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Peter R. King

Publications and source records attributed to Peter R. King.

15 recordsLinked to original sources

Force chains bias the dynamic response to impacts in rubble-pile asteroids

The impact response of rubble-pile asteroids is essential for both elucidating their formation and evolution history and evaluating the efficacy of impact defense strategies. Although state-of-the-art numerical simulations have allowed for the replication of many macroscopic impact characteristics consistent with observations, the understanding of dynamics and response mechanisms within rubble-pile structures remains incomplete and requires further in-depth investigation. Such understanding is critical for assessing the effects and safety of impact defense missions. The loose structure of rubble-pile asteroids affects inhomogeneous internal stress propagation via inherent force chains, which may lead to structural fracturing. We demonstrate this phenomenon here, using a proof-of-principle two-dimensional model of granular aggregates. We find that the velocity response front to impact disturbances preferentially propagates along pre-existing force chains, with particles not in chains responding more slowly. The sites within the response zone where high dynamic stresses manifest are strongly correlated with these initial force chains, and the damages that result are predominantly located within areas enclosed by these chains. The strong correlation between pre-existing force chains and dynamic response is independent of the location, magnitude, direction of the disturbance velocity, or the aggregate's particle size distribution. All evidence suggests that the core reasons for this propagation preference lie in the structural heterogeneity of granular aggregates and the resulting differences in mechanical wave propagation. This investigation provides guidance for future research aimed at quantitatively assessing fragmentation risks based on the statistical properties of force chains.

astro-ph.EP

Determining effective permeability at reservoir scale: Numerical simulations and theoretical modeling

Determining the effective permeability (keff) of geological formations has broad applications to site remediation, aquifer discharge or recharge, hydrocarbon production, and enhanced oil recovery. The objectives of this study are: (1) to explore an approach to estimating keff at the reservoir scale using the critical path analysis (CPA), (2) to evaluate the accuracy of this new approach by comparing the estimated keff to the numerically simulated effective permeability, and (3) to compare the performance of CPA estimates of keff to estimates by three other models i.e., perturbation theory (PT), effective-medium approximation (EMA), and renormalization group theory (RGT). We construct two- and three-dimensional random (uncorrelated) geologic formations based on permeability measurements from the Borden site and assume that the permeability distribution conforms to the log-normal probability density function over a wide range of means and standard deviations. Comparing keff estimated via CPA to keff values derived from numerical flow simulations indicates that CPA provides accurate estimations in both two and three dimensions over a wide range of heterogeneity levels, similar to RGT. Inter-model comparisons show that although PT and EMA provide reasonable keff estimations in rather homogeneous formations, they substantially overestimate the effective permeability in highly heterogeneous formations.

physics.geo-ph

Path integral Monte Carlo method for the quantum anharmonic oscillator

The Markov chain Monte Carlo (MCMC) method is used to evaluate the imaginary-time path integral of a quantum oscillator with a potential that includes both a quadratic term and a quartic term whose coupling is varied by several orders of magnitude. This path integral is discretized on a time lattice and calculations for the energy and probability density of the ground state and energies of the first few excited states are carried out on lattices with decreasing spacing to estimate these quantities in the continuum limit. The variation of the quartic coupling constant produces corresponding variations in the optimum simulation parameters for the MCMC method and in the statistical uncertainty for a fixed number of paths used for measurement. The energies and probability densities are in excellent agreement with those obtained from numerical solutions of Schrödinger's equation.

physics.comp-ph

Path Integral Renormalization of Flow through Random Porous Media

The path integral for Darcy's law with a stochastic conductivity, which characterizes flow through random porous media, is used as a basis for Wilson renormalization-group (RG) calculations in momentum space. A coarse graining procedure is implemented by integrating over infinitesimal shells of large momenta corresponding to the elimination of the small scale modes of the theory. The resulting one-loop $β$-functions are solved exactly to obtain an effective conductivity in a coarse grained theory over successively larger length scales. We first carry out a calculation with uncorrelated Gaussian conductivity fluctuations to illustrate the RG procedure before considering the effect of a finite correlation length of conductivity fluctuations. We conclude by discussing applications and extensions of our calculations, including comparisons with the numerical evaluation of path integrals, non-Gaussian fluctuations, and multiphase flow, for which the path integral formulation should prove particularly useful.

