1 Cemagref, Research Unit Hydrology – Hydraulics
2 Cemagref, Research Unit Spatial Structures and
Systems
Cemagref, 3 bis, quai Chauveau, CP220, F69336 LYON CEDEX 09
Tel.: +33 472208775, Fax: +33 478477875, E-mail:andre.paquier@cemagref.fr
Abstract: Simulating flooding process using 2-D model provides detailed water
depths and velocities. This information may be directly used for assessment of
flood risks and thus became the basis for prevention or emergency measures and
for the development of a specific information system. The three cases described
are typical of the use of 2-D model: complex hydraulic network the topology of
which changes because of flooding for confluence of Oignin River, breaching of
embankment for a flood plain along Loire River and flood passing through the
streets of the town of Nîmes.
Such simulations require
that adapted methods are used. First, a convenient modelling should include a
right description of the topography, which means a computational grid which
takes into account the main features of natural ground level and the main linear
man-made structures. The extreme case is the urban area in which obstacles to
flow (essentially buildings) constitute the major part of the flooded area.
Then, the numerical method should be able to represent both subcritical and
supercritical flows and the transitions between them and to integrate the
description of the flow through hydraulic structures. Such structures should be
as complex as a breaching in an embankment. Code Rubar 20 developed by Cemagref
meets these requirements and seems to simulate such complex situations in a
suitable way.
Keywords: shallow water equations, 2-D modelling, flood risks, embankment, urban
catchment
Flood events occur throughout the world and cause major damages and casualties. The main causes of the increase of the consequent damages are the modified land use in the upper catchments, the urbanisation of flood plains and may be the climatic changes. The more efficient solution to reduce damages or, at least, to reduce the rise of damages consists in managing the use of flood plains. Managing the use of flood plains demand prevention measures and emergency plans. Their efficiency require that reliable information about the parameters of the floods are made available. Such parameters are the maximum extent of the flooded area, the time of arrival of the flood but also the flow velocity, the expected water depth and the evolution of these parameters during the flood period. This information is provided by numerical models which need to have been calibrated on the observations of past events and to be fed by the discharges coming from the upper catchments.
Historically, 1-D numerical models were used first.
They solved de Saint Venant equations in the main channels and included storage
areas for flood plains. If these 1-D models are still widely used for usual
flows, 2-D modelling is replacing them for extreme floods. 2-D models solving
shallow water equations provide water depth and velocity in any plot and any
time which is precisely the required information; as they rely on a 3-D
topographical model, all the changes in velocity directions might be modelled.
Due to the increase in the power of the computers and the advance in numerical
methods, such calculations that lasted several days a few years ago require only
a few hours nowadays. Here below, complex cases are presented, which illustrate
the potentialities of 2-D models to provide more accurate and reliable
information to be used in the management of flood plains. The three cases have
been modelled with the same numerical tool, Rubar 20 which is described first.
Code Rubar 20 has been developed by Cemagref since 1988. It solves 2-D shallow water equations by an explicit finite volume scheme. The second order Godunov-type scheme includes 2 main steps which consists in, first, solving 1-D Riemann problems to estimate the fluxes through edges for the conservative part of the equations, and then, integrating second member on the surface of the cell to add the corresponding contribution (Paquier and Farissier, 1996) . An hydraulic structure is defined as a set of a few cells where fluxes through one edge are computed from the relations for the hydraulic structure linking discharge and water levels upstream and downstream. These relations may be a weir-type relation but also some more complicated relations such as a breaching in an embankment dam modelled in a simplified way (Paquier et al., 1998) . The computation of discontinuities or fronts is included in the scheme as ordinary points so that there is no need of any particular treatment for drying and wetting. The validity of the numerical method was checked (Paquier, 1995) on a lot of tests including 1-D and 2-D situations, steady and unsteady flows, cases with or without analytical solutions, comparisons with laboratory experiments.
To pass from the topographical data to the
computational mesh, the code Mocahy developed by Cemagref is used. This code
organises the topography in cross sections and main topographic linear features
from which a 3-D numerical model of the bottom of the valley is built
(Farissier, 1993)
.
From this model, a grid composed of triangles or quadrilaterals may be drawn
which will respect the main features of the topography. Note that the main
linear features may be embankments or other man made linear structures. After
computation, Mocahy draws the results using the same topographical basis.
At a confluence, main floods often occur when
the upstream floods on the two rivers reach the confluence at the same moment.
