Skip to main content
SpringerLink
Log in

Aspects of Solvers for Large-Scale Coupled Problems in Porous Media

  • Published:
Transport in Porous Media Aims and scope Submit manuscript

Abstract

This work summarizes solution strategies for discrete systems occurring in the simulation of processes in the subsurface. The focus is on scalable solvers for large and coupled systems. The goal of this work is to enable researchers to select suitable algorithms and parameter settings to efficiently solve their problems. The work provides an overview of existing methods, highlighting their features, potential, and also frequent pitfalls. Numerical examples are provided for single phase flow, density driven flow and poroelasticity.Aspects of multiphase flow are discussed briefly; a detailed discussion of reactive transport is beyond the scope of the article. Future trends are discussed.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10

Similar content being viewed by others

Notes

  1. Under certain conditions, (1b) becomes inaccurate and is replaced by nonlinear generalizations. In particular, for high velocities, the Forchheimer correction can be used (Bennethum and Giorgi 1997).

  2. Stencil refers to the nonzero entries in a matrix row, e.g., a 5-point 2D stencil can be thought of as the center, North, South, East and West connections.

References

  • Ainsworth, M., Oden, J.: A posteriori error estimation in finite element analysis. Comput. Methods Appl. Mech. Eng. 142(1), 1–88 (1997)

    Google Scholar 

  • Almani, T., Kumar, K., Dogru, A., Singh, G., Wheeler, M.: Convergence analysis of multirate fixed-stress split iterative schemes for coupling flow with geomechanics. Comput. Methods Appl. Mech. Eng. 311, 180–207 (2016)

    Google Scholar 

  • Amestoy, P.R., Duff, I.S., L’Excellent, J.Y., Li, X.S.: Analysis and comparison of two general sparse solvers for distributed memory computers. ACM Trans. Math. Softw. 27(4), 388–421 (2001)

    Google Scholar 

  • Angermann, L., Knabner, P., Thiele, K.: An error estimator for a finite volume discretization of density driven flow in porous media. Appl. Numer. Math. 26, 179–191 (1998)

    Google Scholar 

  • Angot, P., Boyer, F., Hubert, F.: Asymptotic and numerical modelling of flow in fractured porous media. ESAIM: M2AN Math. Model. Numer. Anal. 43(2), 239–275 (2009). https://doi.org/10.1051/m2an/2008052

    Article  Google Scholar 

  • Ashby, S.F., Falgout, R.D.: A parallel multigrid preconditioned conjugate gradient algorithm for groundwater flow simulations. Nucl. Sci. Eng. 124(1), 145–159 (1996). UCRL-JC-122359

    Google Scholar 

  • 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)

    Google Scholar 

  • Baker, A.H., Gamblin, T., Schulz, M., Yang, U.M.: Challenges of scaling algebraic multigrid across modern multicore architectures. In: Proceedings of the 25th IEEE International Parallel and Distributed Processing Symposium (IPDPS 2011) (2011a)

  • Baker, A.H., Schulz, M., Yang, U.M.: On the performance of an algebraic multigrid solver on multicore clusters. In: Palma, J.M.L.M., et al. (eds.) VECPAR 2010 Lecture Notes in Computer Science, vol. 6449, pp. 102–115. Springer, Berlin (2011b)

    Google Scholar 

  • Baker, A.H., Falgout, R.D., Kolev, T.V., Yang, U.M.: Multigrid smoothers for ultraparallel computing. SIAM J. Sci. Comput. 33(5), 2864–2887 (2011c)

    Google Scholar 

  • Baker, A.H., Falgout, R.D., Kolev, T.V., Yang, U.M.: Scaling Hypre’s Multigrid Solvers to 100,000 Cores, pp. 261–279. Springer, London (2012). https://doi.org/10.1007/978-1-4471-2437-5_13

    Book  Google Scholar 

  • Baker, A., Klawonn, A., Kolev, T., Lanser, M., Rheinbach, O., Yang, U.: Scalability of classical algebraic multigrid for elasticity to half a million parallel tasks. In: Bungartz, H., Neumann, P., Nagel, W. (eds.) Software for Exascale Computing—SPPEXA 2013–2015, Lecture Notes in Computational Science and Engineering, vol. 113, pp. 113–140. Springer, Berlin (2016)

