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

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.

Upscaling the Flow and Transport in an Evolving Porous Medium with General Interaction Potentials

“…The formal, two-scale, asymptotic expansion in a level set framework is also applicable in three space dimensions, [27]. In that case, the results that are derived in section 4 and section 5 are maintained without any changes.…”

“…In such a setting, an appropriate multiscale model has been developed and a numerical simulation in randomly distributed media illustrating the phenomenon of clogging has been performed in [21]. Taking the evolving microstructure into account, first analytical results have recently been derived in [27] for a diffusion problem.…”

“…In the context of evolving microstructures, further examples for geometries that are characterized by a single parameter have been investigated in [27]. In particular, a circular geometry was used in [21].…”

Effective Behavior Near Clogging in Upscaled Equations for Non-isothermal Reactive Porous Media Flow

“…The effective model in (Bringedal et al 2016) is derived for a two-dimensional domain, but Schulz et al (2016) recently derived an effective three-dimensional model for an isothermal reactive model without flow. The extension from two to three spatial dimensions could be performed by defining a tangent plane for the fluid-grain interface in the upscaling procedure, while the resulting effective model have the same appearance as in the two-dimensional case.…”

Biofilm modeling in evolving porous media with Beavers‐Joseph condition

“…In terms of homogenization in non‐rigid porous media, the literature offers only a small number of analytical results: In [], upscaling of an advection–diffusion(–reaction) system in a locally periodic medium, including low and high diffusivities, was considered. Upscaled diffusion–precipitation equations with effective coefficients coupled with a level‐set equation are considered in []. An effective model describing biofilm growth in porous media is already derived and analyzed in [], but neglects chemotactical effects completely, which were involved in [].…”