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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