oct99_routing.pptx

NWS-COMET May 1998 Hydrometeorology Course

1RoutingSimulate the movement of water through physical components of watershed (e.g., channels)Commonly used to predict the magnitudes, volumes, and temporal patterns of flow (often a flood wave) as it moves down a channelPhysical/Hydraulic: conservation of mass and momentumConceptual/Hydrologic: some physics (continuity), but inexact representationsRegression/Empirical: black boxPhysical (Hydraulic) - continuity - momentumConceptual (Hydrologic) - continuityRegression (Empirical)Presented by Dr. Fritz FiedlerCOMET Hydromet 00-1Monday, 25 October 1999

2Continuity Equation The change in storage in a time interval (Dt) equals the difference between inflow (I) and outflow (O) or in its discrete form: The continuity equation in differential form is:

A = the cross-sectional area, Q = channel flow, and q = lateral inflowPhysical (Hydraulic) - continuity - momentumConceptual (Hydrologic) - continuityRegression (Empirical)

3Hydrologic RoutingCombine the continuity equation with some relationship between storage, outflow, and possibly inflowThese relationships are usually assumed, empirical, or analytical in natureAn of example of such a relationship might be a stage-discharge relationshipPhysical (Hydraulic) - continuity - momentumConceptual (Hydrologic) - continuityRegression (Empirical)4Use of Manning EquationStage is also related to the outflow via a relationship such as Manning's equation

Physical (Hydraulic) - continuity - momentumConceptual (Hydrologic) - continuityRegression (Empirical)5Hydraulic RoutingHydraulic routing methods combine the continuity equation with a more realistic relationship describing the actual physics of the movement of the waterThe equation used results from conservation of momentum, assuminguniform velocity distribution (depth averaged)hydrostatic pressuresmall bottom slopeIn hydraulic routing analysis, it is intended that the dynamics of the water or flood wave movement be more accurately describedPhysical (Hydraulic) - continuity - momentumConceptual (Hydrologic) - continuityRegression (Empirical)6Momentum EquationExpressed by balancing the external forces acting on a control section of water as it moves down a channelHenderson (1966) expressed the momentum equation as :

Physical (Hydraulic) - continuity - momentumConceptual (Hydrologic) - continuityRegression (Empirical)

7Forms of Momentum Equations

Sf = SoUnsteady -NonuniformSteady - NonuniformDiffusion or non-inertialKinematicPhysical (Hydraulic) - continuity - momentumConceptual (Hydrologic) - continuityRegression (Empirical)8Routing MethodsModified PulsKinematic WaveMuskingumLag and KMuskingum-CungeDynamicModified PulsKinematic WaveMuskingumMuskingum-CungeDynamicModeling Notes9Modified PulsThe Modified Puls routing method is most often applied to reservoir routingstorage related to outflowThe method may also be applied to river routing for certain channel situationsThe Modified Puls method is also referred to as the storage-indication methodAs a hydrologic method, the Modified Puls equation is described by considering the discrete continuity equation...Modified PulsKinematic WaveMuskingumMuskingum-CungeDynamicModeling Notes10Modified PulsThe solution to the Modified Puls method is accomplished by developing a graph (or table) of O -vs- [2S/t + O]. In order to do this, a stage-discharge-storage relationship must be known (rules) or derived (outlet works). Modified PulsKinematic WaveMuskingumMuskingum-CungeDynamicModeling Notes

Re-writing (substituting O for Q to follow convention)11Modified Puls Example Given the following inflow hydrograph and 2S/Dt + O curve, find the outflow hydrograph for the reservoir assuming it to be completely full at the beginning of the storm. Inflow hydrograph:

12Modified Puls Example

2S/Dt + O curve:13Modified Puls ExampleA table may be created as follows:


Next, using the hydrograph and interpolation, insert the inflow (discharge) values. For example at 1 hour, the inflow is 30 cfs.


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 30.16Modified Puls Example

This is then repeated for the rest of the values in the column.17Modified 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 18Modified 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 provide (darn hard to see!)19Modified 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 = 2020Modified 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 = 7421Modified 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

23Review of the MethodWhat are the critical components?Can this method be used for channel routing?What effect does the choice of time step have?24Kinematic WaveKinematic wave channel routing is the most basic form of hydraulic routingThis method combines the continuity equation with a simplified form of the momentum equation Kinematic wave routing assumes that the friction slope is equal to the bed slopeAdditionally, the kinematic wave form of the momentum equation assumes a simple stage-discharge relationshipModified PulsKinematic WaveMuskingumMuskingum-CungeDynamicModeling Notes25Kinematic Wave Basic Equations

Q = AmAn explicit finite difference scheme in a space-time grid domain is often used for the solution of the kinematic wave procedure.XtModified PulsKinematic WaveMuskingumMuskingum-CungeDynamicModeling Notes26Working Equation

