Some problems with the Muskingum method

Transcription

1 Hydrlgical Sciences - Jurnal- des Sciences Hydrlgiques, 32, 4,12/1987 Sme prblems with the Muskingum methd INTRODUCTION LUO BOKUN Bureau f Hydrlgy, Yangtze Valley Planning Office, Wuhan, Peple's Republic f China QIAN UEWEI Heilngjiang Prvincial Hydrlgical Service, Peple's Republic f China ABSTRACT The Muskingum fld ruting methd is widely used by hydrlgists and gd results are frequently achieved. Hwever, there is still sme dispute abut the Muskingum methd. In this paper it is shwn that the Muskingum methd is an apprximate slutin f the instantaneus unit hydrgraph (IUH) f the lag and rute fld ruting methd. The integratin slutin f the Muskingum methd fr multiple river reaches is als derived. The negative respnse issue is discussed in relatin t the basis f the linear thery f hydrlgical systems. Certains prblèmes rencntrés dans l'applicatin de la méthde de Muskingum RESUME La méthde de Muskingum pur l'étude de la prpagatin de la crue vers l'aval est largement utilisée par les hydrlgues et n abutit suvent à de bns résultats. Cependant il y a encre certaines cntestatins à prps de cette méthde de Muskingum. Dans cet article, n mntre que la méthde de Muskingum est une slutin apprchée du temps de répnse et de la prpagatin de l'nde de crue de 1'hydrgramme unitaire instantané (HUI). On en déduit aussi la slutin d'intégratin de la méthde de Muskingum pur plusieurs biefs de la rivière. Le résultat crrespndant a une répnse négative est discuté en relatin avec les bases de la thérie linéaire des systèmes hydrlgiques. The Muskingum fld ruting methd is based n the assumptin f a linear relatinship between the inflw t and the utflw frm a river reach and the reach strage. This methd uses tw equatins, namely the strage equatin which is written as: S(t) = K[xl(t) + (1 - x)q(t)] (1) and the cntinuity equatin: ds(t)/dt = I(t) - Q(t) (2) Open fr discussin until 1 June

2 486 Lu Bkum & Qian uewei where S(t) is the reach strage between the upstream and the dwnstream ruting sectins, I(t) and Q(t) are the rates f inflw and utflw, respectively, x is a weighting factr and K is called the strage cefficient and has the dimensin f time. Equatins (1) and (2) can be cmbined int: f + Q = I - Kxf (3) If the inflw I(t) is the Dirac delta functin, <5(t), equatin (3) can be integrated t yield (Venetis, 1969): u(t) = K(i - x)» exp^ a^> " T^l ô(t) (4) Equatin (4) is the expressin fr the IUH f the Muskingum methd fr a single river reach. When equatin (3) is slved by a finite difference slutin, the fllwing equatins are easily btained: Q mat = VmAt + C l I (m-l)at + C 2 Q (m-l)at (5) where : C = -(Kx - 0.5At)/(K - Kx + 0.5At) C = (Kx + 0.5At)/(K - Kx + 0.5At) (6) C 2 = (K - Kx - 0.5At)/(K - Kx + 0.5At) where At is the finite difference sample interval and m is an integer which indicates the number f intervals frm the time rigin. The lag and rute fld ruting methd cnsiders that the prpagatin f a fld wave is subjected t the effects f translatin and attenuatin; the impulse respnse is that f a single linear reservir reach delayed by the lag, x (Nash, 1960; Dge, 1973). This mdel has the system respnse: u(t)=iexp(-^ (7) where x and K^ are parameters fr the single reach, If the ruting reach f this mdel is divided int n subreaches each delayed by the lag, x, and having the strage cnstant, K L, the system respnse fr the verall ruting n times is (Lu et al., 1978; Wang, 1982):, s! ft - nx-i ft- nti u(t) = i^a H ^ e*p( "É^-J < 8) In this paper we will derive the integratin slutin f the Muskingum methd fr multiple river reaches. The relatinship between the Muskingum methd and the lag and rute methd fr multiple reaches will be discussed n the basis f the linear thery f hydrlgical systems. The issue f initial negative utflws with the Muskingum methd will als be explred.

