A Statistical Correlation Between Permeability,Porosity, Tortuosity and Conductance

A statistical evaluation of 13,000 numerical simulations of random porous structures is used to establish a correlation between permeability, porosity, tortuosity and conductance. The random structures are generated with variable porosities, and parameters such as the permeability and the tortuosity are determined directly from the structures. It is shown that the prevalent definition of tortuosity, as the ratio of length of the real flow path to the projected path in the overall flow direction, does not correlate with permeability in the general case. Also, the correlation between the conductance of the medium, as an indicator of the accessible cross section of a flow path and permeability is no more reliable than the permeability–porosity correlation. However, if the definition of tortuosity is corrected using the cross-sectional variations, the resulting parameter (i.e., the minimum-corrected tortuosity) has a reliable correlation with permeability and can be used to estimate permeability with an acceptable error for most of the simulations of the random porous structures. The feasibility of extending the conclusions from 2-dimensional to 3-dimensional configurations and the numerical percolation thresholds for random structures are also discussed.     

Pore-Scale Simulation of Shear Thinning Fluid Flow Using Lattice Boltzmann Method

The present work attempts to identify the roles of flow and geometric variables on the scaling factor which is a necessary parameter for modeling the apparent viscosity of non-Newtonian fluid in porous media. While idealizing the porous media microstructure as arrays of circular and square cylinders, the present study uses multi-relaxation time lattice Boltzmann method to conduct pore-scale simulation of shear thinning non-Newtonian fluid flow. Variation in the size and inclusion ratio of the solid cylinders generates wide range of porous media with varying porosity and permeability. The present study also used stochastic reconstruction technique to generate realistic, random porous microstructures. For each case, pore-scale fluid flow simulation enables the calculation of equivalent viscosity based on the computed shear rate within the pores. It is observed that the scaling factor has strong dependence on porosity, permeability, tortuosity and the percolation threshold, while approaching the maximum value at the percolation threshold porosity. The present investigation quantifies and proposes meaningful correlations between the scaling factor and the macroscopic properties of the porous media.     

Tortuosity based on Anisotropic Diffusion Process in Structured Plate-like Obstacles by Monte Carlo Simulation

The diffusion of fluids in porous media, composed of regularly aligned plate-like obstacles, was studied by Monte Carlo simulation. The diffusion coefficients and all diagonal components of the diffusion tensor were estimated for these media. The calculated tortuosities were modeled as a function of porosity by using the Koponen’s equation related to percolation threshold. These results indicated that a media with a homogeneous porosity has a heterogeneous tortuosity, is affected by the alignments of the plate-like obstacles. Furthermore, the calculation results were compared with the experimental results for fixed perpendicular plates of Comiti and Renaud as a function of porosity. The results for tortuosity compared well for porosity larger than 0.86.     

Formation factor and tortuosity of homogeneous porous media

In this paper, volume averaging in porous media is applied to the microscopic electric charge conservation equation (differential form of Ohm's law) and an expression is derived for the formation factor of a homogeneous porous medium saturated with an electrically conductive fluid. This expression consists of two terms; the first term involves the integral of the current density over the fluid volume and the second term involves the integral of the electric potential over the solid-fluid interface. The physical meaning of the two terms is discussed with the help of three idealized porous media. The results for these media indicate a definite relation between the second term and tortuosity. These results also demonstrate the simplistic nature of the classical definition of the tortuosity as a ratio of geometric lengths. An exact relation between the formation factor and tortuosity is presented. It is shown that the assumed equivalence of the electrical and hydraulic tortuosities is not valid. The general application of the expression for the formation factor is discussed briefly.     

A constitutive approach to 3-d nonlinear fluid flow through finite deformable porous continua

The present paper proposes a thermodynamically consistent Forchheimer-type filter law for application in macroscopic porous media theories. The constitutive flow equation is thereby capable of describing the essential nonlinearities during 3-d fluid percolation through deformable porous solids. In particular, tortuosity effects, anisotropic properties, and the indispensable influence of finite distortions of the interconnected pore space are accounted for. However, the common shape of a Darcy-type relation is retained by assigning all nonlinearities to a general permeability tensor. Finally, to show the validity and applicability of the proposed formulation, the filter law is correlated with the data of permeability experiments on a high-porosity polyurethane foam and is used in a 3-d finite element analysis to simulate the pneumatic damping properties of the material.     