    Google Scholar 

  • Balay, S., Abhyankar, S., Adams, M.F., Brown, J., Brune, P., Buschelman, K., Dalcin, L., Eijkhout, V., Gropp, W.D., Kaushik, D., Knepley, M.G., McInnes, L.C., Rupp, K., Smith, B.F., Zampini, S., Zhang, H.: PETSc Web Page (2015). http://www.mcs.anl.gov/petsc

  • Bastian, P., Chen, Z., Ewing, R.E., Helmig, R., Jakobs, H., Reichenberger, V.: Numerical simulation of multiphase flow in fractured porous media. In: Chen, Z., et al. (eds.) Numerical Treatment of Multiphase Flows in Porous Media. Proceedings of the International Workshop, Beijing, China, August 2–6, 1999, Lecture Notes in Physics, vol. 552, pp. 50–68. Springer, Berlin (2000)

    Google Scholar 

  • Bear, J.: Dynamics of Fluids in Porous Media. Dover, New York (1988)

    Google Scholar 

  • Bear, J., Bachmat, Y.: Introduction to Modeling of Transport Phenomena in Porous Media. No. 4 in Theory and Applications of Transport in Porous Media. Springer, Berlin (1990)

    Google Scholar 

  • Bear, J., Tsang, C.F., deMarsily, G.: Flow and Contaminant Transport in Fractured Rocks. Academic Press, New York (1993)

    Google Scholar 

  • Bennethum, L.S., Giorgi, T.: Generalized Forchheimer equation for two-phase flow based on hybrid mixture theory. Transp. Porous Media 26(3), 261–275 (1997)

    Google Scholar 

  • Benzi, M., Golub, G.H., Liesen, J.: Numerical solution of saddle point problems. Acta Numer. 14, 1–137 (2005). https://doi.org/10.1017/S0962492904000212

    Article  Google Scholar 

  • Berre, I., Doster, F., Keilegavlen, E.: Flow in Fractured Porous Media: A Review of Conceptual Models and Discretization Approaches. Transport in Porous Media, New York (2018)

    Google Scholar 

  • 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)

    Google Scholar 

  • Brabazon, K.J., Hubbard, M.E., Jimack, P.K.: Nonlinear multigrid methods for second order differential operators with nonlinear diffusion coefficient. Comput. Math. Appl. 68, 1619–1634 (2014)

    Google Scholar 

  • Brandt, A.: Multi-level adaptive solulions to boundary-value problems. Math. Comput. 31, 333–390 (1977)

    Google Scholar 

  • Brandt, A., McCormick, S.F., Ruge, J.W.: Algebraic multigrid (AMG) for automatic multigrid solution with application to geodetic computations. Technical Report, Institute for Computational Studies, Colorado State University (1982)

  • Brezina, M., Tong, C., Becker, R.: Parallel algebraic multigrid methods for structural mechanics. SIAM J. Sci. Comput. 27(5), 1534–1554 (2006)

    Google Scholar 

  • Briggs, W.L., Henson, V.E., McCormick, S.F.: A Multigrid Tutorial, 2nd edn. Society for Industrial and Applied Mathematics, Philadelphia (2000)

    Google Scholar 

  • Bungartz, H., Neumann, P., Nagel, W. (eds.): Software for Exascale Computing–SPPEXA 2013–2015, Lecture Notes in Computational Science and Engineering, vol. 113, pp. 113–140. Springer, Berlin (2016)

    Google Scholar 

  • Carrayrou, J., Kern, M., Knabner, P.: Reactive transport benchmark of MoMaS. Comput. Geosci. 14(3), 385–392 (2009). https://doi.org/10.1007/s10596-009-9157-7

    Article  Google Scholar 

  • Castelletto, N., White, J.A., Ferronato, M.: Scalable algorithms for three-field mixed finite element coupled poromechanics. J. Comput. Phys. 327, 894–918 (2016)

    Google Scholar 

  • Chen, Z., Huan, G., Ma, Y.: Computational Methods for Multiphase Flows in Porous Media. SIAM, Philadelphia (2006)

    Google Scholar 

  • Clees, T.: AMG Strategies for PDE Systems with Applications in Industrial Semiconductor Simulation. Shaker Verlag GmbH, Herzogenrath (2005)

    Google Scholar 

  • Clees, T., Ganzer, L.: An efficient algebraic multigrid solver strategy for adaptive implicit methods in oil-reservoir simulation. SPE J. 15(03), 670–681 (2010)

    Google Scholar 

  • Courant, R., Friedrichs, K., Lewy, H.: Über die partiellen Differenzengleichungen der mathematischen. Phys. Math. Ann. 100, 32–74 (1928)

