Sustainable Hydraulics in the Era of Global Change – Erpicum et al. (Eds.)
© 2016 Taylor & Francis Group, London, ISBN 978-1-138-02977-4
Estimating probability of dike failure by means of a Monte Carlo approach
R. Van Looveren, T. Goormans & J. Blanckaert
International Marine and Dredging Consultants nv, Antwerp, Belgium
Kristof Verelst & Patrik Peeters
Flanders Hydraulics Research, Antwerp, Belgium
ABSTRACT: The European Flood Directive 2007/60/EC requires Member States to assess the flood risk along
their water courses and to take adequate measures to reduce this risk. Often flood risk maps are developed by
only taking into account dike overtopping. However, floods can also be caused by dike failure. Not taking this
into account underestimates the flood risk. This paper presents a method to estimate the probability of dike
section failure, taking into account several failure mechanisms. The method follows a Monte Carlo approach, in
which all relevant parameters for the hydraulic loads as well as for the considered resisting forces vary according
to specific distributions. Both the loads and failure mechanisms are described using numerical models. This
method has been successfully applied on various dike segments along navigable water courses in Flanders, and
in the Periodic Safety Review of the nuclear power plant of Doel.
1
2 THE METHODOLOGY
INTRODUCTION
The European Flood Directive 2007/60/EC requires
Member States to assess the flood risk along their
water courses and coast lines, to map the flood extent
together with assets and humans at risk in these areas,
and to take adequate measures to reduce this risk.
Often flood risk maps are developed, in which the
probability of the flood extent is quantified by only
taking into account dike overtopping. However, structural dike failure according to different mechanisms
can also cause floods. Not taking these mechanisms
into account underestimates the flood risk.
This paper presents a method to estimate the probability of failure of a dike section, taking into account
several failure mechanisms. The method follows a
Monte Carlo approach, in which all relevant parameters for the hydraulic loads as well as for the considered
resisting forces vary according to specific distributions. Besides these structural parameters, also the
loads vary according specific distributions. Both the
loads and failure mechanisms are described using
physically-based numerical models.
The method has been applied to various dike segments along navigable water courses in Flanders, and
in the Periodic Safety Review of the nuclear power
plant of Doel along the Scheldt estuary. Results of the
method can be used in impact analyses and flood risk
mapping.
This paper first describes the methodology, then
gives some practical examples on the implementation,
and ends with conclusions and recommendations.
2.1 General
The failure of a dike section is evaluated by comparing the loading forces R to the resisting forces S, both
described through numerical models.
The dike section fails if the load is higher than the
resistance, i.e. if R – S > 0 (limit state equation). There
is a different limit state equation for each of the five
failure mechanisms considered here: macro instability (or global instability), micro instability (or local
instability), piping, and erosion of the inner slope and
erosion of the outer slope. Section 2.2.2 explains these
failure mechanisms in more detail.
To take into account the existing uncertainty on
the load as well as on the resistance, each of these
five limit state equations is calculated for different
sets of loads and resistances. At the load side, this
set is constituted by a probabilistic set of synthetic
storms (see sections 3.1.3 and 3.2.3 for examples). At
the resistance side, a stratified sampling technique on
the resistance parameters is performed to compose the
different sets. Since the probability distribution function (pdf) is known, the probability of each set can be
determined.
Physically-based numerical models are then used
(i) to transfer the load at the boundary conditions of a
region to the specific dike section and (ii) to evaluate
the different failure mechanisms.
By calculating each of these five limit state equation
for several combinations of load sets and resistance
sets (Monte Carlo approach), the full spectrum of
896
possible combinations can be considered, and hence
the global probability of failure can be determined.
2.2
Load
2.2.1 General
Figure 1 shows in more detail the methodology at the
load side (R). This figure is explained from bottom to
top in this and the following section.
