The storage-outflow func t ion is ca lcu la ted i n the fol lowing table using t h e same method as i n Example-48.2.1) i n t h e t e x t u i h A t - 2 h r s . for b example , f o r Q = 57 m a / s , 2 s /A t + Q = ( ( 2 x 75 x 10 ) / ( 2 x 3600) + 5 7 = 20,890 m s / s as shown

I n o r d e r t o pe r fo rm t h e l e v e l poo l computations, a compute r program g iven i n T a b l e 8.2.2-1, was used. The i n p u t data is d e s c r i b e d i n t h e READ s ta tements ( l i n e s 9, 11, 12 and 13) and the input va r i ab l e s a r e described on l i n e s 15 th rough 29. The i n p u t data f o r t h i s problem i s i n T a b l e 8.2.2-2 and the o u t p u t is i n Table 8.2.2-3. The method is t h e same as t h a t p r e sen t ed i n Example (8.2.1) i n t h e t e x t . The n i t l a l o u t f l o w is 57 m a / s , b sponding t o an i n i t i a l s to rage of 75 x 10 ma. The r e s e r v o i r reduces

eak f low from 1930 m '1s t o 1 148 m '/a and delays i t by 4 hours.

Table 8.2.2-1 Program For Level Pool Reservoir R o u t i n g

PROGRAM SAMPLE5 ( INPUT, OUTPUT, TAPES-INPUT, TAPE^-OUTPUT) DIMENSION S(30),QS(30),Q(SO),SFUNC(30) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I t

THIS PROGRAM IS FOR RESERVOIR LEVEL POOL ROUTING FOR PROBLEMS 8.2.1 THROUGH 8.2.6.

READ (5,101) SJO,DT,DTT,JSS,JNN,TPM FORHAT(3FlO.O,2I1O,F10*0) READ(5.102) (S(JS),JS=1,JSS) READ (5,102) (QS(JS),JS-~,JSS) READ (5,102) (Q(JN),JNm1,JNN) FORMAT(~FIO.~) *************************************************************

INPUT DATA C,ONV-CONVERSION FACTOR-1 SJO-INITIAL RESERVOIR STORAGE DT-ROUTING INTERVAL, MIN DTT-TIHE INTERVAL FOR HYDROGRAPH, HIN JSS-NUMBER OF VALUES DESCRIBING DISCHARGE-STORAGE RELATION JNN-NUMBER OF VALUES DESCRIBING INFLOW HYDROGRAPH TIN-TOTAL TIHE FOR ROUTING COMPUTATIONS, HIN S( JS 1-RESERV.OIR STORAGE QS(JS)-SPILLWAY DISCHARGE Q( JN)-RESERVOIR INFLOW

DT-DT.60. DTT=DTT*60. TI!4=TIM*60. NTIM-TIH/DT+1 WRITE( 6,200) FORMAT(SX,'LEVEL POOL ROUTING',//

+SX,'DISCHARGE-STORAGE RELATION1,//) WRITE(6.202) F o R M A T ( ~ X , 'STORAGE', 2 X , tDISCHARCEt ,2X, 'STORAGE FUNCTION', ) DO 50 JS-1, JSS SFUNC( JS)-2.*S( JS)/DT+QS( Js) WRITE(6,201) s( Js) 9 QS( JS) ,SFUNC( JS) CONTINUE FORMAT(SX,F12.O,2F1Oo2) SJ-SJO QINI -Q(I) WRITE (6,154) FORMAT(///~X, *TIHE(HfN) ',4X,'I(T) t,4X, 'I(T)+I(T+DT) ',1X,

+~~s(T)/DT-Q(T) ',1X, '2S(T+DT)/3T+Q(T+DT)t,3X, 'Q(T+DT) ' , I

DETERMINE INITIAL DISCHARGE GIVEN INITIAL RESERVOIR STORAGE

DO 30 J-1 ,JSS IF(SJO.LT.S( J+l 1 .AND.SJO.GE.S( J)) GO TO 35 CONTINUE CONTINUE QOUTJ-(QS( J+1 )-as( J) )*(SJO-S( J) )/(S( J+1 1-S( J) )+QS( J) T-0.0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

PERFORM RESERVOIR ROUTING COMPUTATIONS

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . TI -0. DO 100 J-1 ,NTIM T=T+DT CALL INFLOW(JNN,QIN2,T,DTT,DT,Q) QT-QIN1 +QIN2 ST-2.*SJ/DT-QOUTJ SFJJ-QT+ST IF(J.EQ.1) WRITE(6.155) TI ,QIN1 ,QT,ST-,SFJJ,QOUTJ

DETERMINE DISCHARGE .

