Natural Hazards

, Volume 61, Issue 2, pp 673–687

Model for prediction of sea dike breaching initiated by breaking wave impact

Authors

    • Department of Hydraulic EngineeringWestpomeranian University of Technology
  • Hocine Oumeraci
    • Leichtweiß, Institute of Hydraulic Engineering
Original Paper

DOI: 10.1007/s11069-011-0054-8

Cite this article as:
Stanczak, G. & Oumeraci, H. Nat Hazards (2012) 61: 673. doi:10.1007/s11069-011-0054-8

Abstract

A computational model system is proposed for the prediction of sea dike breaching initiated from the seaward side by breaking wave impact with the focus on the application of the model system for the estimation failure probability of the defence structure. The described model system is built using a number of existing models for the calculation of grass, clay, and sand erosion. The parameters identified as those having the most significant influence on the estimation of the failure have been described stochastically. Monte Carlo simulations to account for uncertainties of the relevant input parameters and the model itself have been performed and the probabilities of the breach initiation and of the full dike breaching have been calculated. This will form the basis to assess the coastal flood risk due to dike breaching.

Keywords

Coastal floodsSea dikes breachingUncertainty analysisMonte Carlo simulation

1 Introduction

Flood risk is often defined as the product of the (1) predicted flooding probability obtained from risk sources and risk pathways and (2) expected damages and losses that depend on the vulnerability of the flood-prone areas (Fig. 1). Coastal flood risk sources are described by the probability of occurrence of an extreme wave action, while the assessment of the failure probability of one or more components of the flood defence system is crucial for risk pathways. This requires reliable models for the prediction of loadings and resistance of a defence structure. Although a significant amount of knowledge on the extreme sea states and associated loading on the flood defence structures is available, there is still a gap in the knowledge on the processes that lead to the failure of such structures. In order to satisfy this need, a computational model system for the prediction of the sea dike breaching initiated from the seaward side by breaking wave impact has been developed.
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Fig. 1

Source-pathway-receptor framework (Oumerac 2004)

2 Description of the model system

One may distinguish several causes of dike breaching, depending upon the type of the dike and on the hydraulic conditions. A breach may be initiated either from the landward side by wave overtopping and overflow or from the seaward side by repeated breaking wave impact on a dike slope (Fig. 2). For a dike breach initiated by wave overtopping, an important research work has been done by D’Eliso (2007) or Tuan and Oumeraci (2010) and some outcomes of those researches are applicable also for the dike breaching initiated from the seaward side, since processes related to the breach channel growth are very similar. However, as the processes associated with the first two phases, i.e., breach initiation and breach formation from the seaward side are completely different from those of a breach initiated from the landward side, a computer model for the dike breach initiated from the seaward side has been developed (Stanczak 2008). The complete model system consists of a simpler preliminary model and a more process-oriented, detailed model (see Fig. 3). The preliminary model is a simple and fast tool that needs only a basic set of input data and allows one to roughly estimate the time of breaching and the breach outflow. The detailed model is more sophisticated tool that includes also processes neglected in the preliminary model, but it requires more input data and more computational effort.
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Fig. 2

Locations of a dike breach initiation—seaward side (left side) and landward side (right side) (Oumerac 2004)

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Fig. 3

Tiered approach to model wave-induced dike breaches from the seaward side

2.1 Preliminary model

The general features of the preliminary model are as follows:
  • reproduced structure: coastal dike made of a sandy core protected by clay cover with grass vegetation;

  • simulated processes: initiation, formation and development of the dike breaching induced by repeated breaking wave impacts;

  • type of model: empirical model that should serve as a simple tool for the prediction of breach initiation, formation and development processes;

As the preliminary model has been developed in order to provide a fast, overall picture of the breaching process, only few basic input data are needed:
  • dike parameters: geometry and material properties;

  • sea state at the toe of the dike: wave height and period distribution; mean water level;

  • numerical parameters: time step and grid size

It should be, however, emphasized that the model is developed specially for a specific type of grass-clay dike which are built on the German and Dutch coasts, and therefore, its application for other types of flood defence structures should be considered to be rather indicative.

