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Diffusion with dissolution and precipitation in a porous medium: Mathematical analysis and numerical approximation of a simplified model

Published online by Cambridge University Press:  15 December 2007

Nicolas Bouillard ,
Robert Eymard ,
Raphaele Herbin  and
Philippe Montarnal
Nicolas Bouillard
Affiliation:
CEA/Saclay, DM2S/SFME/MTMS, Bât. 454 p. 18, 91191 Gif-sur-Yvette Cedex, France.
Robert Eymard
Affiliation:
Université de Marne-la-Vallée, 5 boulevard Descartes, 77454 Champs-sur-Marne, France.
Raphaele Herbin
Affiliation:
Université de Provence, LATP, UMR 6632, 39 rue Joliot Curie, 13453 Marseille, France. Raphaele.Herbin@cmi.univ-mrs.fr
Philippe Montarnal
Affiliation:
CEA/Saclay, DM2S/SFME/MTMS, Bât. 454 p. 17, 91191 Gif-sur-Yvette Cedex, France.
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Abstract

Modeling the kinetics of a precipitation dissolution reaction occurring in a porous medium where diffusion alsotakes place leads to a system of two parabolic equations and one ordinary differential equation coupled with a stiff reaction term. This system is discretized by a finite volume scheme which is suitable for the approximation of thediscontinuous reaction term of unknown sign. Discrete solutions are shown to exist and converge towards a weak solution of the continuous problem. Uniqueness is proved under a Lipschitz condition on the equilibrium gap function. Numerical tests are shown which prove the efficiency of the scheme.

Keywords

Diffusiondissolutionprecipitationkineticsfinite volume method.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 41 , Issue 6 , November 2007 , pp. 975 - 1000
Copyright
© EDP Sciences, SMAI, 2007

