Block-partitioned solvers for coupled poromechanics: a unified framework. (English) Zbl 1425.74497
Summary: Coupled poromechanical problems appear in a variety of disciplines, from reservoir engineering to biomedical applications. This work focuses on efficient strategies for solving the matrix systems that result from discretization and linearization of the governing equations. These systems have an inherent block structure due to the coupled nature of the mass and momentum balance equations. Recently, several iterative solution schemes have been proposed that exhibit stable and rapid convergence to the coupled solution. These schemes appear distinct, but a unifying feature is that they exploit the block-partitioned nature of the problem to accelerate convergence. This paper analyzes several of these schemes and highlights the fundamental connections that underlie their effectiveness. We begin by focusing on two specific methods: a fully-implicit and a sequential-implicit scheme. In the first approach, the system matrix is treated monolithically, and a Krylov iteration is used to update pressure and displacement unknowns simultaneously. To accelerate convergence, a preconditioning operator is introduced based on an approximate block-factorization of the linear system. Next, we analyze a sequential-implicit scheme based on the fixed-stress split. In this method, one iterates back and forth between updating displacement and pressure unknowns separately until convergence to the coupled solution is reached. We re-interpret this scheme as a block-preconditioned Richardson iteration, and we show that the preconditioning operator is identical to that used within the fully-implicit approach. Rapid convergence in both the Richardson- and Krylov-based methods results from a particular choice for a sparse Schur complement approximation. This analysis leads to a unified framework for developing solution schemes based on approximate block factorizations. Many classic fully-implicit and sequential-implicit schemes are simple sub-cases. The analysis also highlights several new approaches that have not been previously explored. For illustration, we directly compare the performance and robustness of several variants on a benchmark problem.
MSC:
| 74S05 | Finite element methods applied to problems in solid mechanics |
| 65M22 | Numerical solution of discretized equations for initial value and initial-boundary value problems involving PDEs |
| 65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |
| 74F10 | Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.) |
References:
| [1] | Castelletto, N.; Gambolati, G.; Teatini, P., A coupled MFE poromechanical model of a large-scale load experiment at the coastland of Venice, Comput. Geosci., 19, 1, 17-29 (2015) |
| [2] | Hettema, M. H.H.; Schutjens, P. M.T. M.; Verboom, B. J.M.; Gussinklo, H. J., Production-induced compaction of a sandstone reservoir: The strong influence of stress path, SPE Reservoir Eval. Eng., 3, 4, 342-347 (2000) |
| [3] | Wang, W.; Regueiro, R. A.; McCartney, J. S., Coupled axisymmetric thermo-poro-mechanical finite element analysis of energy foundation centrifuge experiments in partially saturated silt, Geotech. Geolog. Eng., 33, 2, 373-388 (2014) |
| [4] | Dupray, F.; Laloui, L.; Kazangba, A., Numerical analysis of seasonal heat storage in an energy pile foundation, Comput. Geotech., 55, 67-77 (2014) |
| [5] | Rutqvist, J., The geomechanics of \(CO_2\) storage in deep sedimentary formations, Geotec. Geolog. Eng., 30, 525-551 (2012) |
| [6] | Castelletto, N.; Teatini, P.; Gambolati, G.; Bossie-Codreanu, D.; Vincé, O.; Daniel, J.-M.; Battistelli, A.; Marcolini, M.; Donda, F.; Volpi, V., Multiphysics modeling of \(CO_2\) sequestration in a faulted saline formation in Italy, Adv. Water Resour., 62, 570-587 (2013) |
| [7] | White, J. A.; Chiaramonte, L.; Ezzedine, S.; Foxall, W.; Hao, Y.; Ramirez, A.; McNab, W., Geomechanical behavior of the reservoir and caprock system at the In Salah \(CO_2\) storage project, Proc. Natl. Acad. Sci., 111, 24, 8747-8752 (2014) |
