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

摘要

We use three methods to identify power-law scaling of multi-scale log airpermeability data collected by Tidwell and Wilson on the faces of alaboratory-scale block of Topopah Spring tuff: method of moments (M),Extended Self-Similarity (ESS) and a generalized version thereof (G-ESS).All three methods focus on q-th-order sample structure functions ofabsolute increments. Most such functions exhibit power-law scaling at bestover a limited midrange of experimental separation scales, or lags, whichare sometimes difficult to identify unambiguously by means of M. ESS andG-ESS extend this range in a way that renders power-law scaling easier tocharacterize. Our analysis confirms the superiority of ESS and G-ESS over Min identifying the scaling exponents, ξ(q), of corresponding structurefunctions of orders q, suggesting further that ESS is more reliable thanG-ESS. The exponents vary in a nonlinear fashion with q as is typical of realor apparent multifractals. Our estimates of the Hurst scaling coefficientincrease with support scale, implying a reduction in roughness(anti-persistence) of the log permeability field with measurement volume.The finding by Tidwell and Wilson that log permeabilities associated withall tip sizes can be characterized by stationary variogram models, coupledwith our findings that log permeability increments associated with thesmallest tip size are approximately Gaussian and those associated with alltip sizes scale show nonlinear variations in ξ(q) with q, are consistent with aview of these data as a sample from a truncated version (tfBm) ofself-affine fractional Brownian motion (fBm). Since in theory the scalingexponents, ξ(q), of tfBm vary linearly with q we conclude that nonlinearscaling in our case is not an indication of multifractality but an artifactof sampling from tfBm. This allows us to explain theoretically how power-lawscaling of our data, as well as of non-Gaussian heavy-tailed signalssubordinated to tfBm, are extended by ESS. It further allows us to identifythe functional form and estimate all parameters of the corresponding tfBmbased on sample structure functions of first and second orders.