b ma. The reservoir reduces The storage-outflow function is calculated in the following example, for Q = 57 ma/s, 2s/At b 20,890 ms/s as shown

Transcription

1 The storage-outflow function is calculated in the following - table using the same method as in Example ) in the text ui h A t 2 hrs. for example, for Q = 57 ma/s, 2s/At b + Q = ((2 x 75 x 10 )/(2 x 3600) + 57 = 20,890 ms/s as shown In order to perform the level pool computations, a computer program given in Table , was used. The input data is described in the READ statements (lines 9, 11, 12 and 13) and the input variables are described on lines 15 through 29. The input data for this problem is in Table and the output is in Table The method is the same as that presented in Example (8.2.1) in the text. The nitlal outflow is 57 ma/s, sponding to an initial storage of 75 x 10 b ma. The reservoir reduces eak flow from 1930 m '1s to 1148 m '/a and delays i t by 4 hours.

2 Table Program For Level Pool Reservoir Routing PROGRAM SAMPLE5 ( INPUT, OUTPUT, TAPES-INPUT, TAPE^-OUTPUT) DIMENSION S(30),QS(30),Q(SO),SFUNC(30)... It THIS PROGRAM IS FOR RESERVOIR LEVEL POOL ROUTING FOR PROBLEMS THROUGH 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) FoRMAT(~X, 'STORAGE', 2X, 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) 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

3 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 QOUTJ-(QS( J+1 )-as( J) )*(SJO-S( J) )/(S( J+1 1-S( J) )+QS( J) T 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 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 QOUTJJ=(QS(JS+~)-QS(JS))*(SFJJ-SFUNC(JS))/(SFUNC(JS+~)-SFUNC(JS)) + +QS(JS)

4 Table (continued, page 3) RETURN END SUBROUTINE INFLOW(JNN,QINl,T,DTT,DT,QI)... SUBROUTINE USES LINEAR INTERPOLATION TO DETERMINE INFLOW AT DIFFERENT TIMES... DIMENSION QI(50) TX-T TT-0.0 DO 10 JJ-1,JNN-1 IF(TX.GE.TT.AND.TX.LE.(TT+DTT)) GO TO 20 TT=TT+ DTT QIN1 -QI( 1 ) GO TO 25 QINl -(QI( JJ+1 )-QI(JJ))*(T-TT)/DTT+QI(JJ) RETURN END

5 1 Table Input Data for Level Pool Reservoir Routing LEVEL POOL ROUTING ' ; * Table Output Data Prom Level Pool Reservoir Routing I DISCHARGE-STORAGE RELATION STORAGE DISCHARGE STORAGE FUNCTION ;OO 22727;OO ;OO

6 The inflow and outflow hydrographs are shown in Cols. (2) and (3) of Table and plotted in Figure Given X, the value of K is specified by Eq. (8.4.11) In this case, A t - 3 min P 180 sec. The numerator is computed in Col. (4) of Table 8.4.3; the denominator for X = 0.25 is shown in Col. (5). Figure shows a plot of the numerator vs. the denominator of the expression for K. It is nearly a straight line. Similar plotsfor X and 0.3 are shown in Figures and , respectively; both are looped more than for X Choosing values further away from X = 0.25 produces even wider loops. The value of X is thus chosen to be The value of K is computed as. the average of the values in Col. (6) of the table, which is the ratio - of the numerator to the denominator of Eq. (8.4.11) and is K = 599 sec 10 mins. As a check, the values of K = 10 min, X are used in the Muskingum routing procedure (C1 = , C , C3 = 0.667, initial outflow - 0) and the outflow hydrograph obtained following routing is shown in Col. (7) of Table which is very similar to the observed outflow hydrograph in Col. (3).

7 CO~: (1) (2) (31 (4) (5 (6) (7) Time Inflow Outflow Nun Denom K Routed Outflow (rnin) (cf s (cfs) X = 0.25 (sec) (cfs) Table Determination of the Muskingum parameters K and X

8 The values of the Muskingum coefficients are given by Eqs..(8.4.8) to ( ) in the text, with K h, X = Because K < At, it is necessary to interpolate the inflow hydrograph so that the values of C,, C2 and C are all positive and the computations are accurate, A t = 0.25 h is satis 3 actory for this purpose... C, = (At - 2KX)/[ZK(1 - X) + btl

9 C1 + C2 + C3 = = as required. The values of the inflow hydrograph are given in Column (3) of Table where the values at 0, 0.5, 1.0, 1.5 hr are given in the problem description and the remainder obtained by linear interpolation. The computation is performed using Eq. (8.4.7) in the text. For example, for j-1, and Q1 = 739 cfs as specified in the problem description = 825 cfs as shown in Columns (4) to (7) of the table. Succeeding calculations are performed in the same way. The inflow and outflow hydrographs are plotted in Figure where it can be seen that the routing is essentially a translation of the flood wave along the channel in this example. - If A t = 0.5 hr is used, the values of the coefficients are C1 = 0.442, C C3 = These coefficients. with Cg negative, cause an inaccuracy in the calculation that leads a slight increase in discharge as the flow passes down the channel. In the absence of lateral inflow along the channel, the discharge cannot physically increase in this way so the value of A t = 0.25 hrs was chosen for this solution as described previously.

10 Table Flow routing in a stream channel by the Muskingum method. The inflow hydrograph given in the problem has been linearly interpolated at 0.25 hour time intervals. CO~: (1) (2 (3) (4) (5) (6) (7 Time Time Inflow C C Outflow 1 (hr) 1ndex.j (cfs) O.I%O (cfs)

11 Figure Trial curve. for determination of Muskingum parameters Determrnation of K Figure Trial curve for determination of Muskingum parameters.

12 Figure lnflow and Outflow Hydrographs Figure 8.4,.3-2. Final curve for determination of Muskingum parameters Determination of. K X-Q2b

13 (a) The major advantages of the lumped or hydrologic routing methods are that they are simple and have been incorporated into various rainfallrunoff models such as the U.S. Army Corps of Engineers HEC-1 computer program. Hydrologic routing methods only require lumped system parameters, e.g., the Muskingum method only requires K and X for the channel reach. Disadvantages of the hydrologic routing methods include the fact that the description of the process does not consider flowrate, velocity, and depth as distributed variables, id=., functions of space. Little use is made of the fundamental principles of conservation of mass and energy. The dis-

14 advantages of the lumped method are actually advantages of the distributed routing methods. Hydrologic methods require determination of the flowrate and water surface elevation as separate calculations, whereas distributed methods simultaneously compute flowrates and water surface elevations. On the other hand, the distributed methods, especially the full-dynamic model, have the disadvantage that they are more difficult to use and require some knowledge of numerical methods such as finite difference techniques. (b) Limitations of the kinematic wave method stem from the fact that the local and convective acceleration and pressure terms are neglected in the momentum equation, so that backwater effects or downstream disturbances are not considered in the computations. The friction slope is taken as Sf = S,, which neglects the local and convective acceleration and pressure terms. The flood wave properties are described primarily by the equation of continuity, describing the water movement exclusive of the influence of mass and force. In dynamic wave routing these quantities are included. (c) The kinematic wave could be justified for applications where the channels are fairly steep and downstream disturbances cannot propagate upatrtrrm.