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