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Портал Begell Электронная Бибилиотека e-Книги Журналы Справочники и Сборники статей Коллекции
Special Topics & Reviews in Porous Media: An International Journal
ESCI SJR: 0.277 SNIP: 0.52 CiteScore™: 1.3

ISSN Печать: 2151-4798
ISSN Онлайн: 2151-562X

Special Topics & Reviews in Porous Media: An International Journal

DOI: 10.1615/SpecialTopicsRevPorousMedia.2019029274
pages 305-321


Raphael Schulz
Mathematics Department, Friedrich-Alexander University Erlangen-Nürnberg, Cauerstraße 11, 91058 Erlangen, Germany

Краткое описание

In this article, we derive an upscaled model for crystal precipitation and dissolution in a saturated porous medium with a perforated solid matrix. We model the solid matrix itself at the pore scale as a porous medium. Hence, we consider at the pore scale a Darcy–Stokes system, where the Beavers–Joseph boundary condition is proposed at the corresponding interface. By asymptotic expansions we derive an upscaled model describing the process via Darcy's law, a transport equation, and corresponding effective coefficients given by the evolution of the microstructure. Weak solvability of the upscaled model is also investigated.


  1. Arbogast,T. and Lehr, H.L.,HomogenizationofaDarcy-StokesSystemModelingVuggy Porous Media, Comput. Geosci, vol. 10, no. 3, pp. 291-302,2006.

  2. Arqub, O.A. and Al-Smadi, M., Numerical Algorithm for Solving Time-Fractional Partial Integrodifferential Equations Subject to Initial and Dirichlet Boundary Conditions, Numer. Meth. Part. D. E, vol. 34, no. 5, pp. 1577-1597,2018a.

  3. Arqub, O.A., Odibat, Z., and Al-Smadi, M., Numerical Solutions of Time-Fractional Partial Integrodifferential Equations of Robin Functions Types inHilbert Space with Error Bounds and Error Estimates, Nonlin. Dyn, vol. 94, no. 3, pp. 1819-1834,2018b.

  4. Beavers, G.S. and Joseph, D.D., Boundary Conditions at aNaturally Permeable Wall, J. FluidMech, vol 30, no. 1, pp. 197-207, 1967.

  5. Bringedal, C., Berre, I., Pop, I.S., and Radu, F.A., Upscaling of Non-Isothermal Reactive Porous Media Flow with Changing Porosity, Trans. Porous Media, vol. 114, no. 2, pp. 371-393,2016.

  6. Bringedal, C. and Kumar, K., Effective Behavior near Clogging in Upscaled Equations for Non-Isothermal Reactive Porous Media Flow, Trans. Porous Media, vol. 120, no. 3, pp. 553-577,2017.

  7. Hornung,U., Homogenization and Porous Media, New-York, NY: Springer-Verlag, 1996.

  8. Jones, I.P., Low Reynolds Number Flow past a Porous Spherical Shell, Proc. Camb. Philos. Soc., vol 73, no. 1,pp. 231-238,1973.

  9. 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., vol 76, no. 3, pp. 171-185,1995.

  10. Kumar, K., Neuss-Radu, M., and Pop, I.S., Homogenization of a Pore Scale Model for Precipitation and Dissolution in Porous Media, IMA J. Appl. Math, vol. 81, pp. 877-897,2014.

  11. Ladyzenskaja, O.A., Solonnikov, V.A., and Ural'ceva, N.N., Linear and Quasi-Linear Equations of Parabolic Type, Providence, RI: American Mathematical Society, 1968.

  12. Landa-Marban, D., Liu, N., Pop, I.S., Kumar, K., Pettersson, P., Bedtker, G., Skauge, T., and Radu, F.A., A Pore Scale Model for Permeable Biofilm: Numerical Simulations and Laboratory Experiments, Trans. Porous Media, vol. 127, no. 3, pp. 643-660, 2019. DOI: 10.1007/s11242-018-1218-8.

  13. Muntean, A. and van Noorden, T.L., Corrector Estimates for the Homogenization of a Locally Periodic Medium with Areas of Low and High Diffusivity, Euro. J. Appl. Math., vol. 24, no. 5, pp. 657-677,2013.

