Wave overtopping - CORE

Wave overtoppingImpact of water jets on grassed inner slope transitions

Astrid ValkC1143654

Faculty of Civil Engineering& Geosciences

MSc-ThesisDelft, February 2009

Wave overtoppingImpact of water jets on grassed inner slope transitions

DeltaresRotterdamseweg 185

2629 HD Delfttel. +31 [0]15-2858585

Delft University of TechnologyFaculty of Civil Engineering & Geosciences

Section of coastal engineeringStevinweg 1

2628 CN Delfttel. +31 [0]15-2785440

Graduation committee

Prof. dr. ir. M.J.F. StiveFaculty of Civil Engineering & Geosciences

Section of coastal engineeringIr. H.J. Verheij

DeltaresDepartment Inland Water Systems

Ir. H.J. VerhagenFaculty of Civil Engineering & Geosciences

Section of coastal engineeringDrs. W.N.J. Ursem

Faculty of Applied SciencesBotanical Garden TUDelft

Astrid ValkC1143654

Faculty of Civil Engineering & Geosciences

MSc-ThesisDelft, February 2009

Preface

The subject of my MSc-thesis is: Wave overtopping and stability of grass cover layers attransitions on the inner slope of sea dikes. The MSc-thesis is a part of the curriculum of theMSc-study “Hydraulic Engineering” at the faculty of Civil Engineering & Geosciences at the DelftUniversity of Technology in the Netherlands. The work is done in cooperation with Deltares.

Wave overtopping occurs when the water level in front of the dike and the waves are highenough that after breaking on the outer slope, the wave run up exceeds the crest level. Waveovertopping is quantified as the mean discharge per meter of width, q [l/s m-1 or m3/s m-1],that is reaching the inner slope of the dike (or structure). It has a negative effect on thestability of the grass cover layer on the inner slope of the dike and can result in severe damage.

I would like to thank my graduation committee, prof. dr. ir. M.J.F. Stive, ir. H.J. Verheij, ir. H.J.Verhagen and drs. W.N.J. Ursem. In addition, I would like to thank Deltares for the use of thefacilities. Furthermore, I would like to thank the people who helped to correct my writings andsupported me during my study.

Astrid Valkc1143654

Abstract

The safety of a large part of the Netherlands depends on sea dikes. Due to climate change, asea level rise is predicted. Together with stronger storms and more wave attack, waveovertopping on the current dikes will increase. As a response, crest levels of the dikes need tobe raised in order to meet the present regulations for wave overtopping. Alternatively, theseregulations could be lowered if the dikes can be proven strong enough to cope with theincreased wave overtopping. Experiments indicate that higher wave overtopping loads can beallowed in comparison to the present regulations. A better insight in the erosion phenomenaresulting from wave overtopping will make it possible to define more accurate guidelines onallowable loads. This will lead to less unnecessarily disapproval of the present dikes. The SBW(‘Sterkte & Belastingen Waterkeringen’ or ‘Strength and Loads on Water Defences’) project withrespect to grassed inner slopes focuses on the improvement of reliable overtopping criteria forthe present dike structures.

To evaluate the actual strength of the grassed inner slope of a sea dike, experiments arecarried out using a wave overtopping simulator. The wave overtopping simulator is developedto create full scale overtopping conditions on a real inner slope of a dike. Recent experiments(Delftzijl, 2007 and Boonweg and Zeeland, 2008) show initiation of erosion of the grass layerdownstream of the transition of the slope and a horizontal part. Due to the impinging forces ofthe overtopping wave tongue, a scour hole has been formed. After the scour hole has reacheda certain depth headcut erosion is observed. Because nowadays used erosion models forgrassed inner slopes do not take this transition induced erosion into account, modelling of thisprinciple is the focus of this MSc-thesis.

To gain some insight into failure mechanisms, hydraulic boundary conditions and grass coverlayers a literature study is preformed first. This part is followed by a study to analyse thenowadays used erosion models for grass cover layers. After these introductory studies, theerosion process of a grassed inner slope transition is schematized in three stages. These threestages are depicted in Figure 1.

1: Development of the jetscour hole

2: Transition: surface erosion– headcut erosion

3: Headcut erosion

Figure 1: Erosion process of a grassed inner slope transition

The erosion process can be described by two different models. The Transition Model (TM)describes the first stage; the development of the scour hole due to surface erosion. The SSEA(Sites Spillway Erosion Analysis) model describes the development of headcut erosion.

The equation for the TM reads:2

0 ( ) ( )ˆ

c

soil

d ddydt E

Equation 1.1

Where y is the erosion depth and t̂ is the characteristic overtopping time. Calculation of thecritical shear stress ( c) is based on the cubic turf model of Hoffmans. Determination of theapplied shear stresses ( 0) is based on oblique impingement theories of Beltaos. Overtopping

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velocities and overtopping times are as described by the Bosman formulas. The soil parameterEsoil is a function of the critical shear stress and is compared to the overall strength parameterCE as defined by Delft Cluster.

The equation for the SSEA model (headcut erosion) reads:

( )headcut cdx C L Ldt

Equation 1.2

Where dx/dt is the headcut velocity, L (q, H) is the hydraulic load and Lc is the critical hydraulicload which, like coefficient Cheadcut, depends on material characteristics. (q is the specificdischarge and H the headcut height)

The results of the model are compared with results of the experiments (Boonweg and Zeeland,2008). Model and experimental results show some discrepancies. The reason for thesediscrepancies is the fact that irregularities at the real dikes are not taken over in the model.Clinker roads and drainage measurements at a transition have a certain impact on the rootdevelopment and therefore on the erosion resistance. Furthermore, it can be doubted whethergrass strength characteristics at the transition are equal to grass strength characteristics on theslope due to drainage issues. It is recommended to do further research to grass and claycharacteristics at the transition as well as soil characteristics at a certain depth. According tothe defined erosion model, the present guidelines are stern and the actual strength of agrassed inner slope is underrated. On account of the recent research programs, like the SBWproject with respect to grassed inner slopes, assessment guidelines for wave overtopping canbe reformulated. This way dikes can be assessed more accurate. Finally, this will lead to moreaccurate insight in the safety of the Netherlands with regards to grassed sea dikes.

List of contents

1. INTRODUCTION AND STUDY OBJECTIVES 151.1. General introduction 161.2. Problem analysis 17

1.2.1. Problem definition 191.2.2. Objective 19

1.3. Methodology 20

2. BOUNDARY CONDITIONS AND DIKE STRUCTURES 232.1. Failure mechanisms 242.2. Safety standards 262.3. Hydraulic loads 272.4. Wave overtopping 29

2.4.1. Overtopping discharge 292.4.2. Overtopping volumes per wave 302.4.3. Overtopping depth, velocities and time 31

2.5. Dike structure 332.6. The use of clay as dike cover material 342.7. The grass cover layer 35

2.7.1. Grass mat management 362.7.2. Grassland type 372.7.3. Root density 38

2.8. Conclusion on the situation in the Netherlands 39

3. EROSION MODELS 413.1. Introduction 433.2. EPM 44

3.2.1. Modified EPM 443.3. Rajaratnam 493.4. Stein theory 503.5. Stanczak 513.6. Hoffmans 553.7. Canepa 563.8. Young 573.9. SSEA 583.10. Discussion 59

4. EROSION PROCESS 614.1. The initial situation 624.2. Development of the jet scour hole 634.3. Transition: surface erosion – headcut erosion 634.4. Headcut erosion 64

5. DERIVATION EROSION MODEL 65

5.1. Load term 20 ( )d 66

5.1.1. Derivation 0 665.1.2. Depth variability 69

5.2. The strength terms: ( )c d & soilE 725.2.1. Derivation c 725.2.2. Depth variability 72

5.3. Summary of all equations 74

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6. EVALUATION EROSION MODEL 756.1. Evaluation Transition Model 76

6.1.1. Determination parameters 766.1.2. Simulations 806.1.3. Evaluation Transition Model 836.1.4. Sensitivity analysis 846.1.5. Conclusion evaluation TM 90

6.2. Evaluation Headcut erosion model 906.3. Conclusions 92

7. VALIDATION 937.1. Wave Overtopping simulator 947.2. Test simulations 947.3. Test locations 967.4. Results 97

7.4.1. Implementation of Bosman and distribution of waves as given by Van der Meer 997.4.2. Introducing depth dependency 1007.4.3. Validation of the Transition Model 106

7.5. Conclusion 110

8. CONCLUSIONS AND RECOMMENDATIONS 1118.1. Conclusions 1128.2. Recommendations 114

REFERENCES 116

LIST OF PARAMETERS 119

APPENDIXES 123

List of figuresFigure 1: Erosion process of a grassed inner slope transition ......................................................................................7Figure 1-1: KOLB-principle .....................................................................................................................................17Figure 1-2: Transitions at an inner slope .................................................................................................................17Figure 1-3: Special protection measures at St. Philipsland test location .....................................................................18Figure 1-4: Scour hole at test section Kattendijke ....................................................................................................18Figure 1-5: Damage of the inner slope induced by scour hole...................................................................................18Figure 1-6: Scour hole caused by a water jet...........................................................................................................19Figure 2-1: Failure mechanisms dikes (TAW 1998)................................................................................................... 24Figure 2-2: Surface erosion (M) ..............................................................................................................................25Figure 2-3: Shallow slip erosion (N) ........................................................................................................................25Figure 2-4: Headcut erosion ...................................................................................................................................25Figure 2-5: Dike ring areas and safety standards (according to the Dutch Law for water defences ‘wet op de

waterkering’ (original document is translated to English)) ................................................................................26Figure 2-6: Location of measuring stations ..............................................................................................................28Figure 2-7: Exceeding probability curve Hm0 ............................................................................................................28Figure 2-8: Exceeding probability curve Tm-1,0 ..........................................................................................................29Figure 2-9: Overtopping flow depth h at the inner slope [m] ....................................................................................31Figure 2-10: Overtopping time T at the inner slope [s].............................................................................................32Figure 2-11: Max flow velocity u at the inner slope [m/s] .........................................................................................32Figure 2-12: Cross-section of a sea dike (source: Bosman 2007) ..............................................................................33Figure 2-13: Dike covered by grass (EurOtop 2007).................................................................................................33Figure 2-14: Dike covered by asphalt (EurOtop 2007) ..............................................................................................34Figure 2-15: Dike covered by placed block revetment ..............................................................................................34Figure 2-16: Left: Schematisation of the soil structure; right: Soil structure in clay of dikes (TAW 1996b) ...................34Figure 2-17: Definition sketch – grass cover layer (TAW 1999) .................................................................................35Figure 2-18: Sod quality as function of root density (modified version) .....................................................................39Figure 2-19: Necessarily freeboard height according to a specific discharge (with: Hs of 2 m; Tp of 4.7 s, an outer slope

of 1:4 and Tstorm of 6 hours) ...........................................................................................................................40Figure 3-1: Strength schematization of root cohesion...............................................................................................46Figure 3-2: Forces acting upon cubic turf ................................................................................................................47Figure 3-3: Left: observed radial wall jet (source: Kortenhaus); right: schematization radial wall jet ...........................49Figure 3-4: Free water jet falling in a plunge pool ....................................................................................................50Figure 3-5: Test set-up ..........................................................................................................................................52Figure 3-6: Shear Failure model Führböter. .............................................................................................................53Figure 3-7: Predicted and observed clay failure-principle sketches (Stanczak 2007) ...................................................53Figure 3-8: Schematization.....................................................................................................................................55Figure 3-9: Test set-up Canepa ..............................................................................................................................56Figure 3-10: Forces on the slope ............................................................................................................................57Figure 3-11: Shallow slip erosion ............................................................................................................................57Figure 3-12: Headcut erosion .................................................................................................................................58Figure 4-1: Initial situation .....................................................................................................................................62Figure 4-2: Development of jet scour hole...............................................................................................................63Figure 4-3: Transition point: surface erosion - headcut erosion.................................................................................63Figure 4-4: Headcut erosion ...................................................................................................................................64Figure 5-1: Flow regions within a vertically impinging jet .........................................................................................67Figure 5-2: Flow region oblique impinging ( < 30º).................................................................................................68Figure 5-3: Maximum wall shear stress (original graph see (Beltaos 1976))...............................................................68Figure 5-4: Comparison tail water damping models..................................................................................................70Figure 5-5: Influence of variable tail water damping coefficient. ...............................................................................71Figure 5-6: Strength function with respect to depth .................................................................................................73Figure 6-1: Original strength profile ........................................................................................................................77Figure 6-2: Modified strength profile; using cmin =0.21c ...........................................................................................78Figure 6-3: Modified strength profile; using cmin =0.21c and cs =5 [m-1] .................................................................78Figure 6-4: Strength profiles according to three different methods............................................................................79Figure 6-5: Computed erosion process over time (TM) in case of overflowing water ..................................................81Figure 6-6: Wave distribution 10 l/s m-1 and 75 l/s m-1.............................................................................................82Figure 6-7: Computed erosion process 1, 10, 30, 50 and 75 l/s m-1 (TM)...................................................................83Figure 6-8: Comparison TM & EPM/Hoffmans model (q =75 [l/s m-1]).......................................................................84Figure 6-9: Variability Failure Time due to change in q [l/s m-1]................................................................................85Figure 6-10: Sensitivity to the load coefficient (q =75 l/s m-1) ..................................................................................86Figure 6-11: Sensitivity to air-water ratio (q =75 l/s m-1).......................................................................................... 86Figure 6-12: Sensitivity to air-water ratio (q =50 l/s m-1).......................................................................................... 87Figure 6-13: Sensitivity to damping coefficient (q =75 l/s m-1)..................................................................................88Figure 6-14: Sensitivity to suggested clay strength modifications (q =75 l/s m-1) .......................................................89Figure 6-15: Sensitivity to suggested clay strength modifications (q =50 l/s m-1) .......................................................89Figure 6-16: Variability headcut erosion due to variable q [l/s m-1] (kh =0.03 & H =0.8 m).........................................91Figure 6-17: Sensitivity headcut erosion to index number kh (q =10 l/s m-1 & H =0.8 m) ...........................................91Figure 6-18: Sensitivity headcut erosion to headcut height H (q =10 l/s m-1 & kh =0.03) ............................................92Figure 7-1: Principle Wave Overtopping simulator....................................................................................................94

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Figure 7-2: Theoretical and simulated number and volumes of wave events for a 6-hour storm duration ....................95Figure 7-3: Impression experimental test set-up......................................................................................................96Figure 7-4: Root density profiles with respect to depth given as number of roots per m2............................................97Figure 7-5: Computed development of the erosion depth due to the wave events of a 6 hours storm of 1 singular

specific discharge (blue line: using characteristic values for wave distribution, velocity and overtopping time andcyan line: using formulas of Bosman and Van der Meer). .................................................................................99

Figure 7-6: Computed development of soil parameter with respect to depth; method C uses a depth variable profile andmethod B uses a constant strength profile .................................................................................................... 100

Figure 7-7: CE profile of the depth variable strength profile .................................................................................... 101Figure 7-8: CE profile with tr =20*106 [N/m2] and a modified profile with tr =45*106 [N/m2] ..................................... 102Figure 7-9: Computed erosion profile (method C): continues line tr =20*106 [N/m2]; dotted line tr =30*106 [N/m2] and

discontinues line tr =45*106 [N/m2] .............................................................................................................. 103Figure 7-10: Computed erosion profile TM (method D) with tr =45*106 N/m2 and =0; left graph: erosion profile with

unmodified clay parameters; right graph: modified erosion profile: continues line f =0.21 and cs =5 [m-1];discontinues line f =0.21 and cs =20 [m-1]................................................................................................... 104

Figure 7-11: CE profile with tr =20*106 [N/m2] f =0.021 and cs =1.75 [m-1] and a modified profile (CE mod) with tr

=45*106 [N/m2], f =0.21 and cs =20 [m-1] .................................................................................................. 105Figure 7-12: Strength profiles with tr =45*106 N/m2, f =0.21 and cs =20 m-1 left: Using root density according to

Sprangers; right: strength profile for location Kattendijke .............................................................................. 105Figure 7-13: Computed erosion profiles Boonweg section 1: green line; section 2: magenta line; section 3: blue line and

section 4: cyan line ..................................................................................................................................... 106Figure 7-14: Computed erosion profile left: St-Philipsland and right: Kattendijke (K-1)............................................. 107Figure 7-15: Left: computed erosion profile K-2; right: computed erosion profile K-D............................................... 108Figure 7-16: Computed erosion profile for a grass layer with an averaged root density as defined by Sprangers........ 109Figure A-1: Mechanism causing damage due to wave impact. ................................................................................ 124Figure A-2: Erosion mechanisms on water filled crack ............................................................................................ 126Figure B-3: Averaged values for root density (Stanczak) ........................................................................................ 128Figure B-4: Root length [m/dm3] .......................................................................................................................... 129Figure B-5: Root weight [g/dm3]........................................................................................................................... 129Figure B-6: Root density with respect to depth; in black: RVR [%] (equation derived by Stanczak) & in red: root length

[m/dm3] (equation derived by Sprangers) ..................................................................................................... 129Figure B-7: Root density with respect to depth; in black: RVR [%] (equation derived by Stanczak) & in red: RAR [%]

(equation derived by Valk) ........................................................................................................................... 130Figure C-8: Root density profiles with respect to depth given as number of roots per m2.......................................... 133Figure E-9: Boonweg section 1; upper left: initiation after S3; upper right: situation after S5; down left: situation after 3

hours S6; down right: situation after 6 hours S6 ........................................................................................... 140Figure E-10: Boonweg section 2: situation after 2 hours S6.................................................................................... 141Figure E-11: Boonweg section 2: situation after 6 hours S6.................................................................................... 141Figure E-12: Boonweg section 3: situation after 6 hours S6.................................................................................... 142Figure E-13: Boonweg section 3: erosion development from 1:30 - 1:50 hours S6 ................................................... 142Figure E-14: Boonweg section 3; left: situation after S6; right: situation after 6:30 hours S6.................................... 142Figure E-15: Boonweg section 4: Wearing of the tractor trail .................................................................................. 143Figure E-16: Boonweg section 4: Initiation of erosion at the tractor trail ................................................................. 143Figure E-17: Boonweg section 4: erosion development 5:10 - 5:50 hours S6........................................................... 144Figure E-18: St-Philipsland: erosion at connection grass – asphalt road................................................................... 145Figure E-19: St-Philipsland: erosion development S6; upper left: after 2 hours S6; lower right: after 6 hours S6 ....... 146Figure E-20: Kattendijke section 1: Upper left: damage after S4; upper right: situation after 2 hours S6; down: situation

after 6 hours S6 .......................................................................................................................................... 147Figure E-21: Kattendijke section 1: Development headcut erosion; pictures obtained from video material ................. 148Figure E-22: manure injection .............................................................................................................................. 148Figure E-23: Kattendijke section 2: Erosion hole at grind box; each cell of framework in the left figure is 1 by 1 m.... 148Figure E-24: Kattendijke demonstration: Erosion depth 0.5 m............................................................................ 149

List of tablesTable 2-1: Calculated mean and 10-4 percentile wave statistics (source: Diermanse et al. 2005) .................................27Table 2-2: Management specifications for quantification off a grass mat (source: TAW 1999) ....................................36Table 2-3: Qualification of grass species according to management, covering rate and density (source: TAW 1999). ...38Table 2-4: Management specifications for quantification off a grass mat (VTV2006) .................................................. 39Table 3-1: Failure mechanisms due to wave overtopping .........................................................................................43Table 3-2: Indicative values for Dutch grass (HOFFMANS, 2008b) ............................................................................47Table 3-3: Summary of erosion models. ..................................................................................................................59Table 4-1: Erosion process .....................................................................................................................................62Table 5-1: Influence of tail water depth ..................................................................................................................70Table 6-1: Indicative values for strength of clay (r0=0.15 and cmin =0.021c) ..............................................................76Table 6-2: Soil structure development over time......................................................................................................76Table 6-3: Characteristics of an 6-hour storm duration for different q [l/s m-1] ..........................................................82Table 6-4: Averaged Failure Times; variable: air-water ratio ( )................................................................................87Table 6-5: Differences averaged Failure Times differences [%] ................................................................................87Table 6-6: Averaged Failure Times; variable: clay strength.......................................................................................90Table 6-7: Differences averaged Failure Times differences [%] ................................................................................90

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Table 7-1: Total amount of wave events as given in the steering lists .......................................................................95Table 7-2: Characteristics of test locations; with y =characteristic clay layer thickness at the place of failure, L =length

of the inner slope, =slope angle, clay quality c3 Table 6-1: structured clay quality, grass quality characteristicssee Table 3-2................................................................................................................................................96

Table 7-3: Validation steps .....................................................................................................................................98Table 7-4: Computed results method A and B (for explanation methods see Table 7-3) .............................................99Table 7-5: Computed results method B and C (for explanation methods see Table 7-3) ........................................... 100Table 7-6: Characteristic CE values grass, clay and sand (Vroeg et al. 2002)............................................................ 101Table 7-7: Summary table: prediction of erosion depths [m] computed with the TM; X means not executed ............. 109Table 7-8: Summary table of experimental results; erosion depths [m]; X means not executed................................ 109Table C-1: Number [-] and wave Volumes [l/m] as used in the prediction report (Verheij and Hoffmans 2008) (using

characteristic wave volumes) ....................................................................................................................... 132Table C-2: Number [-] and wave Volumes [l/m] as used in the steering lists (using the de Van der Meer formula for

wave volume distribution)............................................................................................................................ 132Table C-3: Number of roots in a soil sample .......................................................................................................... 133

Introduction and study objectives

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1. Introduction and study objectives

1.1 General introduction

1.2 Problem analysis

1.2.1 Problem definition

1.2.2 Objective

1.3 Methodology

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1 Introduction and study objectives

In this chapter the objectives of the MSc-study are given. Starting with a general introductionof sea dikes in the Netherlands, the chapter describes the problem of erosion of grassed innerslope transitions. Photos and schematisation within section (1.2) Problem analysis are shownto introduce the problem. In the subsections: (1.2.1) Problem definition and (1.2.2) Objectivethe purpose of this study is given. The chapter ends with an explanation of the methodologyused and the structure of this report.

1.1. General introductionThe safety of a large part of the Netherlands strongly depends on sea dikes. These dikes canbe found in the Northern part (Groningen, Friesland and Noord-Holland) as well in the south-west part (Zeeland) of the Netherlands. The dikes protect the area from hazards coming fromthe sea. Due to climate change, a sea level rise is predicted which will increase the possiblehazard. Stronger storms and more wave attack is expected which lead to increased waveovertopping on the current dikes. As a response, crest levels of the dikes needs to be raisedto meet the present regulations for wave overtopping. As an alternative, these regulationscould be lowered if the dikes can be proven strong enough to cope with the increased waveovertopping.

In the Netherlands, within two research projects, ComCoast and SBW (‘Sterkte & BelastingenWaterkeringen’ or ‘Strength and Loads on Water Defences’), the consequences were or arebeing investigated regarding more wave overtopping on Dutch sea dikes. Both programs dealwith the strength of the Dutch sea dikes but have different focus areas.

The ComCoast project focused on strengthening of the dike to be able to withstand theincreased loads induced by increased wave overtopping. Strengthening of the dike can bedone by applying innovative measures to provide overtopping-resistant dikes.The SBW project with respect to grassed inner slopes focuses on the improvement of reliableovertopping criteria for the present dike structures, without a reinforcement. Within the SBWproject research is done to investigate erosion of inner dike grass covers. The goal of theprogram is to explain failure mechanisms of grass covers induced by wave overtopping andthe developing of practical assessment and design formula.

The SBW program was started in 2007 and will last until 2011. One goal of SBW grass-protection program is to develop practical assessment formula. Nowadays used assessmentcriteria, the VTV2006 (Voorschriften toetsen veiligheid 2006), will also be validated using theresults of the experiments. The methodology used within the program is based on a cyclicprocess (Verheij and Hoffmans 2008), the so-called KOLB-principle, see Figure 1-1.


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Figure 1-1: KOLB-principle

Each year the whole cycle will be completed. After an evaluation of the 2008 experimentsrecommendations are made for adjustments of the nowadays used prediction models and forthe next round of experiments.

Prediction for the last test experiments (Delftzijl, 2007 and Boonweg and Zeeland, 2008) hasbeen made according to the EPM/Hoffmans model, the model of Young and the SSEA-modelThe three erosion theories mentioned here are explained in Chapter 3: ‘Erosion models’.During the experiments of February to April 2008, some erosion of the grass layer has beennoticed, but this was not in accordance with the EPM/Hoffmans model or other models. Theinitiation of failure of the grass layer was created at the transition of the slope and ahorizontal part. Due to the impinging forces caused by the overtopping wave tongue, a scourhole is formed. After the scour hole reached a certain depth, headcut erosion is observed,finally resulting in serious damage of the inner slope.An analysis of the failure mechanism of the inner slope during the experiments of February toApril 2008, pointed out the need for further research and modelling with respect to the forcesof overtopping waves and the strength of grass layers.

1.2. Problem analysisTransitions can be seen as abrupt changes within a profile see Figure 1-2. Transitions in aprofile may cause weak spots in the dike system. Results of the recent SBW experimentssupport the idea that profile changes within a dike profile, like the toe of the inner slope,which is normally used as a road, form weak spots.

Figure 1-2: Transitions at an inner slope

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During the tests no serious damage occurred on the inner slopes, but at the transitions it did.At the test location St. Philipsland, special measures at the toe were taken to cope with theenhanced forces (see Figure 1-3). At the test location Kattendijke, Zeeland, these kinds ofmeasures were not taken, leading to a big scour hole at the toe (see Figure 1-4). This scourhole finally leaded to serious damage of the inner slope grass layer.

Figure 1-3: Special protection measures at St. Philipsland test location

Figure 1-4: Scour hole at test section Kattendijke Figure 1-5: Damage of the inner slope induced by scour hole

As described, failure of the grassed inner slope has been initiated by failure at the transitionof the slope and a horizontal part. Transition induced erosion starts at the horizontal partdownstream of the transition, in this report this horizontal part of the transition will bereferred to as transition. SBW tests with respect to the strength of grassed inner slopes aredeveloped to test the straight inner slope. Transitions were not a focal point in the originalSBW test program yet. At the preformed experiments scour holes due to transitions wereobserved. To understand the results, investigation of the enhanced forces due to the bendingand impinging of the overtopping water tongue is necessary.

TransitionsAt transitions, the water stream is strongly disrupted. At these particular spots in the dikeprofile the water is not just flowing over the bed like on the slope, but the water flow hits thehorizontal part. The water tongue resembles a jet impinging on the bed (see Figure 1-6).


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Figure 1-6: Scour hole caused by a water jet

The impinging forces on the bed profile and increased turbulence within the stream profile,cause enhanced shear stresses on the bed. When the shear stress induced by the flowexceeds the critical bed shear stress, erosion of the bed occurs and a scour hole is formed(see Figure 1-6). Enhanced shear stresses due to impinging forces and enhanced turbulenceincreases the erosion rates of the bed. Bed material and vegetation properties have a largeinfluence on the erosion resistance of a bed. Modelling of the enhanced erosion rates causedby the water jets at the transitions in the dike profile has not been done yet.

1.2.1. Problem definitionAt a transition erosion might be initiated, stimulating the creation of scour holes and finallycreating serious damage of the inner slope. The overall strength of a dike body depends onits weakest link. Due to additional loads at the transition point of the inner slope and theconnecting horizontal part, this point could be the critical point in the system. To determinethe strength of the total dike system, all kinds of possible failure mechanisms needs to betaken into account and determined. The present models are based on infinite slope, leavingtransition induced scour out of the scope. The fact that enhanced loads and erosion rates attransitions are not negligible can be concluded from the test results shown in the pictures inthis section. The fact that these enhanced loads are undefined means a gap in the knowledgeof the overall stability of the dike system.

1.2.2. ObjectiveThe objective of this MSc. thesis follows from the evaluation of the experiments of Februaryto April 2008. During the evaluation phase the conclusion was drawn that a transition couldbe the weakest link in the dike system. A model is required to predict erosion of grassed innerslopes transitions. The objective of this MSc-Thesis is to develop an erosion model for atransition of a grassed inner slope.

To come to this objective the following activities are distinguished:To gain insight into failure mechanisms, safety standards, hydraulic boundaryconditions and cover materials of dike structuresTo analyse nowadays used erosion models for grass coversTo determine the erosion process of a grass cover layer of an inner slope transitionTo derive a erosion model for grass covers of an inner slope transitionTo evaluate the behaviour of the erosion modelTo calibrate the parameters of the modelTo validate the erosion model for transitions according to field results

After validation, the model should be used to make predictions of the wave overtoppinginduced erosion at grassed inner slope transitions, in order to be further improved.