This situation often creates a local flooding which reduces the peak discharge.
In the South East of France, approximately 100 kilometres East from the town of Lyon, the
possibilities to reduce flooding in such a situation for the 100-year flood
(peak discharge: about 270 m3/s) were investigated
(Ramez, 2000)
.
The study
relates to the zone of confluence between the rivers Lange and Oignin (10
kilometres upstream for Lange, 6 kilometres upstream and 2 kilometres downstream
for Oignin). The exchanges with the Lake of Nantua by the means of a channel
(called “Bras du Lac”) made the hydraulic system more complex (see Figure
1); during a flood, the level in the lake rises more slowly than the rivers and
thus receives water before the peak, water which is sent back to the rivers when
their level is low enough. It was decided to study an additional means to reduce
the peak flow: a diversion channel will link Oignin upstream from the confluence
with Bras du Lac to Oignin downstream from the confluence with Lange. The
100-year discharge hydrographs used were synthetic monofrequency discharge
hydrographs because these hydrographs are a convenient representation of one
return period
(Galéa and Prudhomme, 1997)
. Due
to the extension of the flooded area between the two rivers, 2-D model seems
more convenient. The computational grid (6649 cells) was built from the main
topographic linear features which are the rivers, the channel to the Lake of
Nantua, the future diversion channel and a motorway. The main hydraulic
structures along the rivers were represented by a relation between the discharge
through the structure and the upstream and downstream water levels. The
relations which were used are either a classical relation for a weir or a
relation estimated from measurements on the field. The friction Strickler
coefficients used were 30 for the main beds and 15 for the flood plain.
Fig. 1 shows the maximum water levels provided by the
computation with the diversion channel for the 100-year flood. A first
conclusion was the small decrease of water level relatively to the simulation
without the diversion channel. This point may be explained by the presence of a
narrow reach downstream from the outlet of the diversion channel and by the
interaction with the Lake of Nantua.
Loire River is about 1000 kilometres long. From early
Middle Age, people have protected themselves from the floods by dikes which have
been successively risen. Studies about the whole middle reach (length: about 400
km) were performed with a 1-D model considering generally steady computations
using peak discharge as the flood lasts several days maintaining more than 90%
of that discharge during several hours. They show that the 100-year flood (peak
discharge of about 5000 m3/s) flows between the dikes whereas the 500-year flood
(peak discharge of about 6450 m3/s) starts inundating the flood plain protected
by the dikes to a large extent. However, the most
important threat is the breaching of the dikes: for instance, it is proved by
the three major floods of the 19th century during which 336 breaches
occurred
(Halbecq,
1996)
. Parallel to this study, Cemagref deals with the
development of a GIS (Geographical Information System) in order to optimise the
maintenance and the reinforcement of the dikes as well as emergency measurements
in the protected zones. In order to define the characteristics of such an
information system, a prototype was elaborate on a small zone, the Val de Cisse.
The inundation risk linked to the breaching of the
dikes was included in this prototype
(Maurel et al., 2000)
.
Because the water spreads from the breach in all the
directions due to the flat topography of the flood plain, the estimation of the
flood was performed using a 2-D model. The length of the studied area is about
20 kilometres. In the flood plain, the land use is mainly agriculture with some
dwellings and industries, particularly near the town of Amboise. The topography
included all the main embankments: dikes along Loire River, railways and main
roads. The Cisse River, a tributary flowing in the North part of the flood plain
was also modelled. The structures across the embankments were included in the
topography. The mesh was then built from this description of the topography in
order to represent all these details in a more or less simplified way. Total
number of cells was 13713 of which about 30% for the river between the dikes.
The potential breaches were located at two various places in which the main bed
of Loire river is close to the dikes. Both 100-year and 500-year floods were
considered. Breaching might happen by overtopping or by piping.
Fig. 2 shows the maximum water depths computed for the
500-year flood with a downstream breach by overtopping (by steps of 0.5 m till
1.5 m). It can be noticed that the highest maximum water depths (>1.5 m) are
concentrated along the Cisse river (lowest area of the flood plain) except just
downstream the breach and in the downstream part of the flood plain in which the
embankments of a railway and a motorway rise water levels. The comparison
between the various simulations show that, as the peak of the flood lasts
several days, the flooding of the area behind the dikes was complete till a
level close to the maximum water level in the main bed. A few centimetres may be
saved depending whether breaching occurs early or not. The embankments inside
the flood plain have nearly no influence on maximum water level if they are
parallel to the flow but may cause local rise of water level if they are
perpendicular to the flow. Velocities in the flood plain are high (>1m/s)
only in the first phase of flooding or locally when structures imply
concentration of flow.