cond-mat.stat-mech

Pressure and flow statistics of Darcy flow from simulated annealing

The pressure and flow statistics of Darcy flow through a random permeable medium are expressed in a form suitable for evaluation by the method of simulated annealing. There are several attractive aspects to using simulated annealing: (i) any probability distribution can be used for the permeability, (ii) there is no need to invert the transmissibility matrix which, while not a factor for single-phase flow, offers distinct advantages for the case of multiphase flow, and (iii) the action used for simulated annealing is eminently suitable for coarse graining by integrating over the short-wavelength degrees of freedom. In this paper, we show that the pressure and flow statistics obtained by simulated annealing are in excellent agreement with the more conventional finite-volume calculations.

cs.CE

Pressure statistics from the path integral for Darcy flow through random porous media

The path integral for classical statistical dynamics is used to determine the properties of one-dimensional Darcy flow through a porous medium with a correlated stochastic permeability for several spatial correlation lengths. Pressure statistics are obtained from the numerical evaluation of the path integral by using the Markov chain Monte Carlo method. Comparisons between these pressure distributions and those calculated from the classic finite-volume method for the corresponding stochastic differential equation show excellent agreement for Dirichlet and Neumann boundary conditions. The evaluation of the variance of the pressure based on a continuum description of the medium provides an estimate of the effects of discretization. Log-normal and Gaussian fits to the pressure distributions as a function of position within the porous medium are discussed in relation to the spatial extent of the correlations of the permeability fluctuations.

physics.geo-ph

User's guide to Monte Carlo methods for evaluating path integrals

We give an introduction to the calculation of path integrals on a lattice, with the quantum harmonic oscillator as an example. In addition to providing an explicit computational setup and corresponding pseudocode, we pay particular attention to the existence of autocorrelations and the calculation of reliable errors. The over-relaxation technique is presented as a way to counter strong autocorrelations. The simulation methods can be extended to compute observables for path integrals in other settings.

physics.comp-ph

Evaluation of the path integral for flow through random porous media

We present a path integral formulation of Darcy's equation in one dimension with random permeability described by a correlated multi-variate lognormal distribution. This path integral is evaluated with the Markov chain Monte Carlo method to obtain pressure distributions, which are shown to agree with the solutions of the corresponding stochastic differential equation for Dirichlet and Neumann boundary conditions. The extension of our approach to flow through random media in two and three dimensions is discussed.

physics.comp-ph

Topological Analysis of Foams and tetrahedral structures

In this paper we characterize foams and tetrahedral structures in a unified way, by a simplified representation of both that conserves the system topology. The paper presents a workflow for an automated characterization of the topology of the void space, using a partition of the void space into polyhedral cells connected by windows. This characterization serves as the basic input for the Edwards entropic formalism that deals with the statistical characterization of configurational disorder in granular aggregates and argued to work for foams. The Edwards formalism is introduced and simplified expectation values are calculated.

