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

Nicolas Bouillard; Robert Eymard; Raphaele Herbin; Philippe Montarnal

ESAIM: Mathematical Modelling and Numerical Analysis (2007)

  • Volume: 41, Issue: 6, page 975-1000
  • ISSN: 0764-583X

Abstract

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Modeling the kinetics of a precipitation dissolution reaction occurring in a porous medium where diffusion also takes 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 the discontinuous 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.

How to cite

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Bouillard, Nicolas, et al. "Diffusion with dissolution and precipitation in a porous medium: Mathematical analysis and numerical approximation of a simplified model." ESAIM: Mathematical Modelling and Numerical Analysis 41.6 (2007): 975-1000. <http://eudml.org/doc/250079>.

@article{Bouillard2007,
abstract = { Modeling the kinetics of a precipitation dissolution reaction occurring in a porous medium where diffusion also takes 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 the discontinuous 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. },
author = {Bouillard, Nicolas, Eymard, Robert, Herbin, Raphaele, Montarnal, Philippe},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Diffusion; dissolution; precipitation; kinetics; finite volume method.; finite volume method; uniqueness; weak solution},
language = {eng},
month = {12},
number = {6},
pages = {975-1000},
publisher = {EDP Sciences},
title = {Diffusion with dissolution and precipitation in a porous medium: Mathematical analysis and numerical approximation of a simplified model},
url = {http://eudml.org/doc/250079},
volume = {41},
year = {2007},
}

TY - JOUR
AU - Bouillard, Nicolas
AU - Eymard, Robert
AU - Herbin, Raphaele
AU - Montarnal, Philippe
TI - Diffusion with dissolution and precipitation in a porous medium: Mathematical analysis and numerical approximation of a simplified model
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2007/12//
PB - EDP Sciences
VL - 41
IS - 6
SP - 975
EP - 1000
AB - Modeling the kinetics of a precipitation dissolution reaction occurring in a porous medium where diffusion also takes 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 the discontinuous 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.
LA - eng
KW - Diffusion; dissolution; precipitation; kinetics; finite volume method.; finite volume method; uniqueness; weak solution
UR - http://eudml.org/doc/250079
ER -

References

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  1. P. Aagaard and H.C. Helgeson, Thermodynamic and kinetic constraints on reaction rates among minerals and aqueous solutions, I, Theorical considerations. Am. J. Sci.282 (1982) 237–285.  
  2. B. Bary and S. Béjaoui, Assessment of diffusive and mechanical properties of hardened cement pastes using a multi-coated sphere assemblage model. Cem. Concr. Res.36 (2006) 245–258.  
  3. D. Bothe and D. Hilhorst, A reaction diffusion system with fast reversible reaction. J. Math. Anal. Appl.286 (2003) 125–135.  
  4. 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: .  URIhttp://hal.archives-ouvertes.fr/
  5. K. Deimling, Nonlinear functional analysis. Springer-Verlag, Berlin (1985).  
  6. R. Eymard, T. Gallouët, M. Ghilani and R. Herbin, An error estimate for a finite volume scheme forba nonlinear hyperbolic equation on a triangular mesh. IMA J. Numer. Anal.18 (1998) 563–594.  
  7. R. Eymard, T. Gallouët and R. Herbin, Convergence of finite volume schemes for semilinear convection diffusion equations. Numer. Math.82 (1999) 91–116.  
  8. 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 Proceedings6, Soc. Math. Appl. Indust., Paris (1999) 41–55.  
  9. R. Eymard, T. Gallouët and R. Herbin, Finite Volume Methods, in Techniques of Scientific Computing (Part III), Handbook for Numerical AnalysisVII, P.G Ciarlet and J.L. Lions Eds., North Holland (2000) 713–1020.  
  10. R. Eymard, T. Gallouët, M. Gutnic, R. Herbin and D. Hilhorst, Approximation by the finite volume method of an elliptic-parabolic equation arising in environmental studies. Math. Models Methods Appl. Sci.11 (2001) 1505–1528.  
  11. 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.  
  12. R. Eymard, T. Gallouët, R. Herbin and A. Michel, Convergence of a finite volume scheme for nonlinear degenerate parabolic equations. Num. Math.92 (2002) 41–82.  
  13. J. Ganor, T.J. Huston and L.M. Walter, Quartz precipitation kinetics at 180 °C in NaCl solutions. Implications for the usability of the principle of detailed balancing. Geochim. Cosmochim. Acta69 (2005) 2043–2056.  
  14. 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. Acta66 10 (2002) 1739–1757.  
  15. D. Hilhorst, R. van der Hout and L. Peletier, The fast reaction limit for a reaction-diffusion system. J. Math. Anal. Appl.199 (1996) 349–373.  
  16. A. Holstad, A mathematical and numerical model for reactive fluid flow systems. Comp. Geosc.4 (2000) 103–139.  
  17. U. Hornung, W. Jäger and A. Mikelić, Reactive transport through an array of cells with semipermeable membranes. RAIRO Modél. Math. Anal. Numér.28 (1994) 59–94.  
  18. P. Knabner, C.J. van Duijn and S. Hengst, An analysis of crystal dissolution fronts in flows through porous media. Part 1: Compatible boundary conditions. Adv. Water Res.18 (1995) 171–185.  
  19. D. Langmuir, Aqueous environmental geochemistry. Prentice Hall (1997).  
  20. A.C. Lasaga, Kinetic Theory in the Earth Sciences. Princeton University Press (1998).  
  21. E. Maisse and J. Pousin, Diffusion and dissolution/precipitation in an open porous reactive medium. J. Comp. Appl. Math.82 (1997) 279–290.  
  22. E. Maisse and J. Pousin, Finite element approximation of mass transfer in a porous medium with non equilibrium phase change. Numer. Math12 (2004) 207–231.  
  23. 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.  
  24. 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: .  URIhttp://hal.archives-ouvertes.fr/
  25. J.W. Morse and R.S. Arvidson, The dissolution kinetics of major sedimentary carbonate minerals. Earth Science Reviews58 (2002) 51–84.  
  26. P. Moszkowicz, J. Pousin and F. Sanchez, Diffusion and dissolution in a reactive porous medium: Mathematical modelling and numerical simulations. J. Comp. Appl. Math.66 (1996) 377–389.  
  27. 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.  
  28. D. Planel, J. Sercombe, P. Le Bescop, F. Adenot and J.-M. Torrenti, Long-term performance of cement paste during combined calcium leaching-sulfate attack: kinetics and size effect. Cem. Concr. Res.36 (2006) 137–143.  
  29. J. Pousin, Infinitely fast kinetics for dissolution and diffusion in open reactive systems. Nonlin. Anal.39 (2000) 261–279.  
  30. G. Stampacchia, Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus. Ann. Inst. Fourier15 (1965) 189–258.  
  31. C.I. Steefel and A.C. Lasaga, 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.  
  32. 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.  

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