Explicit Estimates of Arrival Times for Dispersion in Rivers

Explicit formulas for the times of arrival of the leading and trailing edges of a contaminant cloud are developed from a one-dimensional advection-dispersion model of transport. These times—defined as the times at which the concentration is some fraction $\alpha$ of the peak concentration—can be determined easily with iterations using a spreadsheet or other software, but this approach becomes time-consuming if many calculations are needed. Also, solutions to the transient storage zone model and its relatives can be made more efficient by identifying the arrival times of the edges of a contaminant cloud. The expressions for the arrival times depend only the fraction $\alpha$ and the Péclet number $\mathsf{P}$. For $0.01<\alpha <0.1$, the formula for the time of arrival of the leading edge is within 7% for $\mathsf{P}>1$ and 1% for $\mathsf{P}>20$, and the expression for the time of arrival of the trailing edge is within 10–20% for $\mathsf{P}=1$ and less than 1% for $\mathsf{P}>6$. Despite the criticisms of the advection-dispersion model for transport in real waterways, the formulas based on it predict the arrival times at least as well as previously proposed formulas, as long as the mean velocity over the river reach can be estimated.