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Upscaling of a system of diffusion-reaction equations coupled with a system of ordinary differential equations originating in the context of crystal dissolution and precipitation of minerals in a porous medium. (English) Zbl 1472.35029

Summary: In this paper, we consider diffusion and reaction of mobile chemical species, and dissolution and precipitation of immobile species present inside a porous medium. The transport of mobile species in the pores is modeled by a system of semilinear parabolic partial differential equations. The reactions amongst the mobile species are assumed to be reversible. i.e. both forward and backward reactions are considered. These reversible reactions lead to highly nonlinear reaction rate terms on the right-hand side of the partial differential equations. This system of equations for the mobile species is complemented by flux boundary conditions at the outer boundary. Furthermore, the dissolution and precipitation of immobile species on the surface of the solid parts are modeled by mass action kinetics which lead to a nonlinear precipitation term and a multivalued dissolution term. The model is posed at the pore (micro) scale. The contribution of this paper is two-fold: first we show the existence of a unique positive global weak solution for the coupled systems and then we upscale (homogenize) the model from the micro scale to the macro scale. For the existence of solution, some regularization techniques, Schaefer’s fixed point theorem and Lyapunov type arguments have been used whereas the concepts of two-scale convergence and periodic unfolding are used for the homogenization.

MSC:

35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
35K51 Initial-boundary value problems for second-order parabolic systems
35K57 Reaction-diffusion equations
35K58 Semilinear parabolic equations
76S05 Flows in porous media; filtration; seepage
47J35 Nonlinear evolution equations