HYDROLOGIC ROUTING Preamble The flood in River can

HYDROLOGIC ROUTING Preamble § The flood in River can be treated simply as a uniformly progressive flow. “However, if the River is irregular and resistance of flow is high, the configuration of this wave is modified as it moves with time and space”. The determination of this modification is called routing. § Thus, routing is a “process used to determine the variation of flow rate for a flood wave as it moves through water course”. Hydrological routing is an important technique necessary for the complete solution of flood control problems and for the satisfactory operation of a flood prediction service. Routing technique is applied to both river and reservoir. § Routing is classified into following two methods; i. Hydrologic Routing ii. Hydraulic Routing

HYDROLOGIC ROUTING Hydrologic Routing Hydrologic routing is “based on Storage Continuity Equation”. If ‘I’ is the inflow rate, ‘O’ is outflow rate, ‘S’ is the Storage, ‘t’ is the time interval then the continuity equation is; I – O = ΔS …………………. (1) Δt The above equation can be written from the enclosed Fig-’a’ as under ( l 1 + I 2)– (O 1 + O 2) = S 2 – S 1 2 2 Δt [l 1 + I 2]Δt– [O 1 + O 2]Δt = S 2 – S 1 ……. . (2) 2 2 “Almost all storage or Hydrologic routing methods are based on above equation”. § § The peak of inflow Hydrographs are attenuated and delayed as shown by attenuation and lag as shown in the enclosed Fig-’ 1(a)’. Assuming negligible loss or gain of water in the reach the total area under the two hydrographs is equal. The difference between ordinates of inflow and outflow Hydrographs shown by dotted area is the storage up to time ‘t’ Fig-1(b), Fig-1(c) shows storage curve. Peak storage occurs when ‘I’ = ‘O’.

HYDROLOGIC ROUTING Hydrologic Routing (Picture ‘I’ vs ‘O’)

HYDROLOGIC ROUTING Relationship of Outflow and Storage § The enclosed figure shows the plot of outflow ‘O’ against storage ‘S’. When outflow ‘Q’ increases initially, storage also increases. But after sometime, it starts decreasing, thus the “resulting curve represents a loop with “rising and falling limbs as shown in the figure. § “The dotted line between the two limbs represents storage - outflow relationship of steady flow”.

HYDROLOGIC ROUTING Relationship of Outflow and Storage

HYDROLOGIC ROUTING The General Storage Equations § Flow in river is unsteady. “In unsteady flow, storage in a channel reach is dependent on inflow ‘l’ outflow ‘O’, channel geometry, hydraulic characteristics and its control features”. § If ‘y’ is the depth, following relationship of ‘I’, ‘O’, ‘Si’, (Storage at upstream inflow section), So (Storage at downstream outflow section) may be assumed; I = ayn …………. (2) O = ayn …………. (3) Si = bym …………. (4) S o = bym …………. (5) § Substituting the value of ‘y’ from equation (2), (4) Si = b [I/a]m/n …………. (6) Similarly So = b [O/a]m/n …………. (7)

HYDROLOGIC ROUTING The General Storage Equations § Let ‘X’ is the non-dimensional factor that defines the relative weights mto the inflow and outflow in determination of storage volume within the reach. “Then storage ‘S’ at any given time may be expressed as”; S = XSi + (I – X)So ………… (8) § The weighing effect ‘X’ may be defined by the fact that if storage is a sole function of ‘O’ only, then X = 0. When storage has the effect of back water at upstream, ‘X’ is greater than ‘ 0’. In uniform flow, since equal effect of both inflow and outflow is there and therefore, X = 0. 5 § Substituting Si and So from equations (6) & (7) to equation (8), it may be written as; S = K > [(Xlx + (1 – X)Ox] …………. . (9) where K=b a m/n and x = m n

HYDROLOGIC ROUTING Muskingum Routing Equation Muskingum proposed the value of exponents in the equation (9) to be unity. S = K[XI + (1 – X)O] ……… (10) This is called Muskingum Routing Equation. “The values of ‘K’ and ‘X’ are determined by the following method”; Determination of constant ‘K’ and ‘X’ i. Collect flood data of the channel reach from previous year floods. ii. Assume appropriate value of X (i. e. , 0 < X < 0. 3) iii. Calculate values of the term [XI + (1 – X)O] with the chosen value of ‘X” (Say X = 0. 3). iv. Estimate Storage ‘S’ at different times from known values of inflows ‘I’ and outflow ‘O’

