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Erosional hot spots at Delray Beach, Florida

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Title:
Erosional hot spots at Delray Beach, Florida mechanics and probable causes
Series Title:
UFLCOEL-99014
Creator:
Fernández, Guillermo José Simón
University of Florida -- Coastal and Oceanographic Engineering Dept
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Gainesville Fla
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Coastal & Oceanographic Engineering Dept., University of Florida
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English
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ix, 109 p. : ill., maps ; 28 cm.

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Beach erosion -- Florida -- Delray Beach ( lcsh )
Beach nourishment -- Florida -- Delray Beach ( lcsh )
Coast changes -- Mathematical models -- Florida -- Delray Beach ( lcsh )
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bibliography ( marcgt )
theses ( marcgt )
non-fiction ( marcgt )

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Thesis:
Thesis (M.S.)--University of Florida, 1999.
Bibliography:
Includes bibliographical references (p. 108-109).
Statement of Responsibility:
by Guillermo José Simón Fernández.

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UFL/COEL-99/014

EROSIONAL HOT SPOTS AT DELRAY BEACH, FLORIDA: MECHANISMS AND PROBABLE CAUSES by
Guillermo Jos6 Sim6n Fernfindez Thesis

1999




EROSIONAL HOT SPOTS AT DELRAY BEACH, FLORIDA:
MECHANISMS AND PROBABLE CAUSES
By
GUILLERMO josE SIMON FERNANDEZ

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

1999




ACKNOWLEDGEMENT

I would like to express my sincere appreciation to my supervisory committee chairman Dr. Robert G. Dean. His support and advice made this experience an irreplaceable one. I also would like to express appreciation to Dr. Ashish J. Mehta who always showed concern for my development and to Dr. Daniel M. Hanes for their excellent lectures and for serving in my supervisory committee.
I extend my acknowledgement to all other faculty members, including Dr. Michel Ochi, Dr. Robert J. Thieke, and Dr. Hsiang Wang, whose lectures helped me fulfill my coastal engineering interests. Special thanks go to Helen Twedell for her assistance in the archives and her affection, Becky Hudson for her friendship, and to Subarna Malakar for his computer aid.
I would like to express my special gratitude to Dr. Bruce Taylor from Taylor Engineering and to Miguel A. Ydifiez from Consultoria Ydfiez-Taylor, who have encouraged and supported me, ever since I decided to come to the University of Florida to complete my academic and professional skills, and my personal interests.
My friends at the Coastal and Oceanographic Engineering Department made my experience in Gainesville unforgettable. I would like to thank Nicholas Grunnet, Erica Carr, Roberto Liotta, Kevin Barry, Edward Albada, Hugo Rodriguez, Kerry Anne Donohue, Joel Melanson, Al Browder, and Jamie MacMahan, with whom I shared wonderful moments.
Finally, I would like to thank my wonderful mother and father from whom I learned to reach all my goals. The support received from Marcela Ballina is also greatly appreciated.




This work is dedicated to my extraordinary wife and son, whose presence, companionship, and love throughout times of joys and hardships have been an invaluable source of inspiration. Thanks Lului and Rodrigo for your support in all my endeavors.




TABLE OF CONTENTS
A CKN OW LEDGEM EN T ............................................................................................................... 11
LIST OF TA BLES .......................................................................................................................... vi
LIST OF FIGURES ....................................................................................................................... vii
A B S T R A C T .................................................................................................................................... x
C14APTERS
I INTRODU CTION ..................................................................................................................... I
1. 1 Problem Statem ent .............................................................................................................. 1
1.2 Objectives and Scope .......................................................................................................... 3
2 REVIEW OF LITERATURE AND POTENTIAL CAUSES FOR EROSIONAL
H O T S P O T S ............................................................................................................................. 4
2 .1 In tro d u ctio n ....................................................................................................................... 4
2.2 Possible M echanism s for Creating Erosional H ot Spots .................................................... 5
2 .2 .1 R e fra ctio n ................................................................................................................ 6
2.2.2 Breaks in Bars (Diffraction) ................................................................................. 12
2.2.3 Use of Different Sediment Sizes Along the Nouni shed Beach ............................. 13
2.2.4 U se of Different Sand Placem ent Techniques ...................................................... 15
2.2.5 Presence of Coastal Structures .............................................................................. 16
3 DESCRIPTION OF THE DELRAY BEACH NOURISHMENT PROJECT AND
COM PILA TION OF DA TA ................................................................................................... 19
3 .1 D ata S o u rc e s ................................................................................................................... 19
3.2 Site D escription ............................................................................................................... 20
3.2.1 H istorical Evolution .............................................................................................. 20
3.2.2 Delray Beach N ounshm ent Project Desciption .................................................... 23
3.2.3 Hydrodynam ic Conditions .................................................................................... 27
3.2.4 Littoral Transport .................................................................................................. 28
3.2.5 Beach Profiles ....................................................................................................... 29
4 DELRAY BEACH NOURISHMENT PROJECT PERFORMANCE .................................... 30
4.1 Previous Studies at Delray Beach, FL ............................................................................ 30
4.1.1 Shoreline Changes ................................................................................................ 31
iv




4.1.2 Volumetric Profile Changes .................................................................................. 35
4.1.3 Conclusions from the Previous Studies ................................................................ 36
4 .2 A nalysis of the F ield D ata ............................................................................................... 36
4.2.1 Distribution of Fill Volumes Along the Project ................................................... 37
4 .2 .2 Sedim ent Size A naly sis ......................................................................................... 37
4 .2 .3 Sh o relin e C h an g es ................................................................................................ 4 2
4.2.4 Volumetric Profile Changes .................................................................................. 45
4 .2 .5 S u m m a ry ............................................................................................................... 4 9
5 APPLICATION AND MODELING RESULTS .................................................................... 51
5.1 Numerical Model for Beach Planform Evolution ........................................................... 51
5. 1.1 Application of Delray Beach Data for Planform Performance Predictions .......... 52
5.1.2 Predicted Shoreline and Volumetnc Profile Changes .......................................... 55
5.1.3 Influnce of the Sediment Transport Parameter ..................................................... 59
5.2 Comparison Between Measured and Predicted Changes ................................................ 61
5 .2 .1 Sh o relin e C h an g es ................................................................................................ 62
5.2.2 Volumetric Profile Changes .................................................................................. 65
5.2.3 Standard D eviation A nalysis ................................................................................. 67
5.3 Hot Spot Identification and Mitigation Measures ........................................................... 74
5.3.1 H istorical Shoreline Position ................................................................................ 75
5.3.2 Sediment Size Distribution Along the Project ...................................................... 75
5.3.3 Standard Deviation Reference Value .................................................................... 76
5 .3 .4 S u m m a ry ............................................................................................................... 8 0
6 SUMMARY, CONCLUSIONS AND RECOMMENDATIONS ........................................... 81
6 .1 S u m m a ry ......................................................................................................................... 8 1
6 .2 C o n c lu sio n s ..................................................................................................................... 8 2
6 .1 R ecom m en d ation s ........................................................................................................... 84
APPENDICES
A MODEL FOR BEACH PLANFORM EVOLUTION ........................................................... 85
B PROGRAM LISTING AND SAMPLE INPUT AND OUTPUT ........................................... 90
R E F E R E N C E S ............................................................................................................................ 10 5
B IO G R A PH IC A L SK E T C H ....................................................................................................... 109




LIST OF TABLES

Table page
3.1 Volume of sand placed in the Delray Beach Nourishment Project ........................... 23
3.2 Delray Beach renourishment proj ect, forty-eight month monitoring study. History of Dune
Accretion from DNR monument R- 177 to R- 182............................................... 27
3.3 Predicted tidal datums (NGVD) for Delray Beach, Florida.................................... 28
3.4 Available profile data for Delray Beach, FL from the Bureau of Beaches and Coastal
Systems .......................................................................................... 29
5.1 Approximate corresponding values of the sediment transport parameter to selected sediment
sizes (from Dean, 1989) ........................................................................... 55
5.2 Standard deviations computed for the 1975 -1990 span......................................... 74




LIST OF FIGURES

Figure page
2.1 Post-nourishment irregular bathymetry due to mechanic and hydraulic placement ....... 7 2.2 Irregular bathymetry due to dredge spoil placement ......................................................... 8
2.3 Beach planshape due to refraction over a 2 m deep hole, 1220 m offshore, from Motyka and
W illis ( 1 9 7 4 ) ........................................................................................................................... 9
2.4 Shoreline position showing the shoreline displacement after 2 hours, from Horikawa et al.
( 1 9 7 7 ) .................................................................................................................................... 1 0
2.5 Contours of diffraction coefficients for single pit with a/L= 1.0, b/L=0.5, d/h=3, K-h= 0.167,
and 0=00, from McDougal et al. (1995). Waves propagate from left to fight ................. 10
2.6 Wave refraction behind a dredged hole or borrow pit and associated longshore sediment
tra n sp o rt ................................................................................................................................ 1 2
2.7 Influence of borrow pits at Grand Isle, Louisiana, on the shoreline configuration (Date of
the Photography: M ay of 1998) ....................................................................................... 13
2.8 Effect of nourishment scale parameter, AF, on width of resulting dry beach. Four examples
of decreasing AF, with same added volume per unit beach length (Dean, 1991) ............. 14
2.9 Low ering of the profile at seaw alls .................................................................................. 17
2.10 Change in bathymetry due to background erosion under influence of a groin (adapted from
B ri d g e s, 19 9 5 ) ....................................................................................................................... 1 8
3.1 Location m ap for Delray Beach, Florida .......................................................................... 21
3.2 Historical shoreline position from DNR monuments at Delray Beach, FL ...................... 22
3.3 Placement of sand for the different maintenance fills ..................................................... 24
3.4 Location of the borrow area with respect of the fill area ................................................ 26
4.1 Mean high water shoreline changes computed by Beachler (1993) ................................. 33
a) Between 1974 to 1990, and
b) Between 1973 and 1990




4.2 Comparison between mean high water shoreline changes from 1974 to 1990 and mean high
water shoreline changes from 1974 to 1995 (Beachler and Mann, 1996) ........................ 34
4.3 Measured mean high water shoreline changes between January, 1987 and October, 1992
(from G raven s, 1997) ..................................................................................................... . 34
4.4 Volume changes computed by Beachler (1993) between 1974 and 1990 ........................ 35
4.5 Volumes of sand placed along the project for each of the nourishments ......................... 38
4.6 Cumulative volume of sand placed along the project ..................................................... 39
4.7 Longshore distribution of the sedim ent size ..................................................................... 40
a) After second renourishment, and
b) After third renourishment
4.8 Sedim ent size variation w ith tim e ..................................................................................... 41
a) After second renourishment, and
b) After third renourishment
4.9 Shoreline changes from 1975 to 1990 .............................................................................. 43
4.10 Shoreline changes from 1975 to 1998 .............................................................................. 44
4.11 Comparison between the 1/15/75 and 8/1/90 profiles at R-188 ....................................... 46
4.12 Volumetric profile changes from 1975 to 1990 ............................................................... 47
4.13 Volumetric profile changes from 1975 to 1998 .............................................................. 48
5.1 Computational scheme used in computational method ................................................... 53
5.2 Predicted volume and NGVD shoreline changes from 1975 to 1990 with ..................... 57
a) Incoming deep water waves perpendicular to the shoreline, and
b) Incoming deep water waves 200 north from perpendicular to the shoreline
5.3 Predicted volume and NGVD shoreline changes from 1975 to 1998 with ..................... 58
a) Incoming deep water waves perpendicular to the shoreline, and
b) Incoming deep water waves 20' north from perpendicular to the shoreline
5.4 Variation of sediment transport with different sediment sizes ........................................ 61
5.5 Comparison between the predicted and the measured NGVD shoreline changes from 1975
to 1 9 9 0 .................................................................................................................................. 6 3
5.6 Comparison between the predicted and the measured NGVD shoreline changes from 1975
to 1 9 9 8 .................................................................................................................................. 6 4
5.7 Comparison between the predicted and the measured volumetric profile changes from 1975
to 1 9 9 0 .................................................................................................................................. 6 6




5.8 Compari son between the predicted and the measured volumetic profile changes from 1975
to 1998 ........................................................................................... 66
5.9 Values of ou/,,a computed for the fill area for 1975-1998, with.............................. 69
a) Incoming deep water waves perpendicular to the shoreline, and
b) Incoming deep water waves 200 north from perpendicular to the shoreline
5. 10 Values of qpm/um computed for the fill area for 1975-1998, with ............................ 70
a) Incoming deep water waves perpendicular to the shoreline, and
b) Incoming deep water waves 20' north from perpendicular to the shoreline
5.11 Values of p,, computed for the fill area for 1975-1998, with ................................. 71
a) Incoming deep water waves perpendicular to the shoreline, and
b) Incoming deep water waves 20' north from perpendicular to the shoreline
5.12 Values of up, computed for the fill area for 1975-1998, with .................................. 72
a) Incoming deep water waves perpendicular to the shoreline, and
b) Incoming deep water waves 20' north from perpendicular to the shoreline
5.13 Location of erosional hot spots and cold spots for 1975 to 1990, using ..................... 78
a) Shoreline changes differences, and
b) Volume changes differences.
The area shown encompasses the project limits
5.14 Location of erosional hot spots and cold spots for 1975 to 1998, using ..................... 79
a) Shoreline changes differences, and
b) Volume changes differences.
The area shown encompasses the project limits




Abstract of Thesis Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Master of Science EROSIONAL HOT SPOTS AT DELRAY BEACH, FLORIDA: MECHANISMS AND PROBABLE CAUSES By
Guillermo Jos6 Simon Fernandez
August, 1999
Chairman: Dr. Robert G. Dean
Maj or Department: Coastal and Oceanographic Engineering
The Delray Beach restoration project has been nourished four times since 1973. The monitoring of the project, as well as other studies, has demonstrated that the beach fill has performed atypically in some areas along the project, showing higher erosion rates than the project's average.
Being a matter of recent concern, erosional hot spots lack established criteria that would allow them to be clearly identified. An erosional hot spot is an area of the shoreline that is receding faster than the rest of the project and that is not predicted directly from applying available theory. On the other hand, erosional cold spots are areas which accrete considerably faster or erode more slowly than the rest of the fill and are not predicted by available theory.
A detailed analysis of the behavior of the beach fills is performed, based on shoreline and volumetric profile changes. i order to predict the shoreline position after the initial nourishment, a one-dimensional numerical model for beach planform evolution is applied. The model considers a simplified refraction and shoaling of the wave field by assuming straight and parallel contours, and considers that the active profile is displaced seaward or landward without change of form.




The model can include the presence of shore-parallel structures and background erosion; however, the case of Delray Beach is that of an uninterrupted beach.
Erosional hot spots have been identified within the area encompassed by the Delray Beach project as a result of a standard deviation analysis of the shoreline and volume changes. Additionally, the value of the sediment transport parameter is analyzed in detail, and compared to the statistical difference between measured and predicted changes. Measured shoreline and volume changes were also compared to predictions with different incoming deep water wave angle conditions.
The one-line model applied proved to be accurate when predicting shoreline and volume changes. However, in order to achieve higher precision on prediction values, a model is needed that not only accounts for the three-dimensional character of the nearshore processes, but also includes more data such as different sediment characteristics along the project.
The Delray Beach restoration project has been renourished three times since 1973, when the initial nourishment took place. Monitoring of the project has been performed on a yearly basis, providing a rich data set that allowed analysis of its morphological behavior in detail. A total of 3.7 million cubic meters have been placed over a distance of 4.3 kilometers, from a borrow area around 800 meters seaward of Delray Beach. Although an overall successful project, its behavior has been atypical in some areas.
Compilation of data and other studies on Delray Beach, FL, are also included. Even though these contributions are not focused on the location of erosional hot spots, their results have been compared showing a reasonable agreement. The collected data were analyzed and correlated with the results computed from the measured and predicted conditions.




