UFL/COELTR/098
REALISTIC PREDICTION OF
BEACH NOURISHMENT PERFORMANCE
BY
CHULHEE YOO
DECEMBER, 1993
DISSERTATION
I
REALISTIC PREDICTION OF
BEACH NOURISHMENT PERFORMANCE
By
CHULHEE YOO
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN
PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1993
ACKNOWLEDGEMENTS
At the University of Florida, I have met a very exceptional person. For the last
four years, he has been the most frequently used reference for me in my research. His
enthusiasm on the research is simply unbelievable and his personality is purely unaf
fected. I have never seen him look at a book on my numerous questions nor overlook
even an extremely stupid question. Not only academically but also humanistically he
is a perfect person for me. Since I have met him in 1989, I am proud of my relation
ship with him and it makes me very happy at every moment. I sincerely enjoyed my
life with him and regret our separation in a certain sense. He is my adviser, Professor
Robert G. Dean. His encouragement, enthusiasm and directions were essential factors
on my achievement. I would like to express my most grateful thanks to him .
My appreciation is also extended to my committee members, Dr. Hsiang Wang,
Dr. Daniel M. Hanes, Dr. Ashish J. Mehta, and Dr. Ulrich H. Kurzweg. I am also
grateful to Dr. Robert J. Thieke, Dr. D. Max Sheppard and former adviser, Dr.
Charles K. Sollitt, at the Oregon State University for providing useful background on
various subjects. Their efforts and advice became a basis of my work. I have a great
debt to my dear friends Dr. Jung Lyul Lee and Mr. Jorge Abramian who shared
their knowledge from the beginning. Discussions with them were extremely useful
and made it easier to understand various physical problems. From time to time we
worked together all through the night exchanging our knowledge. I believe that they
will succeed in their life just as they did in school. Thanks go to Mr. Subarna Malakar
for the assistance of my PC installation and for his patience in the beginning of my
school when I was not familiar with the system and Mrs. Cynthia Vey for preparing
manuscripts for publication of my papers. I am also indebted to Mrs. Becky Hudson,
Mrs. Sandra Bivins, Mrs. Lucy Hamm, Mrs. Helen Twedell, and Mr. John Davis.
Personal communications with them were also always pleasant.
Many thanks, and my fortune, go to my wife, Mrs. Soonmi Yoo and my lovely
daughter Sookyung Yoo for their smiles, patience, love and for not asking to go back
to my country in the middle of my study. I must thank my parents, Myungja Lee
and Insick Yoo (who passed away in 1983) most of all for their ultimate support
throughout my life and for giving me the wonderful ability to see the real world. If I
did not have such an ability, I would have missed meeting these unforgettable persons.
TABLE OF CONTENTS
ACKNOWLEDGEMENT ............................. ii
LIST OF FIGURES ............ ..................... vi
LIST OF TABLES ................................. xii
ABSTRACT ............ ...................... xiii
CHAPTERS
1 INTRODUCTION ........... ................... 1
1.1 General Description ............................ 1
1.2 Review of Previous Research ......................... 6
1.3 Scope of Study ............................... 12
2 GOVERNING EQUATIONS ......................... 15
3 METHODOLOGY ............................... 23
3.1 Wave Field Calculations ......................... 23
3.1.1 Simplified Wave Refraction and Shoaling ............ 23
3.1.2 Detailed Refraction and Shoaling ................ 25
3.2 Grid System and Transformation of Initial Geometry ........ 27
3.3 Background Erosion ........................... 30
3.4 Numerical Solution of Governing Equations ............... 30
3.5 Effective Wave Height and Period .......... ........ 31
4 MODEL RESULTS .................. ............. 33
5 APPLICATIONS TO LARGE SCALE DATA ................ 55
5.1 The Field Data .............................. 56
5.2 W ave Data ... ...................... ...... .. 59
5.3 Characteristics of Project Evolution ................. .... 61
5.4 Analysis Procedure ............................. 63
5.5 Results ....................... .. ... ........ 64
5.6 Engineering Approach ..................... .... 70
6 BEACH NOURISHMENT IN THE PRESENCE OF A SEAWALL .... 73
6.1 Introduction ............. ................. 73
6.2 Analytic Development ............................. 74
6.2.1 Conservation of Volume ....................... 74
6.2.2 Planform Centroid Migrational Characteristics . ... 76
6.2.3 Planform Variance ....................... 77
6.3 Numerical Treatment ................... .... .... 78
7 EXPERIMENTAL STUDY FOR NOURISHMENT ON SANDY AND SEA
WALLED SHORELINES .. ... .. ............ .... 86
7.1 Introduction ... .... ... ... ..... ....... 86
7.2 Experimental Procedure ................ ......... 86
7.3 Description of the Experiments .................... 88
7.3.1 Experiment ............................ 88
7.3.2 Experiment 2 ........................... 89
7.3.3 Experiment 3 ............................. 94
7.3.4 Experiment 4 ........................... 94
7.3.5 Experiment 5 .................. ........ 99
7.3.6 Experiment 6 ........................... 99
8 ANALYSIS OF LABORATORY RESULTS ............... 105
8.1 Other possible transport reduction factor ............... 105
8.2 Experiment 1 .................... ........... 106
8.3 Experiment 2 ..... ............... ........... 111
8.4 Experiment 3 ... .............. .............. 116
8.5 Experiment 4 .................... ........... 116
8.6 Experiment 5 .................. .............. 121
8.7 Experiment 6 .................... ........... 128
8.8 Behavior of Planform Centroid and Variance .............. 128
8.9 Design Consideration ........................... 138
9 SUMMARY, CONCLUSIONS AND RECOMMENDATIONS FOR FUTURE
RESEARCH ...... ........... ................. 139
9.1 Summary ............................... 139
9.2 Conclusions ......................... .... 140
9.3 Suggestions for Future Research ................. .... .. 142
BIBLIOGRAPHY ................................ 144
BIOGRAPHICAL SKETCH ........................... 150
LIST OF FIGURES
1.1 Processes Affecting Beach Nourishment Projects. . . 2
2.1 Definition Sketch. ................... .... 17
2.2 Three Generic Types of Nourished Profiles. . . ... 21
3.1 Definition Sketch for Effect of Beach Nourishment on Contours. 24
3.2 Definition Sketch for Numerical Model. . . .... 28
3.3 Recommended Ad Hoc Transformation for Modeling Coastal Sys
tems in which Large Perturbations are to be Introduced. . 29
4.1 Example 1. Comparison of Beach Nourishment Evolution for Sim
ple and Detailed Methods of Wave Refraction and Shoaling, Nor
mal Incidence, Ho=0.6 m, T=6.0 sec, ao = 90. No Background
Erosion. . . . . . . . .. 34
4.2 Example 2. Comparison of Planform Evolution Obtained by Two
Methods for 200 Oblique Waves, Ho=0.6 m, T=6.0 sec, ao = 70.
No Background Erosion ................... .... 36
4.3 Example 3. Planform Evolution by Simple Method for Deep Water
Wave Directions, a, = 700, 80,90, Ho=0.6 m, T=6.0 sec. No
Background Erosion ................... ....... 37
4.4 Example 3. Planform Evolution by Detailed Method for Deep
Water Wave Directions, a. = 70, 80, 900, Ho=0.6 m, T=6.0 sec.
No Background Erosion ........... ..... ......... 38
4.5 Example 4. Even and Odd Components of Shoreline Position After
10 years for Deep Water Wave Directions, a0 = 700, 800, Ho=0.6
m, T=6.0 sec. Results Obtained by Detailed Method. ...... .40
4.6 Example 5. Illustration of Wave Height Effect on Rate of Planform
Evolution. Results Based on Simple Method, ao = 900, T=6.0 sec.
Results Shown for 1, 3, 5 and 10 Years. No Background Erosion. 41
4.7 Example 6. Effects of Various Project Lengths and Wave Heights
on Project Longevity. ......................... .. 42
4.8 Example 7a. Effects on Planform Evolution of Two ShoreNormal
Retention Structures of Length Equal to OneHalf the Initial Project
Width. Normal Wave Incidence. No Background Erosion. Ho=0.6
m T=6.0 sec .............................. 44
4.9 Example 7b. Effects on Planform Evolution of Two ShoreNormal
Retention Structures of Length Equal to OneHalf the Initial Project
Width. Normal Wave Incidence. Uniform Background Erosion
Rate at 0.5 m/yr, Zero Background Transport at Project Center
line. Ho=0.6 m, T=6.0 sec ..................... 46
4.10 Example 7c. Effects on Planform Evolution of Two ShoreNormal
Retention Structures of Length Equal to OneHalf the Initial Project
Width. Normal Wave Incidence. Uniform Background Erosion at
0.5 m/yr, Zero Background Transport Located 4,500 m to Left of
Left Structure. Ho=0.6 m, T=6.0 sec . . . .. 47
4.11 Plot of K vs D (Modified from Dean, et al., 1982). . ... 49
4.12 Example 8. Planform Evolution for Nourishment Sand Less Trans
portable than the Native (KF=0.693, KN=0.77). Note Centroid
of Planform Migrates Updrift. Variation of Surface Layer K Val
ues with Time at Locations A, B, C, D and E are presented in
Figure 4.14. Wave and other Project Conditions are the Same as
Example 2. ............................... 50
4.13 Example 8. Planform Evolution for Nourishment Sand More Trans
portable than the Native (KF=0.847, KN=0.77). Note Centroid
of Planform Migrates Downdrift. Variation of Surface Layer K
Values with Time at Locations A, B, C, D and E are presented in
Figure 4.15. Wave and other Project Conditions are the Same as
Example 2. .............................. 51
4.14 Example 8. Variation of Surface Layer Longshore Transport Co
efficient K with Time at the Five Locations Shown in Figure 4.12.
Case of KF=0.693, KN=0.77, Ym;,=2.0 m. Refer to Figure 4.12
for Locations of Points A, B, C, D and E. . ... .. 53
4.15 Example 8. Variation of Surface Layer Longshore Transport Co
efficient K with Time at the Five Locations Shown in Figure 4.13.
Case of KF=0.847, KN=0.77, Ymiz=2.0 m. Refer to Figure 4.13
for Locations of Points A, B, C, D and E . . .... 54
5.1 Plan View of Beach Monitoring Profiles in the Sand Key Project
and Location of Directional Wave Gage . . ... 57
5.2 Initial Shoreline Displacement and Placement Volume Density.
(Monitored Profiles Shown As Dashed Lines) . . ... 58
5.3 Illustration of Nourishment and Subsequent Evolution Over a Two
Year Period for Profile R102G. Note That Initially There Was an
Exposed Seawall at This Profile . . . .. 60
5.4 Three Phases of Observed Sediment Transport in Vicinity of Nour
ished Projects. Note: CrossContour Transport Due to Profile
Disequilibrium (from Dean et al., 1992). . . ... 62
5.5 Variation of Error Measure and Best Fit Transport Coefficient(K =
1.02) for Eq.(1) and Southern Half of Monitored Region .... 66
5.6 Volumetric Change Distributions at Two Years Over Monitored
Area for BestFit Coefficient (K=1.02). Measured vs Predicted.
CrossShore Transport Taken Into Account . . . 67
5.7 Variation of Sediment Transport Coefficient with Sand Size. Re
sults Include Previous Field Studies and That of This Project. 68
5.8 Volumetric Change Distributions at Two Years for "Engineering
Approach". No Consideration of CrossShore Transport, and K =
0.77. Measured vs Predicted. . . .... ...... .. 71
5.9 Shoreline Displacement Change Distributions at Two Years for
"Engineering Approach". No Consideration of CrossShore Trans
port, and K = 0.77. Measured vs Predicted. . . ... 72
6.1 Definition Sketch for Beach Nourishment in the Presence of a Sea
wall.............. ...... ............. .. 75
6.2 Calculated Planform and Volumetric Evolutions of an Initially
Rectangular Beach Nourishment Project Fronting a Seawall. Nor
mal Wave Incidence. Based on Numerical Model, n = 1. ..... .81
6.3 Calculated Planform and Volumetric Evolutions of an Initially
Rectangular Beach Nourishment Project Fronting a Seawall. Deep
Water Waves at 100 to Shore Normal, n = 1 . . .... 82
6.4 Translation Speed of Center of Gravity of Nourishment Volumetric
Anomaly. Deep Water Waves at 100 to Shore Normal, n = 1. 84
6.5 Time Dependence of the Nourishment Volumetric Variance. Deep
Water Waves at 0 and 100 to Shore Normal . . .... 85
7.1 Wave Basin Arrangement. Wave Rays Shown For Normal Wave
Incidence .................. ............ 87
7.2 Case l.a. Contours Showing Planform Evolution For a Seawalled
Shoreline. Normal Wave Incidence. Contours Shown Include +8
cm, +4 cm, 0, and 4 cm Relative to Still Water Level. Sediment
Size=0.2 mm.......... ........ ............. 90
7.3 Case 1.b. Contours Showing Planform Evolution For a Seawalled
Shoreline. Normal Wave Incidence. Contours Shown Include +8
cm, +4 cm, 0, and 4 cm Relative to Still Water Level. Sedi
ment Size=0.2 mm. Note: This is a Repeat of the Test Shown in
Figure 7.2... . ................... .. 91
7.4 Case 2.a. Contours Showing Planform Evolution For a Seawalled
Shoreline. Oblique Wave Incidence (30"). Contours Shown In
clude +8 cm, +4 cm, 0, and 4 cm Relative to Still Water Level.