    Google Scholar 

  • Crouzeix, M.: Une méthode multipas implicite–explicite pour l’approximation des équations d’évolution paraboliques. Numer. Math. 35(3), 257–276 (1980)

    Google Scholar 

  • 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)

    Google Scholar 

  • Davis, T.A., Rajamanickam, S., Sid-Lakhdar, W.M.: A survey of direct methods for sparse linear systems. Acta Numer. 25, 383–566 (2016)

    Google Scholar 

  • 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(3), 215–249 (1997)

    Google Scholar 

  • Deal.II—an open source finite element library. http://www.dealii.org/

  • Delpopolo Carciopolo, L., Formaggia, L., Scotti, A., Hajibeygi, H.: Conservative multirate multiscale simulation of multiphase flow in heterogeneous porous media. Technical Report, Politecnico di Milano (2019). MOX-Report 04/2019

  • Dendy, J.: Black box multigrid. J. Comput. Phys. 48(3), 366–386 (1982)

    Google Scholar 

  • Deuflhard, P.: Newton Methods for Nonlinear Problems: Affine Invariance and Adaptive Algorithms. Springer, Berlin (2004)

    Google Scholar 

  • Deuflhard, P., Weiser, M.: Adaptive Numerical Solution of PDEs. de Gruyter, Berlin (2012)

    Google Scholar 

  • Diersch, H.J.G.: FEFLOW: Finite Element Modeling of Flow, Mass and Heat Transport in Porous and Fractured Media. Springer, Berlin (2014)

    Google Scholar 

  • Diersch, H.J.G., Kolditz, O.: Coupled groundwater flow and transport: 2. Thermohaline and 3D convection systems. Adv. Water Resour. 21(5), 401–425 (1998)

    Google Scholar 

  • Dobrev, V., Kolev, T., Petersson, N., Schroder, J.: Two-level convergence theory for multigrid reduction in time (MGRIT). SIAM J. Sci. Comput. 39, S501–S527 (2016)

    Google Scholar 

  • Eriksson, K., Estep, D., Hansbo, P., Johnson, C.: Introduction to adaptive methods for differential equations. Acta Numer. 4, 105–158 (1995). https://doi.org/10.1017/S0962492900002531

    Article  Google Scholar 

  • Ern, A., Vohralík, M.: Adaptive inexact Newton methods for discretizations of nonlinear diffusion PDEs. II. Applications. Technical Report, HAL Preprint 00681426 (2012a)

  • Ern, A., Vohralík, M.: Adaptive inexact Newton methods for discretizations of nonlinear diffusion PDEs. I. General theory and a posteriori stopping criteria. Technical Report, HAL Preprint 00681422 (2012b)

  • Falgout, R.D., Schroder, J.B.: Non-Galerkin coarse grids for algebraic multigrid. SIAM J. Sci. Comput. 36(3), 309–334 (2014)

    Google Scholar 

  • Falgout, R.D., Friedhoff, S., Kolev, T.V., MacLachlan, S.P., Schroder, J.B.: Parallel time integration with multigrid. SIAM J. Sci. Comput. 36(6), C635–C661 (2014). LLNL-JRNL-645325

    Google Scholar 

  • Flemisch, B., Darcis, M., Erbertseder, K., Faigle, B., Lauser, A., Mosthaf, K., Müthing, S., Nuske, P., Tatomir, A., Wolff, M., Helmig, R.: Dumux: dune for multi-phase, component, scale, physics,. flow and transport in porous media. Adv. Water Resour. 34(9), 1102–1112 (2011). New computational methods and software tools

    Google Scholar 

  • Forum, M.P.: MPI: a message-passing interface standard. Technical Report, Knoxville (1994)

  • Frolkovič, P.: Application of level set method for groundwater flow with moving boundary. Adv. Water Resour. 47, 56–66 (2012). https://doi.org/10.1016/j.advwatres.2012.06.013

    Article  Google Scholar 

  • Fumagalli, A., Scotti, A.: Numerical modelling of multiphase subsurface flow in the presence of fractures. Commun. Appl. Ind. Math. (2012). https://doi.org/10.1685/journal.caim.380

    Article  Google Scholar 

  • Gander, M.J.: 50 Years of time parallel time integration. In: Carraro, T., Geiger, M., Körkel, S., Rannacher, R. (eds.) Multiple Shooting and Time Domain Decomposition, pp. 69–114. Springer, Berlin (2015)