2.1.1 Preliminary hydrodynamic module

The preliminary hydrodynamic module provides the first information on the loading of the dike. The main load consists of the breaking wave impact on the outer slope, but the overflow process is also simulated during the last phase, i.e., when the breach reaches the inner slope. The flow simulation is performed for two cases:
  1. 1.

    The flow conditions on the outer slope of the dike, from the dike toe up to the crest are simulated during the breach initiation, formation and development, i.e., until the breach has reached the inner slope. The simulation is performed essentially using the information provided by available models for wave breaking induced pressures and flow on the slope (see Table 1)

     
  2. 2.

    The flow along the whole dike in cross-shore direction—from the outer up to the inner dike toe of the dike—is simulated in the last phase, after the breach has reached the inner slope and overflow occurred. In this phase, the available model for the flow through the breach channel developed by D’Eliso (2007) is implemented.

     
Table 1

Models implemented in the preliminary hydrodynamic module

Parameter (model)

References

Maximal impact pressure

Führböter and Sparboom (1988)

Location of impact, run-up levels and velocities

Schüttrumpf and Oumeraci (2005)

Shape of impact area

Stive (1983)

Angle of impact incidence

Führböter (1966)

The formula for the calculation of maximal impact pressures proposed by Führböter and Sparboom (1988) as well as the formula for the estimation of the shape of impact area (Stive 1983) were developed after a series of field measurements on slopes 1:3 and 1:4 so that for both flatter and steeper slopes an adjustment of impact pressure according to Führböter (1966) is needed. The method of calculation of impact location as well as of wave run-up velocities and levels is based on a large series of medium-scale experiments in wave flume, applying a wide range of slopes and wave parameters, and therefore, this formulae are used in the presented model without any limitations or corrections. The angle of impact incidence is calculated using a theoretical approach which has not been verified yet, so that it is rather indicative.

2.2 Preliminary morphodynamic module

The main purpose of the preliminary morphodynamic module is the calculation of the temporal breach profile evolution. The information on the loading provided by the hydrodynamic module is used as an input for the calculation of breach initiation, formation and development. During the last phase, i.e., breach deepening and widening up to the total breach, the shape of the breach is updated using the information on the overflow parameters calculated by the hydrodynamic module and sediment transport model selected from the ones implemented in the model (Bagnold-Visser or Bagnold-Bailard formula)

The entire breaching process is divided into the following phases (Fig. 4):
  • Phase 1: erosion of grass cover, surface erosion of the cover directly subject to the repeated action of the breaking waves;

  • Phase 2: discrete local erosion of the clay cover up to the exposure of the sand core to the breaking wave impacts;

  • Phase 3: discrete erosion of the sand core, cliff formation, and development of the horizontal bottom of the breach;

  • Phase 4: continuous breach deepening and widening due to erosion during the overflow The following general assumptions for the preliminary morphodynamic module are made:
    • only the erosion of the seaward slope is calculated during Phases 1 and 2, the possible erosion of the inner slope resulting from the wave overtopping is neglected,

    • during Phases 1 and 2 the shape of the scour hole is calculated as a function of the pressure distribution on the slope (Stive 1983). In Phase 3, the horizontal breach bottom and vertical cliff are assumed. In Phase 4, the breach profile is calculated according to the sediment transport model applied for the rectangular channel cross-section

    • no change in the material properties due to mixing of clay and sand occurs, the median grain size D50 is constant constant during the entire breaching process,

    • during Phases 1, 2, and 3 the simulation is performed in two-dimensional plane x-z. In Phase 4, the simulation becomes 2D + 2D (x-z and x-y planes). The initial conditions for the simulation in x-y plane are assumed based on the observations of historical dike failures,

