UFL/COEL2002/008
THE EFFECT OF OFFSHORE DREDGE PITS ON ADJACENT
SHORELINES CONSIDERING WAVE REFRACTION AND
DIFFRACTION
by
Yuhong Tang
Thesis
2002
THE EFFECT OF OFFSHORE DREDGE PITS ON ADJACENT SHORELINES
CONSIDERING WAVE REFRACTION AND DIFFRACTION
By
YUHONG TANG
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
2002
ACKNOWLEDGMENTS
I would like to express my sincere gratitude to those individuals who have helped me
accomplish this work. I would like to thank my advisor, Dr. Robert G. Dean, for
providing me with the opportunity to pursue this work. He has afforded me tremendous
support, guidance and freedom of work environment in which to pursue my goals. His
suggestions and ideas were of substantial assistance. I also express my appreciation to my
committee members, Dr. Daniel M. Hanes and Dr. Robert J. Thieke, for their excellent
lectures and help in my research and studies. I thank Helen Twedell and Kimberly Hunt
in the Archives, and Becky Hudson for their help and friendship. I appreciate all my
colleagues for helping making my endeavors at the University of Florida both productive
and enjoyable.
The study on which this thesis is based was funded by the Bureau of Beaches and
Wetland Resources of the Florida Department of Environmental Protection, its support
for this work is appreciated greatly.
I would like to thank my husband for his love, support and encouragement, and my
daughter for her lovely smiles. Finally I would like to thank my parents for their endless
love and my friends for their encouragement.
TABLE OF CONTENTS
page
ACKNOWLEDGMENTS ....................................................... ................................... ii
A B STR A C T ................................................................................................................... v
CHAPTERS
1 IN TR O D U CTION ................................................................................................... 1
Problem Statem ent.................................................................................................. 1
O objectives and Scope.............................................................................................. 2
2 EQUILIBRIUM BEACH PROFILE .............................................................................4
3 R CPW A V E M OD EL............................................................................................... 8
Introduction .............................................................................................. . ............ 8
B background of R CPW A V E ....................................................................................... 8
Assumptions and Limiations................................................................................. 9
G governing Equations .............................................................................................. 9
4 DNRBS MODEL FOR CALCULATING LONGSHORE
SEDIMENT TRANSPORT................................................................................. 12
Introduction ........................................................................................................... 12
Governing Equation for DNRBS........................................................................ 12
5 SHORELINE CHANGE CONSIDERING GEOMETRY ........................................17
6 EXAMPLE APPLICATION OF MODEL FOR AN ACTUAL CASE ....................23
Geographic Description of Grand Isle .............................................................. 23
Borrow Pit Characteristics ...................................................... ............................ 24
Results Based on RCPWAVE ............................................................................ 27
Adding Second Term in "CERC" Equation and Comparing the Results ................... 39
7 SOME GENERAL RESULTS FOR IDEALIZED CASES ......................................41
Introduction ........................................................................................................... 4 1
Shoreline Effects Due to Various Pit Parameters .................................... .......... 41
Shoreline Changes as a Function of Distance of Pit from the Shoreline ............ 43
Effects of Pit Depth on Shoreline Changes..................................... ........... ... 47
Effects of Pit Size and Shape on Shoreline Changes.......................................... 49
8 SUMMARY AND CONCLUSION .........................................................................63
BIOGRAPHICAL SKETCH ........................................ ................. ............................67
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
THE EFFECT OF OFFSHORE DREDGE PITS ON ADJACENT SHORELINES
CONSIDERING WAVE REFRACTION AND DIFFRACTION
By
Yuhong Tang
August 2002
Chairman: Dr. Robert G. Dean
Major Department: Civil and Coastal Engineering
Wave climates can be modified by offshore dredge pits and thus alter the equilibrium
shoreline planform, thereby resulting in erosional hot spots (EHSs). The four wave
transformation processes associated with offshore bathymetric changes due to borrow pits
can include wave refraction, diffraction, reflection and dissipation.
Computational models have been used to simulate wave transformation and shoreline
evolution based on the equilibrium beach profile, the RCPWAVE model, and the
DNRBS model. These models account for wave refraction and diffraction only, and can
predict the shoreline change for arbitrary bathymetry of mild slope, including the case of
offshore contours which are not straight and parallel. An actual case at Grand Isle is used
to examine this model and the results from this simulation are opposite to the
observations in nature. Thus, it is not adequate to predict realistic shoreline changes
accounting only for wave refraction and diffraction.
Several cases with a bathymetric anomaly (pit), described by a bivariate normal
distribution with the pit dimensions in the cross shore and longshore directions
represented by the associated standard deviations, ox and oa, respectively, are examined
using these models. It is found that the shoreline changes decrease with dredge pit
distance from the shoreline and increase with pit depth. The smaller the pit dimension in
the cross shore direction (,) and the greater the standard deviation in the longshore
direction (ay), the smaller the shoreline changes. The shoreline changes are largest when
the pit shape is approximated by a circular pit (cr = o, ). Thus the best pit shape is one
with small oc and large ay, i.e. a pit that is elongated with its major axis parallel to the
shoreline.
CHAPTER 1
INTRODUCTION
1.1 Problem Statement
Beaches serve as recreational resources and provide protection to upland structures
from damage by hurricanes and other storms. However beaches are dynamic and many
suffer erosional pressure leading to shoreline retreat. A number of methods have been
employed to prevent the effects of erosion including "hard" structures, such as seawalls,
revetments, groins and offshore breakwaters, thus reducing potential damage due to
hurricanes and wave attack. However, this armoring of the shoreline can interrupt the
longshore transport of sand and adversely impact adjacent beaches. "Softer" approaches,
such as beach nourishment restore the beaches, provide shore protection and possibly
environmental benefits and thus have grown in acceptance.
Beach nourishment typically comprises the placement of sand on eroding beaches to
compensate for erosion, with the nourishment sand source from an offshore or onshore
borrow area. Most beach nourishment projects designed to reduce coastal erosion are
successful. However some beaches can experience local erosional hot spots (EHSs) that
erode more rapidly than anticipated in design or more rapidly than neighboring portions
of the nourished projects. Some of these EHSs can be related to the offshore borrow pits
associated with the project.
A possible linkage between borrow pits and EHSs is that the pits can modify the
wave climate and thus affect shoreline stability. There are four wave transformation
processes including wave refraction, wave diffraction, wave reflection and wave
dissipation which can be caused by altering the offshore bottom topography due to
offshore borrow pits. Therefore one approach to mitigating or avoiding EHSs is to
understand the reasons and physical mechanisms associated with their formation. Toward
this goal, this thesis uses available numerical models to simulate the effects of the borrow
pits on wave transformation, the resulting alongshore sediment transport and formation of
EHSs.
1.2 Objectives and Scope
Horikawa, Sasaki and Sakuramoto (1977), and Motyka and Willis (1974) evaluated
the effects of dredge pits on the shoreline using numerical and physical models. The
numerical model included refraction, but not wave diffraction, reflection or dissipation.
However diffraction effects were clearly visible in the lee of borrow pits of oblique aerial
photography at Grand Isle, LA. After beach nourishment, two large cuspate features
formed in the lee of the two borrow pits and resulted in a narrow beach adjacent to the
cuspate feature (Combe and Soileau, 1987). This outcome was probably due to the
combination of transformation processes as affected by the borrow pit bathymetry,
causing sand to be transported to and deposited in the lees of the two borrow pits.
