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Channel Routing

Channel Routing. Simulate the movement of water through a channel Used to predict the magnitudes, volumes, and temporal patterns of the flow (often a flood wave) as it translates down a channel. 2 types of routing : hydrologic and hydraulic.

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Channel Routing

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  1. Channel Routing • Simulate the movement of water through a channel • Used to predict the magnitudes, volumes, and temporal patterns of the flow (often a flood wave) as it translates down a channel. • 2 types of routing : hydrologic and hydraulic. • both of these methods use some form of the continuity equation. Continuity equation Hydrologic Routing Hydraulic Routing Momentum Equation

  2. Continuity Equation Continuity equation Hydrologic Routing Hydraulic Routing Momentum Equation • The change in storage (dS) equals the difference between inflow (I) and outflow (O) or : • For open channel flow, the continuity equation is also often written as : A = the cross-sectional area, Q = channel flow, and q = lateral inflow

  3. Hydrologic Routing Continuity equation Hydrologic Routing Hydraulic Routing Momentum Equation • Methods combine the continuity equation with some relationship between storage, outflow, and possibly inflow. • These relationships are usually assumed, empirical, or analytical in nature. • An of example of such a relationship might be a stage-discharge relationship.

  4. Use of Manning Equation Continuity equation Hydrologic Routing Hydraulic Routing Momentum Equation • Stage is also related to the outflow via a relationship such as Manning's equation

  5. Hydraulic Routing Continuity equation Hydrologic Routing Hydraulic Routing Momentum Equation • Hydraulic routing methods combine the continuity equation with some more physical relationship describing the actual physics of the movement of the water. • The momentum equation is the common relationship employed. • In hydraulic routing analysis, it is intended that the dynamics of the water or flood wave movement be more accurately described

  6. Momentum Equation Continuity equation Hydrologic Routing Hydraulic Routing Momentum Equation • Expressed by considering the external forces acting on a control section of water as it moves down a channel • Henderson (1966) expressed the momentum equation as :

  7. Combinations of Equations Continuity equation Hydrologic Routing Hydraulic Routing Momentum Equation • Simplified Versions : Unsteady -Nonuniform Steady - Nonuniform Diffusion or noninertial Kinematic Sf = So

  8. Routing Methods • Modified Puls • Kinematic Wave • Muskingum • Muskingum-Cunge • Dynamic Modified Puls Kinematic Wave Muskingum Muskingum-Cunge Dynamic Modeling Notes

  9. Modified Puls • The modified puls routing method is probably most often applied to reservoir routing • The method may also be applied to river routing for certain channel situations. • The modified puls method is also referred to as the storage-indication method. • The heart of the modified puls equation is found by considering the finite difference form of the continuity equation. Modified Puls Kinematic Wave Muskingum Muskingum-Cunge Dynamic Modeling Notes

  10. Modified Puls Modified Puls Kinematic Wave Muskingum Muskingum-Cunge Dynamic Modeling Notes Continuity Equation Rewritten • The solution to the modified puls method is accomplished by developing a graph (or table) of O -vs- [2S/Δt + O]. In order to do this, a stage-discharge-storage relationship must be known, assumed, or derived.

  11. Modified Puls Example • Given the following hydrograph and the 2S/Dt + O curve, find the outflow hydrograph for the reservoir assuming it to be completely full at the beginning of the storm. • The following hydrograph is given:

  12. Modified Puls Example • The following 2S/Dt + O curve is also given:

  13. Modified Puls Example • A table may be created as follows:

  14. Modified Puls Example • Next, using the hydrograph and interpolation, insert the Inflow (discharge) values. • For example at 1 hour, the inflow is 30 cfs.

  15. Modified Puls Example • The next step is to add the inflow to the inflow in the next time step. • For the first blank the inflow at 0 is added to the inflow at 1 hour to obtain a value of 30.

  16. Modified Puls Example • This is then repeated for the rest of the values in the column.

  17. Modified Puls Example • The 2Sn/Dt + On+1 column can then be calculated using the following equation: Note that 2Sn/Dt - On and On+1 are set to zero. 30 + 0 = 2Sn/Dt + On+1

  18. Modified Puls Example • Then using the curve provided outflow can be determined. • In this case, since 2Sn/Dt + On+1 = 30, outflow = 5 based on the graph provided.

