Applicability criteria of the variable parameter Muskingum stage and discharge routing methods

Transcription

1 WATER RESOURCES RESEARCH, VOL. 43, W05409, doi: /2006wr004909, 2007 Applicability criteria of the variable parameter Muskingum stage and discharge routing methods Muthiah Perumal 1 and Bhabagrahi Sahoo 1 Received 18 January 2006; revised 8 January 2007; accepted 17 January 2007; published 4 May [1] The applicability criteria of the variable parameter Muskingum stage hydrograph (VPMS) and the variable parameter Muskingum discharge hydrograph (VPMD) routing methods are assessed and quantified. The assessment is made by studying the propagation characteristics of hypothetical stage hydrographs of the form of a four parameter Pearson type III distribution and its corresponding discharge hydrographs in uniform rectangular and trapezoidal channels using the VPMS and the VPMD methods, respectively, in comparison with the propagation characteristics of the respective hydrographs simulated by solving the Saint-Venant equations, which form the benchmark model. For each of the uniform rectangular and trapezoidal shape channel categories, a total of 2880 numerical experiments covering various combinations of Manning s roughness coefficient, channel bed slope, peak stage and respective time to peak, and shape factors of the stage hydrographs were conducted for each of these VPMS and VPMD methods, respectively, by routing the hypothetical stage and the respective discharge hydrographs for a reach length of 40 km. The routed stage and discharge hydrographs were compared with the corresponding Saint-Venant solutions using four performance measures, namely, percentage variance explained, percentage error in volume, percentage error in peak, and percentage error in time to peak. Considering a maximum error of 5% of these measures, the experiments indicate that the applicability limit of the VPMS method for stage routing is (1/S o )(@y/@x) max 0.79, while for the simultaneous computation of the discharge hydrograph corresponding to the routed stage hydrograph this method can be applied up to (1/S o )(@y/@x) max 0.63 (where S o is the channel slope is the water profile gradient). The VPMD method is applicable up to (1/S o )(@y/@x) max 0.43 for both discharge routing and the corresponding stage computation. The applicability of the VPMD method is compared with the variable parameter Muskingum-Cunge (VPMC) method, an alternative method for discharge hydrograph routing, and it is found that for the same 5% error criteria of the VPMD method, the VPMC method is applicable only up to (1/S o )(@y/@x) max Thus it is seen that the VPMD method has an improved performance and wider applicability range than the VPMC method. Citation: Perumal, M., and B. Sahoo (2007), Applicability criteria of the variable parameter Muskingum stage and discharge routing methods, Water Resour. Res., 43, W05409, doi: /2006wr Introduction [2] Flood routing methods involve the tracking of a flood wave mathematically from an upstream point to a downstream point of a river or channel. The ability to simplify the degree of complexity involved in the computation has been of great interest to hydrologists and field engineers. One of the main simplifications has been to assume the unsteady flow to be one-dimensional and it can be described by the full dynamic wave equations consisting of continuity and momentum equations, known as the Saint-Venant (SV) equations. Over the decades, a plethora of routing methods have been developed and applied, either in the form of 1 Department of Hydrology, Indian Institute of Technology Roorkee, Roorkee, India. Copyright 2007 by the American Geophysical Union /07/2006WR W05409 dynamic wave model by solving the full Saint-Venant equations, or their simplifications in the form of the kinematic wave (KW) model, the noninertia wave model and the quasi-steady dynamic wave model [Henderson, 1966; Woolhiser and Liggett, 1967; Ponce et al., 1978; Yen, 1979; Daluz, 1983; Price, 1985; Tsai and Yen, 2001; Yen and Tsai, 2001]. These simplified hydraulic methods also involve a varied degree of complexity in computation. Flood routing studies based upon simplified methods, derived either directly or indirectly from the Saint-Venant equations, are perceived as inherently less accurate than those based upon the numerical solution of the full Saint- Venant equations [Ferrick, 1985]. However, the simplified models continue to be of valuable tool for river engineers who want rapid access to a broad view of flood behavior in their rivers for planning and operation purposes, especially when sparse information on cross-sectional details are available. Further, numerical problems arise while solving 1of20

2 W05409 PERUMAL AND SAHOO: APPLICABILITY OF MUSKINGUM ROUTING METHODS W05409 Figure 1. Definition sketch of the Muskingum computational reach. the full Saint-Venant equations for studying flood wave movement when the magnitudes of different terms of the momentum equation are widely varying [Ferrick, 1985]. By analyzing different 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 wide spread use of the simplified routing methods like the variable parameter Muskingum-Cunge (VPMC) method [Ponce and Yevjevich, 1978] and its variants [Ponce and Chaganti, 1994; Ponce and Lugo, 2001] in practice [Fread, 1990; U.S. Army Corps of Engineers, 2002]. However, it may be noted that Price [1973, 1985] first proposed the VPMC method and estimated its applicability criterion. Besides the VPMC method and its variants, an alternative version of the variable parameter Muskingum discharge hydrograph (VPMD) routing method has been introduced by Perumal [1994a, 1994b] which enables the routing of a given discharge hydrograph downstream of a uniform river or channel reach to any desired location, and also computes the corresponding stage hydrograph as in the case of the solutions of the Saint-Venant equations. Similarly, a simplified variable parameter Muskingum stage hydrograph (VPMS) routing method having the same form of the routing equation of the Muskingum method has been introduced by Perumal and Ranga Raju [1998a, 1998b] which routes the upstream stage hydrograph to simulate the downstream stage hydrograph in a uniform river or channel reach, and also computes the corresponding discharge hydrograph. Both the VPMS and VPMD routing methods have been derived directly from the Saint-Venant equations. For theoretical details of these routing methods, the readers may refer the works of Perumal [1994a, 1994b] and Perumal and Ranga Raju [1998a, 1998b]. The advantage of these simplified routing methods is that they are capable of estimating both stage and discharge hydrographs simultaneously. However, the major disadvantage of these methods is their inability to handle the presence of downstream disturbances in the routing process, like any other simplified routing methods. [3] Identifying a suitable simplified method(s) for application to a given flood routing problem is a difficult task. Several researchers have attempted to provide criteria for the selection of the appropriate routing methods [Henderson, 1966; Woolhiser and Liggett, 1967;Ponce et al., 1978; Daluz, 1983; Fread, 1985; Ferrick, 1985; Price, 1985; Dooge and Napiorkowski, 1987; Marsalek et al., 1996; Moussa and Bocquillon, 1996; Singh, 1996; Tsai, 2003] for the application to a given routing problem with or without considering any downstream boundary condition. Among these criteria, the one introduced by Ponce et al. [1978] has found its place in standard textbooks [French, 1986; Ponce, 1989; Chaudhry, 1993; Viessman and Lewis, 1996; Singh, 1996]. By using linear stability analysis theory Ponce et al. [1978] developed the applicability criteria for the kinematic flood wave (KW) and diffusive flood wave (DW), respectively, as follows: TS o v o d o 171 ð1þ rffiffiffiffi g TS o 30 d o ð2þ where T is wave period of the inflow hydrograph usually twice the time of rise of the flood wave, S o is bottom slope of the channel, v o is average velocity of flood wave, d o is average flow depth, and g is acceleration due to gravity. Table 1. Various Combinations of Parameters Used for the Test Runs Parameters Values Figure 2. Error involved in the truncation of the binomial expansion of [1 (1/S 0 )] 1/2 in the rising and recession limbs of the input hydrograph. 2of20 Gamma g 1.05, 1.15, 1.25, 1.50 Bed slope S o , , , , , Manning s roughness n 0.01, 0.02, 0.03, 0.04, 0.05, 0.06 Initial stage y b 1.0 m Peak stage y p 5.0 m, 8.0 m, 10.0 m, 12.0 m, 15.0 m Time to peak t p 5.0 h, 10.0 h, 15.0 h, 20.0 h Bottom width b m

