Breach growth formula (Water Overlay): Difference between revisions

Revision as of 08:49, 7 September 2022

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

TODO: THIS SECTION IS OUTDATED, NEEDS UPDATE!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))*(pow(g*H,1.5)/pow(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).

f_{2}

= Constant, set to 0.04.

g

= Gravity constant, defined for the Water Overlay.

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

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

↑ 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)

@@ Line 12: / Line 12: @@
 : <math>\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)</math>
+: <math>\Delta W_{b,t} = ((f_1 * f_2) / ln(10)) * (pow(g * H, 1.5) / pow(cs_b, 2)) * (1 / (1 + (f_2 * g * t) / (3600 * cs_b)))</math>
 The current breach width is then equal to the last calculated breach width, plus the calculated breach width increment.
@@ Line 24: / Line 26: @@
 : <math>w_{e,t}</math> = Outer water level at input area (or external) at time t.
 : <math>\Delta h_{t}</math> = The difference between the height of the water columns on either side of the breach at time t.
-: <math>f_m</math> = Material factor, set to 1.3 (average for sand and clay levees).
+: <math>f_1</math> = Material factor, set to 1.3 (average for sand and clay levees).
+: <math>f_2</math> = Constant, set to 0.04.
 : <math>g</math> = Gravity constant, defined for the Water Overlay.
 * <math>cs_{b}</math> = The critical [[Breach speed (Water Overlay)|BREACH_SPEED]] of the breach (e.g. 0.2 for sand and 0.5 for clay).