Jorge A. Ramírez HOMEWORK NO. 6 - SOLUTION Problem 1.: Use he Sorage-Idcao Mehod o roue he Ipu hydrograph abulaed below. Tme (h) Ipu Hydrograph (m 3 /s) Tme (h) Ipu Hydrograph (m 3 /s) 0 0 90 450 6 50 96 330 1 10 10 80 18 5 108 10 4 300 114 160 30 55 10 110 36 700 16 90 4 1100 13 50 48 1500 138 30 54 1475 144 0 60 1300 150 15 66 1100 156 10 7 900 16 0 78 750 84 600 Iflow (m3/s) Iflow (m3/s) 1600 1400 100 1000 800 600 400 00 0 0 50 100 150 00 Tme (h) Iflow (m3/s) Ths hydrograph flows o a reservor whose sorage ad dscharge characerscs are as preseed he followg able. The al sorage he sysem s 1'000,000 m 3, ad he al ouflow s 9.5 m 3 /s. 1 Jorge A. Ramírez
H (m) O (m 3 /s) S (m 3 ) 130 0 08187.34 131 34 353918.98 13 57 5933336.17 133 96 999987.3 134 16 16863165.96 136 463 48195344.68 137 781 8197114.89 138 1318 137195387.3 139 6 3171391.49 Dscharge (m3/s) 4500 4000 3500 3000 500 000 1500 1000 500 0 18 130 13 134 136 138 140 Elevao (m) Jorge A. Ramírez
300000000 50000000 Sorage (m3) 00000000 150000000 100000000 50000000 0 18 130 13 134 136 138 140 Elevao (m) Reservor or level pool roug refers o roug for sysems whose sorage ad ouflow are relaed by a fuco of he ype S() = f[o()] whch s of he varable ype (uque, ohyserec). These relaoshps mply ha for a gve se of codos (e.g. sage) he ouflow s uque, depede of how ha sage s acheved. Reservors or sysems wh horzoal waer surfaces have S Vs. O relaoshps of he varable ype. Such sysems have a pool ha s wde ad deep compared o s legh he dreco of flow, ad low flow veloces he reservor. For such sysems, he peak ouflow occurs whe he ouflow hydrograph ersecs he flow hydrograph. The Sorage-Idcao mehod s a level pool roug procedure for calculag he ouflow hydrograph of a sysem wh horzoal waer surface, gve s flow hydrograph, ad sorage ouflow characerscs. The soluo volves egrag he couy equao as dcaed below, ad rearragg erms such ha all he ukow quaes are o he lef had sde of he equao. S ( + 1 ) S( ) ds( ) d = S ( S ( ) = I( ) O( ) + 1 + 1 ) ds( ) = I( ) d + 1 O( ) d 3 Jorge A. Ramírez
Δ Δ S ( + ) S( ) = [ I( + 1) + I( )] [ O( + 1) + O( 1 Sorage-Idcao Roug Equao: S( Δ + 1 ) + O( + 1 ) = [ I( + 1 ) + I( S( )] + [ Δ ) O( For a level pool reservor, he sorage s a uque fuco of elevao; ad he ouflow s a uque fuco of elevao. Thus, he lef had sde of he equao above s a uque fuco of elevao he sysem, oly. Usually, he sorage-elevao relaoshp s avalable from opographc surveys, ad he ouflow-elevao relaoshp s avalable from hydraulc cosderaos wh respec o he oule srucures (e.g. spllways, ec.) The soluo volves he developme of he fuco S/Δ + O = f(o) ad he solvg sequeally for every me sep. These seps are llusraed below. A- Develop he fuco S/Δ + O vs. O. Use a Δ of 6 hours, as suggesed by he me erval of he flow hydrograph. 1 3 5 H (m) O (m 3 /s) S (m 3 ) S/Δ + O (m 3 /s) 130 0 08187.34 1.7659574 131 34 353918.98 361.70178 13 57 5933336.17 606.389787 133 96 999987.3 101.76595 134 16 16863165.96 173.40456 136 463 48195344.68 495.531915 137 781 8197114.89 8308.510638 138 1318 137195387. 1401.766 139 6 3171391.5 3680.85106 I he able above, Colums 1-3 are gve. Colums ad 5 correspod o he desred fuco, S/Δ + O vs. O, whch has bee graphed above. )] )] 4 Jorge A. Ramírez