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1.3. MethodologyThe basic idea is to analyse present knowledge about grass layer erosion and jet impact andto bring them together in one formula. An analysis of the nowadays used models andknowledge is done, to evaluate the possible use of these concepts for the specific case: waterjet induced erosion of the grass cover layer at transitions. The methodology used as well asthe structure of the report is as follows:

Chapter 2: Boundary conditions and dike structure: Initial study to the failuremechanisms of dikes, safety standards, hydraulic boundary conditions and thestructure of a sea dike. Besides that, the use of a grass layer as a cover material ondikes is analysed.Chapter 3: Erosion models: Description of and discussion over possible application ofnowadays used erosion formulas.Chapter 4: Erosion process: Description and schematization of the erosion process ofthe grass cover layer.Chapter 5: Derivation of an erosion model for grassed inner slope transitions.Chapter 6: Evaluation of the erosion model, including a determination of parametersand a sensitivity analysis.Chapter 7: Validation of the erosion model. Validation is done using the results of theSBW experiments done in February to April 2008. Since the results of computationswith a variable strength profile and with a constant strength profile model showdiscrepancies, parameters used in the variable strength profile are redefined. This isdone according experimental results for clay and grass layer erosion of Delft Cluster.Chapter 8: Conclusions and recommendations.

The total outline can also be found in the following scheme.


21

1: Introduction and studyobjectives

2: Boundary conditions and dikestructures

3: Erosion models

4: Erosion process

5: Derivation erosion model

6: Evaluation erosion model

7: Validation

20 ( ) ( )

ˆ ( )c

soil

d ddydt E d

8: Conclusions andrecommendations

Conclusions andrecommendations

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23

2. Boundary conditions and dike structures

2.1 Failure mechanisms

2.2 Safety standards

2.3 Hydraulic loads

2.4 Wave overtopping

2.4.1 Overtopping discharge

2.4.2 Overtopping volumes per wave

2.4.3 Overtopping depth, velocities and time

2.5 Dike structure

2.6 The use of clay as dike cover material

2.7 The grass cover layer

2.7.1 Grass mat management

2.7.2 Grassland type

2.7.3 Root density

2.8 Conclusion on the situation in the Netherlands

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2 Boundary conditions and dike structures

Large parts of the Netherlands are protected by a natural or artificial water defence fromwater hazards. This protection can be secured by dunes, dikes, dams or other artificialbarriers. An area, surrounded by a water defence system, is called a dike-ring area. If theadjacent water is seawater, the protection belongs to the primary sea defence. To confine thelimits of this study, it will only enclose dikes which are part of the primary sea defence. In theNetherlands the TAW (Technical Adviescommissie voor Waterkeringen; in English: TechnicalAdvice committee for Water defences) advised about the technical as well as the safetyaspect of water defences. They provided guidelines and reports used nowadays. The work ofthe TAW is continued by the ENW (Expertise Network for Flood Protection).

Both aspects of there expertise, the technical as well as the safety aspect, are covered withinthis chapter. In section 2.2 safety standards of the dike-ring area are outlined, followed bytypical hydraulic boundary conditions for Dutch primary sea defences and overtoppingformula. Within section 2.5 up till 2.7 the technical aspects of dikes, clay and grass layers aretreated. The chapter start with an overview of possible hazards and failure mechanismsconcerning dikes in general.

2.1. Failure mechanismsIn the assessment of the reliability of a dike all kind of failure mechanisms should be takeninto account. Failure of a part of a dike is not necessarily the same as total collapse of a dike.Failure of a part of the dike means weakening of the system, this affected system often cancope with the loads and maintain its function but needs to be repaired to be conform theDutch safety guidelines. The different kind of failure mechanisms are shown in Figure 2-1,and explained shortly.

Figure 2-1: Failure mechanisms dikes (TAW 1998)

The different failure mechanisms mentioned are shortly explained (TAW 1998):A. Inundation of the dike ring area through a combination of high water level and wave

overtopping without the collapse of the defence structure;B. Erosion of the inner slope by the force of the flowing water and by a combination of

high water level and wave overtopping;C. Instability (sliding) of the inner slope, due to either infiltration of the overflowing

water in a combination of high water level and wave overtopping, or water pressureagainst the defence and increased water pressure in the subsoil;

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25

D. Shearing of a soil body, also by water pressure against the defence and increasedwater pressure in the subsoil;

E. Sliding of the outer slope in the case of a rapid fall in the outside water level afterhigh water;

F. Instability of the inner (or outer) slope by exiting seepage water through the soilbody (micro instability) analogous to failure mechanism C, but at lower water levels;

G. Piping as a consequence of seepage flow trough the subsoil so that the erosion startsbehind the dike and soil is borne along (sand boils);

H. Erosion of the outer slope by currents or wave movement;I. Erosion of the toe and foreshore by currents or wave movement;J. Large-scale distortions of the soil body;K. Mechanical threats like ice and shipping;L. Analogous to K;

It can be concluded from this scheme that nowadays wave overtopping is seen as a failuremechanism. Overtopping causes additional loads on the crest and inner slope introducingfailure mechanisms which are nowadays so to say abandoned by the strict rule of “no”overtopping. The “no” overtopping rule means in practice a guideline for the maximumquantity of overtopping water of (less than) 1 l/s m-1 for most sea dikes.

Accepting more wave overtopping introduces new failure mechanisms. These additionalfailure mechanisms are surface erosion and shallow slip erosion.

Figure 2-2: Surface erosion (M) Figure 2-3: Shallow slip erosion (N)

M. Erosion of the surface due to overflowing water;N. Instability (sliding) of the inner slope over a limited depth, due to infiltration of the

overflowing water;

After the erosion hole has reached a certain depth, the upstream soil can become unstable.The erosion hole will develop in upstream direction, see Figure 2-4. This kind of erosion iscalled headcut or retrogressive erosion.

Figure 2-4: Headcut erosion

Macro-instability or deep slip erosion (failure mechanism C in Figure 2-1) is also influenced byovertopping but considered not relevant for primary sea defences because of the gentle

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profile used nowadays in the Netherlands (1V to 3H or flatter). Before suggesting an increaseof the loads on the dike profile, load and strength characteristics for a typical Dutch situationare verified.

2.2. Safety standardsBased on a national risk assessment safety standards for the primary flood protection systemhave been derived. These standards range from 1/1,250 to 1/10,000 a year, depending onthe area’s economic activities, population size, and the nature of the threat (fluvial or coastal)see Figure 2-5 (TAW 1996a). For coastal areas the minimum safety is 1/2,000 a year. Thesestandards were established in legislation in 1996 with the Flood Protection Act (Ministry ofTransport, Public Works and Water Management 1996). The flood levels associated withsafety standards are updated every five years to accommodate sea-level rise and recenttechnical developments. Within dike design hydraulic boundary conditions (flood level, windand wave conditions) are applied to these statutory safety standards according the Dutchstandards.

Figure 2-5: Dike ring areas and safety standards (according to the Dutch Law for water defences ‘wet op dewaterkering’ (original document is translated to English))

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27

2.3. Hydraulic loadsHydraulic boundary conditions are applied to statutory safety standards. The most importanthydraulic boundary condition on a sea defence is the wave climate. Typical parameters forthe wave climate are the significant wave height and wave period (Hs or Hm0 respectivelyTm-1,0 or Tp). These parameters are used to calculate the overtopping quantities on a dikestructure, as explained in section 2.4. The given definitions of the wave parameters are(extracted from TAW 2002):

Wave heightThe wave height used in the wave run-up and wave overtopping formulae is the incident significantwave height Hm0 at the toe of the dike, called the spectral wave height, Hm0 = 4 (m0). Anotherdefinition of significant wave height is the average of the highest one third of the waves, H1/3. This waveheight is thus not used. In deep water, both definitions produce almost the same value, but situations inshallow water can lead to differences of 10-15%.

Wave periodThe wave period used for wave run-up and wave overtopping is the spectral period Tm-1.0 (m-1/m0). Thisperiod gives more weight to the longer period in the spectrum than an average period and, independentof the type of spectrum, gives the corresponding wave run-up or wave overtopping for the same valuesand the same wave heights. In this way, wave run-up and wave overtopping can be easily determinedfor double-peaked and ‘flattened’ spectra, without the need for other difficult procedures.In the case of a uniform spectrum with a clear peak there is a fixed relationship between the spectralperiod Tm-1.0 and the peak period. In this report a conversion factor (Tp = 1.1•Tm-1.0 ) is given for thecase where the peak period is known or has been determined, but not the spectral period.

Where m0 is the total area of the wave spectrum which is equal to the variance.Unfortunately accurate wave measurements for assessment of the sea defence are notalways available. Therefore, other projects of the SBW focus on wave measurement to beable to define more accurate boundary conditions for example for the Wadden Sea. Moremeasurements over a longer time period could bring more accuracy but statistics of extremesalways has to deal with uncertainties; the measuring time is small in comparison with therepeating time statutory to the safety standards. Using a Weibull function fitted on themeasured maxima, an exceeding probability curve of Hm0 and Tm-1,0 can be formulated whichis used for design purposes.

The nowadays used exceeding probability function for the wave climate for the Dutch coast isbased on measurements of 9 buoys taken between the years 1979-2002. The mean valuesand the calculated 10-4 quartile value (P (X x)) for the different locations (see Figure 2-6)are given in Table 2-1. The given values hold for deep water conditions.

Table 2-1: Calculated mean and 10-4 percentile wave statistics (source: Diermanse et al. 2005)

mean 10-4 quartile value

loc. Hm0 Tm-1,0 Tp Hm0 Tm-1,0 Tp

SON 1.89 2.87 2.60 11.07 15.94 19.4

ELD 2.01 2.32 2.66 10.65 15.46 17.32

K13 1.93 2.29 2.61 10.36 15.26 16.46

YM6 2.16 2.96 2.93 9.63 14.56 16.34

MPN 1.76 3.93 3.40 8.94 12.61 15.27

EUR 2.24 4.34 2.77 8.34 11.14 13.32

LEG 2.01 5.06 4.45 8.64 10.77 12.40

SWB 2.32 4.97 4.64 7.36 10.59 11.83

SCW 2.50 4.58 4.71 5.99 10.65 12.05

Note: The means values for Hm0 , Tp may not appear on the same time.

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Figure 2-6: Location of measuring stations

Although the values of Table 2-1 of the 10-4 quartile have 2 decimals, this is an inaccurateinterpolated value containing a certain error-range. In Figure 2-7 & Figure 2-8, the range forthe value of the 10-4 quartile of measuring station SWB is given; the black dots aremeasurements and the grey lines are lines of possible extrapolations.

Figure 2-7: Exceeding probability curve Hm0

Note: Figure 2-7 and Figure 2-8 are written in Dutch; ‘kwartiel’ is the 10-4 quartile value.

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29

Figure 2-8: Exceeding probability curve Tm-1,0

Using the deep sea wave data, wave conditions near the shore can be calculated. When awave approaches the shore, refraction, diffraction and shoaling will occur until it finallybreaks. Software programs like SWAN, Simulation Waves Near Shore, are developed to obtainrealistic estimates of waves near shore. The program requires certain parameters like a givenbottom bathymetry, water level, wind field and current field. The simulation softwareprogram is based on formula given by Battjes (Battjes 2001). The estimates could bevalidated with historical information but these kind of data bases are scarce.

2.4. Wave overtoppingFor many centuries, the consequences of wave overtopping are seen as a possible failuremechanism. Therefore the sea defences has been designed to be almost unovertoppable. Thecrest level of a dike is determined by a certain reference level, which consists of the tidalfluctuation and storm surge with a certain return period. Added to this level is an extra heightcalled the crest freeboard Rc. (A graphical explanation of this parameter is given withinsection 2.5: Dike structure). This crest freeboard height is determined using the wave run-upheight according to the wave height near the shore and some location dependentparameters. Usually the wave run-up height taken is the run-up height that is exceeded by2% of the waves, allowing 2 percent of the waves to overtop.

2.4.1. Overtopping dischargeWave overtopping is quantified as the mean discharge per meter of width q [l/s m-1 or m3/sm-1] that is reaching the inner slope of the dike (or structure). Wave overtopping formulaeare mostly exponential functions with the general formula (EurOtop 2007):

*

300

exp c

mm

RqQ a bHgH

Equation 2.1

According to the TAW (2002) and inclusive values for a and b, equation 2.1 reads:

030 00

0.067 1exp 4.75tan

cb

m b f vm

RqHgH

Equation 2.2

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With a maximum overtopping discharge of:

300

10.2exp 2.6 c

m fm

RqHgH

Equation 2.3

Where:q : average wave overtopping discharge per meter [m3/s/m]Hs : significant wave height (average height of 33% highest waves) [m]Hm0 : significant wave height (based on spectrum) [m]Rc : crest freeboard relative to SWL [m]

: outer slope angle [-]0 : breaker parameter [-] : influence factors [-]

The Dutch guidelines (TAW 2002) suggest the following allowable discharges on the innerslope of river and sea dikes:

0.1 l/s m-1 for sandy soil with a poor grass mat1.0 l/s m-1 for clayey soil with a reasonable good grass mat10.0 l/s m-1 for a clay cover and a grass mat according to the requirements for theouter slope or for an armoured inner slope

Usually crest heights of Dutch sea dikes are designed allowing 2 percent of the waves toovertop, which corresponds to a specific discharge of (less than) 1 l/s m-1. According toexperimental results, the nowadays used guidelines for overtopping discharges could be sternregulations. This leads to unnecessarily disapproval of the current dikes. The SBW projectwith respect to grassed inner slopes focuses on the improvement of reliable overtoppingcriteria for the present dike structures (Meer et al. 2007). The guidelines are based on meandischarges within a certain time period. A volume of 3600 litre in one hour means a meandischarge of 1 l/s. Two examples of cases which cause a mean discharge of 1 l/s are: 1 wavewith a volume of 3600 litre or 3600 waves with a volume of 1 litre. Several small waves mighthave a different effect on the dike than the same volume caused by one single wave. Thismeans it is important to determine the amount and volumes of the overtopping waves, whichwill be explained in the next paragraph.

2.4.2. Overtopping volumes per waveAccording to Van der Meer the overtopping volumes during a storm can be described by aWeibull distribution, (EurOtop 2007):

0.75 0.841 exp ;with mv

ov

T qVP P V V aa P

Equation 2.4

Where:Pv : probability that the overtopping volume per wave V is greater than or the same as

V. [-]V : volume of an overtopping event [m3/m]Tm : mean wave period [s]Pov : probability of an overtopping event [-]

With Pov, probability of an overtopping event, defined as the number of overtopping waves/number of incoming waves (Pov =Now/Nw). The formula for Pov reads:

2

2%

exp ln 0.02 cov

RPz

Equation 2.5

The wave run-up height to be taken is the run-up height that is exceeded by 2% of thewaves, z2%:

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31

2%0

0

1.65 b fm

zH

Equation 2.6

Where 0 is the breaker parameter and are influence factors. The maximum volume by onewave during an event depends on the actual number of overtopping waves Now, and can becalculated by:

4 /3max ln owV a N Equation 2.7

Where a is the same as used in equation 2.4

2.4.3. Overtopping depth, velocities and timeBosman formulated formula for the depth of the overtopping water layer h, the overtoppingvelocity u and the overtopping time T at the inner crest line. He defined the value that isexceeded by 2% as the maximum. The formula reads (Bosman 2007):

32% 2%

2

7 10sin

c

s s s

h z RH H

Equation 2.8

2% 2%0.25sin

c

s ss

u z RHgH

Equation 2.9

,2% 2%

1.0

1.02ovt c

m s s

T z RT H

Equation 2.10

Where s is the influence factor for grass layers, s =1. These formulas are defined for thelandward side of the crest and assumed also valid for the inner slope. The calculation of theloads on the inner slope due to overtopping will be done according to these formulas. Theformulas give the following graphs:

Figure 2-9: Overtopping flow depth h at the inner slope [m]

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Figure 2-10: Overtopping time T at the inner slope [s]

Figure 2-11: Max flow velocity u at the inner slope [m/s]

In the same figure the characteristic values, as used before the Bosman formula wereavailable (used for example in: Verheij and Hoffmans 2008), are given. These characteristicvalues, referred to as ref, are given as function of wave Volume and not, like Bosman does,as function of the wave run up height. Because in general these characteristic values arehigher than the values as can be obtained applying the Bosman formula for the same wavevolumes, calculating with the Bosman formula will probably result in lower erosion rates. Onthe other hand, in the report of Verheij the maximum wave volume is 3500 l/m which is a lotsmaller than the maximum volume as defined by Bosman (V >7000 l/m). This will probablyresult in higher erosion rates. The exact result of application of the Bosman equations instead of the characteristic values will be analysed in a later stadium.

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33

2.5. Dike structureAfter deriving formulas for the loads on the dike, the technical part of the dike structure isexplained. The overview starts with a general explanation of terminology and materials usedin dike design. Thereafter more technical information of clay and grass layers is givenconcluding with guidelines to quantify the strength of grass layer.

Because of the focus on sea dikes and not on river dikes, only the structure of a sea dike willbe explained. A typical dike consists of an outer and inner slope, a crest and berms. Theslope at the seaside is called the outer slope and the one at the landside the inner slope. Theouter slope is affected by the water level and wave attack. The inner slope and crest has tocope with the overtopping water. The dike usually has a berm at MSL, Mean Sea Level and atoe structure at the outer slope of the dike. Important parameters for the height of a sea dikeare Storm Surge Level, SSL, and Crest freeboard, Rc (see also Figure 2-12).

Figure 2-12: Cross-section of a sea dike (source: Bosman 2007)

Characteristic for an outer slope are a gentle slope of 1:4 and an armour protection,constructed with placed blocks, rock or asphalt around SSL. A grass layer usually covers therun-up zone of the outer slope, the crest and the inner slope. The core material is clay orsand. A sand core needs to be covered with a clay layer to prevent erosion of the sand body.This clay layer is protected by an armoured protection (this include also a grass cover layer).

Figure 2-13: Dike covered by grass (EurOtop 2007)

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Figure 2-14: Dike covered by asphalt (EurOtop 2007) Figure 2-15: Dike covered by placed block revetment

2.6. The use of clay as dike cover materialFor many hundreds of years, clay is used as material in top layers and the core of dikes. Thissoil type is known for good erosion resistance and shape retention. The relatively limitedpermeability of the material plays an important role for the selection of clay because of itspositive influence regarding piping. Nevertheless, clay is nowadays not used for the coreanymore because of the limited availability. The application of the material is restricted to atop layer of around 80 cm.

Clay is a cohesive material. Cohesion is a result of the binding forces between the very finesoil particles; these forces are large in relation to the weight of the particles. Erosionresistance is dominated by these binding forces and not by the weight of the particle as iswith non-cohesive material. The power of clay to retain water is due to the fact that watermolecules bind fairly strongly to the surface of the soil particles. Because of the very finepores in the clay, transport of water through the clay layer is strongly inhibit. When no largerpores exist in the clay package, then the clay is still only slightly permeable. This means nolarge building up of water pressure inside the soil structure and less danger from upliftpressures. A second important aspect of clay is the presence of the so-called 'soil structure'.As a consequence of climate effects, pores, also referred as cracks, occur in the outermost 1to 2 meters. Cracking occurs due to shrinking and swelling due to difference in water content.Biological activity, for example, burrowing animals (worms, insects, mice, moles) togetherwith the cracking due to climate effects leads to soil-structure formation. Root penetrationfrom vegetation also lead to soil-structure formation but, by contrast roots also have positiveeffect on the cohesion of the soil. The soil structure consists of larger and smaller mainlyangular lumps, so-called 'soil aggregates'. The smallest aggregates, with dimensions of lessthan 2 mm, are found in and directly under the grass sod. Under certain conditions largeaggregates of sometimes more than 20 cm can occur (TAW 1996b).

Figure 2-16: Left: Schematisation of the soil structure; right: Soil structure in clay of dikes (TAW 1996b)

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The development of a soil structure is a very important factor for the stability of a clay layer.Erosion tests show a strength reduction of clay with the development of a soil structure(Visser 2007). A soil structure will always develop in time, but the amount of developmentdepends on many factors and is difficult to describe or predict quantitatively. According toresults of Delft Cluster (Vroeg et al. 2002) the stability of the clay layer depends on thefollowing factors:

Ground structureDensityPercentage clay/sandSalinity of the groundSalinity of the overtopping waterHumidity of the groundClay mineral structureNa+ densityLime contentOrganic materialOther elements (Fe, Al)TemperaturePresence of rootsPresence of holes of animalsIn-homogeneities (e.g. enclosed sand layers)

Important influences on restricting soil structure development are the way of compacting andthe composition of the soil. Sand inclusions reduce the coherence of the clay, enlarging thepermeability and supply of oxygen and water in the clay which increases the development ofthe soil structure. The coherence of the clay mainly determines the strength of the clay. If thesoil structure development is less clear, the coherence, as well as the strength, of the clay ishigher.

2.7. The grass cover layerIf the erosion of a bare clay layer is compared to a grass covered clay layer, a positive effecton erosion resistance could be noticed. Although there is a positive effect of vegetation onthe development of a soil structure, a higher erosion resistance is obtained by the grass coverlayer. A typical grass cover layer consist of different layers which have there specificcharacteristics, see Figure 2-17.

Figure 2-17: Definition sketch – grass cover layer (TAW 1999)

The increase in erosion resistance is explained by two mechanisms. The sward of grasscontributes to the erosion resistance by covering the clay aggregates during overtopping,providing a sheltered environment. Secondly, the roots of the grass act to bind the soil andcreate a flexible and tough layer that offers significantly higher erosion resistance than baresoil. The roots support the development of the cementing substances which are responsiblefor the binding forces between the clay particles. This results in an elastic network, whichprovides a strong and flexible layer which allows certain deformation without cracking.

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The development of a root system requires clay with a high sand percentage. Thedevelopment of the roots in high sand percentage clay layer is faster and the root structurehas a larger density than in clay with a low sand percentage.Depending on the type of grass the root system is up to a certain depth. Consequently thepositive effect of the roots diminishes with depth. Near the surface of the grass layer thestrength of the grass cover is dominated by the root cohesion, whereas the depth increasesthe amount of roots decreases, the cohesion and internal friction of the clay dominates thestrength. The development of a grass cover with a well developed root system takes severalseasons, on average four years (Hoffmans et al. 2008b)/(Visser 2007).

The conclusion of a positive influence of the grass layer on the stability of clay results fromdifferent kind of experiments. The exact mechanism behind this increase in stability due tothe root system is not well known. The need to define some grass quality guidelines resultedin a strength calculation method based on the measurement of the number of roots in theclay layer. According to the TAW the performance of a grass layer depends on the grasslandmanagement (TAW 1999). Fertilizing, grazing by sheep’s and grass species should have ahigh impact on the grass cover and the root characteristics as on the strength characteristics.Nowadays quantification of a grass mat concerns the management type, grassland type aswell as the root density of the grass layer.

2.7.1. Grass mat managementThe use of fertilizer has a big influence on growth conditions and biological composition ofthe grassland. To reach a sufficient vegetation cover a certain amount of nutrients should beavailable in the ground. Fertilizer is used to increase the amount of nutrients which stimulatethe growing of vegetation. When fertilization has been terminated, productivity decreases,the soil nutrient content declines and the number of species increases. On the other hand,changes in botanical structure are also due to changes in canopy structure. The canopystructure changes by time and frequency and method of aboveground biomass removal.Grazing or mowing with or without hawing, are different methods to remove biomass andhave different consequences concerning growth and germination of seedlings. Every grassmat has a specific grass mat management concerning fertilization and removing of biomass.Four categories of grass mat management has been defined, see Table 2-2, subscriptions ofthese categories are also given (subscriptions are extracted from: TAW 1999).

Table 2-2: Management specifications for quantification off a grass mat (source: TAW 1999)

A Hydraulic management and natural-technical managementB Agrarian and lawn managementC Intensive agrarian managementD Very poor erosion-resistant management

Category A:The management strategies included in this category lead to effective erosion-resistant grassrevetments. The erosion resistance attained is such that few requirements need to be established forlow loading of the substrate. Hydraulic management and natural-technical management are included inthis category. Fertilisation and use of pesticides are definitely omitted from this category. The ecologicalvalue can be high.- Haying: Mowing twice per year; more or less frequent mowing possible depending on production. Innutrient-poor situations mowing once yearly in the fall will suffice. Such a situation can also arisefollowing years of hay management. A characteristic for haying is that after each mowing the cuttingsare removed within about 8 days to, among other things, avoid nutrients being washed out of the hay.In time such hay management will lead to diverse false oat meadows. In the long term a developmenttowards river valley grassland is possible.- Pasturing: Periodic or continuous pasturing with sheep. The number of sheep is always matched tograss production. In continuous pasturing, pasturing takes place for the entire grass growing season(from mid April to mid October) with a low livestock density. Additional mowing will be required forareas where the vegetation has not been grazed.In the long term this management can lead to a diverse crested dog’s-tail grass meadow.

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Category B:This category includes forms of management that produce a well-closed, reasonably erosion resistantgrass cover, but with a sod thickness and rooting that is actually not as much as in category A. For highloading, the strength may be insufficient. The ecological value of this and the following categories islow. This management form includes:- Agrarian management, for example by pasturing with sheep, continuously or periodically. Lightfertilisation (up to 70 kg N/ha. per year) is applied. The difference with category A is that fertilisation isused in this case and livestock density is higher.The result of this management is a non-diverse crested dog’s-tail grass meadow.- Lawn management consists of mowing 7 to 12 times per year and the cuttings are left in place. Thereis no fertilizer application in this category. Lawn management leads to a non-diverse bluegrass-rye grassmeadow.

Category C:The result of this management is a poor to mediocre erosion-resistant grass revetment. The sod ispoorly to moderately rooted. Open areas can arise very rapidly due to intensive pasturing. These areasare hardly to not at all rooted and do not become further closed. Erosion resistance is largely derivedfrom the clay in and under the sod. This is usually not sufficient in case of high loading and the usualcovering layer thickness.Category C consists of:- Intensive agrarian management strategies (usually pasturing), characterized by (intensive) fertiliserapplication. The result for pasturing is a non-diverse bluegrass-rye grass meadow, and for haying, anondiverse false oat meadow.

Category D:Very poor erosion-resistant revetments are formed by:- neglecting annual management,- mowing once to four times a year without removing the cuttings (‘mulch mowing’),- burning from time to time,- pasturing with cattle or horses,- applying extreme amounts of fertiliser and intensive pasturing.The sod is very poorly rooted and has a low covering and many open areas. The first threemanagement strategies produce a vegetation with rough herbage making the clay in the sod consist ofloose, crumbly aggregates that are very easily washed away. Pasturing leads to large open areas thatincrease in size. These forms of management are not suited to water-retaining dikes.

The type of management influences the number of species and the depth of the roots.Specie-rich communities probably have a higher erosion resistance. Species of nutrient-poorsoil conditions that occur in such communities probably have deeper-rooting systems. Thedevelopment of species-rich communities is a long term process and although low productionlevels (obtained by the abandoning of fertilization) are essential for high-species-diversity, itis not a guarantee.

2.7.2. Grassland typeA certain grass land management results in a certain specie-diversity and grass land type.Generally three groups of grassland vegetation can be distinguished: Lolio-Cynosuretum(dog’s tail grasses), Poo-Lolietum (bluegrass-rye grasses) and Arrhenathetum (false oatmeadows) (Sprangers 1999). According to above subscribed management specification andgrassland vegetation type, quantification of the total a grass sod is given in Table 2-3. Acloser description of different types of grassland vegetation and management specificationscan be found in literature written by Sprangers (Sprangers 1989).

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Table 2-3: Qualification of grass species according to management, covering rate and density (source: TAW 1999).

Grassland type Indicative numberof speciesper 25 m2

Erosion resistanceof the sod

Ecologicalvalue

Grasslandmanagement

Streamside valleygrassland ( 1)

30 Very good Very high 1 to 2x haying,unfertilised

False oat meadowwith edge species(Arrhenathetum)

27 Good Very high Irregular haying,unfertilised

False oat meadow,diverse(Arrhenathetum)

32 Very good Very high 1 to 2x haying,unfertilised

False oat meadow,non-diverse(Arrhenathetum)

13 Moderate Low Haying, fertilised

Rough meadowland(Arrhenathetum)

8 – 20 Poor Low Haying, heavilyfertilised, ormulch mowing

Crested dog’s-tailgrass, diverse(Lolio-Cynosuretum)

36 Very good High Pasturing,unfertilised

Crested dog’s-tailgrass, non-diverse(Lolio-Cynosuretum)

15 Good Low Pasturing, lightlyfertilised

Bluegrass-rye grassmeadow(Poo-Lolietum)

12 – 18 Moderate Low Pasturing,heavilyfertilized

( 1) Note: The name Streamside valley grassland is not mentioned in (TAW 1999). Name obtained within (TAW1997). According to this source the grassland type does not occur on sea dikes (only on river dikes).