Sediment Size=0.2 mm. ....................... 92
7.5 Case 2.b. Contours Showing Planform Evolution For a Seawalled
Shoreline. Oblique Wave Incidence. Contours Shown Include +8
cm, +4 cm, 0, and 4 cm Relative to Still Water Level. Sedi
ment Size=0.2 mm. Note: This is a Repeat of the Test Shown in
Figure 7.4 .. .............................. 93
7.6 Case 3.a. Contours Showing Planform Evolution For a Seawalled
Shoreline. Normal Wave Incidence. Contours Shown Include +8
cm, +4 cm, 0, and 4 cm Relative to Still Water Level. Sediment
Size=0.5 mm .............................. 95
7.7 Case 3.b. Contours Showing Planform Evolution For a Seawalled
Shoreline. Normal Wave Incidence. Contours Shown Include +8
cm, +4 cm, 0, and 4 cm Relative to Still Water Level. Sedi
ment Size=0.5 mm. Note: This is a Repeat of the Test Shown in
Figure 7.6.... ... .. .. .. .. ... ... .. .. ... .. 96
7.8 Case 4.a. Contours Showing Planform Evolution For a Seawalled
Shoreline. Oblique Wave Incidence. Contours Shown Include +8
cm, +4 cm, 0, and 4 cm Relative to Still Water Level. Sediment
Size=0.5 mm ............................. 97
7.9 Case 4.b. Contours Showing Planform Evolution For a Seawalled
Shoreline. Oblique Wave Incidence. Contours Shown Include +8
cm, +4 cm, 0, and 4 cm Relative to Still Water Level. Sedi
ment Size=0.5 mm. Note: This is a Repeat of the Test Shown in
Figure 7.8.... ... .. .. .. ... .. .. .. .. ... .. .. 98
7.10 Case 5.a. Contours Showing Planform Evolution For a Sandy
Shoreline. Normal Wave Incidence. Contours Shown Include +9
cm, +5 cm, +1 cm, and 3 cm Relative to Still Water Level.
Sediment Size=0.2 mm. ....................... 100
7.11 Case 5.b. Contours Showing Planform Evolution For a Sandy
Shoreline. Normal Wave Incidence. Contours Shown Include +9
cm, +5 cm, +1 cm and 3 cm Relative to Still Water Level. Sedi
ment Size=0.2 mm. Note: This is a Repeat of the Test Shown in
Figure 7.10 ......................... .. .. 101
7.12 Case 6.a. Contours Showing Planform Evolution For a Sandy
Shoreline. Oblique Wave Incidence. Contours Shown Include +9
cm, +5 cm, +1 cm, and 3 cm Relative to Still Water Level.
Sediment Size=0.2 mm. ..................... 103
7.13 Case 6.b. Contours Showing Planform Evolution For a Sandy
Shoreline. Oblique Wave Incidence. Contours Shown Include +9
cm, +5 cm, +1 cm, and 3 cm Relative to Still Water Level.
Sediment Size=0.2 mm. Note: This is a Repeat of the Test Shown
in Figure 7.12. .... ........................ 104
8.1 Comparison of Calculated Transport Distribution for Normal Waves
Incidence Based on Two Different Sets of Relationship. . 107
8.2 Comparison of Calculated Transport Distribution for Oblique Waves
Incidence Based on Two Different Sets of Relationship. . 108
8.3 Computed Error Surface for Normal Waves Incidence for Case 1.a,
Sand Size=0.2 mm, The First Test ............ ..... 109
8.4 Computed Error Surface for Normal Waves Incidence for Case 1.b,
Sand Size=0.2 mm, The Second Test. . . . .... 110
8.5 Comparison of the Measured and Predicted Transport Rate for
Normal Waves Incidence for Case 1.a, The First Test, Parameters
for Calculated Results: K=0.15, n=3.0. . . ... 112
8.6 Comparison of the Measured and Predicted Transport Rate for
Normal Waves Incidence for Case 1.b, The Second Test, Parame
ters for Calculated Results: K=0.12, n=3.0. . . ... 113
8.7 Computed Error Surface for Oblique Waves Incidence for Case 2.a,
Sand Size=0.2 mm, The First Test ................. 114
8.8 Computed Error Surface for Oblique Wave Incidence for Case 2.b,
Sand Size=0.2 mm, The Second Test. . . . .. 115
8.9 Comparison of the Measured and Predicted Transport Rate for
Oblique Wave Incidence for Case 2.a, The First Test, Parameters
for Calculated Results: K=0.32, n=1.5. . . ... 117
8.10 Comparison of the Measured and Predicted Transport Rate for
Oblique Wave Incidence for Case 2.b, The Second Test, Parame
ters for Calculated Results: K=0.35, n=1.8. . . ... 118
8.11 Comparison of the Measured and Predicted Transport Rate for
Normal Wave Incidence for Case 3.a, The First Test, Parameters
Calculated Results: K=0.02, n=3.0. . . . ... 119
8.12 Comparison of the Measured and Predicted Transport Rate for
Normal Wave Incidence for Case 3.b, The Second Test, Parameters
Calculated Results: K=0.02, n=3.0 . . . .. 120
8.13 Comparison of the Measured and Predicted Transport Rate for
Oblique Wave Incidence for Case 4.a, The First Test, Parameters
Calculated Results: K=0.15, n=1.5 . . ..122
8.14 Comparison of the Measured and Predicted Transport Rate for
Oblique Wave Incidence for Case 4.b, The Second Test, Parame
ters Calculated Results: K=0.14, n=1.5. . . ... 123
8.15 Comparison of the Measured and Predicted Volume Change for
Normal Wave Incidence for Case 5.a, The First Test, Parameter
Calculated Results: K=0.57. ................... 124
8.16 Comparison of the Measured and Predicted Volume Change for
Normal Wave Incidence for Case 5.b, The Second Test, Parameter
Calculated Results: K=0.62. . . . ..... 125
8.17 Computed Error Curve for Normal Waves Incidence for Case 5.a,
Sand Size=0.2 mm, The First Test. . . . .. 126
8.18 Computed Error Curve for Normal Wave Incidence for Case 5.b,
Sand Size=0.2 mm, The Second Test. . . . .... 127
8.19 Comparison of the Measured and Predicted Volume Change for
Oblique Wave Incidence for Case 6.a, The First Test, Parameter
Calculated Results: K=0.38. . . .... .. 129
8.20 Comparison of the Measured and Predicted Volume Change for
Oblique Wave Incidence for Case 6.b, The Second Test, Parameter
Calculated Results: K=0.30 . . . ..... 130
8.21 Computed Error Curve for Oblique Waves Incidence for Case 6.a,
Sand Size=0.2 mm, The First Test ................. 131
8.22 Computed Error Curve for Normal Wave Incidence for Case 6.b,
Sand Size=0.2 mm, The Second Test. . . . ... 132
8.23 Translational Speed of Center of Gravity for a Seawalled Shoreline. 133
8.24 Translational Speed of Center of Gravity for a Sandy Shoreline. 134
8.25 Variation of Volumetric Variance as a Function of Time for a Sea
walled Shoreline. ...................... ....... 135
8.26 Variation of Volumetric Variance as a Function of Time for a Sandy
Shoreline ............................. 136
LIST OF TABLES
5.1 Wave Characteristics. ...... ........... ........ 61
5.2 Comparison of Best Fit Coefficients and Their Respective Non
Dimensional Goodness of Fits, ei,. Based on Two years of Data. 65
5.3 Comparison of Best Fit Coefficients and Their Respective Non
Dimensional Goodness of Fits, eC,2. Based on Second Year's Data. 69
7.1 Experimental Conditions ...................... 88
8.1 Summary of Analysis Results From Planform Evolutions . 137
Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
REALISTIC PREDICTION OF
BEACH NOURISHMENT PERFORMANCE
By
CHULHEE YOO
December 1993
Chairman: Dr. Robert G. Dean
Major Department: Coastal and Oceanographic Engineering
A simple method is developed for representing wave refraction and shoaling in
the vicinity of a beach nourishment project. The method applies for the case of a
oneline model of shoreline evolution in which the active profile is displaced seaward
or landward without change of form. The model can include the presence of shore
perpendicular structures and background erosion.
The simple method is compared to a oneline model which includes a more de
tailed gridbased refraction and shoaling algorithm. For all cases tested, the simple
method of representing refraction and shoaling results in shoreline evolution in good
correspondence with the detailed method. The models are used to illustrate the effects
of several features of beach nourishment projects that are of engineering interest.
For purposes of comparison of the model performance with actual data, a nourish
ment project at Redington Shores, Florida, was monitored to investigate longshore
sediment transport processes. Monitoring was carried out over a twoyear period
and included seven surveys of twentysix profiles, wave measurements and sediment
sampling. These data were analyzed to evaluate three longshore sediment transport
formulae and to determine the bestfit coefficients and relative goodnessoffit.
Beach nourishment, placed along a seawalled shoreline, can exhibit a markedly
different behavior than projects on shorelines with adequate compatible sands to
transport. The most striking effects of engineering significance are: (1) the migration
of the centroid of the nourishment planform anomaly when acted upon by oblique
waves, and (2) a potentially different rate of spreading of the planform anomaly.
Under the most idealized consideration, the speed of centroid migration is shown
to increase as the planform anomaly spreads out under the mobilizing action of the
waves and the rate at which the planform spreading (dispersion) occurs is not affected
by the seawall. Also, for normal incident waves, the seawall does not affect the
planform evolution. Analtyical, numerical and experimental approaches are employed
to demonstrate the aforementioned effects.
CHAPTER 1
INTRODUCTION
1.1 General Description
Beach nourishment, the placement of large quantities of sediment in the nearshore
region to advance the shoreline seaward, is being applied increasingly as a method of
erosion control. Advantages over other methods include maintaining a nearnatural
wide beach for storm protection and recreational purposes. In some localities, the
wide beach also provides suitable nesting areas for several endangered marine crea
tures including sea turtles. Sand placed in this manner along a long, uninterrupted
shoreline represents a perturbation which, over time, tends to be smoothed out by
longshore sediment transport. Additionally, because the sand is usually placed at
a slope which is steeper than the "equilibrium" slope, seaward sediment transport
occurs. That is, if the sand is placed at ar itial profile that is steeper than equi
librium, offshore transport will occur with Isociated narrowing of the dry beach.
Figure 1.1 illustrates these adjustments.
The longshore sediment transport is mostly due to the waveinduced longshore
current, while the crossshore transport is a result of the water particle motion near
the surf zone due to the waves and the gravitational forces associated with the steeper
profile. Realistic prediction of the performance of a beach nourishment project in
cludes representation of both the crossshore and longshore transport processes and
is of considerable importance to a rational evaluation of the economic benefits of the
project. However, the coupling mechanism between the hydrodynamics and sediment
transport to date is not all that well defined.
Reduced Beach
Width Due to
Profile Equilibration
H
SNourished Profile
a) Equilibration of Nourished Beach Profile.
Evolving Planform
PreNourishment
Shoreline "
Initial Nourished
S" Planform
y
/"Spreading Out" Losses
b) Evolving Planform Due to "Spreading Out" of Initial Nourished Profile.
Figure 1.1: Processes Affecting Beach Nourishment Projects.
In spite of the many unknowns, numerous attempts have been made to predict
the performance of beach nourishment projects and also of modifications due to man
made structures within the coastal zone. Analytical and numerical models are often
employed with some simplification of the problem to describe the evolution of a nour
ishment project. With these simplifications and assumptions, a satisfactory model
should still describe the distribution of wave height over the domain of interest, en
ergy dissipation over the surf zone due to wave breaking, bottom friction, turbulence
and the resulting sediment transport relationship.
Methods have been proposed by a number of investigators to predict the perfor
mance of nourishment projects. A beach nourishment manual has been developed
based on experience in the Netherlands (Pilarczyk and Overeem, 1987) which, in
addition to performance, addresses dredge equipment, manner of placement, environ
mental effects, etc. Krumbein and James (1965), James (1974), and Dean (1974) have
proposed ad hoc but quantitative methods of assessing the quality of material. Ap
plication of the James method is described in the Shore Protection Manual (1984). In
general, these methods attempt to establish the required volume of borrow material
which is equivalent to one unit volume of native material. However, other parameters
critical to project performance are not addressed by these procedures.
Pilkey and his coworkers (Pilkey and Clayton, 1989; Leonard, Clayton and Pilkey,
1990; Leonard, Dixon and Pilkey,1990) have conducted "broad brush" analyses of the
performance of many projects along the East and Gulf coast of the United States.
Attempts have been made to define project performance in terms of the actual versus
design lifetimes of the projects. Also, performance was plotted versus project length,
sediment grain size, etc. In general, they found little or no correlation between pre
dicted and actual performances. Rather, the following equation was proposed for
the East Coast of the United States to represent the required renourishment volume,
VR, conducted at intervals of n (in years) for an initial restoration volume,V, and a
project life, N (in years).
VR = NV (1.1)
where n is renourishment interval and N is a project life. It was recommended that
n = 9 for Florida, n = 3 for New Jersey, and n = 5 for the remaining portions of
the East Coast. Eq. (1.1) simply states that it will be necessary to renourish with
an amount equal to the initial restoration volume every n years. The rationale for
Eq. (1.1) is not apparent from sediment transport considerations. The studies by
Pilkey and his coworkers do not recognize explicitly the effect of background erosion,
sediment quality or project length on project performance.
Dean (1983) reviewed available methods of predicting beach nourishment perfor
mance and showed that based on the PelnardConsidere (1956) solution and in the
absence of background erosion, the time, tP, required for a project to lose p percent
of the material placed is
tP H2 (1.2)
in which is the project length and Hb is the height of the breaking waves which mobi
lizes the sediment causing the spreading out beyond the project limits. Recently Dean
(1988b) has shown that the proportionality factors for 50% "loss" are approximately,
for units as shown
t5o% = K" 1 (1.3)
H"
in which t50% is in years and K"=0.172 years m/2/km2 for in kilometers and Hb in
meters, and K"=8.7 years ft5/2/mile2 for i in miles and Hb in feet. It is emphasized
that the material "lost" from the project area is transported alongshore to protect
adjacent areas and continues to provide benefits there (Dean,1988a).
