Volume Conservation Controversy of the Variable Parameter Muskingum Cunge Method

Transcription

1 Volume Conservation Controversy of the Variable Parameter Muskingum Cunge Method Muthiah Perumal 1 and Bhabagrahi Sahoo 2 Abstract: The study analyzes the volume conservation problem of the variable parameter Muskingum Cunge VPMC method for which some remedial solutions have been advocated in recent literature. The limitation of the VPMC method to conserve volume is brought out by conducting a total of 6,400 routing experiments. These experiments consist of routing a set of given hypothetical discharge hydrographs for a specified reach length in uniform rectangular and trapezoidal channels using the VPMC method, and comparing the routed solutions with the corresponding benchmark solutions obtained using the full Saint-Venant equations. The study consisted of 3,200 routing experiments carried out each in uniform rectangular and trapezoidal channel reaches. Each experiment was characterized by a unique set of channel bed slope, Manning s roughness coefficient, peak discharge, inflow hydrograph shape factor, and time to peak. A parallel study was carried out using an alternate physically based variable parameter Muskingum discharge hydrograph VPMD routing method proposed by Perumal in 1994 under the same routing conditions, and the ability of both the VPMC and VPMD methods to reproduce the benchmark solutions was studied. It is brought out that within its applicability limits, the VPMD method is able to conserve mass more accurately than the VPMC method. The reason for the better performance of the former over the latter method is attributed to the physical basis of its development. It is argued that adoption of artificial remedial measures to overcome the volume conservation problem makes the VPMC method semiempirical in nature, thereby losing the fully physically based characteristics of the method. The paper also dwells on the problems of negative initial outflow or dip in the beginning of the Muskingum solution, and the negative value of the Muskingum weighting parameter. Besides, the effect of incorporating the inertial terms in the estimation of Muskingum parameters and their impact on the overall Muskingum routing solutions is addressed by conducting another set of 6,400 numerical experiments using both the VPMC and VPMD methods. DOI: / ASCE :4 475 CE Database subject headings: Hydrographs; Parameters; Conservation; Channels; Routing. Introduction A plethora of simplified flood routing methods are available in the literature for studying flood wave propagation in rivers, in both diagnostic and prognostic modes. The evolution of all these simplified routing methods is based on the argument, apart from the simplicity of their easy understanding by the field engineers and their suitability for application in data deficient situations, that numerical problems arise while solving the full Saint-Venant equations for studying the propagation of flood waves, when the magnitudes of different terms of the momentum equation are widely varying; and, hence, the use of more complete Saint- Venant equations may not yield more accurate river wave simulations for all wave types Ferrick 1985; NERC Among the 1 Associate Professor, Dept. of Hydrology, Indian Institute of Technology Roorkee, Roorkee , India. p_erumal@yahoo.com 2 Scientist, National Academy of Agricultural Research Management ICAR, Rajendranagar, Hyderabad , India; formerly, Research Scholar, Dept. of Hydrology, Indian Institute of Technology Roorkee, Roorkee , India corresponding author. bsahoo2003@ yahoo.com Note. Discussion open until September 1, Separate discussions must be submitted for individual papers. To extend the closing date by one month, a written request must be filed with the ASCE Managing Editor. The manuscript for this paper was submitted for review and possible publication on March 11, 2006; approved on July 26, This paper is part of the Journal of Hydraulic Engineering, Vol. 134, No. 4, April 1, ASCE, ISSN /2008/ /$ different simplified discharge routing methods, the Muskingum Cunge method Cunge 1969; Price 1973 and its improvements, the variable parameter Muskingum Cunge VPMC method, and its variant Price 1973, 1985; Ponce and Yevjevich 1978; Ponce and Theurer 1982; Ponce and Chaganti 1994; Ponce et al. 1996; Ponce and Lugo 2001 have become prominent in the fields of urban and rural flood hydrology. While the Muskingum Cunge method is fully mass conservative Cunge 2001, the VPMC method is unfortunately saddled by a small but perceptible loss of mass Ponce Such systematic volume errors play a crucial role in water resource assessment, especially in water dispute situations. To overcome this volume conservation problem, a number of variants of the VPMC method proposed by Ponce and Yevjevich 1978 have been evolved either by modifying the parameter estimation methods e.g., Ponce and Chaganti 1994; Tang et al. 1999, or by accounting for the inertial terms e.g., Ponce and Lugo 2001 in the computational framework of the VPMC method. Ponce and Chaganti 1994 concluded that the four-point averaging scheme is the best numerical scheme as it increases the accuracy of the VPMC method. Tang et al attributed the systematic nonconservation of volume in the VPMC method to the problem of inappropriate estimation of the reference discharge in each computational cell and, hence, on the overall accuracy of the VPMC method and its variants. Following the study by Cappelaere 1997 that nonconsideration of the longitudinal water surface gradient term, y/ x, in assessing the diffusivity term of the diffusion equation leads to a mass conservation problem, Tang et al proposed a different modified numerical scheme JOURNAL OF HYDRAULIC ENGINEERING ASCE / APRIL 2008 / 475

