Critical Volume Fraction For Percolation Research Articles

Dependence of the Electrical Conductivity on Saturation in Real Porous Media

Archie's Law for the porosity and saturation dependence of electrical conductivity, σ(θ) = aσbϕm(θ/ϕ)n (where σ is electrical conductivity, the subscript b denotes the brine or bulk solution, ϕ is porosity, θ is volume wetness, and a, m, and n are fitting parameters) was recently derived by applying continuum percolation theory to fractal porous media. We have recast Archie's Law in terms of saturation alone to obtain σ(θ) = σ0 (θ − θc)μ, where θc is the critical volume fraction for percolation, and σ0 = aσb/(1 − θc)μ. The value of the exponent, μ = 2.0 for three‐dimensional systems and 1.28 for two, is consistent with theory and simulations. We examined the universality of the exponent's value, and the range of validity of our expression. Drawing on published data, we compared predicted and measured values of σ(θ) across the full range of saturation, and found that the newly derived expression provides good predictions, is robust with respect to secondary effects such as residual salinity and contact resistance, and yields meaningful physical parameters.

Comparing van Genuchten and Percolation Theoretical Formulations of the Hydraulic Properties of Unsaturated Media

Recent results from percolation theory as applied to the hydraulic conductivity of unsaturated porous media are compared with the frequently used phenomenological relationship due to van Genuchten. In applications so far percolation theory has proven to be predictive and unifying, while the van Genuchten phenomenology is intended only to be descriptive. To illustrate: the critical volume fraction for percolation is an analog to the residual moisture content, but a phenomenological result exists for the former in terms of the specific surface area, whereas the latter is intended only as a fit parameter. Second, it is recognized not to be possible to predict the hydraulic conductivity as a function of saturation using merely textural data—enough information about the pore space must be available to distinguish between various potential models. When the data are sufficient to confirm a fractal model, however, percolation theory yields a result that has several similarities to the van Genuchten parameterization. The analysis helps lead to a better understanding of the van Genuchten relationship.

Continuum percolation theory for water retention and hydraulic conductivity of fractal soils: estimation of the critical volume fraction for percolation

Systematic experimental deviations from theoretical predictions for water retention characteristics of fractal porous media develop at a moisture content, θ d. Below θ d larger suction pressures are required than would otherwise be predicted from the implied fractal pore-size distribution. Inferred values of θ d were shown to be compatible with observed values of the volumetric moisture content, θ t, at which solute diffusion vanishes. The comparison was based on an accurate phenomenological result for θ t. The approximate equivalence of the two moisture contents was interpreted in the context of continuum percolation theory (at low moisture contents, below a critical volume fraction of water, α c, connections for capillary flow are interrupted, leading to a near vanishing of solute diffusion and increasing enormously the time required for equilibration to new pressures). The experimental results for θ d are now investigated to determine whether their values are generally compatible with theoretically based estimates for α c.

Wet‐End Deviations from Scaling of the Water Retention Characteristics of Fractal Porous Media

A previous work investigated the dry‐end deviation from fractal scaling of water retention characteristics of a suite of 43 U.S. Department of Energy (USDOE) Hanford site soils in relationship with the vanishing of solute diffusion at a moisture content θt It was found that the deviation from fractal scaling of the water retention set on at a moisture content typically about 0.06 higher than the moisture content at which solute diffusion vanishes. Assuming that the vanishing of solute diffusion resulted from a lack of continuity of the water phase, we interpreted the deviation from fractal scaling in terms of a lack of water‐phase continuity rather than as a breakdown of the fractal model. Now the wet‐end deviations from fractal scaling of the same suite of soils are investigated. It is shown that the wet‐end moisture contents at which the deviation occurs are correlated with the critical volume fraction for percolation and with the dry‐end deviations. However, wet‐end deviations occur at moisture contents closer to full saturation than do dry‐end deviations from zero saturation. This contrast between the wet and dry ends of the water retention curve, h(θ), is suggested to be a result of the role of equilibration in the experimental determination of h(θ), and thus to be traceable to the much lower values of the hydraulic conductivity, K, at the dry end.

Percolative transport in fractal porous media

Application of continuum percolation theory to a fractal pore space model yields results for the constitutive relationships for unsaturated flow in agreement with experiment. This application also unites understanding in that the same dry end moisture content, θt=0.039SAvol0.52 as a function of the surface area to volume ratio, is shown to be associated with the deviation of experimental water retention from fractal scaling as well as with the vanishing of the diffusion constant. Substituted into a critical path analysis (based on continuum percolation theory) for the dependence of the unsaturated hydraulic conductivity, K(θ), on moisture, the same value of θt produces excellent agreement with experimental data (y=1.0015x−0.0065, R2=0.96), with y experiment, x theory and using no adjustable parameters. Though critical path analysis is based on percolation theory, the result obtained for K(θ) is more closely tied to the fractal characteristics of the medium, and the dependence is referred to as a fractal scaling of the hydraulic conductivity. In all three properties, the interpretation of θt is the same; it represents the minimum value for which a continuous interconnected path of capillary flow is possible, making it the critical volume fraction for percolation. This identification means that the low moisture content deviation from fractal predictions in h(θ) does not conflict with fractal models of the pore space, as the deviation is due to dynamics rather than to structure. Critical path analysis does not yield percolation scaling, in which K vanishes as a power of (θ−θt). However, it is shown here that the data for K(θ) and h(θ) are consistent with an interpretation in which the fractal scaling of K at large moisture contents crosses over to a percolation scaling at a moisture content slightly above θt.

Infrared reflectance and magnetorefractive effects in metal–insulator CoFe–Al2O3 granular films

The infrared reflectance of (Co50Fe50)x–(Al2O3)1−x granular films which exhibit tunneling magnetoresistance has been measured in the wavelength range 2–25 μm for compositions both above and below the critical percolation volume fraction. The modifications of the Al2O3 Restrahlen reflection region are well accounted for in an effective medium model. In addition, frequency dependent changes of the reflection have been observed under the application of magnetic fields up to 12 kOe. At the longitudinal optic reflection edge around 9 μm distinctive magnetic signatures are found below and above the percolation threshold φc at x=0.17.

This Week's Letters

A News article about the John Templeton Foundation, which funds studies of religion with the scientific method, drew much comment. A critic of the foundation says, “I have found myself continuously bewildered by the proliferation of programs, initiatives, awards, symposia, and so on proffered by this group….” The systematics of the genus Homo are explored. Overlooked work on the critical volume fraction for percolation of small metal spheres in a compressed powder is highlighted. And how wounds heal is discussed.

Percolation in oil-continuous microemulsions. A dielectric study of aerosol OT/water/isooctane

The percolation transition in AOT/water/isooctane microemulsions has been studied by means of dielectric spectroscopy. At the percolation threshold the conductivity of the system increases sharply and the static dielectric permittivity attains a large maximum. It is shown that the frequency dependence of the permittivity can be well described by simple scaling relations of percolation theory. The critical exponent u of the frequency dependence has been found to be equal to u=0.62±0.02 and independent of temperature and microemulsion composition. The temperature dependence of the critical volume fraction for percolation has been related to the temperature dependence of the permittivity of the system far away from the percolation threshold.