• TABLE OF CONTENTS
HIDE
 Front Cover
 Title Page
 Acknowledgement
 Table of Contents
 List of Figures
 List of Tables
 Abstract
 Introduction
 Literature review
 Field monitoring data
 Morphological development
 Model derivation and solution
 Application and modeling resul...
 Summary and conclusions
 Reference
 Biographical sketch














Group Title: UFLCOEL-99005
Title: Evaluation of beach and shoreface nourishment
CITATION THUMBNAILS PAGE IMAGE ZOOMABLE
Full Citation
STANDARD VIEW MARC VIEW
Permanent Link: http://ufdc.ufl.edu/UF00091074/00001
 Material Information
Title: Evaluation of beach and shoreface nourishment a case study at Torsminde, Denmark
Series Title: UFLCOEL-99005
Physical Description: xi, 126 p. : ill., maps. ;
Language: English
Creator: Grunnet, Nicholas Moulun
University of Florida -- Dept. of Civil and Coastal Engineering
Publisher: Coastal & Oceanographic Engineering Program, Dept. of Civil & Coastal Engineering, University of Florida
Place of Publication: Gainesville Fla
Publication Date: 1999
 Subjects
Subject: Beach erosion -- Denmark -- Torsminde   ( lcsh )
Coast changes -- Denmark -- North Sea Coast   ( lcsh )
Beach nourishment -- Management -- Denmark -- Torsminde   ( lcsh )
Genre: government publication (state, provincial, terriorial, dependent)   ( marcgt )
bibliography   ( marcgt )
theses   ( marcgt )
non-fiction   ( marcgt )
 Notes
Thesis: Thesis (M.S.)--University of Florida, 1999.
Bibliography: Includes bibliographical references (p. 122-126).
Statement of Responsibility: by Nicholas Moulun Grunnet.
 Record Information
Bibliographic ID: UF00091074
Volume ID: VID00001
Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
Resource Identifier: oclc - 43305752

Table of Contents
    Front Cover
        Front Cover
    Title Page
        Page i
    Acknowledgement
        Page ii
        Page iii
    Table of Contents
        Page iv
        Page v
    List of Figures
        Page vi
        Page vii
        Page viii
    List of Tables
        Page ix
    Abstract
        Page x
        Page xi
    Introduction
        Page 1
        Page 2
        Page 3
    Literature review
        Page 4
        Page 5
        Page 6
        Page 7
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        Page 17
        Page 18
    Field monitoring data
        Page 19
        Page 20
        Page 21
        Page 22
        Page 23
        Page 24
        Page 25
        Page 26
        Page 27
        Page 28
        Page 29
        Page 30
    Morphological development
        Page 31
        Page 32
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        Page 78
        Page 79
        Page 80
        Page 81
        Page 82
    Model derivation and solution
        Page 83
        Page 84
        Page 85
        Page 86
        Page 87
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        Page 89
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        Page 95
    Application and modeling results
        Page 96
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    Summary and conclusions
        Page 119
        Page 120
        Page 121
    Reference
        Page 122
        Page 123
        Page 124
        Page 125
        Page 126
    Biographical sketch
        Page 127
Full Text



UFL/COEL-99/005


EVALUATION OF BEACH AND SHOREFACE
NOURISHMENT: A CASE STUDY AT
TORSMINDE, DENMARK



by



Nicholas Moulun Grunnet




Thesis


1999














EVALUATION OF BEACH AND SHOREFACE NOURISHMENT:
A CASE STUDY AT TORSMINDE, DENMARK













By

NICHOLAS MOULUN GRUNNET


A THESIS PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
MASTER OF SCIENCE

UNIVERSITY OF FLORIDA


1999













ACKNOWLEDGMENTS


First and foremost, I wish to express my deepest gratitude and appreciation to my advisor

and supervisory committee chairman, Dr. Robert G. Dean, who provided much more than support

and guidance. His attitude towards the field of coastal engineering is truly contagious. In

possession of both zeal and passion for coastal processes, he faces the challenges with humility

and courage. With such an inspiration throughout my study, I hope to have been influenced

enough to carry on in this spirit. However, Dr. Dean soon enough made me lose the tan, I had

come to get under the Florida sun.

I would also like to thank Dr. Daniel M. Hanes and Dr. Robert J. Thieke for their

participation as supervisory committee members and for their excellent lectures. Appreciation is

extended to all.other faculty members and staff in the department whose contributions have made

the completion of this thesis possible. They include Dr. Ashish J. Mehta and Dr. Michel K. Ochi

for their inspiring lectures, Subarna B. Malakar for computer assistance, Helen Twedell for

helping in the archives and to Becky Hudson and all other secretarial personnel in the department

for their help and kindness.

Special thanks also go to the COWI Foundation for providing the financial support which

made my stay at the University of Florida possible and even more so to COWI Head of Ports,

Waterways and Coastal Engineering Department, Ole Juul Jensen, for encouragement and

continuous support.

The collaboration with the Danish Coastal Authority and in particular their willingness in

making the high quality NOURTEC data available for this study, is gratefully acknowledged.

Thanks are in order to many coastal and oceanographic engineering students, but to name

one would be to omit another. However, special gratitude is due to my friends in the department








Edward Albada, Roberto Liotta and Guillermo Simon Fernandez for their friendship and

assistance while sharing the same office. Thanks are extended to Albert E. Browder for his

patience, valuable discussions and advice at most needed times.

I am forever indebted to Mathilde for sharing this unforgettable experience of joys and

hardships with me. Her presence, both physical and moral, made the efforts less tortuous.

Whether it was me or 1200 miles of sandy Florida beaches that attracted her to Gainesville,

remains unsaid. Intensive pediatric research at Shands University Hospital also made Mathilde's

tan fade away, allowing her to better blend into the hectic environment of the Neonatal Intensive

Care Unit.

Finally, this work is dedicated to my wonderful mother and father. Their attempt in

understanding what coastal engineering research is about, is honorable. "Ask the fish, they must

know," seems to be their valuable contribution to this field.















TABLE OF CONTENTS


ACKNOW LEDGM ENTS ......................................................................................................... ii

LIST OF FIGURES .................................................................................................................. vi

LIST OF TABLES ................................................................................................................... ix

ABSTRACT .............................................................................................................................. x

CHAPTERS

1 INTRODUCTION .............................................................................................................. 1

1.1 Problem Statement ....................................................................................................... 1
1.2 Objectives and Scope................................................................................................... 2

2 LITERATURE REVIEW ......................................................................................................... 4

2.1 Previous Shoreface Nourishment Projects............................. ..................................... 6
2.2 Parameterization of Cross-Shore Sediment Transport.................................. .............. 8
2.2.1 Classification of Berms....................................................................................... 9
2.2.2 Cross-Shore Transport Direction................................... .............................. 10
2.3 Cross-Shore Sediment Transport M odels............................. ................................... 14
2.3.1 Coastal Profile M models ..................................................................................... 14
2.3.2 Other American M ethods ................................................. .......................... 18

3 FIELD M ONITORING DATA ......................................................................................... 19

3.1 Site Description .......................................................................................................... 19
3.1.1 Historical Evolution.......................................................................................... 19
3.1.2 M orphology of Field Site .................................................................................... 23
3.1.3 Nourishment Projects ....................................................................................... 23
3.1.4 Sediment Size Distribution................................................... .......................... 27
3.2 Hydrodynamic Setting................................................................................................ 28
3.2.1 W ave, Tide and Current M easurements ........................................... .......... .... 29
3.2.2 W ind Characteristics......................................................................................... 29
3.2.3 Littoral Drift ..................................................................................................... 29

4 M ORPHOLOGICAL DEVELOPM ENT ........................................................................... 31

4.1 Previous Study on Torsminde Data................................................ .......................... 32
4.1.1 M orphological Analysis .................................................. ........................... 32
4.1.2 Relative Effect of the Two Nourishment Projects............................................. 35








4.1.3 Modeling Results............................................................................................. 38
4.2 Volumetric Profile Changes ....................................................................................... 41
4.2.1 Cross-Shore Sand Transport Rate Distribution ................................... ........... 44
4.2.2 Longshore Volume Changes........................................................................... 51
4.2.3 Cumulative Volume Changes......................................................................... 56
4.3 Longshore Variability................................................................................................. 61
4.3.1 Shoreline Positions ........................................................................................... 63
4.3.2 Sediment Transport Distribution .................................................................... 63
4.4 Beach Nourishment Performance Prediction......................................................... 66
4.4.1 Longevity of Nourishment Project ............................................................... 67
4.4.2 Multiple Nourishments in the Study Area......................................... ........... .. 69
4.4.3 Effect of Distance to Groin Field.................................................................... 75
4.5 Submerged Berm Response........................................................................................ 78
4.5.1 Migration of the Berm ...................................................................................... 78
4.5.2 Breakwater and Feeder Effects................................................................... 80

5 MODEL DERIVATION AND SOLUTION.................................................................. 83

5.1 Model Derivation..................................................................................................... 83
5.1.1 Energetics Transport Model ........................................................................ 84
5.1.2 Wave-Based Model of Berm Migration ........................................... ........... ... 86
5.1.3 Methodology Limitations ................................................. .......................... 90
5.2 Numerical Modeling................................................................................................... 93
5.2.1 The QUICKEST Algorithm for Unsteady Flow................................. ........... 93
5.2.2 Stability Conditions .......................................................................................... 95

6 APPLICATION AND MODELING RESULTS.............................................. ............ 96

6.1 Application of Torsminde Data .................................................................................. 96
6.1.1 Influence of Depth on Berm Behavior............................................................ 96
6.1.2 Representative Profile Evolution.................................................................... 98
6.2 Numerical Modeling Results .................................................................................... 108
6.2.1 Estimate of Diffusion Coefficient......................................................................... 108
6.2.2 Simulation of Morphological Evolution............................................................. 109
6.2.3 Model Assessment................................................................................................ 115

7 SUMMARY AND CONCLUSIONS.................................................................................... 119

7.1 Sum m ary ......................................................................................................................... 119
7.2 C conclusions .............................................................................................................. 120

R E FE R E N C E S ............................................................................................................................ 122

BIOGRAPHICAL SKETCH ........................................................................................127















LIST OF FIGURES


Figure Page


2.1 Prediction of cross-shore movement of outer bar, from Larson and Kraus (1992)......... 11

2.2 Accretion (A) or erosion (E) beach profile condition as a function of
Uc and Ut for field data, from Ahrens and Hands (1998)............................................. 13

2.3 Velocity components calculated by NPM, LITCROSS, UNIBEST and SEDITEL
and 2-D current field calculated by SEDITEL,
from Br0ker Hedegaard et al. (1992) ........................................................... ......... 16

2.4 Cross-shore sediment transport calculated by NPM, LITCROSS, UNIBEST,
SEDITEL and WATAN3, from Broker Hedegaard et al. (1992).................................... 17

2.5 Comparison of measured and calculated coastal profiles after 4.3 hours of exposure,
from Broker Hedegaard et al. (1992) .................................................... ............... 17

3.1 NOURTEC location map, from Laustrup et al. (1997a)................................................ 20

3.2 Coastal protection measures 1978-1992, from Laustrup et al. (1997a)....................... 22

3.3 Survey lines and positions for sediment sampling, from Laustrup et al. (1997b) ........... 24

3.4 Coastal profile surveys from March to June 1993 through ........................................... 26
(a) Shoreface nourishment at survey line 21880,
(b) Beach nourishment at survey line 22130.

3.5 Variation of grain size with depth, from Laustrup et al. (1997b) ................................. 27

3.6 Wind rose for the monitoring period, from Laustrup et al. (1997a)............................ 30

4.1 Volume development in selected cells, from Laustrup et al. (1996) ........................... 33

4.2 Development of the bar system, from Laustrup et al. (1997b).................................... 34

4.3 Definition of design parameters, from Laustrup et al. (1997b) ................................... 36

4.4 Nourishment effect on the design parameters, from Laustrup et al. (1996)................ 37

4.5 Relative effect of the two nourishment methods, from Laustrup et al. (1996) ............ 38








4.6 Cross-shore transport calculated with Mike 21, from Laustrup et al. (1997b)............... 39

4.7 Longshore transport for three different bathymetries, from Laustrup et al. (1997b)....... 41

4.8 A average survey lines ................................................................................................. 43

4.9 Net cross-shore sand transport rate distributions for................................... ........... 45
(a) Pure Migration,
(b) Pure diffusion.

4.10 Net cross-shore volumetric transport past any offshore location assuming no
longshore transport gradients. Results for average profiles 21840, 21870
and 21900 from June 10, 1993 to April 12, 1993..................................... ............ ... 47

4.11 Profile evolution for average profile 21870........................................................... 49

4.12 Net cross-shore volumetric transport past any offshore location assuming no
longshore transport gradients. Results for average profiles 21720, 21770
and 21970 from June 10, 1993 to April 12, 1993...................................... ............. 50

4.13 Longshore volume change from +8 m to -12 m........................................ ............ .. 52

4.14 Longshore volume change from +8 m to -2 m......................................... ............ ... 54

4.15 Longshore volume change from -2 m to -12 m......................................... ........... ... 55

4.16 Cumulative volume change from +8 m to -12 m relative to March 1993................... 57

4.17 Cumulative volume change from +8 m to -2 m relative to March 1993..................... 59

4.18 Cumulative volume change from -2 m to -12 m relative to March 1993................... 60

4.19 Shoreline change relative to June 10, 1993 position ................................................ 62

4.20 Shoreline change relative to March 29, 1993 position .............................................. 64

4.21 Northern 5 km stretch of study area ................................... .................................. 71

4.22 Calculated planform evolution for multiple nourishment projects from 1985 to 1992... 74

4.23 Calculated planform evolution for beach nourishment located 5 km downdrift of
G roin Q ............................................................................................................................ 76

4.24 Calculated planform evolution for beach nourishment located 8 km downdrift of
G roin Q ............................................................................................................................ 77

4.25 Berm development for average profile 21900.......................................... ............ ... 79

4.26 Measured and expected breakwater effects.............................................. ........... .... 81








6.1 Dependency of convection and diffusion coefficients with depth for a range of
3 wave heights in dotted and solid lines, respectively............................ ........... ... 98

6.2 Berm cross-section evolution from June 10, 1993 to July 15, 1993........................... 100

6.3 Berm cross-section evolution from August 26, 1993 to November 22, 1993 ............ 101

6.4 Berm cross-section evolution from November 22, 1993 to January 11, 1994 .......... 102

6.5 Berm cross-section evolution from February 8, 1994 to April 12, 1994..................... 102

6.6 Hydrographic monitoring results from November 26, 1993 to January 11, 1994....... 104
(a) Wave height and period,
(b) Water level.

6.7 Hydrographic monitoring results from February 8, 1994 to April 12, 1994............ 105
(a) Wave height and period,
(b) Water level.

6.8 Initial bathymetry and profile development for average profile 21870..................... 107
(a) From November 26, 1993 to January 11, 1994,
(b) From February 8, 1994 to April 12, 1994.

6.9 Predicted and measured diffused shapes following placement of sand..................... 110

6.10 Comparison of Stokes' second order theory with SFWT for H = 2.0 m, T = 6 s
and h= 4.0 m ......................................................................................................... 112
(a) Near-bottom velocities,
(b) Velocities cubed.

6.11 Comparison of simulated and observed migration of berm in time period .............. 114
(a) November 26, 1993 to January 11, 1994,
(b) February 8, 1994 to April 12, 1994.

6.12 Prediction of direction of cross-shore berm movement............................................ 117
(a) From November 26, 1993 to January 11, 1994,
(b) From February 8, 1994 to April 12, 1994.