500" 000" O"(m3/s)" 1500" 1000" Seres1" 500" 0" 0" 5000" 10000" 15000" 0000" 5000" S/D"+"O"(m3/s)" B - Proceed wh he roug of he flow hydrograph by usg he Sorage-Idcao roug equao sequeally for every me sep: = 0 - = 0. Ial Codos: S o = 1'000,000 m 3 ; O o = 9.5 m 3 /s. = 6 - = 1 (I o + I 1 ) = (0 + 50) m 3 /s = 50 m 3 /s (S o /Δ - O o ) = ( x 1'000,000 m 3 )/(6 x 3600 s) - 9.5 m 3 /s = 83.09 m 3 /s (S 1 /Δ + O 1 ) = (I o + I 1 ) + (S o /Δ - O o ) = 133.09 m 3 /s Usg he relaoshp (S/Δ + O) vs. O developed Par A oba he ouflow O correspodg o he value of (S 1 /Δ + O 1 ) obaed above. Use erpolao as dcaed below. O 1 = 1.51 m 3 /s = 1 - = (I 1 + I ) = (50 + 10) m 3 /s = 170 m 3 /s (S 1 /Δ - O 1 ) = (S 1 /Δ + O 1 ) - x O 1 = 133.09 m3/s - x 1.51 m3/s = 108.07 m3/s 5 Jorge A. Ramírez
(S /Δ + O ) = (I 1 + I ) + (S 1 /Δ - O 1 ) = 78.07 m 3 /s Usg he relaoshp (S/Δ + O) vs. O developed Par A, oba he ouflow O correspodg o he value of (S 1 /Δ + O 1 ) obaed above. Use erpolao as dcaed below. O 1 = 6.14 m 3 /s Proceed as above for every me sep. Resuls are abulaed below. Tme (h) I (m3/s) I + I +1 S /Δ - O I S +1 /Δ + O +1 O S +1 /Δ - O +1 (m 3 /s) (m 3 /s) (m 3 /s) (m3/s) (m3/s) 0 0 0 6 50 50 7.5959 1.596 1.51417 79.5645 1 10 170 79.5645 49.5643 40.80086 167.965 18 5 345 167.965 51.965 8.151 348.713 4 300 55 348.713 873.713 139.053 595.6058 30 55 85 595.6058 140.606 5.175 970.609 36 700 15 970.608 195.61 345.0893 1505.08 4 1100 1800 1505.08 3305.08 513.6837 77.715 48 1500 600 77.715 4877.715 755.40 3367.34 54 1475 975 3367.35 634.34 980.8015 4380.631 60 1300 775 4380.631 7155.631 1105.50 4944.68 66 1100 400 4944.68 7344.68 1133.417 5077.794 7 900 000 5077.794 7077.794 1094.005 4889.785 78 750 1650 4889.785 6539.785 1011.8 4517.33 84 600 1350 4517.33 5867.33 907.658 405.014 90 450 1050 405.014 510.014 789.7861 35.441 96 330 780 35.44 430.441 666.638 969.165 10 80 610 969.165 3579.165 555.3199 468.55 108 10 490 468.55 958.55 461.0377 036.449 114 160 370 036.449 406.449 377.171 165.107 10 110 70 165.107 19.107 303.5941 1314.919 16 90 00 1314.919 1514.919 40.003 1034.878 13 50 140 1034.878 1174.878 186.4873 801.9036 138 30 80 801.9036 881.9036 140.3459 601.119 144 0 50 601.119 651.119 103.9415 443.388 150 15 35 443.387 478.387 76.65971 35.0093 6 Jorge A. Ramírez
Dscharge (m3/s) 1600 1400 100 1000 800 600 400 00 0 0 50 100 150 00 Tme (h) Iflow (m3/s) Ouflow (m3/s) 7 Jorge A. Ramírez