2 known as the VPMC-4H scheme as described later. In this modified four-point averaging scheme of Ponce and Chaganti 1994, the celerity, c and the diffusion coefficient, D employed in the estimation of the Muskingum parameters were modified to account for the effect of y/ x. The main aim of the study by Tang et al. 1999, 2001 was to extend the range of applicability of the VPMC method by introducing an adjustment or correction factor to reduce the volume conservation error in the VPMC method solution. However, Cunge 2001 opined that any such artificial complication of the VPMC method by altering the numerical scheme, aimed at creation of a Muskingum and a half method, is unwarranted. Similarly, Ponce and Lugo 2001 modified the Muskingum parameter relationships by incorporating the inertial terms and the term y/ x in the expression for hydraulic diffusivity in an approximate manner. However, in most natural rivers, the magnitudes of inertial terms in the momentum equation are negligible Henderson 1966; Price Hence, the issue of volume nonconservation of the VPMC method is not actually solved by either of the modifications of Tang et al. 1999, and Ponce and Lugo However, unlike the VPMC method and its variants including the VPMC-4H scheme proposed by Tang et al. 1999, the VPMD method proposed by Perumal 1994a, b is directly derived from the Saint-Venant equations using the appropriate assumptions, including the assumption of approximate linear variation of water surface at any instant of time over the space step x. Though the form of the parameter relationships established for the VPMD method is the same as that of the VPMC method and its variants, without involving the direct use of the term y/ x as has been proposed by Tang et al in their VPMC-4H scheme, the VPMD method is still capable of conserving mass accurately 5% within its applicability limit of 1/S 0 y/ x max 0.43 Perumal and Sahoo 2007a. In a recent study by Perumal and Sahoo 2007b, it was inferred that overall the VPMD method performs better than the VPMC method, including the ability of conserving mass. Perumal 1994a,b attributed the volume conservation ability of the VPMD method to its direct derivation from the full Saint-Venant equations, thus retaining the capability of the full Saint-Venant equations to model the nonlinearity process of flood wave movement in an approximate manner. The governing equations of these flood waves from which the VPMD method has been derived are an approximation to the full Saint- Venant equations, and are known as the approximate convectiondiffusion ACD equations. Perumal and Ranga Raju 1999 studied the characteristics of these wave equations in detail. The procedure of varying the parameters of the VPMD method, which have the same form as that of the VPMC method, enables one to account for the nonlinearity of the routing process in a manner consistent with the nonlinearity inherent in the solution of the full Saint-Venant equations. This paper specifically analyzes the capability of the VPMC method and its variants, and the VPMD method in conserving mass. It also brings out the deficiency of the earlier studies on the volume conservation problem of the VPMC method. These studies were made by conducting a total of 6,400 routing experiments, 3,200 each in rectangular and trapezoidal channels. The routing experiments encompass a wide range of channel and flow characteristics encountered in real world river engineering problems. The study is further extended to assess the effect of incorporating the inertial terms in the estimation of Muskingum routing parameters and, hence, on the overall routing performance of the VPMC and VPMD methods. The study also focuses on the problems of negative initial outflow or dip in the beginning of the Muskingum solution, and the negative value of the Muskingum weighting parameter. For a better understanding of the study dealing with the volume conservation problem of the VPMC and VPMD methods, it is considered appropriate that a brief description of these methods be presented herein. VPMC Method By matching the numerical diffusion of the finite difference representation of the kinematic wave with the physical diffusion of Hayami s 1951 diffusion equation, Cunge 1969 developed a physically based Muskingum routing method, subsequently known as the Muskingum Cunge method NERC The method employs the classical Muskingum routing equation advocated by McCarthy 1938, which is expressed as Q k+1 j+1 = C 1 Q k+1 j + C 2 Q k k j + C 3 Q j+1 1 where j spatial index; k temporal index; and K t C 1 = 2a K t K t C 2 = K t 2b K t C 3 = 2c K t The routing parameters K and, denoting the traveltime and weighting parameters, respectively, are given by K = x 3 c = 1 2 Q 4 2S 0 Bc x where t routing time step; x routing space step; Q discharge; S 0 channel bed slope; and B channel top width. The parameters K and of the Muskingum Cunge method remain constant while routing a given flood wave and, therefore, are incapable of capturing the nonlinearity of the flood wave propagation process. To overcome this limitation of the Muskingum Cunge method, Price 1973 and Ponce and Yevjevich 1978 introduced the variable parameter Muskingum Cunge method in which the parameters K and vary at every routing time step in a systematic manner. In the current practice, the variable routing parameters K and are evaluated using the three-point and four-point averaging schemes given by Ponce and Chaganti Further details about the VPMC method can be found in the works of Ponce and Yevjevich 1978, Ponce and Theurer 1982, Ponce and Chaganti 1994, Ponce et al. 1996, and Ponce and Lugo VPMD Method Perumal 1994a,b developed an alternate physically based variable parameter Muskingum discharge hydrograph VPMD routing method derived directly from the Saint-Venant equations, consisting of the continuity and momentum equations, describing the one-dimensional flood wave propagation in rivers and chan- 476 / JOURNAL OF HYDRAULIC ENGINEERING ASCE / APRIL 2008