LIST OF TABLES


Table Page

2.1 Previous submerged berm projects, adapted from Otay (1994) ...................................... 5

3.1 Wave statistics, March 1993 April 1995, Torsminde............................................... 28














Abstract of Thesis Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Master of Science


EVALUATION OF BEACH AND SHOREFACE NOURISHMENT:
A CASE STUDY AT TORSMINDE, DENMARK

By

Nicholas Moulun Grunnet

May, 1999


Chairman: Dr. Robert G. Dean
Major Department: Coastal and Oceanographic Engineering

The morphological behavior of a beach and shoreface nourishment at Torsminde on the

west coast of Denmark has been studied through analysis of survey data and numerical modeling.

The two nourishment volumes of approximate equal size of 250,000 m3, showed a clear

difference in stability. The beach nourishment was completely eroded within a year thus not

fulfilling its design objectives, whereas most of the shoreface nourishment was still in its initial

position two years following the implementation. The relative success of the shoreface

nourishment and the subsequent morphological changes in the outer nearshore were studied in

detail.

The shoreface nourishment, consisting of coarser than native beach quality sand, was

placed on the offshore slope of the outermost breaker bar in an alongshore submerged berm.

Extensive monitoring of the morphological development of the berm and the nearshore zone

showed that the berm quickly became an integral part of the natural bar system. In the first two

years of monitoring, the Torsminde shoreface nourishment had completely fulfilled all design








objectives through its pronounced breaker berm effect and reduced feeder effect. Not only was

shoreline retreat compensated, a net seaward migration of the coastline was observed.

Volumetric analysis of cross-shore profiles was a useful means to describe the

morphological changes induced by the placement of large volumes of sand in the nearshore zone.

The natural variations of the longshore sediment transport were large compared to the changes

introduced by the nourishment volumes which made it difficult to disassociate the natural

processes re-shaping the nearshore profile from the processes caused by the implementation of

the nourishment. The monitored area was soon found to be characterized by cross-shore as well

as longshore natural variability implying that the nearshore processes are highly 3-dimensional.

With emphasis on the cross-shore redistribution of sediments after construction of the

submerged berm, a model was derived to illustrate the shoreface profile development. The

fundamental concept used was that wave asymmetry is primarily responsible for the onshore

migration of the berm. The form of the berm was observed to migrate landward at a distinct speed

while it diffused and lost height. This observed evolution in time was related to the measured

hydrodynamics and reproduced by numerical simulation of the cross-sectional change of form

and berm crest migration. Reasonable agreement with field observations was obtained.














CHAPTER 1
INTRODUCTION


1.1 Problem Statement

The beach is a dynamic system constantly changing under the action of the wave, tide,

current and wind climate. Beachface accretion/recession is an equilibrium between storage of

sediment on the foreshore and storage of sand offshore through creation of longshore bars that

also act to reduce erosive energy reaching the beach by breaking the incident waves. In the cross-

shore plane, sediment moves between the beach face and bars according to the wave and water

level conditions and the grain size of the beach material. During storms, which are characterized

by higher waves and water levels, sediment moves from the beach face to form bars, whereas

under lower waves, bars tend to move onshore. Sediment also moves alongshore in a direction

mainly controlled by the angle of the incident waves.

The positive effects of bars for promoting beach growth and protecting beaches has made

nearshore placement of sand an increasingly interesting option in the mitigation of shoreline

erosion. Bars or nearshoree berms" are constructed with the intent of either to reduce the

incoming wave energy by inducing breaking over the placed bars and/or to supply the beach with

material. If a nearshore berm is not intended to migrate shoreward, it is referred to as a "stable

berm," whereas if the berm is expected to migrate, it is referred to as an "active" or "feeder berm"

due to its potential in supplying the beach face with material. This supplemental source of

material can either result in an increase in beach volume or a reduction in the ongoing erosion.

To prevent coastal erosion, the earliest manmade changes along the coast were the

construction of hard structures such as groins, jetties and seawalls. Designed to trap sediment

moving alongshore, constrain the flow of water at tidal inlets or protect the foreshore, an








interruption of the natural bar-beach system with adverse effects on adjacent beaches was

inherently associated with these coastal protective measures. Such undesired effects coupled with

the increasing significance of the coastal areas for ecology, economy, culture and recreation, has

promoted soft structures such as beach and nearshore nourishment.

Placing the sand in the nearshore instead of directly on the beach, has been shown to

reduce costs and improve the overall cost-benefit ratio of a beach nourishment project (Laustrup

et al. 1996). Placement of sand in a bar-like alongshore submerged berm, introduces only a small

perturbation to the littoral processes in the cross-shore zone characterized by bars. The

environmental impacts to the beaches are thus also improved with nearshore placement instead of

direct beachfill because the sand moves ashore in the form of migrating sand bars instead of being

placed by construction equipment.

However, as yet no quantitative methods have been developed as useful design and

planning tools for predicting the long-term fate of nearshore berms. A primary design parameter

is the depth of placement, which in the limited number of field nearshore nourishment projects

has been based on rational choices and rules of thumb.



1.2 Objectives and Scope

In 1993, the NOURTEC project, an acronym for Innovative Nourishment Techniques

Evaluation, was launched to study and explain the behavior of shoreface nourishment projects in

different European coastal environments (Hoekstra et al. 1996, Knaack et al. 1996 and Laustrup

et al. 1996). The final objective of the NOURTEC project was to study, determine and explain the

feasibility, effectiveness and optimum design characteristics of shoreface nourishment techniques

for different environmental conditions. The project was also intended to give an improved

understanding of the morphological changes induced by placing large volumes of sand in a

coastal environment.








As part of the NOURTEC project, the Danish Coastal Authority performed a full-scale

test on beach and shoreface nourishment located at Torsminde on the west coast of Denmark. The

hydrographic and bathymetric results of the comprehensive 2-year monitoring survey program

were generously made available for the present study.

Through analysis of survey data, the morphological development of the nearshore can be

studied following the implementation of the nourishment projects. Specifically, an attempt to

disassociate the natural processes re-shaping the profile from the processes caused by the

placement of the nourishments is valuable. However, the changes introduced by the nourishment

volumes must be significant enough so as to disturb the natural system by an order of magnitude

higher than its natural variations, both in time and space.

By analyzing the relative effects and stability of each nourishment project, the better

nourishment option can be quantified in terms of stabilization of coastline, coastal protection and

widening of the beach. At the end of the monitoring period, the extent of fulfillment of these

design objectives forms the basis of a comparison of a beach and shoreface nourishment.

Previous studies have indicated that wave asymmetry can cause net onshore sediment

movement and undertow can cause net offshore movement in the surf zone. The placement of the

present berm is offshore of the outermost breaker bar, thus beyond the day-to-day surf zone,

thereby enhancing the effect of wave asymmetry as the dominant forcing condition. The

derivation of a model based on wave-asymmetry can describe such onshore berm migration.

With application of the Torsminde data, the predictive capability of the model can be

evaluated in terms of agreement between measured and simulated profile evolution.














CHAPTER 2
LITERATURE REVIEW



A well documented history of field experimentation coupled with advances in sediment

response modeling, have made nearshore placement of sand an attractive alternative to direct

placement on the beach. Early nearshore nourishment experiments failed because material was

placed too deep for mobilization by wave motion and other cross-shore flows. With the

recognition of dredged material as a resource for nourishment sand and more recent successes in

berm response with shallower placement, nearshore nourishment has proven its concept and is

now used to mitigate chronic coastal erosion.

It is of fundamental importance to be able to predict the direction of sand transport for the

design of nearshore dredged material berms intended to function as feeder berms. By studying the

response of natural longshore bars with respect to the incident wave conditions, criteria have been

developed for predicting the cross-shore direction of movement of berms placed in the nearshore.

Several criteria for onshore and offshore bar movement are readily available.

A number of European and American coastal profile evolution models have been

developed. Some numerical models are available to specifically simulate long-term berm

movement. Due to the limited knowledge of sediment transport in the coastal environment,

motivation is provided by the need for conceptually simple models to help explain the movement

of sediments under waves. Simpler ways to quickly assess sediment stability have thus also been

developed and may be sufficient in cases where detailed analysis is not needed. Whatever the

different degrees of determinism/empiricism and refinement established in the modeling of











Table 2.1: Previous submerged berm projects, adapted from Otay (1994).


Location Date Placed Water Berm Sand Wave Wave On/Off- Shore Reference
Volume Depth Relief Size Height Period Shore Protection
[m'] [m] [m] [mm] [m] [s] Motion


Santa Barbara, CA
Atlantic City, NJ
Long Branch, NJ
Durban, South Africa
Copacabana Beach, Brazil
Long Island Sound, CN
Lake Erie, OH
New River Inlet, NC
Limfjord Barriers, Denmark
Tauranga Bay, New Zealand
Dam Neck, VA
Sand Island, AL
Fire Island, NY
Jones Inlet, NY
Mobile Outer Mound, AL
Coos Bay, OR
Silver Strand, CA
Kira Beach, Australia
Mt. Maunganui, New Zealand
Port Canaveral, FL.
Perdido Key, FL.
Terschelling, the Netherlands
Torsminde, Denmark


1935 154,000
1942 2,700,000
1948 460,000
1970 2,500,000
1970 2,000,000
1974 1,170,000
1975 18,000
1976 26,750
1976 22,000
1976 2,000,000
1982 650,000
1987 350,000
1987 320,000
1987 300,000
1988 14,300,000
1988 4,000,000
1988 113,000
1988 1,500,000
1988 80,000
1990 120,000
1992 3,000,000
1993 2,100,000
1993 250,000


6.1
4.6-7.6
11.5
7-16
4-6
18.3
17
2-4
4-5
11-17
10-11
5.8
4.9
4.9
10.7-13.7
20-26
4.6-5.5
7-10
4-7
5.3-6.8
5-6
5-7
5-6


1.5 0.18 stable none Hall and Herron (1950)
0.32 stable none
2.1 0.34 1-2 7-9 stable none
0-8.3 0.35 1-2 both indirect Zwambom et al. (1970)
0.4-0.5 0.7 10-14 onshore direct Vera-Cruz (1972)
9.1 silt 0.1 stable none Bokuniewicz et al. (1977)
0.36 silt 0.1 stable none Danek et al. (1978)
1-8 0.49 0.55 7.3 onshore direct Schwartz and Musialowski (1977)
2.1 0.25-0.3 onshore direct Mikkelsen (1977)
9 stable Healy et al. (1991)
3.3 0.08 stable Hands and DeLoach (1984)
1.8-2.1 0.22 onshore direct Hands and Bradley (1990)
2 McLellan et al. (1988)
2 -
6.6 fine 0.3-0.8 3.4-4.6 stable indirect McLellan (1990)
4.6-7.6 0.25-0.3 2.7 11.5 loss none Hartman et al. (1991)
2.1 0.2 0.62 13.1 onshore direct Andrassy (1991)
2 4 8 onshore direct Smith and Jackson (1990)
2 0.5-1.5 5.9 direct Forster et al. (1994)
1.65 1.2 6.3 onshore direct Bodge (1994)
1.75 0.3 0.45 5.7 stable indirect Otay (1994)
1 0.2 1.08 7 onshore direct Hoekstra et al. (1996)
2.1 0.57 1.5 4-6 onshore direct Laustrup et al. (1996)








hydrodynamics, sediment transport and bed level evolution, these models seem to imply the

important influence of wave asymmetry in the motion of sediment.



2.1 Previous Shoreface Nourishment Projects

A limited number of field shoreface nourishment projects have been carried out to

construct bars or nearshore berms from dredged material with the intent of the placed material

serving either as a submerged breaker berm and/or to supply the beach with material. Often, the

nourishment only partly acts as a feeder berm as the actual success of the nourishment is also

based on the impoundment of longshore sediment transport due to the damping of the wave

energy leeward of the berm and the subsequent development of a salient effect. If a nearshore

berm is not intended to move, it is referred to as a stable berm, whereas if the bar is expected to

move, it is called an active or feeder berm with reference to its potential action of supplying

material to the littoral zone of the beach.

The concept of placing dredged sand where the natural cross-shore currents would move

it ashore, has a long history of mixed successes and disappointments. Periodic direct placement of

material on the beach had already proven its abilities and been used as an acceptable means for

restoring littoral drift downdrift of littoral barriers. With half a century of experience, feeder

berms are now sited carefully for maximum exposure to onshore transport processes. A summary

of previous submerged berms is presented together with their characteristics and overall stability

in Table 2.1, adapted from Otay (1994).

Early disappointments with nearshore nourishment projects were reported by the U.S.

Army Corps of Engineers (USACE). In 1935 at Santa Barbara, CA, a berm was built intended to

mitigate extensive erosion downdrift of a newly constructed harbor. After 21 months, with no

measurable movement of the berm or lessening of the coastal erosion, nearshore placement of

sand was abandoned in favor of direct placement on the dry beach. At Atlantic City, NJ, larger

volumes were placed in higher wave energy locations with hopes that they would more readily be








mobilized (Hall and Herron 1950). Monitoring for up to 10 years showed no migration of the

nourishment nor any benefits to adjacent shores. Again, experiments with nearshore nourishment

were dropped in favor of direct placement on the subaerial beach.

Despite the failure of initial nourishment projects, the idea persisted that under the right

circumstances, properly designed offshore deposits could benefit adjacent beaches. A number of

successes at Durban, South Africa (Zwamborn et al. 1970), Copacabana Beach, Brazil (Vera-

Cruz 1972) and Limfjord Barriers, Denmark (Mikkelsen 1977) were reported as successes;

extensive field monitoring programs had confirmed that the offshore berms had been of

considerable benefit to the protection of the leeward beaches. The stable berms were reported to

have reduced wave energy to a non-erosive level and active berms had induced an additional

volumetric gain of the beach material; in some cases, the increasing grain sizes in beach sediment

samples suggested an interaction with the offshore placement sites which were originally

nourished with the coarser sand.

Coupled with the reported successes overseas and expanded mechanical abilities for

precise placement in shallower water, a renewed interest led to a another series of shoreface

nourishment projects performed by the USACE. At New River, NC, placement in 2 to 4 m depths

proved that nourishment volumes could be returned to the littoral zone by shallow disposal

(Schwartz and Musialowski 1977). Off the Virginia coast, nourishment tests were carried out

with poor-quality material and constructed as stable stockpiles in 10 to 11 m water depths thus

offering non-feeder benefits (Hands and DeLoach 1984). The nourishment tests demonstrated

beneficial uses for formerly discarded beach quality material placed in deeper water.

Results from field studies at Kira Beach, Australia (Smith and Jackson 1990) and on the

Mobile Outer Mound off Dauphin Island, AL, (McLellan 1990), showed positive berm responses

in progressively deeper water. These tests proved the feasibility of constructing both stable berms

and feeder deposits as nearshore nourishment had been demonstrated with increasingly deep








placement of medium to fine sand. Also, these studies helped narrow the gap between depths

acceptable for sediment retention and depths suitable for feeding material to the nearshore zone.

In 1993, the NOURTEC program was launched to study and explain the behavior of

shoreface nourishment projects in different European coastal environments. Two of the projects

within this program, at Terschelling in the Netherlands and Torsminde in Denmark completely

satisfied the design objectives in the first 2.5 years following the implementation of the

nourishment volumes (Hoekstra et al. 1996, Laustrup et al. 1996). In Terschelling, the sediment

was supplied to the nearshore zone, filling up the trough between the middle and outer breaker

bar. In Torsminde, the sediment was placed on the offshore slope of the outermost bar. Both

nourishment projects were found to only partly act as feeder berms as approximately only a third

of the total gain of sediment in the inner nearshore could be explained by direct losses in the

nourished zone. The projects illustrated the potential breaker berm function of the combination

shoreface nourishment/nearshore breaker bar.



