Extreme Hydrlgical Events: Precipitatin, Flds and Drughts (Prceedings f the Ykhama Sympsium, July 1993). IAHS Publ. n. 213, 1993. 129 Cmparisn f tw variable parameter Muskingum methds M. PERUMAL Department f Cntinuing Educatin, University frrkee, Rrkee-247667, India Abstract The Variable Parameter Muskingum-Cunge (VPMC) methd and the Multilinear Muskingum (MM) methd are cmpared fr ruting fld hydrgraphs in channels. Bth methds attempt t accunt fr nnlinearity in the fld wave mvement by varying the physically based parameters. They are studied by ruting tw different shapes f hypthetical fld hydrgraphs, estimated using tw different initial flws in the channel, fr a reach length f 40 km in tw unifrm wide rectangular channels with widths f 100 m and 200 m, and each having fur different cmbinatins f bed slpe and Manning's rughness values. Bth methds are evaluated by cmparing their ruted slutins with the crrespnding St Venant slutins. Based n the study, it is recmmended that the MM methd be used fr ruting in natural channels and the VPMC methd fr ruting in urban strm drains. INTRODUCTION Simplified hydraulic fld ruting methds which use linear system mdels with time invariant parameters are based n the assumptin that the flw variatins arund a reference discharge which is used fr estimating the mdel parameters are small. Fr ruting a given fld hydrgraph in a channel reach with a linear mdel, this assumptin implies that a reference discharge is used fr the estimatin f the parameters f the mdel irrespective f the magnitude f variatin f the hydrgraph abut this reference value. This limitatin prduces distrtin in the cmputed utflw hydrgraph when wide variatins in flw variable are realized. Mrever, fld waves are inherently nnlinear in nature as different discharges travel at different celerities. The cnvenience f linear systems analysis can be used fr mdelling nnlinear hydrlgical prcesses by wrking within the limitatin impsed by its assumptin. One simple methd by which the nnlinearity f the fld ruting prcess may be taken int accunt is t use a mdel that respnds linearly t the input at any ne time, but the linear prperties f which may vary frm time t time, with the flw variable cntrlling the phenmenn. Using this cncept, Pnce & Yevjevich (1978) prpsed a fld ruting methd knwn as the variable parameter Muskingum-Cunge (VPMC) methd in which the parameters f the Muskingum methd vary at every ruting time level cnsistently with the established physical relatinships as enumerated by Kundzewicz (1986). Recently, Perumal (1992) has prpsed a Multilinear Muskingum (MM) methd in which the same physically based parameter relatinships as adpted in the VPMC methd is used, but the ruting is carried ut using multilinear mdelling apprach based n time distributin scheme (Kundzewicz, 1984). In the VPMC methd, the slutin is
130 M. Perumal achieved using the cnventinal Muskingum equatin, which is in a recursive frm, and in the case f MM methd it is btained using the cnvlutin apprach. Nte that bth appraches yield the same slutin when the parameters remain cnstant thrughut the ruting prcess, and, therefre, the use f recursive equatin is preferred ver that f cnvlutin apprach due t its better cmputatinal efficiency (Overtn, 1970). Hwever, this uniqueness in the slutins exists n lnger when the parameters are varying in time during the ruting prcess f bth methds. Althugh bth attempt t accunt fr nnlinearity in the fld wave mvement prcess, the apprach used fr varying the parameters in each methd is different. Further, in either methd the technique f varying the parameters is nt physically based, i.e. it is nt cnsistent with the variatins built in the slutin f the St Venant equatins. This aspect leads t the lack f cnservatin f mass while using the VPMC methd fr ruting a fld hydrgraph when its rating curve is characterized by a wider lp. Hwever, the cnservatin f mass is ensured always in the case f the MM methd wherein the ruting is achieved using the cnvlutin technique. This paper evaluates the perfrmance f bth methds by ruting tw different shapes f hypthetical fld hydrgraphs, estimated using tw different initial flws in the channel, fr a reach length f 40 km in tw unifrm wide rectangular channels with widths f 100 m and 200 m, and each having fur different cmbinatins f bed slpe and Manning's rughness values, and cmparing the respective cmputed utflw hydrgraphs with the crrespnding bench mark slutins btained by the numerical slutin f the St Venant equatins. A brief descriptin f bth ruting methds is presented herein fr the prper understanding f the parameter variatin prcedures adpted in these methds. VARIABLE PARAMETER MUSKMGUM-CUNGE METHOD The cnventinal ruting equatin f the Muskingum methd fr the grid cell shwn in Fig. 1 is expressed as: Qn.i = CA^c 2 i n + c 3 Q n (i) n-1 At «< Fig. 1 Space-time discretizatin f Muskingum methd.
