A PRACTICALLY VIABLE SIMPLIFIED HYDRODYNAMIC STAGE-HYDROGRAPH ROUTING METHOD

Abstract

The simplified flood routing methods such as the Muskingum method and its variants are well established in the hydrological literature [Chow et al., 1988], and the modest data requirements of these methods make them suitable for practical applications. Generally routing methods employ discharge as the routing variable. But for flood forecasting purposes and in conditions where the civic authorities are involved in the evacuation of people from flood inundation areas, it is necessary to know the stage of the stream rather than discharge for planning emergency evacuation action plans. Since, the stage information can be collected more easily and economically, sometimes it is advantageous to route a stage hydrograph rather than a discharge hydrograph to know the stage or flow depth along the channel reach at the point of interest. Hence, this study emphasizes on the continued interest in improving the stage-hydrograph routing, especially, in the form of the classical Muskingum method. Over the past decades, a number of simplified discharge flood routing methods have been developed. However, in the literature, only a few researchers, such as Hayami [1951] and Perumal et al. [2007, 2010] have developed the simplified stage-hydrograph flood routing methods. The added advantage of the latter method is that it enables also to estimate the discharge hydrograph corresponding to the routed stage hydrograph, a feature generally not available in simplified routing methods. Perumal et al. [2007] developed the VPMS routing method by directly applying the approximate convection-diffusion (ACD) equation in stage formulation [Perumal and Ranga Raju, 1998], using the finite difference approximations of mixed order of accuracy, which resulted in significant volume error in the solutions of the VPMS method [Perumal et al., 2007,2010]. To overcome this problem, the present study investigates the development of an improved variable parameter Muskingum stage-hydrograph routing ((henceforth, identified with the acronym IVPMS) method by revising the current form of the VPMS method advocated by Perumal et al. [2007, 2010]. In this study, the ACD equation is applied along the mid-time level within the computational grid bounded at the top and bottom by two consecutive computational time levels, and bounded along the spatial direction at the two ends of the computational reach length & by the input and output sections. Such a finite difference grid arrangement enables the application of the governing ACD partial differential equation at the mid-point of the grid using the second order accurate finite difference equation. The parameters of the method vary at every computational time interval of the routing process as adopted in the earlier VPMS method, but with changes in the structure of the parameter relationships and their applications in the routing process at every time-step. The VPMS method developed by Perumal et al. [2007, 2010] uses the prismatic cross-section details along with single Manning's roughness coefficient, n, for routing a given stage-hydrograph. Such an approach is appropriate when routing stage-hydrograph in artificial channels like irrigation canals. However, it is difficult to apply such a method for routing stage-hydrographs in natural rivers as it requires first to approximate the river reach to that of a nearest prismatic section reach as required by the method and then subsequently apply this method for routing. Hence, it is considered necessary to improve the VPMS routing method for routing floods in natural rivers by using only the rating curves information available at the two end sections of a given routing reach along with the information of the associated cross-section geometries available at these end sections. The study is undertaken with the following objectives: 1) To replace the current finite difference scheme employed in the representation of the Approximate Convection-Diffusion equation used in the existing VPMS routing method by the second order finite difference scheme representation of the spatial and temporal derivatives leading to the development of the improved VPMS method; 2) To study the celerity-stage relationship of river flood wave required for field application of the IVPMS method; 3) To enhance the field applicability of the IVPMS routing method by replacing the direct use of the reach cross-section information for routing with the rating curves information available at the end sections and the information of the associated cross-section geometries of these respective gauging stations along a river reach; and 4) To develop the applicability criterion for the IVPMS routing method. The capability of the IVPMS method is demonstrated by routing the given hypothetical input stage hydrographs in different types of hypothetical prismatic channel reaches each characterised by different sets of uniform roughness and bed slopes as applied in the study of VPMS method by Perumal et al. [2007,2010]. The routing was carried out for a -4 specified reach length in each of the hypothetical channels using the IVPMS method, and the routed stage hydrographs and the estimated discharge hydrographs