The purpose of this thesis is to evaluate the wave refraction and diffraction
mechanisms using available numerical models in an attempt to determine the causes of
the EHSs due to offshore pits. If the causes can be determined, it should be possible to
design the borrow pits to minimize their adverse effects. Comparing the results with the
shoreline changes at Grand Isle, the usefulness of numerical models in predicting
shoreline changes can be evaluated. Using this numerical approach to investigate a range
3
of idealized pit geometries, some general shoreline change results are developed for the
effects of offshore dredge pits due to different pit locations, shapes and depths.
CHAPTER 2
EQUILIBRIUM BEACH PROFILE
One useful tool in coastal engineering studies is the equilibrium beach profile that
was proposed originally by Bruun (1954). The nearshore profile from the closure depth
(h,) to the shoreline can be obtained by using equilibrium beach profile (EBP) theory if
the sand sizes are known. In some of present application, EBP methodology will be
employed to represent profile landward of and outside the influence of the borrow pits.
The profiles for a wide variety of beaches can be represented reasonably well by the
simple relation
2
h(y)=Ay3 (2.1)
In which h is the depth at a distance y seaward of the shoreline and A is an empirical
coefficient termed the "profile scale parameter", having the dimensions length Dean
(1977) showed that A is related to the wave energy dissipation per unit water volume in
the surf zone and is based on the median sand size. It is assumed that the turbulence in
the surf zone, created by the breaking process, is the dominant destructive force. Thus,
the energy dissipation per unit volume D (d) may be representative of the magnitude of
the destructive force per unit volume.
l dF
D(d)= (2.2)
h dy
Where F is wave energy flux per unit crest length, and y is the shorenormal coordinate
directed offshore. From linear shallow water wave theory, energy flux can be expressed
as
F 1 pgH2 (2.3)
8
The breaking waves are considered to decay throughout the surf zone according to:
H = Kh (2.4)
Where K is a dimensionless breaking wave constant, usually taken as 0.78. Equation (2.2)
can now be written:
d( pgK2h2 gh)
D,(d) = h (2.5)
hdy
Taking the derivative and simplifying:
D (d)= 5 2pg2 h2 dh (2.6)
16 dy
Which is directly dependent on the beach slope and the square root of the water depth h.
Integrating Equation (2.6), we obtain,
S24D(d) y = AyY (2.7)
5pggK2
It is intuitively clear that larger sand particles can withstand greater destructive forces
and are more stable under the influence of turbulence and bottom friction. Sediments of
larger sizes would be associated with steeper profiles.
The relationship between the scale parameter A and median sediment size has been
developed by Moore (1982) (see Figure 2.1). Later, Dean (1987) simply transformed
Moore's relationship of A versus d (diameter) to w, the fall velocity, and found a linear
(on a loglog plot) relationship as shown in Figure 2.1, and given by
A = 0.067w044 (2.8)
In which w is in cm/s and A is in m A realistic beach comprises a range of sand sizes,
with the sand sizes usually becoming finer in the offshore direction. Dean and Charles
(1994) developed a relationship representing nonuniform sand sizes based on the
assumption of a linearly varying sediment scale parameter A between adjacent sampling
locations.
A,, A
A(y) = A, + ( A)(y y,) (2.9)
Yn+1 Yn
SEDIMENT FALL VELOCITY, w (cm/s)
0 01 0.1 1.0 10.0 100.0
Et 1.0 Suggested Empirical
E Relitionshlp A va. D7
(Moorn) O0
From Hughsrs / Aj0 *
S on Individual Field Field R lsults
E Profiles Where a Rang
S Slland Szes was Glv l
< 0.10
DB oA an' ranstftming
a A v.DCu veUslling
n 0.01 0.1 1.0 10.0 100.0
...
SEDIMENT SIZE, D (mm)
Figure 2.1, Variation of beach profile scale parameter, A, with sediment size, D, and fall
velocity, w,. Dean (1987)
Which is applicable for y,, < y < y,,,. The A values were determined from Table 1 which
range from a sediment size D from 0.1mm to 1.09mm (Dean 1994) based on the median
sediment size. Given the calculated depth h, at one sampling location, y,, the depth at
the next sampling location yn+, is calculated as follows:
hn, = {hK + 2 [A A5]
5m,,
In which m, is the slope of the A vs y relationship between y, and y,, i.e.
(A,,, A)
mn =(1 A)
Yn+, Y,
(2.10)
(2.11)
Thus, the equilibrium beach profile theory can be used to predict the beach profile for
both uniform sand sizes and sand sizes which vary across the nearshore zone.
Table 1: Summary of Recommended A Values (Dean)
0.02 0.03 0.04
0.Q714 0.0756 0.0798
0.106 0.109 0.112
0.129 0.131 0.133
0.1482 0.1498 0.1514
0.1634 0.1646 0.1658
0.1754 0.1766 0.1778
0.1868 0.1877 0.1886
0.1956 0.1964 0.1972
0.2036 0.2044 0.2052
0.2116 0.2124 0.2132
0.05
0.084
0.115
0.135
0.153
0.167
0.179
0.1895
0.198
0.206
0.2140
0.06 0.07
0.0872 0.0904
0.117 0.119
0.137 0.139
0.1546 0.1562
0.1682 0.1694
0.1802 0.1814
0.1904 0.1913
0.1988 0.1996
0.2068 0.2076
0.2148 0.2156
0.08 0.09
0.0936 0.0968
0.121 0.123
0.141 0.143
0.1578 0.1594
0.1706 0.1718
0.1826 0,1838
0.1922 0.1931
0.2004 0.2012
0.2084 0.2092
0.2164 0.2172
(Units of A Parameter are m3 ) (To convert A values to feet units, multiply by 1.5)
D(mm)
0.1
0.2
03
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0.00
0.063
0.100
0.125
0.145
0.161
0.173
0.185
0.194
0.202
0210
0.01
0.0672
0,103
0.127
0.1466
0.1622
0.1742
0.1859
0.1948
0.2028
0.2108
__
e
~
~~
.~..
CHAPTER 3
RCPWAVE MODEL
3.1 Introduction
The Regional Coastal Processes Wave (RCPWAVE) Propagation Model (Ebersole, et
al. 1986) is a shortwave numerical model specifically designed for predicting linear
plane wave propagation over arbitrary bathymetry of mild slope. RCPWAVE adopts
linear wave theory to solve wave propagation problems, and refractive and bottom
induced diffractive effects are included in the model. However, this model does not
consider wave reflection and energy losses outside the surf zone, nonlinear effects, nor a
spectral representation of irregular waves. RCPWAVE is the numerical model employed
in this thesis to investigate wave propagation and transformation over borrow pits. The
output of RCPWAVE was later combined with a shoreline change model to investigate
the impact of the borrow pit on shoreline stability.
