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Strong solvability up to clogging of an effective diffusion–precipitation model in an evolving porous medium

  • R. SCHULZ (a1), N. RAY (a1), F. FRANK (a2), H. S. MAHATO (a3) and P. KNABNER (a1)...
Abstract

In the first part of this article, we extend the formal upscaling of a diffusion–precipitation model through a two-scale asymptotic expansion in a level set framework to three dimensions. We obtain upscaled partial differential equations, more precisely, a non-linear diffusion equation with effective coefficients coupled to a level set equation. As a first step, we consider a parametrization of the underlying pore geometry by a single parameter, e.g. by a generalized “radius” or the porosity. Then, the level set equation transforms to an ordinary differential equation for the parameter. For such an idealized setting, the degeneration of the diffusion tensor with respect to porosity is illustrated with numerical simulations. The second part and main objective of this article is the analytical investigation of the resulting coupled partial differential equation–ordinary differential equation model. In the case of non-degenerating coefficients, local-in-time existence of at least one strong solution is shown by applying Schauder's fixed point theorem. Additionally, non-negativity, uniqueness, and global existence or existence up to possible closure of some pores, i.e. up to the limit of degenerating coefficients, is guaranteed.

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[1] Alt H. W. & Luckhaus S. (1983) Quasilinear elliptic–parabolic differential equations. Math. Z. 183 (3), 311341.
[2] Bader R. & Merz W. (2002) Local existence result of the dopant diffusion in arbitrary space dimensions. Z. Anal. Anwend. 21 (1), 91111.
[3] Bergh J. & Löfström J. (1976) Interpolation Spaces, an Introduction. Springer-Verlag, Berlin, New York, pp. x+207.
[4] Besov O., Il'yin V. P. & Nikol'skii S. M. (1979) Integral Transformations of Functions and Embedding Theorems, Vol. 2. Winston & Sons, Washington, D.C., pp. viii+311.
[5] Capdeboscq Y. & Ptashnyk M. (2012) Root growth: Homogenization in domains with time dependent partial perforations. ESAIM Control Optim. Calc. Var. 18 (3), 856876.
[6] DiBenedetto E. (1991) Degenerate Parabolic Equations. Springer-Verlag, New York, pp. xvi+387.
[7] Eck C. (2005) Analysis of a two-scale phase field model for liquid–solid phase transitions with equiaxed dendritic microstructure. Multiscale Model. Simul. 3 (1), 2849.
[8] Eck C. (2005) Homogenization of a phase field model for binary mixtures. Multiscale Model. Simul. 3 (1), 127.
[9] Giaquinta M. & Martinazzi L. (2005) An Introduction to the Regularity Theory for Elliptic Systems, Harmonic Maps and Minimal Graphs. Ed. della Normale, Pisa.
[10] Hoffmann J. (2010) Reactive Transport and Mineral Dissolution/Precipitation in Porous Media: Efficient Solution Algorithms, Benchmark Computations and Existence of Global Solutions. Doctoral thesis, University of Erlangen–Nürnberg. Available at: https://www.mso.math.fau.de/fileadmin/am1/projects/PhD_Hoffmann.pdf
[11] Knabner P. & van Duijn C. J. (1996) Crystal dissolution in porous media flow. J. Appl. Math. Mech. 76, 329332.
[12] Kräutle S. (2008) General Multispecies Reactive Transport Problems in Porous Media. Habilitation thesis, University of Erlangen–Nürnberg. Available at: http://www.mso.math.fau.de/fileadmin/am1/projects/Habil_Kraeutle.pdf
[13] Meier S. (2008) A homogenisation-based two-scale model for reactive transport in media with evolving microstructure. Comptes Rendus Mécanique 336 (8), 623628.
[14] Mikelic A. & Wheeler M. F. (2012) On the interface law between a deformable porous medium containing a viscous fluid and an elastic body. Math. Models Methods Appl. Sci. 22 (11), 1250031.
[15] Moyne C. & Murad M. A. (2006) Electro–chemo–mechanical couplings in swelling clays derived from a micro/macro-homogenization procedure. Int. J. Solids Struct. 39 (25), 61596190.
[16] Moyne C. & Murad M. A. (2006) A two-scale model for coupled electro-chemo-mechanical phenomena and onsager's reciprocity relation in expansive clays: I homogenization results. Transp. Porous Media 62 (3), 333380.
