UFL/COEL2000/013
RELATIVE SIGNIFICANCE OF BACKGROUND EROSION
AND SPREADING LOSSES:
A BEACH NOURISHMENT DESIGN AID
by
Robert G. Dean
December 25, 2000
Prepared for:
Office of Beaches and Coastal Systems
Department of Environmental Protection
Tallahassee, Florida 323993000
RELATIVE SIGNIFICANCE OF BACKGROUND EROSION
AND SPREADING LOSSES:
A BEACH NOURISHMENT DESIGN AID
December 25, 2000
Prepared for:
Office of Beaches and Coastal Systems
Department of Environmental Protection
Tallahassee, FL 32399
Submitted by:
Robert G. Dean
Department of Civil and Coastal Engineering
University of Florida
Gainesville, FL 32611
TABLE OF CONTENTS
1.0 INTRODUCTION .. ................................................. 1
2.0 M ETHODOLOGY .. ..................................... ........... 1
2.1 General .... .... ... .................. ....... ............ 1
2.2 Methodology Development for Nourishment on a Long Beach .............. 2
2.3 Methodology Development for Nourishment on a Barrier Island With One
End of Nourishment Adjacent to an Inlet ................................... 5
2.4 Range of Variables Expected ........................................ 6
3.0 EXAMPLES ILLUSTRATING APPLICATION OF THE METHODOLOGY .... 7
3.1 Hypothetical Examples ......................................... 7
3.2 Application of the Methodology to Five Projects in Florida ................. 7
4.0 SUMMARY AND CONCLUSIONS ......... ..... .............. .......... 10
5.0 REFERENCES .................. ................. ............... 10
LIST OF FIGURES
FIGURE
PAGE
1 Percentage of Material Remaining in Region Placed vs. the Parameter VGt/! for
Initially Rectangular Nourishment on a Long Straight Beach. Dean and Grant
(1989) ..............................................................3
2 Approximate Estimates of Longshore Diffusivity, G(ft2/s), Around the Sandy
Beach Shoreline of the State of Florida. Dean and Grant (1989) ................... 3
3 Isolines of M(t) versus Spreading and Background Erosion Parameters. Long
Straight Beach ..................... ............................. ...... 4
4 Plan View of Nourishment Adjacent to an Inlet ........................... .... 5
5 Isolines of M(t) versus Spreading and Background Erosion Parameters.
Nourishment Adjacent to an Inlet .......................................... 6
LIST OF TABLES
TABLE
PAGE
1 Characteristics and Results For Two Hypothetical Examples ...................... 8
2 Characteristics of and Results For Five Florida Case Studies ...................... 9
RELATIVE SIGNIFICANCE OF BACKGROUND EROSION
AND SPREADING LOSSES:
A BEACH NOURISHMENT DESIGN AID
1.0 INTRODUCTION
The evolution of beach nourishment projects includes the background (prenourishment) erosion
effects, herein referred to as"background erosion" and the project evolution in the absence of
background erosion, namely the "spreading out" effects and profile adjustment. This report examines
the relative significance of background erosion and volumetric spreading for idealized initial
nourishment planforms. According to approximate beach nourishment theory, it is possible to simply
superimpose these two effects. It is useful in the design process to develop estimates of the
approximate relative contributions of these two components. It will be shown that in addition to the
magnitude of background erosion, the relative effects of background erosion are more significant for
the longer projects and increase with time. The interpretation is that the spreading out losses are less
for the longer projects and, after evolving for some time, the project behaves as a long project and
thus the relative importance of the background erosion is greater under these two scenarios.
Design aids are presented in the form of graphs to assist in evaluating the relative importance of
these two components of beach nourishment project evolution. These aids provide a basis for
determining the significance of background erosion and thus the level of effort warranted in
establishing the background erosion rate as compared to the wave forcing which drives the longshore
spreading processes. For cases in which the background erosion is secondary, the design engineer
can concentrate efforts on the factors which govern the spreading out losses. Two beach nourishment
settings are considered for nourishment with initially rectangular planforms: (1) on a long straight
beach, and (2) on a long barrier island with one end of the nourishment adjacent to an inlet. These
design aids are illustrated with idealized situations and additional examples are presented for five
projects constructed in Florida.
2.0 METHODOLOGY
The methodology is presented for nourishment: (1) on a long straight beach, and (2) on a long barrier
island with one end of the nourishment adjacent to an inlet. In both cases, it is necessary to
approximate the background erosion as uniform along the nourishment project and the adjacent
beaches of interest. The methodology will be illustrated by application to two idealized cases in this
section and to five nourishment projects in a later section of this report.
