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Beach nourishment design : Reduction in beach width due to profile equilibration

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Beach nourishment design : Reduction in beach width due to profile equilibration
Series Title:
Beach nourishment design : Reduction in beach width due to profile equilibration
Creator:
Dean, Robert G.
Place of Publication:
Gainesville, Fla.
Publisher:
Coastal & Oceanographic Engineering Dept. of Civil & Coastal Engineering, University of Florida
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English

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University of Florida
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UFL/COEL-2000/012

BEACH NOURISHMENT DESIGN: REDUCTION IN BEACH WIDTH DUE TO PROFILE EQUILIBRATION
by
Robert G. Dean

September 15, 2000
Prepared for:
Office of Beaches and Coastal Systems Florida Department of Environmental Protection Tallahassee, Florida 32399-3000




UFL/COEL-2000/012

BEACH NOURISHMENT DESIGN:
REDUCTION IN BEACH WIDTH DUE TO PROFILE EQUILIBRATION
September 15, 2000
Prepared for:
Office of Beaches and Coastal Systems
Florida Department of Environmental Protection
Tallahassee, Florida 32399-3000
Prepared by:
Robert G. Dean
Civil and Coastal Engineering Department University of Florida 345 Weil Hall, P. 0. Box 116580 Gainesville, Florida 32611-6580




TABLE OF CONTENTS
LIST OF FIGURES...................................................... iii
LIST OF TABLES........................................................ v
1 INTRODUCTION..................................................1
2 METHODS .......................................................1
2. 1 General .....................................................1
2.2 Cross-Shore Equilibrium......................................... 3
2.3 Planform. Evolution............................................. 8
2.4 Time Variation of Profile Equilibration .............................. 8
2.5 Consideration of Time Varying Shoreline Recession Due to Both Profile
Equilibration and Planform. Evolution .............................. 10
3 EXAMPLES ILLUSTRATING APPLICATION OF THE RESULTS .......... 10
3.1 Example 1 .................................................. 10
3.2 Example 2 .................................................. 12
3.2.1 Initial Placed Beach Width, Ay, ............................. 13
3.2.2 Equilibrium Beach Width, Ayo ............................. 13
3.3 Example 3 .................................................. 14
3.4 Example 4 .................................................. 15
4 SUMMARY ...................................................... 16
REFERENCES ......................................................... 16
APPENDICES
A DEVELOPMENT OF SHORELINE ADVANCEMENT, Ay, FOR.
NOURISHMENT PLACEMENT AT A UNIFORM SLOPE............... A-1
B PLOTS OF Ay/W. (TWO PLOTS) AND AyjAy, (SIX PLOTS) ..............B- 1




LIST OF FIGURES

FIGURE PAGE
I Example of Beach Nourishment with DN = 0.2 mm, DF = 0.18 mm, Volume
Density = 100 yd3/ft, B = 6 ft, h. = 18 ft ...................................... 2
2 Variation of Non-Dimensional Volume Density with Non-Dimensional
Alongshore Distance and Time. No Background Erosion ......................... 3
3 Non-Dimensional Placed Shoreline Displacement, Ayl' (- Ay/W.) Versus NonDimensional Volume Density, V (= V/(BW.)) and Slope Parameter h /(miW.).
h /B = 2 .0 .. . ... . .. . .. . .. ..... . . .. . .. . .. ... .. .. . .... . .. .. . 6
4 Non-Dimensional Placed Shoreline Displacement, Ay,' (= Ayl/W,) Versus NonDimensional Volume Density, V (- V/(BW.)) and Slope Parameter h/(miW ).
h /B = 3 .0 .. . .. . . .. . . .. .. ...... .. . .. .. .. .. .. .. .. ... . . .. .. 6
5 Recommended Distribution of h. and h/B Along the Sandy Shoreline of Florida.
Note h. Distribution is Solid Line (- ) and h/B is Dashed Line ( ---- ) .......... 7
6 Ratio of Equilibrium to Placed Beach Widths, Ay/Ay, for Non-Dimensional
Volume Densities, V', ApIAN. Specific Values for This Plot: hJB = 2.0, hJ(miW.)
= 0 .1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
7 Approximate Estimates of G(ft2/s) Around the Sandy Beach Shoreline of the State
of Florida (From Dean and Grant, 1989) ...................................... 9
A -1 D efinition Sketch ..................................................... A -2
B-1 Non-Dimensional Placed Shoreline Displacement, Ay,' (- Ay1/W.) Versus NonDimensional Volume Density, V' (- V/(BW.)) and Slope Parameter h,/(miW.).
h /B = 2 .0 ............................................................ B -3
B-2 Non-Dimensional Placed Shoreline Displacement, Ay,/ (- Ay,/W.) Versus NonDimensional Volume Density, V' (=- V/(BW,)) and Slope Parameter h /(miW.).
h /B = 3.0 ............................................................ B -3
B-3 Ratio of Equilibrium to Placed Beach Widths for the Following Parameters: h/B
= 2, h.I(miW.) = 0.1. No Volume Change in Profile ........................... B-4
B-4 Ratio of Equilibrium to Placed Beach Widths for the Following Parameters: h.B
= 2, hJ(miW.) = 0.3. No Volume Change in Profile ........................... B-4




