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Fully implicit hybrid two-level domain decomposition algorithms for two-phase flows in porous media on 3D unstructured grids. (English) Zbl 1435.76038
Summary: Simulation of subsurface flows in porous media is difficult due to the nonlinearity of the operators and the high heterogeneity of material coefficients. In this paper, we present a scalable fully implicit solver for incompressible two-phase flows based on overlapping domain decomposition methods. Specifically, an inexact Newton-Krylov algorithm with analytic Jacobian is used to solve the nonlinear systems arising from the discontinuous Galerkin discretization of the governing equations on 3D unstructured grids. The linear Jacobian system is preconditioned by additive Schwarz algorithms, which are naturally suitable for parallel computing. We propose a hybrid two-level version of the additive Schwarz preconditioner consisting of a nested coarse space to improve the robustness and scalability of the classical one-level version. On the coarse level, a smaller linear system arising from the same discretization of the problem on a coarse grid is solved by using GMRES with a one-level preconditioner until a relative tolerance is reached. Numerical experiments are presented to demonstrate the effectiveness and efficiency of the proposed solver for 3D heterogeneous medium problems. We also report the parallel scalability of the proposed algorithms on a supercomputer with up to \(8, 192\) processor cores.

76M10 Finite element methods applied to problems in fluid mechanics
76S05 Flows in porous media; filtration; seepage
76T06 Liquid-liquid two component flows
65Y05 Parallel numerical computation
PETSc; Matlab; libMesh; MRST; METIS
Full Text: DOI
[1] Arbogast, T.; Juntunen, M.; Pool, J.; Wheeler, M. F., A discontinuous Galerkin method for two-phase flow in a porous medium enforcing \(H(d i v)\) velocity and continuous capillary pressure, Comput. Geotech., 17, 1055-1078 (2013) · Zbl 1393.76059
[2] Baggag, A.; Atkins, H.; Keyes, D. E., Parallel implementation of the discontinuous Galerkin method, (Proceedings of Parallel CFD’99 (1999)), 115-122
[3] Balay, S.; Abhyankar, S.; Adams, M. F.; Brown, J.; Brune, P.; Buschelman, K.; Eijkhout, V.; Gropp, W. D.; Kaushik, D.; Knepley, M. G.; McInnes, L. C.; Rupp, K.; Smith, B. F.; Zhang, H., PETSc Users Manual (2019), Argonne National Laboratory
[4] Bastian, P.; Rivière, B., Discontinuous Galerkin methods for two-phase flow in porous media (2004), IWR, University of Heidelberg, Tech. Rep. 2004-28
[5] Bastian, P., A fully-coupled discontinuous Galerkin method for two-phase flow in porous media with discontinuous capillary pressure, Comput. Geosci., 18, 779-796 (2014) · Zbl 1392.76072
[6] Barker, A. T.; Cai, X.-C., Two-level Newton and hybrid Schwarz preconditioners for fluid-structure interaction, SIAM J. Sci. Comput., 32, 2390-2417 (2010) · Zbl 1214.92014
[7] Barker, A. T.; Brenner, S. C.; Sung, L.-Y., Overlapping Schwarz domain decomposition preconditioners for the local discontinuous Galerkin method for elliptic problems, J. Numer. Math., 19, 165-187 (2011) · Zbl 1232.65164
[8] Bernacki, M.; Fezoui, L.; Lanteri, S.; Piperno, S., Parallel discontinuous Galerkin unstructured mesh solvers for the calculation of three-dimensional wave propagation problems, Appl. Math. Model., 30, 744-763 (2006) · Zbl 1101.78009
[9] Cai, X.-C.; Gropp, W. D.; Keyes, D. E.; Melvin, R. G.; Young, D. P., Parallel Newton-Krylov-Schwarz algorithms for the transonic full potential equation, SIAM J. Sci. Comput., 192, 46-65 (1998)