2.1 General
The theoretical basis for the linear superposition of longshoree spreading effects" and background
erosion is the socalled Pelnard Considere equation which is a result of combining the equation of
conservation of sediment and the linearized equation of sediment transport. The conservation of
sediment as applied assumes that the profile is translated seaward or landward without change of
form in response to an increase or decrease of sediment volume, respectively. The Pelnard Considere
equation is linear, thus justifying the linear superposition which can be expressed as
AV = AV + AVBE
in which AVT is the total volume change component, AV, is the volume change component
contribution due to spreading and AVBE is the volume change component contribution due to
background erosion.
2.2 Methodology Development for Nourishment on a Long Beach
In the absence of background erosion, the evolution of an initially rectangular planform of initially
uniform volume density, Vo, and length, Q, with longshore diffusivity, G, on a long straight shoreline
can be represented as:
V = ( V0 erf +1 eerf f ( 2x 1)
2 4 4G J ()
which represents the effects of spreading and is a solution to the Pelnard Considere equation, and
"erf" is the socalled error function. The proportion of volume remaining within the project area for
the case of no background erosion is given by (Dean and Grant, 1989) as
Ms(t) e GfI) +erf 1) (2)
and is presented as Figure 1. Estimates of the longshore diffusivity, G, around the sandy shoreline
of Florida have been provided in graphical form by Dean and Grant (1989) and are reproduced as
Figure 2 of this report. The contribution due to background erosion, AyBE is
AM(t)BE= et/Y (3)
where e represents the background erosion rate, t is time and Y is the initial uniform beach width,
considering that profile equilibration has occurred. The proportion of volume remaining, M(t),
including the contributions of longshore spreading and background erosion is
M(t)= q e +erf Q et/Y (4)
which is plotted in Figure 3 for 0
axis in Figure 3 determines the spreading losses in the absence of background erosion as can be seen
by reference to Eq. (2). The vertical axis represents the proportion of material placed which has been
removed from the project area through background erosion. For the upper limit of e t/Y = 1.0, the
volume of material removed by background erosion equals the total volume placed!
JZW
LL IL.0
zc,0
PE<
X C.)
CL nJO.O 
O X
C.
1 2 3 4 5
Figure 1: Percentage of Material Remaining in Region Placed vs. the Parameter /Gt/ for
Initially Rectangular Nourishment on a Long Straight Beach. Dean and Grant (1989).
0.14
F 0.10 
0 h.i b G(ft2/s)
.02 0.06 0.10 0.14
0.02
F
0.02 0.06 0.10 0.14
G(ft2/S)
Figure 2: Approximate Estimates of Longshore Diffusivity, G(ft2/s),
Around the Sandy Beach Shoreline of the State of Florida. Dean and
Grant (1989).
NGt / 40
F = e P2/YG
1.0
0.9
0.6
0.7
0.6 ,
0.5
0.1 .
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
GtlX
Figure 3: Isolines of M(t) versus Spreading and Background Erosion Parameters.
Long Straight Beach.
There are three sets of isolines present in Figure 3. One set of isolines is the proportion of volume
remaining in the project area as a result of both the spreading out losses and background erosion.
This set of isolines is labelled "M(t)". It is noted that for zero background erosion (e t/Y = 0), the
values of M(t) are the same as in Figure 1.
The second set of isolines in Figure 3 are the "trajectories" along which a project moves during its
evolution. Because the abcissa and ordinate of Figure 3 both contain time, t, it can be shown that a
project will evolve along a line of constant F. The lines defining these trajectories are labeled as
constant values of "F", where
eQ2
F e (5)
YG
For example, a project with a value of F = 0.5 will move along the "trajectory" defined by that line
in Figure 3 during all stages of the evolution of that project.
The third set of isolines present in Figure 3 defines the proportion of the loss of volume which is
attributable to background erosion. These lines are labeled "BE". It is seen that as the project evolves
(following a line of constant F) toward the right in these plots, the proportion of material lost due
to background erosion increases for the reasons discussed earlier.
2.3 Methodology Development for Nourishment on a Barrier Island With One End of
Nourishment Adjacent to an Inlet
In this case, the initial nourishment is placed on a long barrier island with one end of the nourishment
adjacent to an inlet; Figure 4 presents a definition sketch. Our understanding of the evolution of this
project setting is much less certain than that for nourishment on a long straight beach as discussed
earlier. For purposes here, it will be assumed that the shoreline position at the inlet is maintained at
zero displacement, y(0,t) = 0.0. The solution for spreading losses for the case of an initially
rectangular planform is
Vs(Xt) = V erf + 1) + erf x 1 2 erf x (6)
2 V t Q V I ^Gt V I ,v(4GtJ
Nourishment
YTFZ
Inlet
Figure 4: Plan View of Nourishment Adjacent to an Inlet.
and the proportion remaining due to spreading losses without background erosion effects is
Ms(t) = 2 erf erf f
V 4Gt C V4Gt
+ 4Gt2e
V rtic
2 e 
2 2
Thus, the total result for the proportion remaining is Eq. (7) minus the effect due to background
erosion
t f4Gt 4Gt V 2 2
et
Y
I (
F = e 2/YG
1.0
0.9
0.8
0.0 01 02 0.3 04 05 06 07 0.8 0.9 1.0
0.6
0.5
Figure 5: Isolines of M(t) versus Spreading and Background Erosion
Parameters. Nourishment Adjacent to an Inlet.
which is plotted in Figure 5 in the same format as Figure 3 which is the result for nourishment on
a long straight beach as discussed earlier.
