Length scales and power laws in the two-dimensional forest-fire model

We re-examine a two-dimensional forest-fire model via Monte-Carlo simulations and show the existence of two length scales with different critical exponents associated with clusters and with the usual two-point correlation function of trees. We check resp. improve previously obtained values for other critical exponents and perform a first investigation of the critical behaviour of the slowest relaxational mode. We also investigate the possibility of describing the critical point in terms of a distribution of the global density. We find that some qualitative features such as a temporal oscillation and a power law of the cluster-size distribution can nicely be obtained from such a model that discards the spatial structure.     

Relation between conductivity and fluid flow permeability in porous alumina ceramics

Alumina ceramics with various porosity have been used to test the theoretical predictions of a recent model of Wong et al. relating the electrical conductivity and fluid flow permeability of sedimentary rocks. The exponent of Archie's law and modified Kozeny equation are found to be, respectively, m = 1.91 and m′ = 3.82, in agreement with the theoretical value of m′ = 2m. The values of these exponents correspond to a non uniform pore distribution which is indeed what is observed using electron microscopy.     

A difference-fractal model for the permeability of fibrous porous media

In this Letter, a difference-fractal model for the permeability of viscous flow through fibrous porous media is proposed. Since fractal objects have well-defined geometric properties, and are discrete and discontinuous, we apply the difference approach to developing the fractal model. The model of non-dimensional permeability is expressed as a function of porosity and fractal dimension. To verify the validity of the proposed model, the predicted permeability values are compared with those of experimental measurements. A good agreement between the prediction of the fractal model and the existing experimental data from the literature is found.     

Polarization correlations in two-dimensional random media

Abstract

Second-order polarization correlation functions, both theoretical and experimental, are presented for optical waves propagating through a highly random multiple-scattering two-dimensional (2D) medium. For normal incidence and scattering, a 2D medium is found to be fully described by two material parameters, one of which is complex. Simple formulae are developed for these parameters in terms of the anisotropy of the medium and the scattering mean free path. General theoretical expressions are given for polarized and unpolarized correlation functions and also for the intensity statistics of the scattered light for arbitrary input polarization states. Experimental data are presented for both types of correlation function and for the intensity statistics, and are found to be in reasonably good agreement with the theory.     

A random field model for anomalous diffusion in heterogeneous porous media

Heterogeneity, as it occurs in porous media, is characterized in terms of a scaling exponent, or fractal dimension. A feature of primary interest for two-phase flow is the mixing length. This paper determines the relation between the scaling exponent for the heterogeneity and the scaling exponent which governs the mixing length. The analysis assumes a linear transport equation and uses random fields first in the characterization of the heterogeneity and second in the solution of the flow problem, in order to determine the mixing exponents. The scaling behavior changes from long-length-scale dominated to short-length-scale dominated at a critical value of the scaling exponent of the rock heterogeneity. The long-length-scale-dominated diffusion is anomalous.     

Wave-induced fluid flow in random porous media: attenuation and dispersion of elastic waves

A detailed analysis of the relationship between elastic waves in inhomogeneous, porous media and the effect of wave-induced fluid flow is presented. Based on the results of the poroelastic first-order statistical smoothing approximation applied to Biot's equations of poroelasticity, a model for elastic wave attenuation and dispersion due to wave-induced fluid flow in 3-D randomly inhomogeneous poroelastic media is developed. Attenuation and dispersion depend on linear combinations of the spatial correlations of the fluctuating poroelastic parameters. The observed frequency dependence is typical for a relaxation phenomenon. Further, the analytic properties of attenuation and dispersion are analyzed. It is shown that the low-frequency asymptote of the attenuation coefficient of a plane compressional wave is proportional to the square of frequency. At high frequencies the attenuation coefficient becomes proportional to the square root of frequency. A comparison with the 1-D theory shows that attenuation is of the same order but slightly larger in 3-D random media. Several modeling choices of the approach including the effect of cross correlations between fluid and solid phase properties are demonstrated. The potential application of the results to real porous materials is discussed.     

Comments on the limp frame equivalent fluid model for porous media

In this letter, the low and high frequency limits of the effective density characterizing a limp frame porous medium are investigated. These theoretical limits are compared to the ones found for a classical rigid frame porous medium, and to experimental measurements. While the high frequency asymptotic behaviors of both limp and rigid effective densities are usually only slightly different, their low frequency behaviors are significantly different. Compared to experimental measurements performed on a limp frame fibrous layer, only the limp frame effective density yields good correlations over the whole frequency range.     

Ageing of random porous media following fluid deterministic displacement, freezing, thawing

This paper introduces and investigates a simple model of random porous media degradation via several fluid displacing, freezing, and thawing cycles. The fluid transport is based on the deterministic method. The result shows that the topology and the geometry of porous media have a strong effect on displacement processes. The cluster size of the viscous fingering (VF) pattern in the percolation cluster increases with the increase of iteration parameter n. When iteration parameter , the VF pattern does not change with n. When and , the peak value of the distribution increases as n increases; is the normalized distribution of throat sizes after different displacement-damage but before the freezing. The distribution of throat size N(r) after displacement but before freezing damage, shows that the major change, after successive cycles, happens at r>0.9. The peak value of the distribution reaches a maximum when and r=1, where is the normalized distribution of the size of invaded throats for different iterations. This result is different from invasion percolation. The distribution of velocities normal to the interface of VF in the percolation cluster is also studied. When , the scaling function distribution is very sharp. The sweep efficiency E increases along with the increasing of iteration parameter n and decreases with the network size L. And E has a minimum as L increases to the maximum size of the lattice. The VF pattern in the percolation cluster has one frozen zone and one active zone. Received 30 March 1999 and Received in final form 8 August 1999     

Scaling of the electrical conductivity of granular media

We derive the scaling properties of the dependence of the macroscopic electrical conductivity of granular media (e.g., sands) with a surface mechanism of electrical conduction on the grain size, when the medium is subjected to a given mechanical stress. In order to eliminate the influence of the inter-grain junction capacity, the direct electrical current is considered. We determine the applicability restrictions on the theory which disregards the ultimate crushing compression strength, adhesion, and the effect of charge carrier tunneling at grain junctions beyond the contact surface area. Solutions for several regular packings of grains are obtained as well.     

Two—phase flow in correlated pore—throat random porous media

We have constructed a porous media model in which there are percolation clusters with varying percolation probability P and correlated site-bonds. Taking into account both the pore and the throat geometry, the viscous fingering (VF) in porous media has been investigated by using the standard over-relaxed Gauss-Seidel scheme. The simulation results show that the VF structure varies with the correlation parameter ε, the viscosity ratio M and the percolation probability P. The smaller the correlation parameter ε, the greater the deviation of the normalized size distribution of the invaded throat N_inv(r) from the truncated Rayleigh distribution. For a larger viscosity ratio M, the VF pattern looks like a diffusion-limited-aggregation structure in percolation clusters. The fractal dimension D increases with the increase of the percolation probability P and the correlation parameter ε. The velocity distribution f(α) of VF in percolation clusters is of a parabola-like curve. The tail of the distribution (large α) is longer for a larger correlation parameter ε. For a smaller ε, the distribution is very sharp. The sweep efficiency E decreases along with the decrease of the correlation parameter ε and the increase of the network size L_nz. E has a minimum as L_nz increases up to the maximum no matter what the values of P, M and ε. The E～ L_nz curve has a frozen zone and an active zone. The geometry and the topology of the porous media have strong effects on the displacement processes and the structure of VF.