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Flood control in reservoirs and storage pounds
Journal of Hydrology, 19 (1973) 351-359 © North-Holland Publishing Company, Amsterdam - Printed in The Netherlands
F L O O D C O N T R O L IN R E S E R V O I R S AND STORAGE POUNDS E.J. SARGINSON
University of Sheffield, SheffieM (Great Britain (Accepted for publication February 12, 1973)
ABSTRACT Sarginson, E.J., 1973. Flood control in reservoirs and storage pounds. J. Hydrol., 19:351-359. By approximating a natural flood hydrograph to a single mathematical curve, the routing procedure has been reduced to a dimensionless differential equation. The computer solution, in the form of a pair of tabulated functions enables the length of spillway necessary to produce any desired reduction in the peak flood or any desired limiting head on the spillway to be obtained in a single calculation. The paper shows that only a very slight modification of the procedure is necessary to deal with a spillway having a variable coefficient, or with a reservoir or storage pound having a variable surface area.
INTRODUCTION
The conventional procedure of routing a flood of variable intensity through a reservoir is by means of an arithmetical integration. This has the disadvantage that it cannot be used directly to calculate the length of spillway necessary either to reduce the peak flow or limit the maximum water level to a desired value. For either of these purposes it is necessary to take trial lengths of weir, carry out the usual procedure with each trial length and interpolate to get the required length. In a previous paper (Sarginson, 1969), the author found that by approximating the input hydrograph to a single mathematical curve, the "Muskingum" equation for stream-flow routing could be dealt with analytically, with a considerable reduction in the time of computation. In the present paper, a similar procedure is applied to reservoir routing, and the resulting tabulated functions can be used to compute the length of spillway necessary to achieve the desired purpose in a single calculation. The analysis strictly applies to a reservoir of constant surface area controlled by a spillway with a constant coefficient, but only slight modifications are necessary to deal with variations in either of these two quantities.
352
E.J. SARGINSON
ANALYSIS OF RESERVOIR ROUTING
The rise and fall of nearly all natural flood hydrographs resemble a curve having the equation:
Q =k
e -t/~
( 1)
If this is combined with the reservoir routing equation (Q - q = dS/dt) for a reservoir of constant surface area A controlled by a weir of length L and coefficient C, the following equation results:
A ~+CLh3/2 = k(t)n
e -t/~
(2)
Eq. 2 can be made dimensionless by the following substitutions: let t/a = x;
h(CXL2aZ/A2) = y ; and k(C2L2t~3/A 3) =/3; eq. 2 now becomes: dy + y3/2 =/3x n e-X dx
(3)
Eq. 3 has been solved for different values of n and 13by computer, which was programmed to print out values of x, y and y3/2 corresponding to maximum discharge over the spillway.
NOTATION List of symbols used A C h k L n
O q S t X
y ot
0
reservoir surface area coefficient of weir or Venturi flume head on weir or flume flood parameter length of weir or throat width of flume flood parameter inflow to reservoir discharge from reservoir stored volume time dimensionless time-variable (t/a) dimensionless head-variable [h (C'~L2a2/A2)] flood parameter dimensionless flood parameter [k(C2L2aa/Aa)]
FLOOD CONTROL IN RESERVOIRS AND STORAGE POUNDS
353
CALCULATION OF THE FLOOD PARAMETERS
The three quantities k, n and a in eq. 1 may be called the "flood parameters". To evaluate them, the m e t h o d of Nash (1960) may be adapted. Nash derived eq. 1 for the Instantaneous Unit Hydrograph, and showed h o w the three parameters could be obtained by taking the area under the curve to infinity, as well as the first and second moments o f area. Unless a considerable length of the decaying "tail" of the hydrograph is included, the time-scale cannot be assumed as even approximating to infinity, and the author has modified Nash's relations by using a table of the Incomplete Gamma Function. For this purpose, a table of the statistical function "Chi-Square" was found to be suitable (Pearson and Hartley, 1966). The following relations were obtained:
~,Qt2× Z Q _ F1 (n) (XQt) 2 ZQt/~,Q
= aF2 (n)
E,Q
= k a F 3 (n )
(4)
Values o f F 1 (n), F2(n) and F 3 (n) are given in Table I. The values to infinity are the same as those given by Nash. In using eq. 4 with Table I the following points should be noted: (a) The hydrograph data should be tabulated and summated in unit intervals of time. The computed value of a is then in the same unit of time, and k has the same dimensions as Q. (b) To get the correct value of x/n, it is assumed that the maximum value of Q occurs at a time represented by x = n. The data are then tabulated as far as a suitable multiple o f n and the corresponding lines of Table I are entered. (c) It is sufficiently accurate to round off n to the nearest integer. EXAMPLE OF THE USE OF EQ. 4
The full-line hydrograph in Fig. 1 has been taken from Bruce and Clark (1966). The peak flow is at 13½ h, but the published data extends only to 24 h. This has been extended to 27 h (x = 2n) by assuming exponential decay for the descending limb of the curve, i.e., by plotting log Q against t, an almost straight line was obtained; this could easily be extrapolated. In this way, the hydrograph was found to approximate to the equation: Q = 55xSe-X where a = 2.70 h = 9,720 sec. This equation is represented by the broken line in Fig. 1.
