# Weir formula (Water Overlay)

Flow across weirs is calculated differently for free flow and submerged flow.

The height of the water at each end of the weir, relative to the weir, is calculated:

${\displaystyle h_{s}=max(0,max(w_{l},w_{r})-z_{w})}$
${\displaystyle h_{d}=max(0,min(w_{l},w_{r})-z_{w})}$

Based on the relative water heights, the weir is judged to have either a submerged flow or a free flow, based on the following ratio:

${\displaystyle h_{ratio}=h_{d}/h_{s}}$
${\displaystyle C=min(C_{submerged},C_{free})}$ if ${\displaystyle h_{ratio}>0.5}$
${\displaystyle C=C_{free}otherwise}$

${\displaystyle C_{free}=f_{w,d}\cdot c_{w}\cdot w_{w}\cdot (h_{s}-h_{d})^{3/2}}$

For submerged flow, the following calculation is used:

${\displaystyle C_{submerged}=U_{loss}\cdot A\cdot {\sqrt {2\cdot g\cdot (h_{s}-h_{d})}}}$

Finally the actual amount of water flow is calculated:

${\displaystyle \Delta w=\Delta t\cdot C/\Delta x}$

Where:

• ${\displaystyle h_{s}}$ = The height of the water column relative to the top of the weir, on the side with the highest water level.
• ${\displaystyle h_{d}}$ = The height of the water column relative to the top of the weir, on the side with the lowest water level.
• ${\displaystyle w_{l}}$ = The water level on the left side of the weir, relative to datum.
• ${\displaystyle w_{r}}$ = The water level on the right side of the weir, relative to datum.
• ${\displaystyle z_{w}}$ = The WEIR_HEIGHT of the weir.
• ${\displaystyle f_{w,d}}$ = Dutch weir factor, set to 1.7.
• ${\displaystyle c_{w}}$ = The WEIR_COEFFICIENT of the weir.
• ${\displaystyle w_{w}}$ = The WEIR_WIDTH of the weir.
• ${\displaystyle C}$ = The potential rate of water flow across the weir.
• ${\displaystyle h_{ratio}}$ = The ratio of water heights on either side of the culvert.
• ${\displaystyle C_{free}}$ = The potential rate of water flow across the weir, based on a free flow calculation.
• ${\displaystyle C_{submerged}}$ = The potential rate of water flow across the weir, based on a submerged calculation.
• ${\displaystyle U_{loss}}$ = Loss coefficient for submerged weirs, set to 0.9.
• ${\displaystyle A}$ = Flow area, based on the highest water column height relative to the top of the weir, and WEIR_WIDTH of the weir.
• ${\displaystyle g}$ = Acceleration factor of GRAVITY, defined for the Water Overlay.
• ${\displaystyle \Delta w}$ = The water flow which takes place.
• ${\displaystyle \Delta t}$ = Computational timestep.
• ${\displaystyle \Delta x}$ = Cell size.

## Related

The following topics are related to this formula.

Structures
Weir
Models
Surface model