Dr. M. Perumal Professor & Head Department of Hydrology Indian Institute of Technology Roorkee INDIA Co-authors: Dr. B. Sahoo & Dr. C.M.

Transcription

1 Dr.. Perumal Professor & Head Department of Hdrolog Indan Insttute of Tehnolog Roorkee INDIA o-authors: Dr. B. Sahoo & Dr... Rao Dr. Dr... Perumal, Professor & & Head, Dept. Dept. of of Hdrolog, I.I.T. I.I.T. Roorkee, Inda Inda

2 Lneart of hannel routng sheme n SWAT model usng lassal uskngum () method Frameworks of varable parameter uskngum-unge (VP) & uskngum-unge-todn (T) routng methods avalable n lterature Development of varable parameter arth- uskngum (VP) routng method Performane evaluaton rtera Numeral Applaton Trapezodal, Retangular & Trangular hannel setons Performane of, VP, T & VP methods onlusons Presentaton Outlne Dr. Dr... Perumal, Professor & & Head, Dept. Dept. of of Hdrolog, I.I.T. I.I.T. Roorkee, Inda Inda

3 lassfaton of Rver Flood Waves Old lassfaton b Ferrk (985) : Bulk waves Dffuson wave Dnam waves Knemat wave Gravt waves New suggested lassfaton : Bulk waves Dnam waves Gravt waves Dffuson wave AD wave Knemat wave (KW) s a speal ase of AD wave Works n transton range of DW & KW Dr. Dr... Perumal, Professor & & Head, Dept. Dept. of of Hdrolog, I.I.T. I.I.T. Roorkee, Inda Inda VP method 3

4 Development of VP ethod VP- dretl derved from Sant-Venant Equatons. an be used for routng flood waves n sem-nfnte rgd bed prsmat hannels havng an rosssetonal shape. Allows smultaneous omputatons of stage and dsharge hdrographs at an ungauged rver ste. Hpothess used: Durng stead flow n the hannel reah, there ests one-to-one dsharge - depth relatonshp; whh s not vald n ase of unstead flows. Durng unstead flow n the hannel reah, the dsharge observed at an seton has ts orrespondng normal depth at the upstream seton. 4

5 Defnton Sketh of VP ethod u 3 L 3 SB o o u 3 3 d d Routng parameters K V Δ/ Δ L o θ. 5 L Dr. Dr... Perumal, Professor & & Head, Dept. Dept. of of Hdrolog, I.I.T. I.I.T. Roorkee, Inda Inda 5

6 Framework of VP ethod (Perumal & Pone, ) K θ : 3 V o, ( ) S B.5 3, o o, o, θ I ( θ ) 3, Δt ( θ ) ( ) t. K. θ t. K. θ t. K. O I I O. ( )... ( ) t K. θ t. K. θ t. K. θ Δ Numeral sheme of VP method For dervaton detals of VP method, lk here 6

7 Framework of VP ethod (Pone and hagant, 994) O : Kθ O I I 3.5 t K( θ ).5 t Kθ.5 t K( θ ) K( θ ).5 t 3.5 t K( θ ).5 t K θ.5 ( S B ) o In method, the routng parameters reman onstant at ever routng tme step so that 3, makng t mass onservatve. 7

8 8 Framework of T ethod (Todn, 7) : O D D I D D I D D O ) / )( / ( t β ) / )( / ( t β v / β / v β ) /( BS D o β ) /( BS D o β

9 Performane Evaluaton omparatve evaluaton of, VP, T & VP methods wth respet to benhmark solutons of the Sant-Venant equatons. Evaluated these methods b routng a gven hpothetal flood wave n unform prsmat Trapezodal, Retangular and Trangular rossseton hannel reahes of dfferent onfguratons. Dr. Dr... Perumal, Professor & & Head, Dept. Dept. of of Hdrolog, I.I.T. I.I.T. Roorkee, Inda Inda 9

10 Performane Evaluaton rtera η q N ( ) ( ) o N o o Nash and Sutlffe (97) N N EVOL I q t per qper ( q q ) p po ( t t ) qp qpo Negatve ndates underestmaton; Postve ndates over estmaton Dr. Dr... Perumal, Professor & & Head, Dept. Dept. of of Hdrolog, I.I.T. I.I.T. Roorkee, Inda Inda

11 # Total reah length 4 km # km; 5 mn. # Total no. of runs 9847 (for eah of the Sant-Venant, VP & VP methods) Numeral Applaton Parameters Values Gamma, γ.5,.5,.5,.5 hannel bed slope, S o.,.,.8,.5,.4,.,. annng s roughness, n.,.,.3,.4,.5 Intal dsharge, b (m 3 /s). Peak stage, p (m) 5., 8.,.,., 5. Tme-to-peak stage, t p (h) 5.,., 5.,. hannel bottom wdth, b (m). hannel sde slope, z.,., 3., 5. Upstream boundar ondton: b p b t p /( γ ) t t γ t p (, t) ( ( t) ) ep Inflow dsharge hdrographs orrespondng to the nput stage hdrographs n the form of Pearson tpe III dstrbuton are used to generate random szes of dsharge hdrographs.

