# Uncertainty Assessment of Flood Hazard Due to Levee Breaching

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Case Study and Numerical Model

#### 2.1. Model Set Up

**Figure 1.**Study area: (

**a**) domain of interest; (

**b**) zoom area. Red stars indicate breach locations (weak points) and blue dots are points of interest in the floodplain. Colored surfaces correspond to friction coefficient areas. Range of variation of friction coefficients in the floodplain is reported in the legend.

#### 2.2. Hydrodynamic Model

#### 2.2.1. Roughness Coefficient

^{3}/s at Given, 17-year flood) from January 2008 (890 m

^{3}/s at Given, less than 1-year flood) and November 2008 (2320 m

^{3}/s at Given, 4-year flood) for calibration of the main river Strickler coefficients. Observation data are unavailable in the floodplain, and no significant flooding events have been recorded to date. Thus, the floodplain friction areas are identified based on the inventory of Corine Land Cover of 2018 [40]. For each area and following expert knowledge, the Strickler coefficient is taken as the mean value of an interval bounded by physical values depending on soil occupation.

#### 2.2.2. Levee Breaches

## 3. Uncertainty Assessment Methodology

#### 3.1. Uncertainty Quantification

#### 3.1.1. Flood Inundation Scenarios

#### 3.1.2. Uncertain Parameter Characterization

- Roughness coefficient quantification:

^{1/3}s

^{−1}from the calibrated value to be compatible with the calibration accuracy of 20 to 25 cm. In the floodplain, bound values are assigned according to land cover classes as presented in Table 1. A total of 280 friction areas are considered.

- Dike breach controlling coefficient quantification:

_{f}is also assumed to be deterministic and deduced from Equation (6) using ${B}_{f}$ and ${k}_{in}$. The breach depth is assumed to be the same as the height of the levee itself, as non-erodible layers are considered in dike foundations.

#### 3.2. Uncertainty Propagation

**,**that is independent of dimension, i.e., the number of factors ($v$) in the problem.

#### 3.3. Sensitivity Analysis

#### 3.3.1. Permutation Feature Importance

#### 3.3.2. Borgonovo Sensitivity Analysis

## 4. Results

#### 4.1. Global Statistical Analysis

#### 4.1.1. Uncertainty Propagation

#### 4.1.2. Sensitivity Analysis

#### 4.2. Local Statistical Analysis

#### 4.2.1. Uncertainty Propagation

#### 4.2.2. Sensitivity Analysis

## 5. Discussion

## 6. Conclusions and Perspectives

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Scheme profile cross-sectional of a dike (surmounted by an earth ridge) with variables used to identify breach initiation. v (m/s) is mean flow velocity.

**Figure 4.**Probabilistic inundation extent map of 90th percentile water depths for 100, 200, 500 and 1000-year return period floods.

**Figure 5.**Probabilistic inundation map of 90th percentile flow scalar velocity with velocity vectors for 100, 200, 500 and 1000-year return period floods with location of potential dike breaches (

**―**).

**Figure 6.**Borgonovo indices for Saint-Père breach width for 100, 200, 500 and 1000-year return period flood events.

**Figure 7.**Convergence graphs of coefficient of variation (standard deviation to mean ration) and its 90% confidence interval according to sample size, for 100, 200, 500 and 1000-year return period flood scenarios at some control points: (

**a**) Sully-Sur-Loire; (

**b**) Les Places; (

**c**) Les Boutrons; (

**d**) Le Mesnil.

**Figure 8.**Water level probability density function estimated with Monte-Carlo sampling for 100, 200, 500 and 1000-year return period floods at control points: (

**a**) Sully-Sur-Loire; (

**b**) Les Places; (

**c**) Les Boutrons; (

**d**) Le Mesnil.

Land Cover Class | Variation Interval (m ^{1/3}s^{−1}) | Probability Distribution |
---|---|---|

Areas with dense urbanization | [5, 8] | Uniform |

Areas with low urbanization | [8, 10] | |

Areas of shrubs, undergrowth | [8, 12] | |

Low-height agricultural areas | [10, 15] | |

High-height agricultural areas | [15, 20] |

**Table 2.**Breach parameters with associated range of variations, based on measurement uncertainty or historical analysis with geotechnical characteristics [41,42] (Failure Criterion ${E}_{s}$ and final breach width ${B}_{f}$, respectively) or defined by expert knowledge (energy balance criterion $\mathsf{\Delta}E$).

