Multifractal anisotropic scaling of the hydraulic conductivity

Abstract

[1] We analyzed the scaling properties of the hydraulic conductivity K at three sites in Northern America: MADE, Borden, and Cape Cod. We found that K at all sites exhibits multifractality (fractal and multiscaling) in both the vertical and horizontal directions, though the multiscaling was within a range smaller than that of the maximum distance between measurements. In the vertical direction, the K data for MADE, Borden, and Cape Cod were multiscaling from 0.15 to 1.35 m, 0.05 to 0.5 m, and 0.15 to 0.9 m, respectively. They were multiscaling in the horizontal direction from 9 to 45 m, 1 to 10 m, and 1 to 17 m, respectively. The multiscaling was also anisotropic. Evidence of scaling was poorest for the horizontal direction of the MADE site, and it spanned half an order of magnitude. Such results compel one to treat multifractality in the horizontal direction of MADE as supported by heuristic arguments, rather than by pure statistical evaluation. We fitted a multifractal model to the data and estimated its parameters. We found the underlying statistics of all data to be non-Gaussian, and the model capable of reproducing the probability distribution of K data, especially the negative skewness of log K. We also generated two-dimensional isotropic multifractal fields illustrating the role of the parameters of the selected multifractal model.

Statistical scaling of pore-scale Lagrangian velocities in natural porous media

“…Following [11][12][13][14][15][16][17][18][19][20][21], statistical scaling of moments of velocity is investigated through the analysis of directional sample structure functions, i.e., qth-order statistical moments of absolute increments…”

“…Indeed, the statistics of hydrological properties in aquifers and reservoirs exhibit dependences on various characteristic length scales, e.g., measurement volume and resolution, scale of observation, spatial correlation, and size of the sampled domain [10]. Scaling analyses of statistical moments of hydraulic conductivity and permeability data have been presented in many works, including [11][12][13][14][15][16][17][18][19][20]. In all these studies, measurement volumes are associated with a support scale ranging from the millimeter to the meter and data sets are collected on domains at the field (kilometer) or laboratory (decimeter to meter) scale.…”

“…Similar to [11][12][13][14][15][16][17][18][19][20][21], we do so by relying on the analysis of sample structure functions of order q > 0 of such increments. We then compare the scaling behaviors observed for static (i.e., local porosity and surface area) and dynamic (i.e., Lagrangian velocity) quantities, evaluated on the same rock samples.…”

Extended power-law scaling of air permeabilities measured on a block of tuff

“…Linear or near-linear variation of log S q i N with log s is typically limited to intermediate ranges of separation scales, s I < s < s II , where s I and s II are theoretical or empirical lower and upper limits, respectively. Breakdown in power-law scaling is attributed in the literature to noise at lags smaller than s I and to undersampling at lags larger than s II (Tessier et al, 1993). Yet noise-free signals subordinated to tfBm generated by Neuman (2010b) and show power-law breakdown at small and large lags even when sample sizes are large.…”

“…Attempts to explain such scale dependencies have focused in part on observed and/or hypothesized powerlaw behaviors of structure functions of variables such as hydraulic (or log hydraulic) conductivity (e.g. Painter, 1996;Liu and Molz, 1997a,b;Tennekoon et al, 2003), spacetime infiltration (Meng et al, 2006), soil properties (Caniego et al, 2005;Si, 2006, 2007), electrical resistance, natural gamma ray and spontaneous potential (Yang et al, 2009) and sediment transport data (Ganti et al, 2009). Power-law behavior means that a sample structure function ( 2) where Y (x) is the variable of interest (assumed to be defined on a continuum of points x in space or time), Y n (s) is a measured increment Y (s) = Y (x + s) − Y (x) of the variable over a separation distance (lag) s between two points on the x-axis, and N (s) is the number of measured increments.…”