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Realistic economic benefits from beach nourishment

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Title:
Realistic economic benefits from beach nourishment
Series Title:
UFLCOEL
Creator:
Dean, Robert G ( Robert George ), 1930-
University of Florida -- Coastal and Oceanographic Engineering Laboratory
Place of Publication:
Gainesville Fla
Publisher:
Coastal & Oceanographic Engineering Dept., University of Florida
Publication Date:
Language:
English
Physical Description:
29 p. : ill. ; 28 cm.

Subjects

Subjects / Keywords:
Shore protection -- Economic aspects ( lcsh )
Beaches -- Protection -- Economic aspects ( lcsh )
Coastal engineering ( lcsh )
Coastal and Oceanographic Engineering thesis M.S
Coastal and Oceanographic Engineering -- Dissertations, Academic -- UF
Genre:
bibliography ( marcgt )
non-fiction ( marcgt )

Notes

Bibliography:
Bibliography: p. 29.
General Note:
Cover title.
Funding:
This publication is being made available as part of the report series written by the faculty, staff, and students of the Coastal and Oceanographic Program of the Department of Civil and Coastal Engineering.
Statement of Responsibility:
by Robert G. Dean.

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REPORT DOCUMENTATION PAGE
1. Report No. 2. 3. Recipient's Accession No.


4. Title sad Subtitle 5. Report Date
REALISTIC ECONOMIC BENEFITS FROM September 1988
BEACH NOURISHMENT 6.

7. Author(s) 8. Performing Organization Report No.
Robert G. Dean UFL/COEL-88/009
9. Performing organization Name and Address 10. Project/Task/Work Unit No.
Coastal and Oceanographic Engineering Dept.
University of Florida
336 Weil Hall 11. Contract or Grant No.
Gainesville, FL 32611 13. Type of Report 12. Sponsoring Organization Name and Address
Miscellaneous


14.
15. Supplementary Notes



16. Abstract

A method is presented and illustrated with examples to establish appropriate storm damage reduction and recreational benefits from beach nourishment projects. Unlike previous methods, benefits to project adjacent areas are recognized due to sand transport out of the project area and deposition on adjacent beaches. Assuming homogeniety along the shoreline, the character of storm damage reduction and recreational benefit relationships are such that sand transported from
a project area and deposited on adjacent beaches always results in an increase rather than a reduction in benefits. A central element in calculating storm damage reduction benefits is the establishment of a proportional damage curve for upland structures as a function of beach
width and storm return period. To illustrate the method, limiting cases are presented in which (A) all sediment remains within the area placed, and (B) all sediment spreads out immediately over a long segment of shoreline. Using Monte Carlo simulation to represent the random character of the storms, the method is applied to 15 realistic cases with varying project lengths, representative wave heights, added beach widths and interest rates. The present worth storm damage

continued

17. Originator's Key Words 18. Availability Statement
Beach nourishment
Damage reduction
Nourishment benefits
Recreational benefits
Storm damage
19. U. S. Security Classif. of the Report 20. U. S. Security Classif. of This Page 21. No. of Pages 22. Price
Unclassified Unclassified 1 29











reduction and recreational benefits are calculated to demonstrate the effects of the various parameters. It is found that for short project lengths and relatively large wave heights, the benefits from project adjacent areas exceed those in the project area where the sand is placed. Although no littoral control structures, such as jetties are
included in the present application, the method could be extended readily to include their effects.