CALL STORFN~JSS,S~JJ,QOUTJJ,QS,SFUNC) SJ-(SFJJ-QOUTJJ)*DT/2. QOUT J- QOUT J J QINI -QIN2 TT=T/60. WRITE(6,155) TT,QIN1 ,QT,ST,SFJJ,QOUTJJ CONTINUE FORMAT(1X,F10.2,F10.2,1X,F12.2,2X,F12.2,5X,F12.2,2X,F12.2) STOP END SUBROUTINE STORFN(JSS,SFJJ,QOUTJJ,QS,SFUNC) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

SUBROUTINE TO COMPUTE DISCHARGE FROM STORAGE RELATIONSHIP

DIMENSION QS(30), SFUNC(30) IF(SFJJ.LT.SFUNC(1)) SFJJ-SFUNC(1) IF(SFJJ.LT.SFUNC(1 1 ) GO TO 120 DO 100 JS-1, JSS-1 .

IF(SFJJ.LT.SFUNC(JS+l).AND.SFJJ.GE.SFUNC(JS)) GO TO 120 CONTINUE CONTINUE QOUTJJ=(QS(JS+~)-QS(JS))*(SFJJ-SFUNC(JS))/(SFUNC(JS+~)-SFUNC(JS))

+ +QS(JS)

T a b l e 8.2.2-1 (continued, page 3 )

RETURN E N D S U B R O U T I N E INFLOW(JNN,QINl,T,DTT,DT,QI) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

S U B R O U T I N E U S E S LINEAR INTERPOLATION T O D E T E R M I N E INFLOW AT D I F F E R E N T T I M E S

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . DIMENSION QI(50) TX-T TT-0.0 DO 10 JJ-1 ,JNN-1 IF(TX.GE.TT.AND.TX.LE.(TT+DTT)) G O T O 2 0 T T = T T + DTT C O N T I N U E QIN1 -QI( 1 ) G O T O 25 CON T I N U E QINl -(QI( JJ+1 )-QI(JJ))*(T-TT)/DTT+QI(JJ) CON T I N U E C O N T I N U E RETURN E N D

1 T a b l e 8.2.2-2 I n p u t Data f o r L e v e l P o o l R e s e r v o i r R o u t i n g

LEVEL POOL ROUTING 75000000. 120. 120. 5 10 1800.0 75000000. 81 000000. 87500000.100000000.11 0200000.

227. '

57. 519. 1330 2270; 60. 100. 232. * 300. 520. 1 31 0. 1930. 1460. 930. 650.

T a b l e 8.2.2-3 O u t p u t D a t a P r o m L e v e l P o o l R e s e r v o i r R o u t i n g

I DISCHARGE-STORAGE RELATION

STORAGE DISCHARGE STORAGE FUNCTION 75000000. 57.00 20890.33 81 000000. 227;OO 22727;OO . 87500000. 519.00 24824.56 100000000. 1330;OO 29107.78 1 1 0200000. 2270 -00 32881.11

The i n f l o w and o u t f l o w hydrographs a r e shown i n Cols . (2) and (3) of T a b l e 8.4.3 and p l o t t e d i n F i g u r e 8.4.3-1. Given X, t h e v a l u e of K i s spec i f i ed by Eq. (8.4.11)

I n t h i s c a s e , A t - 3 min P 180 sec . The numera to r i s computed i n Col. (4) of Tab l e 8.4.3; t h e denomina tor f o r X = 0.25 is shown i n Col. ( 5 ) . Figure 8.4.3-2 shows a p l o t of the numerator vs. t h e denominator of the expression f o r K. I t is n e a r l y a s t r a i g h t l i n e . S i m i l a r p l o t s f o r X - 0.2 and 0.3 a re shown i n F i g u r e s 8.4.3-3 and 8.4.3-4, r e s p e c t i v e l y ; b o t h a r e looped more than f o r X - 0.25. Choosing values f u r t h e r away from X = 0.25 produces even wider loops . The v a l u e o f X i s t h u s chosen t o be 0.25. The v a l u e of K is computed as. t he average of t h e values i n Col. (6) of t h e t ab le , which is the r a t i o of t h e numera to r t o t h e denomina tor o f Eq. (8.4.11) and i s K = 599 sec - 10 mins.

A s a check, t h e v a l u e s of K = 1 0 min, X - 0.25 are used i n the Muskingum r o u t i n g p rocedu re (C1 = -0.1 11 , C2 - 0.444, C3 = 0.667, i n i t i a l outflow - 0) and the outflow hydrograph obtained fo l lowing rout ing is shown i n Col. ( 7 ) of Tab l e 8.4.3 which is v e r y s i m i l a r t o t h e observed outflow hydrograph i n Col. (3 ) .

C O ~ : ( 1 ) ( 2 ) (31 ( 4 ) ( 5 ( 6 ) ( 7 ) Time Inflow Outflow N u n Denom K Routed Outflow

(rnin) (cf s ( c f s ) X = 0.25 ( s e c ) ( c f s )

Table 8 . 4 . 3 Determination of the Muskingum parameters K and X

The v a l u e s o f t h e Muskingum c o e f f i c i e n t s a r e g i v e n by Eqs..(8.4.8) t o (8.4.1 0) i n t h e t e x t , w i t h K - 0.24 h , X = 0.25. Because K < A t , i t i s necessary t o i n t e r p o l a t e t h e inf low hydrograph s o t h a t t h e va lues of C , , C2 and C a r e a l l p o s i t i v e and t h e c o m p u t a t i o n s a r e a c c u r a t e , A t = 0.25 h is s a t i s 3 a c t o r y f o r t h i s purpose.

