Accuracy of the Muskingum-Cunge method for constant-parameter diffusion-wave. channel routing with lateral inflow

Transcription

1 Accuracy of he Muskngum-unge mehod for consan-parameer dffuson-wave channel roung wh laeral nflow L Wang a, Sergey Lapn c,d,*, Joan. Wu b, Wllam J. Ello e, Frz R. Fedler f a Yakama Naon ERWM, Unon Gap, WA 9890, USA b Washngon Sae Unversy, eparmen of Bologcal Sysems Engneerng, Puyallup Research and Eenson ener, Puyallup, WA 987, USA c Washngon Sae Unversy, eparmen of Mahemacs and Sascs, Pullman, WA 99, USA d Kazan Federal Unversy, Insue of ompuaonal Mahemacs and IT, Kazan, Russa e US eparmen of Agrculure, Fores Servce, Rocky Mounan Research Saon, Moscow, I 88, USA f Unversy of Idaho, eparmen of vl Engneerng, Moscow, I 88, USA * orrespondng auhor E-mal address: slapn@wsu.edu (S. Lapn). Absrac hannel roung s mporan n flood forecasng and waershed modelng. The general consanparameer Muskngum-unge (PM) mehod s second-order accurae and easy o mplemen. Wh specfc dscrezaons such ha he emporal and spaal nervals manan a unue relaonshp, he PM mehod can be hrd-order accurae. In hs paper, we derve he average laeral nflow erm n he second- and hrd-order accuracy PM mehod, and demonsrae ha For spaally and emporally varable laeral nflow, he effec of laeral nflow on smulaed dscharge vares wh spaal and emporal dscrezaons, he value and spaal and emporal

2 varaons of laeral nflow, wave celery, and dffuson coeffcen. omparson of he PM soluon wh he analycal soluon shows ha boh he second- and hrd-order accuracy schemes are more accurae han he smplfed mehod by whch spaal dervaves of laeral nflow are gnored. For small me seps, he hrd-order accuracy PM mehod resuls n hgher accuracy han he second-order scheme even when he hrd-order accuracy creron s no fully me. For large me seps, he emporal and spaal dscrezaon of he hrd- and second-order scheme becomes he same, bu he hrd-order scheme yelds hgher accuracy han he second-order scheme because of he hrd-order accurae esmaon of he laeral nflow erm. Keywords Lnear dffuson-wave channel roung; Muskngum-unge mehod; laeral nflow; order of accuracy; emporal and spaal resoluon; numercal and analycal soluons Abbrevaons onsan-parameer Muskngum-unge (PM); roo-mean-suare error (RMSE); Waer Eroson Predcon Proec (WEPP). Inroducon hannel upsream nflow s usually he mos mporan componen for flood roung. In waershed modelng, however, channel waer ofen comes from laeral nflow. As n he Waer Eroson Predcon Proec (WEPP) model, waer generaed from a hllslope (surface runoff, subsurface laeral flow, and groundwaer base flow) may ener a channel as upsream nflow when he hllslope s a he op of he channel, or as laeral nflow when he hllslope s on he sde of he channel (Fg. ).

3 Top hllslope Sde hllslope n hannel ou Sde hllslope Fgure. Schemac of he relaonshp beween hllslopes and a channel segmen n he Waer Eroson Predcon Proec (WEPP) model. A hllslope can be a he op of a channel only n cases of s -order channels, and would oherwse be on he sde of he channel, wh he op of he channel beng upsream channels or an mpoundmen (Flanagan and Lvngson, 995). In addon, he gan or loss of he sream waer by precpaon, nflraon, and evaporanspraon s ofen ncluded n he laeral nflow erm. In commonly used numercal channel roung mehods, e.g., he Muskngum-unge or he knemac-wave mehod, we need o calculae he average laeral nflow n he channel roung euaon. The order of accuracy of he average laeral nflow erm can be a domnan facor affecng he accuracy of he numercal channel roung n waershed smulaons. Prce (009) developed a second-order accurae nonlnear dffuson-wave scheme and solved usng he Newon-Raphson erave mehod. The auhor also analyzed he effec of bed slope on he accuracy and found he accuracy decreased wh decreasng bed slope. However laeral nflow was consdered o be unformly dsrbued n hs sudy. Todn (007) suded varable parameer Muskngum-unge (VPM) mehod and developed a mass-conservave approach by resolvng he sorage and seady-sae nconssences of he orgnal VPM mehod. In hs sudy, laeral nflow was no consdered. Barry and Baracharya (995) showed ha for channel roung whou laeral nflows, when he me sep and he space nerval mananed a ceran relaonshp so ha he ouran number s 0.5, he Muskngum-unge mehod was hrd-order

