Geometric Tortuosity Research Articles

Percolation Theory Generates a Physically Based Description of Tortuosity in Saturated and Unsaturated Porous Media

Tortuosity is a property of porous media that is invoked and used in the literature on hydrology, soil science, physics, and engineering. It has been defined in a variety of ways, one of which is a purely geometrical concept. In this study, we focused on the geometrical tortuosity and developed a model based on percolation theory and the finite-size scaling approach. Our result, developed for porous media of any saturation, expresses the tortuosity as a power-law function of the water content, the critical water content, and the system size. The model parameters include the fractal dimension of either the backbone (the flow-carrying part of porous media) or the optimal path (for crossing the system between two opposite faces), which have clear physical meaning. The results may be combined with power laws, which percolation theory provides for the hydraulic or electrical conductivity, to develop the appropriate form that expresses the connectivity as a function of the water content of a pore space. Comparison with numerical simulations and experiments reported in the literature indicated that the model estimates the tortuosity accurately. Our model and results leads us to conclude that pore connectivity and tortuosity should be treated as two distinct properties and that significant uncertainty in results for the saturation dependence of the hydraulic conductivity is traceable to the confusion that mixes the two as a single concept.

High performance intermediate temperature micro-tubular SOFCs with Ba0.9Co0.7Fe0.2Nb0.1O3−δ as cathode

In this study, anode supported intermediate temperature micro-tubular solid oxide fuel cells (MT-SOFCs) have been fabricated by combination of phase-inversion, dip-coating, co-sintering and printing method. The MT-SOFC consists of a ∼300 μm wall-thickness Ni–Sc2O3 stabilized ZrO2 (ScSZ) anode tube, ∼10 μm ScSZ dense electrolyte layer, ∼10 μm Ce0.9Gd0.1O2−δ (GDC) membrane buffer layer and ∼50 μm Ba0.9Co0.7Fe0.2Nb0.1O3−δ (BCFN) cathode layer. SEM and electrochemical impedance spectroscopy (EIS) analysis suggested that the novel structured anode can remarkably diminish the porous anode geometrical tortuosity and improve the fuel gas diffusivity. High peak power densities of 0.34, 0.51 and 0.72 W cm−2 have been achieved with humidified hydrogen as the fuel and ambient air as oxidant at 550, 600 and 650 °C, respectively. Further, the cell has demonstrated a very stable performance with no significant cell voltage degradation under a constant current of 0.6 A cm−2 for over 213 h test at 650 °C.

Application of X-Ray CT to Study Diffusivity in Cracked Concrete Through the Observation of Tracer Transport

This paper demonstrates the application of microfocus X-ray computed tomography (CT) to study solute transport in cracked concrete. Cracks in a cylindrical specimen of ordinary Portland cement (OPC) and fly ash mortar were induced using a splitting tensile test. Cesium Carbonate (Cs2CO3) was then used as a tracer in the in-situ diffusion test with the aid of X-ray CT. Image analysis was also employed to measure the 3D crack geometry and tracer diffusivity from these CT images. The geometric tortuosity of the crack was approximately 1.25 irrespective of the crack opening width and whether fly ash was added or not. On the other hand, the constrictivity increased for the fly ash mortar having roughly the equivalent crack opening width. The measured diffusivity in the crack was controlled by both crack opening width and constrictivity. Results obtained from microtomographic images suggest that the entire crack space may not always be filled with the tracer. The diffusive transport of solute in cracks thus can be restricted from microstructure's point of view. Smaller crack opening would increase such restricted diffusion. Indications also suggest that the addition of fly ash would lead to the reduction of diffusivity through uncracked body of the mortar.

