Solutions and verification of a scale-dependent dispersion model

In this paper, analytical solutions are derived for a one-dimensional scale-dependent dispersion model (SDM), considering linear equilibrium sorption and first-order degradation for continuous and pulse contaminant sources, with a constant input concentration in a semi-infinite uniform porous medium. In the SDM model, dispersivity alpha(x) is replaced with a constant epsilon multiplied by the transport distance x. The solution for a pulse source is verified experimentally in the analysis of tritium data obtained from an 8-m-long homogenous pea-gravel column with multiple sampling locations, and the results are compared with those analysed by a commonly used solution of a constant dispersion model (CDM). The SDM predicts concentrations satisfactorily at all sampling locations, while the CDM fits the experimental data well for only one location. Both models are then calibrated for each individual concentration breakthrough curve, using local values for either epsilon in the SDM or alpha(x) in the CDM. Both models give equally good fits for appropriate choices of individual epsilon and alpha(x) values, and both indicate a linear increase in alpha(x) with distance. The epsilon values tend to change little as x increases and are expected to approach a constant at relatively large distances downstream. Hence, predictions from the SDM should become more accurate as x increases.