Hydrodynamic derivation of a variable parameter Muskingum method: 1. Theory and solution procedure

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1 Hydrological Sciences -Journal- des Sciences Hydrologiques,39,5, October Hydrodynamic derivation of a variable parameter Muskingum method: 1. Theory and solution procedure INTRODUCTION MUTHIAH PERUMAL Department of Continuing Education, University of Roorkee, Roorkee , India Abstract An approach is presented for directly deriving a variable parameter Muskingum method from the St. Venant equations for routing floods in channels having any shape of prismatic cross-section and flow following either Manning's or Chezy's friction law. The approach also allows the simultaneous computation of the stage hydrograph corresponding to a given inflow or the routed hydrograph. This first paper also describes the solution procedure for routing the discharge hydrograph. A second paper (Peramal, 1994b) presents a verification of the methodology. Démonstration hydrodynamique d'une méthode de Muskingum à paramètres variables: 1. Théorie et procédure Résumé Cet article présente une méthode permettant d'établir directement à partir des équations de Saint Venant une formule de Muskingum à paramètres variables pour le routage des écoulements de crue dans des biefs présentant des lois de frottement de Manning ou de Chézy. Cette approche permet de calculer dans le même temps le limnigramme correspondant à un débit entrant donné de l'hydrogramme calculé. Ce premier papier décrit également la procédure de calcul de l'hydrogramme. Un second papier (Peramal, 1994b) présente une validation de la méthodologie. Flood routing studies based upon simplified methods derived either directly or indirectly from the St. Venant equations are perceived as inherently less accurate than those based upon the numerical solution of the St. Venant equations. However, Ferrick (1985) pointed out that numerical problems arise while solving the full St. Venant equations for studying flood wave movement when the magnitudes of the different terms of the momentum equation are widely varying. By analysing different river wave types, Ferrick (1985) suggested the use of appropriate wave type equations for obtaining accurate solutions without facing numerical problems, and argued that the use of more complete equations may not yield more accurate river wave simulations for all wave types. This argument is now substantiated by the incorporation of the option for routing dambreak floods in steep reaches using the Variable Parameter Muskingum-Cunge (VPMC) method (Ponce & Yevjevich, 1978) in Open for discussion until 1 April 1995

2 432 Muthiah Perumal the recent version of the DAMBRK model (Fread, 1990) which earlier used only the St. Venant equations. It is apt to quote the remarks found in the UK Flood Studies Report (NERC, 1975) in favour of the simplified methods: "Many of the assumptions [of the simplified methods] may appear crude or severe to those who have no experience of flood routing. The accurate results which can, however, be obtained by even a very simple flood routing method indicate that the assumptions are realistic and that simplicity has much to recommend it. " However, most of the simplified hydraulic flood routing models use constant parameters and were developed based on the assumption of linearity which is in contradiction with the nonlinear behaviour of flood waves. The use of constant parameter models such as the Kalinin-Milyukov method (Apollov et al., 1964), the diffusion analogy method (Hayami, 1951) and the physically based Muskingum models (Apollov et al., 1964; Cunge, 1969; Dooge, 1973; Dooge et al., 1982) for routing floods is appropriate only when an assumption is satisfied, viz. that the flow variation around a reference discharge which is used for estimating the parameters is small. To overcome this limitation, variable parameter simplified flood routing methods such as the variable parameter diffusion method (Price, 1973), the multilinear models (Keefer & McQuivey, 1974; Becker, 1976; Kundzewicz, 1984; Becker & Kundzewicz, 1987; Perumal, 1992, 1994a), and the variable parameter Muskingum models (Ponce & Yevjevich, 1978; Ferrick, 1984) were proposed. Since this paper focuses on the development of a variable parameter Muskingum method, attention is drawn herein only to those existing variable parameter Muskingum methods in the literature. While emphasizing future basic research on simplified flood routing methods, the UK Flood Studies Report (NERC, 1975) recommended the development of a variable parameter Muskingum method and pointed out that, if developed, it may well be preferable to the variable parameter diffusion method proposed by Price (1973). In line with this suggestion, Ponce & Yevjevich (1978) proposed the VPMC method in which the parameters of the Muskingum method vary at every routing time level. It was shown by Younkin & Merkel (1988) that the VPMC method produces acceptable results for over 80% of the US Soil Conservation Service field conditions. Due to its wide applicability to real-world routing problems, the US Army Corps of Engineers has recently added a flood routing option using the VPMC method in the HEC-1 model (HEC-1 Flood Hydrograph Package, 1990). However, the VPMC method is unfortunately saddled by a small but perceptible loss of mass (Ponce, 1983). To overcome this deficiency, Perumal (1992) has recently proposed a Multilinear Muskingum (MM) method in which the same physically based parameters as adopted in the VPMC method are used, but the routing is carried out using the multilinear modelling approach. In a recent comparative study of both the methods, Perumal (1993) showed that the MM method scores better than the VPMC method in reproducing the St. Venant solutions closely.