Cmparisn f tw variable parameter Muskingum methds 131 where Q n and Q n+1 dentes the cmputed utflw at time nat and (n+l)at respectively; I n and I n+1 dentes the crrespnding inflw, and At is the ruting time interval. The cefficients C 1; C 2, and C 3 are expressed as: s-\ L\L jù J\. C7 srs \ 1 = (2a) 2K(1-Q) + At C, = 2 At + 2KQ 2K(1-Q) + At c = 2K(1-Q)-At '5 2K(1-B)+At (2 b ) V ' (2c) in which K and 0 are the strage cefficient and weighting parameter respectively. The parameters K and 0 may be expressed by physically based appraches as: K = Axle ( 3 ) 0 = 1-0 (4) 2 2S 0 BcAx in which Ax is the reach length; c is the average fld wave celerity ver the reach; Q 0 is the reference discharge; B is the width f the channel, and S 0 is the bed slpe. Fllwing Pnce & Yevjevich (1978), there are tw acceptable techniques t vary the ruting parameters: (a) the direct three pint technique, and (b) the iterative fur pint technique. Fr bth techniques, the space step Ax and bed slpe S 0 are specified at the start, and kept cnstant thrughut the cmputatin in time. Pnce & Yevjevich (1978), and Kussis & Osbrne (1986) while cmparing these tw variatin techniques cncluded that iterative fur pint technique des nt ffer any advantage ver the direct three pint technique. Therefre, nly the direct three pint technique is used in this study fr varying the parameters f the Muskingum methd. In the three pint technique, the values f c and Q 0 at grid pints n and (n+1) f the inflw sectin, and at grid pint n f the utflw sectin are used t find the average cell value f c and the reference discharge value Q 0 as: C = ( C n,i +C n + l,i + C n, q )'3 ( 5 ) Q 0 = (I n + l n+1 + Q n )l3 (6) in which c nii and c n+1>i are the wave celerities crrespnding t the inflw at nat and (n+l)at respectively; c nq is the wave celerity crrespnding t utflw at nat. Fr all its advantage in accunting fr the nnlinearity in fld wave mvement, VPMC methd is nt withut pitfalls. Perhaps mre imprtant thugh, is its slight tendency nt t cnserve mass. Pnce (1983) recgnized this deficiency and suggested further research t vercme it.
132 M. Perumal MULTILINEAR MUSKINGUM METHOD One f the ways f vercming this deficiency is t use the multilinear Muskingum methd based n time distributin scheme prpsed by Perumal (1992) wherein the utflw hydrgraph is cmputed using cnvlutin technique which ensures cnservatin f mass. In this methd, each element f inflw Idt is ruted thrugh a Muskingum sub-mdel, fr which the parameters K and 0 are determined by equatins (3) and (4). The reference discharge Q 0 needed in these expressins is estimated as: Q = h + a{i{t)-l b ) where I b is the initial steady flw in the reach befre the arrival f the fld; I(t) is the current value f inflw; and a is a cefficient with limits 0<a< 1. Nte that the expressin given by equatin (7) is slightly mdified frm the earlier expressin adpted by Perumal (1992) in rder t accmmdate the initial steady flw in the reach. The wave celerity is estimated fr a wide rectangular channel, using Manning's frictin law as (Dge et al., 1982): c = 1.67v 0 (8) where v 0 is the velcity crrespnding t Q 0. The reference flw depth y 0, needed in the estimatin f v 0, is cnsidered as the nrmal depth crrespnding t the reference discharge Q 0, whence: v 0 = (Q 0 /Bf 4 S 0 3 n- 0-6 (9) where n is the Manning's rughness cefficient. The rdinates f the discrete unit hydrgraph needed fr cnvlutin with the inflw hydrgraph rdinates can be arrived at by successive applicatin f equatin (1) as: V h 2 V h 4 - = c i -~- c 2 + c 3 Cj ~- c 3 (c 2 + c = C 3 ( C 2 + C 3 Cj) c 3 c i) (10) (V) K = c "~ 2 (c 2 + c s c i) The verall utflw discharge at time nat is estimated by the methd by adding the cmpnent utflws. APPLICATION Bth methds were applied fr ruting hypthetical flds in wide rectangular channels with n lateral inflw within the ruting reach cnsidered. The inflw
