Culvert formula (Water Overlay)

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

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

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

The radius of the culvert:

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

The height of the water column at either end of the culvert, relative to the culvert, is calculated:

w_{l}=\max(0,w_{l}-B_{c})

w_{r}=\max(0,w_{r}-B_{c})

For circular culverts, the flow depth d is:

d=\max(0,\min(D,\max(w_{l},w_{r})-B_{c})))

For rectangular culverts, the flow depth d is:

d=\max(0,\min(H,\max(w_{l},w_{r})-B_{c})))

The loss coefficient for the culvert is calculated:

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

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

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

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

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

A=d*D

Failed to parse (unknown function "\begin{cases}"): {\displaystyle P_w = \begin{cases} D+2 \cdot h, & \text{if }n\text{ d \ll H} \\ 2 \cdot D + 2 \cdot H, & \text{if }n\text{ d \ge H} \end{cases} }

The potential flow through the culvert is then calculated:

C=U\cdot K\cdot {\sqrt {2g\cdot \Vert w_{l}-w_{r}\Vert }}

Finally the actual amount of water flow is calculated:

\Delta f={\frac {\Delta t\cdot C}{\Delta x}}

Where:

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