XtModified PulsKinematic WaveMuskingumMuskingum-CungeDynamicModeling Notes27Kinematic Wave AssumptionsThe method does not explicitly allow for separation of the main channel and the overbanksStrictly speaking, the kinematic method does not allow for attenuation of a flood wave. Only translation is accomplished. There is, however, a certain amount of attenuation which results from the finite difference approximation used to solve the governing equationsBest when inflow, free-surface slope, and inertia terms are small compared to bottom slope and frictionFlow resistance may be estimated via Manning's equation or the Chezy equationCommonly used in overland flow routingModified PulsKinematic WaveMuskingumMuskingum-CungeDynamicModeling Notes28Muskingum MethodSp = K OSw = K(I - O)XPrism StorageWedge StorageCombinedS = K[XI + (1-X)O]Modified PulsKinematic WaveMuskingumMuskingum-CungeDynamicModeling NotesObtained by weighting the storage due to inflow and outflow with X, assuming that discharge and storage are single-valued functions of depth, and that storage responds linearly to discharge. K is a storage factor with units of time.29Muskingum, cont...O2 = C0 I2 + C1 I1 + C2 O1S = K[XI + (1-X)O]

Modified PulsKinematic WaveMuskingumMuskingum-CungeDynamicModeling Notes

Substitute the storage equationinto the continuity equationyields30Muskingum Notes :The method assumes a single stage-discharge relationshipHowever, it is used to handle variable storage-discharge relationshipsinflow exceeds outflow: positive wedgeoutflow exceeds inflow: negative wedgeconstant cross section channel: prism storageModified PulsKinematic WaveMuskingumMuskingum-CungeDynamicModeling Notes31Estimating KK can be estimated as 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 equationUse slope of the XI + (1-X)O vs. S plotbest X is least loopedModified PulsKinematic WaveMuskingumMuskingum-CungeDynamicModeling Notes32Estimating XThe value of X should be between 0.0 and 0.5 The parameter X is a weighting coefficient for inflow and outflow. As inflow becomes less important, the value of X decreases The lower limit of X = 0.0 is indicative of a situation where inflow, I, has little or no effect on the storage A reservoir is an example of a situation where attenuation would be the dominant processValues of X = 0.2 to 0.3 are the most common for natural streams; however, values of 0.4 to 0.5 may be obtained for streams with little or no flood plains or storage effects A value of X = 0.5 represents equal weighting between inflow and outflow and would produce translation with little or no attenuationModified PulsKinematic WaveMuskingumMuskingum-CungeDynamicModeling Notes33More Notes - MuskingumThe 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 hydrographAdditionally, it is recommended that the method be limited to moderate to slow rising hydrographs being routed through mild to steep sloping channelsThe method is not applicable to steeply rising hydrographs such as dam breaksFinally, this method also neglects variable backwater effects caused by downstream dams, constrictions, bridges, and tidal influences

Modified PulsKinematic WaveMuskingumMuskingum-CungeDynamicModeling Notes34Muskingum 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:35Muskingum 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)) = 0.66736Muskingum Example ProblemC2= (1- (1*0.30) - (0.5*1)) / ((1-(1*0.30) + (0.5*1)) = 0.167

Therefore the coefficients in this problem are:C0 = 0.167C1 = 0.667C2 = 0.16737Muskingum Example Problem

The three columns now can be calculated.C0I2 = 0.167 * 5 = 0.835C1I1 = 0.667 * 3 = 2.00C2O1 = 0.167 * 3 = 0.50138Muskingum 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

39Muskingum Example ProblemThis can be repeated until the table is complete and the outflow at each time step is known

40Derived from Muskingum with X = 0

Combining this with the continuity equation yields

This equation assumes pure reservoir action, and the peak of the hydrograph must fall on the receding limb of the inflow hydrographEffect of translation re-introduced by lagging inflow hydrographLag and K can be a function of dischargeImplemented in NWSRFS

Lag and K Routing

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44Muskingum-CungeMuskingum-Cunge formulation is similar to the Muskingum type formulationThe Muskingum-Cunge derivation begins with the continuity equation and includes the diffusion form of the momentum equationThese equations are combined and linearizedModified PulsKinematic WaveMuskingumMuskingum-CungeDynamicModeling Notes45Muskingum-Cungeworking equationwhere :Q=discharget=timex=distance along channelqLat = lateral inflowc=wave celeritym=hydraulic diffusivity

Modified PulsKinematic WaveMuskingumMuskingum-CungeDynamicModeling Notes46Muskingum-Cunge, cont...Method attempts to account for diffusion by taking into account channel and flow characteristicsHydraulic diffusivity is derived to be :

The Wave celerity in the x-direction is :

Modified PulsKinematic WaveMuskingumMuskingum-CungeDynamicModeling Notes47Solution of Muskingum-Cunge A solution can be obtained by discretizing the equations

XtModified PulsKinematic WaveMuskingumMuskingum-CungeDynamicModeling Notes48Calculation of K & X

Q, B, and c are best taken as the average values over the Dx reach and Dt time step Modified PulsKinematic WaveMuskingumMuskingum-CungeDynamicModeling Notes49Muskingum-Cunge - NOTESMuskingum-Cunge formulation is 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 hydrographsProperly 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 distanceRules exist for selecting time and distance stepsModified PulsKinematic WaveMuskingumMuskingum-CungeDynamicModeling Notes50Muskingum-Cunge ExampleThe 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.