Cmparisn f tw variable parameter Muskingum methds 133 hydrgraph, defined by a mathematical functin was ruted in the given channel reach fr a specified distance using bth methds, and their respective utflw hydrgraphs were cmpared with the crrespnding St Venant slutins. The inflw hydrgraph defined by a fur parameter Pearsn type-ill distributin functin expressed by the fllwing equatin was adpted in this study: 7(0 = h + Vp-hWItj,) 1 "»-» exp [(l~t/t p )/(j-l)] (11) where I p is the peak flw (1000 m 3 /s); tp is the time t peak (10 h) and 7 is the skewness factr. Tw different inflw hydrgraphs characterized by the values f the skewness factr 7 = 1.15, and 1.5, respectively, were ruted fr a distance f 40 km in all the test runs. Tw wide rectangular channels with widths 100 m and 200 m, each having fur different cmbinatins f bed slpe and Manning's rughness values were used fr all the ruting studies. Tw different initial steady flws, ne with a lw value (100 m 3 /s) and anther with a high value (500 m 3 /s) were used in the study fr estimating the inflw hydrgraph crrespnding t each skewness factr 7 = 1.15, and 1.5. The descriptin f channel cnfiguratins and the skewness factrs f the inflw hydrgraph adpted crrespnding t different test runs are given in Table 1. Thrughut the ruting studies, the value f cefficient a used in equatin (7) fr cmputing the reference discharge was taken as 0.4. This value was adpted frm the experience f the past study (Perumal, 1992). Table 1 Skewness factrs f the inflw hydrgraph and channel cnfiguratins used in the study. Run B S Manning's N. y (m) n 1 1.15 100 0.0002 0.04 2 1.15 200 0.0002 0.04 3 1.5 100 0.0002 0.04 4 1.5 200 0.0002 0.04 5 1.15 100 0.0002 0.02 6 1.15 200 0.0002 0.02 7 1.5 100 0.0002 0.02 8 1.5 200 0.0002 0.02 9 1.15 100 0.002 0.04 10 1.15 200 0.002 0.04 11 1.5 100 0.002 0.04 12 1.5 200 0.002 0.04 13 1.15 100 0.002 0.02 14 1.15 200 0.002 0.02 15 1.5 100 0.002 0.02 16 1.5 200 0.002 0.02 EVALUATION OF BOTH METHODS A ttal f 32 sets f runs, half f which crrespnded t the assumed lw initial flw in the reach (and the ther half - t high initial flw), were made fr evaluating
134 M. Perumal bth the VPMC and the MM methds in reprducing the crrespnding St Venant slutins. A ruting interval f 15 min was used in all these runs t ensure that the slutins are independent f the time interval used. Since bth methds were cnsidered apprpriate fr ruting a fld hydrgraph (Pnce & Yevjevich, 1978; Perumal, 1992) within their theretical limitatins, it was cnsidered sufficient t cmpare their perfrmance in reprducing sme main features f the St Venant slutins. Tables 2 and 3 present the peak and time t peak f the St Venant slutins btained at the end f 40 km reach f the channels using the inflw hydrgraphs with initial steady flw f 100 m 3 /s and 500 m 3 /s, respectively. The crrespnding estimates btained using bth methds under study are als shwn alng with the factrs which characterize the reprductin f the St Venant slutins and the ability t cnserve mass while ruting. Ruting slutins were btained by dividing the 40 km reach int single, eight and frty sub-reaches. The ability t reprduce the entire St Venant slutin by the methds under study is measured by the variance explained by the methd, dented as R 2, which is estimated in percentage using the Nash-Sutcliffe criterin (Nash & Sutcliffe, 1970). The parameter t evaluate the cnservatin f mass f the ruted slutin was estimated as: N N ERVOL = ill ^ _ 100 (12) where Q es y is the jth estimated utflw rdinate, and N is the ttal number f utflw hydrgraph rdinates. Table 2 shws the ruting results btained using the inflw hydrgraph with the initial flw f 100 m 3 /s. It is seen that the MM methd is slightly better than the VPMC methd in reprducing the peak flw characteristics, when the inflw hydrgraph with a wider lp rating curve is ruted. While ruting such inflw hydrgraphs, the VPMC methd als des nt cnserve mass and this tendency increases with the increase in the number f sub-reaches used fr ruting in a given reach. The VPMC methd results in cmputatinal prblem due t the estimatin f negative reference discharge in the beginning f ruting, when the 40 km reach is cnsidered as a single reach. This clearly indicates the deficiency f the prcedure adpted in the VPMC methd fr cmputing the reference discharge. Hwever, bth methds have the same capability in reprducing the St Venant's slutins when the attenuatin f the ruted hydrgraph is small r, in ther wrds, when the rating curve f the inflw hydrgraph is characterized by a narrw lp. Under such cnditins, the lack f cnservatin f mass by the VPMC methd may be cnsidered very small r negligible fr all practical purpses. Table 3 shws the ruting results btained using the inflw hydrgraph with the initial flw f 500 m 3 /s. When all the channel and inflw hydrgraph characteristics remain the same, except that f the initial flw in the reach, the rating curve f the inflw hydrgraph with lwer initial flw has wider lp than that f the inflw hydrgraph with higher initial flw, i.e., unsteady behaviur is mre prnunced while ruting inflw hydrgraph with lwer initial flw. The nndimensinal rating curves estimated at the inlet f the reach crrespnding t the run