The load at a dike section x (bottom of the figure)
is defined by the water level variation, the position
of the phreatic line within the dike and the wave
conditions (defined by the wave amplitude H s and
wave period T p ). Long-term measurements of all these
variables, necessary for the statistical approach, are
seldom available at the dike location. However, by
means of (physically-based) numerical models the
load at the boundaries (based on a statistical analysis of
long-term measurements) available at other locations
can be transferred to the dike location.
2.2.2 Used models
Water level and flow velocity at the dike segment is
calculated by hydrodynamic models (1D or 2D) using
the relevant boundary conditions.
The position of the phreatic line within the dike,
which is of prime importance for the global and local
stability, can be calculated using a finite element software package, e.g. SEEP/W (Geo-Slope 2012). These
calculations however are time consuming, making it
impractical for the Monte Carlo approach. Therefore
the methodology relies on the creation of response
surfaces based on a priori SEEP/W calculations for
different conditions. This way fast computation time
(in the Monte Carlo simulations) is combined with relatively high accuracy. Separate response curves were
established for tidal and non-tidal rivers. For each river
type the phreatic line was schematised using a limited
number of points. An example for a tidal river is shown
in Figure 2, using four points: one point at the inner
side of the dike (4), two points at the outer side corresponding to high and low water (1, 2) and one point
within the dike body indicating intrusion of the tidal
influence in the dike (3). The position of the phreatic
line depends mainly on the dike geometry, dike permeability and the shape and duration of the storm.
These parameters were varied in numerous SEEP/W
calculations, and the positions of the four points was
expressed as a function of the input parameters. A similar approach was used for non-tidal rivers, of course
resulting in different response curves.
For the calculation of the wave load the formula
of Bretschneider (TAW 1985) can be used. With this
formula wave characteristics (H s and T p ) can be determined from wind speed (ws ), wind direction (wd ),
water depth, and fetch length. It is assumed that wind
data from the nearest station are representative for the
dike location as well, i.e. no model is used to transfer the wind load from the boundary conditions to the
dike location, as opposed to the hydraulic load. It is
Figure 1. Determination of the load (R).
Figure 2. Schematization of the phreatic line for a tidal river.
expected that little gain in accuracy is achieved by calculating the nearby wind velocity field, compared to
the extra effort.
2.2.3 Boundary conditions
The statistical information in the boundary conditions
is based on a multivariate frequency analysis of the
principal variables – i.e. taking into account crosscorrelations when appropriate – while the secondary
variables are kept constant. The distinction between
897
value. The higher the number of samples the better
the accuracy but also the more computation time is
needed. Taking five samples was considered adequate.
Figure 3. Determination of the resistance (S).
principal and secondary variables is based on a sensitivity analysis. This makes it possible to determine the
relative share of each variable in the resulting statistical
distribution.
For the tidally influenced Scheldt estuary, the wind
speed ws (t), wind direction wd (t), and water level h(t)
are principal variables, while the upstream discharge
Q(t), the duration d(t) and the storm surge steepness
s(t) are of secondary importance, and can be treated
conditionally. For the more upstream rivers discharge
is the principal load variable, while the wind speed and
direction are secondary variables.
2.3
Resistance
2.3.1 General
Figure 3 shows more details on the methodology at
the resistance side (S). The value of each geotechnical parameter varies according to a pdf, which can
be approximated by normal or lognormal pdfs for all
considered parameters. Hence it is fully described by
a standard deviation σ and mean value µ.
When data is available the pdf is determined based
on a statistical analysis. In absence of data, often the
case for geotechnical variables, the pdfs are determined based on expert judgment and literature data
(TNO 2003b, Kortenhaus et al. s.d.).
Since it is not possible to consider the full, continuous domain of resistance parameters, the pdf is
discretised using stratified sampling. For each parameter of the considered failure mechanism a discrete
number of samples is taken from its associated pdf.The
probability of occurrence of a combination of resisting
parameters is then calculated from the combination of
the respective probabilities of each single parameter
2.3.2 Used models
The following section gives more information on the
considered failure mechanisms and the models used
to describe them. A more extensive description of failure of dikes can be found in standard text books, e.g.