3 Table 1. Combinations of Channel and Flow Characteristics Used for Experimental Runs Parameters Values Skewness factor, : 1.05, 1.15, 1.25, 1.50 Bed slope, S 0 : , , , , 0.001, 0.002, 0.003, 0.01 Manning s roughness, n : 0.01, 0.02, 0.03, 0.04, 0.05 Initial discharge, Q b m 3 /s : y p m : 5.0, 8.0, 10.0, 12.0, 15.0 Time to peak, t p h : 5.0, 10.0, 15.0, 20.0 Channel bottom width, b m : Fig. 1. Definition sketch of VPMD routing method. Q u, Q M, Q 3, and Q d are discharges at Sections 1, M, 3, and 2, respectively; and corresponding stage variables are y u, y M, y 3, and y d, respectively nels. The variation of the parameters at every routing time interval is consistent with the variation built into the solution of the Saint-Venant equations, accounting for the longitudinal water surface gradient term and the inertial terms in an approximate manner. The model parameters are related to the channel and flow characteristics by the same relationships as established for the VPMC method. The VPMD method is based on the hypothesis that during 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 the reach is uniquely related to the discharge at the same location defining the steady flow rating curve; however, during unsteady flow, the same unique relationship is maintained between the stage and the corresponding steady discharge at any given instant of time, recorded not at the same section, but at a downstream section preceding the corresponding steady stage section midsection of the routing reach. The definition sketch of the routing reach of the variable parameter Muskingum method is shown in Fig. 1. At any instant of time, the dynamic steady discharge section 3 occurs at a distance L from the midsection M of the channel reach given by L = Q 3 2S 0 A c 3 y 3 where Q 3 steady discharge; A/ y 3 top width of the channel at the steady discharge section 3; and c 3 celerity at the steady discharge section 3. The routing parameters K and are evaluated by K = x c 3 = 1 2 L 7 x The VPMD method does not consider the concept of matching the numerical diffusion with the physical diffusion as in case of the VPMC method. However, the same routing Eqs. 1 and 2a 2c of the VPMC method are also used for routing by the VPMD method. Further details of the method with reference to its theoretical basis, development, and field applicability can be found in the studies by Perumal and Ranga Raju 1999, Perumal 1994a, b, and Perumal et al. 2001, respectively. 5 6 Numerical Experiments A total of 6,400 routing experiments based on different combinations of channel characteristics bed slope S 0 ; and Manning s roughness coefficient n, and inflow characteristics peak flow Q p ; time to peak t p ; and skewness factor were conducted for the present study. Two types of channel sections, one with a rectangular channel section having a bottom width of b=100 m and another with a uniform symmetrical trapezoidal channel section with a bottom width of b=100 m and a side slopes of 1.0 vertical to the 1.0 horizontal, were used in the routing experiments. The following input stage hydrograph of the form of the Pearson Type III distribution was used for routing using the Saint-Venant equations y 0,t = y b + y p t y b t 1/ 1 t p exp 1 t/t p 8 1 where y 0,t stage at the upstream end x=0 of the channel reach m ; y b initial steady stage m ; y p peak stage m ; and t time variable. The computed discharge hydrograph at x=0 corresponding to the above input stage hydrograph Eq. 8 was used as the inflow hydrograph for routing using the VPMC and VPMD methods. The initial stage y b is computed corresponding to an initial discharge, Q b of 100 m 3 /s. It may be noted that the estimated input discharge hydrograph corresponding to the input stage hydrograph Eq. 8 is used herein, in lieu of any direct input discharge hydrograph of the form of the Pearson Type III distribution, in order to generate a varied number of discharge hydrographs with different peak flow Q p corresponding to a given peak stage y p. The following upstream and initial conditions were used for each of the hypothetical channel routing cases using the explicit method of the Saint-Venant solutions: The upstream stage hydrograph defined by Eq. 8 formed the upstream boundary condition. A steady flow of 100 m 3 /s was used as the initial flow in the semi-infinite channel reach for all the experimental runs. A total of 6,400 routing experiments were carried out, with 3,200 each in trapezoidal and rectangular cross-section channels, encompassing a varied combination of channel and flow characteristics. Each experiment was characterized by a unique set of channel slope, Manning s roughness coefficient, input shape factor, peak stage, and time to peak. The details of channel configurations used and the pertinent characteristics of input hydrographs employed in these experiments are given in Table 1. For all the experimental run cases, a channel reach length L c of 40 km was considered to be the length of the routing reach. The solutions of the full Saint-Venant equations formed the benchmark solutions for all the experimental runs. The solution of the VPMC method was obtained by using the four-point averaging scheme recommended by Ponce and Chaganti JOURNAL OF HYDRAULIC ENGINEERING ASCE / APRIL 2008 / 477

