Der ehemalige Salzbergbau im Raum Staßfurt führt seit dem 19. Jahrhundert zu Subrosion und teils bruchhaften Deformationen und damit verbundenen Senkungen und Vernässungen im Stadtgebiet. Im Rahmen eines Forschungsverbundvorhabens unter Federführung der BGR sollen in dieser Arbeit die grundlegenden strömungsdynamischen Prozesse im Salinar und Deckgebirge der betroffenen geologischen Formationen aufgeklärt werden. An Bohrkernen aus den entsprechenden Bereichen werden Durchflussexperimente durchgeführt und die Fluiddynamik im Inneren der Proben mittels Positronen-Emissions-Tomographie dreidimensional dargestellt. In Kooperation mit der Bundesanstalt für Materialforschung- und prüfung Berlin und dem Geologischen Institut der Johannes-Gutenberg Universität Mainz werden diese PET-Messungen der Fluiddynamik mit hochauflösenden computertomographischen Messungen der internen Struktur der Proben in Übereinstimmung gebracht. Die beobachteten Fließmuster sollen mittels einer Lattice-Boltzmann-Simulation nachvollzogen und dadurch das grundlegende Verständnis der Strömungsdynamik in diesen Gesteinen erweitert werden. Langfristig soll dies zu einer Verbesserung des Verständnisses der Grundwasserdynamik auf regionaler Ebene führen.:1 Einleitung 1 1.1 Die Stadt Staßfurt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Problemstellung . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 Bildgebende tomographische Verfahren . . . . . . . . . . . . . . . . . . . . . 3 1.4 Die PET in den Geowissenschaften . . . . . . . . . . . . . . . . . . . . . . . 5 1.5 Ziel der Arbeit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2 Geologische Übersicht 9 2.1 Allgemein . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.1.1 Das Zechsteinmeer . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.1.2 Das Subherzyne Becken . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.1.3 Der Staßfurt-Egelner Salzsattel . . . . . . . . . . . . . . . . . . . . . 10 2.2 Stratigraphie und Hydrologie . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.2.1 Stratigraphische Einheiten . . . . . . . . . . . . . . . . . . . . . . . . 11 2.2.2 Anstehende Wässer und deren Migration . . . . . . . . . . . . . . . 15 3 Die Positronen-Emissions-Tomographie 19 3.1 Entstehung des Bildes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3.2 Die Auflösung des Bildes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 3.2.1 Die Grenzen der Ortsauflösung . . . . . . . . . . . . . . . . . . . . . 21 3.2.2 Messfehler und Bildrekonstruktion . . . . . . . . . . . . . . . . . . . 24 3.2.3 Konsequenzen für die Interpretation . . . . . . . . . . . . . . . . . . 25 4 Material und Methoden 27 4.1 Die Bohrkerne . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 4.2 Die Injektionslösungen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 4.3 Probenvorbereitung und Experimentverlauf . . . . . . . . . . . . . . . . . . 28 4.4 Auswertemethoden . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 4.4.1 Visualisierung und Datenverarbeitung . . . . . . . . . . . . . . . . . 33 4.4.2 Bestimmung hydrodynamischer Kenngrößen . . . . . . . . . . . . . . 34 4.4.3 Variographie . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 5 Ergebnisse 47 5.1 Chemische Analysen der Lösungen . . . . . . . . . . . . . . . . . . . . . . . 47 5.2 PET- und CT-Daten und resultierende Durchbruchkurven . . . . . . . . . . 50 5.2.1 Hauptanhydrit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 5.2.2 Unterer Buntsandstein . . . . . . . . . . . . . . . . . . . . . . . . . . 52 5.2.3 Mechanisch belastetes Steinsalz . . . . . . . . . . . . . . . . . . . . . 57 5.2.4 Hohlraumversatz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 5.3 Ermittlung raumbezogener Parameter in der Literatur und mit den vorhandenen Daten . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 5.4 Variographischer Abgleich verschiedener Datensätze . . . . . . . . . . . . . 72 5.4.1 Aussagekraft von Variogrammen . . . . . . . . . . . . . . . . . . . . 72 i 5.4.2 Vergleich von Messung und Modellierung mittels 3D-Variographie . 79 6 Diskussion 93 6.1 Diskussion der PET-Daten und Durchbruchkurven . . . . . . . . . . . . . . 93 6.2 Diskussion der Ermittlung raumbezonener Parameter . . . . . . . . . . . . . 100 6.3 Diskussion des variographischen Abgleichs verschiedener Datensätze . . . . 101 6.4 Auswirkungen auf das hydrologische Modell Staßfurts . . . . . . . . . . . . 102 7 Ausblick 105 Literatur 109 Anhang xi A Bohrpunkte und Lösungen xiii B 3D-Variographie des mechanisch belasteten Salzkerns xv C Hydrodynamische Parameter xvii C.1 Bohrkern aus der Leine Formation z3A3 - Hauptanhydrit . . . . . . . . . . xvii C.2 Bohrkern aus dem Unteren Buntsandstein . . . . . . . . . . . . . . . . . . . xviii C.3 Mechanisch belasteter Bohrkern aus dem z2Na Staßfurt-Steinsalz . . . . . . xix C.4 Bohrkern aus dem Hohlraumversatz des Grubenbaus Leopoldshall I /II . . . xx C.5 Probe aus gepresstem Kaolinit . . . . . . . . . . . . . . . . . . . . . . . . . xxi D Quelltext wichtiger Funktionen xxiii D.1 Datenreduktion - Ausschneiden des Bohrkerns . . . . . . . . . . . . . . . . . xxiv D.2 Vergröberung der Auflösung eines PET-Bildes . . . . . . . . . . . . . . . . . xxiv D.3 Verteilung der Maximalwerte und deren Histogramme . . . . . . . . . . . . xxv D.4 Empirische Variogramme . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxv D.5 Richtungsabhängige Variogramme . . . . . . . . . . . . . . . . . . . . . . . xxvi D.6 Dreidimensionale Variogramme . . . . . . . . . . . . . . . . . . . . . . . . . xxix D.7 Ebenenweise Variogramme . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxi E Liste der Kooperationspartner xxxiii

