Theory of Flood Routing

Abstract

The first equation required for the analysis of unsteady flow in an open channel is the continuity equation. This is usually written in differential form as

$$\frac{{\partial Q}}{{\partial x}} + \frac{{\partial A}}{{\partial t}} = r(x,t)$$

(3.1.1)

where Q(x,t) is flow, A(x,t) is area of flow and r(x,t) is the rate of the lateral inflow per unit length of channel. It is convenient to emphasize the element of forecasting by writing the continuity equation in what is known as the prognostic form i.e. with all time derivatives as unknowns on the left hand side. The prognostic form of the continuity equation is

$$\frac{{\partial A}}{{\partial t}} = - \frac{{\partial Q}}{{\partial x}} + r(x,t)$$

(3.1.2)

It is important to note that the area in Equations (3.1.1) and (3.1.2) is the total storage area and includes any overbank storage as well as the conveyance area of the main channel.