4 It is well known that the accuracy of the routing method depends on the sizes of the space step x and the time step t used in its numerical solution. Therefore, the use of large size x and t would impact the accuracy of the solution. Since the major objective of the study is to investigate the mass conservation abilities of the VPMC and VPMD methods for routing a given flood wave in a channel reach, it was considered appropriate not to study the accuracy of the VPMC and VPMD methods from the perspective of varying x and t. Hence, a uniform t=15 min and a uniform x=1 km were used in all the experimental runs on the consideration that these intervals are small enough not to affect the accuracy of the VPMC and VPMD solutions. In the present study, the volume conservation feature of a routed outflow hydrograph is evaluated by the percentage error in flow volume, which is expressed as N EVOL = i=1 N Q ci I i i=1 where Q ci =ith ordinate of the routed discharge hydrograph by the routing method; 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 mass. A value close to zero indicates the mass conservation ability of the routing method. Results and Discussion Tang et al. 1999, 2001 correctly pointed out that not accounting for the longitudinal water surface gradient term y/ x in the three-point and the four-point parameter variation schemes of the VPMC method is responsible for the volume loss problem encountered while using the VPMC flood routing method. Through numerical experiments, they demonstrated that the volume loss increases with decrease in bed slope, and the difference in volume loss due to the use of three-point and the four-point schemes is indeed very small, especially when x/l c 0.1, though the fourpoint scheme scores better than the three-point scheme. On the basis of the limited numerical experiments, they concluded that volume loss increases with decrease in bed slope, which implies that no other factor is responsible for volume loss except the bed slope. Using the hypothetical inflow hydrograph studied by Perumal and Ranga Raju 1998 for routing in rectangular channels, Tang et al conducted numerical experiments and based on the results they proposed two empirical equations for calculating the volume loss percentage V% as a single value function of S 0 given by V% =1/ ,102,084.6S 2 0, S , V% = exp 17,895.7S 0 S , for the VPMC4-1 and VPMC4-H methods, respectively, where the reference values of c and Q/c required for the estimation of K and of the VPMC4-1 method are expressed, respectively, as 9 c = c i 4= f Q i 4 12 Q Q i /c i c = 4; i = 1,2,3,4 13 Similarly, the reference values of c and Q required for the estimation of K and of the VPMC4-H method are expressed as c = c i 4= f Q i 4; i = 1,2,3,4 14 x 4 Q = Q 2D Q i 1 15 c Q where the values within brackets, that is, c and Q/c reference values for evaluating the Muskingum routing parameters K and ; c= f Q implies that the celerity; c is a unction of discharge Q; and volume loss adjustment factor, estimated based on the results of the numerical experiments for the volume loss. Contrary to the volume loss expressions of Eqs. 10 and 11 given by Tang et al. 1999, numerical experiments conducted by Perumal 1993 on a similar type of comparative study between the four-point averaging scheme of the VPMC method Ponce and Chaganti 1994 and the multilinear Muskingum discharge MMD routing method Perumal 1992b reveal that the volume loss is also influenced by the roughness of the channel, steepness of the inflow hydrograph, width of the channel, and the initial flow in the channel. Further, it can be determined from Figs. 2 a d, which show the variation of percentage error in volume, EVOL by the VPMC method with the channel bed slope, S 0 in uniform rectangular channels, that the volume loss percentage is not a single value function of S 0. Furthermore, the relationships Eqs. 10 and 11, given by Tang et al. 1999, are developed for rectangular channels and these relationships may not be valid for trapezoidal or any other shape of channels. The routing equation proposed by Tang et al was arrived at by using finite difference scheme and, hence, the routing time interval, t and the space step, x adopted would also influence the volume loss. Therefore, the two empirical formulas Eqs. 10 and 11 developed based on the results of the numerical routing experiments conducted in rectangular channels are not going to serve any useful purpose in practice. Consequently, as Eqs. 10 and 11 only include channel slope, ignoring the roughness values, skewness of the inflow hydrograph, magnitude of the peak and time to peak, and baseflow in the channel, these equations may not capture the appropriate volume loss estimates in all cases. In an attempt to eliminate the volume loss problem of the VPMC method completely, Perumal 1992b developed the MMD routing method. Since this method uses the convolution approach to arrive at the outflow hydrograph, there is no volume loss while routing using this method. Despite the fact that it fully conserves the mass, this method also has a deficiency of using a heuristically estimated empirical constant for determining the reference discharge required for the estimation of model parameters. This constant was arrived at by routing a given inflow hydrograph in a specified channel reach and by matching the routed solution with the corresponding solution of the Saint-Venant equations, considered as the benchmark solutions. This limitation of estimating a calibration constant required for using the MMD routing method prompted the development of the VPMD routing method by Perumal 1994a, b, which does not require the use of any 478 / JOURNAL OF HYDRAULIC ENGINEERING ASCE / APRIL 2008