354
E.J. SARGINSON
¢~
¢'q
t",-- t"-- t",-
o~ ~
~O ,..-~ t--,I ¢"q t"q t".- 0 0 ~O
cA
,..; ,-.; ,..d ,-.; ,--;
tt~
,..~
eq
,.-;
t/~
,-.-~ ,-'*
,...;
,-~
lyq ¢'q
¢'q
. . . .
t"q
e¢~ , ~
,...; ,...; ,-..; ,-d ,...;
•-~
¢",1 e,~
.,~
8
,d
,< [..,
a7
FLOOD CONTROL IN RESERVOIRS AND STORAGE POUNDS
~ ,
Q
( .~/s )
Nb
,1 1 ,
"~
- NA:rURAk HYDROGRAPH
,s ~ •
\ ~ k
/
8OO
• d
355
~'~" - Q-
SS=="s c"==
/ 600
~L" ,/
40q
\
/ oJJ
IO
o
20
30
HRS
Fig.l. Hydrograph for problem No. 1. FLOOD ROUTING PROCEDURE
The solution to eq. 3 is most conveniently expressed in the form: max. q -
ky3/2
(5) and max. h = ky . a_-
t~
A
and values o f y3~/j3 and y/[3 have been given for different values of/3 and n in Tables II and III, respectively. The following problem will illustrate their rise.
Problem No. 1 "Given the flood hydrograph of Fig. 1 and a reservoir of surface area 10 7 m 2 : (a) Find the maximum discharge over a spillway weir 200 m long with a coefficient o f 2.10. (b) Find the length o f weir with the same coefficient which will reduce the peak discharge to 800 m 3/sec. (c) Find the length of weir which will limit the head to 1 m " .
356
E.J. S A R G I N S O N
~D
~5
c:;
,::5
,:5
~5
c5
= t.rd ,.d ,< [..,
o t~
:::t
FLOOD CONTROL IN RESERVOIRS AND STORAGE POUNDS
357
Problem (a) _ k ( C L ) 2 a 3 _ 55.0× (420) 2 × (9720) 3 _ 0.0089
A3
(107) 3
If Table II is entered at n = 5 and ~ = 0.0089 (interpolating) it is found that y 3 / 2 / ~ = 17.9.
