ANALYTICAL AND EXPERIMENTAL INVESTIGATION OF STREAMBED VARIATION UPSTREAM OF A DAM

Abstract

The problem of predicting transient bed profiles upstream of a dam in a river is important as well as challenging. These transients result progressive reduction in the capactiy of the reservoir created by the dan and thus gradually reduce the benefits of the project. Earlier attempts wwere confined towards either empirical study of a number of such reservoirs or forming a mathematical model of the phenomenon. Several algorithms for solving such a mathematical model have been proposed in the recent past but no computational success of these algorithms has been shown. It has been found that these algorithms are either unstable or they require such a small time interval that no meaningful results can be obtained. IRCBLEM • _. In the present study the phenomenon of variation of streambed upstream of a dam has been observed under controlled conditions. For this purpose experiments have been conducted in a 32 meter long and 2C cm wide recirculate^ flume. A barrier was introduced at the downstream end and the transient bed profiles were observed with a constant sediment inflow discharge. The phenomenon was observed for times ranging from -11- 14 to 7£ ho-or s. ANALYSIS The experimental results have been used in the correct establishment of the mathematical model which consisted of a system of two coupled non linear partial differential equations with known initial conditions. Applicability of the method of characteristics was explored and it has been found that for getting stable results the time interval that must be used is extremely small and hence the use of this method for solving a practical problem is limited because of prohibitive cost. An iterative procedure for numerical solution of the system of equations has been ,eveloped. The algorithm converges rapidly and more than two itera tions are not required in any case. Moreover it was found to be stable over a large number of cycles. An approximate mathemtical model involving a Single non-linear partial differential equation has been proposed and a numerical algorithm has been developed tc solve the general nonlinear equation c't r 1U' dlP °* It has been found from this approximate analysis that the resulting bed profiles are close to the experi mentally observed bed profiles. -111- It has been found analytically as well as experimentally that the bed profile transients can be represented as a wave phenomenon. ,J-he bed wave has a major peak followed by minor peaks in the up stream side. As the time passes these peaks travel in the downstream direction. The height of the major peak increase- monotonically with the passage of time. At the time the major peak reaches the dam its height is nearly equal to the dam height and by this time a major part of the reserv ir is filled up. This can be termed as the first phase of aggradation. The remaining second phase is slow and at the end of the second phase the equilibrium condition is once again established. This process is accomplished in time t . A closed form solution with some idealised assumptions has been obtained for tQ. This gives a physical idea about the magnitude of the equilibrium time. The experimental investigation reveals that the entire process of aggradation upstream of a dam can be conceived as a movement of a front. Empirical equa tions have been obtained for the location of the height of the front as a function of its distance from the dam and also as a function of the time. The front causes the reduction in the reservoir capacity for which an -IVempirical equation of the form o t/t % + %-) Ko-iK has been obtained which gives the reservoir volume •V^ for time t, VQ bein£ the initial volume. The computed bed profiles show a fairly good agreement with the experimentally observed bed profiles. The computations have been performed for 35 cycles incorporating 70 hours of deposition time