. .

C, = (At - 2KX)/[ZK(1 - X ) + b t l

C1 + C2 + C3 = 0.21 3 + 0.607 + 0.180 = 1.000 a s r e q u i r e d .

The v a l u e s of t h e i n f l o w hydrograph a r e g i v e n i n Column ( 3 ) o f Tab le 8.4.4 where t h e v a l u e s a t 0 , 0.5, 1.0, 1.5 h r a r e g i v e n i n t h e problem d e s c r i p t i o n and t h e remainder obtained by l i n e a r in te rpo la t ion .

The computation is performed using Eq. (8.4.7) i n t h e t e x t .

For example, f o r j -1, Q1 = 739 c f s a s s p e c i f i e d i n t h e problem d e s c r i p t i o n and

= 825 c f s

a s shown i n Columns (4) t o ( 7 ) of t h e t a b l e . S u c c e e d i n g c a l c u l a t i o n s a r e performed i n t h e same way. The inflow and outf low hydrographs a r e p lo t t ed i n F i g u r e 8.4.4 where i t c a n be s e e n t h a t t h e r o u t i n g is e s s e n t i a l l y a t r a n s l a t i o n of t h e f lood wave along t h e channel i n t h i s example.

If A t = 0.5 h r is used , t h e v a l u e s of t h e c o e f f i c i e n t s a r e C1 = 0.442, C2 - 0.721. C3 = -0.163. These c o e f f i c i e n t s . w i t h C g n e g a t i v e , c a u s e an inaccuracy i n t h e c a l c u l a t i o n t h a t leads a s l i g h t i n c r e a s e i n discharge a s t h e f l o w p a s s e s down t h e channel . I n the a b s e n c e o f l a t e r a l i n f l o w along t h e c h a n n e l , t h e d i s c h a r g e c a n n o t p h y s i c a l l y i n c r e a s e i n t h i s way s o t h e value of A t = 0.25 h r s was chosen f o r t h i s s o l u t i o n a s descr ibed previously.

Table 8 .4 .4 Flow routing i n a stream channel by the Muskingum method. The in f low hydrograph g iven i n the problem has been l i n e a r l y interpolated a t 0.25 hour time in terva l s .

C O ~ : ( 1 ) ( 2 ( 3 ) ( 4 ) (5 ) ( 6 ) (7

Time T i m e Inflow C C Outflow 1 ( h r ) 1ndex.j ( c f s ) 0 .213 0.687 O . I % O ( c f s )

Figure 8.4.3-3. Trial curve. for determination of Muskingum parameters Determrnation of K

Figure 8.4.3-4. Trial curve for determination of Muskingum parameters.

Figure 8.4.3-1.lnflow and Outflow Hydrographs

Figure 8.4,.3-2. Final curve for determination of Muskingum parameters

Determination of. K X - Q 2 b

(a ) The major advantages of t h e lumped o r hydrologic rou t ing methods a r e t h a t they a r e s imple and have been incorporated i n t o var ious r a i n f a l l - runoff models such as t h e U.S. Army Corps of Engineers HEC-1 computer pro- gram. Hydrologic r o u t i n g methods o n l y r e q u i r e lumped s y s t e m p a r a m e t e r s , e.g., t h e Muskingum method o n l y r e q u i r e s K and X f o r t h e channe l r each . Disadvantages of t h e hydrologic rou t ing methods include t h e f a c t t h a t t h e d e s c r i p t i o n of t h e process does not consider f l o w r a t e , ve loc i ty , and depth a s d i s t r i b u t e d v a r i a b l e s , id=., f u n c t i o n s of space . L i t t l e u s e i s made of t h e f u n d a m e n t a l p r i n c i p l e s of c o n s e r v a t i o n of mass and energy. The d i s -

advantages of the lumped method a r e ac tua l ly advantages of the d i s t r i bu t ed rout ing methods. Hydrologic methods require determination of the flowrate and water s u r f a c e e l e v a t i o n as s e p a r a t e calculations, whereas d i s t r i b u t e d methods simultaneously compute f lowrates and water surface elevations. On the other hand, the d is t r ibuted methods, especial ly the full-dynamic model, have the disadvantage t h a t they a r e more d i f f i c u l t t o use and require some knowledge of numerical methods such a s f i n i t e difference techniques.

(b) Limitations of the kinematic wave method stem from the fact that the loca l and convective accelerat ion and pressure terms a r e neglected i n t he momentum equation, s o t h a t backwater e f f e c t s or downstream disturbances a r e not considered i n the computations. The f r i c t i o n slope is taken as Sf = S,, which neglects the loca l and convective acceleration and pressure terms. The flood wave p r o p e r t i e s are desc r ibed p r i m a r i l y by t h e equation of continuity, describing the water movement exclusive of the influence of mass and force. In dynamic wave routing these quant i t ies a r e included.

(c) The kinematic wave could be j u s t i f i e d f o r applicat ions where the channels a r e f a i r l y steep and downstream d i s t u r b a n c e s cannot propagate upatrtrrm.