Although qualification guidelines for a grass mat are given, quantification of the influence of agrass layer on erosion resistance can not be obtained from Table 2-3. Beside this, recentresults from the experiments at the Boonweg suggest a limited influence of the managementstrategies (Steendam et al. 2008).

2.7.3. Root densityAlthough the exact influence of a grass-root-system on the erosion resistance is unknown,the VTV subscribes quantification according the root density rate (VTV2006). Grasslandmanagement and grassland type are not explicitly mentioned in this method, but because ofthe influence of both on the root-system there is a certain relationship between thisquantification method and the qualification method as presented in table 2-3.The quantification method is based on the number of roots. The measuring proceduresdescribe counting the number of roots within a soil sample obtained with a ground drill with adiameter of 3 cm. By counting the number of roots every 2.5 cm, measured with respect tothe depth direction, the root density as a function of the depth could be determined. Themeasured root density as function of the depth should be quantified according to Figure 2-18and Table 2-4.

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39

Figure 2-18: Sod quality as function of root density (modified version)

Table 2-4: Management specifications for quantification off a grass mat (VTV2006)

Category root density0 Zero roots1 1 - 5 roots2 6 - 10 roots3 11 - 20 roots4 21 - 40 roots5 > 40 roots

This qualification method based on number of roots results in a grass sod qualification ofgood, averaged, poor or very poor. This sod qualification is used to calculate the erosionresistance of the grass mat (this is explained in section 3.2.1)

2.8. Conclusion on the situation in the NetherlandsGrass-protections on inner slopes of sea dikes are widely used in the Netherlands.Nevertheless, little is known about the actual strength and influence of a grass cover layer. Ifwe allow overtopping of our dikes, the dikes, including the grass-protection on the crest andinner slope, should be proven to be strong enough to cope with the increased forces.Allowing wave overtopping reduces the necessarily crest freeboard of a dike. Usually crestheights of Dutch dikes are designed allowing 2 percent of the waves to overtop, whichcorresponds to a specific discharge of (less than) 1 l/s m-1. Assuming a Hs of 2 m; Tp of 4.7 s,an outer slope of 1:4 and a storm duration of 6 hours, this would mean a necessarilyfreeboard of 3.4 m. To get a specific discharge of 75 l/s m-1, the water level should rise with2.2 m (Figure 2-19).

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Figure 2-19: Necessarily freeboard height according to a specific discharge (with: Hs of 2 m; Tp of 4.7 s, an outerslope of 1:4 and Tstorm of 6 hours)

To get a tenfold increase of the specific discharge (10 l/s m-1 in stead of 1 l/s m-1) the waterlevel should have been raised with 1.3 m.Dutch policy instructions subscribe a periodic assessment every 5 year of the quality of thewater defences, including grass layers. A new assessment of the reliability of the dike-ring ismade according the observations. The derived reliability for the upcoming 5 year periodshould be according the Dutch safety standards.Recently a lot of research programs have been developed concerning wave overtopping.Acceptable overtopping discharges might change in the near future, as could overtoppingformula. Furthermore, results of recent experiments give rise to give doubt to nowadays usedqualification of a grass mats concerning grassland management.

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3. Erosion models

3.1 Introduction

3.2 EPM

3.2.1 Modified EPM

3.3 Rajaratnam

3.4 Stein theory

3.5 Stanczak

3.6 Hoffmans

3.7 Canepa

3.8 Young

3.9 SSEA

3.10 Discussion

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3 Erosion models

This chapter is enclosed to gain some insight into erosion models related to failuremechanism which are introduced by allowing wave overtopping. The first section (3.1) givesan introduction to the concept of erosion and a repetition of the introduced failuremechanisms due to wave overtopping. The next section (3.2) explains the surface erosionmodel referred to as EPM. Primary this EPM is derived to predict erosion on bare spots.Modification of the strength characteristics, as explained in subsection 3.2.1, makes theerosion prediction model applicable for the total grass mat. Because the impinging wavetongue resembles a jet impinging on the bed, flow characteristics of the jet as explained byRajaratnam are described in section 3.3. Stein examined jet diffusion characteristics and theerosion characteristics of impinging jets. His erosion model can be found in section 3.4.Section 3.5 describes the experiments done by Stanczak. Hoffmans derived a scour relationbased on the momentum equation. His model is examined in section 3.6. Section 3.7evaluates the influence of the air content in an air-water mixture jet. Thereafter (section 3.8)the shallow slip erosion model of Young is explained followed by the SSEA model, whichdescribes headcut erosion (3.9). The chapter ends with a discussion (section 3.10), reflectingthe similarities and contradictions between the models.

Parts written in italic are ideas and suggestions of applicability within the developing processand are not based on theory.

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43

3.1. IntroductionErosion occurs when the material becomes unstable, looses its coherence with the otherparticles and is taken away by the flow. Probably the best-known stability formula is the oneby Shields from 1936. Shield gives a relation between dimensionless shear stress and the so-called particle Reynolds-number (Schiereck 2001):

2* *

*c c c

cs w

u u dgd f Re fgd

Equation 3.1

c is usually called the critical Shields parameter. This parameter is defined using a criticalshear stress c, the density of the soil s, density of the water w and a particle diameter d,assuming non-cohesive soil particles and uniform flow. Clay is a cohesive material and theflow due to overtopping waves has a non-uniform character. Concluding, the Shields equationis not well applicable, at least not in this form, to predict scour caused by overtopping waveson grassed inner slopes.

Most erosion studies focus on scour in non-cohesive bed material. Defining the equilibriumscour depth being reached when the sediment particles detached from the bed can not leavethe scour hole. Clay is a cohesive material and act different than non-cohesive material,reinforcement due to the grass layer enhanced the cohesive behaviour of the clay and shouldalso be taken into account. The erosion model should adapt this cohesive behaviour to modelgrass layer erosion in the right way.

Grass erosion can be seen as failure of the grass cover layer. The state just before failureoccurs is a limit state. This limit state can be described within a reliability function. Thegeneral form of the reliability function is:Z R S Equation 3.2With:Z= ReliabilityR= ResistanceS= Solicitation

Within the reliability function the resistance, or strength, (R) characteristics and solicitation,or load, (S) parameters are strictly separated. Some scientists kept the strict separation ofthese parameters within their erosion formulas, others used other methods to quantify thestrength and load characteristics.The newly introduced failure mechanisms due to wave overtopping are the following (seealso Table 3-1):

1. Erosion of clay particles or blocks of clay due to overflowing water, known as surfaceerosion.

2. Sliding of a layer parallel to the slope due to instability caused by infiltration of waterand overflowing water, known as shallow slip erosion.

3. Instability of soil, resulting in erosion in an upstream direction, known as headcuterosion.

Table 3-1: Failure mechanisms due to wave overtopping

Surface erosion Shallow slip erosion Headcut erosion

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The different models describe erosion based on one of the failure mechanisms; in sections3.2-3.7 models are described which are based on surface erosion, section 3.8 describesfailure known as shallow slip erosion and headcut erosion is treated in section 3.9.

3.2. EPMVan den Bos did research to erosion of grass layers and derived the EPM (Erosion PredictionModel) (Bos 2006). He assumes: due to the variability of the grass cover layer, bare spotscould limit the strength of a grassed slope. Van den Bos suggested a “Bare Spot Model” or“Erosiegevoelige plekken model” in Dutch to calculate the surface erosion on the slope. Themodel has been modified by Hoffmans et al (Hoffmans et al. 2008b), this modification isexplained in a separate section.

The Van den Bos model supposes typical bare spots in the order of 1 decimetre in flowdirection. He compares the erosion process of these bare spots with scour downstream of abottom protection. The grass acts as a bottom protection and initiation of erosion will takeplace at a bare spot downstream of this bottom protection. Using a 3-dimensional scourformula, he predicted the development of scour holes in the clay substrate. Calibration of theformula has been done on (permanent) overflow test data obtained on a real dike (1970). Hederived the following formula for cohesive material:

20

1.7cm

m

U Uy tc

Equation 3.3

With:ym : Erosion depth [m]

m : Characteristic length scale (0.2 m) [m]: Erosion intensity coefficient =3 [-]

U0 : Depth-averaged flow velocity [m/s]Uc : Depth-averaged critical flow velocity [m/s]c : 1.3*106 [m2/s]

: Relative density [-]t : Time [s]

The formula is limited for an erosion depth m smaller than the thickness of the grass sods.

Assuming equilibrium flow at the inner slope, Van den Bos investigated loads by waveovertopping. He suggested to translate the in stationary maximum wave overtopping flowvelocity to a characteristic velocity per overtopping event with the same erosive action usingthe following formula:

max12charU U Equation 3.4

With:Uchar : Characteristic permanent flow velocity [m/s]Umax : Maximum flow velocity per overtopping event [m/s]

This translation of a overtopping wave flow velocity in a characteristic permanent flowvelocity results in a characteristic velocity for every single wave. The summation of the singlewave events over the time period, give a prediction of the total load during the period. Thisassumption leads to an erosion prediction model of a grass cover for overtopping events like

Equation 3.3 in which2

0 cU U can be replaced by2

max0.7 cU U .

3.2.1. Modified EPMThe EPM of Van den Bos links the initiation of erosion and the presence of bare spots. Themodel focuses only on the erosion process of existing bare spots. He assumes that thesespots erode easier than places with a grass cover. Hoffmans et al., combined different

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45

parameters to one erosion formula, which is not limited to bare spots within a slope, butapplicable for the total grassed (inner) slope. The EPM of Van den Bos is used as the basicerosion relation, in which they suggest to exclude the erosion intensity coefficient , added aturbulence coefficient [-] and replaced c 1.7 by Esoil [m/s] (Hoffmans et al. 2008b). Besidesthis, they exclude the characteristic length scale of 0.2 m from the formula. The Modified EPMbecomes:

20 c

msoil

U Uy t

E Equation 3.5

with turbulence coefficient: 0(1.5 5 )r Equation 3.6

Formula 3.5 will be referred to as EPM/Hoffmans.

The depth-averaged relative turbulence coefficient r0 follows with:

*0 0 0 0

0

with 1.21gur c c c

U C Equation 3.7

The depth-averaged relative turbulence intensity r0 [-] depends on the bed characteristicsand is estimated at: r0 =0.15 for a horizontal bed (very rough bed); r0 =0.2 for the innerslope of a dike or very steep channel (extremely rough bed). Equation 3.7 gives the relationbetween U0 which is the depth averaged flow velocity for uniform flow and the turbulencecoefficient ro. The flow conditions on the inner slope are not uniform but highly turbulent.Therefore it can be doubted whether this equation can be used for this kind of flowconditions. Further research is needed.

As in the original formula, replacing U0 with Umax and adding a factor 0.7 in the equation leadto the erosion prediction formula for overtopping events. The model describes a process oferosion. The modelling of the erosion hole concentrates only on grow normal to the slope.The total erosion depth is the sum of the erosion of all individual events. The applicability ofthe formula is limited by the characteristic length scale of the root system, which is aboutequal to the limit used by the EPM.

Grass layer resistanceThe EPM/Hoffmans model uses as strength terms the critical velocity, Uc, [m/s] and a soilparameter referred to as Esoil. [m/s]

Calculation method EsoilThe basic scour relation used for derivation of the EPM/Hoffmans model is the scour relationas given by Partheniades. According to Partheniades the erosion rate of cohesive soils reads(source: Hoffmans et al. 2008b):

0 cc

ME Equation 3.8

Where:E : Erosion rate (mass) per unit area of the bed [kg/s m-2]M : Sediment coefficient (0.00001-0.0005) [kg/s m-2]0 : Bed shear stress [N/m2]c : Critical bed shear stress [N/m2]

The measure of erosion of clay and grass (factor M/ c in equation 3.8) could also berepresented by an overall strength parameter CE [m-1s-1] (Hoffmans et al. 2008b):

2

2 with*

aE E E

c c

g dM C CU

Equation 3.9

Where aggregate diameter da =0.004 m, kinematic viscosity of water =1*10-6 [m2/s] andE =1*10-10 [-] (Hoffmans et al. 2008a).

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The formula to obtain the soil parameter reads:1( ) with 1

Esoil soil soilE C Equation 3.10

Calculation method critical velocity Uc

The velocity due to the overtopping waves applies a certain shear stress on the surface,which can bring the soil particles in motion. The minimum velocity needed for initiation ofmotion of the particle is called the critical velocity. The shear stress is proportional to thevelocity squared. Because the clay particles are bounded together in aggregates, holdedtogether in the clay layer by the root system, the Shields equation (equation 5.1; source:Schiereck 2001), where stability is based only on gravity, is not well applicable. To model thestrength of a clay layer the Mohr Coulomb equation is more suitable. The additional strengthof the root system is modelled by an artificial cohesion croots (Hoffmans et al. 2008b). Addingthe root cohesion to the Mohr Coulomb equation the strength of the grass layer can beformulated as:

( ) tanclay roots w effc c p Equation 3.11

Where is the soil shear stress, cclay is the effective clay cohesion, croots is the additionalcohesion obtained by the roots, is the soil normal stress, pw is the pore water pressure and

eff is the effective internal friction angle. Where a root crosses a shear zone the strength canbe resolved into components perpendicular ( r =tr*cos ) and parallel (Tr =tr* sin ) to theshear zone (Figure 3-1). The formula for the artificial root cohesion croots and the normal grassstress roots read:

cos tan sinrroots r

Ac tA

Equation 3.12

cosrroots r

AtA

Equation 3.13

Figure 3-1: Strength schematization of root cohesion

tr : root tensile strength [Nm-2]Ar/A : RAR root area ratio [-]

: root angle of shear rotation [-] : internal angle of friction [-]

The term (cos *tan +sin ) is insensitive to changes in , it is close to 1.2 for a large rangeof values, so the root cohesion can be written as:

1.2 rroots r

Ac tA

Equation 3.14

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To investigate the forces acting on a clay aggregate a cubic turf model has been introduced(Hoffmans et al. 2008b). The forces acting on this control volume are:

Fp : Pressure forceFw : Weight of the volumeFs : Shear forcesFc : Cohesion forceFg : Grass reinforcement

Incipient motion occurs when:

p w s c gF F F F F

Equation 3.15

Figure 3-2: Forces acting upon cubic turf

According to the cubic turf model, assuming cracks at the four side walls, the formula for thecritical shear stress resp. critical depth averaged velocity reads:

,minc s w a clay rootsgd c Equation 3.16

0

,min0

0

20with 0.29

clay rootsc a

cU gd

r

c

Equation 3.17

With aggregate diameter da as derived for the soil parameter, minimum grass tension roots

(equation 3.13 with cos =0.4) and minimum clay cohesion cclay,min =0.021*cclay, using meanclay cohesion cclay. The pressure fluctuation coefficient is, being the inverse of themaximum pressure difference. The pressure difference within this model is explained by theinstantaneous structure of the pressure near the bed under turbulent flow conditions.Emmerling (Hoffmans et al. 2008b) found that the standard deviation of the instaneouspressure ( p,b) on the bed is about three times the mean bed shear stress ( 0) and that thepositive and negative pressure peaks could be up to 6 p,b. According to this explanation, themaximum pressure difference under turbulent flow conditions can be up to pmax =3*6=18* 0.The critical condition of lifting the clay aggregate is reached when 0 = c, so =1/18 =0.056 [-]

Making the following assumptions for Dutch grass: root diameter dr =0.13 mm, root tensilestrength tr =20*106 N/m2 the following root properties of Dutch grassland can be determined:

Table 3-2: Indicative values for Dutch grass (HOFFMANS, 2008b)

Ar/A(mm2/m2)

Numberof rootsNo./m2

Quality grassacc. to VTV

roots,min(kN/m2)

croots(kN/m2)

CE[m-1s-1]

Uc[m/s]

200 15100 very poor 2.8 4.8 0.062·10-4 2.5400 30150 poor 5.6 9.6 0.033·10-4 3.4600 45200 averaged 8.4 14.4 0.022·10-4 4.2800 60300 good 11.2 19.2 0.016·10-4 4.9

During the calculation of the critical velocity of a grass layer the cohesion of the clay (cclay,min)is neglected (or cclay =0). The reason is that the cohesion obtained by the root system ismuch higher than the cohesion obtained by the clay. Application of this method, gives onerepresentative value for the total grass layer. But the density of the root system as well as

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the clay cohesion is not constant over depth. Therefore the strength of the layer is assumedto be not constant over the depth profile but decreases and thereafter increases again.Verheij suggests to implement a depth variable strength profile (Verheij and Hoffmans 2008).In this profile at the surface the cohesion due to the root system should dominate and in thedeeper clay layers the cohesion due to the clay particles.

Strength parametersThe only parameter that causes variation in the soil parameter Esoil is the critical velocity Uc,which is calculated according to the critical shear stress (equation 3.16). The critical shearstress is a function of the cohesion obtained by the clay as well as by the roots and a noncohesive part: *g*d. Because for grass layers, the cohesion obtained by the clay isneglected (cclay =0), the only parameter causing variation in the two strength parameters isthe cohesion obtained by the roots ( roots). The number of roots decrease with respect to thedepth, therefore the value of the soil parameters is only representative for a small layer andthe application of the method is restricted to a characteristic length scale. Contrary tocohesion obtained by the roots, cohesion obtained by the clay will increase with respect todepth.The EPM formula is limited by the characteristic length scale of the root system ( 0.2 m).Implementation of a depth variable strength profile will enlarge the applicability of the model.During the derivation of the erosion model for transitions, depth variability will be taken intoaccount.

ModelsBoth the EPM and EPM/Hoffmans are based on an infinite slope. The EPM defines a bare spotas the weakest link in the slope where erosion will be initiated. The EPM/Hoffmans does notlink initiation of erosion to a particular place but predicts erosion over the total slope.Nevertheless, because a grass layer is irregular and uneven, the load model, based on theexistence of holes, has been taken over from the EPM. The EPM/Hoffmans model is based onthe initiation of erosion due to irregularities and humps on the surface of the grass layer.

Transitions are also weak spots: the water stream is strongly disrupted enhancing jetformation and erosion rates. At this location not the strength (bare spot in the grass layer)points the weak spot, but here the disruption of the water tongue, the hydraulic load,enhanced extra forces, causing a more weak spot in the system. To quantity this process ofenhanced forces, an impact study of water jets is necessary.

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3.3. RajaratnamThe EPM and the EPM/Hoffmans model both are derived to predict erosion on the slope. Flowon the slope has a different character than flow at a transition. At a transition the overtoppingflow is assumed to hit the surface like a water jet which is not comparable to paralleloverflowing water on the slope. Therefore, theories and erosion models about jet impact aresummarized to study the effect of the different kind of impact

Rajaratnam subscribes a detailed treatment of the mean flow characteristics of turbulent jetsfor use by engineers (Rajaratnam 1967). He presents in his work typical experimentalfluctuations and turbulence shear-stress profiles with as main objective to be useful by theconstruction of prediction models based on turbulence properties. The situation of a wavetongue at a transition is mostly related to the case of a turbulent wall jet, issuing intostagnant surroundings, which is also called a impinging jet. A schematization and a picture ofsuch a jet can be found in Figure 3-3.

Figure 3-3: Left: observed radial wall jet (source: Kortenhaus); right: schematization radial wall jet

where b0 is the initial thickness of the jet, U0 is the initial velocity and Um is the maximumvelocity. The momentum flux M0 is an important physical quantity controlling the behaviour ofthe jet. Using the momentum equation, it is possible to quantity the variation of the velocityU. The momentum flux equation reads:

20

0

M U dy Equation 3.18

with the density of the water and the width of the jet y. If we neglect the effect of(molecular) viscosity, we can say:

01/ constantm

MU Cx

Equation 3.19

with the distance from the point of impinging x. Replacing M0 according to formula 3.18, thisbecomes:

100

mU xCbU

Equation 3.20

Equations 3.18 and 3.19 state that the momentum quantity in a jet remains constant.Nevertheless, because the jet spreads (the width (y) of the jet increases with respect to thedistance from the origin (x)), the velocity in the jet decreases (equation 3.20). Because thevelocity of the jet will not increase, the right term in equation 3.20 should be < 1, meaningthere is a limitation of applicability of this formula with respect to distance (x) in the jet. Steinused the jet-theory as given by Rajaratnam in a scour formula. The theory of Stein isexplained in the next section.

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3.4. Stein theoryThe primary objective of the study of Stein is to examine the scour caused by free overfalljets in small scale channels such as rills (gullies) (Stein et al. 1993). The analysis stems fromthe theoretical investigation of jet diffusion characteristics in a plunge pool and the erosioncharacteristics of the impinging jet on both cohesive and non-cohesive particles.

Stein’s theory is based on the principles of a free falling jet entering a plunge pool. This jetenters the pool at an angle with an initial thickness b0, and an average velocity U0 at thewater surface, as can be seen in Figure 3-4.

Figure 3-4: Free water jet falling in a plunge pool(Note: in Figure 3-4 parameter hp =ho and H =y)Free jets, unaffected by a boundary, spread laterally and diffuse throughout the surroundingfluid decreasing in average velocity. The zone in which the centreline velocity remainsconstant at U0 defines the potential core, of length Jp from the jet origin. Beyond thisdistance, the velocity U is maximum along the centreline but the entire velocity field isreduced by diffusion. The generally accepted formulation for the centreline velocity whereJ >Jp reads (also mentioned in the theory of Rajaratnam):

0

0

:d pbU C J J

U J Equation 3.21

Where J is the distance from the origin along the centreline and Cd the diffusion constant.Stein defines a limit boundary Jp up from where the equation is valid. The length of thepotential core, referred to as Jp, can be simply obtained by solving equation 3.21 when U =U0

at J =Jp. The equation to determine the length of the potential core reads:2

0dpJ C b Equation 3.22

Which means the potential core depends on the initial thickness of the jet and the diffusionconstant.

Rate of scourThe rate of scour depends on the maximum average shear stress ( ) of the diffused jet withrespect to the critical shear stress ( c) of the given bed material. The maximum shear stressacting upon the bed in the impingement region can be related to the maximum velocity inthe impingement region (U) by introducing a coefficient of friction Cf.

2fC U Equation 3.23

Combining this formula with equation 3.21, gives the maximum applied bed shear stress ,which within the potential core is constant because of the constant velocity ( = 0).

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20 0

2 2 00

f p

d f p

C U J J

bC C U J JJ

Equation 3.24

The applied shear stress is a maximum when the eroding bed is within the potential core, thismeans the scour hole is increasing. If the depth of the scour hole is beyond the depth of thepotential core, the applied shear stress is inversely proportional to J and decreases. As doesthe erosion rate E (mass/time/area); also known as the sediment detachment rate per unitarea (see also equation 3.8). This detachment rate is formulated as the product of sedimentbulk density (mass of the solid/total volume) times the change in scour depth y with time t.The shear stress, as well as the erosion rate, decreases until it approaches the critical shearstress c for the bed material. This can be formulated with:

cyE Bt

Equation 3.25

where and are experimentally determined constants. This equation is comparable toequation 3.8.

At initiation of erosion, the distance of the jet J is within the potential core where = 0, thechange in scour depth y is given as:

20f c

dy C Udt B

Equation 3.26

In the laboratory experiments a cohesive soil with d =0.0045 mm is used. The jet entered thepool at an angle varying between 30 - 600 with an initial thickness b0 varying between 2-5mm. The average velocity U0 varied between 0.6 – 1.2 m/s. The results indicate that thevalue of the sediment detachment exponent is 1.0 for cohesive soils (and 1.5 for non-cohesive soils). Stein uses a critical shear stress c of 0.32 Pa for cohesive bed material, butgives no background for this value.

Stein uses the theory from Rajaratnam to define a scour formula. Erosion occurs when thebed shear stress due to the impinging jet is higher than the critical shear stress of the bedmaterial ( > c). The jet hits the surface under an angle , after impinging the velocities, aswell as the bed shear stress stay constant within the potential core, beyond this distance thevelocities and the shear stresses decrease. Stein validates his model using result ofexperiments with an initial thickness and velocity much smaller than in the SBW experiments(SBW: b0 in the range of 0.02-0.5 m and U0 in the range of 2.5-7.0 m/s). Because of thedifferent jet dimensions, flow characteristics could be different. Also the bed materials differbetween the different experiments. Both criteria have a large influence on the rate of scour,therefore the theory of Stein could be useful but the values for the critical shear stress, thefriction and diffusion coefficient should not be applied without consideration.

3.5. StanczakIn the framework of the Floodsite experiments, Stanczak investigated surface erosion andshear failure erosion due to wave impact on both clay samples and clay samples reinforcedwith grass when subjected to impact pressures (Stanczak 2007). The Floodsite experimentsare preformed to evaluate the erosion due to wave impact on the outer slope. Stanczakdescribed qualitatively as well as quantitatively the erosion. Husrin, a colleague of Stanczakgave a theoretical evaluation of the effects of a wave impact on clay. Besides that, Husringave a more detailed description of observations in the failure process of a clay layer. A briefsummary of Husrin’s report can be found in appendix A.

ExperimentsThe impact pressures are generated by a mass of water, which is dropped from a given fallheight hf, see Figure 3-5.

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Figure 3-5: Test set-up

A computer controlled system fills the pipe with a given amount of water and releases themass using a valve. The falling mass of water hits the soil sample under an angle of 90°generating an impact pressure on the soil sample. The height of the impact pressure dependsupon the fall height of the water mass. Therefore, the values of the impact pressures andenergy will not be directly measured but only calculated as follows (Pachnio, 2005 (source:Stanczak 2007)):

max

2 2w fh ghp

t Equation 3.27

where:pmax : Maximum impact pressure [Pa]hw : Water level in the tube [m]hf : Fall height [m]t : Impact duration [s]

The tube, with a diameter of 10 cm, can be placed in a range of hf =50 cm up to hf =165 cmabove the soil sample, which means the maximum impact pressure is 24.74 kPa.

Stanczak suggest that clay layers can fail due to shear stress failure. The concept of shearfailure is explained by Führböter (source: Stanczak 2007). Summarized Führböter assumesthat the impact pressures in the cracks are transferred fully to the side walls of cracks in theclay layer (see Figure 3-6a).

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53

Figure 3-6: Shear Failure model Führböter.

The forces are absorbed by the shear strength of the soil provided by the cohesion. Figure3-6b shows the forces acting on a block of soil. Force Fcrack is the acting force due to theimpact pressure, this force provides the shear force S on the failure plane, which iscounterbalanced by the shear strength of the soil W. Failure occurs when S >W, here theblock of soil is pushed out of the layer. Because there is one plane of failure the block of soilis pushed out in one piece. Nevertheless, tests on clay show soil erosion in the form ofparticles and small aggregates instead of soil erosion as a single soil block as Stanczaksuspected from theories of Führböter (see Figure 3-7).

Figure 3-7: Predicted and observed clay failure-principle sketches (Stanczak 2007)

Solving the equation for S =W leads to pmax =2*c, meaning failure occurs for impactpressures higher or equal to twice the cohesion. Stanczak experimentally defined a mean rootcohesion croots of 41 kPa, adding the mean cohesion of the clay, cclay =7 kPa, makes the totalmean shear strength of the grass cover total =48 kPa. According to the model of Führböterthe impact pressure required to induce erosion is twice the shear strength: pmax =2*c = 2*48kPa =96 kPa. As the maximal impact pressure which could be generated using theexperimental set-up is equal to pmax = 24.74 kPa, no shear failure is expected to occur. Thiswas confirmed experimentally.

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Due to technical constrictions it was impossible for Stanczak to induce impact pressures highenough to induce grass layer erosion. Führböter based his theory on the presence of pull-cracks, with a width in the order of centimetres, in grass cover most pull cracks are smaller.This conclusion gives rise to doubt the opportunity to use this shear failure model as principlefor a grass erosion model. On the other hand, the testing method could be useful. Besidesthat, the principle sketches of the predicted and observed clay failure show the importance ofthe aggregates within the clay soil structure.

Besides shear failure erosion, Stanczak also gave a formula to calculate the eroded volumeafter a wave impact which reads:

maxtwh

d dR k p e Equation 3.28

Where:Rd : Volume of eroded soil after a single impact [cm3]kd : Empirical detachability coefficient [cm3/kPa]w : Empirical coefficient [-]ht : Water layer thickness [cm]

The factor e-wh represents the effectiveness of a water layer to damp the impact pressure.The empirical detachability coefficient depends on the type of clay and the presence of agrass layer. The focus of the grass layer research is to determine the dependency of theerosion rates on the root volume ratio (RVR).

To analyse the impact of a grass layer, Stanczak used weak clay, kd,p =1.09 and w =0.25, asthe substrate soil and formulates the following empirical equation for the dependency of thedetachability parameter kd,g,p on the root volume ratio:

,, , 2

, , ,

*d t

d g p crit

d g p d p crit

kk d d

b RVRk k d d

Equation 3.29

Where:

*100 %R

ss

VRVRV

Equation 3.30

VR : Volume of roots [m3]Vss : Volume of soil sample [m3]

As the root volume decreases with respect to depth, the detachability parameter increases.Stanczak defines that the vertical distribution of roots over depth is as follows:

(d-2)RVR=1.58*0.75 Equation 3.31

Influence of the roots becomes negligible when the RVR becomes smaller than 0.5%. Thisdepth is referred to as dcrit, as can be seen in equation 3.29 below this critical depth thedetachability parameter takes the value of the substrate soil. The factor b describes theinfluence of the roots on the erodibility of the grass cover. It takes the value b =5 for thebest-fit function on the results from the experiments.