51Muskingum-Cunge ExampleFirst, K must be determined K is equal to :

Dx = 6 km, while c = 2 m/s52Muskingum-Cunge ExampleThe next step is to determine x

All the variables are known, with B = 25.3 m, So = 0.001 and Dx =6000 m, and using the peak Q taken from the table

53Muskingum-Cunge Example

Since there is no lateral flow, QL = 0. The equation can be simplified to:

54Muskingum-Cunge ExampleA table can then be created in 2 hour time steps similar to the one below:

55Muskingum-Cunge ExampleIt is assumed at time zero, the flow is 10 m3/s at each distance

56Muskingum-Cunge ExampleNext, zero is substituted into for each letter to solve the equation

57Muskingum-Cunge ExampleUsing the table, the quantities can be determined



60Dynamic Wave RoutingThe solution of the Saint Venant equations (all momentum terms) is known as dynamic wave routingThey are coupled, non-linear, first-order partial differential equations of the hyperbolic type that require one initial and two boundary conditions to solveThere is no known general analytical solution - must use numerical methods characteristicsfinite differencefinite elementUseful to describe situations where the relationship between stage and discharge is not a single-valued function (looped rating curves)Modified PulsKinematic WaveMuskingumMuskingum-CungeDynamicModeling Notes61Dynamic Wave EquationsModified PulsKinematic WaveMuskingumMuskingum-CungeDynamicModeling NotesNWS dynamic routing models use an extended form of the Saint Venant equations:

Conservation form of the equations that account for lateral flows, off-channel storage, and sinuositySolved with a weighted four-point implicit finite difference scheme; weighting factor defines stability and convergence properties

62Dynamic Wave SolutionsModified PulsKinematic WaveMuskingumMuskingum-CungeDynamicModeling NotesCharacteristics: original equations transformed into ODEs and solved with simple finite difference techniques; more commonly used for simpler equations (kinematic wave)Finite Difference: implicit 4-point finite difference solutions (Preissmann scheme) are most common, but many work wellFinite Element: not advantageous for one-dimensional routing, but competitive with finite difference methods for two-dimensional routing with complex boundaries

63Two-Dimensional Dynamic Routing

The two-dimensional version of the Saint Venant equations form the basis for the shallow water equations:

Modified PulsKinematic WaveMuskingumMuskingum-CungeDynamicModeling Notes64

Modified PulsKinematic WaveMuskingumMuskingum-CungeDynamicModeling Notes65Some DisadvantagesModel simulation input requirements - dynamic routing techniques generally require boundary conditions at one or more locations in the domain, such as the upstream and downstream sections. These boundary conditions may in the form of known or constant water surfaces, hydrographs, or assumed stage-discharge relationships.Stability - the very complex nature of these methods often leads to numeric instability. Also, convergence may be a problem in some solution schemes. For these reasons as well as others, there tends to be a stability problem in some programs. Often times it is very difficult to obtain a "clean" model run in a cost efficient manner.Computational limitations - Two dimensional routing, and sometimes one-dimensional dynamic routing, is not practical for operational purposes (yet)Modified PulsKinematic WaveMuskingumMuskingum-CungeDynamicModeling Notes6566Routing ReviewModified PulsKinematic WaveMuskingumMuskingum-CungeDynamicModeling NotesContinuity is the heart of all routing methods:

Modified Puls: simple hydrologic method, used mostly for reservoir routing (AKA level pool routing)

Kinematic wave: simplest hydraulic method, uses form of momentum equation where bed slope equals the friction slope and single stage-discharge relationship:Q = Am

67Routing ReviewModified PulsKinematic WaveMuskingumMuskingum-CungeDynamicModeling NotesMuskingum Method: hydrologic, used for river-reach routing, accounts for variable discharge-storage relationship with wedge and prism storage, approximate solution to kinematic wave

Muskingum-Cunge: Muskingum with K and X functions of channel characteristics and flow rate, approximate solution to the diffusion wave, hydrologic or hydraulic?

Dynamic: hydraulic, complex but sometimes necessary

68Hydrologic ModelingLumpedSacramento Soil Moisture AccountingDistributedSHESemi-DistributedTopmodelEventHECContinuousSAC-SMA

69Distributed Modeling

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Distributed Modeling

71Semi-Distributed Modeling

Chart101800

Time (hr)Discharge (cfs)Hydrograph For Modified Puls Example

Sheet1Modified Puls006180100003020045400651100TimeInIn+In+12Sn/t - On2Sn/t + OnOn+1TimeInIn+In+12Sn/t - On2Sn/t + OnOn+1TimeInIn+In+12Sn/t - On2Sn/t + OnOn+1TimeInIn+In+12Sn/t - On2Sn/t + OnOn+1TimeInIn+In+12Sn/t - On2Sn/t + OnOn+1TimeInIn+In+12Sn/t - On2Sn/t + OnOn+1TimeInIn+In+12Sn/t - On2Sn/t + OnOn+1TimeInIn+In+12Sn/t - On2Sn/t + OnOn+1(hr)(cfs)(cfs)(cfs)(cfs)(cfs)(hr)(cfs)(cfs)(cfs)(cfs)(cfs)(hr)(cfs)(cfs)(cfs)(cfs)(cfs)(hr)(cfs)(cfs)(cfs)(cfs)(cfs)(hr)(cfs)(cfs)(cfs)(cfs)(cfs)(hr)(cfs)(cfs)(cfs)(cfs)(cfs)(hr)(cfs)(cfs)(cfs)(cfs)(cfs)(hr)(cfs)(cfs)(cfs)(cfs)(cfs)00003000300030000030000030000030000030001301301309013090301309030513090203051309020305130902030526026026015026015026015026015026015074110182601507411018390390390210390210390210390210390210390210160224324120412041202704120270412027041202704120270412027028437043515051505150330515033051503305150330515033051503304505545261806180618031561803156180315618031561803156180315664780587135713571352257135225713522571352257135225713522585397963890890890135890135890135890135890135890135948107865945945945459454594545945459454594545953108565100100100010001000100010001000870998641101101100110011001100110011007468706212012012001200120012001200120063074658