    The module for the erosion of the dike cover (Phases 1 and 2) as well as for the discrete erosion of the sand core (Phase 3) are based on the wave impact approach that calculates the total erosion depth as a sum of incremental erosion caused by every single wave impact (Larson et al. 2004). During the grass erosion phase, the depth of the soil eroded after a single breaking wave impact is calculated on the basis of the empirical dependency of the depth of erosion after a given time period on the significant height of the waves attacking a dike during this period. Nevertheless, although this formula was developed after large and well-documented full-scale tests, it is limited only to a small number of dike samples and would therefore be significantly enhanced if more data sets would be available. The same remark is valid also for the clay erosion phase. The formulae for the calculation of the sand core erosion and washing-out were developed using a significant number of data sets for the homogeneous dunes and are used in the present context without any constraints. Due to variable flow conditions during the sand core erosion, the simulation of this process is divided into two parts: (1) breach development, i.e., erosion of the front-face of the breach due to repeated action of breaking waves and (2) after the breach has reached the inner slope and overflow occurred breach deepening and widening as the result of the erosion due to overflow according to a sediment transport model selected by the user. In Table 2 the summary of implemented models is provided.
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Fig. 4

Phases of dike breach simulation

Table 2

Models implemented in the preliminary morphodynamic module

Parameter (model)

References

1. Surface erosion of the grass revetment

Larson et al. (2004); Smith et al. (1994)

2. Surface erosion of the clay layer

Larson et al. (2004)

3. Erosion of the sand core

Larson et al. (2004)

4. Breach widening and deepening

D’Eliso (2007)

2.3 Detailed model

The detailed sea dikes breaching model is based on the numerical simulation of wave breaking and on empirical formulae for the simulation of grass and clay erosion. It provides the complete information on the entire breaching process, including breach initiation, formation, and development. The developed model is divided into two main parts: (1) hydrodynamic module and (2) morphodynamic module, which in turn consists of (a) cover erosion module (b) sand core erosion module (up to the inner slope) and (c) sand core wash-out module (breach widening and deepening). As the detailed model has been developed in order to provide a detailed picture of the breaching process, a number of input data are needed:
  • dike parameters: geometry and material properties, including possible cracks in the revetment, permeability of the soil and distribution of the grass roots under the soil surface;

  • sea state at the toe of the dike: wave height distribution including wave period and water level;

  • numerical parameters: time step and grid size

2.3.1 Detailed hydrodynamic module

The detailed hydrodynamic module provides the information of the loading on the dike. As in the case of the preliminary model, the main load consists of the breaking wave impact on the outer slope, but the processes of the overtopping and overflow in the breach channel are also simulated. The flow simulation is performed for the following cases:
  • the flow conditions on the outer slope of the dike, from the dike toe up to the crest are simulated during the breach initiation i.e., until the breach has reached the dike core. The simulation is performed using the information provided by the numerical model COBRAS (Liu and Lin 1997) that is based on the Reynolds Averaged Navier Stokes 2DV equations, with a nonlinear, three-dimensional k-e turbulence model.

  • the flow along the whole dike in cross-shore direction—from the outer up to the inner dike toe of the dike—is simulated during the rest of the simulation. Depending on the breaching phase being simulated either the numerical SBeach model (Larson and Kraus 1989) together with the wave overtopping model or wave overflow model (D’Eliso 2007) is implemented.

Furthermore, during the entire breaching process the water infiltration is simulated with a simplified model

2.3.2 Detailed morphodynamic module

The main purpose of the detailed morphodynamic module is, as in the case of the preliminary model, the calculation of the breach profile evolution in time. The information on the loading provided by the hydrodynamic module is used as an input for the calculation of breach initiation, formation and development. The following general assumptions for the detailed morphodynamic module are made:
  • the breach initiates at the point where the maximal impact pressures occur and/or where the clay layer is not protected sufficiently due to randomly distributed weaker points in the grass cover and/or where the cracks in the clay cover are located [see Stanczak (2008) or Stanczak et al. (2007)] for more details on the local erosion in water-filled fissures;

  • both the erosion of the seaward slope and (after the erosion and consequent decrease of crest level) also possible erosion of the inner slope due to wave overflow are included; in the cover erosion module the shape of the scour hole is calculated as a function of the pressure distribution on the slope that is provided by the hydrodynamic module; in the sand core erosion module the beach profile is introduced while in the sand wash-out module the shape of the breach is calculated according to the selected sediment transport model;

