Using the method of moments to analyze three‐dimensional diffusion‐limited solute transport from temporal and spatial perspectives

Abstract

This paper focuses on the analysis and application of an analytical model that incorporates solute diffusion within immobile regions into the three‐dimensional advection/dispersion solute transport equation. The diffusion term of the model is formulated using either a first‐order rate expression or an expression assuming Fickian diffusion into spherical, cylindrical, or rectangular immobile regions. In order to assist in the analysis of solute transport behavior by means of the models, a modified form of Aris's method of moments is developed, which permits the calculation of the spatial and temporal moments of solute distributions simulated using the three‐dimensional diffusion models, without having to invert the Laplace‐ or Fourier‐transformed solutions. By using this method, the moments of the diffusion models are compared with one another, with the moments of a model that assumes equilibrium advective/dispersive transport, and with the moments of a model that assumes that a first‐order rate law governs mass transfer between the mobile and immobile regions. The method of moments also is used to assess the differences in the spatial and temporal moment behavior of each transport model under discussion.

Macroscopic behavior and random‐walk particle tracking of kinetically sorbing solutes

“…These moments will also be interpreted, in section 3.3, as phase-transfer probabilities for particles starting and ending in the specified phases. For the cases for which analytical expressions were already available, the zeroth-moment equations derived in this work match those presented in the literature Goltz and Roberts, 1987]. The same is true for first-and second-moment expressions.…”

“…The original method of moments was derived for conservative solutes [Aris, 1956]. and Goltz and Roberts [1987] derived the analytical solution for the zeroth and first spatial moments of a kinetically sorbing solute injected into the aqueous phase. The second moment was derived only for a solute starting and ending in the aqueous phase.…”

Use of temporal moments to investigate the effects of nonuniform grain‐size distribution on the transport of sorbing solutes

“…Indeed, this importance is reflected in the recent development of several nonequilibrium sorption models that are designed to account for a continuous distribution of sorption rates [Connaughton et In this paper, we use temporal moment analysis to assess the impact of the distribution of diffusion times ta on the transport behavior of sorbing solute. Temporal moments are physically meaningful descriptors of a solute breakthrough curve and have been used by many researchers to investigate and elucidate the effects of mass transfer phenomena on solute transport [Kucera, 1965;Villermaux, 1981aVillermaux, , b, !987, 1990Valocchi, 1985Valocchi, , 1986Valocchi, , 1990Parker and Valocchi, 1986;Goltz and Roberts, 1987; Sardin et al, 1991; Harvey and Gorelick, 1995]. Here we demonstrate that a distribution of diffusion times ta leads to an increase in the higher-order moments of the breakthrough curve without affecting the lower-order moments.…”

“…Table 2 Application of our method to geometries other than spherical results in qualitatively similar behavior. For example, rather than spherical grains, we could consider layers (sometimes called "slabs") [Goltz and Roberts, 1987;Goltz and Oxley, 1991]. In such a case, the coefficients 1/15 and 2/315 that appear in (16) would be replaced by 1/3 and 2/15, respectively, thereby increasing both the second and third temporal moments.…”

Effects of intraborehole flow on groundwater age distribution

“…Estimates of regional-scale transport characteristics must incorporate smaller-scale variability into models of the system (some examples of upscaling methods include Gelhar and Axness 1983;Li et al 1994;Haggerty and Gorelick 1995;Berkowitz et al 2000;Haggerty et al 2000). Studies have shown that accurate upscaled transport modeling often requires detailed knowledge of the magnitude and the structure of small-scale heterogeneity (e.g., Goltz and Roberts 1987;Li et al 1994;Di Federico et al 1999;LaBolle and Fogg 2001;Becker and Shapiro 2001;Guswa and Freyberg 2002;Zinn and Harvey 2003;Liu et al 2004;Zinn et al 2004).…”