    Google Scholar 

  • Gander, M.J., Neumüller, M.: Analysis of a new space–time parallel multigrid algorithm for parabolic problems. SIAM J. Sci. Comput. 38(4), A2173–A2208 (2016)

    Google Scholar 

  • Garrido, I., Espedal, M.S., Fladmark, G.E.: A convergent algorithm for time parallelization applied to reservoir simulation. In: Barth, T.J. et al. (eds.) Domain Decomposition Methods in Science and Engineering. Lecture Notes in Computational Science and Engineering, vol. 40. Springer, Berlin (2005)

  • 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). https://doi.org/10.1016/j.cma.2017.08.025

    Article  Google Scholar 

  • Gear, C.W., Wells, D.R.: Multirate linear multistep methods. BIT Numer. Math. 24(4), 484–502 (1984)

    Google Scholar 

  • Gmeiner, B., Rüde, U., Stengel, H., Waluga, C., Wohlmuth, B.: Performance and scalability of hierarchical hybrid multigrid solvers for Stokes systems. SIAM J. Sci. Comput. 37(2), C143–C168 (2015). https://doi.org/10.1137/130941353

    Article  Google Scholar 

  • Griebel, M., Oeltz, D., Schweitzer, A.: An algebraic multigrid for linear elasticity. J. Sci. Comp. 25(2), 385–407 (2003)

    Google Scholar 

  • Grillo, A., Lampe, M., Logashenko, D., Stichel, S., Wittum, G.: Simulation of salinity- and thermohaline-driven flow in fractured porous media. J. Porous Media 15(5), 439–458 (2012)

    Google Scholar 

  • Grossmann, C., Roos, H.G., Stynes, M.: Numerical Treatment of Partial Differential Equations. Universitext. Springer, Berlin (2007)

    Google Scholar 

  • Gupta, A., Karypis, G., Kumar, V.: Highly scalable parallel algorithms for sparse matrix factorization. IEEE Trans. Parallel Distrib. Syst. 8, 502–520 (1994)

    Google Scholar 

  • Hackbusch, W.: Parabolic multigrid methods. In: Glowinski, R., Lions, J.-L. (eds.) Computing Methods in Applied Sciences and Engineering, VI (Versailles, 1983), pp. 189–197. North-Holland, Amsterdam (1984)

    Google Scholar 

  • Hackbusch, W.: Multi-grid Methods and Applications. Springer, Berlin (1985)

    Google Scholar 

  • Hairer, E., Wanner, G.: Solving Ordinary Differential Equations II. Stiff and Differential-Algebraic Problems, vol. 14. Springer, Berlin (1996)

    Google Scholar 

  • Hassanizadeh, M., Gray, W.G.: General conservation equations for multi-phase systems: 1. Averaging procedure. Adv. Water Resour. 2, 131–144 (1979a)

    Google Scholar 

  • Hassanizadeh, M., Gray, W.G.: General conservation equations for multi-phase systems: 2. Mass, momenta, energy, and entropy equations. Adv. Water Resour. 2, 191–203 (1979b)

    Google Scholar 

  • Hassanizadeh, M., Gray, W.G.: General conservation equations for multi-phase systems: 3. Constitutive theory for porous media flow. Adv. Water Resour. 3(1), 25–40 (1980)

    Google Scholar 

  • Helmig, R.: Multiphase Flow and Transport Processes in the Subsurface. A Contribution to the Modeling of Hydrosystems. Springer, Berlin (1997)

    Google Scholar 

  • Helmig, R., Flemisch, B., Wolff, M., Faigle, B.: Efficient modeling of flow and transport in porous media using multi-physics and multi-scale approaches. In: Freeden, W., Nashed, M., Sonar, T. (eds.) Handbook of Geomathematics. Springer, Berlin (2013)

    Google Scholar 

  • Herreros, M., Mabssout, M., Pastor, M.: Application of level-set approach to moving interfaces and free surface problems in flow through porous media. Comput. Methods Appl. Mech. Eng. 195(1–3), 1–25 (2006)

    Google Scholar 

  • Heße, F., Prykhodko, V., Attinger, S.: Assessing the validity of a lower-dimensional representation of fractures for numerical and analytical investigations. Adv. Water Resour. 56, 35–48 (2013)

    Google Scholar 

  • Hoffmann, J., Kräutle, S., Knabner, P.: A general reduction scheme for reactive transport in porous media. Comput. Geosci. 16(4), 1081–1099 (2012)