3.2 Background of RCPWAVE
Classical wave ray refraction theory fails to yield adequate solutions in regions of
complex bathymetry where waves are strongly convergent or divergent. Often due to
complexities in the bottom topography, wave tracing diagrams may have intersecting
wave rays which cause interpretation difficulties, as this theory predicts infinite wave
heights at these locations. Combining refraction and bottominduced diffraction effects,
Berkhoff (1972, 1976) derived an elliptic equation that can represent the complete wave
transformation process for linear waves over an arbitrary bathymetry limited only by the
assumption of mild bottom slopes. The model is more efficient than traditional wave ray
models because the governing equations are solved directly on the bathymetric grid in
the horizontal plane rather than by ray shooting and interpolation to the grid.
3.3 Assumptions and Limitations
Application of any model requires a clear understanding of its physical basis and
capabilities of the model to simulate the processes of interest. If the model foundation is
appropriate for the problem of interest, the results produced should be much more
realistic.
RCPWAVE is a linear, monochromatic, short wave model. Therefore, nonlinear
effects and irregular waves cannot be modeled. Model applications are restricted to a
mild bottom slope. Timedependent effects are not modeled since RCPWAVE is a
steadystate model. Wave reflection is assumed negligible, and energy losses due to
bottom friction or breaking outside of the surf zone are also assumed negligible.
These assumptions and limitations are common in many numerical models. The
results from the model are sufficiently accurate to be used in the prediction of longshore
sediment transport rates and shoreline change if wave reflection and energy losses can be
neglected.
3.4 Governing Equations
The mild slope equation is
(CC )+ (CC )+ K2CC = O (3.1)
ax ax y 3 ay
where x and y are two orthogonal horizontal coordinate directions; C is wave
celerity; Cgis group velocity; K is wave number and 0 is the complex velocity
potential.
The velocity potential function is
S= ae (3.2)
where a is a wave amplitude function (a=gH/2 c), H is wave height, o is angular wave
frequency and s is the wave phase function. Substituting the velocity potential into
Equation (3.1) to obtain real and imaginary parts and combining the irrotationality of the
wave phase function gradient (V x (Vs) = 0), three equations were developed (Berkhoff
1976),
1 02a 2a 12 2
{ + + C [Va V(CC )]} + K2 Vs 0 (3.3)
a Q2X Q y CCg
a a
(Vs sin ) (Vs cos0) = 0 (3.4)
ax ay
(a2CCg, Vs cos0)+a2(a2CCg Vs sinO) = 0 (3.5)
and form the basis for RCPWAVE.
Numerical solution of the above equations can yield the wave field outside the surf
zone. The wave breaking model (Dally, Dean and Dalrymple 1984) is used to calculate
wave parameters inside the surf zone. Although the model RCPWAVE includes some
assumptionsmild bottom slope, negligible wave reflection and energy losses, the
model is efficient and very stable. Because the wave height, wave angle and water depth
are represented directly on a grid, these results are easily linked to other engineering
11
models in particular the longshore sediment transport model as described in the following
chapter.
CHAPTER 4
DNRBS MODEL FOR CALCULATING LONGSHORE SEDIMENT TRANSPORT
4.1 Introduction
The waves and the longshore currents that are generated by breaking waves and
arrive at the coast at an oblique angle may move considerable amounts of sand in the
longshore direction. The gradients in longshore sediment transport cause beach erosion
and accretion at various longshore locations. A beach nourishment project constructed on
a long straight beach will spread out along the shoreline due to waves.
Dean and Grant (1989) developed a numerical model called DNRBS to simulate
longshore sediment transport. This model is a onedimensional model based on an
explicit solution scheme. The model assumes straight and parallel bathymetric contours
and ignores energy losses from deep water to wave breaking.
4.2 Governing Equations for DNRBS
The DNRBS model is based on the equations of longshore sediment transport and
continuity. The longshore sediment transport equation is an empirical energy flux model.
One relationship for representing longshore sediment transport is based on the
longshore wave energy flux at wave breaking. The simplest and still one of the most
successful relationships is known as the CERCformula (Coastal Engineering Research
Center, 1984) and is based on an empirical correlation between the longshore component
of energy flux P, in the longshore direction and the longshore sediment transport Q.
KP,
Q KP= (4.1)
(p., p)g( p)
Equation (4.1) is dimensionally correct longshore sediment transport and was
developed by Inman and Bagnold (1963). In Eq. (4.1) P, is the alongshore energy flux
per unit length of beach at breaking and is defined as
P, = F cosOsinO = pgH2C, sin 20b (4.2)
where Ob is the angle the wave ray makes with the onshore (X) direction, p is the
porosity of sediment, typically about 0.35 to 0.4, the dimensionless parameter K is 0.77,
(Komar and Inman 1970) p, and p are the densities of sand and water, respectively.
Equations (4.1) and (4.2) can be combined to yield a simplified form of the CERC
equation. Considering shallow water (Cg = gh) and = Hb/h, where Hb is the
breaking wave height and K is the breaking index, the CERC equation becomes:
KH Y% gK sin(l ab) cos( ab) )
8(s 1)(1 p)
where the subscript 'b' denotes breaking conditions, s is the sediment specific gravity
(p, p p 2.65) of the water in which it is immersed, P represents the azimuth of the
outward normal to the shoreline, and ab represents the azimuth of the direction from
which the breaking waves originate. Figure 4.1 presents a definition sketch for a,,, p/ and
/ in which p/ is the azimuth of general alignment of the shoreline as defined by a
baseline, and
xr x
p =/u  tan ( )
2 ay
0
Q
Reference
Base Line
Shoreline
+Q
(4.4)
Figure 4.1, Definition sketch (for waves and shoreline orientation)
(Dean and Grant, 1989)
The conservation of sediment equation is
8Q 8v
OQ = a (4.5)
By at
In which Q is the alongshore transport rate of total volumetric sediment transport
integrated across the surf zone and V is the volume of sand per unit beach length. It is
assumed that as beaches erode or accrete, the profile moves without change of form in a
landward or seaward direction respectively. Thus after equilibration occurs, the shoreline
change, Ax, associated with a volumetric change, AV, can be expressed as
Ax =A (4.6)
h. +B
Substituting Equation (4.6) into (4.5), the conservation of sediment equation then
becomes
6x 1 Q
+ = 0 (4.7)
at h. +B oy
Assuming energy conservation and Snell's law, Equation (4.3) can be expressed in
terms of deep water conditions as follows
KH.4 C.g2 0.4 cos1.2(0 ao)sin(,0 a.)
Q 0 (4.8)
8(s 1)(1 p)C.ICK04 Cos(,0 ab)
and
C*
a. = ,o sin'[sin(f0 a)] (4.9)
Co
where the subscript "0" denotes deep water conditions and C. is the wave celerity in
water depth, h,.
Because cos(0o ab) is approximately unity, this longshore transport equation
depends almost entirely on deep water parameters. However, the complex problem of
longshore sediment transport is not represented completely by this relationship due to the
lack of an overall description of essential parameters, for instance sediment grain size and
bed morphology which affect longshore sediment transport. An offshore dredged area
affects the wave climate and thus the alongshore sediment transport which is of special
concern here.
To assure numerical stability using DNRBS which solves the transport and continuity
equations explicitly, the following criterion must be satisfied,
1 Ay2
At < (4.10)
2G
where Ax is the longshore grid space and At is time step, the longshoree diffusivity" G
can be expressed in term of breaking condition as
KHYJ2 K
G= KHgb K (4.11)
8(1 p)(s 1)(h. + B)
where ( h. + B) represents the vertical dimension of the active profile.