3 Muskingum method derivation: theory and solution 433 Although the MM method performs better than the VPMC method, it still has limitation such as the characterization of a unique relationship between the 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 the parameters, although systematic, are not physically based. Therefore, the appropriate approach for varying the parameters of the Muskingum model would be to account for water surface slope in their relationships in a manner consistent with the variation built into the solution of the St. Venant equations. In this paper, a variable parameter Muskingum method is developed directly from the St. Venant equations for routing flood waves in semi-infinite rigid bed prismatic channels having any shape of cross-section, and for flow following either Manning's or Chezy's friction law. It is found that this method is able to give a physical justification for the Muskingum method better than the 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 simultaneous computation of the stage hydrograph corresponding to a given inflow or to a routed discharge hydrograph. CONCEPT OF THE METHOD During steady flow in a river reach having any shape of prismatic crosssection, the stage, and hence the cross-sectional area of flow at any point of the reach is uniquely related to the discharge at the same location defining the steady flow rating curve. However, this situation is altered during unsteady flow resulting in the same unique relationship between the stage at a given section and the corresponding steady discharge occurring not at the same section but somewhere downstream from that section. This concept has been used in the development of the Kalinin-Milyukov method (Apollov et al., 1964) to determine the "characteristic reach length", and subsequently to its extension for the physical interpretation of the Muskingum method. This concept is also used in the development of the proposed method herein. DEVELOPMENT OF THE METHOD Flood routing in channels is often carried out on the assumption that the flood wave movement is one-dimensional and governed by the St. Venant equations. For gradually varied unsteady flow in rigid bed channels without considering lateral flow, these equations are written as (Henderson, 1966): dx dt (D

4 434 Muthiah Perumal and S f = S-Q-l*l_l?v (2) f " dx g dx g dt in which t = time; x = distance along the channel; y, v, A and Q are depth, velocity, cross-sectional area and discharge, respectively; g = acceleration due to gravity; S f = friction slope; and S 0 bed slope. The magnitudes of the various other terms in equation (2) are usually small in comparison with S 0 (Henderson, 1966; NERC, 1975) and, therefore, quite often some of them can be eliminated or approximated by some procedures when studying many flood routing problems. Assumptions The proposed method is developed based on the following assumptions: (a) a prismatic channel having any shape of cross-section is assumed; (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 acceleration l/g(dy/dt), and the slope due to convective acceleration vlg(dvldx) all remain constant at any instant of time in a given routing reach; (d) the magnitudes of multiples of the derivatives of flow and section variables with respect to both time and distance are negligible; and (e) at any instant of time during unsteady flow, the steady flow relationship is applicable between the stage at the middle of the reach and the discharge passing somewhere downstream of it. The same assumption is employed in the Kalinin-Milyukov method (Apollov et al., 1964; Miller & Cunge, 1975). Friction slope approximation Figure 1 shows a channel reach of length Ax. According to assumption (e), the stage at the middle of the reach corresponds to the normal depth of that discharge 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 <2 3 and the section where it occurs be denoted as section (3). The inflow and outflow sections are represented as sections (1) and (2) respectively. The discharge at any section of the reach may be expressed as: as: Q = Av ( 3 ) The velocity v can be expressed by Manning's or Chezy's friction law v = C f R m Sf (4) where C f is the friction coefficient (C f = C for Chezy's friction law, and C f = \ln for Manning's friction law); R is the hydraulic radius (A/P); P is the wetted