CHAPTER 1
INTRODUCTION
1. 1 Problem Statement
Beach erosion is one of the most important issues and concerns in the coastal environment. As the sea encroaches upon the coast and anthropogenic activities increase along many beaches of the world, the natural response of the environment oftentimes is for the shoreline to retreat in order to reach a new equilibrium, always governed by waves, currents, tides, and wind. The consequences of not preserving this environment could be costly, as a beachdune system provides storm protection to coastal properties, such as homes, hotels, and roads, provides recreational and tourist areas, and environmental benefits, such as turtle nesting and wildlife refuges.
The most "natural" solution to this problem is the placement of additional sand to restore (or to build) a beach. This process is called beach nourishment and unlike other solutions such as groin fields, headlands, and seawxalls, there is no negative impact to the dovwndrift beaches. Another solution that has proven to produce minimal environmental alteration, is nearshore nourishment, which is the placement of sand on the nearshore to build a berm, and it is applied under different physical settings than beach nourishment.
Beach nourishment projects require periodic maintenance according to the observed behavior. As will be shown in this thesis, at Delray Beach, FL, for example, this periodic maintenance has been accomplished according to its performance at the time. In order to design an economical project, the lifetime of a beach nourishment should be estimated. If the project's sand is lost rapidly, then the project can be regarded as a failure, even if this is produced locally.




When a project experiences localized erosion, which is larger than the rest of the project and that has not been predicted by diffusion theory, this is interpreted as an erosional hot spot in that area. An erosional hot spot is a limited area, although at times hard to identify, characterized by a narrower beach and/or loss of sand from the cross-shore section greater than the rest of the project. They can last months or they can be permanent, or until a new fill is built to renouri sh the area. Furthermore, an erosional hot spot has the characteristic that cannot be predicted directly from applying the diffusion theory, first introduced by Pelnard-Consid~re (1956) or other available theories. The concept of erosional hot spot and cold spot, does not apply for other coastal features such as beach cusps or perhaps daily events. The time scale associated with this analysis is important, as an erosional hot spot could be overlooked if different time spans are considered. Nevertheless, since this thesis is intended to analyze the overall performance of the project, by looking at its localized performance, the time spans consider the first survey and the most recent ones. In addition, long time scale erosion is not considered as responsible for erosional hot spots, since it is considered as its natural behavior, and therefore, can be accounted for through background erosion.
Erosional hot spots are of recent concern, therefore, the related literature is sparse and mostly related to a case-by-case analyses. Most of the research performed to understand localized erosion of the shoreline is focused on the effects that borrow sites have on the wave field.
When a nourished area presents localized erosion, the storm protection, recreational areas, and environmental benefits could be jeopardized. Thus, the entire nourishment can be regarded by some as a failure, even if it has proven to perform as predicted for the rest of project. It is thus necessary to investigate the potential causes leading to erosional hot spots in order to avoid them in future projects and to provide solutions to restore the desired conditions of a nourishment with erosional hot spots. The avoidance of hot spots would lead to more economic projects and to better distribution of the economic resources derived from taxpayers.




1.2 Objectives and Scope
In 1973 the City Council of Deiray Beach decided to restore the city's shoreline using beach nourishment, a technique that started to achieve popularity in Florida during this decade. In order to estimate the lifetime of the beach fill, the Coastal and Oceanographic Engineering Laboratory (1973), provided performance predictions of the beach restoration project for five years, when the first periodic maintenance took place. Although inaccurate, this theory indicated that renourishment should take place after approximately five years of construction.
At present, improved theories and numerical methods are able to predict the performance of a beach fill belier than in the 1970s. These predictions become important in order to determine the lifetime of a beach nourishment project and thus allow belier allocation of economic resources. However, erosional hot spots can significantly diminish the lifetime of a project. The main objective of this thesis is to identify any highly erosive areas that occur as hot spots and to detect potential causes, at the Delray Beach restoration project. Methods and criteria are also developed and applied to identify those areas that have evolved atypically.
The procedure used to identify erosional hot spots is based on shoreline and volumetric profile changes, which can describe the overall and local performance of the beach fill. This procedure not only will be an aid to identify hot spots, but it will also help describe the behavior of the shoreline of the entire project. The predictive capability of the numerical model of planform evolution applied here, can also be evaluated in terms of agreement between the measured and simulated shoreline positions.
Reviews of the potential causes of hot spots and the available literature are also included. Previous studies and contributions that suggest the presence of erosional hot spots in the nourishment project were evaluated, and compared to the results presented here. Finally, conclusions towards the application of the numerical model DNRBSM, as well as its capabilities and limitations, are drawn.




CHAPTER 2
REVIEW OF LITERATURE AND POTENTIAL CAUSES FOR EROSIONAL HOT SPOTS
2.1 Introduction
Beach restoration programs through nourishments have been ongoing as an alternative measure to restore Florida's beaches since the 1970s. As a recent concept, the effectiveness of beach fill projects has been questioned, especially given the fact that, economic resources derive from taxpayers and therefore may become a maj or public concern.
Erosional hot spots can prevent a beach nourishment project to be considered as successful, even though this problem can be regarded as local. Dean and Dalrymple (1999) mention that all nourished beaches have erosional hot spots to some extent, thus, becoming important to analyze and, if possible, to prevent.
Some researchers have proposed different methods of predicting the overall performance of beach fills. Some of these methods include those from Krumbein and James (1965), James (1974), Dean (1974), and Pilarczyk and van Overeem (1987); other approaches like those presented by Dean and Grant (1989), and Hanson and Kraus (1989) consider many more important parameters yielding a more accurate approximation to the shoreline response prediction. These methods allow, to different degrees of detail, the prediction of the planform evolution of a beach fill with time, however, they do not predict the presence of erosional hot spots since, as defined earlier, they are a consequence of some irregularity in the coastal zone that is not accounted for in those methodologies.
Numerical and physical modeling has been carried out by a number of investigators, though not necessarily directed to erosional hot spots. Motyka and Willis (1974) obtained some




preliminary results from the study of beach erosion caused by wave refraction over offshore dredged holes with the aid of mathematical modeling. Horikawa et al. (1977) studied mathematical and laboratory models to examine the effects on shoreline shape due to exploitation of submarine deposits of sand and shingle, using an idealized sandy beach and hindcast waves typical of those on the eastern coast of Japan. These experiments provided a good qualitative agreement with the mathematical and the laboratory tests. A more recent study is that of McDougal et al. (1996), who used linearized shallow-water wave theory to investigate the interaction of surface waves with multiple rectangular submarine pits in water of otherwise uniform depth. The application that McDougal and his coworkers gave to this type of breakwater was on navigation channels, never mentioning the possible relationship with beaches. However, their method provides a good tool to understand the wave diffraction patterns due to propagation over these holes.
2.2 Possible Mechanisms for Creating Erosional Hot Spots The analysis of erosional hot spots requires a case-by-case analysis. While some research has been carried out on the direct effects of dredged holes, no detailed work has been performed to study the effects of placing sand of different sizes along the beach, or the long term effects of mechanically versus hydraulically placed sand on a beach, for example.
A closer examination of the erosional hot spots, was made by Bridges (1995), with particular interest in the effects of residual bathymetry as a probable cause to erosional hot spots, where numerical and physical modeling were utilized. The following mechanisms were identified as potential developers of erosional hot spots:
* refraction due to offshore bathymetry and borrow pits,
* breaks in bars,
* sediment size differentials along the nourished area,




* use of different sand placement techniques, and
* presence of coastal structures.
A brief review of these mechanisms is presented below with a broader explanation of the refraction process, which is the one that has received more attention among researchers. The motivation for presenting these mechanisms is not to give a full understanding of the processes creating hot spots, but to be able to make an intuitive approach to the understanding of a particular case. Headland effects are considered to be mechanisms predictable by the diffusion theory, thus not considered as a potential cause for erosional hot spots.
2.2.1 Refraction
Refraction is clearly a major mechanism that shapes the shoreline. Therefore, after large amounts of sand are placed on the beach, the newly-created bathymetry will play a significant role in the planform evolution.
There are two identified situations that can be regarded as the probable responsibles for changing the refraction pattern. Firstly, an irregular offshore bathymetry or what Dean and Yoo (1992) called "residual" contours, and secondly, the location of the fill's borrow sites.
2.2.1.1 Residual Bathvmetry
This man-made irregular bathymetry is believed to be the consequence of different volumes of sand placed along the nourished area. Then "residual" contours are formed beyond the depth of limiting motion resulting in refraction and shoaling changes. It is important to mention that this sand that has been placed to a depth where waves cannot transport it, is of more concern than that placed shallower since the latter will be "spread out" by the longshore transport. Therefore, while the deep contours will continue to alter the wave field, the shallower contours ones will have no long lasting effect on the wave field nor the beach. Dean and Yoo (1992) suggested an equation to compute the displacement of the shoreline about its mean alignment Ays,




caused from an offshore contour with a displacement about the mean contour alignment AyR. This is
Ays 1 -1-AyR (2.1)
C.
in which C1 and C. are the wave celerities at the "residual" contour and at the depth of limiting motion, respectively. Several mechanisms can be the reason of residual bathymetry, for example, irregular beach fill placement and dredge spoil placement. The first one results from sand that is placed both hydraulically and mechanically. As it will be discussed later, when placed mechanically, the outcome is a beach profile with a steeper slope, thus yielding an uneven bathymetry. Figure 2.1 shows these contours.
Points of discharge
from pipeline
Low tide datum
.................
..... .... ", "................ .... ... .. ... ................. .... ....... '-....... .... .... ..... ... .. ..
> ... ... ..... ...................
.- .... ...... ..... .
..................
Depth of Irregular
closure contours
(exaggerated)
Figure 2.1 Post-nourishment irregular bathymetry due to mechanical and hydraulic placement.
Further description of the process used to build the beach fill is shown in Section 2.2.4.
Another common practice that can lead to an irregular bathymetry is dredge spoil. This would be the case of a beach confined by to inlets. When the navigation channels are dredged, and the sand is placed on the adjacent beach, it can result in a major alteration of the bathymetry, especially due to the fact that the channels are often dredged periodically. It ought to be mentioned that the dredged sand is not distributed evenly on longer segments of the beach for economic and technology reasons. The newly formed refraction pattern will produce waves to