A grid definition sketch is presented in Figure 4.2, where i is a spatial index.
Figure 4.2 Grid definition sketch used in computational method
CHAPTER 5
SHORELINE CHANGE CONSIDERING GEOMETRY
Chapter 4 discussed the DNRBS model which employs the assumption that the
bathymetric contours are straight and parallel. In many applications, the offshore
contours can be considered as straight and parallel which allows computation of wave
refraction from deep to shallow water by Snell's law. However, for the case of interest
here, it is necessary to use a numerical model, which can consider the effects of irregular
bathymetry, refraction and diffraction, to determine the nearshore wave conditions. The
linearwave transformation model RCPWAVE can be used to provide such detailed wave
information and this information can be used in DNRBS to obtain more accurate results
for longshore sediment transport. The RCPWAVE model has the following advantages:
1. The computation is very stable and efficient,
2. The refraction and diffraction effects caused by irregular bathymetry are included,
3. The wave variables are calculated at the grid intersection points and are easy to
output.
In the DNRBS model, the energy flux at the wave breaking point is used to calculate
the longshore sediment transport, Q,
K(ECg cos 0 sin) (5.1)
pg(s )( p)
It is difficult to establish directly, the wave conditions at the breaking point, so the
equation is converted to one expressed in deepwater conditions (Chapter 4). In this
conversion, the bathymetry is considered to be straight and parallel and the energy losses
from deep water to the wave breaking line are neglected.
Figure 5.1 shows the computated wave rays for the irregular bathymetry in Figure 5.2
(Ebersole, B. A., Cialone, M. A. and Prater, M. D., 1986, using RCPWAVE). The data
are from Duck, North Carolina. From Figure 5.1, it is seen that the wave rays are affected
by bathymetry and are not parallel to each other. Thus the waves arrive at the wave
breaker line, with directions which vary in the longshore direction.
Figure 5.3 shows the computed wave height contours at Duck, North Carolina. It can
be seen that the wave breaking position would vary with distance from the shoreline.
These figures have shown the effect of bathymetry on wave breaking characteristics
at different locations. These wave climates then will determine the longshore sediment
transport. To express the sediment transport, Q, we need to use the wave climates at the
breaking line instead of those in deep water.
In our calculations, Equation (5.1) is used at the wave breaking locations to calculate
Q. It has several advantages over the deepwater version.
1. More accurate, because at different positions, the different wave angles, heights,
and water depths are used to obtain Q,
2. It is not necessary to use the deep water direction to find C,,
3. The equation is more physically clear.
FRF
.. DUCK, NC
Wa'sRftraction Diagr
: ., ,. "'' Period (reC) 2
Angle (deg) 20
Figure 5.1, Wave rays at Duck, North Carolina
(Ebersole, Cialone and Prater 1986)
YAXIS
XAXIS DUCK PIER. NORTH CAROUNA BATHYMETRY
DEPTHS ARE METERS BELOW MSL
Figure 5.2, Bathymetric contours at Duck, North Carolina
(Ebersole, Cialone and Prater 1986)
All the terms in Equation (5.1) can be determined from the results of the RCPWAVE
model. A flowchart of the computations is presented in Figure 5.4.
When the shoreline changes due to gradients in longshore sediment transport, the
entire bathymetry will also change and it will affect the wave field determined by
RCPWAVE. Thus, an iteration is required to update the RCPWAVE and DNRBS model.
In the DNRBS model, the time step is relatively small due to the stability requirement
(Eq. (4.10)). At every time step (1000s), the shoreline change is very small and its effect
on RCPWAVE is negligible. Besides the RCPWAVE model computations are time
consuming, it is not reasonable to run RCPWAVE model every time step of DNRBS.
Therefore in the computations we can set a key value 1/2 Ax where Ax is the cell size in
the crossshore direction for shoreline change. When the shoreline change exceeds that
value, the RCPWAVE model is run once and then the wave characteristics are
maintained constant until the shoreline change equals the specified value again.
1200
1100
1000 
900 8027
800 2.50062 2
c 700 .303
1.9 2
600 17
500 
0
400
S. 4 8 0 2 7
300
2.2 156
200
100
0 I ,I I I IrI I I I I i
200 400 600 800 1000 1200
Offshore distance (m)
Figure 5.3, Wave height contours at Duck, North Carolina
(T=12 s, H=2 m, ao =20 degrees)
Figure 5.4: Computational flowchart
CHAPTER 6
EXAMPLE APPLICATION OF THE MODEL FOR AN ACTUAL CASE
6.1 Geographic Description of Grand Isle
Grand Isle is located on the Gulf of Mexico in Jefferson Parish, Louisiana, (See
Figure 6.1), and is about 60 miles south of New Orleans, 45 miles northwest of the
Southwest Pass of the Mississippi River (Gravens and Rosati, 1994). Grand Isle is part of
the Bayou Lafourche barrier system that forms the seaward geologic framework of
eastern Terrebonne and western Barataria basin in Terrebonne, Lafourche, and Jefferson
parishes (Penland et al. 1992). Grand Isle is a lateral migration of the flanking barrier
islands developed by recurved spit processes. It is a lowlying barrier island with natural
ground elevations ranging between 3 and 5 ft above National Geodetic Vertical Datum
(NGVD). Its length is nearly 7.5 miles and its width is approximately 0.75 miles
(Gravens and Rosati, 1994 and Combe and Soileau, 1987). Based on data for a 100year
period (McBride et al. 1992), the shoreline of Grand Isle has retreated on its western end,
kept relatively stationary at its middle part, and advanced on its eastern end.
Recently, a 2600 ft revetment and jetty system has been built at the western end of the
island to try to halt the rapid beach erosion commencing in this area near the end of the
1960's and the beginning of the 1970's. Although these structures achieved the expected
objectives stabilizing the shoreline on the western end, storms had caused large amounts
of sand transport from the beach face with offshore deposition. Due to erosion of this
island, a beach nourishment project was undertaken in 1978.
Figure 6.1, Grand Isle vicinity map (Gravens, and Rosati, 1994)
6.2 Borrow Pit Characteristics
As noted previously, borrow pit locations can interact with waves and lead to a
modification of the wave patterns along the shoreline. A manmade beach and dune 7.5
miles long with a crest elevation of 7 feet NGVD was built in 1976 along the gulf shore
at Grand Isle. For controlling beach erosion and reducing hurricane wave damage, a
beach renourishment project was initiated in September 1983, and was completed one
year later. This project comprised the placement of 5,600,000 cubic yards of sand in the
GULF OF MEXIX)I /
GRAND ISLE, LOUISIANA
SCALE
) 5BOU I0m0oe zm0
*4IU ,q2.40.
