Breach flow formula (Water Overlay)

Flow through breaches is calculated based on the weir formula, including the consideration between free flow and submerged flow situations. The weir formula is supplied with the breach width (as stand-in for the weir width), which is always calculated with the breach growth formula before calculating the weir flow. The downstream water level can optionally be measured with an additionally configured Breach Level Area, or an automatically placed

For the entry area, the area where the water originates from, the following also applies:

• The water level used is defined by either an input area or the external water level.
• In case of an external area, the surface height used is defined by external surface level. This height limits how far the water level can be lowered.
• Optionally a second value can be supplied for the external surface level. In that case, the second value represents the area of the bottom.
• In case of an input area, the average water level is based on the water level on the individual grid cells of the input area.

Each timestep, the external water level is changed based on the amount of water flowing in or out.

$\Delta V=min(h_{e}\cdot A_{s},Q_{w}\cdot \Delta t)$ In case $A_{s}==A_{b}$: $w_{e,t+1}=w_{e,t}-{\frac {\Delta V}{A_{b}}}$ Otherwise, the external water area is represented as a trapezoid prism and the new water level has to be calculated differently:

$l={\sqrt {A_{s}}}$

$w_{s}=l$

$w_{b}={\frac {A_{b}}{l}}$

$a={\frac {w_{s}-w_{b}}{4\cdot h_{0}}}:<math>A_{t,t}=2\cdot (a\cdot h_{t}^{2}+0.5\cdot w_{b})$

$\Delta A={\frac {\Delta V}{l}}$

$c={\frac {-\Delta A*A_{t,t}}{2}}$

$D=({\frac {w_{b}}{2}})^{2}-4\cdot a\cdot c$

if $D<0$: $w_{e,t+1}=w_{e,t}+{\frac {\Delta V}{A_{b}}}$

else: $w_{-}={\frac {-{\frac {w_{b}}{2}}-{\sqrt {D}}}{2\cdot a}}+s_{b}$

$w_{+}={\frac {-{\frac {w_{b}}{2}}+{\sqrt {D}}}{2\cdot a}}+s_{b}$

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle w_{{e,t+1} = \begin{cases} h_{+} & \text {if} \Delta V < 0 \text{and} w_{+} \lte w_{e,t} h_{-} & \text {if} \Delta V < 0 \text{and} w_{+} \gt w_{e,t} h_{+} & \text {if} \Delta V >= 0 \text{and} w_{-} \lt w_{e,t} h_{-} & \text {if} \Delta V >= 0 \text{and} w_{-} \gte w_{e,t} \end{cases} }

Where:

$Q_{w}$ = The calculated water flow which takes place, based on the weir formula.

$Q$ = The limited water flow which takes place, based on the remaining water.

$w_{e,t}$ = The water level of the entry area at time $t$; In case of an external area: EXTERNAL_WATER_LEVEL of the breach.

$A_{s}$ = The size of in the entry area; In case of an external area: EXTERNAL_AREA of the breach.

$A_{b}$ = The (optional) size of the bottom of the entry area; In case of an external area: the second value provided for the EXTERNAL_AREA of the breach. In case only one value is provided, $A_{b}=A_{s}$

Notes

• The water level w_e cannot become lower than the surface height s_e, defined by EXTERNAL_SURFACE_LEVEL in case of an external entry area. In case of an input area within the project, the surface height is determined per grid cell.

Related

The following topics are related to this formula.

Features

Breach

Formulas

Weir formula

Breach growth formula

Models

Surface model

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Surface • Ground • Rain • Evaporation • Infiltration • Elevation • Breach • Sewer • Storage • Tracer flow • Border • Order

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Flow formulas

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