5 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 (for example, m - % for Manning's friction law, and m = x h for Chezy's friction law). Equation (3) is re-written using equation (4) as: Q = AC f R'"Sf (5) Differentiating equation (5) with respect to x and invoking assumption (c) that S f is constant over x gives : Ê9. = i^mp^xv^ dx ) dy dy \ dx The celerity of the flood wave can be arrived at from equation (6) as: dq, f PdR/dy 1 1+m' J 1- J '\ v (7) da \ J J Unlike the kinematic wave which has unique celerity for a given discharge, the flood wave governed by constant water surface slope does not result in unique celerity for the same discharge occurring in the rising and falling limbs of the hydrograph. Differentiating equation (6) with respect to x gives: da DdR dv 3y da dx 2 +mp _ + v + mr vdr dx dx dy dy dx 2 dy dy (8) d 2 A dp dr v d 2 R 3y + V +m +mp dx dxdy dx dy dxdy Using assumptions (c) and (d), equation (8) reduces to: (6) d 2 Q dx 2 0 (9)

6 436 Muthiah Perumal Equation (9) implies that the discharge is also varying linearly over the reach considered. Approximate expression for friction slope Using equations (1), (2), (3) and (6), and assumption (c), the friction slope s f can be expressed as: s f = s;!_ i ay S o dx r- 1- mf ' PdR/dy in which F is the Froude number defined as: 2 ' (10) F = V 2 daldy A (11) Note the term mf[(pdr/dy)/()] denotes the Vedernikov number (Chow, 1959) and it defines the criterion for the amplification of flood wave movement down a channel (Jolly & Yevjevich, 1971; Ponce, 1991). Location of weighted discharge section Using equations (5) and (10), the discharge at the middle of the reach is expressed as: Q-M ~ A M^f R M S o 1-1^1 2 l-m 2 PdR/dy (12) Fl S. dx where the subscript M denotes the mid-section of the reach. The normal discharge Q 3 corresponding to y M occurs at section (3), as shown in Fig. 1, located at a distance / downstream of the middle of the reach, and it is expressed as: Equation (12) is modified using equation (13) as: (13) Q = Q 11-1-^1 U U ^3 s dx\ M o For the sake of brevity, let l-m 2 Fi PdR/dy (14) 1 tyi ~s~ âx' M 1 - m2 Fl PdR/dy = r (15)

7 Muskingum method derivation: theory and solution 437 oaseu on me typical values of S a and dy/dx in natural rivers (Henderson, 1966), it may be considered that \r\ < 1. Under such a condition, expanding equation (14) in a Binomial series and then neglecting the higher order terms of r leads to: Q M = fi, m 2 Fl, PdRldy daldy Since dy/dx is constant at any instant of time over the routing reach: dy i 3x dy I 3x where dy/dx\ 3 is the water surface slope at section (3). Equation (16) may be re-written using equations (6) and (17) as: M dx (16) (17) QM = S? 0 " 1- ^1 7 2 r-2 ' PdRldy ' daldy PdRldy' daldy 2 M 3 V 3 3Q, 3x ' 3 (18) Since the discharge also varies linearly, the term adjunct to dq/dx\ 3 represents the distance / between the mid-section and that downstream section where the normal discharge corresponding to the depth at the mid-section passes at the same instant of time, i.e.: m 2 F 2 M ' PdRldy ' daldy 2 M (19) A, 25^ O «y Ï3 PdRldy ' 3 V 3 Derivation of storage-weighted discharge relationship Using equations (1), (3) and (5) and assumption (c), the following expression is arrived at: dt ' PdRldy daldy dq dx Applying equation (20) at section (3) and rearranging the terms yields: (20) PdRldy daldy dq, 30, 3 dx ' 3 dt ' (21)