  • the interaction between clay and sand is neglected, all the materials are considered to be homogenous

  • the cover and sand erosion modules are based on a two-dimensional plane x-z, but the simulation becomes three-dimensional during the sand wash-out. The initial conditions for the simulation in x-y plane are based on the results of historical dike breaches;

  • the local erosion due to impact pressures and surface erosion due to flow of wave run-up and run-down are calculated separately;

The cover erosion module is based on the wave impact approach (Larson et al. 2004) that calculates the total erosion depth as a sum of incremental erosion caused by every single wave impact. The total progress of erosion during this phase is calculated as a sum of (1) erosion increments resulting from successive impact pressures, that act only on a very limited area and (2) erosion increments resulting from shear stress induced by the cyclic flow associated with wave run-up and run-down that acts on larger areas. For the simulation of the sand core erosion, the results of the small-scale tests on the entire process of dike breaching (Husrin 2007 ; Stanczak et al. 2007) are used. The excess shear approach (Temple et al. 1987) is used for the calculation of erosion due to the flow associated with run-up and run-down. The protective properties of the grass stems are accounted for by the grass cover factor (Cf) thus reducing the effective shear stress on the soil. The surface erosion model developed after the laboratory experiments described in Stanczak et al. (2007) is applied for the calculation of the erosion due to impact pressures. The grass root reinforcement model is applied in order to include the effects of a grass cover. Since the excess shear approach is widely used in practice, only the recent grass erosion model will be addressed in this article. The following formula that was derived from the laboratory experiments forms the basis for the calculation of local surface erosion due to impact pressures (Stanczak et al. 2007):
$$ d_i={k_{d,p,i}}\cdot p_{i}\cdot e^{-wh_i} $$
(1)
where the following notations are used:
  • di: depth of erosion at the ith node resulting from a single impact pressure event (m);

  • kd,p,i: soil detachability coefficient for a unit area calculated at the ith node (m3/N) (see Eq. 2);

  • pi: impact pressure at the ith node (N/m2);

  • w: coefficient representing the damping effectiveness of a water layer (–) (see Stanczak 2008)

  • hi: water layer thickness at the ith node (m) (see Stanczak 2008)

The empirical coefficient kd,p,i depends on the soil parameters—type of clay and on the water content. For the clay of the erosion resistance Category 1 according to the Dutch requirements the erodibility coefficient kd,p can be calculated as the following function of the water content that was derived from laboratory experiments by Husrin (2007):
$$ k_{d,i}=0.35\cdot \hbox {arctan}[110-(wc-0.43)]\cdot10^{-12}\,({\text{m}}^3/{\text{Pa}}) $$
(2)
The values for other types of clay can be found in Stanczak et al. (2007) or Stanczak (2008).
In the uppermost layer of the clay cover the soil is reinforced by the grass roots. The modified erodibility coefficient kd,g,p is a function of (1) the dimensionless parameter b that describes the influence of the roots on the erodibility and (2) of the Root Volume Ratio RVR:
$$ k_{d,g,p,i}=\frac{k_{d,p,i}}{b\cdot {\text {RVR}}^2_i}\,({\text {cm}}^3) $$
(3)
The Root Volume Ratio RVR can be calculated as a function of the depth under the grass surface. Two models that describe the distribution of the root volume ratio RVR underneath the soil surface are available (Sprangers 1999; Stanczak et al. 2007):
$$ {\text {RVR}}=A\cdot D^{(d-d_{\rm cor})}\,(\%) $$
(4)
where AD and dcor are the empirical coefficients that depend on the quality of grass cover while d is the depth under the surface given in centimeters. The coefficients A and D are supposed to have a negative correlation with the clay quality, since stronger clay prevents the grow of a dense root network. In Table 3 the coefficients suggested by different authors are given.
Table 3

Coefficients describing the grass roots distribution and their effect on soil erodibility

A

D

dcor

b

References

2.67

0.8

1.5

Sprangers (1999)

