Lattice Boltzmann simulation of fluid flow in fracture networks with rough, self -affine surfaces

Using the lattice Boltzmann method, we study fluid flow in a two-dimensional (2D) model of fracture network of rock. Each fracture in a square network is represented by a 2D channel with rough, self-affine internal surfaces. Various parameters of the model, such as the connectivity and the apertures of the fractures, the roughness profile of their surface, as well as the Reynolds number for flow of the fluid, are systematically varied in order to assess their effect on the effective permeability of the fracture network. The distribution of the fractures’ apertures is approximated well by a log-normal distribution, which is consistent with experimental data. Due to the roughness of the fractures’ surfaces, and the finite size of the networks that can be used in the simulations, the fracture network is anisotropic. The anisotropy increases as the connectivity of the network decreases and approaches the percolation threshold. The effective permeability K of the network follows the power law K∼〈δ〉β, where 〈δ〉 is the average aperture of the fractures in the network and the exponent β may depend on the roughness exponent. A crossover from linear to nonlinear flow regime is obtained at a Reynolds number Re∼O(1), but the precise numerical value of the crossover Re depends on the roughness of the fractures’ surfaces

Influence of polydispersity on micromechanics of granular materials

We study the effect of polydispersity on the macroscopic physical properties of granular packings in two and three dimensions. A mean-field approach is developed to approximate the macroscale quantities as functions of the microscopic ones. We show that the trace of the fabric and stress tensors are proportional to the mean packing properties (e.g. packing fraction, average coordination number, and average normal force) and dimensionless correction factors, which depend only on the moments of the particle-size distribution. Similar results are obtained for the elements of the stiffness tensor of isotropic packings in the linear affine response regime. Our theoretical predictions are in good agreement with the simulation results.Comment: 13 pages, 13 figures, 2 table

Bandgap Narrowing in Quantum Wires

In this paper we consider two different geometry of quasi one-dimensional semiconductors and calculate their exchange-correlation induced bandgap renormalization (BGR) as a function of the electron-hole plasma density and quantum wire width. Based on different fabrication scheme, we define suitable external confinement potential and then leading-order GW dynamical screening approximation is used in the calculation by treating electron-electron Coulomb interaction and electron-optical phonon interaction. Using a numerical scheme, screened Coulomb potential, probability of different states, profile of charge density and the values of the renormalized gap energy are calculated and the effects of variation of confinement potential width and temperature are studied.Comment: 17 Pages, 4 Figure

Empirical equations for bulk and shear moduli of dry sandstones based on grain-scale numerical simulations

Elastic properties of rocks are largely controlled by parameters of grain contacts. In this paper, many computer 3D models of grain aggregates are generated for different parameters of grain contacts. Then, for each of these models, elastic properties are computed using finite element simulations. Finally, empirical fits are proposed that define elastic moduli of these sandstones as functions of grain contact parameters. The results show good agreement between the numerical simulations and predictions of the empirical fit

Effect of grain shapes in coordination number from micro-CT image analysis of an unconsolidated sand

Coordination number (CN, average number of contacts a grain has with its neighbours inside a granular medium) appears to be one of the important parameters in contact based modelling of elastic properties including seismic velocities in unconsolidated sands. We analyse micro-CT images of two different resolutions of the same unconsolidated quartz sand, in order to find CN and morphology of the grains. Both of the image sets give similar results in CN and size distribution whereas grain morphology differs slightly. We find that the number of grains, to be considered for calculations in the lower resolution image, is sufficient enough to provide a better normal distribution data. However, a decreasing of sphericity and roundness on the grains show a very gentle increasing trend in CNs. Therefore, for modelling of elastic properties, not only CN, but grain morphology should also be considered along with other factors such as pressure, initial porosity, loading unloading history, and prepacking conditions