3 W05409 PERUMAL AND SAHOO: APPLICABILITY OF MUSKINGUM ROUTING METHODS W05409 Figure 3. Variance explained by the VPMC method with the applicability criteria for kinematic and diffusive waves in rectangular and trapezoidal channels for four g values of (a) 1.05, (b) 1.15, (c) 1.25, and (d) Figure 4. Variance explained by the CPMD method with the applicability criteria for kinematic and diffusive waves in rectangular and trapezoidal channels for four g values of (a) 1.05, (b) 1.15, (c) 1.25, and (d) of20

4 W05409 PERUMAL AND SAHOO: APPLICABILITY OF MUSKINGUM ROUTING METHODS W05409 Figure 5. Variance explained by the VPMS method with the applicability criteria for kinematic and diffusive waves in rectangular and trapezoidal channels for four g values of (a) 1.05, (b) 1.15, (c) 1.25, and (d) Figure 6. Variance explained by the VPMD method with the applicability criteria for kinematic and diffusive waves in rectangular and trapezoidal channels for four g values of (a) 1.05, (b) 1.15, (c) 1.25, and (d) of20

5 W05409 PERUMAL AND SAHOO: APPLICABILITY OF MUSKINGUM ROUTING METHODS W05409 Figure 7. Attenuation in peak discharge by the Saint- Venant method in rectangular and trapezoidal channels with the existing applicability criteria for (a) kinematic and (b) diffusive waves. [4] The inequalities in equations (1) and (2) were established on the basis of at least 95 percent accuracy in the wave amplitude when compared with the dynamic wave after one propagation period. The linear stability analysis used in arriving at these criteria considers the first-order approximation of the shallow water wave propagation which is treated as an infinitesimal disturbance imposed to the initially steady uniform flow. The common features of these criteria include the assumptions of a prismatic channel and a sinusoidal wave of arbitrary amplitude. However, in reality, flood waves found in natural rivers differ significantly from the assumption of sinusoidal shape and also they exhibit nonlinear behavior. Therefore the assumptions behind the development of these criteria are inherently contradictory with the characteristics of real life flood waves. Ferrick and Goodman [1998] pointed out that large amplitude flow increases of practical interest must be described by the nonlinear equations. Since linear stability theory is valid for small perturbations from the reference flow, and the real world flood waves are frequently very large in amplitude, the linear analysis used in the development of the applicability criteria (1) and (2) is questionable [Crago and Richards, 2000]. Further, Zoppou and O 0 Neill [1982] tested the criteria of Ponce et al. [1978] for a real life flood routing problem of a 33.2 km reach of the Australian river Yarra between Yarra Grange and Yering, for assessing the applicability of the diffusive and kinematic wave models as approximations to the dynamic wave model. A good agreement was obtained in all cases studied using the kinematic wave model, despite the criteria of Ponce et al. [1978] predicting that it would be unsuitable for routing under these circumstances. On the basis of these considerations, Zoppou and O Neill [1982] cautioned the river engineers and hydrologists about the limitations of these criteria. In light of subsequent development of improved simplified methods, which have moved from the domain of complete linear models to that of variable parameter models, duly accounting for the nonlinear characteristics of a flood wave, this caution seems to have a greater significance. Hence the applicability criteria advocated by Ponce et al. [1978] for identifying appropriate flood wave model for a given routing problem may be replaced by alternative criteria with physical significance, given the nonlinear mechanism of flood wave propagation in real world rivers. [5] One of the logical ways of developing these alternative criteria may be by directly incorporating the magnitude of the scaled water profile gradient (1/S o )(@y/@x), which is used for the classification of flood waves [Henderson, 1966; Natural Environment Research Council (NERC), 1975] as kinematic or diffusive. In fact, such an applicability criterion was advocated by Price [1985] for the simplified routing method developed by him, but it is too restrictive with j(1/s o )(@y/@x)j The hydrograph characterized by the presence of (1/S o )(@y/@x) signifies a diffusive flood wave and its absence signifies a kinematic flood wave [NERC, 1975]. The scaled gradient may be estimated at every routing time level of the given hydrograph at the inlet of the routing reach. The present study is intended to develop and quantify such applicability criteria for the VPMS and VPMD routing methods considering the propagation of flood waves in rectangular and trapezoidal prismatic channels. Four performance measures, namely, percentage variance explained, percentage error in volume, percentage error in peak, and percentage error in time to peak were used in the development of these developed applicability criteria. Further, based on these criteria the performance of the VPMS and VPMD methods were compared with the VPMC routing method [Ponce and Chaganti, 1994]. 2. Criteria Formulation [6] The VPMS and VPMD routing methods have been developed based on the assumptions of approximately linearly varying stage and discharge, respectively, along the computational routing reach. The error introduced due to these assumptions can be minimized by decreasing the length of the computational reach [Perumal, 1994a; Perumal and Ranga Raju, 1998a]. It has been shown by Perumal and Ranga Raju [1999] while deriving the VPMD method from the Saint-Venant equations that the assumption of approximately linearly varying discharge over the Muskingum reach is more restrictive than the assumption of approximately linearly varying stage used in the development of the VPMS method. Accordingly, the VPMS method has a wider applicability in routing stage hydrographs in a given channel reach than the VPMD method in routing the corresponding discharge hydrograph in the same reach, as it would be demonstrated in the subsequent sections. Because of the same reason, it may be inferred that the VPMS method may closely reproduce the stage hydrograph routing solutions of the Saint-Venant equations in comparison with the VPMD 5of20

