Modelling slow sand filtration and solving interface problems with the Transfer Path Method

Filtration through porous media is the oldest form of drinking water treatment and, to this day, constitutes a focal point of research in water engineering. Mathematical models of filtration are often used in conjunction with supplementary models of concurrent phenomena. For instance, both fluid flow inside and outside the filtering medium are of interest, and models of both flow through porous media and of free fluid flow become necessary for comprehensive modelling. Such supplementary models are often coupled together in some manner, leading to extended systems of equations and possibly complicated geometries. Additionally, at low filtration rates, biochemical reactions taking place during the treatment process gain relevance and need to be taken into account to have a complete description of the filtration process.In this work, we are concerned with water filtration from two angles: first, we are interested in comprehensive modelling of Slow Sand Filters (SSFs), in which the action of the ecological community settled in the biofilm growing in the medium constitutes the main filtration mechanism; and second, we are interested in the coupling of free flow with flow through porous media with mismatching discretizing geometries at the interface utilizing the so-called Transfer Path Method (TPM) in the context of interface problems in fluid mechanics.The doctoral thesis is consequently divided into two parts. The first part contains a new one-dimensional model framework of SSFs which involves the water flow in the entire filter, consisting of the sand bed and the supernatant water sitting atop it, along with the evolution of microorganisms therein. A positivity-preserving numerical method under certain assumptions is presented together with a documentation of the software developed for simulations. A two-dimensional model for the supernatant water region is also included; a derivation is presented as well as an invariant-region-preserving numerical method for the entire convection-reaction-Cahn--Hilliard system.In the second part, the TPM is applied to coupled problems in fluid mechanics in the context of dissimilar meshes, that is, geometrical discretizations of two independent domains separated by an interface in which the resulting triangulations may present mismatches in the form of gaps or overlaps. The hybridizable discontinuous Galerkin method is used to discretize the corresponding systems of partial differential equations and present results for Stokes flow, Stokes/Darcy coupling and fluid-structure interaction.