory of three-dimensional consolidation, J. Appl. Phys., 12, 2, 155-164 (1941) · JFM 67.0837.01
[3] Biot, M. A., Theory of elasticity and consolidation for a porous anisotropic solid, J. Appl. Phys., 26, 2, 182-185 (1955) · Zbl 0067.23603
[4] Showalter, R., Diffusion in poro-elastic media, J. Math. Anal. Appl., 251, 1, 310-340 (2000) · Zbl 0979.74018
[5] Phillips, P.; Wheeler, M., 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) · Zbl 1117.74015
[6] Bergamaschi, L.; Ferronato, M.; Gambolati, G., Novel preconditioners for the iterative solution to FE-discretized coupled consolidation equations, Comput. Methods Appl. Mech. Eng., 196, 25, 2647-2656 (2007) · Zbl 1173.76330
[7] Ferronato, M.; Bergamaschi, L.; Gambolati, G., Performance and robustness of block constraint preconditioners in finite element coupled consolidation problems, Internat. J. Numer. Methods Engrg., 81, 381-402 (2010) · Zbl 1183.74271
[8] Gaspar, F. J.; Lisbona, F. J.; Oosterlee, C.; Wienands, R., A systematic comparison of coupled and distributive smoothing in multigrid for the poroelasticity system, Numer. Linear Algebra Appl., 11, 93-113 (2004) · Zbl 1164.65344
[9] Luo, P.; Rodrigo, C.; Gaspar, F. J.; Oosterlee, C. W., On an Uzawa smoother in multigrid for poroelasticity equations, Numer. Linear Algebra Appl., 24, 1 (2017) · Zbl 1424.76041
[10] 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, Texas, 2009 SPE 119084. http://dx.doi.org/10.2118/119084-PA; 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, Texas, 2009 SPE 119084. http://dx.doi.org/10.2118/119084-PA
[11] Kim, J.; Tchelepi, H. A.; Juanes, R., Stability and convergence of sequential methods for coupled flow and geomechanics: fixed-stress and fixed-strain splits, Comput. Methods Appl. Mech. Eng., 200, 13-16, 1591-1606 (2011) · Zbl 1228.74101
[12] Kim, J.; Tchelepi, H. A.; Juanes, R., Stability and convergence of sequential methods for coupled flow and geomechanics: Drained and undrained splits, Comput. Methods Appl. Mech. Eng., 200, 23-24, 2094-2116 (2011) · Zbl 1228.74106
[13] Mikelić, A.; Wheeler, M. F., Convergence of iterative coupling for coupled flow and geomechanics, Comput. Geosci., 17, 455-461 (2013) · Zbl 1392.35235
[14] Both, J. W.; Borregales, M.; Nordbotten, J. M.; Kumar, K.; Radu, F. A., Robust fixed stress splitting for Biot’s equations in heterogeneous media, Appl. Math. Lett., 68, 101-108 (2017) · Zbl 1383.74025
[15] Almani, T.; Kumar, K.; Dogru, A.; Singh, G.; Wheeler, M. F., Convergence analysis of multirate fixed-stress split iterative schemes for coupling flow with geomechanics, Comput. Methods Appl. Mech. Eng., 311, 1, 180-207 (2016) · Zbl 1439.74183
[16] Bause, M.; Radu, F. A.; Köcher, U., Space-time finite element approximation of the Biot poroelasticity system with iterative coupling, Comput. Methods Appl. Mech. Eng., 320, 745-768 (2017) · Zbl 1439.74389
[17] Castelleto, N.; White, J. A.; Tchelepi, H. A., Accuracy and convergence properties of the fixed-stress iterative solution of two-way coupled poromechanics, Int. J. Numer. Anal. Methods Geomech., 39, 1593-1618 (2015)
[18] Gaspar, F. J.; Rodrigo, C., On the fixed-stress split scheme as smoother in multigrid methods for coupling flow and geomechanics, Comput. Methods Appl. Mech. Eng., 326, 526-540 (2017) · Zbl 1439.74413
[19] Borregales, M.; Radu, F. A.; Kumar, K.; Nordbotten, J. M., Robust iterative schemes for non-linear poromechanics, Comput. Geosci., 22, 1021-1038 (2018) · Zbl 1402.65109
[20] Both, J. W.; Nordbotten, J. M.; Kumar, K.; Radu, F. A., Anderson accelerated fixed-stress splitting schemes for consolidation of unsaturated porous media, Comput. Math. Appl., 1-24 (2018)