1.58

0.75

2.0

5

Stanczak et al. (2007)

The erodibility parameter for the whole revetment kd,t,p,i can be then calculated taking the critical erosion depth into account as:
$$ \begin{array}{ll} k_{d,t,p,i}=\frac{k_{d,p,i}}{b\cdot {\text{RVR}}^2}&\quad\hbox {for}\; d<d_{\text{crit}}\\ k_{d,t,p,i}=k_{d,p,i}&\quad \hbox{for}\; d>d_{\text{crit}}\\ \end{array} $$
(5)
Randomly located cracks are generated as a part of the definition of the dike geometry. The maximal crack depth dcrack,max expressed in centimeters is limited to dcrack,max = 3 Vs, where Vs is the soil shrinkage expressed in percent (Richwien 2002). Since the soil shrinkage is in the range of Vs = 5–30%, depending on the soil parameters, the maximal crack depth is limited to dcrack,max = 0.15–0.9 (m). At the cracks, the limit state equation for the shear failure is solved and if the failure occurs, the dimensions of the cracks are updated according to the selected shear failure model (Stanczak et al. 2007). According to the simulations of dike breaching initiated from the seaward side by breaking wave impact (Husrin 2007; Stanczak et al. 2007), the remaining part of the clay cover after breach initiation still plays an important protective role. The assumption made in the preliminary model stating that the entire clay cover is removed from the dike after the breach initiation is therefore not fully consistent. Actually, a transition phase containing the erosion of both clay cover and sand core was observed between the clay erosion phase and the sand core erosion phase.This transition phase begins immediately after the end of the clay erosion phase, i.e., when the eroded hole in the clay layer has reached at least one point of the sand core and ends when the dimensions of the scour hole have grown up to the point when the plunge point is located on the uncovered sand core. In the detailed model, during the transition phase the erosion of clay is calculated as in the preceding phase while the progress of sand core erosion is calculated as in the preliminary model, i.e., by applying the wave impact approach (Larson et al. 2004) for sand dune erosion. Since the progress of sand core erosion is significantly faster than that of the clay layer undermining and consequently the clay layer collapses. The results of the small-scale laboratory tests on the dike breaching (Husrin 2007; Stanczak et al. 2007) show that during the front-face erosion of the sand core rather a beach profile (Fig. 5) is formed than a vertical cliff with a horizontal bottom as assumed in the preliminary model.
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Fig. 5

Formation of a beach profile in the small LWI flume

Based on the analysis of the available beach profile models, the SBEACH model (Storm-induced BEAch CHanges) (Larson and Kraus 1989) is selected for the application in the detailed model. The SBEACH model calculates the wave characteristics across-shore from a specified water depth offshore (dike outer toe) to the break point using a linear wave theory. The obtained wave energy dissipation forms the input data for the sediment transport calculation and consequently for the profile change calculation. The continuous erosion and formation of the sand beach profile results in the lowering of the dike crest which in turn can result in wave overtopping and consequently in the erosion of the inner slope. In order to account for this mechanism, from the beginning of the sand core erosion till the beginning of wave overtopping also the possibility of the inner slope erosion is controlled. The dike cross-section is divided into two parts, with the outer edge of the dike crest being the border. The condition for wave overtopping is controlled by calculating the wave run-up. If overtopping occurs, the flow conditions are calculated by applying the model of D’Eliso (2007) and the profile change is calculated using the excess shear approach. The progressive lowering of the dike crest may lead to the overflow and if the latter occurs the core wash-out begins. The simulation of the core wash-out is essentially performed by using the same two approaches as in the preliminary model.

