Breach growth formula (Water Overlay)

Water can flow through breaches into levee protected areas. These breaches often start small and grow over time^[1].

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

This algorithm is based on the formula of Verheij-Van der Knaap, see references below.

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

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 h_{t} = abs( w_{o,t} - max(w_{i,t}, H_{b,t}))}

Using the height difference, the breach width increase (m/s) is calculated per computational timestep.

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} = ((f_1 * f_2) / ln(10)) * ((g * \Delta h_{t})^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.

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_{b,t} = W_{b,t-1} + (\Delta t / 3600) * \Delta W_{b,t}}

Where:

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_b} = The BREACH_WIDTH of the breach.

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 H_{b,t}} = The BREACH_HEIGHT 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 W_{b,t}} = The calculated breach width, initially equal to W_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_{i,t}} = Inner water level at breach area 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 w_{o,t}} = Outer water level at input area (or external) 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 h_{t}} = The difference between the height of the water columns on either side 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 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.
$\Delta t$ = Computational timestep.

Notes

The breach height H_b,t can be defined over time, by configuring the BREACH_HEIGHT as an attribute array.
The inner water level is to be measured not directly inside the breach area but a few meters away. By default 100m, this can be adjusted via the BREACH_MEASUREMENT_DISTANCE_M attribute.
For more details on breach growth, we also recommend reading this "Quickscan"-report, found at: https://edepot.wur.nl/556012.

References

↑ Verheij, H.J. ∙ Aanpassen van het bresgroeimodel in HIS-OM: Bureaustudie ∙ found at: http://resolver.tudelft.nl/uuid:aedc8109-da43-4a03-90c3-44f706037774 ∙ (last visited 2019-03-08)