[21] Gander, M. J., 50 Years of Time Parallel Time Integration, 69-113 (2015), Springer International Publishing: Springer International Publishing Cham · Zbl 1337.65127
[22] Emmett, M.; Minion, M. L., Toward an efficient parallel in time method for partial differential equations, Commun. Appl. Math. Comput. Sci., 7, 105-132 (2012) · Zbl 1248.65106
[23] J. Lions, Y. Maday, G.A. Turinici, A “parareal” in time discretization of PDE’s, C.R. Math. Acad. Sci. Ser. I Math., 332, 661-668. http://dx.doi.org/10.1016/S0764-4442(00)01793-6; J. Lions, Y. Maday, G.A. Turinici, A “parareal” in time discretization of PDE’s, C.R. Math. Acad. Sci. Ser. I Math., 332, 661-668. http://dx.doi.org/10.1016/S0764-4442(00)01793-6 · Zbl 0984.65085
[24] Falgout, R. D.; Friedhoff, S.; Kolev, T. V.; MacLachlan, S. P.; Schroder, J. B., Parallel time integration with multigrid, SIAM J. Sci. Comput., 16, 635-661 (2014) · Zbl 1310.65115
[25] Horton, G.; Vandewalle, S., A space-time multigrid method for parabolic partial differential equations, SIAM J. Sci. Comput., 16, 4, 848-864 (1995) · Zbl 0828.65105
[26] Lubich, C.; Ostermann, A., Multi-grid dynamic iteration for parabolic equations, BIT Numer. Math., 27, 2, 216-234 (1987) · Zbl 0623.65125
[27] Horton, G.; Vandewalle, S.; Worley, P., An algorithm with polylog parallel complexity for solving parabolic partial differential equations, SIAM J. Sci. Comput., 16, 3, 531-541 (1995) · Zbl 0827.65094
[28] Ženíšek, A., The existence and uniqueness theorem in Biot’s consolidation theory, Apl. Mat., 29, 3, 194-211 (1984) · Zbl 0557.35005
[29] Aguilar, G.; Gaspar, F.; Lisbona, F.; Rodrigo, C., Numerical stabilization of Biot’s consolidation model by a perturbation on the flow equation, Int. J. Numer. Methods Eng., 75, 11, 1282-1300 (2008) · Zbl 1158.74473
[30] Rodrigo, C.; Gaspar, F.; Hu, X.; Zikatanov, L., Stability and monotonicity for some discretizations of the Biot’s consolidation model, Comput. Methods Appl. Mech. Eng., 298, 183-204 (2016) · Zbl 1425.74164
[31] Murad, M. A.; Loula, A. F.D., Improved accuracy in finite element analysis of Biot’s consolidation problem, Comput. Methods Appl. Mech. Engrg., 95, 3, 359-382 (1992) · Zbl 0760.73068
[32] Murad, M. A.; Loula, A. F.D., On stability and convergence of finite element approximations of Biot’s consolidation problem, Internat. J. Numer. Methods Engrg., 37, 4, 645-667 (1994) · Zbl 0791.76047
[33] Murad, M. A.; Thomée, V.; Loula, A. F.D., Asymptotic behavior of semidiscrete finite-element approximations of Biot’s consolidation problem, SIAM J. Numer. Anal., 33, 3, 1065-1083 (1996) · Zbl 0854.76053
[34] Abousleiman, Y.; Cheng, A.-D.; Cui, L.; Detournay, E.; Roegiers, J.-C., Mandel’s problem revisited, Geotechnique, 46, 2, 187-195 (1996)
[35] Mandel, J., Consolidation Des Sols (étude de mathématique), Géotechnique, 3, 287-299 (1953)
[36] Mikelić, A.; Wang, B.; Wheeler, M. F., Numerical convergence study of iterative coupling for coupled flow and geomechanics, Comput. Geosci., 18, 3-4, 325-341 (2014) · Zbl 1386.76115
[37] J.W. Both, U. Köcher, Numerical investigation on the fixed-stress split- ting scheme for Biots equations: Optimality of the tuning parameter, arXivPreprint arXiv:1801.08352; J.W. Both, U. Köcher, Numerical investigation on the fixed-stress split- ting scheme for Biots equations: Optimality of the tuning parameter, arXivPreprint arXiv:1801.08352
[38] Bangerth, W.; Kanschat, G.; Heister, T.; Heltai, L.; Kanschat, G., The deal.II library version 8.4, J. Numer. Math., 24, 135-141 (2016) · Zbl 1348.65187
[39] 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, Texas, Feb. 3-5, 2003. http://dx.doi.org/10.2118/79700-MS; 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, Texas, Feb. 3-5, 2003. http://dx.doi.org/10.2118/79700-MS
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.