Computational Geosciences

, Volume 23, Issue 1, pp 127–148 | Cite as

A global implicit solver for miscible reactive multiphase multicomponent flow in porous media

  • Fabian BrunnerEmail author
  • Peter Knabner
Original Paper


We present a numerical framework for efficiently simulating partially miscible two-phase flow with multicomponent reactive transport in porous media using the global implicit approach. The mathematical model consists of coupled and nonlinear partial differential equations, ordinary differential equations, and algebraic equations. Our approach is based on a model-preserving reformulation using the reduction scheme of Kräutle and Knabner (Water Resour. Res. 43(3), 2007), Hoffmann et al. (Comput. Geosci. 16(4):1081–1099, 2012) to transform the system. Moreover, a nonlinear, implicitly defined resolution function to reduce its size is employed. By choosing persistent primary variables and using a complementarity approach, mineral reactions and the local appearance and disappearance of the gas phase can be handled without a discontinuous switch of primary variables. In each time step of the Euler-implicit time stepping scheme, the discrete nonlinear systems are solved using the Semismooth Newton method for linearization using the global implicit approach. Thus, we obtain an efficient, robust, and stable simulation method allowing for large time steps and avoiding the potential drawbacks of splitting approaches.


Multiphase multicomponent flow Phase transitions Reactive transport Porous media Reduction of problem size Global implicit approach 


