Flood risk mapping using uncertainty propagation analysis on a peak discharge: case study of the Mille Iles River in Quebec

Abstract

This study uses uncertainty propagation in real flood events to derive a probabilistic flood map. The flood event of spring 2017 in Quebec was selected for this analysis, with the computational domain being a reach of the Mille Iles River. The main parameter deemed uncertain in this work is the upstream water discharge; a given value of this discharge is utilized to build a random sample of 500 scenarios using the Latin hypercube sampling method. Simulations were run using CuteFlow-Cuda, an in-house finite volume-based shallow water equations solver, to derive the statistical mean and the standard deviation of the free surface elevation and the water depth at each node. For this real flood case, the initial interface flux scheme had to be adapted, combining a developed version of the scheme introduced by Harten, Lax and van Leer at wet interfaces and the Lax–Friedrichs scheme with additional free surface corrections for wet and dry transitions. Comparisons with results obtained from TELEMAC and from in situ observations show generally close predictions, and overall good agreement with observations. Errors of the free surface prediction relative to observations are less than 2.75%. A map based on the standard deviation of the water depth is presented to enhance the areas most prone to flooding. Finally, a flood map is produced, showing the flooded inhabited areas near the municipalities of Saint-Eustache and Deux Montagnes around the reach of the Mille Iles River as it overflows its natural bed.

Appendix A: Procedure of the scheme order calculation using mesh refinement

We consider the L² error for a given mesh at time t:

$$L(t,h)^{2} = \int\limits_{\Omega } {\left( {u_{h} (x,y,t) - u_{{{\text{exact}}}} (x,y,t)} \right)^{2} {\text{d}}x{\text{d}}y} \approx \sum\limits_{e} {A_{e} } \left( {u_{h}^{e} (t) - u_{exact} \left( {x_{G}^{e} ,t} \right)} \right)^{2}$$

where $A_{e}$ and $x_{G}^{e}$ are, respectively, the area and the barycenter of the element, and h is a measure of the mesh size defined as $h = \max \left( {\sqrt {A_{e} } } \right)$. $u_{h}$ represents the numerical solution (water level or velocity components) and $u_{{{\text{exact}}}}$ is the corresponding exact solution.

We then calculate the error for the finest mesh possible (i.e., $h \approx 0$): $L(t,0)^{2}$.

The theoretical error analysis gives the following relation:

$$L(t,h)^{2} \le a\,\Delta t^{\alpha } + b\,h^{\beta }$$

where $\beta$ is the space-order of the scheme that can be found by calculating:

$$E^{2} = L(t,h)^{2} - L(t,0)^{2}$$

Finally, $\beta$ is obtained as the slope of a log–log plot of $(h,E)$.