Stochastic Modeling and DEM Simulation of Granular Media Subsidence Due To Underground Activity

Many communities around the world have been established in areas of ongoing, as well as ceased, underground mining activity. Ground movements induced by ore extraction methods and the collapse of abandoned cavities have long been recognized as a hazard to surface structures. A number of approaches have been proposed for the prediction of subsidence in underground mining regions, and their integration to Geographic Information Systems (GIS) can produce a powerful risk management tool. Nevertheless, this application is often limited by either a lack of generality or excessive computational cost of the methods available. In this work, the stochastic subsidence model proposed by Litwiniszyn (1964) was investigated. Conceptually, the model assumes the ground mass as a discontinuous medium, in which particle displacement towards a collapsing cavity is treated as a Markovian process. The accumulation of the discrete movements amounts to the Komolgorov diffusion equation which is then employed to compute surface displacements. In order to gain better understanding of the mechanism at granular scale and test the stochastic diffusion model in controlled conditions, subsidence in a granular medium was simulated via the Discrete Element Method (DEM). Using a frictional-elastic constitutive law for inter-particle contact, large three-dimensional assemblies of gravel-size grains were generated with a range of microstructural and bulk properties, these were then subjected to trapdoor experiments. In each simulation, particle displacements, ground surface deflections, as well as stresses and changes in the granular matrix structure were monitored and provided detailed information about the phenomenon. The behavior of the granular matrix undergoing subsidence was shown to be highly dependent on both its microstructural and bulk properties. A thorough evaluation of the impact of porosity, particle size dispersion, inter-particle friction and contact stiffness on behavior of the material is presented. The parameters for the stochastic model were calculated based on the displacements obtained from DEM. The stochastic diffusion model and the DEM experiments were found in very good agreement for medium-dense simulated deposits, while in more contrasted types of either loose or dense materials, discrepancies were observed

A radiative transfer framework for non-exponential media

We develop a new theory of volumetric light transport for media with non-exponential free-flight distributions. Recent insights from atmospheric sciences and neutron transport demonstrate that such distributions arise in the presence of correlated scatterers, which are naturally produced by processes such as cloud condensation and fractal-pattern formation. Our theory accommodates correlations by disentangling the concepts of the free-flight distribution and transmittance, which are equivalent when scatterers are statistically independent, but become distinct when correlations are present. Our theory results in a generalized path integral which allows us to handle non-exponential media using the full range of Monte Carlo rendering algorithms while enriching the range of achievable appearance. We propose parametric models for controlling the statistical correlations by leveraging work on stochastic processes, and we develop a method to combine such unresolved correlations (and the resulting non-exponential free-flight behavior) with explicitly modeled macroscopic heterogeneity. This provides a powerful authoring approach where artists can freely design the shape of the attenuation profile separately from the macroscopic heterogeneous density, while our theory provides a physically consistent interpretation in terms of a path space integral. We address important considerations for graphics including energy conservation, reciprocity, and bidirectional rendering algorithms, all in the presence of surfaces and correlated media

A Radiative Transfer Framework for Spatially-Correlated Materials

We introduce a non-exponential radiative framework that takes into account the local spatial correlation of scattering particles in a medium. Most previous works in graphics have ignored this, assuming uncorrelated media with a uniform, random local distribution of particles. However, positive and negative correlation lead to slower- and faster-than-exponential attenuation respectively, which cannot be predicted by the Beer-Lambert law. As our results show, this has a major effect on extinction, and thus appearance. From recent advances in neutron transport, we first introduce our Extended Generalized Boltzmann Equation, and develop a general framework for light transport in correlated media. We lift the limitations of the original formulation, including an analysis of the boundary conditions, and present a model suitable for computer graphics, based on optical properties of the media and statistical distributions of scatterers. In addition, we present an analytic expression for transmittance in the case of positive correlation, and show how to incorporate it efficiently into a Monte Carlo renderer. We show results with a wide range of both positive and negative correlation, and demonstrate the differences compared to classic light transport