Instead of adding a certain additional strength value for the root system to the strengthparameter, the strength parameter is adjusted according to the RVR. The value of b can beseen as a certain grass tension parameter. Below the critical depth, where the influence ofthe roots is neglectable, the detachability parameter is equal to the substrate soil (weak clay).This method does not take into account an increase in clay cohesion with respect to depth.

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55

3.6. HoffmansHoffmans derived a scour relation according to the momentum equation (Hoffmans 2008a).The momentum equation is a statement of Newton’s law and relates the sum of the forcesacting on an element to its rate of change of momentum. The relation describes the scourdepth in the equilibrium phase. The forces acting on this element of the fluid are, see alsoFigure 3-8:

M1 : Momentum flux in the jet (section 1)M2 : Momentum flux at the outflow section (section 2)F1 & F2 : Hydrostatic forcesG : Weight of waterR : Resultant or dynamic force exerted by the jet on the bed

Figure 3-8: Schematization

(Note: in Figure 3-8 parameter = )Using the basic parameters: mean velocity in the jet U1, and at the outlet U2, angle betweenR and the horizontal , air-water ratio , angle of impingement , application of Newton’s law(per unit width) gives:

1 2 2 1Horizontal direction: cos (1 ) cos 0F F R qU qU Equation 3.32

1Vertical direction: sin (1 ) sin 0G R qU Equation 3.33

Applying Shields ( 0,c = c( s - )g*d) and assuming ~ ~ ( and see Figure 3-8), withcritical angle of shear failure , and a constant hydraulic radius Rh, the critical friction factorfc reads:

90*1

tan c cf D Equation 3.34

with1/3

90* 90 2

gD d Equation 3.35

Where D90* is a dimensionless particle, d90 is the particle diameter for which 90% of thesediment grains is finer than d90, is a coefficient, is the relative density, is thekinematic viscosity and c is the critical Shields parameter. The implementation of the Shieldsparameter and the critical friction factor according to the particle diameter d90, reflects thelimitation to use the strength formula only for loose material. (Nevertheless, the limitationdoes not apply for the load part). For application of the formula for cohesive material theassumptions made by Hoffmans are not valid and will not be explained in further detail. Theformula suggested by Hoffmans reads:

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1

,

2

2

2 2 / 3 1/ 390* 90*

sin

1

20with 1 tan

m e t

v

bv

c

qUgloadh

strengthc

acD D

Equation 3.36

In this formula the parameter , known as the air-water ratio, is used. Application of the air-water ratio is suggested by Canepa, this air-water ratio theory will be explained in the nextsection.

Hoffmans uses the momentum equation to calculate the resultant or dynamic force exertedby the jet on the bed. Because the momentum flux theory is an important quantity controllingthe behaviour of the jet (also used by Rajaratnam en Stein) this is very well applicable to usein a scour models. The derivation of the strength part of the formula is designed for non-cohesive material. This part is not applicable, at least not without certain modification forgrass layers. The total relation is defined to predict the erosion depth in equilibrium phase.The scour depth in equilibrium phase is important to ensure the stability of upstream bottomprotections or hydraulic structures. Failure of an inner slope, or even a total dike body, couldhappen before reaching the equilibrium scour depth. In this case, the total scour depth inequilibrium phase is unimportant but development of the erosion over time is important.Therefore, the new model will not be based on this equilibrium scour relation.

3.7. CanepaCanepa looked at the effect of air entrainment of a water jet running from a spillway andplunging in a pool (Canepa and Hager 2004). He made a test set-up consisting of a fullymixed jet containing water discharge Qwater and air discharge Qair with initial thickness b0 withan angle relative to the horizontal impinging into a water body of thickness ht, see Figure3-9. Three different non-cohesive sediment materials were used with grading between 0.001– 0.056 m. The boundary conditions of his experiments were Qwater,max =50 l/s and Qair,max =47l/s. Both water jets and water-air mixtures where studied with maximum values of

=Qair/Qwater close to 3.

Plunge pool scour can be characterized bythe densimetric particle Froude numberFd =U/(g’*di)0.5 where U is the approachvelocity (Umixture), g’ =[( s- )/ ]*g is thereduced gravitational acceleration and di isthe determining sediment size. To detect theeffect of air content on scour, the air content-dependent densimetric particle Froudenumber has been defined:F =Uwater(1+ )0.25/(g’d90)0.5. This equation isbased on the water velocity Uwater, where theair-water mixture velocity has been defined asUmixture =Uwater(1+ )

Figure 3-9: Test set-up Canepa

(Note: in Figure 3-9 parameter D =b0, h0 =ht, zm =y and = )

Canepa defined the dimensionless scour depth ZM=y/b0 and observed the following relationwith the densimetric particle Froude number F :

Erosion models

57

0.37MZ F for water jets Equation 3.37

0.37MZ F for air-water mixture jets (F <10- 15) Equation 3.38

As can be conclude from pictures, the wave tongue contains a lot of air. Canepa suggest toinsert a parameter to implement the air content impact in a scour formula. Assuming that themeasured velocities during the SBW-experiments are air-water mixture velocities, a factor of1/(1+ ) should be included in the scour relation to account for the air content in the flow.

3.8. YoungBesides surface erosion, allowing wave overtopping could also introduce shallow slip erosionand headcut erosion. In the next sections models are mentioned, which are developed toexplain and predict these kind of erosion mechanisms.

A second erosion mechanism called shallow slip erosion, or macro-instability, is due toinfiltrating water. Due to the groundwater flow and a high freatic line inside the dike, parts ofthe clay layer could become instable. Young developed a sliding prediction model of the turflayer at the inner slope (Young 2005). He defined initiation of erosion at the location wherethe stabilizing forces R are overtaken by acting forces S, leading to instability and sliding ofthe sod.

Figure 3-10: Forces on the slope Figure 3-11: Shallow slip erosion

Young defined both saturated weight of the soil acting down the slope (first part of equation3.39) and the shear stress due to the overtopping flow (second part), as the acting forces onthe ground segment:

2 20 sins wS d U C Equation 3.39

With:C : Chezy coefficient [m1/2/s]d : Depth to failure surface [m]

: Dike inner slope angle [-]U : Overtopping velocity [m/s]

The resistance part used within the model of Young (Young 2005) is comparable with themodified equation of Mohr Coulomb as described by Hoffmans (source: Hoffmans et al.2008b). The equation developed by Wu et al, used by Young reads:

sin * cos * cos tanr rc r r eff eff

A AR t t dA A

Equation 3.40

With:d : Depth to failure surface [m]

eff : Effective soil angle of friction [-]eff : Submerged soil unit weight [kN/m3]

tr : Root tensile strength [kN/m2]Ar/A : RAR (root area ratio) [-]

: Root angle of shear rotation [-]

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According to the stability analysis characteristic depths for shallow slip erosion of the turflayer are 0.2-0.3 cm, but even with the highest wave overtopping stability would be ensured(Akkerman et al. 2007).

According to the stability analysis of Akkerman et al. shallow slip erosion is unlikely to occur.Based on this conclusion, the study focus for the new formula will not be based on theerosion mechanism shallow slip erosion.

3.9. SSEAA third erosion mechanism called headcut erosion, or retrogressive erosion, was investigatedat the Stillwater Laboratories of USDA (source: Verheij and Hoffmans 2008). The formulas arealso used in the SSEA model (Sites Spillway Erosion Analysis), which is used to predicterosion of a spillway. Headcut erosion occurs when the soil becomes unstable due to theshear stresses caused by the overflowing water and its own weight. The headcut formulareads:


Equation 3.41

with13( )L qH Equation 3.42


With:dx/dt : headcut velocity [ft/hr]Cheadcut : material head-cut coefficient [s2]L : hydraulic load [ft3/s]Lc : critical hydraulic load [ft3/s]H : headcut height [ft]q : specific discharge [ft3/s ft-1]

Lc and Cheadcut both depend on material characteristics. The calculation of both parametersuses a headcut erodibility index kh:

0.79ln( ) 3.04headcut hC k Equation 3.430.33

0.5 3.23189 expln(101 )hc

h

L kk

Equation 3.44

The value of kh should be defined according to tables presented in the SSEA manual (USDA2001). In (Verheij and Hoffmans 2008) experiments on bare clay are mentioned (with H =1.0m and q =10 l/s m-1). The kh value for clay of 0.03, resulting in a headcut velocity of 0.3 m/h,is conform the test results. The hydraulic load L depends on the headcut height H. Initiationof erosion at transitions in dike profiles occurs by an H =0. This indicates headcut erosion willnot occur at the initiation of failure, but start after a certain height (or scour depth) isreached.

The headcut erosion occurred at places where a certain height difference is present.Therefore, on a regular slope without head differences, headcut erosion mechanism isnormally not the initiator of erosion, but dominates the erosion process after a certainheadcut height (H) is developed. This conclusion states that the erosion process at atransition should be described by two different models, in the first phase an erosion modelbased on surface erosion and for the second phase an erosion model based on the headcuterosion.

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59

3.10. DiscussionA summary of the different kind of models is given in Table 3-3. The described models aredeveloped after making certain assumptions and validated according different model set-ups.

Table 3-3: Summary of erosion models.

name model notesEPM/Hoffmans

20

1

0.7nc

m wavei soil

U Uy t

E

surface erosion,sum ofindividual waveimpact

Stein20f c

dy C Udt B

jet flow, plungepool scour

Stanczakmax

twhd dR k p e pressure impact,

shear erosionHoffmans

1

,

2

sin

1m e t

v

qUgloadh

strengthc

jet impact,erosion depth inequilibriumstate

Canepa (1 )mixture waterU U air-watercontent impacton velocity

Young

2 22%

sin * cos * cos tan

sin

eff

r rr r eff

s w

A AZ T T dA A

d u C

shallow sliperosion

SSEA( )headcut c

dx C L Ldt

headcut erosion

Shallow slip erosion (model of Young) is not likely to occur, and headcut erosion (SSEAmodel) occurs after surface erosion created a scour hole with a certain depth. Consequentlythe focus of this discussion is primarily based on the similarities and contradictions betweenthe models based on surface erosion.

The approach of the first three equations can be reduced to the stress based detachmentequation developed to estimate erosion of cohesive material by Partheniades, which reads:

0 cc

ME Equation 3.45

If we take ~ U2 similarities between the first two models (EPM/Hoffmans & Stein) and theone of Partheniades are easily noticed. The similarities with the model of Stanczak need someexplanation. However not mentioned by Stanczak himself, but according to his colleague ofFloodsite, who uses the same equation to calculate the amount of eroded soil (Rd), thisequation is based on the following equation (Husrin 2007):

aV d ck

Where the hydraulic load, represented by ~ Ekin ~ pmax and the critical stress c is assumed tobe zero ( c =0). Stanczak assumes there is always erosion (in the form of transportedparticles/grains, crack formation or shape deformation) after a hydraulic impact.

The coefficient that represent the strength of the grass layer (defined by EPM/Hoffmans,Stein resp. Stanczak as Esoil, /B resp. kd,grass) shows contradictions. Where Stein just

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mentions to derive the value of /B experimentally, the other two derive a strengthdependency according to the RAR resp. RVR of 1/Esoil ~1/RAR resp. kd,grass ~1/RVR2. Theparameters RAR and RVR are related according the following relationship RVR =50*RAR.(Derivations of the above described dependencies are given in appendix B.2).

Other essential differences between the EPM/Hoffmans model and the model of Stanczak are:(1) the direction of the jet impact (parallel flow with respect to the slope within the EPM andsplash erosion due to perpendicular impact on the surface within the Floodsite experiments ofStanczak) and (2) the parameters used to express the amount of eroded soil (scour depth y[m] and scour volume Rd [m3]).

The fourth equation, a jet impact prediction formula, developed by Hoffmans is based onequilibrium state. The relation is developed to predict the equilibrium scour depth, that isimportant to ensure stability of structures and it has not been developed to predict erosioncaused by a single impact. This scour relation will therefore not be used further on. However,in his theory he mentions the importance of the air-water ratio. This influence is investigatedby Canepa. He suggests to correct the velocities of an air-water mixture according the givenrelation.

This chapter is included to gain some insight into different erosion models based on thefailure mechanism which are introduced by allowing wave overtopping. The different modelsdescribe erosion processes which resemble erosion as observed in the field and therefore arepossible applicable for a erosion model for inner slope transitions. The observed erosionprocess can be divided in more phases with different dominating failure mechanisms. In thenext chapter the process of erosion of a grassed inner slope is described in more detail.

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4. Erosion process

4.1 The initial situation

4.2 Development of the jet scour hole

4.3 Transition: surface erosion – headcut erosion

4.4 Headcut erosion

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4 Erosion process

The observed erosion process is schematized in four stages (see Table 4-1); starting with theinitial situation, second the development of the scour hole due to surface erosion, third thetransition from surface erosion to domination of headcut erosion and the fourth stage isknown as headcut erosion.

Table 4-1: Erosion process

Initial situation Development of the jet scour hole

Transition: surface erosion – headcut erosion Headcut erosion

4.1. The initial situationIn the initial situation, see Figure 4-1, the clay layer on the inner slope of a dike is assumedto have a thickness of 0.5 -1 m, where in the upper 20 cm the reinforcing effect of the rootsystem is noticeable. The water tongue of the overflowing water hits the horizontal parts; itresembles a water jet hitting or impinging on the bed.

Figure 4-1: Initial situation

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63

4.2. Development of the jet scour holeThe jet causes shear stresses on the bed. At a transition, the shear stresses are enhanceddue to impinging forces of the wave tongue and increased turbulence within the streamprofile. When the shear stress induced by the flow exceeds the critical bed shear stress,erosion of the bed occurs. Because the shear stresses are enhanced at the transition, thegrass layer starts to erode at this place and a scour hole will form (see Figure 4-2).

Figure 4-2: Development of jet scour hole

The scour hole develops due to surface erosion caused by jet impact. The scour holedevelops from y =0 at t =0 to y =y(t) at time t. To predict the development of the erosionhole, an erosion-at-transitions model should be applied. The presented surface erosionmodels are not (direct) applicable in an erosion model for transitions. The derivation of thiserosion-at-transitions model is explained in the next chapter (chapter 5).

4.3. Transition: surface erosion – headcut erosionAfter the scour hole has reach a certain depth, the upstream soil can become unstable andheadcut erosion can occur. Because headcut erosion develops faster than surface erosion,headcut erosion will dominate the erosion process. This point where headcut erosion starts todominate surface erosion, is referred to as the transition point (see Figure 4-3).

Figure 4-3: Transition point: surface erosion - headcut erosion

Typical heights during the development of headcut erosion are in the order of 0.5 – 1.0 m.

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4.4. Headcut erosionThe exact depth where headcut erosion starts is unknown, therefore the assumption is madethat this critical depth is equal to the initial thickness of the clay layer (0.5 - 1 m). Because,after the scour hole reaches this depth the dike core material, often sand, will erode. If thecore starts to erode, the clay layer on the slope will be undermined and the soil becomesinstable and headcut erosion will occur (see Figure 4-4).


The headcut erosion process is modelled in the SSEA model. Because the initial purpose ofthis model is prediction of erosion downstream of spillways, evaluation of the model is donefor application on inner slopes. This evaluation, together with evaluation of the newly derivederosion-at-transitions model, is described in chapter 6.

Derivation erosion model

65

5. Derivation erosion model

5.1 Load term 20 ( )d

5.1.1 Derivation 0

5.1.2 Depth variability

5.2 The strength terms: ( )c d & soilE

5.2.1 Derivation c

5.2.2 Depth variability

5.3 Summary of all equations

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5 Derivation erosion model

In this chapter the derivation of the erosion model is explained. The model describes thedevelopment of the scour hole due to surface erosion. The final equation for the TM(Transition Model) reads:

20 ( ) ( )

ˆ ( )c

soil

d ddydt E d

Equation 5.1

where t̂ =the characteristic overflowing time or overtopping time due to wave overtopping.The load during one characteristic time period is assumed to be constant, meaningoverflowing is seen as one constant impact over time and waves are modelled as blockimpacts over time. Because the depth of the scour hole increases over time, depthdependency can be seen as function of time (d =d (t)).

This TM equation can be divided in two parts:load term: 2

0 ( )dstrength terms: ( )c d & ( )soilE d

The charter starts with an explanation of the first term, the load term (Section 5.1). The firstpart of the section (5.1.1) describes the derivation of the shear stress applied by an obliqueimpinging jet. The second subsection 5.1.2, is about the influence of a water layer above theeroding surface or inside the developing scour hole. Section 5.2 deals with the strengthterms, where subsection 5.2.2 explains the depth variability of the strength profile.

5.1. Load term 20 ( )d

The main difference between an impact on the slope and on a transition is the angle of theflow with respect to the erodible bed. On the slope the flow direction is parallel to the bed; ata transition the flow hits the surface under an angle. Because erosion occurred at thetransition and not on the slope, it can be assumed that the shear stress on the transition islarger than on the slope:

0, 0,transition slope Equation 5.2

The turbulence coefficient depends on the turbulence properties of a grass mat. Thisparameter is assumed not to change with respect to impact on transitions or to depth.Because the turbulence coefficient is given according the relation to the velocity: ( ~ U),using ~ U2 the relation to the shear stress is 2 ~ . The equation for the turbulencecoefficient reads:

0(1.5 5 )r Equation 5.3

with: r0 = 0.2 (extreme rough bed).

5.1.1. Derivation 0

The shear stresses on a transition are assumed to be higher than on the slope. First theinfluence of the impact angle has been analysed according to a comparison of resultsobtained by the EPM/Hoffmans (parallel flow) and results of Stanczak (perpendicular impact).Unfortunately, due to differences between the models (for example the impact time andscour dimension) this did not give applicable result. Therefore, a paper written by Beltaos hasbeen used to define the effect of the angle of impingement (Beltaos 1976).

The objective of the study of Beltaos was to investigate flow properties in the impingementregion. Studies of impingement problems have established the existence of three distinct flowregions:


67

1. Free jet region; extends from the outflow opening to some distance above the wall.Wall effects are neglectable and the flow is practically identical to a the free jet.

2. impingement region; influenced by the wall, where the static pressure rises above theambient and significant pressure gradients are set up within the jet, causing the flowto turn to a direction almost parallel to the wall.

3. wall-jet region; parallel wall flow. This region starts when the turn is wellaccomplished and static pressure drops to ambient values.

Figure 5-1: Flow regions within a vertically impinging jet

Within his study, Beltaos gave special attention to the effect of oblique impinging. Theinvestigated impinging angles lie within the range of 20º to 60º. Beltaos described that in theimpinging zone a combination of significant pressure gradients and wall shear stress causesevere hydrodynamic action on the boundary. He used air as the flow medium, but remarkedthat because the air speed is kept well inside the incompressible range, the results shouldalso apply to submerged liquid jets. The impinging plate used is a 0.635 cm thick Plexiglas,meaning afterwards there should be included some turbulence conversion factor to use theresults in grass models. For the case of impinging at a transition, the angle of obliquelinessis defined as 20º (equal to the slope).

ResultsFor the situation =30º, the pressure observed showed negative values. This is believed tobe a result of vortex formation. If becomes sufficiently small, < 30º, the boundaries of thefree jet and the wall jet region get so close together that some fluid in the upper layers of thewall jet is entrained in the free jet region, completing a circulatory motion and forming avortex (see Figure 5-2).

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Figure 5-2: Flow region oblique impinging ( < 30º)

The impact situation seems to have more similarities with wall flow, likeoverflowing/overtopping water as used within the EPM/Hoffmans model, than with the splashimpact model as used by Stanczak. Therefore, the EPM/Hoffmans model will be used as thebasic model, which will be adapted for the case of oblique impingement. Because Beltaosderived an equation to calculate the shear stress according the stagnation pressure, theequation will be based on the difference between applied shear stress and critical shearstress. The basic equation for the TM (Transition Model) reads:

20

ˆc

soil

dydt E

Equation 5.4

According to the values from his experimental results, Beltaos derived some formula and anumber of graphs. To obtain a value for a dimensionless parameter of the stagnationpressure, he derived equation 5.5, which reads:

20

8sin2

sp HU d

Equation 5.5

For an impinging angle of 20º ( = 20º) this gives a value of 2.7. Values for the maximumwall shear stress are only plotted in a graph. A copy of this graph has been made to show thevalues for a dimensionless parameter of the maximum wall shear stress.

Figure 5-3: Maximum wall shear stress (original graph see (Beltaos 1976))

From this graph a value of 0.045 is obtained. Combining equation 3.49 and the informationfrom Figure 5-3 gives:


69

02 20 0

0.045* 2.7*2 2

s mp H HU d U d

or 00.045* 2.7*s mp

This gives the following relation between the shear stress and the stagnation pressure:

0 0.016* sp Equation 5.6

Comparing this relation with the relation given by Stein (Equation 3.24):2

0 * *fC U and using ps = *U2/2 and Cf =0.004, due to the impinging angle the bed

shear stresses increases with ( 0° =0.004* *U2 & 20° =0.008* *U2=) 100%.

Air-contentDerivation of the shear stress is done according to the relation with the velocity of the flow.Canepa suggests to correct the air-water mixture velocity according the air-water content. Hedefines the air-water ratio: =Qa /Q, in where:Q : water discharge [m3/s]Qa : air discharge [m3/s]using Q ~ q ~U; the air-water ratio should be implemented as =Ua /U;

ConclusionAdding the correction factor for the air-water content and the effect of oblique impingement),to the formula for 0, the improved formula for impinging jets under an angle =20ºbecomes:

2

01 10.016* *2 1

U Equation 5.7

Being the total shear stress applied by a jet impinging under an angle of 20°. Where U is theair-water mixture velocity.

5.1.2. Depth variabilityA jet in a water body spreads and diffuses throughout the receiving water, resulting in adecrease of the velocity, as well as of the shear stress. According to different researchers theresulting shear stress is influenced by the presence of a water layer above the erodiblesurface or in the scour hole. The reduction of the applied shear stress is assumed to beaccording to the following relationship:

0 0( ) * wdd e Equation 5.8

where wde is the damping factor and d is the depth.

The reduction of the applied shear stress is modelled by Stein and Stanczak. Stein defines acore length Jp measured up to the point of impinging where the velocity as well as the shearstress remains constant and equal to the outflow velocity. Stanczak uses a water layerdamping coefficient affecting the jet from the point of impinging, see Table 5-1.

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Table 5-1: Influence of tail water depth

Influence of tail water depthStanczak Stein

maxtwhp p e

with:w: empirical damping coefficient [-]ht: tail water depth [m]

0

2 00

if

if

p

d p

J JbC J JJ

with:0: shear stress within core [N/m2]

Cd: diffusion constant [-]b0: initial jet thickness [m]J: distance along centreline [m]

Stanczak suggests to use a damping coefficient w =0.25 and Stein suggests a diffusionconstant of Cd =2.4. The difference between the two methods is shown in Figure 5-4.Because the dependency of the damping coefficient of Stein on the initial jet thickness, thefunction is shown for 3 different values of b0 (b0= 0.1, b0= 0.2 and b0=0.35. (Values for b0 arebased on the values of the overtopping water flow depth according to the Bosman formula.)

Figure 5-4: Comparison tail water damping models

The velocities in the core zone of Stein can be defined as 0 0( ) * wdd e with w =0. The

length of the core Jp, depends on b0. The core length Jp corresponds in the graph with thepoints of intersection with the line y=1 (y-as reflects the damping factor).From the figure it can be concluded that for a tail water depth in the order of a few metersthe methods give corresponding results. Because of the flow included vortex and the highturbulence, the jet is assumed not to hold the original angle of impinging. Therefore therelation between tail water depth and depth of scour hole is assumed to be y ~ ht. The depth-range which is important for scour holes and top layer instability lies in the order of 0-1meter. Focussing on this part of the graph, the two methods give completely different results.

DiscussionStein defines a core length where the velocity remains maximum. This core has a triangularshape (see section 3.4, Figure 3-4: Free water jet falling in a plunge pool ). The length of thecore, Jp ~ Cd

2*b0, is known to decrease with increasing turbulence in the flow. Because the jetdiffuses, the velocity outside this core decreases. The method only reflects the behaviour of asmall part of the jet. This method should be preferred by deriving the maximum velocity of animpinging jet. Application of the method of Stanczak gives a damping coefficient which canbe seen as an averaged value for the influence over the total width of the jet. Because of thisfact, and taking the flow characteristics (vortex formation, turbulence and air-content) intoconsideration, the method based on an averaged coefficient, where there is a decrease fromthe point of impinging, is assumed to be more representative in this case and will be usedfurther on.


71

Using a reference pressure of p =1 N/m2, the influence of the tail water damping coefficientwde is analysed. In Figure 5-5, three different values for the damping coefficient are used;

the green line has been created using w =0.25, which is assumed representative for weakclay and grass; the red line (w =0.1) representative for moderate clay and the blue line(w = 1) is assumed to be representative for strong clay.

Figure 5-5: Influence of variable tail water damping coefficient.

The variability between the coefficients w for the different clay qualities as derived byStanczak, is large. At a depth of 80 cm, the factor twhe becomes 0.82; 0.92 resp. 0.45 forweak clay and grass; moderate resp. strong clay. The values for the coefficient w are derivedexperimentally, and Stanczak does not give an explanation for this difference. The exactvalue of the damping coefficient could be a point of discussion, but because a value w = 0.25shows good agreement with the method supposed by Stein, this value is taken for furthercalculations.

Conclusion depth effectThe existence of a water layer on top of the erodible surface affects the shear stresses on thesurface. Implementation of depth dependency (equation 5.8) in the TM makes:

20 ( ) c

soil

dyt E

Equation 5.9

With ( )0 0( ) wd td e Equation 5.8

And w =0.25

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5.2. The strength terms: ( )c d & soilEGrass at a transition is assumed not to differ from grass at a slope. This assumption regardingthe characteristic values of grass is already used within the derivation of the turbulencecoefficient and will also be used for the critical shear stress c and the soil parameter Esoil.

The following relations used in the EPM/Hoffmans are defined to calculate the critical velocityand the soil parameter:

Soil parameter:2

2 &( )

soil asoil E E

E c

g dE CC U

Equation 5.10

with soil =1

Critical velocity:

0

,min ,min0

0

20with 0.29

clay rootsc aU gd

r

c

Equation 5.11

Equation 5.10 is used to calculate the soil parameter and can be used directly within equation5.9; but the critical shear stress should be derived according to equation 5.11 and the

relation:2

00.7 r U .

5.2.1. Derivation c

Derivation of the critical velocity within the EPM/Hoffmans has been done according to thecubic turf model. Assuming the clay aggregate is only fixed at the underside, the criticalcondition for lifting reads (Equation 3.16; see section 3.2.1):

,min ,minc s w a clay rootsgd (Equation 3.16)

Equation 5.11 can be transformed into a relation with c:2

,min ,min00.7 clay roots

c agd

20 ,min ,min0.7 ( )c s a clay rootsgd

Equalizing with Equation 3.16 results in:2

00.7

The relation for the grass strength reads: roots,min =0.7*Ar/A* tr with Ar =root area [m2];A =soil area [m2] and tr =root tensile strength [N/m2]. The relation for the clay strengthreads: clay,min = 0.021* clay . But, the EPM suggest to neglect the cohesion obtained by the clay( clay,min =0) for grass layers.

5.2.2. Depth variabilityVerheij states that the strength obtained by the clay cohesion and the roots cohesion (twolast terms in equation 3.16: clay,min and roots,min) is not constant over the depth but decreasesand thereafter increases again over the depth profile. Near the surface the root systemdominates, whereas with increasing depths cohesion and internal friction of the clay


73

dominates (Verheij and Hoffmans 2008). This suggestion has been illustrated with a sketch oftwo variable strength profiles with respect to depth see Figure 5-6; the right profile shows asketch of a well rooted soil and the left shows a badly rooted soil.

Figure 5-6: Strength function with respect to depth

The total shear stress obtained by cohesion can be written as:

total clay rootsd d d Equation 5.12

with:*clay cohesion weight cohesiond d d Equation 5.13

Assuming a exponential decrease of the root density with respect to depth, cohesion obtainedby the roots can be written as:

*,0 * d

roots rootsd e Equation 5.14

Where is the coefficient of root decrease over depth. Application of these formulas meansintegration of the clay cohesion within a grass layer. Besides that, residual strength of theclay layer underneath the grass layer will be taken into account.