Lattice Boltzmann method (LBM) has been developed as a powerful numerical approach to simulate the complex fluid flow and heat transfer phenomena during the past two decades. As a mesoscale method based on the kinetic theory, LBM has several advantages compared with traditional numerical methods such as physical representation of microscopic interactions, dealing with complex geometries and highly parallel nature. Lattice Boltzmann method has been applied to solve various fluid behaviors and heat transfer process like conjugate heat transfer, magnetic and electric field, diffusion and mixing process, chemical reactions, multiphase flow, phase change process, non-isothermal flow in porous medium, microfluidics, fluid-structure interactions in biological system and so on. In addition, as a non-body-conformal grid method, the immersed boundary method (IBM) could be applied to handle the complex or moving geometries in the domain. The immersed boundary method could be coupled with lattice Boltzmann method to study the heat transfer and fluid flow problems. Heat transfer and fluid flow are solved on Euler nodes by LBM while the complex solid geometries are captured by Lagrangian nodes using immersed boundary method. Parallel computing has been a popular topic for many decades to accelerate the computational speed in engineering and scientific fields. Today, almost all the laptop and desktop have central processing units (CPUs) with multiple cores which could be used for parallel computing. However, the cost of CPUs with hundreds of cores is still high which limits its capability of high performance computing on personal computer. Graphic processing units (GPU) is originally used for the computer video cards have been emerged as the most powerful high-performance workstation in recent years. Unlike the CPUs, the cost of GPU with thousands of cores is cheap. For example, the GPU (GeForce GTX TITAN) which is used in the current work has 2688 cores and the price is only 1,000 US dollars. The release of NVIDIA's CUDA architecture which includes both hardware and programming environment in 2007 makes GPU computing attractive. Due to its highly parallel nature, lattice Boltzmann method is successfully ported into GPU with a performance benefit during the recent years. In the current work, LBM CUDA code is developed for different fluid flow and heat transfer problems. In this dissertation, lattice Boltzmann method and immersed boundary method are used to study natural convection in an enclosure with an array of conduting obstacles, double-diffusive convection in a vertical cavity with Soret and Dufour effects, PCM melting process in a latent heat thermal energy storage system with internal fins, mixed convection in a lid-driven cavity with a sinusoidal cylinder, and AC electrothermal pumping in microfluidic systems on a CUDA computational platform. It is demonstrated that LBM is an efficient method to simulate complex heat transfer problems using GPU on CUDA.