[10] Cai, X.-C.; Sarkis, M., A restricted additive Schwarz preconditioner for general sparse linear systems, SIAM J. Sci. Comput., 21, 792-797 (1999) · Zbl 0944.65031
[11] Chen, Z.; Ewing, R., From single-phase to compositional flow: applicability of mixed finite elements, Transp. Porous Media, 57, 225-242 (1997)
[12] Chen, Z.; Huan, G.; Li, B., An improved IMPES method for two-phase flow in porous media, Transp. Porous Media, 54, 361-376 (2004)
[13] Chen, Z.; Huan, G.; Ma, Y., Computational Methods for Multiphase Flows in Porous Media (2006), SIAM: SIAM Philadelphia, PA, USA · Zbl 1092.76001
[14] Coleman, T. F.; Moré, J. J., Estimation of sparse Jacobian matrices and graph coloring problems, SIAM J. Numer. Anal., 20, 187-209 (1983) · Zbl 0527.65033
[15] Cusini, M.; Lukyanov, A. A.; Natvig, J.; Hajibeygi, H., Constrained pressure residual multiscale (CPR-MS) method for fully implicit simulation of multiphase flow in porous media, J. Comput. Phys., 299, 472-486 (2015) · Zbl 1351.76055
[16] Dawson, C. N.; Klíe, H.; Wheeler, M. F.; Woodward, C. S., A parallel, implicit, cell-centered method for two-phase flow with a preconditioned Newton-Krylov solver, Comput. Geosci., 1, 215-249 (1997) · Zbl 0941.76062
[17] Dennis, J. E.; Schnabel, R. B., Numerical Methods for Unconstrained Optimization and Nonlinear Equations (1996), SIAM: SIAM Philadelphia · Zbl 0847.65038
[18] Dogru, A. H.; Fung, L. S.K.; Middya, U.; Al-Shaalan, T. M.; Pita, J. A.; HemanthKumar, K.; Su, H. J.; Tan, J. C.T.; Hoy, H.; Dreiman, W. T.; Hahn, W. A.; Al-Harbi, R.; Al-Youbi, A.; Al-Zamel, N. M.; Mezghani, M.; Al-Mani, T., A next generation parallel reservoir simulator for giant reservoirs, (SPE 119272 Presented at the SPE Reservoir Simulation Symposium. SPE 119272 Presented at the SPE Reservoir Simulation Symposium, Texas, Feb. 2-4 (2009))
[19] 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
[20] Epshteyn, Y.; Rivière, B., Fully implicit discontinuous finite element methods for two-phase flow, Appl. Numer. Math., 57, 383-401 (2007) · Zbl 1370.76085
[21] Epshteyn, Y.; Rivière, B., Analysis of hp discontinuous Galerkin methods for incompressible two-phase flow, J. Comput. Appl. Math., 225, 487-509 (2009) · Zbl 1157.76024
[22] Ern, A.; Mozolevski, I.; Schuh, L., Discontinuous Galerkin approximation of two-phase flows in heterogeneous porous media with discontinuous capillary pressures, Comput. Methods Appl. Mech. Eng., 199, 1491-1501 (2010) · Zbl 1231.76143
[23] Eslinger, O., Discontinuous Galerkin finite element methods applied to two-phase, air-water flow problems (2005), University of Texas at Austin, Ph.D. thesis
[24] Ewing, R.; Heinemann, R., Incorporation of mixed finite element methods in compositional simulation for reduction of numerical dispersion, (Reservoir Simulation Symposium, No. SPE12267 (1983))
[25] Gries, S.; Stüben, K.; Brown, G. L.; Chen, D.; Collins, D. A., Preconditioning for efficiently applying algebraic multigrid in fully implicit reservoir simulations, SPE J., 19, 726-736 (2014)
[26] Griewank, A., Evaluating Derivatives: Principles and Techniques of Algorithmic Differentiation (2000), SIAM: SIAM Philadelphia · Zbl 0958.65028
[27] Gropp, W. D.; Keyes, D. E.; McInnes, L.; Tidriri, M., Globalized Newton-Krylov-Schwarz algorithms and software for parallel implicit CFD, Int. J. High Perform. Comput. Appl., 14, 102-136 (2000)
[28] Gunasekera, D.; Childs, P.; Herring, J.; Cox, J., A multi-point flux discretization scheme for general polyhedral grids, (International Oil and Gas Conference and Exhibition, SPE48855 (1998))