5 has been directly taken into account using a primary assumption that the variation of flow depth over the routing reach is approximately linear. This method has the inherent ability to model the physical diffusion of a flood wave directly without attributing the diffusion exhibited by it to any numerical scheme as theorized by Cunge Further, the procedure involved in the development of the method is able to identify the applicability criterion of the method for routing a flood hydrograph in streams as 1/S 0 y/ x 1, to be satisfied at the inlet section where the given inflow hydrograph is applied. Figs. 3 a c illustrate the variation of the volume error EVOL with the inflow peak Q p, percentage attenuation in peak discharge by the Saint-Venant solution, and 1/S 0 y/ x max, respectively, for all the experimental runs by the VPMC method; and the corresponding variations for the VPMD method are shown in Figs. 3 d f. The percentage attenuation in peak discharge by the Saint-Venant solution is expressed as Fig. 2. Volume error by VPMC method in rectangular channel with different bed slopes and Manning s roughness, n values for: a =1.05; b =1.15; c =1.25; and d =1.50 empirical constant. This method may be considered as the most appropriate physically based hydraulic method for the interpretation of the Muskingum method, due to its direct derivation from the Saint-Venant equations. While deriving this method, y/ x = 1 Q po Q p where Q po peak of the outflow discharge hydrograph of the Saint-Venant solution. Fig. 3 reveals that the VPMC method always estimates negative EVOL values to a maximum of 40% with many runs estimating less than 20% and, thereby, implying that volume loss occurs while routing using the VPMC method, whereas there is always mass gain while routing using the VPMD method, although the volume error is less than 5% for many of the 6,400 experimental runs with few reaching up to 7%. The variation of EVOL with 1/S 0 y/ x max, as shown in Fig. 3 f corresponding to the VPMD runs, reveals that the maximum of 1/S 0 y/ x max calculated for all the successful runs of 6,400 experiments is nearly 0.45, confirming the applicability limit of the VPMD method in an earlier study by Perumal and Sahoo 2007b. The attenuation corresponding to this maximum value of 1/S 0 y/ x max is estimated to be nearly 50%. Furthermore, Figs. 3 a and b also show that there are no systematic variations of EVOL with Q p and while using the VPMC method. Typical hydrograph reproductions by the VPMC and VPMD methods in reproducing the Saint-Venant solutions in different rectangular channels are shown in Figs. 4 a c. Fig. 4 a shows that the VPMC method is very much prone to volume loss due to its inability to reproduce the rising limb of the Saint-Venant solution, whereas the VPMD method is more mass conservative. However, in Figs. 4 b and c, both of the methods reproduce the benchmark hydrographs reasonably well. Subsequently, the volume loss phenomenon has a direct bearing on the Nash Sutcliffe efficiency estimate Nash and Sutcliffe 1970; ASCE Task Committee 1993 used in assessing the overall reproduction of the benchmark Saint-Venant solution by the solution of the VPMC method. The results of the assessment of the Nash Sutcliffe efficiency q in % for the VPMC method are shown in Fig. 5 a, which ranges as low as 30%. The Nash Sutcliffe efficiency estimate is expressed as N q = 1 i=1 2 2 N Q oi Q ci Q oi Q oi i= where Q oi =ith ordinate of the benchmark discharge hydrograph at the outlet obtained by the Saint-Venant solution; Q oi mean of the discharge hydrograph ordinates at the outlet obtained by the Saint-Venant solution; Q ci =ith ordinate of the discharge hydrograph computed using the VPMC method; and N total num- JOURNAL OF HYDRAULIC ENGINEERING ASCE / APRIL 2008 / 479

6 Fig. 3. Variation of error in volume by VPMC and VPMD methods with a and d Q p ; b and e ; and c and f 1/S 0 y/ x max for all 6,400 experimental run cases ber of discharge hydrograph ordinates to be simulated. Similarly, Fig. 5 b illustrates that the q estimated by the VPMD method is more than 95% for most of the experimental run cases, although it ranges up to 90% for a very few cases. Figs. 6 a and b demonstrate the peak flow reproductions at the outlet of the channel reach by the VPMC and VPMD methods, respectively, in comparison with the solution of the Saint-Venant equations in all the 6,400 experimental run cases. Figs. 6 a and b show that in most of the experimental run cases, the peak outflow is underpredicted by the VPMC method which may be linked with the tendency of this method to lose mass. The reason for the problem of loss of mass in the VPMC method may be attributed to the method of varying the parameters at every routing interval, which although