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Convergence analysis for a conformal discretization of a model for precipitation and dissolution in porous media

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Abstract

In this paper we discuss the numerical analysis of an upscaled (core scale) model describing the transport, precipitation and dissolution of solutes in a porous medium. The particularity lies in the modeling of the reaction term, especially the dissolution term, which has a multivalued character. We consider the weak formulation for the upscaled equation and provide rigorous stability and convergence results for both the semi-discrete (time discretization) and the fully discrete schemes. In doing so, compactness arguments are employed.

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References

  1. Rubin, J.: Transport of reacting solutes in porous media: relation between mathematical nature of problem formulation and chemical nature of reaction. Water Resour. Res. 19, 1231–1252 (1983)

    Google Scholar 

  2. Metzmacher, I., Radu, F.A., Bause, M., Knabner, P., Friess, W.: A model describing the effect of enzymatic degradation on drug release from collagen minirods. Eur. J. Pharm. Biopharm. 67(2), 349–360 (2007)

    Article  Google Scholar 

  3. Moszkowicz, P., Pousin, J., Sanchez, F.: Diffusion and dissolution in a reactive porous medium: mathematical modelling and numerical simulations. In: Proceedings of the Sixth International Congress on Computational and Applied Mathematics (Leuven, 1994), vol. 66, pp. 377–389 (1996)

  4. Pawell, A., Krannich, K.-D.: Dissolution effects in transport in porous media. SIAM J. Appl. Math. 56(1), 89–118 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  5. Pousin, J.: Infinitely fast kinetics for dissolution and diffusion in open reactive systems. Nonlinear Anal. 39(3, Ser. A: Theory Methods), 261–279 (2000)

  6. Bouillard, N., Eymard, R., Herbin, R., Montarnal, P.: Diffusion with dissolution and precipitation in a porous medium: mathematical analysis and numerical approximation of a simplified model. M2AN. Math. Model. Numer. Anal. 41(6), 975–1000 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  7. Knabner, P., van Duijn, C.J., Hengst, S.: An analysis of crystal dissolution fronts in flows through porous media. Part 1: compatible boundary conditions. Adv. Water Resour. 18, 171–185 (1995)

    Article  Google Scholar 

  8. van Duijn, C.J., Knabner, P.: Solute transport through porous media with slow adsorption. In Free boundary problems: theory and applications, vol. I (Irsee, 1987). Pitman Res. Notes Math. Ser., vol. 185, pp. 375–388. Longman Sci. Tech., Harlow (1990)

  9. van Duijn, C.J., Knabner, P.: Solute transport in porous media with equilibrium and nonequilibrium multiple-site adsorption: travelling waves. J. Reine Angew. Math. 415, 1–49 (1991)

    MATH  MathSciNet  Google Scholar 

  10. van Duijn, C.J., Knabner, P.: Travelling wave behaviour of crystal dissolution in porous media flow. Eur. J. Appl. Math. 8(1), 49–72 (1997)

    MATH  Google Scholar 

  11. van Duijn, C.J., Pop, I.S.: Crystal dissolution and precipitation in porous media: pore scale analysis. J. Reine Angew. Math. 577, 171–211 (2004)

    MATH  MathSciNet  Google Scholar 

  12. van Noorden, T.L., Pop, I.S.: A Stefan problem modelling crystal dissolution and precipitation. IMA J. Appl. Math. 73(2), 393–411 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  13. Muntean, A., Böhm, M.: A moving-boundary problem for concrete carbonation: global existence and uniqueness of weak solutions. J. Math. Anal. Appl. 350(1), 234–251 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  14. Kumar, K., van Noorden, T.L., Pop, I.S.: Effective dispersion equations for reactive flows involving free boundaries at the microscale. Multiscale Model. Simul. 9(1), 29–58 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  15. van Noorden, T.L.: Crystal precipitation and dissolution in a thin strip. Eur. J. Appl. Math. 20(1), 69–91 (2009)

    Article  MATH  Google Scholar 

  16. Kumar, K., van Noorden, T.L., Pop, I.S.: Upscaling of reactive flows in domains with moving oscillating boundaries. Discrete Contin. Dyn. Sys. Ser. S 7, 95–111 (2014)

    Article  MATH  Google Scholar 

  17. van Noorden, T.L., Pop, I.S., Ebigbo, A., Helmig, R.: An upscaled model for biofilm growth in a thin strip. Water Resour. Res. 46, W06505 (2010)

    Google Scholar 

  18. Ray, N., van Noorden, T.L., Radu, F.A., Friess, W., Knabner, P.: Drug release from collagen matrices including an evolving microstructure. Z. Angew. Math. Mech. 93(10–11), 811–822 (2013). doi:10.1002/zamm.201200196

