# The Cross-Dike Failure Probability by Wave Overtopping over Grass-Covered and Damaged Dikes

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## Abstract

**:**

## 1. Introduction

## 2. Framework for the Failure Probability by Wave Overtopping

#### 2.1. Hydraulic Load

#### 2.2. Analytical Grass-Erosion Model

## 3. Methods

#### 3.1. Study Area

#### 3.2. Cross-Dike Failure Probability

#### 3.3. Additional Load at Damaged Spots

#### 3.4. Failure Probability of Damages on the Landward Slope

## 4. Results

#### 4.1. Cross-Dike Failure Probability

#### 4.2. Additional Load at Damaged Spots

#### 4.3. Failure Probabilities of Damaged Spots

## 5. Discussion

#### 5.1. The Cross-Dike Failure Probability

#### 5.2. The Effect of Damages on the Failure Probability

#### 5.3. Additional Load at Damages

#### 5.4. Comparison to Other Studies

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Convergence of the Failure Probability

**Figure A1.**Convergence of the failure probability ${P}_{f}$ for a constant turbulence parameter of 2.0 and an average grass quality.

## Appendix B. The Depth-Averaged Relative Turbulence Intensity

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**Figure 1.**(

**a**) Dike cover erosion on the landward slope and toe during wave overtopping field tests in the Netherlands (Photo by Juan Pablo Aguilar Lopez). (

**b**) A slope instability of a grass-covered clay dike in the Netherlands [17].

**Figure 2.**Schematization of the dike cover where the grass roots strengthen the topsoil of approximate 20 cm and the clay quality determines the strength of the subsoil.

**Figure 3.**Schematization of the framework to calculate the failure probability conditional to the water level ${P}_{f|h}$.

**Figure 4.**Example of the probability density function of the water level $f\left(h\right)$ (solid line) and a fragility curve (dashed line) showing the conditional failure probability ${P}_{f|h}$ as function of the water level h.

**Figure 5.**(

**a**) Top view of the study site at Millingen a/d Rijn and the location in the Netherlands. Retrieved from Google Earth, earth.google.com and Oppervlaktewater in Nederland, www.clo.nl (accessed on 13 November 2020). (

**b**) The dike geometry with a smooth waterside slope and a grass-covered crest and landward slope including the water level h, the free crest height ${R}_{c}$ and the location of failure ${x}_{fail}$.

**Figure 6.**(

**a**) Small cliff at a damaged spot leads to to a jet forming that impacts in the jet impact zone (white circle) resulting in an additional load. (

**b**) Photo of damage during the overtopping test at St. Philipsland resulting in a small cliff [44].

**Figure 7.**Schematization of the method for the calibration of the turbulence parameter $\omega $ at damaged spots as a function of the critical velocity ${U}_{C}$ using the analytical grass-erosion model (GEM) and the overtopping tests.

**Figure 8.**(

**a**) The cross-dike failure probability ${P}_{f}\left(x\right)$ for three grass qualities and a constant turbulence parameter $\omega $. (

**b**) The cross-dike failure probability ${P}_{f}\left(x\right)$ for an average grass quality with a constant turbulence parameter $\omega $ and the turbulence parameter of Equation (4).

**Figure 9.**The relationship between the turbulence parameter $\omega $ and the critical velocity ${U}_{C}$ based on a linear fit through the failure points of the eight test sections.

**Figure 10.**(

**a**) The fragility curves showing the conditional failure probability ${P}_{f|h}$ as function of the free crest height ${R}_{c}$ for failure at the landward toe and failure at a damage around 10 m from the crest for four cover qualities. (

**b**) The cross-dike failure probability for a damaged spot on the slope ${P}_{f,damage}\left(x\right)$ for the four cover qualities. (

**c**) The ratio $\Delta {P}_{f}\left(x\right)$ for the four cover qualities (Equation (5)).

**Table 1.**The turbulence parameter $\omega $ and the threshold velocity ${U}_{t}$ in the GEM are determined from calibration or measurements of wave overtopping field tests on grass-covered dikes in the Netherlands and Belgium.