Problem. Usg he formao abulaed below for a rver reach, esmae he Muskgum parameers k ad x. The al sorage he reach s 6,000,000 m 3. Use boh he leas-squares approach ad he graphcal mehod. Tme (h) Iflow (m 3 /s) Oupu (m 3 /s) 1 180. 160. 70. 00. 3 40. 80. 4 650. 415. 5 890. 590. 6 1100. 770. 7 170. 950. 8 1360. 1090. 9 1380. 1180. 10 1390. 150. 11 1370. 180. 1 1350. 190. 13 1310. 1300. 14 160. 180. 15 110. 150. 16 1160. 10. 17 1100. 1190. 18 1000. 1150. 19 950. 1100. 0 900. 1040. 1 790. 980. 710. 90. 3 650. 860. 4 590. 790. 5 510. 710. 6 450. 650. 7 380. 590. 8 300. 510. 9 30. 460. 30 180. 410. Fally, usg he parameers esmaed usg he leas squares procedure, esmae C o, C 1, ad C, usg a Δ of 1 hour, ad he roue he orgal flow hydrograph. Compare he observed ouflow wh ha predced usg he Muskgum mehod. 8 Jorge A. Ramírez
300000000" S(vs.(Q( 50000000" 00000000" Sorage((m 3 )( 150000000" 100000000" S"vs."Q" 50000000" 0" 0" 00" 400" 600" 800" 1000" 100" 1400" Flow((m 3 /s)( A. Parameer Esmao Graphcal Procedure: The graphcal procedure cosss geerag graphs of [xi + (1-x)O] vs. S for dffere values of x, arbrarly seleced such ha 0 < x < 0.5. The opmal value of x s seleced as ha whch produces he arrowes ad sraghes loop graph of [xi + (1-x)O] vs. S. The slope of he leas squares lear f o he resulg pos s he esmae of k. a) Geerae accumulaed sorage he sysem. Use couy equao as follows: 0 Tme (days) 1 Iflow (m 3 /s) Ouflow (m 3 /s) 3 Average I (m 3 /s) 4 Average O (m 3 /s) 5 Sorage (m 3 ) xi + (1-x)O (m 3 /s) 0.4 0.1 0.3 0. 1 180 160 60000000 168 16 166 164 70 00 5 180 63888000 8 07 1 14 3 40 80 345 40 7960000 336 94 3 308 4 650 415 535 347.5 89160000 509 438.5 485.5 46 5 890 590 770 50.5 117000 710 60 680 650 6 1100 770 995 680 139488000 90 803 869 836 7 170 950 1185 860 167568000 1078 98 1046 1014 8 1360 1090 1315 100 193056000 1198 1117 1171 1144 9 1380 1180 1370 1135 13360000 160 100 140 10 10 1390 150 1385 115 8048000 1306 164 19 178 11 1370 180 1380 165 37984000 1316 189 1307 198 1 1350 190 1360 185 44464000 1314 196 1308 130 13 1310 1300 1330 195 47488000 1304 1301 1303 130 14 160 180 185 190 47056000 17 178 174 176 15 110 150 135 165 44464000 134 146 138 14 16 1160 10 1185 135 40144000 1196 114 10 108 17 1100 1190 1130 105 33664000 1154 1181 1163 117 18 1000 1150 1050 1170 396000 1090 1135 1105 110 19 950 1100 975 115 10336000 1040 1085 1055 1070 0 900 1040 95 1070 197808000 984 106 998 101 9 Jorge A. Ramírez
