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, Giuseppe Formetta, Michael Grinfeld, Sam Pegler, Sheen Cabaneros JBA Trust April 2017 1/21

Overview Background Model Numerical examples Failures Optimisation 2/21

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? 3/21

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 Figure: Network model for the natural phenomena (e.g. storms) basin ◮ Model for water retention due to the dam 4/21

Governing equation Considering a 1D network: d A i ℓ i d t = Q i − 1 − Q i + q i , i = 1 , . . . , N where ℓ i : Channel length A i : Average cross-sectional area Q i : Flux from upstream q i : In-flow to each segment from rain / runoff from the surrounding land. 5/21

Expression for the cross-sectional area Figure: Diagram of the dam considered in our model For a rectangular cross-section, � wh 0 ≤ h ≤ b , A = wb + w ( h − b ) 2 h > b . 2 S ℓ where h represents the water depth, and w and S are the width and slope of a particular channel. 6/21

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. 7/21

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.  w 5 / 3 h 5 / 3 S 1 / 2 ( w + 2 h ) 2 / 3 n , 0 ≤ h < b ,         Q = w √ 2 g � ( h + b ) 1 / 2 + k ( h − b ) h 1 / 2 � h b b ≤ h < H , ,      w √ 2 g  bh 1 / 2 + k ( H − b ) h 1 / 2 + 2  � 3 ( h − H ) 3 / 2 � , h ≥ H ,  where n represents the roughness, k is the permeability and g is gravity. 7/21

Trapezoidal Generalisation IV� IV� III� II� H b I� � bh 1 / 2 + k ( H − b ) h 1 / 2 + 2 3( h − H ) 3 / 2 + 8 m � � w ( h − H ) 5 / 2 Q = w 2 g . 15 8/21

2D Network �atchment basin River Town N d A i � ℓ i d t = a ji Q j − Q i + q i , i = 1 , . . . , N j =1 9/21

Analysis of the dam q max = 0.02 q max = 0.05 q max = 0.5 1.5 2 7 1.9 6.8 Maximum height Maximum height Maximum height 1 1.8 6.6 1.7 6.4 0.5 1.6 6.2 0 1.5 6 0 0.5 1 0 0.5 1 0 0.5 1 b trunk b trunk b trunk q max = 0.02 q max = 0.5 1.5 7 6.8 Maximum height Maximum height 1 6.6 6.4 0.5 6.2 0 6 0 0.5 1 0 0.5 1 b branch b branch 10/21

Numerical examples - 1D Discharge Q [m 3 /s] Discharge Q [m 3 /s] No dams 5 dams 20 20 10 10 0 0 -5 0 5 10 15 -5 0 5 10 15 3 3 Depth h [m] Depth h [m] 1 2 2 2 3 4 5 1 1 0 0 -5 0 5 10 15 -5 0 5 10 15 Time t [h] Time t [h] 11/21

Numerical examples - herringbone Discharge Q [m 3 /s] Discharge Q [m 3 /s] Dams on trunk Dams on branches 20 20 Max Q = 15.1 Max Q = 14.78 10 10 0 0 -5 0 5 10 15 -5 0 5 10 15 3 3 Depth h [m] Depth h [m] 2 2 1 1 0 0 -5 0 5 10 15 -5 0 5 10 15 Time t [h] Time t [h] 12/21

Failures ◮ Dam collapse ◮ Cascade failure ◮ Scouring 13/21

Cascade failure (Video) 14/21

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 � D � ( τ − τ crit ) 3 / 2 dt = λ 1 − D max where D : Scour hole depth h D max : Maximum depth that can be eroded u τ : Bed shear stress ∝ u 2 � τ crit : Critical stress value above which the bed can be eroded λ : Erosion rate 15/21

Optimisation We are interested in: ◮ Avoiding a flood downstream ◮ The best dam design ◮ Minimise the cost 16/21

Optimisation Cost function: N � min c i ( H i − b i ) H , b i =1 The solution of the model satisfies the constraint h N ≤ h flood 17/21

Optimal Solution 4 Water levels Dams 3.5 3 2.5 Height 2 1.5 1 0.5 0 0 1 2 3 4 5 Channel section 18/21

Alternative approach based on transit times ◮ The length, slope and roughness associated with a segment, define a transit time, t i , which depends on the volume of water, V , flowing through it. ◮ Introduce rules on the collision of water parcels depending on the presence or absence of dams. 19/21

Alternative approach based on transit times ◮ The length, slope and roughness associated with a segment, define a transit time, t i , which depends on the volume of water, V , flowing through it. ◮ Introduce rules on the collision of water parcels depending on the presence or absence of dams. ◮ Optimisation problem: Find where to put m leaky dams in the network to minimise � ∞ ( V r ( t ) − ¯ V r ) + dt . 0 where V r ( t ) is the volume of water passing through the root and ¯ V r is the largest allowable volume that can be stored. 19/21

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. 20/21

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 distribution and time evolution. 21/21