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

We use three methods to identify power law scaling of (natural) log air permeability data collected by Tidwell and Wilson (1999) on the faces of a laboratory-scale block of Topopah Spring tuff: method of moments (M), extended power-law scaling also known as Extended Self-Similarity (ESS) and a generalized version thereof (G-ESS). All three methods focus on qth-order sample structure functions of absolute increments. Most such functions exhibit power-law scaling at best over a limited midrange of experimental separation scales, or lags, which are sometimes difficult to identify unambiguously by means of M. ESS and G-ESS extend this range in a way that renders power-law scaling easier to characterize. Most analyses of this type published to date concern time series or one-dimensional transects of spatial data associated with a unique measurement (support) scale. We consider log air permeability data having diverse support scales on the faces of a cube. Our analysis confirms the superiority of ESS and G-ESS over M in identifying the scaling exponents ξ(q) of corresponding structure functions of orders q, suggesting further that ESS is more reliable than G-ESS. The exponents vary in a nonlinear fashion with q as is typical of real or apparent (Guadagnini and Neuman, 2011; Guadagnini et al., 2011) multifractals. Our estimates of the Hurst scaling coefficient increase with support scale, implying a reduction in roughness (anti-persistence) of the log permeability field with measurement volume. ESS and G-ESS ratios between scaling exponents ξ(q) associated with various orders q show no distinct dependence on support volume or on two out of three Cartesian directions (there being no distinct power law scaling in the third direction). The finding by Tidwell and Wilson (1999) that log permeabilities associated with all tip sizes can be characterized by stationary variogram models, coupled with our findings that log permeability increments associated with the smallest tip size are approximately Gaussian and those associated with all tip sizes scale show nonlinear (multifractal) variations in ξ(q) with q, are consistent with a view of these data as a sample from a truncated version (tfBm) of self-affine fractional Brownian motion (fBm). Since in theory the scaling exponents, ξ(q), of tfBm vary linearly with q we conclude, in accord with Neuman (2010a, 2010b, 2011), that nonlinear scaling in our case is not an indication of multifractality but an artifact of sampling from tfBm. This allows us to explain theoretically how power law scaling of our data is extended by ESS. It further allows us to identify the functional form and estimate all parameters of the corresponding tfBm based on sample structure functions of first and second orders. Our estimate of lower cutoff is consistent with a theoretical support scale of the data.