4 accurae. For consan-parameer dffuson-wave channel flows whou laeral nflow, Baracharya and Barry (997) derved a relaonshp of spaal and emporal seps of, where denoes knemac celery and dffuson coeffcen, o assure a second-order accuracy of Muskngum-unge scheme, a relaonshp of or for hrd-order accuracy, and fed and for fourh-order accuracy. Szel and Gaspar (000), whou consderng laeral nflow, relaed he emporal and spaal nervals o he ouran number r and Pecle number P e, dscussed her effec on he sables of he Muskngum-unge scheme, and found ha he relaonshp of he spaal and emporal seps reured for he hrd-order Muskngum-unge mehod can be smplfed o a dmensonless euaon r 0. P In addon o he relaonshp beween and, Moramarco e al. (999) repored ha he choce of reference dscharge, whch s used o calculae he knemac celery and he dffuson coeffcen, can also affec he accuracy of he channel roung wh laeral nflow. By esng he channel roung whou he upsream nflow, hey found ha he error n channel roung changed wh reference dscharge and bed slope. For a channel wh a relavely genle slope, such as 0.000, he seleced reference dscharge should be larger for a beer accuracy; for a channel wh a raher seep slope, e.g., 0.0, he accuracy of he channel roung was no sensve o he reference dscharge. The laeral nflow n a channel roung euaon was usually reaed as concenraed or unformly dsrbued for smplcy (how e al., 988; Fan and L, 00). When laeral nflow s spaally and emporally varable, s effec on accuracy of numercal channel roung has no e

5 been dscussed. In hs sudy, we wll () derve a second- and hrd-order accurae represenaon for he laeral nflow erm used n he consan-parameer Muskngum-unge (PM) mehod for channel roung, () compare he resuls from he hrd- and he second-order accuracy PM mehods wh analycal soluon, and analyze he effec of he me-sep sze on he accuracy of he PM soluon.. Mehods The consan-parameer dffuson-wave euaon wh laeral nflow can be smplfed as (Lghhll and Whham, 955; Baracharya and Barry, 997; Fan and L, 00; Prce, 009) () where = (,) s dscharge (m s ), s downsream dsance (m), s me (s), = (,) s laeral nflow rae per un lengh (m s ), wh posve represenng flow no, and negave flow ou of, he channel, dr s knemac wave celery (m s ), and da R R s BS BS f 0 he dffuson coeffcen (m s ) where R s he reference dscharge, A he cross-seconal area (m ), B he channel wdh a he waer surface (m), S f he frcon slope, and S 0 he channel bed slope.. Thrd-order accuracy PM mehod The Muskngum-unge mehod solvng E. () numercally s (how e al., 988; Ponce, 995; Baracharya and Barry, 997; Szel and Gaspar, 000) () where he Muskngum-unge coeffcens are gven by () 5

6 () (5) and () and s he average laeral nflow. For unformly dsrbued laeral nflow, was calculaed as (how e al., 988; Append A). The PM mehod s second-order accurae whou resrcons on emporal and spaal dscrezaons (Append A). To acheve he hrd-order accuracy whou changng he represenaons of he Muskngum-unge coeffcens, he spaal and emporal nervals mus sasfy he followng relaonshps (Baracharya and Barry, 997; Szel and Gaspar, 000; Append A) (7) for a gven, or (8) wh fed. Es. (7) and (8) are euvalen o he followng dmensonless euaon (Szel and Gaspar, 000) 0 e r P (9)

7 7 Hence, for a dffuson wave wh e P, he smulaed ouflow s of a hgher-order accuracy f e r P (0) The hrd-order accuracy average laeral nflow can hen be calculaed as (Append A) () We can also show ha, for he second-order accuracy PM (Append A) () Es. () and () show ha, depends no only on laeral nflow and s spaal and emporal varaon as well as he spaal and emporal dscrezaon, bu also on wave celery and he dffuson coeffcen of he channel flow. If he spaal varaon of laeral nflow s neglgble, he hrd- and second-order accuracy average laeral nflow can also be esmaed from a dscree daase (Append A),.e.,,, for ) ( for ) ( 5 8 () for hrd-order accuracy, and 0,, for. () for second-order accuracy.. A numercal epermen To es he accuracy of he PM mehod, we consder a synhec channel flow