DiscussionThe influence of the clay (indicated with clay ), is separated into two parts, cohesion and

weight (d). The first term (cohesion part) is taken constant over the depth and the second termis taken variable due to the weight of the upper clay particles. Taking the idea of surfaceerosion into account, only the uppermost clay particles are assumed to be eroded. Theexplanation of an increase in strength due to the weight of the upper laying clay particleslooks therefore quite doubtful. An explanation that relates the increase in strength to thecompaction caused by the weight of the upper laying particles looks more convincing. Besidethe better compaction of the deeper clay, also the diminishing presence of soil structurewithin the clay layer has a positive effect on the coherence. Therefore, equation 5.13 ischanged into equation 5.15, reading:

,0 (1 * )clay clay csd d Equation 5.15

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Where the parameter cs [m-1] represents the increase of clay cohesion over depth.

ConclusionImplementation of depth variability in the relation for the critical shear stress (equation 3.16 )and taking into account clay,min =0.021* clay results in:

( ) ( )c s w a totald gd d Equation 5.16

with:

,min ,min

,0 ,0

0.021( ) 0.7 *

0.021( (1 * )) *

total clay roots

rclay r

dclay cs roots

d d d

Ad t d

Ad e

Equation 5.17

5.3. Summary of all equations

The TM (Transition model) reads as follows (same as equation 5.1):2

0 ( ) ( )ˆ ( )

c

soil

d ddydt E d

with in the boxes below the various terms in the equation.Load term:

0(1.5 5 )r2

01 10.016* *2 1

U

0 0( ) wdd e

Strength terms:( ) ( )c s w a totald gd d ***

,0 ,0

0.021( )

0.021( (1 * )) *total clay roots

dclay cs roots

d d d

d e

2

2 &*

soil asoil E E

E c

g dE CC U

***

with soil =1 and:

0

0

( )totalc a

dU gdr

***

with0

20 00.29 ( 0.055 1.21)c with and c

(the equations marked with *** return when depth dependency is removed, to theEPM/Hoffmans equations)

Evaluation erosion model

75

6. Evaluation erosion model

6.1 Evaluation Transition Model

6.1.1 Determination parameters

6.1.2 Simulations

6.1.3 Evaluation Transition Model

6.1.4 Sensitivity analysis

6.1.5 Conclusion

6.2 Evaluation Headcut erosion model

6.3 Conclusion

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6 Evaluation erosion model

Understanding of the behaviour of the models is very important for good interpretation of themodel results. Because the total erosion process is described by two different models, (1) theTM (Transition Model) and (2) the Headcut erosion model, the behaviour of both models isanalysed in this chapter; starting with the TM in section 6.1, followed by the evaluation of theHeadcut erosion model in section 6.2.

6.1. Evaluation Transition ModelThe model behaviour is analysed using a number of simulations. In section 6.1.2 the outputof the model is given for two different case settings; (1) overflowing water and (2)overtopping waves. This section is followed by an evaluation of the model behaviour (section6.1.3) and a sensibility analysis (section 6.1.4). But before simulations can be made, thevariables need to be determined, this is done in section 6.1.1.

6.1.1. Determination parametersThe parameters which need to be determined are:

The air water ratioThe clay cohesion clay,0

The root cohesion roots

Besides that, a definition of failure is given, resulting in a parameter referred to as FailureTime. This parameter is used to compare the different simulation outputs as done in thefollowing sections.

Determination air-water ratioBecause no results are available from measurements of air content in the flow mixture, avalue of =0-0.4, is suggested by Verheij. For calculations a value of 0.4 is taken. Thesensitivity of this assumption is evaluated in section 6.1.4.

Determination clay cohesion clay,0

According to Hoffmans, the different types of clay have the following indicative values(Hoffmans 2008b):

Table 6-1: Indicative values for strength of clay (r0=0.15 and cmin =0.021c)

Quality of clay Esoil [m/s] Uc [m/s] c [kN/m2]poor 5.48·104 0.5 0.00structured 14.0·104 0.8 5.0good 26.5·104 1.1 12very good 49.3·104 1.5 26

According to this table, structured clay has a cohesion of 5 kN/m2. There is nothingmentioned about a clay strength variability coefficient cs, as used in equation 5.15. Theincrease in strength depends on the soil structure development over depth, which developsover time. Visser gives quantitative data for soil development over time (Table 6-2) (Visser2007):

Table 6-2: Soil structure development over time.

Time [years] Soil structure developmentover depth [m]

1-3 0.3-0.510-15 0.850 1.6-1.8

Because a clay layer in general has a thickness of about 0.5 to 1.0 m, the soil structure willdevelop over the entire depth. The development of the quality is assumed to increase from


77

structured (5 kN/m2) to good (12 kN/m2) over this depth. According to these assumptions theincrease in coherence over depth, indicated with cs, is 1.75 m-1.

Implementation of the derived values in equation 5.17, gives the following relationship overthe strength profile:

,min 0.021( ) 0.021*(5*(1 1.75* ))clay clayd d d Equation 6.1

Determination root cohesion roots

If there are measurements available concerning the density of the root system the RARshould be derived from this information. If this information is not available, based on data ofSprangers (see appendix B) the RVR [%] and RAR [%] with respect to depth can bedescribed by the following functions:

2.67*0.8 ^ ( 0.015)RVR d Equation 6.2and

( 22.32 )0.0746 /100* drARAR eA

Equation 6.3

Implementation of this relation and using tr =20*106

N/m2 and =22.32 [-] gives:22.32( ) 0.7* *(0.0746 /100* )d

roots rd t e

This results in the following strength profile:

,min ,mintotal clay rootsd d d

0.021( ) 0.7 * rclay r

Ad t dA

,0 ,0 0.021( (1 * )) * dclay cs rootsd e

4 22.32 0.021(5(1 1.75* )) 0.7* *7.46*10 * drd t e

The formula is presented in Figure 6-1. As can beseen within the first 20 cm the clay influence isneglectable, and the residual strength of the deeperlying clay has no significant influence on the totalstrength.

Figure 6-1: Original strength profile

This graph does not look like the sketch in Figure 5-6; implementation of the clay strengthhas no significant influence on the total strength and will therefore not give a significantchange in the erosion rates. This is partly because the applied value for the characteristic claystrength is a minimum boundary value. Because clay is an inhomogeneous material, to beable to guarantee a certain strength, assumptions are made to determine a minimum valuefor the characteristic clay strength. Hoffmans used the formula given by Mirtskhoulava todetermine a characteristic value for the strength of the clay. The formula for the criticalvelocity of Mirtskhoulava reads (Mirtskhoulava 1991):

8.8 0.4log 0.6c s a fa

hU gd Cd

Equation 6.4

With:*f fatique clayC

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The factor fatique is introduced to account for fatigue rupture. This factor fatique is a function ofthe plasticity index, the porosity, the density of the aggregates and water and mass of thesoil particles. In absence of test data a factor fatique of 0.035 can be applied. Application of thefactor 0.035 together with a factor 0.6 as given in equation 6.4, gives a factor 0.021 asapplied by Hoffmans (used in equation 3.16). Mirtskhoulava notes that the nonscouringvelocities for soil in air-dry condition are 2-6 times lower than for the same soils in saturatedconditions. Besides that, he applied a safety coefficient of 2-2.5 to account for theinhomogeneousity of the soil resistance. Application of these safety and fatigue factors couldmean an underestimation of the clay strength in the order of a factor 10.

Therefore Figure 6-2 and 6-3 are made to show the sensitivity to assumptions made indetermination of the strength profile. Figure 6-2 has been obtained using min =0.21* clay instead of using min =0.021* clay. Figure 6-3 has been obtained using min =0.21* clay andcoefficient cs =5 [m-1] in stead of 1.75 [m-1], this value for cs has been derived assumingvery good clay quality at d =80 cm.

Figure 6-2: Modified strength profile; using cmin =0.21c Figure 6-3: Modified strength profile; using cmin

=0.21c and cs =5 [m-1]

Application of a less underestimation factor for the clay cohesion result in a strength profilewhich looks more like the suggested profile (see Figure 5-6) and would result in smallererosion rates.

The root cohesion profiles are made according to the formula of Sprangers. This formulashould be applied, when there are no root density measurements available as subscribed bythe guidelines of the VTV available. Regarding the fact that the root density is measured inlayers of 2.5 cm within the upper 20 cm, implementation of a measured density profile resultsin a discretized strength profile with a step size of 2.5 cm. To analyse the influence of a(discretized) depth strength profile, the root cohesion and total cohesion defined by the threedifferent methods are plotted in Figure 6-4. The green line in the left graph is createdaccording the root density formula of Sprangers (equation 6.3), the black lines show thediscretized strength function which has been created using a fictive RAR profile and could beseen as an example of a VTV-defined profile. Plotted in the same figure with a blue line, isthe non-variable strength profile which could be determined for an averaged grass quality byfollowing qualification guidelines as applied in the EPM/Hoffmans model (see Table 3-2). Theright figure shows the total cohesion; the green line is created by adding clay cohesion to the


79

root cohesion as derived using the averaged root density of Sprangers. Because the VTVneglects the clay cohesion in calculation of the total cohesion, the total cohesion is equal tothe root cohesion.

Figure 6-4: Strength profiles according to three different methods.

From the left picture can be concluded that the root cohesion using the formula of Sprangers(green line) is less than the root cohesion as defined using qualification guidelines (blueline). Even if clay cohesion is added, see the right graph, the blue line gives a higheraveraged value then the total cohesion averaged over the upper 0.2 m (green line).Application of a depth variable strength profile affects the critical shear stress and soilparameter in a way that the erosion calculated with a depth variable strength profile is higherthan in a case with a non variable strength profile.Furthermore, it is interesting to notice the fact that the blue line correspond very well to theblack line in the upper 2.5 cm. Practically this could mean that the root cohesion value asgiven in Table 3-2 is only representative for the upper 2.5 cm and is used in theEPM/Hoffmans model for reasons of simplification as representative for the upper 20 cm.Unfortunately, Sprangers gave only values for averaged grass conditions, meaning they cannot be used for all defined grass qualities.From the figure can be concluded that for the upper 20 cm, the average erosion ratecalculated with a constant strength profile will be lower than for a case with a depth variablestrength profile.

Failure TimeWorking with a continuously variable strength profile makes calculations time consuming andunpractical. Working with a profile with the depth discretized into steps could haveadvantages, therefore the total layer could be divided in smaller layers with a characteristicstrength. Regarding the fact that the root density in a discretized strength profile is measuredin layers of 2.5 cm within the upper 20 cm, it has no sense to work with smaller steps than2.5 cm; making the steps larger would reduce the accuracy, so a step size for the grass layeris defined of 0.025 m. Determination of a good step size for the underlying clay layer is lessclear, because, at least until today, no measuring standards are defined to determine thestrength of clay layers underneath grass layers. Assuming a linear increase of the strengthwithin the clay, the strength in the middle of the clay layer represent the average valueswhich could be used as an average characteristic value representative for the whole claylayer.

Application of this method means defining failure of the total layer after each single layer hasfailed. A single layer fails when the eroded depth within this layer equals the thickness which

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is the same as the step taken (0.025 m for the grass layer for example). To implement thisdefinition of failure the TM is converted to equation 6.5:

21 0

*( ) ( )

nsoil

totali c

y d E dT

d d Equation 6.5

This converted equation makes it possible to define a failure time for the total layer with acharacteristic thickness y (d). The time required to reach an erosion depth equal to thecharacteristic thickness of the layer is referred to as the Failure Time T [h]. Because thethickness of the total clay layer is generally in the order of 0.8 m, this value is taken as thecharacteristic thickness. Matlab will be used to make simulations with the TM. Application ofthis program makes is possible to work with a continuously variable strength profile to thecredit of a minor increase in calculation time.

6.1.2. SimulationsFailure Times for two different case settings are simulated: (1) continues overflowing waterand (2) wave overtopping events. For the case with overtopping flow the loading time isequal to the simulation time (or Failure Time). For a case with overtopping waves the loadingtime is defined for each singular wave event which are randomly generated in time. Anoverview of all equations and parameters used can be found in the box below and the Matlabscripts (see appendix D).

TM EPM/Hoffmans (used in section 6.1.3)2

0

1

( ) ( )( )

nc

i soil

d dy T

E d

with: 0 0( ) * wdd e2

01 10.016* *2 1w U

2

1

0.7 ( )( )

nc

i soil

U U dy T

E d

Formula used in both models:

0(1.5 5 )r

soilsoil

E

EC

&2

2( )a

E Ec

g dCU

& 0

0

totalc a

w

U gdr

( ) ( )c s w a totald gd d &,0 ,0

( )

( (1 * )) *total clay roots

dclay cs roots

d f d d

f d e

with (unless given differently):d =depth [m]; g =9.81 [m/s2]; da =0.004 [m]; =1*10-6 [m2/s]; r0 =0.2 [-]; =1/18 [-];f =0.021 [-]; clay,0 =5 [kN/m2]; cs =1.75 [m-1]; soil =1 [-]; E =1*10-10 [-]; 0 =0.29 [-];

=(( s – w)/ w) [-]; s =2000 [kg/m3]; w =1000 [kg/m3]); =22.32 [-];roots,0 = 0.7*7.46*10-4*tr [N/m2]; tr =20*106 [N/m2]; =0.4 [-]; w =0.25 [-];


81

Overflowing waterThe erosion process for a case with overtopping flow has been simulated for 6 different jetvelocities. The results of these simulations can be seen in Figure 6-5.

Figure 6-5: Computed erosion process over time (TM) in case of overflowing water

The different plots in the figure are obtained using a variable flow velocity of U0 =4; 5; 6; 7;8 and 10 m/s. The first plot, U0 =4 m/s, shows an erosion depth of 0 m, which can beexplained as no initiation of erosion. For initiation of erosion, the shear stresses should behigher than the critical shear stress, this means equation 6.6 should be valid:

0 c Equation 6.6

According to the defined parameters, the boundary velocity for initiation of erosioncorresponds with a flow velocity of 4.96 m/s. Simulating a case with this limit velocity, aFailure Time of 1.65 h is calculated. The upper right graph, U0 =5 m/s, is very close to thislimiting velocity for initiation of erosion, but the decrease in Failure Time, 0.86 h, is quitelarge. It can be concluded that in case of overflowing water, if erosion is initiated, U0 4.96m/s, erosion develops fast and the moment of failure is reached in less than 2 hours.

Overtopping wavesTo make simulations for a case with overtopping waves some more assumptions have to bemade. The overtopping waves are represented as blocks of water with a certain volume,randomly distributed over time according to a uniform distribution. The assumption made forthe distribution of waves over time results in a certain spreading in Failure Time between thecomputed results for one simulation case. To evaluate the influence of this assumption thecomputation for a simulation case is repeated a number of times. Whether the assumption forthe distribution of waves over time is right, should be examined in further research. Thevolumes of the overtopping waves are distributed according a Weibull distribution as

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explained in section 2.4.2 (Overtopping volumes per wave). For a case with a specificdischarge of 10 and 75 l/s m-1 this gives the following results:

Figure 6-6: Wave distribution 10 l/s m-1 and 75 l/s m-1

The upper figures of Figure 6-6 show the number of overtopping waves and their volumes asdefined for a six-hour storm duration. The number of waves (x-as) is a function of the specificdischarge; for a case with q is 10 l/s m-1 a number of 995 overtopping wave events aredefined and for a q of 75 l/s m-1 2790 events. The maximum volume of a wave event alsoincreases with increasing specific discharge (Vmax for q =10 l/s m-1 is 2398 l/m and Vmax for q=75 l/s m-1 is 7726 l/m), see Table 6-3. The figures in the second row of Figure 6-6, showthe distribution of the volumes of waves sorted with respect to volume. The probability ofoccurrence of the wave volumes is shown in the graphs on the third row. On the horizontalaxis the generated volumes are plotted against there probability of occurrence (plotted on thevertical axis). In Table 6-3, the characteristic values for 7 different specific discharges q for a6-hour storm duration are given.

Table 6-3: Characteristics of an 6-hour storm duration for different q [l/s m-1]

Specific discharge [ l/s/m] 0.1 0.46 1 10 30 50 75Probabilityof an overtopping event [%]

0.35 % 2 % 4 % 23 % 42 % 52 % 61 %

Numberof overtopping events [-]

16 92 182 1050 1907 2389 2790

Maximum Volumeof an overtopping event [l/m]

440 718 897 2294 42287 5852 7718

The loading time for a singular wave event is defined equal to the overtopping time accordingto Bosman. The Failure Time represents the period in which the summation of the individual


83

wave impacts has generated a scour hole of 0.8 m. In Figure 6-7 the erosion developmentprocess and Failure Time for five different are depicted.

Figure 6-7: Computed erosion process 1, 10, 30, 50 and 75 l/s m-1 (TM)

For a specific discharge of 1 l/s m-1 (upper left graph) the 6 hour-simulation output is5.86*10-6 m, this can be interpreted as no erosion. The graphs created with q =10; 30; 50and 75 l/s m-1, show a decreasing Failure Time with increasing specific discharge. Becausethe number of waves and there volumes increases, and therefore also the overtoppingvelocities and overtopping times, the erosion process develops faster with increasing specificdischarge. Compared with the case of overflowing water the erosion develops less rapidly.

6.1.3. Evaluation Transition ModelBecause erosion occurred at the transition and not on the slope, equation 5.2 is assumed tobe valid:

0, 0,transition slope (equation 5.2)

Consequently, the results of Failure Time calculations with the TM will give smaller valuesthan comparable calculations done with the EPM/Hoffmans model:

TM EPMT T Equation 6.7

In order to evaluate the TM, the results of TM are compared with results of theEPM/Hoffmans model.

In Figure 6-8 the results of simulations with the TM as well as the EPM/Hoffmans are plottedfor a specific discharge of q =75 l/s m-1. The left graph in the figure shows one simulation ofthe development of the erosion depth over time. The graph in the middle shows the results ofthe erosion development of 11 simulations. To analyse the spreading within the computedresults a histogram of the obtained Failure Times is created. The histogram (right graph)shows the results for the Failure Time of 1000 simulations. The histogram is composed out of

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20 classes, which are defined according 1/20th of the total variation in Failure Times. Thenumber of simulations within these class boundaries can be seen on the y-as.

Figure 6-8: Comparison TM & EPM/Hoffmans model (q =75 [l/s m-1])

The results are clear. The TM gives smaller values for the Failure Time than theEPM/Hoffmans model, meaning that the results are conform the conditions as given inequations 5.2 and 6.7. The erosion process looks reasonable; the erosion velocity iscomparable to the inverse of the strength profile. Because of these facts, as an overallconclusion it can be said that the behaviour of the model is according the expectations.

6.1.4. Sensitivity analysisAlthough the behaviour of the model follows the expectations, the assumptions made todetermine the parameters may have a large impact and therefore there individual influence isanalysed. A sensitivity analysis is done for the different parameters of the two terms of theTM:

load term: 20 ( )d

strength terms: ( )c d & ( )soilE dThe sensitivity analyse will be done for the case with overtopping waves, using histogramsobtained with 1000 simulations composed of 20 classes like done in section 6.1.3.

Parameters of the load termThe load term can be subdivided in (1) a formula to calculate the shear stress at the surface(d =0 m) and (2) a formula to calculate the shear stress at depth d:

1.

2

01 10.016* *2 1

U

2. ( )0 0( ) * wd td e

Therefore, the sensitivity of the model to the following parameters is analyzed:Velocity ULoad coefficient 0.016Air-water ratioTail water damping coefficient w

Velocity UThe velocity of the water tongue depends on the volumes of the single wave events. For aspecified specific discharge of 10, 30, 50 and 75 l/s m-1 the following histograms aregenerated:


85

Figure 6-9: Variability Failure Time due to change in q [l/s m-1]

Variability due to changes in specific discharge can be noticed in two ways. First the changein mean Failure Time, by a shifting of the histogram, and secondly the change in spreadingwithin the results which is noticeable in the shape of the histogram. The histogram obtainedwith simulations with a specific discharge of 10 l/s m-1 shows the biggest spreading and thehighest Failure Times. Under the assumptions made for wave characteristics, the number andvolumes of overtopping waves increase with increasing specific discharge, as does theovertopping time and the velocities. For a case with a high specific discharge the number ofwaves are more regularly distributed over time, which gives a more regular load than in acase with just a few waves over the 6 hour storm period. A decrease in spreading can beaccounted to an increase in regularity of wave events over time. The decrease in Failure Timewith increasing specific discharge has to do with the increasing number of overtopping waveswith respect to time and the increase in wave volume. Spreading in Failure Time due to thedistribution of waves over time is less than the variation in Failure Time due to changes inspecific discharge. This means that variation due to variability in discharge is dominant tospreading due to the wave distribution over time.

Further sensitivity analyses will be done using a specific discharge of 75 l/s m-1. To make acomparison between this one and the other histograms easier, the histogram created using aspecific discharge of 75 l/s m-1 and the original derived parameters is dark blue coloured in allfigures.

Load coefficientA load coefficient of 0.016 follows from the theory of Beltaos (see section 5.1.1). This valuehas been obtained from figures, meaning that there is a certain error in this value.Unfortunately, just one experimental result is mentioned for =20°, meaning the Beltaosdata can not be the basis for a value for the spreading and the possible error in this factor.Because the parameter is influenced by the angle of impinging, the influence of the slopeangle assumption can be analysed. An impinging angle of 20° is derived according to a slopeof 1:2.5, assuming a slope of 1:2 the impinging angle becomes 27°. The load coefficientfactor for an impinging angle of 27° is 0.014. To evaluate this influence, simulations aremade with a factor of 0.016 and 0.014 (difference of more than 12%) which is considered tobe a reasonable factor for errors. The histogram obtained with these simulations are given inFigure 6-10.

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Figure 6-10: Sensitivity to the load coefficient (q =75 l/s m-1)

where lf is the load coefficient. The averaged Failure Times are 0.30 h for lf =0.016 and 0.38h for lf =0.014, an increase of 30%. An error of 12% in the value for the load coefficientfactor gives on error of the mean Failure time of 30%. A remarkable fact is that a steeperslope, 27° in stead of 20°, gives a lower load factor giving smaller erosion rates and a higherFailure Time. It is recommended to do some further research to the shear stresses applied bythe water tongue at the transition.

Air-water ratioBecause no results are available regarding the measurement of air content in the flowmixture, a value of =0-0.4, is suggested. The influence of this air-water ratio is analysedusing four different values for , knowing 0.0; 0.2; 0.4 and 0.6, see Figure 6-11.

Figure 6-11: Sensitivity to air-water ratio (q =75 l/s m-1)

Because the influence is quite large the same plot has been created using a specific dischargeof q =50 l/s m-1.


87

Figure 6-12: Sensitivity to air-water ratio (q =50 l/s m-1)

The values for the average Failure Time for a specific discharge of 75 (see Figure 6-11); 50(see Figure 6-12) and 30 l/s m-1 are given in Table 6-4.

Table 6-4: Averaged Failure Times; variable: air-water ratio ( )

=0.0 =0.2 =0.4 =0.6q= 75 l/s m-1 0.12 0.19 0.30 0.52q= 50 l/s m-1 0.16 0.26 0.42 0.82q= 30 l/s m-1 0.24 0.40 0.69 1.66

The expected influence of the air-water ratio can be derived using the following formula:1/(1+ ) ~U and because T~ (1/U) 2, the Failure Time is expected to follow changes in like:T~ (1+ )2. The values for the air-water ratio squared are given in the next table together withthe changes in percentages with respect to the =0.4 value.

Table 6-5: Differences averaged Failure Times differences [%]

=0.0 =0.2 =0.4 =0.6(1+ )2 1 1.44 1.96 2.56exp increase -49 % -27 % - % +31 %

q= 75 l/s m-1 -59 % -36 % - % +74 %q= 50 l/s m-1 -60 % -38 % - % +95 %q= 30 l/s m-1 -65 % -41 % - % +145 %

Changing the air-water ratio has a relative large effect on the mean Failure Time. This has todo with the increase of the velocities with increasing specific discharge. From the figures wealso observe an increase in spreading in Failure Times with an increasing value of the air-water ratio. Differences in the averaged Failure Time are in the order of -65 to +145 %.

Tail water damping coefficient wThe reduction of the applied shear stress over depth is modelled by Stein and Stanczak.Stanczak suggest to use a coefficient w of 0.25; and Stein suggest a coefficient w of 0 (atleast within the first meter, defined as the core, which is of influence). The influence of thetail water damping coefficient in the total model can be seen in Figure 6-13.

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Figure 6-13: Sensitivity to damping coefficient (q =75 l/s m-1)

The averaged Failure Times are 0.30 h for w =0.25 and 0.29 h for w =0.0. A change of wgives a minor change in erosion rate.

Parameters of the strength termsThe strength terms can be divided in (1) a formula to determine the critical shear stress c

and (2) a formula to determine the soil parameter Esoil :1. ( ) ( )c s w a totald gd d

with:,min ,min

,0 ,0

( )

0.021( (1 * )) *total clay roots

dclay cs roots

d d d

d e

2. 2

2( )

soilsoil

aE

c

Eg dU

Grass and clay at the transition is assumed to be the same as on the slope, therefore valuesfor parameters are unmodified taken from the EPM/Hoffmans model and Sprangers. The onlyvariable parameter in the formula for Esoil (second formula) is the critical velocity Uc. The

critical velocity is related to the critical shear stress as2

00.7 r U . The factor 0.021 (in

the first formula) is included to account for inhomogeneousity of the clay. Because this factorreduces the influence of the clay drastically (see Figure 6-1: Original strength profile), thiscould lead to underestimation of the strength.

Therefore, the sensitivity of the model to the following parameters is analyzed:clay inhomogeneousity factor 0.021 [-] (referred to as f)clay cohesion increase factor over depth cs [m-1]

In section 6.1.1 modification suggestions are defined for the both parameters. The influenceof these modifications on a strength profile is analysed. (The suggested modifications are afactor 0.21 in stead of 0.021 and a value for cs of 5 in stead of 1.75). In this subsection, theinfluence of these modifications on the TM is analysed. Figure 6-14 shows the histogramsobtained using 4 different strength profiles: (1) f =0.021 [-] & cs =1.75 [m-1] (nomodifications); (2) f =0.021 & cs =5, (3) f =0.21 & cs=1.75 and (4) f =0.21 & cs =5.


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Figure 6-14: Sensitivity to suggested clay strength modifications (q =75 l/s m-1)

The influence of assumptions made for clay strength are significant, therefore, the same plothas been created using q =50 l/s m-1.

Figure 6-15: Sensitivity to suggested clay strength modifications (q =50 l/s m-1)

The values for the average Failure Time for a specific discharge of 75 (see Figure 6-14); 50(see Figure 6-15) and 30 l/s m-1 are given in Table 6-6. Table 6-7 shows the differences ofthe averaged Failure Time with respect to the original case.

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Table 6-6: Averaged Failure Times; variable: clay strength

f =0.021 &cs =1.75

f =0.021 &cs =5

f =0. 21 &cs =1.75

f =0. 21 &cs =5

q= 75 l/s m-1 0.30 0.34 0.86 1.49q= 50 l/s m-1 0.42 0.47 1.18 2.06q= 30 l/s m-1 0.68 0.75 1.88 3.28

Table 6-7: Differences averaged Failure Times differences [%]

f =0.021 &cs =1.75

f =0.021 &cs=5

f=0. 21 &cs =1.75

f =0. 21 &cs =5

q= 75 l/s m-1 - % +13 % +184 % +395 %q= 50 l/s m-1 - % +12 % +182 % +391 %q= 30 l/s m-1 - % +10 % +176 % +384 %

The differences between the simulations with the different strength profiles are quite large.Applying only a modification in cs, does not give major changes to the Failure Time (+10 %),but applying a factor of 0.21 in stead of 0.021 does (change: +180 %). When both factorsare changed (cyan histogram) the failure time increases with a factor 4 and the spreading isdoubled. Assuming a factor f of 0.21, increasing the value for cs to 5 in stead of 1.75 (seecolumns 4 and 5), gives an increase in Failure Time of 75%.

6.1.5. Conclusion evaluation TMThe behaviour of the model is conform expectations; the Failing Time calculated for thetransition (with the TM) is smaller than the Failing Time calculated for the slope (with theEPM/Hoffmans model; see Figure 6-8). When erosion is initiated, the erosion processdevelops fast over the total depth. The influence of a change in tail water damping coefficientis negligible. A change of a load coefficient factor or the air-water ratio has significantinfluence in spreading as well as in the computed mean Failure Times. The model is mostsensitive to changes to define the clay cohesion. Therefore it is recommended to do moreresearch to strength characteristics of clay and grass layers.

6.2. Evaluation Headcut erosion modelThe second phase of the erosion process can be modelled with the headcut erosion model.Formulas as used in this model are given in the box below. Variability due to changes inspecific discharge is shown in Figure 6-16.

Formula Headcut erosion model:


with:0.79ln( ) 3.04headcut hC k

13( )L qH

0.33

0.5 3.23189 expln(101 )hc

h

L kk

Values for q, H and kh are given by the figures.


