Muskingkum Routing.pdf

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Hydrological Sciences -Journal- des Sciences Hydrologiques,39,5, October 1994 4 3 1

Hydrodynamic derivation of a variableparameter Muskingum method:1. Theory and solution procedureMUTHIAH PERUMALDepartment of Continuing Education, University of Roorkee, Roorkee 247667,IndiaAbstract An approach is presented for directly deriving a variableparameter Muskingum method from the St. Venant equations for routingfloods in channels having any shape of prismatic cross-section and flowfollowing either Manning's or Chezy's friction law. The approach alsoallows the simultaneous computation of the stage hydrograph corresponding to a given inflow or the routed hydrograph. This first paper alsodescribes the solution procedure for routing the discharge hydrograph.A second paper (Peramal, 1994b) presents a verification of themethodology.Dmonstration hydrodynamique d'une mthode de Muskingum paramtres variables: 1. Thorie et procdureRsum Cet article prsente une mthode permettant d'tablirdirectement partir des quations de Saint Venant une formule deMuskingum paramtres variables pour le routage des coulements decrue dans des biefs prsentant des lois de frottement de Manning ou deChzy. Cette approche permet de calculer dans le mme temps le limni-gramme correspondant un dbit entrant donn de l'hydrogrammecalcul. Ce premier papier dcrit galement la procdure de calcul del'hydrogramme. Un second papier (Peramal, 1994b) prsente une validation de la mthodologie.

I N T R O D U C T I O NFlood routing studies based upon simplified methods derived either directly orindirectly from the S t. Venan t equations are perceived as inherently lessaccurate than tho se based upo n the numerical solution of the St. Venantequations. However, Ferrick (1985) pointed out that numerical problems arisewhile solving the full St. Venant equations for studying flood wave movementwhen the magnitudes of the different terms of the momentum equation arewidely varying. By analysing different river wave types, Ferrick (1985)suggested the use of appropriate wave type equations for obtaining accuratesolutions without facing numerical problems, and argued that the use of morecomplete equations may not yield more accurate river wave simulations for allwave types. This argument is now substantiated by the incorporation of theoption for routing dambreak floods in steep reaches using the VariableParameter Muskingum-Cunge (VPMC) method (Ponce & Yevjevich, 1978) inOpen for discussion until 1 April 1995


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432 Muthiah Perumalthe recent version of the DAMBRK model (Fread, 1990) which earlier usedonly the St. Venant equations. It is apt to quote the remarks found in the UKFlood Studies Report (NERC, 1975) in favour of the simplified methods:

"Many of the assumptions [of the simplified methods] may appearcrude or severe to those who have no experience of lood routing.The accurate results which can, however, be obtained by even avery simple flood routing method ind icate that the assumptions arerealistic and that simplicity has much to recommend it. "However, most of the simplified hydraulic flood routing models useconstant parameters and were developed based on the assumption of linearitywhich is in contradiction with the nonlinear behaviour of flood waves. The useof constant parameter models such as the Kalinin-Milyukov method (Apollovet al., 1964), the diffusion analogy method (Hayami, 1951) and the physicallybased Muskingum models (Apollov et al., 1964; Cunge, 1969; Dooge, 1973;Dooge et al., 1982) for routing floods is appropriate only when an assumptionis satisfied, viz. that the flow variation around a reference discharge which isused for estimating the parameters is small.To overcome this limitation, variable parameter simplified flood routingmethods such as the variable parameter diffusion method (Price, 1973), themultilinear models (Keefer & McQuivey, 1974; Becker, 1976; Kundzewicz,1984; Becker & Kundzewicz, 1987; Perumal, 1992, 1994a), and the variableparameter M uskingum models (Ponce & Yevjevich, 1978; Ferrick, 1984) were

proposed. Since this paper focuses on the development of a variable parameterMuskingum method, attention is drawn herein only to those existing variableparameter Muskingum methods in the literature.While emphasizing future basic research on simplified flood routingmethods, the UK Flood Studies Report (NERC, 1975) recommended thedevelopment of a variable parameter Muskingum method and pointed out that,if developed, it may well be preferable to the variable param eter diffusionmethod proposed by Price (1973). In line with this suggestion, Ponce &Yevjevich (1978) proposed the VPMC method in which the parameters of theMuskingum method vary at every routing time level. It was shown by Younkin& Merkel (1988) that the VPMC method produces acceptable results for over80% of the US Soil Conservation Service field conditions. Due to its wideapplicability to real-world routing problems, the US Army Corps of Engineershas recently added a flood routing option using the VPMC method in theHEC-1 model (HEC-1 Flood Hydrograph Package, 1990). However, theVPMC method is unfortunately saddled by a small but perceptible loss of mass(Ponce, 1983).

