Behzad Ghanbarian

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.

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.

Properties Relevant for Transport and Transport Applications

This chapter addresses the most important of those results from percolation theory relevant for the study of porous media that were omitted from Chap. 1, namely scaling relationships for tortuosity and conduction. In the latter case the limits of the validity of percolation scaling, and the appropriate choice for scaling far from the percolation threshold are also addressed. The latter topic, of great importance for diffusion and thermal conductivity of porous media, is reviewed from a classic paper of Kirkpatrick. Later work (by Sahimi and co-authors) that is only mentioned in passing, indicates that the result for the cross-over of validity from percolation to effective-medium theories is, like the percolation threshold itself, not universal, meaning that this particular question should be a subject of active research.

Fractal Models of Porous Media

As pointed out in the foreword to the first edition of this book, it is the distribution of the fluids within a medium rather than the medium itself that directly controls the properties of flow, conduction, dispersion, and fluid retention. Of course the distribution of the fluids within the medium is influenced importantly by the structural properties of the medium itself. Insofar as they can be assumed to reflect the characteristics of the pore space, one can develop medium models which generate observed pressure-saturation curves. Most of these models have fractal characteristics. The present chapter reviews a number of fractal models that generate similar to identical pressure-saturation curves though with different interpretations.

Applications of the Correlation Length: Scale Effects on Flow

The usefulness of the identification of the correlation length from percolation theory with the length scale that defines a representative elementary volume is shown. This identification makes it possible to demonstrate that the apparent scale effect of the hydraulic conductivity in anisotropic media can more easily be interpreted as a “shape” effect. Applying a concept from stochastic subsurface hydrology to transform an anisotropic medium to an isotropic one by a suitable rescaling of axes, leads to an identical rescaling of experimental volumes, taking a cube to an elongated rectangular prism. Flow in a narrow prism can be one-dimensional. Increasing all dimensions of the prism equally leads to a dimensional cross-over in the flow to three dimensions. The much smaller critical volume fraction for percolation in three dimensions than in one leads to a much larger hydraulic conductivity. The analytical solution obtained is shown to describe experimental results closely.

Hydraulic and Electrical Conductivity: Conductivity Exponents and Critical Path Analysis

The saturation dependences of the electrokinetic current, the electrical conductivity, and the hydraulic conductivity are investigated by applying concepts from critical path analysis to the pore-size variability and universal scaling from percolation theory to the topological inputs from phase continuity. This division makes it possible to define ranges of saturation for which the pore-size or the topological effects on the given property dominate. We find that in natural media the hydraulic conductivity is predominantly influenced by the pore-size distribution, the electrical conductivity is usually controlled by topology, and there is no input from the pore-size distribution to the electrokinetic current. These calculations are based on a particular model of the medium as well as scaling of flow appropriate from Poiseuille’s law. Qualitatively similar, but quantitatively different conclusions result from different model inputs. It is shown that previously derived non-universal scaling results for the conductivity are obtained from critical path analysis in a limiting case, proving that the treatment is a generalization of known physics.

Specific Examples of Critical Path Analysis

Known applications of critical path analysis that demonstrate its successful predictions in the electrical conductivity of amorphous semiconductors are reviewed. Then complications in its application to the saturated hydraulic conductivity are discussed. The saturation dependence of the hydraulic conductivity is then derived. General tendencies of the dependence of the effective conductivity with disorder in local conductance distributions and with dimensionality of conduction are discussed. The Matheron conjecture from stochastic subsurface hydrology is then placed in context. The chapter concludes with a discussion of why power-law averaging techniques can never reproduce the appropriate behavior of the conductivity in systems with a bimodal distribution of local conductance values.