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Figure 6-16: Variability headcut erosion due to variable q [l/s m-1] (kh =0.03 & H =0.8 m)

The headcut velocity increases with increasing specific discharge. Within this model there isone important strength variable, known as the headcut erodibility index kh. The indexrepresents a measure of the resistance of the earth material to erosion, it is a product ofdifferent parameters concerning the compressive strength, the mean block size, the strengthof inter-particle bounds and the shape of a block (USDA 2001). The index guidelinesmentions especially values for rocky soil and it needs to be evaluated whether the method isalso applicable for grass erosion.

After some clay experiments a index value of 0.03 is recommended for clay and grass(Verheij and Hoffmans 2008). To analyse the sensitivity to changes of this index value, theerosion velocity has been calculated according to an index value of 0.03 (recommendedvalue; blue line) and 0.035 (red line).

Figure 6-17: Sensitivity headcut erosion to index number kh (q =10 l/s m-1 & H =0.8 m)

Increasing the index number with 15% gives an decrease in headcut velocity of 13%.Typical heights during headcut erosion are in the order of 0.5 to 1.0 m. The influence of theheadcut height can be seen in Figure 6-18.

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Figure 6-18: Sensitivity headcut erosion to headcut height H (q =10 l/s m-1 & kh =0.03)

Increasing the headcut height with a factor 2 gives an increase in headcut velocity of 66%.The erosion distance is measured in horizontal direction. The erosion velocity with respect tothe direction of the slope is a factor 1/(cos ) larger. This means that for a slope with =20°,the headcut velocity measured parallel to the slope is a factor 1.06 larger.

Both variables, kh and H, cause a change in the steepness of the line. Changing the indexvalue has a approximately inverse proportional effect on the headcut velocity. Increasing theheadcut height has an increasing effect on the headcut velocity.

6.3. ConclusionsTo be able to analyse the behaviour of the Transition Model the term Failure Time isintroduced. The Failure Time is defined as the total time needed for the erosion to developuntil a depth equal to the characteristic thickness of the layer. The Failure Time calculatedwith the TM model (derived for transitions) is smaller than the Failure Time computed withthe EPM/Hoffmans model (derived for the slope). This agrees with the expectation thaterosion rates at a transition are higher than on the slope. Because the wave events arerandomly generated, the computed Failure Times show a certain spreading. Both models, theTM and the Headcut erosion model, show to be very sensitive to changes in strengthcharacteristics (formula for clay cohesion in the TM and kh in the headcut erosion model).

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7. Validation

7.1 Wave Overtopping simulator

7.2 Test simulations

7.3 Test locations

7.4 Results

7.4.1 Implementation of Bosman and distribution of waves as givenby Van der Meer

7.4.2 Introducing depth dependency

7.4.3 Validation of the Transition Model

7.5 Conclusion

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7 Validation

Validation of the erosion model will be done using results of experiments done by Infram inFebruary to April 2008. In the framework of SBW project with respect to grassed innerslopes, they tested inner slopes on three locations in the Netherlands using a waveovertopping simulator. They reported there observations in two reports (Bakker et al. 2008a)and (Bakker et al. 2008b).The Chapter starts with an explanation of the experiments: starting with an explanation ofthe Wave Overtopping simulator (7.1), followed by an explanation of the wave volumes overtime (7.2) and a description of the different test sections (7.3). Section 7.4 describes themodel as well as the experimental results. In section 7.5 the conclusions are given.

7.1. Wave Overtopping simulatorThe classical way of testing dikes is testing in a wave flume. As materials like grass and claycan not be scaled like rock, testing needs to be done on large scale. This asks for speciallarge facilities and the transfer of grass and clay from the original location to the flume.In 2001 van der Meer suggested an other test method using a wave overtopping simulator.The idea behind the new testing methodology is creating full scale overtopping conditions ona real inner slope of a dike.The Wave Overtopping simulator is a container which is filled continuously with a pumpdischarge and is opened with a pneumatic valve after fixed time periods. The water insideflows out; resembling an overtopping wave. The principle of the Wave Overtopping simulatorcan be seen in picture Figure 7-1.

Figure 7-1: Principle Wave Overtopping simulator

Using the pneumatic valve a defined specific discharge and wave volume can be achieved.With a total capacity of 5.5 m3/m or 22 m3 over the total test width of 4 meter, simulationsare possible until a specific discharge of 75 l/s m-1.

7.2. Test simulationsSimulations are done with the same hydraulic boundary conditions as used in Chapter 6; Hs

=2.0 m; Tp = 5.7 s; a maximum storm duration of 6 hours and an outer slope of 1:4. Each

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single inner slope experiment is carried out using an increasing specific discharge; thepredefined simulated specific discharges in increasing order are the following:

• S1: 6 hour storm with a specific discharge of 0,1 l/s m-1;• S2: 6 hour storm with a specific discharge of 1 l/s m-1;• S3: 6 hour storm with a specific discharge of 10 l/s m-1;• S4: 6 hour storm with a specific discharge of 30 l/s m-1;• S5: 6 hour storm with a specific discharge of 50 l/s m-1;• S6: 6 hour storm with a specific discharge of 75 l/s m-1

Due to the small number of waves in the cases S1 (0,1 l/s m-1) and S2 (1 l/s m-1) thesimulations are accelerated with a factor 100 resp. 10. The wave events volumes for thesimulation cases S1-S6 are classified, the theoretical number of events and their volumesresp. the experimentally used numbers and volumes of the wave events can be found inFigure 7-2 and Appendix C.1.

Figure 7-2: Theoretical and simulated number and volumes of wave events for a 6-hour storm duration

The experiments are carried out using an increasing specific discharge. The number of eventsand the time in between wave events are described in steering lists. The total number ofwave events for a singular specific discharge and the summation of the total number of waveevents can be seen in Table 7-1.

Table 7-1: Total amount of wave events as given in the steering lists

S1 S2 S3 S4 S5 S6number of wave events 9 126 749 1275 1524 1686

number of wave events 9 135 884 2159 3683 5369

The transitions between the different cases will be marked in the erosion developmentprofiles. In general, after 2 hours of simulation the experiment was disrupted tophotographically and numerically document the erosion process. A reduced number of thesephotos are used in this report to show the appearing erosion process as seen in theexperiments. Figure 7-1 shows an impression of the experimental test set-up.

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Figure 7-3: Impression experimental test set-up

The steering lists are defined to simulate a typical storm with one set of boundary conditions( Hs =2.0 m; Tp = 5.7 s; Tstorm 6 hours and outer_slope of 1:4). Different locations are exposedto different hydraulic conditions. Assessment of the dike sections should be done accordingthe appearing hydraulic conditions. To be able to assign all dike section adequately, steeringlists should be defined for the local appearing boundary conditions.

7.3. Test locationsIn total seven inner slope dike sections are tested on three different locations: 4 sections onthe ‘Boonweg’ (Friesland), 1 section in St. Philipsland (Zeeland) and 2 sections in Kattendijke(Zeeland). Information about the dike structure and clay and grass qualifications of thedifferent dike sections can be found in Table 7-2.

Table 7-2: Characteristics of test locations; with y =characteristic clay layer thickness at the place of failure, L=length of the inner slope, =slope angle, clay quality c3 Table 6-1: structured clay quality, grass qualitycharacteristics see Table 3-2

Boonwegsection 1-4

St. Philipsland Kattendijkesection 1-2-D

core sand sand sandy [m] 0.6 0.4 0.75L [m] 27 13 15

[-]1:2.9 1:2.4 1:3

clay quality c3 c3 c3grass quality good good goodref code (B1-B4) SP K

The grass quality description ‘good’ is obtained according to the measured number of roots ina soil sample taken at the dike sections (see also Figure 7-4). The measured root densityprofiles can be found in appendix C.2.

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Figure 7-4: Root density profiles with respect to depth given as number of roots per m2

Together with the root density profiles of the sections, the root density curve according toSprangers is given. A major difference between the section profiles and the profile ofSprangers can be noticed. As a reason for this difference is given the fact that Sprangersmeasured grass density of 4 year old grass mats. The grass mats on the dikes tested with theWave Overtopping simulator were probably older. The averaged root density profile accordingto Sprangers, has been used in earlier calculations. Because the ‘measured’ root profilesprovide higher number of roots and therefore higher root cohesion, this will result in a highererosion resistance of the grass layer.

The RAR-profile appears to be fluctuating with respect to the yearly seasons. These rootprofiles are measured in February/March 2008. Within this season, the quality is very lowcompared to the others. Besides the RAR-profile, the covering rate and number of species arealso important for the erosion resistance of a grass layer. Both test sections on Zeeland haveobtained values of more than 70% for the covering rate, which is given in the VTV2006 asthe minimum value for a good quality grass sod. On the other hand, within these twosections a high mice and mol activity is noticed, which could reduce the erosion resistance.For the test sections at the Boonweg, this kind of information is not available. The stage ofthe grass mat has an influence on the erosion resistance and should be kept in mind byanalyzing the experimental results and validation of the model.

7.4. ResultsThe Transition Model is based on formulas as used in the EPM/Hoffmans model. TheEPM/Hoffmans model contains several assumptions where improvements are possible.Alterations and modifications are suggested for improvement. To analyze the impact of thesealterations and modifications, they are introduced in steps. The alterations and modificationswhich are introduced in the EPM/Hoffmans model are the following:

Implementation of the Bosman formulas for overtopping water depth, velocity andtime and the distribution of overtopping volumes as given by the Van der Meerformula;Implementation of depth dependency;Rewriting the erosion formula for the slope to an erosion model for transitions;

The validation starts with an evaluation of the implementation of both the Bosman formulasfor flow velocity and overtopping time as well as the Van der Meer formula for the distributionof wave volumes. Thereafter implementation of the variable strength profile with respect to

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depth is analyzed and validated. At last the rewriting step to a erosion model for transitions isvalidated. The formulas used in the different steps are given in Table 7-3.

Table 7-3: Validation steps

description and formula assumptionsmethod A original EPM/Hoffmans model;

20

1

0.7nc

wavei soil

U Uy t

E

(erosion on the slope)

with=(1.5+5*r0)

r0 =0.2 [-]Esoil =1*107 [m/s]Uc =4.5 [m/s]U0 [m/s] characteristic valuetwave [s] characteristic valuen (number of waves) characteristic events

method B modified EPM/Hoffmans model;Implementation Bosman andvan der Meer formula

2

1

0.7nc

i soil

U Uy T

E(erosion on the slope)

with parameters defined for method A andthe following modifications:U0 U = Bosman formula (Equation 2.9)twave T = Bosman formula (Equation 2.10)n (wave events) Van der Meer formula fordistribution of waves (Equation 2.4)

method C modified EPM/Hoffmans model;Implementation of variablestrength profile with respect todepth

2

1

0.7 ( )( )

nc

i soil

U U dy T

E d

(erosion on the slope)

parameters as used for method B and thefollowing modifications:Uc Uc (d)Esoil Esoil (d)with:

2

2 &( )

soil asoil E E

E c

g dE CC U

0

0

totalc a

w

U gdr

,0

( )

( (1 * )) 0.7 * ( )*

total clay roots

rclay cs r

d f d dAf d d tA

with: d =depth [m]; E =1*10-10 [-];da =0.004 [m]; 0 =0.29 [-]; r0 =0.2 [-];f =0.021 [-]; clay,0 =5 [kN/m2]; cs =1.75[m-1]; soil =1 [-]; Ar/A =RAR =Root AreaRatio; tr =20*106 [N/m2] (and g =9.81[m/s2]; =1*10-6 [m2/s]; =(( s – w)/ w)[-]; s =2000 [kg/m3]; w =1000 [kg/m3])

method D TM2

0

1

( ) ( )( )

nc

i soil

d dy T

E d

(erosion at the transition)

parameters as used for method C and thefollowing modifications:

0 0( ) * wdd e2

01 10.016* *2 1w U

( ) ( )c s w a totald gd d

with: w =0.25 [-]; =0 [-]; =1/18 [-]

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7.4.1. Implementation of Bosman and distribution of waves as given byVan der Meer

An analysis of the implementation of the formulas will be done by comparing the predictions(these can be found in: Verheij and Hoffmans 2008), which are made using characteristicvalues and wave volumes, and results of calculations where the formulas are implemented.The characteristic values for overtopping flow depth, overtopping time and overtoppingvelocity are plotted together with the Bosman formulas in Figure 2-9 to Figure 2-11. Thefigures show a big discrepancy between the characteristic values and the Bosman formulas.The reason for the discrepancy is the fact that the characteristic values are defined for atypical overtopping wave volume and the Bosman formula relates the velocity, overtoppingtime as well as the layer thickness to the wave run-up height and the crest freeboard.Implementation of the Van der Meer formula has effect on the distribution of overtoppingwave volumes. As in the prediction report the simulation cases are composed out of 4-7characteristic volumes, in the steering lists the distribution of wave volumes is according aWeibull distribution as given by Van der Meer. The number and volumes of the wave eventsas used in the prediction report as well as in the steering list can be found in appendix C.The prediction report gives the erosion for a singular specific discharge. To validate theimplementation of the formula, the erosion depth as expected after a singular dischargeevent is given (Table 7-4). The development of the erosion process can be seen in Figure 7-5.

Table 7-4: Computed results method A and B (for explanation methods see Table 7-3)

S1 S2 S3 S4 S5 S6y [m]method A

1.19e-5 9.99e-5 1.44e-3 5.58e-3 10.6e-3 10.6e-3

y [m]method B

1.37e-6 1.85e-4 6.75e-3 21.1e-3 32.0e-3 42.3e-3

Figure 7-5: Computed development of the erosion depth due to the wave events of a 6 hours storm of 1 singularspecific discharge (blue line: using characteristic values for wave distribution, velocity and overtopping time and cyanline: using formulas of Bosman and Van der Meer).

In Figure 7-5 method A (blue line) is made using characteristic values and method B (cyanline) is made using formulas of Bosman and Van der Meer. The expected erosion depthcalculated with method A differs somewhat from the original predictions. These dissimilarities

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are a consequence of extracting r0 in the formula to calculate Esoil (for the exact modificationis referred to the paper (Verheij and Hoffmans 2008)). Because the dissimilarities are smallthey will be neglected.Both methods give erosion depths in the order of centimetres (see Table 7-4). This kind oferosion depths are imperceptible depths on a uneven and irregular grass layer. Thereforethese values could be interpreted as zero erosion or only visible wearing of the grass layer.The dissimilarities between the two methods are not neglectable. Comparing the results ofthe two methods it can be concluded that application of Bosman formulas and the Van derMeer formula give higher erosion rates up to a factor 4.

7.4.2. Introducing depth dependencyImplementation of a depth variable strength profile (Method C; for equations see Table 7-3)gives the following results. (The RAR profile as measured on the section Kattendijke has beenused.)

Table 7-5: Computed results method B and C (for explanation methods see Table 7-3)

S1 S2 S3 S4 S5 S6 y [m] method B 1.37*10-5 1.86*10-4 6.94*10-3 28.0*10-3 60.0*10-3 0.102 y [m] method C 2.03*10-6 8.28*10-4 45.5*10-3 8.19 13.9 18.5

In Table 7-5 method C is a situation with a depth variable profile and method B is the sameas in Table 7-4, only here the table gives the summation of the erosion depths. This is donebecause the experiments are carried out using an increasing specific discharge on each testsection. The erosion depth computed after S6 (q =75 l/s m-1) is the result of S1 to S6.Because ‘nothing’ happened on the slope, the results can not be used for validation, howeverimplementation of a depth variable profile (method C) gives worse results than a method witha constant Esoil value of 1*107 [m/s] (method B).

Reason discrepancyThe reason for the discrepancy between both methods is the discrepancy in the values forEsoil and Uc. At the surface the depth variable profile gives a value for Esoil of 8.2 *105 [m/s]and a critical velocity of 5.62 [m/s]. The constant profile has been obtained assuming Esoil =1*107 [m/s] and Uc =4.5 [m/s]. This means for the depth variable strength profile initiation oferosion will occur at a higher velocity, but when erosion occurs it will develop more rapid. Thedevelopment of the soil parameter with respect to depth is shown in Figure 7-6.

Figure 7-6: Computed development of soil parameter with respect to depth; method C uses a depth variable profileand method B uses a constant strength profile

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Review strength profileBecause even the results obtained with method B (see Table 7-5) are too high compared tothe experimental results, first of all the strength profile is reviewed. Delft Cluster looked aftererosion of grass and clay layers (Vroeg et al. 2002). In their work they mentioned CE values(=1/Esoil), which is an overall strength parameter. According to Delft Cluster 2002,characteristic values for CE are the following:

Table 7-6: Characteristic CE values grass, clay and sand (Vroeg et al. 2002)

Soil CE valuegood grass(upper boundary)

0.5*10-6

bad grass(lower boundary)

3.5*10-6

good clay(upper boundary)

0.5*10-4

bad clay(lower boundary)

3.5*10-4

sand 12.5*10-4

These values are defined for the surface. Whether these values are also representative fordeeper laying layers is unknown. Therefore it is recommended to do some further research todefine strength characteristics for deeper laying layers. Assuming that CE values of deeperlaying soil layers should be in the order of the same kind of soil at the surface, a new CE

profiles is calibrated. The constant depth profile has been obtained assuming Esoil = 1*107

[m/s] so CE =0.1*10-6 [m-1s-1] (According to “old” calculations for Esoil this would be 1.5*10-6

[m-1s-1]). Figure 7-7 shows the CE profile of the variable strength profile with respect todepth.

Figure 7-7: CE profile of the depth variable strength profile

Where CE at d =0 is 1.2*10-6 [m-1 s-1]. Although this is in the range of grass, this is not closeto the upper boundary which can be expected according to the classification ‘good grass’.

The critical shear stress ( c) is a function of the soil parameter CE. Determination of thecritical shear stress is done according to the cubic turf model (Equation 3.16). In this cubicturf model the assumption has been made of cracks on the four side walls and only accounts

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the grass reinforcement at the underside of the clay aggregate. Other assumptions made areincluded in the formula for the root and clay cohesion. Whether the assumption of cracks atthe sidewalls results in an underestimation or the root and clay cohesion itself isunderestimated should be evaluated in further research. To obtain a strength profile whichcorresponds better with the CE values of Table 7-6, new values for strength characteristics inthe formula for the root and clay cohesion are calibrated.

Root cohesionOn the surface the root cohesion dominates the strength of the grass layer. Because thegrass cover has been given the classification ‘good grass’ the intention is to obtain a CE atd =0 m which is close to the upper boundary value of grass layers.

The following values are assumed for the averaged root diameter and the averaged roottensile strength: dr =0.13 mm and tr =20*106 N/m2. These values are considerable lowerthan the values measured by Young (Young 2005). The assumption for the characteristicvalue of the critical root tensile strength (tr) is redefined.

Figure 7-8: CE profile with tr =20*106 [N/m2] and a modified profile with tr =45*106 [N/m2]

Assuming a value of 45*106 [N/m2] for the root tensile strength results in a value of 0.54*10-6

[m-1 s-1] for CE at d =0 m, which is close to the upper boundary value. Application of thisvalue in the EPM/Hoffmans model (method C) results in an erosion depth after S6 (q =75 l/sm-1) of 73.4*10-3 m which is a reasonable result since only visual wearing of the grass layeroccurred and no erosion holes have been created. Figure 7-9 shows the sensitivity to changesof the erosion profile to changes in the value of the tensile strength.

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Figure 7-9: Computed erosion profile (method C): continues line tr =20*106 [N/m2]; dotted line tr =30*106 [N/m2]and discontinues line tr =45*106 [N/m2]

Increasing the root tension strength brings along a shift of the erosion profile. Changes in theprofile itself are neglectable. This is a consequence of the fact that close to the surface theroot cohesion dominates and in deeper laying layer the effect of the root cohesion becomeszero. Looking closer to the development of the profiles, the line according to tr =20*106 and tr

=30*106 N/m2 show a decrease of the erosion rates during S6. The line according tr =45*106

N/m2 show an increase of the erosion rates during S6. Enlargement of the number of waveevents for a case with tr =45*106 N/m2, will bring on an erosion development profile similar tothe other two (made with tr =20*106 resp. 30*106 N/m2) only shifted with respect to the x-as.

Whether it is the right choice to redefine the characteristic root tension strength should beexamined in further research. Besides that, it could be interesting to look at the need todefine characteristic values for different types of grass.

Clay cohesionIn the deeper lying layers the cohesion obtained by the clay dominates the strength of thelayer. The CE -profile (Figure 7-7) has a big jump at d =20 cm. This is exactly the lower depthlimit of the RAR profile with the consequence that at this depth the root cohesion becomeszero.Up till here the strength profile is calibrated according to results on the slope and thereforethe EPM/Hoffmans model has been used. Because clay cohesion becomes interesting in thedeeper lying layers this can not be calibrated using results on the slope (because here erosionwas zero). Calibration of strength characteristics of the clay cohesion formula will be doneaccording the TM model (method D).

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Running of the TM model with tr =45*106 N/m2 gives erosion depths of zero after S6.Assuming the air-water ratio to be zero ( =0) give better results. Running the TM modelgives the following results:

Figure 7-10: Computed erosion profile TM (method D) with tr =45*106 N/m2 and =0; left graph: erosion profile withunmodified clay parameters; right graph: modified erosion profile: continues line f =0.21 and cs =5 [m-1];discontinues line f =0.21 and cs =20 [m-1]

In the report of the experiments is mentioned that after 2 hours of S6, the cabin was movedbecause the erosion hole underneath could cause stability problems when continuing testing.After the erosion hole was filed with mine stone the test was continued. After 6 hours of S6the dimensions of erosion hole where 15 m by 4 m by 1 m (see picture Appendix E). Becausethe erosion hole has been filled during testing without measuring the depth at that time, theexact failure time of the clay layer is unknown. Nevertheless, the depth of 11 meter (leftgraph Figure 7-10) is too high. An other noteworthy fact is that one single wave event causeda erosion depth of 1.04 m This computed value can be seen in Figure 7-10, where aroundwave event number 2930 (halfway S5) the erosion development suddenly increases from 0.1m to 10 m due to a few wave events.The lines in the right graph of Figure 7-10 has been made after changing the clay cohesionparameters (f and cs). This results in a reduced erosion depth and a smaller maximumerosion impact for a single wave event ( ymax =0.06 m). All three erosion profiles show adecline of the erosion rates during S6. This decline is put down to the fact that the shearstress caused by the wave event comes close to the critical shear stress of the soil. Thecritical shear stress first diminishes and thereafter increases with respect to depth. Besidesthat, the load decreases with respect to depth due to the water in the scour hole (see Table7-3; column 3; first formula method D). At a certain depth the critical shear stress is equal tothe shear stress caused by the jet impact, and the equilibrium scour depth is being reached.To derive a value for the increase of clay cohesion with respect to depth ( cs) this equilibriumscour depth could be an interesting fact.The dotted line (Figure 7-10) is in line with the experiments and therefore the value 0.21 for fand 20 [m-1] for cs are taken. Application of these values results in the following CE profile:

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Figure 7-11: CE profile with tr =20*106 [N/m2] f =0.021 and cs =1.75 [m-1] and a modified profile (CE mod) with tr=45*106 [N/m2], f =0.21 and cs =20 [m-1]

A CE value of 0.530*10-6 is returned at d =0 and the tail (d =0.2 to d =1.0 m) has moved tothe left side of the graph. The graph of the modified CE profile looks reasonable comparing tovalues of Table 7-6 as is the maximum erosion impact of a singular wave event ( ymax =0.06m). To execute a reasonable validation for the transition model the erosion profiles for theother test sections are calculated and compared with the test results. This validation will bedone using the last mentioned CE profile (with tr =45*106 N/m2, f =0.21 and cs =20 m-1).Implementation of these values gives the following strength profiles:

Figure 7-12: Strength profiles with tr =45*106 N/m2, f =0.21 and cs =20 m-1 left: Using root density according toSprangers; right: strength profile for location Kattendijke

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Where the total strength is defined as:

total weight clay roots Equation 7.1

with: weight s w agd

Equation 7.1 is comparable to Equation 3.16; see section 3.2.1, only in Equation 7.1 thepressure fluctuation coefficient is excluded. Comparing the strength profile in Figure 7-12with the modified strength profile (see Figure 6-3) there is an increase in strength at d =0.8as well as d =0 m with a factor 3.6.

7.4.3. Validation of the Transition ModelBoonwegThe computed erosion profiles for the different test sections on the Boonweg are as follow:

Figure 7-13: Computed erosion profiles Boonweg section 1: green line; section 2: magenta line; section 3: blue lineand section 4: cyan line

There are only minor differences in the calculated erosion development profile for thedifferent sections on the Boonweg. These differences are a consequences of the variability inthe RAR-profiles. Because the root cohesion is dominant in the upper 0.2 m, small variationsin the root density causes significant variation in the critical shear stress and the soilparameter.In the field, only at section B1 and B2 a scour hole has been developed due to jet impact atthe transition. Nevertheless, the erosion developed differently than the model predicts. Areason for this discrepancy is the fact that at a depth of 10 cm a clinker road was situated.Strength characteristics ( c and Esoil) of the model are defined for grass layers and not forbricks. For a broader application of the model it could be recommended to define alsostrength characteristics for soils and (road) materials other than grass layers.

ZeelandThe computed erosion profiles for the two other locations show some more differences whencomparing these profiles to the profiles of sections of the Boonweg. Erosion profiles for theother locations, Kattendijke and St-Philipsland are given in Figure 7-14.

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107

Figure 7-14: Computed erosion profile left: St-Philipsland and right: Kattendijke (K-1)

Again, the differences can be explained by looking at the RAR-profiles. Both sections,Kattendijke and St-Philipsland, have a higher root density in the upper 0.05 m. As aconsequence the Failing Time of the upper 0.05 m is longer but the tail of the profile (0.2-1.2m) develops comparable.

In the field not all simulation cases are executed on all the dike sections. At the section in St-Philipsland test round S6 has not been executed. At the dike in Kattendijke a total test fromS1 to S6 has been executed (this test will be referred to as K-1) and also a test with only S4and S5 (ref. K-2) and a demonstration test (ref. K-D) of 20 min S6 (100 waves).Consequently the erosion profiles of K-2 and K-D show significant contrasts with the erosionprofile of K-1. In the above given erosion profile of K-1, the erosion depth drasticallyincreases just after starting S6. In the test case with only S4 and S5, this is not expected tooccur. For the demonstration test case with only 100 waves of S6, the erosion will developquite fast but because of the limited number of wave events (100 in stead of 1686) theerosion depth is expected to stay limited. The erosion profiles computed for the sections K-2(S4 & S5) and K-D (20 min S6) are depicted in Figure 7-15.

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Figure 7-15: Left: computed erosion profile K-2; right: computed erosion profile K-D

The calculated erosion profiles follow the expectation of limited erosion depth due to thelimited wave attack compared to the erosion profile as given in Figure 7-14.Comparing the experimental results with the calculations with the TM, they are again not inline. At these locations also irregularities are found in the grass layer at the transition (inKattendijke a drainage tube and a mine stone road on a sand bed and in St-Philipsland aasphalt road)

Headcut erosionAfter development of a scour depth in the order of 1 m at section K-1 headcut erosion hasbeen observed. In the Appendix E, some pictures (obtained from video material) are given,showing the development of the headcut erosion. Because, no numerical information hasbeen mentioned in the report about headcut erosion, this part of the erosion model can notbe validated.

Grass cover layers in generalCohesion obtained by the roots dominates the strength at the surface and consequently alsothe Failure Time of the upper centimetres. The root density profiles as measured for thedifferent test sections are significant higher that the averaged root density as given bySprangers (see Figure 7-4). As a possible cause for this difference is given the fact that thegrass mats on the dikes are probably older. Because for practical reasons also the strength ofjuvenile grass mats are of interest also a erosion profile is calculated according to theaveraged root density of Sprangers (see Figure 7-16).

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Figure 7-16: Computed erosion profile for a grass layer with an averaged root density as defined by Sprangers

Using the averaged root profile as defined by Sprangers results in a drastically change of theerosion profile. Here the erosion starts during S2 (1 l/s m-1) and increases drastically duringS3 (10 l/s m-1).

Summary of ResultsA numerical summary of the above given computed erosion profiles, is given in Table 7-7. Anumerical summary of experimental results from the field is depicted in Table 7-8. Becausethe focus during the experiments was not on erosion at the toe, the erosion process has notbeen monitored. Subsequently, there is hardly information about it. Nevertheless, ashortened version of the factual reports can be found in Appendix E.