bend in such a manner that sand can be transported away from the center of the fill area creating an erosional hot spot. This phenomenon is depicted in Figure 2.2.
SedimentI I
original shoreline transport
S positions i I
S.. ..... ....
I / I.
,Z This area will erode due to the 1
- ~new refraction pattern originated \
~~~offshore IDeg
Offshore bathymetry starts to
build up concave contours
Figure 2.2 Irregular bathymetry due to dredge spoil placement.
In order to test this residual bathymetry theory, Bridges (1995) developed a numerical sediment transport model as well as a physical model. The results found from this modeling are limited, yet encouraging. They are limited due to discrepancies between the numerical and the physical model, since it was difficult to determine which one is more accurate. However, the quantitative results should encourage further research, since it was found that in fact there is a shoreline change as a consequence of residual bathymetry. In addition, the results confirmed that there is a relationship between the shoreline change and the wave celerity ratio as suggested by Dean and Yoo in Equation 2.1.
2.2.1.2 Borrow Pit Location
The influence of dredged holes on the shoreline is a subject that has been studied by several authors with numerical and analytical modeling. This is another case where refraction plays a major role in shaping the beach with an erosional hot spot. A borrow pit is formed when mined for sand, most of the times, for the nourishment project itself. Different authors have found apparently opposite results when analyzing the effects of dredged holes on the shoreline. Motyka and Willis (1974) used a mathematical model to investigate the effect of dredged holes on the coasts of England. Although they consider their conclusions "conservative", they found that there




is retreat in the shoreline due to refraction over the modeled holes. Figure 2.3 shows some of their
results.
POSITION OF
40 EDGED
0 ER It DAYS HOLE P
WAVE Ht 2m o WAVE ANGLE 100
AFTR 3 MTHS
-40-20 WAVE Ht O-41m
AVE ANGLE 20
0 WAVE PERIOD 5s" 20-20 AFTER 6 MTHS
-20 WAVE Ht 036m WAVE ANGLE 0* WMVE PERIOD Ss .2020 AFTER 9 MTHS I
WAVE Ht 0 35m WAVE ANGLE- O .
020WAVE PERIOD 5s AFTER 9 rHS 11 QYSI
20 WAVE H1 I7m J I
WAVE ANGLE 10*
0 0
AFTER 12 MTHSI
WAVE ANGLE 100 / I "1-4.20 MMV Ht, 0,,7m,
WAVE AGLE O WAVE PERIOD .5s /_
-20I
0 500 1000 1500 2000 2500 3000 3500 4000 5OO DISTANCE ALONG SHORE -m
Figure 2.3 Beach planshape due to refraction over a 2 m deep hole, 1220 m offshore, from Motyka and Willis (1974).
Horikawa et al. (1977) presented more results from numerical and physical modeling on
the same topic. Using data typical from the Eastern coast of Japan, their results are exactly the
opposite to those presented above by Motyka and Willis. Instead of erosion, the dredged holes
produce beach accretion, and the reason is argued to be refraction as well. The results are shown
in Figure 2.4 where the numerical and physical results are compared.




McDougal et al. (1995) developed a theoretical model using linearized shallow-water wave theory and a two dimensional Green's function. In this study, the pits have the function of a breakwater, and it is suggested that this mechanism could be applied to protect navigation channels. Figure 2.5 presents an example of their computation results and it shows some interesting results. It can be seen that seaward of the pit a partial standing wave system develops, while in the lee of the pit a shadow zone exists in which wave heights are reduced up to 60%. Even though there is no further comment on the possible effects on the beach due to the alteration of the wave field, their results can be conclusive when trying to understand the refraction mechanism behind a hole.
2 HOLE
4 Observed
2 Pr.dicted
= 0 6"
0
0
-4
0 20 40 60
Longshore distance from center of dredged hole ( cm)
Figure 2.4 Shoreline position showing the shoreline displacement after 2 hours, from Horikawa et al. (1977).
C 7Figure 2.5 Contours of Diffraction Coefficients for Single Pit with a/L=1.0, b/L=0.5, d/h=3, i~h=0.167, and 0?=0, from McDougal et al. (1995). Waves Propagate from Left to Right.




Another study on the impact of dredged holes on the shoreline was conducted by Kojima et al. (1986), studying the case of Kyushu Island, Japan. Kojima and his coworkers are skeptical when affirming that the dredge holes are the reason of the erosion of the beach at Kyushu Island. However, they suggest a mechanism different than refraction or diffraction that may cause the beach to erode. They suggest that the dredged holes at the study site, are gradually refilled with sand coming from the landward side of the hole. This will cause the beach profile to steepen and eventually to lose sand from the beach, creating what we call an erosional hot spot.
It has been shown that there are different theories on how a dredged hole influences the shoreline. The mechanisms that can make a beach erode or accrete because of the presence of a dredged hole are explained below.
There are two phenomena that change the sediment movement alongshore in the presence of dredged holes or borrow pits, related to refraction: "bending" of the wave rays and water level differentials. This refraction can be viewed as an anti-shoal process. When travelling over the hole, the wave celerity will increase, causing the wave rays to bend away from its lee. When arriving at the beach, wave orthogonals will have an angle that will transport the sand in opposite directions as shown in Figure 2.6. In addition, an area of lower wave heights (and less energy) is created behind the borrow pit causing a wave set-up differential. The difference in water elevations will generate longshore currents towards the region behind the borrow pit, thus transporting sand that can form a salient. This is also illustrated in Figure 2.6.
Whether the planform will become a salient or an erosional hot spot, depends on a number of variables. Whichever process is stronger and more able to move sand in or out of the sheltered area, will determine the outcome. The fact that borrow pits could have an important impact on the shoreline is well known. However, the how that impact is uncertain, and in order to carry out predictions, it is necessary to bring into account sediment characteristics, wave characteristics, detailed bathymetry, and detailed configuration as well as location of the borrow pit. Then, refraction and diffraction should be accounted for in order to determine the




hydrodynamics produced by the holes. Another example of salients due to dredged holes is one located at Grand Isle, Louisiana, where two borrow pits produced two salients behind them. The negative impact of this case is that the beach becomes narrower in between the salients, since it lost sand to feed them. Therefore, this area can be identified as an erosional hot spot. An aerial photo of Grand Isle is presented in Figure 2.7.
Sediment transport in opposite directions
~4due to the wave angle at breaking \~
Orgia -hnrlmn
Sediment transport
towards the region 1; Z Sediment transport
behind the borrow pit towards the region
due to the difference in Area of lower set-up due to the behind the borrow pit
set-up sraigof tewv'enrydue to the difference in
(lower wave heights) stu
Area of Area of
higher set-up higher set-up
. . . .. ............. . ........................... ........ Borrow pit, waves are
10 refracted by its
-8m
Wave rays
Figure 2.6 Wave refraction behind a dredged hole or borrow pit and associated longshore sediment transport.
2.2.2 Breaks in Bars (Diffraction)
This mechanism consists in the diffraction of waves when travelling over discontinuous nearshore bars or, in other words, when bars act as submerged breakwaters. Once diffracted as described, waves will move sediment in opposite directions creating areas of intensified erosion of the subaerial beach. However, these features are associated with relative short term processes, such as storms, and therefore they may last only for a few months, and the erosional effect can eventually disappear. This phenomenon is not exclusive of bars. Any other feature acting as an underwater breakwater with a gap or break can actually produce the discussed diffraction with the associated beach erosion.




0' 5MI
01 '1Km
Figure 2.7 Influence of borrow pits at Grand Isle, Louisiana, on the shoreline configuration (Date of the Photography: May of 1998).

2.2.3 Use of Different Sediment Sizes along the Nourished Beach
This mechanism is much less studied than refraction and diffraction from dredged holes and submerged berms and bars. It consists in the non-uniform placement of sand along the beach. The consequence has to be regarded as a local effect, in some cases, as erosional hot spots. When sand is placed along the beach with uniform characteristics it would be expected from theory that the beach will perform without great differences, or at least, to remain within the average performance of the project. When there is an area with higher erosion rates than the rest of the project it could mean that the beach is formed of finer sand in that area. The reason is that profiles with finer sand will move faster to equilibration and have less additional dry beach width per volume placed, resulting in larger shoreline erosion rates. hi other words, according to Dean (199 1), the dry beach width corresponding to a finer sediment is narrower than that of a coarser beach. Figure 2.8 shows the difference in profile equilibration between different sand sizes. This Figure shows that for a finer sand the slope of the beach is milder, and the dry beach narrower




than the one with coarser material. AF and AN are the scale parameters for the fill and for the native conditions, respectively; likewise, DF and DN, are the fill and the native sediment sizes, respectively.
V
h,,m
Z~ ~c PW& 1M, 1. An, A0 m4 45,3f
,, f. r .1 r 1'-3 r
M.-6m
D = O, M 0 2mn
A--V
5 9m1
I I I I I I I
0 100 200 300 400 500 O00
OIsho~ trie (e m)
Figure 2.8 Effect of nourishment scale parameter, AF, on width of resulting dry beach. four examples of decreasing AF, with same added volume per unit beach length (Dean, 1991).
One reason that there may be a longshore vaiability in sediment sizes, is economy in the dredging operations. Fine sand is easier to pump than coarse sand, therefore, it requires less fuel. When the borrow site contains different sediment sizes, the contractor can decide to place finer sand at the most distant point of the project, and coarser sand at the closest location from the borrow area, in order to make the overall operation less costly. This process is called dredge selectivity. A typical scenario for dredge selectivity is when an inlet's ebb tidal shoal is selected




for the borrow site. An inlet's shoal is a mixture of sand from the inlet and sand carried along the coast that encounters the beach. Therefore, there can be a vast vaiety of sediment sizes in an ebb tidal shoal. For example, during ebb conditions, currents will carry sediments out of the inlet depositing on the ebb shoal. However, this deposition is not uniform if we consider the hydrodynamic characteristics of the grains. Assuming that a fine grain will have a smaller fall velocity than a coarse grain, it will be transported for a greater distance, thus, being deposited further away. A coarser grain would be deposited closer to the entrance of the inlet. The dredging contractor can take advantage of this situation to minimize costs recognizing that is cheaper to pump finer sediment.
2.2.4 Use of Different Sand Placement Techniqiues
This concept was briefly introduced in Section 2.2. 1. 1, when the concept of residual bathymetry was explained. There are two ways of placing sand on the beach: mechanical and hydraulically. Sand is placed hydraulically when a dredge pumps a mixture of sand and water. When the sand is first pumped into the beach, it is placed in specific points, and then spread out by bulldozers. Mechanically placed sand is transported dry from the borrow sites using trucks and common earth moving equipment, and then dumped on the beach. Then again, bulldozers will complete the process of filling the design template. Another possibility for mechanically placed sand is the use of barges to transport the sand and then place it in the nearshore. Once sand is in place, the contractor is required to achieve the design template with a specific initial beach slope.
Moving large volumes of sand into the beach with trucks can be costly. For this reason, the sand is placed hydraulically in 98%o of the beach nourishment projects in the United States (Dean and Dalrymple, 1999).
The most important influence on the performance of the project from the two placement methods, is the angle of repose of the material. Bagnold (1963) showed that a mixture of sand and water has different mobility characteristics than sand alone. In addition, when placed




mechanically, the earth moving equipment cannot reach underwater fill areas. These factors become important when the contractor is filling a design template. Oftentimes, the construction slope is different from the design template, impeding the contractor to achieve the requested profile. In order to remedy this problem, more sand is placed on the beach to achieve the desired profile. This overfill practice is common due to the high costs of returning to fill again the same spot. Therefore, when contractors decide to overfill using different placement methods, different volumes will be placed in each section. According to Dean (1991) the dry beach width varies directly with volume per unit width of beach. Thus, when different volumes are placed along the beach, there is the possibility of having different dry beach widths and therefore, the potential for hot spots. i order to achieve the design template with hydraulic fills it is not uncommon to have volumetric overfills of 25%o.
2.2.5 Presence of Coastal Structures
Because beach nourishment projects are usually built as a consequence of ongoing erosion, previous remedial measures include structures such as seawalls and groin fields. These two features can have negative impacts on a beach nouri shment proj ect.
2.2.5.1 Lowering of the Profile at Seawalls
Seawalls are shore-parallel structures that provide protection from encroachment of the sea and wave attack. Beaches in front of seawalls are often more eroded than those without protection due to lowering of the profile. This mechanism is shown in Figure 2.9 and explained below.
The post-nourishment profile has its design dry beach width seaward of the structure. Due to the trend of the background erosion, this dry beach width will get narrower until it reaches the incipient beach profile, and further back. This means that the "origin" of the beach profile (intersection between the sea level and the beach profile, or in other words the shoreline) will




move landward of the seawall. Then it is called a "virtual origin." Dean (1991) states that the beach profile seaward of the structure will keep the same position as if the seawall were not there.
As it is depicted in Figure 2.9, the presence of the seawall will result in a greater water depth at the toe of the structure. Should it be necessary to nouri sh this area, the truncation of the upper part of the equilibrium beach profile will be a main factor in reestablishing an incipient beach. For example, in order to renourish this area with the incipient beach, the required volume will be that between the existing profile seaward of the seawall and the incipient beach profile. This means that, in order for the beach to achieve the desired equilibrium profile, a volume must be added to achieve an incipient dry beach and then a second volume to advance the beach to the desired width. The associated erosion of not considering this volume required to achieve an incipient dry beach can lead to an erosional hot spot.
Shore-parallel
Virtual origin
This volume must beprfl
added to achieve the incipient dry beach
Figure 2.9 Lowering of the profile at seawalls.
2.2.5.2 Residual Structure-Induced Slope
Other structures used to control ongoing erosion are groins. If a nourishment project takes places where groins previously existed, the volume of sand held by the groins can play a significant role in the development of the beach. Oftentimes, the groins are removed as part of the nourishment construction after the new sand is placed. Then, the remnant bathymetry will act together with the new fill bathymetry to form a highly erosive area as explained below and shown in Figure 2. 10.




Groins are built in erosive beaches to store sand from the longshore transport, resulting in an altered bathymetry. However, it is clear that the groin can only stabilize the beach that is within its reach, thus allowing the rest of the profile to continue its eroding tendency (excluding inlets, there is no need to build groins on a beach that is not eroding).
Once the structures are removed, the segment of the profile that was under control of it will recede faster than the normal erosion rates of background erosion and spreading out losses. The reason is that the beach profile is no longer under the equilibrium achieved with the structures in place (milder slope within the structure's reach, and steeper further away), and will go faster to achieve the equilibrium profile characteristic of an uninterrupted beach. This phenomenon is shown in Figure 2. 10.
Original Contour
----Contour influenced by
3 __ _-background erosion
_2 Contours, in meters
Groin to be
removed after sand is placed
Figure 2. 10 Change in bathymetry due to background erosion under influence of a groin (adapted from Bridges, 1995).
Another mechanism causing erosional hot spots associated with groins, is that due rip currents. These currents might be strong enough to mobilize large amounts of sand and create highly erosive areas. However, it is difficult to make further statements towards this process since there is no research providing useful information on this topic, nor surveys to determine volumes of sand deposited in the head of a "rip" current.




CHAPTER 3
DESCRIPTION OF THE DELRAY BEACH NOURISHMENT PROJECT AND COMPILATION OF DATA
3.1 Data Sources
The Delray Beach Nourishment Project is the consequence of severe erosion during the 1960s. As it will be shown later, during this decade the erosion trend demanded further action to protect the city's public and private properties. The City Council of Delray Beach, FL, authorized Arthur V. Strock & Associates, Inc. in the early 70's to conduct the necessary studies to design the Beach Restoration Project for the city's beach and its corresponding monitoring. The first available reports from the Delray Beach area are from Arthur V. Strock & Associates, Inc., which basically include monitoring studies on the performance of the project; after 1985, this company was replaced by Coastal Planning and Engineering, to make the necessary monitoring analyses, providing a valuable source of information. The most important information acquired from these reports are the sediment information, fill volumes, and overall description of the project's performance.