To overcome this deficiency, Perumal (1992) has recently proposed aMultilinear Muskingum (MM) method in which the same physically basedparameters as adopted in the VPMC method are used, but the routing is carriedout using the multilinear modelling approach. In a recent comparative study ofboth the methods, Perumal (1993) showed that the MM method scores betterthan the VPMC method in reproducing the St. Venant solutions closely.


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Muskingum method derivation: theory and solution 433Although the MM method performs better than the VPMC method, itstill has limitation such as the characterization of a unique relationship betweenthe Muskingum parameters and the inflow discharge, notwithstanding whether

the discharge to be routed is on the rising or falling limb of the hydrograph.Further, in both methods (VPMC and MM), the techniques of varying theparameters, although systematic, are not physically based. Therefore, theappropriate approach for varying the parameters of the Muskingum modelwould be to account for water surface slope in their relationships in a mannerconsistent with the variation built into the solution of the St. Venant equa tions.In this paper, a variable parameter Muskingum method is developeddirectly from the St. Venant equations for routing flood waves in semi-infiniterigid bed prismatic channels having any shape of cross-section, and for flow

following either Ma nnin g's o r Ch ezy 's friction law. It is found that this methodis able to give a physical justification for the Muskingum method better thanthe approaches so far available advocated by Apollov et al., (1964), Cunge(1969), Dooge (1973), Strupczewski & Kundzewicz (1980) and Dooge et al.,(1982). An added advantage of this method is that it allows the simultaneouscomputation of the stage hydrograph corresponding to a given inflow or to arouted discharge hydrograph.

C O N C E P T O F T H E M E T H O DDuring steady flow in a river reach having any shape of prismatic cross-section, the stage, and hence the cross-sectional area of flow at any point of thereach is uniquely related to the discharge at the same location defining thesteady flow rating curve. However, this situation is altered during unsteadyflow resulting in the same unique relationship between the stage at a givensection and the corresponding steady discharge occurring not at the samesection but somewhere downstream from that section. This concept has beenused in the development of the Kalinin-Milyukov m ethod (Apollov et al., 1964)to determine the "characteristic reach length", and subsequently to its extensionfor the physical interpretation of the Muskingum method. This concept is alsoused in the development of the proposed method herein.

D E V E L O P M E N T O F T H E M E T H O DFlood routing in channels is often carried out on the assumption that the floodwave movement is one-dimensional and governed by the St. Venant equations.For gradually varied unsteady flow in rigid bed channels without consideringlateral flow, these equations are written as (Henderson, 1966):

dx dt (D


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434 Muthiah Perumaland S f = S-Q-l*l_l?v ( 2 )f " dx g dx g dtin which t = time; x = distance along the channel; y, v, A and Q are depth,velocity, cross-sectional area and discharge, respectively; g = acceleration dueto gravity; S f = friction slope; and S0 bed slope.The magnitudes of the various other terms in equation (2) are usuallysmall in comparison with S0 (Henderson, 1966; NERC, 1975) and, therefore,quite often some of them can be eliminated or approximated by someprocedures when studying many flood routing problems.AssumptionsThe proposed method is developed based on the following assumptions:(a) a prism atic channel having any shape of cross-sec tion is assum ed;(b) there is no lateral inflow or outflow from the reach ;(c) the slope of the water surface dy/dx, the slope due to local accelerationl/g(dy/dt), and the slope due to convective acceleration vlg(dvldx) allremain constant at any instant of time in a given routing reach;(d) the mag nitudes of mu ltiples of the deriva tives of flow and sectionvariables with respect to both time and distance are negligible; and(e) at any instant of time during unsteady flow , the steady flow relation shipis applicable between the stage at the middle of the reach and thedischarge passing somewhere downstream of it. The same assumption isemployed in the Kalinin-Milyukov method (Apollov et al., 1964; Miller& Cunge, 1975).Friction slope approximationFigu re 1 shows a channel reach of length Ax. Acco rding to assumption (e), thestage at the middle of the reach corresponds to the normal depth of thatdischarge which is passing at the same instant of time at an unspecified distance/ downstream from the middle of the reach. Let this discharge be denoted as


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Muskingum method derivation: theory and solution 435

Fig.l Definition sketch of the Muskingum reach.perimeter; m is an exponent which depends on the friction law used (forexample, m - % for Manning's friction law, and m = xh for Chezy's frictionlaw). Equation (3) is re-written using equation (4) as:

Q = AC fR'"Sf (5)Differentiating equation (5) with respect to x and invoking assumption (c) thatS f is constant over x gives :9. = i^mP^Xv^dx ) dy dy \ dxThe celerity of the flood wave can be arrived at from equation (6) as:

dQdA , f PdR/dy 11 + m ' J '1- \ v\ dA/dy J J

(6)

(7)Unlike the kinematic wave which has unique celerity for a given

discharge, the flood wave governed by constant water surface slope does notresult in unique celerity for the same discharge occurring in the rising andfalling limbs of the hydrograph.Differentiating equation (6) with respect to x gives:

dx2+ V

dA DdR +mPdy dyd2A

dv 3y _ + vdx dxdA vdR+ mrdy dy dx2 (8)dP dR v d2R

+m + m Pdxdy dx dy dxdy3ydxUsing assumptions (c) and (d), equation (8) reduces to:

02Qdx2 (9)


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436 Muthiah PerumalEquation (9) implies that the discharge is also varying linearly over the reachconsidered.

Approximate expression for friction slopeUsing equations (1), (2), (3) and (6), and assumption (c), the friction slope s fcan be expressed as:

(10)s f = s ; ! _ i aySo dxr -

1 - mF ' PdR/dydA/dy2

'

in which F is the Froude number defined as:F = V2dAldy A (11)

Note the term mF[(PdR/dy)/(dA/dy)] denotes the Vedernikov number (Chow,1959) and it defines the criterion for the amplification of flood wave movementdown a channel (Jolly & Yevjevich, 1971; Ponce, 1991).Location of weighted discharge sectionUsing equations (5) and (10), the discharge at the middle of the reach isexpressed as:

Q-M ~ AM ^fRM So2 1 - 1 ^ 1S. dx l-m2Fl PdR/dydA/dy

(12)

where the subscript M denotes the mid-section of the reach.The normal discharge Q 3 corresponding to yM occurs at section (3), as

shown in Fig. 1, located at a distance / dow nstream of the middle of the reach,and it is expressed as:

Equation (12) is modified using equation (13) as:

Q = Q 1 1 - 1 - ^ 1U U ^ 3 s dx \Mo

For the sake of brevity, let

l-m 2Fi PdR/dydA/dy

(13)

(14)

1 tyi~ s ~ x ' M 1 - m 2Fl PdR/dydA/dy = r (15)


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Muskingum method derivation: theory and solution 437o a s e u on me typical values of Sa and dy/dx in natural r ivers (Henderson,1966) , it may be considered that \r\ < 1. Under such a condi t ion, expanding

equat ion (14) in a Binomial ser ies and then neglect ing the higher o rder t e rmsof r leads to:

QM = fi,- 25 1 - m2Fl,PdRldydAldy M dx

Since dy/dx is constant at any instant of t ime over the rout ing reach:dy i dy I3x 3x

(16)

(17)w h e r e dy/dx\3 is the water surface slope at sect ion (3).

Equat ion (16) may be re-wri t ten us ing equat ions (6) and (17) as:

QM = 0303

2S?0 "

1 -

^ 17

2 r-2

1 +m

' PdRldy 'dAldyPdRldy'dAldy

2

M

3V3

3 Q ,3x '3 (18)

Since the discharge also varies linearly, the term adjunct to dQ/dx\3represents the distance / between the mid-section and that downstream sectionwhere the normal discharge corresponding to the depth at the mid-sectionpasses at the same instant of time, i.e.:

03

2 5 ^O

1 -

A,y3

m2F2M

1 +m

' PdRldy 'dAldyPdRldy 'dA/dy

2

M

3V 3

(19)

Der i vat i on of s torage-weighted di scharge re lat ionshipUsing equat ions (1), (3) and (5) and assumpt ion (c), the fol lowing express ionis arr ived at:

dt 1 +m' PdRldydAldy dQdx

Apply ing equat ion (20) at sect ion (3) and rear rang ing the t e rms y ie lds :

1 +m PdRldydAldy dQ,3 dx '3 3 0 ,dt '

(20)

(21)


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438 Muthiah PerumalDue to the linear variation of discharge over the routing reach, dQ/dx\3 maybe approximated as:

dQ, _ 3G,dx dxO-I

Ax (22)where, / and O denote the inflow and outflow at sections (1) and (2)respectively, and Ax is the reach length. Due to the linear variation ofdischarge as depicted in Fig. 1, Q 3 may be expressed as:

G, = 0 + (I-O)-_L"~2 Ax

Substitution of equations (22) and (23) in equation (21) gives:

(23)

I-O =1 +m

Ax' PdR/dydA/dy 3

ddtV 3

0 + 1 _2~

Let the weighting parameter be:0 = i - _2 t

I\x

IAx (I-O) (24)

(25)Equ ation (24) is the same as the differential equation governing the M uskingu mmethod with travel time K expressed as:

K = Ax1 +m PdR/dYdA/dy

(26)

and the weighting parameter 6 after substituting for / from equation (19) isexpressed as:

2fi,

2S1 -

13

2r>2

1 +m

PdR/dYdA/dyr PdR/dy 'dA/dy

(27)v3Ax

The parameter relationships given above enable one to reduce equation(24) to the form of the conventional Muskingum differential equation as :

I-O = [K{61+(1-6)0)dtwith the storage in the reach expressed as:

S = K[6I+(\-6)0)

(28)(29)


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Muskingum method derivation: theory and solution 439When a constant discharge is used as the reference discharge, thegeneralized expressions for variable K ana 6 reduce to:K Ax

1 +m f PdR/dYdA/dy(30)

and

12

z 0

25 bA0 dy

1

f 0

-m2F0

1 +m

PdR/dYdA/dyPdR/dy'dA/dy

~i 2

0

0v Ax0

(31)

where the suffix 0 refers to the reference level discharge.The expressions for the parameters K and 6 can be deduced readily forsome special friction laws and for regular prismatic channel cross-sec tions fromthe general expressions given by equations (26) and (27) respectively. Theexpressions for K and d valid for Manning's friction law and applicable for auniform rectangular cross-section channel reach are arrived at as:K = Ax5 4 Y33 3 (S + 2y 3) vi

(32)

and

, 9 M 1 - 2 .yk

( * +2y J (33)25.fi ?33 ( f i + 2y,) v3Ax

When the variables in these expressions are fixed about a referencedischarge, and a wide rectangular cross-section is assumed, these expressionsreduce to those given by Cunge (1969) and Dooge et al. (1982).

S T A G E H Y D R O G R A P H C O M P U T A T I O NThe flow depth yd corresponding to the outflow O is estimated using equation(6) as:
http://localhost/var/www/apps/conversion/tmp/scratch_1/25.fihttp://localhost/var/www/apps/conversion/tmp/scratch_1/25.fi


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440 Muthiah Perumalyd yM

SA,(0

1 +m- f in)' PdR/dy

dA/dy MVM

(34)in which yM is estimated iteratively from the normal discharge relationshipgiven by equation (13). Using the computed flow depths yd and yM in the firstsub-reach, the upstream flow depth corresponding to the inflow discharge canbe estimated using the assumption of a linear variation of water surface.SOLUTION PROCEDUREThe algorithm adopted for routing a given inflow hydrograph using thedescribed method is shown in the flow chart in Fig. 2.

CONCLUSIONSAn approach for directly deriving a variable parameter Muskingum methodfrom the St. Venant equations for routing floods in channels having any shapeof prismatic cross-section and flow following either Manning's or Chezy'sfriction law has been presented. The advantage of this simplified hydraulicrouting method is that it allows the simultaneous computation of discharge aswell as the corresponding stage hydrograph. The solution procedure for routinga discharge hydrograph is also presented.

REFERENCESApollov, B. A., Kalinin, O.P. & Komarov, V. D. (1964) Hydrological forecasting (Translated fromRussian ). Israel Program for Scientific Translations, Jerusalem.Becker, A. (1976) Simulation of nonlinear flow systems by combining linear m odels. In : Mathematical