2.2 Parameterization of Cross-Shore Sediment Transport

The criteria reviewed in this section deal with the parameterization of cross-shore

sediment transport in a deterministic and probabilistic way. Guidelines for a suitable depth range

for placement of nourishment sand were based on observed berm response at different sites and

empirical berm stability was defined as a function of wave and sediment parameters. Detailed

analysis was made of bar movement in order to develop empirical predictive expressions for the

effective placement of nearshore berms. Such berms are placed in the form of long sandbars and

are expected to behave similarly to natural bars, both in their movement and interaction with

waves. The predictive criteria developed from field experiments appeared to have applicability to

any site where longshore bars and constructed berms are modified primarily by wave action.

A general methodology was needed to determine the minimum depth assuring stability

and the maximum depth where dispersion would occur.










2.2.1 Classification of Berms

Hands and Allison (1991) and Hands (1991) have proposed a method to help select

depths appropriate for either dispersion or retention of placed materials. This method relates

sediment stability and overall berm behavior to site specific berm and wave properties. As

previously described, sand berms are classified as either stable or active based on repeated

surveys; stable berms retain most of their original volume and remained at the placement site for

years and active berms show significant movement within a few months.

Eleven nearshore berms on the open-ocean coast of the United States were classified

according to their stability characteristics. The measured responses of the berms were compared

to depth estimates using Hallermeier's (1981) concept of coastal profile zonation providing

guidance on depths of sediment transport and profile variation for wave-dominated coasts. The

inner and outer limit depths proposed, were the seaward limit for the active zone and the

landward limit for the stable zone, respectively. The available berm data confirmed the validity of

Hallermeier's limits as excellent site selection guidelines.

To estimate the stability of deposits at any depths, including the uncertain zone between

Hallermeier's limits, a new limit criterion was introduced based on maximum near-bed

oscillations. The distribution of long-term wave-induced near-bed velocities was shown to

correctly categorize all the documented active and stable berm sites. Calculations using published

wave climatology showed consistently that sand berms should not be expected to remain stable if

the 75-percentile velocity exceeded 0.4 m/s or the 95-percentile velocity exceeded 0.7 m/s. For

the eleven nearshore nourishment sites investigated, grain size, tidal currents and wind were

assumed to not have a critical effect on the stability of the berms.

The Hands and Allison (1991) method is essentially empirical and does not attempt to

explain migration direction. However, the proposed method implies that accounting for the








influence of velocity asymmetry on sand transport is recommended for modeling the fate of

nearshore placed material.



2.2.2 Cross-Shore Transport Direction

Larson and Kraus (1989a, 1989b) presented different criteria for distinguishing between

overall beach accretion and erosion. In the present context, beach accretion implies an onshore

sand transport direction and vice-versa for beach erosion. Larson and Kraus (1992) later refined

these criteria against field data obtained by examining the behavior of the offshore bar at DUCK,

NC. Under the assumption that onshore sediment transport was associated with decreasing

volumes of bar and vice-versa for increasing volumes, more appropriate threshold values were

developed for differentiation between onshore and offshore bar movement. These criteria

included the Dean number or fall velocity parameter H I/wT, deep water wave steepness and

ratio of deep water wave height to ds5, and ratio of sediment fall velocity to gH. where

Ho is the deep water wave height, T is the wave period and w is the sediment fall velocity.

One of the successful criterion was related to the Dean number presented by Dean

(1973). The Dean number was first introduced as the ratio of sediment fall time to the wave

period to describe the on/offshore motion of sediment particles suspended and transported by

waves, thus used as a criterion for cross-shore transport formulations and for the formation of

normal and storm beach profiles. Based on the Dean number, Larson and Kraus (1992) showed

that a reasonable prediction of the cross-shore direction of bar migration could be given by the

simple criterion H, / wT = 7.2. Plotted against the deep water steepness, the prediction of cross-

shore movement of the outer bar for this combination of dimensionless parameters is displayed in

Figure 2.1 for the DUCK experiments. A reasonable separation of accretionary and erosional

events is seen to be achieved by the use of this criterion.





11


0.030
o Onshore
0.025 Offshore
Criterlon o
0.020 ---- H/wT 7.2
0.020 -

O o o0 0o
0.015 0. o. *

0 00
o o o
0 0 0
0.010

o0

0



0.005III I I I
3 4 5 6 7 8 9 10 14
H0/wT



Figure 2.1: Prediction of cross-shore movement of outer bar, from Larson and Kraus (1992).




Using the deep water wave climate time history at Torsminde, this criterion will be used

to determine the wave events with accretionary and erosional potential, respectively.

Ahrens and Hands (1998) presented a different method to estimate the tendency of

sediment to move onshore or offshore under the influence of waves in shallow water. Two

dimensionless velocity parameters were developed to predict the tendency of sediment to move

onshore under wave crests or offshore under wave troughs. The two versions of U were


respectively Uc and U,, defined as the following ratio:





U -d max,crest (2.1)
Ucrit


Ut = -d max,trough (2.2)
Ucr
crit










where u-dmax,crest and u-d max,rough are the instantaneous, maximum onshore near-bottom velocity

under a wave crest and offshore velocity under the trough respectively, and uc,,, is the critical

velocity required to initiate sediment movement.

Near-bottom maximum velocities were estimated using the Stream Function wave theory

as presented by Dean (1974) and led to the following expressions:



U-d maxcrest = (H / TXd / Lo)-579 exp [0.289-0.491(H / d)- 2.97(d / L)] (2.3)

maxrouh =-(H /T)exp[1.966 -6.70(d /Lo)- .73(H /d)+5.58(H / L)] (2.4)



where d is the water depth, H is the wave height, T is the wave period and Lo is the deep

water wave length.

Based on research by Hallermeier (1980), the critical velocity for the initiation of

sediment movement for grain sizes smaller than 2.0 mm, was calculated as follows:



Ucrit = 8 g yd5 (2.5)



where g is the acceleration of gravity, y is the specific gravity of the sediment and d50 is the

median sediment diameter.

Data from 98 beaches around the world tabulated in Kraus and Mason (1991), were used

to develop criteria to discriminate between erosional and accretionary type beach profiles.

Accretionary or erosional-type profiles are denoted in Figure 2.2 as a function of Uc and U,. U,

is seen to be a good discriminator between accretionary and erosional beach profiles at a value









around -2. Ahrens and Hands (1998) proposed using a value of U, = -1.90 as threshold level.

Interestingly, even tests with high values of U, were erosional for U, around -2.


Figure 2.2: Accretion (A) or erosion (E) beach profile condition as a function of Uc
and Ut for field data, from Ahrens and Hands (1998).


Despite the simplicity of the method, the U, parameter showed considerable skill in

predicting net cross-shore sediment movement for a large number of both laboratory and field

observations. The method assumed that sediment movement responds only to near-bottom

velocities produced by wave motion, neglecting other variables such as bottom slope and the

presence of other cross-shore flows. However, it provided valuable insight on the processes that

are important by showing a reasonable level of correlation over a wide range of observations of

sediment movement.


0 f- --..... --.. ....... C o
A\ AA Fldr Bdchi Dta
A
A .
-2 -




4I -4

WE T
4--
\t le ?r I 1-7


UW for Hb








2.3 Cross-Shore Sediment Transport Models

Cross-shore sediment transport models are usually classified in two groups, closed loop

and open loop models. Closed loop models are based on equilibrium, beach profile concepts

(Bruun 1954 and Dean 1977) and assume that a profile will eventually reach equilibrium if

exposed to the same conditions for a long time; cross-shore sediment transport is thus caused by

deviations from an equilibrium profile. Open loop models are not constrained by reaching a final

profile and the sediment transport is determined by sediment concentrations and fluid motions.

Existing transport models will be reviewed briefly hereafter.

A number of European and American cross-shore sediment transport models have been

developed to simulate morphological changes in coastal profile evolutions. However, other

American conceptually simple methods have also been developed, specifically for determining

the cross-shore transport of added sand placed on a coastal profile. These are of particular interest

when considering the fate of constructed berms in the nearshore.



2.3.1 Coastal Profile Models

Early closed loop models only required that the form of the equilibrium profile be known;

by assuming that a beach profile maintained its equilibrium form in immediate response to a

water level rise, no sediment transport relationship was required. These models were in reality

beach profile change models as no sediment transport was explicitly included. By introducing

the concept of equilibrium wave energy dissipation per unit volume of water, wave energy

dissipation related transport models such as EDUNE, Kriebel (1982), Kriebel and Dean (1985)

and SBEACH Larson and Kraus (1989) were developed. According to equilibrium beach profile

concepts, offshore transport continues until the wave energy dissipation per unit volume of water

becomes uniform over the entire surf zone. The sediment transport rates across the nearshore

active zone are thus related to the difference between the actual and equilibrium wave energy

dissipation.








With varying degrees of modeling of the hydrodynamic processes, open loop models

relate cross-shore sediment transport to the detailed physics of the flow field such as sediment

concentration, fluid velocity and bottom shear stress. Open loop sediment transport models are

usually based on one of the three following main transport relationships:

product of flow velocity and sediment concentration,

velocity powers,

product of velocity and bottom shear stress.

The first type of models relate the net time-averaged flux of sediment to the product of

flow velocity and sediment concentration. Such sediment transport models include the model

presented by Dally (1980), Dally and Dean (1984) and LITCROSS developed by the Danish

Hydraulic Institute, Br0ker Hedegaard et al. (1991).

The second type of sediment transport modeling is based on the concepts of stream flow

transport by Bagnold (1963). Bailard and Inman (1981) later applied the Bagnold model to a

coastal environment with time-varying oscillatory flow; a further description of this methodology

will be given in Chapter 5. The following three coastal profile models have been developed based

on Bailard's refined transport model: UNIBEST of Delft Hydraulics, NPM of Hydraulic Research

and SEDITEL of Laboratoire National d'Hydraulique.

The third type of transport models determines the sediment transport in relation to local

wave-current conditions and bottom shear stress. The total transport is thus a sum of contributions

from mean currents and waves. An example hereof is the coastal profile model WATAN3

developed by the University of Liverpool.

In an intercomparison of coastal profile models, Broker Hedegaard et al. (1992)

compared and evaluated five European models for short term coastal profile modeling. The five

models are LITCROSS of Danish Hydraulic Institute, UNIBEST of Delft Hydraulics, NPM of

Hydraulic Research, WATAN3 of University of Liverpool and SEDITEL of Laboratoire National

d'Hydraulique; all of which have been briefly introduced. The models were tested against






16



measured profile evolutions from a large wave flume under dune erosion conditions. The initial


profile was a plane beach with an initial slope of 1:20 and the experiment was carried out with


regular waves. In Figures 2.3 and 2.4, the vertical distribution of velocities and the cross-shore


sediment transport are presented for all models, respectively. Figure 2.5 displays the comparison


of measured and calculated coastal profiles after 4.3 hours of wave exposure. Even though large


differences are seen to exist between the coastal profile models presented, all five models


provided a reasonable agreement between the simulated and measured results. However, through


the inherent uncertainties in sediment transport modeling, the intercomparison underlines the


need for more profound understanding of the cross-shore processes.


z*


I U Ill W
I 1, I
SWI
I I


0 O ,0 I SO I 4. I -0IOz") K



DHI, Litcross
.III- 'ji-"* -







-1. 4 1 "Ol .4 .0. "..0 0.2 O.4 we Do 1.
LI-


1"/1

LNH, Seditel
"' -!E--tH
--


*I o ** *
*III

0 40 0 44 42 40 02 04 f 0.0 1.0
W-4


30
211
to


HR, NPM
... .." .'_ ......-.-'".
--1--/--I I
I xxIr


.1 4.0 4 .44 -0. -6.40 0 0.4 1.

DH, Unibest
g- -- -- --
-IV)


OLDE


O*I.0 -. 48 4. 4, 4 0 .2I .4 0.0 0.0 t.




2.0

VIodks CO W
1C 0' -- ofl 52 111-
i tr LN._r.




:0
*M M 010 0. Il
WAk


.^' :"-'.--.----.




-0 2 ** -0 ... T


Figure 2.3: Velocity components calculated by NPM, LITCROSS, UNIBEST and SEDITEL and
2-D current field calculated by SEDITEL, from Br0ker Hedegaard et al. (1992).


I 'IV

















z
5 -MvswI


I I i i 1 i
100 m


ONI. L'Itcrooss __

I










S L14H. Seditel -___

I,



.4

.4. --
-air r nr I


UL, Waton3


.10
.144
*II
*1


'"~ ;- i i j. i i s t t


Figure 2.4: Cross-shore sediment transport calculated by NPM, L1TCROSS, UNIBEST,

SEDITEL and WATAN3, from Br0ker Hedegaard et al. (1992).


0so NPM






00+--. -- im r


Oo
-1000 0io 3000 5000 10 o 0 ooo(<)
B e UNIBEST TC




s" C -- Nmentl aSULIS

20

0 . .. . .- - f)'M i ts-






-0too00 to.o jo oo 7n O no oon (m)
ft 0 LITCROSS



4 0 ^^'^^^---c- --- "ciSu.1e




1000 1000 3000 51)1 7000 o000 (m)


SEDTEL


2.0







- ----------- -i---~-i--
-1000 O100 3000 .000 7000 90.00 (m)
so WATAN 3





-- NCUUmCHM RESULn



*0ooo on 30oo 00 0ooo 70.00 o.oo (m)




REGULAR WAS
H 1.5m
T 6.5M
d .. 0.33.r ,


Initial slopes: lower beach 1:20
upper beach 1:40


Figure 2.5: Comparison of measured and calculated coastal profiles after 4.3 hours of exposure,

from Br0ker Hedegaard et al. (1992).


I .


L


(1 _1 s a


~-~3il------~


I


Y,


HR, NPM







4








2.3.2 Other American Methods

The other existing American methods for evaluating coastal profile evolution are more

tuned to specifically estimating the long-term fate of sand placed in the nearshore. Motivated by

the need to develop a systematic methodology for predicting the stability of dredged material

disposal sites over long periods of time, methods have been developed so as to provide reliable

predictions of the long-term dispersive or nondispersive characteristics of the site.

Scheffner (1991,1996) proposed linking a sediment transport model with a hydrodynamic

model that includes the influence of waves indirectly through a modified bottom friction term.

The result is a current velocity distribution at the site which is used to calculate the sediment

transport. The migration of a berm south of Dauphin Island, Alabama (Hands and Allison 1991)

was used for validation based on numerical simulation of the current field. Douglass et al. (1995)

reported on a detailed analysis of the measured near-bottom currents at the same Alabama site:

the cross-shore component of the mean near-bottom current was found to be offshore during the

time the berm migrated landward. The sediment response to the mean currents was thus not able

to satisfactorily explain the observed shoreward direction of berm migration. Evidently, Douglass

demonstrated the fundamental problem with a model based on mean currents.

Based partly on these findings, Douglass (1995) proposed a different methodology for

estimating landward migration of nearshore constructed sand berms. The fundamental concept

was that the dominant driving force is the waves, specifically, the landward migration of sediment

is due to the net landward transport by the asymmetry of finite-amplitude waves; the velocity

asymmetry was described by Stokes' second order theory. The model considers net shoreward

sand transport based on Bagnold's bedload sand transport equation. In conjunction with the

equation of sand conservation, this lead to a convection-diffusion equation in which the speed of

berm movement and the diffusion coefficient are expressed in terms of wave and sediment

parameters. This model will form the basis of the present study and is presented in Chapter 5.