12 Evaluaton of, VP, T & VP ethods Nash-Sutlffe Effen η (%) (a) method (/S o )( / ) ma alwas performs wth η 5% for all the ases eept fve runs η 95% for all the runs eept 5 (.7%) ases

13 Evaluaton of, VP, T & VP ethods Nash-Sutlffe Effen η (%) (b) VP method (/S o )( / ) ma alwas performs wth η 9% for all the ases eept fve runs η 95% for all the runs eept 5 (.7%) ases 3

14 Evaluaton of, VP, T & VP ethods Nash-Sutlffe Effen η (%) T alwas performs wth η 8% for all the ases eept fve runs 6 () T method 5 (/S o )( / ) ma

15 Evaluaton of, VP, T & VP ethods Nash-Sutlffe Effen η (%) VP alwas performs wth η 9% for all the ases eept nne runs 6 (d) VP method 5 (/S o )( / ) ma

16 Evaluaton of, VP, T & VP ethods Error n Volume EVOL (%) (a) method (/S o )( / ) ma EVOL (%) (b) VP method (/S o )( / ) ma EVOL (%) - - EVOL (%) -3-4 () T method (/S o )( / ) ma -3-4 (d) VP method (/S o )( / ) ma

17 - Evaluaton of, VP, T & VP ethods Error n Peak Dsharge qper (%) - -3 qper (%) -4 (a) method -5 (/S o )( / ) ma (b) VP method -5 (/S o )( / ) ma qper (%) qper (%) -4-5 () T method (/S o )( / ) ma -4-5 (d) VP method (/S o )( / ) ma

18 Evaluaton of, VP, T & VP ethods Error n Tme-to-peak Dsharge tpqer (%) (a) method (/S o )( / ) ma tpqer (%) (b) VP method (/S o )( / ) ma tpqer (%) () T method (/S o )( / ) ma tpqer (%) (d) VP method (/S o )( / ) ma

19 onlusons VP method onsstentl performs better than the & VP method. VP, T & methods are full volume onservatve, whereas the VP method has the tenden to alwas loose mass. Due to the nonlneart dnams of rver flood onveton, the method should not be used n SWAT model. Sne the VP method has a sound phsal bass, t ma be preferred over the other estng methods for ts norporaton n the SWAT model for dealng wth varous feld problems. 9

20

21 athematal Formulaton of VP ethod Sant-Venant Equatons: ontnut equaton: omentum eqn. A t S f v v v S g g t () () S o bed slope; S f frton slope; / water surfae slope; (v/g)(v/) onvetve aeleraton; (/g)(v/t) loal aeleraton. agntudes of varous terms n eqn. () are usuall small n omparson wth S o [Henderson, 966; NER, 975]. Dr. Dr... Perumal, Professor & & Head, Dept. Dept. of of Hdrolog, I.I.T. I.I.T. Roorkee, Inda Inda Bak to Slde 6

22 Important steps of dervaton: Unstead flow relatonshp: Usng equatons (), () (4), (5), & (6) frton slope: athematal Formulaton of VP ethod (, ) ( ) S f / S ( ) Av onsderng / & / : B elert of the flood wave d PdR d v (Henderson,966; NER,975): da / da d 3 / Velot b the annng s frton law: (3) v R S f n 4 PdR d Sf So F So 9 da d (4) (5) /3 / (6) (7) Where Froude number: Bak to Slde 6 F v da d ga (8) To Eq. (4)

23 3 Usng Eq. (7) n (3), unstead dsharge: Smlarl, unstead velot: Hdraul ontnut Eq. (): Usng Eqs. (9) & () n (): Normal dsharge n Eq. (9) an be rewrtten as:.athematal Formulaton of VP ethod Bak to Slde 6 ( ) 9 4, d dr B P F S o (9) () ( ) 9 4, d dr B P F S v v v o v t () () v t ( ) 9 4 d dr B P F S o o (3)

24 4 Where: Whh s n the form of: Usng bnomal seres epanson n Eq. (3) wth (/S )(/) << & Eq. (4): elert an also be wrtten as: Usng Eqs. (9) & (7) n (5):.athematal Formulaton of VP ethod Bak to Slde 6 d dr B P F B S 9 4 a 9 4 d dr B P F B S a (4) (5) (6) (7) ( ) 9 4, d dr B P F S o a (8) 9 4 d dr B P F B S a Where: (9)

25 5 Usng Eq. (8) n ontnut Eq. (): Epressng Eq. (9) at the entre pont of the fnte dfferene grd:.athematal Formulaton of VP ethod Bak to Slde 6 (9) () v a t Fnte-dfferene grd representaton of the VP method omputatonal sheme Δ Δt,, v a v a t

26 6 Eq. () an be reorganzed as: Whh an be wrtten as:.athematal Formulaton of VP ethod Bak to Slde 6 () (),, t a a v a a v [ ] [ ] t K K ) ( ) ( θ θ θ θ

27 Where:.athematal Formulaton of VP ethod Eq. () an also be wrtten n the form of the lassal uskngum routng equaton as: (3) 3 Where: t. K. θ t. K. θ t. K t. K ( ). (4). θ ( θ ) t. K. ( 3 θ ) t. K. θ Bak to Slde 6 θ F K v, a B ga 3 S,, B 4 9 F P B dr d ( ) 7

28 8 The reah storage at an omputatonal tme level an be gven b: orrespondng downstream stage hdrograph an be omputed usng Eq. (4) as: Where:.athematal Formulaton of VP ethod Bak to Slde 6 [ ] ) ( K S θ K Prsm storage Wedge storage ) ( K θ ) ( B Ths proves that the lassal uskngum method advoated b arth (938) has the phsal bass