InterPore2021

Rigorous derivation of an effective model for reactive transport in evolving porous medium

1 Jun 2021, 11:15

15m

Oral Presentation (MS7) Mathematical and numerical methods for multi-scale multi-physics, nonlinear coupled processes MS7

Speaker

Markus Gahn (IWR, University of Heidelberg)

Description

In this talk we derive a homogenized model for a reaction-diffusion equation describing mineral precipitation/dissolution in an evolving porous micro-domain, consisting of a fluid phase and a solid phase build by periodically distributed spherical solid grains. The evolution of the micro-domain depends on the concentration at the surface of the grains, leading to a free boundary value problem on the micro-scale. The periodicity and the size of the grains is of order $\epsilon$, where the parameter $\epsilon$ is small compared to the size of the whole domain. The radius of every micro-grain depends on the concentration at its surface, leading to a nonlinear problem. The aim is to pass to the limit $\epsilon \to 0$ and rigorously derive a macroscopic model, the solution of which approximates the solution of the microscopic model.

In a first step we transform the problem on the evolving micro-domain to a problem on a fixed periodically perforated domain by using the Hanzawa-transformation, depending on the radius of the grains and therefore the concentration. This leads to a change in the coefficients of the equations, which now depend on the radius and the concentration, leading to a nonlinear problem. We prove existence using the Rothe-method and derive \textit{a priori} estimates for the solutions uniformly with respect to the parameter $\epsilon$. For the derivation of the macroscopic model in the limit $\epsilon \to 0$ we use rigorous homogenization methods like the two-scale convergence. For the treatment of the nonlinear terms we need strong compactness results.