Modelling elastic anisotropy of dry rocks as a function of applied stress

We propose an analytical model for seismic anisotropy caused by the application of an anisotropic stress to an isotropic dry rock. We first consider an isotropic, linearly elastic medium (porous or non-porous) permeated by a distribution of discontinuities with random (isotropic) orientation (such as randomly oriented compliant grain contacts or cracks). The geometry of individual discontinuities is not specified. Instead, their behaviour is defined by a ratio B of the normal to tangential excess compliances. When this isotropic rock is subjected to a compressive stress (isotropic or anisotropic), the specific surface area of cracks aligned parallel to a particular plane is reduced in proportion to the normal stress traction acting on that plane. This effect is modelled using the Sayers-Kachanov non-interactive approximation, which expresses the effect of cracks on the elastic compliance tensor as an integral over crack orientations. This integral is evaluated using the Taylor expansion of the stress dependency of the specific surface area of the cracks. This allows the analytical solution previously derived for small anisotropic stresses to be extended to large stresses. Comparison of the model predictions with the results of laboratory measurements shows a reasonable agreement for moderate magnitudes of uniaxial stress (up to 50 MPa).While the model contains five independent parameters, the variation of the anisotropy pattern (which can be expressed by the ratios of Thomsen’s anisotropy parameters ε, δ and γ) with normalized stress is controlled by only two parameters: Poisson’s ratio ν of the unstressed rock and the compliance ratio B. The model predicts that the ε/γ ratio depends on both ν and B but varies only mildly with stress, while the ε/δ ratio varies between 0.8–1.1 in a wide range of values of ν and B. The latter observation implies that the anisotropy remains close to elliptical even for larger stresses (within the assumptions of the model). The proposed model of stress-induced anisotropy may be useful for differentiating stress-induced anisotropy from that caused by aligned fractures. Conversely, if the cause of seismic anisotropy is known, then the anisotropy pattern allows one to estimate P-wave anisotropy from S-wave anisotropy

Numerical Study of Bandgap Renormalization in V-Grooved Quantum Nanowires

By considering a suitable confinement potential, we calculate the exchange-correlation induced band gap renormalization (BGR) in a V-grooved quantum wire as a function of the electron-hole plasma density and quantum wire width. The leading-order GW dynamical screening approximation is used in the calculation by treating the electron-electron Coulomb interaction and electron-optical phonon interaction. A numerical scheme has been proposed, and the screened Coulomb potential, density of states (profile of charge distribution) and the value of the renormalized gap energy are calculated. We will show that the carrier concentration, the screened confinement potential and the relative band gap renormalization are functions of the ratio of the well width in the x and y directions

Fluid flow and conduction in two-dimensional fractures with rough, self-affine surfaces: A comparative study

We study fluid flow and conduction in a two-dimensional model of a fracture with rough, self-affine internal surfaces. The model consists of two parallel flat plates on which two rough, self-affine surfaces, characterized by a roughness exponent H, are superimposed. The methods that we use for computing the effective flow and transport properties of the fracture include the lattice Boltzmann method for computing the flow properties, a random walk method for determining the effective conductivity of the (fluid-saturated) fracture, and the Reynolds approximation. We also develop an asymptotic expression for the effective conductivity. The aperture of the fracture, as well as the roughness of its rough surface, are systematically varied in order to assess their effect on the effective permeability and conductivity of the fracture, and also test the accuracy and consistency of the methods. For large mean apertures, and all values of the roughness exponent H, all the methods yield essentially the same results. However, as the mean aperture decreases, the differences between the predictions of the methods increase significantly. We find that the Reynolds approximation provides relatively accurate estimates of the (hydraulic or electrical) apertures only if the fracture is at least moderately wide and that, similar to real three-dimensional fractures, the electrical aperture is always smaller than the hydraulic fracture

Wavelet based solution of flow and diffusion problems in digital materials

The computation of physical properties in a digital materials labora- tory requires significant computational resources. Due to the complex nature of the media, one of the most diffcult problems to solve is the multiphase flow problem, and traditional methods such as Lattice Boltzmann are not attractive as the computational demand for the solution is too high. A wavelet based algorithm reduces the amount of information required for computation. Here we solve the Poisson equation for a large three dimensional data set with a second order finite difference approximation. Constraints and factitious domains are used to capture the complex geometry. We solve the discrete system using a discrete wavelet transform and thresholding. We show that this method is substantially faster than the original approach and has the same order of accuracy

On the fabric tensor of polydisperse granular materials in two dimension

The trace of the fabric tensor in static, isotropic, two-dimensional, frictionless, polydisperse granular materials is examined theoretically and numerically. In the monodisperse case, the trace of the fabric tensor equals the product of volume fraction and coordination number-thus the fabrics trace can be seen as contact density. For various size distributions, we obtain a correction factor to the monodisperse observation, which involves the first three moments of the particle size distribution function. The theoretical prediction is found to be in good agreement with numerical simulations of static frictionless systems