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

Siena, Martina De; Guadagnini, Alberto; Riva, Mònica; Neuman, Shlomo P.

doi:10.5194/hess-16-29-2012

Cited by 31 publications

(46 citation statements)

References 32 publications

(39 reference statements)

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“…These cutoffs are intimately related to the breakdown in power-law scaling. Interpretations based on such modeling framework have been proven to be consistent with observations associated with a variety of laboratory and field scale hydrological and pedological data from sedimentary and fractured rocks [17][18][19][20]33]. With reference to ESS, a theoretical basis for Eq.…”

Section: Scaling Of Statisticsmentioning

confidence: 99%

“…With reference to ESS, a theoretical basis for Eq. (3) and its validity for all investigated lags has been provided with reference to (a) the one-dimensional Burger equation [34], (b) TFBM or TFGN [18], and (c) sub-Gaussian random processes subordinated to TFBM or TFGN [27].…”

Section: Scaling Of Statisticsmentioning

confidence: 99%

“…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…”

Section: Scaling Of Statisticsmentioning

confidence: 99%

“…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.…”

Section: Introductionmentioning

confidence: 99%

“…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.…”

Section: Introductionmentioning

confidence: 99%

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

et al. 2014

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We investigate the scaling behavior of sample statistics of pore-scale Lagrangian velocities in two different rock samples, Bentheimer sandstone and Estaillades limestone. The samples are imaged using x-ray computer tomography with micron-scale resolution. The scaling analysis relies on the study of the way qth-order sample structure functions (statistical moments of order q of absolute increments) of Lagrangian velocities depend on separation distances, or lags, traveled along the mean flow direction. In the sandstone block, sample structure functions of all orders exhibit a power-law scaling within a clearly identifiable intermediate range of lags. Sample structure functions associated with the limestone block display two diverse power-law regimes, which we infer to be related to two overlapping spatially correlated structures. In both rocks and for all orders q, we observe linear relationships between logarithmic structure functions of successive orders at all lags (a phenomenon that is typically known as extended power scaling, or extended self-similarity). The scaling behavior of Lagrangian velocities is compared with the one exhibited by porosity and specific surface area, which constitute two key pore-scale geometric observables. The statistical scaling of the local velocity field reflects the behavior of these geometric observables, with the occurrence of power-law-scaling regimes within the same range of lags for sample structure functions of Lagrangian velocity, porosity, and specific surface area.

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Section: Scaling Of Statisticsmentioning

confidence: 99%

Section: Scaling Of Statisticsmentioning

confidence: 99%

Section: Scaling Of Statisticsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

et al. 2014

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Anisotropic statistical scaling of vadose zone hydraulic property estimates near Maricopa, Arizona

et al. 2013

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[1] Fluid flow and mass transport in the vadose zone are strongly influenced by spatial variability in soil hydraulic properties. It has become common to characterize spatial variability geostatistically and to solve corresponding flow and/or transport problems stochastically. The typical (though not only) approach is to treat log saturated hydraulic conductivity, Y 5 log 10 K s , and perhaps other medium properties as statistically homogeneous, isotropic, or anisotropic multivariate-Gaussian random fields with unique variance and autocorrelation scale. A growing body of literature suggests that Y as well as many other variables may possess heavy-tailed, non-Gaussian distributions, and/or scaledependent statistical parameters representing a multiscale, hierarchical structure. Elsewhere, we have demonstrated that these important phenomena are difficult to detect, and are not fully tractable, with standard geostatistical methods. They are however detectable and tractable with a novel geostatistical method of analysis which treats such variables as samples from sub-Gaussian random fields subordinated to truncated fractional Brownian motion (tfBm) or truncated fractional Gaussian noise (tfGn). Each such field is a mixture of Gaussian components having random variances. The purpose of this paper is to explore the extent to which hydraulic parameters of unsaturated soils exhibit heavy-tailed distributions and statistical scaling of the above type. As a test bed we have selected an experimental site near Maricopa, Arizona, at which soil texture data have been collected in boreholes and trenches to a depth of 15 m over an area of 3600 m 2 . Since hydraulic parameters are difficult to measure to such depths we estimate them using a neural network pedotransfer model. Input data include percent sand, silt, and clay. Outputs include estimates of saturated and residual volumetric water content, saturated hydraulic conductivity, and (due to their wide applicability) parameters of the van Genuchten-Mualem constitutive model of unsaturated soil property variations with capillary pressure head. We find indeed that vertical and horizontal increments of our hydraulic parameter estimates exhibit heavy-tailed frequency distributions and statistical scaling which conform closely to our novel geostatistical framework. Its generality and applicability to the Maricopa site has far reaching implications vis-a-vis the geostatistical characterization of vadose zone properties and the stochastic modeling of flow and transport in unsaturated media at other sites.