  14. Osher, S. andFedkiw, R.P., Level Set Methods: An Overview and Some Recent Results, J. Comput. Phys, vol. 169, no. 2, pp. 463-502,2001.

  15. Pop, I.S., Bogers, J., and Kumar, K., Analysis and Upscaling of a Reactive Transport Model in Fractured Porous Media with Nonlinear Transmission Condition, Viet. J. Math, vol. 45, no. 1, pp. 77-102,2017.

  16. Quarteroni, A.M. and Valli, A., Numerical Approximation of Partial Differential Equations, 1st ed., Berlin Heidelberg: Springer-Verlag, 1994.

  17. Ray, N., van Noorden, T.L., Radu, F.A., Friess, W., and Knabner, P., Drug Release from Collagen Matrices Including an Evolving Microstructure, ZAMMZ. Angew. Math. Mech, vol. 93, pp. 811-822,2013.

  18. Ray, N., Rupp, A., Schulz, R., and Knabner, P., Old, New Approaches Predicting the Diffusion in Porous Media, Trans. Porous Media, vol. 124, no. 3, pp. 811-822,2018.

  19. Saffman, P., on the Boundary Condition at the Surface of a Porous Medium, Stud. Appl. Math, vol. 50, pp. 93-101,1971.

  20. Schulz, R., Boundedness in a Biofilm-Chemotaxis Model in Evolving Porous Media, Math. Model. Anal., vol. 22, no. 6, pp. 852-869,2017.

  21. Schulz, R., Biofilm Modeling in Evolving Porous Media with Beavers-Joseph Condition, Z. Angew. Math. Mech., vol. 77, no. 5, pp. 1653-1677,2019.

  22. Schulz, R. and Knabner, P., Derivation and Analysis of an Effective Model for Biofilm Growth in Evolving Porous Media, Math. Meth. Appl. Sci, vol. 40, no. 8, pp. 2930-2948,2017a.

  23. Schulz, R. and Knabner, P., An Effective Model for Biofilm Growth Made by Chemotactical Bacteria in Evolving Porous Media, SIAMJ. Appl. Math, vol. 77, no. 5, pp. 1653-1677,2017b.

  24. Schulz, R., Ray, N., Frank, F., Mahato, H.S., and Knabner, P., Strong Solvability Up to Clogging of an Effective Diffusion- Precipitation Model in an Evolving Porous Medium, Eur. J. Appl. Math., vol. 28, no. 2, pp. 179-207,2017.

  25. Schumer, R., Benson, D.A., Meerschaert, M.M., and Baeumer, B., Multiscaling Fractional Advection-Dispersion Equations and Their Solutions, Water Resour. Res., vol. 39 ,no. 1,p. 1022,2003a.

  26. Schumer, R., Benson, D.A., Meerschaert, M.M., and Baeumer, B., Fractal Mobile/Immobile Solute Transport, Water Resour. Res., vol. 39, no. 10, p. 1296,2003b.

  27. van Duijn, C.J. and Pop, I.S., Crystal Dissolution and Precipitation in Porous Media: Pore Scale Analysis, J. Reine Angew. Math., vol. 577, pp. 171-211,2004.

  28. van Noorden, T.L., Crystal Precipitation and Dissolution in a Thin Strip, Tech. Rep., CASA report 30, Eindhoven University of Technology, 2007.

  29. van Noorden, T.L., Crystal Precipitation and Dissolution in a Porous Medium: Effective Equations and Numerical Experiments, Multiscale Model. Simul., vol. 7, pp. 1220-1236,2009.

  30. van Noorden, T.L., Pop, I.S., and Roger, M., Crystal Dissolution and Precipitation in Porous Media: L1-Contraction and Uniqueness, Discrete and Continuous Dynamical Systems Supplement, pp. 1013-1020,2007.

  31. van Noorden, T.L., Pop, I.S., Ebigbo, A., and Helmig, R., An Upscaled Model for Biofilm Growth in a Thin Strip, Water Resour. Res, vol. 46, pp. 1-14,2010.

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