Pilarczyck (1998).
Global instability of the inner and outer slope is
calculated by Bishop’s method (Bishop 1955, Verruijt
1995), which requires the geotechnical parameters
cohesion (c [kN/m2 ]), friction angle (ϕ [◦ ]), wet soil
weight (γn [kN/m3 ]), dry soil weight (γd [kN/m3 ])
and permeability (k [m/s]) as input. Since the Bishop
calculations are time consuming the number of parameters can be reduced based on a sensitivity analysis. For
a dike with a core of sand it appeared that only parameters ϕ, k and γn should be varied, for a clay core the
parameters c, ϕ, and γd . Global instability is evaluated
at both the inner and outer side.
Local instability, which occurs when the phreatic
pressure is too high, in turn causing the (impermeable)
top layer to be pushed off, is evaluated by verifying
the equilibrium of active and passive forces on the
upper layer of the dike and/or the revetment. Following parameters are considered: thickness (d top [m]),
weight (γtop [kN/m3 ]), cohesion (ctop [kN/m2 ]) and
friction angle (ϕtop [◦ ]) of the upper clay layer and
thickness (d rev [m]) and density (ρrev [kg/m3 ]) of the
revetment (if present). Local instability is evaluated at
both the inner and outer side.
Piping can occur when the water level difference
over the dike is too large, causing a flow underneath
the dike that erodes the particles of the base material. Sellmeyer’s method (Sellmeyer 1988) is used for
evaluating piping, based on the following parameters:
thickness (d top [m]) and specific weight (γtop [kN/m3 ])
of the upper clay layer, sand aquifer thickness (Ds
[m]), permeability (k s [m/s]) and specific weight (γs
[kN/m3 ]) of the porous material, and grain diameter
exceeded by 30% of the grains (d 70 [m]). The rolling
resistance angle (θ [◦ ]) and White’s constant (η [−])
are taken from Sellmeyer (1988).
Water flowing along the dike, or run-up of (windinduced) waves can cause erosion of the outer slope.
The resistance, and the corresponding limit state equation, depend on the revetment, e.g. grass, rip rap or
a combination of both (rip rap at the bottom side of
the dike and grass at the higher side). The limit state
equation for rip rap is based on Pilarczyk’s formula
and uses the d 50 [m] of the rip rap as main parameter
(Pilarczyk 1998). The limit state equation for the grass
revetment is based on the method of Seijffert & Verheij
(1998) and uses the thickness of the grass revetment
(d g [m]) and the underlying clay layer (d k [m]), as
well as the quality of both (expressed by coefficients
cE [ms] and cRK [ms]) as input. Also the Manning
coefficient (n [sm−1/3 ]) of the inner side material and
the friction coefficient (f [−]) of the outer side are
898
to be estimated. In a second phase the methodology
was extended to also incorporate other types of revetments, such as open stone asphalt, concrete layers, and
gabions.
Erosion of the inner slope can occur due to overflow and/or wave overtopping. Overflow is calculated
using the weir equation and wave overtopping using
the TAW methodology (TAW 2002). Likewise the erosion mechanism on the outer slope, the strength and
the limit state equation depend on the material on the
slope. For grass again the method from Seijffert &
Verheij (1998) is used, while formulas for most other
materials are based on Pilarczyk (1998).
