Comparison of Moments for Classical‐, Quasi‐, and Convolution‐Fickian Dispersion of a Conservative Tracer

Abstract

Transport moments up to fourth order for convolution-, quasi-, and classical-Fickian dispersion of a conservative tracer are presented and compared. Convolution-Fickian yields a non-Gaussian distribution while quasi-and classical-Fickian result in Gaussian type distributions regardless of the distribution of a stationary velocity field. The first and second central spatial moments are the same for convolution-and quasi-Fickian dispersion, while the third and fourth central moments differ. Introduction For steady mean flow velocity, Deng et al. [1993] have derived a nonlocal transport equation where the dispersive flux of a conservative tracer is a convolution of a time-space dependent dispersion tensor and the gradient of mean concentration. This type of dispersion will be referred as convolution-Fickian. Several researchers [e.g., Dagan, 1984; Sposito and Barry, 1987;Zhang and Neuman, 1995a, b, c, d] have in effect adopted the assumption that the mean concentration gradient varies in space and time much slower than the dispersion tensor, resulting in a dispersive flux which is the product of a time dependent dispersion tensor and the gradient of mean concentration. This type of dispersion will be referred as quasi-Fickian, and it is generally believed to hold at later times. When the dispersive flux is the product of a constant dispersion tensor with the gradient of mean concentration, it will be referred to as classical-Fickian and it is an asymptotic result. Such is the case for Gelhat and Axness [1983], Winter et al. [1984], and Kapoor and Gelhar [1994]. The objective of this note is to illustrate analytically for a conservative tracer the effect on transport spatial moments of the various types of dispersion flux just mentioned.In passing, we should point out that far more general constitutive equations obtain when constraints (e.g., stationarity, smallness in perturbations, etc.) resulting in the above fluxes are relaxed [Neuman, 1993;Cushman et al., 1994;Cushman and Ginn, 1993; Deng et al., 1993, equation (13)]. These more general results do not admit analytic solutions for the various moments and hence are not studied in this note.

Advection-diffusion in spatially random flows: Formulation of concentration covariance

“…Non-local effects can be incorporated in calculations of ~(x,t) by recognizing the simplicity of (5), for a stationary flow field, in Fourier-Laplace space (as done by Dent et al [1993] and Deng and Cushman, [1995]), or by numerically solving (4) (as first done by Graham and McLaughlin [1989]). Cushman and ttu [1995] provide a discussion on nonlocal representations of the dispersive flux.…”

Concentration fluctuations and dilution in aquifers

“…0043-1397/98/97WR-03608 $ 09.00 neous porous media has received much attention (see Deng and Cushman [1995] and Jaekel and Vereecken [1997] for a comparison of different approaches to theoretically study solute spreading in an aquifer). When R i is larger than the empirically inferred hydraulic conductivity correlation scales hi, Ri and (c) may be described by effective dispersion coefficients Dii, which are typically much larger than the local dispersion coefficients dii.…”

The role of scaling laws in upscaling

“…Note this last form is identical to the case presented previously by Deng and Cushman [22] and Deng et al [21] from the ensemble averaging perspective. Although the results in those works were developed using ensemble averaging methods, scaling law 3 (imposing quasi-ergodicity) makes the ensemble and volume averages interchangeable.…”

“…(100). Deng and Cushman [22], considered this question, and indicated that the concentration can be removed under the conditions where the characteristic length scale of the velocity autocovariance is much smaller than the characteristic length scale for the gradient of the average concentration. Quintard and Whitaker [24] also considered this question in the context of volume averaging, and concluded that the approximation was valid when r 0 ( L. To see this latter result, one begins by generating a Taylor series for rhci as follows: r y hcij y;t 0 ¼ r x hcij x;t 0 þ ðy À xÞ Á r x r x hcij x;t 0 þ Á Á Á :…”