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Time (hr)Discharge (cfs)Unit Hydrograph For Modified Puls Example

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Outflow (cfs)2S/Dt + O (cfs)2S/Dt + O curve for Modified Puls example

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Sheet1Modified Puls006180100003020045400651100TimeTimeTimeTimeTimeTimeTimeTime(hr)(cfs)(cfs)(cfs)(cfs)(cfs)(hr)(cfs)(cfs)(cfs)(cfs)(cfs)(hr)(cfs)(cfs)(cfs)(cfs)(cfs)(hr)(cfs)(cfs)(cfs)(cfs)(cfs)(hr)(cfs)(cfs)(cfs)(cfs)(cfs)(hr)(cfs)(cfs)(cfs)(cfs)(cfs)(hr)(cfs)(cfs)(cfs)(cfs)(cfs)(hr)(cfs)(cfs)(cfs)(cfs)(cfs)00003000300030000030000030000030000030001301301309013090301309030513090203051309020305130902030526026026015026015026015026015026015074110182601507411018390390390210390210390210390210390210390210160224324120412041202704120270412027041202704120270412027028437043515051505150330515033051503305150330515033051503304505545261806180618031561803156180315618031561803156180315664780587135713571352257135225713522571352257135225713522585397963890890890135890135890135890135890135890135948107865945945945459454594545945459454594545953108565100100100010001000100010001000870998641101101100110011001100110011007468706212012012001200120012001200120063074658

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Sheet1Modified Puls006180100003020045400651100TimeInIn+In+12Sn/t - On2Sn/t + OnOn+1TimeInIn+In+12Sn/t - On2Sn/t + OnOn+1TimeInIn+In+12Sn/t - On2Sn/t + OnOn+1TimeInIn+In+12Sn/t - On2Sn/t + OnOn+1TimeInIn+In+12Sn/t - On2Sn/t + OnOn+1TimeInIn+In+12Sn/t - On2Sn/t + OnOn+1TimeInIn+In+12Sn/t - On2Sn/t + OnOn+1TimeInIn+In+12Sn/t - On2Sn/t + OnOn+1(hr)(cfs)(cfs)(cfs)(cfs)(cfs)(hr)(cfs)(cfs)(cfs)(cfs)(cfs)(hr)(cfs)(cfs)(cfs)(cfs)(cfs)(hr)(cfs)(cfs)(cfs)(cfs)(cfs)(hr)(cfs)(cfs)(cfs)(cfs)(cfs)(hr)(cfs)(cfs)(cfs)(cfs)(cfs)(hr)(cfs)(cfs)(cfs)(cfs)(cfs)(hr)(cfs)(cfs)(cfs)(cfs)(cfs)00003000300030000030000030000030000030001301301309013090301309030513090203051309020305130902030526026026015026015026015026015026015074110182601507411018390390390210390210390210390210390210390210160224324120412041202704120270412027041202704120270412027028437043515051505150330515033051503305150330515033051503304505545261806180618031561803156180315618031561803156180315664780587135713571352257135225713522571352257135225713522585397963890890890135890135890135890135890135890135948107865945945945459454594545945459454594545953108565100100100010001000100010001000870998641101101100110011001100110011007468706212012012001200120012001200120063074658TimeInIn+In+12Sn/t - On2Sn/t + On+1On+1(hr)(cfs)(cfs)(cfs)(cfs)(cfs)0123456789101112

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Chart101800

Time (hr)Discharge (cfs)Hydrograph For Modified Puls Example

Sheet1Modified Puls006180100003020045400651100TimeInIn+In+12Sn/t - On2Sn/t + OnOn+1TimeInIn+In+12Sn/t - On2Sn/t + OnOn+1TimeInIn+In+12Sn/t - On2Sn/t + OnOn+1TimeInIn+In+12Sn/t - On2Sn/t + OnOn+1TimeInIn+In+12Sn/t - On2Sn/t + OnOn+1TimeInIn+In+12Sn/t - On2Sn/t + OnOn+1TimeInIn+In+12Sn/t - On2Sn/t + OnOn+1TimeInIn+In+12Sn/t - On2Sn/t + OnOn+1(hr)(cfs)(cfs)(cfs)(cfs)(cfs)(hr)(cfs)(cfs)(cfs)(cfs)(cfs)(hr)(cfs)(cfs)(cfs)(cfs)(cfs)(hr)(cfs)(cfs)(cfs)(cfs)(cfs)(hr)(cfs)(cfs)(cfs)(cfs)(cfs)(hr)(cfs)(cfs)(cfs)(cfs)(cfs)(hr)(cfs)(cfs)(cfs)(cfs)(cfs)(hr)(cfs)(cfs)(cfs)(cfs)(cfs)00003000300030000030000030000030000030001301301309013090301309030513090203051309020305130902030526026026015026015026015026015026015074110182601507411018390390390210390210390210390210390210390210160224324120412041202704120270412027041202704120270412027028437043515051505150330515033051503305150330515033051503304505545261806180618031561803156180315618031561803156180315664780587135713571352257135225713522571352257135225713522585397963890890890135890135890135890135890135890135948107865945945945459454594545945459454594545953108565100100100010001000100010001000870998641101101100110011001100110011007468706212012012001200120012001200120063074658