The modem era provides a reliable high quality data base of shoreline positions and profiles. The Bureau of Beaches and Coastal Systems of the Florida Department of Environmental Protection, devoted to preserve Florida's coastal resources, developed an internet website which contains, among other data, historical shoreline trends, nearshore and offshore bathymetry, profile information, general coastal regulations, and description of projects in Florida. In the case of the Delray Beach project, this website contains considerable profile information, which proved to be very accurate when comparing results with other authors, such as Beachler (1993), Beachler and Mann (1996), and Coastal Planning and Engineering (1997).




Additional information was obtained from other reports, papers, and studies performed at the site of interest.
3.2 Site Description
The city of Delray Beach is located in the middle southern portion of Palm Beach County in the southeast coast of Florida (Figure 3. 1). The beach is located on approximately 4.8 km of the barrier island delimited by South Lake Worth Inlet on the north and Boca Raton Inlet on the south. The beach restoration project occupies approximately 4.3 km of the total beach length and is bounded by the Department of Natural Resources (DNR) monuments R-175 and R-189.
3.2.1 Historical Evolution
Located on the east coast of Florida, Delray Beach had been identified by a highly erosive area for the past decades. This area is also affected by a system of littoral barriers and inlets all along the coast, from Georgia to the Florida Keys. Most of these inlets are influenced by littoral drift, making them very unstable. As population continued to establish along this coast, the necessity to create stable entrances became imperative. Dean (1988) explains that the shoreline erosion on the east coast of Florida is due dominantly to the management of the inlets. While the efforts to make these entrances stable, mainly for navigational purposes have worked, the shoreline has been influenced negatively, retreating substantially.
The database available from the Bureau of Beaches and Coastal Systems includes historical Mean H-igh Water (MHW) shoreline positions for Delray Beach. These historical positions are displayed in Figure 3.2 for each of the monuments within the project area. The earliest shoreline postition for this area dates from 1884. Since then, a slow retreat is recorded in most of the project length except for the northern-third of the project, until the 1920's. The shoreline then recovered faster than it was receding, until the 1940's when it continued accreting slowly. It is from 1962 to 1970 when a fast erosion of the shoreline was noticed, and up to 3.7




m/year of shoreline retreat at R-176 were found. As a result of hard coastal structures, the shoreline position remained more or less in the same location until the nourishment program was started in 1973. This nourishment project helped stop the erosional trend that started in the 1960s.
JAC<5GN.ILLE
ALLA$A93iEL~ LAALAA'fE
TAIJPA
YET PALM BEA5H
DELRAY
' MIAI OvCCH
KEE Ai
UL
iW T C I Cr
NTS I

Figure 3.1 Location map for Delray Beach, Florida.




140 i
- -R-1 5 N urishme
.,c r, shineit
120 1 7-R-1 )
SR-1 7
100 -- -- -R 8
.... . R-179
80 __60
40 __""'20
o . . . .. . ? : . .. .. . ,, ,- ,T . . . . .. . .. .. . . .. .. . .. .. .. .. .. .. .. , , l , . . . . . .
1 70 1E80 1 9 1E00 1 10 1E20 130 140 1 50 1 60 1 0 1 80 190 2C00 2,
-20
Year
140 -- R-1 10 N urishment
120 R--. -i 1'1
1 12C
Ri 2 !
-- 1 3
100 Ri 4
. -..... R -184 5N ris
4o _-____6 0 - - -1
40 _____1E 0 180 180 1 1910 190 1 0 1 0 150 190 190 190 190 200 2
-20
Year
140 ____- R 1 8 5 E r is h mn n t '
120 .. -186--!
-- T-187 /
100 . .R-188
60 10'_ i" . . 1 . ... .. . ........
1170 1 80 1190 1100 1110 1120 1 ,30 1140 1150 1160 1170 1 80 1 90 2(00 2
-20Ye Yea r

Figure 3.2 Historical shoreline position from DNR monuments at Delray Beach, FL.




3.2.2 Delray Beach Nourishment Project Description
The erosion control project was planned to be carried out on a periodic basis, as is necessary in this type of project. After the project's almost 26 years of life, sand was placed nonuniformly (Figure 3.3). The initial nourishment, in 1973, placed approximately 1.25 million cubic meters on the 4.3 km project. Five years later, in 1978, 536,000 cubic meters was placed over two separate segments. The second and the third renourishments placed 994,000 and 780,000 cubic meters respectively, as shown in Figure 3.3. A summary of the volumes of sand distributed along Delray Beach as part of this restoration program, is shown in Table 3.1.
Table 3.1 Volume of sand placed in the Delray Beach Nourishment Project
Period of Construction Length Volume Placed Cumulative Volume
Encompassed [ x 10 i3] [! x 103 in3]
[km]
July -August, 1973 4.27 1250 1250
February May, 1978 1.98 (north section) 404 1654
0.91 (south section) 132 1786
September- October, 1984 4.27 994 2780
November -December, 1992 3.14 780 3560
Source: Coastal Planning and Engineering, 1997.
In addition, the fourth renourishment has been proposed to take place in 1999/2000.
As it can be noticed from Table 3.1, the time span between every nourishment has been gradually growing from 5 to 8 years. This is a natural, positive influence of the project on the adjacent beaches. The reason can be thought as if we were building a longer project, which can be explained from the diffusion theory from Pelnard-Consid6re (1956).
The native sediment on this area has been reported to have a mean grain size of 0.46 mm, however, this sand barely formed a dry beach prior to the year of 1973 when the project took place. Further analysis of the sediment size distribution and its evolution due to the project, will be completed later in this thesis since it might be a potential cause for erosional hot spots, and therefore, has to be examined in detail.




1500 0 1500
GRAPHIC SCALE IN FEET Al
LEGEND
A FDEP SURVEY
MONUMENT

Figure 3.3 Placement of sand for the different maintenance fills.




In additon, it should be mentioned that the project is located where no other coastal project will impact directly in a short term. Both South Lake Worth Inlet (updrift), and Boca Raton Inlet (downdrift), are located sufficiently away so that inlet stabilization and/or dredging operations would not affect the project performance predictions. On the other hand, other nourishment projects located on the same barrier island will not influence the project's performance either, since they are sufficiently distant.
3.2.2.1 Borrow Area
The designated borrow area is shown in Figure 3.4. It is located more or less directly seaward of the restoration area, approximately 760 meters offshore. It includes a no-dredging zone 300 meters wide to protect a sewage outfall. This outfall is located almost directly seaward of Atlantic Avenue, at DNR monument R-180. The offshore borrow site is approximately 300 meters wide, 2740 meters long, and before being dredged it was located a depth of 16 m.
Very long profiles were available for the entire project area for the year of 1990 (reaching a depth of -30 meters), from the Florida Department of Environmental Protection internet website. However, no further data was available for the borrow pit. Thus it was not possible to make any further comparisons for volumes removed from or filling of the borrow pit.
Bathymetric contours in the borrow area were more or less straight before dredging occurred, and after this dredging, the nearshore contours remained practically straight and parallel.
The available material, as analyzed before the initial nourishment, is a fine to medium gray quartz sand, is well sorted, and its diameter is smaller than that of the beach material. As noted, the beach sand was determined to have a 0.46 mm mean diameter, while the borrow material was calculated to have a mean diameter of 0.20 mm (Arthur V. Strock & Associates, Ic, 1973). Therefore, similar to many nourishment projects, this is a case where finer sand is placed on the beach to restore it.




--..-NORTH CONSTRUCTION LIMIT
M2RT7 mvECr LMT

'No dredging
zone
BORROW AREA

SOUTH CO MWTMGMW LIMIT

array Beach SOUmTH PROmJECT
aIMr
ighol

Figure 3.4 Location of the borrow area with respect of the fill area.




3.2.2.2 Dune Restoration
To avoid sand being blown out of the project dunes, in 1974 dune vegetation was planted.
In 1979, a program to evaluate the accretion of the dune in the vicinity of the public beach (see
Figure 3.3) was initiated. A sand fence and additional dune vegetation was placed in this area in
1980. Since then the dune has continued to expand in elevation as well as width (Coastal
Planning and Engineering, Inc., 1997). Table 3.2 shows a summary of the history of dune
accretion at the public beach.
Table 3.2 Delray Beach renourishment project, forty-eight month monitoring study. History of Dune Accretion from DNR monument R-177 to R-182
Date Accretion Comments
[mI ]
July 1973 Original restoration project
May 1979 55,000 First renourishment project
July 1980 Sand fence installation along public beach
October 1984 5,160 Immediately after second renourishment project
November 1985 8,250 First year after second renourishment project
December 1986 1,600 Second year after second renourishment project
March 1988 4,400 Third year after second renourishment project
January 1989 385 Fourth year after second renourishment project
March 1990 3,070 Fifth year after second renourishment project
December 1992 9,600 Immediately after third renourishment project
December 1993 3,840 First year after third renourishment project
December 1994 2,790 Second year after third renourishment project
December 1995 3,950 Third year after third renourishment project
January 1997 5,000 Fourth year after third renourishment project
103,045 Total volume of sand accreated on the public
beaches dunes since project inception in 1973. Source: Coastal Planning and Engineering, Inc., 1997.
Table 3.2 shows a total accreated volume of 103,045 inm3. Thus for the public beach
(encompassing approximately 1.98 km), the dune growth has been 2.3 m3/m/year.
3.2.3 Hydrodynamic Conditions
For the entire monitoring area, the dominant deep water wave direction is 200 north from
the perpendicular to the shoreline and will be considered constant for most of the year. For




modeling purposes, the wave height has been defined in terms of its representative characteristics This means that it will be considered in terms of the effective value of time-varying wave height and period according to the Ralyleigh distribution. Dean and Grant (1989) include a series of plots for the state of Florida in which wave height, period, depth of limiting motion, and the coefficient of diffusivity, are defined upon the location on the Florida coast. The effective deep water wave height and period for Delray Beach is 0.43 meters (1.4 ft) and 6.5 seconds, respectively. In addition, the depth of limiting motion and berm height, (d.+B), is 7.16 meters (23.5 ft).
The tidal planes are also defined for this area. Table 3.3 shows these values with reference on the National Geodetic Vertical Datum (NGVD).

DNR
Monument
R-175 R-176 R-177 R-178 R-179 R-180 R-181 R-182 T-183 R-184 R-185 R-186 T-187 R-188 R-189
Source: Balsil

Table 3.3 Predicted tidal datums
MHHW MHW
[in] [in]
0.591 0.570
0.591 0.570
0.591 0.570
0.591 0.570
0.591 0.570
0.591 0.570
0.591 0.570
0.588 0.570
0.588 0.570
0.588 0.570
0.588 0.570
0.588 0.570
0.588 0.570
0.588 0.570
0.588 0.567

(NGVD) for Delray Beach, Florida.
MTL MLW
[in] []in
0.128 -0.283
0.128 -0.280
0.128 -0.280
0.128 -0.280
0.128 -0.280
0.128 -0.280
0.128 -0.280
0.128 -0.280
0.128 -0.280
0.128 -0.280
0.128 -0.280
0.128 -0.277
0.128 -0.277
0.128 -0.277
0.128 -0.277

lie, 1987.

3.2.4 Littoral Transport
Several authors, including Dombrowski and Mehta (1993) have noted that the longshore sediment transport on the east coast of Florida is towards the south. According to their study,

MLLW [in]
-0.335
-0.335
-0.332
-0.332
-0.332
-0.332
-0.332
-0.332
-0.332
-0.332
-0.332
-0.332
-0.332
-0.329
-0.329




123,000 m3/year of sand leave South Lake Worth Inlet, north of Delray Beach, and 93,000
m3/year of sand arrive at Boca Raton Inlet, just south of Delray Beach. Assuming a linear
distribution of the longshore sediment transport in this area, Delray Beach would have a
longshore sediment transport of 108,000 m3/year.
3.2.5 Beach Profiles
Cross-shore sections are vital to analyze the local behavior of the project. Beach profile
availability is summarized in Table 3.4. These data are available from the Bureau of Beaches and
Coastal Systems.
Table 3.4 Available profile data for Delray Beach, FL from the Bureau of Beaches and Coastal Systems.
Approximate Date of Profiles Available Comments
Survey
1/14/75 R-175 through R-189 Approx. 1000 m long (-15 m deep)
8/1/90 R-175 through R-189 Approx. 1600 m long (-28 m deep)
10/92 (Prenourishment) R-175 through R-189 Approx. 600 m long (-8 m deep)
12/92 (Postnourishment) R-175 through R-189 Approx. 600 m long (-8 m deep) 12/93 R-175 through R-189 Approx. 600 m long (-8 m deep)
12/94 R-175 through R-189 Approx. 600 m long (-8 m deep)
12/95 R-175 through R-189 Approx. 600 m long (-8 m deep)
1/97 R-175 through R-189 Approx. 600 m long (-8 m deep)
1/15/98 R-175 through R-189 Approx. 450 m long (-7 m deep)




CHAPTER 4
DELRAY BEACH NOURISHMENT PROJECT PERFORMANCE
The City of Delray Beach, Florida, has arranged for the monitoring of the nourishment project since 1973. The project was planned, since its conception, as a beach restoration program which would eventually require periodic maintenance, just as any beach nourishment project should be planned. i order to schedule these periodic maintenances, it is necessary to make a first approximation of the overall performance and life of the project.
However, these predictions are difficult to achieve in fine detail, at least at present. The evolution of the project depends upon a set of factors that act together to shape the beach, in both longshore and cross-shore directions. Moreover, the coastal environment is subject to sudden and strong changes that can determine the evolution of the beach, especially during storms.
The fact that it is still difficult to predict the detailed evolution of a beach nourishment project, leads to the necessity to monitor these type of projects, and even to be subject to further research, like the one performed in this thesis.
There are two main tools that will be used to anlayze the evolution of the project locally: shoreline changes and volumetric profile changes.
4.1 Previous Studies of the Delray Beach Nouishment Project
The Delray Beach Nourishment Project has been part of several studies in which the shoreline and volume changes have been modeled and/or measured. There are three main efforts that contributed to available results for this project. The first is formed by the set of monitoring studies performed initially by Arthur V. Strock & Associates, Ic., and later by Coastal Planning




and Engineering, Inc. These reports are focused on annual observations of the project. Beach samples, beach profiles, and high water level position are measured, to then compute shoreline and volume changes, and sediment size analyses. Therefore, these reports do not carry on performance predictions of any kind, but they do constitute a rich set of data.
The second contribution used in this thesis, is the one presented by Beachler (1993) and Beachler and Mann (1996), where the purpose is to show that much of the sand which moves out of the project can actually be accounted for and benefits the neighboring beaches. In this way, the authors extend the study area about 3 kilometers north and south of the project, to analyze the benefits of the Delray Beach Nourishment Project to the adjacent areas.
Finally, the third contribution which includes results on shoreline response at the Delray Beach area, is that presented by Gravens (1997). This paper concerns evaluation of the relative influence of various procedures for developing input wave conditions for use in numerical models of shoreline change. His modeling was performed using GENESIS, a program developed by the U.S Army Corps of Engineers. Gravens selected the Delray Beach site because the physical setting and evolution of the shoreline are expected to conform to the assumptions imposed by one-line theory, or in other words, that the nearshore bathymetry can be regarded as straight and parallel; in addition, this site was chosen because of the nourishment program that has been ongoing since 1973, which provides the necessary data.
4.1.1 Shoreline Changes
Coastal Planning and Engineering (1997), summarizes the Mean High Water (MHW) changes since the last nourishment as follows. The shoreline within the project area advanced 66 m on average, with the largest advance at monuments R-186 and T-187, where it reached 82 m and 83 m respectively. After forty-eight months, the MHW shoreline retreat is 40.5 m on average, with 65 m and 72 m at DNR monuments R-186 and T-187, respectively, as the largest recessions. This large retreat is attributed to the largest shoreline advance during the project construction and




also because it is near the end of the fill area, where losses are expected to be larger. In addition, the majority of the MHW shoreline retreat is due to equilibration after the construction profile (Coastal Planning and Engineering, 1997).
Arthur V. Strock & Associates, Inc. (1984) mentions in regard to the shoreline changes at the earlier stages, that prior to the second renourishment, in 1983, the shoreline retreat rates were larger from DNR monument R-180 to the south with the highest shoreline retreat rate at R-186 (7.3 m/year). In fact, at the first three monuments of study (from R-175 to R-177), the shoreline advanced. It is important to point out that these results were obtained after the first renounshment, in 1978, where sand was placed nonuniformly over two separate areas (see Figure
3.3).
Beachler (1993) focused on the shoreline changes from the years of 1973 and 1974 to the year 1990. Figure 4.1 shows the MHW shoreline changes computed by Beachler (1993). In this figure, a minimum value within the fill area is reached at R-186. In addition, this figure shows positive shoreline changes from R-165 to R-201, which is explained to be due to the spreading out of the Delray Beach project sand. Figure 4. lb includes the effects of the initial nourishment, thus a wider dry beach width is computed. It is noticeable, however, that around R-186 an area with higher erosive rates prevails. Beachler and Mann (1996) extended Beachler's previous work to analyze the changes to 1995 which now include the effects of the 1992 renourishment where sand was placed as previously shown in Figure 3.3. Their results are shown in Figure 4.2. In this graph it is possible to see that the shoreline changes reach a minimum at the first monuments, where no sand was placed in 1992, and another minimum at R-186.