Dependency of Tortuosity and Permeability of Porous Media on Directional Distribution of Pore Voids

Single-phase fluid flow in porous media is usually direction dependent owing to the tortuosity associated with the internal structures of materials that exhibit inherent anisotropy. This article presents an approach to determine the tortuosity and permeability of porous materials using a structural measure quantifying the anisotropic distribution of pore voids. The approach uses a volume averaging method through which the macroscopic tortuosity tensor is related to both the average porosity and the directional distribution of pore spaces. The permeability tensor is derived from the macroscopic momentum balance equation of fluid in a porous medium and expressed as a function of the tortuosity tensor and the internal structure of the material. The analytical results generally agree with experimental data in the literature.     

Predicting Tortuosity for Airflow Through Porous Beds Consisting of Randomly Packed Spherical Particles

This article presents a numerical method for determining tortuosity in porous beds consisting of randomly packed spherical particles. The calculation of tortuosity is carried out in two steps. In the first step, the spacial arrangement of particles in the porous bed is determined by using the discrete element method (DEM). Specifically, a commercially available discrete element package (PFC^3D ) was used to simulate the spacial structure of the porous bed. In the second step, a numerical algorithm was developed to construct the microscopic (pore scale) flow paths within the simulated spacial structure of the porous bed to calculate the lowest geometric tortuosity (LGT), which was defined as the ratio of the shortest flow path to the total bed depth. The numerical algorithm treats a porous bed as a series of four-particle tetrahedron units. When air enters a tetrahedron unit through one face (the base triangle), it is assumed to leave from another face triangle whose centroid is the highest of the four face triangles associated with the tetrahedron, and this face triangle will then be used as the base triangle for the next tetrahedron. This process is repeated to establish a series of tetrahedrons from the bottom to the top surface of the porous bed. The shortest flow path is then constructed geometrically by connecting the centroids of base triangles of consecutive tetrahedrons. The tortuosity values calculated by the proposed numerical method compared favourably with the values obtained from a CT image published in the literature for a bed of grain (peas). The proposed model predicted a tortuosity of 1.15, while the tortuosity estimated from the CT image was 1.14.     

Review on Scale Dependent Characterization of the Microstructure of Porous Media

The paper discusses local porosity theory and its relation with other geometric characterization methods for porous media such as correlation functions and contact distributions. Special emphasis is placed on the charcterization of geometric observables through Hadwigers theorem in stochastic geometry. The four basic Minkowski functionals are introduced into local porosity theory, and for the first time a relationship is established between the Euler characteristic and the local percolation probabilities. Local porosity distributions and local percolation probabilities provide a scale dependent characterization of the microstructure of porous media that can be used in an effective medium approach to predict transport.     

幂律流体在多孔介质中径向渗流的分形模型

基于分形理论及毛细管模型,本文研究了非牛顿幂律流体在各向同性多孔介质中径向流动问题,推导了幂律流体径向有效渗透率的分形解析表达式．研究结果表明,幂律流体径向有效无量纲渗透率模型和Chang and Yortsos’s模型吻合很好;同时还得出幂律流体径向有效渗透率随孔隙率、幂指数的增加而增加,随迂曲度分形维数的增加而减少．     

Effective thermal conductivity of wire-woven bulk Kagome sandwich panels

Thermal transport in a highly porous metallic wire-woven bulk Kagome (WBK) is numerically and analytically modeled. Based on topology similarity and upon introducing an elongation parameter in thermal tortuosity, an idealized Kagome with non-twisted struts is employed. Special focus is placed upon quantifying the effect of topological anisotropy of WBK upon its effective conductivity. It is demonstrated that the effective conductivity reduces linearly as the porosity increases, and the extent of the reduction is significantly dependent on the orientation of WBK. The governing physical mechanism of anisotropic thermal transport in WBK is found to be the anisotropic thermal tortuosity caused by the intrinsic anisotropic topology of WBK.     

Modelling characteristic properties of sandstones

A computer model of a porous medium is described. Written in FORTRAN, it consists of a three-dimensional array of cubes and cylinders representing pores and throats, respectively. The pore-size distribution (p.s.d.) is calculated from the mercury porosimetry curve of Clashac outcrop sandstone, a relatively clay-free, well characterised rock. Surface area and porosity are calculated from the program and the original porosimetry curve is simulated as a check of the p.s.d. Fifty values of the tortuosity factor of the sandstone are then simulated via a weighted random walk through the model. The tortuosity is then defined as the median of the resulting distribution. The tortuosity is simulated for two different cases of an assumed potential field.     

