Application of reactive fluid transport modeling to hydrothermal systems

A one-dimensional multi-component reactive fluid transport algorithm, 1DREACT (Steefel, 1993) was used to investigate different fluid-rock interaction systems. A major short coming of mass transport calculations which include mineral reactions is that solid solutions occurring in many minerals are not treated adequately. Since many thermodynamic models of solid solutions are highly non-linear, this can seriously impact on the stability and efficiency of the solution algorithms used. Phase petrology community saw itself faced with a similar predicament 10 years ago. To improve performance and reliability, phase equilibrium calculations have been using pseudo compounds. The same approach is used here in the first, using the complex plagioclase solid solution as an example. Thermodynamic properties of a varying number of intermediate plagioclase phases were calculated using ideal molecular, Al-avoidance, and non-ideal mixing models. These different mixing models can easily be incorporated into the simulations without modification of the transport code. Simulation results show that as few as nine intermediate compositions are sufficient to characterize the diffusional profile between albite and anorthite. Hence this approach is very efficient, and can be used with little effort. A subsequent chapter reports the results of reactive fluid transport modeling designed to constrain the hydrothermal alteration of Paleoproterozoic sediments of the Southern Lake Superior region. Field observations reveal that quartz-pyrophyllite (or kaolinite) bearing assemblages have been transformed into muscovite-pyrophyllite-diaspore bearing assemblages due to action of fluids migrating along permeable flow channels. Fluid-rock interaction modeling with an initial qtz-prl assemblage and a K-rich fluid simulates the formation of observed mineralogical transformation. The bulk composition of the system evolves from an SiO2-rich one to an Al2O3+K2O-rich one. Simulations show that the fluid flow was up-temperature (e.g. recharge) and that fluid was K-rich. Pseudo compound approach to include solid solutions in reactive transport models was tested in modeling hydrothermal alteration of Icelandic basalts. Solid solutions of chlorites, amphiboles and plagioclase were included as the secondary mineral phases. Saline and fresh water compositions of geothermal fluids were used to investigate the effect of salinity on alteration. Fluid-rock interaction simulations produce the observed mineral transformations. They show that roughly the same alteration minerals are formed due to reactions with both types of fluid which is in agreement with the field observations. A final application is directed towards the remediation of nitrate rich groundwaters. Removal of excess nitrate from groundwater by pyrite oxidation was modeled using the reactive fluid transport algorithm. Model results show that, when a pyrite-bearing, permeable zone is placed in the flow path, nitrate concentration in infiltrating water can be significantly lowered, in agreement with proposals from the literature. This is due to nitrogen reduction. Several simulations investigate the efficiency of systems with different mineral reactive surface areas, reactive barrier zone widths, and flow rates to identify the optimum setup.

The grain-scale distribution and behaviour of melt and fluid in crystalline analogue systems

The production, segregation and migration of melt and aqueous fluids (henceforth called liquid) plays an important role for the transport of mass and energy within the mantle and the crust of the Earth. Many properties of large-scale liquid migration processes such as the permeability of a rock matrix or the initial segregation of newly formed liquid from the host-rock depends on the grain-scale distribution and behaviour of liquid. Although the general mechanisms of liquid distribution at the grain-scale are well understood, the influence of possibly important modifying processes such as static recrystallization, deformation, and chemical disequilibrium on the liquid distribution is not well constrained. For this thesis analogue experiments were used that allowed to investigate the interplay of these different mechanisms in-situ. In high-temperature environments where melts are produced, the grain-scale distribution in “equilibrium” is fully determined by the liquid fraction and the ratio between the solid-solid and the solid-liquid surface energy. The latter is commonly expressed as the dihedral or wetting angle between two grains and the liquid phase (Chapter 2). The interplay of this “equilibrium” liquid distribution with ongoing surface energy driven recrystallization is investigated in Chapter 4 and 5 with experiments using norcamphor plus ethanol liquid. Ethanol in contact with norcamphor forms a wetting angle of about 25°, which is similar to reported angles of rock-forming minerals in contact with silicate melt. The experiments in Chapter 4 show that previously reported disequilibrium features such as trapped liquid lenses, fully-wetted grain boundaries, and large liquid pockets can be explained by the interplay of the liquid with ongoing recrystallization. Closer inspection of dihedral angles in Chapter 5 reveals that the wetting angles are themselves modified by grain coarsening. Ongoing recrystallization constantly moves liquid-filled triple junctions, thereby altering the wetting angles dynamically as a function of the triple junction velocity. A polycrystalline aggregate will therefore always display a range of equilibrium and dynamic wetting angles at raised temperature, rather than a single wetting angle as previously thought. For the deformation experiments partially molten KNO3–LiNO3 experiments were used in addition to norcamphor–ethanol experiments (Chapter 6). Three deformation regimes were observed. At a high bulk liquid fraction >10 vol.% the aggregate deformed by compaction and granular flow. At a “moderate” liquid fraction, the aggregate deformed mainly by grain boundary sliding (GBS) that was localized into conjugate shear zones. At a low liquid fraction, the grains of the aggregate formed a supporting framework that deformed internally by crystal plastic deformation or diffusion creep. Liquid segregation was most efficient during framework deformation, while GBS lead to slow liquid segregation or even liquid dispersion in the deforming areas.