[17] Muntean A. & van Noorden T. (2013) Corrector estimates for the homogenization of a locally periodic medium with areas of low and high diffusivity. Eur. J. Appl. Math. 24, 657677, 10.
[18] Peter M. (2009) Coupled reaction–diffusion processes inducing an evolution of the microstructure: Analysis and homogenization. Nonlinear Anal. 70 (2), 806821.
[19] Peter M. (2007) Homogenisation in domains with evolving microstructure. Comptes Rendus Mécanique 335 (7), 357362.
[20] Peter P. (2007) Homogenisation of a chemical degradation mechanism inducing an evolving microstructure. Comptes Rendus Mécanique 335 (11), 679684.
[21] Pop I. S. & Schweizer B. (2009) Regularization schemes for degenerate richards equations and outflow conditions. Math. Models Methods Appl. Sci. 21 (8), 16851712.
[22] Radu F. A., Pop I. S. & Knabner P. (2008) Error estimates for a mixed finite element discretization of some degenerate parabolic equations. Numer. Math. 109 (2), 285311.
[23] Raviart P. A. & Thomas J. M. (1977) A mixed finite element method for 2nd order elliptic problems. Mathematical aspects of finite element methods (Proc. Conf., Consiglio Naz. delle Ricerche (C.N.R.), Rome, 1975), pp. 292–315. Lecture Notes in Math., Vol. 606, Springer, Berlin, 1977.
[24] Ray N., Elbinger T. & Knabner P. (2015) Upscaling the flow and transport in an evolving porous medium with general interaction potentials. SIAM J. Appl. Math. 75 (5), 21702192.
[25] Ray N., van Noorden T., Frank F. & Knabner P. (2012) Multiscale modeling of colloid and fluid dynamics in porous media including an evolving microstructure. Transp. Porous Media 95 (3), 669696.
[26] Ray N. van Noorden T., Radu F. A., Friess W. & Knabner P. (2013) Drug release from collagen matrices including an evolving microstructure. ZAMM Z. Angew Math. Mech. 93 (10–11), pp. 811822.
[27] Redeker I. S., Pop M. & Rohde C. (2014) Upscaling of a tri-phase phase-field model for precipitation in porous media. In: CASA Report, No. 14-31 (2014).
[28] Schmuck M., Pavliotis G. & Kalliadasis S. (2014) Effective macroscopic interfacial transport equations in strongly heterogeneous environments for general homogeneous free energies. Appl. Math. Lett. 35, 12–17
[29] Schmuck M., Pradas M., Pavliotis G. & Kalliadasis S. (2013) Derivation of effective macroscopic stokes–cahn–hilliard equations for periodic immiscible flows in porous media. Nonlinearity 26 (12), 32593277.
[30] Schmuck M., Pradas M., Pavliotis G. A. & Kalliadasis S. (2012) Upscaled phase-field models for interfacial dynamics in strongly heterogeneous domains. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 468 (2147), 37053724.
[31] Schulz R. & Knabner P. (2016) Modeling and analyzing of biofilm growth in saturated porous media. Math. Method Appl. Sci., submitted, 2016. Preprint Series Angewandte Mathematik, ISSN: 2194-5127, No. 391.
[32] Sethian J. A. (1999) Level set methods and fast marching methods, Cambridge, pp. xx+378.
[33] Taniuchi Y. (2006) Remarks on global solvability of 2-d boussinesq equations with non-decaying initial data. Funkcialaj Ekvacioj 49 (1), 3957.
[34] van Duijn C. J. & Pop I. S. (2004) Crystals dissolution and precipitation in porous media: pore scale analysis. J. Reine Angew. Math. 577, 171211.
[35] van Noorden T. (2009) Crystal precipitation and dissolution in a porous medium: effective equations and numerical experiments. Multiscale Model. Simul. 7 (3), 1220– 1236.
[36] van Noorden T. & Pop S. (2008) A stefan problem modelling crystal dissolution and precipitation. IMA J. Appl. Math. 73 (2), 393411.
[37] van Noorden T. & Muntean A. (2011) Homogenization of a locally-periodic medium with areas of low and high diffusivity. Eur. J. Appl. Math. 22 (5), 493516.
[38] van Noorden T., Pop S., Ebigbo A. & Helmig R. (2010) An upscaled model for biofilmgrowth in a thin strip. Water Resour. Res. 46 (6), 114.
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European Journal of Applied Mathematics
  • ISSN: 0956-7925
  • EISSN: 1469-4425
  • URL: /core/journals/european-journal-of-applied-mathematics
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