B-5 Ratio of Equilibrium to Placed Beach Widths for the Following Parameters: h./B
= 2, hJ(mW.) = 0.5. No Volume Change in Profile ........................... B-5
B-6 Ratio of Equilibrium to Placed Beach Widths for the Following Parameters: h./B
= 3, h.I(miW.) = 0.1. No Volume Change in Profile ........................ B-5
B-7 Ratio of Equilibrium to Placed Beach Widths for the Following Parameters: h.JB
= 3, h./(m1W.) = 0.3. No Volume Change in Profile ........................... B-6
B-8 Ratio of Equilibrium to Placed Beach Widths for the Following Parameters: h./B
= 3, h./(miW .) = 0.5. No Volume Change in Profile ........................... B-6




LIST OF TABLES
TABLE PAGE
I Summary of Recommended A Values (ft"3) for Diameters from 0. 10 to 1.09 mm
(From D ean, 2000) ....................................................... 5
2 Parameters of Plots Provided in Figures B-3 through B-8 in Appendix B ............. 9
3 Characteristics of Examples Illustrating Application of Methodology .............. I I
B-1 Parameters of Plots Provided in Figures B-3 through B-8 in Appendix B .......... 13-2




BEACH NOURISHMENT DESIGN:
REDUCTION IN BEACH WIDTH DUE TO PROFILE EQUILIBRATION
1 INTRODUCTION
Profiles placed as beach nourishment are usually steeper than equilibrium profiles. The evolution of the project includes both profile equilibration and spreading losses, each of which
cotrbute to shoreline recession. Shoreline recession is a significant design and prediction issue since unanticipated shoreline recession may lead to misinterpretation of the project performance. Thus the subject of anticipated shoreline recession should be addressed in the design documents and conveyed to the Project Sponsor(s).
This report considers nourished profiles to be placed at a uniform slope seaward of the berm and to intersect with the native profile. Considering equilibrium beach profile concepts and idealized beach profiles typical of the east and west coasts of Florida, and various nourishment volume densities, the recession of the shoreline due to equilibration of the nourished profile is developed and presented in graphical forms. The methods and design aids presented in this report are considered appropriate for preliminary design and checking of and comparison with results from more detailed methods which may be employed for the final design.
2 METHODS
2.1 General
Figures la and lb depict placed and equilibrated profiles considering conservation of sand within the profile (Figure l a) and a 25% reduction of sand volume due to longshore spreading (Figure 1b), respectively. In addition to the recession associated with profile equilibration, the time required for equilibration is a significant design and performance prediction parameter. Most present design methodologies consider profile equilibration to occur instantaneously under the argument that the cross-shore equilibration times are considerably shorter than those for plan-form evolution.
Idealized planform evolution of an initially rectangular planform. on an open coast is schemnatized in Figure 2 where, considering compatible sand, in which even for the case of obliquely incident waves the initially symmetric planform. remains symmetric and the planform, centroid remains fixed.
The objectives of this report are to present simple approximate procedures for estimating the initial and equilibrium shoreline advancements and shoreline recession versus time for the profile equilibration component and to discuss procedures for including the contribution of both components.




-3 0 -I I I I I ,
-200 0 200 400 600 800 1000 1200 1400 1600 1800 2000 Distance From Original Shoreline (ft)
a) Example Native, Placed and Equilibrated Shorelines for Case of No
Volumetric Change in Profile.

-30
-200

0 200 400 600 800 1000 1200 1400 1600 1800 2000
Distance From Original Shoreline (ft)

b) Example Native, Placed and Equilibrated Shorelines. Case of a 25%
Reduction in Volume. Note the Placed Profile Corresponding to the
25% Volume Reduction Is Not Shown in this Plot.
Example of Beach Nourishment with DN = 0.2 mm, DF = 0.18 mm, Volume Density
= 100 yd3/ft, B = 6 ft, h. = 18 ft.