UFL/COEL-88/009


REALISTIC ECONOMIC BENEFITS FROM BEACH NOURISHMENT by


Robert G. Dean










TABLE OF CONTENTS

PAGE

LIST OF TABLES............................................... 3

LIST OF FIGURES.............................................. 4

ABSTRACT............................................................. ... 6

INTRODUCTION............................ .. ........... 7

CONCEPTS.. ....................................... 7

METHODOLOGY...... .......oo......... .... ........ o.... ... ... 11

Shoreline Evolution Model...................... oo-o-o.. 15

Storm Damage Reduction Benefits......................... 15

Idealized Cases...... ... .... .... .. ..... ......... 20

Case (A).,... ...... ......... ...... ............ ..... 22

Case (B)... .............. .... o....... ....... ...... ... 22

Case (A) ............. ....o ... ..o -....... ...... ...........o 22

Case (B)..... . . . . .. . . . ..... 22

RESULTS.. . . . .. . . . .. . . ..... 24

SUMMARY AND CONCLUSIONS............ ............. 28

ACKNOWLEDGEMENTS... .. .............. ... .... ... ....... .o. 29

REFERENCESo............ ...... ....... ....... -............. 29


2










LIST OF TABLES


TALE


PAGE


I. PRESENT WORTH DAMAGE FUNCTION, PWDF(w,>) VERSUS
BEACH WIDTH, w, FOR ALL FUTURE DAMAGE..................

II. PRESENT WORTH STORM DAMAGE AND RECREATIONAL BENEFITS
FOR VARIOUS WAVE AND PROJECT CONDITIONS................


20 25


3









LIST OF FIGURES


FIGURE PAGE

1. Structural Damages Due to Hurricane Eloise (1975)
and Example of Reduced Damages by Beach
Nourishment Advancing the Shoreline Seaward by
Fifty Feet .......................................... 9

2. Damage Reduction Per Structure Resulting from a
One Foot Wide Additional Beach, as a Function of
Structure Location Relative to Control Line. Based
on Hurricane Eloise Data....................0-0..... 10

3. Hypothetical Usage and Recreational Benefits of
Sandy Beaches ....................................... 12

4. Annual Recreational Benefits Per Additional Foot of
Beach Width as a Function of Initial Beach Width,
Per Foot of Beach Length. Developed from
Figure 3b........................................... 13

5. Schematic of Erosion of Nourished Area and Deposition
in Project Adjacent Areas........................... 14

6. Example Solution of Evolution of Initially
Rectangular Beach Planform. Pelnard Considere
Method. Wave Height, Hb = 2.0 ft, Initial
Nourished Beach Width = 100 ft, Fill Length,
X = 4 miles, t = time............................... 16

7. Hypothetical Proportional Storm Damage, D, as a
Function of Storm Return Period, TR, and Beach
Width, w....... ............... ............. ..oo....... 17

8. Flow Chart of Methodology........................... 18

9. Expected Damage D(w) Due to a Single Storm as a
Function of Beach Width, w.......................... 21

10. Interpretation of Damage Reduction Benefits if
Sand Remains Where Placed (Case A) and if Sand
Spreads Out Immediately (Case B) ................... 21

11. Ratio of Storm Damage Benefits, RSD, vs Initial
Beach Width, wo, and Additional Beach Width, Aw.
RSD is Ratio of Storm Damage Benefits for Sand Which
Spreads Out Immediately to Those for Which Sand
Remains Where Placed...... . ...... .. ... ....o.. 23


4










PAGE


12. Ratio of Recreational Benefits, RRB, vs Initial
Beach Width, wo, and Additional Beach Width, Aw.
RRB is Ratio of Recreational Benefits for Sand
Which Spreads Out Immediately to Those for Which
Sand Remains Where Placed ........................... 23

13. Present Worth Storm Damage Reduction and Recreational
Benefits. Hb = 2.0 ft, z = 2.0 miles, wo = 0.0,
Aw = 100.0 ft., Interest Rate = 8%, Run No. 5....... 26

14. Present Worth Storm Damage Reduction and Recreational
Benefits. Hb = 1.0 ft, = 4.0 Miles, wo = 0.0,
Aw = 100.0 ft, Interest Rate = 8%, Run No. 7........ 26


5









REALISTIC ECONOMIC BENEFITS FROM BEACH NOURISHMENT by
Robert G. Dean


ABSTRACT


A method is presented and illustrated with examples to establish appropriate storm damage reduction and recreational benefits from beach nourishment projects. Unlike previous methods, benefits to project adjacent areas are recognized due to sand transport out of the project area and deposition on adjacent beaches. Assuming homogeniety along the shoreline, the character
of storm damage reduction and recreational benefit relationships are such that sand transported from a project area and deposited on adjacent beaches always results in an increase rather than a reduction in benefits. A central element in calculating storm damage reduction benefits is the establishment of a proportional damage curve for upland structures as a function of beach width and storm return period. To illustrate the method, limiting cases are presented in which (A) all sediment remains within the area placed, and (B) all sediment spreads out immediately over a long segment of shoreline. Using Monte Carlo simulation to represent the random character of the storms, the method is
applied to 15 realistic cases with varying project lengths, representative wave heights, added beach widths and interest rates. The present worth storm damage reduction and recreational benefits are calculated to demonstrate the effects of the various parameters. It is found that for short project lengths and relatively large wave heights, the benefits from project adjacent areas exceed those in the project area where the sand is placed. Although no littoral control structures, such as jetties
are included in the present application, the method could be extended readily to include their effects.