6 W05409 PERUMAL AND SAHOO: APPLICABILITY OF MUSKINGUM ROUTING METHODS W05409 Figure 8. Error in volume by the (a) VPMC, (b) CPMD, (c) VPMS, and (d) VPMD methods in rectangular and trapezoidal channels for all the 5760 test runs with the existing applicability criteria. method in reproducing the corresponding discharge hydrograph solutions. This inference would be verified subsequently while conducting numerical experiments on these methods. Further, in the derivation of these routing methods it is assumed that 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 [Perumal, 1994a]. This assumption is also employed in the Kalinin- Milyukov method [Apollov et al., 1964; Miller and Cunge, 1975]. Accordingly, the normal discharge Q 3 occurring at the downstream of the midsection of the computational reach, as shown in the definition sketch (Figure 1), is uniquely related to the stage y M occurring at the midsection as [Perumal, 1994a] ( Q M ¼ Q " )1 1 m 2 FM 2 P@R=@y 2 2 ð3þ M where the subscript M denotes the midsection of the reach, Q M is discharge at the middle of the reach, A is water flow area, P is wetted perimeter, R is hydraulic radius, x is distance along the channel, m is exponent which depends on the friction law used (e.g., m = 2 = 3 for Manning s friction law, and m = ½ for Chezy s friction law), and F M is Froude number at the middle of the routing reach. [7] Perumal and Ranga Raju [1998a] have stated that the inertia approximated in terms of [F(P@R/@y)/(@A/@y)] in equation (3) can be neglected, as it has a minor contribution M toward the computational accuracy of discharge. Hence equation (3) can be modified as Q M ¼ Q ð4þ S Under the assumption that when the magnitude of (1/S o is less than unity, equation (4) may be expanded in binomial series and after neglecting the higher-order terms of (1/S o )(@y/@x), it is approximated as M Q M ¼ Q 3 Q 2S Figure 2 shows the truncation error involved in approximating equation (4) by equation (5) and this approximation error increases rapidly for the recession limb of the hydrograph, wherein (1/S o )(@y/@x) is positive, as compared to the rising limb of the hydrograph, wherein (1/S o )(@y/@x) is negative. Therefore the criteria to be developed for the VPMS and the VPMD methods would be restricted by the positive magnitude of (1/S o )(@y/@x) subject to the requirement of j(1/s o )(@y/@x)j < 1 for binomial series expansion of equation (4). [8] Perumal and Ranga Raju [1999] derived the approximate convection-diffusion (ACD) equations directly from the Saint-Venant s equations for the stage and discharge formulations, respectively, ¼ 0 M ð5þ ð6þ 6of20

7 W05409 PERUMAL AND SAHOO: APPLICABILITY OF MUSKINGUM ROUTING METHODS W05409 Figure 9. Variance explained by the VPMS and VPMD methods for stage reproduction (VarexY) and by the VPMS, VPMD, and VPMC methods for discharge reproduction (VarexQ) forg values of (a) 1.05, (b) 1.15, (c) 1.25, and (d) 1.50 (in rectangular channel ¼ 0 where c is celerity of the flood wave, given by It was brought out by Perumal and Ranga Raju [1999] that equations (6) and (7), using which the VPMS and VPMD methods, respectively, have been developed are capable of modeling the flood waves in the transition of the diffusive and the kinematic waves, including the latter. Therefore one may expect that within the applicability range of the VPMS and VPMD methods given by the developed criteria, it would be possible to closely model the kinematic flood waves in all situations and the diffusive flood waves in situations restricted by the assumption of approximately linearly varying stage and discharge at any instant of time over the routing reach. ð7þ ð8þ [9] Assuming that the inertial terms are insignificant, the discharge at any location of the channel may be expressed as [Henderson, 1966] Q ¼ Q o 1 1=2 ð9þ S where Q o = normal flow corresponding to flow depth y. Rearranging equation (9), the applicability criteria for both the VPMS and VPMD routing methods can be expressed as S ¼ 1 Q 2 ð10þ Q o [10] To quantify these criteria for both the VPMS and VPMD routing methods, a representative input stage hydrograph, defined by a four parameter Pearson type III distribution equation expressed as yð0; t t 1= ðg 1 Þ ¼ y b þ y p y b t p Þ exp 1 t=t p g 1 ð11þ Table 2. Applicability Limits of Different Methods for Discharge Reproduction With 95% Accuracy Level in Variance Explained and 5% Error in Volume for Different g Values (in Rectangular Channel Reaches) (1/S o )(@y/@x) max at VarexQ 95% (1/S o )(@y/@x) max at EVOL 5% g VPMS VPMD VPMC VPMS VPMD VPMC of20

