FALL HOMEWORK NO. 6 - SOLUTION Problem 1.: Use the Storage-Indication Method to route the Input hydrograph tabulated below.

Transcription

1 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) Iflow (m3/s) Iflow (m3/s) 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

2 H (m) O (m 3 /s) S (m 3 ) Dscharge (m3/s) Elevao (m) Jorge A. Ramírez

3 Sorage (m3) 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( ) ) ds( ) = I( ) d + 1 O( ) d 3 Jorge A. Ramírez

4 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 H (m) O (m 3 /s) S (m 3 ) S/Δ + O (m 3 /s) 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

5 500" 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) m 3 /s = m 3 /s (S 1 /Δ + O 1 ) = (I o + I 1 ) + (S o /Δ - O o ) = 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 ) = ( ) m 3 /s = 170 m 3 /s (S 1 /Δ - O 1 ) = (S 1 /Δ + O 1 ) - x O 1 = m3/s - x 1.51 m3/s = m3/s 5 Jorge A. Ramírez

6 (S /Δ + O ) = (I 1 + I ) + (S 1 /Δ - O 1 ) = 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) Jorge A. Ramírez

7 Dscharge (m3/s) Tme (h) Iflow (m3/s) Ouflow (m3/s) 7 Jorge A. Ramírez

8 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) 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

9 " S(vs.(Q( " " Sorage((m 3 )( " " S"vs."Q" " 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) Jorge A. Ramírez

10 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 # # Sorage((m 3 )( # # # # R²#=# # R²#=#0.9435# R²#=# # 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 = 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) Jorge A. Ramírez

11 E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E O (m 3 /s) I (m 3 /s) OI (m 3 /s) SO (m 6 /s) SI (m 6 /s) Toal E E+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 = s B = s k = A+B = s = h x = A/(A + B) = 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

12 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 = ; C 1 = ; ad C = 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) Jorge A. Ramírez

13 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