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A global strategy for solving reactive transport equations. (English) Zbl 1173.76040
Summary: Reactive transport models are complex nonlinear partial differential algebraic equations, coupling the transport engine with the geochemical operator. We propose an efficient and robust global numerical method, based on a method of lines and differential algebraic equations solvers, combined with a Newton method using a powerful sparse linear solver. Numerical experiments show the performance of the method. We also propose a unified framework to describe classical methods such as sequential non-iterative approach, sequential iterative approach, direct substitution approach, and compare them with our global method.

76M25 Other numerical methods (fluid mechanics) (MSC2010)
76V05 Reaction effects in flows
65M20 Method of lines for initial value and initial-boundary value problems involving PDEs
Full Text: DOI
[1] Bethke, C.M., Geochemical reaction modeling: concepts and applications, (1996), Oxford University Press
[2] Walter, A.; Frind, E.; Blowes, D.; Ptacek, C.; Molson, J., Modeling of multicomponent reactive transport in groundwater. 1. model development and evaluation, Water resour. res., 30, 11, 3137-3148, (1994)
[3] Zysset, A.; Stauffer, F.; Dracos, T., Modeling of chemically reactive groundwater transport, Water resour. res., 30, 7, 2217-2228, (1994)
[4] C.I. Steefel, K.T. MacQuarrie, Approaches to modeling of reactive transport in porous media, in: P. Lichtner, C. Steefel, E. Oelkers (Eds.), Reactive Transport in Porous Media, vol. 34 of Rev. Mineral., Mineral. Soc. Am., 1996, pp. 83-125.
[5] Yeh, G.T.; Tripathi, V.S., A critical evaluation of recent developments in hydrogeochemical transport models of reactive multichemical components, Water resour. res., 25, 93-108, (1989)
[6] Carrayrou, J.; MosT, R.; Behra, P., Operator-splitting procedures for reactive transport and comparison of mass balance errors, J. contam. hydrol., 68, 3-4, 239-268, (2004)
[7] Valocchi, A.; Street, R.; Roberts, P., Transport of ion-exchanging solutes in groundwater: chromatographic theory and field simulation, Water resour. res., 17, 5, 1517-1527, (1981)
[8] Miller, C.; Benson, L., Simulation of solute transport in a chemically reactive heterogeneous system: model development and application, Water resour. res., 19, 381-391, (1983)
[9] Cirpka, O.; Helmig, R., Comparison of approaches for the coupling of chemistry to transport in groundwater systems, Notes numer. fluid dynam., 59, 102-120, (1997) · Zbl 0884.76074
[10] Saaltink, M.; Ayora, C.; Carrera, J., A mathematical formulation for reactive transport that eliminates mineral concentrations, Water resour. res., 34, 7, 1649-1656, (1998)
[11] Saaltink, M.; Carrera, J.; Ayora, C., On the behavior of approaches to simulate reactive transport, J. contam. hydrol., 48, 213-235, (2001)
[12] Molins, S.; Carrera, J.; Ayora, C.; Saaltink, M.W., A formulation for decoupling components in reactive transport problems, Water resour. res., 40, W10301, (2004)
[13] Hammond, G.; Valocchi, A.; Lichtner, P., Application of Jacobian-free newton – krylov with physics-based preconditioning to biogeochemical transport, Adv. water resour., 28, 359-376, (2005)
[14] KrSutle, 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, W03429, (2007)
[15] Morel, F.; Morgan, J., A numerical method for computing equilibria in aqueous chemical systems, Environ. sci. tech., 6, 1, 58-67, (1972)
[16] Morel, F.; Hering, J., Principles and applications of aquatic chemistry, (1993), Wiley New York
[17] Lucille, P.L.; Burnol, A.; Ollar, P., Chemtrap: a hydrogeochemical model for reactive transport in porous media, Hydrol. process., 14, 2261-2277, (2000)
[18] Kirkner, D.; Reeves, H., Multicomponent mass transport with homogeneous and heterogeneous chemical reactions: effect of the chemistry on the choice of numerical algorithm. 1. theory, Water resour. res., 24, 1719-1729, (1988)
[19] N. Bouillard, P. Montarnal, R. Herbin, Development of numerical methods for the reactive transport of chemical species in a porous media: a nonlinear conjugate gradient method, in: Int. Conf. Comput. Method Coupled Probl. Sci. Eng. Coupled Probl. 2005, ECCOMAS Conference, Santorini Island, Greece, 2005.
