OVERLAND FLOW MODELLING USING APPROXIMATE CONVECTION-DIFFUSION EQUATIONS

Abstract

This study deals with the development of variable parameter Muskingum overland flow routing methods as an extension of the variable parameter Muskingum discharge hydrograph (VPMD) river routing method advocated by Perumal in 1994 and the variable parameter Muskingum stage hydrograph (VPMS) river routing method developed by Perumal and Ranga Raju in 1998, after duly accounting for the occurrence of uniform rainfall over the plane. Both these VPMD and VPMS overland flow routing methods, like the corresponding channel routing methods, have been developed from the approximate convection-diffusion (ACD) equations in discharge and flow depth formulations, respectively, which are directly derived from the Saint-Venant equations (SVE). The routing parameters ofthese methods expressed in terms offlow and plane characteristics are varied by accounting for the longitudinal gradient of the water depth in their relationships in a way consistent with the variation built into the solutions of the SVE. The Hortonian overland flow modelling is accomplished by coupling the Green-Ampt (GA) infiltration model with the VPMD and VPMS overland flow routing model frameworks. All these methods are studied herein for their strengths and weaknesses in reproducing the benchmark solutions of the full SVE and its variants. The present study is conducted with the following four objectives: 1) To extend the VPMD channel routing method for overland flow modelling studies and to compare its performance with those of other currently used methods; 2) To extend the VPMS channel routing method for overland flow modelling studies and to compare its performance with those ofother currently used methods, and with that of the corresponding VPMD overland flow routing method; 3) To develop the applicability criteria for the above methods; and 4) To apply the above methods for Hortonian overland flow generation and to compare the performance of these methods with those of other currently used methods. The VPMD overland flow routing method is also applied as a component model for runoff generation from a level V-catchment wherein the channel routing is accomplished using the VPMD channel routing method. The interception loss required for runoff computation ofthe V-catchment is accounted using a </> -index type method. The operational performance evaluation of the VPMD method is extensively carried out using the hypothetical overland flow data available in literature, besides the hypothetical numerical solutions obtained from the use of SVE, the laboratory and field overland flow data. Further, the VPMD solutions are evaluated by comparing with the corresponding analytical solutions of the kinematic wave (KW) equation. The numerical study of the VPMD method reveals that it is not necessary to strictly follow the Courant condition Cun * 1 as used in the conventional solution methods for preserving the numerical stability. However, to preserve the solution accuracy and efficient mass conservation (EVOL« 5%), the Courant condition may vary in the range 0.1 < Cun < 10. The proposed VPMD method is found to be advantageous over the currently available numerical overland flow simulation methods because of its unconditional numerical stability, high accuracy level, and higher degree of flexibility in the selection of the computational spatial and temporal grid sizes; thus, making it amenable for coupling with various land surface schemes (LSSs) available for meso and micro-scale catchment modelling studies for assessing the impact of land use and climate changes on catchment runoff. The operational performance of the VPMS method is also extensively evaluated using the same data set as used for evaluating the VPMD overland flow routing method. The efficacy of the VPMS solutions are compared with the corresponding VPMD solutions to verify the merits and demerits of using flow depth as an operating variable in the overland flow models in lieu of using the discharge variable. The results reveal that the VPMS method provides comparatively more flexibility in selecting the computational time interval and slightly more accuracy level in reproducing the overland flow depth hydrographs than the VPMD and KW methods. Hence, the VPMS method is amenable for meso and microscale catchment modelling, especially dealing with sediment erosion problem, by coupling with various LSSs. In order to be consistent with the criterion used for the classificationof one-dimensional flood waves derived from the SVE and their applicability limits, a novel applicability criterion based onthe magnitude of scaled longitudinal flow depth gradient (1/s0)(dy/dx)e is developed for both the VPMD and VPMS overland flow routing methods. In practice, the applicability limits ofthe variants ofthe SVE for overland flow modelling are commonly assessed using the kinematic wave number (k) and (kF?p)e( =l/ju) (where, Frp is the Froude number). In this study, the physical basis of the applicability criterion ' //' is established as: // =(m +1) (1 / s0)(dy 18x)e, where mis the exponent of the Manning's (%) or Chezy's (y2) friction law. Atotal of2268 numerical experiments, each for the VPMD andVPMS methods were conducted to formulate the applicability criteria. The applicability limits of the VPMD and VPMS overland flow routing methods are assessed and quantified by comparing the routing results arrived at the outlet ofan overland flow plane for various n hypothetical cases, comprising of different combinations of rainfall intensities, overland flow plane lengths and slopes, and Manning's roughness coefficients, with the corresponding benchmark solutions of the full SVE and ACD equations. Such evaluation reveals that at 95% accuracy level of these performance evaluation measures, the applicability limits of the VPMD and VPMS overland flow methods can be fixed at: (l/s0)(dy/dx) < 0.6 and (\/sQ)(dy/dx) < 0.35, respectively. Hence, the VPMD method can be successfully used over the entire applicability range of the KW and to a greater extent of the applicability limits of the diffusive wave models. However, the applicability limit of the VPMS method is restricted as compared to that of the VPMD method for overland flow modelling, which is contrary to the applicability limits of the corresponding channel routing methods. Further, for simulating the Hortonian overland flow process, the GA infiltration model is coupled with the VPMD and VPMS overland flow routing methods through sink/source type coupling to develop the VPMD-GA and VPMS-GA models, respectively. These two methods are well-tested by using the numerical experiment data accounting for the spatial heterogeneity in the model framework. The simulation results reveal that these overland flow routing methods are capable of closely reproducing the solutions of the hydrodynamic and characteristic-based KWmodels. Although, the source/sink type coupling is not able to ensure full volume conservation as compared to the full dynamic, sequential-iterative, and decoupling approaches, it provides liberty from the numerical complexity and ensuring simplicity in modelling. Furthermore, these models are very simple to formulate, unconditionally stable, accurate, CPU-run time efficient, and provide relatively wide flexibility in the computational grid sizes selection. Hence, these VPMD-GA and VPMSGA coupled overland flow routing methods can also be used for basin modelling of different scales.