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  1. 1.
    Abadpour, A., Panfilov, M.: Method of negative saturations for modeling two-phase compositional flow with oversaturated zones. Transp. Porous Media 79(2), 197–214 (2009)CrossRefGoogle Scholar
  2. 2.
    Ahusborde, E., Kern, M., Vostrikov, V.: Numerical simulation of two-phase multicomponent flow with reactive transport in porous media: application to geological sequestration of CO2. ESAIM: Proc. Surv. 50, 21–39 (2015)CrossRefGoogle Scholar
  3. 3.
    Amir, L., Kern, M.: A global method for coupling transport with chemistry in heterogeneous porous media. Comput. Geosci. 14(3), 465–481 (2010)CrossRefGoogle Scholar
  4. 4.
    Bear, J.: Dynamics of Fluids in Porous Media. Elsevier, New York (1972)Google Scholar
  5. 5.
    Ben Gharbia, I., Jaffré, J.: Gas phase appearance and disappearance as a problem with complementarity constraints. Math. Comput. Simul. 99, 28–36 (2014)CrossRefGoogle Scholar
  6. 6.
    Bethke, C.: Geochemical Reaction Modeling: Concepts and applications. Oxford University Press, New York (1996)Google Scholar
  7. 7.
    Bourgeat, A., Granet, S., Smaï, F.: Compositional two-phase flow in saturated-unsaturated porous media: benchmarks for phase appearance/disappearance. In: Bastian, P., Kraus, J., Scheichl, R., Wheeler, M. (eds.) Simulation of Flow in Porous Media, volume 12 of Radon Series on Computational and Applied Mathematics, pp. 81–106. Walter de Gruyter (2013)Google Scholar
  8. 8.
    Bourgeat, A., Jurak, M., Smaï, F.: On persistent primary variables for numerical modeling of gas migration in a nuclear waste repository. Comput. Geosci. 17(2), 287–305 (2013)CrossRefGoogle Scholar
  9. 9.
    Brunner, F., Frank, F., Knabner, P.: FV upwind stabilization of FE discretizations for advection–diffusion problems. In: Fuhrmann, J., Ohlberger, M., Rohde, C. (eds.) Finite Volumes for Complex Applications VII–Methods and Theoretical Aspects, volume 77 of Springer Proceedings in Mathematics & Statistics, pp. 177–185. Springer International Publishing (2014)Google Scholar
  10. 10.
    Buchholzer, H., Kanzow, C., Knabner, P., Kräutle, S.: The semismooth newton method for the solution of reactive transport problems including mineral precipitation-dissolution reactions. Comput. Optim. Appl. 50(2), 193–221 (2010)CrossRefGoogle Scholar
  11. 11.
    Carrayrou, J., Hoffmann, J., Knabner, P., Kräutle, S., de Dieuleveult, C., Erhel, J., van der Lee, J., Lagneau, V., Mayer, K.U., MacQuarrie, K.T.B.: Comparison of numerical methods for simulating strongly nonlinear and heterogeneous reactive transport problems—the MoMaS benchmark case. Comput. Geosci. 14(3), 483–502 (2010)CrossRefGoogle Scholar
  12. 12.
    Carrayrou, J., Kern, M., Knabner, P.: Reactive transport benchmark of MoMaS. Comput. Geosci. 14 (3), 385–392 (2010)CrossRefGoogle Scholar
  13. 13.
    Class, H., Helmig, R., Bastian, P.: Numerical simulation of nonisothermal multiphase multicomponent processes in porous media.: 1. An efficient solution technique. Adv. Water Resour. 25(5), 533–550 (2002)CrossRefGoogle Scholar
  14. 14.
    de Cuveland, R.: Two-Phase Compositional Flow Simulation with Persistent Variables. PhD thesis, Universität Heidelberg (2015)Google Scholar
  15. 15.
    de Dieuleveult, C., Erhel, J.: A global approach to reactive transport: application to the MoMaS benchmark. Comput. Geosci. 14(3), 451–464 (2010)CrossRefGoogle Scholar
  16. 16.
    de Dieuleveult, C., Erhel, J., Kern, M.: A global strategy for solving reactive transport equations. J. Comput. Phys. 228(17), 6395–6410 (2009)CrossRefGoogle Scholar
  17. 17.
    de Luca, T., Facchinei, F., Kanzow, C.: A Theoretical and numerical comparison of some semismooth algorithms for complementarity problems. Comput. Optim. Appl. 16(2), 173–205 (2000)CrossRefGoogle Scholar
  18. 18.
    Duan, Z., Møller, N., Weare, J.H.: An equation of state for the CH4-CO2-H2O system: I. Pure systems from 0 to 1000C and 0 to 8000 bar. Geochim. Cosmochim. Acta 56(7), 2605–2617 (1992)CrossRefGoogle Scholar
  19. 19.
    Eck, C., Garcke, H., Knabner, P.: Mathematical Modeling. Springer International Publishing, Berlin (2017)CrossRefGoogle Scholar
  20. 20.
    Fan, Y., Durlofsky, L.J., Tchelepi, H.: A fully-coupled flow-reactive-transport formulation based on element conservation, with application to CO2 storage simulations. Adv. Water Resour. 42, 47–61 (2012)CrossRefGoogle Scholar
  21. 21.
    Fenghour, A., Wakeham, W.A., Vesovic, V.: The viscosity of carbon dioxide. J. Phys. Chem. Ref. Data 27(1), 31–44 (1998)CrossRefGoogle Scholar
  22. 22.
    Forsyth, P.A., Simpson, R.B.: A two-phase, two-component model for natural convection in a porous medium. Int. J. Numer. Methods Fluids 12(7), 655–682 (1991)CrossRefGoogle Scholar
  23. 23.
    García, J. E.: Density of Aqueous Solutions of CO2. Lawrence Berkeley National Laboratory (2001)Google Scholar
