Vittorio Di Federico

Scaling of random fields by means of truncated power variograms and associated spectra

An interpretation is offered for the observation that the log hydraulic conductivity of geologic media often appears to be statistically homogeneous but with variance and integral scale which grow with domain size. We first demonstrate that the power (semi)variogram and associated spectra of random fields, having homogeneous isotropic increments, can be constructed as weighted integrals from zero to infinity (an infinite hierarchy) of exponential or Gaussian variograms and spectra of mutually uncorrelated homogeneous isotropic fields (modes). We then analyze the effect of filtering out (truncating) high- and low-frequency modes from this infinite hierarchy in the real and spectral domains. A low-frequency cutoff renders the truncated hierarchy homogeneous with an autocovariance function that varies monotonically with separation distance in a manner not too dissimilar than that of its constituent modes. The integral scales of the lowest- and highest-frequency modes (cutoffs) are related, respectively, to the length scales of the sampling window (domain) and data support (sample volume). Taking each relationship to be one of proportionality renders our expressions for the integral scale and variance of a truncated field dependent on window and support scales in a manner consistent with observations. The traditional approach of truncating power spectral densities yields autocovariance functions that oscillate about zero with finite (in one and two dimensions) or vanishing (in one dimension) integral scales. Our hierarchical theory allows bridging across scales at a specific locale, by calibrating a truncated variogram model to data observed on a given support in one domain and predicting the autocovariance structure of the corresponding multiscale field in domains that are either smaller or larger. One may also venture (we suspect with less predictive power) to bridge across both domain scales and locales by adopting generalized variogram parameters derived on the basis of juxtaposed hydraulic and tracer data from many sites.

Transport in multiscale log conductivity fields with truncated power variograms

Di Federico and Neuman [this issue] investigated mean uniform steady state groundwater flow in an unbounded domain with log hydraulic conductivity that forms a truncated multiscale hierarchy of statistically homogeneous and isotropic Gaussian fields, each associated with an exponential autocovariance. Here we present leading-order expressions for the displacement covariance, and dispersion coefficient, of an ensemble of solute particles advected through such a flow field in two or three dimensions. Both quantities are functions of the mean travel distance s, the Hurst coefficient H, and the low- and high-frequency cutoff integral scales λl and λu. The latter two are related to the length scales of the sampling window (region under investigation) and sample volume (data support), respectively. If one considers transport to be affected by a finite domain much larger than the mean travel distance, so that s ≪ λl < ∞, then an early preasymptotic regime develops during which longitudinal and transverse dispersivities grow linearly with s. If one considers transport to be affected by a domain which increases in proportion to s, then λl and s are of similar order and a preasymptotic regime never develops. Instead, transport occurs under a regime that is perpetually close to asymptotic under the control of an evolving scale λl ∼ s. We show that if, additionally, λu ≪ λl, then the corresponding longitudinal dispersivity grows in proportion to λl1+2H or, equivalently, s1+2H. Both these preasymptotic and asymptotic theoretical growth rates are shown to be consistent with the observed variation of apparent longitudinal Fickian dispersivities with scale. We conclude by investigating the effect of variable separations between cutoff scales on dispersion.

Anisotropy, lacunarity, and upscaled conductivity and its autocovariance in multiscale random fields with truncated power variograms