Thermal relaxation for particle systems in interaction with several bosonic heat reservoirs

The present thesis is concerned with the study of a quantum physical system composed of a small particle system (such as a spin chain) and several quantized massless boson fields (as photon gasses or phonon fields) at positive temperature. The setup serves as a simplified model for matter in interaction with thermal "radiation" from different sources. Hereby, questions concerning the dynamical and thermodynamic properties of particle-boson configurations far from thermal equilibrium are in the center of interest. We study a specific situation where the particle system is brought in contact with the boson systems (occasionally referred to as heat reservoirs) where the reservoirs are prepared close to thermal equilibrium states, each at a different temperature. We analyze the interacting time evolution of such an initial configuration and we show thermal relaxation of the system into a stationary state, i.e., we prove the existence of a time invariant state which is the unique limit state of the considered initial configurations evolving in time. As long as the reservoirs have been prepared at different temperatures, this stationary state features thermodynamic characteristics as stationary energy fluxes and a positive entropy production rate which distinguishes it from being a thermal equilibrium at any temperature. Therefore, we refer to it as non-equilibrium stationary state or simply NESS. The physical setup is phrased mathematically in the language of C*-algebras. The thesis gives an extended review of the application of operator algebraic theories to quantum statistical mechanics and introduces in detail the mathematical objects to describe matter in interaction with radiation. The C*-theory is adapted to the concrete setup. The algebraic description of the system is lifted into a Hilbert space framework. The appropriate Hilbert space representation is given by a bosonic Fock space over a suitable L2-space. The first part of the present work is concluded by the derivation of a spectral theory which connects the dynamical and thermodynamic features with spectral properties of a suitable generator, say K, of the time evolution in this Hilbert space setting. That way, the question about thermal relaxation becomes a spectral problem. The operator K is of Pauli-Fierz type. The spectral analysis of the generator K follows. This task is the core part of the work and it employs various kinds of functional analytic techniques. The operator K results from a perturbation of an operator L0 which describes the non-interacting particle-boson system. All spectral considerations are done in a perturbative regime, i.e., we assume that the strength of the coupling is sufficiently small. The extraction of dynamical features of the system from properties of K requires, in particular, the knowledge about the spectrum of K in the nearest vicinity of eigenvalues of the unperturbed operator L0. Since convergent Neumann series expansions only qualify to study the perturbed spectrum in the neighborhood of the unperturbed one on a scale of order of the coupling strength we need to apply a more refined tool, the Feshbach map. This technique allows the analysis of the spectrum on a smaller scale by transferring the analysis to a spectral subspace. The need of spectral information on arbitrary scales requires an iteration of the Feshbach map. This procedure leads to an operator-theoretic renormalization group. The reader is introduced to the Feshbach technique and the renormalization procedure based on it is discussed in full detail. Further, it is explained how the spectral information is extracted from the renormalization group flow. The present dissertation is an extension of two kinds of a recent research contribution by Jakšić and Pillet to a similar physical setup. Firstly, we consider the more delicate situation of bosonic heat reservoirs instead of fermionic ones, and secondly, the system can be studied uniformly for small reservoir temperatures. The adaption of the Feshbach map-based renormalization procedure by Bach, Chen, Fröhlich, and Sigal to concrete spectral problems in quantum statistical mechanics is a further novelty of this work.