In this equation 7 is the number of failure mechanisms considered here (global and local instability on
the inner and outer side counting as separate mechanisms). The total failure probability of the dike section,
p, is calculated by:
2.4
2.5
Global failure
Each failure mechanism can be described as a function of a number of ‘resistance’ parameters which can
vary according to associated probability distributions,
resulting in a statistical variation of the resistance S
of the dike section for each failure mechanism. If the
probability distribution of each parameter is known,
also the probability of a certain combination of parameters is known. Also the probability of the load R –
through the synthetic events (see sections 3.1.3 and
3.2.3 for examples) – is known. By combining a resistance S with a load R, the probability of failure can be
calculated – if the limit state equation is exceeded –
based on the probability of both R and S. When doing
this for all possible combinations of R and S and for
each failure mechanism, the whole range of probabilities is covered and the global failure probability of the
considered dike section is calculated.
The above will now be explained in a more formal
way. Denote the probability of a hydraulic boundary
condition by pRj , with j between 1 and the number of
synthetic load events (m), and denote the probability
of a set of geotechnical parameter values associated
with failure mechanism k by pSki , with i between 1 and
the number of geotechnical parameter sets of failure
mechanism k (nk ), with k between 1 and the number
of failure mechanisms (7 in this study).
The size of nk depends on (i) the number of stratified samples (5 in our study) and (ii) the number of
parameters l k used to describe failure mechanism k:
The number of parameter sets hence is different for
each failure mechanism k.
For each hydraulic load condition j and for each
failure mechanism k the limit state equation associated
with mechanism k is verified. The resulting failure
probability pjk is the sum of the probabilities of those
geotechnical parameter sets that lead to dike failure:
The failure probability of the boundary condition j
for all failure mechanisms pj is then calculated as:
The resulting return period T of dike failure can be
calculated as:
BRES software
To execute the necessary calculations a software tool
called BRES (Dutch for ‘breach’) was developed
as part of project WL_11_28, funded by Flanders
Hydraulics Research (FHR).
BRES streamlines the many calculations that are
required for the Monte Carlo approach. Its input and
output consists of simple text files; graphical information is available as well. A graphical user interface
enables a more intuitive set-up of the input and output.
3 APPLICATIONS
The method elaborated above was applied to different dike segments in Flanders. The following sections
describe only a few aspects of these applications in
detail. A full account of the investigations is written in
IMDC (2014a, b, 2015a) for the Meuse River, and in
IMDC (2015b) for the Scheldt River at Doel.
3.1
Meuse river
3.1.1 Context
In this study, commissioned by FHR, the methodology was applied to the stretch of the Meuse River at
the border between Belgium and The Netherlands, the
so-called Grensmaas (Fig. 4). It should be noted that
the considered dikes are located at the border of the
winter bed of the river Meuse. Consequently the distance between the river and the dike can be large (up
to more than 1 km).
The tested locations (six in total) were identified
based on an inventory of general data of the dike,
such as dimensions, presence of a revetment, and core
material (IMDC 2014a).
3.1.2 Available data
Topographically measured cross sections every 100 m
were available. Data of some sections had to be completed with the digital elevation model (DEM) of Flanders created by the Flanders Geographical Information
Agency (AGIV in Dutch) (AGIV 2004), because of the
width of the winter bed. Data from cone penetration
tests (CPTs) at regular spatial intervals of the dike were
available, as well as borehole data. Rijkswaterstaat
899
Figure 5. Synthetic events for the Meuse River. The
depicted hydrographs are the mean hydrograph plus normal
variation.
Figure 4. Overview of the study area (Meuse River in
Flanders). Image source: Google, TerraMetrics.
(the Dutch waterways authority) provided the discharge time series from the station of Borgharen, just
upstream of the study area. Wind data from the Royal
Dutch Meteorological Institute (KNMI) were available
from the station of Beek.
3.1.3 Load
To determine the water level and flow velocity at
the studied locations, Flanders Hydraulics Research
performed simulations with a 2D hydrodynamic
WAQUA model of the Meuse, using SIMONA version 2011. The WAQUA model was made available by
Rijkswaterstaat.
The model boundary conditions consisted of synthetic hydrographs, derived from a statistical analysis
of the available discharge time series.