HYDROLOGIC ROUTING v. Plot ‘S’ vs [XI + (1 – X)O] for different assumed ‘X’ values (say, X = 0. 3, 0. 25, 0. 20, 0. 1, etc. ) vi. It is seen that for values of ‘X’ equal to 0. 3, 0. 2, …. . “Plot forms a loop”. At some particular value of ‘X’, curve rises and then traces back almost the same path say (OB). “The value of ‘X’ at which this situation arises is the value of ‘X’ to be determined”. vii. “For example, let in our case it happens when ‘X’= 0. 1 i. e. , ‘X’ value is equal to 0. 1. Extending ‘OB’ to ‘A’ the slop of ‘OA’ is the value of ‘K’. viii. It is seen that the unit ‘K’ is time (hrs or days), and “it is approximately equal to travel time of flood wave through the reach or basin lag”.

HYDROLOGIC ROUTING Muskingum Routing Equation

HYDROLOGIC ROUTING Muskingum Method of Routing From the Muskingum Routing equation, determine the storage at different intervals; S = K[XI + (1 – X)O] S 1 = K[XI 1 + (1 – X)O 1] ………………… (11) S 2 = K[XI 2 + (1 – X)O 2] ……………. …. . (12) S 2 – S 1 = K[X(I 2 – I 1)(1 – X)(O 2 - O 1)] ……… (13) Also from equation (2) S 2 – S 1 = [{I 1 + I 2}Δt – {O 1 + O 2}] Δt ……. (13 -A) 2 2 Equating the above Equation S 2 – S 1 [I 1 + I 2] Δt – [O 1 +O 2] Δt = K[X(I 2 – I 1) + (1 – X) (O 2 – O 1)] 2 2

HYDROLOGIC ROUTING Simplifying further, Muskingum Routing Equation is obtained as; O 2 = C 1 I 2 + C 2 I 1 + C 3 O 1 …………. (14) § § C 1 = 0. 5Δt – KX K – KX + 0. 5Δt …………. (15) C 2 = 0. 5Δt + KX K – KX + 0. 5Δt …………. (16) C 3 = K – KX - 0. 5Δt K – KX + 0. 5Δt …………. (17) C 1 + C 2 + C 3 = 1 …………. (18) In the equation (14), initial outflow O 1, I 2, C 1, C 2 and C 3 are known (as K, X, Δt are known), so O 2 can be computed. Next to compute O 3 = C 1 I 2 + C 2 I 2 + C 3 O 2, RHS is known so O 3 is computed. Similarly with known O 3, O 4 is computed. Thus, other values of O 5, O 6, …, On On may be computed. “Hence outflow Hydrograph (O vs t) is known, and thus the flood is routed. “The routed Hydrograph is shown in the enclosed figure”.

HYDROLOGIC ROUTING Inflow and Outflow Routed Hydrograph (Picture)

HYDROLOGIC ROUTING Other Methods of Computing ‘K’ Clark’s Method An empirical equation given by Clark is; K = CL √s Here L = Length of main stream in miles s = mean slope of the channel C = constant, varies from 0. 8 to 2. 2

HYDROLOGIC ROUTING Linsley’s Method He suggested that K = b. L√A √s Where A = Drainage area in km 2 b = varies from 0. 01 to 0. 03 when L is in km. L = Length of main stream in miles s = Mean Slope of the Channel

HYDROLOGIC ROUTING Example of Inflow and Outflow Hydrograph (Routed)

HYDROLOGIC ROUTING Example of Inflow and Outflow Hydrograph (Routed) Contd….

HYDROLOGIC ROUTING Example of Inflow and Outflow Hydrograph (Routed) iii) Now Plot S i. e. (∑ΔS) vs [XI + (1 – X) O] And S, i. e. (∑ΔS) vs [XI + (1 – X) O] when X = 0. 3 when X = 0. 25 “From Tables 12. 1 and 12. 2. The above data is taken and the graph is plotted as per enclosed figure”. iv) For X = 0. 25, Slope of OA, K = S [XI + (1 – X) O] = 450 = 14. 06 = 14 hrs. Hence for X = 0. 25 K = 14 hrs.