Parameter | Relation | Method |
---|---|---|

Turbulence parameter | ${\omega}_{toe}=2.75$ | Calibration using the measured erosion depth at 7 field tests [19,34] |

${\omega}_{crest}=2.00$ | Determined from measured pressure fluctuations at Millingen a/d Rijn [35] | |

${\omega}_{slope}=2.35$ | Determined from measured pressure fluctuations at Millingen a/d Rijn [35] | |

Threshold velocity | ${U}_{t}=2.4{U}_{C}$ | Calibration using the measured erosion depth at 7 field tests [19,34] |

**Table 2.**The critical velocity ${U}_{C}$ and the inverse cover strength parameter ${C}_{E}$ of the dike cover for three grass qualities and a good clay quality together with the coefficient of variation $CV$ used for the distribution of the critical velocity.

Grass | Clay | |||||
---|---|---|---|---|---|---|

Good | Average | Poor | Good | Source | ||

${U}_{C}$ | [m/s] | 6.5 | 4 | 2.5 | 0.85 | Aguilar-López et al. [15] |

$CV$ | [-] | 0.3 | 0.3 | 0.3 | 0.1 | Aguilar-López et al. [15] |

${C}_{E}$ | [s/m] | 1 $\times {10}^{-6}$ | 2 $\times {10}^{-6}$ | 3 $\times {10}^{-6}$ | 50 $\times {10}^{-6}$ | Verheij et al. [36] |

**Table 3.**The location of failure ${x}_{fail}$ measured from the start of the landward slope for the eight test sections in the Netherlands and Belgium. The test conditions resulting in failure are described by the critical velocity ${U}_{C}$, the average overtopping discharge q and the simulated storm duration ${t}_{storm}$. The fractions for q indicates that the cover failed at a fraction of the storm duration for that specific discharge.

Test Section | ${\mathit{x}}_{\mathit{f}\mathit{a}\mathit{i}\mathit{l}}$ [m] | ${\mathit{U}}_{\mathit{C}}$ [m/s] | q [L/s/m] | ${\mathit{t}}_{\mathit{s}\mathit{t}\mathit{o}\mathit{r}\mathit{m}}$ [h] | Source |
---|---|---|---|---|---|

Afsluitdijk 2 | 2.8 | 4.0 | 1, 10 | 6 | Bakker et al. [45] |

Tielrodebroek 1 | 1.9 | 1.2 | 1, 10, 30($\frac{1}{3}$) | 2 | Peeters et al. [46] |

Tielrodebroek 2 | 1.9 | 1.6 | 1, 10, 30($\frac{1}{6}$) | 2 | Peeters et al. [46] |

Wijmeers 1 | 1.7 | 3.5 | 1, 5, 25 | 2 | Pleijter et al. [47] |

Wijmeers 3 | 1.3 | 3.0 | 25 | 2 | Pleijter et al. [47] |

Kattendijke 2 | 6.6 | 6.5 | 30, 50 | 6 | Bakker et al. [44] |

St. Philipsland | 6.5 | 6.5 | 0.1, 1, 10, 30, 50 | 6 | Bakker et al. [44] |

Tholen 3 | 6.5 | 0.0 | 1, 5 ($\frac{2}{3}$) | 6 | Bakker et al. [48] |

**Table 4.**Comparison between the model results and the results of Aguilar-López et al. [15] for the maximum failure probability for the grass-covered dike profile of Millingen a/d Rijn without any damages.

Poor | Average | Good | |
---|---|---|---|

Our framework | 3.9 $\times {10}^{-4}$ | 9.0 $\times {10}^{-5}$ | 5.8 $\times {10}^{-6}$ |

Aguilar-López et al. [15] | 8.2 $\times {10}^{-5}$ | 5 $\times {10}^{-5}$ | ≤10^{−6} |

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**MDPI and ACS Style**

van Bergeijk, V.M.; Verdonk, V.A.; Warmink, J.J.; Hulscher, S.J.M.H.
The Cross-Dike Failure Probability by Wave Overtopping over Grass-Covered and Damaged Dikes. *Water* **2021**, *13*, 690.
https://doi.org/10.3390/w13050690

**AMA Style**

van Bergeijk VM, Verdonk VA, Warmink JJ, Hulscher SJMH.
The Cross-Dike Failure Probability by Wave Overtopping over Grass-Covered and Damaged Dikes. *Water*. 2021; 13(5):690.
https://doi.org/10.3390/w13050690

**Chicago/Turabian Style**

van Bergeijk, Vera M., Vincent A. Verdonk, Jord J. Warmink, and Suzanne J. M. H. Hulscher.
2021. "The Cross-Dike Failure Probability by Wave Overtopping over Grass-Covered and Damaged Dikes" *Water* 13, no. 5: 690.
https://doi.org/10.3390/w13050690