Allen G. Hunt

Percolation theory for flow in porous media

Percolation Theory: Topology and Structure.- Properties Relevant for Transport and Transport Applications.- Porous Media Primer for Physicists.- Specific Examples of Critical Path Analysis.- Hydraulic and Electrical Conductivity: Conductivity Exponents and Critical Path Analysis.- Other Transport Properties of Porous Media.- Pressure–Saturation Curves and the Critical Volume Fraction for Percolation: Accessibility Function of Percolation Theory.- Applications of the Correlation Length: Scale Effects on Flow.- Applications of the Cluster Statistics.- Properties based on Tortuosity.- Effects of Multi-Scale Heterogeneity.

Applications of percolation theory to porous media with distributed local conductances

Critical path analysis and percolation theory are known to predict accurately dc and low frequency ac electrical conductivity in strongly heterogeneous solids, and have some applications in statistics. Much of this work is dedicated to review the current state of these theories. Application to heterogeneous porous media has been slower, though the concept of percolation was invented in that context. The definition of the critical path is that path which traverses an infinitely large system, with no breaks, which has the lowest possible value of the largest resistance on the path. This resistance is called the rate-limiting, or critical, element, Rc. Mathematical schemes are known for calculating Rc in many cases, but this application is not the focus here. The condition under which critical path analysis and percolation theory are superior to other theories is when heterogeneities are so strong, that transport is largely controlled by a few rate-limiting transitions, and the entire potential field governing the transport is influenced by these individual processes. This is the limit of heterogeneous, deterministic transport, characterized by reproducibility (repeatability). This work goes on to show the issues in which progress with this theoretical approach has been slow (in particular, the relationship between a critical rate, or conductance, and the characteristic conductivity), and what progress is being made towards solving them. It describes applications to saturated and unsaturated flows, some of which are new. The state of knowledge regarding application of cluster statistics of percolation theory to find spatial variability and correlations in the hydraulic conductivity is summarized. Relationships between electrical and hydraulic conductivities are explored. Here, as for the relationship between saturated and unsaturated flows, the approach described includes new applications of existing concepts. The specific case of power-law distributions of pore sizes, a kind of ‘‘random’’ fractal soil is discussed (critical path analysis would not be preferred for calculating the hydraulic conductivity of a regular fractal). ” 2001 Elsevier Science Ltd. All rights reserved.

Application of critical path analysis to fractal porous media: comparison with examples from the Hanford site

Abstract Critical path analysis from percolation theory is used to calculate the unsaturated hydraulic conductivity, K ( S ), of soils with pore space compatible with a (sometimes complex) fractal description. The fractal descriptions are chosen in accord with particle-size distributions of two soils at the US Department of Energy Hanford Site. One of the two soils exhibits a bimodal particle-size distribution, and is treated as a “dual” fractal. The results are then compared with measured hydraulic properties of these two soils. The analysis yields excellent agreement with experiment over 4–6 orders of magnitude in most investigated properties without use of fitting parameters. It is possible to show that such unusual phenomena as a sudden increase in the spread of K values with reduction of matric potential can be traced to effects of a bimodal distribution of pore sizes. The least certain parameter for calculation of K is the “critical volume fraction”, α c , which describes the minimum water content for which an interconnected network of capillary flow exists. The values deduced for α c , however, allow consistent interpretation in both soils investigated (in contrast to fitted values of a “residual moisture content” obtained by application of the van Genuchten function). Further, values of α c obtained correspond well with threshold moisture contents for solute diffusion reported elsewhere, evidence for the relevance of percolation to dispersion.