CHAPTER 3
FIELD MONITORING DATA


As part of the NOURTEC project, the Danish Coastal Authority (DCA) performed a full-

scale test on beach and shoreface nourishment located at Torsminde on the west coast of

Denmark. A comprehensive 2-year monitoring survey program was initiated with the objective of

a continuous record of hydrographic conditions during the period as well as numerous

bathymetric surveys and sediment sampling. Waves, currents, water levels and winds were

recorded from March 1993 to April 1995. The topographic and hydrographic characteristics of

the monitored area presented in this section are taken from Laustrup et al (1997a).

All elevations are determined with respect to the Danish Normal Zero (DNN), related to

the mean water level (MWL) by the relationship MWL = +0.14 m DNN.



3.1 Site Description

The Danish North Sea coast north of Torsminde was selected as an attractive test site due

to severe profile retreat and regular bathymetry and few structures interfering in the longshore

sediment transport immediately adjacent to the nourishment project area. Regular surveys in

nearly 5 decades furthermore provided an extensive knowledge on the background coastal

development in the area.



3.1.1 Historical Evolution

The Torsminde project area is located approximately 30 km south of Thybor0n inlet on

the west coast of Denmark. The inlet was formed naturally during a severe storm in 1862 when

waves breached the narrow coastal barrier which separated the North Sea from the Limfjord. In
























N

A








Groyne O



Nourtec
monitoring
area


Figure 3.1: NOURTEC location map, from Laustrup et al. (1997a).


'~f~ 1""








the years following the opening of the channel, the adjacent coastlines experienced substantial

erosion which still today threatens the town of Thyboron. The net sediment transport into the inlet

has been estimated to be in the order of 500,000 m3/year; this sediment is deposited on large flood

shoals and is thereby lost from the littoral system.

Since the completion in 1875 of the first groin at the Thybor0n tidal inlet, an extensive

groin field comprising 58 groins was constructed south of the inlet; the groin field stretches from

Thybor0n to Fjaltring as illustrated in Figure 3.1. The terminal groin Q, situated 3 km north of the

NOURTEC project site, was built in the 1960's. With a considerable southerly net littoral drift,

downdrift erosion south of groin Q was inevitable.

To counteract the erosional development, a series of detached beach breakwaters and

revetments in combination with nourishment were implemented from the late 1970's. A total of

5.8 km revetment and 28 breakwaters were built on a 3 km stretch south of groin Q. The shoreline

retreat on this stretch, was thereby reduced from approximately 10 m/year prior to the coastal

protection works to at present approximately 2.4 m/year.

Today, nourishment is the preferred solution. The continuing coastal erosion has been

controlled by a comprehensive nourishment program for many years The nourishment volume

has gradually increased and has now reached a level of about 3 million m3/year distributed along

the west coast of Denmark, Laustrup et al. (1996). The chronological and spatial extent of all the

coastal protection measures implemented in the NOURTEC area since 1978 and prior to the 1993

nourishment projects, are displayed in Figure 3.2.

Prior to 1993, shoreface nourishment had only been used occasionally. The aim of the

NOURTEC project was to improve the understanding of the morphological changes induced by

placing large volumes of sand on the shoreface. Following the positive experience with the

NOURTEC shoreface nourishment, DCA in 1997 placed 1,25 million m3 sand as shoreface

nourishment. This volume accounted for one third of the total nourishment volume in 1997.












Coastal protection measures:
Breakwaters


00
CD )

= 0
0


co
CD
r-

3
CD


1991
I I


Revetment


Nourishment


1989
I


1983-85


1986 1984/85
1988 1987 1985 1985 1986 1987


1980
1982-84
---H

1985
1987 1979


30.ooom3


73,OC0m3(shf.)
1983H


24,000m3


4i


1987 78,000m3 142,00(
19871 ,OO 27,00
1988 101,000m3 ,27,00
19881


136,000m3

1990j-- 233,00m3
199O ---{----


2,000m3
1979 -----
1980 --Om
70,000m3 + 23,000m3(shf.)
1981-----

73,000m1(shf.) 73,0 (shf.)
I-I I73.n3(h.)
S 11,000m3
1985107,
107,000m3


24,000m3
34.000m3
38.,000m
61, 0.m3


)m3
)m3


9,0qm3


19911 310,000m3 182.000m3
19911------------------- -------------I
1992 265,000m3 + 170,000m3 (shf.) 221,000m3
0 19921---I

5-- West coast profiles numbers 5060 5180 10 m


- --. _- - ---.. .,- - - - - - -------- -- ------ - -- ,. 4 m


I C..... . . .

i4 4 m.., - . .,
$"---Am
~"~r~4 m
-44 x.-i. i'








3.1.2 Morphology of Field Site

There are no permanent offshore structures such as breakwaters or groins at the field site.

However, a series of beach breakwaters extend from groin Q to approximately 500 m North of

the beach nourishment. These breakwaters are normally covered by sand as the shoreline retreat

has been arrested due to the mentioned nourishment program. Revetments are also present on the

beach face extending from groin Q to approximately halfway between the beach and the

shoreface nourishment; the revetments are always buried by the nourishment sand and thus are

never evident on profile elevation surveys.

The cross-shore profiles reveal dunes, a beach and 1-3 longshore bars. The top of the

dunes is at +8 m DNN with a dune foot located between +3.5 and +4.0 m DNN. MWL is located

at +0.14 m DNN. The beach is approximately 100 m wide and consists of sediment ranging from

fine sand to pebbles. The longshore bars are located shoreward of the 6 m depth contour with

outer bar heights up to 3 m. The orientation of the shoreline at the field site is 3-1830 with

respect to true North.

The beach slope is approximately 1:20. The depth contours are straight and parallel to the

shoreline and the shoreface slope is approximately 1:100. The depth of closure on the west coast

of Denmark has been established at a depth of -16 m. However, the active profile zonation

extends from +8 m to -12 m, from dune top to well beyond the day-to-day surf zone processes.

Only on a very long time scale is there an interaction between the deep offshore region and

shallower innershore region.



3.1.3 Nourishment Projects

Two nourishment projects, one beach nourishment and one shoreface nourishment, were

located north of Torsminde in front of a 300-500 m wide barrier island and were implemented in

April-May 1993. The alongshore dimension of the beach and shoreface nourishment were each 1

km separated by a distance of 2 km. The projects consisted of a beach and a shoreface















5080




5090


5100





511D





5120






5130





5140



5150





5160 .


D I 1KM


Figure 3.3: Survey lines and positions for sediment sampling, from Laustrup et al. (1997b).








nourishment of approximately equal volumes on the order of 250,000 m3 of sand distributed

alongshore with a density of 250 m3/m.

The beach nourishment was placed directly on the dry beach from the dune foot to the

shoreline. The shoreface nourishment was located on the offshore side of the outermost bar. The

pre-existing depth at the middle of the shoreface nourishment was approximately -5.5 m, located

500 m seaward of the shoreline. The shoreface nourishment resulted in a berm with a maximum

relief of approximately 2.1 m.

The monitoring area was delimited by a southern boundary 2 km south of the shoreface

nourishment and a northern boundary 1 km north of the beach nourishment, covering a total of 7

km alongshore. The alongshore spacing between the survey lines was 100 m. The cross-shore

extent was 1500 m from the top of the dune at +8 m DNN to a water depth of approximately -12

m. A total of 17 survey campaigns was carried out during the 2-year monitoring period.

The southernmost survey line was line number 21620 and the northernmost was 22320.

The shoreface nourishment was located approximately between lines 21820 and 21920 with the

centerline of the berm at line 21870. The beach nourishment was placed between lines 22120 and

line 22220. Figure 3.3 shows the extent of the monitoring area with the survey lines, the positions

for sediment sampling and the location of the two nourishment projects.

A system of global survey lines is also shown on the same figure. These survey lines,

with a spacing of 800-1000 m, have been surveyed regularly since 1938 and are used in

establishing the historical development of the west coast of Denmark. A total of 13 of such

profiles are located within the NOURTEC area, thereby providing information on background

coastal retreat rates.

On Figure 3.4, two representative profiles through the beach and shoreface nourishment

respectively, are shown. The initial bottom elevation of March 29, 1993 is shown together with

the first post-nourishment survey, carried out on June 10, 1993. The profiles are seen to extend

from 1000 to 2500 m seaward of the reference line behind the dunes.













10

8 Mar-93
Jun-93
6 ___MWL

4----- ---- ----- -- --------------------








-4 .... .. ------- ... ..

-6 -- ----93



-1------------...-------.-...-.-- -- ---- -- ------- --- --------
z 2


O -2




-6

-8

-10
(a)
-12
1000 1100 1200 1300 1400 1500 1600 1700 1800 1900 2000 2100 2200 2300 2400 2500
Distance Offshore [ m ]





10

8 ar-93
-Jun-93

6-- ------ -- --- - T- -MW

4 --- -




o -2

[ -4

-6

-8 -

-10
(b)
-12 i ,
1000 1100 1200 1300 1400 1500 1600 1700 1800 1900 2000 2100 2200 2300 2400 2500
Distance Offshore [ m ]



Figure 3.4: Coastal profile surveys form March to June 1993 through
(a) Shoreface nourishment at survey line 21880,
(b) Beach nourishment at survey line 22130.










3.1.4 Sediment Size Distribution


Sediment samples were collected after each bathymetric survey in order to describe the


variation in mean grain size and sorting along and perpendicular to the coast. Figure 3.5 shows


the mean grain diameter d50% plotted against the water depth. The natural grain size is seen to


vary between 0.2 and 0.45 mm, with the coarsest sand found on the beach face. As the offshore


distance increases, the sand tends to be finer around 0.2 mm. However at the bar locations


between 4 to 6 m water depth, a minor coarsening of sand is observed. This is also true at the


seaward end of the surveyed profile. In the vicinity of the outer bar, a seasonal variation in mean


grain size is observed, with coarser sand during winter than summer.


The mean grain size d5o% for the beach nourishment sand was 0.32 mm and 0.57 mm for


the shoreface nourishment.


Dso Microns
S n . ......... .......... .. ... ... -.. 1400 . --- ...
1 .191
*191
--------- --------------- ----------- 1200 19
-------. - ............. .............. .. ........ .... ........... . 1200 19
S191
*19!
19
. 1000 1


S--- --- --- ------- ------- 800 --- -



@ , *t .
...* : . 0 ..
I

... .... ........... .... --. ... ... .-- o ---..... .. ........








-12.0 -10.0 -8.0 -6.0 -4.0 -2.0 0.0 2.0





Figure 3.5: Variation of grain size with depth, from Laustrup et al. (1997b).


33.01
33.05
93.09
34.03
34.05
34.07
35.01


4.0 m
Level









3.2 Hydrodynamic Setting

The wave height, period and direction were recorded by a waverider just north of the

survey area in a water depth of 20 m. Wave-data were recorded for 20. minutes every 3 hours.

Currents were measured by a current meter, initially placed at 14 m depth just seaward of the

monitoring area and later at 4 m depth on the shoreface nourishment; the currents were recorded

once every 15 minutes. The water level was recorded at a groin at Torsminde and at a groin 5 km

north of the survey area once every 15 minutes. Wind speed and direction were recorded at

Torsminde once every 15 minutes.



Table 3.1. Wave statistics, March 1993 April 1995, Torsminde.


Time Period


March 1993
April 1993
May 1993
June 1993
July 1993
August 1993
September 1993
October 1993
November 1993
December 1993
January 1994
February 1994
March 1994
April 1994
May 1994
June 1994
July 1994
August 1994
September 1994
October 1994
November 1994
Decemberl994
January 1995
February 1995
March 1995
April 1995


Mean H,
[m]
N/A
0.81
0.91
1.31
1.39
1.31
0.91
N/A
N/A
1.93
2.27
0.90
2.01
1.15
0.85
1.54
0.67
1.18
1.38
1.20
N/A
2.17
2.16
2.39
2.05
1.68


Max. H,
[m]
N/A
2.64
2.95
3.25
3.55
3.18
2.36
N/A
N/A
5.68
6.34
3.32
4.33
4.04
3.60
4.03
1.89
4.43
3.38
2.33
N/A
4.28
8.97
8.57
6.27
6.00


Mean T
[s]
N/A
3.8
4.2
4.3
4.6
4.4
3.9
N/A
N/A
5.3
5.5
4.1
5.3
4.5
4.1
4.7
3.7
4.3
4.5
4.3
N/A
5.2
5.4
5.4
5.0
4.9


Mean Direction
[deg.]
N/A
272
293
301
291
287
290
N/A
N/A
280
277
285
273
284
281
281
285
281
281
228
N/A
262
271
276
273
294


----








3.2.1 Wave. Tide and Current Measurements

For the entire monitoring period, the dominant deep water waves were from the

Northwest. However, events of high waves above 2.0 m were most frequent from the West-

northwest. The 100-year significant wave height is 8.1 m. The wave climate statistics for the

monitoring period at Torsminde are displayed in Table 3.1. The wave statistics refer to the

energy-based significant wave height, H,, and spectral period, T, measured in 20 m water depth.

Both wave height and period were found from Fast Fourier Transform (FFT) analyses.

The tide is semi-diurnal with a tidal range of 0.6 m varying between mean low water

level (MLW), MLW = -0.16 DNN and mean high water level (MHW), MHW = +0.44 m DNN.

The 100-year water level is +3.40 m DNN. During the monitoring period, the water level was

above +2 m DNN once, namely +2.05 m DNN on December 20, 1993.

The time-averaged currents were measured 1 m above the bottom. Due to the semi-

diurnal tide, the current direction changes 4 times a day. At 13 m depth, the dominant current was

the tidal current with an amplitude of 0.4 m/s with a residual northerly directed current of 0.1 m/s.

At 3 m depth, the current is also significantly influenced by the waves; the mean current is 0.7

m/s with a maximum velocity of 1.34 m/s.



3.2.2 Wind Characteristics

Wind from the Northwest is dominant and the 100-year wind speed is 33 m/s. The wind

rose for the monitoring period is displayed in Figure 3.6.



3.2.3 Littoral Drift

DCA has evaluated the annual littoral drift at Torsminde by two different approaches.

First by accumulating the profile erosion from the nodal point for the littoral processes drift south

of Thybor0n and secondly by use of the CERC formula. The results yield a south-going littoral

drift estimated in the interval 550,000 to 850,000 m3/year.













Period: 290393-240495


5 % of the lime


- Above 15 ms
O-10 5ms

2- 5m.s
iii Below 2 ms
rrTIn Coasiline


Figure 3.6: Wind rose for the monitoring period at Torsminde, from Laustrup et al. (1997a).


Groin Q at Fjaltring is not a perfect littoral barrier as suggested by the net littoral drift;

sediment is bypassed on the bar outside of the groin. Construction of the groin field north of

Torsminde has certainly reduced the rate of shoreline retreat. However, the tips of the groins are

under severe pressure from the receding coastal profile, leading to their slow but progressive

collapse.















CHAPTER 4
MORPHOLOGICAL DEVELOPMENT



The positive effects of placing large volumes of sand in the nearshore region has been

shown to be an attractive way to protect the coast. However, at present no quantitative method

exists to predict the long-term performance of the nearshore berms. Describing the morphological

development following nearshore nourishment projects, can be valuable as to disassociate the

natural processes re-shaping the profile from the processes caused by the implementation of the

nourishment.

Profile survey data were analyzed to yield representative profiles to be used in the further

analysis. The shapes of the chosen neighboring profiles were similar. However, longshore

variability in volumetric changes was investigated to evaluate the applicability of a two-

dimensional approach. It was found that the monitored area is characterized by cross-shore as

well as alongshore natural variability.