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Sheet1Modified Puls006180100003020045400651100TimeInIn+In+12Sn/t - On2Sn/t + On+1On+1TimeInIn+In+12Sn/t - On2Sn/t + OnOn+1TimeInIn+In+12Sn/t - On2Sn/t + OnOn+1TimeInIn+In+12Sn/t - On2Sn/t + OnOn+1TimeInIn+In+12Sn/t - On2Sn/t + OnOn+1TimeInIn+In+12Sn/t - On2Sn/t + OnOn+1TimeInIn+In+12Sn/t - On2Sn/t + OnOn+1TimeInIn+In+12Sn/t - On2Sn/t + OnOn+1(hr)(cfs)(cfs)(cfs)(cfs)(cfs)(hr)(cfs)(cfs)(cfs)(cfs)(cfs)(hr)(cfs)(cfs)(cfs)(cfs)(cfs)(hr)(cfs)(cfs)(cfs)(cfs)(cfs)(hr)(cfs)(cfs)(cfs)(cfs)(cfs)(hr)(cfs)(cfs)(cfs)(cfs)(cfs)(hr)(cfs)(cfs)(cfs)(cfs)(cfs)(hr)(cfs)(cfs)(cfs)(cfs)(cfs)00003000300030000030000030000030000030001301301309013090301309030513090203051309020305130902030526026026015026015026015026015026015074110182601507411018390390390210390210390210390210390210390210160224324120412041202704120270412027041202704120270412027028437043515051505150330515033051503305150330515033051503304505545261806180618031561803156180315618031561803156180315664780587135713571352257135225713522571352257135225713522585397963890890890135890135890135890135890135890135948107865945945945459454594545945459454594545953108565100100100010001000100010001000870998641101101100110011001100110011007468706212012012001200120012001200120063074658TimeInIn+In+12Sn/t - On2Sn/t + OnOn+1(hr)(cfs)(cfs)(cfs)(cfs)(cfs)0123456789101112

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Sheet1Modified Puls006180100003020045400651100TimeInIn+In+12Sn/t - On2Sn/t + OnOn+1TimeInIn+In+12Sn/t - On2Sn/t + On+1On+1TimeInIn+In+12Sn/t - On2Sn/t + OnOn+1TimeInIn+In+12Sn/t - On2Sn/t + OnOn+1TimeInIn+In+12Sn/t - On2Sn/t + OnOn+1TimeInIn+In+12Sn/t - On2Sn/t + OnOn+1TimeInIn+In+12Sn/t - On2Sn/t + OnOn+1TimeInIn+In+12Sn/t - On2Sn/t + OnOn+1(hr)(cfs)(cfs)(cfs)(cfs)(cfs)(hr)(cfs)(cfs)(cfs)(cfs)(cfs)(hr)(cfs)(cfs)(cfs)(cfs)(cfs)(hr)(cfs)(cfs)(cfs)(cfs)(cfs)(hr)(cfs)(cfs)(cfs)(cfs)(cfs)(hr)(cfs)(cfs)(cfs)(cfs)(cfs)(hr)(cfs)(cfs)(cfs)(cfs)(cfs)(hr)(cfs)(cfs)(cfs)(cfs)(cfs)00003000300030000030000030000030000030001301301309013090301309030513090203051309020305130902030526026026015026015026015026015026015074110182601507411018390390390210390210390210390210390210390210160224324120412041202704120270412027041202704120270412027028437043515051505150330515033051503305150330515033051503304505545261806180618031561803156180315618031561803156180315664780587135713571352257135225713522571352257135225713522585397963890890890135890135890135890135890135890135948107865945945945459454594545945459454594545953108565100100100010001000100010001000870998641101101100110011001100110011007468706212012012001200120012001200120063074658TimeInIn+In+12Sn/t - On2Sn/t + OnOn+1(hr)(cfs)(cfs)(cfs)(cfs)(cfs)0123456789101112

