Breach growth formula (Water Overlay)

Water can flow through breaches into levee protected areas. These breaches often start small and grow over time.

TODO: THIS SECTION IS OUTDATED, NEEDS UPDATE!

The water flowing through breaches can originate from an external area outside the project area or an input area within the project area.

First, the difference in height of the water on either side of the breach is calculated.

$\Delta h=abs(max(0,w_{e,t}-H_{b,t})-max(0,w_{b,t}-H_{b,t}))$

Using the height difference, the breach width increase is calculated.

$\Delta W_{b,t}=f_{m}\cdot ({\sqrt {g}}\cdot {\sqrt {\Delta h^{3}}}/cs_{b})\cdot \log _{10}(1+(0.04\cdot g/cs_{b})\cdot \Delta t/3600)$

$\Delta W_{b,t}=((f_{1}*f_{2})/ln(10))*(g*H)^{1}.5/cs_{b}^{2})*(1/(1+(f_{2}*g*t)/(3600*cs_{b})))$

The current breach width is then equal to the last calculated breach width, plus the calculated breach width increment.

$W_{b,t}=W_{b,t-1}+\Delta W_{b,t}$

Where:

$W_{b}$ = The BREACH_WIDTH of the breach.

$H_{b,t}$ = The BREACH_HEIGHT of the breach at time t.

$W_{b,t}$ = The calculated breach width, initially equal to W_b.

$w_{b,t}$ = Inner water level at breach area at time t.

$w_{e,t}$ = Outer water level at input area (or external) at time t.

$\Delta h_{t}$ = The difference between the height of the water columns on either side of the breach at time t.

$f_{1}$ = Material factor, set to 1.3 (average for sand and clay levees).

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 f_2} = Constant, set to 0.04.

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 g} = Gravity constant, defined for the Water Overlay.

• 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 cs_{b}} = The critical BREACH_SPEED of the breach (e.g. 0.2 for sand and 0.5 for clay).
• 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 \Delta W_{b,t}} = The calculated width increase of the breach at time t.
• 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 \Delta t} = Computational timestep.

Notes

• The breach height H_b,t can be defined over time, by configuring the BREACH_HEIGHT as an attribute array.

See also

For an example of the breach growth, take a look at the Demo Breach Project available in all domains.

Related

The following topics are related to this formula.

Features

Breach

Formulas

Breach flow formula

Models

Surface model

References

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