This article contains basic notes on the "Reservoir and Channel Routing" topic of the "Hydrology & Irrigation" subject.
Flood routing is the technique of determining the flood hydrograph at a section of a river by utilizing the data of flood flow at one or more upstream sections. The hydrologic analysis of problems such as flood forecasting, flood protection, reservoir design and spillway design invariably include flood routing. In these applications, two broad categories of routing can be recognized. These are:
- Reservoir routing (Storage is a unique function of Outflow Discharge, S = f(Q) ), and
- Channel routing (Storage is a function of both Inflow & Outflow, here we use Muskingam Method)
A variety of routing methods are available and they can be broadly classified into two categories as:
1. Hydrologic routing:
Hydrologic-routing methods employ essentially the equation of continuity. In its simplest form, inflow to the river reach is equal to the outflow of the river reach plus the change of storage:
- I is average inflow to the reach during
- O is average outflow from the reach during ; and
- S is the water currently in the reach (known as storage)
The hydrologic models (e.g. linear and nonlinear Muskingum models) need to estimate hydrologic parameters using recorded data in both upstream and downstream sections of rivers and/or by applying robust optimization techniques to solve the one-dimensional conservation of mass and storage-continuity equation.
2. Hydraulic routing: Hydraulic methods, on the other hand, employ the continuity equation together with the equation of motion of unsteady.
Prism Storage: It is the volume that would exist if the uniform flow occurred at the downstream depth. i.e., the volume formed by an imaginary plane parallel to the channel bottom drawn at the outflow section water surface.
Wedge Storage: It is the wedge-like volume formed between the actual water surface profile and the top surface of the prism storage.
S = Sp + Sw
Where, S = Total storage in the channel.
Sp = Prism storage= if (Q) = function of outflow discharge.
Sw = Wedge storage= f(I) = function of inflow discharge.
X = Weighting factor
when X = 0
S = KQm (Function of Outflow)
m = Constant
= 0.6 for rectangular channels
= 1.0, S = KQ (for nature channels/reservoirs)
K = storage time constant
Flood Routing Analysis
Attenuation - The peak of the outflow hydrograph will be smaller than of the inflow hydrograph. This reduction in the peak value is called attenuation.
Time Lag - The peak of the outflow occurs after the peak of the inflow; the time difference between the two peaks is known as lag. The attenuation and lag of a flood hydrograph at a reservoir are two very important aspects of a reservoir operating under flood-control criteria.
Channel Routing: Muskingum Method
We want to simulate the propagation of a flood wave along a channel. Storage is the function of both I and O.
Profiles of water flowing in a channel reach during the rising limb (a) and recession limb (b) of a flood wave
Assume that storage can be approximated as;
S = K[xI + (1 - x)O]
Where K [s] is a constant, and x is a weighting factor. K is approximately equal to the “residence time” of the flood wave within the stream reach. K has a unit of time, and is a rough measure of the residence time of flood peak in the channel reach. Change in channel morphology may change the value of K.
The weighting factor x represents the degree of attenuation. For example;
a) x = 0. In this case of a reservoir. Large attenuation.
b) x = 0.5. In this case, S = K(I/2 + O/2) . But we also have
(I1 + I2)/2 - (O1 + O2)/2 = (S2 - S1)/∆t
When ∆t = K, we can show that O2 = I1. The wave is simply translated with a lag time K. No attenuation.
For most river channels, 0.1 < x < 0.3. The values of K and x have to be determined in each channel reach. One can measure I and O in the reach during a storm, calculate S from cumulative I and O, and plot S against xI + (1 - x)O for several values of x.
When a suitable value of x is used, the plot appears to follow a straight line (see the figure below), otherwise it appears to be a loop. The slope of the straight line is K.
Once K and x are determined, we can rout flood waves using the Muskingum method. The balance equation;
(I1 + I2)/2 - (O1 + O2)/2 = (S2 - S1)/∆t
But we also have
S2 - S1 = K[x(I2 - I1) + (1 - x) (O2 - O1)].
Combining the two equations and re-arranging terms,
O2 = C0 I2 + C1 I1 + C2 O1
C0 = -(Kx - 0.5∆t)/(K - Kx + 0.5∆t)
C1 = (Kx + 0.5∆t)/(K - Kx + 0.5∆t)
C2 = (K - Kx - 0.5∆t)/(K - Kx + 0.5∆t)
C0 + C1 + C2 = 1
Methods of calculation
(1) I is given for all time interval. O is equal to I at t = 0.
(2) Determine K, x, C0, C1, and C2 .
(3) Using I1 (Cell-B3), I2(B4) and O1(F3), calculate O2 and enter it in Cell-F4.
Storage is a unique function of outflow discharge (Q or O).
∴ S = f (O)
The relationship is given by a discharge rating curve.
In reservoir routing, we re-arrange the balance equation;
(S2/∆t + O2/2) = (S1/∆t - O1/2) + (I1 + I2)/2
Given, I1, I2, and (S1/∆t - O1/2), we want to estimate the left-hand side.
The first step is to determine the relationships between the water stage (H), S and O. Topographic or bathymetric survey can be used to obtain the H-S relationship. Stream discharge measurement is conducted to determine the H-O relationship.
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