Table 7-7: Summary table: prediction of erosion depths [m] computed with the TM; X means not executed

B1 B2 B3 B4 SP K-1 K-2 K-D SprangersS1 0 0 0 0 0 0 X X 0S2 0 0 0 0 0 0 X X 0.3*10-3

S3 0.8*10-3 0.8*10-3 0.8*10-3 0.8*10-3 0.8*10-3 0.8*10-3 X X 0.89S4 14*10-3 14*10-3 14*10-3 14*10-3 14*10-3 14*10-3 14*10-3 X 1.2S5 1.2 1.2 1.2 1.2 57*10-3 52*10-3 50*10-3 X 1.2

S6 1.3 1.3 1.3 1.3 1.3 1.3 Xafter 20

min:6.5*10-3

1.3

Table 7-8: Summary table of experimental results; erosion depths [m]; X means not executed

B1 B2 B3 B4 SP K-1 K-2 K-DS1 0 0 0 0 0 0 X XS2 0 0 0 0 0 0 X XS3 0 0 0 0 0 0 X XS4 0 0 0 0 0 0 0 X

S5 0.1 m 0 0 0local

erosionat spots

0

overall 0-10cm; at the

drainagetube 0.3 m

X

S6 0.3 m 0.3 m 0 0 X 1 m Xafter 20

min:0.5 m

ComparisonAs explained earlier, erosion depths in the order of a few centimetres or smaller areimperceptible depths on an uneven and irregular grass layer. Therefore, erosion depths

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predicted by the TM model in this order (Table 7-7) could be interpreted as zero erosion inthe experimental results (Table 7-8). The average root density over the upper 5 cm for thelocation Kattendijke and St-Philipsland are higher than for the sections on the Boonweg. Thisresults in a longer computed Failure Time for the upper 5 cm (as can be seen in line S5;Table 7-7). The total erosion depths computed after S6 are equal for the different sections(as can be seen in line S6; columns 1 to 6; Table 7-7). The erosion profiles (see Figure 7-13& Figure 7-14) show a decline of erosion rates at d >1 m. This indicates that the shear stresscaused by the wave tongue approaches the critical shear stress of the soil. Since theassumptions for the clay cohesion are the same for every section, the critical shear stress at adepth > 0.2 m is similar for every section. This similarity results in an similar computederosion development process below 0.2 m and an equally predicted value for the erosiondepth after S6. The computed results show some discrepancies with the experimental results:(1) The experimentally defined erosion depths after S6 for sections B1-4, SP and K are notsimilar. (2) The scour depths in the field after S6 are, except for section K-1, not in the orderof 1 m. (3) The computed depths for the sections K-2 and K-D are lower than theexperimental results. (4) The computed erosion depths for the sections on the Boonweg arehigher than the experimental depths. Because the test round S6 has not been executed atsection SP, the values in line S5 should be compared. As can be seen in Table 7-7 and Table7-8, these values correspond quite well.

In summary, computed results are not in line with results of the experiments, at least not foreach section. This discrepancy can have various possible causes. First of all, the irregularitiesat the transitions are not implemented in the model. A clinker road (at the Boonweg) or adrainage tube 10 cm underneath the surface (at Kattendijke) has influence on rootdevelopment. Secondly, there are reasons to doubt whether grass quality at the transitions isequal to grass quality on the slopes due to drainage issues. Third, introduction of a depthvariable strength profile introduces some assumptions: 1) values of strength parameters aredefined for the surface and it is unknown whether these values (for example CE values asderived by Delft Cluster (Vroeg et al. 2002)) hold for a certain depth and 2) it is unknownwhether assumptions made in the cubic turf model (Hoffmans 2008b)) hold for a certaindepth. Besides that, velocities and overtopping times as calculated with the formulas ofBosman should be applied with a certain care since they have not actually been measured onthe slopes.

7.5. ConclusionThe TM is based on formulas as used in the EPM/Hoffmans model. Alterations andmodifications are suggested to the EPM/Hoffmans model to predict erosion at the transition.To analyze the impact of these alterations and modifications, they are introduced in steps.Four different computation methods (see Table 7-3) have been applied and results arecompared internally. Comparing the results of methods A and B (section 7.4.1; Table 7-4 &Figure 7-5) shows that application of Bosman formulas and the Van der Meer formula for thedistribution of overtopping waves instead of characteristic values, give higher erosion rates upto a factor 4. Introducing a depth variable strength profile, by method C (section 7.4.2) gaveinitially worse results than method B, which assumes a constant strength profile (see Table7-5). After review of the strength profile, the following recommendations are done: (1) theroot tensile strength, tr is 45*106 [kN/m2] instead of 20*106 [kN/m2], (2) the clay cohesionfactor, f is 0.21 [-] instead of 0.021 [-] and (3) the increase in clay cohesion with respect todepth, cs is 20 [m-1] instead of 1.75 [m-1]. When the computed results are compared withthe experimental results from the field (Table 7-7 & Table 7-8) discrepancies appear. Areason for these discrepancies can be found in the fact that grass at a transition is differentfrom grass on the slope. The Transition Model as described can not be calibrated andvalidated sufficiently using available test results. This is mainly because the focus during theexperiments was not at the toe and the irregularities found at the transitions are not takenover in the model. The model in his present state is a preliminary model and interpretation ofmodel results needs cautiousness.

Conclusions and recommendations

111

8. Conclusions and recommendations

8.1 Conclusions

8.2 Recommendations

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8 Conclusions and recommendations

As introduced in section 2.4: ‘Wave overtopping’, experimental results show that nowadaysused guidelines for wave overtopping could be stern restrictions. In order to asses thestrength of a dike, the grass cover layer at the transition of the slope to a horizontal part ofthe inner slope of a sea dike turned out to be a most vulnerable location. Therefore, anerosion model for grassed inner slope transitions due to wave overtopping is developed inthis thesis. This is achieved by deriving a load model of an impinging jet and defining a depthvariable strength profile for the grass and the underlying clay layers. The most importantconclusions and recommendations for further research following from this study arepresented in this chapter.

8.1. ConclusionsIn this subsection, first the erosion process is described in theory. Afterwards, theexperimental results are compared with the theoretical model.

Erosion processAllowing wave overtopping on a Dutch sea dike introduces two failure mechanisms onthe grass cover layer of the inner slope of the sea dike. These mechanisms areknown as surface erosion and headcut erosion. Due to surface erosion a scour holedevelops. After the scour hole has reached a certain depth, it increases in upstreamdirection due to headcut erosion. The erosion development process of the grassedinner slope transition can be schematized in three stages:1. Development of the scour hole due to surface erosion;2. Transition from surface erosion to domination of headcut erosion;3. Erosion of the inner slope due to headcut erosion.

The transition model (TM), is derived to predict the development of the scour holedue to surface erosion. The TM does not focus on erosion on the slope, but at thetransition from slope to horizontal part. The SSEA (Sites Spillway Erosion Analysis)model is used in this thesis to predict headcut erosion. Below the two models areexplained:

1. The equation for the TM reads:

20 ( ) ( )

ˆ ( )c

soil

d ddydt E d

Equation 8.1

With in the boxes below the various terms in the equation.Load term:

0(1.5 5 )r2

01 10.016* *2 1w U

0 0( ) wdd e

with: r0 =0.2 [-]; w =1000 [kg/m3]; =0 [-]; U =overtopping velocity [m/s]; w =0.25 [-] andd =depth [m].


113

Strength terms:( ) ( )c s w a totald gd d

with:,0

( )

( (1 * )) 0.7 * * ( )

total clay roots

rclay cs r

d f d dAf d t dA

2

2 &*

soil asoil E E

E c

g dE CC U

0

0

( )totalc a

w

dU gdr

with0

20 00.29 ( 0.055 1.21)c with and c

with recommended values:=1/18 [-]; s =2000 [kg/m3]; w =1000 [kg/m3]; g =9.81 [m/s2]; f =0.21 [-]; clay,0 =5

[kN/m2]; cs =20 [m-1]; Ar/A =RAR =Root Area Ratio [-]; tr =45*106 [kN/m2]; soil =1;E =1*10-10 [-]; da =0.004 [m]; =1*10-6[m2/s]; r0 =0.2 [-]; =(( s – w)/ w) [-] and

d =depth [m].

2. The equation for the SSEA model reads:


Equation 8.2

With in the box below the various terms in the equation.0.79ln( ) 3.04headcut hC k

13( )L qH

0.33

0.5 3.23189 expln(101 )hc

h

L kk

with: q =specific discharge [m3/s m-1]; H =0.5 to 1.0 [m] and kh =0.03 [-].

ExperimentsIn February to April 2008 SBW experiments with respect to grassed inner slopes have beenconducted in Friesland and Zeeland. Since the focus during these SBW experiments was notat the toe, the models can not be calibrated and validated sufficiently. Thereforeinterpretation of the conclusions needs certain cautiousness. Nevertheless the followingconclusions can be drawn by comparing the model and the field results:

Implementation of the Van der Meer formula for the distribution of overtoppingwaves and the Bosman formulas in as well the TM as the EPM/Hoffmans modelresulted in higher erosion rates compared to using characteristic values.Erosion rates calculated with the EPM/Hoffmans model were too high. Therefore itshould be concluded that strength assumptions of the grass layer need to bereconsidered. This resulted in a recommended value for the root tensile strength (tr)of 45*106 [N/m2] in stead of 20*106 [N/m2].Erosion rates calculated with the TM were too high. Therefore it can be concludedthat the strength assumptions for the clay layer should also be reconsidered. Thisresulted in a recommended value for the clay factor (f) of 0.21 [-] in stead of 0.021[-] and an increase factor for clay cohesion with respect to depth ( cs) of 20 [m-1] in

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114

stead of 1.75 [m-1]. The obtained strength profile with respect to depth according tothese values can be seen in Figure 7-12.

A lot of irregularities have been found at the transitions (for example a clinker roadand drainage box 10 cm underneath the surface). Because these irregularities willhave their consequences on root development, it can be doubted whether theassumption holds that strength characteristics of grass at a transition do not differfrom grass on the slope.

According to the defined erosion model for transitions, the present guidelines are stern andthe actual strength of a grassed inner slope is underrated. Nevertheless there are still al lot ofuncertainties about the erosion resistance of a grass layer and further research is needed.

8.2. RecommendationsThe following recommendations are given for further research:

ExperimentsBecause the focus during the experiments of February to April 2008 was not on erosion at thetoe, the erosion process has been monitored insufficiently. Subsequently, there is hardlyinformation about it. In order to further improve of the erosion model, it is important thatduring the coming experiments, erosion at the toe is reported as well. This should includevisual as well as numerical data.

Strength profileThe critical shear stress is determined by the cubic turf model. Since the model results wherenot in line with the experimental results, some strength characteristic are redefined.Nevertheless, assumptions made in the cubic turf model can also be doubted. For instancethe assumption to only account the grass reinforcement at the underside of the clayaggregate. It should be checked whether assumptions made in the cubic turf model aredisputable or the strength characteristics were defined inaccurate.

Calibrating the strength profile with CE values (as derived by Delft Cluster) resulted in astronger strength profile. These CE values are defined for the surface and therefore probablynot representative for deeper laying layers. It is recommended to do some further research todefine strength characteristics for deeper laying layers.

Grass strength characteristics at transitions are assumed not to differ from strengthcharacteristics on the slope. Because many irregularities are found at transitions justunderneath the surface, it can be doubted whether this assumption holds. Furthermore,drainage issues could have consequences on root development and erosion resistance of thegrass layer. Defining RAR profiles for both the slope and the transition could help to rule outthis doubt. To include the irregularities in the model, strength characteristics for soils and(road) materials other than grass layers should be defined. Besides that, it could beinteresting to look whether there is a need to define grass strength characteristic for differenttypes of grass.

The Root density profiles obtained from the different test sections are considerably higherthan the averaged root density of Dutch grass according to Sprangers. A reason for thisdiscrepancy could be the difference in age of the grass mats. Because the ‘measured’ rootprofiles provides a higher number of roots and therefore higher root cohesion, there is ahigher erosion resistance of the grass layer. Further research is necessarily to investigate thediscrepancy to confirm the assumption or to provide with other explanations.

Load coefficientA load coefficient of 0.016 follows from the theory of Beltaos (see section 5.1.1). Since thisvalue has been obtained from figures, there is a certain error in this value. Besides that, justone experimental result is mentioned for an impinging angle of 20°. It is recommended to


115

do further research to the shear stresses applied by the water tongue at the transition. Thispart of the model could be validated separately by doing experiments on bed material withknown erosion characteristics like small rocks.

Hydraulic conditionsThe number of waves and their volumes are defined to simulate a typical storm with thefollowing conditions: Hs =2.0 m; Tp =5.7 s; Tstorm =6 hours and outer_slope =1:4. Furthermore,the distribution of the waves over time is assumed. These conditions have an impact on theloads on the dike. Assessment of the dike sections should be done according to the appearinghydraulic conditions. Therefore, hydraulic boundary conditions should be derived for everydike section separately and research into the distribution of overtopping waves over timeshould be conducted.

Economical and safety issuesAllowing more overtopping means reducing the effective freeboard. It consequently providesa higher demand for water management in the area. The overtopping water needs to bedrained and intrusion of the salt water could have a large impact on land utilization. Besidesthat, the safety of people on the dike needs to be regarded. An interesting question iswhether dike managers should be able to check the state of the dikes under allcircumstances. Research studies related peak volumes with ‘safe‘ limits for people. Thisresulted in a guideline of a maximum wave volume of 1000 – 2000 l/m for trained and safety-equipped staff (EurOtop 2007). This value is already overrated by the maximum wave eventof a 10 l/s m-1 specific discharge condition. Therefore, besides technical research, moreresearch is needed concerning economical and safety issues. Examples of studies concerningallowing more wave overtopping are:

A cost-benefit analysis to define construction cost, maintenance cost and spaceutilization costs for both options (either raising the dikes or allowing overtopping onthe current dikes).A study into the influence of salt overtopping water on the behaviour and erosionresistance of the grass mat.A study into the drainage requirements of the overtopping water and the influence ofthe salt water on nature.The possibility to conduct reparation works during storm conditions and theavailability of reparation material.

River dikesBesides allowing overtopping on sea dikes, allowing overtopping on river dikes could also beinteresting. Because the hydraulic conditions at and structure of river dikes are totallydifferent from sea dikes, different failure mechanisms could be of influence. To assesswhether the present river dikes can cope with increased wave overtopping this is also aninteresting point for further research.

Application of the modelA lot of different research projects with respect to overtopping are carried out simultaneously.This had had its consequence of application of superseded formula. Besides that, by definingseveral variables and dependencies to be applied in the model, several assumptions havebeen used. The model in his present state is a preliminary model, a kind of –release, andshould be applied with certain cautiousness.

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References

Akkerman, G. J., Gerven, V. K. A. J., Schaap, H. A., and Meer, v. d., J.W. (2007)."Workpackage 3: Development of alternative overtopping-resistant sea defences,Phase 3: wave overtopping erosion tests at Groningen sea dyke." RWS, Delft.

Bakker, J. J., R.J.C., M., and Steendam, G. J. (2008a). "Factual Report; Wave overtoppingexperiments Friesland, NL (Golfoverslagproeven Friese Waddenzeedijk written inDutch)." Infram.

Bakker, J. J., R.J.C., M., and Steendam, G. J. (2008b). "Factual Report; Wave overtoppingexperiments Zeeland, NL (Golfoverslagproeven Zeeuwse zeedijken written in Dutch)."Infram.

Battjes, J. A. (2001). "Short waves (Korte golven; written in Dutch)," lecture notes, TUDelft.Beltaos, S. (1976). "Oblique Impingement of Plane Turbulent Jets." Journal of Hydraulic

Engineering, 14(1), pp. 17-36.Bos, W. v. d. (2006). "Erosionresistance of a grass revetments during wave overtopping

(Erosiebestendigheid van grasbekleding tijdens golfoverslag; written in Dutch)," MScthesis, TUDelft, Delft.

Bosman, G. (2007). "Velocity and flow depth variations during wave overtopping," MSc thesisTUDelft, Delft.

Canepa, S., and Hager, W. H. (2004). "Effect of Jet Air Content on Plunge Pool Scour."Journal of Hydraulic Engineering, 129 (5)(5), 358-365.

Diermanse, F., Weerts, A. H., Wenneker, I., and Groeneweg, J. (2005). "Analyse ofwavestatistics on deep water ", WL | delft hydraulics, Delft.

EurOtop. (2007). "Wave Overtopping of Sea Defences and Related Structures: AssessmentManual." EA/ENW/KFKI.

Hoffmans, G. J. C. M. (2008a). "Closure problem in jet scour." Ministry of Transport, publicWorks and Water Management; Road and Hydraulic Engineering Institute, Delft.

Hoffmans, G. J. C. M. (2008b). "Erosion/scour of foreland ", draft report.Hoffmans, G. J. C. M., Akkerman, G. J., Verheij, H. J., and Hoven, A. v. (2008a). "The

erodibility of grassed inner dike slopes against wave overtopping." Deltares, Delft.Hoffmans, G. J. C. M., Verheij, H. J., and Lindenberg, J. (2008b). "Erodibility of Clay and

Grass." Deltares, Delft.Husrin, S. (2007). "Laboratory Experiments on the Erosion of Clay." Floodsite report.Meer, J. W. v. d., Verheij, H. J., Lindenberg, J., and Hoven, A. v. (2007). "Wave overtopping

and strenght of inner slopes of dikes, prediction report SBW project (Golfoverslag ensterkte binnentaluds van dijken. Rapport predictiespoor SBW written in Dutch)."RWS.

Mirtskhoulava, T. E. (1991). "Scouring by flowing water of cohesive and noncohesive beds."Journal of Hydraulic Research, 29(3)(3), 341-354.

Rajaratnam, N. (1967). Turbulent jets Elsevier, Amsterdam.Schiereck, G. J. (2001). Introduction to Bed, bank and shore protection, DUP Blue Print, Delft.Sprangers, J. T. C. M. (1989). "Vegetation on Dutch Sea dikes (Vegetatie van Nederlandse

zeedijken, plantgemeenschappen in relatie tot standplaatsfactoren; written inDutch)." RWS, Wageningen.

Sprangers, J. T. C. M. (1999). "Vegetation dynamics and erosion resistance of sea dykegrassland," PhD thesis, Wageningen Agricultural University Wageningen.

Stanczak, G. (2007). "Laboratory Tests on the Erosion of Clay Revetment of Sea Dike withand without a Grass Cover Induced by Breaking Wave Impact." Floodsite report.

Steendam, G. J., Hoven, A. v., Vries, W. d., Meer, J. W. v. d., Raat, G. d., and Frissel, J. Y.(2008). "Influence of management and maintenance on erosive impact of waveovertoppiong on grass covered slopes of dikes; Tests." FLOODRISK 2008, Flood riskmanagement: research and practice; proceedings of the European conference, ed.,CRC Press / Balkema, Oxford, UK, 30 September - 2 October 2008.

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Stein, O. R., Julien, P. Y., and Alonso, C. V. (1993). "Mechanics of jet scour downstream of aheadcut." Journal of Hydraulic Research, 31(6)(6), 723-737.

TAW. (1996a). "Flooding risk The Netherlands (written in English)." RWS.TAW. (1996b). "Technical Report: Clay for Dikes (written in English)." RWS.TAW. (1997). "Technical Report: Erosion resistance of grassland as dike covering (written in

English)." RWS.TAW. (1998). "Fundamentals on Water Defences (English translation of the Dutch Guidelines '

Grondslagen voor Waterkeren')." RWS.TAW. (1999). "Grass cover as a dike revetment (English translation of TAW brochure '

Grasmat als dijkbekleding')." RWS.TAW. (2002). "Technical Report Wave Run-up and Wave Overtopping at Dikes." RWS, Delft.USDA. (2001). "National Engineering handbook, part 628 dams, chapter 52: Field procedures

guide for the headcut erodibility index." US Department of Agriculture.Verheij, H. J., and Hoffmans, G. J. C. M. (2008). "Golfoverslag en Sterkte Grasbekkleding;

Modelaanpassing Boonweg, graserosiemodel en predictie graserosie." Q4471,Deltares.

Visser, M. M. d. (2007). "A Clay Layer as a Revetment for Sea Dikes The Behaviour of Clayunder Wave Loading," MSc thesis, TUDelft, Delft.

Vroeg, J. H. d., Gent, M. R. A. v., and Kruse, G. A. M. (2002). "Processes related to breachingof dikes: erosion due to overtopping and overflow." Delft Cluster, 2002.

VTV2006. (2007). "Guidelines safety assessment for primary sea defences ", Ministry ofTransport, Public Works and Water Management.

Young, M. J. (2005). "Wave Overtopping and Grass Cover Layer Failure on the Inner Slope ofDikes," MSc thesis, IHE, Delft.

List of parameters

119

List of parameters

a Root quantity at a depth of 50 cm [-]

A Surface area of the ground [m2]

Ar Root area [m2]b Root quantity at surface [-]

b0 Initial jet thickness [m]

B Soil bulk density [kg/m3]

c Coefficient =1.3*106 [m2/s]

cr Root soil cohesion [N/m2]

cs Cohesion of the soil [N/m2]

C Chezy coefficient [m1/2/s]C Coefficient [-]

CE Overall strength parameter [m-1s-1]

Cheadcut Material headcut coefficient [s2]d Depth to failure surface [m]d Particle diameter [m]

dr Root thickness [m]dx/dt headcut velocity [ft/hr] or [m/h]

D90* Dimensionless particle diameter [-]E Erosion rate per unit area [kg/s m-2 ]

Esoil Soil erosion parameter [m/s]f Clay factor [-]

fc Critical friction factor [-]F Hydrostatic forces [N/m]

Fd Densimetric particle Froude number [-]F Air content-dependent densimetric particle Froude number [-]

g Acceleration of gravity [m/s2]

g' Reduced gravitational acceleration [m/s2]G Weight of water [N/m]h Depth of the overtopping water layer [m]

hf Fall height [m]

ht Tail water depth [m]

hw Water level in the tube [m]H Headcut height [ft]

Hm0 Significant wave height (based on spectrum) [m]

Hs Significant wave height (average height of 33% highest waves) [m]J distance along the centreline [m]

Jp Length of potential core jet [m]

kd,g,p Empirical detachability coefficient grass layer [cm3/J]

kd,p Empirical detachability coefficient [cm3/J]

kh Headcut erodibility index [-]

L Hydraulic load [ft3/s]

Lc Critical hydraulic load [ft3/s]

m0 Area of wave spectrum [m2]M Momentum flux [N/m]

M Sediment coefficient (0.00001-0.0005) [kg/s m-2]

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120

M0 Momentum flux [N/m]MSL Mean Sea Level [m]

Nov Number of overtopping waves [-]

Nw Number of waves [-]

pmax Maximum impact pressure [Pa]

pw Soil pore water pressure [N/m2]

Pov Probability of an overtopping event [-]

Pv Probability of an overtopping volume [-]

q Average wave overtopping discharge per meter [m3/s m-1] or [l/s m-1]

Q Water discharge [m3/s]

Qa Air discharge [m3/s]r Degree of root decrease [-]

r0 Depth averaged relative turbulence intensity [-]R Resultant or dynamic force exerted by the jet on the bed [N/m]R Resistance [var]

Rc Crest freeboard relative to SSL [m]

Rd Volume of eroded soil after a single impact [cm3]Re* Particle Reynolds-number [-]

Rh Hydraulic radius [m]RAR Root Area Ratio [-]RVR Root Volume Ratio [-]

S Sediment transport [m3/s]S Solicitation [var]SSL Storm Surge Level [-]t Time [s]

tr Root tensile stress [N/m2]T Failure Time [h]

Tm Mean period [s]

Tm-1,0 Spectral period [s]

Tovt Overtopping time [s]

Tp Peak period [s]u Overtopping velocity [m/s]

u*c Critical bed shear velocity [m/s]U Diffused jet velocity [m/s]

Um Maximum diffused jet velocity [m/s]

U0 Depth-averaged flow velocity [m/s]

Uc Depth-averaged critical flow velocity [m/s]

V Volume of an overtopping event [m3/m]

Vr Volume of roots [m3]

Vss Volume of soil sample [m3]w Tail water damping coefficient [-]x Distance in the direction of the jet [m]

y Root density [g/dm3] or [m/dm3]y Scour depth [m]

z2% Run-up height exceeded by 2% of the waves [m]

ZM Dimensionless scour depth [-]

List of parameters

121

Dike inner slope angle [-]Coefficient [-]

cs Clay cohesion increase over depth [m-1]Coefficient [-]Upstream scour angle [-]Downstream scour angle [-]Coefficient [-]

eff Submerged soil unit weight [kN/m3]Angle between R and the horizontal [-]

Relative density ( =( s - w )/ w ) [-]t Impact duration [s]

Air-water ratio [-]

Amount of eroded soil per single impact [m3]Root angle of shear rotation [-]

m Characteristic length scale [m]

Kinematic viscosity [m2/s]

0 Breaker parameter [-]

s Density of sediment [kg/m3]

w Density of water [kg/m3]

Soil normal stress [N/m2]

g,min Normal component of the minimum grass tensile stress [N/m2]

roots Strength obtained by the roots [N/m2]

Shear stress [N/m2]

0 Bed shear stress [N/m2]

c Critical bed shear stress [N/m2]

clay Strength obtained by the clay [N/m2]Internal friction angle [-]

eff Effective internal friction angle [-]Jet impact angle [-]Mobility parameter [-]

c Shields parameter (critical mobility parameter) [-]Turbulence coefficient [-]

~ proportional to

identical to

Appendixes

123

Appendixes

A Husrin; failure process of a clay layer 124

B Root characteristics 127

B.1 kd vs. Esoil 127

B.2 Root densities RAR / RVR 127

B.2.1 Sprangers 128

B.2.2 Relations 129

B.3 Root density [g/m3] 130

C Experiments 132

C.1 Wave Volumes [l/m] 132

C.2 Root density profiles 133

D Matlab scripts 134

D.1 main file 134

D.2 Calculation functions 136

E Test results 140

E.1 Boonweg 140

E.2 St. Philipsland 145

E.3 Kattendijke 146

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A. Husrin; failure process of a clay layer

This Appendix has been included to give more insight in the failure process of a clay layer.Husrin worked at the same experiments for the Floodsite project as Stanczak. Although, it isonly Husrin who described a detailed development of the failure process of a clay layer. Heobserved the development of cracks in the walls of cracks and explains erosion of blocks dueto the development of cracks. Application of this observation temps to an improveddescription of the failure mechanism of a clay layer. It also temps to define the understandingof the reinforcing mechanism due to grass roots.

The work done by Husrin and Stanczak within the Floodsite project was primary intended todescribe the effect of breaking wave impact on clay with and without cracks. Nevertheless,there observations could be adopted in a much broader field where water flows hits a claysurface. Four mechanisms causing damage of a clay cover are distinguished, explained as(Stanczak 2007):

breaking wave impact pressures directly on the slope (represented as force A in FigureA-1,b) which may lead to surface erosion;washing-out of soil particles and aggregates due to pressures acting from within thedike (force B in Figure A-1,b);water movement over the dike slope following the expansion of the water jet hittingthe slope (Forces C and D in Figure A-1,b and Figure A-1,c);effect of impact pressures acting on water-filled cracks (force E in Figure A-1,c);

Figure A-1: Mechanism causing damage due to wave impact.

In the case of clay with no significant pull-cracks, usually only surface erosion due to impactpressures (mechanism A/B) and flow induced erosion by wave run-up and run-down(mechanism C/D) occurs. The method to describe this kind of erosion is explained within thesection about Stanczak. The effect of impact pressures on clay with water-filled cracks(mechanism E; Figure A-1) is, at least according to Husrin and Stanczak, not included withinthere detachability parameter. The effect of this mechanism is described according the basicconcept of shear failure as defined by Führböter.

AppendixesHusrin; failure process of a clay layer

125

Basic concept of shear failure (extracted from: Husrin 2007)Führböter (1966): When a water-filled crack is hit by the impact, the impact pressures will be instantlydistributed on both side of the surface wall of the crack with an equivalent speed of sound (pressurepropagation) of 1485 m/s. The model calculates the forces (Fcrack) acting along both wall surface of thecrack as a result of the instantly transferred impact pressure (pmax). The shear stress force (S) acts ascounterforce against the Fcrack holding it from failure. The Fcrack can remove the soil body along theshear failure angles ( ) as it overrides critical situation (Fcrack >S).

The force acting on the wall of a crack is defined as follow:

max * *crackF p a LWhere a [m] is the depth of the crack and L [m] is the length of the crack or a*L is theeffective area [m2].

To define the maximum pressures within the crack, Husrin looked closer to the pressurepropagation and distribution in vertical direction within the crack as well as in horizontaldirection next to the gap. He explains that pressure propagation inside the water-filled crackdepends on its dimension, magnitude of impacts, and air-water mixture ratio. Unfortunately,the pressure magnitude at the surface known as the reference pressure cannot be treated asthe maximum impact pressure (pmax ) inside the water-filled crack. The increases of pressuremagnitude are quite significant. Because he has no data/model to estimate the pressuremagnitude inside the water-filled he restrains himself from giving a quantitative analyze butgives a qualitative description of failure based on his observations.