HYDROLOGIC ROUTING Inflow and Outflow Hydrograph (Routed)

HYDROLOGIC ROUTING Reservoir Routing § The routing technique when applied to reservoir, is called reservoir or storage routing. § In reservoir, inflow l(t) comes from river which is known. Outflow O(t) from reservoir is controlled by spell way gates. Thus both storage S(t) and outflow O(t) vary with time which causes the variation of stage or elevation H(t). When I(t) is known. § Determination of S(t), O(t) and H(t) is called reservoir routing. § Essential data required for reservoir routing are; i) Field data to establish relationship among outflow O(t), Storage S(t) and stage or reservoir elevation H(t). ii) Inflows I(t) at any time step Δt i. e. inflow Hydrograph. iii) Known initial storage S 1, outflow O 1 at the start, i. e. , at t = 0 as boundary condition.

HYDROLOGIC ROUTING Methods of Reservoirs Routing The most common and simple methods that are frequently used by field engineers are; i) Goodrich Method ii) Modified Pul’s Method Goodrich Method In Goodrich method, continuity equation is written as; [I 1 + I 2] – [O 1 + O 2] = S 2 – S 1 ………. . (19) 2 2 Δt or [2 S 2 + O 2] = I 1 + I 2 + [2 S 1 – O 1] …. (20) Δt Δt “This equation is called Goodrich equation of reservoir routing”.

HYDROLOGIC ROUTING Stepwise Procedure to solve the above equation is as under; i. Select time increment Δt. ii. From previous field data, plot stage or elevation H vs [2 S/Δt + O] curve and elevation ‘H’ vs outflow ‘O’ curve (as per figure below)

HYDROLOGIC ROUTING iii. In the beginning of the routing, the data of storage S 1, outflow O 1 and elevation are known so we can calculate [2 S 1/Δt + O 1] iv. In equation (20) R. H. S. is known, so [2 S 2/Δt + O 2] term is known. v. From known [2 S 2/Δt + O 2], find out corresponding H(m) (is known) and from ‘H’, find O 2 from the curve ‘O’ vs ‘H’ vi. Calculate [2 S 2/Δt - O 2] term subtracting 2 O 2 from [2 S 2/Δt + O 2] and again R. H. S. is known for calculating [2 S 3/Δt – O 3] vii. Keep on repeating the procedures until all outflows O 2, O 3, O 4, …… On are obtained, i. e. all floods are routed.

HYDROLOGIC ROUTING Modified Pul’s Method From equation of continuity, [I 1 + I 2] – [O 1 + O 2] = S 2 – S 1 2 2 Δt or ………. (21) ½(I 1 + I 2)Δt + [S 1– O 1 Δt] = [S 2+ O 2 Δt] …… (22) 2 2 Modified Pul’s method is based on above equation. In the Pul’s method, data of storage ‘S’, elevation ‘H’ and outflow ‘O’ are required to prepare the routing curves ‘O’ vs ‘E’ and [S + O/2Δt] vs ‘E’ as shown (figure enclosed)

HYDROLOGIC ROUTING Methods of Reservoirs Routing Steps required in Routing i. Time interval Δt is selected within 20% to 40% of time of rise of inflow Hydrograph. ii. From the available past data, routing curves ‘O’ vs ‘H’ and [S 2 + O/2Δt] vs ‘H’ are plotted as shown (figure enclosed).

HYDROLOGIC ROUTING iii. Initial outflow O 1, S 1, Δt and all inflows are known. Therefore L. H. S. of equation (22) is known. Hence [S 2 + O 2/2 Δt] is known iv. Determine corresponding ‘H’ from known [S 2 + O 2/2 Δt] from the routing curve of [S + O/2 Δt] vs ‘H’. v. Determine O 2 from ‘O’ vs ‘H’ curve. vi. Subtract O 2 Δt from known [S 2 + O 2/2 Δt] to [S 2 - O 2/2 Δt] vii. Again L. H. S. is known to compute [S 3 + O 3/2 Δt] viii. Repeat the process until all the outflows O 4, O 5. . . , On are computed

HYDROLOGIC ROUTING Other Method i) Runge-Kutta Method ii) Linear Reservoir Modelling Method. iii) Composite Model Method