Percolation Thresholds in 2-Dimensional Prefractal Models of Porous Media*

Considerable effort has been directed towards the application of percolation theory and fractal modeling to porous media. We combine these areas of research to investigate percolation in prefractal porous media. We estimated percolation thresholds in the pore space of homogeneous random 2-dimensional prefractals as a function of the fractal scale invariance ratio b and iteration level i. The percolation thresholds for these simulations were found to increase beyond the 0.5927l... porosity expected in Bernoulli (uncorrelated) percolation networks. Percolation in prefractals occurs through large pores connected by small pores. The thresholds increase with both b (a finite size effect) and i. The results allow the prediction of the onset of percolation in models of prefractal porous media and can be used to bound modeling efforts. More fundamental applications are also possible. Only a limited range of parameters has been explored empirically but extrapolations allow the critical fractal dimension to be estimated for a large combination of b and i values. Extrapolation to infinite iterations suggests that there may be a critical fractal dimension of the solid at which the pore space percolates. The extrapolated value is close to 1.89 – the well-known fractal dimension of percolation clusters in 2-dimensional Bernoulli networks.     

Pore Network Analysis of Resistivity Index for Water-Wet Porous Media

Pore network analysis is used to investigate the effects of microscopic parameters of the pore structure such as pore geometry, pore-size distribution, pore space topology and fractal roughness porosity on resistivity index curves of strongly water-wet porous media. The pore structure is represented by a three-dimensional network of lamellar capillary tubes with fractal roughness features along their pore-walls. Oil-water drainage (conventional porous plate method) is simulated with a bond percolation-and-fractal roughness model without trapping of wetting fluid. The resistivity index, saturation exponent and capillary pressure are expressed as approximate functions of the pore network parameters by adopting some simplifying assumptions and using effective medium approximation, universal scaling laws of percolation theory and fractal geometry. Some new phenomenological models of resistivity index curves of porous media are derived. Finally, the eventual changes of resistivity index caused by the permanent entrapment of wetting fluid in the pore network are also studied.Resistivity index and saturation exponent are decreasing functions of the degree of correlation between pore volume and pore size as well as the width of the pore size distribution, whereas they are independent on the mean pore size. At low water saturations, the saturation exponent decreases or increases for pore systems of low or high fractal roughness porosity respectively, and obtains finite values only when the wetting fluid is not trapped in the pore network. The dependence of saturation exponent on water saturation weakens for strong correlation between pore volume and pore size, high network connectivity, medium pore-wall roughness porosity and medium width of the pore size distribution. The resistivity index can be described succesfully by generalized 3-parameter power functions of water saturation where the parameter values are related closely with the geometrical, topological and fractal properties of the pore structure.     

Comment on ‘fractal and superdiffusive transport and hydrodynamic dispersion in heterogeneous porous media’, by Muhammad Sahimi

Two fundamental questions regarding the application of percolation theory to transport in porous media are addressed. First, when critical path arguments (based on a sufficiently wide spread of microscopic transition rates) are invoked (in analogy to the case of transport in disordered semiconductors) to justify the application of percolation theory to the determination of relevant transport properties, then for long time scales (compared to the inverse of the critical percolation rate), the fractal structure of the critical path is relevant to transport, but not at short time scales. These results have been demonstrated concretely in the case of disordered semiconductors, and are in direct contradiction to the claims of the review. Second, the relevance of deterministic or stochastic methods to transport has been treated heretofore by most authors as a question of practicality. But, at least under some conditions, concrete criteria distinguish between the two types of transport. Percolative (deterministic) transport is temporally reproducible and spatially inhomogeneous while diffusive (stochastic) transport is temporally irreproducible, but homogeneous, and a cross-over from stochastic to percolative transport occurs when the spread of microscopic transition rates exceeds 4–5 orders of magnitude. It is likely that such conditions are frequently encountered in soil transport. Moreover, clear evidence for deterministic transport (although not necessarily percolative) exists in such phenomena as preferential flow. On the other hand, the physical limitation of transport to (fractally connected) pore spaces within soils (analogously to transport in metal-insulator composites) can make transport diffusive on a fractal structure, rather than percolative.Transport in Porous Media13 (1993), 3–40.     

Jots - a mathematical model of microscopic fluid flow in porous media

The model described in this paper is an approach to simulating flow through porous media on a microscopic scale. It is based on a variation of diffusion limited aggregation. The model is shown to match coreflood average saturation profiles and production histories as predicted by Darcy's equations while generating saturation distributions resembling viscous fingering. The model also is shown to simulate the limiting cases of infinite mobility ratio and zero flow rates as previously modeled by diffusion limited aggregation and percolation theory. With some simplifying assumptions, differential equations very similar to Darcy's equations are derived from the microscopic interpretation of fluid behavior in porous media used in this model.     

