Three-dimensional phase-field study of crack-seal microstructures - insights from innovative post-processing techniques

Numerical simulations of vein evolution contribute to a better understanding of processes involved in their formation and possess the potential to provide invaluable insights into the rock deformation history and fluid flow pathways. The primary aim of the present article is to investigate the influence of a “realistic” boundary condition, i.e. an algorithmically generated “fractal” surface, on the vein evolution in 3-D using a thermodynamically consistent approach, while explaining the benefits of accounting for an extra dimensionality. The 3-D simulation results are supplemented by innovative numerical post-processing and advanced visualization techniques. The new methodologies to measure the tracking efficiency demonstrate the importance of accounting the temporal evolution; no such information is usually accessible in field studies and notoriously difficult to obtain from laboratory experiments as well. The grain growth statistics obtained by numerically post-processing the 3-D computational microstructures explain the pinning mechanism which leads to arrest of grain boundaries/multi-junctions by crack peaks, thereby, enhancing the tracking behavior

The Compact Implicit Integration Factor Scheme For the Solution of Allen-Cahn Equations

In this thesis we apply the compact implicit integration factor (cIIF) scheme towards solving the Allen-Cahn equations with zero-flux or periodic boundary conditions. The Allen-Cahn equation is a second-order nonlinear PDE which has been the focus of many applications spanning a wide range of fields, such as in material science where it was first introduced to model the phase separation of two metallic alloys, and in biology to study population dynamics, just to name a few. The compact implicit integration method works by first transforming the PDE into a system of ODEs by discretizing the spatial derivatives using the central differencing scheme. This yields a semi-discretized form which produces a nice compact representation to the original PDE. The resulting system is then integrated with respect to time, thereby treating the linear component of the PDE exactly. The nonlinear portion which represents the integrand is then approximated by a Lagrange interpolation polynomial of order r and then integrated exactly, with r = 0,1,2 in our study . Altogether, this a fully discrete scheme which is second-order accurate in space and (r + 1)-order accurate in time. Experiments are also performed to numerically demonstrate the stability and convergence properties of the proposed scheme

Phase-field modeling of epitaxial growth of polycrystalline quartz veins in hydrothermal experiments

Mineral precipitation in an open fracture plays a crucial role in the evolution of fracture permeability in rocks, and the microstructural development and precipitation rates are closely linked to fluid composition, the kind of host rock as well as temperature and pressure. In this study, we develop a continuum thermodynamic model to understand polycrystalline growth of quartz aggregates from the rock surface. The adapted multiphase-field model takes into consideration both the absolute growth rate as a function of the driving force of the reaction (free energy differences between solid and liquid phases), and the equilibrium crystal shape (Wulff shape). In addition, we realize the anisotropic shape of the quartz crystal by introducing relative growth rates of the facets. The missing parameters of the model, including surface energy and relative growth rates, are determined by detailed analysis of the crystal shapes and crystallographic orientation of polycrystalline quartz aggregates in veins synthesized in previous hydrothermal experiments. The growth simulations were carried out for a single crystal and for grain aggregates from a rock surface. The single crystal simulation reveals the importance of crystal facetting on the growth rate; for example, growth velocity in the c-axis direction drops by a factor of ~9 when the faceting is complete. The textures produced by the polycrystal simulations are similar to those observed in the hydrothermal experiments, including the number of surviving grains and crystallographic preferred orientations as a function of the distance from the rock wall. Our model and the methods to define its parameters provide a basis for further investigation of fracture sealing under varying conditions

Non-local Allen-Cahn systems: Analysis and a primal dual active set method

We show existence and uniqueness of a solution for the non-local vector-valued Allen-Cahn variational inequality in a formulation involving Lagrange multipliers for local and non-local constraints. Furthermore, we propose and analyze a primal-dual active set method for local and non-local vector-valued Allen-Cahn variational inequalities. Convergence of the primal-dual active set algorithm is shown by interpreting the approach as a semi-smooth Newton method and numerical simulations are presented demonstrating its efficiency

Phase-field modeling of multi-domain evolution in ferromagnetic shape memory alloys and of polycrystalline thin film growth

The phase-field method is a powerful tool in computer-aided materials science as it allows for the analysis of the time-spatial evolution of microstructures on the mesoscale. A multi-phase-field model is adopted to run numerical simulations in two different areas of scientific interest: Polycrystalline thin films growth and the ferromagnetic shape memory effect. FFT-techniques, norm conservative integration and RVE-methods are necessary to make the coupled problems numerically feasible