3 Sme prblems with the Muskingum methd 487 THE INTEGRATION SOLUTION OF THE MUSKINGUM METHOD FOR MULTIPLE RIVER REACHES Taking the Laplace transfrm f equatin (3) and the initial cnditins f Q(0) = 0 and Q'(0) = 0, we btain: L[Q] = s L(I) (9) S + In equatin (9), if the inflw, I(t), is a Birac delta functin, ô(t), then the utflw, Q(t), is the IUH. Taking L[6(t)] = 1, equatin (9) becmes: L[u(t)] = x When equatin (10) is applied t multiple river reaches, i.e. the ttal ruting reach is divided int n identical subreaches, then we have a system cmpsed f n subsystems in series with the identical values, K and x, fr each subreach. Thus,, by using the utflw f a preceding subreach fr the inflw t the succeeding ne and perfrming utflw rutings ne after anther frm the first subreach dwn t the n-th subreach, equatin (10) becmes: (10) L[u(t)] = 2 S + (ID Expanding equatin (11) by means f the binmial therem, and evaluating the inverse image f each term, we can btain: u(t) L i=0 1 II (-1) (1 ) n_i+1 r(n) n-i-1 exp + (-D <5(t) (12) Equatin (12) is the IUH f the Muskingum methd fr multiple identical river reaches, i.e. the integratin slutin f the Muskingum methd by successive ruting. It will be prved that the fllwing necessary cnditin hlds fr equatin (12): Since: J^u(t) dt = 1 (13) n-l (-l) 1 (i n ) x 1 I u(t)dt = I. L J 0 i=0 (1 _ x)n r(n _ ±) J n-i-1 exp + (-1) ^1 - x J jtô(t)dt

4 488 Lu Bkun & Qian uewei and: / <5(t)dt = 1 and: CO J f n-i-1 n-i-i -m dm exp r(n - i) ' where m = t/[], therefre: r u(t)dt = i n ~l ( - 1)1(in) n xi + ^ ^ J i=0 0 1=0 (1., - x),n,, n (1 - x) 1 n (-DW n i=0 (1 - x) = 1 (14) The first three mments f the integratin slutin f the multiple reach Muskingum methd can be shwn t be: (1) M u nk N (2) = nk 2 (l - 2x) u N (3) = 2nK 3 (3x 2-3x + 1) u Where M^D is the first mment abut the time rigin, and N< r > is the rth (central) mment abut the mean. Since the first three mments f an inflw and crrespnding utflw can be cmputed frm the given inflw and utflw, the values f n, K and x defining the IUH can be fund frm equatins (15), (16) and (17). (15) (16) (17) THE IUH OF THE MUSKINGUM METHOD AS AN APPROIMATE SOLUTION OF THE LAG AND ROUTE METHOD Derivatin f frmulae fr the IUH f the Muskingum methd and the lag and rute methd by system analysis The relatinship between the inflw and the utflw f the generalized linear hydrlgical system can be expressed by (Chw, 1975): «m +l m _ ad +a,d a D Q(t) = m ïii-1 O. _n+l n Kt) (18) b D + b nd + + b n n-1 Q D + 1 Q<t> = }i(t> (19)

5 Sme prblems with the Muskingum methd 489 where D = d/dt is the differential peratr and aj, bj are cefficients. If bth the a^ and the bj are cnstants r independent f I(t) and Q(t), equatin (18) r (19) gverns the behaviur f a linear system. The IUH f the Muskingum methd and that f the lag and rute methd can be derived frm these frmulae (Qian & Lu, 1981). Fr a single river reach Let M(D) = e~ a D and N(D) = b 0 D + 1. Equatin (19) can then be reduced t: Q(t) = - * e~ a D I(t) (20) b Q D + 1 The functin I(t - a 0 ) can be expanded in a Taylr's series and written as: I(t - a Q ) = e _a D I(t) (21) When the inflw is a Dirac delta functin, 6(t), the utflw becmes the IUH, i.e.: u(t)= 4-^W e ~ a Dô(t) and s: u(t) =^- p + (\ /b) 6(t-a Q ) (22) Using the Laplace transfrm f equatin (22): 1 * ~ a u(t) = exp( ) a 0 i t b (23) u(t) =0 a Q > t can be btained. When a 0 = T, b 0 = K L, equatin (23) becmes: u(t) = ~- exp(- :^^L) (24a) K L &L which is the expressin f the IUH fr a single river reach with bth the translatin effect and the attenuatin effect f flw (Qian & Lu, 1981) as shwn in equatin (7). When a 0 = Kx, b Q =, equatin (23) becmes: u(t) _ 2 exp[- t -~ ~1 (24b) v vt ' J K ' This is a versin f the expressin fr the IUH f a single river reach. It als embdies these tw effects f translatin and attenuatin except that the translatin effect is reflected in the prduct f the tw cefficients K and x. If the expressin fr the expanded functin f e is truncated t the secnd rder, that is, the inflw is such that its secnd and higher rder time derivatives are small enugh in magnitude t be negligible, then: e~ a D «1 - a Q D

