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

## Authors

- First Online:

- Received:
- Accepted:

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

## 2 Description of the model system

### 2.1 Preliminary model

*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;

*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

#### 2.1.1 Preliminary hydrodynamic module

- 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.
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.

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)

**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: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.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 D

_{50}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,

Models implemented in the preliminary morphodynamic module

### 2.3 Detailed model

*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 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.

#### 2.3.2 Detailed morphodynamic module

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;

*C*

_{f}) 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}: depth of erosion at the*i*th node resulting from a single impact pressure event (m);*k*_{d,p,i}: soil detachability coefficient for a unit area calculated at the*i*th node (m^{3}/N) (see Eq. 2);*p*_{i}: impact pressure at the*i*th node (N/m^{2});*w*: coefficient representing the damping effectiveness of a water layer (–) (see Stanczak 2008)*h*_{i}: water layer thickness at the*i*th node (m) (see Stanczak 2008)

*k*

_{d,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

*k*

_{d,p}can be calculated as the following function of the water content that was derived from laboratory experiments by Husrin (2007):

*k*

_{d,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:

*A*,

*D*and

*d*

_{cor}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.

*k*

_{d,t,p,i}can be then calculated taking the critical erosion depth into account as:

*d*

_{crack,max}expressed in centimeters is limited to

*d*

_{crack,max}= 3

*V*

_{s}, where

*V*

_{s}is the soil shrinkage expressed in percent (Richwien 2002). Since the soil shrinkage is in the range of

*V*

_{s}= 5–30%, depending on the soil parameters, the maximal crack depth is limited to

*d*

_{crack,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.

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

time of grass erosion

*t*_{g};time of clay erosion

*t*_{c};total time of breaching

*t*_{b};final breach width

*B*_{b};peak outflow discharge

*Q*_{p};

*N*= 10,000 realisations are performed.

Input parameter | Symbol and unit | Mean | σ | σ′ |
---|---|---|---|---|

Mean root volume ratio | RVR (–) | 0.55 | 0.41 | 0.74 |

Root reinforcement coefficient |
| 5.00 | 1.20 | 0.24 |

Grass cover factor |
| 0.75 | 0.1 | 0.13 |

Critical grass erosion depth |
| 0.08 | 0.02 | 0.25 |

Damping coefficient |
| 2.5 | 0.5 | 0.20 |

Saturated water content | θ | 0.42 | 0.06 | 0.15 |

Clay percentage |
| 30.00 | 7.50 | 0.25 |

Internal friction angle | ϕ (°) | 32.00 | 3.20 | 0.10 |

Sediment size |
| 0.20 | 0.02 | 0.1 |

Internal friction angle | ϕ (°) | 32.00 | 3.20 | 0.10 |

Soil porosity |
| 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 |
| 0.03 | 0.006 | 0.2 |

SBeach coefficient |
| \(1.4\times10^4\) | \(0.4 \times10^4\) | 0.28 |

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(m | 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 |

**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.

**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).

**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.

**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.

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 10^{8} 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).