A hydraulic model for the catchment-stream problem: I. Kinematic-wave theory

Abstract

Linear mathematical models describing stream or river outflow due to storm runoff do not explain several important observed features, such as the change in shape of the discharge hydrograph and the non-linear variation of peak discharge rate with variation of rainfall intensity. In the present paper, analytical solutions for a hydraulic model are obtained by the method of characteristics, firstly, for flow over a plane V-shaped catchment under a constant uniformly-distributed rainfall of finite duration, and secondly, for the stream outflow arising from the catchment discharge. The advantage of this simplified two-component model lies in the fact that only four parameters are involved; these comprise two dimensionless indices in the power-law equations of motion assumed for catchment and stream flow, a scale of (i) rainfall intensity or (ii) total rainfall or (iii) rainfall duration and a dimensionless parameter which represents a ratio of suitably-defined time constants for stream and catchment respectively. The calculated stream hydrograph – for either stage or discharge rate – is found to be a smooth curve which contains up to six discontinuities in curvature, the locations of these discontinuities, and hence the shape of the function, depending upon the values of the parameters noted above. It is noted that the rising part of the curve depends initially only upon the integral of the catchment outflow. In this region various segments of the curve exhibit power-law behavior, but this is, in fact, a consequence of assuming power-law depth-discharge relationships. For a similar reason, the falling part of the curve exhibits a power-law decay. This paper is the first in a series of three. Part II discusses model solutions for a steady rainfall of finite duration, which could be relevant to work with sprinkled plots. In Part III, the model is applied to three natural catchments. Additional problems arising from infiltration are discussed in Parts II and III.