Storage Equation
  • 23 Aug 2022
  • 2 Minutes to read
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Storage Equation

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The basis of the flood routing procedure can be seen by considering the region between upstream and downstream points as a storage unit. The inflow hydrograph flowing into the storage unit is known whereas the outflow is to be determined.

The principle of continuity of fluid flow is used to provide the basic equation for the solution of the problem. The equation can be expressed as:

(Inflow volume in time increment dt) - (Outflow volume in time dt) = (change in volume of water stored).

In differential form the expression is:

 

(1)

where :

dS/dt = rate of change in reach storage with respect to time.

I = inflow to the reach

O = outflow from the reach

To provide a form more convenient for computational purposes it is usual to express this equation as a function of average flows. Then:

 

(2)

where the subscripts refer to the values at the start and end of the time step Dt and it is assumed that the hydrograph is a straight line during the timestep Dt.

The time step Dt must be chosen with care to ensure that all important features of the hydrograph are retained. In particular the routing period must be less than the time of travel of the flood wave through the reach.

Routing in river channels is complicated by the fact that storage is not a function of outflow alone, as the water surface is not always parallel to the channel bed. When the flow is increasing, the water surface has a greater slope than the channel bottom, while the opposite is the case when the flow decreases. Thus the relation between outflow and storage for a given reach is not easily known and will be different depending upon whether the hydrograph is rising or falling.

This phenomenon can be illustrated by plotting storage against flow, as shown below. The curve results in a wide loop indicating greater storage for a given outflow during rising stages than when stages are falling.

Figure 1: Plotting Storage Against Flow

 The storage can be seen as split into two parts, the first is referred to as "prism storage", which is the volume beneath a line parallel to the stream bed, and the second is termed "wedge storage", which is the volume of water between that line and the water profile. This is shown in the figure below.

In reservoirs it can often be assumed that the water surface slope is constant. However in channels, a relationship that represents the wedge storage is required. This is derived by including inflow as a parameter in the storage equation.

Figure 2: Prism and Wedge Storage

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