Block-preconditioned Newton-Krylov solvers for fully coupled flow and geomechanics.(English)Zbl 1367.76034
Summary: The focus of this work is efficient solution methods for mixed finite element models of variably saturated fluid flow through deformable porous media. In particular, we examine preconditioning techniques to accelerate the convergence of implicit Newton-Krylov solvers. We highlight an approach in which preconditioners are built from block-factorizations of the coupled system. The key result of the work is the identification of effective preconditioners for the various sub-problems that appear within the block decomposition. We use numerical examples drawn from both linear and nonlinear hydromechanical models to test the robustness and scalability of the proposed methods. Results demonstrate that an algebraic multigrid variant of the block preconditioner leads to mesh-independent convergence, good parallel efficiency, and insensitivity to the material parameters of the medium.
MSC:
| 76M10 | Finite element methods applied to problems in fluid mechanics |
| 76S05 | Flows in porous media; filtration; seepage |
| 74F10 | Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.) |
| 86A05 | Hydrology, hydrography, oceanography |
| 74L10 | Soil and rock mechanics |
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References:
| [1] | Gawin, D., Baggio, P., Schrefler, B.A.: Coupled heat, water and gas flow in deformable porous media. Int. J. Numer. Methods Fluids 20, 969–987 (1995) ·Zbl 0854.76052 ·doi:10.1002/fld.1650200817 |
| [2] | Ehlers, W., Graf, T., Ammann, M.: Deformation and localization analysis of partially saturated soil. Comput. Methods Appl. Mech. Eng. 193(27–29), 2885–2910 (2004) ·Zbl 1067.74543 ·doi:10.1016/j.cma.2003.09.026 |
| [3] | Young, Y.L., White, J.A., Xiao, H., Borja, R.I.: Tsunami-induced liquefaction failure of coastal slopes. Acta Geotech. 4, 17–34 (2009) ·doi:10.1007/s11440-009-0083-6 |
| [4] | Borja, R.I., White, J.A.: Continuum deformation and stability analyses of a steep hillside slope under rainfall infiltration. Acta Geotech. 1–14 (2010) |
| [5] | Ferronato, M., Bergamaschi, L., Gambolati, G.: Performance and robustness of block constraint preconditioners in finite element coupled consolidation problems. Int. J. Numer. Methods Eng. 81(3), 381–402 (2010) ·Zbl 1183.74271 |
| [6] | Wan, J.: Stabilized Finite Element Methods for Coupled Geomechanics and Multiphase Flow. Ph.D. thesis, Stanford University (2002) |
| [7] | Minkoff, S.E., Stone, C.M., Bryant, S., Peszynska, M., Wheeler, M.F.: Coupled fluid flow and geomechanical deformation modeling. J. Pet. Sci. Eng. 38(1–2), 37–56 (2003) ·doi:10.1016/S0920-4105(03)00021-4 |
| [8] | 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) ·doi:10.1007/s11440-007-0033-0 |
| [9] | Hayashi, K., Willis-Richards, J., Hopkirk, R.J., Niibori, Y.: Numerical models of HDR geothermal reservoirs–a review of current thinking and progress. Geothermics 28(4–5), 507–518 (1999) ·doi:10.1016/S0375-6505(99)00026-7 |
| [10] | Johnson, J.W., Nitao, J.J., Morris, J.P.: Reactive transport modeling of cap rock integrity during natural and engineered CO2 storage. In: Benson, S. (ed.) CO2 Capture Project Summary, vol. 2. Elsevier, Amsterdam (2004) |
| [11] | Rutqvist, J., Birkholzer, J.T., Tsang, C.F.: Coupled reservoir–geomechanical analysis of the potential for tensile and shear failure associated with CO2 injection in multilayered reservoir–caprock systems. Int. J. Rock Mech. Min. Sci. 45(2), 132–143 (2008) ·doi:10.1016/j.ijrmms.2007.04.006 |