| [8] | Chiaramonte, L.; White, J. A.; Trainor-Guitton, W., Probabilistic geomechanical analysis of compartmentalization at the Snøhvit \(CO_2\) sequestration project, J. Geophys. Res.-Solid Earth, 120, 2, 1195-1209 (2015) |
| [9] | Cowin, S. C., Bone poroelasticity, J. Biomech., 32, 217-238 (1999) |
| [10] | Schulthess, T. C., Programming revisited, Nat. Phys., 11, 369-373 (2015) |
| [11] | Prévost, J. H., Partitioned solution procedure for simultaneous integration of coupled-field problems, Commun. Numer. Methods. Eng., 13, 239-247 (1997) · Zbl 0878.73073 |
| [12] | Prévost, J. H., Two-way coupling in reservoir-geomechanical models: vertex-centered Galerkin geomechanical model cell-centered and vertex-centered finite volume reservoir models, Internat. J. Numer. Methods Engrg., 98, 612-624 (2014) · Zbl 1352.86002 |
| [13] | Jha, B.; Juanes, R., A locally conservative finite element framework for the simulation of coupled flow and reservoir geomechanics, Acta Geotech., 2, 3, 139-153 (2007) |
| [14] | Bergamaschi, L.; Ferronato, M.; Gambolati, G., Novel preconditioners for the iterative solution to FE-discretized coupled consolidation equations, Comput. Methods Appl. Mech. Engrg., 196, 25-28, 2647-2656 (2007) · Zbl 1173.76330 |
| [15] | Kim, J.; Tchelepi, H. A.; Juanes, R., Stability, accuracy and efficiency of sequential methods for coupled flow and geomechanics, SPE J., 16, 2, 249-262 (2011) · Zbl 1228.74106 |
| [16] | White, J. A.; Borja, R. I., Block-preconditioned Newton-Krylov solvers for fully coupled flow and geomechanics, Comput. Geosci., 15, 4, 647-659 (2011) · Zbl 1367.76034 |
| [17] | Mikelić, A.; Wheeler, M. F., Convergence of iterative coupling for coupled flow and geomechanics, Comput. Geosci., 17, 3, 455-461 (2013) · Zbl 1392.35235 |
| [18] | Gambolati, G.; Teatini, P.; Baú, D.; Ferronato, M., Importance of poroelastic coupling in dynamically active aquifers of the Po River Basin, Italy, Water Resour. Res., 36, 9, 2443-2459 (2000) |
| [19] | Mandel, J., Consolidation des sols, Étude Math. Géotech., 3, 7, 287-299 (1953) |
| [20] | Cryer, C., A comparison of the three-dimensional consolidation theories of Biot and Terzaghi, Quart. J. Mech. Appl. Math., 16, 4, 401-412 (1963) · Zbl 0121.21502 |
| [21] | 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. Engrg., 200, 13-16, 1591-1606 (2011) · Zbl 1228.74101 |
| [22] | 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. Engrg., 200, 23-24, 2094-2116 (2011) · Zbl 1228.74106 |
| [23] | Wang, H. F., Theory of Linear Poroelasticity (2000), Princeton University Press: Princeton University Press Princeton, NJ |
| [24] | Coussy, O., Poromechanics (2004), John Wiley & Sons: John Wiley & Sons New York |
| [25] | Brezzi, F.; Fortin, M., Mixed and Hybrid Finite Element Methods (1991), Springer-Verlag: Springer-Verlag Berlin · Zbl 0788.73002 |
| [26] | White, J. A.; Borja, R. I., Stabilized low-order finite elements for coupled solid-deformation/fluid-diffusion and their application to fault zone transients, Comput. Methods Appl. Mech. Engrg., 197, 49-50, 4353-4366 (2008) · Zbl 1194.74480 |
| [27] | Settari, A.; Mourits, F. M., A coupled reservoir and geomechanical simulation system, SPE J., 3, 3, 219-226 (1998) |
| [28] | Di Pietro, D. A.; Eymard, R.; Lemaire, S.; Masson, R., Hybrid finite volume discretization of linear elasticity models on general meshes, (Fořt, J.; etal., Finite Volumes for Complex Applications VI—Problems & Perspectives (2011), Springer-Verlag: Springer-Verlag Berlin), 331-339 · Zbl 1246.76051 |
| [29] | Nordbotten, J. M., Finite volume hydromechanical simulation in porous media, Water Resour. Res., 50, 5, 4379-4394 (2014) |
| [30] | Phillips, P. J.; Wheeler, M. F., 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 |
| [31] | 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, 2, 145-158 (2007) · Zbl 1117.74016 |
| [32] | Ferronato, M.; Castelletto, N.; Gambolati, G., A fully coupled 3-d mixed finite element model of Biot consolidation, J. Comput. Phys., 229, 12, 4813-4830 (2010) · Zbl 1305.76055 |
| [33] | Saad, Y., Iterative Methods for Sparse Linear Systems (2003), SIAM · Zbl 1002.65042 |