Then qmax = 17.9X 55.0 = 985 m3/sec (with a max. head of 1.72 m). An arithmetical solution gives a maximum discharge of 1,012 m 3/sec which differs from the analytical solution by only 2%%. P r o b l e m (b)
For a peak discharge of 800 m 3/sec, y3/2//3 = 800/55 = 14.56. From Table II (interpolating),/3 = 0.00185. Then 0.00185 = (2"10L)2 × 55 × (9720) 3 1021 giving L = 91.2 m (with a max. head of 2.6 m). Problem (c)
For a maximum head of 1 m y/(3= (1 × 107)/(55 X 9 7 2 0 ) = 18.74. From Table III (interpolating),/3 = 0.062. Then 0.062 = (2"10L)2 × 55X (9720) 3 1021 giving L = 527 m and qmax = 1,106 m 3/sec. It will be appreciated that problem (a) could have been solved as easily as this by the usual arithmetical method, but the direct computation o f the lengths of weir in problems (b) and (c) could not have been carried out by any other routing methods known to the author. When the reservoir area is c o n s t a n t b u t the weir coefficient varies
With any spillway likely to be used in practice, it is very unlikely that the weir coefficient will be constant. For example, a form of spillway crest in frequent use in the U.S.A. is the Waterways Experimental Station (W.E.S.) profile, which is designed to avoid negative pressures at the crest and at the same time to have as high a coefficient as possible. The coefficients at different heads are given by Chow (1959). Plotted logarithmically, they approximate to a straight line with the equation:
358
E.J. SARGINSON
C = 2.225
h)0.13
HDD
(6)
If this is inserted into eq. 2, then by a similar procedure to the previous one, a dimensionless differential equation similar to eq. 3 is obtained in which the power o f y = 1.63 instead of 1.50. The computer programme was modified to solve this and a set of data similar to Tables II and III obtained. However, the application o f this data to several problems showed that the same results could be obtained with a spillway having a constant coefficient of 2.17. When the weir coefficient & constant but the reservoir area varies For a spillway controlling the discharge from a large impounding reservoir, the change in surface area as the head rises is not large, so the assumption of a constant surface area is justified. However, in works designed specifically to control flooding, the surface area is likely to change substantially with changes in the water level. For instance, at Hemel Hempstead New Town (Carden, 1960), the surface-water r u n o f f from part of the town was discharged into a small fiver, and flooding was prevented by enlarging a reach of the river to form a storage pound, which was controlled by a standing-wave Venturi flume. The pound had a trapezoid cross-section, so that the surface area increased as the head on the flume rose. In the design of such a pound, it will usually be necessarY to reduce the peak flood to below the bank-full discharge of the river downstream of the pound, and at the same time, the maximum water level in the pound will determine the maximum head on the flume. So the dimensions of the flume are determined, and the problem becomes one of calculating the capacity of the pound. The foregoing theory can be used to calculate the effective surface area of the pound, and it may be assumed that this occurs at 80% of the maximum depth in the pound. The following problem will illustrate the procedure. Problem No. 2 "Using the full-line hydrograph shown in Fig. 2, design a pound to reduce the peak flood to 6.25 m3/sec at a head of 1.5 m. The flume coefficient is 1.70 and the sides of the pound should slope at 1/ 1.5". Solution (1) The required discharge and head require a flume 2 m wide. (2) By working in intervals of 5 m i n - 4 5 min the flood hydrograph is found to approximate to the formula: Q = 10.75 x26e -x, where a = 1.352 (X 5 min = 405 sec)
FLOOD CONTROL IN RESERVOIRS AND STORAGE POUNDS IO
=.,
:,
~
\ \~=. -o=
~Ts
2
6
359
~
/ 8
/
Q
(m3/s)
°
!
,
,f
E2.S ,= -
,,,,
/
- .-,,. %,
oJ
I0
20
30
40
SO
6 0 MINS.
Fig. 2. Hydrograph for problem No. 2.
(3)y3/2/f3
= 6 . 2 5 / 1 0 . 7 5 = 0.581. F r o m Table II,/3 = 0.190. (4) F r o m Table III, y/[3 = 1.215 = (A × 1 . 5 0 ) / ( 1 0 . 7 5 × 405), giving A = 3,530 m 2 . (5) Select a suitable length o f p o u n d , say 300 m, and assume t h a t the m e a n width o f 11.8 m occurs at 0.8 o f the m a x i m u m d e p t h , i.e., at 1.2 m. T h e n the bed w i d t h is 8.2 m. (6) C h e c k the design b y the usual arithmetical m e t h o d , and find t h a t the m a x i m u m discharge is 6.0 m 3/sec, with a m a x i m u m h e a d o f 1.46 m.
REFERENCES Bruce, J.P. and Clark, R.H., 1966. Introduction to Hydrometeorology. Pergamon, Oxford, p. 201. Carden, T.H., 1960. Hemel Hempstead New Town - Engineering Works. J. Inst. Munic. Eng., 87: 51-64. Chow, V.T., 1959. Open.channel Hydraulics. McGraw-Hill, New York, N.Y., p. 366. Nash, J.E., 1960. A unit hydrograph study, with particular reference to British Catchments. Proc. Inst. Or. Eng. (London), 17: 249-282. Pearson, E.S. and Hartley, H.O., 1966. Biometrika Tables for Statisticians, 1. Cambridge University Press, pp. 128-135. Sarginson, E.J., 1969. Streamflow routing analysis~.Civ. Eng. andPub. Wks. Rev., 64: 9 8 2 - 9 8 3 .