    Google Scholar 

  • Holm, E.J., Langtangen, H.P.: A method for simulating sharp fluid interfaces in groundwater flow. Adv. Water Resour. 23(1), 83–95 (1999)

    Google Scholar 

  • Holzbecher, E.O.: Modeling Density-Driven Flow in Porous Media. Principles, Numerics, Software. Springer, Berlin (1998). https://doi.org/10.1007/978-3-642-58767-2

    Book  Google Scholar 

  • Hou, T.Y., Li, Z., Osher, S., Zhao, H.: A hybrid method for moving interface problems with application to the Hele–Shaw flow. J. Comput. Phys. 134, 236–252 (1997)

    Google Scholar 

  • Hundsdorfer, W., Verwer, J.: Numerical Solution of Time-Dependent Advection–Diffusion–Reaction Equations. Springer, Berlin (2003)

    Google Scholar 

  • IPARS: A New Generation Framework for Petroleum Reservoir Simulation. http://csm.ices.utexas.edu/ipars/

  • Ippisch, O., Blatt, M.: Scalability test of \(\mu \varphi \) and the parallel algebraic multigrid solver of dune-ISTL. In: Mohr, B., Frings, W. (eds.) Jülich Blue Gene/P Extreme Scaling Workshop 2011, Technical Report FZJ-JSC-IB-2011-02 (Apr. 2011)

  • Jäger, W., Kačur, J.: Solution of porous medium type systems by linear approximation schemes. Numer. Math. 60(1), 407–427 (1991)

    Google Scholar 

  • Johannsen, K.: On the validity of the Boussinesq approximation for the Elder problem. Comput. Geosci. 7(3), 169–182 (2003)

    Google Scholar 

  • Keyes, D.E.: Multiphysics simulations: challenges and opportunities. Int. J. High Perform. Comput. Appl. 27(1), 4–83 (2013)

    Google Scholar 

  • Kim, J., Tchelepi, H., 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)

    Google Scholar 

  • Knabner, P., Angerman, L.: Numerical Methods for Elliptic and Parabolic Partial Differential Equations. No. 44 in Texts in Applied Mathematics. Springer, New York (2003)

    Google Scholar 

  • Kolditz, O., Bauer, S., Bilke, L.: OpenGeoSys: an open-source initiative for numerical simulation of thermo-hydro-mechanical/chemical (THM/C) processes in porous media. Environ. Earth Sci. 67(2), 589–599 (2012)

    Google Scholar 

  • Koric, S., Gupta, A.: Sparse matrix factorization in the implicit finite element method on petascale architecture. Comput. Methods Appl. Mech. Eng. 302, 281–292 (2016)

    Google Scholar 

  • Kröhn, K.P.: Qualifying a computer program for simulating fracture flow. Comput. Vis. Sci. 15(1), 29–37 (2012)

    Google Scholar 

  • Lagneau, V., van der Lee, J.: HYTEC results of the MoMas reactive transport benchmark. Comput. Geosci. 14(3), 435–449 (2009). https://doi.org/10.1007/s10596-009-9159-5

    Article  Google Scholar 

  • Lang, J.: Adaptive Multilevel Solution of Nonlinear Parabolic PDE Systems: Theory, Algorithm, and Applications. Springer, Berlin (2001)

    Google Scholar 

  • Lee, S., Wheeler, M.F., Wick, T.: Iterative coupling of flow, geomechanics and adaptive phase-field fracture including level-set crack width approaches. J. Comput. Appl. Math. 314, 40–60 (2017)

    Google Scholar 

  • Libpfasst: Parallel full approximation scheme in space and time library. http://libpfasst.readthedocs.io/en/latest/index.html#

  • Lichtner, P.C., Steefel, C.I., Oelkers, E.H. (eds.): Reactive Transport in Porous Media, Reviews in Mineralogy, vol. 34. Mineralogical Society of America, Washington, DC (1996)

    Google Scholar 

  • Lie, K.A., Krogstad, S., Ligaarden, I.S., Natvig, J.R., Nilsen, H.M., Skaflestad, B.: Open-source matlab implementation of consistent discretisations on complex grids. Comput. Geosci. 16(2), 297–322 (2012). https://doi.org/10.1007/s10596-011-9244-4

    Article  Google Scholar 

  • Lions, J.L., Maday, Y., Turinici, G.: Résolution d’EDP par un schéma en temps pararéel. C. R. Acad Sci. Paris Sér. I Math. 332, 661–668 (2001)

