*Antonis Mavritsakis*^{1}*, Casper Gies*^{2}*, Ilse Hergarden ^{2}, Herman Jaap Lodder^{3 }*

^{1}*Deltares, Delft, Nederland*

^{2 }*Royal HaskoningDHV, Amersfoort, Nederland*

^{3 }*Waterschap Rivierenland, Tiel, Nederland*

**SAMENVATTING**

De Waalbandijk Neder-Betuwe is de primaire waterkering gelegen tussen Wolferen en Tiel. Deze 20km lange dijk is afgekeurd en dient versterkt te worden. Royal HaskoningDHV is in opdracht van Waterschap Rivierenland (WRSL) verantwoordelijk voor het ontwerp in de Planuitwerkingsfase (VO, DO) en werkt daarbij nauw samen met Deltares. Dit artikel gaat in op het probabilistische ontwerp en toetsing van het binnentalud voor wat betreft de macrostabiliteit. Met probabilistische stabiliteitsberekeningen is aangetoond dat voor de situatie tijdens hoogwater zonder golfoverslag (overwegend diepe glijvlakken) ten opzichte van de semi-probabilistische aanpak een optimalisatie mogelijk is. Dit resulteert in kleinere stabiliteitsbermen en dus minder ruimtebeslag.

Daarnaast is gevonden dat probabilistische berekeningen voor de situatie met golfoverslag voor deze dijk tot een beter maar conservatiever resultaat leidt. De voorgeschreven semi-probabilistische aanpak (WBI 2017 kalibratielijn) is voor dit mechanisme (met ondiepe glijvlakken) te optimistisch. Ook dit resulteert in ontwerpaanpassingen; er is gekozen voor flauwere binnentaluds om dit effect te mitigeren. Cruciaal in deze analyse is de tijdsafhankelijke ontwikkeling van de waterspanning in de zanddijken als gevolg van infiltratie van het overslaande water. Deze kan vooralsnog alleen numeriek worden onderbouwd.

Met de probabilistische berekeningen is voor de binnenwaartse stabiliteit meer inzicht verkregen in het maken van een ‘slim’ ontwerp, gericht op het aanpakken van de problemen die de faalkans domineren (bij significante golfoverslag).

**KERNWOORDEN: ****Dike Probabilistic Design, Dike Reinforcement Optimization, Dike Reliability, Overtopping**

### INTRODUCTION

The dike “Waalbandijk Neder-Betuwe”, which protects the hinterland against flooding from the river Waal, needs to be improved over a length of approximately 20 km. Royal HaskoningDHV (RHDHV) was appointed by Waterschap Rivierenland (WSRL) to prepare a design for this dike improvement. This paper describes the use of probabilistic calculations in the assessment and design of the inner slope of the dike. RHDHV collaborated with Deltares on this topic. This is the second project in the Netherlands in which this type of calculation is used on a larger scale for macro-stability of the inner slope. In another project of WSRL, Streefkerk-Ameide-Fort Everdingen (SAFE), probabilistic design calculations lead to optimization of the required berm length as described in Tao et al, 2021. The methods used to assess and design the inner slope using probabilistic calculations are similar for both projects. This study focuses on the impact of the overtopping event.

The design approach for this project is that the crest level will be increased, if necessary, to a level at which the overtopping is 10 L/m/s. With this amount of overtopping, the phreatic line schematization within the dike is quite unfavorable. Plaxflow is used to investigate the impact of the duration of large amounts of overtopping. Based on non-stationary models, the phreatic line can increase to a level only just below the surface of the inner slope within 48 hours of overtopping. Therefore, in the stability analyses, the dike has been modelled as fully saturated for the event with large amounts of overtopping.

The trajectory of Neder-Betuwe (NeBe) is divided into multiple sections. For each section, semi-probabilistic stability calculations have been done to assess the current situation and design the inner slope. Schematization and set-up of the profile was done using PaCE, a parametric design tool. Probabilistic calculations were performed for a selection of profiles, making use of the semi-probabilistic D-Stability file. In the first round of calculations FORM-analyses were performed, while in the second round Monte Carlo Importance Sampling (MCIS) and the Probabilistic Toolkit (Deltares, 2021) were used.

### PROBABILISTIC ASSESSMENT

#### Problems with overtopping

A set of probabilistic calculations was performed for 10 profiles of the NeBe trajectory in order to assess the reliability of the inner slope of the dike and optimize the reinforcement design. The calculation profile set also included locations where the semi-probabilistic analysis indicated that no reinforcement was required. The probabilistic assessment results showed that 9 out of 10 profiles were not reliably safe, even in cases where the semi-probabilistic assessment indicated otherwise.