Influences of deep convective cloud systems on tropospheric trace gases and photochemistry over the tropical West Pacific

A numerical model for studying the influences of deep convective cloud systems on photochemistry was developed based on a non-hydrostatic meteorological model and chemistry from a global chemistry transport model. The transport of trace gases, the scavenging of soluble trace gases, and the influences of lightning produced nitrogen oxides (NOx=NO+NO2) on the local ozone-related photochemistry were investigated in a multi-day case study for an oceanic region located in the tropical western Pacific. Model runs considering influences of large scale flows, previously neglected in multi-day cloud resolving and single column model studies of tracer transport, yielded that the influence of the mesoscale subsidence (between clouds) on trace gas transport was considerably overestimated in these studies. The simulated vertical transport and scavenging of highly soluble tracers were found to depend on the initial profiles, reconciling contrasting results from two previous studies. Influences of the modeled uptake of trace gases by hydrometeors in the liquid and the ice phase were studied in some detail for a small number of atmospheric trace gases and novel aspects concerning the role of the retention coefficient (i.e. the fraction of a dissolved trace gas that is retained in the ice phase upon freezing) on the vertical transport of highly soluble gases were illuminated. Including lightning NOx production inside a 500 km 2-D model domain was found to be important for the NOx budget and caused small to moderate changes in the domain averaged ozone concentrations. A number of sensitivity studies yielded that the fraction of lightning associated NOx which was lost through photochemical reactions in the vicinity of the lightning source was considerable, but strongly depended on assumptions about the magnitude and the altitude of the lightning NOx source. In contrast to a suggestion from an earlier study, it was argued that the near zero upper tropospheric ozone mixing ratios which were observed close to the study region were most probably not caused by the formation of NO associated with lightning. Instead, it was argued in agreement with suggestions from other studies that the deep convective transport of ozone-poor air masses from the relatively unpolluted marine boundary layer, which have most likely been advected horizontally over relatively large distances (both before and after encountering deep convection) probably played a role. In particular, it was suggested that the ozone profiles observed during CEPEX (Central Equatorial Pacific Experiment) were strongly influenced by the deep convection and the larger scale flow which are associated with the intra-seasonal oscillation.

Existence of solutions and asymtptotic limits of the Euler-Poisson and the quantum drift-diffusion systems

My work concerns two different systems of equations used in the mathematical modeling of semiconductors and plasmas: the Euler-Poisson system and the quantum drift-diffusion system. The first is given by the Euler equations for the conservation of mass and momentum, with a Poisson equation for the electrostatic potential. The second one takes into account the physical effects due to the smallness of the devices (quantum effects). It is a simple extension of the classical drift-diffusion model which consists of two continuity equations for the charge densities, with a Poisson equation for the electrostatic potential. Using an asymptotic expansion method, we study (in the steady-state case for a potential flow) the limit to zero of the three physical parameters which arise in the Euler-Poisson system: the electron mass, the relaxation time and the Debye length. For each limit, we prove the existence and uniqueness of profiles to the asymptotic expansion and some error estimates. For a vanishing electron mass or a vanishing relaxation time, this method gives us a new approach in the convergence of the Euler-Poisson system to the incompressible Euler equations. For a vanishing Debye length (also called quasineutral limit), we obtain a new approach in the existence of solutions when boundary layers can appear (i.e. when no compatibility condition is assumed). Moreover, using an iterative method, and a finite volume scheme or a penalized mixed finite volume scheme, we numerically show the smallness condition on the electron mass needed in the existence of solutions to the system, condition which has already been shown in the literature. In the quantum drift-diffusion model for the transient bipolar case in one-space dimension, we show, by using a time discretization and energy estimates, the existence of solutions (for a general doping profile). We also prove rigorously the quasineutral limit (for a vanishing doping profile). Finally, using a new time discretization and an algorithmic construction of entropies, we prove some regularity properties for the solutions of the equation obtained in the quasineutral limit (for a vanishing pressure). This new regularity permits us to prove the positivity of solutions to this equation for at least times large enough.