On the holes interaction and heterogeneity distribution effects on the acoustic properties of air-cavity backed perforated plates

Most of the available approaches used to model the acoustical behaviour of perforated plates are based on the assumption of no interaction effect between the holes. In practice, the holes are generally closely separated such that the interaction effect can affect their acoustical response. In this paper, the effects of holes interaction and heterogeneity distribution on the acoustic properties of air-cavity backed perforated plates are studied. The classical functions accounting for the holes interaction effects are theoretically reviewed and illustrated. An expression for the geometrical tortuosity, originally derived for granular materials, is used to properly account for both the holes interaction and heterogeneity distribution effects. This expression is integrated in the characteristic impedance expression following Atalla and Sgard [J Sound Vib 2007;303:195–208] model. Perforated plate specimens with different centre-to-centre holes distances (pitches) are built and tested using an impedance tube. It is particularly shown that in the presence of these effects, the resonance frequency is shifted towards the low frequency ranges without degradation of the absorption curve. These effects on the surface impedance and absorption coefficient are discussed in details. A very good agreement is observed between the simulation and the experiments.

Three‐dimensional pore structure and ion conductivity of porous ceramic diaphragms

The ion conductivity of two series of porous ceramic diaphragms impregnated with caustic potash was investigated by electrochemical impedance spectroscopy. To understand the impact of the pore structure on ion conductivity, the three‐dimensional (3‐D) pore geometry of the diaphragms was characterized with synchrotron x‐ray absorption tomography. Ion migration was calculated based on an extended pore structure model, which includes the electrolyte conductivity and geometric pore parameters, for example, tortuosity (τ) and constriction factor (β), but no fitting parameters. The calculated ion conductivities are in agreement with the data obtained from electrochemical measurements on the diaphragms. The geometric tortuosity was found to be nearly independent of porosity. Pore path constrictions diminish with increasing porosity. The lower constrictivity provides more pore space that can effectively be used for mass transport. Direct measurements from tomographs of tortuosity and constrictivity opens new possibilities to study pore structures and transport properties of porous materials.

The extracellular matrix and diffusion barriers in focal cortical dysplasias

Focal cortical dysplasias (FCDs) of the brain are recognized as a frequent cause of intractable epilepsy. To contribute to the current understanding of the mechanisms of epileptogenesis in FCD, our study provides evidence that not only cellular alterations and synaptic transmission, but also changed diffusion properties of the extracellular space (ECS), induced by modified extracellular matrix (ECM) composition and astrogliosis, might be involved in the generation or spread of seizures in FCD. The composition of the ECM in FCD and non-malformed cortex (in 163 samples from 62 patients) was analyzed immunohistochemically and correlated with the corresponding ECS diffusion parameter values determined with the real-time iontophoretic method in freshly resected cortex (i.e. the ECS volume fraction and the geometrical factor tortuosity, describing the hindrances to diffusion in the ECS). The ECS in FCD was shown to differ from that in non-malformed cortex, mainly by the increased accumulation of certain ECM molecules (tenascin R, tenascin C, and versican) or by their reduced expression (brevican), and by the presence of an increased number of astrocytic processes. The consequent increase of ECS diffusion barriers observed in both FCD type I and II (and, at the same time, the enlargement of the ECS volume in FCD type II) may alter the diffusion of neuroactive substances through the ECS, which mediates one of the important modes of intercellular communication in the brain - extrasynaptic volume transmission. Thus, the changed ECM composition and altered ECS diffusion properties might represent additional factors contributing to epileptogenicity in FCD.

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 car- ried 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 consecutivetetrahedrons.Thetortuosityvaluescalculatedbytheproposednumericalmethod comparedfavourablywiththevaluesobtainedfromaCTimagepublishedintheliteraturefor 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.

Random geometric graphs for modelling the pore space of fibre-based materials

A stochastic network model is developed which describes the 3D morphology of the pore space in fibre-based materials. It has the form of a random geometric graph, where the vertex set is modelled by random point processes and the edges are put using tools from graph theory and Markov chain Monte Carlo simulation. The model parameters are fitted to real image data gained by X-ray synchrotron tomography. In particular, they are specified in such a way that the distributions of vertex degrees and edge lengths, respectively, coincide to a large extent for real and simulated data. Furthermore, the network model is used to introduce a morphology-based notion of pores and their sizes. The model is validated by considering physical characteristics which are relevant for transport processes in the pore space, like geometric tortuosity, i.e., the distribution of shortest path lengths through the material relative to its thickness.