Ideally, verification of a model would be established by comparison with data
obtained from actual projects monitored at proper intervals and with measured wave
conditions nearby. The performance of such a project is of considerable engineering
interest and provides a basis to improve the model capability. Of the many beach
5
nourishment projects constructed over the past approximately three decades, few
projects have been monitored adequately to allow detailed and quantitative evalua
tion of calculation procedures. Fortunately, the University of Florida and University
of South Florida jointly monitored a beach nourishment project at Redington Shores,
Florida, over a twoyear period. The project commenced in July, 1988 with the place
ment of approximately 405,000 m3 of sand along approximately 2.6 km of shoreline.
Project documentation included a wave gage located nearby at Clearwater, Florida.
Data collected included beach and offshore profiles and sediment characteristics. The
beach profiles were surveyed a total of seven times with the first and second surveys
occurring immediately prior to and following nourishment. Detailed analysis of these
project data and model performance will provide a great opportunity for study by
coastal engineers.
Certain features of beach nourishment evolution are well represented by analytical
models and may be somewhat counter intuitive. For example, if the nourishment and
native sands are hydrodynamically the same, an initially symmetrical planform will
remain nearly symmetric even though the waves approach the coast at an angle.
Moreover, the centroid of the planform anomaly remains almost fixed even under
oblique wave incidence. If the placed sediment is less or more transportable than
the native, the centroid will migrate in directions counter to and in the direction of
wave propagation, respectively (Dean and Yoo, 1992). A limiting case of mismatch of
the placed sediments and prenourished conditions is that in which the nourishment
is placed on a seawalled or rocky coast in which there are no native sediments to
be transported. This case, which will be the main subject of this dissertation, may
arise in areas where past efforts to limit erosion through shoreline hardening are
being replaced by beach nourishment. Although it may not be readily anticipated,
when beach nourishment is placed along a seawalled shoreline, the behavior of the
planform evolution is significantly different than that on a shoreline with adequate
6
compatible sand. Unlike the sandy coast case, the planform change pattern can be
extremely sensitive to the direction and the rate of spreading out of the planform
anomaly. When sand is placed in front of a seawall, near the ends of the project,
the actual sediment transport rate will be reduced significantly at the underwater
portions of the placed sand. Although, in these locations, a crude relationship for
sediment transport may be assumed for simplicity, quantifying this actual sediment
transport rate is important to the project performance and economic assessment. A
series of experiments has been conducted to focus on the project end conditions and
to generally address this problem.
In summary, the primary objectives of this study are 1) to develop a numerical
model for beach nourishment on both a sandy and seawalled shoreline, 2) verification
of model performances with large scale data for the sandy beach case, 3) to examine
a broad range of conditions for overall behavior of nourishment projects in the pres
ence of a seawall, and 4) to develop design recommendations. This paper describes
the methodology of numerical modeling, the results of physical modeling and the
associated analysis, and engineering applications to beach nourishment.
1.2 Review of Previous Research
For more than the past half century, many beach nourishment projects have been
constructed to protect eroding beaches. Of the many projects constructed, few were
monitored at all after construction or monitored for only a short time. Only a few
projects were monitored adequately to allow quantification of their evolution. Many
of the studies included measurements of beach evolution only to the extent that could
be established from wading profiles. However, there was very little information on
the subaqueous portion of beach volume change or of the waves causing the evolution.
Additionally, the waves which govern the evolution rate of the project were generally
not monitored. As a result, early monitoring projects often resulted in only qualitative
descriptions of project evolution and a general assessment of their performance.
In 1952, Watts reported on the behavior of beach fill at Virginia Beach, Virginia,
and concluded that basically the beach fill and subsequent periodic nourishment had
served to stabilize the shoreline and maintenance of the beach by artificial nour
ishment is the most economical method of providing stability of the shoreline. He
also commented that the data did not indicate the magnitude or dominant direction
of longshore transport, but did indicate that there is a substantial onshoreoffshore
movement of material which was probably related to seasonal and storm changes in
the wave characteristics in the area. Later (1956), he investigated the cause of the
erosion along the Ocean City beach front and the shore of Great Egg Inlet, New Jer
sey, and recommended beach nourishment for the prevention of further erosion and
the restoration of eroded beaches. Through a comprehensive analysis of the prob
lem, he was led to the conclusion that the most suitable plan of protection for the
problem area consisted of placement of suitable sand on the shore and the extension
of seven existing groins. Also he demonstrated the fineness of the beach fill material
had resulted in increasing the rate of loss of the beach. Similar conclusions (1958)
were made for a nourishment project in Harrison County, Mississippi. He indicated
that neither the beach fill nor seawall alone would provide complete protection from
severe storms. A combination of both would be required for adequate protection to
the shore during such conditions.
Hall (1952) introduced the outline of the design criteria pertinent to the artifi
cially nourished beaches and provided a brief history of five areas where four different
types (the offshore dumping method, stockpiling method, continuous supply method
and direct placement method) of artificial nourishment have been tried. He also
summarized many beach nourishment projects and sand bypassing.
In the nourishment project of the Island of Norderney in Germany, Kramer (1960)
noticed that the fill material at the beginning of the placement had been carried
off much quicker than later on. During the first several months, the unnaturally
8
steep beach above mean water level which was highly exposed to the wave attack
decreased to a large extent. In his subsequent studies (1972), including renourishment
of the same project, he observed the importance of seasonal variation of the shoreline
response and regional distribution of sand characteristics resulted in different behavior
of the nourishment evolution.
A study by Perdikis (1961) was conducted on the behavior of beach nourishment
projects on ten New England beaches. Although for most cases the offshore survey
extent was not specified, detailed quantitative descriptions of volume change and high
and low water shoreline positions were discussed. Beach slope comparisons were also
made for the region between mean high and low water level at selected profiles before
and after nourishment. The difficulty in predicting the behavior of beach fills from
the behavior of the original beaches was addressed in his report. In many cases,
fill material was eroded much faster after nourishment due primary to the fact that
the placed material was finer than the native which resulted in offshore sediment
transport.
A beach monitoring program between 1962 and 1972 at Atlantic City, New Jersey
was designed by Everts et al. (1974). Monitoring was conducted using repetitive
beach surveys down to mean sea level. Their results demonstrate that following
replenishment, losses of the fill material above mean sea level were approximately ten
times the losses measured in adjacent areas outside the nourishment area and loss
rates were largest at the updrift end of the nourished area. They observed that about
two times more material appeared to move in the onoffshore direction than moved
permanently alongshore. They recommended that future fills be placed as often as
possible rather than as large volumes at longer intervals. For the same site, Sorensen
et al. (1988) monitored the fill for a period of eighteen months including two winters
after placement of fill. The loss rate was observed to decrease in the second year
due to a milder wave climate for this period. Comparison of behavior of the former
9
project and the latter project demonstrated an improvement in fill behavior due to
repair of some of the shore structures, which reduced the alongshore loss of sand to
the inlet.
Shemdin et al. (1976) monitored a beach nourishment project in which a total
of 3.4 million cubic yards of sand was placed along the coast of Jupiter Island, FL.
The monitoring program included seasonal hydrographic surveys of the beach, and
wind and current information, wave climate and sediment characteristics. The survey
limit was extended to 3000 feet offshore. The results indicate that beach restoration
has a groin effect in the sense of producing favorable changes in littoral drift due to
shoreline alignment changes. A net accretion occurred updrift of the nourished area.
They concluded that shoreline recessions do not reflect the true state of erosion or
accretion following a beach restoration. The total sand volume contained in a restored
area is critically dependent on the offshore profile and the groin effect increases the
half life of the restored beach.
Dette (1977) predicted in a qualitative manner the evolution of an artificial "sand
groin" on the Island of Sylt in North Sea and found that the nourished material was
mainly distributed in the beach zone from the beach profile data which was monitored
for nearly a four year period. As a coastal protection measure, he recommended that
repeated nourishment would be more economical than the building and/or mainte
nance of existing manmade coastal structures.
Phillips et al. (1984) studied the impact of beach nourishment at South Beach,
Sandy Hook, New Jersey. A field study encompassing beach surveys, sediment anal
ysis and wave observations was conducted during the monitoring period. The vol
umetric changes of sand was computed based on six profiles extending 4.0 m below
mean sea level. It was found that the longshore transport of sediment is primarily
responsible for sediment loss, and wind losses from the flat, unvegetated expanse of
new beaches are also important. It was concluded that offshore losses of sediment
10
were of relatively less importance. They recommended that since beach material was
sometimes lost at a rapid rate during stormy winter periods, renourishment should
take place during spring and summer months when immediate storm losses are less
likely.
Giardino et al. (1987) studied by qualitative manners the cause of shoreline
retreat for a relatively smallscale nourishment project (460 m) in Galveston, Texas
where 11,500 cubic meters of material was placed in front of seawall. During the
seventeen months monitoring program, approximately sixteen percent of the material
placed was lost. The cause of erosion was a result of both geomorphic processes and
human interaction with environment. Human interaction through the construction of
dams upstream, continual dredging of the mouth of channel and destruction of the
dune had contributed to the erosion of the shoreline by either restricting or removing
the sand supply. However, they did not provide detailed profile change patterns.
Combe et al. (1987) described the evolution of a nourished beach at Grand Isle,
Louisiana. Sand was obtained from two offshore borrow areas located approximately
one and a half miles from the beach. They observed that shortly after completion of
construction, cuspate bars began to form in the lee of the borrow areas with erosion
occurring adjacent to the newly formed cuspate bars and once formed, the cuspate
bars appear to be fairly permanent features of the shore. They interpreted these
cuspate bar formations as due to the combination of width, depth and proximity of
the offshore borrow pits.
Stauble (1988) described guidelines for beach nourishment monitoring and a
project design, in which fill placement monitoring, borrow area, shoreline change,
biological assessment and littoral environmental monitoring were emphasized to as
sure a more organized effort to identify project response and improve design guidance
for future projects. Recommendations for detailed but qualitative descriptions of data
collection and analysis techniques were also included.
11
Recently, as the number of projects increased, many reports on beach nourishment
projects have been published (Aubrey et al. (1988) for Jupiter Island, Florida, Jarrett
(1988) for Wrightsville Beach, North Carolina, Jones et al. (1988) for Myrtle Beach,
South Carolina, Chu et al. (1989) for Tybee Island, Georgia, Skrabal et al. (1990)
for Fenwick Island, Delaware). Grain size analysis, volumetric change computations,
beach fill design criteria as well as cost analysis were included in their reports. Some
results indicated that initial loss of the beach material can be significant depending
upon the characteristics of the borrow material and native sand and terminal struc
tures are an effective means of reducing loss of fill material where longshore transport
is significant.
DeKimpe et al. (1991) approximated the behavior of the nourishment material by
one dimensional analytical models for the two different boundary conditions ( a filled
rectangular compartment between two groins and a filled rectangular compartment
with one terminal structure). The solution followed the oneline approach of Pelnard
Considere (1956) but did not give realistic predictions for the data obtained from Dead
Neck Barrier Beach, Massachusetts. The best fit coefficient of "alongshore diffusivity"
between the observed data and the model results was 0.08 ft2/sec for this particular
case.
Lin and Dean (1990) presented reports on a beach nourishment project in Red
ington Shores, Florida. The project comprised placement of 405,000 cubic meter over
a shoreline length of 2.6 kilometer. Monitoring was carried out over a two year period
and included seven surveys of twenty six profile lines. Wave data, current and tide
data were collected from a directional wave gage in Clearwater, Florida. Monitor
ing of beach fill evolution was undertaken jointly by the University of South Florida
and the University Florida. Data also included sediment characteristics at various
locations.
Hall and Herren (1950) studied the subaqueous beach nourishment project at Long
12
Branch, New Jersey and showed little evidence of any substantial movement of the
sand from the stockpile to the shore. Depth of sand placement was approximately 40
feet of water. They also found that nourishment material must be placed in shallower
depths to expect shoreward movement. Wave conditions, sand characteristics, sand
movement diagram, shoreline changes as well as depth contour changes were included
in their study. Similar conclusions were made by Harris (1954) for a subsequent study
of the same location. He suggested that the placement of material should be in less
than 20 feet water depth.
1.3 Scope of Study
As mentioned briefly earlier, the main objective of this investigation is to provide
a framework for predicting the performance of beach nourishment projects on long
straight beaches and also in the presence of a seawall. Two oneline models but with
different schemes of wave field calculations will be presented here. Various aspects
of coastal engineering interest will be investigated from the models and field data
as well. The models can include the presence of shore perpendicular structures and
background erosion which is formulated in terms of crossshore and longshore sediment
transport.
In the oneline model, the crossshore component of sediment transport is not
included explicitly. A complicated mechanism of sediment motion exists and the
distribution of longshore sediment transport over the surf zone is not well defined.
In spite of this deficiency, the crossshore sediment transport rate can be extracted
by computing the volume change between elevations and utilizing equilibrium beach
profile concepts. Also the model will be capable of simulating the effect of nourishing
with material that has sediment transport characteristics different than the native
material. It will be shown that if the nourishment material is less transportable than
the native material, the nourishment project will act as an erodible barrier causing
accretion and erosion on the updrift and downdrift side of the project, respectively.
13
For the case in which the nourishment materials are more transportable than the na
tive materials, the shorelines updrift and downdrift of the project both accrete with
the greater accretion occurring on the downdrift side. When different fill materials
are placed on the beach, the native and fill material will mix within a layer of cer
tain thicknesses and the resulting shoreline displacement and profile pattern are of
engineering interest.
A limiting case of mismatch of the placed sediment and prenourished conditions
is that in which the nourishment is placed on a seawalled coast in which there are
no native sediments to be transported. This case may arise in areas where past
efforts to limit erosion through shoreline "hardening" are being replaced by beach
nourishment. In this case, the centroid of the planform anomaly will move down the
coast with initially increasing speed.
