Two-scale models for reactive transport and evolving microstructure

Abstract

Reactive transport in materials with a complex microstructure is a phenomenon that frequently occurs in nature and in technical applications. An example are chemical reactions taking place in the pores of concrete structures. Since these processes are highly influenced by the geometry of the microstructure, it is a difficult task to understand and predict their behaviour on a macroscopic scale. It has been shown by periodic homogenisation that for certain situations two-scale models are appropriate. Periodic homogenisation is an averaging method that is limited to a uniform and constant microstructure. However, in general a porous medium contains multiple phases that evolve in time. For instance, the air-water mixture in an unsaturated porous medium varies due to chemical reactions or non-constant boundary conditions. In certain applications, such effects need to be captured in a mathematical model. In this thesis, two-scale models are formulated for reactive transport in a porous medium with an evolving phase configuration. The reaction-diffusion problem is motivated by a chemical degradation process in concrete structures. The model equations are formally derived from a pore-scale model by homogenisation, but are defined on a time-dependent geometry. Two different settings are considered: In the first one, the evolution of the pore geometry is independent of the reaction-diffusion process and can be considered as a-priori given. This leads to a semilinear two-scale system of parabolic PDEs. In the second setting, the pore water evolves due to the reaction itself, which results in a quasilinear model. Using a transformation to a reference configuration, it is proven that both two-scale models are well-posed. A simple numerical implementation that works with both settings is suggested and validated. The qualitative behaviour of the solutions of the two-scale models are illustrated for an academic example. In summary, the results give a solid theoretical fundament for further important tasks such as the identification of model parameters or the design of efficient numerical algorithms suitable for real-world applications.