It has been shown by Di Federico and Neuman [1997, 1998a, b] that observed multiscale behaviors of subsurface fluid flow and transport variables can be explained within the context of a unified stochastic framework, which views hydraulic conductivity as a random fractal characterized by a power variogram. Such a random field is statistically nonhomogeneous but possesses homogeneous spatial increments. Di Federico and Neuman [1997] have demonstrated that the power variogram and associated spectra of a statistically isotropic fractal field can be constructed as a weighted integral from zero to infinity (an infinite hierarchy) of exponential or Gaussian variograms and spectra of mutually uncorrelated fields (modes) that are homogeneous and isotropic. We show in this paper that the same holds true when the field and its constituent modes are statistically anisotropic, provided the ratios between principal integral (spatial correlation) scales are the same for all modes. We then analyze the effect of filtering out (truncating) modes of low, high, and intermediate spatial frequency from this infinite hierarchy in the real and spectral domains. A low-frequency cutoff renders the truncated hierarchy homogeneous. The integral scales of the lowest- and highest-frequency cutoff modes are related to length scales of the sampling window (domain) and data support (sample volume), respectively. Taking the former to be proportional to the latter renders expressions for the integral scale and variance of the truncated field dependent on window and support scale (in a manner previously shown to be consistent with observations in the isotropic case). It also allows (in principle) bridging across scales at a specific locale, as well as among locales, by adopting either site-specific or generalized variogram parameters. The introduction of intermediate cutoffs allows us to account, in a straightforward manner, for lacunarity due to gaps in the multiscale hierarchy created by the absence of modes associated with discrete ranges of scales (for example, where textural and structural features are associated with distinct ranges of scale, such as fractures having discrete ranges of trace length and density, which dissect the rock into matrix blocks having corresponding ranges of sizes). We explore mathematically and graphically the effects that anisotropy and lacunarity have on the integral scale, variance, covariance, and spectra of a truncated fractal field. We then develop an expression for the equivalent hydraulic conductivity of a box-shaped porous block, embedded within a multiscale log hydraulic conductivity field, under mean-uniform flow. The block is larger than the support scale of the field but is smaller than a surrounding sampling window. Consequently, its equivalent hydraulic conductivity is a random variable whose variance and spatial autocorrelation function, conditioned on a known mean value of support-scale conductivity across the window, are given explicitly by our multiscale theory.

Flow in multiscale log conductivity fields with truncated power variograms

In a previous paper we offered an interpretation for the observation that the log hydraulic conductivity of geologic media often appears to be statistically homogeneous but with variance and integral scale which grow with the size of the region (window) being sampled. We did so by demonstrating that such behavior is typical of any random field with a truncated power (semi)variogram and that this field can be viewed as a truncated hierarchy of mutually uncorrelated homogeneous fields with either exponential or Gaussian spatial autocovariance structures. The low- and high-frequency cutoff scales λl and λu are related to the length scales of the sampling window and data support, respectively. We showed how this allows the use of truncated power variograms to bridge information about a multiscale random field across windows of different sizes, either at a given locale or between different locales. In this paper we investigate mean uniform steady state groundwater flow in unbounded domains where the log hydraulic conductivity forms a truncated multiscale hierarchy of Gaussian fields, each associated with an exponential autocovariance. We start by deriving an expression for effective hydraulic conductivity, as a function of the Hurst coefficient H and the cutoff scales in one-, two-, and three-dimensional domains which is qualitatively consistent with experimental data. We then develop leading-order analytical expressions for two- and three-dimensional autocovariance and cross-covariance functions of hydraulic head, velocity, and log hydraulic conductivity versus H, λl and λu; examine their behavior; and compare them with those corresponding to an exponential log hydraulic conductivity autocovariance. Our results suggest that it should be possible to bridge information about hydraulic heads and groundwater velocities across windows of disparate scales. In particular, when λl ≫ λu, the variance of head is infinite in two dimensions and grows in proportion to λl2+2H in three dimensions, while the variance and longitudinal integral scale of velocity grow in proportion to λl2H and λl, respectively, in both cases.

Theoretical interpretation of a pronounced permeability scale effect in unsaturated fractured tuff

Received 10 May 2001; revised 8 January 2002; accepted 8 January 2002; published 27 June 2002. [1] Numerous single-hole and cross-hole pneumatic injection tests have been conducted in unsaturated fractured tuff at the Apache Leap Research Site (ALRS) near Superior, Arizona. Single-hole tests have yielded values of air permeability at various locations throughout the tested rock volume on a nominal scale of � 1 m. Cross-hole tests have yielded equivalent air permeabilities (and air-filled porosities) for a rock volume characterized by a length scale of several tens of meters. Cross-hole tests have also provided high-resolution tomographic estimates of how air permeability (and air-filled porosity), defined over grid blocks having a length scale of 1 m, vary throughout a similar rock volume. The results have revealed a highly pronounced scale effect in permeability (and porosity) at the ALRS. We examine the extent to which the permeability scale effect is amenable to interpretation by a recent stochastic scaling theory, which treats the rock as a truncated random fractal. INDEX TERMS: 1869 Hydrology: Stochastic Processes; 1875 Hydrology: Unsaturated Zone; 3260 Mathematical Geophysics: Inverse Theory; 3250 Mathematical Geophysics: Fractals and Multifractals; KEYWORDS: scaling, permeability, fractals, fractured rocks