Figure 1.




1.0
*OgI o t= Gt/i?
>0.8 --
~0.6
0.2 '.5
X..
0250
0.0 1 ___40___0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
X'= x/
Figure 2. Variation of Non-Dimensional Volume Density with NonDimensional Alongshore Distance and Time. No Background Erosion.
2.2 Cross-Shore Equilibration
Applying equilibrium beach profile concepts and considering the native and nourished sands to be each characterized by a single size, the non-dimensional additional dry beach width due to beach nourishment, Ay., has been shown to be a function of three non-dimensional variables (Dean, 1991) as
4. f(BW. B I (1)
where V is the nourishment volume density expressed in volume added per unit length of beach, B is berm height, W., the distance to the depth of closure, h., on the native profile and AF and AN are the profile scale parameters of the nourishment and native sands, respectively. It can be shown that three types of nourished profiles (intersecting, non-intersecting and submerged) can occur. These results are based on equilibrium beach profiles of the following simple form
h =Ay 23 (2)
in which h is the water depth at a distance y from the shoreline and A is the profile scale parameter as mentioned above. The methods leading to Eq. (1) have been described previously in a report (Dean, 2000) to the Office of Beaches and Coastal Systems (OBCS) and will not be repeated here.




Table 1 presents the profile scale parameters, A, for the range of sediment sizes 0.1 mm< D < 1.09 mm.
Referring to Figure la, the initial dry beach width, Ay,, of the placed profile with uniform slope, rn1, can be expressed in non-dimensional form as
Ay, V h. h.
- f )(3) W BW.' B miW.
It is noted that the native profile characteristics are inherent in the term h./(miW.) which can also be
3/2
AN
written as M The nourishment scale factor A F' does not appear in Eq. (3) as the nourished
profile is placed according to geometric rather than equilibrium profile considerations. The development of Eq. (3) is presented in Appendix A. Thus, comparing Eq. (3) with Eq. (1), there is one additional variable (h./miW.) which complicates the presentation in compact graphical form for general conditions. Limiting the conditions to those appropriate for Florida's east and west coasts, the ranges of variables are reduced. For this purpose, the following parameters will be considered
V
V = variable, ranging from 0.02 to 5.0
BW
h
-2 and 3
B
AF (4)
- = variable, ranging from 0.8 to 1.8 AN
h
-0.1, 0.3 and 0.5
miW,
Figures 3 and 4 present for hJB = 2.0 and 3.0, respectively, the variation of AyI/W. versus V
(- )for three values of hd(miW.) ranging from 0.1 to 0.5. With the limited selection .of
BW
variables above, the results are presented on these two plots.
Examining Eqs. (1) and (3), it is seen that the ratio of equilibrated to initial widths, Ayo/Ayj depends on the four non-dimensional parameters below
y f v h. AF h(
Aym BW B'A)




Table I
Summary of Recommended A Values (ftll) for Diameters from 0. 10 to 1.09 mm (From Dean, 2000)

D(mm) 0.00 0.01 [0.02 0.03 j0.04 0.05 0.06 J0.07 J0.08 00
0.1 0.0936 0.0999 0.1061 0.1123 0.1186 0.1248 0.1296 0.1343 0.1391 0.1438
0.2 0.1486 0.1531 0.1575 0.1620 0.1664 0.1709 0.1739 0.1768 0.1798 0.1828
0.3 0.1858 0.1887 0.1917 0.1947 0.1976 0.2006 0.2036 0.2066 0.2095 0.2125
0.4 0.2155 0.2178 0.2202 0.2226 0.2250 0.2274 0.2297 0.2321 0.2345 0.2369
0.5 0.2392 0.2410 0.2428 0.2446 0.2464 0.2482 0.2499 0.2517 0.2535 0.2553
0.6 0.2571 0.2589 0.2606 0.2624 0.2642 0.2660 0.2678 0.2696 0.2713 0.2731
0.7 0.2749 0.2762 0.2776 0.2789 0.2803 0.2816 0.2829 0.2843 0.2856 0.2869
0.8 0.2883 0.2895 0.2907 0.29 19 0.2930 0.2942 0.2954 0.2966 0.2978 0.2990
0.9 0.3002 0.3014 0.3025 0.3037 0.3049 0.3061 0.3073 0.3085 0.3097 0.3109
1.0 0.3121 0.3132 110.3144 10.3156 10.3168 0.3180 0.3192 0.3204 10.3216 10.3228




1.0 0.9
0.8
0.7 0.6 0.5
0.4 0.3
0.2 0.1
/.
0.0 0.0

0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

Non-dimensional Volume, V = V / (B W.)
Figure 3. Non-Dimensional Placed Shoreline Displacement, Ayi/
AyI/W.) Versus Non-Dimensional Volume Density, V,
(- V/(BW.)) and Slope Parameter h /(miW.). h*/B =2.0.