This dissertation is organized as follows. Chapter 2 discusses the governing equa
tions and some useful results from linear wave theory. The onedimensional transport
equation, the socalled KomarInman relationship, will be introduced. Linearizing
the transport equation and combining it with the equation of continuity yields the
familiar one dimensional diffusion equation which has many analytic solutions for ide
alized geometry and boundary conditions. Chapter 3 provides a detailed explanation
of the methods and equations used throughout this study including the formulations
of long term background erosion and numerical scheme. Two different methods for
wave field calculations will also be introduced in this chapter. By comparing both
methods, the performance of the simple method will be shown to yield suitable results
and to require much less computing time. In Chapter 4, several aspects of shoreline
change pattern with different conditions will be discussed and illustrated for a simple
idealized initial planform. Chapter 5 presents the comparison of model results with
the prototype scale data. A broad range of evolutionary characteristics will be ex
tracted from the Redington Shores, Florida project. Documentation including beach
14
profiles, wave conditions and sediment characteristics will be investigated in Chapter
5. A comparison of data with predicted values will also be presented. Chapter 6
will extend the study of nourishment projects to include the presence of a seawall.
The oneline model will be modified with appropriate boundary conditions. The re
sults of a series of experiments will be presented for an intuitive understanding of the
problem. Chapter 7 presents the purposes, procedures, and qualitative interpretation
of the each test for several different experimental conditions. Chapter 8 presents a
broad range of conditions for overall behavior of nourishment projects in the presence
of a seawall focusing on the transport rate for the underwater portion of sediment.
Chapter 9 contains an investigative summary and will include the conclusions of this
study and suggestions for further research.
CHAPTER 2
GOVERNING EQUATIONS
The bases for predicting the performance of beach nourishment projects are the
equation of continuity and transport. In general, these two equations may be used to
develop a oneline model in which only one contour (usually the mean water line) is
used to represent shoreline changes (e.g. Bakker,1968; Le Mehaute and Soldate,1977;
Perlin,1978; Walton and Chiu,1979; Perlin and Dean,1985; Hanson,1989; Hanson and
Kraus,1989). Bakker (1968) has developed a twoline model which, for example, allows
for profile steepening and flattening updrift and downdrift of a structure, respectively.
Hanson and Kraus (1987) have compared the results of oneline numerical models to
analytical solutions applicable for several beach nourishment initial planforms. An
nline model has been developed by Perlin and Dean (1985) in which an arbitrary
number (n) of contour lines is used to represent the beach profile.
A restriction of the multiline models developed to date is that the profile rep
resented must to be monotonic. An alternative formulation method would be to
represent the topography by a grid system as is commonly done in hydrodynamic
modeling thereby eliminating the monotonic requirement. For models which repre
sent the profile by more than one contour, it is necessary to specify a relationship for
crossshore sediment transport. Such models have been developed by Kriebel (1982)
and Kriebel and Dean (1985) for profiles which must vary monotonically and Larson
(1988) and Larson and Kraus (1989) for models not requiring monotonicity.
The equation of sediment conservation can be expressed in three dimensions as
qh + Oqy
S= V = + (2.1)
Qt zx By
16
in which h is the water depth relative to a fixed datum, t is time, V is the horizontal
vector differential operator (V() = i +), qis the horizontal sediment transport
vector (q = iqx + jqy) as presented in Figure 2.1, and Z and j are the unit vectors in
the x and y directions, respectively. Integrating Eq. (2.1) in the crossshore direction
from a landward location, yl, where the crossshore transport (q,) is zero to a seaward
location, y2, where qy is similarly zero, yields
9 /M Q ry2
SJ' hdy qdy = 0 (2.2)
The first term is recognized as minus the time rate of change of volume of sand, V
and the second integral is the total longshore sediment transport, Q. Making these
substitutions yields
av aQ
+  = 0 (2.3)
at ax
In onedimensional model formulation, it is usually assumed that accretion or erosion
of a profile is associated with a seaward or landward displacement, respectively, of
the profile without change of form. The vertical extent of this change is from some
depth, h,, of limiting sediment motion up to the berm elevation, B. Thus the change,
Ay, in any contour associated with a change in volume, AV, is
1
Ay = BAV (2.4)
h. + B
which when substituted in Eq. (2.3) yields the onedimensional equation for conser
vation of sand,
By 1 8Q
+ 1 O 0 (2.5)
at h.+B ax 
The onedimensional equation of sediment transport can be expressed as (Komar
and Inman, 1970)
I = KPt, (2.6)
in which I is the immersed weight sediment transport rate, P1, is the longshore energy
flux factor and K is a nondimensional sediment transport proportionality factor.
17
Depth
Contours
,A
 'qy
'6 Wave Crest
x \
B
h* (x,y)
Depth of Limiting Motion
Figure 2.1: Definition Sketch.
These two quantities can be expanded as
I = Qpg(s 1)(1 p) (2.7)
Pt, = EbCGb sin b cos Ob (2.8)
in which s is the ratio of mass densities of sediment to water( 2.60), g=gravitational
acceleration, p=inplace sediment porosity (taken here as 0.35), E=wave energy den
sity, CG=wave group velocity, 0=angle between the wave crests and the bottom con
tours, and the subscript "b" denotes that the subscripted variable is to be evaluated
at the breaker position. Based on smallamplitude shallow water wave theory and
assuming that the breaking wave height, Hb is proportional to the breaking water
depth, hb(Hb = Khb)
H
Eb = pg9 (2.9)
CGb = Cb = gh = gHb/K (2.10)
in which K is a proportional constant (w 0.78) and Cb=wave celerity at breaking.
Substituting Eqs. (2.7), (2.8), (2.9) and (2.10) into Eq. (2.6),
K H J5/2 /
Q = (8/ sinOb cos b (2.11)
8 (s 1)(1 p)
Eqs. (2.5) and (2.11) form the basis for a onedimensional numerical model.
PelnardConsidere (1956) has shown that by linearizing Eq. (2.11) with respect to
perturbations in the predominant shoreline alignment and combining the result with
Eq. (2.5), the oneline model is transformed into the heat conduction equation
Oy 92y
Gy (2.12)
at aX2
in which G is the longshoree conductivity", defined as
KH2= 8( 1
G = (2.13)
8(s 1)(1 p)(h. + B)
19
Eq. (2.12) has a number of solutions for various initial and boundary condi
tions. A solution of particular interest here is for the evolution of a beach nourish
ment project of an initially rectangular planform of uniform displacement, Y centered
within a length, i. For this idealized situation, the analytic solution can be expressed
as the sum of two error functions as
Y 2x 2x
2 4 ONf 41O I
y(x, t) = {erf[ 1 erf[ 1)]} (2.14)
in which "erf{}" denotes the error function defined as
erf(z) = 2 f e" du (2.15)
Based on Eqs. (2.13) and (2.14), it is possible to infer general performance char
acteristics of beach nourishment projects. The most significant parameters include
project length, wave height, sediment characteristics, and background erosion. Defin
ing the project longevity as the time required for a certain proportion of the sediment
to be transported from the placement area, the effect that each of these parameters
has on project life is reviewed briefly below.
In the absence of background erosion, the longevity of a project increases with
the square of its length. Thus, other conditions being equal, a project with a length
twice that of another project will have a greater longevity by a factor of four. The
"spreading out" of the project is due to the gradients in longshore sediment transport
and since the transport depends on the breaking wave height to the 2.5 power, the
longevity varies inversely with height to the same power. The effect of sediment size
is twofold. The first is through the longshore sediment transport dependency on sed
iment size. According to Dean et al. (1982), coarse sediments are less transportable
than finer sediments thereby increasing longevity in the same proportion as that of
the K values of the native and nourishment sediments. The second effect of sediment
size is due to the wellknown relationship of increasing beach slope with sediment
size. If a sediment size is used which is finer than the native, the resulting profile,
20
when equilibrated, will form a narrower dry beach than if a coarser sediment had
been used. Based on equilibrium profile concepts, Dean (1991) has shown that, de
pending on the relative sizes of the native and nourishment sediments and the volume
of sand placed per unit length of beach, three types of equilibrium profiles can result
as shown in Figure 2.2 and described as follows: (1) intersecting profiles in which
the nourished and original profiles intersect; this type requires the nourishment ma
terial to be coarser than the native although coarser nourishment material does not
ensure intersection; (2) nonintersecting profiles, which always occur if the sediment
is the same size as or finer than the native and may occur if the sediment is coarser
than the native; and (3) submerged profiles characterized by the entire nourished
profiles being subaqueous; this type occurs for placement of relatively small amount
of sediment which is finer than the native. Usually the background erosion affecting
the nourishment project is assumed to continue at the same rate and with the same
distribution as before nourishment. However, since the background erosion is usually
considered to be due to longshore transport, there may be justification in modifying
the background erosion rates by the ratio of the K factors of the nourishment and
native sediments.
If Eq. (2.12) is formulated in finite difference form and solved by an explicit
method, it can be shown that a critical time increment, Ate, exists which, if exceeded,
will cause the numerical solution to become unstable,
1 Az2
At = (2.16)
2G
Although the equation (Eq.(2.12)) for which this criterion is valid is linear, this
criterion applies quite well for the primitive equations of continuity and transport
(Eqs.(2.5) and (2.11)), the later being nonlinear. For multiline models, it is necessary
to specify relationships for the distributed longshore (q,) and crossshore (q,) sediment
transport. Kriebel (1982) and Kriebel and Dean (1985) have proposed the following
w. 
^I eiaT
Added Sand
a) Intersecting Profile: Nourishment Sand Coarser Than Native.
Added Sand
b)NonIntersecting Profile: Nourishment Sand Finer or Coarser Than Native.
Ay<0O
"" "W I
'*j
Added Sand
c)Submerged Profile: Nourishment Sand Finer Than Native.
Figure 2.2: Three Generic Types of Nourished Profiles.
22
for the crossshore transport rate
q, = K'(D VD.) (2.17)
in which ) and V. are the actual and equilibrium values of wave energy dissipation
per unit water volume and K' is a proportionality factor.
CHAPTER 3
METHODOLOGY
3.1 Wave Field Calculations
Only oneline models are discussed here. Two models using quite different method
ologies for representing wave refraction and shoaling will be presented, applied, and
the results will be compared. In the first model, refraction and shoaling will be
presented by a very simple onestep procedure whereas in the other a detailed grid
solution is used.
3.1.1 Simplified Wave Refraction and Shoaling
We start by showing several results from linear wave theory using the wave and
contour directions shown in Figure 3.1. The difference AP, between the nourished
contour orientation, P,(h < h.), and the deep water contour orientation, /3(h > h.),
can be considered small since the ratio of nourished beach width, Y, to length, is
generally on the order of 0.02 at most. Applying conservation of wave energy flux
from deep water to a water depth, h.+
EoCG, cos(p, ao) = E., C.+ cos(o a.,) = E.CG, cos(3, a,) (3.1)
in which the subscripts "+" and "*" indicate conditions just seaward and landward
of the depth transition, respectively (cf. Figure 3.1). The energy flux just landward
of the transition may be equated to that at breaking by
E.CG. cos(/, a.) = EbCG, cos(, ab) (3.2)
Since P,(x) = o3 + Af(x)
E s ) = ECG, cos(o0 ao) EbCG, sin(p, ab) sin(A) (33
EC cos( cos(A) (3.3)
cos(Ap)
North
v i.L
h =h.
Shoreline
A
Region Influenced
by Beach Nourishment
x
Contours 
Deep Water
Contour
N
\9,Ol
~//
A
_2
a) Planform Showing Perturbed Contours to Depth h,
Original' h.
Profile
Nourished Profile
b) Profile Through AA
Figure 3.1: Definition Sketch for Effect of Beach Nourishment on Contours.
y
25
Utilizing shallow water linear wave theory and the wave height breaking proportion
ality factor, neglecting terms modified by sin(A/) sin(p, ab) and approximating
cos(A/3) by unity (At is small),
b gHC. cos(/, a,) .(34)
X2 COS( a()34
which will be useful later. Applying Snell's law in a similar manner across the tran
sition and to breaking,
sin(#, a.+) sin(#, a.) sin(#, ab) 3
(3.5)
C.+ C. Cb
and a,+ can be determined by applying Snell's law from deep water to h.+. Eqs.
(3.3), (3.4) and (3.5) will be used to express the transport relationship (Eq. (2.11))
primarily in terms of deep water wave conditions. With a limited amount of algebra,
we obtain
SKH^4C.20.4 cos1.2(0 ao)
Q = K 0 C' sin(#, a.) (3.6)
8(s 1)(1 p)C. .4 s( 
in which terms modified by sin(Af) sin(f, ab) have been neglected and cos(Afp) has
been approximated by unity. It is interesting to note that if Eq. (3.6) is linearized in
the form of Eq. (2.12), the appropriate value of longshore diffusivity ,G, in terms of
deep water wave conditions can be expressed as
KH^4 /1Cg904 cos12(fo a,) cos 2(fo a.)
8(s 1)(1 p)C.oI04(h. + B) cos(o a.) ()
Eq. (3.6) is the relationship for the longshore sediment transport rate that completes
the simple model development.
3.1.2 Detailed Refraction and Shoaling
The detailed method of solution differs from the simple method as refraction and
shoaling are carried out by a gridbased detailed procedure. This method as first
presented by Noda (1972), and employed by Perlin and Dean (1983) and Dalrymple
(1988), is based on the irrotationality of the wave number and conservation of wave
26
energy. The reader is referred to the papers noted above for further detail. The
irrotationality of wave number is
Vx = 0 (3.8)
in which k is vector wave number. Eq. (3.8) can be expanded to
a 8
(k sin 0) (k cos 0) (3.9)
In the present model, a twodimensional grid is established and Eq. (3.9) is expressed
in finite difference form as
ST(k cos )il,j+i + (1 27)(k cos 0),j+l
01+' = cos1 +r(k cos O)i+1,j+l (3.10)
((k sin 0)+i (k sin 0)_i,j)
in which 7 is a smoothing factor taken as 0.25 in application here and all k and 0
values on the right hand side are understood to be at the (n + 1) time step.