An essential point and assumption in his explanation is the existence of pressure up buildingwithin the pores in the clay layer. Therefore the general accepted idea of an impermeabilityof a clay layer needs to be reconsidered. The impermeable character of clay is inherent to thevery fine pores and the power of clay to bind the water molecules. In a strongly structuredsoil, larger pores exist in the soil, in where surface water infiltrate rapidly and a considerableamount of water can be drained of the surface. The overall conclusion is an increase inpermeability in the clay due to the soil structure. This increase means larger up building andpenetration of water and water pressure inside the clay layer.

Because of the transparent walls around his clay sample Husrin could observe a detaileddevelopment of the failure process of the clay layer, including failure due to pressure upbuilding within the vertical pores. He gives the following description of failure according to hisobservations:

After several impacts, the entrance wall of the crack starts to erode. At the same time, the crack widthis getting wider and the block walls of the clay in both side of the crack are lifted (Figure A-2, figure b)After being hit several times by the impacts, cracks were observed inside the water-filled crack walls. Atthe same time, at the surface, next to the water-filled crack, horizontal cracks are also observed as thereaction against the repeatedly impact pressures (Figure A-2, figure d). Furthermore, the cracksformation along the crack wall and at the surface becomes the weak points where the blocks of soil arefinally removed (Figure A-2, figure e).

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Figure A-2: Erosion mechanisms on water filled crack

A repetition of impacts weakens the soil gradually until erosion occurs after some time.Because the weakening of the soil start by the first impact he assumes the critical stress c tobe zero ( c =0). When a block of soil looses its cohesion due to the crack formation, it isremoved. Only if the blocks of soil are finally removed, we speak about erosion.

Subsequently, the impact of the grass roots is explained by holding the loosened blocks ofsoil together. After several more impacts events, more crack formation occur and finally theblocks of soil extricate themselves from the cohesion obtained by the roots system. Besidesthe reinforcing influence due to the root system, also the observation of outflow of water(and air), which occurred just before parts of the soil where removed, could be explained; Airleft inside the pores and cracks within the soil is pushed out by the suddenly increase inwater pressure inside the soil structure.

DiscussionBecause neither Stanczak nor Husrin quantifies the pressure inside the water-filled crack, theeffects of these impact pressures are unknown. Nevertheless, because normally cracks andpores occur in natural clay, it is quite reasonable that the effect has been included already.Therefore there is a reason to doubt whether the impact of pressures inside the cracks is notjet included in the model parameters. Nevertheless, a good quantitative description of failureis important to understand the behaviour of a soil. Therefore, the observations of Husrincontribute to the understanding of erosion of a clay layer and should not be neglected. But,because there is no need to describe the effect of each erosion mechanism separately, theeffect of pressure forces acting on the walls of a crack will be included in the model but notdefined separately.

AppendixesExperiments

127

B. Root characteristics

B.1 kd vs. Esoil

The detachability parameter kd used by Stanczak can be seen as the inverse of the erosionrate parameter Esoil used within the EPM/Hoffmans.

Table B-1: kd vs. Esoil

EPM/Hoffmans Stanczakbasic equation:

0E cE C

wheresoil

Esoil

CE

withCE ~ 1/Esoil ~ 1/Uc

2 ~ 1/ grass ~1/ r ~ 1/ RAR

basic equation:

d ck

where c =0 & ~ p max

withkd, grass~1/ RVR2

B.2 Root densities RAR / RVR

Both parameters, the RAR as well as the RVR, are used as root density parameters. Only thevalues are not alike. In this section the relation between the to parameters is analyzed.

Equations for RVR & RAR:

RVR (Root Volume Ratio) typical values between 1.58% (max 3.3% at z =2 cm) and 0%3

3

[ ] [%] *100% [ ]

r

ss

V mRVRV m

Equation B-1

WhereVr : Root volumeVss : Volume soil sample

RAR (Root Area Ratio) typical values between 0.08-0.02%2

2

[ ] [%] *100% [ ]

rA mRARA m

Equation B-2

WhereAr : Root areaA : Area of soil sample

Assumptions:Root diameter dr =0.13 mm; Aroot = /4*D2 = /4 0.000132 =1.33*10-8 m3;The root has the same length as the height of the soil sample;Root volume Vroot = /4*D2h= /4*0.000132*0.02=2.65*10-10 m3;

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B.2.1 Sprangers

Stanczak refers to measurements of Sprangers in his report see Figure B-3. Sprangers didresearch to the vertical distribution of roots, but the formula (see Figure B-3) cannot be foundin his literature.

Figure B-3: Averaged values for root density (Stanczak)

Sprangers described the change of root density y [m/dm3 and g/dm3] with depth d [cm]. Theroot density measurements described an exponential decreasing function over depth, theequation reads:

( 1.5)( ) dy a b a r Equation B-3Where:y : root densitya : is the root quantity at a depth of 50 cmb : is the root quantity at surfacer : degree of root degrease

According to his measurements, he derived averaged values for different kind of managementstrategies (A-F and ref). These values are given in Table (Sprangers 1999)). The rightcolumn represents the averaged values of the derived averages. These mean valuesrepresent the mean root density for grassland on a Dutch dike situation. Because we onlyfocus on these mean values, the different management styles are not explained in detail butcan be found in the reference literature.

Table B-2: With b is the root quantity at surface, a is the root quantity at a depth of 50 cm and r is the degree ofroot decrease.

Management style A B C D E F ref meanRoot Length b 1937 1916 2056 1790 1668 1662 2366 1914

a 154 160 187 129 195 160 175 166r 0.71 0.74 0.7 0.78 0.77 0.83 0.79 0.76

Root Weight b 7.6 8.6 7.2 6.9 8.6 10.9 8.9 8.4a 0.55 0.47 0.56 0.81 0.79 0.68 0.59 0.64r 0.67 0.72 0.66 0.72 0.76 0.75 0.78 0.72

Using the mean values the following root density profiles (length as well as weight) areobtained:


129

Figure B-4: Root length [m/dm3]

Figure B-5: Root weight [g/dm3]

B.2.2 Relations

Plotting the formula of Sprangers (Equation B-3) together with the equation of Stanczak forthe RVR (see Figure B-3), which is based on the data of Sprangers, in a figure gives thefollowing graph:

Figure B-6: Root density with respect to depth; in black: RVR [%] (equation derived by Stanczak) & in red: rootlength [m/dm3] (equation derived by Sprangers)

From the picture can be concluded that in the upper 10 cm the equations give correspondingresults. In the lower part the two equations diverge from each other. The reinforcement ofthe root system can be attributed to the fact that root system hold together the clayaggregates. In a low density root length layer, not all clay aggregates are reinforced by root

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system and the effect of the roots on the total plane diminishes. Therefore, Stanczak choiceof zero root density at d =20 cm and lower is taken over.

Multiplying the root length with the Root area, Aroot = 1.33*10-8 m3 , gives the values for theRAR (see Table 3-2: Indicative values for Dutch grass (HOFFMANS, 2008b)). Becausecalculation of the root tension strength is based in the RAR, an equation is derived for theRAR with respect to depth. The equation for the RAR is plotted together with the formula ofStanczak for the RVR in Figure and reads:

( 22.32 )0.0746 /100* drA eA

Equation B-4

Figure B-7: Root density with respect to depth; in black: RVR [%] (equation derived by Stanczak) & in red: RAR [%](equation derived by Valk)

The lines correspond very well and the following relation between the RVR and RAR isobtained:

50*RVR RAR Equation B-5

According to this relation the following table can be obtained:

Table B-3: Column 1/3/4: extracted from (Hoffmans 2008b); Column 2: as given in Figure

Number of rootsNo./m2

RVR [%] RAR [%]Ar/A

Quality grassacc. to VTV

15100 1 0.02 very poor30150 2 0.04 poor45200 3 0.06 averaged60300 4 0.08 good

B.3 Root density [g/m3]

According to the values given by Sprangers, the root density can be calculated using valuesfrom Figure B-4 and Figure B-5:

Dimensions used in the figures:3

3

mean weight [g/dm ] mean weight g=mean length [m/dm ] root length m

Equation B-6

Filling in some characteristic values (obtained from the figures at d =1.5 cm)


131

3

3

8.4 [g/dm ] g =0.00441914 [m/dm ] m

with Aroot = /4*D2= /4*0.000132 =1.33*10-8 m2 gives a root density of:

-8 2

g0.0044 [ ]m = 330 [g/m3]

1.33*10 [m ]

This gives a (mean) root density of 330 g/m3.

In addition to obtaining the relation between the RVR and the RAR, the values given bySprangers could be used to measure the root density and serve as average values for dikedesign. Working with averaged values could be interesting but nevertheless, Sprangers didonly mention mean values and no standard deviations to make probalistic calculationspossible.

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C. Experiments

C.1 Wave Volumes [l/m]

The wave events volumes for the simulation cases S1-S6 are classified. The theoreticalnumber of wave events and their volumes are given in Table C-1. The experimentally usednumbers and volumes of wave events are given in Table C-2.

Table C-1: Number [-] and wave Volumes [l/m] as used in the prediction report (Verheij and Hoffmans 2008) (usingcharacteristic wave volumes)

50 150 400 700 1000 1500 2500 3500 4500 5500 max totalS1 3 3 2 1 - - - - - - 700 9S2 54 54 9 6 3 - - - - - 1000 126S3 384 252 147 57 33 9 6 - - - 2500 888S4 - 687 325 - 206 73 25 8 - - 3500 1324S5 - - 684 234 174 138 42 42 - - 3500 1314S6 - - 684 234 174 138 42 42 - - 3500 1314

Table C-2: Number [-] and wave Volumes [l/m] as used in the steering lists (using the de Van der Meer formula forwave volume distribution)

<50 51-150

151-400

401-700

701-1000

1001-1500

1501-2500

2501-3500

3501-4500

4501-5500

max total

S1 2 2 3 1 1 - - - - - 770 9S2 46 35 31 10 3 1 - - - - 1177 126S3 - 350 229 97 39 24 9 1 - - 2674 749S4 - - 756 240 120 93 54 12 3 - 3790 1275S5 - - 699 327 180 159 114 30 12 3 5182 1524S6 - - - 936 237 225 186 63 21 18 5500 1686

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133

C.2 Root density profiles

The root density profiles for the six different test sections: four profiles for the sections on theBoonweg, Friesland (B1-B4), one for the location St-Philipsland, Zeeland (SP) and one for thelocation Kattendijke, Zeeland (K). Table C-3 shows the number of roots in a soil sample witha diameter of 3 cm.

Table C-3: Number of roots in a soil sample

section B1 B2 B3 B4 SP K0.0-2.5 cm 60 60 60 60 60 602.5-5.0 cm 40 40 50 50 60 605.0-7.5 cm 30 40 40 50 35 507.5-10.0 cm 15 35 20 27 25 3510.0-12.5 cm 17 20 20 20 18 3012.5-15.0 cm 17 17 10 13 15 1815.0-17.5 cm 15 10 10 7 15 1517.5-20.0 cm 9 7 7 4 13 15

This result in the following density profile with respect to depth given as number of roots perm2:

Figure C-8: Root density profiles with respect to depth given as number of roots per m2

Note: This graph is also given in the main report

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D. Matlab scripts

Because of the frequently repetitions in the Matlab scripts, only a shortened version of thescripts are given in this appendix. The total script is available only digitally. The main file startwith a list of all the parameters. The calculation in this main file contains several references tosub-files or function files. These function-files are given in the second section. The filesdepicted, contain the original values and wave distribution as used for computations inChapter 6.

The used script files are given by the digital version of the report on the CD-rom.

D.1 main file

Two main files are created. The first one, called main file, is used for the calculations inChapter 6: Evaluation erosion model. The file called exp_main_file is used in Chapter 7:Validation. The differences between these files are the calculation scheme and the definitionof the wave events. Calculations made for chapter 6 makes uses of random generated waveevents which are randomly distributed over time. Calculations made for chapter 7 makes useof the steering lists. Because of the random generated wave events (chapter 6), the erosionprocess between different simulations can differ somewhat. Therefore, calculation of theerosion process is repeated a few times (see repeatnumber and the fact that the calculationsare written in a loop). The main script as used for Chapter 7, does not have a loop to makethe same calculation a few times but here a loop is created to summarize the erosion depthsof the different test simulations (S1 until S6). In both files it is necessarily to choose theerosion model, so that the right function file is taken (given in section D.2). In addition tothis, in the exp_main_file; the section and the method to define the velocity/overtopping timeshould be chosen (formulaV_hvT).

% strength parametersg=9.81; % gravity constant [m/s2]da=0.004; % aggregate diameter [m]vis=1e-6; % kinematic viscosity [m2/s]r0=0.2; % turbulence coefficient [-]alfatau=1/18; % pressure fluctuation [-]factorclay= 0.021; % clay factor [-]tc0=5e3; % clay cohesion (tau clay at d=o) [N/m2]alfacs=1.75; % clay cohesion variability over depth [N/m2/d]tr= 20e6; % root tensile strength [N/m2]rhosoil=2000; % soil density [kg/m3]rhowater=1000; % water density [kg/m3]alfasoil=1; % soil constant [-]alfae=1e-10; % coefficient [-]

% application of averaged root density according to Sprangersbetaroots= 22.32; % Decrease root density over depthRAR0=0.0746/100; % RAR (at d=o)

% load parameterse=0.4; % air-water ratio [-]turb=(1.5+5*r0); % turbulence coefficient [-]w=0.25; % tail water damping coefficient [-]

% Hydraulic boundariesHs=2; % Significant wave height [m]Tp=5.7; % peak wave period [s]Tm=5.7; % mean wave period [s]slope=1/4; % outer slope angle [-]Tstorm=6*60*60; % Duration storm [sec]

% Headcut erosion parameterskh=0.03; %material characteristicH=0.8; %Headcut height [m]

AppendixesTest results

135

conv=0.3048; %conversion m-ftslope_deg=20;slope_drad=slope_deg*2*pi/360;

toolboxerror=2; %Analyze whether toolbox function available; thanchoose 1(yes; TU); else 2 (no; Deltares)

%% Calculations main_filekeuze=2 ; % choose erosion model 1 =TM / 2 =EPMy0=0;x0=0;d0=y0;t0=0;

repeatnumber=1;for i=1:repeatnumber;

if keuze ==1; %'waveovertopping' q_interface=30; [X,ttot]= waveovertopping(y0,d0,t0,q_interface);

elseif keuze==2 %'EPM' q_interface=30; [X,ttot]= EPM(y0,d0,t0,q_interface);

elseif keuze==3; %'waveoverflow' U_interface=[5 6 7 8 10]; X=waveoverflow(y0,d0,t0,U_interface);

elseif keuze ==4 %Headcut erosion q_interface=30; [X,ttot]=headcut_erosion(x0,t0,q_interface);

end;

Xe=X(1:i);Te=ttot(1:i);end;

%% Calculations exp_main_file

section=6; % Location/section number (1-4=b1-b4;5=SP;6=K & 7=Spr)model=2; % 1 =TM / 2 =EPMformulaV_hvT=1; % 1 =Bosman formula / 2 =characteristicvalues

qvec=[0.1 1 10 30 50 75 ];for i10=1:6 q=qvec(i10);

if q==0.1; y0=0; xx0=0;

else y0= Xa(end); %=0 by individual q calculation xx0=xx(end); %=0 by individual q calculation

end subplotteller=i10; d0=y0; t0=0;

%% TM or EPMif model==1

[Xa,eventnumber,Esoil_exp]=Boonwegresult(y0,t0,q,xx0,subplotteller); xx=eventnumber;

elseif model==2 [Xa,eventnumber]=exp_EPM(y0,t0,q,xx0,subplotteller); xx=eventnumber;

end

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B1Depth(i10)=Xa(end) i10=i10+1;end

%(Note: the Matlab functions “rand” and “wblpdf” are used. These functionsare not available in the standard packages of Matlab as used withinDeltares. Defining the value for the “toolboxerror” makes is possible to use1 set of randomly distributed wave data for the wave characteristics)

D.2 Calculation functions1. function [waveovertopping,ttot]=waveovertopping(y0,d0,t0,q_interface) d=(Tbos*(tau0_d(d,U)-tauc_d(d,t))/Esoil_d(d,t));

2. function [EPM,ttot]=EPM(y0,d0,t0,q_interface)d=(Tbos*((0.7*turb*U-Uc_d(d,t))^2)/Esoil_d(d,t));

%% Calculation Wave characteristics - Overtopping formulaL0=g*Tp^2/(2*pi);sop=Hs/L0;irr=slope/(sqrt(sop)); % Irribaren number/breaking parameterz2=1.65*irr*Hs; % 2% Run-Up heightq2=1000*0.067*sqrt(g*Hs^3)/sqrt(slope)*irr*exp(-4.75*z2/(Hs*irr));q1=[0.1 q2 1 10 30 50 75]; % specific discharge l/s/m;

%% collect Agolf= characteristic values for ..q l/s/mif q_interface==q(1) cq=1;elseif q_interface==q(2) cq=2;elseif q_interface==q(3) cq=3;elseif q_interface==q(4) cq=4;elseif q_interface==q(5) cq=5;elseif q_interface==q(6) cq=6;elseif q_interface==q(7) cq=7;end

A=golfA(q,cq); % golfA=[ time b Ru hbos' ubos' Tbos' ];

%% Calculation y=y0; d=d0; t=t0; n=Tstorm;

for i=1:n; x=find (A(:,1)==t); nwsize=(size(x)); nw=nwsize(1);

for i2=1:nwif x>=1;

U=A(x(i2),5); Tbos=A(x(i2),6);

if (tau0_d(d,U)-tauc_d(d,t))>0 d=(Tbos*(tau0_d(d,U)-tauc_d(d,t))/Esoil_d(d,t));

%% insert function formula for delse;

d=0;end

else d=0;


137

end; y=y+d; d=y;

if y>5.80;break;

endend

t=t+1; toterosion(i)=y; time(i)=t/3600;

if y>5.80;break;

end;end;if toterosion(end)<=0;

toterosion(1:i)=0;end; Waveovertopping / EPM = toterosion; ttot=time;

3. function [waveoverflow]=waveoverflow(y0,d0,t0,U_interface)%% Waveoverflow

for U=[U_interface] y=y0; d=d0; t=t0; n=10000; % maximale tijd

for i=1:n; t=t+1;

if tau0_d(d,U)-tauc_d(d,t)<0 time =(0:1) erosion=0+1e-9999*time

else y=y+(1*(tau0_d(d,U)-tauc_d(d,t))/Esoil_d(d,t)); d=y; erosion(i)=y; taucritical(i)=tauc_d(d,t); time(i)=t/3600;

endif y>0.80;

break;end

endend;

waveoverflow= [u;h];

4. function [headcut_erosion,ttot]=headcut_erosion(x0,t0,q_interface)

Lc=(189*(kh^(0.5))*exp(-3.23/(log(101*kh))))^(0.33);Chc=-0.79*log(kh(1))+3.04;x=x0;t=t0;

for i=1:Tstorm; % Tstorm; t=t+1; k=find (A(:,1)==t); nwsize=(size(k)); nw=nwsize(1);

for i2=1:nw;if k>=1;

hbos=A(k(i2),4);

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Ubos=A(k(i2),5); Tbos=A(k(i2),6); q_m=hbos*Ubos; q_ft=q_m/((0.3048)^2); L=(q_ft*H).^(1/3); xt=cos(slope_drad)*(Tbos/3600*Chc*(L-Lc));

else; xt=0;

end; x=x+xt;

end; erosiondistance(i)=x*0.3048; %transformation ft. to m. time(i)=t/3600;

if x*0.3048>20; %max slope length.break;

end;end;

headcut_erosion=erosiondistance; ttot=time; headcut_velocity=max(erosiondistance)/6

function [golfA_exp]=golfA_exp(q)% Implementation of the steering lists; collecting wave volumes% The steering lists are given in the excel worksheet -->delete theinformation between brackets and define the right source.

if q==0.1A= xlsread('C:\Documents and Settings\valk_ad\MyDocuments\Deltares\matlab model\stuurlijst','q01' );

elseif q==1A= xlsread('C:\Documents and Settings\valk_ad\MyDocuments\Deltares\matlab model\stuurlijst','q1' );





end

function [golfA]=golfA(q,cq) % derivation wave distributionfunction

% Implementation of randomly defined wave events

%% Overtopping formula (part2)Rc1=log(q/1000*sqrt(slope)/(sqrt(g*Hs^3)*0.067*irr))*Hs*irr/-4.75;

% Run-up height of overtopping wavePov=1*exp(-(sqrt(-log(0.02))*Rc/z2).^2); %Probability of overtoppingn=floor (Tstorm/Tm); % Number of waves in a stormNow=floor(Pov*n); % Number of overtopping wavesa=0.84*Tm*q./Pov; % Weibull distribution constantVmax=a.*(log(Now).^(4/3)); % Maximum Overtopping Volume


139

%% Calculations golfA and golfA_exp

%% Bosman constantsch=7e-3/(slope)^2; % flow depth constantcu=0.25/sin(slope); % velocity constantcT=1.02; % overtopping time constant

%% Calculation wave-distribution matrix AwblA=a(cq);b=wblrnd(wblA,0.75,Now(cq),1);p=wblpdf(b,wblA,0.75);Ru=sqrt(log(p)/log(0.02))*z2;

teller=1;for i=1:Now(cq)

if Ru(teller)>Rc(cq); hbos(teller)=ch*(Ru(teller)-Rc(cq)); ubos(teller)=cu*sqrt(g*(Ru(teller)-Rc(cq))); Tbos(teller)=cT*sqrt((Ru(teller)-Rc(cq))/Hs)*Tm;

else hbos(teller)=0;ubos(teller)=0;Tbos(teller)=0;end

teller=teller+1;end

time=ceil(Tstorm*rand(Now(cq),1));

golfA=[ time b Ru hbos' ubos' Tbos' ];

function [tau0_d]=tau0_d(d,U) % applied shear stresstau0_d=turb^2*0.016/2*rhowater*((1/(1+e))*U).^2*exp(-w*d);

function [tauc_d]=tauc_d(d,t)tc=tc0*(1+alfacs*d); % clay cohesion (d)ttotal=sroots(d,t)+factorclay*tc; % total cohesion part(d)tauc_d=alfatau*((rhosoil-rhowater)*g*da+ttotal); %total critical shearstress(d)

function [sroots]=sroots(d,t)

if RAR==UNDEFINED sroots0=0.7*tr*RAR0;

sroots=sroots0*exp(-betaroots*d); % root cohesion Sprangers(d)else

sroots=0.7*tr*RAR/100; % root cohesion DEFINED(d)end

function [Uc_d]=Uc_d(d,t) % critical velocity [m/s]Uc_d=(sqrt(tauc_d(d,t)/(0.7*rhowater)))/r0;

function [Esoil_d]=Esoil_d(d,t) % soil parameter [m/s]Esoil_d= alfasoil/(alfae*g^2*da/(vis*(Uc_d(d,t))^2));

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E. Test results

In the framework of SBW project with respect to grassed inner slopes, inner slopes on threelocations in the Netherlands have been tested. The observations are reported in two reports(Bakker et al. 2008a) and (Bakker et al. 2008b). Here, the information is depicted about theobserved erosion development of the transition.As a general note is given the fact that small waves (<150 l/m) in the beginning of theexperiments did not reach the toe of the dike because they infiltrated in the grass or claylayer.

E.1 Boonweg

At the location Boonweg 4 sections are tested. In the toe of the dike, underneath the grasslayer, at a depth of approximately 0.1 m a clinker road is present. Here the density andquality of the root system is less. This road became visual during the tests. The results of thetests are as follow:

Section 1

ResultS1 & S2:

no visual signs of erosion

Result S3: little wearingResult S4: initiation of erosion Figure E-9

Result S5: After 4 hours: grass layer erosion in the middle. A clinkerroad became visible.After 6 hours: Track development of the erosion hole

Figure E-9

Result S6: Three separated erosion holes/tracks. A lot of clickers has beeneroded.

Figure E-9

Figure E-9: Boonweg section 1; upper left: initiation after S3; upper right: situation after S5; down left: situationafter 3 hours S6; down right: situation after 6 hours S6


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Section 2

ResultS1 & S2:


Result S3: loose grass clumpsResult S4: little wearingResult S5: wearing of the grass layerResult S6: After 2 hours: At the upstream side of the clicker road,

on the right side, a erosion hole/ track developed.After resuming: The erosion track developed and anumber of clickers are eroded, but contradictory tosection 1 only at the right side of the section a trackdeveloped.

Figure E-10

Figure E-11

Figure E-10: Boonweg section 2: situation after 2 hours S6


Section 3

ResultS1 & S2:


Result S3: little wearingResult S4: wearingResult S5: ongoing wearing processResult S6: Hole in the grass layer on the slope which developed

downstream and caused erosion at the transition.After the 6 hours, it is decided to continue the test with 30minutes.

Figure E-12Figure E-13Figure E-14

Note: erosion at the slope is caused by the erosion mechanism lifting or ‘opbolling’. This is a‘unknown’ failure mechanism for a grass layer.

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Figure E-13: Boonweg section 3: erosion development from 1:30 - 1:50 hours S6

Figure E-14: Boonweg section 3; left: situation after S6; right: situation after 6:30 hours S6


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Section 4

Note: Visual tractor trails in the slope, these trails resembles a transition of the slope and thetoe.ResultS1 & S2:

little wearing

Result S3: wearing of the tractor trails Figure E-15

Result S4: ongoing wearing process but no initiation of erosionResult S5: erosion at the places of the tractor trails in the order of a few

centimetresResult S6: Erosion hole due to the failure mechanism lifting or “Opbolling”

at the place of a tractor trail.Figure E-16Figure E-17

Figure E-15: Boonweg section 4: Wearing of the tractor trail

Figure E-16: Boonweg section 4: Initiation of erosion at the tractor trail

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Figure E-17: Boonweg section 4: erosion development 5:10 - 5:50 hours S6


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E.2 St. Philipsland

At this location, directly on the transition point tractor tails are noticeable and at a distance of2 m measured from this transition point a asphalt road is situated.

ResultS1 & S2:

wearing

Result S3: increased wearing process and first signs of local erosionResult S4: Ongoing wearing processResult S5: After 2 hours: At a ‘irregularity’ on the slope initiation

of erosion. The irregularity caused a jump of thewater tongue.After 4 hours: Erosion development at the jump wenton. Furthermore, local erosion holes at the connectionbetween grass and asphalt road were initiated. Theseholes has been filled with mine stone to protect theasphalt road.After 6 hours: Erosion at the place of the irregularitywent on, leading to a big erosion hole which finallyreached the sand core underneath the clay layer. Afterthe scour hole reached the sand, the erosion processdeveloped much faster. Because of the damage,experiments with 75 l/s/m did not take place.

Figure E-18

Figure E-19

Figure E-18: St-Philipsland: erosion at connection grass – asphalt road

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Figure E-19: St-Philipsland: erosion development S6; upper left: after 2 hours S6; lower right: after 6 hours S6

E.3 Kattendijke

The horizontal part of the transition in Kattendijke is used as a maintenance road,constructed with mine stones on a sand bed. The distance between the transition point andthe mine stone road is approximately 2 meters. In comparison to the sections of theBoonweg the grass layer looks worse, mainly due to the number of mole holes(approximately 70 in the total slopes or 1/m2). The test results are as follow:

Section 1:

Result S1 &S2:


Result S3: small damage on the right sideResult S4: erosion holes; road is damaged Figure E-20Result S5: big damage roadResult S6: After 2 hours: measuring cabinet moved and

erosion hole filled with mine stone.After resuming; a few big waves eroded the fillmaterial and after 6 hours the dimensions of theerosion hole where 15 by 4 by 1 m. Headcuterosion damaged the inner slope.

Figure E-20

Figure E-21

Notes: In the toe of the dike, a drainage tube became visible which is surrounded by a gravelbox.


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Figure E-20: Kattendijke section 1: Upper left: damage after S4; upper right: situation after 2 hours S6; down:situation after 6 hours S6

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Figure E-21: Kattendijke section 1: Development headcut erosion; pictures obtained from video material

Section 2:

Note: manure injection over 4 meter of the toe and crown; directly started with S4 (30 l/s m-1).Result S4: Visual wearing at the transition and despite a protection measure

at the maintenance road; an erosion hole at this place.Result S5: Development of the erosion hole; gravel box became visible.

Because of the results of section 1 by S6, experiment stoppedFigure E-23

Mole holes until a depth of 50 cm within the clay layer

Figure E-22: manure injection

Figure E-23: Kattendijke section 2: Erosion hole at grind box; each cell of framework in the left figure is 1 by 1 m.


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Demonstration test:

Note: as demonstration, for 20 minutes a new dike section is tested with a specific dischargeof 75 l/s m-1.Result(20 min) S6:

Scour hole in the order of 0.5 m Figure E-24

.

Figure E-24: Kattendijke demonstration: Erosion depth 0.5 m

Wave overtopping - CORE

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