F L O O D C O N T R O L IN R E S E R V O I R S AND STORAGE POUNDS E.J. SARGINSON
University of Sheffield, SheffieM (Great Britain (Accepted for publication February 12, 1973)
ABSTRACT Sarginson, E.J., 1973. Flood control in reservoirs and storage pounds. J. Hydrol., 19:351-359. By approximating a natural flood hydrograph to a single mathematical curve, the routing procedure has been reduced to a dimensionless differential equation. The computer solution, in the form of a pair of tabulated functions enables the length of spillway necessary to produce any desired reduction in the peak flood or any desired limiting head on the spillway to be obtained in a single calculation. The paper shows that only a very slight modification of the procedure is necessary to deal with a spillway having a variable coefficient, or with a reservoir or storage pound having a variable surface area.
INTRODUCTION
The conventional procedure of routing a flood of variable intensity through a reservoir is by means of an arithmetical integration. This has the disadvantage that it cannot be used directly to calculate the length of spillway necessary either to reduce the peak flow or limit the maximum water level to a desired value. For either of these purposes it is necessary to take trial lengths of weir, carry out the usual procedure with each trial length and interpolate to get the required length. In a previous paper (Sarginson, 1969), the author found that by approximating the input hydrograph to a single mathematical curve, the "Muskingum" equation for stream-flow routing could be dealt with analytically, with a considerable reduction in the time of computation. In the present paper, a similar procedure is applied to reservoir routing, and the resulting tabulated functions can be used to compute the length of spillway necessary to achieve the desired purpose in a single calculation. The analysis strictly applies to a reservoir of constant surface area controlled by a spillway with a constant coefficient, but only slight modifications are necessary to deal with variations in either of these two quantities.
352
E.J. SARGINSON
ANALYSIS OF RESERVOIR ROUTING
The rise and fall of nearly all natural flood hydrographs resemble a curve having the equation:
Q =k
e -t/~
( 1)
If this is combined with the reservoir routing equation (Q - q = dS/dt) for a reservoir of constant surface area A controlled by a weir of length L and coefficient C, the following equation results:
A ~+CLh3/2 = k(t)n
e -t/~
(2)
Eq. 2 can be made dimensionless by the following substitutions: let t/a = x;
h(CXL2aZ/A2) = y ; and k(C2L2t~3/A 3) =/3; eq. 2 now becomes: dy + y3/2 =/3x n e-X dx
(3)
Eq. 3 has been solved for different values of n and 13by computer, which was programmed to print out values of x, y and y3/2 corresponding to maximum discharge over the spillway.
NOTATION List of symbols used A C h k L n
O q S t X
y ot
0
reservoir surface area coefficient of weir or Venturi flume head on weir or flume flood parameter length of weir or throat width of flume flood parameter inflow to reservoir discharge from reservoir stored volume time dimensionless time-variable (t/a) dimensionless head-variable [h (C'~L2a2/A2)] flood parameter dimensionless flood parameter [k(C2L2aa/Aa)]
FLOOD CONTROL IN RESERVOIRS AND STORAGE POUNDS
353
CALCULATION OF THE FLOOD PARAMETERS
The three quantities k, n and a in eq. 1 may be called the "flood parameters". To evaluate them, the m e t h o d of Nash (1960) may be adapted. Nash derived eq. 1 for the Instantaneous Unit Hydrograph, and showed h o w the three parameters could be obtained by taking the area under the curve to infinity, as well as the first and second moments o f area. Unless a considerable length of the decaying "tail" of the hydrograph is included, the time-scale cannot be assumed as even approximating to infinity, and the author has modified Nash's relations by using a table of the Incomplete Gamma Function. For this purpose, a table of the statistical function "Chi-Square" was found to be suitable (Pearson and Hartley, 1966). The following relations were obtained:
~,Qt2× Z Q _ F1 (n) (XQt) 2 ZQt/~,Q
= aF2 (n)
E,Q
= k a F 3 (n )
(4)
Values o f F 1 (n), F2(n) and F 3 (n) are given in Table I. The values to infinity are the same as those given by Nash. In using eq. 4 with Table I the following points should be noted: (a) The hydrograph data should be tabulated and summated in unit intervals of time. The computed value of a is then in the same unit of time, and k has the same dimensions as Q. (b) To get the correct value of x/n, it is assumed that the maximum value of Q occurs at a time represented by x = n. The data are then tabulated as far as a suitable multiple o f n and the corresponding lines of Table I are entered. (c) It is sufficiently accurate to round off n to the nearest integer. EXAMPLE OF THE USE OF EQ. 4
The full-line hydrograph in Fig. 1 has been taken from Bruce and Clark (1966). The peak flow is at 13½ h, but the published data extends only to 24 h. This has been extended to 27 h (x = 2n) by assuming exponential decay for the descending limb of the curve, i.e., by plotting log Q against t, an almost straight line was obtained; this could easily be extrapolated. In this way, the hydrograph was found to approximate to the equation: Q = 55xSe-X where a = 2.70 h = 9,720 sec. This equation is represented by the broken line in Fig. 1.
