Breaching of coastal dikes. Detailed breaching model

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Breaching of coastal dikes: detailed model

LWI REPORT NR. 937

Summary of Contents:

Based on an extensive literature study on the most relevant processes associated with dike breaching and the models available, a modelling strategy was developed which consists in the development of a preliminary and a detailed breaching model (LWI Report 910, FLOODsite Report ). Results from the preliminary model give an overview of the breaching process, but indicate that improvements into a more process-oriented detailed model are required (LWI Report 927, FLOODsite Internal Report T06-06-01).

This progress report aims to address the development and implementation of the detailed model.

May 2007

Co-ordinator: Paul Samuels, HR Wallingford, UK Project Contract No: GOCE-CT-2004-505420

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D

OCUMENT

I

NFORMATION

Title Breaching of coastal dikes: detailed breaching model Lead Author Claudia D’Eliso

Contributors Hocine Oumeraci, Andreas Kortenhaus Distribution Task 6 Partners

Document Reference

D

OCUMENT

H

ISTORY

Date Revision Prepared by Organisation Approved by Notes

29/05/07 CD LWI 1st_Draft

D

ISCLAIMER

This report is a contribution to research generally and third parties should not rely on it in specific applications without first checking its suitability.

In addition to contributions from individual members of the FLOODsite project

consortium, various sections of this work may rely on data supplied by or drawn from sources external to the project consortium. Members of the FLOODsite project consortium do not accept liability for loss or damage suffered by any third party as a result of errors or inaccuracies in such data.

Members of the FLOODsite project consortium will only accept responsibility for the use of material contained in this report in specific projects if they have been engaged to advise upon a specific commission and given the opportunity to express a view on the reliability of the material concerned for the particular application.

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Bericht Nr. 937

Breaching of coastal dikes

- Detailed breaching model -

Progress report Nr. 3

Dipl.-Ing. C. D’Eliso

Prof. Dr.-Ing. H. Oumeraci

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Abstract

Coastal dikes are used as defence structures against flooding in lowland areas and where high storm surge levels occur. The development of a breach induced by wave overtopping has been one of the most frequent causes of dike failure associated with disastrous damages. Therefore, dike breaching is closely related both to the dynamics of a protected coast and to the flood risk assessment and management. Despite the importance of dike breaching, the underlying processes, their simulation and prediction are still not well understood.

Based on an extensive literature study on the most relevant processes associated with dike breaching and the models available, a modelling strategy was developed which consists in the development of a preliminary and a detailed breaching model (progress report N. 1, LWI Report 910). Results from the preliminary model give an overview of the breaching process, but indicate that improvements into a more process-oriented detailed model are required (progress report N. 2, LWI Report 927, FLOODsite Internal Report T06-06-01).

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List of figures

Fig. 1.1: Overview of the detailed model... 2 Fig. 2.1: Infiltration sources at a real sea dike ... 5 Fig. 2.2: Mass instability processes as breach initiation at the landside induced by

infiltration... 5 Fig. 2.3: Computational domain for the VOF model... 6 Fig. 2.4: Flow velocity and wave overtopping discharge calculated with the VOF

model ... 8 Fig. 2.5: Flow velocity and overtopping discharges at the landward of the dike

crest ... 9 Fig. 2.6: Computational domain for infiltration: red (grass), green (clay), yellow

(sand) ... 10 Fig. 2.7: Stationary phreatic line with different landside toe protections ... 17 Fig. 2.8: Calculation of the transient phreatic line in the dike: definition sketch . 18 Fig. 2.9: Calculation of the infiltration due to wave overtopping with and without

overflow in the dike: definition sketch... 20 Fig. 2.10: Volumetric water content (θ) at the landside of the dike (Weißmann

model)... 25 Fig. 2.11: Comparison between the selected models (saturated and infiltration

water fronts) ... 26 Fig. 2.12: Simulated breach growth in coastal dikes induced by wave overtopping

... 28 Fig. 2.13: Definition sketch of weaknesses along the dike profile considered in the

model ... 30 Fig. 2.14: Definition sketch of mass instability in the cover layer ... 32 Fig. 2.15: Mass instability of the cover layer... 36 Fig. 2.16: Influence of material parameters (inner slope, soil cohesion and friction

angle) on the cover stability ... 42 Fig. 2.17: Influence of the position of the saturated zone on the factor of safety,

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Fig. 2.19: Headcut migration modes under wave overtopping conditions ... 45

Fig. 2.20: Definition sketch for flow at the overfall and headcut erosion ... 47

Fig. 2.21: Equilibrium scour depth ... 50

Fig. 2.22: Definition sketch for mass instability of the overfall during headcut erosion ... 51

Fig. 2.23: Comparison between three modes of headcut advance though mass instability (shearing, bending and turning)... 55

Fig. 2.24: Flow pattern and shear stresses at the overfall base ... 56

Fig. 2.25: Headcut erosion in a two-layered material ... 57

Fig. 2.26: Definition sketch of the scour erosion model in two-layered material. 58 Fig. 2.27: Sediment pick up volume (Vp) and deposited volume (Vd) in the scour hole ... 62

Fig. 2.28: Scour hole development ... 62

Fig. 2.29: Initial breach channel in the dike core... 64

Fig. 3.1: Prototype dike for example application... 71

Fig. 3.2: Headcut height HH, width BH and cumulated headcut advance dX with time (Phase 3a) ... 72

Fig. 3.3: Headcut width BH and cumulated headcut advance dX with time (Phase 3b)... 73

Fig. 4.1: Influence of soil cohesion and clay content on time of core erosion tce [hr] ... 79

Fig. 4.2: Influence of initial saturated water front zsi and saturated hydraulic conductivity of clay ks on time of core failure tcf [hr] ... 79

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List of tables

Tab. 2.1: Main input parameters of the VOF model ... 7

Tab. 2.2: Typical values of the saturated hydraulic conductivity (ks) ... 12

Tab. 2.3: Typical values of saturated volumetric water content (θs)... 13

Tab. 2.4: Typical values of residual volumetric water content (θr) ... 14

Tab. 2.5: Initial volumetric water content (θi)... 14

Tab. 2.6: Typical values of air entry value (hb)... 15

Tab. 2.7: Typical values of pore size distribution index (N) ... 15

Tab. 2.8: Typical values of coefficients α and n for the van Genuchten soil-water retention curve... 16

Tab. 2.9: Implemented models for the infiltration due to wave overtopping with and without overflow... 17

Tab. 2.10: Material properties used in the calculation of the infiltration due to wave overtopping ... 25

Tab. 2.11: Correspondence between description of the initial dike surface and incipient breaching and initiation ... 31

Tab. 2.12: Shear strength of grass root after (Cazzuffi & Crippa, 2005), (Wu et al., 1979) and (Stanczak et al., 2006) ... 34

Tab. 2.13: Overview of mass instability processes implemented in the detailed model ... 41

Tab. 2.14: Conclusions about wave overtopping, combined flow and overflow at a real sea dike for purposes of simulation of breaching initiated at the landside 65 Tab. 2.15: Conclusions about water infiltration in a real sea dike for purposes of simulation of breaching initiated at the landside... 65

Tab. 2.16: Conclusions about breach location and initiation for purposes of simulation of breaching initiated at the landside... 67

Tab. 2.17: Conclusions about turf set-off and clay cover instability for purposes of simulation of breaching initiated at the landside... 67

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Tab. 2.19: Conclusions about headcut erosion in two-layered material for

purposes of simulation of breaching initiated at the landside ... 69

Tab. 3.1: Outcomes from the model system ... 71

Tab. 4.1: Model parameters included in the sensitivity analysis ... 75

Tab. 4.2: Model outcomes considered for the sensitivity analysis ... 76

Tab. 4.3: Sets of parameters and outcomes for Level II sensitivity analysis ... 76

Tab. 4.4: Initial scenarios tested to assess the influence of weaknesses along the dike profile ... 77

Tab. 4.5: Initial scenarios tested to assess the influence of weaknesses along the dike profile ... 77

Tab. 4.6: Influence of the initial scenario on the model output ... 77

Tab. 4.7: Influence of the position of the weakest point on the model output... 77

Tab. 4.8: Suggested values of clay cohesion (c) ... 80

Tab. A.1: List of input parameters: sea and geometrical parameters... 84

Tab. A.2: List of input parameters: material parameters ... 87

Tab. A.3: List of input parameters: numerical parameter and user choice ... 89

Tab. A.4: List of outcomes... 90

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1. Introduction

1.1. Objective

Catastrophic floods in coastal zones are often due to breaching of dikes, e.g. storm surge of 1953 in The Netherlands and 1962 in Germany.