0.0 L* 0.0

0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6
Non-dimensional Volume, V = V / (B W.)

1.8 2.0

Figure 4. Non-Dimensional Placed Shoreline Displacement, Ay
- Ayl/W*) Versus Non-Dimensional Volume Density, V'
(- V/(B W.)) and Slope Parameter h./(miW.). h./B = 3.0.




Approximate values of h. and hJB for the Florida sandy shoreline are presented in Figure 5. Note in applying all results presented here, it is necessary to use consistent units.
-- -- h -B h. (Feet)
16 "" -- ".. .
L 12 16 20 24
JA
h-- ------ 2 3 T MA
cc
CI I J 1
M1
1Z 16 20 24
h. (Feet) 2 3
-" ""h,/B -----Figure 5. Recommended Distribution of h. and hdB Along the Sandy Shoreline of Florida. Note h. Distribution is Solid Line (-) and hJB is Dashed Line ( ---- ).
Figure 6 presents, as an example, the ratio Ayo/AyI for ranges of the non-dimensional parameters, V = V/(BW.) and AF/AN for values of h.IB = 2.0 and h.I(m W.) = 0.1. There is a total of six such plots presented in Appendix B with the characteristics listed in Table 1.
A discussion of Figure 6 and some of the other figures will assist in interpreting the results. First consider the non-dimensional volume, V' = 0.2. For sediment sizes smaller than the native (AIAN 0.88), there will be no additional equilibrium dry beach width, Ay., i.e., the profile is submerged. For very coarse nourishment sand (AlAN > = 1.8), the average slope of the equilibrium profile is approximately equal to that of the placed profile (m. = 0.05) and thus there will be little change in the position of the constructed shoreline during equilibration. Considering next the variation of AyjAy, with non-dimensional volume, V, for sediment which is finer than the native (Al'AN < 1), it is seen that the ratio AyoAy, increases with increasing non-dimensional volumes. The reason is that for small relative sediment sizes, small non-dimensional volumes can result in submerged profiles (Ayo = 0). However, it is interesting from Figure 6 that for the larger relative nourishment sediment sizes, the ratio AyoAy, is less for the larger non-dimensional volumes. The




explanation is that whereas the placed profiles are intersecting (by definition), the equilibrium profiles are non-intersecting and thus have a decreasing Ay0/Ay, with increasing V'.

0.0 0.8

1.0 1.2 1.4 1.6 1.8 2.0

AF/AN
Figure 6. Ratio of Equilibrium to Placed Beach Widths, Ayo/Ayl, for NonDimensional Volume Densities, V', ApIAN. Specific Values for This
Plot: h/B = 2.0, h./(miW.) = 0.1.

2.3 Planform Evolution
For the case of nourishment with compatible sand forming an initially rectangular planform on a straight shoreline, the volumetric density evolution is represented in non-dimensional form as presented in Figure 2. The non-dimensional time parameter e = 16 Gt/p2, in which t is time, P is the project length and G is the so-called "longshore diffusivity". Recommended values of G are presented in Figure 7 for the sandy beach shorelines of Florida. With an estimate of the local volume density, V(x,t), available, this value can be employed in Eq. (3) or Figures 3 and 4 to determine Ay, and AyJAyI in the six plots presented in Appendix B for the characteristics presented in Table 2.
2.4 Time Variation of Profile Equilibration
Although relatively little is known regarding profile equilibration time scales, the following is recommended

Ay(t) = Ayo + (Ayi-Ayo)e -Kt

i.e., the profile approaches equilibrium exponentially. In non-dimensional form




0.02 0.06 0.10 0.14
G(ft2 IS)

G(ft2ls)
0.02 0. 0.1 0.14
I I
lit

o

Figure 7. Approximate Estimates of G(ft2/s) Around the Sandy Beach
Shoreline of the State of Florida (From Dean and Grant, 1989).