| [12] | Morris, J.P., Detwiler, R.L., Friedmann, S.J., Vorobiev, O.Y., Hao, Y.: The large-scale effects of multiple CO2 injection sites on formation stability. Energy Procedia 1(1), 1831–1837 (2009) ·doi:10.1016/j.egypro.2009.01.239 |
| [13] | Bramble, J.H., Pasciak, J.E.: A preconditioning technique for indefinite systems resulting from mixed approximation of elliptic problems. Math. Comput. 50(181), 1–17 (1988) ·Zbl 0643.65017 ·doi:10.1090/S0025-5718-1988-0917816-8 |
| [14] | 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 ·doi:10.1016/j.jcp.2007.09.026 |
| [15] | 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) ·doi:10.1016/j.pepi.2008.07.036 |
| [16] | Burstedde, C., Ghattas, O., Stadler, G., Tu, T., Wilcox, L.C.: Parallel scalable adjoint-based adaptive solution of variable-viscosity Stokes flow problems. Comput. Methods Appl. Mech. Eng. 198(21–26), 1691–1700 (2009) ·Zbl 1227.76027 ·doi:10.1016/j.cma.2008.12.015 |
| [17] | Toh, K.C., Phoon, K.K., Chan, S.H.: Block preconditioners for symmetric indefinite linear systems. Int. J. Numer. Methods Eng. 60, 1361–1381 (2004) ·Zbl 1065.65064 ·doi:10.1002/nme.982 |
| [18] | Biot, M.A.: General theory of three-dimensional consolidation. J. Appl. Phys. 12(2), 155–164 (1941) ·JFM 67.0837.01 ·doi:10.1063/1.1712886 |
| [19] | Biot, M.A.: Theory of elasticity and consolidation for a porous anisotropic solid. J. Appl. Phys. 26(2), 182–185 (1955) ·Zbl 0067.23603 ·doi:10.1063/1.1721956 |
| [20] | Borja, R.I.: On the mechanical energy and effective stress in saturated and unsaturated porous continua. Int. J. Solids Struct. 43(6), 1764–1786 (2006) ·Zbl 1120.74446 ·doi:10.1016/j.ijsolstr.2005.04.045 |
| [21] | Terzaghi, K.: Theoretical Soil Mechanics. Wiley, New York (1943) |
| [22] | Bishop, A.W.: The principle of effective stress. Tekn. Ukebl. 39, 859–863 (1959) |
| [23] | Skempton, A.W.: Effective stress in soils, concrete and rocks. In: Pore Pressure and Suction in Soils, pp. 4–16. Butterworths, London (1961) |
| [24] | Nur, A., Byerlee, J.D.: An exact effective stress law for elastic deformation of rock with fluids. J. Geophys. Res. 76, 6414–6419 (1971) ·doi:10.1029/JB076i026p06414 |
| [25] | Borja, R.I., Koliji, A.: On the effective stress in unsaturated porous continua with double porosity. J. Mech. Phys. Solids (2009). doi: 10.1016/j.jmps.2009.04.014 ·Zbl 1425.74310 |
| [26] | van Genuchten, M.T.: A closed-form equation for predicting the hydraulic conductivity of unsaturated soils. Soil Sci. Soc. Am. J. 44(5), 892–898 (1980) ·doi:10.2136/sssaj1980.03615995004400050002x |
| [27] | Cryer, C.W.: A comparison of the three-dimensional consolidation theories of Biot and Terzaghi. Q. J. Mech. Appl. Math. 16(4), 401–412 (1963) ·Zbl 0121.21502 ·doi:10.1093/qjmam/16.4.401 |
| [28] | Brezzi, F.: On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers. RAIRO Anal. Numer. 8, 129–151 (1974) ·Zbl 0338.90047 |
| [29] | Brezzi, F.: A discourse on the stability conditions for mixed finite element formulations. Comput. Methods. Appl. Mech. Eng. 82(1–3), 27–57 (1990) ·Zbl 0736.73062 ·doi:10.1016/0045-7825(90)90157-H |
| [30] | Arnold, D.N.: Mixed finite element methods for elliptic problems. Comput. Methods Appl. Mech. Eng. 82, 281–300 (1990) ·Zbl 0729.73198 ·doi:10.1016/0045-7825(90)90168-L |
| [31] | Murad, M.A., Loula, A.F.D.: On stability and convergence of finite element approximations of Biot’s consolidation problem. Int. J. Numer. Methods Eng. 37, 645–667 (1994) ·Zbl 0791.76047 ·doi:10.1002/nme.1620370407 |