The study performed by Gravens (1997) is aimed at analyzing the influence of different wave conditions on predictions of the shoreline changes. The modeling starts in January 1987 and predicts shoreline positions for October 1992, immediately before the third renourishment. Figure 4.3 depicts an example of his modeling, where only the measured conditions are of interest. The figure is bounded by the limits of the project. Gravens (1997) mentions that a comparatively




higher erosion rate of almost 2.0 m/yr occurs around 2000 m from the origin. The first 2500 m of the project average 1.6 m/yr of erosion and the remainder of the modeled reach, only 0.8 m/yr. Even though this paper is not directed to identifying erosional hot spots, it does refer to the mentioned area with 2.0 m/yr of recession, as an erosional hot spot. This highly erosive area corresponds to the location of an opening in an offshore reef and to a 300-m-wide no-dredging zone where an outfall pipeline lies.
1974 1990 SHORELINE CHANGES
200
1973 1990 SHORELINE CHANGES
IAV~ERAl
SMREIM~ !itaE IFEETI
r R MOIO ENT
Figure 4.1 Mean high water shoreline changes computed by Beachler (1993) a) Between 1974 to 1990, and b) Between 1973 and 1990.




R or
4i:

20

'L~ i- I 1- JA
165 17 In it' 10 175 177 179 11 1 M63 183 187 189 191 13 '105 17 19t 201
168 1~8 170 17 174 176 178 1 0 112 14 1N I 1 92 19 190 206 FEP MONJUMENT
m1,74 TO 1ne m 1974 TO 195
FI, AREA 15 FROM MONUMENT 175 7O MONLAUENT 1
Figure 4.2 Comparison between mean high water shoreline changes from 1974 to 1990 and mean
high water shoreline changes from 1974 to 1995 (Beachler and Mann, 1996).

0 500 L00 IMeIS SO 25O 3000
DilmanA ElmI odig()W

3300 4M00 4W I00

Figure 4.3 Measured mean high water shoreline changes between January, 1987 and October, 1992 (from Gravens,1997).




4.1.2 Volumetric Profile Changes
The latest monitoring report from Coastal Planning and Engineering, Inc. corresponds to the 48-month monitoring after the third renourishment. Volume changes are computed in this report from the monument (onshore) to the -24 foot contour (-7.3 in). Even though these changes are not analyzed locally, it has been calculated that, the area with the highest erosion trends is that encompassed by monuments R-186 and T-187. Their results also show that outside the 1992 project limits, there has been a positive impact after sand has spread out. Again, there has been a larger volume of sand placed towards the southern limits of the project (T-1 87).
The results from Beachler (1993) also show a highly erosive trend at approximately R186, with respect to its surroundings. This is depicted in Figure 4.4, where volume changes were computed to the -18 foot contour (-5.5 in). From this figure, it becomes somewhat obvious that at R-186 an erosional hot spot exists; however, further research is necessary to try to predict or explain the potential causes behind this phenomenon.
VOLUME CHANL3CA CUOIC YARAS ,Th n p
I An1

so 40

0
-20
44

- 1U ARRA

III

II till

I'

Figure 4.4 Volume changes computed by Beachler (1993) between 1974 and 1990.

a .........~w1.a~. .......... - - i
TooM 7 T is Onto Frill R t 15% A201 ONR MONUMENT




One more study that analyzes data from the Delray Beach Nourishment Project was carried out by Dean and Abramian (1991). This report studies techniques for evaluating potential sands for beach nourishment projects and is another source of data for grain sand distribution.
4.1.3 Conclusions from the Previous Studies
It is important to mention that none of the previous studies was focused on the identification and analysis of erosional hot spots, except for the monitoring studies which should be able to detect such a problem if it happened to exist; however, none of the monitoring reports mention the presence of erosional hot spots at all. Only the study by Gravens (1997) has included some comments in what he considers the presence of an erosional hot spot. This erosional hot spot is said to be found around DNR monument R-180, landward of the reef opening, and where a no-dredge zone is located; nevertheless, he considers a somewhat different time span for modeling, as he studies shoreline positions from January of 1987 through October of 1992, and the other two studies consider the changes between 1973 and 1990, and 1992 to 1997. It is possible then that, for some reason, only during this lapse a higher erosive trend was acting in this area.
Further analysis of the results presented by Beachler (1993), Beachler and Mann (1996), and Coastal Planning and Engineering, Inc. (1997), suggest that it is possible to determine the areas with larger erosion trends. Both set of reports coincide that the erosion rates are larger at R186 and T-187, between 1974 and 1990, and between 1974 and 1995. Of course, this does not necessarily mean that there is an erosional hot spot in this area.
4.2 Analysis of the Field Data
From the profile data it is possible to study the behavior of the beach locally through shoreline and volume changes. Shoreline changes are important to be analyzed because they are the most visible factors in evaluating beach profile fluctuations, and because the beach width




provides natural protection from storms absorbing the energy of the waves. In addition, the beach width represents available recreational area for beach goers. Volumetric profile changes are here referred to as the change in volume per unit width at a location. The importance of volumetric profile changes is that they account for most of the profile equilibration as it moves offshore to form the bar system. Volume changes are important to compute since it is necessary to study the amount of sand that remains within the project after construction completion.
4.2.1 Distribution of Nourishment Volumes Along the Project
Every nourishment has been designed different, according to the observed need prior to each nourishment. This is why, only the initial nourishment and the second renourishment, encompassed the entire study limits, from R-175 to R-189. Likewise, sand has not been placed uniformly every time a nourishment takes place. The outcome is an irregular placement of sand that may help explain why some areas have higher erosion rates than others.
According to the monitoring reports, Figure 4.5 depicts the longshore distribution of sand placement for each of the nourishments. Figure 4.6 shows the total volume per unit width placed in the Delray Beach Nourishment Project. This information was obtained from the monitoring and construction reports from Arthur V. Strock & Associates, Inc. and from Coastal Planning and Engineering, Inc. Except for the second renourishment in 1984, where no detailed information was available, the volume of sand per unit length placed is significantly non-uniform along the project length (see Figures 4.5 and 4.6).
4.2.2 Sediment Size Distribution Along the Project
The monitoring reports provide results of sediment collection and analysis. However, data are relatively complete after the second renourishment. Composite analyses of sand samples were considered to study the sediment size distribution. Figure 4.7 portrays the mean grain size distribution along the project since October of 1984, when the second renourisment took place.




The same data is plotted showing the mean grain size evolution over time in Figure 4.8. Sand samples were collected at DNR monuments R-177, R-180, R-184, and T-187, starting form the top of dune to the -20 foot contour (-6.1 in), covering the beach and nearshore zones.
I I I u
--1973 --- 1978
1984 1992
450
150
0- -0.50 -0.0 .0 .
o I I
I 15I
_ I
I I
Fiur 4.7 shw th vaiaio of th sdmnsieaogtepjctaftrtescn nI.I I
2.50 -2.00 -1.50 -1.00 -0.50 0.00 0.50 1.00 1.50 2.00 2.50
Longshore Distance [kin]
Figure 4.5 Volumes of sand placed along the project for each of the nourishments.
Figure 4.7a shows the variation of the sediment size along the project after the second renounshment. Even though there is no information available for the first month at DNR monument T-187 for the second renourishment, the influence of the fill material in the first month is clear, when the mean grain size ranges from 0.21 to 0.29 mm approximately. After 15 months there is a large increase in the mean grain size over the entire project where the sediment size
th ,d
ranges from 0.38 to 0.39 mm. From the 27h month to the 52 the mean grain size has less variation, with an average around 0.35 mm. It is possible to argue that, for the first month, coarser sand was found on the updrift side of the project, however, these differences disappear after the




39
first year and these fluctuations have become more stable. Figure 4.8a depicts the convergence of the mean grain size to a value around 0.35 mm.
-V e Pe s149
-Volume Placed since 1973
Volume Placed since 1974
1200 A
E
800
E
_ 400_200
-2.50 -2.00 -1.50 -1.00 -0.50 0.00 0.50 1.00 1.50 2.00 2.50
Longshore Distance [km]
Figure 4.6 Cumulative volume of sand placed along the project.
The sediment size variation along the project after the third renourishment is shown in Figure 4.7b, where after one year the project has the finer sand, ranging from 0.24 to 0.33 mm. After two years, the sand reaches its larger mean diameter and ranges from 0.38 mm on the updrift side to 0.28 mm on the downdrift side. From the mean grain size evolution depicted in Figure 4.8b it is remarkable that at T-187 the mean grain size has the finer sand with respect to the rest of the project.
Therefore, these two figures describe the manner in which the mean grain size has evolved since 1984 within the entire project. Since the native mean grain size in 1973 was 0.46 mm and the fill mean grain size is around 0.20 mm, there is a tendency to reach an equilibrium, as shown in the figures; however, it is noticeable how the interaction of the fill sand, the native sand,




40
and the longshore sediment transport interact together to yield a nonuniform sediment size along the beach, thus, causing the beach to evolve differently from one location to the other.
Sediment Size Variation along Delray Beach Project 2nd Renourishment (October, 1984)

U oUU
S.................. ........
"'" ."" "" 0.375 . "" .-,. ,
0.350
0. 300
9 0-27
A A LA
R-17 R-18 1 R-184 T187
2 -1.75 -1.5 -1.25 -1 -0.75 -0.5 -0.25 0 0.25 0.5 0.75 1 1.25 1.5 1.75 2
Longshore Distance [km]
Sediment Size Variation along Delray Beach Project 3rd Renourishment (December, 1992)
b)-3.426
.- ).375
). .7. - .- -
R-17. R- R- 84 T-187

---1 month
---- 15months
- 27 months
41 months
- - 52 months
12 months
---- 24 months
- 36 months
49 months

-2 -1.75 -1.5 -1.25 -1 -0.75 -0.5 -0.25 0 0.25 0.5 0.75 1 1.25 1.5 1.75 2
Longshore Distance, [km]
Figure 4.7 Longshore distribution of the sediment size a) After second renourishment, and b) After third renourishment.




41
Mean Grain Size Evolution (1984 Renourishment)

-U.0
0426 a)
0.400 ___
0.375 .. ...-........ ...
0.325 w0D.300
0.275- "
0.250
0.225
0.200
0.175
0.150
0 6 12 18 24 30 36 42 48 54 6
Months after Nourishment
Mean Grain Size Evolution (1992 Renourishment)
0.450
0.425 b)
0.400
0.375
0.350 __0.3265-__0.300 _____"
0.275 ,,- -'"_0.2500.225
0.2007
n 171

-. R-177
--*-- R-180
- R-184
T-187

- R-177
--o-R-180
- R-184
T-187

0 6 12 18 24 30 36 42 48 54 60
Months after Nourishment

Figure 4.8 Sediment size variation with time
a) After second renourishment, and
b) After third renourishment.

(5

(0

0




4.2.3 Shoreline Changes
As explained before, shoreline changes are important because the dry beach width provides protection from storms to the coastal region, it is the most visible factor to analyze the changes of the shoreline, and provides recreational area. Instead of shoreline positions, shoreline changes are presented here in order to identify the areas with the largest deviation, and therefore to identify possible erosional hot spots and cold spots.
Based on the profiles provided by the Bureau of Beaches and Coastal Systems of the State of Florida through their internet website, it was possible to calculate the shoreline changes of the beach throughout the life span of the project. Changes within two periods of time are presented here: from January 14, 1975, to August Is', 1990, and from January 15, 1975, to January 15, 1998.
These shoreline changes are shown in Figure 4.9 and 4. 10. In order to study the beach response in the vicinity of the project, shoreline changes between 1975 and 1990 are shown from monument R- 165 through monument R-20 1, this is, about 3 kilometers north and 3 kilometers south of the project limits. Shoreline changes between 1975 and 1998 encompass the project length only, since no more data were available.
From 1975 to 1990 the measured values of the shoreline change show that there is a large fluctuation along the study area. During this period, the project area has been renourished twice, in 1978 and in 1984 as shown in Figure 3.3. The middle part of the project (approximately from R-178 to R-185) has advanced around 30 mn, which is the projected dry beach width to maintain through the beach nourishments. The areas outside of updrift and downdrift sides of the project show an advance of approximately 15 mn. In addition, Figure 4.9 shows that there are two minima at R-177 and at R-186 of 10 mn. The fact that the ends of the project have more erosion than the middle area, can be explained from the diffusion theory first introduced by Pelnard-Consid~re (1956). Therefore, the shoreline (and volume) changes in these areas should be predictable.




Fill area
20
10
.0 -6.0 -5.0 -4.0 -3.0 -2.0 -1.0 00 10 20 30 40 50 6 7
10
AAA A AA A, AA A
10 0 ~ 0)(N ~ 0 ~ 0 (N MO ~ 0 (0 M M 00 (N MO 1M( 0
0 0 (0 0 (0 Nk Nk N Nk Nk N E 0) 0) 0) 0) 0)0 0) 0)0 0
n & & n

Longshore Distance [knm]

Figure 4.9 Shoreline changes from 1975 to 1990.




4- Fill area
40
35
30
25
20
15
10
A X A A A A A A A A A A A A A
L.O .0 r'- 00 M 0 N' M3 I- .O (0 r'- CO M
0- 0- 0- 00 CO CO CO CO
____ 7T '7 ______of of of of of of 6 of of of of of of
of _f

Longshore Distance [knm]

Figure 4.10 Shoreline changes from 1975 to 1998.




After fifteen years, the updrift area of the project has accreted, although large differences within short distances are present. Same differences are present on the downdrift side, however, much less shoreline advancement is computed.
For the period 1975 to 1998, where only the project area is analyzed (Figure 4. 10), it can be seen that from monument R-175 to R-180, the shoreline increases its width from 15 to 50 mn; at R-186, the shoreline advanced only 15 mn, while at the previous and at the next survey lines the advancements are 43 and 30 mn, respectively. Finally, at the middle portion of the project, the shoreline advanced between 36 and 50 meters.
4.2.4 Volume Changes
Volume changes represent the sand remaining within the project. These volumes are computed per unit width of beach. Because some of the profiles consist of very long surveys (reaching -28 in), it is necessary to establish a datum plane of reference, to which volumes are to be computed. This is very important in order to exclude negative volume changes associated with the borrow areas. As an example, Figure 4.11 shows the survey lines at R-188 in 1975 and 1990. The depression shown in this figure from -15 mn to -20 mn represents the borrow pit as surveyed in 1990, which means that the third renourishment in 1992 is not accounted for.
This borrow pit is bounded seaward by the aforementioned shore-parallel reef. Whether some influence exists from the borrow pit or not, is difficult to establish; however, Figure 4.11 shows that, from -8 mn to -15 mn, the profile has steepened probably due to sand lost to the borrow pit. This process can be regarded as a natural tendency of the profile to compensate for the sand lost in the dredging operation. Therefore, if this volume is accounted for in the volumetric profile changes, there will be an error, given that this material is not the consequence of wave and current action. In fact, some authors such as Kojima et al. (1986), have noted the possiblity that this steepening of the profile due to replenishment of the borrow pit, can reach the shoreline if the




hole is located close enough to the beach (thus becoming another potential cause for erosional hot spots).
10
Cross-shor Distance [m
20 40 600 80 100 1200 1400 1600 100
-5 _______ _______ \____-0-1/15/75
R -ef
8/1/90
Z -15 ____-20 Borrow pit
-25 ____-30
-35
Figure 4. 11 Comparison between the 1/15/75 and 8/1/90 profiles at R-188.
In this context, volumetric profile changes were computed out to different depths. Figure 4.12 shows the volume changes between 1975 and 1990, and Figure 4.13 between 1975 and 1998. From both plots it is noted that the values converge around -7 m. Note in Figure 4.12 that volume changes indicate large loss of volume for depths greater than 9 m in the fill area. Computations for volume changes will then be computed from the monument to the -7 m datum plane.
Figure 4.12 shows that updrift of the project, despite large fluctuations, there is a volume accumulation after 15 years, while downdrift of the project area, there are large fluctuations showing both loss and gain of volume. Throughout the project area, there is an approximate




800
A Fill area
600 400
.0 -6.0 -5.0 -4.0 -3.0 -1. 0 0 O 1 0 40 5 0 0 7
100
600 1
800 1l
I
1 1000
14A I
Longshore Distance [kn]
-5 m - - --6 m -7 nm -9 m -12.5 m 1 7.5 m -20 m - -22.5 m

Figure 4.12 Volumetric Profile Changes from 1975 to 1990.

-7
E
a)
C
O E0)
E
-)




150I
Fill area
400
350 -----...
.-200 .,
150 a ,
100
000 0 0 0 00
"- 00 20 00 00 00 00 00 00 00
0: F & FL
07 7 7 _7 7" 7 7 "

Longshore Distance [knm]

------ 5 m W -6 m -6.5 m -7 m -7.3 m
Figure 4.13 Volumetric profile changes from 1975 to 1998.




average increase of 150 m3/m except around monument R-186 where the volume changes are close to zero, indicating that all the sand placed in this area has been lost. This result is consistent to that computed by Beachler (1993), shown in Figure 4.4. From T-187 to R-189 the volume increase is on the order of 100 m3/m.
For the other period of time, that is, 1975 to 1998, Figure 4.13 shows large fluctuations as well. The largest difference is present, again, at R-186, where the volume gained after three renounshments is 90 m3/m, while an approximate average for the volume change in the rest of the project is around 250 m3/m. Also, for the updrift side of the project, there is a small gain of volume compared to the rest of the project, which is due to the third renourishment which was placed between R-180 and R-188.
4.2.5 Summary
The analysis of the field data yields information on grain size, shoreline and volume changes, profile information, and description of the volumes placed throughout the life of the project. Previous contributions describe the effects that different wave conditions have on the shoreline response, and the positive influence of Delray Beach nourishment on adjacent beaches.