1 790 980 845 1010 18355000 904 961 93 94 710 90 750 950 1667000 836 899 857 878 3 650 860 680 890 14818000 776 839 797 818 4 590 790 60 85 130416000 710 770 730 750 5 510 710 550 750 113136000 630 690 650 670 6 450 650 480 680 95856000 570 630 590 610 7 380 590 415 60 78144000 506 569 57 548 8 300 510 340 550 60000000 46 489 447 468 9 30 460 65 485 4099000 368 437 391 414 Colums 1 & are gve. Colums 3 & 4 are he average flow flux (I +1 + I )/ ad ouflow flux (O +1 + O )/, respecvely. Colum 5 s he cumulave sorage he sysem obaed usg he couy equao below. S +1 = S + Δ (I +1 + I ) Δ (O +1 + O ) Colums 6-9 are he values of he weghed average flux [xi + (1-x)O] for dffere values of x. The graph of Colums 6-9 vs. Colum 5 s show below. 300000000# 50000000# Sorage((m 3 )( 00000000# 150000000# 100000000# 50000000# R²#=#0.93376# R²#=#0.9435# R²#=#0.94186# y#=#18893x##4e+06# R²#=#0.945# 0# 0# 00# 400# 600# 800# 1000# 100# 1400# xi+(11x)o((m 3 /s)( x=.4# x=.1# x=.3# x=.# Lear#(x=.4)# Lear#(x=.1)# Lear#(x=.3)# Lear#(x=.)# Based o hese resuls, a value of x = 0. s seleced. The bes f o he correspodg pos yelds a value of k = 18893 s =.187 d. Leas Squares Procedure Tme(days) Iflow(m 3 /s) Ouflow(m 3 /s) Sor (m 3 ) O (m 3 /s) I (m 3 /s) OI (m 3 /s) SO (m 6 /s) SI (m 6 /s) 1 180 160 60000000 5600 3400 8800 9600000000 10800000000 70 00 63888000 40000 7900 54000 1777600000 1749760000 3 40 80 7960000 78400 176400 117600 048800000 3064300000 4 650 415 89160000 175 4500 69750 37001400000 57954000000 10 Jorge A. Ramírez
5 890 590 117000 348100 79100 55100 6640480000 999080000 6 1100 770 139488000 59900 110000 847000 1.07406E+11 1.53437E+11 7 170 950 167568000 90500 161900 106500 1.5919E+11.1811E+11 8 1360 1090 193056000 1188100 1849600 148400.10431E+11.6556E+11 9 1380 1180 13360000 139400 1904400 168400.51765E+11.94437E+11 10 1390 150 8048000 156500 193100 1737500.8506E+11 3.16987E+11 11 1370 180 37984000 1638400 1876900 1753600 3.046E+11 3.6038E+11 1 1350 190 44464000 1664100 18500 1741500 3.15359E+11 3.3006E+11 13 1310 1300 47488000 1690000 1716100 1703000 3.1734E+11 3.409E+11 14 160 180 47056000 1638400 1587600 161800 3.163E+11 3.1191E+11 15 110 150 44464000 156500 1464100 151500 3.0558E+11.95801E+11 16 1160 10 40144000 1488400 1345600 141500.9976E+11.78567E+11 17 1100 1190 33664000 1416100 110000 1309000.7806E+11.5703E+11 18 1000 1150 396000 13500 1000000 1150000.5679E+11.396E+11 19 950 1100 10336000 110000 90500 1045000.3137E+11 1.99819E+11 0 900 1040 197808000 1081600 810000 936000.057E+11 1.7807E+11 1 790 980 18355000 960400 64100 77400 1.79881E+11 1.45006E+11 710 90 1667000 846400 504100 65300 1.597E+11 1.18053E+11 3 650 860 14818000 739600 4500 559000 1.739E+11 968300000 4 590 790 130416000 64100 348100 466100 1.0309E+11 76945440000 5 510 710 113136000 504100 60100 36100 8036560000 57699360000 6 450 650 95856000 4500 0500 9500 6306400000 4313500000 7 380 590 78144000 348100 144400 400 46104960000 969470000 8 300 510 60000000 60100 90000 153000 30600000000 18000000000 9 30 460 4099000 11600 5900 105800 1885630000 948160000 O (m 3 /s) I (m 3 /s) OI (m 3 /s) SO (m 6 /s) SI (m 6 /s) Toal 593165 6389300 5665750 4.7898E+1 4.77515E+1 O A = =1 =1 I =1 S I O I S O =1 =1 =1 O [ O I ] =1 B = Usg he above equaos yelds: =1 A = 34917.74035 s B = 150149.469 s k = A+B = 185066.9873 s =.1419790 h x = A/(A + B) = 0.1886764 B. Muskgum Roug I =1 =1 I S O O I S I =1 =1 =1 O [ O I ] Use he Muskgum roug procedure o roue he orgal hydrograph. =1 11 Jorge A. Ramírez