8 438 Muthiah Perumal Due to the linear variation of discharge over the routing reach, dq/dx\ 3 may be approximated as: dq, _ 3G, dx dx O-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 of discharge as depicted in Fig. 1, Q 3 may be expressed as: G, = 0 +!-_L" (I-O) ~2 Ax Substitution of equations (22) and (23) in equation (21) gives: I-O = Ax d _ I (I-O) dt 2~ Ax ' PdR/dy V 3 3 Let the weighting parameter be: 0 = i-_ 2 t I \x (23) (24) (25) Equation (24) is the same as the differential equation governing the Muskingum method with travel time K expressed as: K = Ax PdR/dY (26) and the weighting parameter 6 after substituting for / from equation (19) is expressed as: 2 2S fi, r>2 PdR/dY r PdR/dy ' v 3 Ax (27) 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) dt with the storage in the reach expressed as: S = K[6I+(\-6)0) (28) (29)

9 Muskingum method derivation: theory and solution 439 When a constant discharge is used as the reference discharge, the generalized expressions for variable K ana 6 reduce to: K Ax f PdR/dY (30) and 1 2 z 0 25 b A 0 dy 1 -m 2 F 0 f 0 PdR/dY PdR/dy' ~i v Ax 0 (31) where the suffix 0 refers to the reference level discharge. The expressions for the parameters K and 6 can be deduced readily for some special friction laws and for regular prismatic channel cross-sections from the general expressions given by equations (26) and (27) respectively. The expressions for K and d valid for Manning's friction law and applicable for a uniform rectangular cross-section channel reach are arrived at as: K = Ax 5 4 Y (S + 2y 3 ) v i (32) and Ô, 9 M 1-2. y k (* + 2yJ (33) 25.fi?3 3(fi + 2y,) v 3 Ax When the variables in these expressions are fixed about a reference discharge, and a wide rectangular cross-section is assumed, these expressions reduce to those given by Cunge (1969) and Dooge et al. (1982). STAGE HYDROGRAPH COMPUTATION The flow depth y d corresponding to the outflow O is estimated using equation (6) as:

10 440 Muthiah Perumal yd y M SA, (0 -fin) ' PdR/dy M V M (34) in which y M is estimated iteratively from the normal discharge relationship given by equation (13). Using the computed flow depths y d and y M in the first sub-reach, the upstream flow depth corresponding to the inflow discharge can be estimated using the assumption of a linear variation of water surface. SOLUTION PROCEDURE The algorithm adopted for routing a given inflow hydrograph using the described method is shown in the flow chart in Fig. 2. CONCLUSIONS An approach for directly deriving a variable parameter Muskingum method from the St. Venant equations for routing floods in channels having any shape of prismatic cross-section and flow following either Manning's or Chezy's friction law has been presented. The advantage of this simplified hydraulic routing method is that it allows the simultaneous computation of discharge as well as the corresponding stage hydrograph. The solution procedure for routing a discharge hydrograph is also presented. REFERENCES Apollov, B. A., Kalinin, O.P. & Komarov, V. D. (1964) Hydrological forecasting (Translated from Russian). Israel Program for Scientific Translations, Jerusalem. Becker, A. (1976) Simulation of nonlinear flow systems by combining linear models. In: Mathematical Models in Geophysics (Proc. IUGG Assembly, Moscow, August 1971), IAHS Publ. no Becker, A. & Kundzewicz, Z. W. (1987) Nonlinear flood routing with multilinear models. Wat. Resour. Res. 23(6), Chow, VenTe (1959) Open - Channel Hydraulics. McGraw-Hill, New York, USA. Cunge, J. A. (1969) On the subject of a flood propagation computation method (Muskingum method). /. Hydraul. Res.JAHR 7(2), Dooge, J. C. I. (1973) Linear theory of hydrologie systems. ARS Tech. Bull. no. 1468, US Agriculture Res. Serv., Washington, DC, USA. Dooge, J. C. I., Strupczewski, W. G. & Napiorkowski, J. J. (1982) Hydrodynamic derivation of storage parameters of the Muskingum model, /. Hydrol. 54, Ferrick, M. G. (1984) Modeling rapidly varied flow in tailwaters. Wat. Resour. Res. 20(2), Ferrick, M. G. (1985) Analysis of river wave types. Wat. Resour. Res. 21(2), Fread, D. L. (1990) DAMBRK: The NWS Dam-Break Flood Forecasting Model. National Weather Service, Office of Hydrology, Silver Spring, Maryland, USA. Hayami, S. (1951) On the Propagation of Flood Waves. Bull. No.l, Disaster Prevention Research Institute, Kyoto University, Japan. HEC-1 (1990) Flood Hydrograph Package: User's manual. US Army Corps of Engineers, Hydrologie Engineering Center, Davis, California, USA. Henderson, F. M. (1966) Open Channel Flow. MacMillan & Co., New York, USA. Jolly, J. P. & Yevjevich, V. (1971) Amplification criterion of gradually varied, single peaked waves, Hydrol. Pap. no. 51, Colorado State University, Fort Collins, USA.