[29] Hoteit, H.; Firoozabadi, A., Numerical modeling of two-phase flow in heterogeneous permeable media with different capillarity pressures, Adv. Water Resour., 31, 56-73 (2008)
[30] Hou, J.; Chen, J.; Sun, S.; Chen, Z., Adaptive mixed-hybrid and penalty discontinuous Galerkin method for two-phase flow in heterogeneous media, J. Comput. Appl. Math., 307, 262-283 (2016) · Zbl 1382.76166
[31] Karypis, G.; Kumar, V., MeTis: Unstructured Graph Partitioning and Sparse Matrix Ordering System, Version 4.0 (2009), University of Minnesota: University of Minnesota Minneapolis, MN
[32] Kirk, B. S.; Peterson, J. W.; Stogner, R. H.; Carey, G. F., libMesh: a C++ library for parallel adaptive mesh refinement/coarsening simulations, Eng. Comput., 22, 237-254 (2006)
[33] Klieber, W.; Rivière, B., Adaptive simulations of two-phase flow by discontinuous Galerkin methods, Comput. Methods Appl. Mech. Eng., 196, 404-419 (2006) · Zbl 1120.76327
[34] Knoll, D. A.; Keyes, D. E., Jacobian-free Newton-Krylov methods: a survey of approaches and applications, J. Comput. Phys., 193, 357-397 (2004) · Zbl 1036.65045
[35] Kou, J.; Sun, S., Convergence of discontinuous Galerkin methods for incompressible two-phase flow in heterogeneous media, SIAM J. Numer. Anal., 51, 3280-3306 (2013) · Zbl 1282.76124
[36] Kou, J.; Sun, S., Upwind discontinuous Galerkin methods with conservation of mass of both phases for incompressible two-phase flow in porous media, Numer. Methods Partial Differ. Equ., 30, 1674-1699 (2014) · Zbl 1308.76169
[37] Lie, K. A., An Introduction to Reservoir Simulation Using MATLAB/GNU Octave: User Guide for the MATLAB Reservoir Simulation Toolbox (MRST) (2019), Cambridge University Press: Cambridge University Press Cambridge · Zbl 1425.76001
[38] Liu, H.; Wang, K.; Chen, Z., A family of constrained pressure residual preconditioners for parallel reservoir simulations, Numer. Linear Algebra Appl., 23, 120-146 (2016) · Zbl 1413.65045
[39] Liu, L.; Keyes, D. E.; Sun, S., Fully implicit two-phase reservoir simulation with the additive Schwarz preconditioned inexact Newton method, (SPE Reserv. Charact. Simul. Conf. Exhib. (2013), Society of Petroleum Engineers)
[40] Luo, L.; Shiu, W. S.; Chen, R.; Cai, X.-C., A nonlinear elimination preconditioned inexact Newton method for blood flow problems in human artery with stenosis, J. Comput. Phys., 399, Article 108926 pp. (2019)
[41] Michel, A., A finite volume scheme for the simulation of two-phase incompressible flow in porous media, SIAM J. Numer. Anal., 41, 1301-1317 (2003) · Zbl 1049.35018
[42] Monteagudo, J. E.P.; Firoozabadi, A., Comparison of fully implicit and IMPES formulations for simulation of water injection in fractured and unfractured media, Int. J. Numer. Methods Eng., 69, 698-728 (2007) · Zbl 1194.76160
[43] Nair, R. D.; Choi, H. W.; Tufo, H. M., Computational aspects of a scalable high-order discontinuous Galerkin atmospheric dynamical core, Comput. Fluids, 30, 309-319 (2009) · Zbl 1237.76129
[44] Nayagum, D.; Schäfer, G.; Mosé, R., Modelling two-phase incompressible flow in porous media using mixed hybrid and discontinuous finite elements, Comput. Geosci., 8, 49-73 (2004) · Zbl 1221.76119
[45] Peaceman, D. W., Fundamentals of Numerical Reservoir Simulation (1977), Elsevier: Elsevier Amsterdam
[46] Reichenberger, V.; Jakobs, H.; Bastian, P.; Helmig, R., A mixed-dimensional finite volume method for two-phase flow in fractured porous media, Adv. Water Resour., 29, 1020-1036 (2006)