This thesis focuses on two problems lying within the field of soft condensed matter: the viscous fingering or Saffman-Taylor instability and nematic liquid crystals in confinement. Whenever a low viscosity fluid displaces a high viscosity fluid in a porous medium, for example water pushing oil out of oil reservoirs, the interface between the two fluids is rendered unstable. Viscous fingers develop, grow and compete until a single finger spans all the way from inlet to outlet. Here, using a free energy lattice Boltzmann algorithm, we examine the Saffman-Taylor instability for two different wetting situations: (a) when neither of the two fluids wet the walls of the channel and (b) when the displacing fluids completely wets the walls. We demonstrate that curvature effects in the third dimension, which arise because of the wetting boundary conditions, can lead to a novel suppression of the instability. Recent experiments in microchannels using colloid-polymer mixtures support our findings. In the second part of the thesis we examine nematic liquid crystals confined in wedge-structured geometries. In these systems the final stable configuration of the liquid crystal system is controlled by the complex interplay between confinement, elasticity and surface anchoring. Varying the wedge opening angle this competition leads to a splay to bend transition mediated by a defect in the bulk of the wedge. Using a hybrid lattice Boltzmann algorithm we study the splay-bend transition and compare to recent experiments on {em fd} virus particles in microchannels. Our numerical results, in quantitative agreement with the experiments, enable us to predict the position of the defect as a function of opening angle, and elucidate its role in the change of director structure. This has relevance to novel energy saving, liquid crystal devices which rely on defect motion and pinning to create bistable director configurations.

Micrometer injection moulding is a type of moulding in which moulds have geometrical design features on a micrometer scale that must be transferred to the geometry of the produced part. The difficulties encountered due to very high shear and rapid heat transfer of these systems has motivated this investigation into the fundamental mathematics behind polymer heat transfer and associated processes. The aim is to derive models for polymer dynamics, especially heat dynamics, that are considerably less approximate than the ones used at present, and to translate this into simulation and optimisation algorithms and strategies, Thereby allowing for greater control of the various polymer processing methods at micrometer scales.

Title: Non-newtonian fluid flow simulation using lattice Boltzmann method Author: Bc. Pavel Kuriščák Department: Mathematical Institute, Charles University Supervisor: RNDr. Ing. Jaroslav Hron Ph.D. Supervisor's e-mail address: Jaroslav.Hron@mff.cuni.cz Abstract: The aim of this thesis is to find and estabilish a modification to the Lattice Boltzmann Method, allowing it to simulate non-newtonian behaviour of fluids. In the theoretical part of thesis, there is introduced a derivation, based on the work of [22], that is capable of arriving to macroscopical Navier-Stokes equa- tions completely a priori from the Boltzmann equation, utilizing the Hermite basis expansion. This derivation is afterwards applied to the method suggested by [11], that uses the changed equilibrium distribution to fine-tune the local fluid viscosity according to the non-newtonian model. In the last part of thesis, this method is implemented in the form of lattice kinetic scheme and tested on three sample problems. Keywords: Lattice Boltzmann Method, non-newtonian fluids, Hermite expansion, lattice kinetic scheme