The distribution of volumes along the project has been shown to be irregular, that is, the volume of sand per unit width of beach placed varies along the project. In addition, every nourishment has had different limits.
The mean sediment size is also analyzed here. Perhaps, as a consequence of the irregularity of the volumes placed along the beach, the mean grain size proved to be also irregular. Two patterns have been identified: coarser sand is located on the updrift part of the project, and the slow natural tendency of the project to reach an equilibrium of the mean grain size.
Volumetric and shoreline changes show large fluctuations. A highly erosive zone has been identified around monument R-186, which is also consistent with the previous studies




50
performed at Delray Beach. Whether this area corresponds to an erosional hot spot can only be determined after performance predictions are applied, in this case through the one-line numerical model DNRBSM.




CHAPTER 5
APPLICATION AND MODELING RESULTS
5.1 Numerical Model for Beach Planform Evolution Dean and Grant (1989) developed a one-line model for calculating the shoreline response in the vicinity of beach nourishment projects. The original purpose of the model was to establish thirty-year shoreline position predictions for a beach nourishment project. The original numerical procedure did not consider the fact that most nourishment projects are periodically maintained according to the observed losses. A new version of this methodology is applied in this thesis, and it is able to compute the shoreline response over a period of time, including renourishments, if any. Additonally, Appendix A contains the description of the theory and governing equations used to develop the model and, since no written literature that accounted for the modifications performed to Dean and Grant (1989) methodology exist, Appendix B contains the program listing and a sample input and output. However, this thesis is exclusively directed to the analysis of the performance of the Delray Beach Nourishment Project, and therefore, the development of the latest version of the program is not included.
This methodology was developed in Fortran language. The name of the program is DNRBSM and stands for Department of Natural Resources, Beaches and Shores, which is the entity that first sponsored the development of this methodology. The letter 'in' was later added to stand for multiple nourishments. For further detail in the development of the program and its applications, including description of variables, numerical procedure and detailed capabilities of the model, the reader is referred to Dean and Grant (1989).




5. 1.1 Application of the DeIray Beach Data for Planform Performance Predictions
Prior to using the program, certain data are required. These data include mean grain size, volume of sand placed along the project per unit width, wave characteristics, background erosion, location of coastal structures, boundaries of the project, etc. The compilation of these data was mostly taken from the monitoring reports from Arthur V. Strock & Associates, hic., and later Coastal Planning and Engineering, hic.
5. 1. 1.1 Sediment Size
From several sand samples collected throughout the life span of the project, it has been seen that the mean sediment size varies both with time and location along the project. This is due to the large difference between the fill grain size and the native grain size. The variation of the mean grain size with time along the project has been previously described in Figures 4.7 and 4.8.
Given the importance that the sediment size has in the sediment transport equation (through the sediment transport parameter, K), it was necessary to determine a sediment size representative of the actual conditions. However, the large fluctuations along the project show that there is no apparent representative mean grain size, and even an average is probably unrealistic. To account for this situation, several runs varying the mean sediment size were performed. These results will be shown later in this chapter.
It seems, however, that the tendency of the mean grain size is to reach an "equilibrium" around 0.3 to 0.35 mm. From the information compiled, a mean grain size of 0.32 mmn was used to compute shoreline and volumetric changes. According to Dean (1989) the sediment transport parameter, K, corresponding to this diameter is approximately 1. 10.
5.1.1.2 Wave Characteristics
The effective deep water wave height of 0.43 meters (1.4 ft), and a period of 6.5 seconds is used. The dominant deep water wave direction adopted in the model is 200 north from the perpendicular to the beach, and is assumed to be constant throughout the entire year. These




characteristics were previously discussed in Section 3.2.3. In addition, the sum of the depth of limiting motion and berm height (h. B) has been set to 7.16 meters (23.5 ft) as mentioned before.
5.1.1.3 Model Set-up
The numerical solution applied here is based on an explicit scheme in which the equations for sediment transport and continuity are solved sequentially. The one-line numerical method uses a grid or computational scheme as depicted in Figure 5.1, where the shoreline positions are held constant for a time step, At, while the sediment transport is computed. After this computation, the sediment transport is held constant and the shoreline positions are updated. This process is then repeated until the time of desired modeling has been accomplished. The DNRBSM model assumes straight and parallel contours seaward of h., and contours parallel to the nourished shoreline landward of h.. The project length encompasses 56 cells, 76.2 meters (250 ft) wide each, for a total of 4270 meters. The total modeled length was 200 cells or 15240 meters (50000 ft).
Qa,1
Q = sediment transport
Yi y shoreline position
Qi% '+l time step
AIXY+ Ax =cell width =76.2 m

Figure 5.1 Computational scheme used in computational method.




The grid system applied in this case requires definition of the boundary conditions at both ends in order to solve the continuity and sediment transport equations. The case of Delray Beach is that of an uninterrupted shoreline, which means that the shoreline position is specified at both ends of the computational domain for all times, and the initial shoreline is also specified.
The stability criterion for this numerical procedure is given by
(At) G (5.1)
mx 2 G
in which G is the "alongshore diffusivity" coefficient in the so-called diffusion or heat conduction equation. The same graphical procedure used to estimate the effective deep water wave height and period, and the depth of limiting motion discussed in Section 3.2.3, is available for the coefficient of alongshore diffusivity, from Dean and Grant (1989). According to the location of the project, this coefficient is approximately 4.18x10- m2/s (0.045 ft2/s). Therefore, a time step of 86400 seconds (1 day), which is a reasonable value for this type of modeling, meets the criterion established for numerical stability. Additionally, it has been found that the area encompassed by Delray Beach does not show background erosion (Dean et al. 1998).
The initial conditions are specified from the volume per unit width placed in the first nourishment. These volumes are input for each one of the nourishments at the corresponding time step. In order to account for shoreline changes, the nourished profile is assumed to be of the same form as the pre-nourished form, but displaced seaward. Once the volume change, depth of limiting motion and berm height are known, the shoreline change is computed using the following expression,
AV
Ay (5.2)
k +B
in which, Ay indicates the shoreline change, and AV the volume change per unit beach length. Notice from Equation 5.2, that a constant profile is assumed, thus not accounting for the profile equilibration immediately after sand has been placed.




5.1.2 Predicted Shoreline and Volumetric Profile Changes
Shoreline changes are calculated here using the National Geodetic Vertical Datum (NGVD) as reference. The predicted shoreline and volume changes between 1975 and 1990, are shown in Figure 5.2. Because predicted shoreline and volume changes are proportional to (h, B) (see Equation 5.2), they can be plotted in the same figure. Additionally, these changes show the influence that the sediment transport parameter, K, has for modifying the shoreline displacements. According to Dean (1989), the sediment diameter, D, is related to the sediment transport parameter, K, as shown in Table 5. 1. This table, includes selected grain sizes that will be used later in this thesis.
Table 5.1 Approximate corresponding values of the sediment transport parameter to selected sediment sizes (from Dean, 1989). Sediment Diameter, Sediment Transport Parameter,
D [mm] K
0.40 0.95
0.35 1.04
0.33 1.10
0.29 1.18
0.27 1.24
0.24 1.30
0.22 1.36
0.20 1.41
0.18 1.45
0.17 1.50
0.15 1.55
Even though it is known from refraction theory that waves "Wrap" around a beach nourishment proj ect, thus spreading out sand almost independently from the wave angle, conditions with normal and oblique w~aves were tested.
5.1.2.1 Changes from 1975 to 1990
Figure 5.2 shows the predicted shoreline and volume changes at the Delray Beach Nourishment Project. It can be seen from these figures that, regardless of the sediment transport




parameter, these changes are characterized by three maxima and two minima. The maxima correspond to the center part of the project which is the area that erodes least compared to the limits of the project. The two other maxima, together with the minima, are the consequence of a rapid erosion at the project ends. While the sharp gradients at the ends of the fill planform experiment high sediment transport rates, the areas immediately outside of the project rapidly accrete.
The importance of the sediment transport parameter, K, will be examined later in Section
5.1.3, since the variation of this value can highly alter the results.
Finally, this model predicts that there are more shoreline and volume changes from using an incoming wave angle of 200 in the area encompassed by the project. On the other hand, in the adjacent beaches to the project, the shoreline has less changes with the incoming angle of 200. This could possibly be regarded as if the project was acting as an erodible bafier to the longshore transport, and therefore, sand is stored within the project limits; however, this mechanism is discarded given the dimensions of the project, which makes it a small perturbation compared to the length of the spreading out limits. It is important to mention that, even though spreading out losses change with oblique waves, this change is relatively small.
5.1.2.2 Changes from 1975 to 1998
Shoreline and volumetric profile changes for this period are shown in Figure 5.3. This figure describes the predicted changes for different sediment transport parameters and wave angles. The difference between these changes and those from 1975 to 1990 is mainly given by the 1992 nourishment which took place from approximately R-180 to R-188. Even though it had been mentioned before that this period would be analyzed only within the project limits due to lack of data outside the limits, a broader analysis of the theoretical predictions may help understand the general performance of the project.
On the updrift side of the project, the greater erosion due to large spreading out losses, as well as the accretion immediately updrift from it, still remain similar to those seen for the period




57
of 1974 to 1990. On the downdrift side of the project, although this same pattern is not clearly observed, it is possible to notice that it is starting to develop.

-6.0 -5.0 -4.0 -3.0 -2.0 -1.0 0.0 1.0
Longshore Distance [kin]
- K=0.95 --- K=1.10

-6.0 -5.0 -4.0 -3.0 -2.0 -1.0 0.0 1.0
Longshore Distance [kin]

2.0 3.0 4.0 5.0 6.0

K=1.30

300
275 250
225 200 175 150 125 3o 100
75 50
25

2.0 3.0 4.0 5.0 6.0

- K=0.95 ---- K=1.10 K=1.30
Figure 5.2 Predicted volume and NGVD shoreline changes from 1975 to 1990 with
a) Incoming deep water waves perpendicular to the shoreline, and
b) Incoming deep water waves 200 north from perpendicular to the shoreline.




70 500
65
60 a) 450
55 --400
50 / "- -350
45
0 ,300
40 3
a
35 250
30_
25 .-2003
20 / 150
15 -/Fill area 100
100
5 ..10 .-._- _ __ "500
-6.0 -5.0 -4.0 -3.0 -2.0 -1.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0
Longshore Distance [knm]
- K=0.95 .....- K=1.10 K=1.30
70 500
65 ___60 b) ---_.450
55- 400
50 350
45 2<
300
40
a
35 ____250
25 -200
20 '} 150
Fill area
15 100
10 ____ __0 50
-6.0 -5.0 -4.0 -3.0 -2.0 -1.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0
Longshore Distance [knm]
- K=0.95 .....- K=1.10 K=1.30
Figure 5.3 Predicted volume and NGVD shoreline changes from 1975 to 1998 with
a) Incoming deep water waves perpendicular to the shoreline, and
b) Incoming deep water waves 200 north from perpendicular to the shoreline.




It is interesting that the maxima achieved immediately outside of the project ends due to large accretion, have almost the same value regardless of the values of K and wave angle, for both periods. These results can be regarded as if the beach had a maximum natural capacity to accrete, after which sand is more readily transported away from the project.
In addition, the largest changes are seen at the middle of the 1992 renourishment, that is, around monument R-184, which is the area that erodes slower compared to the rest of the project. In Figure 4.5, where the volumes of sand placed for each nourishment are shown, it can be seen that the fluctuations in volumes placed alongshore as a result of the construction procedures, are not as important as the project's sharp ends, and according to theory, they are smoothed out faster.
5.1.3 Influence and Importance of the Sediment Transport Parameter
The equation of longshore sediment transport used in this model, which is also widely applied, is developed in terms of the energy available in the waves arriving at the beach. This relationship has been developed by Inman and Bagnold (1963) as Q = KPI(5.3)
Q=(P,. P )g(I P)(.)
where Q is the longshore sediment transport, p. and p are the density of the sand and water, respectively, g is the acceleration of gravity, and p the porosity. P1 is the alongshore energy flux per unit of beach width, and K a dimensionless parameter later adopted as the sediment transport parameter. Therefore, the longshore sediment transport is directly proportional to the constant K for the same beach geometry. The correct value of this parameter is particularly important in beach nourishment projects since the larger the parameter, the larger the longshore sediment transport, and thus the smaller the life of the project.
Komar and Inman (1970) introduced a K value of 0.77. Other studies such as those from Kraus et al. (1982), Dean et al. (1982) and Caldwell (1956) have found that K has the value of




0.2, 1.23, and 2.2, respectively. There is still no general consensus as to what the value of K is, or as to whether it is constant or possibly vanes with other parameters, such as diameter, fall velocity, beach profile, or angle of incidence of waves. From several field experiments, Dean (1989) suggested that the sediment transport parameter depends on the grain size. This dependency shows that larger sediment sizes are associated with lower values of K, which is expected from intuition as coarser sand is less transportable. The corresponding values of K to selected grain sizes are shown in Table 5. 1.
It has been mentioned that the Delray Beach Nounshment Project has a somewhat large variation in the sediment size, both alongshore and with time. Therefore, it is of great importance that a representative sand diameter is chosen. To study the effects that modeling with finer or coarser sand would have on the predictions, Figure 5.4 describes how these changes would be if only longshore sand transport is considered.
It is clear, as illustrated in Figure 5.4, that there is more sediment transport expected from considering finer sand, and less from coarser. The ends of the project are the areas where more differences are expected from modeling with different grain sizes than the actual conditions. This analysis shows that a different grain size could alter the predictions not only by using one sediment size along the entire project, but also in a particular location, what could develop into an erosional hot spot. Therefore, these theoretical assumptions must be considered when the measured and predicted conditions are compared.
The effects of the sediment transport parameter on the performance predictions from Figures 5.2 and 5.3, are more important within the project limits. However, when changing the grain size from 0.40 to 0.24 mm, the shoreline retreated only about 7 and 10 meters more in the center of the project for 1990 and for 1998, respectively. Adjacent to the beach nourishment project, the difference between changes with different sediment transport parameters are minimal, however, from conservation of sand, they extend over a longer distance.




,lIncoming
waves

-=F Finer sand
.+ ,Sand used in
model
----> Coarser sand

Original/ shoreline

- Finer sand
-Sand used in
model
--------- Coarser sand

4

v aQ
OIt OIx

Note: the size of the arrow indicates
the amount of sand transport.
Beach Fill

A VpredictedA Vmeasured
Figure 5.4 Variation of sediment transport with different sediment sizes.
5.2 Comparison Between Measured and Predicted Changes The comparisons presented below are divided into shoreline changes and volumetric profile changes, which have been the tools used throughout this thesis to analyze the beach evolution.
This analysis, however, will not compare the curves previously shown for predicted and measured changes directly. In order to be able to recognize the "natural" erosion of the

-------------- -------------- ---------------k
-------------------

// __ __ 1,

,.-, !x




shoreline, the volumes placed or added beach widths, will be substracted from the volumetric and shoreline changes, respectively. The volumes substracted are those presented in Chapter 4 in Figure 4.5 which were added in between the periods analyzed.
The same two periods considered before are examined here. However, for the 19751998 period, only the area encompassed by the project is studied, since no data were available for the vicinity of the project.
5.2.1 Shoreline Changes
5.2.1.1 Changes from 1975 to 1990
Over this period, two renourishments have been carried out to maintain Delray Beach: the first in 1978, and the second in 1984, which encompassed two separate areas as described before.
Although it is difficult to specify one sediment diameter representative for the entire project, it is possible to perform a first approach. From the analysis of the grain size distribution along the project in Section 4.2.2, it has been determined that the average mean grain size is 0.33 mm. Coincidentally, the average between the native and the fill mean grain sizes (0.46 and 0.20 mm, respectively), is also 0.33 mm, which corresponds to a sediment transport parameter of K=1.10. Additionally, grain sizes tend to range between 0.30 and 0.35 mm. Even though other sediment sizes will be accounted for later in this thesis, the comparisons between measured and shoreline changes are based on K=I. 10.