systematic, is not physically based. Therefore, the physically based VPMD method, which is directly derived from the Saint-Venant equations, is more volume conservative. Furthermore, as revealed from Figs. 7 a and b, the VPMC method either underpredicts or overpredicts the time to peak of the routed hydrographs, whereas the VPMD method has the capability to reproduce the time to peak outflow very well. Contrary to the artificial correction suggested by Tang et al to overcome the volume conservation problem of the VPMC method due to nonconsideration of the longitudinal water surface gradient in the diffusivity term of the convection-diffusion equations Hayami 1951, the VPMD method accounts for the variation of y/ x in the parameter estimation in a physically based manner using the ACD equation Perumal and Ranga Raju 1999 wherein the method of variation of the model parameters enables one to account for the nonlinear dynamics of the routing mechanism in a manner consistent with the nonlinearity inherent in the solution of the full Saint-Venant equations. The VPMC method and its variants have been developed based on the extension of the Muskingum Cunge method to vary the parameters K and at every routing time interval in a systematic manner which is, however, not physically based. It should be noted that the relationships of K and used in both the VPMC and VPMD methods are the same. Hence, the volume loss problem of the VPMC method can be attributed to the method of varying the routing parameters which is not physically based. Tang et al showed that the VPMC4-H method definitely reduced the volume loss and estimates improved outflow peak closer to the Saint-Venant solution for the case of S 0 = and n=0.04 see Tang et al. 1999, p However, it is pertinent to note that the estimated flood peak magnitude de- 480 / JOURNAL OF HYDRAULIC ENGINEERING ASCE / APRIL 2008

7 Fig. 5. Nash Sutcliffe efficiency, q estimated by: a VPMC; b VPMD methods for all 6,400 experimental run cases Fig. 4. Hydrograph reproductions by Saint-Venant, VPMC, and VPMD methods in rectangular channel with: a S 0 =0.002, n=0.05, =1.05; b S 0 =0.002, n=0.02, =1.15; and c S 0 =0.0002, n =0.02, =1.25 creases with the decrease of the space step size, indicating that the application of the VPMC4-H method may not be appropriate for this case. The VPMD method Perumal 1994b when applied for this case produced poor results for the cases of single reach, two subreaches, and three subreaches solutions, and the solution could not be obtained when the 40 km reach was divided into four and higher subreaches. Perumal 1994b reasoned that violation of the assumptions used in the derivation of the VPMD method is responsible for the failure of this method when the channel is subdivided into four or more subreaches. The point to be emphasized here is that though the VPMC method and its variations including the VPMC4-H method do not indicate any limitation on the application of the method, in reality, these methods produce very poor results in comparison with the observed one. For the present case of S 0 = and n=0.04 see Tang et al. 1999, p. 617, the Saint-Venant peak estimate is 765 m 3 /s and the corresponding estimate using the VMC4-H method yields a value of 670 m 3 /s resulting in an error of 12.5%, which is clearly unacceptable for a hypothetical routing problem. Therefore, the claim made by Tang et al that the VPMC4-H is applicable for routing a flood hydrograph in a channel with S 0 = is misleading the field engineers about the capability of the method, as for such ranges of channel bed slope, the VPMC4-H would result in poor reproduction of the entire outflow hydrograph and its peak. Problem of Dip in Rising Limb of Hydrograph The problem of leading-edge dip in the routed flood hydrograph has been addressed in much of the past literature. A comprehensive review of the literature was presented by Perumal 1992a who brought out the cause of the formation of negative outflow or dip in the beginning of the Muskingum solution. This cause was, further, reconfirmed by Perumal 1994a, b while studying the VPMD method. Although many remedial measures based on numerical considerations have been suggested to eliminate the defect, the problem is not really solved, but is merely skipped as pointed out by Kundzewicz Moreover, a good numerical system should produce a result which agrees closely with the theoretical result and, therefore, all good numerical approximations to the Muskingum method should also produce the well known reduced output or dip at the beginning of the solution under certain conditions. One can, of course, fudge this by using a bad numerical approximation to the original equations, particu- JOURNAL OF HYDRAULIC ENGINEERING ASCE / APRIL 2008 / 481