8 , sn sn (m s ) (5) L T for 0 L and 0 T wh L = 0,000 m and T = 0,000 s. The wdh of he recangular channel s m, bed slope 0.0 so ha he effec of he slope seepness on reference dscharge can be negleced (Moramarco e al., 999), and, Mannng s roughness coeffcen The mnmum nflow from E. (5), b = m s, and he peak nflow p = m s. We can calculae he reference dscharge followng Ponce and hagan (99) b p m R s () We hen oban knemac wave celery =.57 m s, and dffuson coeffcen = 50.0 m s. From E. (8), wh he spaal nerval L, he me sep for he hrd-order accuracy PM s = 7 s, and he ouran number s r. 00. Bu wh hs me sep, here are only a few pons whn range of he smulaon me, and much nformaon on emporally and spaally varable dscharge would be los. For easy comparson of he PM wh he analycal soluon, we may choose dfferen me-sep szes, e.g.,,, 5, 0, 0, 50, 00, 00, 500, and 000 s, and dvde he channel no mulple segmens (n s ). For he second-order accuracy PM, we need o keep r as close o as possble n our spaal dscrezaon for any specfc me-sep sze (Ponce, 995, p. 9). For he hrd-order accuracy PM, we dvde he channel no mulple segmens followng E. (7). If he channel lengh s no dvdable by he reured spaal nerval for he hrd-order accuracy, we would make as close as praccal, and n hs case he accuracy would be slghly lower han hrd order. From E. (5), we have cos (7) T T 8

9 cos (8) L L and sn (9) L L Incorporang Es. (7) (9) no () and smplfyng, we oban he laeral nflow, cos cos sn (0) T T L L L L So our channel roung problem s composed of E. (), nal condon 0 boundary condon,, sn, L 0 sn, and laeral nflow calculaed by E. (0). The channel T roung resuls from second- and hrd-order accuracy PM mehod are compared wh he analycal soluon calculaed by E. (5) a =L. The resuls of PM mehod wh average laeral nflow calculaed by E. () are also compared wh he analycal soluon. Snce he laeral nflow (,) defned by E. (0) s no unformly dsrbued, we name he mehod of calculang by E. () as he smplfed mehod. In he smplfed mehod, we sll use he acual values of laeral nflow ha are varable wh space and me, bu he spaal dervaves of laeral nflow are negleced. For unformly dsrbued laeral nflow, hs smplfed mehod recovers he second-order accuracy. To calculae n he PM usng E. () or (), we also need he followng dervaves of (,) sn () T T cos () T T 9

10 sn cos () L L L L cos sn () L L L L sn cos (5) L L L L 5 cos sn () L L L L 5 0 (7) 0 (8). Resuls The smulaed me o peak ( p ) by he second- and hrd-order PM mehods compare well wh he analycal soluon of 500 s for 00 s (Tables and )., s n s, m RMSE, m s p, m s p, s r p, m s E E E E E E E E E E E E E E E E E E E E 0 0

11 Table. Accuracy of he second-order PM channel roung wh laeral nflow for dfferen me-sep szes., s n s, m RMSE, m s p, m s p, s r p, m s 80.5.E E E E E E E E 05.57E E E 05.E E E E E E 05.0E E E 0.08E E E 0.8E E E 0 5.9E E E 0.E 0 r P e Table. Accuracy of he hrd-order PM channel roung wh laeral nflow for dfferen me-sep szes. For =00, 500, and 000 s, he second-order PM resuled n smaller p. The hrd-order PM led o smaller p for =00 and 000 s. Boh mehods adeuaely esmaed he peak dscharge ( p, m s ). The RMSE for he hrd-order PM soluon s 8 mes smaller han for he second-order PM for each correspondng (Table and ). The RMSE for boh mehods decreases wh for 0 s, and remans nearly consan for < 0 s (Fg. ) For large me seps, he spaal dscrezaons of second- and hrd-order accuracy scheme are he same (Tables and ). For small me seps, however, he hrd-order accuracy scheme.

12 Fgure. Roo-mean-suare errors (RMSE) of he smplfed, second-, and hrd-order PM. need fewer spaal seps o reach an mproved accuracy han he second-order accuracy scheme, even he hrd-order accuracy creron r 0 or r s no fully me. P P e For he smplfed mehod, he average laeral nflow are calculaed by gnorng he spaal dervaves of laeral nflow bu sll accounng for dfferen laeral nflow values a dfferen locaons. The smulaed p was underesmaed for 0 s and overesmaed for 50 s (Table ). The smulaed p was comparable wh he hrd-order accuracy scheme for 0 s bu over-esmaed for 0 s. The RMSE of he smplfed mehod was generally larger han ha of he second- or hrdorder scheme (Fg. ). The RMSE was smalles when was close o 0 s, and ncreased wh decreasng and ncreasng. ne ecepon was when =000 s, he resuls were more accurae han when =00 or 500 s, and were comparable wh he hrd-order accuracy soluon. Hence, e, s n s, m RMSE, m s p, m s p, s r p, m s 7..E E 0

