A natural gradient experiment on solute transport in a sand aquifer: Spatial variability of hydraulic conductivity and its role in the dispersion process

Abstract. It has been shown by Di Fededco 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 weigh. ted 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, an d 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.

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confidence: 99%

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

Federico

Neuman

Tartakovsky

1999

“…This is the same type of autocovariance function that was used by Sudicky [1986] However, he rather arbitrarily chose to average the process o.…”

Section: Cr(s) _ Or2 Exp [_(sl 2 + $22 (/H)2 + (;-••2) (6)mentioning

confidence: 99%

Comment on “A natural gradient experiment on solute transport in a sandy aquifer: Spatial variability of hydraulic conductivity and its role in the dispersion process” by E. A. Sudicky

Kemblowski¹

1988

Sudicky's [1986] article presents an analysis of the spatial variability of hydraulic conductivity at the Borden site. The results of this analysis are consequently used to calculate dispersion parameters, based on the three-dimensional transport theory developed by Gethar and Axness [1983] and the vertically averaged, two-dimensional approach developed by Dagan [1982Dagan [ , 1984. The Borden experiment is very unique and valuable to the groundwater community, and the contribution of the researchers involved in it can hardly be overestimated. Thus it is important to clarify all the components of We will now examine the applicability of Dagan's [1984] vertically averaged, two-dimensional approach to the Borden site data. We first note that this theory was developed for the vertically averaged flow properties. In fact, Dagan [1984] defines such properties as averaged over the aquifer depth. Since

“…They demonstrate that the interpolation procedure adopted The same argument applies to the statistical analysis of the measured log conductivity [Sudicky, 1986], which also has been the object of recent debates. Again, the values derived for tr2r, I v, and I h by Sudicky [1986] should be regarded as estimates, which are subjected to estimation errors.…”

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confidence: 99%

“…

The tracer test at the Borden Site [Mackay et al, 1986;Sudicky, 1986;Freyberg, 1986] has generated considerable interest, being one of the first field experiments in which both concentration distributions and formation properties have been measured extensively. A few attempts have been made in the past in order to compare field measurements with existing theoretical formulae which relate the concentratio½ to aquifer heterogeneity.

The author's [Naff et al, 1988] main thesis is that the measured particles displacement variance in the longitudinal direction X•(t), evaluated by Freyberg [1986], can be fitted equally well by a few theoretical solutions.

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confidence: 99%

Comment on “A note on the recent natural gradient tracer test at the Borden site” by R. L. Naff, T.‐C. Jim Yeh, and M.W. Kemblowski

Dagan¹

1989

The tracer test at the Borden Site [Mackay et al., 1986;Sudicky, 1986;Freyberg, 1986] has generated considerable interest, being one of the first field experiments in which both concentration distributions and formation properties have been measured extensively. A few attempts have been made in the past in order to compare field measurements with existing theoretical formulae which relate the concentratio½ to aquifer heterogeneity.The author's [Naff et al., 1988] main thesis is that the measured particles displacement variance in the longitudinal direction X•(t), evaluated by Freyberg [1986], can be fitted equally well by a few theoretical solutions. In particular, this is true for a two-dimensional transport model [Dagan, 1982[Dagan, , 1984[Dagan, , 1987Freyberg, 1986] or a three-dimensional one for a anisotropic log conductivity covariance [Gelhar and Axness, 1983]. This writer completely agrees with the authors reasoning and herein are some additional arguments, based on recent publications.In a theoretical study of the statistical moments of the solute particles displacements [Dagan, 1988]