MATHEMATICAL MODEL

16.8 MATHEMATICAL MODEL

A linear reservoir attenuates the peak of an inflow hydrograph and a linear channel translates inflow hydrograph in time, which are representative of the physical action performed by the catchment. Hence, a model of a linear reservoir connected in series with a linear channel may

be selected in this study. A linear channel is defined by the delay time or the time of travel of the flood wave and is approximately determined with the help of time to peaks of flood events on the upstream and downstream stations of a river reach. Muskingum equation defines the linear reservoir by its storage-discharge relation as

...(16.34) From the principle of continuity

S = K[xI + (1 – x)O]

HG ...(16.35)

2 KJ HG 2 KJ

From Eq. (16.34)

1 S = K[xI 1 + (1 – x)O 1 ]

2 S = K[xI 2 + (1 – x)O 2 ]

...(16.36) From Eqs. (16.35) and (16.36), O 2 =C 0 I 2 +C 1 I 1 +C 2 O 1 ...(16.37)

S 2 –S 1 = K[x(I 2 –I 1 ) – (1 – x)(O 2 –O 1 )]

C 2 = ...(16.37a)

The routing period t (time interval between O 1 and O 2 ) should be equal to or less than the time of travel through the reach.

Eq. (16.37) for successive time intervals may be written as O 3 =C 0 I 3 +C 1 I 2 +C 2 O 2 ...(16.38) O 4 =C 0 I 4 +C 1 I 3 +C 2 O 3 ...(16.39)

Eq. (16.38) – (16.37) gives:

...(16.40) Eq. (16.39) – (16.38) gives:

O 3 –O 2 =C 0 (I 3 –I 2 )+C 1 (I 2 –I 1 )+C 2 (O 2 –O 1 )

...(16.41) Assuming the stage (gauge height)—discharge curves as a straight line (as a first ap- proximation), if the slopes of the curve on upstream and downstream are 1/a and 1/b,

O 4 –O 3 =C 0 (I 4 –I 3 )+C 1 (I 3 –I 2 )+C 2 (O 3 –O 2 )

Fig. 16.18 (a) then

and

H – Upstream gauge

G – Downstream

gauge

Actual rating

curve

Assumed

Actual rating

Assumed

rating curve

curve

1 rating curve

3 a Slope a

(m)

H 2 H 2 –H 1 1 (m) G 3

G 2 G–G 2 1 1 Stage

1 O–O = 2 1 b

Stage

Inflow I (cumec)

Outflow O (cumec)

a. Stage – discharge rating curves assumed linear H – Upstream gauge

G – Downstream gauge Actual rating

Actual

(m) curve

a 3 1 rating curve

H (m)

1 Assumed

a 2 G rating

1 Assumed

b 2 rating curve Stage

curve

1 Stage

Inflow I (cumec)

Outflow O (cumec)

b. Stage–discharge curves divided into three linear parts

Fig. 16.18 Stage-discharge-rating curves Assume

Substituting these in Eq. (16.40)

b (G 3 –G 2 )=C 0 a (H 3 –H 2 )+C 1 a (H 2 –H 1 )+C 2 b (G 2 –G 1 )

or

b 3 –H 2 )+C 1 b (H 2 –H 1 ) Similarly, substitution in Eq. (16.41) gives

–H 3 )+C 1 (H 3 –H 2 ) ...(16.43)

Eqs. (16.42) and (16.43) may be written as

Thus, a number of equations can be obtained from the observed data and solved for x 1 , x 2 and x 3 by the least square technique. Since the number of such equations are very large, from large sets of data, a computer can be used. For example, the equations developed for a straight reach between Muzaffarpur and Rossera on Burhi-Gandak are

Rising stage:

G 3 –G 2 = – 0.401 (G 2 –G 1 ) + 2.826 (H 2 –H 1 ) – 0.94 (H 3 –H 2 ) ...(16.44)

MATHEMATICAL MODELS IN HYDROLOGY

Falling stage:

G 3 –G 2 = – 0.1413(G 2 –G 1 ) + 0.3018(H 2 –H 1 ) + 0.6082(H 3 –H 2 ) ...(16.45) As a further refinement the stage discharge curves may be divided into linear parts, say

three, and the slopes denoted according to the linear ranges in which the stages lie, as shown in Fig. 16.18 (b), and the equations solved by using multiple regression technique.

Contribution due to a major tributary between the base station and the forecasting station has to be taken into account. The results of flood routing between Sikanderpur (Muzaffarpur) and Rossera by differ- ent methods during the 1975 floods are given in Table 16.3 for comparison. It can be seen that the Muskingum method has given more consistent results. A clear picture of forecast and comparison with past values have been found possible only in graphical correlation and minor adjustment based on experience can be done in the predicted value. These are not possible in the mathematical model and also the model cannot give better results in rivers having large scale fluctuations due to existance of control structures, their operation or flashy nature of the stream.

Contribution due to Tributary Contribution due to a major tributary between the base station and the forecasting station has

to be taken into account. For example, three major tributaries of Ganga affect the gauge down- stream at Patna, other than its own (Fig. 16.19).

Table 16.3 Comparison of flood forecast results

Date Time

Level

Level downstream (m)

(hr) upstream

Date

Time

Graphi- Musk in- Mathe- Observed

(m)

(hr)

cal cor-

45.139 44.583* *The large variation in the last observation is due to number of breaches in the embankment

between Muzaffarpur and Rossera .

I, H

ib utar

Trib utar

River

Trib y 2

Downstream gauge (at Patna)

Fig. 16.19 Tributary effect on gauge downstream

HYDROLOGY

In such a case, the modified Muskingum equation can be written as O 2 =C 2 O 1 +C 1 I 1 +C 1 ′I 1 ′+C 1 ″I 1 ″+C 1 ′″I 1 ′″ + C 0 I 2 +C 0 ′I 1 ′

...(16.46) Similarly, O 3 can be written and O 3 –O 2 can be evaluated. Aproximating, its stage (gauge

+C 0 ″I 2 ″+C 1 ′″C 2 ′″

height)-discharge curve, to a straight line, the ultimate equation will be of the form (writing

G 3 –G 2 as G 3.2 and so on)

G 3.2 =x 1 G 2.1 +x 2 H 2.1 +x 3 H 3.2 +x 4 H ′ 2.1 +x 5 H ′ 3.2 +x 6 H ″ 2.1 +x 7 H ″ 3.2 +x 8 H ″′ 2.1 +x 9 H ″′ 3.2 ...(16.47) Number of equations have been formed like this from the observed data and solved for

the constants by the least square technique by using a computer. The constants obtained for the forecasting site at Patna for the 1975 floods are given in Table 16.4, and the levels reached in Table 16.5 (compared with values obtained by graphical correlation, which has given better results).

Table 16.4 Constants for forecasting site at Patna Constant

Rising stage

Falling stage

Table 16.5 Flood forecast results at Patna during 1975

Date Time

Level downstream (m) (hr)

Mathe- Observed

matical model