0.42.4 Range of Variables Expected
0.3
0.2
Prior to illustrating the application of Figures 3 and 5 with applications, it is informative to examine
representative values of the two axes in the figures under discussion for typical projects in Florida.
For this purpose, we consider the following ranges of variables:
Project Length, : 1 mile to 5 miles
0.07
Longshore Diffusivity, G: 0.04 to 0.10 ft (0.04 0.5 0.6milesyear to 0.7 0.8 0.9 1.0mi
h.: 12 to 18 feet
Background ErosFigure 5: Isolines of M(t) versus Spreading and Background Erosion
BermParameters. Nourishment Adjacent to an Inlet.11 feet
which is plotted in Figure 5 in the same format as Figure 3 which is the result for nourishment on
Time,a long straight bt: 1 each as discussed earlier.6 years
2.4 Range of Variables Expected
Prior to illustrattial Shoreling the application of Figures 3 and 5 with applications, it is informative to examine200 ft
representative values of the two axes in the figures under discussion for typical projects in Florida.
For this purpose ranges above, the associated ranges in and e t/Y are, approximately:variables:
Project Length, 0: 1 mile to 5 miles
Longshore Diffusivity, G: 0.04 to 0.10 ft2/s (0.045 miles2/year to 0.113 miles2/year)
h: 12 to 18 feet
Background Erosion Rate, e: 0.5 to 5 ft/year
Berm Height, B: 6 to 11 feet
Time, t: 1 years to 6 years
Initial Shoreline Displacement (Equilibrated), Y: 40 ft to 200 ft
For the ranges above, the associated ranges in lGQ and e t/Y are, approximately:
vlG/ : 0.04 to 0.8
e t/Y: 0.0025 to 0.75
The ranges of the two axes in Figures 3 and 5 have been selected to encompass these ranges.
3.0 EXAMPLES ILLUSTRATING APPLICATION OF THE METHODOLOGY
This section presents both hypothetical examples and actual case examples from the State of Florida.
For the latter, reasonably representative values of relevant project characteristics and parameters will
be employed as provided in some of the figures presented earlier.
3.1 Hypothetical Examples
Two hypothetical examples will be presented with the characteristics of and results for as shown in
Table 1.
Only the geomorphic settings for these two projects differ with Case 1 located on a long straight
beach and Case 2 located adjacent to an inlet. The other characteristics have been chosen to be the
same to illustrate the effects of the inlet in reducing the longevity of the project. The calculated value
of the trajectory defined by F is approximately 2.0 (actually, F = 1.99). For Case 1 which is the
project that is constructed on a long straight beach, the proportion remaining after 8 years is 0.52
with the proportion lost due to background erosion being 33%. For comparison, the proportion
remaining for Case 2 after 8 years is 0.37 with background erosion accounting for 25% of the
reduction. Referring to Table 1, it is clear that the percentage reduction due to background erosion
increases with time for both cases. The lines of "BE" can be used to determine the approximate
percentages of reduction due to background erosion. A second approach is to recognize that the
ordinate, e t/Y represents the proportion of the volume placed lost due to background erosion. Thus,
an alternate and more accurate procedure to obtain BE is: BE (%) = (e t/Y)/(1.0M)x 100% since the
term (1.0 M) represents the total reduction in volume. As an example, after 8 years, the proportion
volume remaining is 0.52, thus the total volume reduction is 0.48 of which the background erosion,
e t/Y = 0.16 or 0.16/0.48 = 0.33 (or 33%) of the total reduction (see Table 1).
