| The volumetric flow rate of water in open channels (discharge) is commonly estimated from the measured water-surface elevation (stage) using empirical relations fit to many measurements of discharge and concurrent stage. While these relations, often referred to as ratings, have been used for over a century, many supplemental empirical corrections have been developed because the discharge often is not uniquely defined by the stage. Despite the extensive application of stage-discharge relations, this research represents an unprecedented examination of hydraulic factors that affect these ratings based on the hydrodynamics of unsteady, nonuniform flow. Dimensional analysis of factors controlling flow past a cross section, combined with an examination of open-channel flow equations, provides understanding of the equations and important independent variables that describe relations among stage, discharge, and other parameters. Theoretical reasons behind the limited success of common empirical corrections for conditions where the discharge is not uniquely defined by the stage are presented. A basis for theoretical development of discharge ratings that reflect the hydrodynamics of unsteady, nonuniform flow is presented. Procedures to develop discharge ratings based on the theoretical analysis of this research and that use commonly measured explanatory variables are presented. Theoretical ratings were developed for five channels, including two flume studies, a navigation canal, a small meandering channel, and a large natural river. Discharges from the theoretical ratings explained much of the variance in discharges measured in these channels, and also indicated areas where further research could improve the agreement between rated and measured discharges. Uncertainty in the relations that comprise the theoretical rating, as well as in discharges determined from the rating, were examined for one of the flume studies, providing an example of applying methods commonly used to examine the reliability of engineering systems to discharge ratings. These methods provide better definition of the uncertainty associated with ratings than the traditional methods, which are limited to statistical analysis of the goodness-of-fit between the rating curve and the measurements from which it was derived. |