  19. Modified Puls Example • To obtain the final column, 2Sn/Dt - On, two times the outflow is subtracted from 2Sn/Dt + On+1. • In this example 30 - 2*5 = 20

  20. Modified Puls Example • The same steps are repeated for the next line. • First 90 + 20 = 110. • From the graph, 110 equals an outflow value of 18. • Finally 110 - 2*18 = 74

  21. Modified Puls Example • This process can then be repeated for the rest of the columns. • Now a list of the outflow values have been calculated and the problem is complete.

  22. MuskingumMethod Modified Puls Kinematic Wave Muskingum Muskingum-Cunge Dynamic Modeling Notes Sp = K O PrismStorage Sw = K(I - O)X WedgeStorage S = K[XI + (1-X)O] Combined

  23. Muskingum,cont... Substitute storage equation, S into the “S” in the continuity equation yields : Modified Puls Kinematic Wave Muskingum Muskingum-Cunge Dynamic Modeling Notes S = K[XI + (1-X)O] O2 = C0 I2 + C1 I1 + C2 O1

  24. Muskingum Notes : • The method assumes a single stage-discharge relationship. • In other words, for any given discharge, Q, there can be only one stage height. • This assumption may not be entirely valid for certain flow situations. • For instance, the friction slope on the rising side of a hydrograph for a given flow, Q, may be quite different than for the recession side of the hydrograph for the same given flow, Q. • This causes an effect known as hysteresis, which can introduce errors into the storage assumptions of this method. Modified Puls Kinematic Wave Muskingum Muskingum-Cunge Dynamic Modeling Notes

  25. Estimating K • K is estimated to be the travel time through the reach. • This may pose somewhat of a difficulty, as the travel time will obviously change with flow. • The question may arise as to whether the travel time should be estimated using the average flow, the peak flow, or some other flow. • The travel time may be estimated using the kinematic travel time or a travel time based on Manning's equation. Modified Puls Kinematic Wave Muskingum Muskingum-Cunge Dynamic Modeling Notes

  26. Estimating X Modified Puls Kinematic Wave Muskingum Muskingum-Cunge Dynamic Modeling Notes • The value of X must be between 0.0 and 0.5. • The parameter X may be thought of as a weighting coefficient for inflow and outflow. • As inflow becomes less important, the value of X decreases. • The lower limit of X is 0.0 and this would be indicative of a situation where inflow, I, has little or no effect on the storage. • A reservoir is an example of this situation and it should be noted that attenuation would be the dominant process compared to translation. • Values of X = 0.2 to 0.3 are the most common for natural streams; however, values of 0.4 to 0.5 may be calibrated for streams with little or no flood plains or storage effects. • A value of X = 0.5 would represent equal weighting between inflow and outflow and would produce translation with little or no attenuation.

  27. Did you know? Lag and K is a special case of Muskingum -> X=0!

  28. More Notes - Muskingum Modified Puls Kinematic Wave Muskingum Muskingum-Cunge Dynamic Modeling Notes • The Handbook of Hydrology (Maidment, 1992) includes additional cautions or limitations in the Muskingum method. • The method may produce negative flows in the initial portion of the hydrograph. • Additionally, it is recommended that the method be limited to moderate to slow rising hydrographs being routed through mild to steep sloping channels. • The method is not applicable to steeply rising hydrographs such as dam breaks. • Finally, this method also neglects variable backwater effects such as downstream dams, constrictions, bridges, and tidal influences.

