Review of Flood Routing, Chapter 4

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Review of Flood Routing
Philip B. Bedient
Rice University
Lake Travis and Mansfield Dam
Lake Travis
Mansfield Dam, Hill Country of Texas
Lake Livingston
Lake Conroe
Addicks and Barker Reservoirs
Barker Reservoir Watershed
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.
River vs. Reservoir Routing
Lake Conroe Weir
Reservoir Routing
~ Reservoir acts to store
water and release through
control structure later
Max Storage = A = C
~ Inflow hydrograph
~ Outflow hydrograph
~ S - Q Relationship
~ Outflow peaks are reduced
~ Outflow timing is delayed
Inflow and Outflow
ds
I −Q =
dt
Numerical Equivalent
Assume I1 = Q1 initially
I1 + I2 – Q1 + Q2
2
2
=
S2 – S1
Δt
Numerical Progression
1.
I1 + I2 – Q1 + Q2
2
2.
I2 + I3 – Q2 + Q3
2
3.
2
2
I3 + I4 – Q3 + Q4
2
2
=
=
=
S2 – S1
Δt
S3 – S2
Δt
S4 – S3
Δt
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 Equation
~ Orifices - Orifice Equation
~ Weirs or combination outflow structures - Weir Equation
Weir Flow
Orifice/pipe
Determining Outflow
Q = CA 2gH 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
S (acre-ft)
Pipe/Weir
Pipe
Q (cfs)
Reservoir Routing
1. LHS of Equation is
known
2. Know S as function of Q
3. Solve Equation for RHS
4. Solve for Q2 from S2
5. Repeat each time step
⎛ 2S1
⎞ ⎛ 2S2
⎞
I1 + I 2 + ⎝
− Q1 ⎠ = ⎝
+ Q2 ⎠
dt
dt
Example Reservoir Routing Storage Indication
Storage Indication Method
STEPS
1. Storage - Indication
2. Develop Q (orifice) vs h
3. Develop Q (weir) vs h
4. Develop A and Vol vs h
5. 2S/dt + Q vs Q where Q is
sum of weir and orifice
flow rates.
NOTES
~ Outlet consists of weir
and orifice
~ Weir crest at h = 5.0 ft
~ Orifice at h = 0 ft
~ Area (6000 to 17,416 ft2)
~ Volume ranges from 6772
to 84006 ft3
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
Area
102 ft
Cum Vol
103 ft
Q total
cfs
2S/dt +Qn
cfs
0
6
0
0
0
1
7.5
6.8
13
35
2
9.2
15.1
18
69
3
11.0
25.3
22
106
4
13.0
37.4
26
150
5
15.1
51.5
29
200
7
17.4
84.0
159
473
Storage-Indication
Storage Indication Tabulation
Time
In
In + In+1
(2S/dt - Q)n
(2S/dt +Q)n+1
Qn+1
0
0
0
0
0
0
10
20
20
0
20
7.2
20
40
60
5.6
65.6
17.6
30
60
100
30.4
130.4
24.0
40
50
110
82.4
192.4
28.1
50
40
90
136.3
226.3
40.4
60
30
70
145.5
215.5
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
Equation
River Reaches
River Rating Curves
~ Inflow and outflow are complex
~ Wedge and prism storage
occurs
~ Peak flow Qp greater on rise
limb than on the falling limb
~ Peak storage occurs later than
Qp
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 Equation
S = K {x I + (1-x)Q}
~ Parameters are:
~ x = weighting coefficient
~ K = travel time or time between peaks
~x = ranges from 0.2 to about 0.5 (pure trans)
~ Assume that initial outflow = initial inflow
Muskingum Method - 1938
~ Continuity Equation
I - Q = dS / dt
~ Storage Equation
S = K {x I + (1-x)Q}
~Combine 2 equations using finite differences for I,
Q, S
S2 - S1 = K [x(I2 - I1) + (1 - x)(Q2 - Q1)]
~ Solve for Q2 as function of all other parameters
Muskingum Equations
Q2 = C0I2 + C1I1 + C2Q1
Where
C0 = (– Kx + 0.5Δt) / D
C1 = (Kx + 0.5Δt) / D
C2 = (K – Kx – 0.5Δt) / D
D = (K – Kx + 0.5Δt)
Repeat for Q3, Q4, Q5 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
Qp = 1.49 A (R2/3) S1/2
Where Qp = flow rate
n = roughness
A = cross sect A
R=A/P
S = Bed Slope
n
Hydraulic Shapes
~Circular pipe diameter D
~Rectangular culvert
~Trapezoidal channel
~Triangular channel
n
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