Numerical studies of colloidal systems

The interplay of hydrodynamic and electrostatic forces is of great importance for the understanding of colloidal dispersions. Theoretical descriptions are often based on the so called standard electrokinetic model. This Mean Field approach combines the Stokes equation for the hydrodynamic flow field, the Poisson equation for electrostatics and a continuity equation describing the evolution of the ion concentration fields. In the first part of this thesis a new lattice method is presented in order to efficiently solve the set of non-linear equations for a charge-stabilized colloidal dispersion in the presence of an external electric field. Within this framework, the research is mainly focused on the calculation of the electrophoretic mobility. Since this transport coefficient is independent of the electric field only for small driving, the algorithm is based upon a linearization of the governing equations. The zeroth order is the well known Poisson-Boltzmann theory and the first order is a coupled set of linear equations. Furthermore, this set of equations is divided into several subproblems. A specialized solver for each subproblem is developed, and various tests and applications are discussed for every particular method. Finally, all solvers are combined in an iterative procedure and applied to several interesting questions, for example, the effect of the screening mechanism on the electrophoretic mobility or the charge dependence of the field-induced dipole moment and ion clouds surrounding a weakly charged sphere. In the second part a quantitative data analysis method is developed for a new experimental approach, known as "Total Internal Reflection Fluorescence Cross-Correlation Spectroscopy" (TIR-FCCS). The TIR-FCCS setup is an optical method using fluorescent colloidal particles to analyze the flow field close to a solid-fluid interface. The interpretation of the experimental results requires a theoretical model, which is usually the solution of a convection-diffusion equation. Since an analytic solution is not available due to the form of the flow field and the boundary conditions, an alternative numerical approach is presented. It is based on stochastic methods, i. e. a combination of a Brownian Dynamics algorithm and Monte Carlo techniques. Finally, experimental measurements for a hydrophilic surface are analyzed using this new numerical approach.

Hydrodynamics and solid dosage form disintegration/dissolution : immediate release tablets and novel in situ polyelectrolyte gastroretentive drug delivery systems

Solid oral dosage form disintegration in the human stomach is a highly complex process dependent on physicochemical properties of the stomach contents as well as on physical variables such as hydrodynamics and mechanical stress. Understanding the role of hydrodynamics and forces in disintegration of oral solid dosage forms can help to improve in vitro disintegration testing and the predictive power of the in vitro test. The aim of this work was to obtain a deep understanding of the influence of changing hydrodynamic conditions on solid oral dosage form performance. Therefore, the hydrodynamic conditions and forces present in the compendial PhEur/USP disintegration test device were characterized using a computational fluid dynamics (CFD) approach. Furthermore, a modified device was developed and the hydrodynamic conditions present were simulated using CFD. This modified device was applied in two case studies comprising immediate release (IR) tablets and gastroretentive drug delivery systems (GRDDS). Due to the description of movement provided in the PhEur, the movement velocity of the basket-rack assembly follows a sinusoidal profile. Therefore, hydrodynamic conditions are changing continually throughout the movement cycle. CFD simulations revealed that the dosage form is exposed to a wide range of fluid velocities and shear forces during the test. The hydrodynamic conditions in the compendial device are highly variable and cannot be controlled. A new, modified disintegration test device based on computerized numerical control (CNC) technique was developed. The modified device can be moved in all three dimensions and radial movement is also possible. Simple and complex moving profiles can be developed and the influence of the hydrodynamic conditions on oral solid dosage form performance can be evaluated. Furthermore, a modified basket was designed that allows two-sided fluid flow. CFD simulations of the hydrodynamics and forces in the modified device revealed significant differences in the fluid flow field and forces when compared to the compendial device. Due to the CNC technique moving velocity and direction are arbitrary and hydrodynamics become controllable. The modified disintegration test device was utilized to examine the influence of moving velocity on disintegration times of IR tablets. Insights into the influence of moving speed, medium viscosity and basket design on disintegration times were obtained. An exponential relationship between moving velocity of the modified basket and disintegration times was established in simulated gastric fluid. The same relationship was found between the disintegration times and the CFD predicted average shear stress on the tablet surface. Furthermore, a GRDDS was developed based on the approach of an in situ polyelectrolyte complex (PEC). Different complexes composed of different grades of chitosan and carrageenan and different ratios of those were investigated for their swelling behavior, mechanical stability, and in vitro drug release. With an optimized formulation the influence of changing hydrodynamic conditions on the swelling behavior and the drug release profile was demonstrated using the modified disintegration test device. Both, swelling behavior and drug release, were largely dependent on the hydrodynamic conditions. Concluding, it has been shown within this thesis that the application of the modified disintegration test device allows for detailed insights into the influence of hydrodynamic conditions on solid oral dosage form disintegration and dissolution. By the application of appropriate test conditions, the predictive power of in vitro disintegration testing can be improved using the modified disintegration test device. Furthermore, CFD has proven a powerful tool to examine the hydrodynamics and forces in the compendial as well as in the modified disintegration test device. rn