The breaching process is important for several aspects associated with flood risk assessment and management (D'Eliso et al., 2005):

• Dike failure assessment; • Estimate of the warning time;

• Definition of initial conditions for flood propagation modelling.

Nevertheless, a reliable prediction of breach initiation and development for a real sea dike is still not available. Only the breaching of dikes with homogeneous material such as sand has been modelled yet (Visser, 1998).

The existing models for dam engineering cannot be applied to coastal dikes because many of the physical processes involved are different, i.e. (i) sea waves as primary load and (ii) sand-clay dikes with a protective layer of grass.

A review of the state of the art can be found in D’Eliso & Oumeraci (2005).

A model system for the simulation of the breaching process induced by wave overtopping, optionally including combined wave overtopping and overflow, has been developed. The model system consists in a preliminary

simplified model and a detailed process-oriented model.

The development of a simplified preliminary model is particularly needed in order (i) to explore and identify the problems and the most important issues to be improved in the development of the detailed model, (ii) to get familiar with the simulated processes (iii) to start quantifying the uncertainties. The preliminary model is extensively described in LWI, Progress Report N. 927 (D'Eliso et al., 2006).

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1.2. Brief outline

The detailed model is based on the preliminary model (D'Eliso et al., 2006), but several simplifying assumptions are removed and it provides a more process-oriented description of the entire breaching process.

The model includes all processes involved in the breach growth, which are described by a set of modules (Fig. 1.1).

A - DETAILED FLOW MODEL (LEVEL II) ALONG PLANE SLOPE

Phases 1Cov, 2Cov, 3Cov, 1Cor Schümttrumpf et al. (2002) vertical velocity and viscous effects are neglected

IN THE IMPINGING JET REGION Phase 3Cov

Schümttrumpf et al. (2002), Temple & Hanson (1994) the flow is still calculated over a plane slope

OVERFLOW DISCHARGE Phases 2Cor, 3Cor Bleck et al. (2000), Visser (1998) stationary broad-crested weir formulae FLOW THROUGH THE BREACH CHANNEL

Phases 2Cor, 3Cor Hassan (2002), Rozov (2003)

steady non uniform flow WAVE OVERTOPPING

Empirical formulae

WAVE OVERTOPPING +OVERFLOW Empirical formulae, energy balance

Different evaluation of the energy slope Duration of each event Modification of the formulae Each wave is treated as regular

Duration of each event Each overtopping event is equivalent to an overflow event

No flow calculation downstream the plunging jet Wave-averaged description of

the discharge, regular waves

Wave-averaged energy balance

INFILTRATION DUE TO OVERFLOW WATER Phases 1Cov, 2Cov, 3Cov, 1Cor Wang (2000), Wang (2003), Weißmann (2003)

Darcy flow, simplified solutions of Richard eq. _{Non uniform porous medium} (grass, clay and sand) SEEPAGE FLOW

All phases

Mishra & Singh (2005), Wang, (2002)

Darcy flow Saturated and unsaturated soil

2D water infiltration Validity of Darcy flow WATER INFILTRATION

Empirical formulae, Darcy’s law

B) DETAILED MORPHODYNAMIC MODEL (LEVEL II)

GRASS EROSION AND SOD STRIPPING Phase 1Cov

Temple et al. (1987)

uniform flow, definition of a grass covering factor

HEADCUT EROSION Phase 3Cov

Scour erosion: Robinson (1992), Stein et al. (1993) Headcut advance: Robinson (1992), Darby (1996),

Hassan et al. (2001)

discrete approach, bending

LOCAL EROSION Phase 2Cov Temple & Hanson (1994)

uniform flow

SITES Model NRCS (1997)

SEDIMENT TRANSPORT MODEL Phase 1Cor, 2Cor Visser (1998), Galappatti (1983)

formulae not derived under breaching conditions

MASS STABILITY Phase 2Cor Darby (1996), Hassan et al. (2001)

discrete approach, force balance, shearing

CONTINUOUS EROSION Phase 1Cor, 2Cor Visser (1998), Hassan et al. (1999)

1D Exner equation for dA

SEDIMENT TRANSPORT MODEL Phase 3Cor Smart (1984), van Rijn (1984)

formulae not derived under breaching conditions

CONTINUOUS EROSION Phase 3Cor Visser (1998), Hassan et al. (1999)

1D Exner equation for dA, two base behaviours

MASS STABILITY Phase 3Cor Darby (1996), Hassan et al. (2001)

discrete approach, force balance, shearing

HR BREACH, BRES Models Hassan et al. (1999), Visser (1998)

GRASS COVER FAILURE

Excess shear stress relation Empirical parameters for the grass

CLAY LAYER FAILURE

Excess shear stress relation Discrete approach

FULL BREACH FORMATION

Sediment transport discharge Discrete approach

BREACH DEVELOPMENT

Sediment transport discharge Discrete approach

Final erosion hole width Influence of critical shear stress Hypothesis on the whole grass

cover removal

Vertical failure plane Rectangular shape after failure Hypothesis for the failure time

Assumptions on scour shape

Vertical failure plane Rectangular-trapezoidal shape

after failure Initial rectangular shape

from the cover failure 1Cor: deepening 2Cor: deepening and widening Influence of the bed-load transport

Analysis of Galappatti approach

Different sand erosion rates (dike and base) Inclusion of suction pressure

Inclusion of suction pressure Transition to trapezoidal shape Simple relation for the failure time

Definition of different landside surface scenarios

SHEARING OF THE COVER LAYER

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The dike reproduced in the model has a sand core and a clay revetment with a grass cover, as in the preliminary model. The cross-section of the dike is selected as simple as possible, i.e. without toe berms, toe protections or ditches. The breaching process is induced by wave overtopping at the landside of the dike, but optionally also by combined wave overtopping and overflow.

Compared to the preliminary model, the detailed model aims at introducing and improving the simulation of the following processes:

1. Introduction of new processes • Water infiltration in the dike;

• Scour hole in the clay cover and sand core; • Sliding of the cover layer;

2. Improvement of process already included in the preliminary model • Wave overtopping discharge at the dike;

• Incipient breach location; • Headcut erosion and advance;

In Chapter 2 the assumptions that have been made and the equations that have been used for the detailed model are presented and discussed in terms of the physical processes that they describe.

In Chapter 3 a description of the detailed model, with examples of inputs and outputs are presented.

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2. Mathematical formulation

2.1. Overview and major assumptions

Past dike failures due to breaching formation and growth at the landside, as well as laboratory and field tests reported in the literature, show that the breaching process is a complex 3D physical process. From the incipient point of failure up to the final breach, several mechanisms occur, simultaneously or successively (in cascade).

The proposed breach model describes the whole process, but several assumptions are still imposed:

• The process is still reduced to 2D + 2D. The covering layer fails according to a completely 2D model, while for the core erosion also the third dimension is added;

• The hydrodynamic and the morphodynamic modules are not coupled.

