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Flood Risk Management Measures Deployed on River Networks Clint - - PowerPoint PPT Presentation
Flood Risk Management Measures Deployed on River Networks Clint - - PowerPoint PPT Presentation
Flood Risk Management Measures Deployed on River Networks Clint Wong, Attila Kovacs, Kris Kiradjiev, Raquel Gonzalez, Alissa Kamilova, Federico Danieli, Teague Johnstone, Ian Hewitt, Graham Sander, Barry Hankin, Ann Kretzschmar,
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Background
◮ Small-scale runoff
attenuation features (RAFs)
◮ Holds back flow in a river
basin
◮ Mitigate flood risks
Figure: Leaky dam
Questions:
◮ How to model flow in a river basin using networks? ◮ How to model its interaction with RAFs? ◮ How effective are they? What happens if they fail?
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Model
◮ Network model (rooted tree) for global behaviour
◮ Vertices (V) - channel ◮ Edges (E) - flow between channels ◮ For each channel, we consider physical properties such
as dimensions, slope, and roughness
◮ Dams: one for each vertex
with prescribed permeability
◮ Flow: lumped description
for each channel
◮ Consider inflow from
natural phenomena (e.g. storms)
◮ Model for water
retention due to the dam
- Figure: Network model for the
basin
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Governing equation
Considering a 1D network: ℓi dAi dt = Qi−1 − Qi + qi, i = 1, . . . , N where ℓi : Channel length Ai : Average cross-sectional area Qi : Flux from upstream qi : In-flow to each segment from rain / runoff from the surrounding land.
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Expression for the cross-sectional area
Figure: Diagram of the dam considered in our model
For a rectangular cross-section, A = wh 0 ≤ h ≤ b, wb + w(h−b)2
2Sℓ
h > b. where h represents the water depth, and w and S are the width and slope of a particular channel.
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Expression for the flux
◮ Before the water reaches the bottom of the dam, the flux
is governed by Manning’s law.
◮ In the other cases, we use Bernouilli’s theorem and
include a leakage and an overflow term.
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Expression for the flux
◮ Before the water reaches the bottom of the dam, the flux
is governed by Manning’s law.
◮ In the other cases, we use Bernouilli’s theorem and
include a leakage and an overflow term. Q = w 5/3h5/3S1/2 (w + 2h)2/3n, 0 ≤ h < b, w√2g
- b
h (h+b)1/2 + k(h − b)h1/2
, b ≤ h < H, w√2g
- bh1/2 + k(H − b)h1/2 + 2
3(h − H)3/2
, h ≥ H, where n represents the roughness, k is the permeability and g is gravity.
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Trapezoidal Generalisation b H I II III IV IV
Q = w
- 2g
- bh1/2 + k(H − b)h1/2 + 2
3(h − H)3/2 + 8 15 m w (h − H)5/2
- .
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2D Network
atchment basin Town River
ℓi dAi dt =
N
- j=1
ajiQj − Qi + qi, i = 1, . . . , N
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Analysis of the dam
btrunk 0.5 1 Maximum height 0.5 1 1.5 qmax = 0.02 btrunk 0.5 1 Maximum height 1.5 1.6 1.7 1.8 1.9 2 qmax = 0.05 btrunk 0.5 1 Maximum height 6 6.2 6.4 6.6 6.8 7 qmax = 0.5 bbranch 0.5 1 Maximum height 0.5 1 1.5 qmax = 0.02 bbranch 0.5 1 Maximum height 6 6.2 6.4 6.6 6.8 7 qmax = 0.5
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Numerical examples - 1D
- 5
5 10 15
Discharge Q [m3/s]
10 20
No dams Time t [h]
- 5
5 10 15
Depth h [m]
1 2 3
1 2 3 4 5
- 5
5 10 15
Discharge Q [m3/s]
10 20
5 dams Time t [h]
- 5
5 10 15
Depth h [m]
1 2 3
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Numerical examples - herringbone
- 5
5 10 15
Discharge Q [m3/s]
10 20
Dams on trunk
Max Q = 15.1
Time t [h]
- 5
5 10 15
Depth h [m]
1 2 3
- 5
5 10 15
Discharge Q [m3/s]
10 20
Dams on branches
Max Q = 14.78
Time t [h]
- 5
5 10 15
Depth h [m]
1 2 3
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Failures
◮ Dam collapse ◮ Cascade failure ◮ Scouring
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Cascade failure
(Video)
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Scouring
We can consider the Meyer-Peter-M¨ uller model for the erosion happening below the dam due to a flow with velocity u: dD dt = λ
- 1 −
D Dmax
- (τ−τcrit)3/2
where D : Scour hole depth Dmax : Maximum depth that can be eroded τ : Bed shear stress ∝ u2 τcrit : Critical stress value above which the bed can be eroded λ : Erosion rate
- u
h
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Optimisation
We are interested in:
◮ Avoiding a flood downstream ◮ The best dam design ◮ Minimise the cost
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Optimisation
Cost function: min
H,b N
- i=1
ci (Hi − bi) The solution of the model satisfies the constraint hN ≤ hflood
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Optimal Solution
1 2 3 4 5
Channel section
0.5 1 1.5 2 2.5 3 3.5 4
Height
Water levels Dams
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Alternative approach based on transit times
◮ The length, slope and roughness associated with a
segment, define a transit time, ti, which depends on the volume of water, V , flowing through it.
◮ Introduce rules on the collision of water parcels depending
- n the presence or absence of dams.
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Alternative approach based on transit times
◮ The length, slope and roughness associated with a
segment, define a transit time, ti, which depends on the volume of water, V , flowing through it.
◮ Introduce rules on the collision of water parcels depending
- n the presence or absence of dams.
◮ Optimisation problem: Find where to put m leaky dams
in the network to minimise ∞ (Vr(t) − ¯ Vr)+ dt. where Vr(t) is the volume of water passing through the root and ¯ Vr is the largest allowable volume that can be stored.
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Conclusion
◮ We have derived a lumped network model for the
description of water flow through channels with RAFs, and presented an expression for flux and stream cross-sectional area.
◮ Numerical experiments showed the effect of dams in
delaying peaks.
◮ The model has lots of scope for investigating potential
failure mechanisms and for optimising dam design.
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Future work
◮ Explore this scope. ◮ Extension of the model to a real-life network. ◮ Couple to a 2D model of the overland flow. ◮ Optimisation algorithm generalised for random rainfall