This behavior was due to the increased influence of overtopping. The probabilistic analysis lead to significantly low reliability scores when performed conditional to the overtopping schematization. In addition, deciding to not increase the crest height of the dike meant that the probability of overtopping with a rate greater than 10 L/m/s, as calculated by HYDRA-NL (Rijkswaterstaat, 2020), was significantly high. As a result, overtopping had significant contribution to the probability of failure and managed to depreciate the overall reliability results of the profiles.

Apparently, the effect of overtopping was not perceived by the semi-probabilistic analysis, as it approaches the matter of overtopping in a different fashion. According to the basis of design, the semi-probabilistic analysis estimates the required reliability in the overtopping situation as described in the KPR guideline (Kennisplatform Risicobenadering, 2018). Subsequently, the reliability requirement is converted to a safety factor requirement through the calibration line provided by WBI 2017. Figure 1 presents the relationship between the reliability index and the safety factor in the overtopping situation for the examined profile. Moreover, the WBI calibration line is plotted for the sake of comparison. It becomes apparent that for this project the WBI calibration line overestimates the reliability associated to a safety factor value at such reliability index levels. As a result, the semi-probabilistic analysis indicated that the profiles were reliably safe, as it was not properly equipped to perceive the influence of overtopping.

The findings of the first round of probabilistic analysis lead to the revision of the dike reinforcement calculation. While the initial scope of the probabilistic analysis included only the assessment of the current dike state, a reliability driven-design component was added. In this way, the impact of overtopping would be properly incorporated into the calculations and therefore in the design. Moreover, probabilistic analysis of the dike in the non-overtopping situations was expected to avoid the inherent conservatism of the standards adopted by the semi-probabilistic analysis, thus providing a more accurate estimation of the reliability of the dike and possibly leading to less conservative results.

### PROBABILISTIC DESIGN

The semi-probabilistic reinforcement design focused on extending the berm in dike profiles where the estimated safety factor in the WBN situation (Waterstand Bij de Norm) was lower than the requirement. Reinforcement optimization is achieved when the required reinforcement groundworks are minimized, while the reliability target is met. By highlighting the importance of overtopping, as well as indicating that the semi-probabilistic design approach might be conservative for non-overtopping situations, the probabilistic design approach allows for reinforcement optimization by reducing the dike slope steepness to 1:3.5. This measure specifically aims to increase the reliability of the profile in overtopping, by stabilizing the shallow failure mechanism forming on the dike slope, which is in most cases the normative failure mode in overtopping. After reducing the slope steepness, the probabilistic design approach checks the reliability of the dike overall situations. In case the profile in not reliably safe, the profile’s berm is extended until the overall reliability is sufficiently high. Essentially, the reinforcement design becomes reliability – driven.

An example of the design procedure is given in Figure 1 for the profile shown in Figure 2, demonstrating a typical pattern of behavior met in probabilistic reinforcement design. At first, both the safety factor in the WBN situation and the overall reliability index of the profile are below their respective requirements. The semi-probabilistic assessment and design identify the lack of a sufficiently high factor of safety in the WBN situation and mends it by extending the berm. However, the extended berm does not stabilize the shallow failure mechanism at the dike slope and so leads to no improvement to the overtopping behavior of the profile. Thus, the reliability index of the profile stays below the requirement, even though mitigative measures have been applied. On the other hand, the probabilistic assessment and design approach recognizes the impact of overtopping and tackles it by reducing the dike slope steepness. This measure leads to an increase of the overtopping and overall reliability indices, while having little effect on the safety factor and reliability index of the WBN situation. Hence, the reliability index is increased to levels higher than the requirement. The difference between the achieved reliability index and the requirement represents the room for reinforcement optimization. Following, a sequence of probabilistic assessments is carried out as part of the probabilistic design, each one of them evaluating the overall reliability of the profile with different berm lengths. In the optimized situation, the overall reliability index is equal to the requirement, even if this means that the WBN safety factor is lower than the respective requirement. Such a situation is acceptable, because the true requirement of dike safety is expressed in terms of reliability, while the safety factor is a construct aiming at practicality, loosely resembling the reliability of the dike.

Lastly, some profiles with a sand dike core scored barely below the overall reliability requirement. Such cases acted as motivation for further investigating the strength of the sandy dike material, which was expected to be stronger than modelled, based on experience. Additional laboratory testing provided evidence that the sandy dike material was considerably stronger than originally modelled. Therefore, the friction angle of the sandy dike material was increased in the analyses.