6









INTRODUCTION

Policies and methodologies should evolve continuously to
remain consistent with modern understanding of coastal processes and the true equities of those residing along the shoreline. Several changes have occurred in the last few decades that argue for an examination and modifications of present economic analysis
procedures relating to beach nourishment: (1) It is now clear that on a long, uninterrupted shoreline, good quality sand placed in a beach nourishment project will eventually be transported out of the region placed, but will remain within the active nearshore system, (2) Sand transported from a project area and deposited on project adjacent areas provides not only continuing damage reduction and recreational benefits, but provides enhanced
benefits, and (3) With increasing concern over the use of "hard structures" as a means of shoreline control, beach nourishment will play an increasing future role.

This paper considers the economic consequences of sand eroded from a beach nourishment project area and deposited on project adjacent areas. Realistic damage reduction relationships
and recreational benefits for a widened beach are utilized to demonstrate that this evolution process actually results in a net increase in project benefits. Benefits from simple limiting cases
are examined in which (1) the sand remains in the area placed, and (2) the sand spreads out immediately. A direct procedure is
presented to account for total present worth project benefits. The procedure utilizes Monte Carlo simulation to faithfully represent the probability of storm occurrences.

Although the methodology presented here is not applicable to
shorelines which include features which would cause longshore sediment transport interruptions, the concepts could be extended readily for such cases.


CONCEPTS

There are two simple concepts which are critical to the methodology presented here:


7









(1) Good quality sand placed in a beach nourishment project will
be eroded from the area placed but will remain indefinitely
in the active nearshore region, and


(2) The greatest storm damage and recreational benefits are
generally realized for the initially narrower beaches.


The first concept will be considered as valid without much discussion. Although "good quality sand" is a matter of degree,
here it refers to sand that is greater than 0.14 mm or so in diameter and that is coarser than or as coarse as the material originally present on the beach. For those nourishment materials
in which the above is not the case, this paper refers to that sand fraction which is compatible. Monitoring results from a number of beach nourishment projects have demonstrated the first
concept, for example at Port Canaveral, FL (Dean, 1988) and Captiva Island, FL (Tackney and Associates, 1983).

The second concept is illustrated by Figure la which
represents a survey (by Shows, 1978) of the structural damage caused by Hurricane Eloise (1975) in Bay County, FL as a function of proximity of the structures relative to a jurisdictional control line which is generally parallel to the shoreline. Of particular significance in Figure la is the steeply sloped portion of the damage curve near its seaward end and the relatively mild slope near its landward end. It is instructive to consider
the effect of a beach nourishment project which displaces the beach seaward by a certain amount such as 50 ft as shown in Figure lb. It is seen that due to the slope characteristics discussed above, the greatest damage reductions occur for those structures which initially have very little beach in front of them. Figure 2 presents the damage reduction per structure associated with an additional one foot of beach width. For the narrower initial beach widths, the reduction is approximately
$3,000 per structure whereas for greater initial beach widths, the damage reduction per structure is less than $500. In summary, the damage reduction benefits are greater for beaches which are initially much narrower.
8









200
00

0 160


< 120': 80- CL



> I m
1- 40
C,,

-150 -100 -50 0 50 100 150 Seaward 0 Landward DISTANCE FROM CONTROL LINE (ft)
a) Damage to Structures in Relation to their Location with
Control Line (Resulting from Study of 540 Structures in Bay
County after Hurricane Eloise, by Shows, 1978).


0 S 200
I-0
0
o 0 160


LU120w = Reduced Damages

o 80 \
cc Damage Curve W MShifted by 50 ft
> -40 0-
150 100 50 0 50 100 150

Seaward IFLandward
DISTANCE FROM CONTROL LINE (ft)

b) Damage Reduction Due to Beach Nourishment Advancing the
Profile Fifty Feet Seaward.

Figure 1. Structural Damages Due to Hurricane Eloise (1975) and Example of Reduced Damages by Beach Nourishment Advancing the Shoreline Seaward by Fifty Feet.