The above equation is solved by considering the wave direction, 0, at the offshore
boundary as known and the wave directions along the lateral boundaries given by
Snell's Law. These conditions along the lateral boundaries should not affect the
interior solution if the lateral boundaries are sufficiently distance from the area of
interest.
With the refraction solution for a given time step, the conservation of wave en
ergy equation is solved to determine the wave height field, Hi,. The equation for
conservation of wave energy (Eq. (3.1)), is expressed in finite difference form as
r(H2C sin O);ij+l + (1 27)(H2C sin O)i,+l 1/2
H"+ ' 1 +r(H2CG sin O)ir+.l.+0 1
2((H2CG cos 0)i,+Ij (H2CG cos 0)l,j)
(3.11)
As with wave refraction, in solving Eq. (3.11), the wave height along the outer
boundary is considered as known and along the lateral boundaries, it is assumed that
27
= 0. Eq. (3.11) is iterated until convergence of the solution is obtained to within
an arbitrary predefined limit.
The wave height calculations are carried landward to a location where the inequal
ity H,. > Kh,j occurs. The transport, Q is then calculated by transforming the
wave conditions at i, j' 1 to the breaking location much as was done for the simple
method in transforming wave conditions from deep water to the breaking point.
3.2 Grid System and Transformation of Initial Geometry
The continuity and transport equations (Eqs. (2.5) and(2.11)) were solved using
an explicit method with grid system shown in Figure 3.2. The shoreline displacements
are maintained fixed while the transport is computed and in the second part of the
same time step the transport is held constant while the shoreline displacements are
computed. Thus although the primitive equations are employed, the limiting time
step specified by Eq. (2.16) is still valid. In areas of special interest where greater
resolution is required, smaller grid elements, Ax, can be used in the grid system
presented in Figure 3.2.
In conducting numerical modeling of generally straight shorelines where a large
perturbation has been introduced, it is recommended, in general, that the original
shoreline and offshore bathymetry be represented by straight and parallel contours.
This is equivalent to the ad hoc transformation shown in Figure 3.3. The ratio
nale for the transformation is that, with the exception of the longterm background
erosion that will be discussed subsequently, the nearshore system has approached a
nearequilibrium, the details of which present modeling techniques cannot represent
adequately. The equilibrium may depend on rather subtle and perhaps nonlinear
threedimensional wave transformation (refraction, diffraction and shoaling) over mi
nor bathymetric features. Usually, if a model is applied to a shoreline situation such
as Figure 3.3a, it is found that the shoreline and offshore contours will approach unre
alistically straight alignments and/or the changes will occur over time scales that are
28
Note:
..7 b I i b
Figure 3.2: Definition Sketch for Numerical Model.
Con
a) Initial Actual Shoreline
and Contours.
Ad Ho
Transform
SContours
c
action
b) Initial Shoreline and Contours
to be Modeled. (Empirically Based
Background Erosion Included)
Figure 3.3: Recommended Ad Hoc Transformation for Modeling Coastal Systems in
which Large Perturbations are to be Introduced.
S
30
much shorter than actual. Thus numerical modeling is much more effective in cases
where substantial perturbations (humaninduced or natural) have placed the sys
tem out of balance. The transformation presented in Figure 3.3 recognizes that the
changes caused by the perturbation introduced will be characterized by much shorter
time scales than those associated with the preexisting nonequilibrium causative fea
tures. Subsequent to the modeling, the results can be transformed back to the actual
system.
3.3 Background Erosion
Longterm background shoreline changes must be accounted for appropriately if
the nourishment evolution is to be predicted with good accuracy. This is especially
the case if stabilizing structures are to be installed. Longterm background changes
could be caused by divergences in longshore or crossshore transport (cf Eq. (2.1)).
Thus, the general treatment considers the longterm background changes composed
of a longshore transport component, YB,L, and a crossshore transport component,
YB,C
9YB 8YB,L 9yB,c (31
+ (3.12)
at at Bt
and the background change, , is expressed as background longshore transport,
according to Eq. (2.5),
QB,L(X) = QB,L(XR) (h. + B) dx (3.13)
JxR Ot
in which QB,L(XR) denotes the background transport at some reference location, XR,
to be based on local knowledge or other calculation procedures not discussed here.
3.4 Numerical Solution of Governing Equations
The azimuth, /i, of the shoreline normal at the n~h time level, is established to
represent the value at the grid line associated with Qt (see Figure 3.2)
S= tan'l(y+ y) (3.14)
2 Zi+1 Xi
31
The background transport, QB,L, (Eq. (3.13)), is added to the longshore transport
resulting from the planform anomaly to yield the total transport, QT,
Q, = Q7 + QB,L, (3.15)
Finally, the shoreline position is updated from the n^h to the (n + 1)th time level
+ At A + tC (3.16)
YiU+I = n + A(h + B) Q' ,) + At (3.16)
In most applications, it is reasonably correct to consider ybc=0. An example of an
exception would be sediment losses to a submarine canyon such as might occur off
California.
3.5 Effective Wave Height and Period
Although the wave height and period change continuously with time, it is evident
that in the absence of littoral barriers, there is an effective constant wave height and
period that will produce the same spreading out of the beach nourishment material
as the actual timevarying values.
In the following development, a Rayleigh distribution f(H) for wave heights will
be assumed
f(H) = e(H/Hrm)2 (3.17)
in which Hrm, is the rootmeansquare wave height. Referring to the sediment trans
port equation (Eq. (3.6)) expressed in terms of deep water conditions, for a given sea
state characterized by the significant deep water wave height H,, the effective deep
water wave height, Ho,,f, is given by
Ho,0, = [o H"2p(H)dH] (3.18)
Evaluating Eq. (3.18) numerically and using the approximate Rayleigh distribution
relationship between significant and rootmeansquare wave height (H, /2Hr,,ms),
it can be shown that
Hoe,, = Krm,H.m, = KH,
(3.19)
32
where Kr,, = 1.04 and K, = 0.735. Thus the longterm effective wave height Ho.,1
at a particular location is
1
Ho.,f = E(K[Ha)24 (3.20)
in which H,, is the significant wave height of the nh record in a series of N records
encompassing the time period of interest. Examining Eq. (3.6), it is clear that a
somewhat more appropriate but more cumbersome value of effective wave height,
Ho',, is given by
LEN (KH_ )2.4H21
[ Nl ni (3.21)
N n=1 c I
and the effective value of C 12/C. (or equivalently, wave period) to be used in Eq.
(3.6) is the denominator of Eq. (3.21) raised to the 2.4 power.
This concludes description of the simple and detailed procedures for calculating
shoreline evolution following a nourishment project. While the required computer
times are not large for either procedure, the times for the simple procedure were
approximately 1/250 of those for the more detailed procedure.
CHAPTER 4
MODEL RESULTS
Several examples are investigated to demonstrate the effect of various design op
tions and to establish, by comparison with the detailed procedure, the relative validity
of the simple procedure. In the examples, the background erosion will be taken as zero
unless otherwise stated. For those examples including effects of background erosion,
only longshore transport contributions will be represented. Unless noted otherwise,
the grid spacing, (Ax), and project length for all examples were 150 m and 6000
m, respectively. At the lateral boundary conditions, the shoreline position changed
according to the background erosion rate and the domain was selected to be suffi
ciently large that the perturbation effects at the ends of the domain were insignificant.
Example 1: Initially Rectangular Planform, f=6000 m, Y=30 m, Ho=0.60
m, T=6.0 s, a, =/o = 900, h,=5.5 m, B=2.5 m
The results of this example comparing the two methods are presented in Figure 4.1
for 1,3,5 and 10 years after nourishment. The simple and detailed procedures yield
very similar results which is somewhat surprising in view of the extremely simple
algorithm employed to represent wave refraction and shoaling. The interpretation of
this is that the aspect ratio of the nourishment project (Y/1) is quite small. Therefore,
the effects of linearization are small.
50
Simple Method
S 40 ....Detailed Method
i Initial Planform
30 ..........
~5 Years
20
S20 1 10 Years
0
S 0
0
0 3000 6000 9000 12000 15000 18000
Longshore Distance, (m)
Figure 4.1: Example 1. Comparison of Beach Nourishment Evolution for Simple and
Detailed Methods of Wave Refraction and Shoaling, Normal Incidence, Ho=0.6 m,
T=6.0 sec, ao, = 900. No Background Erosion.
35
Example 2: Initially Rectangular Planform,f=6000 m, Y=30 m, Ho=0.60
m, T=6.0 s, ao= 700, P~ = 90 h.=5.5 m, B =2.5 m
Conditions for this example are the same as for Example 1, except the deep water
wave is at a 200 angle to the shoreline. Reference to Figure 4.2 demonstrates that
the two methods are again in quite good agreement with surprisingly little planform
asymmetry due to the oblique wave direction. This asymmetry is examined in greater
detail in the next two examples.
Example 3: Same as Example 2, Except Response to Different Wave Di
rections
Figures 4.3 and 4.4 present shoreline planforms at 1, 3, 5 and 10 years after
nourishment for wave directions of 700, 800 and 900, i.e. 200, 10 and 0" obliquity
to the general shoreline alignment. It is seen that the results obtained by the two
methods are quite similar and that the effects of wave direction are relatively small.
Although slight, the major effect seems to be that for the more oblique angles, there
is less wave energy flux toward the shoreline resulting in less longshore sediment
transport. A small asymmetry is evident.
Example 4: Same as Example 2, Except Detailed Examination of Planform
Asymmetry for Oblique Wave Directions
The purpose of this example is to examine the asymmetry resulting from an
oblique wave acting on a nourished planform which is initially symmetric. Figure 4.5
presents the results ten years after nourishment for deep water wave directions, ao,
of 700 and 80", i.e. 200 and 10" obliquity to the prenourished shoreline alignment.
36
50
Simple Method
S40 ....Detailed Uethod
Initial Planform
1
30 1 Year
I s3 Years
Q, K 5 Years
20 "10 Years
10
0,
0 I
0 3000 6000 9000 12000 15000 18000
Longshore Distance, (m)
Figure 4.2: Example 2. Comparison of Planform Evolution Obtained by Two Methods
for 200 Oblique Waves, H,=0.6 m, T=6.0 sec, ao = 700. No Background Erosion.
0 3000 6000 9000 12000 15000
18000
Longshore Distance, (m)
Figure 4.3: Example 3. Planform Evolution by Simple Method for Deep Water Wave
Directions, ao = 70, 80, 90, Ho=0.6 m, T=6.0 sec. No Background Erosion.
0 3000 6000 9000 12000 15000
18000
Longshore Distance, (m)
Figure 4.4: Example 3. Planform Evolution by Detailed Method for Deep Water
Wave Directions, ao = 70, 800, 900, Ho=0.6 m, T=6.0 sec. No Background Erosion.
39
In order to quantify the asymmetry, the planforms were separated into even and odd
components, ye(x') and yo(x') where x' is the longshore coordinate with origin at
the project centerline. Denoting yT(X') as the total shoreline displacement, it can be
shown that the even and odd components are determined as
ye(x') = [YT(Xr ) + YT(X)] (4.1)
1
yo(') = 1[YT(x') YT(X')] (4.2)
The maximum seaward displacement associated with the odd components is 1 m.
The reason that the odd components are small is that the aspect ratio (Y/l) of
the nourished planform is so small, in this case 30 m/6000 m=1/200. Thus the
asymmetries are small and are primarily due to the nonlinearities in the sin 20 term
of the transport equation.
Example 5: Effect of Wave Height
Figure 4.6 presents the planforms calculated by the simple method at 1, 3, 5
and 10 years after nourishment and for 0.4 and 0.8 m wave heights and normal wave
incidence. Of particular interest is the major role of wave height in causing spreading
out of the nourished planform. This effect is also evident from Eq. (1.2) which
indicates that the spreading out is proportional to H/ .
Example 6: Effect of Various Wave Heights and Project Lengths
Earlier examples have demonstrated (for homogeneous sediment conditions) the
relative insignificance of wave direction. The simple model was exercised to demon
strate the effect on the fraction of sand remaining in the project area for a wide range
of wave heights and project lengths. These results are presented in Figure 4.7 where
40
50
0o = 700
40 a = 800
40 Initial Planform
30
g 30 .........................................
SEven Components
0
10 i i1 i
0 3000 6000 9000 12000 15000 18000
Longshore Distance, (m)
Figure 4.5: Example 4. Even and Odd Components of Shoreline Position After 10
years for Deep Water Wave Directions, ao = 700, 800, Ho=0.6 m, T=6.0 sec. Results
Obtained by Detailed Method.
0 5000 10000 15000 20000
Longshore Distance, (m)
25000 30000
Figure 4.6: Example 5. Illustration of Wave Height Effect on Rate of Planform
Evolution. Results Based on Simple Method, ao = 900, T=6.0 sec. Results Shown
for 1, 3, 5 and 10 Years. No Background Erosion.
1.0
HO .3m, =24000m
S.. Ho=0.6 m, = 24000 m
0.8 t Ho=1.2 m, ,= 24000 m
\ .. Ho=0.3 m, J= 6000 m
0.6 \ * ...
SHo=0.6 m, .= 6000
0.4 Ho=1.2 m, = 6000 m
H0=O.9m, 2=1560;m
0.2 
0 Ho=0.6m, 9=1 50
Ho=1.2m, = 1500 m
I II I I I I I II I
2 4 6 8 10 12 14
YEARS AFTER NOURISHMENT
16 18
Figure 4.7: Example 6. Effects of Various Project Lengths and Wave Heights on
Project Longevity.
