Distribution of hydraulic conductivity in single scale anisotropy

Abstract

The cluster statistics of percolation theory are used to find the distributions of hydraulic conductivity, K, of anisotropic (truncated) random fractal media. Rescaling of variables to transform anisotropic to isotropic media also produces deformations of, for example experimental volumes, and the resulting non-equidimensional shapes may generate interesting size effects on K. Previously, the most likely value of K was obtained by comparing the correlation length from percolation theory with system dimensions, a procedure analogous to those developed for hopping conduction in disordered systems to calculate the longitudinal conductivity of thin films. The result probably explains the frequent tendencies of measurements of K in anisotropic fracture networks and agricultural soils to increase with the scale of measurement, similarly to how the longitudinal conductivity of a thick film would be larger than the corresponding conductivity of a thin film (three-rather than two-dimensional conduction). However, the same procedure applied to the conductivity in the perpendicular direction (analogous to the transverse electrical conductivity of a thin film) shows a diminishing function of spatial scale. Collectively, these 'scale effects' disappear if the shape of the experimental volume is selected to maintain the relationships of conduction in the various directions as the scale of the experiment is increased analogously to equidimensional volumes in isotropic media. The increase in K is, thus, merely due to an increase in the dimensionality of conduction from one to three with increasing system size. The paper, thus, provides a solid argument against a common assumption in the porous media communities that the connectivity of highly conducting regions of a medium should increase with increasing scale of measurement.

Flow, Transport, and Reaction in Porous Media: Percolation Scaling, Critical‐Path Analysis, and Effective Medium Approximation

“…In addition to showing that W p ( g | x ) is in accord with the experimental data for K (Hunt et al, ), it was possible to use the same general results, but with a laboratory‐determined width of the local conductance distribution (Hunt & Skinner, ) and a field‐determined length scale, to predict both tension for which the chief technetium transport would occur and, at which horizontal spatial scale, vertical solute transport would become important in the Hanford subsurface. In Hunt and Skinner () it was predicted that this would occur at a length scale of 100 m, but it was erroneously stated there that the actual value was 1–10 km, apparently falsifying the prediction.…”

“…A second, and related, goal is to be able to predict the spreading of solute plumes, both in the direction of flow and transverse to it (Berkowitz & Scher, ; Gelhar et al, ; Neuman, ; Sahimi et al, ; Sahimi, Davis, et al, ; Sahimi, Heiba, et al, ; Sahimi, Hughes, et al, ; Sahimi & Imdakm, ). This problem will be taken up later in this review, where we describe how the percolation concepts used to predict the hydraulic conductivity distribution (Hunt, ; Hunt et al, ) can also be used to predict the longitudinal dispersion coefficient, the arrival time distribution of solute, the dispersivity, and the scaling of chemical reaction rate (Hunt, Skinner, et al, ; Hunt, , ). Although quantitative prediction requires a particular model, for many purposes, including dispersion and the scaling of chemical reaction rates, the details of the porous medium are less important than the parameters provided by percolation theory that are related most directly to the connectivity of the fluid flow paths through the medium (Lee et al, ; Sahimi et al, ; Sheppard et al, ).…”

Predicting dispersion in porous media

“…In other work [57], we applied results from Ref. 54 to predict experimental distributions [58,59] of K values in anisotropic fracture networks.…”

Saturation dependence of dispersion in porous media

“…In the case of anisotropy, an observed increase of K with increasing length scale [36,37] is more properly interpreted as a shape, rather than as a scale, effect [53,54]. Thus, with increasing system size, the experimental volume (which effectively is highly non-equidimensional) accommodates a cross-over in flow path structure from one-dimensional to three-dimensional, with corresponding reduction in the critical volume fraction for percolation and associated increase in K. These relationships are best understood within the framework of critical path analysis from percolation theory [55], which clarifies that, at large length scales, important flow and dominant solute transport paths are controlled by the topology of percolation.…”