hydrology note 5

7/27/2019 Hydrology Note 5

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FLOW ROUTING

Introduction Reservoir routing Channel routing

IntroductionFlow Routing

Flow routing is a procedure to determine the time and magnitude of flow(i.e, the flow hydrograph) at a point on a watercourse from known or

assumed hydrographs at one or more points upstream (Chow et.al, 1988)

If the flow is a flood, the procedure is specifically known as flood routing. In a broad sense, flow routing may be considered as an analysis to trace the

flow through a hydrologic system, given the input.

Lumped and Distributed

Chow et al. (1988) divided the flow routing to the lumped and distributedrouting. A lumped system model: the flow is calculated as a function of time alone at

a particular location. It is called hydrologic routing.

A distributed system model: the flow is calculated as a function of space andtime through the system. It is called hydraulic routing.

Hydrologic Routing

Hydrologic routing methods employ essentially the equation of continuity. Hydraulic methods, on the other hand, employ the continuity equation

together with the equation of motion of unsteady flow.

Hydrologic routing computes the outflow hydrograph corresponding to agiven inflow hydrograph .


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The hydrologic analysis of problems such as flood forecasting, floodprotection, reservoir design and spillway design invariably include floodrouting.

Hydrologic routing often called storage routing. Two commonly appliedmethods:(a)reservoir routing and (b) channel routing

In reservoir routing the effect of a flood wave entering a reservoir isstudied. Knowing the volume-elevation characteristic of the reservoir andthe outflow-elevation relationship for the spillways and other outlet

structures in the reservoir, the effect of a flood wave entering the reservoir

is studied to predict the variations of reservoir elevation and outflow

discharge with time.

Reservoir Routing

The reservoir routing is essential in:

1) Design of the capacity of spillways and the other reservoir outlet structures

2) Determine the location and sizing of the capacity of reservoirs to meet specificrequirements.

Channel Routing

In the channel routing the change in the shape of hydrograph as it travelsdown a channel is studied.

The method can predict the flood hydrograph at various sections of thechannel/river reach by considering a channel reach and an input hydrograph

at upstream end.

It is useful in flood-forecasting operations and flood-protection works.Hydraulic Routing

The flow water through the soil and stream channels of a watershed is a distributed

process because the flow rate, velocity, and depth vary in space throughout thewatershed.

Estimate of the flow rate or water level at important locations in the channelsystem can be obtained using a distributed flow routing model .

This type of model is based on partial differential equations (Saint-Venant

equations for one dimensional flow) that allow the flow rate and water level to be


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computed as functions of space and time, rather than of time alone as in the

lumped models.

One dimensional distributed routing methods have been classified as kinematicwave routing, diffusion wave routing, and dynamic wave routing.

Kinematic waves govern the flow when the inertial and pressure forces are notimportant, that is, when the gravitational force of the flow is balanced by the

frictional resistance force. It is applicable for the channel slopes are steep andbackwater effects are negligible.

When pressure forces become important but innertial forces remain unimportant, adiffusion wave model is applicable.

Both the kinematic wave model and the diffusion wave model are helpful indescribing downstream wave propagation when the channel slope is greater than

0.01 percent and there are no waves propagating upstream due to disturbances such

as tides, tributary inflows, or reservoir operations.

Kinematic wave routing is a simplified hydraulic routing procedure often used for

overland flow, where the momentum equation is replaced with the Manningequation .

The dynamic wave routing method is required when both inertial and pressure

forces are important, such as in mild sloped rivers, and backwater effects fromdownstream disturbances are not negligible.

Dynamic routing is based on the complete Saint Venant equations representing

conservation of mass and momentum for unsteady flow in open channels.

This involves the numerical solution of the full Saint-Venant equations.Basic Equations

For a hydrologic system, the amount of water stored S may related to the rates of

inflow I and outflow Q by the integral equation of continuity:

Where, I = inflow rate, Q = outflow rate, S = storage

For a time interval , the continuity equation may be written in terms of averageinflow and outflow as:


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If O replaces Q represents the outflow, the equation is,

Referring to figure (inflow and outflow hydrograph), the area between the inflow

storage volume.

If the average inflow is greater than the average outflow during , thechange in storage will be positive.S decreases if is less than .The effect of storage in decreasing the peak flow and broadening the time base ofthe hydrograph is called attenuation.

Basic Equations


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For computational purposes, the equation is rewritten as follows

Where subcripts 1 and 2 refer to the beginning and end of the computational timeinverval . Ift is relatively small, the assumption of linear variation in dischargerate is adequate. The routing computations step through time. For each time step,

the inflows ( ) and beginning storage () are known. The two unknownare and .

Basic Equations

Hydrologic routing is based on combining the above equation with a relationship

between storage and discharge.

The storage-outflow approach is based on the premise that storage (S) is a uniquefunction of outflow (O). It is associated with reservoirs.

Muskingum routing was developed specifically for streams and rivers. It relates

storage S to a linear function of weighted inflow (I) and outflow (O).Reservoir Routing

The basic premise of the storage-outflow method is that outflow is known for any

amount of storage.

A storage-outflow relationship is combined with the above equation give theapproach called modified Puls routing. It is often called level-pool routing when

used with a storage-outflow relationship for a reservoir.