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Theoretical analysis of non‐Gaussian heterogeneity effects on subsurface flow and transport

Riva

Guadagnini

Neuman

2017

Water Resources Research

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Much of the stochastic groundwater literature is devoted to the analysis of flow and transport in Gaussian or multi‐Gaussian log hydraulic conductivity (or transmissivity) fields, Ytrue(boldxtrue)=ln⁡Ktrue(boldxtrue) (x being a position vector), characterized by one or (less frequently) a multiplicity of spatial correlation scales. Yet Y and many other variables and their (spatial or temporal) increments, ΔY, are known to be generally non‐Gaussian. One common manifestation of non‐Gaussianity is that whereas frequency distributions of Y often exhibit mild peaks and light tails, those of increments ΔY are generally symmetric with peaks that grow sharper, and tails that become heavier, as separation scale or lag between pairs of Y values decreases. A statistical model that captures these disparate, scale‐dependent distributions of Y and ΔY in a unified and consistent manner has been recently proposed by us. This new “generalized sub‐Gaussian (GSG)” model has the form Ytrue(boldxtrue)=Utrue(boldxtrue)Gtrue(boldxtrue) where Gtrue(boldxtrue) is (generally, but not necessarily) a multiscale Gaussian random field and Utrue(boldxtrue) is a nonnegative subordinator independent of G. The purpose of this paper is to explore analytically, in an elementary manner, lead‐order effects that non‐Gaussian heterogeneity described by the GSG model have on the stochastic description of flow and transport. Recognizing that perturbation expansion of hydraulic conductivity K=eY diverges when Y is sub‐Gaussian, we render the expansion convergent by truncating Y's domain of definition. We then demonstrate theoretically and illustrate by way of numerical examples that, as the domain of truncation expands, (a) the variance of truncated Y (denoted by Yt) approaches that of Y and (b) the pdf (and thereby moments) of Yt increments approach those of Y increments and, as a consequence, the variogram of Yt approaches that of Y. This in turn guarantees that perturbing Kt=eYt to second order in σYt (the standard deviation of Yt) yields results which approach those we obtain upon perturbing K=eY to second order in σY even as the corresponding series diverges. Our analysis is rendered mathematically tractable by considering mean‐uniform steady state flow in an unbounded, two‐dimensional domain of mildly heterogeneous Y with a single‐scale function G having an isotropic exponential covariance. Results consist of expressions for (a) lead‐order autocovariance and cross‐covariance functions of hydraulic head, velocity, and advective particle displacement and (b) analogues of preasymptotic as well as asymptotic Fickian dispersion coefficients. We compare these theoretically and graphically with corresponding expressions developed in the literature for Gaussian Y. We find the former to differ from the latter by a factor k = true〈U2true〉/〈U〉2 ( true〈true〉 denoting ensemble expectation) and the GSG covariance of longitudinal velocity to contain an additional nugget term depending on this same factor. In the limit as Y becomes Gaussian,...

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

Cited by 31 publications

References 32 publications

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

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

Anisotropic statistical scaling of vadose zone hydraulic property estimates near Maricopa, Arizona

Theoretical analysis of non‐Gaussian heterogeneity effects on subsurface flow and transport

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