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# Triangular Weir Effective Storage

The old method of determining the effective storage of a weir assumes that the river channel has a rectangular cross section and that the surface area of the reach is constant. Generally natural rivers do not have this sort of geometry and the surface area increases with flow. Hydrodynamic modelling also reveals a more complicated relationship between the weir and reach surface area that is inconsistent with the old assumptions. By representing the weir reach as having a triangular cross section and a constant bed slope the estimated effective surface area of the weir can be considered as more realistic.

Assumptions of the revised method are:

• The river channel is a triangular uniform shape
• The channel slope is constant
• When the water in the weir is at the highest level in the weir table, the backwater just reaches back to the start of the reach
• Flow related storage has a constant depth over the reach and occupies the bottom of the channel

The maximum storage on the weir table has height H, width W and length L.

Figure 1. Triangular Weir, general representation of weir reach

The maximum volume of the weir is calculated by:

 Equation 1

Where:                 Sm = Maximum Storage Volume

W = Maximum Width of the water body

H = Maximum Height of the water body

When the height of the weir storage is a fraction (wf) of the height (H), the width of the channel will be wf W and the extent of the weir pool upstream of the weir wall will be wf L, and the weir storage will be:

 Equation 2

or;

 Equation 3

The fraction of the reach upstream (wf) that the weir storage extends can therefore be calculated as:

 Equation 4

and the location of the start of the weir pool will be 1 – wf.

The geometry of the flow related storage (Sf) is presented in Figure 2.

Figure 2. Triangular Weir, representation of flow related reach storage

The cross-sectional area of the flow related storage reaches can be calculated as:

 Equation 5

When ff is the ratio of the Flow related storage height to H

 Equation 6

Therefore:

 Equation 7

thus,

 Equation 8

If wf > ff, the effective storage (Se) will be the trapezoidal prism on the top of the flow related storage (component B) as shown figure 2. The effective storage will have a height of (wf – ff) H, a width which reduces from wf W at the weir to ff W at the most upstream extent and a length of (wf – ff) L. The upper extent of the effective storage will be at (1 – (wf – ff)).

The effective storage is equal to the weir storage less the weir storage upstream of the effective extent (component A) less the flow related storage downstream of the effective extent (component B):

 Equation 9

The revised method of determining the Effective Storage does not change the values at the extremes of the weir, but as shown in the example (Figure 3), the revised method results in a higher Effective Storage volume for mid-range flows.

Figure 3. Effective storage's as calculated by the old and new methods

Once the Effective Storage has been determined, it is distributed between sub-reaches as follow:

• At the most downstream reach, all the Effective Storage is upstream of the downstream face
• Calculate the Effective Storage upstream of the upstream face using;
• Effective Storage Upstream = Weir Storage Upstream – Weir Storage Upstream of the Effective Extent – Flow Related Storage between upstream end of sub-reach and the Effective Extent
 Equation 10
• Sub-reach Storage = Flow Related Sub -reach Storage + Effective D/S – Effective U/S
• Sub-reach outflow = Initial Outflow – Effective Storage Upstream
• Sub-reach Inflow = Initial Inflow – Effective Storage Downstream
• Proceed upstream to the extent of the Effective Storage to distribute the Effective Storage across all sub-reach sections

The revised method will result in the weir storage being held closer to the weir as demonstrated figure 4.

Figure 4. Change in distribution of weir storage between sub-reaches

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