Jerome Dubois,
Jean-Louis Boillat, Anton Schleiss
Laboratory of Hydraulic Constructions, LCH, Swiss Federal
Institute of Technology Lausanne
EPFL-LCH,
CH-1015 Lausanne, Switzerland
Phone:
++41 21 693 23 85
Fax:
++41 21 693 22 64
e-mail:
secretariat.lch@epfl.ch
internet: http://lchwww.epfl.ch
Abstract: The actual possibilities of numerical simulation allow a deterministic and physically based approach of natural phenomena. Hydrology is also concerned and develops models based on hydraulic equations to compute the flood generation and routing on watersheds. The equation of movement imposes the knowledge of a head loss coefficient in order to be solved. This coefficient is the only unknown physical parameter for the flood routing on watersheds, and the quality of its estimation directly influences the quality of the model. The classical head loss formulations, empirically developed for turbulent and rough flows in rivers, are not applicable to surface runoff. In order to find a more appropriate friction law, laboratory experiments were conducted in order to define the relationship between the mean velocity and the surface roughness.
The numerical model "Faitou", was
developed for the simulation of the generation and routing of floods on steep
river basins. With the new head loss equation, it solves the 2D kinematic wave
equation over the catchment topography with the finite volume method. This
computation of surface runoff is coupled with a hydrodynamic modeling of the
river network. The determination of the watershed limits, the mesh of finite
volumes and the river network are automatically generated from a digital
elevation model. This software, associated with the new head loss equation,
demonstrates its potential and qualities by application on an alpine watershed.
Keywords: fluvial system, flood routing, head loss equation, macro-roughness, flood, network, hydrology, watershed, numerical simulation
The design discharge or design flood is a determinant parameter for projects related to flood protection. In the past, it was obtained through the application of empirical relations including the catchment area and the developed length of the hydrographic network. Unfortunately, these formula produce a strong dispersion which makes their use hazardous (LCH 1998; Elektro-Watt 1960). The statistical approach was thereafter abundantly used for the adjustment of measured discharges. However, this method doesn't solve the problem of extrapolation to extreme values of which the confidence interval strongly increases with the return period. An other process is proposed by the global hydrological models, like the unit hydrograph, which defines a total flood event from the related rainfall. The weakness of these models rests in a lack of significance of the physical parameters and in the fact that the calibrated values are conserved when extrapolating.
The last generation of hydrological models belongs to the physically based and spatially distributed category. These models implement the equations of the physical phenomena and deliver a solution at any point. The kinematic wave approximation is generally applied to compute the rainfall-discharge transfer, by modelling the topography with the help of planes, where the only parameter concerning the run-off is the roughness coefficient. The first models using this technique only considered one or two planes connected to the effluent, (Hager 1984) or used a geomorphologic description for modelling the catchment area (Bérod 1995). In both cases, the Manning-Strickler equation was used for the head loss computation. The experience acquired shows that the calibration of this type of models lead to very low Strickler coefficients, comprised between 0.2 and 2 m1/3/s, which go largely out of the application domain of the formula. The present contribution tends to explain the drift of these models and proposes a new concept for the physical modelling of flows.
The topographical modelling of a watershed with
a few planes constitutes a crude simplification of the water advance. This
modelling default is fully compensated by the roughness coefficient, the only
free parameter of the flow. This partially explains the observed drift of the
roughness coefficient in the kinematic wave models over planes. Today, detailed
topographic information is accessible due to the digital elevation models, with
meshes in the order of 25 to 250 meters, on which the runoff computation tends
to reduce the modelling default mentioned above. This facility allows to follow
the water advance contrary to the global models.
The model “Faitou”,
developed in the frame of a research project related to extreme floods make use
of this detailed definition of the watershed when considering the general effect
of the flows schematised on figure 1.
Fig.1 Schematisation of the different flow types considered by the hydrodynamic model “Faitou”
At first, this model solves the two-dimensional equations of the kinematic wave over the topography as described by the digital elevation model till the runoff reaches a river or a reservoir. Then, a one-dimensional hydrodynamic model compute the river routing down to the outlet. In this way, "Faitou" achieves the coupled modelling of the surface and river flows.
The more accurate description of the flowing ways issued from the digital elevation model reduces the modelling skew when compared with the simulation over a single plane. However, the Manning-Strickler equation considers a constant head loss coefficient, independently of the running discharge. This relation doesn't take into account the Reynolds Number and, when compared with the Colebrook-White law developed for pressure flows, it should only be applied in the case of rough turbulent flows. Therefore, the typical low water depths and velocities of the runoff usually correspond to laminar or weakly turbulent flows. Moreover, the uneven elements forming the surface roughness are in the same order of magnitude as the height of the running wave. So many things which compromise the validity of that empirical law when used in hydrological models.
