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Water-Resources Investigations 1996–4013

Prediction of Traveltime and Longitudinal Dispersion in Rivers and Streams

By Harvey E. Jobson

Thumbnail of and link to report PDF (433 KB)Summary

The possibility of a contaminant being accidentally or intentionally spilled upstream from a water supply is a constant concern to those diverting and using water from streams and rivers. Although many excellent models are available to estimate traveltime and dispersion, none can be used with confidence before calibration and verification to the particular river reach in question. Therefore, the availability of reliable input information is usually the weakest link in the chain of events needed to predict the rate of movement, dilution, and mixing of contaminants in rivers and streams.

Measured tracer-response curves produced from the injection of a known quantity of soluble tracer provide an efficient method of obtaining the necessary data. The purpose of this report is to use previously presented concepts along with extensive data collected on time of travel and dispersion to provide guidance to water-resources managers and planners in responding to spills. This is done by providing methods to estimate (1) the rate of movement of a contaminant through a river reach, (2) the rate of attenuation of the peak concentration of a conservative contaminant with time, and (3) the length of time required for the contaminant plume to pass a point in the river. Although the accuracy of the predictions can be greatly increased by performing time-oftravel studies on the river reach in question, the emphasis of this report is on providing methods for making estimates where few data are available.

Results from rivers of all sizes can be combined by defining the unit concentration as that concentration of a conservative pollutant that would result from injecting a unit of mass into a unit of flow. Unit-peak concentrations are compiled for more than 60 different rivers representing a wide range of sizes, slopes, and geomorphic types. Analyses of these data indicate that the unitpeak concentration is well correlated with the time required for a pollutant cloud to reach a specific point in the river. The variance among different rivers is, of course, larger than for a specific river reach. Other river characteristics that were compiled and included in the correlation included the drainage area, the reach slope, the mean annual discharge, and the discharge at the time of the measurement. The most significant other variable in the correlation was the ratio of the river discharge to mean annual discharge.

The prediction of the traveltime is more difficult than the prediction of unit-peak concentration; but the logarithm of stream velocity can be assumed to be linearly correlated with the logarithm of discharge. More than 980 subreaches for about 90 different rivers were analyzed and prediction equations were developed based on the drainage area, the reach slope, the mean annual discharge, and the discharge at the time of the measurement. The highest probable velocity, which will result in the highest concentration, is usually of concern after an accidental spill. Therefore, an envelope curve for which more than 99 percent of the velocities were smaller was developed to address this concern.

The time of arrival of the leading edge of the pollutant indicates when a problem will first exist and defines the overall shape of the tracer-response function. The traveltime of the leading edge is generally about 89 percent of the traveltime to the peak concentration.

The area under a tracer-response function (a known value when unit concentrations are used) can be closely approximated as the area under a triangle with a height of the peak concentration and a base extending from the leading edge to a point where the concentration has reduced to 1C percent of the peak. Knowing the time of the leading edge and the peak, the peak concentration, and the time when the response function has reduced to 10 percent of its peak value allows the complete response function to be sketched with fair accuracy.

Four example applications are included to illustrate how the prediction equations developed in this report can be used either to calibrate a mathematical model or to make predictions directly.

First posted May 6, 2010

For additional information contact:
USGS, Office of Surface Water
415 National Center
Reston, VA 20192
Telephone: 703-648-5301

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Suggested citation:

Jobson, H.E., 1996, Prediction of traveltime and longitudinal dispersion in rivers and streams: U.S. Geological Survey Water-Resources Investigations Report 96–4013, 69 p.;




Background and Techniques

Analysis of Existing Data and Development of Prediction Equations

Example Applications

Extension to Continuous Sources by Use of the Superposition Princ



Appendix A. Basic Da

Appendix B. Other Data Available

Appendix C. Symbols

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