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Sheet20000


Sheet3

Sheet1Modified Puls006180100003020045400651100TimeInIn+In+12Sn/t - On2Sn/t + OnOn+1TimeInIn+In+12Sn/t - On2Sn/t + OnOn+1TimeInIn+In+12Sn/t - On2Sn/t + On+1On+1TimeInIn+In+12Sn/t - On2Sn/t + OnOn+1TimeInIn+In+12Sn/t - On2Sn/t + OnOn+1TimeInIn+In+12Sn/t - On2Sn/t + OnOn+1TimeInIn+In+12Sn/t - On2Sn/t + OnOn+1TimeInIn+In+12Sn/t - On2Sn/t + OnOn+1(hr)(cfs)(cfs)(cfs)(cfs)(cfs)(hr)(cfs)(cfs)(cfs)(cfs)(cfs)(hr)(cfs)(cfs)(cfs)(cfs)(cfs)(hr)(cfs)(cfs)(cfs)(cfs)(cfs)(hr)(cfs)(cfs)(cfs)(cfs)(cfs)(hr)(cfs)(cfs)(cfs)(cfs)(cfs)(hr)(cfs)(cfs)(cfs)(cfs)(cfs)(hr)(cfs)(cfs)(cfs)(cfs)(cfs)00003000300030000030000030000030000030001301301309013090301309030513090203051309020305130902030526026026015026015026015026015026015074110182601507411018390390390210390210390210390210390210390210160224324120412041202704120270412027041202704120270412027028437043515051505150330515033051503305150330515033051503304505545261806180618031561803156180315618031561803156180315664780587135713571352257135225713522571352257135225713522585397963890890890135890135890135890135890135890135948107865945945945459454594545945459454594545953108565100100100010001000100010001000870998641101101100110011001100110011007468706212012012001200120012001200120063074658TimeInIn+In+12Sn/t - On2Sn/t + OnOn+1(hr)(cfs)(cfs)(cfs)(cfs)(cfs)0123456789101112

Sheet1000


Sheet20000


Sheet3

Sheet1Modified Puls006180100003020045400651100TimeInIn+In+12Sn/t - On2Sn/t + OnOn+1TimeInIn+In+12Sn/t - On2Sn/t + OnOn+1TimeInIn+In+12Sn/t - On2Sn/t + OnOn+1TimeInIn+In+12Sn/t - On2Sn/t + On+1On+1TimeInIn+In+12Sn/t - On2Sn/t + OnOn+1TimeInIn+In+12Sn/t - On2Sn/t + OnOn+1TimeInIn+In+12Sn/t - On2Sn/t + OnOn+1TimeInIn+In+12Sn/t - On2Sn/t + OnOn+1(hr)(cfs)(cfs)(cfs)(cfs)(cfs)(hr)(cfs)(cfs)(cfs)(cfs)(cfs)(hr)(cfs)(cfs)(cfs)(cfs)(cfs)(hr)(cfs)(cfs)(cfs)(cfs)(cfs)(hr)(cfs)(cfs)(cfs)(cfs)(cfs)(hr)(cfs)(cfs)(cfs)(cfs)(cfs)(hr)(cfs)(cfs)(cfs)(cfs)(cfs)(hr)(cfs)(cfs)(cfs)(cfs)(cfs)00003000300030000030000030000030000030001301301309013090301309030513090203051309020305130902030526026026015026015026015026015026015074110182601507411018390390390210390210390210390210390210390210160224324120412041202704120270412027041202704120270412027028437043515051505150330515033051503305150330515033051503304505545261806180618031561803156180315618031561803156180315664780587135713571352257135225713522571352257135225713522585397963890890890135890135890135890135890135890135948107865945945945459454594545945459454594545953108565100100100010001000100010001000870998641101101100110011001100110011007468706212012012001200120012001200120063074658TimeInIn+In+12Sn/t - On2Sn/t + OnOn+1(hr)(cfs)(cfs)(cfs)(cfs)(cfs)0123456789101112

Sheet1000


Sheet20000


Sheet3

Chart202004001100


Sheet1Modified Puls006180100003020045400651100TimeTimeTimeTimeTimeTimeTimeTime(hr)(cfs)(cfs)(cfs)(cfs)(cfs)(hr)(cfs)(cfs)(cfs)(cfs)(cfs)(hr)(cfs)(cfs)(cfs)(cfs)(cfs)(hr)(cfs)(cfs)(cfs)(cfs)(cfs)(hr)(cfs)(cfs)(cfs)(cfs)(cfs)(hr)(cfs)(cfs)(cfs)(cfs)(cfs)(hr)(cfs)(cfs)(cfs)(cfs)(cfs)(hr)(cfs)(cfs)(cfs)(cfs)(cfs)00003000300030000030000030000030000030001301301309013090301309030513090203051309020305130902030526026026015026015026015026015026015074110182601507411018390390390210390210390210390210390210390210160224324120412041202704120270412027041202704120270412027028437043515051505150330515033051503305150330515033051503304505545261806180618031561803156180315618031561803156180315664780587135713571352257135225713522571352257135225713522585397963890890890135890135890135890135890135890135948107865945945945459454594545945459454594545953108565100100100010001000100010001000870998641101101100110011001100110011007468706212012012001200120012001200120063074658