2.2. Hydrodynamic description

The hydrodynamic module includes the simulation of the free surface flow along the dike profile from the seaside toe up to the landside toe and the water infiltration into the dike body due to overflow water over the dike crest and seepage. As in the preliminary model, wave overtopping is the main hydrodynamic load, but also combined wave overtopping and overflow (called hereafter “combined flow”) can be simulated (D'Eliso et al., 2005).

As illustrated in Fig. 2.1, the infiltration of water in a real sea dike originates from four sources (D'Eliso et al., 2005):

• Mean water level (seaside, quasi-steady source);

• Wave overtopping with and without overflow (seaside above the mean water level, crest and landside, intermittent source);

• Rain (seaside above the mean water level, crest and landside, unsteady source);

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LAND

MWL

b) Wave Overtopping

SEA _{Phreatic line} h_p

Seepage flow

LAND

MWL

a) Wave impact

SEA _{Phreatic line} _h

p

Seepage flow

Rain Run-up – Run-down

Rain Mean water level

Infiltration front

Wave overtopping Mean water level

Infiltration front

Fig. 2.1: Infiltration sources at a real sea dike

The infiltration process represents the main cause of breach initiation as it may reduce the strength of the dike due to:

• Cover layer instability (Fig. 2.2a), see Section 2.3; • Inner slope instability (Fig. 2.2b).

LAND

b) Inner slope instability due to seepage

MWL

a) Cover instability due to infiltration caused by wave overtopping and overflow

SEA

h_p

LAND

MWL

SEA Phreatic line

Seepage flow Infiltration front

Sliding

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Moreover, the infiltration modifies water content and suction pressure in the dike, thus influencing the headcut erosion and the stability of the breach side slopes (Section 2.3).

2.2.1. From preliminary to detailed model

The preliminary model uses an empirical wave module, which can be substituted by a number of more process-oriented models (D'Eliso et al., 2005). A proposal of how to include a Volume of Fluid Model (VOF) in the detailed breaching model will be suggested at the end of the study.

In the preliminary model the infiltration process was totally neglected, although it may represent the primary cause of breach initiation. The detailed model includes the calculation of the infiltration, both due to overflow water over the dike crest and to seepage flow. Results of the infiltration module are used as input in stability modules (Section 2.3).

2.2.2. Wave overtopping

Within the available models to simulate wave overtopping at the dike (D'Eliso et al., 2005), the Volume of Fluid Model (VOF) developed at Cornell University is one of the more efficient and process-oriented (Liu & Lin, 1997).

This model (COBRAS) is a bidimensional numerical model that solves the Reynolds Averaged Navier-Stokes (RANS) 2DV equations, with a three-dimensional nonlinear turbulence model. It is able to simulate propagation and breaking of regular waves also in presence of obstacles, i.e. structures (Liu & Lin, 2003).

All details about the model are given in (Liu & Lin, 1997).

The model has been used to simulate wave overtopping at the dike. Flow velocity and thickness calculated in the model, can be used for the purposes of the breaching model, up to the landward end of the crest. With the present version of the model, it is not possible to simulate the flow at the landside of the dike, because the mean water level is set constant throughout the whole calculation domain (Fig. 2.3).

Fig. 2.3: Computational domain for the VOF model

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A modification of the model code is required in order to calculate the flow also at the landside. Nevertheless, the overtopping discharge at the dike crest can be properly estimated and this represents a step forward with respect to the preliminary model.

In the detailed model, as an alternative to the simplified model proposed in (D'Eliso et al., 2006), the wave overtopping flow at the dike can be calculated as follow:

• VOF model at the seaside and at the crest; • Schüttrumpf model at the landside.

In Tab. 2.1, the input used in the VOF model for a prototype dike and sea parameters are listed.

Dike geometry Hd [m] Bd [m] m [1] n [1] 7.00 3.00 6 3 Sea parameters Hs [m] Tp [s] MWL [m] 2.00 8.00 5.50 Numerical parameters dx [m] dz [m] dt [s] tmax [s] 0.15 0.075 0.1 100.0

Tab. 2.1: Main input parameters of the VOF model

Flow velocity v (Fig. 2.4a) and overtopping discharge q (Fig. 2.4b) obtained with the VOF model are plotted over time. From a qualitative point of view, the discharge calculated with the VOF model has approximately a triangular shape, slightly asymmetrical, with an almost vertical increasing reach and a rapidly decreasing reach. Looking at the results in Fig. 2.4, the assumptions made in the preliminary model (D'Eliso et al., 2006) concerning the shape of the wave overtopping discharge within a wave cycle, seem to be then reasonable.

The overtopping water flows over the dike for a shorter time than the associated wave period.

The model can also simulate the interaction between two successive overtopping waves. Due to this interaction, the overtopping flow of a regular wave, as in the example, is not constant, but varies from one wave to the following (Fig. 2.4). Information about flow velocity v (Fig. 2.5a) and wave overtopping discharge q (Fig. 2.5a) at the landward side of the dike crest, as results of the VOF model have been synthesized by calculating for each overtopping wave:

• The peak value (purple line); • The mean value (pink line).

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a) Flow velocity at the dike crest 0.00 1.00 2.00 3.00 4.00 5.00 6.00 7.00 0 20 40 60 80 100 Time [s] F lo w v elo ci ty [m /s ]

b) Flow discharge at the dike crest

0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.60 0 20 40 60 80 100 Time [s] O verto p p in g d isch arg e [m 3 /sm ]

Fig. 2.4: Flow velocity and wave overtopping discharge calculated with the VOF model

2.2.3. Infiltration process and its simulation: overview

The infiltration is a 3D non-stationary process and involves non-homogeneous materials. Non-homogeneity of the materials may have important consequences on the process. The formation of entrapped air regions below the dike crest are responsible of the formation of cracks that may lead to breaching (Elela, 1996; Zaradny, 1994). In fact they are:

• Almost impermeable to water;

• Cause high saturation of soils in the upper part of the dike;

• May have an air pressure much higher than the atmospheric pressure, In the detailed model, the infiltration process is reduced to a 2D model with the following assumptions:

• The free surface flow model and the infiltration model are uncoupled; • The interaction between the infiltration due to wave overtopping with and

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• The wave run-up and run-down does not influence the breaching process initiated at the landside and is therefore neglected;

• The infiltration due to rain storms is neglected. In fact, rain storms are likely to occur together with wind storms that generates high mean water level and waves (sea storm), resulting in a negligible contribution to the infiltration. The influence of rain is only roughly included in the selection of the initial water content in the dike (Section 2.2.4)

• As we assume that the seaward side of the dike is not damaged, the infiltration due to mean water level is very slow and doesn’t really affect the dike stability (Section 2.2.5)

a) Flow velocity at the dike crest

0.00 2.00 4.00 6.00 8.00 F lo w v elo ci ty [ m /s] Peak value 4.01 2.76 5.84 3.24 2.75 4.53 3.55 5.17 1.83 0.88 Averaged peak value 3.45 3.45 3.45 3.45 3.45 3.45 3.45 3.45 3.45 3.45 Mean value 0.79 0.55 0.75 0.55 0.50 0.56 0.66 0.72 0.31 0.07 Averaged mean value 0.57 0.57 0.57 0.57 0.57 0.57 0.57 0.57 0.57 0.57

1 2 3 4 5 6 7 8 9 10

b) Flow discharge at the dike crest

0.00 1.00 2.00 3.00 Fl ow di sc ha rg e [ m 3 /sm ] Peak value 0.90 0.62 2.19 0.73 0.62 1.02 0.81 1.35 0.55 0.13 Averaged peak value 0.89 0.89 0.89 0.89 0.89 0.89 0.89 0.89 0.89 0.89 Mean value 0.17 0.10 0.22 0.10 0.09 0.11 0.14 0.16 0.06 0.01 Averaged mean value 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.12

1 2 3 4 5 6 7 8 9 10

Fig. 2.5: Flow velocity and overtopping discharges at the landward of the dike crest

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• Mass instability at the dike toe due to seepage flow is not included, while sliding and erosion of the cover layer are both considered.