Table 2
of Plots Provided in Figures B-3 through B-8

in Appendix B

Parameters

[Case I Reference Figure No. h.IB (hJmW.)
1 Standard 1 6, B-3 2 0.1
2 Standard 1, Except hJmiW. = 0.3 B-4 2 0.3
3 Standard 1, Except h./miW. = 0.5 B-5 2 0.5
4 Standard 2 B-6 3 0.1
5 Standard 2, Except hJmiW. = 0.3 B-7 3 0.3
6 Standard 2, Except hJmiW. = 0.5 B-8 3 0.5




Ay(t) =_Y0 + Ay e -Kt (7)
Ay, Ay, Ay)
in which K' is a time scale. For present purposes, based on analysis of a limited number of nourishment projects, the following range of values is recommended
0.1/yr < K < 0.3/yr (8)
consistent with a profile response of 63% to equilibrium in 10 years and 3.3 years respectively, for K = 0.1/yr and 0.3/yr.
2.5 Consideration of Time Varying Shoreline Recession Due to Both Profile Equilibration
and Planform Evolution
The shoreline position due to both profile equilibration and volumetric spreading is determined by first calculating the volume density remaining, V(x,t), at a particular location from the volumetric evolution model (Figure 2) or a numerical model. With V(x,t) available, nondimensional placed shoreline displacement, Ayi'(=Ay1/W.) is determined from Figures 3 and 4 (also Figures B- 1 and B-2, respectively in Appendix B) and the ratio of equilibrium to placed shoreline displacements determined from Figures B-3 to B-8 in Appendix B. The time varying value of Ay is calculated from Eq. (6) using K values of 0.1/yr and 0.3/yr to bracket the results.
3 EXAMPLES ILLUSTRATING APPLICATION OF THE RESULTS
Four examples are presented below which illustrate application of the methodology. The characteristics of these examples are summarized in Table 3.
3.1 Example 1
This is a hypothetical example, i.e., it does not apply to any particular location. The following non-dimensional quantities are calculated
h. 18 ft
-- -3
B 6ft
DN =0.2 mm, AN= 0.149 ft", Table 1
DF = 0.18 mm, AF= 0.139 ft", Table 1
AF/AN = 0.93




Table 3
Characteristics of Examples Illustrating Application of Methodology
Parameters Specified by Designer Example Location D, Volume Placed Project Objective of Example
No. (mm) Density Slope Length
(yd3/ft) rn1 (miles)
I Hypothetical 0.18 100 1:20 NA For a Native Sediment Size of 0.20 mm, h.= 18
ft, B=6 ft, Calculate the Initial (Ay,) and Equilibrium (Ay.) Shoreline Position
2 Martin County, FL 0.21 120 1:30 2 For Martin County, FL Conditions, Calculate
Ay, and Ay.
3 Martin County, FL 0.21 120 1:30 2 Extend Example 2 to Include Ay at Project
Centerline and at One-half Mile from Centerline After 3 Years, Accounting for Profile Equilibration
4 Martin County, FL 0.21 120 1:30 2 Extend Example 3 to Account for a Background
Erosion Rate of 2 ft/year

W = (h /AN)'.5 = (18/0.149)'" = 1328 ft.

18
(0.05)(1328)

= 0.271

V/ V (100)(27) = 0.339
BW. (6)(1328)
The ratio AylW. is determined from Figure 4 (which applies for h./B = 3.0) as
Ay,
- 0.18
W
yielding a value of the placed shoreline additional width, Ay, = (1328)(0.18)=239 ft and the values of Ayo/Ay, from Figures B-6 and B-7 for As/AN = 0.93 are both approximately 0.25, which results in an equilibrium value, Ayo, without any volume density change of

Ay. = (0.25)(239) = 60 ft The actual values are presented in Figure la.




A second part to this example would be to consider the equilibrium shoreline position after a reduction of 25% of the nourishment volume due to transport to adjacent beaches,
V = 2700 x 0.75 = 2025 ft3/ft V, 2025 0.254
(6)(1328)
Referring to Figure 4 (applicable for h/B = 3.0), the non-dimensional placed width, Ay/W. is approximately 0.14. The associated value Ay, is thus 186 ft. From Figures B-6 and B-7, for AE/AN, the values of Ayo/AyI are the approximate same value: Ayo/AyI = 0.18. Thus the equilibrated additional beach width, Ay., at this time is approximately
Ay. = (0.18)(186) = 33.5 ft
The actual value based on the equations is 34.9 ft as presented in Figure lb.
3.2 Example 2
Example 2 pertains to Martin County, FL. For purposes of this example, the following will be considered:
V = 120 yd3/ft