| [34] | Bramble, J. H.; Pasciak, J. E., A preconditioning technique for indefinite systems resulting from mixed approximation of elliptic problems, Math. Comp., 50, 181, 1-17 (1988) · Zbl 0643.65017 |
| [35] | Toh, K. C.; Phoon, K. K.; Chan, S. H., Block preconditioners for symmetric indefinite linear systems, Internat. J. Numer. Methods Engrg., 60, 1361-1381 (2004) · Zbl 1065.65064 |
| [36] | Benzi, M.; Golub, G. H.; Liesen, J., Numerical solution of saddle-point problems, Acta Numer., 14, 1, 1-137 (2005) · Zbl 1115.65034 |
| [37] | Elman, H.; Howle, V. E.; Shadid, J.; Shuttleworth, R.; Tuminaro, R., A taxonomy and comparison of parallel block multi-level preconditioners for the incompressible Navier-Stokes equations, J. Comput. Phys., 227, 3, 1790-1808 (2008) · Zbl 1290.76023 |
| [38] | May, D. A.; Moresi, L., Preconditioned iterative methods for Stokes flow problems arising in computational geodynamics, Phys. Earth Planet. Inter., 171, 1-4, 33-47 (2008) |
| [39] | Pestana, J., On the eigenvalues and eigenvectors of block triangular preconditioned block matrices, SIAM J. Matrix Anal. Appl., 35, 2, 517-525 (2014) · Zbl 1304.65122 |
| [40] | Griebel, M.; Oeltz, D.; Schweitzer, M. A., An algebraic multigrid method for linear elasticity, SIAM J. Sci. Comput., 25, 2, 385-407 (2003) · Zbl 1163.65336 |
| [41] | Baker, A. H.; Kolev, T. V.; Yang, U. M., Improving algebraic multigrid interpolation operators for linear elasticity problems, Numer. Linear Algebra Appl., 17, 2-3, 495-517 (2010) · Zbl 1240.74027 |
| [42] | Koric, S.; Lu, Q.; Guleryuz, E., Evaluation of massively parallel linear sparse solvers on unstructured finite element meshes, Comput. Struct., 141, 19-25 (2014) |
| [43] | Janna, C.; Ferronato, M.; Sartoretto, F.; Gambolati, G., FSAIPACK: a software package for high performance factored sparse approximate inverse preconditioning, ACM Trans. Math. Software, 41, 2 (2015) · Zbl 1369.65052 |
| [45] | Silvester, D.; Wathen, A., Fast iterative solution of stabilized Stokes systems, Part II: Using general block preconditioners, SIAM J. Numer. Anal., 31, 5, 1352-1367 (1994) · Zbl 0810.76044 |
| [46] | Elman, H. C.; Silvester, D. J.; Wathen, A. J., Iterative methods for problems in computational fluid dynamics, (Iterative Methods in Scientific Computing (1997), Springer-Verlag: Springer-Verlag Singapore), 271-327 · Zbl 0957.76071 |
| [47] | Castelletto, 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, 14, 1593-1618 (2015) |
| [48] | Bergamaschi, L.; Ferronato, M.; Gambolati, G., Mixed Constraint Preconditioners for the iterative solution of FE coupled consolidation equations, J. Comput. Phys., 227, 9885-9897 (2008) · Zbl 1154.65015 |
| [49] | 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 |
| [50] | Mason, D.; Solomon, A.; Nicolaysen, L., Evolution of stress and strain during the consolidation of a fluid-saturated porous elastic sphere, J. Appl. Phys., 70, 9, 4724-4740 (1991) |
| [51] | Bangerth, W.; Hartmann, R.; Kanschat, G., Deal. II—a general-purpose object-oriented finite element library, ACM Trans. Math. Software, 33, 4, 24 (2007) · Zbl 1365.65248 |
| [52] | Heroux, M. A., An overview of the Trilinos project, ACM Trans. Math. Software, 31, 3, 397-423 (2005) · Zbl 1136.65354 |
| [53] | Hughes, T. J.R, The Finite Element Method: Linear Static and Dynamic Finite Element Analysis (1987), Dover: Dover New York · Zbl 0634.73056 |
| [54] | Borja, R. I.; Liu, X.; White, J. A., Multiphysics hillslope processes triggering landslides, Acta Geotech., 7, 4, 261-269 (2012) |
| [55] | Choo, J.; White, J. A.; Borja, R. I., Hydromechanical modeling of unsaturated flow in double porosity media, Int. J. Geomech. (2015) |
| [56] | Jha, B.; Juanes, R., Coupled multiphase flow and poromechanics: A computational model of pore-pressure effects on fault slip and earthquake triggering, Water Resour. Res., 50, 5, 3776-3808 (2014) |
| [57] | Mikelić, A.; Wang, B.; Wheeler, M. F., Numerical convergence study of iterative coupling for coupled flow and geomechanics, Comput. Geosci., 18, 3, 325-341 (2014) · Zbl 1386.76115 |
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