## Fachbereich Mathematik

This thesis deals with the numerical study of multiscale problems arising in the modelling of processes of the flow of fluid in plain and porous media. Many of these processes, governed by partial differential equations, are relevant in engineering, industry, and environmental studies. The overall task of modelling and simulating the filtration-related multiscale processes becomes interdisciplinary as it employs physics, mathematics and computer programming to reach its aim. Keeping the challenges in mind, the main focus is to overcome the limitations of accuracy, speed and memory and to develop novel efficient numerical algorithms which could, in part or whole, be utilized by those working in the field of porous media. This work has essentially four parts. A single grid basic algorithm and a corresponding parallel algorithm to solve the macroscopic Navier-Stokes-Brinkmann model is discussed. An upscaling subgrid algorithm is derived and numerically tested for the same model. Moving a step further in the line of multiscale methods, an iterative Mutliscale Finite Volume (iMSFV) method is developed for the Stokes-Darcy system. Additionally, the last part of the thesis deals with ways to incorporate changes occurring at different (meso) scale level. The flow equations are coupled with the Convection-Diffusion-Reaction (CDR) equation, which models the transport and capturing of particle concentrations. By employing the numerical method for the coupled flow and transport problem, we understand the interplay between the flow velocity and filtration.

In this thesis, the coupling of the Stokes equations and the Biot poroelasticity equations for fluid flow normal to porous media is investigated. For that purpose, the transmission conditions across the interfaces between the fluid regions and the porous domain are derived. A proper algorithm is formulated and numerical examples are presented. First, the transmission conditions for the coupling of various physical phenomena are reviewed. For the coupling of free flow with porous media, it has to be distinguished whether the fluid flows tangentially or perpendicularly to the porous medium. This plays an essential role for the formulation of the transmission conditions. In the thesis, the transmission conditions for the coupling of the Stokes equations and the Biot poroelasticity equations for fluid flow normal to the porous medium in one and three dimensions are derived. With these conditions, the continuous fully coupled system of equations in one and three dimensions is formulated. In the one dimensional case the extreme cases, i.e. fluid-fluid interface and fluid impermeable solid interface, are considered. Two chapters of the thesis are devoted to the discretisation of the fully coupled Biot-Stokes system for matching and non-matching grids, respectively. Therefor, operators are introduced that map the internal and boundary variables to the respective domains via Stokes equations, Biot equations and the transmission conditions. The matrix representation of some of these operators is shown. For the non-matching case, a cell-centred grid in the fluid region and a staggered grid in the porous domain are used. Hence, the discretisation is more difficult, since an additional grid on the interface has to be introduced. Corresponding matching functions are needed to transfer the values properly from one domain to the other across the interface. In the end, the iterative solution procedure for the Biot-Stokes system on non-matching grids is presented. For this purpose, a short review of domain decomposition methods is given, which are often the methods of choice for such coupled problems. The iterative solution algorithm is presented, including details like stopping criteria, choice and computation of parameters, formulae for non-dimensionalisation, software and so on. Finally, numerical results for steady state examples, depth filtration and cake filtration examples are presented.