6 490 Lu Bkum & Qian uewei If a Q = Kx, b 0 equatin (22)., then equatin (4) can be derived frm Fr multiple river reaches When the length f a river reach between the upstream sectin and the dwnstream sectin is quite lng, the reach shuld be divided int n identical subreaches and the fld ruting carried ut reach by reach. Frm equatin (22), the fllwing equatin hlds: u(t) = b n V D + <l/b 0 )J -na D Using the Laplace transfrm, equatin (25) becmes: u(t) =^7(nT H-g^H 6(t) (25) 1 A ~ na 0. n-l exp f- (26) If a Q = T, b Q = K L, equatin (26) can be reduced t the frmula fr the IUH f the lag and rute methd fr multiple reach reaches (viz. equatin (8)): u(t) ft - nti n-l K L r(n) ^ K L exp I t - nr-i K T - 1 (27a) If a Q = Kx, b Q =, equatin (26) can be reduced t anther frm f the lag and rute methd fr multiple river reaches: u(t) = T(n) t - nkx n-l exp t - nkx (27b) Similarly, the Laplace transfrmatin may be applied t equatin (25) where e -a nd 1 - a 0 D is taken. If a 0 = Kx, b 0 =, the integratin slutin fr the Muskingum methd fr multiple river reaches (equatin (12)) can be btained. Relatinship between the strage equatins fr the Muskingum methd and the lag and rute methd Changing equatin (2) int (b Q D + l)q(t) = e -a u I(t), and expanding e -a in a Taylr's series, a simultaneus slutin with the cntinuity equatin can be perfrmed. If a 0 = Kx, b 0 =, we btain: a a 2 S(t) = b Q Q(t) + ~ I(t) - ^y DI(t) (-1) D mi I(t) = Q(t) + Kxl(t) - ^-jf- DI(t) + + (.!)»-! HELL- D»- 1 I(t) +. m! (28)

7 Sme prblems with the Muskingum methd 491 This is the strage equatin crrespnding t the lag and rute methd. Cmparing equatin (28) with equatin (1), bviusly the latter is an apprximate expressin fr the frmer when its secnd and higher rder time derivatives are neglected. Tw imprtant prperties f the lag and rute methd The lag and rute methd becmes the mdel f linear reservirs in series in which pure attenuatin is cnsidered when either T = 0 in equatins (24a) and (27a) r x = 0 in equatins (24b) and (27b). If x = 1, then a 0 = Kx = K, and b 0 = K(l x) = 0. Substitute a 0 and b 0 int equatin (20), whence Q(t) = e " "I(t). Slving simultaneusly with equatin (21), finally we get: Q(t) = I(t - K) (29) Fr T = Kx = K: Q(t) = I(t - T) (30) and its IUH can be expressed as u(t) = <5(t - T) (31) which indicates the cncept f a single linear channel in which there is pure translatin effect but n attenuatin effect exists. The parameter, x, is used fr mdifying the shape f a fld hydrgraph and is representative f the translatin effect The channel characteristics can be visualized accrding t the range f values f x, as shwn in Table 1, s that a chice between different methds f fld ruting can be made. Table 1 x range Characteristics Cmmn ruting methd x = 0 <x<y 2 x = % y 2 < x < 1 x = 1 Pure attenuatin effect Attenuatin effect is dminant Attenuatin effect and translatin effect are the same Translatin effect is dminant Pure translatin effect Nash mdel Muskingum methd Lag and rute methd Lag and rute methd Lag and rute methd In general, if the inflw t a reach is gentle and the duratin f the rising limb f the inflw appraches the travel time between

