The French Riviera is very often threatened by flash floods. These hydro-meteorological events, which are fast and violent, have catastrophic consequences on life and property. The development of forecasting tools may help to limit the impacts of these extreme events. Our purpose here is to demonstrate the possibility of using b-flood (a subset of the Basilisk library

The south of France is very often affected by flash floods, strong and rapid events that arise particularly in the summer and autumn due to slow-moving convective storms bringing moisture from the Mediterranean Sea, with the induced rainfall amplified by topographic influences

In Sect. 2 we present the model, the numerical method, and the adaptive mesh technique. In Sect. 3 we first exhibit the ability of the Basilisk software to catch different types of flow regimes for analytical solutions developed in

Floods have horizontal length scales much larger than the vertical one. This observation is used as an hypothesis for the model. This gives a pressure that is hydrostatic as a first approximation. Integrating the Navier–Stokes equations over the thin flow depth then gives the following classical Saint-Venant equations

The shallow-water equations (Eqs.

We use a predictor–corrector scheme as the time-stepping algorithm: it is second-order in time, and the source terms are dealt with using a time-split method; see Sect.

To compute the numerical flux

We treat all the other source terms with the time-splitting technique. If we call

We will describe in the following paragraphs the different terms that can be modelled in this source term

The rain is simply treated as

We have integrated the infiltration source term of the Green–Ampt model (e.g.

Two different friction models are implemented, using the Manning or Darcy–Weisbach relations. These terms are explicitly written as follows:

As recommended in a previous study

When it rains on a very steep topography, waterfalls can occur. In this case, the physics describing the phenomenon is very different from the laws of turbulent friction that we use. It can therefore appear in the simulations that speeds that are too high and do not correspond to any physical reality. This is why we have developed a simple velocity transformation that prevents the norm of the vector velocity field from exceeding a certain threshold value set by the user. It is written as follows:

The b-flood software takes advantage (in addition to other techniques) of the adaptive mesh refinement (AMR) technique developed on Basilisk by S.Popinet

Passing from level

Passing from level

In practice, the b-flood software is a sub-component of the open-source Basilisk software created by

It is important to ensure that the numerical schemes we use are consistent. We test them on complete benchmarks, i.e. the transition from subcritical flow, when the wave velocity is higher than the flow velocity to supercritical flow, as well as the transition from supercritical to subcritical flow, which is characterized by the presence of a shock. The following test cases can be found in the software SWASHES published in

In this benchmark, we test the transition from a subcritical to a supercritical regime with Manning's law of friction. A constant flow

Comparison of the water height profiles between the simulation with the analytical solution of the Manning friction test case for two different resolutions.

The water heights profile at resolutions

Convergence of the different error norm when the resolution is increasing.

In this benchmark, we validate the transition from a supercritical flow to a subcritical flow, which is characterized by the presence of a shock. The domain is 100 m long and a constant discharge of

Comparison between the simulation with the analytical solution of the transonic test case for two different resolutions.

As done before, the water depth profile at resolutions

Convergence of the different error norms when the resolution is increasing.

In order to validate b-flood, we use a flood experiment recreated by researchers: the Toce Valley model. The Toce model was designed to study the capabilities of different numerical models to accurately represent flow characteristics. This model was done for the CADAM project by

The first case is done on the entire river which is 50 m long and 11 m large. The DTM is at a resolution of 5 cm, note that this includes the reproduction of houses. The topography with the position of the 21 gauge stations can be seen in Fig.

Topography of the fluvial case with the position and name of the gauge stations. The inflow is coming from the left border of the domain.

Hydrograph of the imposed inflow on the left boundary. Comparison of the volume of water in the simulation and the experimental one.

We reproduce the exact same case with b-flood with a minimal cell size of

Picture of the water depth

To quantify the performance of b-flood, we define the following different numbers, starting with

We also define

Note that unlike

Finally, we define the arrival time delay as the time between the two instants when at least 5 mm of water arrives at the measuring station in the real case and in the case simulated by b-flood. This arrival time delay is a good metric to quantify the capacity of b-flood to mimic the dynamics of the experimental case. Note that a positive arrival time delay corresponds to the case where water arrives first in the numerical case: arrival time delay is positive when b-flood is early and negative when b-flood is late.