8 W05409 PERUMAL AND SAHOO: APPLICABILITY OF MUSKINGUM ROUTING METHODS W05409 Figure 10. Volume errors by the VPMS, VPMD, and VPMC models for g values of (a) 1.05, (b) 1.15, (c) 1.25, and (d) 1.50 (in rectangular channel reaches). is considered for routing in uniform rectangular and trapezoidal channels, where y(0, t) is stage at the upstream end (x = 0) at any instant of time t, y b is initial stage corresponding to initial steady flow, y p is peak stage, t p is time to peak, and g is shape factor. [11] Differentiating equation (11) with respect to t tþ ¼ yð0; tþ y b 1 t g 1 ð12þ Equations (8), (10) and (12) can be used to calculate the absolute limiting value of the scaled dimensionless ACD equation in stage form, up to which the VPMS method is applicable, i.e., S o þ S ¼ e y max ð13þ where e y = tolerance limit for stage routing which tends to zero for kinematic waves. [12] Similarly for the VPMD method, the scaled dimensionless ACD equation in discharge form can be derived using the Saint-Venant s continuity equation as [Perumal, 1994b] S o Bc þ S ¼ e q max ð14þ where B is the free surface flow width and e q is tolerance limit for discharge routing which tends to zero for kinematic waves. [13] These formulations show that the applicability criteria (1/S o )(@y/@x) is a function of the characteristics of the inflow stage (or discharge) hydrograph and the channel, i.e., ð Þ ¼ f S o ; n; y b ðor Q b ð1=s o Þ; y p or Q p ; tp ; g ð15þ where n is Manning s roughness coefficient, Q b is initial steady discharge, Q p is peak discharge of the inflow hydrograph, and f{.} denotes a function. 3. Numerical Application [14] The above criteria were evaluated by applying the VPMS and VPMD methods for routing the stage and discharge hydrographs, respectively, in prismatic semiinfinite rectangular cross-section channel reaches for different combinations of the parameters described in equation (15). The different parameters of the input stage hydrograph are illustrated in Table 1. The hypothetical rectangular channel reach has a bottom width of 100 m and all the hydrographs were routed for a fixed reach length of 40 km. The results were also verified by routing the stage and discharge hydrographs in prismatic semi-infinite uniform trapezoidal cross-section channel reaches for different combinations of the parameters described in equation (15). The uniform trapezoidal channel reach has a bottom width of 100 m and side slopes of 1.0 horizontal to 1.0 vertical. No lateral flow is considered in this study. [15] One of the best ways to test the efficacy of these routing methods is by comparing the hydrographs obtained by routing the given hypothetical input stage and discharge hydrographs with the corresponding stable routed hydrograph 8of20

9 W05409 PERUMAL AND SAHOO: APPLICABILITY OF MUSKINGUM ROUTING METHODS W05409 Figure 11. Error in peak stage by the VPMS and VPMD methods for stage reproduction (y per ) and error in peak discharge by the VPMS, VPMD, and VPMC methods for discharge reproduction (q pe )for g values of (a) 1.05, (b) 1.15, (c) 1.25, and (d) 1.50 (in rectangular channel reaches). solutions obtained by using the Saint-Venant equations (solved by using the explicit scheme), which are considered as the benchmark solutions. Further, the discharge hydrograph estimated at the inlet of the reach by the solution of the Saint-Venant equations corresponding to the same input stage hydrograph used in the VPMS routing method, was used as the input discharge hydrograph for routing using the VPMD method and the four point averaging scheme of the VPMC method, recommended by Ponce and Chaganti [1994]. The discharge Q b computed corresponding to the initial stage y b is considered as the initial discharge for these methods. For all the test runs, the same input stage hydrograph shape was used, however, with differing g, y p and t p values, so that all the methods described herein could be evaluated under identical flow and channel characteristics. A total of 2880 test runs, each for the rectangular and trapezoidal channels, formed by different combinations of parameters as given in Table 1, were used for studying each of the VPMS, VPMD and VPMC methods and to quantify the applicability potential of these routing methods. [16] While solving the Saint-Venant equations for arriving at the benchmark solutions, and applying the VPMS and VPMD methods to reproduce these benchmark solutions, there exist the possibility of numerical errors creeping into the solutions to mask the physical characteristics of the actual solutions and their reproductions by the VPMS and VPMD routing methods. To overcome this problem, it is ensured in the study that the benchmark solutions obtained using the explicit numerical scheme very well satisfied the Courant s condition and the routed solutions were fully mass conservative. Since no external condition was imposed for conservation of mass, the successfully arrived at explicit numerical solutions of the Saint-Venant equations were found to be fully mass conservative and thus ensuring the benchmark solutions to be free from numerical errors including the aspect of attenuation characteristics. To overcome such a numerical problem with the VPMS and VPMD routing methods, solutions were obtained for a small space step of 1 km and a time step of 15 minutes for all the test runs made in this study. When these space and time steps were further reduced, the same routing solutions were obtained by the VPMS and VPMD routing methods, thus ensuring numerical error free solutions. 4. Performance Evaluation [17] The following evaluation measures were adopted for assessing and quantifying the applicability criteria of the VPMS, VPMD, and VPMC methods in simulating the corresponding benchmark solutions Variance Explained [18] The closeness with which the VPMS, VPMD, and VPMC routing methods reproduce the benchmark solutions, including the closeness of shape and size of the hydrograph, can be measured using the criterion of variance explained advocated by Nash and Sutcliffe [1970] and recommended by the ASCE Task Committee on Definition of Criteria for Evaluation of Watershed Models of the Watershed Management Committee, Irrigation and Drainage Division [1993], popularly known as the Nash-Sutcliffe criterion. The vari- 9of20

10 W05409 PERUMAL AND SAHOO: APPLICABILITY OF MUSKINGUM ROUTING METHODS W05409 Table 3. S o n y p t p Combinations of Input Stage Hydrograph for Which the Error in Peak Stage y per > 5% Due to Stage Routing by the VPMS Method, While Fixing the Applicability Criteria of (1/S o )(@y/@x) max 0.79 (in Rectangular Channel Reaches) g S o n y p,m t p,h y per,% ance explained in percentage for stage and discharge hydrograph reproductions are, respectively, given as VarexY ¼ 1 XN X ðy oi y ci Þ 2 N ðy oi y o Þ! ð16þ i¼1 i¼1 VarexQ ¼ 1 XN i¼1 S max X ðq oi Q ci Þ 2 N ðq oi Q o Þ! i¼1 ð17þ where VarexY is variance explained estimate in reproducing benchmark stage hydrograph, VarexQ is variance explained estimate in reproducing benchmark discharge hydrograph, y oi is ith ordinate of the benchmark stage hydrograph at the outlet, y o is mean of the observed stage hydrograph ordinates at the outlet, y ci is ith ordinate of the routed or computed stage hydrograph, Q oi is ith ordinate of the benchmark discharge hydrograph at the outlet, Q o is mean of the benchmark discharge hydrograph ordinates at the outlet, Q ci is ith ordinate of the routed or computed discharge hydrograph corresponding to y ci, and N is total number of stage or discharge hydrograph ordinates to be simulated. Table 4. S o n y p t p Combinations of Input Stage Hydrograph for Which the Error in Peak Discharge q per > 5% Due to Discharge Computation by the VPMS Method, While Fixing the Applicability Criteria of (1/S o )(@y/@x) max 0.79 (in Rectangular Channel Reaches) g S o n y p,m t p,h q per,% S max of 20