The predicted and measured shoreline changes, including the subtraction of the added beach width, are portrayed in Figure 5.5. This figure shows that the predicted quantities are in good agreement with the measured values within the project limits. On the updrift side, there appears to be an overprediction of the shoreline changes, while, on the downdrift side, despite the large fluctuations of the measured values, it seems to be underpredicted. Overprediction as used here refers to the predicted changes being smaller than those found in the field, and by underprediction, the opposite.




Shoreline changes "without" added beach width between 1/14/75 and 1/15/90
40
Fill area
29
T ......
U),
100
_ _~i lag
Longshore distance [kin]
Predicted Measured ---- Difference [p-m] Figure 5.5 Comparison between the predicted and the measured NGVD shoreline changes from 1975 to 1990.
Analyzing closer the area encompassed by the project, the differences between predicted and measured values are less than 10 meters except for three areas around monuments R-178, R184, and R-186. The first two are overpredicted while the other one is underpredicted. Although the presence of these three peaks may suggest the presence of cold or hot spots, it is necessary to establish a criterion to determine such behavior. This will be examined in Section 5.3.
A standard deviation analysis is also presented later in this thesis (see Section 5.2.3), which will be an indicator of the overall performance of the nourishment, and will be an aid for locating erosional hot spots (and eventually cold spots).
5.2.1.2 Changes from 1975 to 1998
This period considers one more nourishment, as in October of 1992, sand was placed over the southern part of the project, roughly, from R-180 to R-188. Shoreline changes between




January 1975 and January 1998 are illustrated in Figure 5.6. The results shown in this figure have the same degree of agreement between predicted and measured conditions as those from Figure 5.5, and they can also be regarded as a good approximation. In this case, shoreline
positions were predicted within a 10-meter "error" for almost the entire project, except for the downdrift side where, at R-186, the shoreline was underpredicted almost 40 meters, and at T-187 about 20 meters.
It can also be noted that from the updrift end to R-180, the shoreline changes were overpredicted, while for the rest of the project, they were underpredicted. The area that is
presented here as underpredicted coincides with the area where the 1992 renourishment took place. Therefore, it can be said that it eroded faster than predicted, however, only the downdrift end proved to be poorly approximated, since up to 37 meters were underpredicted.
Shoreline changes "without" added beach width between 1/14/75 and 1/15/98
-Fill area
20
40
z
1 00
Longshore distance [knm]
- Predicted Measured ...... Difference [p-m]
Figure 5.6 Comparison between the predicted and the measured NGVD shoreline changes from 1975 to 1998.




Comparing Figures 5.5 and 5.6 some similarities can be found within the project limits, where the effects of the last renourishment are readily seen. Quantitatively, both comparisons predict shoreline changes within 10 meters approximately, except for the aforementioned areas. Qualitatively, both predictions were best at the middle of the project, and found largest differences at R-178 and at R-186.
5.2.2 Volumetric Profile Changes
5.2.2.1 Changes from 1975 to 1990
Volumetric profile changes over this period are portrayed in Figure 5.7. This prediction proved to be less accurate than that for shoreline changes. The entire project length was predicted within approximately 150 m3/m of "error". The largest differences were around R-181 and R-186, where 140 and 160 m3/m were underpredicted, and around R-178, where 120 m3/m were overpredicted.
On the updrift vicinity of the fill area predictions were very close to the actual measured values, except around R-170, where a cold spot might be present. On the downdrift side there is a large fluctuation in the volume changes so that it is not possible to predict accurately such changes with this model. However, the prediction in this area agreed on average with the measured conditions. Therefore, differences presented in Figure 5.7 can also be regarded as relatively small and as a good performance prediction.
5.2.2.2 Changes from 1975 to 1998
The analysis of the volumetric changes for this interval, which includes the presence of one more nourishment than the previous case, is shown in Figure 5.8. It can be noted from this figure, that volumetric changes for this period were predicted more accurately than for the 19751990 span. The entire project area is predicted within nearly 100 m3/m, with the exception of the downdrift side, where a large underprediction of 275 m3/m was found. At R-178 where 120 m3/m were found for the 1975-1990 term, an overprediction is again found, this time of 140 m3/m.




Volume changes "without" volume placed
between 1/14/75 and 1/15/90

A A AA AA A AA A AA Longshore distance [km]
-Predicted Measured ---- Difference Figure 5.7 Comparison between the predicted and the measured volumetric profile changes from 1975 to 1990.

Volume changes "without" volume placed
between 1/14/75 and 1/15/98

Longshore distance [kin]
--Predicted Measured ---- Difference
Figure 5.8 Comparison between the predicted and the measured volumetric profile changes from 1975 to 1998.

Fill area

2\

.J

I-

I-ht




It is also remarkable that comparing Figures 5.6 and 5.8, which correspond to shoreline and volume changes for the 1975-1998 span, respectively, the predictions are very similar. In other words, both plots show under and overpredictions at approximately over the same locations. Particularly, at R-186 where the largest differences were found, these differences follow the Equation 5.2 almost exactly. The implication is that the nourished profiles were displaced approximately with the same form as the pre-nourished profile.
5.2.3 Standard Deviation Analysis
In order to describe whether predictions agree or disagree with field measurements, the standard deviation is computed here for both, shoreline and volumetric changes. Given also the importance and uncertainty of the sediment transport parameter, several grain sizes were modeled, as well as different wave conditions. Only the area encompassed by the project is accounted for, since the purpose of this thesis is to identify erosional hot spots.
The standard deviation is given by
" = -LAy (5.4)
where N is the number of points and Ay the shoreline changes. Obviously, the same expression is applicable to any other data types such as volume changes. Equation 5.4 provides the root mean square (rms) deviation of the data from the origin, while Equation 5.5, shown below, provides the rms deviation from a reference value, in this case, the measured changes,
Spm(AY AY (5.5)
where p denotes the predicted data, and m the measured data.
After several runs of the DNRBSM model, varying the sediment transport parameter and incoming deep water wave direction, a set of plots was obtained to summarize these results. Of particular interest is the relationship qm/qp, which means that, when this expression approaches




zero, a good overall planform performance prediction has been made. In general, it can be said that if upm/,ap is smaller than 0.4 to 0.45 a good prediction has been achieved. Likewise, Upm/ r, upm, and p were computed and summarized in graphic forms. Figure 5.9 shows computed values of upm/ap, Figure 5.10 the values of qpm/ and Figures 5.11 and 5.12 show upm and p values, respectively. These data represent computations only for the period of 1975-1998, since these are the actual conditions of the project, and because predictions proved to be less accurate for this interval. However, for K=1.10, the standard deviation is computed for the 1975-1990 span.
Figures 5.9 and 5.10 describe the overall performance of the project based on the standard deviations computed from the differences between measured and predicted values for 1975-1998. The results illustrated in Figure 5.9a show that, for an increasing K, there is apparently better agreement between predicted and measured values. However, this does not mean that the actual value of K should be larger than the one previously considered of 1.10. The curves appear to converge to a value around 0.22 and 0.26 for shoreline and volume changes, respectively, when K=2.0, whose corresponding diameter is not determined by Dean (1989). According to the sediment size analysis, and considering the relationship between the sediment size and the sediment transport parameter given in Table 5.1, if it is assumed that K=I. 10 as the representative conditions for this project, it is then found that Upm/ap = 0.3 for the shoreline changes, and 0.33 for the volume changes. Therefore the project has a good overall performance.
For an incoming deep water wave angle of 20', Figure 5.9b shows that Upm/ap values also decrease with larger K, not showing, however, convergence to any specific value. For a value of K=I. 10, which was assigned for Delray Beach, it has been found that upm/ap = 0.44 for shoreline changes, and upm/ap = 0.46 for volumetric changes. These values are considered fairly good for performance predictions.




69
Values of Gpm / Op (deep water incident wave angle = 0)

036 M
- - -'--hrln Chage --Vlm Changes
Values of Gpm / Op (deep water incident wave angle = 20 deg)

065
0.4 0.3
0.2 0.1

Sediment Transport Parameter, K

---Shoreline Changes -U*-Volume Changes
Figure 5.9 Values of qm,/p computed for the fill area for 1975 -1998, with
a) Incoming deep water waves perpendicular to the shoreline, and
b) Incoming deep water waves 200 north from perpendicular to the shoreline.




70
Values of Gpm / Gm (deep water incident wave angle = 0)

E
- 0.25
E
0.23
0.21 0.19 0.17
a)
0.15
Sediment Transport Parameter, K
--s-Shoreline Changes --Volume Changes
Values of Gpm / Gm (deep water incident wave angle = 20 deg)

C m-shoreli le = 52. m SdmnTr movPlumE = 71. m3m
b)
Sediment Transport Parameter, K

--.--Shoreline Changes -Volume Changes
Figure 5.10 Values of (qm,,/m,, computed for the fill area for 1975-1998, with a) Incoming deep water waves perpendicular to the shoreline, and
b) Incoming deep water waves 200 north from perpendicular to the shoreline.

0.4
0.35 0.3
E
?0.25
E
0.2
0.15 0.1




71
Values of Gpm (deep water incident wave angle = 0)

Sediment Transport Parameter, K
--Shoreline Changes --U-Volume Changes

Values of Gpm (deep water incident wave angle = 20 deg)

Sediment Transport Parameter, K
--*--Shoreline Changes ---Volume Changes
Figure 5.11 Values of upm computed for the fill area for 1975-1998, with
a) Incoming deep water waves perpendicular to the shoreline, and
b) Incoming deep water waves 200 north from perpendicular to the shoreline.




72
Values of Gp (deep water incident wave angle = 0)

55 50
45
0)6
35
60 55
0
46
40
35 30

430
410
___ 390
370
350
_330
__ 310
290
Ulm-sho elin = 52.E m 270 270
a) -vol me 3 '1.9 m
260
. . . . . .
Sediment Transport Parameter, K
----Shoreline Changes -U-Volume Changes
Values of Gpm (deep water incident wave angle = 20 deg)
370
350
/ _330
310
290
, 270
;, O-shorelin. = 52. m 270
b) ----voi me = 31. m3
260

Sediment Transport Parameter, K
---Shoreline Changes ---Volume Changes
Figure 5.12 Values of p computed for the fill area for 1975-1998, with
a) Incoming deep water waves perpendicular to the shoreline, and
b) Incoming deep water waves 200 north from perpendicular to the shoreline.

Q C)
C
0
2
CD CD




From Figure 5.9, it is clear that the overall performance of the project has been apparently better predicted for perpendicular incoming waves. The reason is that when considering an incoming wave angle different than zero, a longshore sediment transport is induced in the vicinity of the project, thus interacting with the beach fill. These conditions can also be modeled, however, it is necessary to establish a different sediment transport parameter for the vicinity of the project. At present, DNRBSM is only configured to manage one sediment size for the entire area, therefore, the results in which spreading out effects are investigated alone (perpendicular waves), are assumed to be more accurate.
Unlike upm/ap, the values of pm/um are referred to the measured standard deviation, which is a constant. Therefore, this parameter can also be considered as an overall indicator of the project performance, as shown in Figure 5.10. In Figure 5. 10a, a minimum appears to be achieved around K=1.50 with pm/Up=0.21, and qpGm/Um=0.25 for shoreline and volumetric changes, respectively. Thus, the values of qpm/um are closer to zero than those illustrated in Figure 5.9 for upm/ap, as they range from about 0.30 to 0.21. Furthermore, the same differences of modeling with two different wave angles found before, are found here. In general, there is a better prediction using those values calculated from a perpendicular wave angle.
The other two figures, 5.11 and 5.12, show the values used to plot Figures 5.9 and 5.10, and therefore, they have the same characteristics and differences among them. The standard deviation of the measured values is 52.6 m and 371.9 m3/m, for shoreline and volume changes, respectively.
In conclusion, it has been found that the project has a good overall performance. This allows to identify any highly erosive areas, as it is expected that most of the project do not have a large deviation from the predicted values. In addition, it has been shown that the value of upm/ap becomes smaller with larger sediment transport parameters, and with waves perpendicular to the original shoreline.




Although not illustrated in a graphic form, the standard deviations for the 1975-1990 period, were computed for only three different sediment transport parameters, since they predictions proved to be very similar to the measured conditions (see Table 5.2).
Table 5.2 Standard deviations computed for the 1975-1990 span.
Data Shoreline Changes Volume Changes
Kp a u P P/ U/U p Urn U/UP U/U
K a Up Upm Upm/U p~a U!Urn pmlp pml
[ mn] [ mn] [ mn3 / m] [ mn / mn]
0.95 00 58.7 7.3 0.124 0.120 420.5 88.6 0.211 0.183
0.95 200 55.6 8.2 0.148 0.135 398.5 106.5 0.267 0.220
1.10 00 60.3 7.5 0.125 0.123 432.0 81.4 0.188 0.170
1.10 200 57.0 7.6 0.133 0.124 408.5 97.7 0.239 0.202
1.30 00 62.2 8.3 0.134 0.136 445.8 75.1 0.169 0.155
1.30 200 58.8 7.3 0.124 0.119 420.8 88.3 0.210 0.182
Note: a is the deep water wave angle of approach; Um-shoreline = 61.0 m; Um-volume = 484.7 m3/m.
For the 1975-1990 period we have that, for shoreline changes, pm/Up = 0.125 when K
1.10 and waves perpendicular to the beach, and for volume changes, pm/Up = 0.188. Therefore, these changes have been better predicted than those for the 1975-1998 period, and the overall performance of the project can be considered as very good, from 1975 through 1990.
5.3 Hot Spot Identification and Mitigation Measures As it has been defined herein, an erosional hot spot has the characteristic that can not be predicted directly from applying diffusion theory. Based on this, three criteria will be considered in order to identify erosional hot spots, and eventually cold spots: historical shoreline changes, the sediment size distribution, and the standard deviation reference value. Although these tools are used to identify and locate erosional hot spots, they do not exclude other techniques that may be more suitable for other situations, such as wave refraction and diffraction analyses. The method of study will depend upon the potential causes of the erosional hot spots.




5.3.1 Historical Shoreline Positions
This criterion is aimed at identifying the areas that historically present more erosion than others. To achieve this, the erosive trend of the shoreline encompassed between DNR monuments R-175 and R-189 has been previously described in Section 3.2.1.
Throughout the project area, an overall advance of the shoreline has been identified since 1884 until the early 1960's. It is during this decade when a large erosive trend has affected Delray Beach, mainly as a result of inlet management on the east coast of Florida (Dean, 1988). From 1962 to 1970, the beach had an average shoreline change of 1.86 m/year. After this year the shoreline was stabilized by hard coastal structures, which were still not enough to withstand the encroaching sea. From R-175 to R-180 the erosive rates were larger than the average, including the worst case at R-176 with 3.7 m/year. From R-181 through the downdrift end of the project, the shoreline retreat was below the average with the lowest shoreline retreat at R-186 (0.65 m/year).
From the historical shoreline positions, it can be observed that, should any erosional hot spot existed at Delray Beach prior to the nourishments, that would be within the first third of the project, where the largest erosion rates were found. However, it is only during an eight-year interval, when the erosive trend is found, and it has been identified as a natural adjustment due to inlet management Prior to this decade, it is an accreting process which dominates the shoreline changes. It is concluded, therefore, that there is no evident highly erosive trend, within the project area that would lead to an erosional hot spot.
5.3.2 Sediment Size Distribution Along the Project
As described in the literature review, having different sediment sizes along the project, causes the beach to equilibrate differently. For the second renourishment, there is a larger mean grain size on the updrift side, however, the rest of the project suggest a fairly even distribution of the grain size, thus, not suggesting any large fluctuations that may cause large differences in the




shoreline changes. At R-177, on the updrift side of the project, a larger sediment size was found compared to the rest of the project. This may be an indicator of why the beach presented a higher erosion rate between R-180 and R-188, causing the authorities to decide to restore only this segment in the third renourishment. The sediment size variation is depicted in Figures 4.7 and 4.8.
The last renourishment took place only from R-180 to R-188. Since 1992, the mean grain size has proven to be finer at the downdrift side of the project, which is a potential factor that may cause an erosional hot spot, due to the narrower beach width associated with finer grain sizes. In order to determine the existence of an erosional hot spot, it is necessary to apply the performance predictions included in this chapter.
5.3.3 Standard Deviation Reference Value
To establish a criterion to determine the presence of an erosional hot spot, the standard deviation is used. The method considers that those areas with higher deviations from the standard are erosional hot spots or cold spots.