8 Fig. 6. Peak outflow reproductions by: a VPMC; b VPMD methods in comparison with Saint-Venant solutions for all experimental run cases Fig. 7. Time to peak outflow reproductions by: a VPMC; b VPMD methods in comparison with Saint-Venant solutions for all experimental run cases larly by choosing a time interval of sufficient length, but surely this is not a proper way to proceed. Certainly, one should not attempt to correct the deficiencies in the theoretical equations by a further error in the numerical approximation even if by doing so one avoids the undesirable negative ordinates J. E. Nash, personal communication, However, it has been demonstrated by Perumal 1994a, b that the problem of initial dip is reduced or eliminated when a given routing reach is divided into many smaller subreaches for the application of the VPMD method. Negative Weighting Parameter Regarding negative weighting parameter values in the Muskingum routing method, it was shown by Perumal 1994b, based on theoretical consideration, that the value becomes negative when the outlet section of the Muskingum reach is located well upstream of the steady discharge section 3 where the normal discharge corresponding to the flow depth at the middle of the reach passes see Fig. 1, that is, when the routing reach length is small. The results obtained by Szilagyi 1992 using field data corroborates this inference. Therefore, truncating the negative values is not consistent with the theoretical basis of the Muskingum method and it will have perceptible effect on the shape of the hydrograph as concluded by Tang et al based on numerical experiments. Effect of Inertial Terms on Volume Conservation Ponce and Lugo 2001 extended the VPMC model to the realm of looped dynamic ratings by reformulating the Muskingum Cunge algorithm to use the local water surface slope and the Vedernikov number in the expression for hydraulic diffusivity. Ponce and Lugo 2001 claimed that this makes possible the simulation of looped ratings, which more closely resemble the actual flood wave propagation, thereby increasing the accuracy of the routed flood hydrograph. Since in most natural rivers, the inertial terms in the momentum equation are negligible Henderson 1966; Price 1985, this argument is not convincing. To test the validity of this argument another 6,400 experimental runs, 3,200 each in rectangular and trapezoidal channels, with different channel and flow characteristics as given in Table 1 were conducted by considering the inertial terms in the framework of the VPMC and VPMD methods, which are termed, hereafter, as the VPMCFR and VPMDFR methods, respectively. Fig. 8 a shows the percentage error in volume by the VPMC and VPMCFR methods for all the 6,400 experimental run cases. Similarly, Fig. 8 b shows the percentage error in volume estimate by the VPMD and VPMDFR methods for all the 6,400 experimental run cases. It can be evidenced from Figs. 8 a and b that there is no significant impact of the inertial terms on the volume conservation capability of the VPMCFR and VPMDFR routing methods. Figs. 9 a and b illustrate the peak outflow reproductions by the VPMCFR and VPM- DFR methods, respectively, in comparison with the solutions of the VPMC and VPMD methods. Similarly, Figs. 10 a and b il- 482 / JOURNAL OF HYDRAULIC ENGINEERING ASCE / APRIL 2008

9 Fig. 8. Error in volume estimates by: a VPMC and VPMCFR; b VPMD and VPMDFR methods for all experimental run cases lustrate the time to peak outflow reproductions by the VPMCFR and VPMDFR methods, respectively, in comparison with the solutions of the Saint-Venant method. The comparison of Fig. 9 a with Fig. 6 a shows that with the consideration of the inertial terms in the framework of the original VPMC method, namely the VPMCFR method, most of the peak outflow reproductions deviate from the 1:1 line in the same pattern as that by the VPMC method, except in a few experimental run cases with high peak discharges and =1.05. By comparing Fig. 10 a with Fig. 7 a, a similar inference can be obtained for the time to peak reproductions by the VPMCFR method. However, as inferred from Figs. 9 b and 10 b, there is no or little impact of the inertial terms on the routed peak and time to routed peak reproductions of the VPMD method. Consequently, the consideration of the inertial terms, which are negligible as far as the real world river routings are concerned, does not affect the volume conservation, peak outflow, and time to peak outflow reproductions of the VPMC method. Hence, these routing experiments do not substantiate the claim made by Ponce and Lugo 2001 that consideration of the inertial terms in the estimation of K and improves the accuracy of the VPMC solution. Conclusions The problem of volume conservation in the widely used VPMC flood routing method is studied. The study highlights the limitation of the recent attempts to overcome this problem by either modifying the form of the Muskingum parameter relationships or Fig. 9. Peak outflow reproductions by: a VPMC and VPMCFR; b VPMD and VPMDFR methods for all experimental run cases explicitly incorporating the inertial terms in these relationships. The VPMC routing results were compared with an alternate physically based variable parameter Muskingum method, namely the VPMD method, which shows that the latter routing method is more volume conservative than the former method. Conclusively, the volume loss problem of the VPMC method is linked to the method of varying the parameters which, although it is systematic, is not physically based. This deficiency is overcome by the VPMD method as demonstrated by its routing results. The VPMD method is directly derived from the full Saint-Venant equations, without involving the use of the concept of matching the numerical diffusion with the physical diffusion. Acknowledgments The writers gratefully acknowledge the valuable comments offered by the anonymous reviewers in improving the technical contents of this paper. Notation The following symbols are used in this paper: A cross-sectional area of flowing water L 2 ; B channel top width L ; b channel bottom width L ; c wave celerity LT 1 ; JOURNAL OF HYDRAULIC ENGINEERING ASCE / APRIL 2008 / 483