Non-Debye relaxation and the glass transition

A review of some long-standing problems, as well as some new theoretical results regarding non-Debye relaxation and the glass transition is given. The question of whether a phase transition below (or above) the glass transition temperature, Tg, exists is considered. The result that the glass transition temperature and jumps in the dynamic heat capacity and volume expansion coefficient can be calculated if the relevant frequency-dependent linear response functions are known makes the question of the origin of non-Debye relaxation even more important. In effective medium theories, certainly valid at high temperatures, non-Debye relaxation is apparently due to dynamic effects of interactions. At lower temperatures, energy disorder is larger compared with kT, and percolation-based theories may be more appropriate. The question as to the actual magnitude of disorder in energies cannot be conclusively resolved at this time. Certainly, it has been shown that systems exist in which non-Debye relaxation is due exclusively to disorder but, in many systems, an important component of the energy disorder is a manifestation of the influence of topological disorder on interactions. Thus, in these systems, an increase in the magnitude of dynamic interactions relative to kT accompanies the increase in the disorder. The question as to whether universal properties imply universality in the mechanism of non-Debye relaxation is explored in some depth. The present article reaches the tentative conclusion that the relative importance of ‘disorder’ and ‘interactions’ may be dependent on the type of glass considered, and possibly even on specific systems. Certainly, the relative importance of dynamic interaction effects increases as the frequency of an applied ‘force’ is reduced. If their respective influences ‘crossover’ in importance at some finite frequency, ω∗, the relevant question is whether ω∗ is above, at, or below the relaxation peak frequency, ω∗. If universality exists, it relates to the role of disorder, but such a conclusion would require that ω∗ ≤ αc always, the generality of which has not been established.

Ac hopping conduction: Perspective from percolation theory

Abstract The origin of the sublinear frequency dependence of the ac conductivity is discussed using percolation theory. Different physical bases for calculation in different frequency regimes are discussed. Both the crossover frequencies separating the frequency regimes and the different results for the conductivity are given. The appropriate analogies between one- and three-dimensional systems are summarized. Some important comparisons with results from effective-medium theories are given. Results for the ac conductivity in one dimension and three dimensions are summarized in tabular and graphical (schematic) forms.

Upscaling in Subsurface Transport Using Cluster Statistics of Percolation

Transport/flow problems in soils have been treated in random resistor network representations (RRN’s). Two lines of argument can be used to justify such a representation. Solute transport at the pore-space level may probably be treated using a system of linear, first-order differential equations describing inter-pore probability fluxes. This equation is equivalent to a random impedance network representation. Alternatively, Darcy’s law with spatially variable hydraulic conductivity is equivalent to an RRN. Darcy’s law for the hydraulic conductivity is applicable at sufficiently low pressure ‘head’ in saturated soils, but only for steady-state flow in unsaturated soils. The result given here will have two contributions, one of which is universal to any linear conductance problem, i.e., requires only the applicability of Darcy’s (or Ohm’s) law. The second contribution depends on the actual distribution of linear conductances appropriate. Although nonlinear effects in RRN’s (including changes in resistance values resulting from current, analogous to changes in matric potential resulting from flow) have been treated within the framework of percolation theory, the theoretical development lags the corresponding development of the linear theory, which is, in principle, on a solid foundation. In practice, calculations of the nonlinear conductivity in relatively (compared with soils) well characterized solid-state systems such as amorphous or impure semiconductors, do not agree with each other or with experiment. In semiconductors, however, experiments do at least appear consistent with each other.In the limit of infinite system size the transport properties of a sufficiently inhomogeneous medium are best calculated through application of ‘critical rate’ analysis with the system resistivity related to the critical (percolating) resistance value, Rc. Here well-known cluster statistics of percolation theory are used to derive the variability, W (R,x) in the smallest maximal resistance, R of a path spanning a volume x3 as well as to find the dependence of the mean value of the conductivity, 〈σ(x)〉. The functional form of the cluster statistics is a product of a power of cluster size, and a scaling function, either exponential or Gaussian, but which, in either case, cuts off cluster sizes at a finite value for any maximal resistance other than Rc. Either form leads to a maximum in W (R,x) at R=Rc. When the exponential form of the cluster statistics is used, and when individual resistors are exponential functions of random variables (as in stochastic treatments of the unsaturated zone by the McLaughlin group [see Graham and MacLaughlin (1991), or the series of papers by Yeh et al. (1985, 1995), etc.], or as is known for hopping conduction in condensed matter physics), then W (R,x) has a power law decay in R/Rc (or Rc/R, the power being an increasing function of x. If the statistics of the individual resistors are given by power law functions of random variables (as in Poiseiulle’s Law), then an exponential decay in R for W (R,x) is obtained with decay constant an increasing function of x. Use, instead, of the Gaussian cluster statistics alters the case of power law decay in R to an approximate power, with the value of the power a function of both R and x.