    Google Scholar 

  19. Kumar, K., Pop, I.S., Radu, F.A.: Numerical analysis for an upscaled model for dissolution and precipitation in porous media. In: Cangiani, Andrea, Davidchack, Ruslan L., Georgoulis, Emmanuil, Gorban, Alexander N., Levesley, Jeremy, Tretyakov, Michael V. (eds.) Numerical Mathematics and Advanced Applications 2011, pp. 703–711. Springer, Berlin (2013)

    Chapter  Google Scholar 

  20. Barrett, J.W., Knabner, P.: Finite element approximation of the transport of reactive solutes in porous media. I. Error estimates for nonequilibrium adsorption processes. SIAM J. Numer. Anal. 34(1), 201–227 (1997)

    Google Scholar 

  21. Barrett, J.W., Knabner, P.: Finite element approximation of the transport of reactive solutes in porous media. II. Error estimates for equilibrium adsorption processes. SIAM J. Numer. Anal. 34(2), 455–479 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  22. Radu, F.A., Pop, I.S.: Newton method for reactive solute transport with equilibrium sorption in porous media. J. Comput. Appl. Math. 234(7), 2118–2127 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  23. Radu, F.A., Pop, I.S.: Mixed finite element discretization and newton iteration for a reactive contaminant transport model with nonequilibrium sorption: convergence analysis and error estimates. Comput. Geosci. 15, 431–450 (2011)

    Google Scholar 

  24. Radu, F.A., Pop, I.S., Attinger, S.: Analysis of an Euler implicit-mixed finite element scheme for reactive solute transport in porous media. Numer. Methods Partial Differ. Equ. 26(2), 320–344 (2010)

    MATH  MathSciNet  Google Scholar 

  25. Radu, F.A., Muntean, A., Pop, I.S., Suciu, N., Kolditz, O.: A mixed finite element discretization scheme for a concrete carbonation model with concentration-dependent porosity. J. Comput. Appl. Math. 246, 74–85 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  26. Chalupecky, V., Fatima, T., Muntean, A., Kruschwitz, J.: Macroscopic corrosion front computations of sulfate attack in sewer pipes based on a micro-macro reaction-diffusion model. In Multiscale Mathematics: Hierarchy of collective phenomena and interrelations between hierarchical structures (Fukuoka, Japan, December 8–11, 2011). COE Lecture Note Series, vol. 39 pp. 22–31. Fukuoka: Institute of Mathematics for Industry, Kyushu University (2012)

  27. Klöfkorn, R., Kröner, D., Ohlberger, M.: Local adaptive methods for convection dominated problems. In: ICFD Conference on Numerical Methods for Fluid Dynamics (Oxford, 2001) Int. J. Numer. Methods Fluids 40(1–2):79–91 (2002)

  28. Ohlberger, M., Rohde, C.: Adaptive finite volume approximations for weakly coupled convection dominated parabolic systems. IMA J. Numer. Anal. 22(2), 253–280 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  29. Cariaga, E., Concha, F., Pop, I.S., Sepúlveda, M.: Convergence analysis of a vertex-centered finite volume scheme for a copper heap leaching model. Math. Methods Appl. Sci. 33(9), 1059–1077 (2010)

    MATH  MathSciNet  Google Scholar 

  30. Rivière B., Wheeler, M.F.: Non conforming methods for transport with nonlinear reaction. In: Fluid flow and transport in porous media: mathematical and numerical treatment (South Hadley, MA, 2001). Contemp. Math., vol. 295, pp. 421–432. American Mathematical Society, Providence (2002)

  31. Dawson, C.: Analysis of an upwind-mixed finite element method for nonlinear contaminant transport equations. SIAM J. Numer. Anal. 35(5), 1709–1724 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  32. Dawson, C., Aizinger, V.: Upwind-mixed methods for transport equations. Comput. Geosci. 3(2), 93–110 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  33. Eymard, R., Hilhorst, D., Vohralík, M.: A combined finite volume-nonconforming/mixed-hybrid finite element scheme for degenerate parabolic problems. Numer. Math. 105(1), 73–131 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  34. Hilhorst, D., Vohralík, M.: A posteriori error estimates for combined finite volume-finite element discretizations of reactive transport equations on nonmatching grids. Comput. Methods Appl. Mech. Eng. 200(5–8), 597–613 (2011)

    Article  MATH  Google Scholar 

  35. Ciavaldini, J.F.: Analyse numerique d’un problème de Stefan à deux phases par une methode d’éléments finis. SIAM J. Numer. Anal. 12, 464–487 (1975)