Selec a Δ = 1 h, as suggesed by he flow daa. However, check ha wh he seleced Δ, parameer values mee resrcos: x < 0.5 Δ/k < 1 - x For hs case, we have ses of parameers. Boh ses mee he parameer resrcos. Proceed wh roug, by obag C o, C 1, ad C. I wha follows, he Leas Squares parameers are used. kx + 0.5Δ kx + 0.5Δ k(1 x) 0.5Δ C o = C 1 = C = k(1 x) + 0.5Δ k(1 x) + 0.5Δ k(1 x) + 0.5Δ Ths yelds: C o = 0.04835748; C 1 = 0.404040; ad C = 0.5531403. Usg hese values he Muskgum roug equao: O +1 = C o I +1 + C 1 I + C O oba he ouflow hydrograph as abulaed below. Tme(days) Iflow(m3/s) Oobs (m3/s) Co x I+1 C1 x I (m3/s) C x O (m3/s) Opred (m3/s) (m3/s) 1 180 160 160 70 00 11.5656501 7.743391 88.5043681 17.79417 3 40 80 17.9910143 109.0864859 95.57843494.655935 4 650 415 7.8433631 169.6900891 13.1599551 30.693806 5 890 590 38.1381587 6.6156141 177.388355 478.17785 6 1100 770 47.119399 359.5813794 64.471713 671.174154 7 170 950 54.40140018 444.46439 371.54643 870.080884 8 1360 1090 58.5661751 513.1105076 481.764109 105.643536 9 1380 1180 59.1133348 549.476696 58.594877 1190.84549 10 1390 150 59.54168996 557.55315 658.704548 175.799388 11 1370 180 58.68497499 561.593390 705.6959671 135.97433 1 1350 190 57.886003 553.519098 733.4497473 1344.790917 13 1310 1300 56.1148301 545.43494 743.8579573 1345.40517 14 160 180 53.9730469 59.714685 744.197751 137.446 15 110 150 51.831559 509.070674 734.617184 195.16341 16 1160 10 49.68946788 488.8690663 716.406893 154.96547 17 1100 1190 47.119399 468.667865 694.171865 109.959053 18 1000 1150 4.83574817 444.46439 669.7709 1156.53901 19 950 1100 40.69396076 404.04017 639.783599 1084.44634 0 900 1040 38.5517335 383.8807 599.8508994 10.5893 1 790 980 33.8404105 363.616196 565.434659 96.896165 710 90 30.413381 319.178977 53.616585 88.089433 3 650 860 7.8433631 86.8570554 487.985579 80.6855496 4 590 790 5.730914 6.6156141 443.9976696 731.886375 5 510 710 1.8463157 38.374178 404.835798 665.05604 6 450 650 19.7608668 06.05511 367.8693408 593.1976786 7 380 590 16.775843 181.8108098 38.115004 56.098945 8 300 510 1.8507445 153.59183 91.067861 457.4477148 1 Jorge A. Ramírez
9 30 460 9.85079 11.07065 53.03734 384.09168 The resulg hydrographs are graphed below. 1600" 1400" 100" Flow%(m 3 /s)% 1000" 800" 600" 400" 00" 0" Iflow(m3/s)"" Ou8low:observed"(m3/s)" Ou8low:predced"(m3/s)" 1" " 3" 4" 5" 6" 7" 8" 9" 10" 11" 1" 13" 14" 15" 16" 17" 18" 19" 0" 1" " 3" 4" 5" 6" 7" 8" 9" Tme%(days)% 13 Jorge A. Ramírez