Module 5. Lecture 3: Channel routing methods

Transcription

1 Lecture 3: Channel routing methods

2 Hydrologic flow routing 2. Channel Routing In very long channels the entire flood wave also travels a considerable distance resulting in a time redistribution and time of translation as well. Thus, in a river, the redistribution due to storage effects modifies the shape, while the translation changes its position in time. In reservoir, the storage is a unique function of the outflow discharge S = f(o). Storage in the channel is a function of both outflow and inflow discharges and hence a different routing method is needed. The water surface in a channel reach is not only parallel to the channel bottom but also varies with time.

3 Hydrologic flow routing 2. Channel Routing Contd The total volume in storage for a channel reach having a flood wave can be considered as prism storage + wedge storage. Prism storage: The volume that would exist if 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. At a fixed depth at a downstream section of a river reach the prism storage is constant while the wedge storage changes from a positive value at an advancing flood to a negative value during a receding flood.

4 Hydrologic flow routing 2. Channel Routing Contd Prism Storage: It is the volume that would exits if 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.

5 Hydrologic flow routing 2. Channel Routing Contd Wedge storage : It is the wedge-like volume formed between the actual water surface profile and the top surface of the prism storage.

6 Hydrologic flow routing 2. Channel Routing Contd At a fixed depth at a downstream section of a river reach, the prism storage is constant while, the wedge storage changes from a positive value for advancing flood to a negative value during a receding flood. Total storage in the channel reach can be expressed as : where k and x are coefficients and m= a constant exponent. It has been found that m varies from 0.6 for rectangular channels to a value of about 1.0 for natural channels, Q = outflow

7 Channel routing Assuming that the cross sectional area of the flood flow section is directly proportional to the discharge at the section, the volume of prism storage is equal to KQ where K is a proportionality coefficient, and the volume of the wedge storage is equal to KX(I- Q), where X is a weighing factor having the range 0 < X < 0.5. The total storage is therefore the sum of two components S = KQ + KX ( I Q) It is known as Muskingum storage equation representing a linear model for routing flow in streams.

8 Channel routing Contd S S = Prism Wedge KQ = KX ( I Q) K is a proportionality coefficient, X is a weighing factor on inflow versus outflow (0 X 0.5) X = Natural stream Advancing Flood Wave I > Q I I Q Q I Q Q Q S S = = KQ + KX ( I Q) K[ XI + (1 X ) Q] Receding Flood Wave Q > I I Q I I

9 Channel routing Contd S = K[ XI + (1 X ) Q] The value of X depends on the shape of the modeled wedge storage. It is zero for reservoir type storage (zero wedge storage or level pool case S = KQ) and 0.5 for a full wedge. In natural streams mean value of X is near 0.2. The parameter K is the time of travel of the flood wave through the channel reaches also known as storage time constant and has the dimensions of time.

10 Channel routing Contd From the Muskingum storage equation, the values of storage at time j and j+1 can be written as and So, change in storage over time interval t is, From the continuity equation the storage for the same time interval t is,

11 Channel routing Contd Equating these two equations, Collecting similar terms and simplifying This is the Muskingum s routing equation for channels

12 Channel routing Contd Muskingum s routing equation for channels: where For best results, the routing interval t should be so chosen that K> t>2kx. If t<2kx, the coefficient C1 will be negative. Generally negative values of coefficients are avoided by choosing appropriate values of t.

13 Channel routing Contd To use Muskingum equation to route a given inflow hydrograph through a channel reach: K, X and Oj should be known. Procedure: (i)knowing K and X, select an appropriate value of t (ii) calculate C1, C2, and C3 (iii) starting from the initial conditions known inflow, outflow calculate the outflow for the next time step. (iv) Repeat the calculations for the entire inflow hydrograph.