  29. Muskingum Example Problem • A portion of the inflow hydrograph to a reach of channel is given below. If the travel time is K=1 unit and the weighting factor is X=0.30, then find the outflow from the reach for the period shown below:

  30. Muskingum Example Problem • The first step is to determine the coefficients in this problem. • The calculations for each of the coefficients is given below: C0= - ((1*0.30) - (0.5*1)) / ((1-(1*0.30) + (0.5*1)) = 0.167 C1= ((1*0.30) + (0.5*1)) / ((1-(1*0.30) + (0.5*1)) = 0.667

  31. Muskingum Example Problem C2= (1- (1*0.30) - (0.5*1)) / ((1-(1*0.30) + (0.5*1)) = 0.167 • Therefore the coefficients in this problem are: • C0 = 0.167 • C1 = 0.667 • C2 = 0.167

  32. Muskingum Example Problem • The three columns now can be calculated. • C0I2 = 0.167 * 5 = 0.835 • C1I1 = 0.667 * 3 = 2.00 • C2O1 = 0.167 * 3 = 0.501

  33. Muskingum Example Problem • Next the three columns are added to determine the outflow at time equal 1 hour. • 0.835 + 2.00 + 0.501 = 3.34

  34. Muskingum Example Problem • This can be repeated until the table is complete and the outflow at each time step is known.

  35. Muskingum Example - Tenkiller Look at R-5

  36. Reach at Outlet K = 2 hrs X = 0.25

  37. K = 2 & 4 Hrs.X = 0.25

  38. K = 2, 4, & 6 Hrs.X = 0.25

  39. Let’s Alter XK = 2 hrs. K = 2 hrs X = 0.25 K = 2 hrs X = 0.5

  40. And Again.... K = 2 hrs X = 0.5 K = 2 hrs X = 0.05

  41. Muskingum-Cunge • Muskingum-Cunge formulation is similar to the Muskingum type formulation • The Muskingum-Cunge derivation begins with the continuity equation and includes the diffusion form of the momentum equation. • These equations are combined and linearized, Modified Puls Kinematic Wave Muskingum Muskingum-Cunge Dynamic Modeling Notes

  42. Muskingum-Cunge“working equation” where : Q = discharge t = time x = distance along channel qx = lateral inflow c = wave celerity m = hydraulic diffusivity Modified Puls Kinematic Wave Muskingum Muskingum-Cunge Dynamic Modeling Notes

  43. Muskingum-Cunge, cont... • Method attempts to account for diffusion by taking into account channel and flow characteristics. • Hydraulic diffusivity is found to be : Modified Puls Kinematic Wave Muskingum Muskingum-Cunge Dynamic Modeling Notes • The Wave celerity in the x-direction is :

  44. t X Solution of Muskingum-Cunge • Solution of the Method is accomplished by discretizing the equations on an x-t plane. Modified Puls Kinematic Wave Muskingum Muskingum-Cunge Dynamic Modeling Notes

  45. Calculation of K & X Modified Puls Kinematic Wave Muskingum Muskingum-Cunge Dynamic Modeling Notes Estimation of K & X is more “physically based” and should be able to reflect the “changing” conditions - better.

  46. Muskingum-Cunge - NOTES • Muskingum-Cunge formulation is actually considered an approximate solution of the convective diffusion equation. • As such it may account for wave attenuation, but not for reverse flow and backwater effects and not for fast rising hydrographs. • Properly applied, the method is non-linear in that the flow properties and routing coefficients are re-calculated at each time and distance step • Often, an iterative 4-point scheme is used for the solution. • Care should be taken when choosing the computation interval, as the computation interval may be longer than the time it takes for the wave to travel the reach distance. • Internal computational times are used to account for the possibility of this occurring. Modified Puls Kinematic Wave Muskingum Muskingum-Cunge Dynamic Modeling Notes

  47. Muskingum-Cunge Example • The hydrograph at the upstream end of a river is given in the following table. The reach of interest is 18 km long. Using a subreach length Dx of 6 km, determine the hydrograph at the end of the reach using the Muskingum-Cunge method. Assume c = 2m/s, B = 25.3 m, So = 0.001m and no lateral flow.

  48. Muskingum-Cunge Example • First, K must be determined. • K is equal to : Is “c” a constant? • Dx = 6 km, while c = 2 m/s

  49. Muskingum-Cunge Example • The next step is to determine x. • All the variables are known, with B = 25.3 m, So = 0.001 and Dx =6000 m, and the peak Q taken from the table.

  50. Muskingum-Cunge Example • The coefficients of the Muskingum-Cunge method can now be determined. Using 7200 seconds or 2 hrs for timestep - this may need to be changed

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