Universal scaling of the formation factor in porous media derived by combining percolation and effective medium theories

The porosity dependence of the formation factor for geologic media is examined from the perspective of universal scaling laws from percolation and effective medium theories. Over much of the range of observed porosity, the expected percolation scaling is observed, but the values of the numerical prefactor do not conform to the simple predictions from percolation theory. Combining effective medium and percolation theories produces a numerical prefactor whose value depends on both the threshold porosity and the porosity above which the formation factor crosses from percolation to effective medium scaling. This change allows extraction of a numerical value of the prefactor, which is reasonably close to experimental values. Subsequent evaluation of the porosity dependence of the formation factor shows that difficulties in prior comparisons of theory and experiment are largely removed when percolation scaling is allowed to transition to effective medium scaling far above the percolation threshold.

Saturation-Dependence of Dispersion in Porous Media

In this study, we develop a saturation-dependent treatment of dispersion in porous media using concepts from critical path analysis, cluster statistics of percolation, and fractal scaling of percolation clusters. We calculate spatial solute distributions as a function of time and calculate arrival time distributions as a function of system size. Our previous results correctly predict the range of observed dispersivity values over ten orders of magnitude in experimental length scale, but that theory contains no explicit dependence on porosity or relative saturation. This omission complicates comparisons with experimental results for dispersion, which are often conducted at saturation less than 1. We now make specific comparisons of our predictions for the arrival time distribution with experiments on a single column over a range of saturations. This comparison suggests that the most important predictor of such distributions as a function of saturation is not the value of the saturation per se, but the applicability of either random or invasion percolation models, depending on experimental conditions.

Universal scaling of gas diffusion in porous media

Gas diffusion modeling in percolation clusters provides a theoretical framework to address gas transport in porous materials and soils. Applying this methodology, above the percolation threshold the air-filled porosity dependence of the gas diffusion in porous media follows universal scaling, a power law in the air-filled porosity (less a threshold value) with an exponent of 2.0. We evaluated our hypothesis using 71 experiments (632 data points) including repacked, undisturbed, and field measurements available in the literature. For this purpose, we digitized Dp/D0 (where Dp and D0 are gas diffusion coefficients in porous medium and free space, respectively) and e (air-filled porosity) values from graphs presented in seven published papers. We found that 66 experiments out of 71 followed universal scaling with the exponent 2, evidence that our percolation-based approach is robust. Integrating percolation and effective medium theories produced a numerical prefactor whose value depends on the air-filled porosity threshold and the air-filled porosity value above which the behavior of gas diffusion crosses over from percolation to effective medium.

Longitudinal dispersion of solutes in porous media solely by advection

The purpose of this work is to predict the transport of non-sorbing solutes through water flow in the subsurface. We derive what we consider to be the first reliable calculations of the entire distribution of arrival times, W(t), for non-sorbing solutes in advective flow in strongly disordered porous media. Solutes treated can be contaminant plumes from any source or radioactive tracers, both experimentally and naturally generated. Our approach is microscopic and based on effects of disorder. It generates longitudinal dispersion (in the direction of flow) in the absence of diffusion. Effects on dispersion from a single capillary tube velocity distribution, known to produce long-tailed arrival time distributions, are also neglected. On the other hand, our calculations are based on effects generated from real porous media, such as wide pore-size distributions and complex connectivity. In particular, the calculation of the distribution of arrival times is based on a distribution of conserved fluxes and the known tortuosity of the associated geometrical paths. The results are found to be predictive when compared with simulations of two-dimensional flow on percolation structures, and appear to have relevance for experiments as well.