Fig. 2 Graphic presentation of the new head loss law related to free surface flows where V: mean flow velocity, J0: bottom slope, h: mean water depth, D: mean diameter of the roughness elements, p: marble cover density
In order to bring an answer to these questions, a fundamental research was achieved concerning the hydraulic behaviour of thin flows over macro-roughness surfaces (Dubois 1998). A first experimental set-up, principally constituted of a plate covered with a variable density of marbles, allowed to establish the water depth-velocity relation in uniform and permanent flow conditions over a plane. The results issued from 273 tests allowed to develop a new relationship between water depth and velocity for this particular type of runoff. This formula was then validated with experimental tests conducted under a rainfall simulator for non uniform and unsteady flow conditions. The obtained results, presented graphically in figure 2, reveal original behaviours for free surface flows.
Hence, it appears that the velocity is independent of the water depth so long as the latter remains lower than the roughness elements and higher than 0.1 this dimension. This behaviour is similar to that of flows in porous media which don't depend on the water depth but especially on the hydraulic gradient. As soon as the runoff submerges the roughness elements, the velocity increases abruptly in function of the water depth.
The new proposed law is also valid for laminar and for turbulent flows and ensures a continuous transition from one type to the other. It connects, on one hand with the classical relations for free surface flows as soon as the water depth to roughness size ratio increases, and on the other hand, to the analytical solution of laminar flow over a plane for low water depths and velocities.
The Toules arch dam is situated in Switzerland
on the historic route of the Great St. Bernard Pass. The upper divide of
its watershed are the summit of the Valais Alps forming the border between
Switzerland and Italy. Its main runoff direction is from south to north. The
Toules reservoir and its natural watershed can be seen on the shaded relief map
in figure 3. The detailed description of the hydroelectric power scheme can
be found in Boillat et al. (1999).
In order to apply the modelling concept presented in figure 1, the "Faitou" model offers an automatic calculation domain generating tool. The pre-processing consists in establishing the watershed contour, to localise the hydrographical network based on the DEM, discretizing the watershed surface into triangles in accordance to the river network and establishing the topological relations as a whole between the different elements. Figure 3 presents the result of the automatic generation of the computer model for the watershed of the Toules reservoir. With a mesh size of 25 meters, the catchment area counts approximately 100'000 elements.
Fig.
3 Shaded
relief map of the region around the Toules reservoir obtained using the
“Faitou” model (left) and the result of the automatic generation of the
computer model (right)
The automatic generation tool is therefore capable to prepare the model for the numerical simulation, based on the DEM, for an outlet chosen without restraint. In figure 3, each element of the catchment area is coloured according to the slope. The arborescence of the rivers illustrates the complexity of the 1D network to simulate.
As an example, the historic floods of 1993 were simulated in the modelled watershed. This event is presented graphically on figure 4, with precipitation registered every 10 minutes at the metrological station of Great St. Bernhard. Although this hasn’t been mentioned yet, the “Faitou” Model is able to consider a spatially distributed infiltration function. This explains the quasi-zero response of the watershed during the first day of precipitations.
Fig. 4 Simulation of the September 1993 flood with the “Faitou” Model
Considering that no gauging station exists at the entry of the reservoir, the flood hydrograph was reconstituted numerically on the basis of registered reservoir water levels and exploitation data from the powerhouse (to pump and turbine). The precision of the water level measurements explains the presence of level stages with constant discharge in the reconstituted flood hydrograph.
Figure 4 also presents furthermore the virtual hydrograph entering the river network. This hydrograph cannot be measured in reality, because it corresponds to the exit of the 2D surface runoff model. It is computed by simultaneous addition of all the individual flow contributions entering the rivers or directly into the lake. It is generated here for comparison with the inflow hydrograph of the reservoir, named "Faitou simulation" in figure 4, with the aim of measuring the routing effect of the hydrographic network.
This graphical representation shows the capacity of the "Faitou" model to reproduce accurately the behaviour of the watershed.
The richness of the spatially referred information can be used today and integrated in the hydrological modelling. The spatially distributed models provide results not only at the outlet of the watershed but at any other point of interest. Every modification in Nature, either of anthropic or climatic origin, can easily be introduced into the physically based parameters of the model. The computation will therefore be able to put in evidence the effects of these modifications.
The “Faitou” model proposes not only the functionality suitable to the spatially distributed models but also integrates a new head loss relation for low water flows over macro roughness. This law, experimentally validated using rainfall simulator tests, offers a wide utilisation domain, ranging from the laminar flows up to turbulent conditions. This law insures the validity of results when extrapolating the calibrated model to extreme rainfall conditions.
The last point is of particular interest for the examination of measures dedicated to the mitigation of flood disasters. In this case, the physical behaviour of runoff will be modified depending on the surface flow conditions. A significant difference can be expected compared to the use of a classical law like Manning-Strickler, where the calibrated head loss coefficient is maintained constant whatever the rainfall intensity. The presented example shows another domain of application related to dimensioning of dam spillways where we need to define extreme flood conditions.
References
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