Effect of inhomogeneities in an explosive on the critical detonation diameter

The effect of inhomogeneities in an explosive on the critical detonation diameter is theoretically analyzed. On the assumption that the inhomogeneities are centers of the reaction behind the detonation front, a formula is obtained for determining the critical diameter as a function of the concentration and size of the inhomogeneities. It is shown that when the inhomogeneities are particles of an inert substance added to the explosive, the dependence of the critical diameter on the mass fraction and size of the inert particles may have a minimum. The results of the theoretical analysis are in qualitative agreement with the experimental data.In conclusion the author thanks R. Kh. Kurbangalina for formulating and discussing'the problem and L. N. Stesik for his useful remarks.     

Electrical Conductivity and Percolation Aspects of Statistically Homogeneous Porous Media

A method of 3-D stochastic reconstruction of porous media based on statistical information extracted from 2-D sections is evaluated with reference to the steady transport of electric current. Model microstructures conforming to measured and simulated pore space autocorrelation functions are generated and the formation factor is systematically determined by random walk simulation as a function of porosity and correlation length. Computed formation factors are found to depend on correlation length only for small values of this parameter. This finding is explained by considering the general percolation behavior of a statistically homogeneous system. For porosities lower than about 0.2, the dependence of formation factor on porosity shows marked deviations from Archie's law. This behavior results from the relatively high pore space percolation threshold (0.09) of the simulated media and suggests a limitation to the applicability of the method to low porosity media. It is additionally demonstrated that the distribution of secondary porosity at a larger scale can be simulated using stochastic methods. Computations of the formation factor are performed for model media with a matrix-vuggy structure as a function of the amount and spatial distribution of vuggy porosity and matrix conductivity. These results are shown to be consistent with limited available experimental data for carbonate rocks.     

DEM simulation of particle percolation in a packed bed

The phenomenon of spontaneous particle percolation under gravity is investigated by means of the discrete element method. Percolation behaviors such as percolation velocity, residence time distribution and radial dispersion are examined under various conditions. It is shown that the vertical velocity of a percolating particle moving down through a packing of larger particles decreases with increasing the restitution coefficient between particles and diameter ratio of the percolating to packing particles. With the increase of the restitution coefficient, the residence time and radial dispersion of the percolating particles increase. The packing height affects the residence time and radial dispersion. But, the effect can be eliminated in the analysis of the residence time and radial dispersion when they are normalized by the average residence time and the product of the packing height and packing particle diameter, respectively. In addition, the percolation velocity is shown to be related to the vertical acceleration of the percolating particle when an extra constant vertical force is applied. Increasing the feeding rate of percolating particles decreases the dispersion coefficient.     

DEM simulation of particle percolation in a packed bed

The phenomenon of spontaneous particle percolation under gravity is investigated by means of the discrete element method. Percolation behaviors such as percolation velocity,residence time distribution and radial dispersion are examined under various conditions. It is shown that the vertical velocity of a percolating particle moving down through a packing of larger particles decreases with increasing the restitution coefficient between particles and diameter ratio of the percolating to packing particles. With the increase of the restitution coefficient,the residence time and radial dispersion of the percolating particles increase. The packing height affects the residence time and radial dispersion. But,the effect can be eliminated in the analysis of the residence time and radial dispersion when they are normalized by the average residence time and the product of the packing height and packing particle diameter,respectively.In addition,the percolation velocity is shown to be related to the vertical acceleration of the percolating particle when an extra constant vertical force is applied. Increasing the feeding rate of percolating particles decreases the dispersion coefficient.     

Explanation of scaling phenomenon based on fractal fragmentation

A significant ‘size effect' is observed in tensile strength of solid particles, such as ice, rock, ceramics and concrete: the tensile strength is not independent of the fragment size, but decreases with increasing size. The Weibull statistical theory was universally used to calculate the size effect observed in solid particles. Recent developments in fractal theory suggest that fractals may provide a more realistic representation of solid particles. In this paper, the scaling phenomenon of ice mechanics is explained using the fractal model for ice particle fragmentation. The Weibull statistics is modified using the fractal crushing of ice, and is compared with the conventional one. Goodness-of-fit statistics show that the modified Weibull statistics fits the experimental data of ice much better than the conventional one. The modified Weibull statistics has only one parameter, the fractal dimension of the fragment size distribution, which has a general value of 2.50 for the ice fragmentations.     

