On the Dagan Model of solute transport in groundwater: Application to the Borden Site

Abstract

In the first part of this study, Sposito and Barry (1987) derived an ensemble-mean convectiondispersion equation (CDE) for tracer solute transport subject to a random velocity field. It was shown that the model dispersion coefficients originally presented by Dagan (1984) could be derived from the general expression for the dispersion coefficients in this mean CDE. Under the assumption of ergodicky, the Dagan model is used in this paper to predict chloride and bromide concentrations in the welldocumented Borden aquifer experiment reported by Roberts and Mackay (1986). Because of a possible influence on the solute from the upper aquifer boundary, it was appropriate to apply the twodimensional form of the model. A number of steps was necessary to reduce the three-dimensional raw data to a two-dimensional form, the main ones being integration over the vertical axis and the use of a gridding algorithm to form a two-dimensional solute concentration surface. Incomplete sampling of the solute plume during the early sampling sessions, as well as the assumptions made with respect to the data analysis, produce a rather large degree of uncertainty in the specification of the initial solute plume. These factors hinder a thorough experimental evaluation of the Dagan model. Data from some of the later sampling sessions were more complete, however, and the model predictions appeared to agree well with the field concentration data, especially in the preasymptotic region for the longitudinal dispersion coefficient. INTRODUCTION A number of field experiments and theoretical descriptions o[ solute dispersion in groundwater aquifers has demonstrated the inadequacy of using a constant dispersion coefficient in the convection-dispersion equation (CDE) for the purpose of modeling field scale solute transport [Anderson, 1979; Matheron and de Marsily, 1980; Pickens and Grisak, 1981; Sudicky et al., 1983; Da•Tan, 1984, 1986; Giiven et al., 1984; Thompson et al., 1984; Roberts and Mackay, 1986; Frind and Hokkanen, 1987; Frind et al., 1987]. One theoretical approach to overcoming this problem, although it creates significant unresolved mathematical questions itself, has been to model the solute convection velocity in an aquifer as a single realization of a random function, such that the CDE becomes a stochastic partial differential equation [Sposito et al., 1986]. This approach was adopted in the first part of the present study [Sposito and Barry, 1987] to derive a CDE for the ensemble mean concentration of a conservative solute transported in an incompressible liquid whose velocity in a porous medium is a wide-sense stationary, stochastic process parameterized by space and time coordinates. The cumulant expansion technique used to derive the mean CDE led to a general, time-dependent expression for the solute dispersion coefficients in terms of' the solute velocity autocorrelation function. This expression, by virtue of different choices of model for the autocorrelation function, could be specialized to reproduce the equations for time-depen...

Numerical simulations of transport of non-ergodic solute plumes in heterogeneous aquifers

“…One may conclude that the Dagan's ®rst-order theory provides a fairly accurate approximation to the longitudinal second spatial moment of an ergodic plume, X 11 , for the value of r 2 Y at least up to 1. This conclusion is consistent with several previous studies (e.g., Barry et al, 1988;Rubin, 1990;Chin and Wang, 1992;Bellin et al, 1992). The ®rst-order theory of Dagan's (1984) signi®cantly underestimate the transverse moment in more heterogeneous media, as illustrated in Figure 9b where the simulated X 22 for r 2 Y 0X5 and 1.0 are much higher than their respective linear solutions (the solid curves), even though the simulated X 22 for r 2 Y 0X1 ®t well to the theoretical one.…”

Numerical simulations of non-ergodic solute transport in three-dimensional heterogeneous porous media

“…6, we may conclude that Dagan's firstorder theory provides a fairly accurate approximation to the longitudinal second spatial moments of an ergodic plume, X 0 11 t 0 ð Þ, for the value of r 2 Y up to 0.46. This conclusion is consistent with several previous studies (e.g., Barry et al 1988;Rubin 1990;Chin and Wang 1992;Bellin et al 1992;Zhang and Lin 1998).…”

Three-dimensional field study of solute transport through unsaturated, layered, porous media. 1. Methodology, mass recovery and mean transport

“…Secondly, the method of moments The procedure chosen for computation of the moments was a three-dimensional interpolation of the data using an inverse distance weighting scheme. As pointed out by Barry et al [1988], the calculated moments are sensitive to the weighting factor used in the interpolation procedure. Crossvalidation [Geisser, 1975] was the criteria used to select the weighting factor in the present study.…”