# Culvert formula (Water Overlay)

Flow through culverts is based on an open channel flow calculation.

The actual datum height of the culvert is at least the height of the terrain on either end of the culvert and the provided threshold height:

${\displaystyle B_{c}=\max(T_{c},B_{l},B_{r})}$

For circular culverts, the flow height ${\displaystyle h_{f}}$ is:

${\displaystyle h_{f}=\max(0,\min(D,\max(w_{l},w_{r})-B_{c})))}$

For rectangular culverts, the flow depth h is:

${\displaystyle h_{f}=\max(0,\min(H,\max(w_{l},w_{r})-B_{c})))}$

The loss coefficient for the culvert is calculated:

${\displaystyle U={\sqrt {\frac {1}{1+{\frac {2g\cdot n^{2}\cdot length}{R_{h}^{\frac {4}{3}}}}}}}}$

The hydraulic radius ${\displaystyle R_{h}}$ is calculated as:

${\displaystyle R_{h}={\frac {A}{P_{w}}}}$

For circular culverts, the flow area ${\displaystyle A}$ and the wetted perimeter ${\displaystyle P_{w}}$ is calculated using the formula's in the image below.

Where the radius of the culvert:

${\displaystyle r={\frac {D}{2}}}$

For rectangular culverts, the flow area and wetted perimeter is calculated as followed:

${\displaystyle A=h_{f}\cdot D}$

${\displaystyle P_{w}={\begin{cases}D+2h_{f},&{\text{if }}h_{f}

The potential flow through the culvert is then calculated:

${\displaystyle Q=U\cdot K\cdot {\sqrt {2g\cdot \Vert w_{l}-w_{r}\Vert }}}$

Finally the actual amount of water flow is calculated:

${\displaystyle \Delta f={\frac {\Delta t\cdot Q}{\Delta x}}}$

Where:

• ${\displaystyle D}$ = The CULVERT_DIAMETER attribute of the culvert, representing either the inside diameter of a circular culvert or the inside width of a rectangular culvert.
• ${\displaystyle H}$ = The CULVERT_RECTANGULAR_HEIGHT attribute of the culvert, representing the inside height of a rectangular culvert.
• ${\displaystyle h_{f}}$ = The (flow) height of the water inside the culvert.
• ${\displaystyle T_{c}}$ = The CULVERT_THRESHOLD attribute of the culvert.
• ${\displaystyle B_{c}}$ = The datum height of the base of the culvert.
• ${\displaystyle w_{l}}$ = The water level on the left end of the culvert, relative to datum.
• ${\displaystyle w_{r}}$ = The water level on the right end of the culvert, relative to datum.
• ${\displaystyle R_{h}}$ = The hydraulic radius in the culvert[1].
• ${\displaystyle R_{w}}$ = The wetted perimeter.
• ${\displaystyle A}$ = The flow area.
• ${\displaystyle K}$ = Circular flow area, based on the height of the water in the (circular) culvert.
• ${\displaystyle g}$ = the acceleration due to gravity, set to 9.80665.
• ${\displaystyle L}$ = The length of the culvert, calculated as the distance between the culvert's endpoints.
• ${\displaystyle U}$ = Loss coefficient for culverts.
• ${\displaystyle n}$ = The CULVERT_N attribute of the culvert.
• ${\displaystyle Q}$ = The potential rate of water flow through the culvert in ${\displaystyle m^{3}s^{-1}}$.
• ${\displaystyle \Delta f}$ = The water flow which takes place.
• ${\displaystyle \Delta t}$ = Computational timestep in seconds.
• ${\displaystyle \Delta x}$ = Cell size in meters.

## Related

The following topics are related to this formula.

Structures
Culvert
Models
Surface model