beach and dune system. The dune dimensions were 10foot crown at elevation 11.5 feet
NGVD, with side slopes equal to 1 vertical to 5 horizontal extending to a natural ground
elevation on the protected side and to elevation 8.5 feet NGVD where it intersected the
beach berm (Combe and Soileau, 1987). Sand was obtained from two offshore borrow
areas located about 3000 feet Gulfward. The original water depth in the vicinity of the
borrow pit was 15 feet, the depth of excavation below the sea bed was 20 feet near both
ends and 10 feet at the center. As a consequence, the borrow areas are described as a
"dumbbell" shape with the centroids of the bells located about 4,500 feet apart and
considerably deeper than the middle portion (see Figure 6.2). After a short time, a series
of shoreline changes became evident. By August 9, 1985, the two large cuspate features
along the nourished beach, which had formed landward of the ends of the primary borrow
areas adjacent to the center of the island were clearly visible as shown in Figure 6.3. The
sand in these features was drawn from the neighboring beaches, leaving these areas quite
narrow. Gravens and Rosati (1994) examined this situation using the numerical models
RCPWAVE and GENESIS which was used to compute longshore sediment transport
rates and shoreline change, and have concluded that wave refraction led to lower wave
energy at the shoreline directly landward of the pits and higher wave energy adjacent to
the pits. This results in larger wave setup at the shoreline on the sides of two dredged pits
which could drive currents into the sheltered area result in sediment transport and
localized deposition which forms the salients. Thus, the offshore borrow pits at Grand
Isle were likely the cause of the observed shoreline changes.
Because longterm wave measurements were not available in the vicinity of Grand
Isle, Gravens and Rosati (1994) used a 20year hindcast for the Gulf of Mexico coast that
1885
Figure 6.2, Pit geometry at Grand Isle
27
was summarized in the U.S. Army Corps of Engineers Wave Information Study database.
The average incident wave conditions in deep water conclude that the representative
wave height, wave angle and wave period, are 1.5 ft, 6.5 degrees west of shorenormal
and 5.6 seconds respectively. A contour plot of the bathymetry grid that causes an
irregular longshore breaking wave field produced by the borrow pit is indicated in Figure
6.4. (Gravens and Rosati, 1994)
* ..
(Combe and Soileau, 1987)
Figure 6.4, Contour plot of Grand Isle nearshore bathymetry grid
(Gravens, and Rosati, 1994)
6.3 Results Based on RCPWAVE
The Grand Isle borrow pit geometry and shoreline were used to evaluate the
methodology described in this thesis (RCPWAVE and DNRBS). The sediment size is
0.2mm and the base conditions for the following cases are a water depth, h, of 15 feet,
incident waves with a wave height of 1.5 feet and a period of 5.6 seconds. The profile
landward of borrow pit was represented as an equilibrium beach profile. The initial
computational domain is shown in Figure 6.5. The length in the longshore direction
encompasses 343 cells, 35 m each of length, for a total of 12,005 m, and the modeled
length in the crossshore direction comprises 80 cells, 28.75 m width each, for a total of
2300 m. The time step is 1000 seconds. Two cases with different wave angles were
examined.
1. Wave angle =0.0 degree
2. Wave angle=6.5 degree
z
X
Figure 6.5 Computational domain for bathymetry and wave direction
The length in the longshore direction encompasses 343 cells, 35 m each of length, for
a total of 12,005 m, and the modeled length in the crossshore direction comprises 80
cells, 28.75 m width each, for a total of 2300 m at Grand Isle
12000
11000
10000
9000
, 8000
7000
6000
5000
4000
3000
2000
1000
0
I I
I I I
I I I I I
10 0 10
Shoreline Change (m)
Figure 6.6 Shoreline changes for a normal incident wave before smoothing
(Pit centerline at 6000 m offshore)
(Water depth, h, of 15 feet, incident waves with a wave height of 1.5 feet
and a period of 5.6 seconds at Grand Isle)
Figure 6.6 shows the calculated shoreline evolution with time for a normally incident
wave. It is seen that the shoreline changes are quite irregular and this irregular pattern
remains even for large time. Comparison with the real shoreline demonstrates that this
result is not physically correct. To examine the cause of this effect, the breaking wave
conditions were examined. Figure 6.7 shows the wave angle change at the breaker line
where is seen that the wave angle varies smoothly, thus this is not the source of the
~c
31
irregularity. The wave angle here has been averaged using three points to reduce any
oscillations, i.e.
(6.1)
12000
11000
10000
9000
8000
7000
6000
5000
4000
3000
2000
1000
10
Pit Centerline
`~_ _
<^^" ~______~~
I I I I i i
I I I I I I I I I I
5 0 5 11
Breaking Wave Angle (degree)
Figure 6.7 Wave angle at the breaker line before smoothing
(Water depth, h, of 15 feet, incident waves with a wave height of 1.5 feet
and a period of 5.6 seconds at Grand Isle)
Figure 6.8 shows the wave height at the breaking line where it is evident that the
wave height varies significantly over small longshore distances. This type of oscillation is
believed to be physically unrealistic, because the large gradients will be smoothed by
ai = 1/ 3(a,_ + a, + a,,)
32
diffraction and possibly other processes. In the RCPWAVE model, there is no smoothing
although diffraction is represented. This character suggests that to obtain correct results,
these large gradients should be smoothed.
12000
11000
10000
9000
8000
7000
6000
5000
4000
3000
2000
1000
.
Pit Centerline
, I , I
0.4 0.6 0.8
Breaking Wave Height (m)
Figure 6.8 Breaking wave height before smoothing at Grand Isle
(Water depth, h, of 15 feet, incident waves with a wave height of 1.5 feet
and a period of 5.6 seconds at Grand Isle)
. 1 1 1 ,,1 ,,,
IIIIHII
ri l l ll.l.l l T
2
I
Figure 6.2 shows the pit geometry used to conduct the computations. There are sharp
changes in the beach bathymetry and these sharp changes may cause large gradients in
the solution. To verify this possibility, the beach bathymetry is smoothed using the
following equation
h,j = 1/ 8(h,_1 + 4h,, + hi+, + h,_j + h,,J) (6.2)
To obtain smoother results, the original bathymetry is averaged 100 times using
Equation (6.2). This equation will not change the total amount of sand excavated from the
pit; however, the maximum depth is decreased.
Figure 6.9 shows the computational domain after the smoothing and Figure 6.10 shows
the wave height at the breaker line where it is seen that the wave height is much smoother
than in Figure 6.8. Figure 6.11 shows the wave angle at the breaker line which is very
similar to that in Figure 6.7. It is quite interesting that smoothing the bathymetry seems to
have a considerable smoothing effect on the breaking wave height but a relatively small
effect on the wave angle. Figure 6.12 shows the calculated shoreline change. Because the
RCPWAVE model only includes the effects of wave refraction and diffraction, and not
wave reflection and dissipation, this result is opposite to that appearing in Figure 6.3 for
Grand Isle. Figure 6.13 shows the shoreline change for an incident wave angle equal to
6.5 degrees which is also opposite to the actual case which has two erosional cold spots
and three erosional hot spots.
Figure 6.14 depicts wave directions and wave rays in the vicinity of the smoothed pit.
It can be noted that in the center of pit the direction of wave rays is towards inside which
associated with an erosional cold spot and the direction of wave ray near the ends of pit is
toward outside which related to forming erosional hot spots.