Average representative profiles were used in the analysis by averaging three neighboring

profiles. If only one profile survey was selected for modeling, significant variability or deviation

in calculated and measured results could be expected.

In the following, no attempt has been made to duplicate the modeling efforts previously

conducted on the same data set, by using other equivalent coastal profile models. Rather, trying

to supplement the existing analyses using a different approach to combine all the results, has been

the motivation.








Although the nourishment projects consisted of a beach and shoreface nourishment of

approximately equal volumes, focus is made on the evolution and processes around the nearshore

berm.



4.1 Previous Study on Torsminde Data

An extensive study on the interpretation of the Torsminde data has previously been

performed by the Danish Coastal Authority (DCA) in cooperation with the Danish Hydraulic

Institute (DHI); the results hereof are presented in Laustrup et al. (1996) and Laustrup et al.

(1997a). By using numerical modeling systems developed by DHI, part of the aim of the study

was an attempt to reproduce numerically the long-term profile evolution over the monitoring

period. Only the most important findings of these modeling results are mentioned in this section.



4.1.1 Morphological Analysis

The volume changes in the monitored area have been analyzed by the use of a system of

cells with fixed horizontal boundaries. In Figure 4.1 the system of cells is shown together with the

calculated volume development in selected cells. Immediately following the completion of the

shoreface nourishment in cell H, a volume increase of approximately 100,000 m3 is observed.

Landward of the shoreface nourishment, a gradual increase in volume is seen in cells A, B and C

throughout the monitoring period. This increase is significant downdrift of the nourishment and

less accentuated updrift. Although the volume in cell H containing the shoreface nourishment

appears constant, the berm migrated 50-75 m onshore and lost approximately 0.5 m of its height

over the monitoring period. The beach nourishment sand in cell E is seen to be eroded in

approximately 3/ year. Tracer results indicated that part of this sand migrated towards the south in

the shoreline region.

The horizontal development of the bar system is shown in Figure 4.2. Just after

completion of the nourishment, it appears that the berm becomes an integral part of the bar










system. South of the shoreface nourishment, the bar becomes continuous due to increased wave

breaking over the very wide and shallow berm which in return enhances southerly sediment

transport on the bar. This bar remains continuous for the rest of the monitoring period.


Volume 1000 m3


.* .


* S
* -

U ~ .5


0 __ o 0 0
FnH
A*B-C I


Figure 4.1: Volume development in selected cells, from Laustrup et al. (1996)


At the end of the monitoring period, the bar had migrated shoreward 50-75 m. A

simultaneous southern migration of the berm was also observed together with a weakening of the

northern end of the berm. A continuous bar developed landward of the shoreface nourishment.

The bar development in the northern half of the monitoring area is less clear, as the bars are

fragmented. A tendency towards alignment with the bar developed landward of the shoreface

nourishment is however observed. Following the nourishment works, the lack of a consistent bar

outside the beach nourishment probably contributed to the fast erosion of the beach nourishment.

Earlier surveys in the Torsminde monitoring area have documented the average vertical

erosion in the offshore end of the survey profile at 12 m depth to be approximately 20 cm/year

over the entire longshore stretch. Although the amount of erosion at the offshore end of the


M P ao o Ir
o I I I K













Just before nourishment


~- ~%


Just after
, ,= ='="- -


8 months after


10 months after


_ 18 months after

__won


23 months after


>m-z


0 0,5 1 km


Figure 4.2: Development of the bar system, from Laustrup et al. (1997b).


Q~i~sBar


~LCI


I ,








surveyed area is approximately equal, the cross-shore sediment transport is not evenly distributed

in the longshore direction. This is probably due to the presence of migrating bed forms. However,

the longshore variation in cross-shore transport has not yet been successfully correlated with the

migrating bed forms.

Results from sediment sampling in the nourishment area show that 60% of the

nourishment sand was still in the initial position at the end of the monitoring period. The 40%

loss of the nourishment sand only accounted for 1/3 of the total accretion shoreward of the

nourishment. The shoreface nourishment had therefore an obvious breakwater effect. Similarly to

a detached breakwater, the nourishment reduced the longshore transport landward resulting in an

accretion at the shoreline.

Overall a considerable erosion took place even though 0.5 million of m3 sand had been

supplied to the area.



4.1.2 Relative Effect of the Two Nourishment Projects

In order to obtain an improved understanding of the morphological changes induced by

placing large volumes of sand in a coastal environment, general design objectives were chosen in

the planning of the nourishment projects. The design objectives of the nourishment projects were

-stabilization of the shoreline,

coastal protection,

widening of the beach.

The effects of each of the nourishment projects on the coastal profile development as well

as the relative effect of the two types of nourishment have been analyzed based on their ability to

fulfil the design objectives. In Figure 4.3, the definition of the corresponding design parameters

are shown. The design parameters are obtained by transforming the volumes to an average

horizontal position by dividing by the height of the zone used in the volume calculations. The

result is a horizontal distance to the chosen reference line on the landward side of the dune top.









Dune top
Dune foot

Beach width 5
44








-B




Figure 4.3: Definition of design parameters, from Laustrup et al. (1997b).



The level +4 m DNN corresponds to the beach level at the dune foot. The longshore

extension of the nourishment effect has been determined to be approximately 3 km located

symmetrically around the two project areas. Together with the natural development of the coastal

profile, the development of the three design parameters for the southern and the northern 3 km

stretches is shown in Figure 4.4.

It appears that the beach nourishment stabilized the coastline and improved the coastal

protection but the beach width had not been improved. On the other hand, the shoreface

nourishment fulfilled all three design objectives. By the end of the monitoring period, the

shoreface nourishment had an effect on the coastal stability of 10.4 m for the 3 km stretch while

the similar effect of the beach nourishment was 9.0 m. Similarly the effect on the coastal

protection was 9.0 m for the 3 km stretch for the shoreface nourishment and 5.4 m for the beach

nourishment.

Figure 4.5 compares the relative effects of the two nourishment projects in terms of the

design parameters. The net effects thereby obtained show again that the shoreface nourishment is

the better option for all parameters, even though the beach nourishment in the first 3-year was










more effective. The better relative effects of the shoreface nourishment is even increased as the

actual price to nourish on the shoreface was 30% cheaper than to nourish on the beach.


3 km with shoreface nourishment 3 km with beach nourishment
Position of the coastline coastal stabilty Posfion of the costlUne coastal stablity
m Oistance from reference tine m Oistance from reference in
225 175

200 50ISO
-~iOn ----..-- ,, -0- 2.8
175 125
93 94 95 96Year 93 94 95 96Year

Position of the upper part of the profnr coastal proectflon Position of the upper part of the profile coastal protection
m Distance tom reference .n m Distance from reference line
25 25

0 0 -------- ------

-25 T -25
93 94 95 96Year 93 94 95 96Year

Beach vwdth widenlng of the beach Beech width widening of the beach
m Width m With
125 100

100 7S me=-\ -4.3

75 50
93 94 9 96Year 93 94 9S 96 Year

Figure 4.4: Nourishment effect on the design parameters, from Laustrup et al. (1996).


Shoreface nourishment seems to be a more stable solution than the beach nourishment.


This is partly due to the coarser sand in the shoreface nourishment. But even more contributory to

the stability of the shoreface nourishment, is its placement in a natural bar zone where the

nourishment immediately becomes an integrated part of the bar system instead of being eroded


away. Placing sand on a natural outer bar, thereby increasing its volume, does not bring a

considerable change to the system and will therefore not disturb the natural processes in a


significant way.













Relative effect on the design parameters
Posllion of the colellne coastal stablllty
m Southern nonrtrn section
20

10

0

-10
93 04 o9 96Year

PolUlon or the upper part of the profile coastal protection
m Southemr northern section
20

10




-10
*3 94 05 96Y.ar

Seuch width widening of the beach
m Southern norhenmn second
20

to







-20
93 94 D5 96 Year

Figure 4.5: Relative effect of the two nourishment methods, from Laustrup et al. (1996).


4.1.3 Modeling Results


The computational modeling of hydrodynamics and sediment transport processes was


based on the use of the models LITPACK and MIKE 21, both models developed by DHI.


LITPACK is a 1-dimensional deterministic modeling system for littoral processes and coastline


kinetics. MIKE 21 is a 2-dimensional deterministic modeling system for coastal waters and seas.


The purpose of using the models was to analyze the effect of coarser grain size in the shoreface


nourishment and to calculate sediment budgets.


Due to the dominant role of the longshore sediment transport, no cross-shore transport


was considered in the use of LITPACK. After calibrating LITPACK to the annual south-going


net littoral drift estimated to be in the interval of 550,000-850,000 m3/year, the longshore littoral













Shoreface nourishment



After:


Before:


Bathymetry : 93.01
Wave ongle : 3 dog
Grain size: 0.32 mm
8 m contour
170







130

120
110-



ato
-15000-l00o -5000 0 o000
transport (m'/m/y.r)


-- H,. 1.50 m
_- H, 2.25 m
- H_. 3.00 m


Bothymetry : 93.05
Wave angle : 313 deg
Grain ize : 0.32 mm
8 m contour
170




.' -140


30

120

110

100
-1soo0-000t -5000 0 S000
troanport (m'/m/yeor)


Beach nourishment



After:


Before:


Botyetry : 93.01
Wave angle : 313 deg
Grain ize : 0.32 mm
8 m contour
310
300 ---

2W----

:2 IO- -- -- -- --





240
230 -- --- -- ;--

15000-1 000 100 500
transport (m/m/near)


H... 1.50 m
H... 2.25 m
--- H. 3.00 m
4 m contour




-- --- ^- -
I----








-1:o00-tooo -500o0 500
transport (m'/m/yoer)


Bathymet : 93.05
Wave angle : 313 dog
Grain iiz : 0.32 mm
8 m contour
310

300- -





.260- -- -- -- --
502
S2170



250

240-
-1 000-1000-500 0 500
transport (m'/m/year)


H, 1.50 m
-- H,, 2.25 m
*-- H,. 3.00 m

4 m contour



;I




//




-15000-10000 -5000 0 5000
transport (m'/m/year)


Figure 4.6: Cross-shore transport calculated with Mike 21, from Laustrup et al. (1997b).






transport across the survey lines was determined by modeling. Better agreement was found with


the shoreface nourishment than with the beach nourishment. The LITPACK results were then


applied to the calibration of MIKE 21.


H,. 1.50 m
-- H 2.25 m
-*-- H, 3.00 m








The simulated conditions with the MIKE 21 model were limited to 3 wave heights and

corresponding water levels, 2 wave directions, 3 bathymetries and 4 different types of sediments.

For both nourishment projects, significant wave heights of 1 to 2 m were found to cause most of

the sediment transport; this range of wave heights was thus chosen in the modeling. Wave

directions from Southwest and Northwest were chosen, both 400 off the coast normal. The

bathymetries chosen were an initial bathymetry before nourishment, a "summer" bathymetry with

nourishment and a "winter" bathymetry with nourishment.

In Figure 4.6, the effect of the beach and shoreface nourishment on the potential cross-

shore transport is shown as calculated by MIKE 21. The beach nourishment is seen to have nearly

no influence on the cross-shore transport at 4 m depth and deeper. In the case of the shoreface

nourishment, the increased sand volume in the berm and the local breaking of incoming waves,

induced a significant increase in the cross-shore transport. In both cases, bathymetric variations

due to the presence of the nourishment caused a longshore variation of the cross-shore transport.

The results for the longshore transport presented in Figure 4.7, show that even small

morphological changes are highly reflected in the sediment transport pattern. The modeling

results show a significant longshore variation of the longshore transport. The wave-driven current

field is very sensitive to even small bathymetric irregularities, such as longshore variations in the

bar system. These irregularities in turn have an enhanced effect in the longshore sediment

transport pattern. The changes in the longshore transport introduced by the beach and shoreface

nourishment appear to be nearly insignificant compared to the natural longshore bathymetric

variations. The volume and location of the shoreface nourishment are such that the stability of the

total bathymetry has not been disturbed significantly.

Although helpful in understanding details of some of the processes which contribute to

the natural re-shaping of the monitored area, the modeling results could not be combined in such

a way to satisfactorily reproduce the observed losses and gains of sediment. The coastal profile









development could not be reproduced and thus could not be used in a predictive way. The

modeling efforts showed that the processes in the nearshore region are highly 3-dimensional.


Longshore transport
Effect of nourishment only

F-2
2


Longshore transport
'Summer' and 'winter'


) C




i i c
..





SS t 8iii t -

S' S '


I t i C i




n o o



Figure 4.7: Longshore transport for three different bathymetries, from Laustrup et al. (1997b).



4.2 Volumetric Profile Changes

In the following, emphasis was given to the changes of the nourishment volumes

themselves and to the changes caused by placing the nourishment sand. The first year of

monitoring was of greater interest than the second year since the effect of the disturbance due to

the implementation of the nourishment projects, decreases with time as the natural processes tend

to restore an equilibrated profile.








The reference line for all the surveys was chosen at the top of the dune and nearly parallel

to the shoreface nourishment. With the chosen reference line, the initial position of the shoreline

landward of the shoreface nourishment is approximately 1175 m. The.cross-shore bathymetry

perpendicular to the shoreline and through the shoreface nourishment is characterized by two

offshore bars nearly parallel to the shoreline. The inner bar is located in the interval 1250-1400 m

with a crest at approximately 1300 m and the outer bar in the interval 1400-1600 m with a crest at

approximately 1525 m. The starting and ending points of the bars are here to a large degree

defined arbitrarily. The shoreface nourishment was placed on the offshore slope of the outer bar,

located in the interval 1575-1750 m with its crest at approximately 1650 m. The above locations

relate to the time just after completion of the nourishment works.

Due to the large amount of survey lines, only representative sections of the monitored

area were selected. Figure 4.8 shows the selected survey lines, three of which represent one third

of the shoreface nourishment. To serve as control beaches, three additional survey lines were

used, one survey line 500 m north on the updrift side and two lines 500 m and 1000 m south on

the downdrift side of the nourishment. Only one survey line north of the shoreface nourishment

was chosen due to the migration of the eroded beach nourishment sand towards the south. This

survey line was thus probably not affected much by the beach nourishment sand. The survey

dates used were selected as being the most characteristic to describe the development of the

coastal profile during the elapsed time between the surveys. These were the initial June 10, 1993

survey and the April 12, 1994 survey.

Prior to the NOURTEC nourishment project, the average coastal retreat was 2.4 m/year

corresponding to a volume loss of 48 m3/m/year between the dune top at +4 m DNN and

estimated depth of closure at -16 m DNN, Laustrup et al. (1997a). This erosion volume must be

considered in comparison with the nourishment volume of approximately 250 m3/m.





































21970





21900


21870

21840





21770



21720


Figure 4.8: Average survey lines.


0 0.5 1 KM








4.2.1 Cross-shore Sand Transport Rate Distribution

In the following, a right-hand coordinate system was defined with the x -axis oriented

alongshore to the right when looking seaward and the y -axis oriented offshore. The cross-shore

sand transport rate distributions were determined from the development of consecutive profiles by

integrating the mass conservation equation



ah aSq sy (
-+ qx + +s=0
t ax y (4.1)



where h represents the water depth, q, and q, are the longshore and cross-shore transport rates,

respectively and s is the volume rate of sand added per unit plan area.

In an initial attempt to two-dimensionalize the sediment transport pattern by neglecting

gradients in the longshore transport, the second term in the Eq. (4.1) was reduced to zero and s is

also taken as zero. By choosing a stable reference point on the dune top where no changes occur

between consecutive profile surveys, the cross-shore rate transport distribution was obtained as




q (y)= y (4.2)
o t (4.2)



where q, is the accumulated cross-shore transport until a seaward distance y.

