Chord-distribution functions of three-dimensional random media: approximate first-passage times of Gaussian processes

A review on reactive transport model and porosity evolution in the porous media

This work comprehensively reviews the equations governing multicomponent flow and reactive transport in porous media on the pore-scale, mesoscale and continuum scale. For each of these approaches, the different numerical schemes for solving the coupled advection-diffusion-reactions equations are presented. The parameters influenced by coupled biological and chemical reactions in evolving porous media are emphasised and defined from a pore-scale perspective. Recent pore-scale studies, which have enhanced the basic understanding of processes that affect and control porous media parameters, are discussed. Subsequently, a summary of the common methods used to describe the transport process, fluid flow, reactive surface area and reaction parameters such as porosity, permeability and tortuosity are reviewed.

On the excursion area of perturbed Gaussian fields

We investigate Lipschitz-Killing curvatures for excursion sets of random fields on ℝ2under a very specific perturbation, namely a small spatial-invariant random perturbation with zero mean. An expansion formula for mean curvatures is derived when the magnitude of the perturbation vanishes, which recovers the Gaussian Kinematic Formula at the limit by contiguity of the model. We develop an asymptotic study of the perturbed excursion area behaviour that leads to a quantitative non-Gaussian limit theorem, in Wasserstein distance, for fixed small perturbations and growing domain. When letting both the perturbation vanish and the domain grow, a standard Central Limit Theorem follows. Taking advantage of these results, we propose an estimator for the perturbation variance which turns out to be asymptotically normal and unbiased, allowing to make inference through sparse information on the field.

Review of pore network modelling of porous media: Experimental characterisations, network constructions and applications to reactive transport

Pore network models have been applied widely for simulating a variety of different physical and chemical processes, including phase exchange, non-Newtonian displacement, non-Darcy flow, reactive transport and thermodynamically consistent oil layers. The realism of such modelling, i.e. the credibility of their predictions, depends to a large extent on the quality of the correspondence between the pore space of a given medium and the pore network constructed as its representation. The main experimental techniques for pore space characterisation, including direct imaging, mercury intrusion porosimetry and gas adsorption, are firstly summarised. A review of the main pore network construction techniques is then presented. Particular focus is given on how such constructions are adapted to the data from experimentally characterised pore systems. Current applications of pore network models are considered, with special emphasis on the effects of adsorption, dissolution and precipitation, as well as biomass growth, on transport coefficients. Pore network models are found to be a valuable tool for understanding and predicting meso-scale phenomena, linking single pore processes, where other techniques are more accurate, and the homogenised continuum porous media, used by engineering community.

How fast can the chord length distribution decay?

The modeling of random bi-phasic, or porous, media has been, and still is, under active investigation by mathematicians, physicists, and physicians. In this paper we consider a thresholded random process X as a source of the two phases. The intervals when X is in a given phase, named chords, are the subject of interest. We focus on the study of the tails of the chord length distribution functions. In the literature concerned with real data, different types of tail behavior have been reported, among them exponential-like or power-like decay. We look for the link between the dependence structure of the underlying thresholded process X and the rate of decay of the chord length distribution. When the process X is a stationary Gaussian process, we relate the latter to the rate at which the covariance function of X decays at large lags. We show that exponential, or nearly exponential, decay of the tail of the distribution of the chord lengths is very common, perhaps surprisingly so.

How fast can the chord length distribution decay?

The modeling of random bi-phasic, or porous, media has been, and still is, under active investigation by mathematicians, physicists, and physicians. In this paper we consider a thresholded random process X as a source of the two phases. The intervals when X is in a given phase, named chords, are the subject of interest. We focus on the study of the tails of the chord length distribution functions. In the literature concerned with real data, different types of tail behavior have been reported, among them exponential-like or power-like decay. We look for the link between the dependence structure of the underlying thresholded process X and the rate of decay of the chord length distribution. When the process X is a stationary Gaussian process, we relate the latter to the rate at which the covariance function of X decays at large lags. We show that exponential, or nearly exponential, decay of the tail of the distribution of the chord lengths is very common, perhaps surprisingly so.

A simple model for the pore size distribution in random fibre networks

A heuristic approach based upon excluded volume arguments is developed for modelling the distribution of pore sizes in isotropic networks of randomly distributed cylindrical fibres. Our formalism accounts for the finite hard core diameters of the fibres, and leads to compact, analytically tractable expressions that span the complete range of volume fractions. Results are presented for the mean and mean-squared pore radii as functions of the fibre volume fraction, and for the partition coefficient of a spherical tracer particle into such a network under conditions such that steric effects are dominant.