    Article  MATH  MathSciNet  Google Scholar 

  36. Kumar, K., Pop, I.S., Radu, F.A.: Convergence analysis of mixed numerical schemes for reactive in a porous medium. SIAM J. Numer. Anal. 51, 2283–2308 (2013)

    Google Scholar 

  37. Devigne, V.M., Pop, I.S., van Duijn, C.J., Clopeau, T.: A numerical scheme for the pore-scale simulation of crystal dissolution and precipitation in porous media. SIAM J. Numer. Anal. 46(2), 895–919 (2008)

    Google Scholar 

  38. Mikelić, A., Devigne, V., van Duijn, C.J.: Rigorous upscaling of the reactive flow through a pore, under dominant Peclet and Damkohler numbers. SIAM J. Math. Anal. 38, 1262–1287 (2006)

    Google Scholar 

  39. Girault, V., Riviere, B., Wheeler, M.F.: A splitting method using discontinuous galerkin for the transient incompressible Navier–Stokes equations. ESAIM: M2AN 39(6):1115–1147 (2005)

    Google Scholar 

  40. Knobloch, P., Tobiska, L.: On the stability of finite-element discretizations of convection–diffusion-reaction equations. IMA J. Numer. Anal. 31(1), 147–164 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  41. van Noorden, T.L., Pop, I.S, Röger, M.: Crystal dissolution and precipitation in porous media: \(L^1\)-contraction and uniqueness. Discrete Contin. Dyn. Syst., (Dynamical Systems and Differential Equations. Proceedings of the 6th AIMS International Conference, suppl.), 1013–1020 (2007)

  42. Ciarlet, P.G.: The finite element method for elliptic problems. In: Studies in Mathematics and its Applications, vol. 4. North-Holland Publishing Co., Amsterdam (1978)

  43. Ladyzhenskaya, O.A., Ural’tseva, N.N.: Linear and quasilinear elliptic equations. Academic Press, New York (1968) (Translated from the Russian by Scripta Technica, Inc. Translation editor: Leon Ehrenpreis)

  44. Simon, J.: Compact sets in the space \(L^p(0, T;B)\). Ann. Mat. Pura Appl. 146(4), 65–96 (1987)

  45. Brezis, H.: Functional Analysis. In: Sobolev Spaces and Partial Differential Equations. Springer, New York (2011)

  46. Lenzinger, M., Schweizer, B.: Two-phase flow equations with outflow boundary conditions in the hydrophobic–hydrophilic case. Nonlinear Anal. 73(4), 840–853 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  47. Temam, R.: Navier–Stokes equations. In: Studies in Mathematics and its Applications, vol. 2, 3rd edn. North-Holland Publishing Co., Amsterdam (1984) (Theory and numerical analysis. With an appendix by F, Thomasset)

  48. Pop, I.S., Radu, F., Knabner, P.: Mixed finite elements for the Richards’ equation: linearization procedure. J. Comput. Appl. Math. 168(1–2), 365–373 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  49. Elliott, C.M.: Error analysis of the enthalpy method for the Stefan problem. IMA J. Numer. Anal. 7(1), 61–71 (1987)

    Article  MATH  MathSciNet  Google Scholar 

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Acknowledgments

K. Kumar would like to thank the Technology Foundation STW for the financial support through the Project 07796, “Second Generation of Integrated Batteries”. The authors are members of the International Research Training Group NUPUS funded by the German Research Foundation DFG (GRK 1398) and by the Netherlands Organisation for Scientific Research NWO (DN 81-754). Part of the work was carried out when K. Kumar visited the Institute of Mathematics, University of Bergen. The support is gratefully acknowledged. K. Kumar would also like to thank Dr. Thomas Wick and Dr. Ahmed El Sheikh (UT Austin) for their feedback on the numerical computations. F.A. Radu acknowledges the support of Statoil through the Akademia Grant 2012–2013.

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Authors and Affiliations

  1. Center for Subsurface Modeling, The University of Texas at Austin, Austin, USA

    K. Kumar

  2. Department of Mathematics and Computer Science, Eindhoven University of Technology, Eindhoven, The Netherlands

    I. S. Pop

  3. Institute of Mathematics, University of Bergen, Bergen, Norway

    I. S. Pop & F. A. Radu

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  1. K. Kumar
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Correspondence to I. S. Pop.

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Kumar, K., Pop, I.S. & Radu, F.A. Convergence analysis for a conformal discretization of a model for precipitation and dissolution in porous media. Numer. Math. 127, 715–749 (2014). https://doi.org/10.1007/s00211-013-0601-1

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  • DOI: https://doi.org/10.1007/s00211-013-0601-1

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