The transport rate distribution thereby obtained represents an average net rate for the

period between the two profile surveys. By calculating the cross-shore transport rate distribution

between such two profiles, a picture of the overall morphological response within the profile is

obtained.









(a) pure migration


z[m]





Sq,


qy[ m3/m]


(b) pure diffusion


y[m]
















Sy[m]







y[m]


Figure 4.9: Net cross-shore sand transport rate distributions for
(a) Pure Migration,
(b) Pure Diffusion.








The general characteristics of the transport rate distributions can be classified as the

following three main types:

- Accretionary; the accumulated transport rate at the seaward end of the profile is negative,

- Erosional; the accumulated transport rate at the seaward end of the profile is positive,

- Mixed accretionary and erosional; the transport is directed onshore along the seaward part of

the profile and offshore along the shoreward part of the profile or vice-versa.

Equilibrium distributions can be obtained if there is conservation of sand within the

consecutive profiles indicating transport only in the cross-shore direction. At an unchanged point

in time, corresponding to the stable established reference line on land, the transport is zero and if

the profile experiences conservation of sand, a zero transport rate at the seaward extremity of the

survey follows. Having neglected the effect of the longshore transport in Eq. (4.2), an equilibrium

distribution should not be expected. The net transport rate value at the seaward extremity of the

survey at 2500 m, will thus be an indicator of the gradient of longshore transport at the profile

cross section. This transport rate value quantifies the integrated loss/gain of sand volume over the

entire length of the surveyed profile. The volume is either lost to or gained from neighboring

profiles. By performing such net cross-shore sand transport rate distributions for adjacent profiles

along the shoreline, the longshore variation of the longshore sediment transport can be obtained.

Characteristics of net cross-shore sand transport rate distributions are presented in Figure

4.9, for the case of pure convection and pure diffusion respectively, for the reshaping of a berm

under wave action. Conservation of sand is seen to be assumed as no loss/gain of sand volume

occurred in the berm cross-section.

Figure 4.10 illustrates the net cross-shore sand transport rate distributions for the

representative shoreface nourishment profiles, average profile 21840, 21870 and 21900

respectively. The general characteristics of all three transport rate distributions are a shoreward

migration of the berm and vertical erosion of the sea bed over the entire offshore stretch seaward















200 -- ---





oo .100 ...... .. .. ... -...v......P


50


E 0


-50 .


-100"

Average Profile 21840
-150
Average Profile 21870
-Average Profile 21900
-200........ ........ .
Shoreface Nourishment

-250 1 -
1000 1100 1200 1300 1400 1500 1600 1700 1800 1900 2000 2100 2200 2300 2400 2500
Distance Offshore [ m ]



Figure 4.10: Net cross-shore volumetric transport past any offshore location assuming no longshore transport gradients. Results for average
profiles 21840, 21870 and 21900 from June 10, 1993 to April 12, 1993.








of the nourishment. The total amount of erosion varies significantly with a volume change of

approximately -90 m3/m, -191 m3/m and +33 m'/m, respectively.

A decreasing transport rate in the interval 1550-1625 m, indicates a deposition of

sediment shoreward of the shoreface nourishment. An increasing transport rate in the interval

1625-1750 m immediately seaward hereof, illustrates the loss of height of the shoreface

nourishment. Altogether, this corresponds to the observed 50-75 m shoreward migration of the

berm during the first year. On the offshore side of the shoreface nourishment, a decreasing

transport rate illustrates the effect of the strong diffusive processes; the natural re-shaping of the

seaward side of the berm is obtained by sediment moving offshore and depositing into a milder

equilibrated slope.

Seaward of 1900 m, a steady increase over the entire length of the shoreface nourishment

illustrates the ongoing vertical erosion. This offshore erosion appears to be independent of the

morphological changes occurring in the nearshore.

The northern 1/3 of the shoreface nourishment, average profile 21900, is seen to

experience an accretion at the shoreline until the inner bar. This does not occur as consequently in

the central and southern 1/3 of the profile, implying a temporary stronger deposition of sediment

in the northern part. However, the deposited sand is probably eroded sediment from the beach

nourishment and not a result of the breakwater effect of the shoreface nourishment. Sampling of

sediment in the nearshore at neighboring profile 21920, did show a concentration of fluorescent

tracer originally mixed with the beach fill, thus confirming the southern movement of the beach

fill into the shoreface nourishment area, Laustrup et al (1997a).

In conjunction with corresponding profile surveys, the cross-shore morphological

changes are best illustrated. Figure 4.11 shows a typical profile evolution across the shoreface

nourishment, here chosen as the central average profile 21870.

The net cross-shore transport rates for the three control beaches, average profile 21720,

21770 and 21970, are displayed in Figure 4.12. The same graph scaling as for the shoreface

















SMar-93
6
6 --Jun-93
S Apr-94
4 I- Apr-95





t I 1I
-MWL

2


0z 0


2 . .. . .




-6




-8


-10


-12 i
1000 1100 1200 1300 1400 1500 1600 1700 1800 1900 2000 2100 2200 2300 2400 2500
Distance Offshore [ m ]


Figure 4.11: Profile evolution for average profile 21870.












200 .... averagee r-unle L /u
Average Profile 21770
Average Profile 21970
Shoreface Nourishment









-20 -- ------------------I i-- ---------------------------------------i-
50.. ...

E 0

-50 -




-150-
+ i -I







1000 1100 1200 1300 1400 1500 1600 1700 1800 1900 2000 2100 2200 2300 2400 2500
Distance Offshore [ m ]



Figure 4.12: Net cross-shore volumetric transport past any offshore location assuming no longshore transport gradients. Results for average
profiles 21720, 21770 and 21970 from June 10, 1993 to April 12, 1993.








nourishment profiles in Figure 4.10 was chosen, in order to facilitate an intercomparison of the

magnitude of the transport rates.

In the nearshore zone of the two neighboring downdrift profiles, a smaller amount of

activity is seen to occur. This relates to the fact that no immediate disturbance of the system was

introduced at that location. No parallel development is seen in the profiles downdrift of the

shoreface nourishment. In the nearshore zone, longshore fluctuations in the transport rates are

attributed in part to profile changes introduced by small variations in forcing conditions.

North of the shoreface nourishment, the bathymetry is developing from a profile with one

bar to a profile with two bars. The increased transport rate seen in average profile 21970 relates to

this change in bathymetry. As mentioned previously, the beach face accretion corresponding to

the decreasing transport rate in the interval 1175-1300, is not a sheltering effect due to the

shoreface nourishment, but sand transported from the beach nourishment. The negative

accumulated transport rate at the end of the profile is seen to be caused by the gain of sediment in

the 500 m nearshore. No impact of migrated beach nourishment sand is seen in the offshore,

where vertical erosion is predominant, as earlier described for neighboring profiles.

By superimposing the two downdrift profiles, it can be seen that the cross-shore sediment

transport in the nearshore is unevenly distributed in the longshore direction implying a transport

direction that is not so distinct.



4.2.2 Longshore Volume Changes

The longshore volumetric changes were computed directly from the surveyed coastal

profiles by integrating the gains/losses over the entire length of the profiles, from dune top to the

offshore limit of the profile. In order to determine the cross-shore variation of longshore

volumetric changes, the profiles were also divided into two zones, a nearshore zone in the interval

from +8 m to -2 m and an offshore zone from -2 m to -12 m. The bathymetry just after

nourishment was used as the reference profile. Two following bathymetries were used to describe











400


200


0-


Figure 4.13: Longshore volume change from +8 m to -12 m.


7T


-200


-400


-600


-800-


-10001 I I
21620 21670 21720 21770 21820 21870 21920 21970 22020 22070 22120 22170 22220 22270 22320
Survey Line No.








the development during the monitoring period. The selected surveys were April 12, 1994 and the

last survey April 24, 1995.

The overall longshore volumetric changes from dune top to offshore limit of the surveyed

area are presented in Figure 4.13. Considerable fluctuations alongshore are observed. Erosion and

accretion appear to follow a spatial periodicity of approximately 500 m. In time, longshore

migration of the erosional and accretionary effects is observed; no distinct migration directions

seem to prevail. However, extensive erosion is characteristic in the beach nourishment segment.

By integrating the volumetric changes over the entire length of the profile, the breakwater effect

of the shoreface nourishment is not very accentuated as the nearshore accretion is added to the

offshore erosion.

Looking at the nearshore changes displayed in Figure 4.14, three characteristics appear:

the downdrift accretion adjacent to the shoreface nourishment, the erosion at the beach

nourishment and the mixed accretion/erosion fluctuations in the remaining shoreline stretches. All

three effects seem to intensify during the monitoring period.

During the first year, the downdrift accretion over a length of 1000 m adjacent to the

shoreface nourishment is observed with an average gain of 70 m3/m/yr. During the second year,

this effect seems to grow as the accretion migrated further downdrift to a total length of 1500 m

with a supplemental average gain of 65 m3/m/yr. The breakwater effect is clear.

The beach nourishment sand is seen to be totally eroded during the first year and this

erosion continues in the second year. The major losses from beach fills occur through the

diffusive losses at the project ends by longshore sediment transport. However, the nearshore

fluctuations are seen to be a small part of the total fluctuations shown in Figure 4.13, as a

significant offshore loss is observed also to have taken place seaward of the beach nourishment. It

is of interest that at the end of the 2-year monitoring period, the total loss in the beach

nourishment area exceeds almost by a factor of 2 the amount of beach nourishment, leading to a

considerable profile steepening.






























S -00 -
-600 - ---- -. ------- .................. ............... .......... ....... ... ....... .
I I i


-600 i -


jun93-apr94
-800- - --jun93-apr95
nourishment

I Shoreface iNourshm nt Beach No rishment
-1000 I I --- i i --- --
21620 21670 21720 21770 21820 21870 21920 21970 22020 22070 22120 22170 22220 22270 22320
Survey Line No.


Figure 4.14: Longshore volume change from +8 m to -2 m.












400-




200-




0-



-200
E


-400-




-600




-800


Figure 4.15: Longshore volume change from -2 m to -12 m.


-1000I I I- ,
21620 21670 21720 21770 21820 21870 21920 21970 22020 22070 22120 22170 22220 22270 22320
Survey Line No.








Even in the offshore zone, a great amount of fluctuation describes the longshore changes,

see Figure 4.15. The same spatial periodicity as in Figure 4.13 is observed. However, a general

erosional trend can be seen from one year to the next. Again, it appears that most of the ongoing

erosion in the beach nourishment stretch, took place in the offshore zone.



4.2.3 Cumulative Volume Changes

The longshore volumetric changes previously calculated were integrated alongshore.

Again in order to determine the cross-shore variation of volumetric changes, the profiles were

also divided into two zones, a nearshore zone in the interval from +8 m to -2 m and an offshore

zone from -2 m to -12 m. The initial bathymetry before nourishment was used as the reference

profile. Three bathymetries were used to describe the development during the monitoring period.

The selected surveys were again June 10, 1993 and April 12, 1994 to which the last survey April

24, 1995 was added.

Figure 4.16 shows the measured cumulative volumetric changes for the entire 7 x 1.5 km

monitored area, starting from the southernmost survey line 21620. Integrating over the entire

length of the profile, the overall morphological changes are highlighted. Approximately

1,000,000 m3 of material was lost from the survey area over the two-year monitoring period. The

accretionary effect in the profiles containing the shoreface nourishment, seem to be slightly

decreasing from the first to the second year. However, the erosional effect in the profile

containing the beach nourishment seems to be increasing. A further division in cross-shore

regions will be more precise in showing where on the profile these volume changes occurred.

Also displayed in Figure 4.16, are the expected cumulative volumetric changes when

only considering the established local background erosion rate of 48 m3/m/yr and the added

nourishment volumes. That is, not taking into consideration the changes introduced in the

nearshore littoral processes by the nourishment projects. For all three time periods shown, the

expected changes differ somewhat from the measured. The offshore deposition of sand in the















700


600


500


400


S300
E

o 200
x

< 100


0


-100


-200


-300


Figure 4.16: Cumulative volume change from +8 m to -12 m relative to March 1993.


-400 1 i t I
21620 21670 21720 21770 21820 21870 21920 21970 22020 22070 22120 22170 22220 22270 22320
Survey Line No.








berm area is not accounted for, but the greatest difference is however related to the significant

erosion in the beach nourishment area, also not accounted for. The comparison between the

measured and expected cumulative volume changes underlines both the significant impact of the

nourishment project in disturbing the natural littoral processes and as well the natural longshore

variability in these processes.

The cumulative volume changes in the nearshore zone are presented in Figure 4.17. In

two consecutive years following the completion of the nourishment projects, the breakwater

effect of the shoreface nourishment is clearly seen; a gradual accretion immediately downdrift of

the shoreface nourishment is observed. A total of approximately 150,000 m3 sand was deposited

over a distance of 1500 m at the end of the monitoring period. Directly landward of the shoreface

nourishment, only a slight accretion occurred just after completion of the nourishment; this effect

was not present in the remaining time of the monitoring. The beach nourishment sand is seen to

be completely eroded after a year. After the first and the second year, there appears to be a small

periodic fluctuation between accretion and erosion at the shoreline but no general trend is seen.

Similarly, the cumulative volume changes in the offshore zone are presented in Figure

4.18. The overall characteristics in the offshore zone appear to be similar to the characteristics for

the entire profile, suggesting that by far most of the erosion took place in the deeper end of the

profile thus creating a steeper profile. An immediate volume increase downdrift of the shoreface

nourishment following its completion is seen; the longshore transport on the berm is increased

due to extensive wave breaking over the berm. After two years, the bar system seems to be

stabilizing as the alongshore volume changes diminish.

A very extensive erosion seaward of the beach nourishment at an amount much greater

than the nourishment volume itself is observed. The bathymetry in front of the beach nourishment

is characterized by the presence of only one bar, an inner bar not very pronounced and

discontinuous. As was shown in Figure 4.2, the nearshore area in this coastline stretch has not the

benefit of an outer bar to dissipate the incoming wave energy. A loss of approximately 100,000














- Jun-93
- - -Apr-94
...... Apr-95
Nourishment




-I

; o- j'


500-


400-


300


200


100-
E
0
o 0
x

-100-


-200


-300


-400


i


i.

--t-----.
-- -----


,1


-horeface Nounshment


j..,*---- -1 c


`--L. --'


Beach Ndurishment


Figure 4.17: Cumulative volume change from +8 m to -2 m relative to March 1993.


I-- ---


-500-I I II
21620 21670 21720 21770 21820 21870 21920 21970 22020 22070 22120 22170 22220 22270 22320
Survey Line No.


i


i


I ; J


-- -


i


I



"'"~`--
''
'"

,
L
)"
i
















400-
4 0 0~ *-~ *- ***- - i- - -- ** *-- ....... ,^ ^ ..........---- -..--.... ........ ..... ... ..- .. ..-- . ... .... ... ;i- --
- -Apr-94
......- Apr-95
3 0 0 -. ... ... .... ........ ..... .... ......... .. .. .. .. ................ ......................
Nourishment ..

200 -- - -.


100 i
E



20 --- -- --.......--..,....... ...,
"0 0..'
oo "'-..-" ___'_'




-I '

-400 - f 4
-300


-400 i

Shoreface Nourishment Beach Nourishment -... ..*
-500 I i
21620 21670 21720 21770 21820 21870 21920 21970 22020 22070 22120 22170 22220 22270 22320
Survey Line No.


Figure 4.18: Cumulative volume change from -2 m to -12 m relative to March 1993.








m3 occurred during the first year followed in the second year by a loss of approximately 300,000

m3. This increase in erosion from the first to the second year was probably also caused by the

extremely strong erosion in the winter in 1994-95 outside the beach nourishment.