Ultimately, Figure 2 shows an example profile with the base assumptions of the Neder-Betuwe project. The left figure shows the semi-probabilistic design result, which includes a dike slope of 1:3 and a berm of 29.3m, while not achieving the required level of reliability. In contrast, the right figure shows the optimization achieved with probabilistic design, which has a slope of 1:3.5 and a berm shorter by 5.8m, while also meeting the reliability requirement.

### Conclusions

The insights gained in the NeBe dike reinforcement project highlight the benefits of applying probabilistic analysis. Firstly, the probabilistic dike assessment has revealed overtopping as an impactful contributor to failure for many of the dike profiles considered. In several cases, the results of the semi-probabilistic approach were not meeting the reliability requirement, even though the safety factor requirement was met. Thus, the semi-probabilistic approach would have approved profiles that were not reliably safe, solely based on the safety factor being sufficient in the WBN situation. Secondly, probabilistic design was able to achieve reinforcement optimization, by recognizing an impactful hazard and implementing an efficient tailor-made solution against it. Specifically, probabilistic design managed to tackle overtopping by reducing the dike slope steepness, and then if needed, optimize the berm length in a reliability-driven approach. Eventually, the probabilistic approach always leads to a better design than the semi-probabilistic approach, either by removing redundant conservatism, or by preventing insufficiently safe semi-probabilistic designs. Overall, from the 7 profiles that were semi-probabilistically designed to a safety factor above 1.35, only 1 that passed the integrated beta requirement. After applying an inner dike slope of 1:3.5, the berm length could be reduced on an average with 48%.

In Figure 3 the reliability index is plotted against the safety factor, comparing the results of the NeBe probabilistic analysis to data used in the WBI calibration. The reliability index presented in the calibration chart is calculated by integrating all river water levels, while the safety factor responds to the WBN river water level. The NeBe data is split into two categories: results for overtopping and results without incorporating overtopping. The latter should be similar to the WBI data, which did not include any overtopping analysis. For lower reliability indices, the overtopping results plot well above the WBI 2017 calibration line and suggest that a different line would capture better the behavior of the NeBe data. Subsequently, given a reliability index requirement for overtopping, the equivalent safety factor requirement as estimated by the WBI calibration line, will be lower than the safety factor indicated by the NeBe data trend. This leads to profiles passing the overtopping requirement in the semi-probabilistic analysis and not recognizing the extent of the impact of overtopping.

However, this insight does not mean that the WBI calibration line is inaccurate. The data used for forming the line has been gathered from analyses of profiles over the entire Netherlands. The calibration line captures the general trend, while every location is expected to have its own line, as the relationship of reliability to safety factor is defined by location-specific attributes, such as the stratigraphy, soil strength, dike geometry, etc. Therefore, the results of NeBe define a line that is specific to this location and leads to more accurate estimations.

The insights gained in the NeBe probabilistic analysis can provide some pieces of advice for future dike reinforcement projects in the Netherlands. Firstly, additional preparation for the probabilistic analysis can increase its effectiveness and accuracy. For example, setting up the SU-table model for the undrained dike material in a probabilistic mindset and investing more into determining the friction angle of the sandy dike material has led to more accurate and less conservative results, which in turn prompted to shorter berms. Secondly, probabilistic analysis can be utilized to identify the critical failure mechanisms of the dike system. In this way, the advantages and shortcomings of the reinforcement design practice were revealed. Moreover, probabilistic design can provide tailor-made reinforcement measures, which address the actual threat to dike integrity. Lastly, the reliability index-safety factor calibration line plays a fundamental role in the semi-probabilistic dike assessment. It has been shown that the difference between the location-specific and general calibration lines can be critical to the accuracy and effectiveness of the semi-probabilistic approach. Therefore, setting up the location-specific calibration line before performing the semi-probabilistic analysis can appear as an attractive alternative to adopting a full-extent probabilistic analysis, in case the application of the latter is not feasible.

**BRONVERMELDING**

- Ir. P.Y. Tao. Ing. L. Kwakman, Ir. A.W. Van Der Meer, Ir. J.A. Van Zuylen, Ir. C.J. Van Veen, PROBABILISTISCH ONTWERPEN VOOR STABILITEIT BIJ DIJKVERSTERKING SAFE, LAND+WATER NR.5 – MEI 2021
- Deltares, PROBABILISTIC TOOLKIT 2.1.18 and D-STABILITY 2021.01 – 2021
- Rijkswaterstaat, Water, Verkeer en Leefomgeving, HYDRA-NL versie 2.8 – 2020
- Kennisplatform Risicobenadering, M. De Visser, R. Jongejan, KPR FACTSHEET WERKWIJZE MACROSTABILITEIT I.C.M. GOLFOVERSLAG, OI2014v4, 2018