Z
x
Figure 6.9 Bathymetry after smoothing at Grand Isle
The length in the longshore direction encompasses 343 cells, 35 m each of length, for a
total of 12,005 m, and the modeled length in the crossshore direction comprises 80 cells,
28.75 m width each, for a total of 2300 m
12000
11000
10000
9000
E 8000
S7000
5 6000
0
S5000
= 4000
3000
2000
1000
Pit Centerline
I I I
3
f
? I I I I I I r I I I T I
. I, I . I 1 , ,
I I
%0
0.5
Breaking Wave Height (m)
Figure 6.10 Breaking wave height after smoothing at Grand Isle
(Water depth, h, of 15 feet, incident waves with a wave height of 1.5 feet
and a period of 5.6 seconds)
12000
11000
10000
9000
E 8000
S7000
5 6000
5000
=. 4000
CO
3000
2000
1000
nt1
Pit Centerline
<"_
SI
. .I I I I I I I
10 5 0 5 10
Breaking Wave Angle (degree)
Figure 6.11 Breaking wave angle after smoothing at Grand Isle
(Water depth, h, of 15 feet, incident waves with a wave height of 1.5 feet
and a period of 5.6 seconds)
i I
12000
11000
10000
9000
E 8000
r 7000
S6000
0
S5000
. 4000
3000
2000
1000
0
MM
0
Shoreline Change (m)
Figure 6.12 Shoreline change for a normal incident wave after smoothing
(Pit centerline at 6000 m offshore)
(Water depth, h, of 15 feet, incident waves with a wave height of 1.5 feet
and a period of 5.6 seconds at Grand Isle)
i n I I I I I I I
SI
I
12000
11000
10000
9000
8000
S7000
a 6000
5000 o
4000
3000
2000
1000
15 10 5 0 5 10
Shoreline Change (m)
Figure 6.13 Shoreline change for incident wave angle equal to 6.5 degree after smoothing
(Pit centerline at 6000 offshore)
(Water depth, h, of 15 feet, incident waves with a wave height of 1.5 feet
and a period of 5.6 seconds at Grand Isle)
8000
7000
0 a (Positive)
6000
6. a (Negative)
5000
4000
0 1000 2000 3000 4000 5000
Offshore Distance (m)
Figure 6.14 Wave rays for smoothed pit at Grand Isle and definition sketch
for wave angle direction at Grand Isle
(Water depth, h, of 15 feet, incident waves with a wave height of 1.5 feet
and a period of 5.6 seconds)
6.4 Adding Second Term in "CERC" Equation and Comparing the Results
Chapter 4 presented a form of the CERC equation which included only one term
which represents sediment transport due to obliquely incident breaking waves. However,
a second term represents the transport contribution produced by longshore currents
resulting from a gradient in wavesetup if breaking wave heights vary in the longshore
direction (Ozasa and Brampton, 1980). The full CERC transport equation as follows,
KH5 glc sin(Oab)cos(Oa) K2Hb,5 g/Kcos(O ab) dHb
8(s 1)(1 p) 8(s 1)(1 p) tan(7) dx
where K2 is the sediment transport coefficient of the second term and tan y is the slope
from nearshore to depth of closure. Figure (6.15) presents shoreline change results of
applying Equation (6.3) with K2= 0.77, K1=0.78 and tany= 0.02. The resulting
shoreline changes including the second term are only slightly smaller than that with only
the first term included.
12000
11000
10000
9000
S8000
7000
6000
5000
4000
3000
2000
1000
 Including First Term Only
  Including First and Second Terms
Shoreline Change (m)
Figure 6.15 Comparing the results of simplified CERC and full CERC equations normal
incident waves at Grand Isle
(Water depth, h, of 15 feet, incident waves with a wave height of 1.5 feet
and a period of 5.6 seconds)
CHAPTER 7
SOME GENERAL RESULTS FOR IDEALIZED CASES
7.1 Introduction
As much as 90 percent of the sand for beach nourishment projects is obtained from
offshore sources. In these areas, sediment sizes are often reasonably compatible with the
beach that is to be nourished. These offshore areas can be located small or large (tens of
kilometers) distances from the beach to be nourished. The excavated depths of borrow
pits are also different. Regardless of their geometries and locations, borrow pits can affect
shoreline stability. Because borrow pits can change the offshore bathymetry significantly,
the bathymetric pattern will impact the shoreline due to wave refraction, diffraction
reflection and dissipation. To obtain generalized effects of offshore borrow areas, a series
of idealized pit geometries represented as a bivariate normal distributions were calculated
using the numerical simulation methods described in the previous section.
7.2 Shoreline Effects Due to Various Pit Parameters
Chapter 3 has discussed the requirement of a mild bottom slope for the model
RCPWAVE. For purposes here, we represent the pit geometry by an idealized bivariate
normal distribution as follows,
(xxo)2 (yyo)2
h(x, y) = h + h e 2<'x 2y2
In which is the water depth without the pit, Ah is the maximum depth of offshore
borrow pit relative to the ambient depth, x0 is the distance from the shoreline base line to
the center of pit, yo is the distance from the center of pit to the X axis (offshore
direction) and o, and ay are the distances from the center of pit to the respective
standard deviation distances, respectively (see Figure 7.1). The profile landward of the pit
is represented by equilibrium beach profile based on a sand size of 0.2 mm.
15000
14000
13000
12000
.11000
10000
S9000 1
,S 8000 i I __ a (Positive)
a 7000 
S7000 a (Negative)
6000 
o 5000
4000
3000 xo
2000
1000 
5000 10000 15000
Offshore Distance X (m)
Figure 7.1 Sketch for normal distribution pit
(ox = 400 m, o=1100 m and x0=3600 m)
7.2.1 Shoreline Changes as a Function of Distance of Pit from the Shoreline
Beach nourishment is less costly if the borrow pits are located near the shoreline.
However, the shoreline will be affected more for small distances of the pit from the
shoreline. In the present study, we examine the general effect on shoreline changes as
related to the distance from the pit to the shoreline. Thus in these cases, all the parameters
are kept constant except for x, which is the distance from the center of the pit to the
shoreline. The computational domain is 320 by 300, ie 300 cells in longshore direction
(y) and 320 cells in cross shore (x) direction. The grid sizes in the x and y directions are
Ax = 20 meters and Ay = 60 meters. The base incident wave parameters for the following
trials are that the relative wave angle P a seaward of the pit is equal zero, the wave
height Ho is one meter and the wave period equals 10 seconds. In the simulation, h = 8
m, Ah = 6 m, x = 200 m, r,= 1,200 m and y = 9,000 m. Eight different cases are
calculated at x0 = 1,400 m, 1,600 m, 1,800 m, 2,000 m, 2,200 m, 2,400 m, 2,600 m and
3,600 m.
Figure 7.2 shows the shoreline changes for x0=1,400 m. It is seen that the largest
shoreline accretion occurs at y= 12,000 m and y=6,000 m and the maximum erosion
occurs at y= 9,000 m, ie directly landward of the pit. Figure 7.3 shows the shoreline
change for x0 =2,200 m. From Fig. 7.2 and 7.3 it is seen that although for this different
x0, the position of largest erosion is not changed, the maximum shoreline changes
decrease with increasing pit distance from the shoreline.