13 8..09E E E E E E E E E E E E E E E E E E 0 Table. Accuracy of he PM channel roung wh smplfed calculaon of laeral nflow (assumng unformly dsrbued) for dfferen me-sep szes. for hs specal eample, he spaal dervaves of laeral nflow can be negleced f and were se as 000 s and 000 m, respecvely. The larges errors for PM soluons of dfferen order of accuracy occur a dfferen mes. The larges errors for he second-order PM mehod occur before and afer he peak, beng overesmaes before, and underesmaes afer, he peak (Fg. ). The larges error for he hrdorder or he smplfed mehod occurs only around he peak. The smulaon resuls by he second- and hrd-order PM mehods mached he analycal soluon well for <500 s, and by he smplfed mehod for <00 s.

14 Fgure. Analycal soluon and dfferences n dscharge beween numercal and analycal soluons. (a) second-order accuracy PM, (b) hrd-order accuracy PM, and (c) smplfed mehod. Noe dfferen scales used for dfference n dscharge n (a), (b), and (c).. onclusons

15 For consan-parameer Muskngum-unge dffuson-wave channel roung wh spaally and emporally varable laeral nflow, he accuracy of laeral nflow calculaon s an mporan facor affecng he overall channel roung accuracy. In hs sudy, we derved he average laeral nflow erm n he second- and hrd-order accuracy PM mehods for channel roung. The derved euaons ndcaed ha for spaally and emporally varable laeral nflow, he effec of laeral nflow on smulaed dscharge depended no only on he value of laeral nflow, s spaal and emporal dervaves, he spaal and emporal dscrezaons, bu also on wave celery and dffuson coeffcen of he channel flow. The second-order PM mehod led o ncreased accuracy wh decreasng me-sep szes, and kep relavely consan for furher decreased me-sep szes. Usng larger me-sep szes s compuaonally more effcen, bu wh hgher rsk of mssng he eac peak dscharge pon by as much as one me sep. The accuracy of he hrd-order PM soluon ncreased wh decreasng me-sep szes, and was hgher han he second-order PM mehod, even when he hrd-order accuracy PM mehod reuremen r 0 was no fully sasfed because of lmaon of consan P e emporal and spaal nervals used. Is compuaonal coss can be much lower han he secondorder PM mehod for smaller me-sep szes when reured few spaal seps. For larger me seps, s spaal dscrezaon became he same as for he second-order scheme. Ths suggesed ha for a fed me sep, we can ge second-order accuracy PM mehod by mananng a ouran number of as close o as praccal, and oban a hgher accuracy by usng a larger spaal sep or a smaller ouran number, r, wh he condon ha P e. P When we gnore he spaal dervaves of he laeral nflow as n he smplfed mehod, he RMSE of he numercal channel roung resuls was generally larger han ha of he second- and e 5

16 hrd-order accuracy schemes. I was smalles for a me sep of 0 s, and ncreased wh boh decreasng and ncreasng of he me-sep sze. nly for a specal dscrezaon, he smplfed mehod led o he same resul wh he hrd-order accuracy scheme. The second-order accuracy PM led o overesmaon before and underesmaon afer, he me of peak dscharge. The hrd-order accuracy PM and he smplfed mehod only led o over- or underesmaon near he me of peak dscharge. Acknowledgemens Ths work was suppored by he USA SREES EAP Gran (No ) and Washngon Sae Unversy. References Baracharya, K., Barry,.A., 997. Accuracy crera for lnearsed dffuson wave flood roung. J. Hydrol. 95, Barry, A, Baracharya, K., 995. n he Muskngum unge flood roung mehod. Envron. In.,, how, V.T., Madmen,.R., Mays, L.W., 988. Appled Hydrology, McGraw-Hll, New York. Fan, P., L, J.., 00. ffusve wave soluons for open channel flows wh unform and concenraed laeral nflow. Adv. Waer Resour. 9, Flanagan,.., Lvngson, S.J. (Eds.), 995. USA-Waer Eroson Predcon Proec: User Summary. NSERL Rep. No., Nal. Sol Eroson Res. Lab., USA ARS, Wes Lafayee, IN, 9 pp. Hayam, S., 95. n he propagaon of flood waves. saser Prev. Res. Ins., Kyoo Unv., Bull., : -.