Most realistic fluid flow problems are characterised by high Reynolds numbers and complex boundaries. Over the last ten years, immersed boundary methods (IBM) that are able to cope with realistic geometries have been applied to Lattice- Boltzmann methods (LBM). These methods, however, have normally been applied to low Reynolds number problems. In the present work, an iterative direct forcing IBM has been successfully coupled with a multi-domain cascaded LBM in order to investigate viscous flows around rigid, moving and wilfully deformed boundaries at a wide range of Reynolds numbers. The iterative force-correction immersed boundary method of (Zhang et al., 2016) has been selected due to the improved accuracy of the computation, while the cascaded LB formulation is used due to its superior stability at high Reynolds numbers. The coupling is shown to improve both the stability and numerical accuracy of the solution. The resulting solver has been applied to viscous flow (up to a Reynolds number of 100000) passed a NACA-0012 airfoil at a 10 degree angle of attack. Good agreement with results obtained using a body-fitted Navier-Stokes solver has been obtained. At moving or deformable boundary applications, emphasis should be given on the influence of the internal mass on the computation of the aerodynamic forces, focusing on deforming boundary motions where the rigid body approximation is no longer valid. Both the rigid body and the internal Lagrangian points approximations are examined. The resulting solver has been applied to viscous flows around an in-line oscillating cylinder, a pitching foil, a plunging SD7003 airfoil and a plunging and flapping NACA-0014 airfoil. Good agreement with experimental results and other numerical schemes has been obtained. It is shown that the internal Lagrangian points approximation accurately captures the internal mass effects in linear and angular motions, as well as in deforming motions, at Reynolds numbers up to 4 • 104. Finally, an expanded higher-order immersed boundary method which addresses two major drawbacks of the conventional IBM will be presented. First, an expanded velocity profile scheme has been developed, in order to compensate for the discontinuities caused by the gradient of the velocity across the boundary. Second, a numerical method derived from the Navier-Stokes equations in order to correct the pressure distribution across the boundary has been examined. The resulting hybrid solver has been applied to viscous flows around stationary and oscillating cylinders and examined the hovering flight of elliptical wings at low Reynolds numbers. It is shown that the proposed scheme smoothly expands the velocity profile across the boundary and increases the accuracy of the immersed boundary method. In addition, the pressure correction algorithm correctly expands the pressure profile across the boundary leading to very accurate pressure coefficient values along the boundary surface. The proposed numerical schemes are shown to be very efficient in terms of computational cost. The majority of the presented results are obtained within a few hours of CPU time on a 2.8 GHz Intel Core i7 MacBook Pro computer with a 16GB memory.

Modelling the ice accretion in glaze regime for the supercooled large droplets is one of the most challenging problems in the aircraft icing field. The difficulties are related to the presence of the liquid water film on the surface in the glaze regime and also the phenomena associated with SLD conditions, specifically the splashing and re-impingement. The steady improvement of simulation methods and the increasing demand for highly optimised aircraft performance, make it worthwhile to try to get beyond the current level of modelling accuracy. A semi-empirical method has been presented to characterize the thin water film in the icing problem based on both analytical and experimental approaches. The experiments have been performed at the Cranfield icing facilities. Imaging techniques have been used to observe and measure the features of the thin water film in the different conditions. A series of numerical simulations based on an inviscid VOF model have been performed to characterize the splashing process for different water film to droplet size ratios and impact angles. Based on these numerical simulations and the proposed methods to estimate the thin water film thickness, a framework has been presented to model the effects of the splashing in the icing simulation. These effects are the lost mass from the water film due to the splashing and the re-impingement of the ejected droplets. Finally, a new framework to study the solidification process of the thin water film has been explored. This framework is based on the lattice Boltzmann method and the preliminary results showed the capabilities of the method to model the dynamics, thermodynamics and the solidification of the thin water film.