cond-mat.mtrl-sci

Flow Between Two Sites on a Percolation Cluster

We study the flow of fluid in porous media in dimensions $d=2$ and 3. The medium is modeled by bond percolation on a lattice of $L^d$ sites, while the flow front is modeled by tracer particles driven by a pressure difference between two fixed sites (``wells'') separated by Euclidean distance $r$. We investigate the distribution function of the shortest path connecting the two sites, and propose a scaling {\it Ansatz} that accounts for the dependence of this distribution (i) on the size of the system, $L$, and (ii) on the bond occupancy probability, $p$. We confirm by extensive simulations that the {\it Ansatz} holds for $d=2$ and 3, and calculate the relevant scaling parameters. We also study two dynamical quantities: the minimal traveling time of a tracer particle between the wells and the length of the path corresponding to the minimal traveling time ``fastest path'', which is not identical to the shortest path. A scaling {\it Ansatz} for these dynamical quantities also includes the effect of finite system size $L$ and off-critical bond occupation probability $p$. We find that the scaling form for the distribution functions for these dynamical quantities for $d=2$ and 3 is similar to that for the shortest path but with different critical exponents. The scaling form is represented as the product of a power law and three exponential cutoff functions. We summarize our results in a table which contains estimates for all parameters which characterize the scaling form for the shortest path and the minimal traveling time in 2 and 3 dimensions; these parameters are the fractal dimension, the power law exponent, and the constants and exponents that characterize the exponential cutoff functions.

cond-mat.stat-mech

Dependence of Conductance on Percolation Backbone Mass

On two-dimensional percolation clusters at the percolation threshold, we study $<σ(M_B,r)>$, the average conductance of the backbone, defined by two points separated by Euclidean distance $r$, of mass $M_B$. We find that with increasing $M_B$ and for fixed r, $<σ(M_B,r)>$ asymptotically {\it decreases} to a constant, in contrast with the behavior of homogeneous sytems and non-random fractals (such as the Sierpinski gasket) in which conductance increases with increasing $M_B$. We explain this behavior by studying the distribution of shortest paths between the two points on clusters with a given $M_B$. We also study the dependence of conductance on $M_B$ slightly above the percolation threshold.

cond-mat.stat-mech

Scaling of the distribution of shortest paths in percolation

We present a scaling hypothesis for the distribution function of the shortest paths connecting any two points on a percolating cluster which accounts for {\it (i)} the effect of the finite size of the system, and {\it (ii)} the dependence of this distribution on the site occupancy probability $p$. We test the hypothesis for the case of two-dimensional percolation.

cond-mat.stat-mech

Traveling time and traveling length for flow in porous media

We study traveling time and traveling length for tracer dispersion in porous media. We model porous media by two-dimensional bond percolation, and we model flow by tracer particles driven by a pressure difference between two points separated by Euclidean distance $r$. We find that the minimal traveling time $t_{min}$ scales as $t_{min} \sim r^{1.40}$, which is different from the scaling of the most probable traveling time, ${\tilde t} \sim r^{1.64}$. We also calculate the length of the path corresponding to the minimal traveling time and find $\ell_{min} \sim r^{1.13}$ and that the most probable traveling length scales as ${\tilde \ell} \sim r^{1.21}$. We present the relevant distribution functions and scaling relations.

cond-mat.stat-mech

Spontaneous Stratification in Granular Mixtures

Granular materials size segregate when exposed to external periodic perturbations such as vibrations. Moreover, mixtures of grains of different sizes spontaneously segregate in the absence of external perturbations: when a mixture is simply poured onto a pile, the large grains are more likely to be found near the base, while the small grains are more likely to be near the top. Here, we report a spontaneous phenomenon arising when we pour a mixture between two vertical plates: the mixture spontaneously stratifies into alternating layers of small and large grains whenever the large grains are rougher than the small grains. In contrast, we find only spontaneous segregation when the large grains are more rounded than the small grains. The stratification is related to the occurrence of avalanches; during each avalanche the grains comprising the avalanche spontaneously stratify into a pair of layers through a "kink" mechanism, with the small grains forming a sublayer underneath the layer of large grains.

cond-mat.stat-mech

Quantitative Characterization of Permeability Fluctuations in Sandstone

Sedimentary rocks have complicated permeability fluctuations arising from the geological processes that formed them. These permeability fluctuations significantly affect the flow of fluids through the rocks. We analyze data on two sandstone samples from different geological environments, and find that the permeability fluctuations display long-range power-law correlations characterized by an exponent $H$. For both samples, we find $H \approx 0.82-0.90$.

cond-mat