• Entrapped air is neglected in the model and only the influence of existing cracks at the beginning of the simulation is addressed.

The computational domain for the infiltration process is a rectangular 2D grid, where three homogeneous layers of different materials (grass, clay and sand) are described (Fig. 2.6). Water can flow all over in the dike.

Fig. 2.6: Computational domain for infiltration: red (grass), green (clay), yellow (sand)

2.2.4. Richard equation’s and analytical solutions

The 2D Richard equation (Richards, 1931) states that the difference in the flow flux (H) entering and leaving an element volume at a given time is equal to the change in the volumetric water content (θ):

x y H H k k Q x x y y t ⎛ ⎞ ∂ ⎛ ∂ _{⎞ +} ∂ ∂ _{+ =}∂θ ⎜ ⎟ ⎜ ⎟ ∂ ⎝ ∂ ⎠ ∂ ⎝ ∂ ⎠ ∂ (2.1) Where:

• H is the total head:

w u H z g = + ρ (2.2)

Where z is the elevation [m] and u is the pore pressure [N/m2]

• k is the hydraulic conductivity in x and y direction: k=

(

k , k m s_x _y

)

[ ]

The hydraulic conductivity k, in the unsaturated zone, is function of the total head H.

• Q is an applied boundary flux [1/s]

• θ is the volumetric water content: w w

[ ]

c,d V w 1 V ρ θ = = ρ (2.3)

with Vw volume of water and V volume of the soil.

Whilst the volumetric water content, θ, is most used in soil science, the gravimetric water content, w, is more commonly used in geotechnical engineering practice.

Eq. (2.1) is based on Darcy’s law applied to both saturated and unsaturated soils: = ⋅

v k i (2.4)

Sand

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The Richard equation can be solved when two soil functions are known: • Hydraulic conductivity function (k-u curve);

• Soil-water characteristic curve (θ-u curve), see Sections 2.2.4.1 and 2.2.4.2.

Numerical solutions of the Richard equation are then typically obtained using finite element techniques.

For the detailed model, a selection of simplified solutions of the Richard equation is implemented in order to achieve a faster solution of the infiltration problem. A commercial solver of the Richard equation (SEEPW) is also applied to simple idealized overflow problems in order to suggest an alternative to the selected simplified models.

2.2.4.1 Soil-water characteristic curves

The soil water characteristic curve (or soil-water retention curve) is the relationship between volumetric water content θ, and matrix suction pressure u, for a given soil, when the volumetric water content is decreasing from the saturated value θs, to the residual value θr (desorption curve).

Several empirical relations have been proposed in the literature. For a detailed review see (Fredlung & Xing, 1994). In the detailed model, two soil-water characteristic curves, among the more commonly applied, have been selected. Both curves give the suction head in cm:

a) Brooks and Corey (1964)

( )

N 1 N r r b s r b s r h h h h − − ⎛ θ − θ ⎞ ⎛₌ ⎞ _⇒ _{θ =} ⎛ θ − θ ⎞ ⎜_{θ − θ} ⎟ ⎜ ⎟ ⎜_{θ − θ} ⎟ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ (2.5)

Where N is the pore-size distribution index and hb is the bubbling head, expressed

in cm (Section 2.2.4.2). b) van Genuchten (1980)

( )

1 n m _{1 m} r r n s r s r 1 1 h 1 1 h − ⎡ ⎤ ⎡ ⎤ ⎛ θ − θ ⎞ _⎢⎛ θ − θ ⎞ _⎥ =⎢ ⎥ ⇒ θ = − ⎜_{θ − θ} ⎟ _α ⎜_{θ − θ} ⎟ + α ⎢ ⎥ ⎢ ⎥ ⎝ ⎠ _⎣ _⎦ _⎣⎝ ⎠ _⎦ (2.6)

1 From soil-water retention data m

1 1 n Mualem's model ⎧

= ⎨ ₋

⎩ (2.7)

Where α, expressed in cm-1, is the inverse of the bubbling head, and n is related to the pore-size distribution (Section 2.2.4.2).

In both cases, in saturated soil: s

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Large deviations between the two models occur as the volumetric water content θ, approaches the saturation θs: while the Brooks and Corey curve tends

asymptotically to suction equal to the air-entry pressure, the van Genuchten curve tends asymptotically to zero suction. For a detailed derivation and comparison of the curves, see (van Genuchten, 1980).

2.2.4.2 Estimate of soil parameters for infiltration

All infiltration models require several soil parameters:

• Saturated hydraulic conductivity (ks);

• Saturated volumetric water content (θs);

• Residual volumetric water content (θr);

• Initial volumetric water content (θi);

• Brooks and Corey’s curve parameters: Pore-size distribution index (N);

Air entry value or bubbling suction head (hb);

• Van Genuchten’s curve coefficients:

Inverse of air entry value or bubbling suction head (α); Pore-size distribution index parameter (n).

These parameters are all empirically determined, according to procedures that are described in the literature, e.g. (Tzimopoulos et al., 2005; Wang et al., 2003; Weißmann, 2003). If specific in-situ measurements are not available, measured values reported in the literature can be used in the detailed model.

Typical values of the parameters reported in Tab. 2.2-Tab. 2.8 are reported in (Weißmann, 2003) and (EPA, 1998), without any further explanation on their selection and use. For detailed information on how the parameters have been derived, in Tab. 2.2-Tab. 2.8, the original references are indicated.

Saturated hydraulic conductivity

The saturated hydraulic conductivity (ks) is the hydraulic conductivity of the

saturated soil. It increases from clay to sandy soils (Tab. 2.2)

Saturated hydraulic conductivity ks [m/s]

Type of soil (Rawls et al., 1992) (Weißmann, 2003)

Sand 5.833·10-5_8.250·10-5

Loamy Sand 1.697·10-5_4.053·10-5

Sandy Loam 7.194·10-6_1.228·10-5

Loam 3.667·10-6_2.000·10-6

Silty Loam 1.889·10-6_6.944·10-7

Sandy Clay Loam 1.194·10-5 1.000·10-6 Clay Loam 6.389·10-7 3.639·10-6 Silty Clay Loam 4.147·10-7 7.222·10-7 Sandy Clay 3.333·10-7_1.944·10-7

Silty Clay 2.500·10-7_3.333·10-7

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The effects of the grass on the soil permeability can be due to: • Variation in the soil pore distribution;

• Development of preferential infiltration pathways given by the grass roots. Nevertheless, to the authors’ knowledge, only very rough empirical formulae are available for the estimate of the permeability of a grassed soil as a function of the soil and grass type. In the detailed model, if specific in-situ measurements are not available, one of the two following relations can be optionally used:

• -7 1.4

s,grass s

k =k +2.78 10 a⋅ ⋅ (Holtan, 1961) (2.9)

Where a is percentage basal area of the vegetation and depends on the type of vegetation. Values of the coefficient a are given in the literature for agricultural field vegetation and forests (Holtan, 1961; Styczen & Morgan, 1995). A value of 0.40-0.50 is suggested for grass used as sea dike protection;

• 5

s,grass

k ₌10 m s− _{(Pilarczyk, 2003)} _(2.10)

During the infiltration process at the dike, at least three layer of different permeability (grass, clay and sand) are penetrated by water.