Reactive Transport Models (RTMs) provide quantitative tools to analyze the interaction between transport and biogeochemical processes in subsurface environments such as aquatic sediments and groundwater flow. A tremendous amount of research has shown the role and impact of scaling behavior of the reactive systems which stems from geologic heterogeneity. Depending on the kinetics of the reactions, different types of formulations have been proposed to describe reactions in RTMs. We introduce a novel quantitative criteria on the range of validity of local equilibrium assumption (LEA) in aquatic sediments with irreversible heterogeneous sorption reactions. Then we present a one-dimensional (1-D) early diagenetic module, MATSEDLAB, developed in MATLAB. The module provides templates for representing the reaction network, boundary conditions and transport regime, which the user can modify to fit the particular early diagenetic model configuration of interest. We describe the theoretical background of the model and introduce the MATLAB pdepe solver, followed by calibration and validation of the model by a number of theoretical and empirical applications. Finally, we introduce a new pore-scale model using lattice Boltzmann (LB) approach. It uses an iterative scheme for the chemical transport-reaction part and recent advances in the development of optimal advection-diffusion solvers within the lattice Boltzmann method framework. We present results for the dissolution and precipitation of a porous medium under different dynamical conditions, varying reaction rates and the ratio of advective to diffusive transport (Pe, Peclet number) for linear reactions. The final set of calculations considers sorption reactions on a heterogeneous porous medium. We use our model to investigate the effect of heterogeneity on the pore-scale distribution of sorption sites and the competition between three different sorption reactions.

Modelling the ice accretion in glaze regime for the supercooled large droplets is one of the most challenging problems in the aircraft icing field. The difficulties are related to the presence of the liquid water film on the surface in the glaze regime and also the phenomena associated with SLD conditions, specifically the splashing and re-impingement. The steady improvement of simulation methods and the increasing demand for highly optimised aircraft performance, make it worthwhile to try to get beyond the current level of modelling accuracy. A semi-empirical method has been presented to characterize the thin water film in the icing problem based on both analytical and experimental approaches. The experiments have been performed at the Cranfield icing facilities. Imaging techniques have been used to observe and measure the features of the thin water film in the different conditions. A series of numerical simulations based on an inviscid VOF model have been performed to characterize the splashing process for different water film to droplet size ratios and impact angles. Based on these numerical simulations and the proposed methods to estimate the thin water film thickness, a framework has been presented to model the effects of the splashing in the icing simulation. These effects are the lost mass from the water film due to the splashing and the re-impingement of the ejected droplets. Finally, a new framework to study the solidification process of the thin water film has been explored. This framework is based on the lattice Boltzmann method and the preliminary results showed the capabilities of the method to model the dynamics, thermodynamics and the solidification of the thin water film.

Three-dimensional large-eddy simulations (LES) of single and multiple jets in cross-flow (JICF) were conducted using the 19-bit Lattice Boltzmann Equation (LBE) method coupled with a conventional Navier-Stokes (NS) finite-volume scheme. In this coupled LBE-NS approach, the LBE-LES was employed to simulate the flow inside jet nozzles, while the NS-LES was used to simulate the cross-flow. The key application area was to study the micro-blowing technique (MBT) for drag control similar to recent experiments at NASA/GRC. A single jet in the cross-flow case was used for validation purposes, and results were compared with experimental data and full LBE-LES simulation. Good agreement with data was obtained. Transient analysis of flow structures was performed to investigate the contribution of flow structures to the counter-rotating vortex pair (CRVP) formation. It was found that both spanwise roller (at the lee side of the jet) and streamwise vortices (at the jet-side) contribute to the generation of the CRVP. Span-wise roller at the corner of the jet experiences high spanwise vortex compression as well as high streamwise vortex stretch. As a result, they get realigned, mix with the jet-side streamwise vortices, and eventually generate the CRVP. Furthermore, acoustic pulses were used to test the proper information exchange from the LBE domain to the NS domain, and vice-versa. Subsequently, MBT over a flat plate with porosity of 25 percent was simulated using nine jets in a compressible cross-flow at a Mach number of 0.4. Three cases with injection ratios of 0.003, 0.02 and 0.07 were conducted to investigate how the blowing rate impacts skin friction. It is shown that MBT suppressed the near-wall vortices and reduced the skin friction by up to 50 percent. This is in good agreement with experimental data.