9

















4


X
a-s

z8 3
0
F

0W 2




I


01
150 100 50 0 50 100 150

Seaward I Landward DISTANCE FROM CONTROL LINE (ft)




Figure 2. Damage Reduction Per Structure Resulting from a One Foot Wide Additional Beach, as a Function of Structure Location Relative to Control Line. Based on Hurricane Eloise Data.


10









The same concepts demonstrated above for damage reduction benefits apply for recreational benefits. Figure 3 presents the hypothetical usage and associated recreational benefits for beaches of varying widths. The number of people using the beach
will increase with beach width; however, the rate of increase decreases for the greater widths. The results in Figure 3b are based on a visitation value of $6.00 per visitor per day and a plan area visitation requirement of 200 square feet. The annual recreational benefits associated with an additional foot of beach width versus initial beach width, based on Figure 3, are presented in Figure 4. As before, it is seen that the greatest benefits occur for the initially narrower beaches.

Referring to Figure 5, the significance of greater benefits for initially narrower beaches is that as a beach nourishment project evolves with the beach fronting the project area narrowing and the project adjacent beaches widening, benefits are lost in the initially wider project area. This loss of benefits is small compared to the gain of relatively large benefits in the initially narrow project adjacent areas. Assuming that the value of the upland structures protected by the project and the initial
beach widths in project adjacent areas are uniform along the beach, there is always a net gain in storm reduction benefits as a result of project evolution. Similarly with respect to recreational benefits, assuming that the need for and access to recreational beaches are uniform, etc., the net effect of project evolution is a gain in recreational benefits.


METHODOLOGY

The methodology will be described and illustrated for idealized cases of no project evolution and rapid project evolution and general cases of benefits due to project evolution over realistic time frames.


11

















Cl,


300
20 Weekend 200 -Visitation w4
m 100

Z 0o 50 100 150 BEACH WIDTH, w(ft) a) Number of Days Per Year that Full Beach Width Is Used




Z 500400300
-- 200100
z 0
0 50 100 150 BEACH WIDTH, w(ft) b) Annual Recreational Benefits vs Beach Width Per
Foot of Beach Length

Figure 3. Hypothetical Usage and Recreational Benefits of Sandy Beaches.


12





















cn C',
U
z


z 5
0
I
cc


-j

z Z 0


0


50


100


150


INITIAL BEACH WIDTH (ft)





Figure 4. Annual Recreational Benefits Per Additional Foot of Beach Width as a Function of Initial Beach Width, Per Foot of Beach Length. Developed from Figure 3b.


13


-




















Benefits Gained Benefits Lost Benefits Gained


a) Initial Nourished Planform


b) Eroded Nourished Planform
with Material Deposited on
Adjacent Beaches


Figure 5. Schematic of Erosion of Nourished Area and Deposition in Project Adjacent Areas.


14


/l~.
/
/
// I ~ /1


-:I









Shoreline Evolution Model


The shoreline evolution model adopted here will be that due
to Pelnard-Consider for an initially rectangular planform as presented in Figure 6. The factor G is the so-called "longshore diffusivity" and for small angles of wave incidence is K H 5/2
8(s -1) (1 -p) (h*+ B)

in which K is the sediment transport factor usually taken as 0.77, Hb is the representative breaking wave height, g is gravity, K is the spilling breaker ratio (on the order of 0.8), s is the ratio of sediment specific gravity to that of the water in
which transport is occurring, p is the in situ porosity and (h*+ B) is the vertical extent of beach profile response.


Storm Damage Reduction Benefits

Development of storm damage reduction benefits commences with the establishment of the relationship of a proportional storm damage factor, D, as a function of beach width fronting the structure, w, and storm return period, TR. Figure 7 presents one
example of such a relationship which has been used in the state of Florida Beach Management Plan. Development of this
relationship is by no means trivial and should be based on an analysis of the expected damage to a range of representative structures as well as calibration with available storm results if such data are available. The proportional storm damage factor, D, depends on the foundation and structural and elevation characteristics of the buildings as well as the beach morphology, and presence and integrity of coastal protection structures, etc.

With the availability of D(w,TR) it is possible to predict the present worth damage reduction benefits PWDRB(N) during N years by the general procedure described in Figure 8 and the following equation


15














DISTANCE FROM ORIGINAL
SHORELINE, y (ft)


-


4


17


--


2


100



60

40

20


0


2


-Original Nourished Beach Planform
-- Planform After 3 Months
-10 Months
---7 Years
30 Years
130 Years Pre-Nourished
- Shoreline


4


6


8


ALONGSHORE DISTANCE, x (miles)








Figure 6. Example Solution of Evolution of Initially Rectangular Beach Planform. Pelnard Consider Method. Wave Height, Hb = 2.0 ft, Initial Nourished Beach Width = 100 ft, Fill Length, k = 4 miles, t = time.