    Google Scholar 

  • Lu, B., Wheeler, M.: Iterative coupling reservoir simulation on high performance computers. Pet. Sci. 6, 43–50 (2009)

    Google Scholar 

  • Lubich, C.: Linearly implicit extrapolation methods for differential-algebraic systems. Numer. Math. 55, 197–211 (1989)

    Google Scholar 

  • Luo, P., Rodrigo, C., Gaspar, F.J., Oosterlee, C.W.: Uzawa smoother in multigrid for the coupled porous medium and Stokes flow system. SIAM J. Sci. Comput. 39(5), S633–S661 (2017)

    Google Scholar 

  • MacNamara, S., Strang, G.: Operator Splitting, pp. 95–114. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-41589-5_3

    Book  Google Scholar 

  • Mayer, K.U., MacQuarrie, K.T.B.: Solution of the MoMaS reactive transport benchmark with MIN3p—model formulation and simulation results. Comput. Geosci. 14(3), 405–419 (2009). https://doi.org/10.1007/s10596-009-9158-6

    Article  Google Scholar 

  • Mikelić, A., Wheeler, M.F.: Convergence of iterative coupling for coupled flow and geomechanics. Comput. Geosci. 17(3), 455–461 (2013)

    Google Scholar 

  • Morin, P., Nochetto, R., Siebert, K.: Convergence of adaptive finite element methods. SIAM Rev. 44(4), 631–658 (2002). https://doi.org/10.1137/S0036144502409093

    Article  Google Scholar 

  • Nägel, A., Wittum, G.: Scalability of a parallel monolithic multilevel solver for poroelasticity. In: Nagel W., Kröner D., Resch M. (eds.) High Performance Computing in Science and Engineering ’ 18, pp. 427–437. Springer, Cham. https://doi.org/10.1007/978-3-030-13325-2_27

    Chapter  Google Scholar 

  • Nägel, A., Vogel, A., Wittum, G.: Evaluating linear and nonlinear solvers for density driven flow. Comput. Methods Appl. Mech. Eng. 292, 3–15 (2015)

    Google Scholar 

  • Nägel, A., Deuflhard, P., Wittum, G.: Efficient stiff integration of density driven flow problems, ZIB Report 18-54 (2018)

  • Nield, D.A., Bejan, A.: Convection in Porous Media. Springer, New York (2017)

    Google Scholar 

  • Nield, D.A., Simmons, C.T.: A Brief Introduction to Convection in Porous Media. Transport in Porous Media, New York (2018)

    Google Scholar 

  • Nützmann, G., Moser, H.: Elemente Einer Analytischen Hydrologie. Springer, Berlin (2016)

    Google Scholar 

  • Pop, I., Radu, F., Knabner, P.: Mixed finite elements for the Richards’ equation: linearization procedure. J. Comput. Appl. Math. 168(1), 365–373 (2004)

    Google Scholar 

  • Putti, M., Paniconi, C.: Picard and Newton linearization for the coupled model for saltwater intrusion in aquifers. Adv. Water Resour. 18(3), 159–170 (1995)

    Google Scholar 

  • Puzyrev, V., Koric, S., Wilkin, S.: Evaluation of parallel direct sparse linear solvers in electromagnetic geophysical problems. Comput. Geosci. 89, 79–87 (2016)

    Google Scholar 

  • Radu, F.A.: Convergent mass conservative schemes for reactive solute transport and flow in variably saturated porous media. Habilitation Thesis, University of Erlangen-Nuremberg, Germany (2013)

  • Radu, F.A., Nordbotten, J.M., Pop, I.S., Kumar, K.: 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)

    Google Scholar 

  • Reiter, S., Vogel, A., Heppner, I., Rupp, M., Wittum, G.: A massively parallel geometric multigrid solver on hierarchically distributed grids. Comput. Vis. Sci. 16(4), 151–164 (2013)

    Google Scholar 

  • Reiter, S., Logashenko, D., Stichel, S., Wittum, G., Grillo, A.: Models and simulations of variable-density flow in fractured porous media. Int. J. Comput. Sci. Eng. 9(5/6), 416–432 (2014). https://doi.org/10.1504/IJCSE.2014.064527

    Article  Google Scholar 

  • Reiter, S., Logashenko, D., Vogel, A., Wittum, G.: Mesh generation for thin layered domains and its application to parallel multigrid simulation of groundwater flow. Comput. Vis. Sci. 16, 151–164 (2017)