8 492 Lu Bkun & Qian uewei the inflw sectin and the utflw sectin, the Muskingum methd is custmarily used; cnversely, if the inflw rises r recedes steeply and the duratin f the rising limb f the inflw is much shrter than the travel time between the inflw sectin and the utflw sectin, it is nt apprpriate t use the Muskingum methd. In this case, hwever, the Muskingum methd f successive ruting thrugh subreaches can be used. Of curse, it is mre apprpriate t use the lag and rute methd in this case since the methd cannt nly avid any negative utflw, but als guarantees certain cmputatinal accuracy. PROBLEM OF THE NEGATIVE RESPONSE Negative respnse f the integratin slutin f the Muskingum methd The integratin slutin f the Muskingum methd is as stated abve. Nw cnsider the fllwing prperties f the IUH f the Muskingum methd. Rts f the IUH f the Muskingum methd Since u(t) in equatin (12) is nt equal t zer at t =0, i.e. u(0) 0, then t = 0 is nt a rt f u(t) = 0. The rts f u(t) = 0 must be fund frm values f t ther than t = 0. In additin, based n the prperties f the impulse functin (<5(t) = at t = 0, and ô(t) = 0 at t? 0), the term 6(t) in equatin (12) must be zer in determining the rts f u(t) = 0. Therefre the rts f u(t) are btained prvided that the fllwing equatin f the (n - l)th rder is slved: u(t) = r. n-l (-DV11 n-i-1 ) i=0 T(n - i) x 1 = 0 (32) The number f rts indicates the number f times that the respnse functin changes sign. Prperties f the IUH f the Muskingum methd at t = 0 As indicated abve, u(0) ^ 0. Hwever, when t appraches zer frm the psitive directin, that is t ~> 0 +, the terms including t in equatin (12) can be cnsidered as zer; the term 6(t) in equatin (12) is equal t zer based n the same prperties f the impulse functin, ô(t), as described in the preceding subparagraph. The fllwing equatin can be derived frm equatin (12): n-l n (-1) (n 1) x n-l u(0+) =, n+1 The impulse f u(t) at t = 0 is (-l) n [x/(l - x)] n 6(t). When n is an dd number, a negative impulse emerges at t =0, which indicates that psitive and negative impulses f u(t) at t = 0 emerge alternatively. The value f u(t) at t = 0 can be expressed by: (33)

9 Sme prblems with the Muskingum methd 493 / " i \ n - 1, n -i \ n - 1 n u(0) = Azl> <n- Dx + (_ 1>n rj^ 6(t) (34) l t As the result f exp[-. _ ]-»- 0 fr t-> <*>, s u(t)-> 0 fr t -*. The abve prperties are shwn in Table 2. Obviusly, the IUH f the Muskingum methd is an scillatry curve alng the t-axis. The curve intersects the t-axis at the beginning f the curve. The number f intersectin pints n the t-axis increases with n and is equal t (n - 1). Negative utflw f the finite difference slutin f the Muskingum methd The frmulae fr this methd are equatins (5) and (6). The frmulae fr the finite difference slutin f successive ruting subreaches are given in the frm (East China Cllege f Hydraulic Engineering, 1977; Yangtze Valley Planning Office, 1979): in P n P mn which c n En i= =l B i A = = C 1 fr cn-: + C m = lc i A : and _ n! (m - 1)! i _ i! (i - 1)! (n - i)! (m - i)! fr m > 0 and (m - i) > 0 (35) where n = the number f subreaches; m = an integer which indicates the number f ruting steps. When x > 0 and At < 2Kx, the frmula fr the Muskingum methd, equatin (5), may yield a negative utflw, since C 0 is negative. Further, when n is an dd number, the frmula fr the Muskingum methd, equatin (35), P n = C < 0. Thus, this methd may als yield a negative utflw. When x > 0 and At > 2K(1 - x), the tail f the utflw hydrgraph defined by equatin (5) may prduce negative utflw r pulsative phenmena. Similarly, the tail f the utflw hydrgraph described by equatin (35) may prduce negative utflw. Therefre, in rder t avid negative utflw, the fllwing restrictins shuld be cnsidered in selecting the ruting perid 2K(1 - x) S At S 2Kx (36) The reasn that a negative respnse is prduced under sme circumstances is that the strage equatin f the Muskingum methd cannt actually reflect the fact that the river flw des nt vary linearly with time r alng the river. As shwn in Fig.l, bth at t = tx and t = t2 the inflw and utflw are unvarying. Althugh at these times the water surface prfiles and hence the strages in the river channel are different, the