I-J


6













1.0


0

z


0


0.81


0.6


0.4


0.21


0.0


1


10


100


1000


RETURN PERIOD, TR (Years)




Figure 7. Hypothetical Proportional Storm Damage, D, as a Function of Storm Return Period, TR, and Beach Width, w.


H


-4






FLOW CHART OF


Damage Reduction Benefits
(a) In Project Area
(b) In Project Adjacent Areas
IniialNouishd BachPlanform Evolution/
Planform (Pelnard-Considered)
(Rectangular) Recreational Benefits
(1) (2) (a) In Project Area
(b) In Project Adjacent Areas
(3)
-- -- -----------------------------------------------------------


Damage Reduction Benefits


x
y


Y



(1)


x



(2) y


(2)


y =L Ierf[A(1+ )]+ erf [A( 2x

A-


(a) Random Storm Magnitudes in Accordance
with Actual Distribution
(b) Damage Reduction vs. Storm Retum Period
and Beach Width
(c) Present Worth Damage Reduction Benefits


Recreational Benefits

(a) Recreational Benefits vs. Beach Width


(b)


Present Worth Recreational Benefits


Figure 8. Flow Chart of Methodology.


H~.


METHODOLOGY









PWDRB(N) =


N I
1 V(x,n)[D(w(x,n)TR(n))-D(w ,TR(n))]dx
n=1 (1+I) Project
Area
(1)
N I
+ I V(x,n)[D(w(x,n),TR(n))-D(wo,TR(n))]dx
n=1 (1+I) Project
Adjacent
Areas
in which I is the interest rate and V(x,n) represents the structure value at a location, x, at a time n years into the future. The two integrals differ only in their respective intervals of integration and are written separately here to illustrate the contributions from the two areas.

Eq. (1) accomplishes the objective of providing methodology for quantifying storm damage reduction. However, it is
instructive to develop concepts further. Referring to Figure 7 which presents the proportional storm damage factor, D, the
expected damage by a single storm D(w) as a function of beach width, w, is


D(w) = f 0D(w,TR)p(D)dD (2)


in which p is the probability density function and is related to the cumulative probability distribution P by

p(D) dP (3)

and noting that

TR = -~(4)
TR

Eq. (2) simplifies to

I dP 1
D(w) = f D(w,TR) dD = f D(w,TR)dP (5)
0 0


19









Figure 9 presents D as a function of beach width as developed from Eq. (5). It is noted that this distribution is qualitatively similar to damages experienced in Hurricane Eloise (Figure 1) which was approximately a 70 year storm.

Assuming that the value of the upland structures remain constant with time and that damaged structures are rebuilt to the same standards (both considerable assumptions), the present worth
damage factor, PWDF(w) as a function of beach width for N years into the future is

N
PWDF(w,N) = D(w)= 1- N]D(w) (6)
n=1 (1+I)n (1+1)

and again, I is the interest. The bracketed factor in Eq. (6) approaches unity with large N. Table I presents values of PWDF(w,o) for several beach widths interest rates. It is noted that the present worth damage factor can range as high as 1.31 for the case of zero beach width and an interest rate of 8%.


TABLE I

PRESENT WORTH DAMAGE FUNCTION, PWDF(w,o)
VERSUS BEACH WIDTH, w, FOR ALL FUTURE DAMAGE

Present Worth Damage Function, PWDF(w,-) Interest For Beach Width, w Rate O ft 50 ft 100 ft 150 ft

6% 1.75 0.67 0.47 0.35 8% 1.31 0.50 0.35 0.26 12% 0.88 0.33 0.23 0.18


Idealized Cases

In contrasting project benefits realized within the project
area to those outside the project area, it is instructive to consider two simple cases:


20














0.10

LUO


( 0.05
z.




X
w 0
0 50 100 150 BEACH WIDTH, w(ft)


Figure 9. Expected Damage D(w) Due to a Single Storm as a Function of Beach Width, w.


