Variable Parameter Flood Routing Methods for Hydrological Analyses of Ungauged Basins
by
Dr. Bhabagrahi Sahoo
1. Introduction
Solving the problem of predictions in ungauged basins is one of the biggest contemporary challenges in hydrological science [Zehe et al., 2006]. However, the predictions in ungauged or poorly gauged basins for water resources planning and also the predictions of global climate change impact even in gauged basins seem hardly possible with the kind of predictive tools or models that we currently have which rely heavily on calibration [Sivapalan et al., 2003]. Modeling of all the hydrological component processes are subjected to the same calibration problem. Among these component processes, flood routing is an important process which forms the main focus of this study. Flood routing problems are solved using the hydraulic and hydrological methods. However, the hydraulic-based methods, which generally use the full Saint-Venant equations, are of limited use primarily due to non-availability of topological and hydrological inputs required at smaller spatial scales and also due to computational limitations of the numerical schemes adopted in the solution procedure. An alternative way of overcoming this data and computational problem is by using the simplified methods which are derived from the Saint-Venant equations, but at the same time they are not data intensive. Many researchers [e.g., NERC, 1975] argued in favor of using the simplified flood routing models. Ferrick [1985] showed that numerical problems arise while solving the full Saint-Venant equations when the magnitudes of different terms of the Saint-Venant’s momentum equation are widely varying; and in such a case, use of simplified flood routing methods is imminent.
Simplified flood routing methods such as the Muskingum method and its variants have been successfully used for routing flood waves in basin-scale and continental-scale models [Richey et al., 1989; Elias and Cavalcante, 1983; Arora et al., 1999; Zhao and Liu, 1995; Nawarathna et al., 2005]. Hence, the Muskingum flood routing method has the immense potential for meso- and macro-scale field applications, and it is particularly well suited to applications where a combination of physically based parameters and computational efficiency are required. Real-time flood forecasting is one of the field applications where all the attributes of this routing method are desirable [e.g., Moramarco and Singh, 2001]. In the distributed physically based modeling of continental-scale river basins, where computational efficiency is again of major importance if such models are to be linked to the general circulation models (GCMs), the variable parameter Muskingum (VPM) flood routing methods [Cunge, 1969; Price, 1973; Ponce and Chaganti, 1994; Perumal, 1994a, 1994b; Perumal and Ranga Raju, 1998a, 1998b] have an immense potential. The variable parameter Muskingum discharge hydrograph (VPMD) [Perumal, 1994a, 1994b] and the variable parameter Muskingum stage hydrograph (VPMS) [Perumal and Ranga Raju, 1998a, 1998b] routing methods can also be suitably coupled with the hydrological land-surface schemes of the climate change models as the flood routing component models. An added advantage of using the VPMD and VPMS routing methods is that both stage and discharge variables can be computed simultaneously at any river section similar to the Saint-Venant method and, hence, these methods can be suitably used to develop rating curves at any ungauged river section.
However, simplified flood routing methods such as the Muskingum method are considered to be unsuitable for applications when the flood inundates the floodplain within the routing reach [Chow et al., 1988]. An approach to extend the applicability of the VPM method could be accounting for the floodplain flow explicitly in the model framework of the VPM method. Apart from the VPM methods, the multilinear Muskingum flood routing methods, a variant of the variable parameter based methods which use the convolution approach in their model structures, can suitably be used for flood forecasting in ungauged river basins.
The river runoff in the GCMs, even on an annual basis, is predicted relatively poorly for the world’s major river basins with large uncertainties and discrepancies. In some cases, this is partly due to problems in predicting rainfall; but, in many other cases, it is the land-surface scheme which may be at fault [Naden et al., 1999]. Over the decades, although, a number of simplified flood routing methods have been developed, the variable parameter Muskingum-Cunge (VPMC) method [Price, 1973; Ponce and Yevjevich, 1978; Ponce and Chaganti, 1994] is widely used in practical flood hydrology problems. However, Heatherman [2004] has pointed out that more insight in the Muskingum method can be gained using the interpretation proposed by Perumal [1994a, 1994b] rather than that given by Cunge [1969]. Further, the VPMC method is saddled with the volume conservation problem [Tang et al., 1999]. Although the theories proposed by Cunge [1969] and Perumal [1994a, 1994b] attempt to give physical interpretation to the classical Muskingum routing method, the interpretation given by the latter is able to explain all the features of the classical Muskingum method including the formation of the well-known “initial dip” in the beginning of the Muskingum solution.