Name | $\mathbf{Dike}\mathbf{Crest}{\mathit{Z}}_{\mathit{b}}\left(0\right)$ (m NGF N) | $\mathbf{Failure}\mathbf{Criterion}{\mathit{E}}_{\mathit{s}}$ (m) | $\mathbf{Failure}\mathbf{Criterion}\mathsf{\Delta}\mathit{E}$ (m) | $\mathbf{Final}\mathbf{Breach}\mathbf{Width}{\mathit{B}}_{\mathit{f}}$ (m) | Dike Material |
---|---|---|---|---|---|

Fusible du déversoir | 120.80 | [−0.25, 0.3] | [0.3, 1.5] | [50, 850] | clay-dominated |

Les ormes | 119.72 | [0.05, 0.6] | [0.3, 1.5] | [50, 950] | sandy-loamy |

Embouchure de la Sange | 118.82 | [0.05, 0.6] | [0.3, 1.5] | [50, 950] | sandy-loamy |

Saint-Père | 116.66 | [0.05, 0.6] | [0.3, 1.5] | [50, 950] | clay-dominated |

Prouteaux | 115.60 | [−0.25, 0.3] | [0.3, 1.5] | [50, 950] | sandy-loamy |

Bouteille | 114.46 | [−0.25, 0.3] | [0.3, 1.5] | [50, 950] | sandy-loamy |

Les Boutrons | 112.08 | [0.05, 0.6] | [0.3, 1.5] | [50, 950] | sandy-loamy |

Sigloy | 111.54 | [0.05, 0.6] | [0.3, 1.5] | [50, 950] | sandy-loamy |

**Table 3.**90% confidence intervals of the coefficient of variation obtained from the Monte Carlo sample set of 3000 computations for 100, 200, 500 and 1000-year return period flood scenarios at some control points.

Control Points | Sully-Sur-Loire | Les Places | Les Boutrons | Le Mesnil |
---|---|---|---|---|

100-years flood scenario | $\left[0.375,0.391\right]$ | $\left[0.821,0.849\right]$ | $\left[0.242,0.25\right]$ | $\left[0.147,0.150\right]$ |

200-years flood scenario | $\left[0.309,0.315\right]$ | $\left[0.242,0.252\right]$ | $\left[0.0676,0.0704\right]$ | $\left[0.0452,0.0465\right]$ |

500-years flood scenario | $\left[0.0834,0.0865\right]$ | $\left[0.0747,0.779\right]$ | $\left[0.0342,0.036\right]$ | $\left[0.0479,0.0518\right]$ |

1000-years flood scenario | $\left[0.0562,0.0579\right]$ | $\left[0.0623,0.0641\right]$ | $\left[0.0379,0.0394\right]$ | $\left[0.0851,0.0877\right]$ |

**Table 4.**Sensitivity analysis at Les Places control point for 100, 200, 500 and 1000-year return period flood events: Borgonovo indices and permutation feature importance with their corresponding asymptotic confidence intervals, typically with a confidence level of 90%. PK and FR indicates, respectively, the friction coefficients for the riverbed and the floodplain (as reported in Figure 1).