Nîmes is a small town (about 120, 000 inhabitants)
located in the south of France. On October 3, 1988, a heavy storm coming from
Mediterranean Sea produced about 300 mm in about three hours over the whole
catchment upstream the town. The flow concentrated in the beds of several
streams crosses a large part of the town, which results in high damages (500
millions $) and 10 casualties.
Inside a French national research program “Risques
d'inondations” (flood risks), a comparison was performed between a 1-D model
and a 2-D model on the case of the flooding of a district of Nîmes during this
event. The 2-D model has the advantage of making possible the description of the
detailed topography of the crossroads and of the areas (squares, gardens, etc)
in which water may be stored or diverted. The results presented here below refer
to 2-D simulations in which only the streets are supposed to transfer water. The
cross section of one street was described by 11 points in order to represent the
pavements and gutter and the whole grid includes 20498 cells. Although the space
step varies from 0.2 to 50 metres between two adjacent cells, the stability of
the computation was obtained for a time step which corresponds to a maximum
Courant number of about 1 for the smaller cells. The inputs are defined by two
discharge hydrographs computed from rain in the upstream catchments.
Fig. 3 shows the computational grid with the
calculated maximum water depths (by steps of 0.5 m up to 1.5 m). The comparison
of the computational results with the maximum water levels of the 1988's flood
showed some differences with higher or lower levels depending of the points of
observations. This discrepancy seems to result from the uncertain downstream
condition and from the errors in the topography. However, in spite of these
local discrepancies, the limits of the flooded area and the dynamics of the
flood are simulated in a convenient way.
The three cases described here above proved that the
simulation of large floods using a 2-D model is possible for engineering
purposes. Although, the computations still remain very long, the advantages
appear clearly when it is difficult to determine precisely the direction of the
flow and, in particular, when this direction changes during the event, which is
often the case during the flooding process of large areas. The use of an adapted
numerical scheme has permitted to use the same 2-D model to simulate the water
levels in the streets in which the flow passes from subcritical to
supercritical.
References
Farissier, P., 1993, Etude d'un modèle cartographique adapté à la
simulation des écoulements en rivière [Ph D thesis], Université Claude
Bernard Lyon 1. (in French)
Galéa,
G., and Prudhomme, C., 1997, Notions de base et concepts utiles pour la compréhension
de la modélisation synthétique des régimes de crue des bassins versants au
sens des modèles QdF: Revue des Sciences de l'Eau, p. 83-101. (in French)
Halbecq,
W., 1996, Approche géomorphologique des brèches dans les levées de la Loire
(Geomorphologic approach of breaches in Loire levees) [DEA thesis]: Orléans,
Université d'Orléans.(in French)
Maurel,
P., Mériaux, P., Tourment, R., Paquier, A., Pardo, C., and Chryat, M., 2000,
Analyse et prototypage d'un SIRS générique pour aider à la gestion intégrée
des zones fluviales endiguées, Colloque national sur les risques naturels: 28
septembre 2000, Grenoble (France), Cemagref, 13 p.(in French)
Paquier, A., 1995, Modélisation et simulation de la propagation de l'onde de rupture de barrage (Modelling and simulating the propagation of dam-break wave) [PhD thesis], Université Jean Monnet de Saint Etienne.(in French)
Paquier,
A., and Farissier, P., 1996, Use of a 2-D model for simulating the flooding of a
plain, in Muller, A., ed.,
Hydroinformatics' 96, Volume 1: Zurich, Switzerland, Balkema, Rotterdam, p.
129-136.
Paquier,
A., Nogues, P., and Herledan, R., 1998, Model of piping in order to compute dam
break wave, 2nd CADAM meeting:
Munchen, Germany, European Commission, Science research Development.
Ramez,
P., 2000, Expertise des crues du Lange et de l'Oignin - Etude Hydraulique -
Cartographie des zones inondables, Direction Départementale de l'Agriculture et
de la Forêt de l'Ain.
(in
French)
Fig. 1 Maximum water depths for the confluence between Lange and Oignin rivers.
Fig.2 Maximum water depths and main linear features for Val de Cisse
Fig. 3 Maximum water
depths and computational grid for Nîmes