In the detailed model, the saturated hydraulic conductivity is set according to the behaviour of a series system:

• Infiltration in the grass layer: k_s =k_s,grass

• Infiltration in the clay layer and in the sand core:

(

)

s s,soil s,sand s,soil

k =min k , k =k (2.11)

Saturated volumetric water content

The saturated water content (θs) is the water content at saturation. A review of

values found in the literature is in Tab. 2.3.

Saturated volumetric water content θs [m3/m3]

Type of soil (Brakensiek _{et al., 1981)} (Pajian, ₁₉₈₇₎ _{Parrish., 1988)}(Carsel & (Weißmann, ₂₀₀₃₎

Sand 0.35 0.38 0.43 0.43

Loamy Sand 0.41 0.43 0.41 0.41

Sandy Loam 0.42 0.44 0.41 0.41

Loam 0.45 0.44 0.43 0.43

Silty Loam 0.48 0.49 0.45 0.46

Sandy Clay Loam 0.41 0.48 0.39 0.45

Clay Loam 0.48 0.47 0.41 0.39

Silty Clay Loam 0.47 0.48 0.43 0.41

Sandy Clay - - - 0.43

Silty Clay 0.48 0.49 0.36 0.38

Clay 0.48 0.49 0.38 0.36

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Residual volumetric water content

The residual water content (θr) is the water content for which a large suction range

is required to remove additional water from the soil. It decreases from clay to sandy soils. A review of values found in the literature is in Tab. 2.4.

Residual volumetric water content θr [m3/m3]

Type of soil (Brakensiek _{et al., 1981)} (Pajian, ₁₉₈₇₎ _{Parrish., 1988)}(Carsel & (Weißmann, ₂₀₀₃₎

Sand 0.054 0.020 0.045 0.045

Loamy Sand 0.060 0.032 0.057 0.057

Sandy Loam 0.118 0.045 0.065 0.065

Loam 0.078 0.057 0.078 0.078

Silty Loam 0.038 0.026 0.067 0.034

Sandy Clay Loam 0.188 0.093 0.100 0.067

Clay Loam 0.185 0.107 0.095 0.100

Silty Clay Loam 0.155 0.089 0.089 0.095

Sandy Clay - - - 0.089

Silty Clay 0.182 0.102 0.070 0.100

Clay 0.226 0.178 0.068 0.120

Tab. 2.4: Typical values of residual volumetric water content (θr)

Initial volumetric water content

The residual water content (θi) is the water content at the beginning of the

simulated infiltration event. It is in the range of the residual water content and the saturated water content and its value depends on the initial dike conditions (site-specific value). Factors that mainly determine the value of the initial water content are rain rate, temperature, variation in the mean water level, occurrence and rate of the overtopping events in the previous days.

If specific field measurements are not available, three classes of initial volumetric water content are suggested (Tab. 2.5).

Soil conditions Residual volumetric water content θr [m3/m3]

Dry (high temperature, low rain rate) θr ÷ θr+0.3(θs θr) Medium wet θr+0.3(θs θr) ÷ θr+0.6(θs θr) Wet (low temperature, high rain rate) θr+0.6(θs θr) ÷ θs

Tab. 2.5: Initial volumetric water content (θi)

Brooks and Corey’s curve parameters

The Brooks and Corey curve (1964) includes two soil parameters.

The air entry value (or bubbling suction head) (hb) is the matrix suction head

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(

a w b

)

b w u u h g − = ρ (2.12)

Where the matrix suction (suction or capillary pressure) is:

a w

u u= −u (2.13)

With ua pore-air pressure and uw pore-water pressure

The air entry value decreases from clay to sandy soils.

The pore size distribution index (N) summarises information about the pore size distribution in the soil. It increases from clay to sandy soils.

A review of values found in the literature is given in Tab. 2.6 and Tab. 2.7.

Air entry value (hb) [cm]

Type of soil (Brakensiek et al., 1981)

(Pajian, 1987)

(Carsel & Parrish., 1988) Sand 35.30 3.58 6.90 Loamy Sand 15.85 1.32 8.06 Sandy Loam 29.21 9.01 13.33 Loam 50.94 19.61 27.78 Silty Loam 69.55 31.25 50.00

Sandy Clay Loam 46.28 7.81 16.95

Clay Loam 42.28 31.25 52.63

Silty Clay Loam 57.78 30.30 100.00

Sandy Clay - - -

Silty Clay 41.72 15.87 200.00

Clay 63.96 10.00 125.00

Tab. 2.6: Typical values of air entry value (hb)

Pore size distribution index (N) [1]

Type of soil (Brakensiek et al., ₁₉₈₁₎ (Pajian, ₁₉₈₇₎ (Carsel & Parrish., ₁₉₈₈₎

Sand 0.57 0.40 1.68

Loamy Sand 0.46 0.47 1.28

Sandy Loam 0.40 0.52 0.89

Loam 0.26 0.40 0.56

Silty Loam 0.22 0.42 0.41

Sandy Clay Loam 0.37 0.44 0.48

Clay Loam 0.28 0.40 0.31

Silty Clay Loam 0.18 0.36 0.23

Sandy Clay - - -

Silty Clay 0.21 0.38 0.09

Clay 0.21 0.41 0.09

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van Genuchten curve parameters

The Brooks and Corey curve (1964) includes two soil parameters (α and n), that have no physical meaning. The parameter α decreases from clay to sandy soils, while the parameter n increases from clay to sandy soils. A review of values found in the literature is in Tab. 2.8 (Simúnek et al., 1986).

Type of soil α [1/cm] n [1] Sand 0.145 2.68 Loamy Sand 0.124 2.28 Sandy Loam 0.075 1.89 Loam 0.036 1.56 Silty Loam 0.016 1.37

Sandy Clay Loam 0.020 1.41

Clay Loam 0.059 1.48

Silty Clay Loam 0.019 1.31

Sandy Clay 0.010 1.23

Silty Clay 0.027 1.23

Clay 0.005 1.09

Tab. 2.8: Typical values of coefficients α and n for the van Genuchten soil-water retention curve

2.2.4.3 Selection of simplified model

Numerous analytical solutions of the Richard equation have been proposed, mostly in case of soil infiltration due to rain.

The infiltration due to mean water level is calculated using Darcy’s law for the saturated soil. The infiltration in the unsaturated soil is solved extending the model proposed by Wang (Wang et al., 2002) to the infiltration through the dike body (Section 2.2.5). As we assume that the seaward side of the dike is not damaged, the infiltration due to mean water level is very slow and doesn’t really affect dike stability. The final version of the detailed model doesn’t include this kind of infiltration.

To the authors’ knowledge only two models are available for infiltration due to wave overtopping or overflow (Wang, 2000a; Weißmann, 2003). Nevertheless both models are only able to calculate the infiltration in the saturated soil. Making use of simplifying assumptions and other models that also calculate the infiltration in the unsaturated soil (Wang et al., 2003; Wang, 2000a), the detailed model is able to describe the infiltration in both saturated and unsaturated soil (Section 2.2.6).

The outputs of the infiltration model are: • Saturated water front (zs)

• Infiltration water front or wetting front (zw)

• Volumetric water content (θ) • Suction pressure (u)

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Saturated front zs [m]

Infiltration rate vinf [m/s]

Infiltration front zw [m] Water content θ(z) [-] Suction pressure u(z) [N/m2_]

Weißmann (2003) Weißmann (2003) Linear: θ∈[θs, θi] van Genuchten

Weißmann (2003) Wang, Q. (2003) Wang, Q. (2003) Brooks and Corey Wang, Z. (2000) Wang, Z. (2000) Linear: θ∈[θs, θi] van Genuchten

Wang, Z. (2000) Wang, Z. (2000) Linear: θ∈[θs, θi] Brooks and Corey

Wang, Z. (2000) Wang, Q. (2003) Wang, Q. (2003) Brooks and Corey Wang, Q. (2003) Wang, Q. (2003) Wang, Q. (2003) Brooks and Corey

Tab. 2.9: Implemented models for the infiltration due to wave overtopping with and without overflow

2.2.5. Infiltration due to mean water level

The infiltration due to variation in the mean water level is separately calculated for the saturated and the unsaturated soil (Sections 2.2.5.1and 2.2.5.2).