1>
UI>
zec
on
gC
zo
c.c
o)
OUJ
0
0
n
i
43
the proportion remaining is shown over a 20 year time period for nine combinations of
wave heights and project lengths. For a small wave height and long project (Ho=0.3
m, =24,000 m), it is seen that at the end of a 20 year period, over 95% of the material
remains. By contrast, for a fourfold larger wave height and a length onesixteenth of
the former, the sand remaining after one year is less than 40% of that placed. This
illustrates the significance of project length and wave height. It has been noted ear
lier that the longevity of a project varies with the square of the project length and
inversely with the wave height to the 2.5 power. It is emphasized that the results
presented in Figure 4.7 do not include effects of background erosion which can be
of considerable magnitude, especially when evaluating project performance over long
time periods.
Example 7: Effect of Retention Structures
Several examples will be presented illustrating the effects of retention structures,
with and without background erosion. The algorithm for the transport boundary con
dition at a structure is fairly complex and will not be described fully here. In general
the algorithm is consistent with Eqs. (3.6),(3.14),(3.15) and (3.16) with transport de
termination based on relative positions of structure tip and the shorelines on the two
adjacent grid cells and the background erosion transport rate components. Diffraction
was not included in the examples presented here.
Example 7a and Figure 4.8 present results for no background erosion and normal
wave incidence. Structures onehalf the length of the initial project width (30 m)
are present at the two ends of the project. Here, the structure length should be
interpreted as the "effective" structure length. Consistent with intuition, it is seen
that the shorelines immediately adjacent to the project advance as sand from the
project area "spills" around structures; however as the project recedes approaching
44
the structure length, less and less sand is transported to the project adjacent areas
and the planform will eventually evolve toward one of shoreline segments straight and
parallel to the incoming waves with the beach width within the project area equal to
the lengths of the stabilizing structures.
0 5000 10000 15000 20000 25000
Longshore Distance, (m)
30000
Figure 4.8: Example 7a. Effects on Planform Evolution of Two ShoreNormal Re
tention Structures of Length Equal to OneHalf the Initial Project Width. Normal
Wave Incidence. No Background Erosion. Ho=0.6 m, T=6.0 sec.
45
Example 7b and Figure 4.9 present results for the same conditions as Example
7a, except there is a uniform background erosion of 0.5 m/year and the reference
background erosion is taken as zero at the project centerline. The transient results
are qualitatively similar to those of Example 7a. However, because of the background
erosion, an equilibrium planform exists only within the confines of the stabilization
structures; within this region, the equilibrium planform is concave outward and sym
metric about the project centerline. For equilibrium conditions within this region,
the background erosion transport is exactly balanced locally by the planform orien
tation transport due to the waves. This accounts for the character of the planform
in Figure 4.9. At a great distance from the structures, the shoreline retreats as the
background rate of 0.5 m/year; however, immediately adjacent to the retention struc
tures, the erosion rate would be greater to compensate for the effect of the reduced
erosion (at later times) within the project area.
The final example (7c) that will be presented illustrating the effects of structures
is the same as Example 7b, except now the zero reference background transport is
located 4,500 m to the left of the left structure in Figure 4.10. It is seen that updrift of
the left structure, transport is toward the structure and the shoreline accretes there.
Inside the two structures, sand is transported initially in both directions, but some
what later the positive transport prevails and sand is carried past only the right hand
structure. Eventually, as before, the planform within the structures will be aligned
for equilibrium and both the downdrift and updrift shorelines will erode. For zero
reference transport located much farther to the left than for the example shown here,
sand would bypass the updrift structure, flow past the infrastructure segment and
onto the downdrift shoreline. However, both the updrift and downdrift shorelines at
great distances from the project would continue to erode at the background rate and
only the updrift shoreline immediately adjacent to the project area would experience
a net accretion.
40
4 Initial Planform
S 30 ....... .... .... 1 Year
0 A31 Year
/ 3 Years
# 5 Years
.0 20
10 Years
10
0
0 .................... ... .. .
Structure Structure
0 5000 10000 15000 20000 25000 30000
Longshore Distance, (m)
Figure 4.9: Example 7b. Effects on Planform Evolution of Two ShoreNormal Re
tention Structures of Length Equal to OneHalf the Initial Project Width. Normal
Wave Incidence. Uniform Background Erosion Rate at 0.5 m/yr, Zero Background
Transport at Project Centerline. Ho=0.6 m, T=6.0 sec.
40
Initial Planform
jI
a1 Year
'. 3 Years
20 5 Years
10 Years
S10
0
Structure 3 eStructure
10 L 
0 5000 10000 15000 20000 25000 30000
Longshore Distance, (m)
Figure 4.10: Example 7c. Effects on Planform Evolution of Two ShoreNormal Reten
tion Structures of Length Equal to OneHalf the Initial Project Width. Normal Wave
Incidence. Uniform Background Erosion at 0.5 m/yr, Zero Background Transport
Located 4,500 m to Left of Left Structure. Ho=0.6 m, T=6.0 sec.
Example 8: Effect of Different Transport Characteristics for Native and
Nourishment Sand
Dean, et al. (1982) have examined sediment transport factors,K(Eq. (2.11)),
determined from various field programs and have proposed the dependency of K
on sand diameter,D, shown in Figure 4.11. Intuition suggests that the effects of
nourishing with KF # KN could have a significant effects on the adjacent shorelines.
The subscripts "F" and "N" denote fill and native, respectively. This effect will be
greater for the case of oblique waves.
Figure 4.12 and 4.13 present the case of deep water waves at a 200 obliquity acting
on nourishment projects with KF/KN=0.9 and 1.1, respectively. It is seen from Fig
ure 4.12 that for the case in which the nourishment material is less transportable than
the native, the nourishment planform acts as an erodible barrier with an associated
accretion and erosion on the updrift and downdrift sides of the barrier, respectively.
For the case in which the sand is more transportable than the native, Figure 4.13,
the pattern is qualitatively a mirror image of that noted.
As the project evolves, the calculation procedure for transport requires local deter
mination of the degree to which the sand exposed to the waves is of nourishment and
native character. This was accomplished by the following algorithm. A mixed layer of
minimum thickness,Ymix, was assumed. If, during a time increment, erosion occurred
such that the mixed layer was less than Y,mi thick, the remaining material within the
mixed layer was mixed with the underlying sand to reestablish a thickness of Ymiz.
The character (i.e. K) of the mixed layer was calculated. If deposition occurred,
the mixed layer thickness and character were determined based on the thickness and
character at the previous time step and the character of the material in the littoral
stream. Obviously, it was necessary to calculate the character of the sand flowing
49
2.0
I*
O '
< \
 1.0 
_ ",
SO
0 I I
0 0.5 1.0
DIAMETER, D (mm)
Figure 4.11: Plot of K vs D (Modified from Dean, et al., 1982).
40
0 AB C DE
E 30
(. Initial Planform
S 20 1 Year
S3 Years
1 5 Years
o H 10 Years
0
10 I i
0 5000 10000 15000 20000 25000 30000
Longshore Distance, (m)
Figure 4.12: Example 8. Planform Evolution for Nourishment Sand Less Trans
portable than the Native (KF=0.693, KN=0.77). Note Centroid of Planform Mi
grates Updrift. Variation of Surface Layer K Values with Time at Locations A, B,
C, D and E are presented in Figure 4.14. Wave and other Project Conditions are the
Same as Example 2.
40A B C DE
S30 ..... Initial Planform
0 1 Year
3 Years
20 
a 2 5 Years
10 Years
S10 \
0
10
10 I I I I
0 5000 10000 15000 20000 25000 30000
Longshore Distance, (m)
Figure 4.13: Example 8. Planform Evolution for Nourishment Sand More Trans
portable than the Native (KF=0.847, KN=0.77). Note Centroid of Planform Mi
grates Downdrift. Variation of Surface Layer K Values with Time at Locations A, B,
C, D and E are presented in Figure 4.15. Wave and other Project Conditions are the
Same as Example 2.
52
into and out of a cell; this was accomplished by starting with the updrift grid line as
a boundary condition and, on cells where erosion occurred, modifying the magnitude
and character of the material in the littoral stream. In the nourished area, the K
value was set equal to KN when the nourished thickness reached zero. Figure 4.14
and 4.15 show the composite surface layer K values at the five locations indicated
in Figure 4.12 and 4.13, respectively. It is noted that the centroids of the planform
anomalies migrate updrift and downdrift for KF < KN and KF > KN, respectively
and that this profile migrational signature could possibly be used in conjunction with
a field monitoring program to establish the relative sediment transport coefficients,
KF and KN.
53
0.80
0
) 0.70 
0
0.75 ............ ... "
SAt
0.65  At
UAt E
S, 0.705 /
S.
0.605 II I At C
0 1 2 3 4 5 6 7 8 9 10
Years After Nourishment
Figure 4.14: Example 8. Variation of Surface Layer Longshore Transport Coeffi
cient K with Time at the Five Locations Shown in Figure 4.12. Case of KF=0.693,
KN=0.77, Yi,=2.0 m. Refer to Figure 4.12 for Locations of Points A, B, C, D and
E.
54
0
0
0
0.85
0.5 ..... At 
E
0 0
KN=0.77, Y8i0=2.0 m. Refer to Figure 4.13 for Locations of Points A, B, C, D and
0E.
  
.......... At A
0.75  At B
At C
. At D
SAt E
O
0.70 1   ii
0 1 2 3 4 5 6 7 8 9 10
Years After Nourishment
Figure 4.15: Example 8. Variation of Surface Layer Longshore Transport Coeffi
cient K with Time at the Five Locations Shown in Figure 4.13. Case of KF=0.847,
KN=0.77, Ymiz=2.0 m. Refer to Figure 4.13 for Locations of Points A, B, C, D and
E.
CHAPTER 5
APPLICATIONS TO LARGE SCALE DATA
Two numerical methods were developed and have been described for calculating
shoreline evolution subsequent to a beach nourishment project. These methods fall
within the class of oneline shoreline models in which the active vertical portion of the
profile is represented by only one contour line. One model is quite simple, with wave
refraction and shoaling from deep water to breaking conditions and the associated
transport occurring in a closed form. In the second method, wave refraction and
shoaling are carried out on a twodimensional grid using conditions of wave number
irrotationality and conservation of wave energy, respectively. Comparison of results of
applying the two models has established that they yield essentially the same evolution.
Thus only the simple model which requires much less computing time and preparation
effort will be applied and compared with prototype scale data.
As discussed in chapter 2, the bases for predicting the effects of longshore spread
ing are the equations of longshore transport and continuity which may be expressed
in the "oneline" form. The three alternate transport formulae evaluated include
KH2.sTj
Q1 = sin(/ ab) cos(( ab) (5.1a)
8(s 1)(1 p)
Q2 = K' SY (5.1b)
pg(s 1)(1 p)
where Sx is the socalled the radiation stress and
Q3 = K"6.4x104HbTp.m5 75 D 0.25 [sin(2(p/ab))]6sign(lab) (m3/yr) (5.1c)
The common continuity equation is
y 1 9Q (5.2)
at h.+Bx (.
56
Eq. (5.1a) is the socalled "KomarInman" (1970) relationship, Eq. (5.1b) was
evaluated by Dean et al. (1982) and Eq. (5.1c) is a formulation proposed recently
by Kamphius based on an analysis of laboratory and field data. The terms in Eq.
(5.1c) are defined as follows: Hb is the significant breaking wave height, Tp is the
peak period of the offshore wave spectrum, mb is the beach slope at the breaking
zone, D50 is the median grain size, and ab is the breaking wave angle. In Kamphius'
formulation, K" = 1, this coefficient is included as an index of the difference between
his results and that obtained here.
5.1 The Field Data
The Redington Shores beach nourishment project commenced in July, 1988 with
the placement of approximately 405,000 m3 of sand along approximately 2.6 km of
shoreline. Prior to the nourishment, the shoreline had been receding at an approxi
mate uniform rate over the project areas of 0.3 m/yr. Project monitoring included
a wave gage at the location indicated in Figure 5.1. During the first year, a pres
sure gage was installed, thus providing only a onedimensional spectrum. During the
second year, a gage was installed comprising a pressure sensor and a biaxial current
meter which provided a basis for determining a measure of wave direction. Monitoring
of the beach fill evolution was undertaken jointly by the University of South Florida
and the University of Florida. Data collected included beach and offshore profiles and
sediment characteristics. The profiles were surveyed a total of seven times with the
first and second surveys occurring immediately prior to and following nourishment.
Various aspects of the project and its performance have been reported by Davis (1991)
and Davis et al. (1991).
One interesting aspect of the project that proved beneficial to the analysis pre
sented herein was that for reasons that are not clear, the shoreline displacement and
sand placement volume density varied along the shoreline as shown in Figures 5.2a and
5.2b, respectively. This nonuniform induced gradients in longshore transport that
County
PROJECT ARER
Figure 5.1: Plan View of Beach Monitoring Profiles in the Sand Key Project and
Location of Directional Wave Gage.
Wave Gage
E
Project Area 
58
100
Bieakivater
VJ
1 75 I
CU
i 50
S25
0
0
0 T l 1_  L l  I 
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000
 North
Longshore Distance ) North
a) Initial Shoreline Displacement.
300
3oo iiiiiiiiiii ii 
.... .i i eakiyater
0
S200 
t1oo
0 f I f
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000
Longshore Distance (m) North
b) Initial Placement Volume Density.