3 Probability of dike breaching

A dike breach is initiated when the depth of erosion on the outer slope resulting from the wave attack has reached a critical erosion depth, which is usually defined as the depth of grass reinforced clay revetment, consequently uncovering the sand core of a dike. Generally, a dike breach is considered to be initiated, as soon as the duration of a storm surge will be longer than the time needed for the dike revetment to fail under given loading conditions. Although a number of probability density functions for the storm surge duration are available (Kortenhaus 2003), there is still a lack of the information on the predicted revetment failure time under given loading conditions. In order to gain knowledge on the probability density function of the whole breaching process, including initiation, Monte Carlo simulations were performed with the described model for dike breaching. For the simulation of dike breaching, the following key outputs are selected:
  • time of grass erosion tg;

  • time of clay erosion tc;

  • total time of breaching tb;

  • final breach width Bb;

  • peak outflow discharge Qp;

The simulation is performed for both preliminary and detailed model. The most uncertain parameters together with their mean values μ, standard deviations σ and coefficients of variations (σ′ = σ/μ) are given in Table 4. All the parameters are assumed to be normally distributed. For each model N = 10,000 realisations are performed.
Table 4

Uncertainties related to the input parameters for the morphodynamic module (partially after D’Eliso 2007; Kortenhaus 2003)

Input parameter

Symbol and unit

Mean

σ

σ′

Mean root volume ratio

RVR (–)

0.55

0.41

0.74

Root reinforcement coefficient

b (–)

5.00

1.20

0.24

Grass cover factor

Cf (–)

0.75

0.1

0.13

Critical grass erosion depth

dcrit (m)

0.08

0.02

0.25

Damping coefficient

w (–)

2.5

0.5

0.20

Saturated water content

θs (m3/m3)

0.42

0.06

0.15

Clay percentage

c% (–)

30.00

7.50

0.25

Internal friction angle

ϕ (°)

32.00

3.20

0.10

Sediment size

D50 (mm)

0.20

0.02

0.1

Internal friction angle

ϕ (°)

32.00

3.20

0.10

Soil porosity

n (–)

0.40

0.112

0.28

Initial breach channel width

\(B_{ini} (n\cdot r_h)\)

2.00

0.5

0.25

Breach growth coefficient

Kvl (–)

0.03

0.006

0.2

SBeach coefficient

K (m4/N)

\(1.4\times10^4\)

\(0.4 \times10^4\)

0.28

Although the input parameters are assumed normally distributed, due to the nonlinearity of equations implemented in both preliminary and detailed model, the probability density functions of all outcomes are asymmetric with clearly wider right tail. Generally, the preliminary model provides more conservative results, as intended. Table 5 provides the mean values (μ), standard deviations (σ) and coefficients of variation (σ′) provided by the preliminary and detailed model. It should be, however, emphasized that the times of clay failure given by the preliminary and detailed model cannot be compared directly, as the models are differently sub-divided, with two additional transition phases included in the detailed model.
Table 5

Main outcomes from the Monte Carlo simulation

Outcomes

Preliminary model

Detailed model

μ

σ

σ′

μ

σ

σ′

Grass erosion time (h)

13.61

10.08

0.74

28.57

20.68

0.72

Clay erosion time (h)

11.06

9.04

0.81

4.12

1.55

0.37

Core failure time (h)

11.96

11.67

0.98

5.87

0.82

0.14

Total breaching time (h)

36.54

21.60

0.59

38.53

21.68

0.56

Peak outflow discharge(m3/s)

1,287

378

0.29

1,241

179

0.15

Final breach width (m)

81.22

21.4

0.26

63.19

12.49

0.20

The mean time of grass erosion (Fig. 6)—is much longer in the detailed model (μ = 28.57 h) when compared to the results given by the preliminary model (μ = 13.61 h). This difference occurred most probably due to the new model for the calculation of the grass root reinforcement and grass erosion resistance that was applied in the detailed model.
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Fig. 6