Multiscale method for characterization of porous microstructures and their impact on macroscopic effective permeability

AbstractRecent technology advancements on X‐ray computed tomography (X‐ray CT) offer a nondestructive approach to extract complex three‐dimensional geometries with details as small as a few microns in size. This new technology opens the door to study the interplay between microscopic properties (e.g. porosity) and macroscopic fluid transport properties (e.g. permeability). To take full advantage of X‐ray CT, we introduce a multiscale framework that relates macroscopic fluid transport behavior not only to porosity but also to other important microstructural attributes, such as occluded/connected porosity and geometrical tortuosity, which are extracted using new computational techniques from digital images of porous materials. In particular, we introduce level set methods, and concepts from graph theory, to determine the geometrical tortuosity and connected porosity, while using a lattice Boltzmann/finite element scheme to obtain homogenized effective permeability at specimen‐scale. We showcase the applicability and efficiency of this multiscale framework by two examples, one using a synthetic array and another using a sample of natural sandstone with complex pore structure. Copyright © 2011 John Wiley & Sons, Ltd.

Air entry–based characteristic length for estimation of permeability of variably compacted earth materials

The permeability k of a porous medium is a geometrical pore space attribute required for quantifying fluid flows and transport processes for hydrological, civil, agriculture, and petroleum engineering applications. Permeability is often expressed as proportional to a characteristic length squared and inversely proportional to factors accounting for porosity, pore shape, and tortuosity effects. Compaction and diagenesis reduce porosity, mean pore size, and connectivity, resulting in decrease in k. Permeability prediction relies on suitable selection of a characteristic length that may vary from simple hydraulic radius approximation of Kozeny‐Carman to more complex critical path analysis to identify flow‐limiting pore size. For porous media undergoing significant changes in pore space (e.g., compaction due to anthropogenic activities), the proper choice of a robust characteristic length is particularly challenging. We propose using the air entry pressure, a natural characteristic length that gauges the largest drainable pore size. This choice of a characteristic length is compatible with the Aissen formula that provides robust estimates of k for complex pore shapes. Additionally, the model considers geometrical (tortuosity) factors and links relative changes in porosity to concurrent changes in k. The model was tested against experimental data for sands, sandstones with different cementing agents, and unconsolidated soils. For unconsolidated sands and soils the model provides reasonable predictions of permeability for the entire range of porosities determined in laboratory or field experiments. However, for sandstones, and especially those containing cementing agents such as clay, the model is valid up to a critical porosity where hydraulic connectivity is lost, resulting in drastic reduction in k. The geometrical factor for soils was influenced by silt‐to‐clay ratio, while for sands, it was correlated with mean grain diameter. The model offers improvement in predicting k and provides a means for incorporating critical pore size and connectivity information in addition to porosity.

Mathematica Programs for the Analysis of Three-Dimensional Pore Connectivity and Anisotropic Tortuosity of Porous Rocks using X-ray Computed Tomography Image Data

Understanding of the transport properties of porous rocks is important for safe nuclear waste disposal because harmful contaminated groundwater can migrate along pore spaces over long distances. We developed three original Mathematica® version 5.2 programs to calculate the transport properties (porosity, pore connectivity, surface-to-volume ratio of the pore space, and anisotropic tortuosity of the pore structure) of porous rocks using three-dimensional (3-D) 8-bit TIFF or BMP X-ray computed tomography (CT) images. The pre-processing program Itrimming.nb extracts a 3-D rectangular region of interest (ROI) from the raw CT images. The program Clabel.nb performs cluster-labeling processing of the pore voxels in the ROI to export volume, surface area, and the center of gravity of each pore cluster, which are essential for the analysis of pore connectivity. The random walk program Rwalk.nb simulates diffusion of non-sorbing species by performing discrete lattice walks on the largest (i.e., percolated) pore cluster in the ROI and exports the mean-square displacement of the non-sorbing walkers, which is needed to estimate the geometrical tortuosity and surface-to-volume ratio of the pore. We applied the programs to microfocus X-ray CT images of a rhyolitic lava sample having an anisotropic pore structure. The programs are available at http://www.jstage.jst.go.jp/browse/jnst/44/9/ and http://staff.aist.go.jp/nakashima.yoshito/progeng.htm to facilitate the X-ray CT approach to groundwater hydrology.

