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CIVE 541 Engineering Hydrology Notes

Chapter 5: Flood Routing

5 Flood Routing1 Flow routing consists of predicting the outflow hydrograph given the inflow hydrograph. The inflow hydrograph is the runoff from the watershed (ref. chapter 4) and the outflow hydrograph is either the outflow from a detention reservoir or from a channel segment (figure 5.1). The outflow from a channel can also be the inflow into a reservoir and vice-versa. The movement of a flood down a river and through a reservoir system is an important part of any flood study. Flow routing methods can be used to predict the outflow from a watershed system composed of various reservoirs and channels.

Figure 5.1 Reservoir & River Routing Concepts (Bedient, 2008 – Figure E4-2(a))

5.1

Reservoir Routing & Design of Detention Reservoirs Detention reservoirs provide a mean to control floods (often due to urban development which results in an increase of the volume of runoff and flow rates, which is the cause of frequent flooding and severe soil erosion downstream) by detaining runoff and releasing it later at a desirable rate. They are therefore provided for reducing the peak flows, thus assuring that the outflow from the reservoir is not more than the permitted downstream rate (figure 5.2). Storage routing methods can be used to translate inflow hydrographs through a reservoir or detention basin of particular size to attenuate peak flow.

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Figure 5.2 Reservoir Concepts (Bedient, 2008 – figure 4-1)

References: - Basha, H. Hydrology Notes. February 2007 - Bedient, P.B., Huber, W.C., & B.E. Vieux, Hydrology & Floodplain Analysis, 4th ed., Pearson Education – Prentice Hall, NJ, 2008.

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Chapter 5: Flood Routing

5.1.1 Reservoir Routing Equation The routing procedure is based on the application of the continuity equation in conjunction with the flow equation for the outlet. The continuity equation is expressed by:

where dS/ dt is the rate of change of storage S with respect to time t and I(฀t)฀฀and Q(฀t) ฀฀are the rates of inflow and outflow from the reservoir at any given time. This equation expresses the change in storage during time dt as the difference between the inflow and outflow from the reservoir assuming that the water surface in the reservoir is horizontal and that the net losses due to evaporation, seepage and rainfall are negligible. It is also assumed that the outflow and inflow begin at the same time, which implies that the inflow passes through the reservoir instantaneously regardless of its length. The outflow equation, which relates the outflow Q and the water depth h at the outlet, can be written in the following form:

where K and d are constants dependent on the type, size and shape of the outlet, and h is measured from the centerline of the orifice or the base of the weir, depending on the outlet. For a culvert outlet

For a rectangular or trapezoidal weir spillway outlet d =฀1.5

Cd is the discharge coefficient n is the number of pipes D is their diameter Equation (5.3) is applicable only when the pipes are flowing full which corresponds to h≥฀0.7d. Furthermore, it is assumed that there is no downstream influence and that all outlets are at the same level.

Cw is the weir discharge coefficient L is the length of the spillway crest

d = ฀0.5

The storage of the reservoir is function of the reservoir configuration and the water depth h. The relationship between the storage and water depth could be assumed to be of the following form:  where A and b are constants dependent on the reservoir shape and h is the water depth. For a constant area reservoir, b is equal to 1 and A is the constant surface area, and in general, the value of b lies between 1 and 1.5. Combining equation (5.2) with equation (5.5), a direct relationship between storage and outflow is obtained:

The value of a ranges from a minimum value of 0.25 to a maximum value of 1.5 for a trapezoidal and rectangular weir spillway outlet or a maximum value of 2.5 for a triangular notch spillway.

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Chapter 5: Flood Routing

Substituting (5.6) into the continuity equation (5.1), one obtains

where S0 is the initial storage in the pond that could be negative for dead or permanent storage beneath the outlet's level. The shape of the inflow function I฀(t) ฀฀can be assumed to be triangular, a commonly assumed shape in engineering design and analysis:

I

Ip tp

t

0 < t < tp (5.8)

I  Ip 

Ip tb  t p

(t  t p ) tp < t < tb

where Ip is the peak inflow, t p is the time to peak inflow and tb is the duration time.

5.1.2 Numerical Solution of the Reservoir Routing Equation Writing equation (5.7) in the following form:

One can then apply the method of Runge-Kutta 4th order to solve for the storage at every time increment:



where

  Starting from t=0 and the known initial condition S=0, one can calculate ki and determine the storage S at the next time step. The above steps are repeated for each time increment.

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5.1.3 Design of Detention Reservoirs There are several considerations involved in the design of stormwater detention reservoirs (figure 5.3). These include: inflow, outflow, storage, surface area, depth, culvert diameter, weir length ... etc. In order to check the adequacy of a proposed design, an inflow hydrograph is routed through the reservoir and the maximum outflow is determined and checked against the permitted discharge rate. This procedure is repeated until the optimum design is found. The routing procedure is based on the application of the continuity equation in conjunction with the flow equation for the outlet. Various numerical methods are available for solving the reservoir routing equation, e.g. Runge-Kutta method.

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Chapter 5: Flood Routing

Figure 5.2 Example of a Reservoir (Bedient, 2008 – Figure E4-5(b))

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Chapter 5: Flood Routing

5.2 River Routing River routing differs from reservoir routing in that storage in a river depends on the outflow as well as the inflow because the storage in the river depends on whether stages are rising or falling (figure 5.4). The river routing procedure is based on the application of the continuity equation in conjunction with the storage relationship with the inflow and outflow. The finite difference approach is used to give a recurrent algebraic equation known as the Muskingum method. The Muskingum method is function of two parameters: k which is a measure of the travel time of the flood wave in the reach and x which is a weighting factor of the inflow and outflow in determining the storage volume in the reach. River routing consists of estimating the downstream hydrograph given the upstream hydrograph. One simple routing procedure is based on the Muskingum method, which is derived from the continuity equation in its spatially lumped form and a linear storage discharge relationship of the following form

Figure 5.3 River Routing (Bedient, 2008 – Figure 4-2)

where k and x are the routing parameters. Physically, k is considered as the average travel time for the reach, and x is the relative effect of the inflow and outflow on reach storage (a weighting factor that varies from 0 to 0.5 for a given reach – a typical range for most natural streams is x = 0.2 to 0.3): Substituting (5.12) in the continuity equation (5.1) and expressing the resulting equation in finite difference form, one obtains

where

Note that K and Δt must have the same units, and 2Kx  Δt ≤ K for numerical accuracy, and that the coefficients C0, C1 and C2 sum to 1.0.

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