The synthetic hydrographs were scaled versions of a
unit hydrograph, determined from the 20 most extreme
events in Borgharen. This number was decided to be a
suitable compromise between using sufficient extreme
hydrographs on the one hand, and using hydrographs
that are sufficiently extreme on the other hand. The
variation in the hydrographs of these events was taken
into account by applying a normal variation on the
‘mean’ hydrograph, for each value of the discharge.
Figure 5 shows an example. More information about
the derivation of the synthetic hydrographs is described
in IMDC (2014b).
In total, 55 different hydrographs were simulated
with the WAQUA model. Since no correlation was
Figure 6. Profile of CPT results along the Meuse dike at
location MM174.
found between the occurrence of extreme discharges
and extreme wind speeds, it was sufficient to only
simulate the synthetic hydrographs, and combine the
resulting water levels and velocities with the 12 wind
direction classes and 11 wind speed classes (different
for each direction). This added up to a total of 7200
synthetic events for the dike sections (7200 and not
7260 because in the discretization of the multivariate
distribution, the combination of the lowest wind and
discharge class is not considered), each with its own
probability of occurrence.
The phreatic line and wave load were calculated
with the methods described in section 2.2.2.
3.1.4 Resistance
The geotechnical parameters were derived from the
CPTs and the lab results of the borehole samples.
For some locations, several CPTs were executed close
enough, enabling to derive a mean value and variation instead of having to rely on literature values. For
instance, Figure 6 shows a profile of CPT results in
the vicinity of MM174. The presence of a layer with
higher resistance (indicated between the red lines) is
clear. The layer can be classified as coarse sand/gravel.
For other locations however, values from other locations along the Meuse, or from literature, had to be
assumed.
900
Table 1.
Failure probability of section MM89, Meuse River.
Failure
mechanism
No. of
computations
p
1/year
T
year
global instability outer slope
global instability inner slope
piping
erosion outer slope
erosion inner slope
global failure
53 × 7200 = 9.0E+05
53 × 7200 = 9.0E+05
58 × 7200 = 2.8E+09
54 × 7200 = 4.5E+06
55 × 7200 = 4.5E+06
5.8E−04
0
0
3.4E−03
8.6E−04
4.8E−0.3
1.72E+03
∞
∞
2.97E+02
1.16E+03
2.08E+02
The models used to calculate the limit state are
described in section 2.3.2. For the Meuse case, the
failure mechanism of local instability was not considered because the dike sections are homogeneous, i.e.
there is no top layer that can be pushed of by excessive
phreatic pressure.
3.1.5 Results
Table 1 shows the results of the BRES calculations for
one of the studied sections (MM89).
The insensitivity to piping could be expected since
the underground along the Meuse dike is characterised
by a layer of coarse sand/gravel, which is beneficial for
its resistance to piping.
The relatively high probability of failure for erosion
at the outer slope is due to the fact that there is no
variation for wind taken into account. Since the events
last for almost 4 days, the assumption of a constant
extreme wind speed – and correspondingly extreme
waves – is likely to be conservative.
Eventually, it can be seen that the inner slope is stable, a result found in the other sections as well. The
dikes along the Meuse winter bed are relatively low
on the inner side. Moreover the dike core consists of a
material described as ‘sand-loam’, a mixture of sand
and loam, which has both a low hydraulic conductivity and high resistance characteristics (ϕ and c), two
elements resulting in a very stable slope.
3.2
Scheldt river at Doel
3.2.1 Context
The Maritime Access Division of the Flemish Government is preparing the construction of a new area with
controlled reduced tide (CRT) at the left bank of the
Scheldt River, the so-called CRT area ‘Doelpolder’.
The newly designed ‘ring dike’ of the area is close
to the nuclear power plant (NPP) of Doel (15 km
northwest from Antwerp). Figure 7 shows the study
area.