Sheet1


Sheet2


Sheet3

Sheet1Modified Puls006180100003020045400651100TimeInIn+In+12Sn/t - On2Sn/t + OnOn+1TimeInIn+In+12Sn/t - On2Sn/t + OnOn+1TimeInIn+In+12Sn/t - On2Sn/t + OnOn+1TimeInIn+In+12Sn/t - On2Sn/t + OnOn+1TimeInIn+In+12Sn/t - On2Sn/t + On+1On+1TimeInIn+In+12Sn/t - On2Sn/t + OnOn+1TimeInIn+In+12Sn/t - On2Sn/t + OnOn+1TimeInIn+In+12Sn/t - On2Sn/t + OnOn+1(hr)(cfs)(cfs)(cfs)(cfs)(cfs)(hr)(cfs)(cfs)(cfs)(cfs)(cfs)(hr)(cfs)(cfs)(cfs)(cfs)(cfs)(hr)(cfs)(cfs)(cfs)(cfs)(cfs)(hr)(cfs)(cfs)(cfs)(cfs)(cfs)(hr)(cfs)(cfs)(cfs)(cfs)(cfs)(hr)(cfs)(cfs)(cfs)(cfs)(cfs)(hr)(cfs)(cfs)(cfs)(cfs)(cfs)00003000300030000030000030000030000030001301301309013090301309030513090203051309020305130902030526026026015026015026015026015026015074110182601507411018390390390210390210390210390210390210390210160224324120412041202704120270412027041202704120270412027028437043515051505150330515033051503305150330515033051503304505545261806180618031561803156180315618031561803156180315664780587135713571352257135225713522571352257135225713522585397963890890890135890135890135890135890135890135948107865945945945459454594545945459454594545953108565100100100010001000100010001000870998641101101100110011001100110011007468706212012012001200120012001200120063074658TimeInIn+In+12Sn/t - On2Sn/t + OnOn+1(hr)(cfs)(cfs)(cfs)(cfs)(cfs)0123456789101112

Sheet1000


Sheet20000


Sheet3

Sheet1Modified Puls006180100003020045400651100TimeInIn+In+12Sn/t - On2Sn/t + On+1On+1TimeInIn+In+12Sn/t - On2Sn/t + On+1On+1TimeInIn+In+12Sn/t - On2Sn/t + On+1On+1TimeInIn+In+12Sn/t - On2Sn/t + On+1On+1TimeInIn+In+12Sn/t - On2Sn/t + On+1On+1TimeInIn+In+12Sn/t - On2Sn/t + On+1On+1TimeInIn+In+12Sn/t - On2Sn/t + On+1On+1TimeInIn+In+12Sn/t - On2Sn/t + On+1On+1(hr)(cfs)(cfs)(cfs)(cfs)(cfs)(hr)(cfs)(cfs)(cfs)(cfs)(cfs)(hr)(cfs)(cfs)(cfs)(cfs)(cfs)(hr)(cfs)(cfs)(cfs)(cfs)(cfs)(hr)(cfs)(cfs)(cfs)(cfs)(cfs)(hr)(cfs)(cfs)(cfs)(cfs)(cfs)(hr)(cfs)(cfs)(cfs)(cfs)(cfs)(hr)(cfs)(cfs)(cfs)(cfs)(cfs)00003000300030000030000030000030000030001301301309013090301309030513090203051309020305130902030526026026015026015026015026015026015074110182601507411018390390390210390210390210390210390210390210160224324120412041202704120270412027041202704120270412027028437043515051505150330515033051503305150330515033051503304505545261806180618031561803156180315618031561803156180315664780587135713571352257135225713522571352257135225713522585397963890890890135890135890135890135890135890135948107865945945945459454594545945459454594545953108565100100100010001000100010001000870998641101101100110011001100110011007468706212012012001200120012001200120063074658TimeInIn+In+12Sn/t - On2Sn/t + On+1On+1(hr)(cfs)(cfs)(cfs)(cfs)(cfs)0123456789101112

Sheet101800


Sheet202004001100


Sheet3

Sheet1Modified Puls006180100003020045400651100TimeTimeTimeTimeTimeTimeTimeTime(hr)(cfs)(cfs)(cfs)(cfs)(cfs)(hr)(cfs)(cfs)(cfs)(cfs)(cfs)(hr)(cfs)(cfs)(cfs)(cfs)(cfs)(hr)(cfs)(cfs)(cfs)(cfs)(cfs)(hr)(cfs)(cfs)(cfs)(cfs)(cfs)(hr)(cfs)(cfs)(cfs)(cfs)(cfs)(hr)(cfs)(cfs)(cfs)(cfs)(cfs)(hr)(cfs)(cfs)(cfs)(cfs)(cfs)00003000300030000030000030000030000030001301301309013090301309030513090203051309020305130902030526026026015026015026015026015026015074110182601507411018390390390210390210390210390210390210390210160224324120412041202704120270412027041202704120270412027028437043515051505150330515033051503305150330515033051503304505545261806180618031561803156180315618031561803156180315664780587135713571352257135225713522571352257135225713522585397963890890890135890135890135890135890135890135948107865945945945459454594545945459454594545953108565100100100010001000100010001000870998641101101100110011001100110011007468706212012012001200120012001200120063074658Time(hr)(cfs)(cfs)(cfs)(cfs)(cfs)0123456789101112