11 W05409 PERUMAL AND SAHOO: APPLICABILITY OF MUSKINGUM ROUTING METHODS W05409 Table 5. Applicability Limits of Different Methods With Maximum Error Level of 5% in Peak Discharge q per, Time to Peak Discharge t pqer, and Time to Peak Stage t pyer for Different g Values (in Rectangular Channel Reaches) (1/S o )(@y/@x) max at q per 5% (1/S o )(@y/@x) max at t pqer 5% (1/S o )(@y/@x) max at t pyer 5% g VPMS VPMD VPMC VPMS VPMD VPMC VPMS VPMD Peak Reproduction [19] The percentage errors in simulating peaks of the stage and discharge hydrographs are, respectively, given as y per ¼ q per ¼ y pc y po q pc q po ð18þ ð19þ where y pc is routed or computed peak of the stage hydrograph at the outlet, y po is peak of the benchmark stage hydrograph at the outlet, q pc is routed or computed peak of the discharge hydrograph at the outlet, and q po is the peak of the benchmark discharge hydrograph at the outlet. The positive values of y per and q per indicate overestimation of the respective peaks from the benchmark values, and the negative values of y per and q per indicate their underestimation Time to Peak Reproduction [20] The percentage errors in time to peak stage and discharge, respectively, are given as t pyer ¼ t ypc ð20þ t ypo t pqer ¼ t qpc t qpo ð21þ where t ypc is time corresponding to routed or computed peak stage at the outlet, t ypo is time corresponding to benchmark peak stage at the outlet, t qpc is time corresponding to routed or computed peak of the discharge hydrograph at the outlet, and t qpo is time corresponding to the peak of the benchmark discharge hydrograph at the outlet Volume Conservation [21] The percentage error of volume conservation is expressed as 0 1 X N Q ci i¼1 EVOL ¼ 1 X N C A 100 ð22þ I i i¼1 where Q ci = ith ordinate of the routed or computed discharge hydrograph; and I i = ith ordinate of the inflow discharge hydrograph. A negative value of EVOL indicates loss of mass and a positive value of EVOL indicates gain of Table 6. S o n y p t p Combinations of Input Stage Hydrograph for Which the Error in Peak Discharge q per > 5% due to Discharge Computation by the VPMD Method, While Fixing the Applicability Criteria of (1/S o )(@y/@x) max 0.43 (in Rectangular Channel Reaches) g S o n y p,m t p,h q per,% S max of 20

12 W05409 PERUMAL AND SAHOO: APPLICABILITY OF MUSKINGUM ROUTING METHODS W05409 Figure 12. Error in time to peak stage by the VPMS and VPMD methods for stage reproduction (t pyer ) and error in time to peak discharge by the VPMS, VPMD, and VPMC methods for discharge reproduction (t pqer )forg values of (a) 1.05, (b) 1.15, (c) 1.25, and (d) 1.50 (in rectangular channel reaches). mass. A value close to zero suggests mass conservation ability of the method Attenuation in Peak Discharge [22] The percentage attenuation in peak discharge is expressed as x ¼ 1 Q pc 100 Q p ð23þ where Q pc is peak of the routed or computed discharge hydrograph and Q p is peak of the inflow discharge hydrograph. made using the VPMC, CPMD, VPMS, and VPMD routing methods in both channel types. The lines showing the minimum limits of these criteria are also shown therein. It is inferred from Figures 3 6 that there are a large number of test runs (region below the DW limit line), which neither satisfy the applicability criteria of the kinematic wave nor the diffusive wave. However, for most of these test runs in this region, the variance explained criterion exceeded 95% suggesting a good overall reproduction. Although, both the CPMD method and the applicability criteria (1) and (2) have been developed based on linear theory, many test runs of 5. Results and Discussion 5.1. Evaluation of the Existing Applicability Criteria [23] The existing applicability criteria (1) and (2) were evaluated by applying the VPMC, constant parameter Muskingum discharge (CPMD), and VPMD methods for routing the discharge hydrograph, and the VPMS method for routing the stage hydrograph in prismatic semi-infinite rectangular and trapezoidal cross-section channel reaches for different combinations of parameters of the input stage hydrograph as illustrated in Table 1. A total of 2880 numerical experiments (test runs), each for the uniform rectangular and trapezoidal channel reaches, were made following the approach described in section 3.1. Figures 3a 3d, 4a 4d, 5a 5d, and 6a 6d show the variation of Ponce et al. criteria estimated using equations (1) and (2) with the corresponding estimated variance explained criterion using equations (16) and (17), respectively, depending on stage or discharge as the operating variable, for all the 5760 test runs 12 of 20 Figure 13. Comparison of the discharge hydrographs reproduced by the VPMS, VPMD, and VPMC methods for S o = 0.001, n = 0.06, y b =1m(Q b = m 3 /s), y p =15m (Q p = 4, m 3 /s), t p = 5 h, and g = 1.15 (in rectangular channel reach).

