A probabilistic framework is developed to calculate the cross-dike failure probability by overtopping waves on grass-covered dikes. The cross-dike failure probability of dike profiles including transitions and damages can be computed to find the most likely location of failure and quantify the decrease in the failure probability when this location is strengthened. The erosion depth along the dike profile is calculated using probability distributions for the water level, wind speed and dike cover strength. Failure is defined as the exceedance of 20 cm erosion depth when the topsoil of the grass cover is eroded. The cross-dike failure probability shows that the landward toe is the most vulnerable location for wave overtopping. Herein, the quality of the grass cover significantly affects the failure probability up to a factor 1000. Next, the failure probability for different types of damages on the landward slope are calculated. In case of a damage where the grass cover is still intact and strong, the dike is most likely to fail at the landward toe due to high flow velocity and additional load due to the slope change. However, when the grass cover is also damaged, the probability of failure at the damage is between 4 and 125 times higher than for a regular dike profile.

The wave overtopping flow can exert high hydraulic loads on the grass cover of dikes leading to failure of the cover layer on the crest and the landward slope. Hydraulic variables such as the near bed velocity, pressure, shear stress and normal stress are important to describe the forces that may lead to cover erosion. This paper presents a numerical model in the open source software OpenFOAM^® to simulate the overtopping flow on the grass-covered crest and slope of individual overtopping waves for a range of landward slope angles. The model provides insights on how the hydraulic forces change along the profile and how irregularities in the profile affect these forces. The effect of irregularities in the grass cover on the overtopping flow are captured in the Nikuradse roughness height calibrated in this study. The model was validated with two datasets of overtopping tests on existing grass-covered dikes in the Netherlands. The model results show good agreement with measurements of the flow velocity in the top layer of the wave, as well as the near bed velocity. The model application shows that the pressure, shear stress and normal stress are maximal at the wave front. High pressures occur at geometrical transitions such as the start and end of the dike crest and at the inner toe. The shear stress is maximal on the lower slope, and the normal stress is maximal halfway of the slope, making these locations vulnerable to cover failure due to high loads. The exact location of the maximum forces depends on the overtopping volume. Furthermore, the model shows that the maximum pressure and maximum normal stress are largely affected by the steepness of the landward slope, but the slope steepness only has a small effect on the maximum flow velocity and maximum shear stress compared to the overtopping volume. This new numerical model is a useful tool to determine the hydraulic forces along the profile to find vulnerable points for cover failure and improve the design of grass-covered flood defences.

Wave overtopping can cause erosion on the landward slope due to high flow velocities and turbulence that cause high stresses on the cover. Innovative block revetments such as Grassblocks protect the subsoil of the dike against erosion. The blocks are permeable, which reduces the flow velocity and the pressures along the landward slope. The performance of these blocks is assessed in physical tests, which provides insights into the stability of the blocks. However, such experiments are expensive and accurate measurements are difficult due to highly turbulent conditions. Therefore, the goal of this study is to determine the hydrodynamic conditions at the dike cover caused by the wave run-up on the seaward slope and by the overtopping flow over the crest and landward slope. The geometry and wave conditions from the physical test at the Deltares Delta flume are implemented in an OpenFOAM^® numerical model. Using the porousWaveFoam solver, a porous layer on the crest and landward slope is implemented, where the flow resistance of this porous layer largely depends on the resistance coefficients $\alpha$ [-] and $\beta$ [-]. The numerical model is calibrated based on resistance coefficients as introduced earlier in the literature, which showed that the resistance coefficients of $\alpha =500$ and $\beta =2.0$ performed best for the peak flow velocities and the peak pressures. The numerical model is evaluated by using these resistance coefficients in other time series of the physical tests. The evaluated model is then used to determine the hydrodynamic conditions on the landward slope, which showed that the pressure was the most influential hydrodynamic condition at the time of failure. Finally, the model showed that a porosity of $n=0.6$ and the porous layer thickness $\eta =36\mathrm{mm}$ reduced the peak pressure the most.