354
E.J. SARGINSON
¢~
¢'q
t",-- t"-- t",-
o~ ~
~O ,..-~ t--,I ¢"q t"q t".- 0 0 ~O
cA
,..; ,-.; ,..d ,-.; ,--;
tt~
,..~
eq
,.-;
t/~
,-.-~ ,-'*
,...;
,-~
lyq ¢'q
¢'q
. . . .
t"q
e¢~ , ~
,...; ,...; ,-..; ,-d ,...;
•-~
¢",1 e,~
.,~
8
,d
,< [..,
a7
FLOOD CONTROL IN RESERVOIRS AND STORAGE POUNDS
~ ,
Q
( .~/s )
Nb
,1 1 ,
"~
- NA:rURAk HYDROGRAPH
,s ~ •
\ ~ k
/
8OO
• d
355
~'~" - Q-
SS=="s c"==
/ 600
~L" ,/
40q
\
/ oJJ
IO
o
20
30
HRS
Fig.l. Hydrograph for problem No. 1. FLOOD ROUTING PROCEDURE
The solution to eq. 3 is most conveniently expressed in the form: max. q -
ky3/2
(5) and max. h = ky . a_-
t~
A
and values o f y3~/j3 and y/[3 have been given for different values of/3 and n in Tables II and III, respectively. The following problem will illustrate their rise.
Problem No. 1 "Given the flood hydrograph of Fig. 1 and a reservoir of surface area 10 7 m 2 : (a) Find the maximum discharge over a spillway weir 200 m long with a coefficient o f 2.10. (b) Find the length o f weir with the same coefficient which will reduce the peak discharge to 800 m 3/sec. (c) Find the length of weir which will limit the head to 1 m " .
356
E.J. S A R G I N S O N
~D
~5
c:;
,::5
,:5
~5
c5
= t.rd ,.d ,< [..,
o t~
:::t
FLOOD CONTROL IN RESERVOIRS AND STORAGE POUNDS
357
Problem (a) _ k ( C L ) 2 a 3 _ 55.0× (420) 2 × (9720) 3 _ 0.0089
A3
(107) 3
If Table II is entered at n = 5 and ~ = 0.0089 (interpolating) it is found that y 3 / 2 / ~ = 17.9.