Methods | Borgonovo Indices | Permutation Importance |
---|---|---|

100-years flood scenario | Saint-Père overtopping level: 0.22 ± 0.011 | Fusible du déversoir overtopping level: 0.73 ± 0.028 |

Fusible du déversoir overtopping level: 0.20 ± 0.011 | Saint-Père overtopping level: 0.72 ± 0.024 | |

PK403-414: 0.12 ± 7.2 × 10^{−3} | PK403-414: 0.33 ± 0.02 | |

Saint-Père breach width: 0.10 ± 4.2 × 10^{−3} | PK414-420: 0.099 ± 6 × 10^{−3} | |

Fusible du déversoir breach width: 0.091 ± 3.9 × 10^{−3} | Saint-Père breach width: 0.018 ± 1.4 × 10^{−3} | |

FR-64977: 0.09 ± 4.3 × 10^{−3} | FR-64977: 0.0083 ± 6.9 × 10^{−4} | |

200-years flood scenario | Saint-Père breach width: 0.22 ± 9.9 × 10^{−3} | Saint-Père overtopping level: 0.91 ± 0.039 |

Saint-Père overtopping level: 0.21 ± 0.011 | PK403-414: 0.57 ± 0.028 | |

PK403-414: 0.13 ± 8.29 × 10^{−3} | Saint-Père breach width: 0.24 ± 9.7 × 10^{−3} | |

Prouteaux overtopping level: 0.097 ± 6 × 10^{−3} | FR-64977: 0.081 ± 5.6 × 10^{−3} | |

FR-64977: 0.083 ± 7.8 × 10^{−3} | Fusible du déversoir breach width: 0.045 ± 4 × 10^{−3} | |

Fusible du déversoir breach width: 0.078 ± 7 × 10^{−3} | Prouteaux overtopping level: 0.01 ± 1 × 10^{−3} | |

500-years flood scenario | Saint-Père breach width: 0.32 ± 0.011 | Saint-Père breach width: 1.12 ± 0.042 |

FR-65537: 0.1 ± 0.01 | FR-65537: 0.30 ± 0.018 | |

Les Ormes overtopping level: 0.085 ± 7.02 × 10^{−3} | Prouteaux overtopping level: 0.14 ± 0.011 | |

Prouteaux overtopping level: 0.081 ± 9.1 × 10^{−3} | Les Ormes overtopping level: 0.12 ± 0.01 | |

PK403-414: 0.071 ± 7.65 × 10^{−3} | PK403-414: 0.048 ± 4.4 × 10^{−3} | |

Les Ormes breach width: 0.058 ± 9 × 10^{−3} | Prouteaux breach width: 0.047 ± 5.8 × 10^{−3} | |

1000-years flood scenario | FR-65537: 0.22 ± 0.014 | FR-65537: 0.68 ± 0.028 |

Saint-Père breach width: 0.19 ± 9.84 × 10^{−3} | Saint-Père breach width: 0.63 ± 0.023 | |

Prouteaux overtopping level: 0.10 ± 0.012 | Prouteaux overtopping level: 0.23 ± 0.011 | |

PK403-414: 0.098 ± 0.011 | PK403-414: 0.14 ± 6.2 × 10^{−3} | |

Prouteaux breach width: 0.062 ± 9.55 × 10^{−3} | Prouteaux breach width: 0.10 ± 8.4 × 10^{−3} | |

Les Ormes breach width: 0.054 ± 7.1 × 10^{−3} | Les Ormes breach width: 0.08 ± 8.9 × 10^{−3} |

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**MDPI and ACS Style**

Goeury, C.; Bacchi, V.; Zaoui, F.; Bacchi, S.; Pavan, S.; El kadi Abderrezzak, K. Uncertainty Assessment of Flood Hazard Due to Levee Breaching. *Water* **2022**, *14*, 3815.
https://doi.org/10.3390/w14233815

**AMA Style**

Goeury C, Bacchi V, Zaoui F, Bacchi S, Pavan S, El kadi Abderrezzak K. Uncertainty Assessment of Flood Hazard Due to Levee Breaching. *Water*. 2022; 14(23):3815.
https://doi.org/10.3390/w14233815

**Chicago/Turabian Style**

Goeury, Cédric, Vito Bacchi, Fabrice Zaoui, Sophie Bacchi, Sara Pavan, and Kamal El kadi Abderrezzak. 2022. "Uncertainty Assessment of Flood Hazard Due to Levee Breaching" *Water* 14, no. 23: 3815.
https://doi.org/10.3390/w14233815