3.2 Application of the Methodology to Five Projects in Florida
This section presents the application of the methodology to five projects in Florida. The
characteristics of these projects and the associated results of applying the methodology presented
here are summarized in Table 2. An attempt has been made to use as representative characteristics
of these projects as possible; however, the values for the uniform background erosion rates have all
been taken as 2.0 feet/year which is believed to represent an approximate upper limit. More detailed
calculations would require correcting for the nourishment grain size (a sediment transport
coefficient, K value of 0.77 was used here which is appropriate for a grain size of approximately 0.48
mm). The value of the longshore diffusivity has been obtained from Figure 2, the time considered
is the approximate length of time since project construction, the value of h. is taken from Dean and
Grant (1989) from a figure of the same format as Figure 2 of this present report, and the initial
shoreline displacement, Y, is determined from
V
Y TO (9)
(h. +B)Q
Table 1
Characteristics of and Results For Two Hypothetical Examples
Background
Project Erosion Time
G Length, V h. + B Rate, e F= Considered
Case (ft2/s) (Miles) (ft) (ft/year) e 2/GY (Years) VGt/ e t/Y M BE
0 0 0 1.0 NA
L
O
N
G 1 0.100 0.02 0.87 15%
S
H 1 0.08 3.0 20.0 2.0 1.99 2 0.142 0.04 0.79 19%
R
E
L 4 0.200 0.08 0.69 26%
N
E
8 0.283 0.16 0.52 33%
0 0 0 1.00 NA
I
N
L 1 0.100 0.02 0.81 11%
c 4 0.200 0.08 0.58 19%
E
T
A 2 0.08 3.0 20.0 2.0 1.99 2 0.142 0.04 0.71 14%
D
J
A
c 4 0.200 0.08 0.58 19%
N
T 8 0.283 0.16 0.37 25%
Table 2
Characteristics of and Results For Five Florida Case Studies
Time Volume Project Background
Year G (ft'/s) Considered (Millions (h. + B) Length, 9 Erosion Rate, e Y
Project Constructed (Fig. 2) (Years) of yd3) (ft) (miles) (ft/yr) ~Gt/ (ft) et/Y M BE
Martin 1996 0.075 4.0 1.50 22.0 3.75 2 0.155 93.0 0.086 0.74 33%
County
Anna 1993 0.080 7.0 2.21 20.7 4.90* 2 0.162 111.4 0.126 0.69 41%
Maria
Key
Bay 1998 0.095 2.0 8.00 21.0 18.0 2 0.026 108.2 0.037 0.93 53%
County
Midtown 1995 0.060 5.0 0.88 20.9 1.02 2 0.570 211.1 0.047 0.40 8%
Beach**
Perdido 19891990 0.115 8.0 5.36 23.0 4.66 2 0.219 255.7 0.063 0.58 15%
Key ***
* Including a 0.7 mile transition at the south end of the project.
** This project included 11 groins for stabilization. The stabilizing effects of these groins are not accounted for by the methodology
presented here.
*** This project adjacent to Pensacola Pass on east end and thus Figure 5 is to be used whereas Figure 3 is used for all other projects.
in which VTOT represents the total volume placed and, of course, consistent units are used. The first
four projects were constructed on beaches that are reasonably long and thus Figure 3 applies and the
last project (Perdido Key, FL) was constructed adjacent to an entrance (Pensacola Pass) and thus
Figure 5 was applied. It is seen that for long projects, the relative role of background erosion is
greater simply because the spreading losses are quite small. For example, the total predicted losses
from the Bay County project after two years are only 9%; however the assumed background erosion
represents 41% of this amount. For the shorter projects, the background erosion plays a relatively
minor role simply because the spreading losses are relatively rapid. In general for the five projects
examined, the role of background erosion is relatively small.
4.0 SUMMARY AND CONCLUSIONS
Convenient graphical aids have been presented which allow rapid determination of the proportionate
volumetric losses due to background erosion and longshore spreading for projects constructed: (1)
on a long uninterrupted beach, and (2) on a long barrier island with one end of the nourishment
adjacent to an inlet. The background erosion is considered uniform in the alongshore direction. The
theoretical basis for this methodology is the socalled Pelnard Considere equation which is linear and
thus allows superposition of these two effects. This methodology and associated graphical aids
presented in this report allow the designer to concentrate his or her design efforts on the most
significant processes and related variables governing project evolution. Two hypothetical projects
are examined and used to illustrate application of the graphical aids. The methodology is also applied
to five projects constructed in Florida with lengths ranging from 1 to 18 miles and which have been
in place for periods ranging from 2 to 8 years. It is found that the proportional reduction in volume
due to background erosion is greater for the longer projects which have small spreading losses. For
most beach nourishment projects in Florida, the longshore spreading losses will govern over
reasonable evolution times.
5.0 REFERENCES
Dean, R. G. and J. Grant (1989) "Development of Methodology for ThirtyYear Shoreline
Projections in the Vicinity of Beach Nourishment Projects", UFL/COEL89/026, Coastal and
Oceanographic Engineering, University of Florida, Gainesville, FL.
Pelnard Considere, R. (1956) "Essai de Theorie de l'Evolution des Formes de Rivate en Plages de
Sable et de Galets," 4'h Jourees de l'Hydraulique, Les Energies de la Mar, Question Il, Rapport No.
1 (in French).