Monte Carlo simulation results for the time of grass erosion

The coefficients of variation are however very similar (σ′ = 0.74 for the preliminary model and σ′ = 0.72 for the detailed one) indicating a similar relative level of the uncertainties of model outputs.The uncertainties of the outputs given by the detailed model however are related rather to the input parameters than to the model parameters and the detailed model itself is considered to be more reliable. The mean values of the clay erosion time (Fig. 7)—obtained from the preliminary and the detailed model cannot be directly compared, as a different phase subdivision is used. In the detailed model, the transition phases between (1) grass and clay and (2) clay and sand erosion are included. However, even assuming that the transition phases between clay and sand erosion are included in the clay erosion time, in the detailed model (μ = 4.12 h), it is still significantly shorter than in the preliminary model (μ = 11.06 h). The most probable reason might be the cracks in the clay layer. In the preliminary model they were fully neglected, while their presence and the calculation of possible shear failure due to impact pressures are implemented in the detailed model. Moreover, the prediction of the clay cover erosion time given by the detailed model is subject to relatively smaller uncertainties (σ′ = 0.37 as compared with σ′ = 0.81 for the preliminary model).
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Fig. 7

Monte Carlo simulation results for the time of clay erosion

The mean total breaching times (Fig. 8) obtained from the preliminary and detailed model are comparable (μ = 36.54 h for the preliminary model and μ = 38.53 h for the detailed model). The standard deviation and consequently the coefficient of variation that are observed in the case of the detailed model (σ′ = 0.56) indicate that it is subject to slightly smaller uncertainties, compared with σ′ = 0.59 for the preliminary model. However, the levels of the uncertainties indicated by both models are similar, which suggest that rather the variations in the input parameters, than the model formulation have the most important effect on the overall model performance.
https://static-content.springer.com/image/art%3A10.1007%2Fs11069-011-0054-8/MediaObjects/11069_2011_54_Fig8_HTML.gif
Fig. 8

Monte Carlo simulation results for the time of the full breaching

The application of the alternative volume-averaged approach for the calculation of the breach channel growth results in the reduction of the uncertainties related to the final breach width (Fig. 9) and peak outflow discharge (Fig. 10). The following differences are observed: reduction of the coefficient of variation σ′ for the final breach width from σ′ = 0.26 obtained from the preliminary model to σ′ = 0.20 from the detailed model. In the case of the peak outflow discharge, the coefficient of variation is reduced from σ′ = 0.29 to σ′ = 0.15. The second reason of this improvement is the better prediction of the boundary conditions for the overflow simulation. In the preliminary model, they were assumed (with a given range of variation), while in the detailed model they are directly calculated, including the changes of the inner slope profile due to the erosion which results from wave overtopping.
https://static-content.springer.com/image/art%3A10.1007%2Fs11069-011-0054-8/MediaObjects/11069_2011_54_Fig9_HTML.gif
Fig. 9

Monte Carlo simulation results for the final breach width

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Fig. 10

Monte Carlo simulation results for the peak outflow discharge

In order to calculate the probability of breach initiation and total breaching, the fitted lognormal probability density functions have been compared to the probability density function of the storm surge duration time for the German coast of the North Sea as given by Kortenhaus (2003). In Fig. 11 all the probability density functions are given.
https://static-content.springer.com/image/art%3A10.1007%2Fs11069-011-0054-8/MediaObjects/11069_2011_54_Fig11_HTML.gif
Fig. 11

Probability density functions of storm surge duration and dike breach initiation time for both preliminary and detailed model

The probability of breach initiation calculated using a convolutional integral is equal \(P_f=5.16\times10^{-5}\) for the preliminary model and \(P_f=3.55\times10^{-5}\) for the detailed one. In both cases the obtained values are larger as the values provided in existing studies for similar dike and loading parameters (\(P_f=8.3\times10^{-6}\) given by Kortenhaus (2003), for instance). It should be, however, emphasized that in the presented case the calculation is based on detailed simulation of the breaching process. Although the obtained curves are very case specific and may be subject to strong variations due to differently assumed distributions of input parameters, the provided example illustrates that the comparison of these two probability density functions is a possible way to the formulation of limit state equations. Nevertheless, since such Pf usually requires about 108 of Monte Carlo simulations, the presented results should be considered rather to be tentative.

Acknowledgments

The financial support of the German Research Foundation (DFG) within the International Graduate College IGC802 is gratefully acknowledged. This research is also a part of the FLOODsite project (Contract Number: GOCE-CT-2004-505420).

Copyright information

© Springer Science+Business Media B.V. 2011