However, this method will always include areas with larger deviations than the standard, even if the predictions are within an allowable range. For example, if the difference between predicted and measured changes followed the normal probability distribution, only 68.3%o of the points are considered to be within one standard deviation, leading to a 31.700 of the remaining area, as erosional hot spots and cold spots, even if the project had performed well.
To account for this, an erosional hot spot will be considered where an area with larger deviations than the standard exists over a substantial longshore length, and it will not be considered a hot spot, where only a spike or peak exceeds the standard deviation.
Since an erosional hot spot has been defined as an area that does not perform as predicted by theory, the standard deviation to be considered must be that of the differences between measured and predicted quantities.




The differences between predicted and measured changes with K = 1. 10 and deep water waves perpendicular to the shoreline, are depicted in Figures 5.13 and 5.14. On these figures, erosional hot spots and cold spots have also been identified. In order to identify these spots it is necessary to correlate shoreline and volumetric profile changes, as both are indicators of the performance of the beach.
Figure 5.13 depicts the differences previously shown between predicted and measured quantities for the 1975-1990 period. The shoreline change differences shown in Figure 5.13a portray two cold spots and one erosional hot spot. The hot spot found around R-186, has the same location as one of the hot spots identified from the volume differences (Figure 5.13b). The other hot spot identified by the volumetric changes, encompasses monuments R-180 and R-181, but there is no evidence whatsoever, that an erosional hot spot can be identified at the same location using shoreline changes (Figure 5.13a). It is at this location where Gravens (1997) has identified the presence of an erosional hot spot, probably due to a sewage outfall and a 300-mwide no-dredging zone. However, the time span used by Gravens is from 1987 to 1992. Both shoreline and volume changes identify an erosional cold spot at R-178.
The 1975-1998 span is illustrated in Figure 5.14. Both plots included in this figure, clearly locate an erosional hot spot at the downdrift end of the project. It is remarkable that, for the shoreline changes, the entire project has a deviation smaller than one standard deviation, except for what has been identified as an erosional hot spot. Additionally, both shoreline and volume change differences, have more or less the same areas with accretion or erosion, that is, both plots identify overprediction within the first third, underprediction within the middle third, and another underprediction over the last third, the last one, leading to an erosional hot spot.
Finally, the comparison between the two intervals of time, shows that differences in the 1975-1998 are much larger than the other. The hot spots found for the 1975-1990 span, reach up to 2, while for 1975-1998, the hot spot at R-186 almost extents to 3a.




78
Erosional hot spot identification (1975-1990)

Longshore Distance [kin]

Erosional hot spot identification (1975-1990)

Longshore Distance [kin]

Figure 5.13 Location of erosional hot spots and cold spots for 1975 to 1990, using a) Shoreline changes differences, and b) Volume changes differences. The area shown encompasses the project limits.




79
Erosional hot spot identification (1975-1998)

Longshore Distance [knm]

Erosional hot spot identification (1975-1998) b)
_ _200
_ t150
- -------- ---- -- ----- 103 16
-2.5 -2.0 -1.5 -10 -C.5 00 0 0 1 5 2 0 2
- 103. 16 .. .. -7 ..... ....-. .,-103. 16
Longshore Distance [knm]
Figure 5.14 Location of erosional hot spots and cold spots for 1975 to 1998, using a) Shoreline changes differences, and b) Volume changes differences. The area shown encompasses the project limits.




5.3.4 Summary
Figures 5.13 and 5.14 are consistent in identifying the presence of erosional hot spots at the downdrift end. The potential cause of this highly erosive zone is attributed to the finer sand located in this area. It is also consistent that the updrift third of the project, where the coarser sediment was found, presents slower erosive rates than the rest of the fill. It has been mentioned also, that the nourishment project possibly may act as a natural barrier to the longshore sediment transport, situation that was not accounted for in the final results. However, this mechanism was discarded given the relative dimensions of the project, which tends to smooth out with time, and represents only a short distance seaward compared to a few kilometers over which the project is extended. The finer sand located in the vicinity of monument T-187, is believed to cause the erosional hot spot located in this area. Moreover, the location of this hot spot is the same as that identified from Beachler (1993).
No other large systems of hot spots or cold spots were found. The hot spot described by Gravens (1997), which is located around R-180 may have disappeared after the 1992 renourishment. The reason is that the location of the hot spot was immediately downdrift of the fill, where large accretion rates are expected.
Hot spots created by sediment size differentials along the coast, could be the result of dredge selectivity, as explained in Chapter 2. Stricter controls in dredge operations are suggested in order to achieve a more uniform grain size along the project. Likewise, another remedial measure is simply to place enough fill to achieve the equilibrium design, which gives the contractor more freedom as to where to mine. Therefore, different template sections can be considered along the project as a function of the grain size. Both remedial measures are likely to increase the cost of the dredging operation, and therefore, the cost of the beach nourishment; however, every project should be analyzed carefully to consider the options. Borrow sites where large variation in the grain size is found, are more likely to produce erosional hot spots due to dredge selectivity.




CHAPTER 6
SUMMARY, CONCLUSIONS, AND RECOMMENDATIONS
6.1 Summary
A numerical model for beach planform evolution was used to compute shoreline changes and volumetric profile changes. This methodology is similar to a one-dimensional model based on the linearization of the equations of sediment transport and continuity, first introduced by Pelnard-Considbre (1956). The result is transformed into the heat-conduction equation (see Appendix A). In addition, the model accounts for renourishments throughout the life of the project.
The hydrodynamic conditions were represented by effective parameters. Initial volumes were considered distributed along the project in accordance with their actual placement, and the sediment size representative of the project was taken as the mean grain size. This mean sediment size proved to be converging to a value near the average between the borrow zone and the native beach, and was taken as 0.33 mm, although, given the variability of this parameter, other diameters were evaluated. This experimentation allowed conclusions to be developed towards this essential characteristic of the beach. According to Dean et al. (1998) this area presents negligible background erosion.
Two time intervals were considered, the first being from 1975 to 1990, and the second from 1975 to 1998. This permitted separation of the effects of the last renourishment to analyze the behavior of the fill prior to this nourishment. Shoreline and volumetric profile changes were computed and compared to the actual measured conditions. Finally, a criterion based on the




standard deviation of the difference between measured and predicted conditions was established to identify areas with accentuated erosion or accretion.
6.2 Conclusions
Erosional hot spots and cold spots were identified within the Delray Beach limits, with different degrees of erosion or accretion according to the time spans examined. Even though the 1975-1990 time span proved to be better simulated with the numerical modeling, it shows at least three areas in which the actual trends present large deviation from the predicted values. The standard deviation computed from the differences between measured and predicted values, was smaller in the case of 1975-1990, which means better prediction, thus belier performance of the project. For this interval, the shoreline changes identify two cold spots and one hot spot, while volume changes identify two hot spots and one cold spot. The location of the updrift cold spot and the downdrift hot spot coincide in the locations obtained from both analyses. Moreover, the location of this hot spot, agrees with the results provided by Beachler (1993), around DNR monument No. R-186. The hot spot identified around R-181, does not show any signs of reappearance after the 1992 renourishment, perhaps, because high accretion rates are expected in this region (immediately outside of the fill).
From the 1975-1998 analysis, both shoreline and volume changes were predicted with good accuracy from R-175 through R-185. Almost all the points exceeding the value of the standard deviation occupy the area from R-1 86 to the downdrift end of the project. This has been identified as an erosional hot spot, given the fact that it has been repeatedly found as such in several analyses and other studies, and has not been mitigated. The reason, although not obvious, appears to be the smaller sediment size found within these limits. Since DNRBSM does not account for sediment size differentials along the beach, any area with a large variation of the sediment size would not be predicted accurately.




Therefore, two time spans have been examined. Nonetheless, further considerations towards the time scales can be determined as needed, since they could become a factor in identifying highly erosive areas. For instance, in his study, Gravens (1997) used a different period than those analyzed herein, and found an erosional hot spot in the middle portion of the project. Other coastal features such as beach cusps are not considered as erosional hot spots.
The DNRBSM model, however, has been applied successfully in the prediction of the shoreline positions. It is also important to mention that shoreline retreat due to equilibration of the cross-shore section is not accounted for. The reliability of this program can be increased if different sediment sizes can be represented. Nonetheless, the model has proved successful in predicting both shoreline and volumetric changes with acceptable accuracy. A numerical model that could account for cross-shore changes at the same time, can also increase its accuracy. It is still shown that shoreline and volume changes are difficult to predict in fine detail, thus additional criteria have to be applied to interpret the results.
The influence of the sediment transport parameter is readily observed from the analysis performed in Chapter 5. Varying the sediment transport parameter for different sizes, predictions appeared to be better. This could mean that even though the diameter of the sediment is 0.33 mm, the actual sediment transport parameter could differ from 1. 10, the value that corresponds to the relationship proposed by Dean (1989). However, it is suggested that consideration of the sediment size (and the sediment transport parameter) should be computed differently than the simple average of the mean grain sizes found. It would be necessary then to apply some statistical method to account for more representative conditions. The overall performance of the project has been described as u,m/qp = 0.3 for shoreline changes and upm/qp = 0.33 for volume changes. which are reasonable values. Likewise, the values of qpjm/um for shoreline and volumetric profile changes were 0.25 and 0.28, respectively, with K =1. 10 and a perpendicular incoming waves.




6.3 Recommendations
Improvements to the numerical model can be performed to achieve a finer detail of prediction. Sand variability is among the most important parameters that are needed in order to better predict shoreline and volume changes, according to the results obtained from this thesis. Cross-shore motion of sediments is also important in order to account for equilibration after project completion. hi order to predict erosional hot spots, it would be necessary to develop a large and rich set of data, and perhaps a somewhat complex model, able to put together all the variables that will define the shoreline performance. All these requirements imply a much more expensive approach to the project, but a much precise one. DNR13SM provides with a reasonable level of accuracy.
Further research is also necessary in order to understand the potential causes creating hot (and cold) spots, including the consequences of different placement techniques, and sediment size variations along the beach. Application of refraction and diffraction models are also suggested for the case of Delray Beach.




APPENDIX A
MODEL FOR BEACH PLANFORM EVOLUTION
Introduction
A beach nourishment project represents an alongshore and cross-shore perturbation, which is equilibrated by waves and currents. While the shoreline retreats due to equilibration of the profile and the possible formation of a bar system, sand is also eroded away from the fill area "flattening out" this anomaly due to longshore sediment transport. Realistic prediction of the shoreline evolution should include longshore and cross-shore transport, however, cross-shore motion is still less predictable and, oftentimes, considered the less dominant from the two transport directions. (Such assumptions are applied in this thesis for the analysis of the performance of the Delray Beach restoration project.) Therefore, it is valid to apply numerical methods to predict only the spreading out losses of beach nourishment projects, thereby providing a valuable estimate of the overall performance.
Background
The numerical procedure applied throughout this thesis was developed by Dean and Grant (1989) and is called DNRBS. The vesion applied in this thesis accounts for multiple nourishments, and was named DNRBSM. It consists of a one-dimensional model in which the shoreline position is computed. This type of model is the simplest approach to describe a beach nourishment planform evolution, and requires that the profile form is unchanged as the spreading out losses occur. A three-dimensional model would be required to account for profile changes to cross-shore transport.




Governing Equations
The bases for predicting beach nourishment project changes are the equations of longshore sediment transport and continuity. The three-dimension continuity equation is ah aq., aq,
+- (A-i)
at ax ay
in which h is the water depth relative to a fixed datum, t is time, and q, and qy are the sediment transport components in the longshore and cross-shore directions, respectively. Since the proposed method calculates only transport in the longshore direction, qy = 0. The integration of Equation A-I with respect to y, yields
a Y2hdy J q dy (A-2)
in which y, and Y2 are a landward location and a seaward location, respectively, in which the cross-shore transport is zero. These locations are considered to be the top of dune of height B, and the depth of limiting motion, h.. In addition, volumes are computed per unit width of beach. The integral on the left hand side of the equation represents the total volume in the system, V, or volume of the water column. Therefore, -a Vat can be regarded as the time rate of change of volume of sand instead of water. The integral on the right hand side is the total longshore sediment transport Q. The continuity or conservation of sand equation then becomes
-v + -Q = 0 (A-3)
at ax
It has been noted that the beach profile is considered constant with respect with time, assuming that a seaward or landward displacement is associated with accretion or erosion, respectively. This shoreline displacement, Ay, associated with a volume change, AVis
1
Ay = I-AV (A-4)
h,+B




Substituting Equation A-4 into A-3 yields the one-dimensional equation for conservation of sand
L_ + 1 0 (A-5)
at h +B ax
The longshore sediment transport equation is an empirically based energy flux model. The final form of this equation is
KH/2 g/Kcsin(3-ajb)cos(3-a) (A-6)
8(s -1- p) (Ain which K is the longshore sediment transport parameter, Hb is the wave height, g is time, K is the ratio of breaking wave height to water depth (usually assumed as 0.78), /3 is the azimuth of the outward normal to the shoreline, ab is the azimuth of the direction from which the breaking waves originate, s is the specific gravity of the sediment (approximately 2.65), and p is the porosity of the sand (usually taken as 0.35). (f3-ab) is the angle between the wave crest and the shoreline at breaking conditions.
The longshore sediment transport equation (A-6) can be linearized and then combined with the one-dimensional conservation of sand equation (A-5), to yield the classical heatconduction equation proposed by Pelnard-Considbre (1956) ay = G azy (A-7)
at aX2
in which G is the longshore diffusivity coefficient, and is defined as JKf5 /2 Jg /K
8 (s-lX1-pXh. + B) (a-8)
Equation A-7 describes the planform evolution of a beach nourishment and different boundary conditions, like those for uninterrupted shorelines or littoral barriers, can be simulated.
In order to predict with good accuracy the shoreline position it is necessary to include long-term background erosion, ayBlJt. To incorporate this into the numerical solution,




background erosion rates are translated into background transport rates, as shown by the following equation
QB(x) QB(xo)-(h. + B)Jf yBdx (A-9)
0 dt
in which xo is a reference shoreline location at which a reference transport QB(xo) is specified. QB(x), is the total sediment transport along the beach including the associated background erosion.
If the bathymetric contours are regarded as straight and parallel, it is possibly to apply simplified wave refraction and shoaling to express the transport in terms of deep water conditions. Applying Snell's Law and foregoing the algebra, the sediment transport is computed by
KH 2.4 0.6T0.2 cos12 0 sin 0A
Q = 8(0 X )1./ o. (A-lO0)
in which the subscript 'o' denotes deep water conditions. If Equation A-10 is also linearized in the form of Equation A-7, the appropriate value of the longshore diffusivity coefficient, G, defined in deep water wave conditions is KH24T02g06"Cs41.2 (o -ao)COS2(A -a.)
G 8(S-- IXI p 2K/ t ) o-(o (A- 11)
in which the subscript .' denotes conditions at the depth of limiting motion.
The finite-difference solution applied here, requires the following relationship to be achieved, which, if exceeded, will cause numerical instability Atmax = G -- (A-12)
2 G
in which Ax is the alongshore gid spacing. This expression shows that the smaller the grid spacing and the larger the wave height, the smaller the allowable time step.