Mathematica Programs for the Analysis of Three-Dimensional Pore Connectivity and Anisotropic Tortuosity of Porous Rocks using X-ray Computed Tomography Image Data

Understanding of the transport properties of porous rocks is important for safe nuclear waste disposal because harmful contaminated groundwater can migrate along pore spaces over long distances. We developed three original Mathematica® version 5.2 programs to calculate the transport properties (porosity, pore connectivity, surface-to-volume ratio of the pore space, and anisotropic tortuosity of the pore structure) of porous rocks using three-dimensional (3-D) 8-bit TIFF or BMP X-ray computed tomography (CT) images. The pre-processing program Itrimming.nb extracts a 3-D rectangular region of interest (ROI) from the raw CT images. The program Clabel.nb performs cluster-labeling processing of the pore voxels in the ROI to export volume, surface area, and the center of gravity of each pore cluster, which are essential for the analysis of pore connectivity. The random walk program Rwalk.nb simulates diffusion of non-sorbing species by performing discrete lattice walks on the largest (i.e., percolated) pore cluster in the ROI and exports the mean-square displacement of the non-sorbing walkers, which is needed to estimate the geometrical tortuosity and surface-to-volume ratio of the pore. We applied the programs to microfocus X-ray CT images of a rhyolitic lava sample having an anisotropic pore structure. The programs are available at http://www.jstage.jst.go.jp/browse/jnst/44/9/ and http://staff.aist.go.jp/nakashima.yoshito/progeng.htm to facilitate the X-ray CT approach to groundwater hydrology.

Acoustical determination of the parameters governing viscous dissipation in porous media

Analytical solutions are derived to extract from dynamic density the macroscopic parameters governing viscous dissipation of sound waves in open-cell porous media. While dynamic density is obtained from acoustical techniques, the analytical solutions are derived from the model describing this dynamic density. Here, semiphenomenological models by Johnson et al. and by Wilson are investigated. Assuming dynamic density, open porosity, and static airflow resistivity known, analytical solutions derived from the Johnson et al. model yield geometrical tortuosity and viscous characteristic dimension. For the Wilson model, only dynamic density needs to be known. In this case, analytical solutions yield-for the first time-Wilson's density parameter and vorticity-mode relaxation time. To alleviate constraints on the Johnson et al. model, an extrapolation approach is proposed to avoid prior knowledge of static resistivity. This approach may also be used to determine this latter parameter. The characterization methods are tested on three materials covering a wide range of static airflow resistivities (2300-150 100 Ns/m4), frame rigidities (soft and rigid), and pore geometries (cells and fibers). It is shown that the analytical solutions can be used to assess the validity of the descriptive models for a given material.

Open Celled Material Structural Properties Measurement: From Morphology To Transport Properties

Metallic foams are highly porous materials which present complex structure of three-dimensional open cells. The effective transport properties determination is essential for these widely used new materials. The aim of this work is to develop morphology analysis tools to study the impact of foams structure on physical transport properties. The reconstruction of the solid-pore interface allows the visualization of the 3D data and determination of specific surface and porosity. We present an original method to measure the geometrical tortuosity of a porous media for the two phases. A centerline extraction method allows us to model the solid matrix as a network of linear connected segments. The thermal conductivity of metallic foams is determined by solving energy equation over the solid phase skeleton. Results obtained on a set of nickel foams covering a wide range of pore size are discussed.

Diffusivity measurement of heavy ions in Wyoming montmorillonite gels by X-ray computed tomography