A model for permeability evolution during volcanic welding

Volcanic ash and pyroclasts can weld when deposited hot by pyroclastic density currents, in near-vent fall deposits, or in fractures in volcano interiors. Welding progressively decreases the permeability of the particle packs, influencing a range of magmatic and volcanic processes, including magma outgassing, which is an important control on eruption dynamics. Consequently, there is a need for a quantitative model for permeability evolution during welding of ash and pyroclasts under the range of conditions encountered in nature. Here we present in situ experiments in which hydrous, crystal-free, glassy pyroclasts are imaged via x-ray tomography during welding at high temperature. For each 3D dataset acquired, we determine the porosity, Darcian gas permeability, specific surface area, and pore connectivity. We find that all of these quantities decrease as a critical percolation threshold is approached. We develop a constitutive mathematical model for the evolution of permeability in welding volcanic systems based on percolation theory, and validate the model against our experimental data. Importantly, our model accounts for polydispersivity of the grainsize in the particle pack, the pressures acting on the pack, and changes in particle viscosity arising from degassing of dissolved H2O during welding. Our model is theoretically grounded and has no fitting parameters, hence it should be valid across all magma compositions. The model can be used to predict whether a cooling pyroclast pack will have sufficient time to weld and to degas, the scenarios under which a final deposit will retain a permeable network, the timescales over which sealing occurs, and whether a welded deposit will have disequilibrium or equilibrium H2O content. A user-friendly implementation of the model is provided

The permeability of magma mush

Models for the evolution of magma mush zones are of fundamental importance for understanding magma storage, differentiation in the crust, and melt extraction processes that prime eruptions. Mush mobilisation and melt segregation are predominant mechanisms that control mush evolution yet are to date still insufficiently understood and models for these are poorly constrained. These models are underpinned by calculations of the permeability of the evolving crystal frameworks in the mush, which controls the rate of melt movement relative to crystals. To date, no mush permeability model accounts for the shape of the crystals that form the crystal-framework in the mush. Herein, we assume that mush crystals are approximately cuboidal, and using that geometric approximation, we present new models for the permeability of mush in which crystal shape parameters are a key input. First, we present an extension of the Kozeny-Carman permeability law specifically for crystal packs at their maximum packing, for which the axis lengths of the crystals are the primary input. Second, we present a model for the evolution of magma mush permeability that is valid from maximum packing down to low melt fractions, ideal for simulating permeability as mush crystalises. In all cases we use a combination of numerical approaches to generate packs of cuboids for analysis, and experimental approaches to create digital 3D scans of anisotropic crystal shapes as an analogue for crystal mush. Using a combination of Avizo 3D image analysis, and a lattice-Boltzmann simulation technique, we constrain the permeability of both the numerical and experimental samples; these data then validate our models across a wide range of parameter space applicable to real magma mush. Furthermore, we propose and validate innovative solutions for permeability that can be found using only 2D data (for example, using a thin section scan), which is useful for common situations where full 3D information may not be available for analysis. In general, our results show that if we consider melt percolation in magma mush akin to fluid flow through porous media, the complexity and anisotropy are well represented by the specific surface area of the crystals. Knowledge of the crystal shape and size are essential variables in our proposed permeability model, unless the mush displays overgrowth textures at low melt fraction, in which case the effect of shape becomes less important. Our results have key implications for melt extraction timescales and cumulate textures as well as for crustal melt segregation processes and reactive flow on the scale of mush reservoirs

Modern computational studies of the glass transition

The physics of the glass transition and amorphous materials continues to attract the attention of a wide research community after decades of effort. Supercooled liquids and glasses have been studied numerically since the advent of molecular dynamics and Monte Carlo simulations in the last century. Computer studies have greatly enhanced both experimental discoveries and theoretical developments and constitute an active and continually expanding research field. Our goal in this review is to provide a modern perspective on this area. We describe the need to go beyond canonical methods to attack a problem that is notoriously difficult in terms of time scales, length scales, and physical observables. We first summarise recent algorithmic developments to achieve enhanced sampling and faster equilibration using replica exchange methods, cluster and swap Monte Carlo algorithms, and other techniques. We then review some major recent advances afforded by these novel tools regarding the statistical mechanical description of the liquid-to-glass transition as well as the mechanical, vibrational and thermal properties of the glassy solid. We finally describe some important challenges for future research

CFD Modeling of Fluidized Beds

We review the mathematical modeling of fluidized suspensions with focus on the Eulerian (or multifluid) approach. After a brief survey of different modeling approaches adopted for multiphase flows, we discuss the Eulerian equations of motion for fluidized suspensions of a finite number of monodisperse particle classes, obtained by volume averaging. We present the problem of closure for the stress tensors and the interaction forces between the phases and report some of the constitutive relations used for them in the literature. Finally, we explain briefly the population balance modeling approach, which allows handling suspensions of particles continuously distributed over any of their properties of interest