17 Moramarco, T., Fan, Y., Bras, R.L., 999. Analycal soluon for channel roung wh unform laeral nflow. ASE, J. Hydraul. Eng. 5, Moussa, R., 99. Analycal Hayam soluon for he dffusve wave flood roung problem wh laeral nflow. Hydrol. Process. 0, Moussa, R., Bocullon,., 99. Algorhms for solvng he dffusve wave flood roung euaon. Hydrol. Process. 0, 05. Prce, R. K., 009, Volume-conservave nonlnear flood roung, J. Hydr. Eng. 5, Ponce, V. M., 995. Engneerng Hydrology, Prncples and Pracces, Prence Hall, Englewood lffs, NJ. Ponce, V. M., hagan, P.V., 99. Varable-parameer Muskngum-unge mehod revsed, J. Hydrol., 9. Szel, S. and Gaspar,., 000. n he negave weghng facors n Muskngum-unge scheme, J. Hydraul. Res., 8(), 99 0 Sngh, V. P., 99. Knemac Wave Modelng n Waer Resources, Surface-Waer Hydrology. Wley, London. Tang, X., Kngh,. W. and Samuels, P. G., 999. Volume conservaon n varable parameer Muskngum-unge mehod, J. Hydraulc Eng. (ASE), 5(), 0 0. Thomas, J. W., 995. Numercal Paral fferenal Euaons: Fne fference Mehods, Sprnger-Verlag, New York. Todn, E., 007. A mass conservave and waer sorage conssen varable parameer Muskngum-unge approach. Hydrol. Earh Sys. Sc.,,

18 Append A ervaon of he hrd-order accuracy PM mehod for consan-parameer dffusonwave channel roung wh laeral nflow In order o oban he rd -order accurae soluon of E. (), we calculae he dervaves of respec o space and me, and represen hem as he dervaves of space only. Frs, we rearrange E. () as (A) where,,,,,,, and, are used for brevy, and smlar noaons are used for he followng dervaons. The dervaves are hen (A) (A) (A) 5 (A5) (A) 5 (A7) 8

19 (A8) Epand he erm, n E. () respec o (, ) o he hrd order Taylor seres, neglec he superscrp n he dervaves a (, ) for brevy, and drop he hgher order erms, (A9) Incorporang he dervave erms and rearrangng gves (A0) Smlarly, (A) (A)

20 0 Incorporang E. (A0), (A), and (A) no (), we have (A) Euae he coeffcen erms relaed o,,,, and, respecvely, : ` (A) : (A5) : (A) : (A7)

21 : (A8) Solvng he sysem euaons of (A) (A) gves he Muskngum-unge coeffcens (A9) (A0) and (A) The coeffcens are he same as ha gven by how e al. (988) and Ponce (995). And shows ha he PM mehod whou furher resrcon s second-order accurae. Incorporang E. (A0) and (A) no (A7) and smplfyng, we have 0 (A) solvng for, we have (A) or solvng for, we have

22 (A) Es. (A) and (A) are he relaonshps beween and reured o manan he hrdorder accurae for he PM mehod, and has been derved by Baracharya and Barry (997) and Szel and Gaspar (000) for PM mehod solvng he dffuson-wave channel roung whou laeral nflow. vdng boh sdes of (A) by and nroducng r and P e, we can smplfy (A) o a dmensonless euaon reured for elmnang he dsperson error o oban he hrd-order PM mehod (Szel and Gaspar, 000): 0 e r P. From E. (A), n order for o be real, we mus have (A5) From E. (A8), we oban (A) Leng (A7) we ge

23 (A8) And ncorporang E. (A0) no (A7) and smplfyng resuls n (A9) Es. (), (A9) (A), (A9), and (A8) ogeher form he hrd-order PM mehod wh spaal or emporal lmaons defned by E. (A) and (A), respecvely. If he spaal varaon of laeral nflow s neglgble so ha he dervaves of wh respec o vansh n E. (A8), he average laeral nflow can also be esmaed smply from a dscree daase. Snce (Thomas, 995),, for ) ( 0 for ) ( (A0) and,, for ) ( 0 for ) ( (A) Incorporang Es. (A0) and (A) no (A8) and smplfyng, we have,, for ) ( for ) ( 5 8 (A) Followng he same procedure, he second-order accuracy laeral nflow erm can be obaned as

24 (A) If we gnore he spaal dervaves, and use, E. (A) can be smplfed o (how e al., 988) 0,, for (A) For knemac wave channel roung, = 0, E. (A) s smplfed o (A5) If we esmae he dervaves and, respecvely, by and, E. (A5) becomes. (A)