4.3 Longshore Variability

Placing nearly half a million m3 of sand in a littoral system might seem to be a large

nourishment volume capable of a considerable perturbation to a system in dynamic equilibrium.

However, the system experiences natural background "noise" and as the effect of the perturbation

decreases with time due to the re-shaping of the profile, it becomes even more difficult to

disassociate the natural processes from nourishment introduced processes as the natural

background noise increases with time. The above inevitably raises the question of whether the

perturbations introduced by the nourishment volumes were great enough in a system with such a

high degree of variability.

The longshore volumetric changes in the whole monitoring area suggest a lack of

coherency in the longshore processes. A considerable spatial and temporal variability in the

littoral processes seems to characterize the area. This natural variability will also be found in the

alongshore variation of shoreline positions. The development of the shoreline position will thus

illustrate the effect of the nourishment projects.

The dominant role of the longshore sediment transport pattern has been shown. Both the

cross-shore and longshore sediment transport seem to be very sensitive to the bathymetric

variations in the nearshore region. The implementation of the nourishment volumes in addition to

the lack of uniformity in the coastal profiles will therefore result in a variable alongshore

distribution of the sediment transport pattern.











50 -

Apr-94
40 - Apr-95
*- Nourishment

30 1 1

\/




I-o 1


20--
\ **




S30 I

-1 0 NI










oreface Nurishment Beah No rishment
-30-


-40 i --------I- i


-50
21620 21670 21720 21770 21820 21870 21920 21970 22020 22070 22120 22170 22220 22270 22320
Survey Line No.

Figure 4.19: Shoreline change relative to June 10, 1993 position.








4.3.1 Shoreline Positions

The changes in shoreline positions between consecutive profiles were calculated by

determining the change of volume in the zone MLW -0.25 m and MLW +0.25 m. The obtained

volume was then divided by 0.50 m which yielded horizontal distances from the reference line to

the updated shoreline.

The shoreline changes relative to the June 10, 1993 survey, just after completion of the

nourishment works, are given in Figure 4.19. A main characteristic of the development of the

shoreline position during the two years of monitoring, seem to be a spatial as well as a temporal

variation. An alongshore fluctuation between shoreline retreat and shoreline advance is present

over the entire stretch. No distinct direction of migration of the fluctuating pattern seems to

prevail.

The considerable retreat of the shoreline at the beach nourishment location indicates the

eroded nourishment volume by the end of the first year. However during the second year, the

retreat becomes much greater than the imposed shoreline advance due to the nourishment volume.

This is highlighted in Figure 4.20, showing the shoreline changes relative the March 29, 1993

survey before nourishment. It supplements the former figure by showing that the natural

alongshore variability in shoreline position, seem to follow the same pattern at the beach

nourishment as for the rest of the stretch.

The positive effect of the shoreface nourishment is a distinct average shoreline advance

of approximately 30 m over a distance of 1000 m downdrift. This effect is visible at the end of the

second year and correlates with the accretion previously observed in the longshore volumetric

changes.



4.3.2 Sediment Transport Distribution

The cross-shore distribution of the longshore sediment transport was shown to be very

sensitive to the local shape of the shoreface nourishment on the outer bar. Even small














I Apr-94
40 ---Apr-95


30 .. .. .


20 / '





10


S-10 I



-30 1
o I I








Survey Line No.
-50
21620 21670 21720 21770 21820 21870 21920 21970 22020 22070 22120 22170 22220 22270 22320

Survey Line No.


Figure 4.20: Shoreline change relative to March 29, 1993 position.








morphological changes were reflected in the sediment transport pattern, making it difficult to

distinguish between the sediment transport caused by the natural variations and the sediment

transport caused by the nourishment volumes. In an attempt to eliminate all other changes but the

ones caused by the nourishment volumes, Laustrup et al. (1996) integrated the longshore

transport in depth intervals of +4 m to -4 m and of -4 m to -8 m; these intervals correspond to

the shoreline zone and the shoreface nourishment zone respectively. Figure 4.7 showed examples

of calculated 2-dimensional sediment transport fields and integrated longshore sediment transport

for these intervals. It appears that the changes introduced by the nourishment volumes are nearly

insignificant compared to the natural variations in both space and in time.

The sediment transport distribution was shown to be not particularly affected by the

beach nourishment since it extended only a small distance seaward.

The modeling results showed that for both the shoreface and the beach nourishment areas

during the monitoring period, waves with a significant wave height of 1 to 2 m caused most of the

sediment transport with the net sediment transport being towards South.

Migrating bed forms in deep water in the interval 15-20 m, influence the direction of the

incoming waves by refraction and thus through focusing, have an effect on the local longshore

sediment transport. The migration rate of these bed forms has been established at a velocity up to

80 m/year directed towards the North. This migration furthermore induces a yearly variation in

incoming wave field. However, the correlation between the instability of the sediment transport

and the migrating bed forms in the seaward end of the coastal profile has not been found to be

significant enough to draw conclusions.

Unequal amounts of alongshore volumetric changes would suggest that even with similar

forcing conditions along the region examined, differentials in longshore sediment transport were

important. In the immediate vicinity of the nourishment projects, coastal structures were not

present in such a way as to create differentials in longshore transport; i.e. during the monitoring

period, the revetments were always buried on the beach and the beach breakwaters normally








covered by nourishment sand so as not to protrude in the nearshore. However, the overall

erosional state of the coastline is inherently related to the groin field 3 km updrift, but the

variability in longshore transport was probably not due to localized processes associated with the

updrift groin field as the variability is on a much smaller spatial and temporal scale than would be

expected if the groin field were responsible.



4.4 Beach Nourishment Performance Prediction

Beach nourishment projects are generally placed with profiles which are steeper than the

natural profiles for the size of sediment that is used in the beach nourishment project. Over the

years, an offshore transport of sediment results as the profile will tend to equilibrate to its natural

shape, consistent with the grain size used in the nourishment. The qualitative effect of grain size

in terms of the dry beach width for the same added volume per unit length of beach has been

investigated by Dean (1998).

In addition to the cross-shore profile equilibration, the placement of a beach nourishment

project results in a planform anomaly which interacts with the waves to result in sediment

transport away from this anomaly. This process of mobilization of sediments by waves has been

referred to as "spreading out" losses in which the transport is occurring away from the anomaly.

These losses relate to a longshore redistribution of the sediment and not a total loss to the system

but rather a loss from the region in which the sediment initially was placed. The spreading out

losses from the nourished area are thus manifested by a relative gain of sediment volume in areas

adjacent to the beach nourishment.

Simple analytical methods are readily developed for predicting the performance of beach

nourishment projects, Dean (1983). These methods are valuable in providing an estimate of

overall beach nourishment performance such as project longevity and remaining volume in the

project area. To allow for greater flexibility in the modeling of shoreline evolution, Dean and

Grant (1989) have developed methodologies for calculating the response of shorelines in the








vicinity of beach nourishment projects. Their numerical approach provides the capability to

include site specific features such as arbitrary initial shoreline position, background erosion rates

and littoral barriers. The study area at Torsminde has such features and will thus also be modeled

numerically.



4.4.1 Longevity of Nourishment Project

One-line models of shoreline evolution have been proposed by many investigators and

these models have been shown to perform effectively in many different beach nourishment

projects of engineering interest. A one-line model is the simplest model to describe the time

history of a nourishment project along a shoreline. In the absence of background erosion, Pelnard-

Considere (1956) combined the conservation equation and the linearized transport equation which

resulted in the classical heat-conduction equation



y =G 2 y (4.3)
at ax2



in which the longshore diffusivity, G, is defined as



KHb512g-j (4.4)
8(s -1)(1 p)(h. + B)



where K is a sediment transport parameter, Hb is the breaking wave height, g is the

acceleration of gravity, Ki is a proportionality constant relating breaking wave height to water

depth, s is the ratio of mass densities of sediment to water, p is the in-place sediment porosity,

h. is the depth of limiting sediment motion and B is the berm elevation.








The above one-line model can be used to predict the planform evolution of a beach fill. It

is thus possible to obtain information on the proportion of sand remaining at the placement site at

any time following the nourishment works. For estimating the "half-life".of a beach nourishment

project on a long straight shoreline with no background erosion, corresponding to half the initial

fill volume still within the project length, Dean (1983) defined the half-life tso, as



12 (4.5)
ts0% = 0.21-
G



where l is the project length and G the shoreline diffusivity defined in Eq. (4.4).

In determining the longshore diffusivity of the present beach nourishment project,

common constants and coefficients were set to known values, such as K = 0.78, s = 2.65 and

p = 0.35. Other site specific parameters were arbitrarily chosen as best estimates based on the

available data, such as K= 1.10 based on the beach fill mean sediment grain size 0.32 mm,

Hb = 1.5 m, h = 16 m and B = 6 m based on monitoring survey results. (It is noted that the

relationship between the sediment parameter and the sediment grain size is based on effective

wave height as defined by a Rayleigh distribution. The present representative significant wave

characteristics are based on energy spectrum, thus requiring a correction of the wave height by a

factor of 0.735).

The estimated longevity yields a half life of t50S = 3.1 months for the beach nourishment

at Torsminde. The monitoring results showed that the beach fill was totally eroded after 9

months. As the erosion rate of planform anomaly tends to decrease as the remaining nourishment

volume diminishes, the predicted longevity is found to be in order of magnitude in agreement

with the measured performance.








The rapid erosion of the entire beach fill suggests the importance of the short project

length of only 1 km. As can be deduced from Eq. (4.5), t50o is seen to be proportional to the

square of the project length, thus doubling the project length would result in extending its

longevity by a factor of 4.



4.4.2 Multiple Nourishments in the Study Area

As described in Chapter 3, a system of coastal profiles has been surveyed regularly since

1938, 13 of which are located within or adjacent to the NOURTEC area, thus providing the

historical development of the shoreline position in the study area. Decadal variation in wind and

wave conditions and implementation of extensive coastal protection measures at different times,

has altered the shoreline trend lines for separate periods. DCA has selected the period 1985-1992

to determine the representative development of the area. The choice of 1985 relates to the

significant commencement of nourishment volumes. Figure 3.2 shows the coastal measures

implemented in the NOURTEC area since 1978; specifically the chronological and spatial extent

of the multiple nourishment projects is of interest. With knowledge of the annual shoreline

retreat, the wave characteristics and the added nourishment volumes, the effect of the multiple

nourishments can be modeled.

The numerical model DNRBS (Department of Natural Resources, Beaches and Shores)

developed by Dean and Grant (1989) predicts shoreline change in the vicinity of a beach

nourishment projects. The shoreline modeling utilizes a deep water wave equivalent for the one-

line model presented briefly in the previous section. A slightly modified version of the DNRBS

model also includes the effects of subsequent multiple nourishment projects on the same updated

shoreline.

The DNRBS model assumes straight and parallel contours seaward of the depth of

closure, h,, and contours parallel to the nourished shoreline landward of h,. By considering the








ratio of the wave celerity at h., C., and the deep water wave celerity, Co, the longshore

diffusivity is entirely expressed in terms of deep water wave quantities, with the exception of a.,

the incoming wave angle at h., and C.. The longshore diffusivity is now expressed as



S KH2.4CGol2g 0.4 cos.2 (1o -ao )cos2(o, -a.)]
G = (
8(s -1)(1- p)C.CO.4 (h. + B) cos(,3 -a.) (4.6)



where the subscript "o" denotes deep water conditions; Ho is the deep water wave height, CGo

is the deep water group celerity, P, is the azimuth of the outward normal to the ambient depth

contours in deep water and co is the deep water incoming wave angle.

Starting from the sediment transport equation at the breaker line, using conservation of

energy considerations coupled with Snell's Law and the continuity equation, the transport

equation used for numerical analysis was derived to yield



KHo 2.4 CG .20.4 COSI.2 (P -
Q"= sin(fl ,)
8(s -l1)( p)C.rc0.4 (4.7)



where fi is the azimuth of the outward normal to the depth contours within the depth limit

affected by the nourishment project. With the exception of the trigonometric term involving

(,9 a.) and the term C., all quantities are again expressed in terms of deep water conditions.

The DNRBS model was applied to the NOURTEC area over a 10 km longshore stretch

immediately south of groin Q. Figure 4.21 displays the northern 5 km of the study area with the

shoreline indentation south of groin Q, the location of the beach nourishment, the revetments and

the multiple beach breakwaters located at the shoreline; the southern 5 km half of the modeled






71



A


Groin Q --











Beach Breakwater











Shoreline












Revetment

,-r
















A A

Figure 4.21: Northern 5 km stretch of study area.








stretch has no manmade features or irregularities in its nearly uniform shoreline and is thus not

shown on the figure.

The model is capable of including the downdrift erosion adjacent to a partial littoral

barrier such as groin Q. In order to include the proper amount of sediment bypassing the groin, an

annual sediment budget for the 10 km stretch was developed. Provided that the longshore

volumetric sand transport rate, Q, for any alongshore location on a beach, x, is known, beach

changes can be predicted by the continuity equation


aQ av (4.8)
+-T=s
ax )t



where s is an additional source term indicating any material added per unit length per unit time

and V is the volumetric change per unit beach length over a time period t. After profile

equilibration occurs under the assumption that as beaches erode or accrete, the profile moves

without change of form in a landward or seaward direction, respectively, this volumetric change

can be shown to be given by the Bruun rule



1 av (4.9)
S(h + B) at



where Ay is the shoreline change for a given volumetric change AV. In the considered period

from 1985 to 1992, the annual shoreline retreat at Torsminde is 1.4 m/year, accounting for the

average nourishment-related shoreline advance of 1.0 m/year, Laustrup et al. (1997a).

The sediment transport capacity of the incident waves can be computed from Eq. (4.7).

Using representative wave and site characteristics for the monitoring period: Ho= 1.5 m, =








2730 and to = a.= 2830, the annual net southerly sediment transport is estimated to be on the

order of 1.5 million m3/year.

In the time period 1985 to 1992, an average of approximately 250,000 m3/year was

supplied to the study area as nourishment, largely as beach nourishment. The small amount of

volume placed as shoreface nourishment before 1992, represents 6% of the total added volume

and was thus included in the average annual supply.

Under the assumption that the extensive annual nourishment prior to 1993 nourishment

have established a uniform alignment of the shoreline and following integration of the continuity

equation, Eq. (4.8), over the 10 km stretch, the amount of sediment bypassing groin Q was found

to be approximately 750,000 m3/year. This volume was included in the model as a constant

sediment supply at the location of the groin Q.

The DNRBS model neglects wave diffraction in the lee of a littoral barrier or more

precisely in this modeled case, downdrift of a sediment sink as the amount of sand bypassing the

groin is less than the transport capacity of the waves. Thus, accentuated erosion immediately

downdrift of groin Q should be expected in the modeling of planform evolution. In reality, this

erosion is limited locally and distributed downdrift as the shoreline stretch south of the groin

features several beach-parallel breakwaters and continuous revetments.

Figure 4.22 displays on a distorted scale the calculated planform evolution from 1985 to

1992 for the 10 km modeled stretch. The shoreline progresses from South to North from the left

of the plot at 0 km to the right at 10 km at the location of groin Q. All 18 nourishment projects

from 1985 to 1992 have been included at their respective locations and with their actual

distributed densities, although only selected years are displayed for clarity.