Medical X-ray computed tomography (CT) was applied to the measurement of the diffusion coefficients of heavy ions in an artificial barrier material for the disposal of nuclear wastes. Cs +, Sr 2+, I −, and Br − are the heavy ions measured and the barrier used is the water-rich gel of Wyoming montmorillonite (86.5–100 wt.% H 2O). X-ray CT yields an inevitable artifact (beam-hardening) in the obtained images. Before the diffusion experiments, the polychromatic primary X-ray spectrum of the CT scanner was measured by a CdZnTe detector, and the effects of the artifact were examined for an aqueous CsCl solution sample. The results show that the beam-hardening artifact derived from the polychromatic photon energy distribution can be suppressed by applying a special image reconstruction method assuming the chemical composition of samples. The transient one-dimensional diffusion of heavy ions in a plastic container filled with the gel was imaged nondestructively by the X-ray CT scanner with an in-plane resolution of 0.31 mm and slice thickness of 2 mm. The results show that diffusivities decrease with increasing clay weight fraction. The degree of the diffusivity decrease was high for cations (Cs + and Sr 2+) and low for anions (I − and Br −). The quantitative decomposition of the contribution of the geometrical tortuosity and of the sorption to the diffusivity was performed by subtracting the diffusivity of nonsorbing I − from the measured diffusivities. The results show that the contribution of the sorption is large for Cs +, Sr 2+ and small for Br −. Because X-ray CT allows nondestructive and quick measurements of diffusivities, the technique would be useful particularly for measuring the diffusive migration of harmful radioactive elements.

Study and application of a mathematical model for the provisional assessment of areas and nasal resistance, obtained using acoustic rhinometry and active anterior rhinomanometry.

Nasal resistance (NR) depends on the geometrical features and tortuosity of the nasal airway and on the air flow. Knowing the longitudinal distribution of cross-sectional areas (CSAs) in the nasal cavity (which can be obtained using acoustic rhinometry) and the laminar nasal resistance (obtainable by processing the rhinomanometric results), it is possible to calculate, utilizing a mathematical model elaborated on the basis of fluid dynamics, the differential nasal resistance (NRdiff) and the cumulative nasal resistance (NRcum), thus localizing the position at which the highest resistance is concentrated and the related longitudinal distribution. Using a mathematical model, we integrated the sigmoid curves DeltaP/Q of rhinomanometry with the cross-sectional areas obtained using acoustic rhinometry, thus obtaining the normal distribution of differential and cumulative nasal resistances. Afterwards, we empirically reduced the cross-sectional areas corresponding to the head, body, tail and the whole inferior turbinate, recalculating the differential and cumulative nasal resistance distribution curves. The results show that reduction of up to 50% of cross-sectional areas does not substantially affect the resistivity role of the nasal valve, while greater reductions move the highest resistivity point to an area at the junction of the body and the head of the inferior turbinate. The study of the differential nasal resistance trend curves as a function of the reduction of cross-sectional areas shows that the resistance variation of the body and the whole inferior turbinate prevail with reductions of up to 40%, while the variation of cross-sectional areas of the body bordering the inferior turbinate head is predominant with higher reductions. The cross-sectional areas of the nasal airway cavity with highest resistivity are mainly located in an anterior position, where the differential nasal resistances are higher, but there are substantial variations produced by reducing the cross-sectional area of the posterior nasal airway. A similar model can produce provisional values for the results obtainable with functional nasal surgery.

Pulsed field gradient proton NMR study of the self-diffusion of H2O in montmorillonite gel: Effects of temperature and water fraction

Self-diffusion coefficients of water molecules (H 2 O) in Na-montmorillonite gel were measured as a function of water fraction (54.8 to 100 wt%) and temperature (30.4 to 60.0 °C) using proton nuclear magnetic resonance (NMR). Spin-echo pulse sequences with magnetic field gradient pulses for the diffusion measurement were applied to the montmorillonite gel at the Larmor frequency of 20 MHz. The self-diffusion coefficient, D , of H 2 O in the clay gel is expressed phenomenologically by ln( D / D 0 ) = 1.77{exp[0.0798( w − 100)] − 1}, where D 0 is the water diffusivity in bulk water and w is the water fraction of the gel (wt%). The data for w > 84.7 wt% can be explained by a theoretical diffusion model for the randomly distributed clay grains. The activation energy of the water diffusivity in the montmorillonite gel was nearly equal to that in bulk water, so the normalized diffusivity, D/D 0 , obeys the temperature-independent master curve. The transition from the free diffusion to the restricted diffusion was not observed for gradient pulse intervals ranging from 5 to 120 ms. This indicates that the average pore size of the gel is much smaller than a few tens of micrometers, so the random walk trajectory of water molecules in the gel is geometrically restricted by the packing of clay mineral grains. Water diffusivity higher than that of the present NMR study was found by computer simulations and neutron scattering experiments in which effects of bound water were considered but those of the tortuosity of the grain packing were neglected. Thus, the predominant factor controlling the diffusivity in the NMR experiments is not the bound water near the clay surface but the geometrical tortuosity of the packing of clay mineral grains.