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References

Aagaard, P. and Helgeson, H.C., Thermodynamic and kinetic constraints on reaction rates among minerals and aqueous solutions, I, Theorical considerations. Am. J. Sci. 282 (1982) 237–285. CrossRef
Bary, B. and Béjaoui, S., Assessment of diffusive and mechanical properties of hardened cement pastes using a multi-coated sphere assemblage model. Cem. Concr. Res. 36 (2006) 245–258. CrossRef
Bothe, D. and Hilhorst, D., A reaction diffusion system with fast reversible reaction. J. Math. Anal. Appl. 286 (2003) 125–135. CrossRef
N. Bouillard, P. Montarnal and 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. on Computational Methods for Coupled Problems in Science and Engineering, Coupled Problems (2005), M. Papadrakis, E. Onate and B. Schreffer Eds., CIMNE, Barcelona, Spain, p. 229, ISBN: 84-95999-71-4, available on CD. See also: http://hal.archives-ouvertes.fr/.
K. Deimling, Nonlinear functional analysis. Springer-Verlag, Berlin (1985).
Eymard, R., Gallouët, T., Ghilani, M. and Herbin, R., An error estimate for a finite volume scheme forba nonlinear hyperbolic equation on a triangular mesh. IMA J. Numer. Anal. 18 (1998) 563–594. CrossRef
Eymard, R., Gallouët, T. and Herbin, R., Convergence of finite volume schemes for semilinear convection diffusion equations. Numer. Math. 82 (1999) 91–116. CrossRef
R. Eymard, T. Gallouët, R. Herbin, D. Hilhorst and M. Mainguy, Instantaneous and noninstantaneous dissolution: approximation by the finite volume method, in Actes du 30ème Congrès d'Analyse Numérique, ESAIM Proceedings 6, Soc. Math. Appl. Indust., Paris (1999) 41–55.
R. Eymard, T. Gallouët and R. Herbin, Finite Volume Methods, in Techniques of Scientific Computing (Part III), Handbook for Numerical Analysis VII, P.G Ciarlet and J.L. Lions Eds., North Holland (2000) 713–1020.
Eymard, R., Gallouët, T., Gutnic, M., Herbin, R. and Hilhorst, D., Approximation by the finite volume method of an elliptic-parabolic equation arising in environmental studies. Math. Models Methods Appl. Sci. 11 (2001) 1505–1528. CrossRef
R. Eymard, D. Hilhorst, R. van der Hout and L.A. Peletier, A reaction diffusion system approximation of a one phase Stefan problem, in Optimal Control and Partial Differential Equations, IOS Press (2001) 156–170.
Eymard, R., Gallouët, T., Herbin, R. and Michel, A., Convergence of a finite volume scheme for nonlinear degenerate parabolic equations. Num. Math. 92 (2002) 41–82. CrossRef
Ganor, J., Huston, T.J. and Walter, L.M., Quartz precipitation kinetics at 180 °C in NaCl solutions. Implications for the usability of the principle of detailed balancing. Geochim. Cosmochim. Acta 69 (2005) 2043–2056. CrossRef
E.R. Giambalvo, C.I. Steefel, A.T. Fisher, N.D. Rosenberg and C.G. Wheat, Effect of fluid-sediment reaction on hydrothermal fluxes of major elements, Eastern flank of the Juan de Fuca Ridge. Geochim. Cosmochim. Acta 66 10 (2002) 1739–1757.
Hilhorst, D., van der Hout, R. and Peletier, L., The fast reaction limit for a reaction-diffusion system. J. Math. Anal. Appl. 199 (1996) 349–373. CrossRef
Holstad, A., A mathematical and numerical model for reactive fluid flow systems. Comp. Geosc. 4 (2000) 103–139. CrossRef
Hornung, U., Jäger, W. and Mikelić, A., Reactive transport through an array of cells with semipermeable membranes. RAIRO Modél. Math. Anal. Numér. 28 (1994) 59–94. CrossRef
Knabner, P., van Duijn, C.J. and Hengst, S., An analysis of crystal dissolution fronts in flows through porous media. Part 1: Compatible boundary conditions. Adv. Water Res. 18 (1995) 171–185. CrossRef
D. Langmuir, Aqueous environmental geochemistry. Prentice Hall (1997).
A.C. Lasaga, Kinetic Theory in the Earth Sciences. Princeton University Press (1998).
Maisse, E. and Pousin, J., Diffusion and dissolution/precipitation in an open porous reactive medium. J. Comp. Appl. Math. 82 (1997) 279–290. CrossRef
Maisse, E. and Pousin, J., Finite element approximation of mass transfer in a porous medium with non equilibrium phase change. Numer. Math 12 (2004) 207–231. CrossRef
E. Maisse, P. Moszkowicz and J. Pousin, Diffusion and dissolution in a reactive porous medium: modeling and numerical simulations, in Proceedings of the Mathematical modelling of flow through porous media: proceedings of the conference, A.P. Bourgeat, C. Carasso, S. Luckhaus and A. Mikelic Eds., World Scientific (1995), p. 515, ISBN: 981-02-2483-4.
P. Montarnal, A. Dimier, E. Deville, E. Adam, J. Gaombalet, A. Bengaouer, L. Loth and C. Chavant, Coupling methodology within the software platform Alliances, in Int. Conf. on Computational Methods for Coupled Problems in Science and Engineering, Coupled Problems 2005, M. Papadrakis, E. Onate and B. Schreffer Eds., CIMNE, Barcelona, Spain, p. 229, ISBN: 84-95999-71-4, available on CD. See also: http://hal.archives-ouvertes.fr/.
Morse, J.W. and Arvidson, R.S., The dissolution kinetics of major sedimentary carbonate minerals. Earth Science Reviews 58 (2002) 51–84. CrossRef
Moszkowicz, P., Pousin, J. and Sanchez, F., Diffusion and dissolution in a reactive porous medium: Mathematical modelling and numerical simulations. J. Comp. Appl. Math. 66 (1996) 377–389. CrossRef
C. Mügler, P. Montarnal, A. Dimier and L. Trotignon, Reactive transport modelling in the Alliances software platform, in Proceeding of CMWR, 13–17 June (2004), Chapel Hill, USA (2004) 1103–1115.
Planel, D., Sercombe, J., Le Bescop, P., Adenot, F. and Torrenti, J.-M., Long-term performance of cement paste during combined calcium leaching-sulfate attack: kinetics and size effect. Cem. Concr. Res. 36 (2006) 137–143. CrossRef
Pousin, J., Infinitely fast kinetics for dissolution and diffusion in open reactive systems. Nonlin. Anal. 39 (2000) 261–279. CrossRef
Stampacchia, G., Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus. Ann. Inst. Fourier 15 (1965) 189–258. CrossRef
Steefel, C.I. and Lasaga, A.C., A coupled model for transport of multiple chemical species and kinetic precipitation/dissolution reactions with application to reactive flow in single phase hydrothermal systems. Am. J. Sci. 294 (1994) 529–592. CrossRef
C.J. van Duijn and I.S. Pop, Crystal dissolution and precipitation in porous media: pore scale analysis. J. Reine Angew. Math. 577 (2004) 171–211.