The largest shoreline recession and advancement for different x0 are plotted in
Figures 7.4 and 7.5 respectively. These results indicate the principal trends of shoreline
change which are that the greater the distance from the shoreline base line to the center of
pit, the smaller the effect on the shoreline. The trend of shoreline change is nearly linear
when the pit is close to the shoreline. It is interesting to note that there is a small region
where the effect of the pit on the shoreline dramatically decreases with increasing
distance from the pit to the shoreline. If the borrow pit is sufficiently far from the
shoreline, very small shoreline changes result.
18000
16000
14000
E
12000
10000
e 8000
o 6000
4000
2000
S .
Vy/t
r
I I I I i i I i i i
I I I II
U
3 2 1 0 1 2
Shoreline Change (m)
Figure 7.2 Shoreline changes at x0 =1400 m, pit centerline at y=9000 m
(h= 8 m, Ah= 6 m, o = 200 m, y= 1,200 m, T=10 s, H =1 m)
18000
16000
14000
12000 
S 000
S8000
o 6000
C,,
4000
2000 
0.4 0.2 0 0.2
Shoreline Change (m)
Figure 7.3 Shoreline changes at x0 =2200 m, pit centerline at y=9000 m
(h= 8 m, Ah=6m, = 200 m, a= 1,200 m, T=10 s, H =1 m)
0 
1*
0.5
S1.5
,
0
E 2
0/
2.5 
3 I I l I i lI i l i Il i i I , I , , I ,
0 500 1000 1500 2000 2500 3000 3500
Offshore Distance, X (m)
Figure 7.4 The largest shoreline recession for different x0
(h= 8 m, Ah= 6 m, = 200 m, o= 1,200 m, T=10 s, H =1 m)
1.5
S0.5
E
0 500 1000 1500 2000 2500 3000 3500
Offshore Distance, X (m)
Figure 7.5 The largest shoreline advancement for different x0
I
0 500 1000 1500 2000 2500 3000 3500
Offshore Distance, X (m)
Figure 7.5 The largest shoreline advancement for different x0
(h= 8 m, Ah= 6 m, o = 200 m, o= 1,200 m, T=10 s, H =1 m)
7.2.2 Effects of Pit Depth on Shoreline Changes
The pit depth is related to the shoreline change magnitude. To quantify this effect, in
the simulations to be discussed, all other parameters were kept constant except for Ah
which is the maximum pit depth. The computational domain is 380 by 380 with 380 grids
in both longshore (y) and cross shore (x) directions. The grid size in the x direction, Ax,
is 20 meters and Ay is 30 meters. The base conditions as described previously for wave
characteristics and water depth h were used. In these calculations, cr and o, were
maintained at 200 m and 600 m, xo and yo were 3,600 m and 5,700 m respectively.
Seven different cases are simulated with Ah equals 2 m, 4 m, 6 m, 8 m, 10 m, 12 m and
15 m. Figures 7.6 and 7.7 show the maximum shoreline recession and advancement
respectively. It is seen that the shoreline increases with increasing pit depth.
I I I I I
I I I I I I I I I
5 10 1V
Depth of Pit (m)
Figure 7.6 The maximum shoreline recession versus pit depth
(h = 8 m, cr = 200 m, o = 600 m, T=10 s, Ho =1 m)
10
9 
8
E 7
) 6
5
0 4
E 
0 5 10 15
Depth of Pit (m)
Figure 7.7 The maximum shoreline advancement versus pit depth
3: y/
)4 /
S /
I /
0 5 10 15
Depth of Pit (m)
Figure 7.7 The maximum shoreline advancement versus pit depth
(h= 8 m, = 200 m, o= 600 m, T=10 s, H =1 m)
7.2.3 Effect of Pit Size and Shape on Shoreline Changes
The borrow pit size and shape will affect the shoreline changes. By changing ox and
ay for several cases, basic trends can be obtained showing the effect on shoreline change
due to different pit size and shape. For the numerical model, the base conditions for water
depth and wave criteria are the same as previous cases. x0 = 3600 meters, Ah equals 6
meters. Seventy different cases are calculated with cx ranging from 200 m to 800 m, and
C, ranging from 200 m to 1200 m. Figures 7.8 and 7.9 show the maximum shoreline
recessions and advancement respectively. Figure 7.8 indicates that there are two
recessional peaks for ac equals 800 m, ac equals approximately 600 m and 200 m.
Beach recession is largest for o, larger than ay,. Figure 7.9 shows that there are two
maximum shoreline advancement peaks located on x = 800 m and oy =700 m and
1200
1100
1000
900
800
700
600
500
400
300
200
500 1000
o (m)
Figure 7.8 The maximum shoreline recession versus the sizes of the pit in the x and y
dimensions (x = 3600 m, h= 8 m, Ah= 6 m, T=10 s, H0 =1 m)
a,=400 m. Again shoreline advancement is largest if o, is larger than oy. For ox
less than 200 m and cr greater than 800 m, the associated beach recession and
advancement are quite small. Therefore the best pit shape is one with small Cx and
large o, ie a pit that is elongated with its major axis parallel to the coastline.
1200 
1100 
1000 
900
800
z 700 14"
2:!* IP INV
700
500
400
300
200
500 1000
ox (m)
Figure 7.9 The maximum shoreline advancement versus the sizes of the pit in the x and y
dimensions (x = 3600 m, h= 8 m, Ah= 6 m, T=10 s, H =1 m)
It is instructive to examine in greater detail two of the seventy cases resulting in
Figures 7.8 and 7.9. Case 1 is for ox = 400 m and ca,=1,100 m, and Case 2 is ox= 800 m
and ca,=600 m. In Case 1, the computational domain is 320 by 300, ie 300 cells in
longshore direction (y) and 320 cells in cross shore (x) direction. The grid sizes in the x
and y directions are Ax = 20 meters and Ay = 50 meters and y =7,500 m. Figure 7.10
shows wave rays and the wave angle directions. The wave rays diverge slightly as they
pass over the pit. Figure 7.11 presents the breaking wave angle which varies between
1.2 deg. and 1.2 deg. Figure 7.12 presents the breaking wave height which changes
from 1.2 m to 1.4 m. Figure 7.13 shows shoreline evolution with a maximum shoreline
advancement and recession of 3.1 m and 4.4 m respectively. The maximum erosion
occurs immediately landward of the pit, and this result is similar to the results obtained
by Motyka and Willis(1974). Figure 7.14 presents wave height contours, and Figure 7.15
presents the final shoreline changes considering the first term in the CERC equation and
both terms. It is seen that using the full CERC equation causes smaller maximum
shoreline changes, i.e. including this second term causes changes which reduce slightly
the differences between these computed results and those at Grand Isle. In Case 2, the
computational domain is 380 by 320, ie 320 cells in longshore direction (y) and 380 cells
in cross shore (x) direction. The grid sizes in the x and y directions are Ax = 20 meters
and Ay = 30 meters and y0=4800 m. Figure 7.16 presents the wave rays where it is seen
that the deviation due to the pit are much larger than for Case 1. Figure 7.17 shows that
the breaking wave angle varies between 11 deg. and 11 deg. Figure 7.18 presents
breaking wave height which varies from 0.6 m to 2.6 m. Figure 7.19 shows shoreline
evolutions with a maximum shoreline advancement of 30 m and a maximum recession of
29 m. These plots show large variations in breaking wave angle and wave height which
result in large shoreline changes, the shoreline changes are even larger if o is smaller
than y .
