Abstract
In this article, an approach for the efficient numerical solution of multi-species reactive transport problems in porous media is described. The objective of this approach is to reformulate the given system of partial and ordinary differential equations (PDEs, ODEs) and algebraic equations (AEs), describing local equilibrium, in such a way that the couplings and nonlinearities are concentrated in a rather small number of equations, leading to the decoupling of some linear partial differential equations from the nonlinear system. Thus, the system is handled in the spirit of a global implicit approach (one step method) avoiding operator splitting techniques, solved by Newton’s method as the basic algorithmic ingredient. The reduction of the problem size helps to limit the large computational costs of numerical simulations of such problems. If the model contains equilibrium precipitation-dissolution reactions of minerals, then these are considered as complementarity conditions and rewritten as semismooth equations, and the whole nonlinear system is solved by the semismooth Newton method.
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Amir, L., Kern, M.: Newton-krylov methods for coupling transport with chemistry in porous media. In: CMWR—XVI International Conference on Computational Methods in Water Resources. Copenhagen, Denmark (2006)
Bethke, C.M.: Geochemical Reaction Modeling. Oxford University Press, New York (1996)
Bourgeat, A., Bryant, S., Carrayrou, J., Dimier, A., Duijn, C.V., Kern, M., Knabner, P., Leterrier, N.: GDR MoMaS benchmark reactive transport. http://www.gdrmomas.org/ex_qualifications.html
Carrayrou, J., Hoffmann, J., Knabner, P., Kräutle, S., de Dieuleveult, C., Erhel, J., Van der Lee, J., Lagneau, V., Kern, M., Amir, L., Mayer, K., McQuarrie, K.: A synthesis of the MoMaS reactive transport results. Comput. Geosci. (2009, submitted)
Carrayrou, J., Kern, M., Knabner, P.: Reactive transport benchmark of MoMaS. Comput. Geosci. (2009). doi:10.1007/s10596-009-9157-7
Hoffmann, J.: Ein Entkopplungsverfahren für Systeme von Transportreaktionsgleichungen in porösen Medien: Algorithmische Realisierung und Simulation realistischer 2D-Szenarien. in German, Friedrich-Alexander-Universität Erlangen-Nürnberg, Lehrstuhl für Angewandte Mathematik I, Deutschland (2005). http://www.am.uni-erlangen.de/am1/en/theses.html
Hoffmann, J.: Results of the GdR MoMaS reactive transport benchmark with RICHY2D. Preprint No. 326, Department of Mathematics, University of Erlangen-Nuremberg, Erlangen, Germany (2008). ISSN 1435-5833: http://www.am.uni-erlangen.de/papers/pr326.pdf
Kanzow, C.: Inexakt semismooth newton methods for large-scale complementarity problems. Optim. Methods Softw. 19(3–4), 309–325 (2004)
Kräutle, S.: General multi-species reactive transport problems in porous media: Efficient numerical approaches and existence of global solutions. Habilitation thesis, University Erlangen–Nuremberg (2008). http://www.am.uni-erlangen.de/am1/en/theses.html
Kräutle, S., Knabner, P.: A new numerical reduction scheme for coupled multicomponent transport-reaction problems in porous media: generalization to problems with heterogeneous equilibrium reactions. Water Resour. Res. 43, W03429 (2007). doi:10.1029/2005WR004465
Kräutle, S., Knabner, P.: A new numerical reduction scheme for fully coupled multicomponent transport-reaction problems in porous media. Water Resour. Res. 41, W09414 (2005) doi:10.1029/2004WR003624
Lichtner, P.C., Steefel, C.I., Oelkers, E.H. (eds.): Reactive Transport in Porous Media. Reviews in Mineralogy 34. Mineralogical Society of America, Washington (1996)
Molins, S., Carrera, J., Ayora, C., Saaltink, M.W.: A formulation for decoupling components in reactive transport problems. Water Resour. Res. 40(10), W10,301 (2004). doi:10.1029/2003WR002970
Prechtel, A., Hoffmann, J., Kräutle, S., Knabner, P.: Reaktive Mehrkomponentenprobleme: Sicherung von Effizienz und Zuverlässigkeit. In: Modellierung und Prognose von Natural Attenuation-Prozessen im Untergrund. Statusseminar des KORA-TV 7, 08.06.2006, Gemeinsame Mitteilungen des Dresdner Grundwasserforschungszentrums e.V., pp. 75–90. Eigenverlag des Hrsg., Dresden (2006). http://www.dgfz.de/Neue_DGFZ-Seite/Dateien/TV7-Statusseminar.pdf
Saaltink, M.W., Ayora, C., Carrera, J.: A mathematical formulation for reactive transport that eliminates mineral concentrations. Water Resour. Res. 34(7), 1649–1656 (1998)
Wieners, C.: Distributed point objects. a new concept for parallel finite elements. In: Kornhuber, R., Hoppe, R., Priaux, J., Pironneau, O., Widlund, O., Xu, J. (eds.) Domain Decomposition Methods in Science and Engineering. Lecture Notes in Computational Science and Engineering, vol. 40, pp. 175–183. Springer, New York (2005)
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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, 421–433 (2010). https://doi.org/10.1007/s10596-009-9173-7
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DOI: https://doi.org/10.1007/s10596-009-9173-7