Error norms and arrival time delay with respect to the gauge stations.

We report the values of the different norms in Fig.

Water depth for experiment and b-flood for gauge stations P2, P10, and P24.

In conclusion, we can say that b-flood correctly models this fluvial case of flood on impermeable soil with imposed inflow and the presence of houses.

The second case of validation is reproducing “the Model city flooding experiment benchmark” presented in

Zoom on the first 10 m of the topography of the urban model with positions of buildings and gauge stations.

Hydrograph of the imposed inflow on the left boundary. Comparison of the volume of water in the simulation and the experimental one.

We reproduce this case using b-flood. The domain consists of cells of sizes between

Picture of the water depth

Metrics with respect to the gauge stations and their mean values.

We record in the simulation the water heights at the exact locations where the measurement stations are in the experiment. Following this, for each of these stations we calculate the norms

Water depths for the experiment and b-flood for gauge stations P5 and P2.

The produced results allow us to conclude on the validity of our simulations in this case of urban flooding on impermeable soil.

Here we demonstrate the possibility of using b-flood, a software based on shallow-water equations and mesh refinement, in a real flash-flood situation in a small watershed (less than

To simulate this event with b-flood, we use a digital terrain model at a resolution of 1 m, courtesy of the IGN (National Institute of Geographic and Forestry Information of France) (RGE-ALTI). We also use the digital surface model (DSM) to add buildings. The buildings are simulated thanks to an elevation of the topography corresponding to their real heights. The total size of the domain is 7 km

We use Manning's law and the infiltration source term. IGN also provides soil plant occupation maps (BD TOPO). We can see the zones of high vegetation in Fig.

We use the source term of rain to add the precipitation measured by Météo-France; these are provided free of charge (RADAR PANTHERE). These data are at 5 min time steps, and the pixels are of 1 km

We fix the threshold value on the celerity to

Flood extent of the event simulated by b-flood.

This paper presented b-flood, an open-source Saint-Venant model for simulations of surface flows in two dimensions using adaptive refinement. The code is completely free and open source like the Basilisk software from which it is derived. The model uses a well-balanced scheme that does not prevent water from flowing over steep topography.

The validity of the numerical scheme has been tested on two analytical benchmarks. The convergence of the scheme has been observed with a good order of convergence. The code has been then tested on two experimental cases in the Toce Valley, one fluvial and the other urban. The results of the simulation gave satisfactory agreement with the experimental results. Finally, we demonstrated the practical effectiveness of b-flood on a real case of flash flooding on a small watershed in the south of France: the October 2015 flooding of the city of Cannes in the French Riviera. This event caused 20 fatalities and a lot of material damage. The city of Cannes faced 200 mm of precipitation over less than 3 h. In the upstream area, the soil was already saturated by a heavy rain that occurred on 2 October, and in the city the storm water system was also saturated. This has demonstrated the feasibility of using a software based on shallow-water equations and mesh refinement for flash-flood simulation on small watersheds (less than

Future work will focus on (1) implementing hydraulic structures such as culverts, gates, and weirs and (2) coupling this overland flow model with a storm water network model. This will improve b-flood's capability when performing more complete flash-flood simulations, particularly in southern French watersheds.

The address of the relevant Zenodo folder is as follows:

The movie representing the water height during the Cannes flood (Cannes-height.mpg) and the video representing the refinement level of the cells of the same simulation (Cannes-level.mpg) can be downloaded from the following address:

GK developed the code for b-flood and took care of the technical part. OD took care of the test cases and brought his expertise on the different competing codes of b-flood. PYL supervised the project and brought his expertise on the different physical models. SP developed the codes on which b-flood is based (basilisk) and brought his technical expertise on the technical and algorithmic part. CJ was in charge of the acquisition of the funds and supervised the project. GK prepared the manuscript with contributions from all co-authors.

The authors declare that they have no conflict of interest.

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

We would like to thank François Bourgin for the initial idea of this paper. We would like to thank the IGN for providing us with the topographic data and Météo-France for providing us with the data from their RADAR PANTHERE. We also thank the AXA Foundation for Research fund for their financial support at the very beginning of this study.

This paper was edited by Bethanna Jackson and reviewed by two anonymous referees.