Extended power-law scaling of heavy-tailed random air-permeability fields in fractured and sedimentary rocks

Abstract

Abstract. We analyze the scaling behaviors of two fieldscale log permeability data sets showing heavy-tailed frequency distributions in three and two spatial dimensions, respectively. One set consists of 1-m scale pneumatic packer test data from six vertical and inclined boreholes spanning a decameters scale block of unsaturated fractured tuffs near Superior, Arizona, the other of pneumatic minipermeameter data measured at a spacing of 15 cm along three horizontal transects on a 21 m long and 6 m high outcrop of the Upper Cretaceous Straight Cliffs Formation, including lowershoreface bioturbated and cross-bedded sandstone near Escalante, Utah. Order q sample structure functions of each data set scale as a power ξ(q) of separation scale or lag, s, over limited ranges of s. A procedure known as extended self-similarity (ESS) extends this range to all lags and yields a nonlinear (concave) functional relationship between ξ(q) and q. Whereas the literature tends to associate extended and nonlinear power-law scaling with multifractals or fractional Laplace motions, we have shown elsewhere that (a) ESS of data having a normal frequency distribution is theoretically consistent with (Gaussian) truncated (additive, self-affine, monofractal) fractional Brownian motion (tfBm), the latter being unique in predicting a breakdown in power-law scaling at small and large lags, and (b) nonlinear power-law scaling of data having either normal or heavy-tailed frequency distributions is consistent with samples from sub-Gaussian random fields or processes subordinated to tfBm or truncated fractional Gaussian noise (tfGn), stemming from lack of ergodicity which causes sample moments to scale differently than do their ensemble counterparts. Here we (i) demonstrate that the above two data sets are consistent with sub-Gaussian random fields subordinated to tfBm or tfGn and (ii) provide maximum likelihood estimates of parameters characterizing the corresponding Lévy stable subordinators and tfBm or tfGn functions.

Recent advances in scalable non-Gaussian geostatistics: The generalized sub-Gaussian model

“…A multifractal is a non-Gaussian fractal with Hurst exponent H that varies continuously with q. Nonlinear power-law scaling is also exhibited by fractional Laplace functions (Meerschaert et al, 2004;Kozubowski et al, 2006Kozubowski et al, , 2013) such as those applied to sediment transport data by Ganti et al (2009). Yet Neuman (2010) for observed non-Gaussian heavy-tailed behavior, we (Neuman, 2010a(Neuman, , 2010b(Neuman, , 2011Guadagnini et al, 2011Guadagnini et al, , 2012aGuadagnini et al, , 2012bGuadagnini et al, , 2013Guadagnini et al, , 2014Guadagnini et al, , 2015Riva et al , 2013bRiva et al , 2013c s  ) lags. We thus have a crossover between two diverse power-law regimes at distance scales 1.5 -1.8 m delineated in Figure 2 by a dashed red line.…”

Comment on “Is unique scaling of aquifer macrodispersivity supported by field data?” by A. Zech et al.

“…Evidently, Zech et al . [] are unaware that hydraulic fractured rock properties tend to exhibit a multiscale hierarchical structure very similar to that of porous media [Vickers et al, ; Neuman , ; Guadagnini et al ., ; Riva et al ., ] or that flow as well as transport in fractured rock environments are often amenable to stochastic continuum representation by flow and transport equations similar to those developed for porous media [ Neuman , ; Ando et al ., ; Neuman and Di Federico , ]. A glance at my Figure 1 [ Neuman , ] demonstrates this vividly.…”

Hydraulic conductivity fields: Gaussian or not?

“…This is due to the well-known fact that a mixture of Gaussian random variates with different mean and/or standard deviation cannot be normally distributed. Indeed, many non-Gaussian distributions that have been used to model ln K data, including the Laplace and symmetric stable, are Gaussian mixtures [ Kotz et al , 2001; Guadagnini et al , 2012; Riva et al , 2013a]. We conclude that GPR facies are useful in this simulation, as they provide a data-based procedure for delineating statistically distinct regions of K values, leading to the more sharply peaked and non-Gaussian profile evident in Figure 4 (d).…”