The ring dike was submitted to the same type of
failure analysis as the already existing dikes around
the NPP, which were analysed in the framework of the
period safety review, the so-called ‘stress test’ (Electrabel 2011, FANC-Bel V 2011). The study also took
into account a sequential failure of the Scheldt dike
and the ring dike.
Figure 7. Planned layout for the CRT area Doelpolder. The
Scheldt dike is currently existing, the to-be-constructed ‘ring
dike’ will surround the CRT area.
The results will be used in the new stress test of
the plant, and will serve as input for the design of the
surrounding water evacuation infrastructure.
3.2.2 Available data
A local DEM of the projected (designed) situation was
available. Outside the project area, the DEM of Flanders (AGIV 2004) was used (Fig.7). Data from CPTs
along the alignment of the future ring dike were available, as well as some borehole data, approx. 1–1.5 km
southwest of the area.The structure of the ring dike was
derived from available design plans. Wind data were
taken from the measurements inVlissingen, performed
by the KNMI.
3.2.3 Load
To determine the hydraulic load on the Scheldt dike,
synthetic events (see further below) were simulated
with the existing Sea Scheldt model (IMDC et al.
2003a, b, IMDC 2010). This model is implemented
in the Mike11 software (DHI, Denmark). For the ring
dike, a Mike11 model, coupled to this Scheldt model,
was made available by Flanders Hydraulics Research
(Coen et al. 2013).
The boundary conditions for the hydrodynamic
model consisted of 544 synthetic storm events, derived
as explained in IMDC (2007) and Blanckaert et al.
(2015). Basically, each event was characterized by
a water level class, a wind speed class, and a wind
direction class. For the Sea Scheldt, there is a strong
901
correlation between the downstream (extreme) water
level and the wind speed and direction. A perfect
correlation was assumed, so each of the 34 chosen
water level classes corresponded to one wind speed
value. The wind direction was taken separately into
account, divided into 16 classes, resulting into a total of
544 synthetic events. The probability of occurrence of
each event was derived from the directional frequency
curves of the water levels. As an example, Figure 8
shows the downstream boundary water levels of the
model for western winds.
As in the Meuse case, the wave load on the Scheldt
dike was calculated using Bretschneider’s formula
(TAW 1985). For the ring dike results from a SWAN
model (TUDelft 2010), set up by FHR (Coen et al.
2013), could be used.
3.2.4 Resistance
The angle of repose ϕ was derived from the CPT data.
Also, because more than one CPT was performed, the
distribution of ϕ could be estimated as well, instead
of having to rely on literature. The resulting values
were: ϕ = 27.9◦ (σ = 1.67◦ ), c = 0 kN/m2 (σ = 0◦ ). Of
course, for the to-be-constructed ring dike no data
were available. Values were taken from the design
calculations: ϕ = 30.8◦ (σ = 3.4◦ ), c = 0 kN/m2 (no
variation). Other geotechnical parameters, such as particle size, specific weight or hydraulic conductivity,
were derived from laboratory analyses of the borehole
samples, performed in the framework of the project,
or using ‘typical values’ from literature if necessary.
Figure 8. Downstream boundary water levels for synthetic
storm events, in this case corresponding to western winds.
Table 2.
The models used to calculate the limit state are
described in Section 2.3.2.
3.2.5 Results
Table 2 shows the most important results of the BRES
calculations for the Scheldt dike.
It can be argued whether it was necessary to perform
the calculations for the failure mechanisms showing zero probability, because from beforehand it was
known that the resistance was many times higher than
the load. However because of the proximity of the NPP
Doel it was requested to explicitly consider all failure
mechanisms.
Also, the global failure might appear low. However,
a sequential failure analysis on the ring dike segments ‘EAST’, ‘SOUTH’ and ‘WEST’ was performed
as well. This resulted in a return period of failure
of respectively 4.84E+05, 5.43E+02 and 3.45E+04
year. A consequence analysis was performed if a
breach would develop in the southern section, using
a 2D surface runoff model (IMDC 2015b). From that
analysis, it appeared that the key parts of the NPP site
are not compromised, because the site’s platform level
is at 8 m TAW and higher. Non-crucial parts, such as
the visitors parking, do flood during extreme events.