DN = 0.21 mm, AN 0.153 DF = 0.23 mm, AF 0.162

h = 15.6 ft h/B = 2.6

ft 1/3
ft 13 AF/AN = 1.06

I Figure 5

.'. B = 15.6/2.6 = 6.0 ft G = 0.075 ft2/s, Figure 7 Placed Beach Slope = 1:30 Background Erosion Rate = 2 ft/year Project Length, P = 2 miles




3.2.1 Initial Placed Beach Width, Ay,
Calculate, W.
W =(h./AN)'15 = 1030 ft
Calculate, h.(m/W.) and Non-Dimensional Volume, V/(BW.) h. 15.6
h. -= =1. 0.454
miW. (.0333)(1030) V (120)(27) = 0.523 B W. (15.6/2.6)(1030)
Determine Ay, From Figures 3 and 4
From Figure 3 (h./B = 2.0), Ay/W.= 0.27 From Figure 4, (hJB = 3.0), Ay/W. = 0.22
Interpolating, we find Ay/W. = 0.24
Ay, = 0.24 (1030) = 247 ft
3.2.2 Equilibrium Beach Width, Ayo
Determine Ayo by interpolation from Figure B-4 (hJ/B = 2.0, hJ/(miW.) = 0.3) and Figure B-5 (h/B = 2.0, hJ(miW.) = 0.5) and Figure B-7 (h.IB = 3.0, hJ(miW.) = 0.3) and Figure B-8 (hJB =
3.0 and hJ(miW,) = 0.5).
First, interpolating between Figures B-3 and B-5.
Figure B-4 (hJ/B = 2.0, hJ/(miW.) = 0.3): Ayj/AyI = 0.69 Figure B-5 (hJB = 2.0, hJ(miW.) = 0.5): AyJAyj = 0.81
The interpolated value from the above for hJ(mW.) = 0.45 is AyjAyj = 0.78.




Second, interpolating between Figures B-7 and B-8

Figure B-7 (h./B = 3.0, hJ./(miW.) = 0.3): Ay/AyI = 0.65 Figure B-8 (h./B = 3.0, hJ(miW.) = 0.5): Ay/Ay, = 0.80
The interpolated value for hJ.(miW.) = 0.45 is Ay/Ay, = 0.76. Thus the interpolated value for h/JB = 2.6 and hJ(miW.) = 0.45 is Ay/Ay, = 0.77. Thus Ayo = (0.77)(247) = 190 ft.
3.3 Example 3
The basic conditions for this example are the same as for Example 2; however, now we wish to calculate the actual shoreline displacement after 3 years at the project centerline (x/Q = 0) and at a distance of 0.5 miles (x/Q = 0.25) from the centerline. Using consistent units, the value of nondimensional time t' is

I Gt Gt Q2

0.075 ft
s

365 days 24 hrs 3600 s years x
year day hr

/( 5280 ft
2 miles i ) =0.064
mile

Referring to Figure 2,

V(0,0.64) 0.8

V(0.25,0.064) =

0.72

Thus

V(x '=0, t '=0.06) = 0.8 (120x27) = 2,592 ft 3/ft V(x'=0.25, t'=0.06) = 0.72(120x27) = 2,333 ft3/ft
V'(x/'=0,t'=0.06) = V 2,592 = 0.42 BW. (6)(1030)
V '(x '=0.25, t '=0.06)- = V 2,333 = 0.38 BW (6)(1030)




Repeating the same double interpolation procedure as in Example 2, the initial and equilibrium values of shoreline displacement are
x'=0, f=0.06 Ay, = 195 ft Ayo = 148 ft
X'=0.5, t=0.06
Ay, = 185 ft Ayo = 128 ft
The ranges of anticipated shoreline advancement at t = 3 years are (from Eq. (6)) Ay1(0,3 years) = 148 + (195-148)e -( 1)(3) = 183 ft Ay2(0,3 years) = 148 + (195-148)e -(3)(3) = 167 ft and the subscripts 1 and 2 apply for k1 = 0.1 year' and k2 = 0.3 year-, respectively. Similarly for x' = 0.5 (x = 0.5 miles) Ay, (0.5 miles, 3 years) = 170 ft Ay2 (0.5 miles, 3 years) = 151 ft
3.4 Example 4
This is an extension of Example 3 to include the effect of a background erosion rate of 2 ft/yr. This effect is accounted for by a simple reduction in the shoreline position by an amount equal to the product of the shoreline change rate and time, in this case a total of six feet [(2 ft/yr)(3 years) = 6 ft)]. Thus, the values of Ay, and Ay2 for x = 0 and 0.5 mile at 3 years are
x=0, t=3 years
Ay1 = 177 ft Ay2 = 161 ft
x = 0.5 mile, t = 3 years
Ay, = 164 ft Ay2 = 145 ft