Mean field calculation of effective permeability based on fractal pore space

Effective permeability for porous rocks is calculated using mean field theory. We make two simplifying assumptions about the internal conductances in a network representation of the porous rock: (i) Pore space is characterized by a uniform fractal scaling; (ii) the internal conductances depend only on the characteristic pore sizes. Within these approximations, it is possible to derive a simple probability density for the internal conductances which is used for calculating effective permeability. Good agreement between calculations and experimental data of permeability vs. porosity is achieved.     

Percolation with a limiting gradient in a medium with random inhomogeneities

The coverage of a medium by percolation and the effective permeability of a medium with stagnant zones are determined. It is shown that effective permeability is a function of external conditions, particularly the average pressure gradient. Three-, two-, and one-dimensional flows are discussed. The theory of overshoots of random functions and fields beyond a prescribed level [1, 2] is used for the investigation. Overshoots of elements of the percolation field in media with random inhomogeneities are studied. Overshoots of energy being dissipated in a volume are discussed in particular; this permits an approximate determination of the coverage of an inhomogeneous porous medium by migration during percolation with a limiting gradient, i.e., in the case of formation of stagnant zones chaotically disseminated in the flow region.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 159–165, September–October, 1970.The authors thank V. M. Entov for discussing the article and useful comments.     

岩土介质的分形孔隙和分形粒子

岩土介质是晶粒状材料,存在大量的孔隙。这些孔隙的存在严重地影响岩土介质的力学、物理和化学性能。大量的研究表明,岩土介质的孔隙几何从原子尺度到晶粒尺寸范围内均表现出分形特征。目前分形几何已被广泛应用来研究岩土的孔隙率,输运特性和渗透性等等。本文从5个方面来讨论岩土介质的分形孔隙和分形粒子:①分形几何简介;②孔隙介质的分形模型;③岩土介质的分形孔隙特性和它们的分形量测方法;④岩土的分形粒子;⑤水土保持估计中的分形毛细管模型。      

电流变液体黏性变化趋势的试验研究

以微观试验和流变性能试验为手段，分别研究零电场下和在电场作用下的电流变液体黏性变化规律．研究结果表明：零电场下电流变液体的黏性与Krieger-Dougherty公式具有很好的拟合效果，其中逾渗临界值强依赖于悬浮液体中固体颗粒的性质并随工作温度变化．在电场作用下，电流变悬浮液体的黏度随剪切速率的变化规律分为3个阶段：即呈线性的启动段、非线性的幂定律模型流动段和宾汉模型流动段．研究结果为电流变效应工程应用提供依据．     

Effects of Bubbles on the Hydraulic Conductivity of Porous Materials – Theoretical Results

In a porous material, both the pressure drop across a bubble and its speed are nonlinear functions of the fluid velocity. Nonlinear dynamics of bubbles in turn affect the macroscopic hydraulic conductivity, and thus the fluid velocity. We treat a porous medium as a network of tubes and combine critical path analysis with pore-scale results to predict the effects of bubble dynamics on the macroscopic hydraulic conductivity and bubble density. Critical path analysis uses percolation theory to find the dominant (approximately) one-dimensional flow paths. We find that in steady state, along percolating pathways, bubble density decreases with increasing fluid velocity, and bubble density is thus smallest in the smallest (critical) tubes. We find that the hydraulic conductivity increases monotonically with increasing capillary number up to Ca 10^–2, but may decrease for larger capillary numbers due to the relative decrease of bubble density in the critical pores. We also identify processes that can provide a positive feedback between bubble density and fluid flow along the critical paths. The feedback amplifies statistical fluctuations in the density of bubbles, producing fluctuations in the hydraulic conductivity.     

Statistical Synthesis of Imaging and Porosimetry Data for the Characterization of Microstructure and Transport Properties of Sandstones

The microstructure of a suite of sandstone samples is quantitatively analyzed using a method which combines information from thin section micrographs of the pore space with mercury injection porosimetry in a statistical framework. This method enables the determination of a continuous distribution of pore sizes ranging from few nanometre to several hundred micrometre. The data obtained unify fractal and Euclidean aspects of the void space geometry, yield estimates of the pore-to-throat aspect ratio and challenge the ability of commonly used network models to describe fluid percolation in multiscale porous media. Application of critical path analysis to the prediction of flow permeability and electrical conductivity of sandstone core samples using the new information produces results comparable to those obtained by the classical approach—a fact attributed to the presence of macroscopic heterogeneity at the scale of several millimetres.     