13 W05409 PERUMAL AND SAHOO: APPLICABILITY OF MUSKINGUM ROUTING METHODS W05409 Figure 14. Comparison of the discharge hydrographs reproduced by the VPMS, VPMD, and VPMC methods for S o = , n = 0.03, y b =1m(Q b = m 3 /s), y p =5m (Q p = m 3 /s), t p = 10 h, and g = 1.15 (in rectangular channel reach). routing using the CPMD method which failed to satisfy both these criteria could reproduce the benchmark solutions with the variance explained criterion exceeding 95%. This aspect demonstrates the deficiency of the Ponce et al. [1978] criteria (1) and (2). This inference is further strengthened from the results presented in Figures 7a and 7b wherein the KW criterion given by equation (1) and the DW criteria given by equation (2) are plotted with the percentage attenuation in peak discharge estimated from the solutions of the Saint-Venant equations for all the test runs made. It was found that x 1% for 22% of test runs and x 2% for 32% of test runs. If it may be assumed that the flood waves with x <1%arekinematic, then it is inferred from Figure 7a that almost all these 22% of test runs fail to satisfy the KW criterion (1). Similarly, almost all the 32% of test runs with x 2% fail to satisfy the KW criterion. On the similar basis, it is inferred from Figure 7b that many test runs with perceptible attenuation which could be reproduced using these discharge routing methods with variance explained criterion being greater than 95%, failed to satisfy the DW criterion (2). Further, it is inferred from Figure 7b that for some test runs with x 0, the estimated values of the DW criterion were well above 30, thus inferring a kinematic wave to be a diffusive wave. This contradiction further demonstrates the failure of the Ponce et al. criteria (1) and (2). Alternatively, considering a criterion of 5% acceptable error in volume reproduction of the model considered, Figures 8a 8d evidently show that the criteria (1) and (2) are erroneous for nearly all the test runs of different routing models, except for few test runs of the VPMC routing method. It is evident from these discussions that the Ponce et al. criteria (1) and (2), developed based on the linear stability theory, are not appropriate for checking the applicability of variable parameter routing methods. Hence the need to develop alternative applicability criteria suitable for the considered variable parameter stage and discharge routing methods is established Evaluation of the VPMS, VPMD, and VPMC Methods for Routing in Rectangular Channel Reaches to Establish the Applicability Criteria [24] Test runs were made for each channel type configuration characterized by a unique set of values of S o and n by routing the given hydrographs for a length of 40 km of reach, with the reach divided into forty equal subreaches. Figures 9a 9d show the variation of (1/S o )(@y/@x) max estimate at the inlet of the reach of each test run versus the corresponding test run s variance explained estimated by the VPMS, VPMD, and VPMC methods in reproducing the routed stage and discharge hydrographs of the full Saint- Venant solutions for the input stage hydrograph shape factors of g = 1.05, g = 1.15, g = 1.25, and g = 1.50, respectively. It is inferred from Figure 9 that the variance explained in reproducing the benchmark stage hydrograph solutions (VarexY) by the VPMS method is more than 96.75%, 97.86%, 96.33%, and 96.23% for g = 1.05, g = 1.15, g = 1.25, and g = 1.50, respectively. The test run results and Figure 9 reveal that corresponding to the limits of 95% of variance explained, the VPMS method has the applicability limits of (1/S o )(@y/@x) max 0.81 for g = 1.05; (1/S o )(@y/@x) max 0.81 for g = 1.15; (1/S o )(@y/@x) max 0.84 for g = 1.25; and (1/S o )(@y/@x) max 0.79 for g = Further, these results reveal that beyond these applicability Figure 15. Comparison of the discharge hydrographs reproduced by the VPMS, VPMD, and VPMC methods for S o = , n = 0.06, y b =1m(Q b = m 3 /s), y p =15m (Q p = 3, m 3 /s), t p = 5 h, and g = 1.50 (in rectangular channel reach). 13 of 20 Figure 16. Comparison of the discharge hydrographs reproduced by the VPMS, VPMD, and VPMC methods for S o = , n = 0.06, y b =1m(Q b = m 3 /s), y p =10m (Q p = 1, m 3 /s), t p = 15 h, and g = 1.50 (in rectangular channel reach).