4 SUMMARY

Beach nourishment projects are usually constructed with approximately uniform sloped profiles, the forms of which are relatively unrelated to the equilibrium profiles associated with the nourishment sediment. Subsequent to the nourishment, wave action commences the equilibration process which is accompanied by beach width reduction. Significant parameters in the design of beach nourishment projects are the anticipated amount of beach width reduction and the associated time scales.
This report has presented methodology, based on equilibrium beach profile concepts, for determining the initial beach width, Ay,, and the equilibrium beach width, Ay,. A series of graphs is provided which allows these results to be approximated with a hand calculator. A method is also recommended for calculating the time-varying additional beach width from the time of placement to the equilibrated state. The effects of evolution of an initially rectangular planform and background erosion, both of which reduce equilibrium beach width can be taken into consideration using graphical forms presented for the sandy beaches of Florida.
Application of the methodology is illustrated by examples. The methods and design aids presented here are considered appropriate for preliminary design and checking of and comparison with results from more detailed methods which may be employed for the final designs.
REFERENCES
Dean, R.G., and J. Grant (1989) "Development of Methodology for Thirty-Year Shoreline
Projections in the Vicinity of Beach Nourishment Projects," Report No. UFICOEL -8 9/026, Coastal and Oceanographic Engineering Department, University of Florida, Gainesville, FL.
Dean, R. G. (1991) "Equilibrium Beach Profiles: Characteristics and Applications," Journal of
Coastal Research, Vol. 7, No. 1, Winter, 1991, pp. 53-84.
Dean, R. G. (2000) "Beach Nourishment Design: Consideration of Sediment Characteristics,"
Report No. UFLICOEL-2000/002, Coastal and Oceanographic Engineering Program of the
Department of Civil and Coastal Engineering, University of Florida, Gainesville, FL.




APPENDIX A
DEVELOPMENT OF SHORELINE
ADVANCEMENT, Ay, FOR NOURISHMENT PLACEMENT AT A UNIFORM SLOPE




APPENDIX A
DEVELOPMENT OF SHORELINE ADVANCEMENT, Ay, FOR NOURISHMENT PLACEMENT AT A UNIFORM SLOPE
Introduction
The purpose of this appendix is to present the development of the equations for the initial shoreline displacement due to placement of a volume density, V, at a uniform slope, mi. Equilibrium beach profile concepts are used. The geometry of interest is presented in Figure A-1.
10 j
> 0
z
0
, B
.>.
Co -10
0
a. -20
-3 0 ijI
-200 0 200 400 600 800 1000 1200 1400 1600 Distance From Original Shoreline (ft)
Figure A-1. Definition Sketch.
Development
Referring to Figure A-i, the volume density added is expressed as
V = BAy, + fY ANy2/3dy I(yV-AyI)-mi (A-2)
3 AN5/3 1 BAy + 2(YI-AY)mi (A-2)

A-2




in which B is the berm height, AN, is the profile scale parameter for the native sediment, y, is the distance from the original shoreline to the intersection point of the original and placed profiles and mi is the slope of the placed profile.
There are two unknowns in Eq. (A-1). An auxiliary equation is obtained by equating the intersection depth of the two profiles.
hi = ANYi3 = mi(yI-Ayi) (A-3)
Substituting Eq. (A-3) into (A-2) to eliminate Ay, A NYi 3 3 5/3 1 2 4/3 V = B Y my. + -A y ANYI (A-4)
m. 5 2 m.
which contains only one unknown. Defining the following non-dimensional quantities Yi = YI/w.
V/= V/(BW.) (A-6)
Ayi= Ayr/W
the following can be shown
, / h. /2/3 3 h. /5/3 1 h. h. /4/3
V/=y YI + ---Yi -- Yi (A-6)
m. W 5 B 2 B m.
which can be solved by iteration. Once y,/ is determined, the non-dimensional value of Ay,/ can be determined from the non-dimensional form of Eq. (A-3) as
/ /h. /213
Ay, = y m y (A-7)
It is noted thaty andy are functions of three non-dimensional variables
It is noted thatAy,/ andy,/ are functions of three non-dimensional variables




Yi =f V ,(A-8)
AYI/ =f2( V / h. h(A-9)
The non-dimensional equilibrium shoreline displacement, Ayo/, is also a function of three nondimensional variables
yo=V/' AFh.
Ay,, (A- 10)
ttAN B A -IO)
It is noted that two of the three non-dimensional variables are common for Ay,/ and Ay'/