arrived at the end of the reach using the IVPMS and the VPMS method were compared with the corresponding benchmark hydrographs obtained using the MIKE 11 model [DHJ, 2008]. The study demonstrates that the hypothetical routing results of the IVPMS method closely reproduce the benchmark solutions of the stage-hydrographs. However, when the longitudinal gradient of the water depth I(1/So) (oy/ax)Imax >0.4 estimated at the inlet of the routing reach, the discharge hydrograph estimated using the routed stage hydrograph of the IVPMS routing method is not able to reproduce the benchmark discharge hydrographs better than the corresponding estimated discharge hydrographs obtained using the VPMS routing method. Besides, it is seen that the mass conservation capability of the method is far better than that of the earlier version of the VPMS method with most of the simulations estimating less than one percent mass conservation error. In general, simplified flood routing methods employ two routing parameters, viz., the travel time which accounts for the convection dynamics of the flood wave, and a parameter accounting for its diffusion characteristics. The parameter characterizing the travel time is directly linked to the wave celerity, i.e., the velocity with which a given stage or discharge propagates along the given routing reach under unsteady flow condition. The celerity- discharge relationship of natural rivers was studied by Price [1975], and Wong and Laurenson [1983, 1984] carried out for UK and Australian rivers, respectively. As no celerity-stage relationship has been studied earlier, the same has been developed on the similar lines as investigated by Wong and Laurenson [1983, 1984] for Australian rivers. The purpose of developing such a celerity-stage relationship for a given routing reach is for its use in hypothetical and practical flood routing studies using the IVPMS routing method. The study based on the hypothetical routing results of the IVPMS method demonstrates that the routing procedure developed based on the IVPMS method using only the reach end sections rating curves and associated cross-section geometries information are capable of closely reproducing the benchmark solutions of the stage-hydrographs. The IVPMS routing method is able to reproduce the benchmark discharge hydrographs closely when I(1/So)(ay/ax)Imax estimated at the inlet of the reach is less than 0.4. For such a verification, two different types of hypothetical prismatic channel reaches, such as: (i) Ackers' [1993] hypothetical river channels consisting of two-step compound trapezoidal section with single and multiple roughness values at a section and (ii) hypothetical channels of Price [2009] with a given Maiming's roughness for the main channel and a different Maiming's roughness for the floodplain section of the channel were employed. The practical usefulness of this method is also demonstrated by routing different flood events in two channel reaches of the Upper Tiber River in Central Italy for river reach lengths of 15 km and 50 km. It is pertinent to point out herein that the applicability of the IVPMS method is restricted by the assumptions of no lateral flow and no effect of downstream disturbances in the study reach. The routed stage hydrographs arrived at the end of these two channel reaches using the IVPMS routing method and the corresponding estimated discharge hydrographs reproduce the respective observed hydrographs more closely for many of the events better than the respective reproductions obtained using the VPMS method [Perumal et al., 2010]. This in essence clearly confirms the efficacy of the IVPMS routing method over that of the VPMS routing method studied earlier by Perumal et al. [2007, 2010]. The applicability criterion of the VPMS method was established by Perumal and Sahoo [2007] based on the limit of longitudinal water surface gradient obtained using a number of hypothetical routing experiments and was compared with the corresponding benchmark solutions obtained using the solution of the Saint-Venant equations. In the same manner, the applicability criteria of the IVPMS routing method are also assessed and quantified based on the magnitudes Of(1/So)(ÔY/ÔX)max. Considering a 95% level of model performance, the experiments indicate that the applicability limit of the IVPMS method for stage routing is (1/So)(ay/ax),,,l .0 ; while for the simultaneous computation of the discharge hydrograph corresponding to the routed stage hydrograph, this method can be applied up to (1/So)(3y/3x),0.86, which reveals a significant improvement of the applicability criteria over the VPMS method. The IVPMS routing method is advantageous over all the simplified flood routing methods for its capability to route the stage hydrograph and enable the estimation of concurrent discharge hydrograph by closely reproducing the benchmark solutions. Also the method has better mass conservative ability than the VPMS method and, thus, can be used by the river engineers and field hydrologists in practice. The study reveals that this simplified hydraulic routing method has ample scope for its extension for meso-scale river basin modelling, especially, as a component module of the land-surface scheme of the climate change models.