Modeling of scale‐dependent dispersion in hydrogeologic systems

Abstract

Pickens and Grisak [1981] present an interesting analysis of modeling of scale-dependent dispersion in hydrologic systems. They conclude that for systems that exhibit a constant dispersivity at large times or large mean travel distances, the importance of scale-dependent dispersion at early times or short travel distances is minimal in long-term predictions of solute transport. The purpose of this comment is to point out that this conclusion may not be appropriate.The effect of scale-dependent dispersivity on spreading of the contaminant plume is self-evident. A graph (Figure 1) of spatial concentration variances of contaminant plumes resulting from a constant dispersivity and a scale-dependent dispersivity versus mean travel distance or time illustrates the effect. The spatial concentration variance of a contaminant plume is a measure of spreading of the plume. Thus the value Arrx 2 in Figure 1 represents the difference between the sizes of the plumes resulting from a constant dispersivity and a scale-dependent dispersivity. This value depends on the mean displacement distance, the development distance or time of the asymptotic dispersivity, and the magnitude of the dispersion coefficient (D = a U) or the asymptotic macrodispersivity a if the velocity, U is constant. This figure shows that asymptotic macrodispersivities with large values of development time will produce much smaller variances (curve 1) at large values of distance than the constant dispersivity (curve 2). Also, the difference in the spatial concentration variances increases with the magnitude of the asymptotic dispersivity (Arrx 2' > Arrx2). However, at this moment there is no adequate information to quantify the relationship between spatial variances and mean travel distances and the magnitude of asymptotic macrodispersivity. It is premature to conclude that the scale-dependent dispersion at early times or short travel distances has little effect in long-term predictions of solute transport, although the relative error in such predictions may diminish with distances. It should also be pointed out that a well-defined scale-dependent dispersivity function does not warrant the use of the classical diffusion equation in the prediction of large-scale dispersion processes [Dagan, 1982]. (2) (1) distance x Fig. 1. Spatial concentration variance versus distance. Unless experimental data from various geological environments are available, the effect of scale-dependent dispersivity remains at large. REFERENCES Dagan, G., Stochastic modeling of groundwater flow by unconditional and conditional probabilities, 2, The solute transport, Water Resour. Res., 18(4), 835-848, 1982. Pickens, J. F., and G. E. Grisak, Modeling of scale-dependent dispersion in hydrogeologic systems, Water Resour.

The Characteristics of Hydrogeological Parameters of Unconsolidated Sediments in the Nakdong River Delta of Busan City, Korea

“…This fact was in the agreement with the relationship presented by Pickens and Grisak (1981). Thus, longitudinal dispersivity was dependent on the scale of column test, and all hydrogeological parameters were low to high values due to the sand quantity of sediments.…”

Channel model of flow through fractured media

“…In fact, there is much field evidence that the dispersivity is not a constant due to medium heterogeneities, but depends on the travel distance and/or scale of the system. Several workers have taken different approaches to incorporate the scale dependent dispersion in modeling studies (e.g., Sauty et al,1979;Gelhar et al,1979;Pickens and Grisak, 1981).…”

An effective scale-dependent dispersivity deduced from a purely convective flow field

“…Such a procedure has no predictive capability since it requires the knowledge of a xl (t{), say from observations, and cannot be used for any other time. Others have used equation (11) as if representing dispersion (Pickens & Grisak, 1981;Yates, 1990) with a dispersivity function of space (or equivalently time). Either procedure is an approximation as equation (1) or (11) is not applicable.…”