[20] Hundsdorfer, W.; Verwer, J., Numerical solution of time-dependent advection – diffusion – reaction equations, Springer series in computational mathematics, vol. 33, (2003), Springer-Verlag Berlin · Zbl 1030.65100
[21] E. Hairer, G. Wanner, Solving Ordinary Differential Equations. II. Stiff and Differential-Algebraic Problems, No. 14 in SCM, Springer-Verlag, 1991. · Zbl 0729.65051
[22] D. Parkhurst, C. Appelo, User’s guide to PHREEQC (version 2)—a computer program for speciation, batch-reaction, one-dimensional transport, and inverse geochemical calculations, Tech. Rep., U.S. Geological Survey Water-Resources Investigations Report 99-4259, 1999.
[23] Appelo, C., Some calculations on multicomponent transport with cation exchange in aquifers, Groundwater, 32, 968-975, (1994)
[24] Appelo, C.; Postma, D., Geochemistry groundwater and pollution, (2005), A.A. Balkema
[25] Hoteit, H.; Ackerer, P.; MosT, R.; Erhel, J.; Philippe, B., New two-dimensional slope limiters for discontinuous Galerkin methods on arbitrary meshes, Int. J. numer. method eng., 61, 2566-2593, (2004) · Zbl 1075.76575
[26] Barry, D.A.; Miller, C.T.; Culligan, P.J.; Bajracharya, K., Analysis of split operator methods for nonlinear and multispecies groundwater chemical transport models, Math. comput. simul., 43, 331-341, (1997)
[27] J. vander Lee, L. deWindt, CHESS tutorial and cookbook, updated for version 3.0, user’s manual, Tech. Rep. LHM/RD/02/13, École des Mines de Paris, France, 2002.
[28] Kanney, J.F.; Miller, C.T.; Kelley, C.T., Convergence of iterative split operator approaches for approximating nonlinear reactive transport problems, Adv. water resour., 26, 247-261, (2003)
[29] Saaltink, M.; Carrera, J.; Ayora, C., A comparison of two approaches for reactive transport modelling, J. geochem. explor., 69-70, 97-101, (2000)
[30] Ortega, J.; Rheinboldt, W., Iterative solution of nonlinear equations in several variables, Computer science and applied mathematics, (1970), Academic Press · Zbl 0241.65046
[31] J. Carrayrou, ModTlisation du transport de solutTs rTactifs en milieu poreux saturT, ThFse de doctorat, UniversitT Louis Pasteur, Strasbourg, 2001.
[32] Montarnal, P.; Dimier, A.; Deville, E.; Adam, E.; Gaombalet, J.; Bengaouer, A.; Loth, L.; Chavant, C., Coupling methodology within the software platform alliances, ()
[33] Holstad, A., A mathematical and numerical model for reactive fluid flow systems, Comput. geosci., 4, 103-139, (2000) · Zbl 0997.76094
[34] L. Amir, M. Kern, Newton-Krylov methods for coupling transport with chemistry in porous media, in: P. Binning, P. Engesgaard, H. Dahle, G. Pinder, W.G. Gray (Eds.), XVI International Conference on Computational Methods in Water Resources (CMWR XVI), 2006. <http://proceedings.cmwr-xvi.org/>.
[35] Shen, H.; Nicolaidis, N., A direct substitution method for multicomponent solute transport in groundwater, Groundwater, 35, 1, 67-78, (1997)
[36] A.C. Hindmarsh, P.N. Brown, K.E. Grant, S.L. Lee, R.Serban, D.E. Shumaker, C.S. Woodward, SUNDIALS: Suite of nonlinear and differential/algebraic equation solvers, ACM Trans. Math. Softw. 31 (2005) 363-396 (also available as LLNL Technical Report UCRL-JP-200037). · Zbl 1136.65329
[37] R. Eymard, T. Galloudt, R. Herbin, Finite volume methods, Handbook of Numerical Analysis, vol. VII, North-Holland, Amsterdam, 2000, pp. 713-1020. · Zbl 0981.65095
[38] C. de Dieuleveult, An efficient and robust global method for reactive transport, Ph.D. Thesis, University of Rennes 1, December 2008.
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