2.2.5.1 Saturated zone

The saturated water front is given by the combination of the initial stationary phreatic line and the transient phreatic line.

Stationary phreatic line

The stationary phreatic line, which represents the initial condition for the mean water level, is first calculated as in the preliminary model (D'Eliso et al., 2005), but two different dike toes on the landward side are considered:

• Dike ditch (Fig. 2.7a)

• Toe filter and protection (Fig. 2.7b)

h H, T 1 m H_d B_d 1 n β α h_sea x C _B A d Dike ditch a) Dike ditch h H, T 1 m H_d B_d 1 n β α h_sea C _B A d

b) Toe filter and protection

x

Protection against seepage

h_f

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The main assumptions are briefly reminded: (i) validity of Darcy’s law, laminar flow in saturated soil (Darcy, 1856), (ii) stationary flow in the dike, (iii), homogeneous materials, (iv) Dupuit hypotheses (Dupuit, 1863): the hydraulic gradient is constant along each vertical section and equal to the slope of the phreatic line and laminar flow, (v) the dike base is impermeable.

Under these assumptions, the phreatic line can be determined applying Eqs. (2.14) -(2.16) (Casagrande, 1940; Mishra & Singh, 2005):

CB 0.3AC= (2.14) 2 2 sea S= d +h −d (2.15) 2 f h = 2Sx S+ (2.16)

Where (see Fig. 2.7):

• hf is the position of the phreatic line in the dike (m)

• S is the focal distance of the parable (m);

• hsea is the position of the phreatic line at the entrance point (m)

It is then imposed that the phreatic line enters the dike in the point B (Fig. 2.7):

( )

f sea

h B =h (2.17)

The stationary phreatic line is finally approximated as linear for the calculation of the transient phreatic line (Fig. 2.7).

Transient phreatic line

The transient phreatic line, which changes its position in the dike depending on the mean water level hydrograph (MWL(t)), is calculated applying Darcy’s law throughout the dike body.

For every time step of the mean water level hydrograph, the infiltration path of each entrance point of the water in the dike is assigned (Fig. 2.8).

X

H, T_{MWL (t)}

A, B, C, D: Entrance points E: Exit point

AE, BE, CE, DE: Infiltration paths

Stationary phreatic line

(initial condition) Transient phreatic line_{(time t)} MWL (0) A D C B E Saturated soil (time t)

Transient phreatic line (time t + dt) l_s(t)

d_ls(t + dt) Z

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The saturated water front is given by the infiltrated distance from the entrance point along the infiltration path:

inf s s inf s dh dh v k dl v dt k dt dx dx = − ⇒ = = − (2.18) dh MWL z= − (2.19) E Entrance dx x= −x (2.20)

(

)

( )

(

)

s s s l t dt+ =l t +dl t dt+ (2.21)

This calculation is qualitatively in agreement with experimental measurements of infiltration in homogeneous dikes (Nakayama & Sako, 2004; Scheuermann & Brauns, 2001).

The effect of a crack along the seaside can be roughly included in the model assigning as input, its length and position. In order to simulate the preferential pathway for infiltration flow given by the crack, the dike portion under the crack is assumed saturated at the beginning of the simulation.

The volumetric water content is set equal to the saturated value within the saturated water front.

The water pore pressure is assumed hydrostatic and then calculated according to the vertical distance from the phreatic line:

w w s

u = ρ g z −z (2.22)

Where zs is the elevation of the phreatic line and z is the vertical coordinate.

2.2.5.2 Unsaturated zone

The infiltration in the unsaturated soil is calculated adapting an analytical solution of the Richard equation for horizontal infiltration (Wang et al., 2002). This model is based on the following assumptions:

• Mono-dimensional horizontal infiltration;

• Brooks and Corey’s (1964) soil-water characteristic curve;

• The soil water suction has a non-linear distribution along the horizontal infiltration path.

In order to apply this model to sea dikes, the following assumptions have to be added:

• First Dupuit hypothesis (Section 2.2.5.1);

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Solving the Richard equation under the specified assumptions, it yields (Wang et al., 2002):

Infiltration water front: s b

(

)

w

s r

ak h M 1 1 N

x = − − t

θ − θ (2.23)

Volumetric water content:

( )

(

)

2 N r s r w ax x 1 x ⎛ ⎞ θ = θ + −_⎜ _⎟ θ − θ ⎝ ⎠ (2.24) Where:

• ks is the saturated hydraulic conductivity of the soil [m/s]

• a [1] is a parameter given by:

2 1 N i r s r a 1_{= − ⎜}⎛θ − θ ⎞_⎟ θ − θ ⎝ ⎠ (2.25)

This parameter is close to 1 if the initial water content θi is very low.

• N is the pore size index [1]

• M 2 3N= + [1] (2.26)

The suction pressure (u) is calculated with the Brooks and Corey soil-water characteristic curve (Eq. (2.5)).

2.2.6. Infiltration due to wave overtopping with and

without overflow

The infiltration due to wave overtopping with and without overflow is calculated optionally applying one of selected models (Tab. 2.9) that are presented in the following sections (Sections 2.2.6.1-2.2.6.3). The definition sketch of the infiltration due to overtopping flow is in Fig. 2.9.

H, T

MWL (t)

Saturated water front z_s (time t) Unsaturated soil (time t) E Saturated soil (time t)

Infiltration water front z_w (time t) Z

X h (t)

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The effect of a crack along the crest and landside can be roughly included in the model assigning as input, its length and position as for the infiltration at the seaside. In order to simulate the preferential pathway for infiltration flow given by the crack, the dike section corresponding to the location of the crack is assumed saturated at the beginning of the simulation.

2.2.6.1 Weißmann’s model

The Weißmann model (Weißmann, 2003) is derived from laboratory experiments of wave overtopping and overflow on a sea dike (without grass cover) and calculates the saturated water front solving Darcy’s law under this assumptions:

• Validity of Darcy’s law;

• Homogeneous and isotropic material and constant saturated hydraulic conductivity (ks);

• The hydraulic conductivity (k(θ)) is constant overall in the clay cover and equal to the saturated hydraulic conductivity (ks);

• The calculation domain is limited to the saturated soil only;

• The effects of the wave overtopping tongues of irregular and regular waves and of combined flow depth can be summarized in an overflow layer thickness constant over time and equal to the mean value h over the simulation time ts:

( )

s t 0 s s h t dt h t t =

∫

(2.27)

• The hydraulic gradient (i) is constant along the saturated water depth; • The saturated infiltration front advances parallel to the dike surface (pores

have the same size along the infiltration depth);

• van Genuchten’s (1980) soil-water characteristic curve;

Solving Darcy’s law under the specified assumptions yields (Weißmann, 2003):

Saturated water front:

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( )

1 n 1 m s r i i r 1 h θ = ⎡⎢⎛_⎜θ − θ ⎞_⎟ −1⎤⎥ α_⎢_⎣_⎝θ − θ _⎠ _⎥_⎦ (2.31)

In the detailed model, the saturated water front over time is then calculated solving iteratively Eq. (2.28) for each time t.