If reservoir storage is large compared with the ouflow rate, the water surface isnearly horizontal, storage and outflow are uniquely related without considering

inflow, and the basic premise of storage-outflow routing is most nearly valid.

Routing Algorithm

The computational algorithm is rewritten

[ ] [ ]

At each timestep, the terms to the right of the equal sign are known, and the

[2S/ + O ]term on the left is computed.


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As the computational algorithm advances to the next time step, - O] isdetermined as

[ ] [ ]

A relationship between the term on the left side and outflow O

[ ]

Procedures for developing the [2S/t + O] versus O relationship differ betweenreservoirs and stream reaches.

The computational algorithm can also rewritten as

[ ] [

] [

]

At each timestep, the terms to the left of the equal sign are known, and the [S+

Ot/2] term on the right is computed.

As the computational algorithm advances to the next time step, [S - Ot/2] isdetermined as

[ ] [

]

is determined from the graph of [S+Ot/2] vs O

SpillwayIf an uncontrolled spillway is provided in a reservoir, typically

Or if Q is assumed as the outflow it can be written as


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The procedure and calculation stepFrom the known storage-elevation and discharge-elevation data, prepare a curve of

[2S/t + O] versus Outflow (O) or [S + Ot/2] versus Outflow (O) . Here t is anychosen interval, approximately 20 to 40% of the time of rise of the inflow

hydrograph.

EXAMPLE 1: RESERVOIR ROUTING

1. A dam has an uncontrolled weir spillway 10 m wide with a crest elevation of548.0 m and a discharge coefficient of 0.45.

2.

The reservoir water surface elevation versus area relationship provided inthe table and was developed from a topographic map. The elevation versus

discharge relationship is computed using the weir equation.

3. The inflow hydrograph provided in the table is to be routed throughreservoir.

4. Use the starting surface elevation in the reservoir is 544.0 m and t is 3,600seconds.

Example 1: reservoir routing

The discharge rate through the spillway is computed from the following equation

Refer to column (5) in the table

The discharge rate through the spillway:


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Reservoir storage is computed based on

Where = 2 m Refer to column (3) in the table

Reservoir storage is computed based on


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Example 1: reservoir routing

(

)


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Hydrologic Channel Routing

In reservoir routing, the storage was a unique function of the outflow discharge S = f(Q)or S = (O).

However, in channel routingthe storage is a function of both outflow and inflowdischarges S = (I, O).

The flow in a river during a flood belongs to the category of gradually varied unsteadyflow .

The water surface in a channel is not flat. It is not only parallel to the channel bottom butalso varies with the time.

During flood flow, the channel reach can be considered under two categories as prismstorage and wedge storage.

Prism storage is the volume that would exist if uniform flow occurred at the downstreamdepth.

The wedge volume formed between the actual water surface profile and the top surface ofthe prism storage.


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Tree conditions of wedge storage

Wedge storage negative


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Sprisma is function of Outflow ; Swedge is function of Inflow and Outflow

The total storage in the channel reach can be expressed as:

WhereKandxare coefficients and mis a constant exponent. It has been found the the value ofm

varies from 0.6 for rectangular channels to a value of about 1.0 for natural channel.

Using m = 1.0 reduces to a linear relationship for S in terms of I and Q as:

And this relationship is known as the Muskingum equation. In this the parameterx

is known as weighting factor and takes a value between 0 and 0.5.

Whenx = 0 , obviously the storage is a function of discharge only and the equation reduces to:

Such a storage is known a linear storage orlinear reservoir. Whenx = 0.5, both the inflow and

outflow are equally important in determining the storage. The coefficientKis known asstorage-

time constantand has the dimensions of time. It is approximately equal to the time of travel of a

flood wave through the channel reach.


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The outflow peak does not occur at the point of intersection of the inflow and outflow

hydrographs.

The increment in storage at any time t and time element t can be calculated. Summation of the

various incremental storage values enable one to find the channel storage S vs time relationship.


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If an inflow and outflow hydrograph set is available for a given reach, values of S at various time

intervals can be determined by the above technique. By choosing a trial value of x, values of S at

any time t are plotted against the corresponding [xI + (1x)Q] values.

If the value of x is chosen correctly, a straight-line relationship will result. However, if an

incorrect value of x is used, the plotted points will trace a looping curve. By trial and error, avalue of x is determined. The inverse slope of this straight-line will give the value of K.

Normally for natural channels, the value of x lies between 0 to 0.3. For a given reach, the values

of x and K are assumed to be constant.

Storage refer to conditions before:


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Storage refer to conditions after:

For a given channel reach by selecting a routing interval t and using the Muskingum equation,

the change in storage is,

The continuity equation for the reach is,

( ) (

)

Rearrange gives equation where is evaluated:


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Example: Muskingum routing

Route the following hydrograph through a river reach for which K = 12 h and x = 0.2. At the

start of the inflow flood, the outflow is 10 m/s.

Since K = 12 h and x = 0.2.; 2 x Kx X = 4.8 h; t should be such that 12 h > t > 4.8 h. In the

present case t = 6 h is selected to suit the given inflow hydrograph ordinate interval.

For the first time interval 0 to 6 h.


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For the next time step, 6 to 12 h, O1 = 10.48 m/s. The procedure is repeated for the entire

duration of the inflow hydrograph. The computations are done in a tabular form.


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Storage , S = flow x time interval


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Cumulative Storage, Si


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hydrology note 5

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