Transcript Lecture 5 – Ch 4a
Review of Flood Routing
Philip B. Bedient Rice University
Lake Travis and Mansfield Dam
Lake Travis
LAKE LIVINGSTON
LAKE CONROE
ADDICKS/BARKER RESERVOIRS
Storage Reservoirs - The Woodlands
Detention Ponds
These ponds store and treat urban runoff and also provide flood control for the overall development. Ponds constructed as amenities for the golf course and other community centers that were built up around them.
DETENTION POND, AUSTIN, TX
LAKE CONROE WEIR
Comparisons: River vs. Reservoir Routing
Level pool reservoir River Reach
Max Storage Reservoir Routing •
Reservoir acts to store water and release through later. control structure
• Inflow hydrograph • Outflow hydrograph • S - Q Relationship • Outflow peaks are
reduced
• Outflow timing is delayed
Inflow and Outflow
I
Q
dS dt
Numerical Equivalent
I 1 Assume I 1 = Q 1 initially + I 2 – Q 1 + Q 2
=
2 2
S 2 – S 1
D
t
Numerical Progression
1.
2
.
3
.
I 1 + I 2 – Q 1 + Q 2
=
2 2
S 2 – S 1
D
t I 2 + I 3
2 2
– Q 2 + Q 3
2
I 3 + I 4 – Q 3 + Q 4
2
S 3 – S 2
D
t S 4 – S 3
D
t
DAY 1 DAY 2 DAY 3
Determining Storage
• •
Evaluate surface area at several different depths Use available topographic maps or GIS based DEM sources (digital elevation map)
•
Storage and area vary directly with depth of pond
Elev Volume Dam
Determining Outflow
• Evaluate area & storage at several different depths • Outflow Q can be computed as function of depth for Pipes Manning’s Eqn Orifices - Orifice Eqn Weirs or combination outflow structures - Weir Eqn Weir Flow Orifice/pipe
Determining Outflow
Q
CA
2
gH
for orifice flow
Q
CLH
3/2
for weir flow
Weir H Orifice H measured above Center of the orifice/pipe
Typical Storage -Outflow
•
Plot of Storage in acre-ft vs. Outflow in cfs
•
Storage is largely a function of topography
•
Outflows can be computed as function of elevation for either pipes or weirs
Pipe/Weir S Pipe Q
Reservoir Routing
I
1
I
2 2
S
1
dt
Q
1 2
S
2
dt
Q
2
1. LHS of Eqn is known 2. Know S as fcn of Q 3. Solve Eqn for RHS 4. Solve for Q 2 from S 2
Repeat each time step
Example Reservoir Routing --------- Storage Indication
Storage Indication Method
Note that outlet consists of weir and orifice.
Weir crest at h = 5.0 ft Orifice at h = 0 ft Area (6000 to 17,416 ft 2 ) Volume ranges from 6772 to 84006 ft 3 STEPS Storage - Indication Develop Q (orifice) vs h Develop Q (weir) vs h Develop A and Vol vs h
2S/dt + Q vs Q
where Q is sum of weir and orifice flow rates.
Storage Indication Curve
• • •
Relates
Q
and storage indication, (
2S / dt + Q)
Developed from topography and outlet data Pipe flow + weir flow combine to produce Q (out)
Only Pipe Flow Weir Flow Begins
Storage Indication Inputs
height h - ft
0 1 2 3 4 5 7
Area 10 2 ft
6 7.5
9.2
11.0
13.0
15.1
17.4
Cum Vol 10 3 ft
0 6.8
15.1
25.3
37.4
51.5
84.0
Q total cfs
0 13 18 22 26 29 159
2S/dt +Q n cfs
0 35 69 106 150 200 473 Storage-Indication
Storage Indication Tabulation
Time 0 10 20 30 40 50 60 I n 0 20 40 60 50 40 30 I n + I n+1 0 20 60 100 110 90 70 (2S/dt - Q) n 0 0 5.6
30.4
82.4
136.3
145.5
(2S/dt +Q) n+1 0 20 65.6
130.4
192.4
226.3
215.5
Q n+1 0 7.2
17.6
24.0
28.1
40.4
35.5
Time 2 Note that 20 - 2(7.2) = 5.6 and is repeated for each one
S-I Routing Results
I > Q Q > I See Excel Spreadsheet on the course web site
S-I Routing Results
I > Q Q > I Increased S
RIVER FLOOD ROUTING
CALIFORNIA FLASH FLOOD
River Routing
Manning’s Eqn
River Reaches
River Rating Curves
•
Inflow and outflow are complex
•
Wedge and prism storage occurs
•
Peak flow Q p greater on rise limb than on the falling limb
•
Peak storage occurs later than Q p
Wedge and Prism Storage
• • •
Positive wedge
I > Q
Maximum S when
I = Q
Negative wedge
I < Q
Actual Looped Rating Curves
Muskingum Method -
1938
• Continuity Equation
I - Q = dS / dt
• Storage Eqn
S = K {x I + (1-x)Q}
• Parameters are
x
= weighting Coeff K = travel time or time between peaks x = ranges from 0.2 to about 0.5 (pure trans) and assume that initial outflow = initial inflow
Muskingum Method -
1938
• Continuity Equation
I - Q = dS / dt
• Storage Eqn
S = K {x I + (1-x)Q}
• Combine 2 eqns using finite differences for
I, Q, S S 2 - S 1 = K [ x(I 2 - I 1 ) + (1 - x)(Q 2 - Q 1 ) ]
Solve for Q2 as fcn of all other parameters
Muskingum Equations
Where C 0 Q
2
C
0
I
2
C
1
I
1
= ( – Kx + 0.5
D
t) / D
C
2
Q
1
Where C 1 = (Kx + 0.5
D
t) / D C 2 = (K – Kx – 0.5
D
t) / D D = (K – Kx + 0.5
D
t) Repeat for Q 3 , Q 4 , Q 5 and so on.
Muskingum River X
Select X from most linear plot
Obtain K from line slope
Manning’s Equation
Manning’s Equation used to estimate flow rates
Q p = 1.49 A (R 2/3) S 1/2 n
Where
Q
p = flow rate
n
= roughness
A
= cross sect A
R = A / P S
= Bed Slope
Hydraulic Shapes
•
Circular pipe diameter D
•
Rectangular culvert
•
Trapezoidal channel
•
Triangular channel