4
CONCLUSIONS AND
RECOMMENDATIONS
A probabilistic method was presented to determine
the probability of dike failure. This method considers
seven different failure mechanisms, and uses multivariate analyses and stratified sampling as statistical
instruments. Moreover, it uses several physicallybased numerical models (or response curves derived
from calculations with these models) for the failure
evaluation and the transformation of boundary conditions to hydraulic loads at the location of the dike
section. The methodology was successfully applied at
different dike segments in Flanders, both on tidal and
non-tidal rivers.
The probabilistic approach calculates the global
probability of failure of the dike section, but also the
probability related to the different failure mechanisms
Failure probability of the considered section of the Scheldt dike.
Failure
mechanism
No. of
computations
p
1/year
T
year
global instability outer slope
global instability inner slope
local instability outer slope
local instability inner slope
piping
erosion outer slope
erosion inner slope
global failure
53 × 544 = 6.8E+04
53 × 544 = 6.8E+04
55 × 544 = 1.7E+06
55 × 544 = 1.7E+06
56 × 544 = 8.5E+06
54 × 544 = 3.4E+05
55 × 544 = 3.4E+05
2.29E−03
0
9.80E−06
6.76E−10
0
0
0
2.30E−03
4.37E+02
8
1.02E+05
1.48E+09
8
8
8
4.34E+02
902
separately. In addition, the full spectrum of load and
resistance is considered, instead of only the most conservative ones. These elements provide a clearer view
on those parts of the dike that require additional focus
for possible reinforcement.
The methodology considers many phenomena in
detail. Still, further improvement is possible. At the
load side ship-induced waves and currents as well as
a varying wind profile could be incorporated. At the
resistance side, residual strength could be considered
(at this moment the dike is already considered as ‘failing’ at initiation of the failure mechanisms), additional
dike revetment materials could be included, or the phenomenon of water infiltration into the dike could be
considered.
REFERENCES
AGIV. 2004. Digitaal Hoogtemodel Vlaanderen.
Bishop, A.W. 1955. The use of the slip circle in the stability
analysis of earth slopes. Géotechnique, 5(1): 7–17.
Blanckaert, J., Leysen, G., Franken, T., Gullentops, C., Fang,
Z., Pereira, F. & Swings, J. 2015. Generation of sets of
synthetic events by multivariate stratified sampling. Proc.
36th IAHR World Congress, 28 June–3 July 2015. The
Hague, The Netherlands.
Coen, L., Suzuki, T., Altomare, C., Plancke, Y., Peeters,
P., Taverniers, E., Verwaest, T. & Mostaert, F. 2013.
Advies dijkhoogtes Doelpolder: Deelrapport 1 – Aanvullende scenarioberekeningen met betrekking tot de
veiligheid tegen overstromingen in het kader van de
aanleg van het GGG Doelpolder. Version 2_0. WL Rapporten, 12_130 (in Dutch). Flanders Hydraulics Research,
Antwerp, Belgium.
Electrabel. 2011. Kerncentrale Doel: Rapport weerstandstesten – Bijkomende veiligheidsherziening, 31 October
2011 (in Dutch).
FANC-Bel V. 2011. Belgian Stress tests: National Report
for Nuclear Power Plants. Federal Agency for Nuclear
Control.
Geo-Slope. 2012. SEEP/W. Groundwater Seepage Analysis.
IMDC. 2007. Onderzoek naar maximale waterstanden van
de Schelde te Doel. Study for Tractebel Engineering.
I/RA/14098/06.152/JBL (in Dutch).
IMDC. 2010. PSR Doel 2010 Flood Hazard: Technisch
Rapport. Study for Tractebel Engineering. I/RA/14157/
10.198/JBL (in Dutch).