Models in Geophysics 135-142. (Proc. IUGG Assembly, Moscow, August 1971), IAHS Publ.no. 116.Becker, A. & Kundzewicz, Z. W. (1987) Nonlinear flood routing with multilinear models. Wat.Resour. Res. 23(6), 1043-1048.Chow, VenTe (1959) Open - Channel Hydraulics. McGraw-Hill, New York, USA.Cunge, J. A. (1969) On the subject of a flood propagation computation method (M uskingum m ethod)./ . Hydraul. Res.JAHR 7(2), 205-230.Dooge, J. C. I. (1973) Linear theory of hydrologie systems. ARS Tech. Bull. no. 1468, US AgricultureRes. Serv., Washington, DC, USA.Dooge, J. C. I., Strupczewski, W. G. & Napiorkowski, J. J. (1982) Hydrodynamic derivation ofstorage parameters of the Muskingum mod el, /. Hydrol. 54, 371-387.Ferrick, M . G. (1984) Modeling rapidly varied flow in tailwaters. Wat. Resour. Res. 20(2), 271-289.Ferrick , M. G. (1985) Analysis of river wave types. Wat. Resour. Res. 21(2), 209-220.Fread, D. L. (1990) DAMBRK: The NWS Dam-Break Flood Forecasting Model. National WeatherService, Office of Hydrology, Silver Spring, Maryland, USA.Hayami, S. (1951) On the Propagation of Flood Waves. Bull. No.l, Disaster Prevention ResearchInstitute, Kyoto University, Japan.HEC-1 (1990) Flood Hydrograph Package: User's manual. US Army C orps of Engineers, HydrologieEngineering Center, Davis, California, USA.Henderson, F. M. (1966) Open Channel Flow. MacMillan & C o., N ew York, USA.Jolly, J. P. & Yevjevich, V. (1971) Am plification criter ion of gradually v aried , single peaked waves,Hydrol. Pap. no. 51, Colorado State University, Fort Collins, USA.


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Muskingum method derivation: theory and solution 441

A

NO

NO

Routing Step J = 1Estimate initial K and 60 using Equations (30) & (31)

TJ = J + lIteration step 1= 1

Estimate C^ C2& C3-Kd+At/2C, = K(l-d)+At/2

C = K0+At/22 K(l-6)+t/2C, = K(l-6)-Atl2K(l-6)+At/2

TEstimate O, = C,/, + C 2 /^, + C 3 0 , ^Estimate Q, = 61 j + (1 - 0)0 ,TEstimate y using Newton-Raphson Method from

Estimate QM =Estimate F M = (Q &

(I, + Oj)l21(dA/dy)\M)lgAi,1

Estimate v,j3 = yM + (G, - 2 ) / ( ^ % ) I {i + [(paR/a>)/o-4/ay)]M}vIEstimate Ay corresponding to y,IEstimate v3 = 3M3

Estimate revised K and 6 for the present routing step using equations (26) & (27)1 = 1+1

ISI > 2YE S

Estimate y, = yM + ( 0 , - eM)/OA%)| {1 + m [ ( /> 3 % ) /0 ^% ) ] M } v

IS J > N steps?Y ES

STOPFig. 2 Solution procedure.


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442 Muthiah PerumalKeefer, T. N. & McQuivey, R. S. (1974) Multiple linearization flow routing mod el. J. Hydraul. Div.ASCE 100(7), 1031-1046.Kundzewicz, Z. W. (1984) Multilinear flood routing. Acta Geophys. Pol. 32(4), 419-445.Miller, W. A. & Cunge, J. A. (1975) Simplified equations of unsteady flow . In: Symp. on UnsteadyFlow in open Channels, Water Resources Publications, Fort Collins, Colorado, USA.Natural Environment Research Council (1975) Flood studies report, volume III - Flood routing studies.London, UK.Perum al, M. (1992) Multilinear Muskingum flood routing method. /. Hydrol., 133, 259-272.Perumal, M. (1993) Comparison of two variable parameter Muskingum methods In: ExtremeHydro logical Events: Precipitation, Floods and Droughts, 129-138. (Proc. Yokohama Sym p.,July 1993) IAHS Publ. no. 213.Perum al, M . (1994a) Multilinear discrete cascade model for channel routing. /. Hydrol., (in press)Perumal, M. (1994b) Hydrodynamic derivation of a variable parameter Muskingum method: 2.Verification. Hydrol. Sci. J. 39(5) (this issue)Ponce, V. M . (1983) Accuracy of physically based coefficient m ethods of floo d routing. Tech. ReportSDSU Civil Engineering Series No. 8315 0, San Diego State University, San D iego, C alifornia,USA.Ponce, V. M. (1991) New perspective on the Vedernikov number. Wat. Resour. Res. 27(7), 1777-1779.Ponce, V. M. & Yevjevich, V. (1978) Muskingum-Cunge method with variable parameters. /.Hydraul. Div. ASCE, 104 (12), 1663-1667.Price, R. K. (1973) Variable parameter diffusion method for flood routing. Report no. INT 115,Hydraulics Research Station, Wallingford, UK.Strupczewski, W. G. & Kundzewicz, Z . W. (1980) Muskingum method revisited. / . Hydrol. 48 , 327-342.Younkin, L. M. & Merkel, W. H. (1988) Evaluation of Diffusion Models for Flood Routing. Proc.ASCE H ydraul. Div. Annual Conference, Colorado Spring, C olorado, USA.Received 8 February 1993; accepted 7 April 199 4

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