Investigation of a fully coupled micro-macro model for mineral reactions - Gärttner, Stephan (Friedrich-Alexander-Universität Erlangen-Nürnberg, 2022)
Micro-macro models pose a powerful tool to mathematically describe reactive flow and transport phenomena in porous media research. Honoring the inherent multiscale spatial structure of porous media such as natural rock, these approaches hold the potential to combine the well-known benefits of pure microscopic (pore-scale) and pure macroscopic (continuum-) models. As such, the micro-macro ansatz aims at achieving computational efficiency comparable to macroscopic models while ensuring a detailed and comprehensive description of all relevant physical/chemical processes typical of pore-scale representations. This thesis contributes to the elimination of current restrictions to micro-macro approaches by considering a model incorporating the additional physics arising from the presence of two different mineral phases complemented by a powerful numerical scheme allowing for simulations of realistic complexity. In a first step, a new sharp interface micro-macro model for reactive flow and transport in two-mineral evolving porous media is derived generalizing existing single-mineral models. As such, our approach is capable of accurately capturing covering and encapsulation phenomena as typically occurring due to dissolution/precipitation reactions. Being obtained from an underlying pore-scale model by methods of formal periodic homogenization, the model consists of effective equations for flow and transport on the macro-domain supplemented by auxiliary cell-problems from which the required effective parameters are derived. Demanding conservation of mass, macroscopic concentration fields in turn dictate geometry evolution within the reference cells. Secondly, analytical studies are performed proving the local-in-time existence of strong solutions to a simplified version of our micro-macro model. Smooth parameter dependence results obtained along the way indicate stability of the multiscale coupling and provide a theoretical motivation for the methods employed in our numerical approach. Finally, an iterative solution scheme capable of efficiently handling the tight nonlinear coupling between both spatial scales is presented. Employing a specialized version of the Voronoi implicit interface method, a proper description and evolution of the resulting three-phase geometry is ensured. Besides the development of a suitable adaptivity scheme significantly reducing the number of evaluated auxiliary cell-problems, data-driven methods are employed to further decrease computational effort. More precisely, convolutional neural networks are trained in two and three spatial dimensions to predict the permeability directly from the representative cell geometry, evading costly solutions of pore-scale flow problems at acceptable loss of accuracy. The power of our approach is demonstrated by conducting numerical experiments of challenging self-enforcing processes such as wormholing phenomena. Moreover, the methods are verified by direct comparison to microscopic simulation results as well as to a related diffuse interface model being implemented in an independent numerical framework.
Keywords: level-set method; machine learning; micro-macro model; porous media