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Cmparisn f tw variable parameter Muskingum methds 137 number 1 f Tables 2 and 3 are shwn in Fig. 2 which demnstrates the varying unsteady flw aspects due t the variatin f initial flw nly in the reach. The stage values were rendered nndimensinal using the maximum stage f the crrespnding inflw hydrgraph in the given channel. The ruting results crrespnding t the inflw hydrgraph with higher initial flw as presented in Table 3 shw that bth methds have the same capability in reprducing the St Venant slutin in all its aspects. All the deficiencies f the VPMC methd bserved while ruting the inflw hydrgraph, with lwer initial flw value f 100 m 3 /s, disappear in this case. 0.2 I I I I I I I I I 100 200 300 «00 500 600 700 600 «00 1000 DISCHARGE (»V 1 ) Fig. 2 Rating curves f the inflw hydrgraphs with different initial flws. All the successful runs using the VPMC methd indicate that its cmputatinal efficiency is better than that f the crrespnding runs f the MM methd, specifically when the number f sub-reaches increases fr ruting in a given length f reach. Hwever, the slutins btained fr 8 and 40 sub-reaches f the given 40 km reach are very clse, implying negligible imprvement in the ruted slutin when the number f sub-reaches increases beynd a certain limit. This suggests that the use f the MM methd is cmputatinally nly slightly less efficient than the VPMC methd. Frm the cnsideratin f cmputatinal efficiency, ne may prefer the use f VPMC methd prvided the magnitude f ERVOL is within certain acceptable limits. Hwever, the decisin n such a limit is subjective. T avid this subjectivity, ne may use the MM methd fr ruting in channels f natural basins irrespective f the attenuatin f peak flw. The VPMC methd may be used fr ruting in urban strm drains, wherein the attenuatin f the hydrgraph is negligibly small.
138 M. Perumal CONCLUSIONS The results f the present study indicate that the MM methd is slightly better than the VPMC methd in reprducing the peak flw characteristics, when the inflw hydrgraph subjected t larger attenuatin is ruted. While ruting such inflw hydrgraphs, the VPMC methd als des nt cnserve mass and this tendency increases with the grwth f the number f sub-reaches used fr ruting in a given reach. Ruting using the Multilinear Muskingum (MM) methd is free f this deficiency due t the use f the cnvlutin apprach in arriving at the slutin. But the MM methd is cmputatinally slightly less efficient than the VPMC methd and this shrtcming is prnunced when the number f sub-reaches increases. Frtunately, very few sub-reaches are sufficient fr ruting in a given reach using the MM methd in rder t get a slutin clse t the bserved ne. T avid subjectivity n the decisin regarding the acceptable errr in the cnservatin f mass while ruting using VPMC methd, it is recmmended that the MM methd be adpted fr ruting in natural channels and the VPMC methd fr ruting in urban strm drains, wherein the attenuatin is small. REFERENCES Dge, J. C. I., Strupczewski, W. G. & Napirkwski, J. J. (1982) Hydrdynamic derivatin f Strage parameters f the Muskingum mdel. J. Hydrl. 54, 371-387. Kussis, A. D. & Osbrne, B. J. (1986) A nte n nnlinear strage ruting. Wat. Resur. Res. 22 (13), 2111-2113. Kundzewicz, Z. W. (1984) Multilinear fld ruting. Acta Gephys. Pl. 32(4), 419-445. Kundzewicz, Z. W. (1986) Physically based hydrlgical fld ruting methds, Hydrl. Sci. J. 31(2), 237-261. Nash, J. E. & Sutcliffe, J. V. (1970) River flw frecasting thrugh cnceptual mdels part I-a discussin f principles, J. Hydrl. 10, 282-290. Overtn, D. E. (1970) Rute r cnvlute? Wat. Resur. Res. 6(1), 43-52. Perumal, M. (1992) Multilinear Muskingum fld ruting methd. J. Hydrl. 133, 259-272. Pnce, V. M. (1983) Accuracy f physically based cefficient methds f fld ruting. Tech. Reprt SDSU Civil Engineering Series N. 83150, San Dieg State University, San Dieg, Califrnia, USA. Pnce, V. M., & Yevjevich, V. (1978) Muskingum-Cunge methd with variable parameters. /. Hydraul. Div. ASCE, 104(12), 1663-1667.