The infiltration water front can in principle not be calculated. Making use of the coefficient f calibrated on FE model of the Richard equation (Weißmann, 2003) and extending its validity to the calculation of the unsaturated soil, in the detailed model, the infiltration water front is tentatively calculated as:

s w s w s s z 1 z z z z 1 z 15z f f ⎛ ⎞ ∆ = − = ⇒ = +_⎜ _⎟ ≅ ⎝ ⎠ (2.32)

The infiltration water front gives higher values than the other models (Section 2.2.6.4).

The volumetric water content can be calculated with the Wang Q. model (Section 2.2.6.3) or by simply assuming a linear distribution along the vertical infiltration path with values:

• θ

( )

z_s = θ_s (2.33)

• θ

( )

z_w = θ_i (2.34)

The suction pressure (u) is calculated with the van Genuchten soil-water characteristic curve (Eq. (2.6)).

2.2.6.2 Wang’s Z model

The Wang’s Z. model (Wang, 2000a) calculates the saturated water front solving Darcy’s law under this assumptions:

• Validity of Darcy’s law;

• The calculation domain is limited to the saturated soil;

• The hydraulic gradient is only function of the free surface water depth; • The influence of unsaturated soil on infiltration is concentrated in an

empirical parameter (α).

Solving Darcy’s law under the specified assumptions yields (Wang, 2000a):

Saturated water front: z t_s

( )

= 2 kα _{s 0}

_∫

th t dt

( )

(2.35) Where:

4.5

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Validation of the model is only given by comparison with numerical results obtained from a FE model of Richard equation.

In order to account for the variation of the water depth with time, in the detailed model, the saturated water front over time is calculated rewriting Eq. (2.35) as:

(

)

2

( )

(

)

s s s z t dt+ = z t +dz t dt+ (2.37)

(

)

( )

s s dz t dt+ = α2 k h t dt (2.38)

The infiltration water front can be calculated using an equation similar to Eq. (2.35) (Wang, 2000a):

Infiltration water front:

( )

t

( )

0

z t = 2β

_∫

h t dt (2.39)

Validation of equation (2.39) is still lacking. Moreover, there are no precise indications on the values of the coefficient β.

Making the ratio of Eqs. (2.35) and (2.39), the coefficient β can be calculated as a function of the ratio between the infiltration water front zw and the saturated water

front zs: w s s z k z β = α (2.40)

Making use of the coefficient f as already tentatively proposed for the Weißmann model (Section 2.2.6.1, Eqs. (2.32)), in the detailed model, the coefficient β is tentatively calculated as:

s 15 k

β = α (2.41) In order to account for the variation of the water depth with time, in the detailed

model, the infiltration water front over time is calculated rewriting Eq. (2.39) as:

(

)

2

( )

(

)

w w w z t dt+ = z t +dz t dt+ (2.42)

(

)

( )

w dz t dt+ = β2 h t dt (2.43)

The infiltration water front is substantially lower than that obtained with the Weißmann model (Section 2.2.6.4).

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The suction pressure (u) is calculated either calculated with the Brooks and Corey soil-water characteristic curve (Eq. (2.5)) or with the van Genuchten soil-water characteristic curve (Eq. (2.6)).

2.2.6.3 Wang’s Q model

The Wang’s Q. model (Wang et al., 2003) solves analytically the Richard equation for vertical infiltration and results have been compared with soil water infiltration data (Tzimopoulos et al., 2005; Wang et al., 2003). The model is based on the following assumptions:

• Mono-dimensional vertical infiltration;

• Brooks and Corey’s (1964) soil-water characteristic curve;

• The analytical solution of the Richard equation is obtained starting from a solution proposed in the literature (Parlange, 1971) and using a Taylor’s series method.

Solving the Richard equation under the specified assumptions, it yields (Wang et al., 2003):

Infiltration water front:

(

s

)

i w

(

w

)

s ln z 1 t z 1 k β + ⎛ ⎞ θ − θ = _⎜ − _⎟ + α _⎝ β _⎠ (2.44)

Volumetric water content:

( )

_r

(

_s _r

)

w z z 1 z α ⎛ ⎞ θ = θ + −_⎜ _⎟ θ − θ ⎝ ⎠ (2.45) Where: • N; M 2 3N M α = = + • b M ; a 1 ah β = ≅

The saturated water front is not strictly defined by this model because the volumetric water content immediately decreases starting from the infiltration surface (z = 0). For this reason the saturated water front can be optionally introduced in two ways:

• z_s = according to the Wang Q. model; 0

• zs is calculated with the Weißmann model or with the Wang Z. model and

the infiltration water front zw obtained with the Wang Q. model is referred

to the calculated saturated water front zs.

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2.2.6.4 Comparative analysis of the models

The three models (Sections 2.2.6.1-2.2.6.3) in the six combinations (Tab. 2.9) provide very different results. An idealized simulation case with a significant wave height Hs of 3 m and wave period TP of 10 s, an overtopping freeboard of

0.50 m (MWL = 6.50 m), a dike 7 m high, with a landside profile of 1:6 slope is used for comparisons. Material properties are relative to soils of type sand and clay with grass, according to the classification of Tab. 2.2-Tab. 2.7 (Carsel & Parrish., 1988; Weißmann, 2003), see Tab. 2.10.

MATERIAL PROPERTIES CLAY SAND

Saturated hydraulic conductivity ks [m/s] 10-6 10-5

Saturated water content θs [1] 0.48 0.35

Residual water content θr [1] 0.226 0.054

Initial water content θi [1] 0.25 0.07

Air entry value hb [cm] 63.96 35.3

Pore size distribution index N [1] 0.21 0.57

Pore-size distribution index parameter n 1.41 2.68

Inverse of air entry value α [1/cm] 0.005 0.145

Tab. 2.10: Material properties used in the calculation of the infiltration due to wave overtopping

As an example of the results, in Fig. 2.10, the volumetric water content calculated with the Weißmann model at the landside of the dike. A crack of 0.5 m depth has been assumed at the landside of the dike (section A and B in Fig. 2.10). Typical material parameters for sand and clay have been used (Section 2.2.4.2), see Tab. 2.10. Saturated and infiltration water fronts are constant along the dike profile, apart from the dike sections with the two cracks and so does the volumetric water content (Fig. 2.10). These results indicate that the wave overtopping flow induces a constant infiltration all over a uniform dike profile. Variation in the infiltration fronts are only due to weak dike sections.

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In Fig. 2.11, the saturated and infiltration water fronts are compared. The Weißmann model predicts the highest values and the Wang Z. model the lowest.

a) Weißmann’s model

b) Wang Z.’s model

c) Wang Q.’s model

Fig. 2.11: Comparison between the selected models (saturated and infiltration water fronts)

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2.3. Morphodynamic description

The morphodynamic module includes the simulation of breach initiation and growth through the following three phases: breach initiation, breach formation and breach development (D'Eliso et al., 2005). The evolution of the dike profile and of the breach cross-section with time is given. In Fig. 2.16, an overview of the simulated processes is given. In Sections 2.3.1-2.3.5, only processes that have been further developed with respect to the preliminary model or included for the first time are presented. For all other processes, a detailed description can be found in (D'Eliso et al., 2006).

2.3.1. From preliminary to detailed model

The breaching process, from a 3D process, is reduced to a 2D + 2D process, as in the preliminary model.