In light of the above discussion, the scope of investigating the variable parameter flood routing methods for hydrological analyses of ungauged basins can be briefly summarized as below.
a) The flood routing methods used in the conventional hydrological land-surface schemes are mostly in linear modes which are not physically based. The use of inappropriate flood routing model structure is one of the main causes of poor performance of the climate change models based on the output of the GCMs.
b) The full Saint-Venant equations cannot be used for flood routing studies in ungauged basins as well as in meso- and macro-scale basin modeling due to the computational problem and large data requirement.
c) The classical Muskingum method and its improved variants including the VPMC method have been used in the land-surface schemes and basin-scale models for flood routing studies. However, the VPMC method has the limitations such as inability to appropriately model the nonlinearity in the routing process resulting in volume conservation problem and inability to explain all the features of the classical Muskingum method. Hence, there is a need to thoroughly investigate this method for its applicability and limitations.
d) The currently available simplified methods only route discharge downstream and they do not have the feature to simultaneously compute the stage variable corresponding to the routed discharge. However, the physically based VPMD and VPMS methods route discharge and stage downstream, respectively, and they compute the corresponding stage or discharge at any river cross section similar to the Saint-Venant solutions. Subsequently, the question that arises is whether the VPMD and VPMS methods have better routing capabilities than the well-known VPMC method?
e) The multilinear Muskingum flood routing methods can suitably be used for hydrological analyses of ungauged basins.
f) Study of the literature shows that the Muskingum method is not suitable for over-bank flood routing studies. Hence, the VPMD, VPMS, and multilinear Muskingum methods could be investigated by extending these routing methods to the floodplain flow condition.
2. Objectives
Taking into consideration some of the above perspectives of simplified flood routing methods, it is proposed to focus attention on the study of extension of the physically based variable parameter Muskingum discharge and stage routing methods [Perumal, 1994a, 1994b; Perumal and Ranga Raju, 1998a, 1998b] suitable for performing hydrological analyses of ungauged or poorly gauged and large-scale basin problems. The basic aim of this study is to develop and verify different simplified VPM and multilinear Muskingum flood routing methods in stage and discharge formulations explicitly accounting for the floodplain channel flow in the model structure. These developed methods should have the capability to realistically simulate streamflow at various ungauged locations of the river reach. The different objectives of this study are:
1) Study of the limitations of the existing simplified flood routing methods, exclusively the VPMC and VPMD methods;
2) Development of an applicability criterion for the variable parameter Muskingum flood routing methods;
3) Extension of the physically based variable parameter Muskingum flood routing methods to the realm of floodplain flow;
4) Development of multilinear Muskingum flood routing methods accounting for the floodplain flow; and
5) Comparative study of these developed simplified flood routing methods.
3. Reanalysis of the VPMC and VPMD Routing Methods
Corresponding to the first objective, this study analyzes the volume conservation problem of the VPMC method for which some remedial solutions have been advocated in recent literature [Tang et al., 1999]. The limitation of the VPMC method to conserve volume is brought out by conducting a total of 6400 routing experiments in uniform rectangular and trapezoidal channels wherein the routed solutions by the VPMC method are compared with the corresponding benchmark solutions obtained using the full Saint-Venant equations. A parallel study was carried out using the VPMD routing method 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 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 as done by Tang et al. [1999] to overcome the volume conservation problem makes the VPMC method semi-empirical in nature, thereby, loosing the fully physically based characteristics of the method. The study also dwells on the negative value of the Muskingum weighting parameter. Besides, the effect of incorporating the inertial terms in the estimation of Muskingum routing parameters and their impact on the overall Muskingum routing solutions is addressed by conducting another set of 6400 numerical experiments using both the VPMC and VPMD methods.
4. Development of Applicability Criteria of the Muskingum Flood Routing Methods
Corresponding to the second objective, it is proved that the widely used applicability criteria developed by Ponce et al. [1978] [e.g., Singh, 1996] cannot be applicable for nonlinear flood waves. Subsequently, the applicability criteria of the VPMS and VPMD routing methods are assessed and quantified using a new applicability criterion (1/So)(∂y/∂x)max. 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 full Saint-Venant equations, which form the benchmark model. Considering a 95% level of model performance, the experiments indicate that the applicability limit of the VPMS method for stage routing is (1/So)(∂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/So)(∂y/∂x)max ≤ 0.63 (where So = channel slope and ∂y/∂x = longitudinal water surface gradient). The VPMD method is applicable up to (1/So)(∂y/∂x)max ≤ 0.43 for both discharge routing and the corresponding stage computation. The applicability of the VPMD method is compared with the VPMC method and it is found that for the same 95% level of model performance of the VPMD method, the VPMC method is applicable only up to (1/So)(∂y/∂x)max ≤ 0.11. Thus it is seen that the VPMD method has an improved performance and wider applicability range than the VPMC method. Hence, the VPMC method is not considered herein for its possible extension to the floodplain flows.