    Google Scholar 

  • Reusken, A.A.: A note on multigrid methods for nonlinear problems. Reports on Applied and Numerical Analysis RANA 95-07, Eindhoven University of Technology (1995)

  • Rudi, J., Ghattas, O., Malossi, A.C.I., Isaac, T., Stadler, G., Gurnis, M., Staar, P.W.J., Ineichen, Y., Bekas, C., Curioni, A.: An extreme-scale implicit solver for complex PDEs. In: Proceedings of the International Conference for High Performance Computing, Networking, Storage and Analysis on—SC’15. ACM Press, New York (2015)

  • Ruge, J.W.: AMG for problems of elasticity. Appl. Math. Comput. 19, 293–309 (1986)

    Google Scholar 

  • Ruge, J.W., Stüben, K.: Algebraic multigrid (AMG). In: McCormick, S.F. (ed.) Multigrid Methods, Frontiers Applied Mathematics, pp. 73–130. SIAM, Philadelphia (1987)

    Google Scholar 

  • Rybak, I., Magiera, J., Helmig, R., Rohde, C.: Multirate time integration for coupled saturated/unsaturated porous medium and free flow systems. Comput. Geosci. 19(2), 299–309 (2015)

    Google Scholar 

  • Saad, Y.: Iterative Methods for Sparse Linear Systems. SIAM, Philadelphia (2003)

    Google Scholar 

  • Saad, Y., van der Vorst, H.A.: Iterative solution of linear systems in the 20th century. J. Comput. Appl. Math. 123(1), 1–33 (2000). https://doi.org/10.1016/S0377-0427(00)00412-X. Numerical Analysis 2000. Vol. III: Linear Algebra

    Article  Google Scholar 

  • Saaltink, M., Carrera, J., Ayora, C.: A comparison of two approaches for reactive transport modelling. J. Geochem. Explor. 69–70, 97–101 (2000)

    Google Scholar 

  • Schenk, O., Gärtner, K.: Solving unsymmetric sparse systems of linear equations with pardiso. J. Future Gener. Comput. Syst. 20(3), 475–487 (2004)

    Google Scholar 

  • Schneider, A., Kröhn, K.P., Püschel, A.: Developing a modelling tool for density-driven flow in complex hydrogeological structures. Comput. Visual Sci. 15(4), 163–168 (2012)

    Google Scholar 

  • Schneider, A., Zhao, H., Wolf, J., Logashenko, D., Reiter, S., Howahr, M., Eley, M., Gelleszun, M., Wiederhold, H.: Modeling saltwater intrusion scenarios for a coastal aquifer at the German north sea. E3S Web Conf. 54, 00031 (2018)

    Google Scholar 

  • Settari, A., Walters, D.A.: Advances in coupled geomechanical and reservoir modeling with applications to reservoir compaction. SPE J. 6(03), 334–342 (2001)

    Google Scholar 

  • Slodička, M.: A robust and efficient linearization scheme for doubly nonlinear and degenerate parabolic problems arising in flow in porous media. SIAM J. Sci. Comput. 23(5), 1593–1614 (2002)

    Google Scholar 

  • Smith, B.F., Bjørstad, P.E., Gropp, W.D.: Domain Decomposition: Parallel Multilevel Methods for Elliptic Partial Differential Equations. Cambridge University Press, New York (1996)

    Google Scholar 

  • Speck, R., Ruprecht, D., Emmett, M., Minion, M., Bolten, M., Krause, R.: A multi-level spectral deferred correction method. BIT Numer. Math. 55(3), 843–867 (2015)

    Google Scholar 

  • Steinbach, O., Yang, H.: Comparison of algebraic multigrid methods for an adaptive space–time finite-element discretization of the heat equation in 3D and 4D. Numer. Linear Algebra Appl. 25(3), e2143 (2018)

    Google Scholar 

  • De Sterck, H., Yang, U., Heys, J.: Reducing complexity in parallel algebraic multigrid preconditioners. SIAM J. Matrix Anal. Appl. 27, 1019–1039 (2006)

    Google Scholar 

  • De Sterck, H., Falgout, R.D., Nolting, J.W., Yang, U.M.: Distance-two interpolation for parallel algebraic multigrid. Numer. Linear Algebra Appl. 15(2–3), 115–139 (2008). Special issue on Multigrid Methods. UCRL-JRNL-230844

    Google Scholar 

  • Stichel, S.: Strömung in geklüftet porösen Medien. Dissertation, Universität Frankfurt (2014)

