Review of Flood Routing

Philip B. BedientPhilip B. Bedient

Rice UniversityRice 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 These ponds store and treat urban runoff and also provide flood control for the overall development. provide flood control for the overall development.

Ponds constructed as amenities for the golf course Ponds constructed as amenities for the golf course and other community centers that were built up and other community centers that were built up around them.around them.

DETENTION POND, AUSTIN, TX

LAKE CONROE WEIR

Comparisons:River vs. ReservoirRouting

Level pool reservoir

River Reach

Reservoir Routing

• Reservoir acts to store      water and release through      control structure later.

• Inflow hydrograph

• Outflow hydrograph

• S - Q Relationship

• Outflow peaks are reduced

• Outflow timing is delayed

Max Storage

Inflow and Outflow

€

I − Q =dSdt

Numerical EquivalentAssume I1 = Q1 initially

I1 + I2 – Q1 + Q2 S2 – S1

2 t2=

Numerical Progression

I1 + I2 – Q1 + Q2 S2 – S1

2 t2=

I2 + I3 – Q2 + Q3 S3 – S2

I3 + I4 – Q3 + Q4 S4 – S3

t

t2

22

2

DAY 1

DAY 2

DAY 3

1.

2.

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

Volume

Elev

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

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Q = CA 2gH for orifice flow

Q = CLH 3/2 for weir flow

Weir

Orifice H measured above

Center of the orifice/pipe

H

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

Q

Pipe/Weir

Pipe

Reservoir Routing

1. LHS of Eqn is known

2. Know S as fcn of Q

3. Solve Eqn for RHS

4. Solve for Q2 from S2

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I1 + I2 +2S1

dt− Q1

⎛ ⎝

⎞ ⎠ =

2S2

dt+ Q2

⎛ ⎝

⎞ ⎠

Repeat each time step

Example

Reservoir

Routing

----------

Storage

Indication

Storage Indication Method

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.

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 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)

Weir Flow Begins

Only Pipe Flow

Storage Indication Storage Indication InputsInputs

heighheightt

h - fth - ft

AreaArea

101022 ft ftCum Cum

Vol 10Vol 1033 ftft

Q totalQ total

cfscfs2S/dt 2S/dt +Q+Qnn

cfscfs

00 66 00 00 00

11 7.57.5 6.86.8 1313 3535

22 9.29.2 15.115.1 1818 6969

33 11.011.0 25.325.3 2222 106106

44 13.013.0 37.437.4 2626 150150

55 15.115.1 51.551.5 2929 200200

77 17.417.4 84.084.0 159159 473473Storage-Indication

Storage Indication Storage Indication TabulationTabulation

TimeTime IInn IInn + I + In+1n+1 (2S/dt - Q)(2S/dt - Q)nn (2S/dt (2S/dt +Q)+Q)n+1n+1

QQn+1n+1

00 00 00 00 00 00

1010 2020 2020 00 2020 7.27.2

2020 4040 6060 5.65.6 65.665.6 17.617.6

3030 6060 100100 30.430.4 130.4130.4 24.024.0

4040 5050 110110 82.482.4 192.4192.4 28.128.1

5050 40 40 9090 136.3136.3 226.3226.3 40.440.4

6060 3030 7070 145.5145.5 215.5215.5 35.535.5

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

River Reaches

Manning’s Eqn

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 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

S2 - S1 = K [x(I2 - I1) + (1 - x)(Q2 - Q1)]

Solve for Q2 as fcn of all other parameters

Muskingum Equations

Where C0 = (– Kx + 0.5t) / D

C1 = (Kx + 0.5t) / D

C2 = (K – Kx – 0.5t) / D

Where D = (K – Kx + 0.5t)

Repeat for Q3, Q4, Q5 and so on.

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Q2 = C0I2 + C1I1 + C2Q1

Muskingum River X

Obtain K from line slope

Select X from most linear plot

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

Manning’s Equation

n

• Circular pipe diameter D

• Rectangular culvert

• Trapezoidal channel

• Triangular channel

Hydraulic Shapes