Polynomial chaos expansion for global sensitivity analysis applied to a model of radionuclide migration in a randomly heterogeneous aquifer

We perform global sensitivity analysis (GSA) through polynomial chaos expansion (PCE) on a contaminant transport model for the assessment of radionuclide concentration at a given control location in a heterogeneous aquifer, following a release from a near surface repository of radioactive waste. The aquifer hydraulic conductivity is modeled as a stationary stochastic process in space. We examine the uncertainty in the first two (ensemble) moments of the peak concentration, as a consequence of incomplete knowledge of (a) the parameters characterizing the variogram of hydraulic conductivity, (b) the partition coefficient associated with the migrating radionuclide, and (c) dispersivity parameters at the scale of interest. These quantities are treated as random variables and a variance-based GSA is performed in a numerical Monte Carlo framework. This entails solving groundwater flow and transport processes within an ensemble of hydraulic conductivity realizations generated upon sampling the space of the considered random variables. The Sobol indices are adopted as sensitivity measures to provide an estimate of the role of uncertain parameters on the (ensemble) target moments. Calculation of the indices is performed by employing PCE as a surrogate model of the migration process to reduce the computational burden. We show that the proposed methodology (a) allows identifying the influence of uncertain parameters on key statistical moments of the peak concentration (b) enables extending the number of Monte Carlo iterations to attain convergence of the (ensemble) target moments, and (c) leads to considerable saving of computational time while keeping acceptable accuracy.

Optimal Scheduling of Replacement and Rehabilitation in Wastewater Pipeline Networks

To fulfill the objective of providing acceptable level of service to customers, the water managers have to plan how to operate, maintain, and rehabilitate the system under budget constraints. The model presented in this paper uses risk cost as an appropriate framework to define the optimal replacement time prediction based on the balance between investment for replacing and expenditures for maintaining the asset. An economic analysis compares the costs associated with maintaining an existing pipe in service, being completely depreciated or not, to the cost of replacing or rehabilitating the pipe. On this basis, the right time in the future to rehabilitate the pipeline can be determined. The costs associated with an existing pipe include direct operational and maintenance costs and indirect costs, such as those associated with risk of failure. The optimal replacement time is identified as the year in which the cost to maintain the existing stock of pipes exceeds the investment to replace it. A dynamic prog...

Solute transport in three‐dimensional heterogeneous media with a Gaussian covariance of log hydraulic conductivity

The analytical solutions for the velocity covariance, uij one particle displacement covariance Xij, and the macrodispersivity tensor αij defined as (0.5/μ)(dXij/dt), were derived in three-dimensional heterogeneous media. A Gaussian covariance function (GCF) of logarithmic hydraulic conductivity, log K, was used, assuming uniform mean flow and first-order approximation in log-conductivity variance, where μ is the magnitude of the mean flow velocity μ. Based on these solutions, the time-dependent ensemble averages of the second spatial moments, Zij ≡ 〈Aij〉 − Aij∥0) = Xij − Rij and the effective dispersivity tensor γij defined as (0.5/μ)(d〈Aij〉/dt, were evaluated for a finite line source either normal or parallel to μ, where Aij(0) is the initial value of the second spatial moment of a plume, Aij and Rij is the plume centroid covariance. The results obtained in this study were compared with previous results for an exponential covariance function (ECF). It was found that in a stationary log K field the spreading of a solute plume depends not only on the variance and integral scale of the log K field but also on the shape of its covariance function. The more strongly correlated the hydraulic conductivities at short separation distances are, the faster Zii and γii grow at early time. Also, the earlier that γii approaches its asymptote or peak, and the higher the peak is, the larger the negative transverse dispersivity. More importantly, the ergodic limits for GCF are reached faster than those for ECF, as the initial size of a plume increases. The ergodic limit X11 for GCF is slightly larger than that for ECF, but X22 and X33 are significantly smaller than those for ECF even though the asymptotic αii for GCF is the same with that for ECF.

Non-Newtonian power-law gravity currents propagating in confining boundaries

The propagation of viscous, thin gravity currents of non-Newtonian liquids in horizontal and inclined channels with semicircular and triangular cross-sections is investigated theoretically and experimentally. The liquid rheology is described by a power-law model with flow behaviour index