The whole process is still divided in six phases: three phases for the breach growth in the cover layer (Fig. 2.12a) and three phases for the breach growth in the core (Fig. 2.12b):

• Initial condition: Incipient overtopping flow (or combined flow) over the dike;

• Phase 1: Failure of the grass cover due to erosion or sod stripping; • Phase 2: Local erosion in the clay cover layer;

• Phase 3a: Headcut erosion and discrete advance in the clay cover layer up to the core exposure to the flow action;

• Phase 4a: Headcut erosion and discrete advance of the sand-clay scour up to the erosion or sliding of the cover layer;

• Phase 4b: Crest shortening to erosion and mass failure;

• Phase 5a: Crest lowering and breach widening due to erosion and mass failure, progressive failure of the cover layer at the seaside (wave overtopping);

• Phase 5b: Crest lowering and breach widening due to erosion and mass failure, progressive failure of the cover layer at the seaside (combined flow);

• Phase 6a: Full breach, breach widening and deepening (in case of erodible base) due to erosion and mass failure (super-critical overflow);

• Phase 6b: Breach widening and deepening (in case of erodible base) due to erosion and mass failure up to the equilibrium final breach (sub-critical overflow).

The following processes are mainly improved or introduced for the first time in the detailed:

• Breach initiation (Section 2.3.2);

• Grass and cover layer sliding (Section 2.3.3); • Headcut erosion and advance (Section 2.3.4);

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Loading type: wave overtopping

Main assumption: each wave of the time series is assumed to be a regular wave with a certain height (H) and period (T)

Loading type: combined wave overtopping and overflow

infiltration due to wave overtopping or combined flow

Simulated process: headcut erosion and advance

Erosion: cumulated excess shear stress, shear stress at the base of an

overfall

Leading parameter: erodibility coefficient

Headcut advance: discrete approach, quasi-3D mass instability Leading parameter: clay cohesion

Initial condition: incipient wave overtopping

MWLH, T

Phase 1: Grass cover erosion

Phase 2: Local clay erosion

Phase 3a: Headcut erosion

MWLH, T

Simulated process: erosion of the grass cover

Erosion: cumulated excess shear stress, the grass erosion up to failure is due to the repetition of the flow action

Leading parameter: Overall Manning roughness (Section)

Formation of spots in the grass cover

MWLH, T Concentrated erosion in the spots MWLH, T Headcut formation and advance

Simulated process: concentrated clay erosion is small rills and holes

Erosion: cumulated excess shear stress Leading parameter: erodibility coefficient

a) Cover layer failure

b) Mass instability b) Vertical erosion a) Trapezoidal section Breach widening Phases 4, 5, 6a b) Undermining erosion c) Mass instability a) Rectangular section

Simulated process: breach widening up to the equilibrium

breach

Flow: transition between super-critical and sub-critical overflow

Breach morphology: sloped sides, vertical erosion and mass instability from the sides

Erosion and mass stability as in Phase 4 and 5.

Phase 4: Headcut erosion in sand-clay Possible cover sliding

Phase 5a: Dike lowering (wave overtopping)

Phase 5b: Dike lowering (combined flow)

Phase 6: Full breach

MWLH, T

Simulated process: headcut erosion in two-layered materials, sliding Erosion: non-equilibrium sediment transport model

Leading parameter: sediment size

Headcut advance: discrete approach, 2D mass instability Leading parameter: soil tensile strength

Sliding: 2D mass instability

Leading parameters: cohesion, inner slope of the dike

Simulated process: erosion of the sand core and breach

channel formation

Flow: transition between wave overtopping and combined flow and between combined flow and overflow

Breach morphology: vertical sides (rectangular shape), undermining erosion and mass instability from the sides Erosion: sediment transport capacity and sediment mass conservation (vertical and lateral erosion)

Mass stability: simplified stability analysis

Leading parameters: friction factor, sediment transport calculation

b) Failure of the sand core

Phase 6b

MWLH, T

Headcut formation and advance

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2.3.2. Breaching initiation and location

Initial breach location and breach initiation still represents an open question (D'Eliso et al., 2005).

In the detailed model the following assumptions are imposed:

• The distribution of incipient breaches along the dike defence line is neglected and the incipient points of breaching are only given along ideal dike cross-section;

• Non-homogeneity of the dike materials are only considered along the dike surface;

• The incipient point of breaching is located on a “weak point”. Types and severity of weak points are given as inputs and not predicted.

Based on observations on real sea dikes (TAW, 2000), breach initiation is likely to occur in one of the following cases (Fig. 2.13):

• Areas with damaged grass cover;

• Cracks due to shrinkage and holes dig by animals in the clay cover;

• Areas with strongly fractured soil due to weathering or sandy-clay lenses due to non-homogeneities in the materials.

In the detailed model, the incipient point of breaching and breach initiation are simulated based on a qualitative description of the dike surface.

In Fig. 2.13, the definition sketch of weaknesses along the dike profile which are included in the model is reported.

Each weakness, together with its severity, corresponds to a different initial scenario and influences one or both erosion and cover instability (Tab. 2.11). Depending on the severity of the weakness (where included in the inputs), the affected material properties (Tab. 2.11) are locally modified as follows:

• Severity R: 3 f ,l s,l s R ,l l p,l 75,l 75 C =0; k =10 k ; c =0; c =0; I =0; D =1.60D • Severity L: 2 f ,l f s,l s R,l R l p,l p 75,l 75 C =1 4C ; k =10 k ; c =1 4c ; c =1 4c; I =1 4 I ; D =1.40D • Severity M: 1.5 f ,l f s,l s R,l R l p,l p 75,l 75 C =2 4C ; k =10 k ; c =2 4c ; c =2 4c; I =2 4 I ; D =1.25D • Severity H: f ,l f s,l s R,l R l p,l p 75,l 75 C =3 4C ; k =10k ; c =3 4c ; c =3 4c; I =3 4 I ; D =1.10D Where Cf,l, ks,l, cg,l, cl, Ip,l and D75,l are the local values of the grass cover factor,

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Breaching initiation: initial scenario approach

W

S _S L

S

Dike slope on landside

Type Position Length L Width W Depth D Level

Single crack X - - X -

Hole X - X X -

Damaged grass X - - - X

Fractured soil X X - X X

Clay-sandy lenses X X - X X

Length: extension of the weakness along the profile Width: representative width of the hole at the surface Depth: length of the weakness normal to the profile

Level: severity of the weakness (R, L, M, H from maximum to minimum severity): R = locally removed grass and no soil strength

L = low grass resistance and low soil strength with respect to the average value M = half grass resistance and half soil strength with respect to the average value

H = slight reduction of grass resistance and soil strength with respect to the average value Fig. 2.13: Definition sketch of weaknesses along the dike profile considered in the model

This procedure also automatically includes the much simpler technique used in the preliminary model, in order to reproduce breach initiation (D'Eliso et al., 2005).

Fractured clay

Sand lens Damaged grass

Breaching of coastal dikes. Detailed breaching model

Breaching of coastal dikes: detailed model

LWI REPORT NR. 937

May 2007

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Breaching of coastal dikes

- Detailed breaching model -

Progress report Nr. 3

Dipl.-Ing. C. D’Eliso

Prof. Dr.-Ing. H. Oumeraci

Abstract

Table of contents

List of figures

List of tables

1. Introduction

1.1. Objective

1.2. Brief outline

2. Mathematical formulation

2.1. Overview and major assumptions

2.2. Hydrodynamic description

2.2.1. From preliminary to detailed model

2.2.2. Wave overtopping

2.2.3. Infiltration process and its simulation: overview

2.2.4. Richard equation’s and analytical solutions

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2.2.4.1 Soil-water characteristic curves

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2.2.4.2 Estimate of soil parameters for infiltration

Saturated hydraulic conductivity

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Saturated volumetric water content

Residual volumetric water content

Initial volumetric water content

Brooks and Corey’s curve parameters

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van Genuchten curve parameters

2.2.4.3 Selection of simplified model

2.2.5. Infiltration due to mean water level

2.2.5.1 Saturated zone

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2.2.5.2 Unsaturated zone

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2.2.6. Infiltration due to wave overtopping with and

without overflow

2.2.6.1 Weißmann’s model

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2.2.6.2 Wang’s Z model

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