The Role of Tortuosity in Upscaling

The concept of tortuosity is an integral part of models that describe transport in multiscale systems. Traditionally, tortuosity is defined as the ratio of an effective path length to the shortest path length in the microstructure. While the shortest path length can be unambiguously specified, the same is not true for the effective path length, since it changes from one type of transport to another. Consequently, it is possible to have different values of tortuosity for different transport processes taking place in the same system. This is convenient since, under this approach, different transport processes can involve the same type of filters of the microscale information, but the nature of such information is what characterizes each type of transport process. In order to avoid running into unclear interpretations, a set of tortuosity rules are proposed, which relate this concept only to the microscale geometry. On the basis of these rules, we examine the pertinence of introducing the tortuosity concept in mass transport. In particular, we study mass diffusion with and without chemical reaction and convection in porous media. Of all these cases, our analysis indicates that the concept of tortuosity is only adequate for passive diffusion, since in the other cases there is an unavoidable coupling of the transport phenomena that determine the effective path of the solute.     

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.     

Permeability coefficient tensor in the Kozeny-Carman capillary model

The representation of the permeability coefficient tensor for capillary models of porous media displaying isotropic and anisotropic flow properties is considered. The representation proposed is compared with the Kozeny-Carman formula. It is shown that in general, as distinct from the widely accepted representation of the Carman constant in the form of a product of the form factor and tortuosity, this constant is equal to a combination of three coefficients, namely the form factor, the tortuosity, and the structure coefficient. The presence of the latter is due to the fact that in periodic capillary models the porosity is not equal to the surface porosity. It is shown that the Carman constant, the form factor, and the structure coefficient are not universal and their intervals of variation are calculated. The results obtained make it possible to explain and interpret numerous experimental data on the determination of the Carman constant in various porous media.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 4, pp. 96–104, July–August, 1996.     

Fractal porous media I: Longitudinal stokes flow in random carpets

Two-dimensional porous media whose random cross-sections are derived from site percolation are constructed. The longitudinal flow of a Newtonian fluid in the Stokes approximation is then computed and the longitudinal permeability is obtained. Two methods are used and yield the same result when porosity is low. The Carman equation is shown to apply within ±7% when porosity is within the range from 0 to 0.75. Finally, random structures derived from stick percolation are investigated; results are qualitatively the same, but the Carman equation yields a poorer approximation.     

Thermal stretching in two-phase porous media: Physical basis for Maxwell model

An alternate yet general form of the classical effective thermal conductivity model (Maxwell model) for two-phase porous materials is presented, serving an explicit thermo-physical basis. It is demonstrated that the reduced effective thermal conductivity of the porous media due to non-conducting pore inclusions is caused by the mechanism of thermal stretching, which is a combination of reduced effective heat flow area and elongated heat transfer distance (thermal tortuosity).     

孔隙渗流对环面复层含油轴承润滑性能的影响

含油轴承基体中油液的渗流特性对轴承油膜润滑性能影响显著. 以不同孔隙率分布的环面复层含油轴承为研究对象，利用达西定律建立复层含油轴承基体中流体渗流的控制方程，在极坐标下建立环面复层含油轴承系统渗流润滑模型，研究复层环面副系统中油膜压力分布规律，分析轴承结构参数及孔隙渗流行为对油膜润滑性能的影响. 结果表明:复层含油轴承的流体动压力主要发生在环形摩擦面间，从摩擦界面到轴承底面，流体压力逐渐由外缘向圆心部位传导，流体动压力作用面积逐渐增大，压力峰值逐渐降低；随着倾角增大，摩擦界面间的油膜动压效应增强，油膜润滑性能变好；随着表层渗透率或厚度减小，摩擦界面间的油膜的渗流效应减弱，油膜润滑性能提高；与普通单层含油轴承相比，含致密表层的复层含油轴承能降低整体孔隙率，防止过多轴承间隙油液渗入多孔介质，提高轴承润滑性能. 研究工作为明晰环面复层含油轴承渗流行为及润滑机理提供一定理论依据.      

Percolation models of the permeability properties of media

Models that describe the permeability of media with allowance for the structure of the pore space are considered. It is proposed to use percolation theory to describe the topology of the pore space. If the distribution of the pore channels in the medium is random, percolation theory makes it possible to determine the percolation threshold, and also to estimate the fluid conductivity of the cluster that then results. Results obtained for models of granular, porous, and cracked media are given.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 81–86, May–June, 1984.     

Effective Correlation of Apparent Gas Permeability in Tight Porous Media

Gaseous flow regimes through tight porous media are described by rigorous application of a unified Hagen–Poiseuille-type equation. Proper implementation is accomplished based on the realization of the preferential flow paths in porous media as a bundle of tortuous capillary tubes. Improved formulations and methodology presented here are shown to provide accurate and meaningful correlations of data considering the effect of the characteristic parameters of porous media including intrinsic permeability, porosity, and tortuosity on the apparent gas permeability, rarefaction coefficient, and Klinkenberg gas slippage factor.