  • Stüben, K.: Algebraic multigrid (AMG): An introduction with applications. In: Trottenberg, U., Oosterlee, C., Schueller, A. (eds.) Multigrid. Academic Press, Berlin (2000)

    Google Scholar 

  • Toselli, A., Widlund, O.B.: Domain Decomposition Methods—Algorithms and Theory. Springer, Berlin (2005)

    Google Scholar 

  • Trottenberg, U., Oosterlee, C.W., Schüller, A.: Multigrid. Academic Press, San Diego (2001). Contributions by A. Brandt and P. Oswald and K. Stüben

    Google Scholar 

  • Vaněk, P., Mandel, J., Brezina, M.: Algebraic multigrid by smooth aggregation for second and fourth order elliptic problems. Computing 56, 179–196 (1996)

    Google Scholar 

  • Verfürth, R.: A Posteriori Error Estimation Techniques for Finite Element Methods. Numerical Mathematics and Scientific Computation. Oxford University Press, Oxford (2013)

    Google Scholar 

  • Vogel, A., Reiter, S., Rupp, M., Nägel, A., Wittum, G.: UG 4: a novel flexible software system for simulating PDE based models on high performance computers. Comput. Vis. Sci. 16(4), 165–179 (2013)

    Google Scholar 

  • Vohralík, M., Wheeler, M.F.: A posteriori error estimates, stopping criteria, and adaptivity for two-phase flows. Comput. Geosci. 17(5), 789–812 (2013)

    Google Scholar 

  • Voss, C.I., Souza, W.R.: Variable density flow and solute transport simulation of regional aquifers containing a narrow freshwater–saltwater transition zone. Water Resour. Res. 23(10), 1851–1866 (1987)

    Google Scholar 

  • Wallis, J., Kendall, R., Little, T.: Constrained residual acceleration of conjugate residual methods. In: SPE Reservoir Simulation Symposium. Society of Petroleum Engineers (1985)

  • Wang, H.F.: Theory of Linear Poroelasticity with Applications to Geomechanics and Hydrogeology (Princeton Series in Geophysics). Princeton University Press, Princeton (2017)

    Google Scholar 

  • White, J.A., Castelletto, N., Tchelepi, H.A.: Block-partitioned solvers for coupled poromechanics: a unified framework. Comput. Methods Appl. Mech. Eng. 303, 55–74 (2016)

    Google Scholar 

  • White, J.A., Castelletto, N., Klevtsov, S., Bui, Q.M., Osei-Kuffuor, D., Tchelepi, H.A.: A two-stage preconditioner for multiphase poromechanics in reservoir simulation. Comput. Methods Appl. Mech. Eng. 357, 112575 (2019)

    Google Scholar 

  • Wohlmuth, B.I.: Discretization Methods and Iterative Solvers Based on Domain Decomposition. Springer, Berlin (2001)

    Google Scholar 

  • XBraid: Parallel multigrid in time. http://llnl.gov/casc/xbraid

  • Xu, J.: Iterative methods by space decomposition and subspace correction. SIAM Rev. 34(4), 581–613 (1992)

    Google Scholar 

  • Yang, U.: Parallel algebraic multigrid methods—high performance preconditioners. In: Bruaset, A., Tveito, A. (eds.) Numerical Solution of Partial Differential Equations on Parallel Computers, vol. 51, pp. 209–236. Springer, Berlin (2006)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Goethe-Center for Scientific Computing, Goethe-University, Frankfurt, Germany

    Arne Nägel

  2. Computer, Electrical and Mathematical Sciences and Engineering Division (CEMSE), King Abdullah University of Science and Technology, Thuwal, Saudi Arabia

    Dmitry Logashenko

  3. Department of Mathematics and Statistics, University of New Mexico, Albuquerque, USA

    Jacob B. Schroder

  4. Center for Applied Scientific Computing (CASC), Lawrence Livermore National Laboratory, Livermore, CA, USA

    Ulrike M. Yang

Authors
  1. Arne Nägel
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Dmitry Logashenko
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jacob B. Schroder
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Ulrike M. Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Arne Nägel.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

U.M. Yang: Portions of this work were performed under the auspices of the US Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Nägel, A., Logashenko, D., Schroder, J.B. et al. Aspects of Solvers for Large-Scale Coupled Problems in Porous Media. Transp Porous Med 130, 363–390 (2019). https://doi.org/10.1007/s11242-019-01323-w

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11242-019-01323-w

Keywords

Search

Navigation