15000
14000
13000
12000
11000
E 10000
8 000
S8a (Positive)
7000 a (Negative)
S6000
0
o 5000
4000 
3000
2000
1000
5000 10000 15000
Offshore Distance (m)
Figure 7.10 Wave rays and wave angle direction for Case 1
(h= 8 m, Ah= 6 m, xo= 3600 m, a,= 400 m and a,=1,100 m, T=10 s, H0=1 m)
15000
14000
13000
12000
11000
E10000 
S9000
. 8000
S7000
S6000
o
S. 5000
4000
3000
2000
1000
01
1.5
Pit Centerline
1 0.5 0 0.5 1 1.i
Breaking Wave Angle (degree)
Figure 7.11 Breaking wave angle for Case 1
(h = 8 m, Ah= 6 m, xo= 3600 m, c, = 400 m and o=1,100 m, T=10 s, Ho=1 m)
K

15000
14000
13000
12000
11000
E10000
1 9000
C
a 8000
S7000
S6000
o
.. 5000
4000 
3000
2000
1000
.15
1.2 1.25 1.3 1.35 1.4
Breaking Wave Height (m)
Figure 7.12 Breaking wave height for Case 1
(h= 8 m, Ah= 6 m, xo= 3600 m, o,= 400 m and o=1,100 m, T=10 s, H =1 m)
jj
(
I I i I
Pit Centerline
I I I I I
1.45 1.5
I I I I I
i l li i i i i i iil i i il . . . .
15000
14000
13000
12000
S11000
'10000
9000
o 5000 
4 4000
3000
2000
1000
0 5000 ) U2
3000
2000
1000
0
5 4 3 2 1 0 1 2 3 4
Shoreline Change (m)
Figure 7.13 Shoreline change for Case 1
(h= 8 m, Ah= 6 m, xo= 3600 m, o= 400 m and o=1,100 m, T=10 s, H =1 m)
15000
14000
13000 i
12000 Il,l
I10000 375 551'
12000
11000 . . 0
S70000
S6000 .
L
5 8000 0
3000
2000
000 5 10 0
4000 i 102625
3000
2000 !
1000
0 2000 4000 6000 8000 10000 12000 14000 16000
X
Figure 7.14 Wave height contours for Case 1
(h= 8 m, Ah= 6 m, x0= 3600 m, o= 400 m and oy=1,100 m, T=10 s, Ho=1 m)
Including First Term Only
  Including First and Second Terms
15000
14000
13000
12000
.11000
E
10000
>
S9000
2 8000
a 7000
S6000
o 5000
S4000
3000
2000
1000
0
5
2 1 0 1
Shoreline Change (m)
Figure 7.15 Comparing the results of simplified CERC and full CERC equations normal
incident waves for Case 1
(h= 8 m, Ah= 6 m, x0= 3600 m, o,= 400 m and cry=1,100 m, T=10 s, H =1 m)
9000
8000
7000
6000
. 5000 a~ (Positive)
0 :a (Negative)
4000
C) 3000
2000
1000
0 2000 4000 6000 8000 10000
Offshore Distance (m)
Figure 7.16 Wave rays and wave direction for Case 2
(h= 8 m, Ah= 6 m, x= 3600 m, o,= 800 m and o =600 m, T=10 s, H0=1 m)
9000
8000
7000
E
6000
.T 5000
 4000
o
c 3000
2000
1000
, I ,. , I .
~~~c~~~'~
Figure 7.17 Breaking wave angle for Case 2
(h = 8 m, Ah = 6 m, xo = 3600 m, r, = 800 m and a, =600 m, T=10 s, Ho =1 m)
. i i r i l l l. l l lr. Ir.i l l l l r l l
Pit Centerline
I . . I . . I I I
15
On
5 0 5
Breaking Wave Angle (degree)
9000
8000
7000 
a 6000 
. 5000 f Pit Centerline
= 4000
S3000
2000
1000
%.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8
Breaking Wave Height (m)
Figure 7.18 Breaking wave height for Case 2
(h= 8 m, Ah= 6 m, x = 3600 m, o, = 800 m and o,=600 m, T=10 s, Ho =1 m)
9000
8000 
7 000 .
6 4000
5000 
3000
u,
2000 ..
1000
30 20 10 0 10 20 30
Shoreline Change (m)
Figure 7.19 Shoreline change for Case 2
(h= 8 m, Ah= 6 m, x = 3600 m, o,= 800 m and coy=600 m, T=10 s, H =1 m)
CHAPTER 8
SUMMARY AND CONCLUSION
A computational model for beach planform evolution as affected by offshore borrow
pits has been developed based on the equilibrium beach profile, RCPWAVE model and
DNRBS model. This new method presented here can predict the shoreline change for
arbitrary bathymetry of mild slope including the case of offshore contours which are not
straight and parallel. The model described in this study accounts for wave refraction and
diffraction only and does not account for wave reflection or dissipation which can affect
shoreline changes significantly. In this model, shoreline changes were predicted based on
equilibrium beach profiles which depend on the sand size and the basic borrow pit
geometry, which served as input data for the RCPWAVE model. After obtaining the
wave conditions from the RCPWAVE model, the DNRBS model is used to calculate
shoreline change. The RCPWAVE model and DNRBS model have been incorporated
into one code, and data exchange between the RCPWAVE model and the DNRBS model
occurs inside the code. In simulating shoreline change, the wave conditions at the wave
breaking points are used, which differs from the general way in which DNRBS was
developed originally which used deep water conditions as input. This change was
required because of the lack of straight and parallel bottom contours as required by the
original DNRBS model. To avoid unnecessary oscillations in this model, averaging of the
wave height and angle at breaking has been applied.
A test for Grand Isle conditions, used to qualitatively evaluate the present model, shows
that the shoreline change results from the present method are opposite to the actual case,
(see Chapter 6) probably because the RCPWAVE model does not consider wave
reflection or dissipation. It is worth mentioning that a smoothing procedure is necessary
for Grand Isle bathymetry to obtain reasonable results. The full CERC equation (adding
the second term) was introduced to simulate shoreline change, and the effect is
reasonably small for the conditions examined.
To systematically study the effect of pits on shoreline change, different cases with
bathymetric anomalies represented by a bivariate normal distribution were investigated
using the present method. Conditions varied included the distance from the pit center to
the shoreline, the pit depth, and the size and shape of the pit. It was found that the
shoreline changes attributable to the pit decrease with increasing pit distance from the
shoreline and increase with pit depth. The pit shape also affects shoreline recession and
advancement. The smaller the pit standard deviation in the cross shore direction (rx)
and the greater the standard deviation in the longshore direction (a,), the smaller the
effect on shoreline change. When the pit shape is approximated by circular contours,
shoreline changes become large. Therefore the best pit shape is one with small ox and
large cry, ie a pit that is elongated with its major axis parallel to the coastline.
Future investigation of the effect of borrow pits on shoreline changes should include
the effects of wave reflection and dissipation.
65
The results presented in this thesis demonstrate conclusively that for pits of realistic
geometry, the combination of wave refraction and diffraction are not sufficient to predict
realistic shoreline responses.
Additional field data relating shoreline response to borrow pit geometry and wave
characteristics will assist in evaluating present capabilities and contribute to the
development of improving future predictive procedures.
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