10 494 Lu Bkun & Qian uewei " 7 Q :*-? CO II c / ( j r-u ee 3 II " O -rj CC -S -M c~ c r-> «- *' Il II 2 "t " c i z> II. <* O 09 si TO H > ï= C t 0) 3 -C LU.2. c O II 1 > O II t + > t 1 ^ CO > c t + Ci > 1 >- * + > O

11 Sme prblems with the Muskingum methd 495 Fig, 1 Strage varying with time in a river reach. strage equatin at these times remains the same. In summary, an unreliable strage equatin is used in the Muskingum methd; in additin this methd is an apprximate slutin f a generalized linear hydrlgical system mdel. Therefre, bth the integratin slutin and the finite difference slutin f the Muskingum methd may have a negative respnse. In rder t avid a negative respnse withut lwering the ruting accuracy the ruting perid shuld be restricted. T eliminate the negative respnse f the Muskingum methd, ther methds may be adpted. Fr example, the Nash mdel (which can be regarded as the best amng the versins fr the IUH f the Muskingum methd) can eliminate the negative respnse by setting x t zer. The lag and rute methd is anther methd t eliminate the negative respnse by means f the translatin effect. In essence, these methds can be used t eliminate the negative respnse by mdifying their strage equatins. CONCLUSION Bth the Muskingum methd and the lag and rute methd are derived frm the equatin fr a linear hydrlgical system, and it is shwn frm their strage equatins that the Muskingum methd is an apprximate slutin fr the lag and rute methd. The integratin slutin fr the Muskingum methd f successive ruting thrugh subreaches is derived. The range f x in equatin (27b) is frm 0 t 1, and its value can be used as an index in chsing amng the different fld ruting methds. Because the strage equatin f the Muskingum methd cannt accurately describe the change f strage in the river reach, the negative respnse f the Muskingum methd is unavidable frm the mathematical derivatin, but it is physically unrealistic. Nevertheless, the negative respnse can be avided by limiting the finite difference interval, At.

12 496 Lu Bkun & Qian uewei ACKNOWLEDGEMENTS Special thanks are given t Yang Ganghe fr his suggestins in English and a critical review f this paper. REFERENCES Chw, V.T. (1975) Hydrlgie mdeling. Selected Wrks in Water Resurces, IWRA, March. Dge, J.C.I. (1973) Linear thery f hydrlgical systems. USDA Rgric. Res. Service Tech. Bull US Gvernment Printing Office, Washingtn, DC, USA. East China Cllege f Hydraulic Engineering (ECCHE) (1977) Fld Frecasting Methd fr Humid Regins f China. ECCHE, Nanking, China. Lu, B.K., Yang, G.H. et al. (1978) An applicatin f Nash's IUH t fld ruting in channels (in Chinese). Selected Wrks in Techniques and Experiences f Hydrlgie Frecasting. China Water Resurces and Electric Pwer Press. Nash, J.E. (1960) A unit hydrgraph study, with particular reference t British catchments. Prc. Instn Civ. Engrs 17, Qian,.W. & Lu, B.K. (1981) Relatinship and review between the IUH f lag and rute methd and the IUH f Muskingum methd (in Chinese). Yangtze River, N.2. Venetis, C. (1969) The IUH f the Muskingum channel reach. J. Hydrl. 4, Wang, Qinliang (1982) The mdel f lag-instantaneus flw cncentratin. Hydrlgy, Beijing, China, N.l, Yangtze Valley Planning Office (1979) Hydrlgie Frecast Methds in China (in Chinese). China Water Resurces and Electric Pwer Received 18 April 1986; accepted 21 April 1987.