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Hydrology and Earth System Sciences An interactive open-access journal of the European Geosciences Union
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Volume 16, issue 9
Hydrol. Earth Syst. Sci., 16, 3249–3260, 2012
© Author(s) 2012. This work is distributed under
the Creative Commons Attribution 3.0 License.
Hydrol. Earth Syst. Sci., 16, 3249–3260, 2012
© Author(s) 2012. This work is distributed under
the Creative Commons Attribution 3.0 License.

Research article 10 Sep 2012

Research article | 10 Sep 2012

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

A. Guadagnini1, M. Riva1, and S. P. Neuman2 A. Guadagnini et al.
  • 1Dipartimento di Ingegneria Idraulica, Ambientale, Infrastrutture Viarie e Rilevamento, Politecnico di Milano, Piazza L. Da Vinci 32, 20133 Milano, Italy
  • 2Department of Hydrology and Water Resources, University of Arizona, Tucson, Arizona 85721, USA

Abstract. We analyze the scaling behaviors of two field-scale 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 lower-shoreface 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.

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