[47] Rivière, B.; Wheeler, M. F.; Banas, K., Part II. Discontinuous Galerkin method applied to a single phase flow in porous media, Comput. Geosci., 4, 337-349 (2000) · Zbl 1049.76565
[48] Rivière, B.; Wheeler, M. F., Miscible displacement in porous media, (Hassanizadeh, S. M.; Schotting, R. J., Proceedings of the XIV International Conference on Computational Methods in Water Resources (2002), Elsevier: Elsevier Amsterdam) · Zbl 1060.76072
[49] Saad, Y., A flexible inner-outer preconditioned GMRES algorithm, SIAM J. Sci. Comput., 14, 461-469 (1993) · Zbl 0780.65022
[50] Saad, Y., Iterative Methods for Sparse Linear Systems (2003), SIAM: SIAM Philadelphia, PA · Zbl 1002.65042
[51] Shadid, J. N.; Tuminaro, R. S.; Devine, K. D.; Hennigan, G. L.; Lin, P. T., Performance of fully coupled domain decomposition preconditioners for finite element transport/reaction simulations, J. Comput. Phys., 205, 24-47 (2005) · Zbl 1087.76069
[52] Smith, B.; Bjørstad, P.; Gropp, W., Domain Decomposition: Parallel Multilevel Methods for Elliptic Partial Differential Equations (1996), Cambridge University Press · Zbl 0857.65126
[53] Sun, S.; Wheeler, M. F., Discontinuous Galerkin methods for coupled flow and reactive transport problems, Appl. Numer. Math., 52, 273-298 (2005) · Zbl 1079.76584
[54] Sun, S.; Wheeler, M. F., Symmetric and nonsymmetric discontinuous Galerkin methods for reactive transport in porous media, SIAM J. Numer. Anal., 43, 195-219 (2005) · Zbl 1086.76043
[55] Toselli, A.; Widlund, O., Domain Decomposition Methods - Algorithms and Theory (2005), Springer: Springer Berlin · Zbl 1069.65138
[56] Wang, K.; Liu, H.; Chen, Z., A scalable parallel black oil simulator on distributed memory parallel computers, J. Comput. Phys., 301, 19-34 (2015) · Zbl 1349.76833
[57] Yang, C.; Cai, X.-C., Parallel multilevel methods for implicit solution of shallow water equations with nonsmooth topography on cubed-sphere, J. Comput. Phys., 230, 2523-2539 (2011) · Zbl 1316.76016
[58] Yang, C.; Cai, X.-C., A scalable fully implicit compressible Euler solver for mesoscale nonhydrostatic simulation of atmospheric flows, SIAM J. Sci. Comput., 36, S23-S47 (2014) · Zbl 1305.86005
[59] Yang, H.; Cai, X.-C., Parallel fully implicit two-grid Lagrange-Newton-Krylov-Schwarz methods for distributed control of unsteady incompressible flows, Int. J. Numer. Methods Fluids, 72, 1-21 (2013)
[60] Yang, H.; Yang, C.; Sun, S., Active-set reduced-space methods with nonlinear elimination for two-phase flow problems in porous media, SIAM J. Sci. Comput., 38, B593-B618 (2016) · Zbl 1383.76384
[61] Yang, H.; Sun, S.; Yang, C., Nonlinearly preconditioned semismooth Newton methods for variational inequality solution of two-phase flow in porous media, J. Comput. Phys., 332, 1-20 (2017) · Zbl 1378.76115
[62] Yang, H.; Sun, S.; Li, Y.; Yang, C., A scalable fully implicit framework for reservoir simulation on parallel computers, Comput. Methods Appl. Mech. Eng., 330, 334-350 (2018)
[63] Yang, H.; Sun, S.; Li, Y.; Yang, C., A fully implicit constraint-preserving simulator for the black oil model of petroleum reservoirs, J. Comput. Phys., 396, 347-363 (2019)
[64] Yang, H.; Sun, S.; Li, Y.; Yang, C., Parallel reservoir simulators for fully implicit complementarity formulation of multicomponent compressible flows, Comput. Phys. Commun., 244, 2-12 (2019)
[65] Zidane, A.; Firoozabadi, A., An implicit numerical model for multicomponent compressible two-phase flow in porous media, Adv. Water Resour., 85, 64-78 (2015)
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