10 y b initial stage corresponding to initial steady discharge, Q b L ; y p peak stage corresponding to peak inflow, Q p L ; t routing time step T ; x space step L ; q Nash Sutcliffe efficiency - ; spatial weighting coefficient of Muskingum routing - ; and percentage attenuation in peak discharge by the Saint-Venant solution -. References Fig. 10. Time to peak outflow reproductions by: a VPMC and VPMCFR; b VPMD and VPMDFR methods for all experimental run cases c 3 wave celerity at steady discharge section 3 of Fig. 1 LT 1 ; D diffusion coefficient of flood wave LT 2 ; EVOL percentage error in volume - ; I i ordinate of inflow hydrograph at ith time step L 3 T 1 ; K travel time factor of Muskingum equation T ; L dynamic distance between midsection and normal discharge section of routing subreach for discharge routing by VPMD method L ; L c total length of routing channel reach L ; N total number of ordinates of hydrograph - ; n Manning s roughness coefficient - ; Q inflow discharge at any instant of time L 3 T 1 ; Q b initial steady discharge L 3 T 1 ; Q ci ordinate of outflow hydrograph at ith time step L 3 T 1 ; Q p peak inflow L 3 T 1 ; Q po peak of outflow discharge hydrograph of Saint-Venant solution L 3 T 1 ; Q 3 steady discharge at section 3 of Fig. 1 L 3 T 1 ; S 0 channel bed slope - ; t time T ; t p time to peak of input hydrograph T ; V% volume loss/gain percentage of outflow by Tang et al ; x longitudinal space vector L ; ASCE Task Committee on Definition of Criteria for Evaluation of Watershed Models of the Watershed Management Committee, Irrigation and Drainage Division Criteria for evaluation of watershed models. J. Irrig. Drain. Eng., 119 3, Cappelaere, B Accurate diffusive wave routing. J. Hydraul. Eng., 123 3, Cunge, J. A On the subject of the flood propagation computation method Muskingum method. J. Hydraul. Res., 7 2, Cunge, J. A Volume conservation in variable parameter Muskingum-Cunge method: Discussion. J. Hydraul. Eng., 127 3, 239. Ferrick, M. G Analysis of river waves types. Water Resour. Res., 21 2, Hayami, S On the propagation of flood waves. Bull. Disaster Prevention Res. Inst., Kyoto, Japan, 1 1, Henderson, F. M Flood routing. Open channel flow, Chap. 9, Macmillan, New York. Kundzewicz, Z. W Approximate flood routing methods: A review Discussion. J. Hydr. Div., , McCarthy, G. T The unit hydrograph and flood routing. Proc., Conf. North Atlantic Division, U.S. Army Corps of Engineers, New London, Conn. Nash, J. E., and Sutcliffe, J. V River flow forecasting through conceptual models. Part I A Discussion of principles. J. Hydrol., 10 3, Natural Environment Research Council NERC Flood studies report. Flood routing studies, Vol. III, London. Perumal, M. 1992a. Cause of negative initial outflow with the Muskingum Method. Hydrol. Sci. J., 37 4, Perumal, M. 1992b. Multilinear Muskingum flood routing method. J. Hydrol., , Perumal, M Comparison of two variable parameter Muskingum methods. Proc., Symp. on Extreme Hydrological Events, Precipitation, Floods and Droughts, International Association of Hydrological Sciences, IAHS Publication No. 213, Yokohama, Japan, Perumal, M. 1994a. Hydrodynamic derivation of a variable parameter Muskingum method: 1. Theory and solution procedure. Hydrol. Sci. J., 39 5, Perumal, M. 1994b. Hydrodynamic derivation of a variable parameter Muskingum method: 2. Verification. Hydrol. Sci. J., 39 5, Perumal, M., O Connell, P. E., and Ranga Raju, K. G Field applications of a variable parameter Muskingum method. J. Hydrol. Eng., 6 3, Perumal, M., and Ranga Raju, K. G Variable parameter stagehydrograph routing method: II. Evaluation. J. Hydrol. Eng., 3 2, Perumal, M., and Ranga Raju, K. G Approximate convectiondiffusion equations. J. Hydrol. Eng., 4 2, Perumal, M., and Sahoo, B. 2007a. A critical evaluation of two variable parameter Muskingum routing methods. Proc., 18th Int. Association of Science and Technology for Development (IASTED) Conf. on Modeling and Simulation (MS 2007), R. Wamkeue, ed., ACTA Press, Montreal, Quebec, Canada, / JOURNAL OF HYDRAULIC ENGINEERING ASCE / APRIL 2008

11 Perumal, M., and Sahoo, B. 2007b. Applicability criteria of the variable parameter Muskingum stage and discharge routing methods. Water Resour. Res., 43 5, W05409, Ponce, V. M Accuracy of physically based coefficient method of flood routing. Tech. Rep. SDSU Civil Engineering Series No , San Diego State Univ., San Diego. Ponce, V. M., and Chaganti, P. V Variable-parameter Muskingum-Cunge method revisited. J. Hydrol., , Ponce, V. M., Lohani, A. K., and Scheyhing, C Analytical verification of Muskingum-Cunge routing. J. Hydrol., , Ponce, V. M., and Lugo, A Modeling looped ratings in Muskingum Cunge routing. J. Hydrol. Eng., 6 2, Ponce, V. M., and Theurer, F. D Accuracy criteria in diffusion routing. J. Hydr. Div., 108 6, Ponce, V. M., and Yevjevich, V Muskingum Cunge method with variable parameters. J. Hydr. Div., , Price, R. K. 1973, Flood routing methods for British rivers. Proc. Inst. Civ. Eng., Waters. Maritime Energ., 55 12, Price, R. K Flood routing. Developments in hydraulic engineering, P. Novak, ed., Vol. 3, Elsevier, New York, Szilagyi, J Why can the weighing parameter of the Muskingum channel routing method be negative. J. Hydrol., , Tang, X.-N., Knight, D.-W., and Samuels, P. G Volume conservation in variable parameter Muskingum Cunge method. J. Hydraul. Eng., 125 6, Tang, X.-N., Knight, D.-W., and Samuels, P. G Volume conservation in variable parameter Muskingum Cunge method: Closure. J. Hydraul. Eng., 127 3, JOURNAL OF HYDRAULIC ENGINEERING ASCE / APRIL 2008 / 485