14 W05409 PERUMAL AND SAHOO: APPLICABILITY OF MUSKINGUM ROUTING METHODS W05409 Table 7. Maximum Attenuation in Peak Discharge for Different Routing Methods With Variance Explained in 95% Peak Discharge Reproduction (in Rectangular Channel Reaches) g VPMS, % VPMD, % VPMC, % SV, % levels, the simulation runs by the VPMS routing method terminates automatically due to the computation of high negative values of q indicating violation of assumptions built into the model [Perumal and Ranga Raju, 1998b]. Hence, given the 95% accuracy limit of variance explained for stage routing, the VPMS method can be applied up to (1/ S o )(@y/@x) max [25] Similarly, the discharge hydrograph computed using the routed stage hydrograph by the VPMS method could reproduce the respective benchmark discharge hydrograph (VarexQ) by more than 97.55%, 98.17%, 96.90%, and 96.48% for g values of 1.05, 1.15, 1.25, and 1.50, respectively. Therefore, with an accuracy level of 95% for discharge hydrograph reproduction, the VPMS method has the same applicability limit of (1/S o )(@y/@x) max 0.79 as that for the stage routing. [26] Similarly, the test run results of the VPMD method successful in reproducing the benchmark discharge hydrographs reveal that the minimum values of the variance explained (VarexQ) for the VPMD method are 96.70%, 97.27%, 96.08%, and 96.45% for g values of 1.05, 1.15, 1.25, and 1.50, respectively; the corresponding values for the VPMC method are 77.45%, 75.99%, 72.77%, and 69.48%. Numerical experiments reveal that the successfully arrived at routing results using the VPMS and VPMD methods always reproduce the stage and discharge hydrographs, respectively, with an accuracy level of more than 95%, and the failure of these methods for the remaining test runs indicates the violation of the assumptions built into these methods during the process of routing. However, the VPMC method successfully routes all the input discharge hydrographs, resulting in poor reproduction of many benchmark discharge hydrograph solutions. Hence the VPMS and VPMD methods may be preferred for routing the given stage and discharge hydrographs due to the built-in capability of the method to fail when the assumptions of the methods are violated during routing. Consequently, one can place more confidence on the VPMS and VPMD methods over that of the VPMC method for their application to routing problems in ungauged river basins. With the accuracy level of 95%, the applicability criteria estimated for the VPMS, VPMD, and VPMC methods are given in Table 2. Figures 9a 9d show that the VarexQ by the VPMC method is very much scattered as compared to the VPMS and VPMD methods. This analysis shows that considering the accuracy of VarexQ 95%, the applicability criteria for discharge hydrograph estimation by the VPMS, VPMD and VPMC methods can be fixed at (1/S o )(@y/@x) max 0.79, (1/S o )(@y/@x) max 0.43 and (1/S o )(@y/@x) max 0.20, respectively. [27] Further, for stage hydrograph computation from the routed discharge hydrograph by the VPMD method, the variance explained in reproducing stage hydrograph (VarexY) is more than 97.78%, 98.29%, 97.65%, and 97.90% for g values of 1.05, 1.15, 1.25, and 1.50, respectively. Therefore, for an accuracy level of >95% for stage hydrograph reproduction using the VPMD method, the same applicability limit as that for discharge routing is applicable. [28] Figures 10a 10d show the percentage error in volume (EVOL) estimated by equation (22) for all successful test runs studied in simulating the benchmark discharge hydrographs using the VPMS, VPMD and VPMC methods. The maximum EVOL values that the VPMS, VPMD and VPMC methods exhibit corresponding to the input stage hydrographs characterized by the g values of 1.05, 1.15, 1.25, and 1.50, respectively, are 1.57%, 1.86%, 4.54% and 8.08%; 1.59%, 2.30%, 2.34% and 4.95%; and 21.27%, 23.40%, 19.39% and 23.62%. When g = 1.50, the VPMS method estimates the volume error less than 5%, except for four cases with S o n y p t p combinations of m 20 h (EVOL = 5.27%), m 20 h (EVOL = 5.55%), m 20 h (EVOL = 6.26%), and m 20 h (EVOL = 8.08%). In case of the VPMD method the error in volume is well within 5% for all the successfully simulated cases. Therefore, with EVOL 5% being the criterion, the applicability limits of the VPMS, VPMD and VPMC methods can be fixed at 0.79, 0.43, and 0.11, respectively (see Table 2). Hence the VPMS and VPMD methods are more mass conservative than the VPMC method. [29] It is seen from Figures 11a 11d that the maximum percentage errors in peak stage reproduction, y per exhibited by the VPMS method are 6.20%, 7.55%, 8.02%, and 7.90% for g values of 1.05, 1.15, 1.25, and 1.50, respec- Table 8. Summary of the Applicability Limits of Different Methods With 95% Accuracy Level of Model Performance in Trapezoidal Channel Reaches g Stage Routing/Computation Discharge Routing/Computation VPMS VPMD VPMS VPMD VPMC Variance Explained 95% EVOL 5% Error in Peak 5% Error in Time to Peak 5% of 20

15 W05409 PERUMAL AND SAHOO: APPLICABILITY OF MUSKINGUM ROUTING METHODS W05409 Figure 17. Variation of scaled dimensionless ACD equation with the water profile gradient by the VPMS method for g = 1.05 (in rectangular channel reaches). tively. If the applicability criterion for the VPMS method is fixed at (1/S o )(@y/@x) max 0.79, then this method mostly estimates the peak stage within 95% accuracy except for three test run cases when g = 1.05, seven test run cases when g = 1.15, nine test run cases when g = 1.25 and fifteen test run cases when g = 1.50 (see Table 3). However, for all the test run cases with y per 5%, the applicability criterion of the VPMS method can be fixed at (1/S o )(@y/@x) max [30] Similarly, the maximum percentage errors in peak discharge q per, exhibited by the VPMS method are 9.99%, 11.79%, 12.08%, and 12.41% corresponding to the input stage hydrographs characterized by g values of 1.05, 1.15, 1.25, and 1.50, respectively; and the corresponding values exhibited by the VPMD method are 6.44%, 7.29%, 8.04%, and 8.21%; and that by the VPMC method are 41.15%, 39.97%, 40.03%, and 46.58%, respectively. Using the applicability criterion of (1/S o max 0.79 for the VPMS method, it is seen that q per is greater than 5% in 10 cases when g = 1.05, 9 cases when g = 1.15, 15 cases when g = 1.25, and 16 cases when g = 1.50 (see Table 4). However, q per 5% for all cases when (1/S o )(@y/@x) max Table 5 shows the applicability criteria for different methods for the condition q per 5%. When (1/S o )(@y/@x) max 0.43, the VPMD method shows an error in peak discharge q per > 5% in 5 cases when g = 1.05, 11 cases when g = 1.15, 14 cases when g = 1.25, and 17 cases when g = 1.50 (see Table 6). The VPMC method gives wide scattered results of error in peak. [31] The maximum percentage errors in peak stage y per, computed by the VPMD method are 1.49%, 2.46%, 2.56%, and 2.74% for g values of 1.05, 1.15, 1.25, and 1.50, respectively. These values are well below 5% error level and it can be inferred from Figures 11a 11d that for stage computation from the routed discharge hydrograph, the VPMD method has the same applicability range as for the discharge hydrograph routing. [32] The maximum percentage errors in time to peak stage t pyer, exhibited by the VPMS method are 6.86%, 9.94%, 10.73% and 11.93% for g values of 1.05, 1.15, 1.25 and 1.50, respectively (Figures 12a 12d). In case of stage hydrograph computation by the VPMD method, the maximum percentage errors in time to peak stage corresponding to g values of 1.05, 1.15, 1.25 and 1.50 are 3.80%, 5.74%, 7.60% and 9.70%, respectively. The applicability limits of the VPMS method for stage routing and that of the VPMD method for stage computation for different g values are shown in Table 5. However, with an applicability criterion of (1/S o )(@y/@x) max 0.79 for the VPMS method for stage hydrograph routing, the value of t pyer is less than 5% except in 2.21%, 4.37%, 4.11%, and 6.90% cases of all successful test runs for g values of 1.05, 1.15, 1.25, and 1.50, respectively. Further, with an applicability criterion of (1/S o )(@y/@x) max 0.43 for the VPMD method for stage computation, t pyer > 5% in 0.00%, 1.79%, 3.48%, and 5.15% cases of all successful test runs for g values of 1.05, 1.15, 1.25, and 1.50, respectively. [33] Similarly, the maximum percentage errors in time to peak discharge t pqer, exhibited by the VPMS, VPMD and VPMC methods for g values of 1.05, 1.15, 1.25 and 1.50, respectively, are 4.84%, 7.49%, 10.11% and 9.70%; 4.35%, 6.23%, 8.70% and 10.15%; and 16.03%, 20.93%, 25.00% and 27.83%. The applicability values of (1/S o )(@y/@x) max for all the methods and for different g values are illustrated in Table 5. With the applicability criterion of (1/S o )(@y/@x) max 0.43, the VPMD method for discharge routing performs 15 of 20