Effective thermal conductivity of partially saturated porous rocks

The present work is concerned with the determination of the effective thermal conductivity of porous rocks or rock-like composites composed by multiple solid constituents, in partially saturated conditions. Based on microstructure observations, a two-step homogenization scheme is developed: the first step for the solid constituents only, and the second step for the (already homogenized) solid matrix and pores. Several homogenization schemes (dilute, Mori–Tanaka, the effective field method and Ponte Castañeda–Willis technique) are presented and compared in this context. Such methods are allowing: (i) to incorporate in the modellization the physical parameters (mineralogy, morphology) influencing the effective properties of the considered material, and the saturation degree of the porous phase; (ii) to account for interaction effects between matrix and inhomogeneities; (iii) to consider different spatial distributions of inclusions (spherical, ellipsoïdal). An orientation distribution function (ODF) permits simultaneously to incorporate in the modelling the transverse isotropy of pore systems. Appearing as homogeneous at the macroscopic scale, it is showed that the effective conductivity depends on the physical properties of all subsidiary phases (microscopic inhomogeneities). By considering the solution of a single ellipsoïdal inhomogeneity in the homogenization problem it is possible to observe the significant influence of the geometry, shape and spatial distribution of inhomogeneities on the effective thermal conductivity and its dependence with the saturation degree of liquid phase. The predictive capacities of the two-step homogenization method are evaluated by comparison with experimental results obtained for an argillite.     

A micromechanical model for effective conductivity in granular electrode structures

Optimization of composition and microstructure is important to enhance performance of solid oxide fuel cells （SOFC） and lithium-ion batteries （LIB）. For this, the porous electrode structures of both SOFC and LIB are modeled as a binary mixture of electronic and ionic conducting particles to estimate effective transport properties. Particle packings of 10000 spherical, binary sized and randomly positioned particles are created numerically and densified considering the different manufacturing processes in SOFC and LIB： the sintering of SOFC electrodes is approximated geometrically, whereas the calendering process and volume change due to intercalation in LIB are modeled physically by a discrete el- ement approach. A combination of a tracking algorithm and a resistor network approach is developed to predict the con- nectivity and effective conductivity for the various densified structures. For SOFC, a systematic study of the influence of morphology on connectivity and conductivity is performed on a large number of assemblies with different compositions and particle size ratios between 1 and 10. In comparison to percolation theory, an enlarged percolation area is found, es- pecially for large size ratios. It is shown that in contrast to former studies the percolation threshold correlates to varying coordination numbers. The effective conductivity shows not only an increase with volume fraction as expected but also with size ratio. For LIB, a general increase of conductivity during the intercalation process was observed in correlation with increasing contact forces. The positive influence of cal- endering on the percolation threshold and the effective conductivity of carbon black is shown. The anisotropy caused by the calendering process does not influence the carbon black phase.     

Modelling the Aggregation and Break-up of Fractal Aggregates in a Shear Flow

A population balance model has been proposed to describe simultaneous coagulation and fragmentation of fractal aggregates during shear-flocculation induced by means of a Couette-flow system. Given enough time, a floc-size distribution achieves the steady state, which reflects the balance between coagulation and fragmentation forces. Experimental results obtained recently show that higher shear rate values shift the steady state to smaller aggregate sizes. Also, for a fixed value of the shear rate, when the volume fraction of the particles increases, the steady state size increases if the flow is laminar, and decreases if the flow is turbulent. In order to model the fragmentation of the aggregates, a power dependence between the breakage rate coefficient and the shear rate has been proposed. The model is adjusted by using two different parameters: the effective probability of success for collisions _eff, and the break up coefficient for shear fragmentation, B. The dependence of these two parameters on the shear rate and the volume fraction of the particles is discussed.     

Prediction of Porosity and Permeability of Caved Zone in Longwall Gobs

The porosity and permeability of the caved zone (gob) in a longwall operation impact many ventilation and methane control related issues, such as air leakage into the gob, the onset of spontaneous combustion, methane and air flow patterns in the gob, and the interaction of gob gas ventholes with the mining environment. Despite its importance, the gob is typically inaccessible for performing direct measurements of porosity and permeability. Thus, there has always been debate on the likely values of porosity and permeability of the caved zone and how these values can be predicted. This study demonstrates a predictive approach that combines fractal scaling in porous medium with principles of fluid flow. The approach allows the calculation of porosity and permeability from the size distribution of broken rock material in the gob, which can be determined from image analyzes of gob material using the theories on a completely fragmented porous medium. The virtual fragmented fractal porous medium so generated is exposed to various uniaxial stresses to simulate gob compaction and porosity and permeability changes during this process. The results suggest that the gob porosity and permeability values can be predicted by this approach and the presented models are capable to produce values close to values documented by other researchers.