Then qmax = 17.9X 55.0 = 985 m3/sec (with a max. head of 1.72 m). An arithmetical solution gives a maximum discharge of 1,012 m 3/sec which differs from the analytical solution by only 2%%. P r o b l e m (b)
For a peak discharge of 800 m 3/sec, y3/2//3 = 800/55 = 14.56. From Table II (interpolating),/3 = 0.00185. Then 0.00185 = (2"10L)2 × 55 × (9720) 3 1021 giving L = 91.2 m (with a max. head of 2.6 m). Problem (c)
For a maximum head of 1 m y/(3= (1 × 107)/(55 X 9 7 2 0 ) = 18.74. From Table III (interpolating),/3 = 0.062. Then 0.062 = (2"10L)2 × 55X (9720) 3 1021 giving L = 527 m and qmax = 1,106 m 3/sec. It will be appreciated that problem (a) could have been solved as easily as this by the usual arithmetical method, but the direct computation o f the lengths of weir in problems (b) and (c) could not have been carried out by any other routing methods known to the author. When the reservoir area is c o n s t a n t b u t the weir coefficient varies
With any spillway likely to be used in practice, it is very unlikely that the weir coefficient will be constant. For example, a form of spillway crest in frequent use in the U.S.A. is the Waterways Experimental Station (W.E.S.) profile, which is designed to avoid negative pressures at the crest and at the same time to have as high a coefficient as possible. The coefficients at different heads are given by Chow (1959). Plotted logarithmically, they approximate to a straight line with the equation:
358
E.J. SARGINSON
C = 2.225
h)0.13
HDD
(6)
If this is inserted into eq. 2, then by a similar procedure to the previous one, a dimensionless differential equation similar to eq. 3 is obtained in which the power o f y = 1.63 instead of 1.50. The computer programme was modified to solve this and a set of data similar to Tables II and III obtained. However, the application o f this data to several problems showed that the same results could be obtained with a spillway having a constant coefficient of 2.17. When the weir coefficient & constant but the reservoir area varies For a spillway controlling the discharge from a large impounding reservoir, the change in surface area as the head rises is not large, so the assumption of a constant surface area is justified. However, in works designed specifically to control flooding, the surface area is likely to change substantially with changes in the water level. For instance, at Hemel Hempstead New Town (Carden, 1960), the surface-water r u n o f f from part of the town was discharged into a small fiver, and flooding was prevented by enlarging a reach of the river to form a storage pound, which was controlled by a standing-wave Venturi flume. The pound had a trapezoid cross-section, so that the surface area increased as the head on the flume rose. In the design of such a pound, it will usually be necessarY to reduce the peak flood to below the bank-full discharge of the river downstream of the pound, and at the same time, the maximum water level in the pound will determine the maximum head on the flume. So the dimensions of the flume are determined, and the problem becomes one of calculating the capacity of the pound. The foregoing theory can be used to calculate the effective surface area of the pound, and it may be assumed that this occurs at 80% of the maximum depth in the pound. The following problem will illustrate the procedure. Problem No. 2 "Using the full-line hydrograph shown in Fig. 2, design a pound to reduce the peak flood to 6.25 m3/sec at a head of 1.5 m. The flume coefficient is 1.70 and the sides of the pound should slope at 1/ 1.5". Solution (1) The required discharge and head require a flume 2 m wide. (2) By working in intervals of 5 m i n - 4 5 min the flood hydrograph is found to approximate to the formula: Q = 10.75 x26e -x, where a = 1.352 (X 5 min = 405 sec)
FLOOD CONTROL IN RESERVOIRS AND STORAGE POUNDS IO
=.,
:,
~
\ \~=. -o=
~Ts
2
6
359
~
/ 8
/
Q
(m3/s)
°
!
,
,f
E2.S ,= -
,,,,
/
- .-,,. %,
oJ
I0
20
30
40
SO
6 0 MINS.
Fig. 2. Hydrograph for problem No. 2.
(3)y3/2/f3
= 6 . 2 5 / 1 0 . 7 5 = 0.581. F r o m Table II,/3 = 0.190. (4) F r o m Table III, y/[3 = 1.215 = (A × 1 . 5 0 ) / ( 1 0 . 7 5 × 405), giving A = 3,530 m 2 . (5) Select a suitable length o f p o u n d , say 300 m, and assume t h a t the m e a n width o f 11.8 m occurs at 0.8 o f the m a x i m u m d e p t h , i.e., at 1.2 m. T h e n the bed w i d t h is 8.2 m. (6) C h e c k the design b y the usual arithmetical m e t h o d , and find t h a t the m a x i m u m discharge is 6.0 m 3/sec, with a m a x i m u m h e a d o f 1.46 m.
REFERENCES Bruce, J.P. and Clark, R.H., 1966. Introduction to Hydrometeorology. Pergamon, Oxford, p. 201. Carden, T.H., 1960. Hemel Hempstead New Town - Engineering Works. J. Inst. Munic. Eng., 87: 51-64. Chow, V.T., 1959. Open.channel Hydraulics. McGraw-Hill, New York, N.Y., p. 366. Nash, J.E., 1960. A unit hydrograph study, with particular reference to British Catchments. Proc. Inst. Or. Eng. (London), 17: 249-282. Pearson, E.S. and Hartley, H.O., 1966. Biometrika Tables for Statisticians, 1. Cambridge University Press, pp. 128-135. Sarginson, E.J., 1969. Streamflow routing analysis~.Civ. Eng. andPub. Wks. Rev., 64: 9 8 2 - 9 8 3 .