| [32] | 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. Eng. 197(49–50), 4353–4366 (2008) ·Zbl 1194.74480 ·doi:10.1016/j.cma.2008.05.015 |
| [33] | Pastor, M., Li, T., Liu, X., Zienkiewicz, O.C., Quecedo, M.: A fractional step algorithm allowing equal order of interpolation for coupled analysis of saturated soil problems. Mech. Cohes.-Frict. Mater. 5(7), 511–534 (2000) ·doi:10.1002/1099-1484(200010)5:7<511::AID-CFM87>3.0.CO;2-S |
| [34] | Truty, A., Zimmermann, T.: Stabilized mixed finite element formulations for materially nonlinear partially saturated two-phase media. Comput. Methods Appl. Mech. Eng. 195, 1517–1546 (2006) ·Zbl 1116.74067 ·doi:10.1016/j.cma.2005.05.044 |
| [35] | Dohrmann, C.R., Bochev, P.B.: A stabilized finite element method for the Stokes problem based on polynomial pressure projections. Int. J. Numer. Methods Fluids 46, 183–201 (2004) ·Zbl 1060.76569 ·doi:10.1002/fld.752 |
| [36] | Bochev, P.B., Dohrmann, C.R.: A computational study of stabilized, low-order C 0 finite element approximations of Darcy equations. Comput. Mech. 38, 323–333 (2006) ·Zbl 1177.76191 ·doi:10.1007/s00466-006-0036-y |
| [37] | Burman, E.: Pressure projection stabilizations for Galerkin approximations of Stokes’ and Darcy’s problem. Numer. Methods Partial Differ. Equ. 24(1), 127–143 (2007) ·Zbl 1139.76029 ·doi:10.1002/num.20243 |
| [38] | White, J.A.: Stabilized Finite Element Methods for Coupled Flow and Geomechanics. Ph.D. thesis, Stanford University, Stanford, CA (2009) |
| [39] | Benzi, M., Golub, G.H., Liesen, J: Numerical solution of saddle point problems. Acta Numer. 14, 1–137 (2005) ·Zbl 1115.65034 ·doi:10.1017/S0962492904000212 |
| [40] | Verfürth, R.: Error estimates for a mixed finite element approximation of the Stokes equations. RAIRO. Anal. Numér. 18(2), 175–182 (1984) ·Zbl 0557.76037 |
| [41] | Elman, H.C., Silvester, D.J., Wathen, A.J.: Iterative methods for problems in computational fluid dynamics. In: Iterative Methods in Scientific Computing, p. 271 (1997) ·Zbl 0957.76071 |
| [42] | Bangerth, W., Hartmann, R., Kanschat, G.: Deal.II–a general purpose object oriented finite element library. ACM Trans. Math. Softw. 33(4), 24 (2007) ·Zbl 1365.65248 ·doi:10.1145/1268776.1268779 |
| [43] | Burstedde, C., Wilcox, L.C., Ghattas, O.: p4est: scalable algorithms for parallel adaptive mesh refinement on forests of octrees. SIAM J. Sci. Comput. (in press, 2011) ·Zbl 1230.65106 |
| [44] | Bangerth, W., Burstedde, C., Heister, T., Kronbichler, M.: Algorithms and data structures for massively parallel generic adaptive finite element codes. ACM Trans. Math. Softw. (submitted, 2011) ·Zbl 1365.65247 |
| [45] | Heroux, M.A., Bartlett, R.A., Howle, V.E., Hoekstra, R.J., Hu, J.J., Kolda, T.G., Lehoucq, R.B., Long, K.R., Pawlowski, R.P., Phipps, E.T., et al.: An overview of the Trilinos project. ACM Trans. Math. Softw. 31(3), 397–423 (2005) ·Zbl 1136.65354 ·doi:10.1145/1089014.1089021 |
| [46] | Sala, M., Heroux, M.: Robust algebraic preconditioners with IFPACK 3.0. Technical Report SAND-0662, Sandia National Laboratories (2005) |
| [47] | Gee, M.W., Siefert, C.M., Hu, J.J., Tuminaro, R.S., Sala, M.G.: ML 5.0 smoothed aggregation user’s guide. Technical Report SAND2006-2649, Sandia National Laboratories (2006) |
| [48] | Eisenstat, S.C., Walker, H.F.: Choosing the forcing terms in an inexact Newton method. SIAM J. Sci. Comput. 17, 16–32 (1996) ·Zbl 0845.65021 ·doi:10.1137/0917003 |
| [49] | Verruijt, A.: Theory of Consolidation. In: An Introduction to Soil Dynamics, pp. 65–90 (2010) |
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