Narrow Results

The calculation of proppant pack properties under load is useful to determine the optimal material parameters and grain size distribution. A simple, yet effective model for simulation of both mechanical deformation and permeability of the pack in specific case of polymeric proppant is presented. The mechanical deformation modeling is similar to the discrete elements method, where permeability is calculated with the lattice Boltzmann method. The simulation imitates the standard procedure for measuring the conductivity of a proppant. Permeabilities for different loads and different grain size distributions are obtained.

Pore-scale simulations of fluid flow and mass transport offer a direct means to reproduce and verify laboratory measurements in porous media. We have compared lattice-Boltzmann (LB) flow simulations with the results of NMR spectroscopy from several published flow experiments. Although there is qualitative agreement, the differences highlight numerical and experimental issues, including the rate of spatial convergence, and the effect of signal attenuation near solid surfaces. For the range of Reynolds numbers relevant to groundwater investigations, the normalized distribution of fluid velocities in random sphere packings collapse onto a single curve, when scaled with the mean velocity. Random-walk particle simulations in the LB flow fields have also been performed to study the dispersion of an ideal tracer. These simulations show an encouraging degree of quantitative agreement with published NMR measurements of hydrodynamic and molecular dispersion, and the simulated dispersivities scale in accordance with published experimental and theoretical results for the Peclet number rangek 1 ≤Pe≤1500. Experience with the random-walk method indicates that the mean properties of conservative transport, such as the first and second moments of the particle displacement distribution, can be estimated with a number of particles comparable to the spatial discretization of the velocity field. However, the accurate approximation of local concentrations, at a resolution comparable to that of the velocity field, requires significantly more particles. This requirement presents a significant computational burden and hence a numerical challenge to the simulation of non-conservative transport processes.

It is shown that there is no quasi-sphere packing of the lattice grid ℤ^d+1 or a co-compact hyperbolic lattice of or the 3-regular tree × ℤ, in ℝ^d, for all d. A similar result is proved for some other graphs too. Rather than using a direct geometrical approach, the main tools we are using are from nonlinear potential theory.

How can we understand the origins of highly symmetrical objects? One way is to characterize them as the solutions of natural optimization problems from discrete geometry or physics. In this paper, we explore how to prove that exceptional objects, such as regular polytopes or the E₈ root system, are optimal solutions to packing and potential energy minimization problems.