IMDC, Tractebel Engineering & Jan Maertens bvba.
2010. Onderzoek naar de bresgevoeligheid van de
Vlaamse winterdijken – Deelopdracht 5: Opstellen
van een wetenschappelijk verantwoorde en praktisch
haalbare methode. For Flanders Hydraulics Research.
I/RA/11279/08.009/RVL (in Dutch).
IMDC. 2014a. Het berekenen van de faalkansen van
de winterdijken langs de Maas – Deelopdracht 1:
Inventarisatie Maas. For Flanders Hydraulics Research.
I/RA/11400/13.111/MOE (in Dutch).
IMDC. 2014b. Het berekenen van de faalkansen van de
winterdijken langs de Maas – Deelopdracht 2: Verbetering probabilistische methode. For Flanders Hydraulics
Research. I/RA/11400/13.181/TGO (in Dutch).
IMDC. 2015a. Het berekenen van de faalkansen van
de winterdijken langs de Maas – Deelopdracht 3:
Probabilistische faalkansberekening van de winterdijken langs de Maas. For Flanders Hydraulics Research.
I/RA/11400/13.301/TGO (in Ducth).
IMDC. 2015b. Stresstest Kerncentrale Doel i.h.k.v. GGG:
Gevolgstudie bij dijkfalen. Study for Maritime Access
Division. I/RA/14209/14.217/TGO (in Dutch).
Kortenhaus, A., Weissmann, R., Oumerai, H. & Rihwein, W.
s.d. Probabilistische Bemessungsmethode für seedeiche:
Prodeich (in German).
Pilarczyk, K.W. 1998. Dikes and Revetments. Rotterdam:
Balkema.
Seijffert, J.W. & Verheij, H. 1998. Grass covers and reinforced
measurements. In Krystian W. Pilarczyk (ed.), Dikes and
Revetments: 289–302. Rotterdam: Balkema.
Sellmeyer, J.B. 1988. On the mechanism of piping under
impervious structures, PhD thesis Delft.
TAW. 1985. Leidraad voor het ontwerpen van rivierdijken: Deel 1, Bovenrivierengebied. Staatsuitgeverij, ’sGravenhage, ISBN: 90-12-05169-X (in Dutch).
TAW. 1998. Technisch rapport erosiebestendigheid van
grasland (in Dutch).
TAW. 2001. Technisch rapport TR-19: Technisch Rapport
Waterkerende Grondconstructies (in Dutch).
TAW. 2002. Technisch rapport golfoploop en golfoverslag bij
dijken (in Dutch).
TAW. 2004. Technisch rapport waterspanningen bij dijken (in
Dutch).
THV Sigma. 2003b. Actualisatie van het Sigmaplan –
Deelopdracht 3: Hydrologische en hydraulische modellen, Volume 2a: Hydraulica Scheldebekken. Study for
the Sea Scheldt division of the Ministry of the Flemish
Community. I/RA/11199/03.003/SME (in Dutch).
THV Sigma. 2003c. Actualisatie van het Sigmaplan –
Deelopdracht 3: Hydrologische en hydraulische modellen,
Volume 2b: Hydrodynamisch model Rupelbekken. Study
for the Sea Scheldt division of the Ministry of the Flemish
Community. I/RA/11200/03.002/SME (in Dutch).
TNO. 2003a. Theoriehandleiding PC-Ring. Versie 4.0. Deel
A: Mechanismenbeschrijving. (in Dutch).
TNO. 2003b. Theoriehandleiding PC-Ring. Versie 4.0. Deel
B: Statistische modellen. (in Dutch).
TUDelft. 2010. SWAN (Simulating WAves Nearshore):
a third generation wave model. Copyright © Delft
University of Technology. Available free of charge at
http://www.swan.tudelft.nl.
Verruijt A. 1995. Computational Geomechanics. Dordrecht:
Kluwer Academic Publishers.
903