11 Muskingum method derivation: theory and solution 441 Routing Step J = 1 Estimate initial K and 6 0 using Equations (30) & (31) T J = J + l Iteration step 1=1 Estimate C^ C 2 & C 3 -Kd+At/2 C, = K(l-d)+At/2 C = 2 K0+At/2 K(l-6)+àt/2 C, = K(l-6)-Atl2 K(l-6)+At/2 T Estimate O, = C,/, + C 2 /^, + C 3 0,^ Estimate Q, = 61 j + (1-0)0, T Estimate y using Newton-Raphson Method from A Estimate Q M = (I, + Oj)l2 1 Estimate F M = (Q& ()\ M )lgai, 1 Estimate v, j3 = y M + (G, - 2«) / (^%) I {i + «[(par/a>)/o-4/ay)] M }v I Estimate A y corresponding to y, I Estimate v 3 = Ô 3 M 3 Estimate revised K and 6 for the present routing step using equations (26) & (27) 1 = 1+1 NO ISI > 2 YES Estimate y, = y M + (0, - e M )/OA%) {1 + m [(/>3«%)/0^%)] M }v NO IS J > N steps? Fig. 2 Solution procedure. STOP YES

12 442 Muthiah Perumal Keefer, T. N. & McQuivey, R. S. (1974) Multiple linearization flow routing model. J. Hydraul. Div. ASCE 100(7), Kundzewicz, Z. W. (1984) Multilinear flood routing. Acta Geophys. Pol. 32(4), Miller, W. A. & Cunge, J. A. (1975) Simplified equations of unsteady flow. In: Symp. on Unsteady Flow 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. Perumal, M. (1992) Multilinear Muskingum flood routing method. /. Hydrol., 133, Perumal, M. (1993) Comparison of two variable parameter Muskingum methods In: Extreme Hydro logical Events: Precipitation, Floods and Droughts, (Proc. Yokohama Symp., July 1993) IAHS Publ. no Perumal, 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 methods of flood routing. Tech. Report SDSU Civil Engineering Series No , San Diego State University, San Diego, California, USA. Ponce, V. M. (1991) New perspective on the Vedernikov number. Wat. Resour. Res. 27(7), Ponce, V. M. & Yevjevich, V. (1978) Muskingum-Cunge method with variable parameters. /. Hydraul. Div. ASCE, 104 (12), 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, Younkin, L. M. & Merkel, W. H. (1988) Evaluation of Diffusion Models for Flood Routing. Proc. ASCE Hydraul. Div. Annual Conference, Colorado Spring, Colorado, USA. Received 8 February 1993; accepted 7 April 1994