Figure 5.2: Initial Shoreline Displacement and Placement Volume Density. (Moni
tored Profiles Shown As Dashed Lines)
59
proved useful in determining the sediment transport coefficient, K. As is the case in
most beach nourishment projects, the sand was placed at a considerably steeper slope
than equilibrium. Thus with time, there was a gulfward flow and deposition of sand
in water depths greater than normally influenced by the waves. Figure 5.3 presents
an example of the nourishment and subsequent evolution of a profile at which there
was initially an exposed seawall. A feature of the project that caused difficulties in
the analysis was the presence of a detached breakwater some 130 m in length located
in a water depth of 1.3 m near the northern end of the project. This breakwater had
been constructed in early 1986 and was the first of five planned elements for the pur
pose of stabilizing the overall beach nourishment project of which the nourishment
project considered here was the first phase. Additional information describing the
offshore breakwater is available in Dean and Pope (1987), Terry and Howard (1988),
and Terry and Martin (1989).
5.2 Wave Data
As noted, wave characteristics were measured by a pressure gage (First year)
and a directional gage (Second year) located as shown in Figure 5.1. Significant
wave heights varied from 0.1 m to 0.5 m for most of the period with an occasional
maximum of 1.5 m. The variations of wave period and wave direction were relatively
small. Although the wave height, period and direction change continuously with time,
the effective wave heights were calculated by Eq. (3.20).
1 2
Hi = [ E(KH,)2"4] (5.3)
in which H,, is the significant wave height at time, t, and K, (= 0.735) is based on
the Rayleigh distribution for wave heights. The overall effective wave height, Hff
and the average wave period and direction were calculated to be 0.27 m, 4.6 s and
950, respectively where the wave direction is measured clockwise and 900 signifies
normal to the shoreline. In the project evolution computations, the effective wave
heights were calculated and applied for each intersurvey period. The effective wave
I Seawall
July 1988
May 1988
50 0 50 100 150 200
Distance from Monument (m)
I Seawall
July 1988
Jan 1989
50 0 50 100 150 200
Distance from Monument (m)
1 Seawall
July 1988
July 1989
I I I
50 0 50 100 150 200
Distance from Monument (m)
A r
50 0 50 100 150 200
Distance from Monument (m)
1 Seawall
July 1988
Apr. 1989
I I I
50 0 50 100 150 200
Distance from Monument (m)
50 0 50 100 150 00
Distance from Monument (m)
6
60 0 50 100 150 200
Distance from Monument (m)
Figure 5.3: Illustration of Nourishment and Subsequent Evolution Over a Two Year
Period for Profile R102G. Note That Initially There Was an Exposed Seawall at This
Profile.
1 Seawall
 Z
July 1988
Jul. 1990
4 1
Table 5.1: Wave Characteristics.
Intersurvey Duration Effective Effective Wave Direction
Period (Month) Wave Height' (m) Wave Period (Sec) (Degrees)2
1 3 0.26 4.5 Not Available
2 3 0.27 4.8 Not Available
3 3 0.30 4.6 Not Available
4 3 0.29 4.6 Not Available
5 6 0.26 4.5 96.2
6 6 0.24 4.5 94.9
1. Calculated Deep Water Values.
2. Measured Clockwise With 900 Representing Normal to Nominal Shoreline Orientation.
characteristics for each intersurvey period are summarized in Table 5.1. Rosati (1989)
reported on the results of a directional wave gauge installation in 5.2 m of water
some 1250 m off Redington Shores. Data were collected over a one year period from
February 1986 to February 1987. Although over different time periods, the wave
heights were similar to those found here with a range of 0.1 to 1.2 m and a wave
period of approximately 5 seconds. The wave direction was generally from the south.
Wave heights were compared with data from the University of Florida nondirectional
gage at Clearwater and were generally found to be within 0.1 m.
5.3 Characteristics of Project Evolution
Figure 5.4 illustrates the general variations in transport during evolution of a
nourishment project. In the case of our field data, through examination of the mon
itored profiles, several features were noted that were relevant to and influenced the
analysis and interpretation of the data and may also be important to the prediction of
performance of a beach nourishment project. Three such features are discussed. As
expected, the more gulfward portions of the nourishment (Figure 5.2a) retreated most
rapidly with a portion of the sand transported to the adjacent profiles. Because the
profiles within the nourished region were oversteepened relative to equilibrium, the
paths of the transported sand were characterized by an offshore component. Thus
62
a0
\ \ 0 2 Q
N N. C
"rtep P s C fl
NNI
U.==
..,, ... ._
0e a
LO
7~
.0 CL
z i<
2u
Fir 5Zohne of OverSteepened Transpor n V
Projects.INote: CrosC r T t D
et al., 1992).
63
profiles adjacent to the more gulfward shoreline segments experienced less offshore
deposition than did the oversteepened profiles. This effect was also illustrated in
Figure 1.1a, depicting profile equilibration.
Realignment of the initially irregular nourished shoreline occurred in part due to
the seaward transport of sediment. Thus, even without longshore sediment transport,
the tendency for alongshore sediment transport was reduced and it was necessary to
account for this effect when modeling the longshore sediment transport.
5.4 Analysis Procedure
The primary objective of the analysis procedure was, through correlation of calcu
lated and measured project evolution, to develop estimates of the longshore sediment
transport coefficients for the three models tested. The steps involved in this procedure
are described below.
The crossshore sediment transport at each profile and for each intersurvey period
was determined directly from the data for each intersurvey period. In the calcula
tions, the rate of change over the intersurvey period due to crossshore transport
was assumed constant. Thus at each time step, the shoreline change, Aye, due to
crossshore transport only was calculated from the integrated equation of continuity
as
AYc (5.4)
h. + B
in which AVC was the prorated change in volume over the time step due to measured
crossshore transport above the depth, h.. The corresponding change due to gradients
in longshore transport was calculated based on Eqs. (5.1) and (5.2). This process
was repeated for all profiles and all six intersurvey periods using the effective wave
heights, periods and directions associated with those intersurvey periods (Table 5.1).
For the first and second transport models, the wave directions measured during the
second year were assumed to apply in the same sequence as the first year when they
were not measured. It can be shown that this is approximately correct since in the
64
absence of structures, the evolution of a beach nourishment project is very weakly
dependent on wave direction (Dean and Yoo, 1992). The third (Kamphius) model is
quite dependent on wave direction and thus it was possible to apply the procedure only
over the second year. The effect of background erosion was included by comparing the
calculated and measured volumes of sediment remaining within the monitored area
after the final (two year) survey. The background erosion was considered to account
for this difference and the calculated results adjusted assuming that the background
erosion was uniform over the region. It is noted that the background erosion thus
determined was generally found to be relatively small (equivalent to approximately
0.3 m/yr compared to the rootmeansquare shoreline change of 10.4 m/yr). The
pattern of the thus calculated volumetric changes due to longshore transport was then
compared with the measured for varying K values and the process repeated until a
best fit K value was obtained. This procedure may be viewed as using the overall
volume change to determine the background erosion (for each candidate transport
coefficient) whereas the pattern of volumetric change provides a basis for determining
the longshore sediment transport coefficient, K.
5.5 Results
In order to provide a quantitative basis for evaluating fits, the following measure
of normalized error was defined.
6
GAVmi,,, AY,,
E2 = 1 6 (5.5)
j=1
in which AVm,,, and AVp,i, are the best fit volumetric changes at the ith profile and
the th intersurvey period. It is seen that 0 < e2 < 1, with e2 = 0 indicating a
perfect fit and e2=1 pertaining for a transport coefficient of 0. It was found that
the patterns of calculated and measured shoreline changes agreed quite well in the
southern half of the monitored region; however, the fit attainable was not as favorable
Table 5.2: Comparison of Best Fit Coefficients and Their Respective
NonDimensional Goodness of Fits, ein. Based on Two years of Data.
Transport Relationship Best Fit Coefficient fei
Q = KHi sin(P a) cos(p ab) K = 1.02 0.147
Q = K' s K' = 2.01 0.150
in the northern portion. We interpret this to be due to the influence of the detached
breakwater in the latter area as shown in Figure 5.2. Thus the bestfit transport
coefficients are based on the southern onehalf of the monitored area. The variation
62(K) is presented in Figure 5.5 and shows a welldefined minimum at K = 1.02.
The measured and predicted distributions of volumetric changes for K = 1.02 and
the full two year period are presented in Figure 5.6. It is seen that although the fit is
reasonable throughout the entire region, it is far better in the area least affected by
the detached breakwater.
The bestfit K value and the associated sediment size were compared with a
composite relationship developed by Dean et al. (1982) and augmented by del Valle
et al. (1991) in Figure 5.7. The sediment characteristics at the Redington Shores
beach nourishment project were characterized by examining the sizes on the beach
face and at the 3 m contour within the nourishment area. The approximate averages
for these two locations were 0.45 mm and 0.16 mm, respectively. A size of 0.30
mm is considered as an approximate representative size. This K coefficient was also
compared with an earlier plot of K vs ( by Dean and Walton (1985) (w=sediment
fall velocity was defined earlier) and was found to agree well.
Several approaches were employed to extract values of transport coefficients for
the three transport formulae discussed earlier. These involved determining the best
fit coefficient based on the volume change distribution over the entire two year pe
0.18
0.17
2p 0.16
0.15
0.14
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
K
Figure 5.5: Variation of Error Measure and Best Fit Transport Coefficient(K = 1.02)
for Eq.(1) and Southern Half of Monitored Region.
I I I~ 1 I
i._.ij~reddted i Leekw~ater
I I
I
0 1000 2000 3000 4000 5000 6000 7000 8000
, North
Longshore (m) North
Figure 5.6: Volumetric Change Distributions at Two Years Over Monitored Area
for BestFit Coefficient (K=1.02). Measured vs Predicted. CrossShore Transport
Taken Into Account.
6 50
o
25
a)
50
25
0
50
75
100
Dean, et al.(1982)
................ del Valle, et al.(1991)
C DEAN
O BRUNO
a DUANE
+ WATTS
0 KOMAR
X INMAN
X KNOTH
)K CALDWELL
X ADRA
f MOORS & COLE
* THIS PAPER
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
Sediment Size D (mm)
Figure 5.7: Variation of Sediment Transport Coefficient with Sand Size. Results
Include Previous Field Studies and That of This Project.
1.5
1.4
1.3
1.2
1.1
1.0
0.9
K 0.8
K
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
I I I I I I I 1
Table 5.3: Comparison of Best Fit Coefficients and Their Respective
NonDimensional Goodness of Fits, em. Based on Second Year's Data.
Transport Relationship Best Fit Coefficient in
Q = ) sin(p ab) cos(p ab) K = 0.32 0.333
Q = K s K)' = 0.61 0.336
Q = K"6.4 x 104 T15 0.75D0.25 [sin(2( ab))]0.6 K" = 1.10 0.329
riod and over the first and second years, separately. In addition, the coefficients were
determined which agreed with the net volumetric changes inside and outside the nour
ished areas. It is worth noting that the project evolution associated with transport
relationship given by Eqs. (5.1a) and (5.1b) are very insensitive to wave direction
(Dean and Yoo, 1992) and thus computations could be carried out over the full two
year period. Project evolution due to transport given by Eq. (5.1c) is quite sensitive
to wave direction and thus evolution was possible only over the second year. Finally,
as a measure of our capability to predict project evolution, standard engineering ap
proaches were applied. Because planform evolution based on Eq. (5.1c) is sensitive
to wave direction, the best fit coefficients for all three relationships were recalculated
for the second year monitoring data for which wave direction data are available, and.
the results are presented in Table 5.3.
Although Eq. (5.1c) yields a slightly superior least squares fit (Table 5.3) to the
data compared to Eq. (5.1a) to the data, the measures of fit are very nearly the
same, (within 2 %) and thus not considered significant. The reasons appear to be
that the wave directions relative to the shoreline (i3 ab) occurring in these data
were in the approximate range I P Cab 1< 50. It can be shown that sin 2(3 ab) and
6[sin 2(# ab)]0.6sign(P ab) do not differ significantly over this range if 6 is allowed
as a free variable. Since K" = 1.10, this indicates good quantitative agreement with
Kamphuis' formulation.
5.6 Engineering Approach
In addition to the analysis of sediment transport relationships to determine K, as
described above, it is of engineering interest to evaluate how well the usual approach
(based on Eqs. (5.1a) and (5.2)) for calculating nourishment performance would
apply. Differences here are threefold: (1) the time varying berm height was neglected
and based on the data, a constant h. + B=2.4 m was used, (2) crossshore sediment
transport was not taken into account, and (3) a K value of 0.77 was selected as that
recommended by Komar and Inman (1970) and the most frequently referenced.
The results in terms of the volumetric and shoreline changes are presented in
Figures 5.8 and 5.9. The volume change distribution in Figure 5.8 is to be compared
with that in Figure 5.6 for which crossshore sediment transport was included. It
is seen that on an overall basis, the fit is somewhat better for the more detailed
calculations (e2 = 0.147) than for the engineering calculations (e2 = 0.23). Within
the nourishment area and in the southern half of the project, the volumetric changes
are in reasonably good agreement (Figure 5.8). South of the nourishment area, the
volumetric changes are overpredicted substantially. The shoreline changes (Figure 5.9)
are not predicted well inside or outside the nourished area. The reasons for the
differences are due, in part, to the effect of crossshore transport which diminished
the transport from the nourished area and also caused an associated shoreline retreat
within the project area. The smaller K factor (0.77 vs 1.02) caused the volumetric
changes in the nourished area to be approximately the same as measured.
71
100
75 s_ ,esud Bekwater
 Prediclted
50o
25
75
50 I
100
0 1000 2000 3000 4000 5000 6000 7000 6000
,North
Longshore ,(m) North
Figure 5.8: Volumetric Change Distributions at Two Years for "Engineering Ap
proach". No Consideration of CrossShore Transport, and K = 0.77. Measured vs
Predicted.
72
50
40 Mesu
PreditedI
30 Breakwater
o 0
o i
W 20
o \
30
40
50
0 1000 2000 3000 4000 5000 6000 7000 8000
Longshore ,(m) North
Figure 5.9: Shoreline Displacement Change Distributions at Two Years for "Engi
neering Approach". No Consideration of CrossShore Transport, and K = 0.77.