Sheet1


Sheet2


Sheet3

Sheet1Modified Puls006180100003020045400651100TimeTimeTimeTimeTimeTimeTimeTime(hr)(cfs)(cfs)(cfs)(cfs)(cfs)(hr)(cfs)(cfs)(cfs)(cfs)(cfs)(hr)(cfs)(cfs)(cfs)(cfs)(cfs)(hr)(cfs)(cfs)(cfs)(cfs)(cfs)(hr)(cfs)(cfs)(cfs)(cfs)(cfs)(hr)(cfs)(cfs)(cfs)(cfs)(cfs)(hr)(cfs)(cfs)(cfs)(cfs)(cfs)(hr)(cfs)(cfs)(cfs)(cfs)(cfs)00003000300030000030000030000030000030001301301309013090301309030513090203051309020305130902030526026026015026015026015026015026015074110182601507411018390390390210390210390210390210390210390210160224324120412041202704120270412027041202704120270412027028437043515051505150330515033051503305150330515033051503304505545261806180618031561803156180315618031561803156180315664780587135713571352257135225713522571352257135225713522585397963890890890135890135890135890135890135890135948107865945945945459454594545945459454594545953108565100100100010001000100010001000870998641101101100110011001100110011007468706212012012001200120012001200120063074658Time(hr)(cfs)(cfs)(cfs)(cfs)(cfs)0123456789101112

Sheet1


Sheet2


Sheet3

Sheet1MuskingumTimeInflowOutflow03315210384655

Sheet2

Sheet3

Sheet1MuskingumTimeInflowOutflow03315210384655TimeInflowOutflow030.8352.000.501315210384655

Sheet2

Sheet3

Sheet1MuskingumTimeInflowOutflow03315210384655TimeInflowOutflow030.8352.000.501315210384655TimeInflowOutflow030.8352.000.5013153.34210384655

Sheet2

Sheet3

Sheet1MuskingumTimeInflowOutflow03315210384655TimeInflowOutflow030.8352.000.501315210384655TimeInflowOutflow030.8352.000.5013153.34210384655TimeInflowOutflow030.8352.000.5013151.673.340.5573.342101.346.670.935.57381.005.341.498.94460.8354.001.317.83553.341.036.14

Sheet2

Sheet3

Sheet100.150.030.40.10.60.240.80.51Time (hr)010112218328.545057861077134.58147915010146111291210513781459154516331724181719122010

Sheet1

xDx/(cDt)Curve for Dx/(cDt)

Sheet2

Sheet3

Sheet100.150.030.40.10.60.240.80.51Time (hr)Time (hr)010010112112218218328.5328.5450450578578610761077134.57134.581478147915091501014610146111291112912105121051378137814591459154515451633163317241724181718171912191220102010

Sheet100000


Sheet2

Sheet3

Sheet100.150.030.40.10.60.240.80.51Time (hr)Time (hr)Time (hr)0 km6 km12 km18 km010010010112112218218218450328.5328.5610745045081475785781014661076107121057134.57134.51459814781471633915091501817101461014620101112911129221012105121052410137813782610145914592810154515451633163317241724181718171912191220102010

Sheet1


Sheet2

Sheet3

Sheet100.150.030.40.10.60.240.80.51Time (hr)Time (hr)Time (hr)0 km6 km12 km18 km010010010112112218218218450328.5328.5610745045081475785781014661076107121057134.57134.51459814781471633915091501817101461014620101112911129221012105121052410137813782610145914592810154515451633163317241724Time (hr)0 km6 km12 km18 km18171817010101010191219122182010201045061078147101461210514591633181720102210241026102810

Sheet1


Sheet2

Sheet3

Sheet100.150.030.40.10.60.240.80.51Time (hr)Time (hr)Time (hr)0 km6 km12 km18 km010010010112112218218218450328.5328.5610745045081475785781014661076107121057134.57134.51459814781471633915091501817101461014620101112911129221012105121052410137813782610145914592810154515451633163317241724Time (hr)0 km6 km12 km18 kmTime (hr)0 km6 km12 km18 km1817181701010101001010101019121912218218201020104504506107610781478147101461014612105121051459145916331633181718172010201022102210241024102610261028102810

Sheet1


Sheet2

Sheet3

Sheet100.150.030.40.10.60.240.80.51Time (hr)Time (hr)Time (hr)0 km6 km12 km18 km010010010112112218218218450328.5328.5610745045081475785781014661076107121057134.57134.51459814781471633915091501817101461014620101112911129221012105121052410137813782610145914592810154515451633163317241724Time (hr)0 km6 km12 km18 kmTime (hr)0 km6 km12 km18 kmTime (hr)0 km6 km12 km18 km181718170101010100101010100101010101912191221821821813.8920102010450450450610761076107814781478147101461014610146121051210512105145914591459163316331633181718171817201020102010221022102210241024102410261026102610281028102810

Sheet1


Sheet2

Sheet3

Sheet100.150.030.40.10.60.240.80.51Time (hr)Time (hr)Time (hr)0 km6 km12 km18 km010010010112112218218218450328.5328.5610745045081475785781014661076107121057134.57134.51459814781471633915091501817101461014620101112911129221012105121052410137813782610145914592810154515451633163317241724Time (hr)0 km6 km12 km18 kmTime (hr)0 km6 km12 km18 kmTime (hr)0 km6 km12 km18 kmTime (hr)0 km6 km12 km18 km181718170101010100101010100101010100101010101912191221821821813.8921813.892010201045045045045034.5161076107610761078147814781478147101461014610146101461210512105121051210514591459145914591633163316331633181718171817181720102010201020102210221022102210241024102410241026102610261026102810281028102810

Sheet1


Sheet2

Sheet3

oct99_routing.pptx

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