APPENDIX B
PLOTS OF Ay/W. (TWO PLOTS) AND Ayj/Ay, (SIX PLOTS)




APPENDIX B

PLOTS OF Ay1/W. (TWO PLOTS AND Ay0/Ay (SIX PLOTS)
Introduction
Two types of graphs are presented in this appendix. The first type presents the nondimensional values of the initial beach width, Ay1/W. for three values of hI(miW.). These graphs were presented earlier as Figures 3 and 4 in the main body of the report and are repeated here as Figures B-1 and B-2 for convenience for hJB values of 2.0 and 3.0, respectively. The second type of graph depicts the ratio of equilibrium dry beach Ayo to the initial beach width, Ay,. Each of these graphs includes a range of ratios AiJAN and non-dimensional volumes, V/(BW.). A total of six plots is presented which include combinations of two values of hJB (= 2.0 and 3.0) and three values of h.I(miW.) (= 0.1, 0.3 and 0.5). The characteristics of these six plots are presented in Table 2 in the main body of this report which is repeated in this appendix as Table B- I for convenience and the six plots are presented as Figures B-3 through B-8.

Table B-1
Parameters of Plots Provided in Figures B-3

through B-8 in Appendix B

B-2

Case [ Reference Figure No. Jh,/B (h/miW.)
1 Standard 1 6, B-3 2 0.1
2 Standard 1, Except hdmiW, = 0.3 B-4 2 0.3
3 Standard 1, Except h./miW. = 0.5 B-5 2 0.5
4 Standard 2 B-6 3 0.1
5 Standard 2, Except h/miW, = 0.3 B-7 3 0.3
6 Standard 2, Except h./miW, = 0.5 B-8 3 0.5




1.0
0.9 0.8 0.7 0.6 0.5
0.4
0.3
0.2 0.1
0.0
0.0

Figure B-1.

0.0
0.0

Figure B-2.

0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
Non-dimensional Volume, V = V / (B W.)
Non-Dimensional Placed Shoreline Displacement, Ay,' ( Ay,/W.) Versus Non-Dimensional Volume Density, V' (- V/(B W.)) and Slope Parameter h. /(mi W,). h /B =2.0.

0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6
Non-dimensional Volume, V = V / (B W.)

1.8 2.0

Non-Dimensional Placed Shoreline Displacement, Ay,/ ( Ayj/W*) Versus Non-Dimensional Volume Density, V' (- V/(BW,)) and Slope Parameter h /(miW.). h,/B= 3.0.

B-3




1.4
1.2
1.0
0.8 0.6
0.4 0.2
0.0
0.8

1.0 1.2 1.4 1.6 1.8 2.0

AF/AN
Figure B-3. Ratio of Equilibrium to Placed Beach Widths for the Following
Parameters: h./B = 2, h.J(mW.) = 0.1. No Volume Change in
Profile.

0.0 0.8

1.0 1.2 1.4 1.6 1.8

AF/AN

Figure B-4.

Ratio of Equilibrium to Placed Beach Widths for the Following Parameters: h./B = 2, hJ(miW.) = 0.3. No Volume Change in Profile.




2.0
1.8 1.6
1.4 1.2 1.0
0.8 0.6 0.4
0.2 0.0
0.8

Ratio of Equilibrium to Placed Beach Widths for the Following Parameters: hJB = 2, hd(miW.) = 0.5. No Volume Change in Profile.

1.0 1.2 1.4 1.6 1.8 2.0
AF/AN
Ratio of Equilibrium to Placed Beach Widths for the Following Parameters: hJB = 3, hJ(miW.) = 0.1. No Volume Change in Profile.

1.0 1.2 1.4 1.6 1.8
AF/AN

Figure B-5.

0.0
0.8

Figure B-6.




1.4
1.2 1.0
0.8 0.6
0.4 0.2 0.0
0.8

Ratio of Equilibrium to Placed Beach Widths for the Following Parameters: h./B = 3, h./(m,W.) = 0.3. No Volume Change in Profile.

1.0 1.2 1.4 1.6 1.8 2.0

AF/AN

Figure B-8. Ratio of Equilibrium to Placed Beach Widths for the Following
Parameters: hJB = 3, hJ(miW.) = 0.5. No Volume Change in
Profile.

1.0 1.2 1.4 1.6 1.8
AF/AN

Figure B-7.

2.0
1.8 1.6
1.4 1.2
1.0
0.8
0.6
0.4 0.2
0.0 0.8