Measured vs Predicted.
CHAPTER 6
BEACH NOURISHMENT IN THE PRESENCE OF A SEAWALL
6.1 Introduction
In this chapter, the behavior of a beach nourishment placed along a seawalled
coast will be discussed and approximate methods will be recommended for represen
tation. From the previous chapters, we have seen that if the nourishment and native
sands are hydrodynamically the same, an initially symmetrical planform will remain
nearly symmetric even though the waves approach the coast at an angle. Moreover,
the centroid of the planform anomaly remains almost fixed even under oblique wave
incidence. If the placed sediment is less or more transportable than the native, the
centroids will migrate in directions counter to and in the direction of wave propaga
tion, respectively (See Figures 4.13 and 4.14).
A limiting case of mismatch of the placed sediments and prenourishment condi
tions is that in which the nourishment is placed on a seawalled coast such that there
are no native sediments to be transported. This case may arise in areas where past
efforts to limit erosion through shoreline "hardening" are being replaced by beach
nourishment. In this case, the subject of this chapter, it will be shown analytically,
numerically and experimentally, that, under the most idealized considerations, the
centroid of the planform anomaly moves down the coast at an approximate speed,
Uc, given by
Vc = (6.1)
in which Qo is the ambient potential sediment transport, V is the volume of sediment
added by nourishment, and e is the effective length of the project. Because the length,
4,, initially increases with time, the speed,Uc, of the centroid increases also. Under
73
74
these idealized considerations, the rate of spreading of the project or equivalently the
rate of increase of the variance will be shown to be unaffected by the seawall; however,
actually the rate of spreading is dependent on the transport relationships near the
ends of the project which are presently poorly understood.
6.2 Analytic Development
Consider a beach nourishment project for which the shoreline displacement is
given by y(x,t). It is assumed that as the shoreline position, y(z,t), changes, the
entire profile moves without change of form over the active vertical dimension, h. +B,
where h. is the depth to which the active profile extends and B is the berm height, see
Figure 6.1. Thus, the displacement rate of change, 9y/9t, associated with a gradient
of transport ,8Q/Ox, is
By 1 9Q
Oy 1 OQ (6.2)
t (h. + B) Ox
In the case of a seawall present where only a portion of the profile is active, it is
more appropriate to utilize the following form
Ov 8Q
v = Q (6.3)
Ot ~z
in which v is the volume of sand per unit length of shoreline.
6.2.1 Conservation of Volume
We begin by illustrating the procedure to be used by demonstrating the rather
obvious result that the total volume of sediment, V, is conserved. The total volume
is
V = (h. + B) y(x, t)dz (6.4)
The time rate of change of the total volume is
9V "0 9y(x, t)
= (h. + B) y(,t)d (6.5)
Wt 1 at
and, substituting Eq.(6.2) or more directly
OV Q d (6.6)
9t L 9xx
Seawall
Dry Sandy
Beach
B
C
1__
I I I
1 % \
%1 %
, \ Ii t Contours
i.i I t
," I / I
.''I I I A
^// I
II I U
IIIr~
a) Planform
c) Profile at Section BB
d) Profile at Section CC
Equivalent to PreNourishment
Conditions
Figure 6.1: Definition Sketch for Beach Nourishment in the Presence of a Seawall.
which is integrated directly to
9V
at
= Qoo Qoo = 0 (6.7)
since the transport at great distances from the nourishment are unaffected by the
nourishment and therefore presumed equal. This result holds for sandy and seawalled
shorelines (In the seawalled case, the transports at 00 are zero). Thus we have
demonstrated that the volume is conserved.
6.2.2 Planform Centroid Migrational Characteristics
The centroid, xz, of the planform anomaly is defined as
_ y(x,t)xdx
xZ(t) = (6.8)
( y(x, t)dx
and the velocity of the centroid is
Uc.= (6.9)
which can be established from Eq.(6.8) as
8rz Ie a dz Iy(x,;t)xdzx ydx
c c (6.10)
at L 0y(x,t)dx [L y(x,t)dx]2
oo oo
The last integral in the numerator is zero by conservation of sediment (Eq.(6.7)) and
the resulting equation may be integrated by parts, yielding
I dx 01 o
C _Jo Vx {[xQ] [xQ]oo} + V Qdz (6.11)
In interpreting the last equation, for a seawalled case, the first two terms are zero
when evaluated at sufficient distances from the nourishment that the transport is
zero. It is worthwhile to note that in the case of a shoreline composed of compatible
sand, the first two terms would nearly balance the remaining integral resulting in
very little movement of the center of gravity. Returning to the seawalled case, there
is a contribution to the last integral in Eq.(6.11) only over the region of the shoreline
where sand is available to transport. Although the sand transport, Q, will vary over
the active length, the most simple consideration is that the average transport is nearly
equal to Qo, over the effective length, e,, which yields
U xc Q (6.12)
5t V
As noted previously, since the length, f,, increases initially with time, the translational
speed of the centroid also increases with time. In reality, the transport will be less
than Qo near the ends of the project where the sand volume is so small that sand is
present over only a portion of the potentially active profile, h. + B. This situation is
shown in Figure 6.1c. The effect of this locally reduced transport can cause e to be
a constant or even decrease in the later evolutionary stages.
6.2.3 Planform Variance
The rate at which the nourishment planform spreads is examined next. The time
rate of change of the planform variance, 8a2/8t, is defined as
2 9y(x xx)2dx 2 y(x )dx
J y(xtc)dd
0 Jo Ot &tJO
S (6.13)
f 0 y(x, t)dx z y(x,t)dx
where the numerator of the last term is zero by definition. To proceed further, it is
probably best to shift our x coordinate to the centroid, xc. Defining x' = x z,,
substituting Eq.(6.2) and integrating once by parts
a2 1 2 0oo
= [(Qx)oo (Q )ool + zx'Qdx' (6.14)
The first term (in brackets) is zero since Qoo = Qoo. In order to carry out the
remaining integration, it is helpful to refer to the linearized transport equation
Q = Qo + Q' = Qo (h. + B)G (6.15)
Ox
in which as noted previously, Qo is the ambient potential transport on an unperturbed
shoreline and Q' is the transport associated with the nourishment project. The G
term is referred to as the longshoree diffusivity" defined in Eq.(2.13).
78
The remaining term in Eq.(6.14) can be written as
Bea2 2 /00 By
t= 2/ x'[QO (h. + B)Ga x]dx' (6.16)
The term involving Qo is antisymmetric and thus does not survive the integration.
Integrating the last term by parts yields
002 2G(h + B) _
2G( [(x'y) (x'y)o] 1 y(x', t)dz'} = 2G (6.17)
The terms in the square brackets are zero and the remaining integral is simply the
total plan area which, upon examination of the above development for the case of a
long sandy beach, will be found to be the same regardless of whether or not a seawall
is present.
However, in the case of a seawalled beach, the transport as given by Eq. (6.15)
will not be valid near the ends of the project where all of the sand to be nourished
is underwater and at the very ends is submerged to a depth, h. (Figures 6.1c and
1d). In this case, it is expected that the rate of increase of the variance could differ
significantly from the sandy shoreline value, 2G.
In summary, we have shown that in contrast to the nearly stationary centroid
of a nourishment project on a sandy shoreline, simple consideration of the evolution
of a nourishment project in the presence of a seawall results in a translation of the
centroid of the planform and that the rate of "spreading" of the planform will depend
strongly on the transport conditions near the project ends. We will return later to
discuss in more detail the effects of a local transport reduction near the project ends.
6.3 Numerical Treatment
The two dimensional sediment transport model described in Chapter 3 was mod
ified to account for the presence of a seawalled shoreline. As noted previously this
explicit model utilizes a simple refraction procedure in the vicinity of a nourishment
anomaly that has been compared to a more detailed gridbased model and found to
79
yield comparable results. The original form of the model has been described in de
tail by Dean and Grant (1989). The model solves the continuity equation (a modified
form of Eq.(6.2)) and the following nonlinear form of the sediment transport equation
as contrasted to the linearized form (Eq.(6.15))
Q = sin(#, ab) cos(P, ab) (6.18)
8(s 1)(1 p)
in which Hb is breaking wave height, g is gravity, K is the proportionality factor
between breaking wave height and water depth (; 0.78), and /, and ab are azimuths
of the outward normal to the shoreline and the direction from which the wave is
propagating, respectively, as shown in Figure 3.1.
For the case of an equilibrium profile given by h = Ay2/3 (Bruun, 1954 Dean,
1977, 1991) and for the horizontal bottom assumed for this example, the threshold
volume required to yield an incipient dry beach can be shown to be
h2.5
VT = 0.4A. (6.19)
The system represented is shown in Figure 6.1c where a "threshold" volume, VT, of
405m3/m is required to establish a dry sandy beach for the conditions below.
K = 0.77
H0 = 0.5m
T = 8.0s
h. = 4.0m
A = 0.10m1/3
B = 2.0m
3o cao = 00 and 100
s = 2.65
p = 0.35
t = 6000m
I
80
in which 3o pertains to the ambient shoreline, co refers to the wave direction in deep
water, and I is the initial project length. In those locations (near the ends of the
project), it was assumed for simplicity that the actual transport, QA, is related to
the calculated transport, Q, by
Q( ", <1
QA = (6.20)
Q, >1
and QA is used in applying the continuity equation. For larger values of n, there
is less transport near the project ends. Also, in determining the effective shoreline
orientation, &,, the effective shoreline displacement, ye, is defined as
VyT (6.21)
h. + B
which is seen to be negative for v < VT.
To provide examples of evolution, the model was applied for both sandy and
seawalled shorelines and the volumes, lengths, centroid locations and variances of the
planform anomalies calculated annually. For the seawalled case, two limits of the
project length, were determined which bracketed the effective project length, ei:
first, the length over which the volume exceeds the critical, fl and second, the total
length over which nourishment sand is present, 2. The results for the planforms,
centroids and variances are presented in Figures 6.2, 6.3, 6.4 and 6.5.
Figures 6.2 and 6.3 present, for n = 1 (Eq.(6.21)), the calculated planform and
volumetric evolutions for initially rectangular planforms and for deep water waves
of normal incidence and 100 obliquity, respectively. It is seen that although the
evolution is symmetric for the case of normal wave incidence, an asymmetric and
migrating planform occurs for oblique wave incidence. The physical interpretation
of the migration is that since there is no sediment inflow on the updrift side of the
planform, this end of the nourishment is "cannibalized" and the sand transported
to the downdrift side of the project where it is deposited. Although the planforms
5000 10000 15000 20000
25000 30000 35000 40000 45000
0 5000 10000 15000 20000 25000 30000 35000 40000 45000
Longshore Distance, (m)
b) Volume Densities at Years: 0, 1, 3, 5, 10, 15
Figure 6.2: Calculated Planform and Volumetric Evolutions of an Initially Rectangu
lar Beach Nourishment Project Fronting a Seawall. Normal Wave Incidence. Based
on Numerical Model, n = 1.
Longshore Distance, (m)
a) Planforms at Years: 0, 1, 3, 5, 10, 15
20
0
0
1200
1000
800
S600
Co
E 400
78
0 5000 10000 15000 20000 25000 30000 35000 40000 45000
Longshore Distance, (m)
a) Planforms at Years: 0, 1, 3, 5, 10, 15
1200
1000
800
Threshold Volume
"600 Density, VT = 405m3/m
2 400  
400
200
0 5000 10000 15000 20000 25000 30000 35000 40000 45000
Longshore Distance, (m)
b) Volume Densities at Years: 0, 1, 3, 5, 10, 15
Figure 6.3: Calculated Planform and Volumetric Evolutions of an Initially Rectan
gular Beach Nourishment Project Fronting a Seawall. Deep Water Waves at 100 to
Shore Normal, n = 1.
83
and volumes for the case of a nourished beach on a sandy shoreline have not been
presented here, and they are scaled versions of each other, we noted that the centroid
remains nearly fixed and the planform is much more continuous near the project ends
(c.f. Figure 4.1).
The translational speed of the nourishment volume (Figure 6.4) is seen to increase
rapidly at first and later to increase at a slower rate. The increase is consistent with
the earlier discussions (under "Analytical Development") and the knowledge that in
diffusion processes such as this the length scale is proportional to the square root of
the time scale. For still larger time, the lower limit, Qol/V would decrease to zero
as all of the placed sand spreads out under water. In Figure 6.4, the translational
speed as determined from the last term in Eq.(6.11) is bracketed by the two measures
of effective length, l and 12.
Figure 6.5 presents the time dependence of the normalized nourishment volume
variance. It is seen that the rates of variance increase for the case of nourishment on a
sandy beach are in accord with the theoretical value of 2G. For a seawalled shoreline,
as expected the rate of increase of variance is greater for the smaller values of the
exponent n in Eq.(6.21). It is surprising that for the seawalled case, the time rate
of variance increase is substantially larger (a factor of 2.5 for n=0.5) for the oblique
waves than for normal incidence.
S2.0
co
0.5
I
0
.o1.0
0.5
0 3 6 9 12 15
Time,(Year)
Figure 6.4: Translation Speed of Center of Gravity of Nourishment Volumetric
Anomaly. Deep Water Waves at 100 to Shore Normal, n = 1.
85
10.0
S ao = 800
Sa8.0  ao = 900
S
6.0 n=0.5
> n =0.5
z 2.0 n0 n 2.0
J 4.0 =1.0
o 20J  
S 2G and Sand n= 2.0 n= 1.0
0.0
0 3 6 9 12 15
Time,(year)
Figure 6.5: Time Dependence of the Nourishment Volumetric Variance. Deep Water
Waves at 00 and 100 to Shore Normal.