The shoreline is seen to retreat downdrift of groin Q and advance at the location of the

greatest nourishment density. The computed alongshore average of shoreline variation was found

to be -1.34 m. In other words, all the different volumetric changes within the modeled stretch












20

Q


10




0
1 2 3 4 5 6 7



-10




-20




-30

1987

1989
-40 -
--1991

1993


-50
Shoreline Length [ km ]

Figure 4.22: Calculated planform evolution for multiple nourishment projects from 1985 to 1992.








were reasonably accounted for. A -1.4 m average shoreline change was expected as it resulted

from the sediment budget in which the annual erosion was not incorporated.

The resulting -1.4 m shoreline change infers that the 10 km shoreline stretch downdrift of

the groin field was in approximate volumetric sediment equilibrium with a background coastal

retreat rate of 2.4 m/year and with a nourishment rate of 250,000 m3/year corresponding to a

shoreline advance rate of 1.0 m/year. Thus, the remaining shoreline retreat rate of 1.4 m/year

should be expected with the actual annual nourishment rate.



4.4.3 Effect of Distance to Groin Field

The proximity of the updrift groin field with its terminal groin Q and the inevitable

downdrift erosion into the study area, might be in part responsible for the rapid erosion of the

beach nourishment. Although the previously predicted project longevity was in reasonable

agreement with the observed lifetime without taking the effect of the adjacent groin field into

account, the effect of the distance from the beach fill relative to groin Q, can be significant.

Placement of sand in an area prone to extensive erosion, for example, downdrift of a partial

littoral barrier, is certainly where it is most needed but at the same time where it is inherently

most unstable. By varying the location of the beach nourishment, the effect of the distance to the

groin field can be estimated.

Figures 4.23 and 4.24 display the planform evolution for the centerline of the beach

nourishment located 5 km and 8 km downdrift of groin Q, respectively. Both nourishment

volumes were 250,000 m3 of sand distributed alongshore with a density of 250 m3/m. The initial

dry beach width was chosen as 45 m from consecutive surveyed profile elevations before and

after implementation of the nourishment.

The computed percentages of remaining volume after the first as well as the second year

were for each location nearly equal; 23% remained after the first year and 17% after the second

year. However, these percentages are not in agreement with the observed evolution of the beach



















40





20




E
0
0



o
S-20





-40





-60





-80


Shoreline Length [ km ]


Figure 4.23: Calculated planform evolution for beach nourishment located 5 km downdrift of groin Q.





























S1 2 3 4 5 6 7 9



0

















80
I-20










-60 Year 1






Shoreline Length [ km ]



Figure 4.24: Calculated planform evolution for beach nourishment located 8 km downdrift of groin Q.








nourishment project, which eroded entirely within the first year. This lack in predictive capability

is probably related to the assumption that the volume of beach nourishment sand is evenly

distributed over the entire height of the active profile, from dune top at.+6 m DNN to depth of

closure at -16 m DNN. The beach nourishment project was placed with much steeper profile

from +4 m DNN to -2 m DNN and was likely eroded before it ever nearly reached equilibrium in

the cross-shore direction.

Nevertheless, the present results suggest that the relative performance of the beach

nourishment compared with shoreface nourishment is not related to its proximity to the groin

field. As noted previously, the presence of the shore armoring would tend to distribute some of

the predicted erosion immediately downdrift of groin Q, further downdrift and would place

greater erosion on the beach nourishment.



4.5 Submerged Berm Response



4.5.1 Migration of the Berm

The shoreface nourishment was constructed as a nearly symmetrical berm. Immediately

following its completion, the berm cross-section became asymmetrical with a steeper landward

slope. In part, this asymmetry reflects the initial diffusive processes or the spreading of sediment

down-slope into the approaching waves. As the berm migrated, the asymmetry continued

throughout the monitoring period. This asymmetry reflects the shoreward transport as waves

carry sand over the crest to deposit on the leeward side of the berm.

During the first year of monitoring, the location of the initial trough between the inner

and outer bar became the location of the new crest as the berm migrated shoreward. The inner bar

also migrated shoreward. The berm responded along its entire length in the same persistent

shoreward fashion; a typical profile development in the region of the nearshore berm is presented




























































Figure 4.25: Berm development for average profile 21900.


z
z

c
0

(
El


-8 Iii I I
1400 1450 1500 1550 1600 1650 1700 1750 1800 1850 1900 1950
Distance Offshore [ m ]


2000








in Figure 4.25 for average profile 21900. Over the monitoring period, the berm migrated 50-75 m

onshore and the level of the top of the berm was reduced by approximately 0.5 m.



4.5.2 Breakwater and Feeder Effects

The shoreface nourishment was constructed with the intent of both reducing incoming

wave energy and causing breaking over the berm and to supply material to the littoral zone of the

beach. Both effects were found to be occurring. The breakwater effect of the berm was most

pronounced by visibly increasing beach volume on some stretches and by reducing erosion on

other stretches. The feeder effect was determined by corings taken on the beach face and analyzed

for the coarser shoreface nourishment material.

The performance of an offshore breakwater is based on the principle of reducing the

amount of wave energy that reaches a coastline in order to reduce shoreline erosion. As in the

case of a shoreface nourishment, the breakwater can be effective even if it is not emergent. Wave

diffraction landward of the breakwater, will lead to an equilibrating shoreline planform

characterized by a depositional feature behind the breakwater, referred to as a salient. If the

incoming wave angle is nearly normal to the shoreline, the salient is located symmetrically

around the breakwater. The reduction of wave height by diffraction implies a decreasing local

transport, resulting in impoundment of sediment. Ultimately there is a downdrift erosion due to

this impoundment of sand by the advanced shoreline.

According to the hydrographic results for the whole monitoring period, the annual

average wave direction was oriented approximately 2830 relative to true North. The coast normal

is 2730 along the entire shoreline, in other words, the average incident deep water wave angle can

arbitrarily be taken as 100 off the shore normal.

The nearshore in the surveyed area is characterized by straight and parallel offshore

contours, hence each wave refracts similarly along the beach in the longshore direction. As

incoming waves shoal, this change in wave direction further reduces the incoming wave angle to














3330


N






2830


Figure 4.26: Measured and expected breakwater effects.








the shoreline, thereby justifying the nearly symmetrical depositional feature behind an offshore

breakwater.

The measured breakwater effects induced by the shoreface nourishment are given in

Figure 4.26. Together with these monitored effects, are superimposed the breakwater effects as

would be expected by the measured hydrographic results and as described above. As mentioned

in the previous section, a shoreline accretion was seen to occur directly downdrift of the

nourishment and not behind the nourishment. This would correspond to an incoming wave angle

approximately 500 off the actual measured wave angle. The dotted line corresponds to the

expected shoreline development under the measured average wave direction; wave refraction is

not taken into account, but if so, would increase the discrepancy.

The observed accretionary and erosional effects on the shoreline behind the nourishment

can not be attributed completely to the breakwater effect of the shoreface nourishment. The

shoreline development as it occurred is not typical of a pattern caused by an offshore breakwater.

Other processes must have strongly interacted, such as those induced by the considerable

southgoing littoral drift, and might in part be responsible for the observed development.

The effect of the berm serving as a supplemental source of feeder material, was much less

pronounced. Through sediment sampling, it was found that the 40% loss of the nourishment sand

at the end of the monitoring period, only accounted for 1/3 of the total accretion landward of the

nourishment. This relates to the placement of sand on the outer bar where the nourishment

quickly becomes an integral part of the natural bar system. Instead of being eroded away, the

nourishment behaves like a longshore bar in dynamic equilibrium with the nearshore littoral

processes. The feeder effects are thus reduced.














CHAPTER 5
MODEL DERIVATION AND SOLUTION



A model for the migration of submerged berms constructed offshore of the day-to-day

surf zone but within the depths disturbed by waves during storm events was proposed by

Douglass (1995). The model assumes that landward migration of the nourishment is due primarily

to the net landward transport by the velocity asymmetry of finite-amplitude waves, and is based

on Bagnold's bedload-sand transport equation and Stokes' second-order wave theory.

Conservation of sand considerations in the cross-shore direction lead to the classical convection-

diffusion model equation. In the present study, this model is derived and forms the basis for

further investigation.

The combined convection-diffusion model equation presented, is a classical nonlinear

partial differential equation. By using readily established methods in computational fluid

dynamics, the equation can be solved numerically. Given a specified sea state in form of a wave

climate time history, the equation predicts the behavior of a berm; it implies that the form of a

berm will migrate landward while it diffuses.



5.1 Model Derivation

To illustrate the coastal profile development under onshore-offshore transport from a

basic sediment transport model, the equations proposed by Bagnold (1963) have been used.

Bagnold's approach has been applied fairly successfully in a wide variety of sedimentological

problems and will thus also be used in this study in an attempt to determine quantitative

predictions of onshore-offshore transport.








Bailard and Inman (1981) applied Bagnold's transport model to a coastal environment as

a basis for the development of a total load model of time varying sediment transport over a plane

sloping bed. This model will be used in the cross-shore direction under, the assumption that all

water motion is directly onshore or offshore.

Neashore placement of sand is typically placed seaward of the surf zone. A fundamental

concept for the model is that in such depths, the dominant driving force is the waves. Specifically,

the landward migration of sediment is due to the net landward transport by the asymmetry of

finite-amplitude waves. Douglass (1995) applied Stokes' second order wave theory to describe

this velocity asymmetry.

Consideration of conservation of sand with the wave theory and transport model will lead

to a wave-based model of berm migration.



5.1.1 Energetics Transport Model

Current sediment transport models reflect two distinct approaches: an energetic

approach and a traction approach. In both cases, these approaches are based on adaptation of

stream flow sediment transport models. The energetic approach, which is used in the present

study, is based on the stream transport model developed by Bagnold (1963).

Bagnold's energetics-based sediment transport model assumes that the sediment is

transported in two distinct modes, each differing according to the support of the sediment grains.

Sediment transported as bedload is supported on the bed via grain to grain interactions, while

sediment transported as suspended load is supported by the stream fluid via turbulent diffusion. In

both modes, energy is dissipated by the stream by transporting the sediment load. Bagnold,

comparing the stream to a machine, defined the sediment transport efficiency as the ratio of the

rate of energy expended in transporting the bed load, divided by the total rate of energy

production of the stream.








For steady, two-dimensional stream flow, Bagnold (1963) developed the following total

load sediment transport equation:



L b +
= ib + b, E, s
i, = tano -tanf P (WI)-tan p) (5.1)



where i, is the total immersed weight sediment transport rate composed of the sum of the bedload

transport rate, ib, and the suspended load transport rate, i,, (0 is the rate of energy production of

the stream, i is the mean velocity of the stream, tan P is the slope of the stream bed, 0 is the

internal angle of friction of the sediment, W is the fall velocity of the sediment and ,b and E,

are the bedload and suspended load efficiencies respectively.

For oscillatory flows, such as those found in the surf zone, Inman and Bagnold (1963)

further developed a conceptual sediment transport model, currently the most widely accepted and

equivalent to the equation recommended by the Shore Protection Manual (1984). Based on

Bagnold's models, Bailard and Inman (1981) presented a total load sediment transport model

developed for time-varying flow over an arbitrarily sloping planar bed. The model predicts the

local, near-bottom sediment transport rate as a function of the near-bottom water velocity vector.

Following Bailard and Inman (1981), by integrating the instantaneous bedload and

suspended load sediment transport equations in time, the total load sediment transport is obtained

as

= p C Eb (l\-\2l- tan 3 .l-l3%
('t)=pC a t--no[ tan (0i

+ pC ((l f )- tan l ) < (5.2)








where p is the density of water, C, is a friction coefficient, the velocity vector ii has an


arbitrary orientation to the y -axis, j is a unit vector pointing shoreward and ( ) means average

in time over one wave period.

In both bedload and suspended load, the transport vectors are found to be composed of a

velocity-induced component directed parallel to the instantaneous velocity vector and a gravity-

induced component in the cross-shore direction.



5.1.2 Wave-Based Model of Berm Migration

The success of a shoreface nourishment project is directly dependent on the depth in

which the nourishment is placed. If placed within the outer limit of sand-transport initiation by

waves, the nourishment is expected to migrate onshore. Nearshore nourishment of sand is

typically placed offshore of the location of the outer bar in depths beyond the day-to-day surf

zone but within depths where the wave orbital velocities are capable of moving sediment during

higher waves and storm events. In these water depths, wave breaking-induced turbulence is

minimal. As turbulence is the most important factor in the suspension of sediment, bedload

transport can reasonably be assumed to be the dominant transport mechanism outside the surf

zone. Thus, in the following, only bedload transport is considered in the cross-shore sediment

transport. This assumption simplifies greatly the sediment transport equation, as the second term

in Eq. (5.2) reduces to zero.

In order to enhance the longevity of a shoreface nourishment project and the overall

performance of the project, nourishment material was chosen coarser than the native material at

the same cross-shore location. The suspension of such coarse material would require a highly

turbulent flow field, which is not present at the nourishment location. Hence, the use of coarse

sediment further justifies bedload as being the main cross-shore sediment transport mode outside

the surf zone.








Another simplifying assumption can be made, by only considering water motion in the

cross-shore direction. This assumption is consistent with the interest in the onshore migration of

the nourishment and the corresponding cross-shore transport. By incorporating all the above

assumptions into Eq. (5.2), the bedload transport rate becomes




(i)= Cf E (tan-tanfl)(u3)
tan 2 (5.3)



Flows induced by dissipation of wave energy in the nearshore, play a dominant role in

sediment transport. A variety of cross-shore flows, such as asymmetric oscillatory flow, breaking

induced turbulent flow and momentum decay-induced undertow have been identified by Roelvink

and Stive (1989), with the aim of revealing the role of each of the flow mechanisms in the two-

dimensional case of modeling bar generation. One of these flows is the breaking-induced

undertow, the role of which in the bar generation was first described qualitatively by Dyhr-

Nielsen and S0rensen (1970). A study of Stive (1986) shows that it is important to account for the

effects of wave asymmetry. These studies indicate that some of the main bar-generating cross-

shore flow mechanism are identified.

As the shoreface nourishment was, placed on the offshore flank of the outermost bar

beyond the day-to-day surf zone, the effects of undertow are reduced and wave asymmetry

mechanism dominate. Considering depths beyond the day-to-day surf zone, this fundamental

concept is used by Douglass (1995) in assuming that the wave asymmetry mechanism dominates

the water motion.

In intermediate depths, the velocity profile of a finite-amplitude wave is characterized by

larger forward velocities under the crest than the backward velocities under the trough; the wave

profile is much more peaked at the wave crest and flatter at the wave trough. Douglass proposed

using Stokes' second-order wave theory, the lowest order wave theory with wave asymmetry, to








describe the orbital water velocities. Stokes' second order wave theory predicts near-bottom

velocities of

HgT cos(2tl/r) 37r2H2 cos(4rt/T)
U= +
2L cosh(27hlL) 4 LT sinh4(2rh/L) (5.4)



where u is the near-bottom water velocity, H is the wave height, L is the wave length, T is the

wave period and h is the water depth. Substituting Eq. (5.4) into Eq. (5.3) and following

integration with respect to time, the bedload transport rate is


9Qr2 pg2C/e H4T 2
(ib) 2 r fb (tan -tan )-- -sech2 (2r h L)csch4(27r h L)
64 tan 2 Le (5.5)



The volumetric bed load sediment transport is obtained as

ibPs
P (P P)g (5.6)



Substitution of Eq. (5.5) into Eq.(5.6) yields


9Q2 p gCEb HT
Sq=- gC b (tan -tan P)-T sech2(27rh/L)csch4(27rh/L)
64 (p, -p)atan2 L3 (5.7)



where a' is the ratio of total volume to volume of solids and p, is the sand density.

Considering the one-dimensional conservation of sand in the cross-shore direction,

h =_ q (5.8)
at ay




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