Hostname: page-component-78c5997874-s2hrs Total loading time: 0 Render date: 2024-11-10T13:30:59.806Z Has data issue: false hasContentIssue false
  • English
  • Français

Degenerate equations in a diffusion–precipitation model for clogging porous media

Part of: Representations of solutions Parabolic equations and systems Diffusion and convection

Published online by Cambridge University Press:  18 December 2019

RAPHAEL SCHULZ
RAPHAEL SCHULZ*
Affiliation:
Friedrich–Alexander Universität Erlangen–Nürnberg, Department of Mathematics, Cauerstraße 11, 91058Erlangen, Germany, email: raphael.schulz@math.fau.de
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this article, we consider diffusive transport of a reactive substance in a saturated porous medium including variable porosity. Thereby, the evolution of the microstructure is caused by precipitation of the transported substance. We are particularly interested in analysing the model when the equations degenerate due to clogging. Introducing an appropriate weighted function space, we are able to handle the degeneracy and obtain analytical results for the transport equation. Also the decay behaviour of this solution with respect to the porosity is investigated. There a restriction on the decay order is assumed, that is, besides low initial concentration also dense precipitation leads to possible high decay. We obtain nonnegativity and boundedness for the weak solution to the transport equation. Moreover, we study an ordinary differential equation (ODE) describing the change of porosity. Thereby, the control of an appropriate weighted norm of the gradient of the porosity is crucial for the analysis of the transport equation. In order to obtain global in time solutions to the overall coupled system, we apply a fixed point argument. The problem is solved for substantially degenerating hydrodynamic parameters.

Keywords

Evolving porous mediadegenerate equationcloggingweighted spacesdecay behaviour

MSC classification

Primary: 35D30: Weak solutions
Secondary: 35K65: Degenerate parabolic equations 76R50: Diffusion
Type
Papers
Information
European Journal of Applied Mathematics , Volume 31 , Issue 6 , December 2020 , pp. 1050 - 1069
Check for updates
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s) 2019

References

Arbogast, T. & Taicher, A. L. (2016) A linear degenerate elliptic equation arising from two-phase mixtures. SIAM J. Numer. Anal. 54(5), 3105–3122.Google Scholar
Arendt, W. & Kreuter, M. (2018) Mapping theorems for Sobolev-spaces of vector-valued functions. Studia Mathematica 240, 275–299.Google Scholar
Bringedal, C., Berre, I., Pop, I. S. & Radu, F. A. (2016) Upscaling of non-isothermal reactive porous media flow with changing porosity. Transp. Porous Med. 114(2), 371–393.Google Scholar
Bringedal, C. & Kumar, K. (2017) Effective behavior near clogging in upscaled equations for non-isothermal reactive porous media flow. Transp. Porous Med. 120(3), 553–577.Google Scholar
Grafakos, L. (2009) Classical Fourier analysis, Graduate Texts in Mathematics, 2nd edn., vol. 249, Springer, New York.Google Scholar
Jikov, V. V., Kozlov, S. M. & Oleinik, O. A. (1994) Homogenization of Differential Operators and Integral Functionals, Springer-Verlag, Berlin Heidelberg.Google Scholar
Kumar, K., Pop, I. S. & Radu, F. A. (2013) Convergence analysis of mixed numerical schemes for reactive flow in a porous medium. SIAM J. Numer. Anal. 51(4), 2283–2308.Google Scholar
Mitra, K. & Pop, I. S. (2019) A modified L-scheme to solve nonlinear diffusion problems. Comput. Math. Appl. 77(6), 1722–1738.Google Scholar
Muntean, A. & van Noorden, T. L. (2013) Corrector estimates for the homogenization of a locally periodic medium with areas of low and high diffusivity. European J. Appl. Math. 24, 657–677.Google Scholar
Ray, N., van Noorden, T. L., Radu, F. A., Friess, W., Knabner, P. (2013) Drug release from collagen matrices including an evolving microstructure. ZAMM Z. Angew. Math. Mech. 93, 811–822.Google Scholar
Ray, N., Rupp, A., Schulz, R. & Knabner, P. (2018) Old and new approaches predicting the diffusion in porous media. Transp. Porous Med. 124(3), 803–824.Google Scholar
Schulz, R., Ray, N., Frank, F., Mahato, H. & Knabner, P. (2017) Strong solvability up to clogging of an effective diffusion-precipitation model in an evolving porous medium. Eur. J. Appl. Math. 28(2), 179–207.Google Scholar
Schulz, R. & Knabner, P. (2017) Derivation and analysis of an effective model for biofilm growth in evolving porous media. Math. Meth. Appl. Sci. 40(8), 2930–2948.Google Scholar
Schulz, R. & Knabner, P. (2017) An effective model for biofilm growth made by chemotactical bacteria in evolving porous media. SIAM J. Appl. Math. 77(5), 1653–1677.Google Scholar
Schulz, R. (2017) Boundedness in a biofilm-chemotaxis model in evolving porous media. Math. Model. Anal. 22(6), 852–869.Google Scholar
Schulz, R. (2020) Degenerate equations for flow and transport in clogging porous media. J. Math. Anal. Appl. 483(2), 1–24. DOI: 10.1016/j.jmaa.2019.123613.Google Scholar
van Noorden, T. L. (2007) Crystal precipitation and dissolution in a thin strip, Technical Report, CASA report 30, Eindhoven University of Technology.Google Scholar
van Noorden, T. L. (2009) Crystal precipitation and dissolution in a porous medium: Effective equations and numerical experiments. Multiscale Model. Simul. 7, 1220–1236.10.1137/080722096CrossRefGoogle Scholar
van Noorden, T. L. & Muntean, A. (2011) Homogenization of a locally-periodic medium with areas of low and high diffusivity. Eur. J. Appl. Math. 22, 493–516.10.1017/S0956792511000209CrossRefGoogle Scholar