Geometric and viscous components of the tortuosity of the extracellular space in the brain.

To understand the function of neuro-active molecules, it is necessary to know how far they can diffuse in the brain. Experimental measurements show that substances confined to the extracellular space diffuse more slowly than in free solution. The diffusion coefficients in the two situations are commonly related by a tortuosity factor, which represents the increase in path length in a porous medium approximating the brain tissue. Thus far, it has not been clear what component of tortuosity is due to cellular obstacles and what component represents interactions with the extracellular medium ("geometric" and "viscous" tortuosity, respectively). We show that the geometric tortuosity of any random assembly of space-filling obstacles has a unique value ( approximately 1.40 for radial flux and approximately 1.57 for linear flux) irrespective of their size and shape, as long as their surfaces have no preferred orientation. We also argue that the Stokes-Einstein law is likely to be violated in the extracellular medium. For molecules whose size is comparable with the extracellular cleft, the predominant effect is the viscous drag of the cell walls. For small diffusing particles, in contrast, macromolecular obstacles in the extracellular space retard diffusion. The main parameters relating the diffusion coefficient within the extracellular medium to that in free solution are the intercellular gap width and the volume fraction occupied by macromolecules. The upper limit of tortuosity for small molecules predicted by this theory is approximately 2.2 (implying a diffusion coefficient approximately five times lower than that in a free medium). The results provide a quantitative framework to estimate the diffusion of molecules ranging in size from Ca2+ ions to neurotrophins.

Tortuosity: a guide through the maze

Abstract Despite its widespread use in petrophysics, tortuosity remains a poorly understood concept. Tortuosity can have various meanings when used by physicists, engineers or geologists to describe different transport processes taking place in a porous material. Values for geometrical, electrical, diffusional and hydraulic tortuosity are in general different from one another. Electrical tortuosity is defined in terms of conductivity whereas hydraulic tortuosity is usually defined geometrically, and diffusional tortuosity is typically computed from temporal changes in concentration. A better approach may be to define tortuosity in terms of the underlying flux of material or electrical current with respect to the forces which drive this flow. Unsteady transport processes, including diffusion, can be described only by a population of tortuosities corresponding to the different flow paths taken by particles traversing the medium. In measurements of steady flow (e.g., those normally used to obtain resistivity or permeability), information about particle travel times is lost, and so the multiple values of tortuosity are homogenised. It can be shown that the maximum amount of information about pore structure is embedded in transport processes that combine advective and diffusive elements. Most existing formulations of tortuosity are model-dependent, and cannot be correlated with independently measurable pore-structure properties. Nevertheless, tortuosity underpins the rigorous relationships between transport processes in rocks, and ties them with the underlying geometry and topology of their pore spaces. Tortuosity can be redefined in terms of the energetic efficiency of a flow process. The efficiency is related to the rate of entropy dissipation (or isothermally, energy dissipation) with respect to a simple, non-tortuous model medium using the postulates of non-equilibrium thermodynamics. Through Onsager’s reciprocity relation for coupled flows it is possible to inter-relate efficiency for pairs of transport processes, and so go some way towards unifying tortuosity measures. In this way we can approach the goal of predicting the value of one transport parameter from measurements of another.

Electrical conduction and formation factor in isotropic porous media

A macroscopic form of Ohm's law is obtained for isotropic porous media saturated with an electrically conductive fluid by using volumetric averaging concepts. Closure of the macroscopic charge transport equation is aided by approximative modelling of the average geometric structures of three different types of isotropic porous media, namely foamlike materials, granular media and crossflow over prismatic bundles. Modelling of the microscopic charge transport necessitated the introduction of a representative interstitial flux of charge carriers and required quantification of the geometric tortuosity applicable to transport phenomena in general. Deterministic expressions for the formation factor are obtained and compare favourably with experimental results.