Reconstruction of three-dimensional porous media using a single thin section

The purpose of any reconstruction method is to generate realizations of two- or multiphase disordered media that honor limited data for them, with the hope that the realizations provide accurate predictions for those properties of the media for which there are no data available, or their measurement is difficult. An important example of such stochastic systems is porous media for which the reconstruction technique must accurately represent their morphology--the connectivity and geometry--as well as their flow and transport properties. Many of the current reconstruction methods are based on low-order statistical descriptors that fail to provide accurate information on the properties of heterogeneous porous media. On the other hand, due to the availability of high resolution two-dimensional (2D) images of thin sections of a porous medium, and at the same time, the high cost, computational difficulties, and even unavailability of complete 3D images, the problem of reconstructing porous media from 2D thin sections remains an outstanding unsolved problem. We present a method based on multiple-point statistics in which a single 2D thin section of a porous medium, represented by a digitized image, is used to reconstruct the 3D porous medium to which the thin section belongs. The method utilizes a 1D raster path for inspecting the digitized image, and combines it with a cross-correlation function, a grid splitting technique for deciding the resolution of the computational grid used in the reconstruction, and the Shannon entropy as a measure of the heterogeneity of the porous sample, in order to reconstruct the 3D medium. It also utilizes an adaptive technique for identifying the locations and optimal number of hard (quantitative) data points that one can use in the reconstruction process. The method is tested on high resolution images for Berea sandstone and a carbonate rock sample, and the results are compared with the data. To make the comparison quantitative, two sets of statistical tests consisting of the autocorrelation function, histogram matching of the local coordination numbers, the pore and throat size distributions, multiple-points connectivity, and single- and two-phase flow permeabilities are used. The comparison indicates that the proposed method reproduces the long-range connectivity of the porous media, with the computed properties being in good agreement with the data for both porous samples. The computational efficiency of the method is also demonstrated.

Boolean reconstructions of complex materials: Integral geometric approach

We show that for the Boolean model of random composite media one can, from a single image of a system at any particle fraction, define a set of parameters which allows one to accurately reconstruct the medium for all other phase fractions. The morphological characterization is based on a family of measures known in integral geometry which provides powerful formulas for the Boolean model. The percolation thresholds of either phase are obtained with good accuracy. From the reconstructions one can subsequently predict property curves for the material across all phase fractions from the single three-dimensional image. We illustrate this for transport and mechanical properties of complex Boolean systems and for experimental sandstone samples.

Pore-network extraction from micro-computerized-tomography images

Network models that represent the void space of a rock by a lattice of pores connected by throats can predict relative permeability once the pore geometry and wettability are known. Micro-computerized-tomography scanning provides a three-dimensional image of the pore space. However, these images cannot be directly input into network models. In this paper a modified maximal ball algorithm, extending the work of Silin and Patzek [D. Silin and T. Patzek, Physica A 371, 336 (2006)], is developed to extract simplified networks of pores and throats with parametrized geometry and interconnectivity from images of the pore space. The parameters of the pore networks, such as coordination number, and pore and throat size distributions are computed and compared to benchmark data from networks extracted by other methods, experimental data, and direct computation of permeability and formation factor on the underlying images. Good agreement is reached in most cases allowing networks derived from a wide variety of rock types to be used for predictive modeling.

Using convex quadratic programming to model random media with Gaussian random fields

Excursion sets of Gaussian random fields (GRFs) have been frequently used in the literature to model two-phase random media with measurable phase autocorrelation functions. The goal of successful modeling is finding the optimal field autocorrelation function that best approximates the prescribed phase autocorrelation function. In this paper, we present a technique which uses convex quadratic programming to find the best admissible field autocorrelation function under a prescribed discretization. Unlike previous methods, this technique efficiently optimizes over all admissible field autocorrelation functions, instead of optimizing only over a predetermined parametrized family. The results from using this technique indicate that the GRF model is significantly more versatile than observed in previous studies. An application to modeling a base-catalyzed tetraethoxysilane aerogel system given small-angle neutron scattering data is also presented.

Uniqueness of reconstruction of multiphase morphologies from two-point correlation functions

The restoration of the spatial structure of heterogeneous media, such as composites, porous materials, microemulsions, ceramics, or polymer blends from two-point correlation functions, is a problem of relevance to several areas of science. In this contribution we revisit the question of the uniqueness of the restoration problem. We present numerical evidence that periodic, piecewise uniform structures with smooth boundaries are completely specified by their two-point correlation functions, up to a translation and, in some cases, inversion. We discuss the physical relevance of the results.

Stochastic Reconstruction of Particulate Media from Two-Dimensional Images

In this contribution we address the problem of reconstructing particulate media from limited morphological information that may be readily extracted from 2D images of their microstructure. Sixty-five backscatter SEM images of the microstructure of a lightly consolidated pack of glass spheres are analyzed to determine morphological descriptors, such as the pore-pore autocorrelation function and pore and solid phase chord distributions. This information is then used to constrain the stochastic reconstruction of the glass sphere packing in two dimensions using a simulated annealing method. The results obtained demonstrate that the solid-phase chord distribution contains additional information that is critical for the reconstruction of the morphology of particulate media exhibiting short-range order. We further confirm this finding by successfully reconstructing the microstructure of a pack of irregular silica particles.

Efficient reconstruction of multiphase morphologies from correlation functions

A highly efficient algorithm for the reconstruction of microstructures of heterogeneous media from spatial correlation functions is presented. Since many experimental techniques yield two-point correlation functions, the restoration of heterogeneous structures, such as composites, porous materials, microemulsions, ceramics, or polymer blends, is an inverse problem of fundamental importance. Similar to previously proposed algorithms, the new method relies on Monte Carlo optimization, representing the microstructure on a discrete grid. An efficient way to update the correlation functions after local changes to the structure is introduced. In addition, the rate of convergence is substantially enhanced by selective Monte Carlo moves at interfaces. Speedups over prior methods of more than two orders of magnitude are thus achieved. Moreover, an improved minimization protocol leads to additional gains. The algorithm is ideally suited for implementation on parallel computers. The increase in efficiency brings new classes of problems within the realm of the tractable, notably those involving several different structural length scales and/or components.