5. Extension of the VPMD and VPMS Flood Routing Methods for Floodplain Flows
Corresponding to the third objective, based on the extension of the VPMD routing method [Perumal, 1994a, 1994b], a routing method for discharge computation in channels with floodplains is proposed herein. The upstream discharge hydrograph is routed using this extended VPMD method in different two-stage symmetrical trapezoidal compound cross section channel reaches, each having different size of floodplains. The stage hydrograph corresponding to the routed discharge hydrograph is also estimated by the extended VPMD method. The performance of the proposed VPMD method extension is evaluated by conducting 72 systematically planned numerical experiments and comparing the results with the routing results obtained using the MIKE-11 hydraulic model, which is used as the benchmark model in this study. Further, the suitability of the proposed extended VPMD routing method is verified using six sets of field data of the Tiber River in central Italy. The results reveal that the proposed method is capable of accurately routing the discharge hydrographs, and for establishing the rating curves at downstream ungauged river sites which are not affected by any downstream effects. Furthermore, based on the Tiber River data, the form of the wave speed-discharge relationship developed using the extended VPMD method is found to be in compliance with the earlier studies by Wong and Laurenson [1983, 1984] and Tang et al. [2001].
Similarly, another methodology for estimating discharges and development of rating curves at ungauged river sites is presented by employing a routing method, which is an extension of the VPMS routing method developed by Perumal and Ranga Raju [1998a, 1998b], for routing a given upstream stage hydrograph in a channel reach characterized by trapezoidal compound cross section to arrive at the stage hydrograph at the downstream site. Further, the VPMS method enables one to estimate the discharge hydrographs at the upstream and downstream sites. It is assumed that there is no lateral flow within the routing reach. The proposed approach of developing the rating curve using the extended VPMS method is verified for a number of hypothetical data sets as used for the VPMD method above. Furthermore, the appropriateness of the proposed extended VPMS routing method is verified using two sets of experimental data on unsteady flows, obtained from a laboratory channel with rectangular compound flow section. The methodology is also field tested using six sets of concurrent stage hydrograph data obtained at the upstream and downstream sites of a 15 km reach length of the Tiber River in central Italy, out of which only one set of data was used for calibrating the reach-averaged Manning’s roughness coefficient. The close reproductions of the rating curves and discharge hydrographs recorded at the upstream and downstream sites demonstrate that the proposed methodology can be confidently used for rating curve development and discharge estimation at ungauged sites, thus avoiding the manual discharge measurement at any river site not subjected to backwater effects.
6. Multilinear Muskingum Flood Routing Methods for Floodplain Flows
Corresponding to the fourth objective, two alternative variable parameter hydrograph routing methods based on the improved time-distribution scheme of the multilinear modeling approach, namely the multilinear Muskingum discharge hydrograph (MMD) and multilinear Muskingum stage hydrograph (MMS) routing methods operating on the discharge and stage variables, respectively, are investigated for studying the flood wave propagation process in uniform rectangular and trapezoidal channels considering floodplain flows. In both the extended MMD and the MMS routing methods, the Muskingum-type method is used as the sub-model of the multilinear method, wherein the parameters of the Muskingum sub-model are related to the channel and flow characteristics by adopting the same relationships as established for the corresponding parameters of the VPMD and VPMS routing methods, respectively. The routing study reveals that the proposed MMD and MMS routing methods reproduce their respective benchmark Saint-Venant solutions closely when the rating curve corresponding to the input discharge or stage hydrograph is characterized by a narrow loop. The routing capability of the extended MMD method is studied by using the field data of the Tiber River in central Italy; and the MMS method is used to simulate a laboratory compound channel stage routing experiment and the field data of the Tiber River.
7. Conclusions
The comparative study of the four different variable parameter flood routing methods developed in this study reveal that the VPMD and VPMS routing methods are more amenable for flood routing in both diagnostic and prognostic modes for hydrological analyses of ungauged basins. An added advantage of these two routing methods is their capability to establish the normal rating curve and celerity-discharge relationship at any ungauged downstream river site. The VPMS and VPMD methods have a wide applicability limit and, hence, can be used in an extensive range of flow conditions between the dynamic wave and the kinematic wave, including the latter. With the drastic decline in in-situ streamflow gauging networks day-by-day in the real world river basins due to high operational and maintenance costs, the VPMD and VPMS routing methods are of special relevance as far as the predictions in ungauged basins is concerned. The MMD and MMS routing methods can only suitably be used in prognostic modes for hydrological analyses of ungauged basins. Each of these three-parameter-based MMD and MMS routing methods has a parsimonious model structure in which only one parameter is being calibrated and the other two parameters are computed based on the physically measurable channel and flow characteristics. Simplicity in the model structures of the stage and discharge routing methods are maintained between the VPMD-VPMS and MMD-MMS routing methods by using the same model parameters and similar model frameworks. The developed simplified routing methods also hold a great promise for their use in the meso- and macro-scale basin modeling. The other applications involve the automatic canal operation, rainfall–runoff modeling, and climate change impact assessment in the domains of general circulation models (GCMs) and regional climate models (RCMs).
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