  24. 24.
    Hammond, G., Lichtner, P., Lu, C.: Subsurface multiphase flow and multicomponent reactive transport modeling using high-performance computing. J. Phys. Conf. Ser. 78(1), 012025 (2007)CrossRefGoogle Scholar
  25. 25.
    Hao, Y., Sun, Y., Nitao, J.J.: Overview of NUFT: a versatile numerical model for simulating flow and reactive transport in porous media. In: Zhang, F., Yeh, G.-T., Parker, J.C. (eds.) Groundwater Reactive Transport Models, pp. 212–239. Bentham Science Publishers (2012)Google Scholar
  26. 26.
    Hoffmann, J.: Reactive Transport and Mineral Dissolution/Precipitation in Porous Media, Efficient Solution Algorithms, Benchmark Computations and Existence of Global Solutions. PhD Thesis, friedrich-alexander-universität erlangen-nürnberg (2010)Google Scholar
  27. 27.
    Hoffmann, J., Kräutle, S., Knabner, P.: A parallel global–implicit 2-D solver for reactive transport problems in porous media based on a reduction scheme and its application to the MoMaS benchmark problem. Comput. Geosci. 14(3), 421–433 (2010)CrossRefGoogle Scholar
  28. 28.
    Hoffmann, J., Kräutle, S., Knabner, P.: A general reduction scheme for reactive transport in porous media. Comput. Geosci. 16(4), 1081–1099 (2012)CrossRefGoogle Scholar
  29. 29.
    Knabner, P., Angermann, L.: Numerical Methods for Elliptic and Parabolic Partial Differential Equations. Springer, New York (2003)Google Scholar
  30. 30.
    Kräutle, S.: General Multi-Species Reactive Transport Problems in Porous Media: Efficient Numerical Approaches and Existence of Global Solutions. Habilitation thesis, Friedrich-Alexander-Universität Erlangen-Nürnberg (2008)Google Scholar
  31. 31.
    Kräutle, S.: The semismooth Newton method for multicomponent reactive transport with minerals. Adv. Water Resour. 34(1), 137–151 (2011)CrossRefGoogle Scholar
  32. 32.
    Kräutle, S., Knabner, P.: A new numerical reduction scheme for fully coupled multicomponent transport-reaction problems in porous media. Water Resour. Res. 41(9), W09414 (2005)Google Scholar
  33. 33.
    Kräutle, S., Knabner, P.: A reduction scheme for coupled multicomponent transport-reaction problems in porous media: Generalization to problems with heterogeneous equilibrium reactions. Water Resour. Res. 43(3), W03429 (2007)Google Scholar
  34. 34.
    Lauser, A., Hager, C., Helmig, R., Wohlmuth, B.: A new approach for phase transitions in miscible multi-phase flow in porous media. Adv. Water Resour. 34(8), 957–966 (2011)CrossRefGoogle Scholar
  35. 35.
    Marchand, E., Müller, T., Knabner, P.: Fully coupled generalized hybrid-mixed finite element approximation of two-phase two-component flow in porous media Part I: formulation and properties of the mathematical model. Comput. Geosci. 17(2), 431–442 (2013)CrossRefGoogle Scholar
  36. 36.
    Millington, R.J., Quirk, J.P.: Permeability of porous solids. Trans. Faraday Soc. 57, 1200–1207 (1961)CrossRefGoogle Scholar
  37. 37.
    Neumann, R., Bastian, P., Ippisch, O.: Modeling and simulation of two-phase two-component flow with disappearing nonwetting phase. Comput. Geosci. 17(1), 139–149 (2013)CrossRefGoogle Scholar
  38. 38.
    Qi, L., Sun, J.: A nonsmooth version of newton’s method. Math. Program. 58(3), 353–367 (1993)CrossRefGoogle Scholar
  39. 39.
    Saaltink, M.W., Carrera, J., Ayora, C.: A comparison of two approaches for reactive transport modelling. J. Geochem. Explor. 69–70, 97–101 (2000)CrossRefGoogle Scholar
  40. 40.
    Saaltink, M.W., Vilarrasa, V., De Gaspari, F., Silva, O., Carrera, J., Rötting, T.S.: A method for incorporating equilibrium chemical reactions into multiphase flow models for CO2 storage, vol. 62 (2013)Google Scholar
  41. 41.
    Spycher, N., Pruess, K.: CO2-H2O mixtures in the geological sequestration of CO2. II partitioning in chloride brines at 12–100C and up to 600 bar. Geochim. Cosmochim. Acta 69(13), 3309–3320 (2005)CrossRefGoogle Scholar
  42. 42.
    Valocchi, A.J., Malmstead, M.: Accuracy of operator splitting for advection-dispersion-reaction problems. Water Resour. Res. 28(5), 1471–1476 (1992)CrossRefGoogle Scholar
  43. 43.
    Xu, T., Sonnenthal, E., Spycher, N., Zhang, G., Zheng, L., Pruess, K.: TOUGHREACT: a simulation program for subsurface reactive chemical transport under non-isothermal multiphase flow conditions. In: Zhang, F., Yeh, G.-T., Parker, J.C. (eds.) Groundwater Reactive Transport Models, pp. 74–95. Bentham Science Publishers (2012)Google Scholar
  44. 44.
    Yeh, G.-T., Tripathi, V.S.: A critical evaluation of recent developments in hydrogeochemical transport models of reactive multichemical components. Water Resour. Res. 25(1), 93–108 (1989)CrossRefGoogle Scholar

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© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Erlangen-NürnbergErlangenGermany

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