16 W05409 PERUMAL AND SAHOO: APPLICABILITY OF MUSKINGUM ROUTING METHODS W05409 Figure 18. Variation of scaled dimensionless ACD equation with the water profile gradient by the VPMS method for g = 1.15 (in rectangular channel reaches). well within t pqer 5% except in 0.00%, 1.84%, 2.27%, and 5.15% cases of all successful test runs for g values of 1.05, 1.15, 1.25, and 1.50, respectively; while the VPMS method for discharge computation with the applicability criterion of (1/S o )(@y/@x) max 0.63 performs well within t pqer 5% for all the cases. The foregoing discussion reveals the better performance of the VPMS and VPMD methods over the VPMC method, and the typical comparison of discharge hydrograph reproductions shown in Figures 13, 14, 15, and 16 conclusively support this inference. It is pertinent to point out herein that the parameters of all the three methods, namely, VPMS, VPMD and VPMC, have the same relationships with channel and flow characteristics. However, the reason for the better performance of the former two methods over the latter method may be attributed to the method of varying the parameters from one routing time interval to another in a physically based manner consistent with the variation built into the solution of the Saint-Venant equations. However, in the case of the VPMC method, the technique of varying the parameters, although systematic, are not physically based. [34] The maximum attenuation percentages of the peak discharge for different methods corresponding to 95% of variance explained criteria are given in Table 7. This reveals that the VPMS method is more capable of routing a stage hydrograph characterized by large attenuation of peak discharge, followed by the VPMD and VPMC methods Verification of the Applicability Criteria for Routing in Trapezoidal Channel Reaches [35] A similar analysis as described in section 5.2 was carried out for 2880 test runs in a uniform trapezoidal channel reach. The applicability limits for this analysis are shown in Table 8. These results show that the applicability criteria, obtained by routing in the rectangular channel reaches, are the lower limits for routing in the trapezoidal channel. Hence the same criteria as given for the rectangular channel reach is also applicable for the trapezoidal channel reach Characteristics of the Applicability Criteria in Relation to the ACD Equation [36] Figures 17, 18, 19, and 20 illustrate the variation of (1/S o )(@y/@x) versus the scaled dimensionless ACD equation estimate given by equation (13) corresponding to g = 1.05, 1.15, 1.25 and 1.5, respectively, in the rectangular channel reach. It is seen from Figures that test runs characterized by increase in the range of (1/ S o )(@y/@x) also exhibit larger increase in the range of the estimates of the scaled dimensionless ACD equation [(1/ S o )(@y/@x) + (1/(S o c))(@y/@t)] away from zero value. This aspect brings out the reason for the inapplicability of the ACD equation [(1/S o )(@y/@x) + (1/(S o c))(@y/@t)] for test runs with large range of (1/S o )(@y/@x) and hence the VPMS method which uses the ACD equation. Further, Figures also reveal that with increase in g values, the value of [(1/S o )(@y/@x) + (1/(S o c))(@y/@t)] gradually becomes more positive for the same value of (1/S o A similar inference could also be made from the typical plot (Figure 21) of (1/S o )(@y/@x) versus the scaled dimensionless ACD equation estimate given by equation (14) for the VPMD method corresponding to the case of g = The different combinations for S o n y p t p plotted in Figures are given in Table 9. It is also inferred from Figures that the higher the Manning s roughness value and smaller the bed slope, the 16 of 20

17 W05409 PERUMAL AND SAHOO: APPLICABILITY OF MUSKINGUM ROUTING METHODS W05409 Figure 19. Variation of scaled dimensionless ACD equation with the water profile gradient by the VPMS method for g = 1.25 (in rectangular channel reaches). Figure 20. Variation of scaled dimensionless ACD equation with the water profile gradient by the VPMS method for g = 1.50 (in rectangular channel reaches). 17 of 20

18 W05409 PERUMAL AND SAHOO: APPLICABILITY OF MUSKINGUM ROUTING METHODS W05409 Figure 21. Variation of scaled dimensionless ACD equation with the water profile gradient by the VPMD method for g = 1.15 (in rectangular channel reaches). summation terms in the ACD equation, that is, e y and e q, deviate farther away from zero. 6. Summary and Conclusions [37] This study attempts to develop the applicability criteria of the variable parameter Muskingum stage hydrograph (VPMS) and the variable parameter Muskingum discharge hydrograph (VPMD) routing methods. The study also attempts to compare the performance of the VPMD and VPMC methods. The appropriateness in using the scaled water profile gradient of the input stage hydrograph as the applicability criterion for determining the applicability range of the VPMS, VPMD, and VPMC methods is brought out. This is verified through a number of numerical experiments covering a wide range of combinations of channel characteristics (i.e., channel bed slope and Manning s roughness) and upstream flow characteristics (i.e., peak, time to peak and shape factor). To develop the applicability criteria based on the performance of the VPMS, VPMD and VPMC methods in reproducing the benchmark solutions, eighty different upstream input stage hydrographs were used for routing using these methods in uniform rectangular and trapezoidal channels under the same routing conditions as that used for arriving at the benchmark Saint-Venant solutions. It is noted herein that the VPMD and VPMC methods use the same upstream discharge hydrographs computed by using the solutions of the full Saint-Venant equations corresponding to the considered eighty stage hydrographs. The applicability range of all these methods were determined using 95% accuracy level of the model performance in explaining the variance of the benchmark solution by the test routing method, and 5% level of errors Table 9. S o n y p t p Combinations Used in Figures 17, 18, 19, 20, and 21 Test Run S o n y p,m t p,h sim sim sim sim sim sim sim sim sim sim sim sim sim sim sim sim sim sim sim sim sim sim sim sim sim sim sim sim sim sim sim sim of 20