In the Netherlands, dikes are the most commonly used structures to retain water and provide safety against flooding. In order to ensure the safety of these dikes against a number of different failure mechanisms such as backwards erosion piping, wave overtopping and macro stability, a detailed safety assessment has been created. The focus of this thesis is on the failure mechanism known as piping. Which consists of three different sub-mechanisms; uplift, heave and backwards erosion piping. The safety assessment against piping consists of a multi-step approach. First an elementary assessment is done, followed by a detailed assessment of the different sub-mechanisms and lastly a final analysis is done using more detailed data and software. Currently, Sellmeijer’s design rule is used in the detailed safety assessment to design against the sub-mechanism backwards erosion piping. His design rule is the starting point of this thesis. Before Sellmeijer was able to derive his design rules, he had to make a number of different assumptions and simplifications. The most significant being, simplifying the three-dimensionality of the piping process to a two-dimensional process, by neglecting the meandering nature of the pipe and excluding the effect of the lateral groundwater flow on the pipe progression. By limiting the problem to twodimensions Sellmeijer derived a mathematical model on which his design rule was based. In his design rule Sellmeijer relates the progression of a pipe to a hydraulic gradient. Over the years, Sellmeijer’s design rule has been adapted three different times. Each adaptation resulted in a more accurate design rule accounting for more and more parameters. However, with each adaptation, Sellmeijer’s design rule did not become more transparent. During the derivation of his design rule, Sellmeijer fails to account for the effect of a leaky layer on the critical hydraulic gradient for pipe progression. This limitation in his design rule is the focus of this thesis. In order to determine if leakage affects the critical hydraulic head for pipe rogression, a study is done using a two dimensional groundwater model in which his design rule is implemented. From these simulations it was observed that the presence of a leaky layer results in an increase in the critical hydraulic head. In the simulations studied, a maximum of 12 % increase is observed. Next an analysis is done on the results of these simulations. On the one hand an attempt is made to determine if and how leakage can be included in Sellemijer’s design rule. The simulations show that the effect of leakage on increasing the critical hydraulic gradient for pipe progression is largely dependent on the hydraulic conductivity of the cover layer. However, in order for a leakage term to be determined, and included in Sellmeijer’s design rule, additional simulations should be done. In MSeep it was also observed that leaky layers significantly impact the groundwater flow under the dike. The effect of this was quantified by looking at the effective depth. Which is defined as the portion of the aquifer which is responsible for groundwater flow towards the pipe and uplift channel. In order to further analyse the effect of leakage on groundwater flow, a number of quasi 3D and 3D scenarios were modelled in iMOD. For these simulations the effective depth, the effective width (in 3D) and the discharge in the well is determined. In this way, the effect of leakage can be quantified. In all three of the models similar trends are observed: a increase in the hydraulic conductivity of the cover layer resulted in the decrease of the effective depth. However, in the full 3D model the impact of leakage on the groundwater flow resulted in a larger range in the results. The results of this thesis, show that derivation of a leakage term would greatly benefit the output of the design rule. As a ’quick fix’ it is recommended to derive a leakage term which can be included in Sellmeijer’s design rule. However, further developments of a 3D model which correctly implements the effect of a pipe on the groundwater flow is also beneficial.

In most current dike assessments only the stationary water levels are investigated in the assessment of the stability of the inner slope, while there are differences for all kind of dikes between the stationary and transient pore water pressures and therefore in the stability. This results in a conservative probability of failure, while determination of a probability of failure should not be conservative but should be as realistic as possible. When time dependency is included in a calculation, an average flood duration is used, while the flood duration is highly variable. The following research question is defined to address the problem: “What is the influence of the flood duration on slope stability and in what degree affects the flood duration the design?” The degree of influence of time dependency on the pore water pressures and slope stability depends on dike characteristics, flood wave characteristics and the delay in failure. The basis for answering the research question is the software SEEP/W to model the time dependent pore water pressures and the software SLOPE/W to calculate the safety factor for the stability of the inner slope. In the research theoretical dike are used and there is focused on the flood waves in the Rhine and Meuse. A correlation analysis is performed to get insight in the contribution of different flood wave shape variables to the safety factor. And a probabilistic analysis is performed using transient and stationary water levels to know the differences in probability of failure between taking the shape of a flood wave into account or not. In both probabilistic analyses is varied in the permeability and the strength of the material; the shape of the flood waves is varied in the transient analysis. In this way the contribution of the flood to the probability of failure can be quantified. Dike characteristics The differences in pore water pressure are especially large for dikes that consist of an impermeable material such as clay. When only the subsoil consists of clay, larger differences are expected than when only the dike body consist of clay. However, large differences in pore water pressures do not necessary lead to large differences in the safety factor. The largest differences in safety factor are obtained when uplifting of the hinterland takes place during the stationary state and/ or during the passage of a flood wave. A transient calculation is therefore most useful for dikes with an aquifer and a thin (thinner than 5 m) weak (low POP values) hinterland. Flood wave characteristics The differences in safety factor during a permanent water level and the passage of a flood wave are large when no stationary conditions are reached during the passage of a flood wave. This is the case for high and short flood waves. Both in the Rhine and Meuse, the amount of short waves (< 7days) is high, which increases the influence of a time dependent calculation. Also, the importance of a time dependent calculation increases when the response to the increased pore water pressures is delayed caused by the permeability of the material. The influence of the height of a flood wave on the stability increases when the soil is permeable. Delay in failure Time dependency causes failure of the embankment to not occur simultaneously with the maximum wave height. The flood wave is decisive for the dike failure, but the permeability and the strength of the dike determines the moment of failure. Influence on design Taking time dependency into account leads to higher safety factors and lower probabilities of failure with exception for dikes that consist completely out of sand. For these types of dikes, the probability of failure and safety factors are the same order of magnitude. This could affect the design, because the dikes are safer when time dependency is considered. The strength of the material is the largest contributor to the distribution of safety factors and therefore to the probability of failure (60-95%). Whereas the contribution of the permeability to the probability of failure is small (2-12%), the variation in the height and duration of a flood wave contribute for 2-20% to the probability of failure. In a permeable dike this contribution is mainly determined by the height of a flood wave, while in an impermeable dike the duration of a flood wave is of importance. Considering the influence of time in stability probabilities of failure, this research proved that probabilities of failure taking the duration into account differ significantly from stationary calculations. It is therefore useful to take time dependency into account when determining the correct safety factor for impermeable dikes, but it is not useful in determining the correct safety factor for permeable dikes, because a stationary calculation is sufficient. In clay dikes it is useful to take the variation in height and duration into account, while for a sand dike it is sufficient to only consider the variation in height of a flood wave. When the variation of the duration of a flood wave is not considered, it is recommended to use a representative duration of a flood wave; that results in the same total probability of failure as when the variation of the duration is included. At Lobith the duration of the representative flood wave varies from 13 - 16 days. At Borgharen the representative duration varies between the 10 – 11 days for different dike types.