0.1

Damage Reduction if F- Sand Remains Where Placed (Case A)
L -Damage Reduction if Sand Spreads Out immediately z (Case B)
Lial
w aw 1
r4-AW Additional Damage Reduction
0 I Due to Sand Spreading t I Out Immedlately W 0 w0 50 w +Aw 100 150

BEACH WIDTH, w(ft)


Figure 10. Interpretation of Damage Reduction Benefits if Sand Remains Where Placed (Case A) and if Sand Spreads Out Immediately (Case B).


21









Case (A). All sediment remains within the area placed, and


Case (B). The sediment placed spreads out immediately over a
long segment of shoreline.


Case (A).

The expected storm damage reduction benefits due to a single storm are


(SDRB)A = w + Aw) D(w )]z (7)


Case (B).

Denoting the (long) distance over which the sediment has been distributed as and the associated additional width as Aw', we have


(SDRB)B w Aw'2' (8)
0

and since sediment is conserved Aw = Aw'k',


(SDRB)B %)w Awk (9)
0

The ratio, RSD, of storm damage reduction benefits for the case of sand spreading out immediately to the case in which sand remains where placed is


(D)~ Aw
RSD = 0 (10)
[D(w0 + Aw) D(w0)]

It is noted that the ratio RSD is always greater than unity. As shown in Figure 10, the interpretation is simple with the numerator representing the tangent of the damage curve at wo and the denominator the secant slope between wo and Wo + Aw. Due to the character of the curve, the ratio will always exceed unity. Figure 11 presents the ratio RSD vs wo for several values of Aw.
22













0


- w LL W 0 M


0 W
c


5.0

4.0 3.0

2.0 1.0


0


0


50


100


150


INITIAL BEACH WIDTH, wo (ft)

Figure 11. Ratio of Storm Damage Benefits, RSD, vs Initial Beach width, wi, and Additional Beach Width, Aw. RSD is Ratio of Storm Damage Benefits for Sand Which Spreads Out Immediately to Those for Which Sand Remains Where Placed.


z
0 I



LL
z
LLW 0rD
0


3.


Ar


~1


2.0 1.0


OL
0


50


100


150


INITIAL BEACH WIDTH, wo (ft)
Figure 12. Ratio of Recreational Benefits, RRB, vs Initial Beach Width, 70, and Additional Beach Width, Aw. RRB is Ratio of Recreational Benefits for Sand Which Spreads Out Immediately to Those for Which Sand Remains Where Placed.


23


o--Aw=150 ft


Aw=100 ft




-


*---Aw=150 ft




Aw=100 ft


\Aw-=50 ft

-----------------------------


.









The same general discussion presented above applies to recreational benefits relationships. The ratio of benefits RRB for a project that spreads out immediately to one that remains in place is


(aw w
RRB (1
[R(w 0 + Aw) R(w )]


and this ratio will always exceed unity by the same argument as for the damage reduction benefits. For the recreational benefits
shown in Figure 3, values of the ratio, RRB, are presented in Figure 12.


RESULTS

Prior to presenting results for the general case, in which the beach planform evolves with time, it is worthwhile to
consider the variables which will tend to favor Case A (sand remains in place) or Case B (sand spreads out immediately). Case A conditions would tend to dominate for:


Large Beach Fill Lengths, z
Low Wave Height, Hb
Small Transport Coefficient, K
Small Additional Beach Widths, Aw
High Interest Rates, I


and vice versa for Case B.

The methodology described in the previous section was
incorporated into a computer program which was "exercised" for the variable values shown in Table II. Results will be presented in two different forms. A schematic of the methodology has been presented as Figure 8.

Figures 13 and 14 present variations of storm damage and recreational benefits with time for Runs 5 and 7, respectively. The relatively large wave height and short beach fill associated


24










TABLE II. PRESENT WORTH STORM DAMAGE AND RECREATIONAL BENEFITS FOR VARIOUS
CONDITIONS


WAVE AND PROJECT


Characteristics Storm Damage Reduction* Recreational Benefits**

In In
Initial Added In Project In Project Project Wave Beach Beach Interest Project Adjacent Total Project Adjacent Total Run Length, Height Width Width Rate Area Area Area Area
X, (miles) Hb (ft) wo (ft) Aw (ft) I

1 1.0 1.0 0 100 0.08 0.368 1.037 1.405 4.4 12.7 17.1 2 1.0 2.0 0 100 0.08 0.176 1.434 1.610 2.1 17.8 19.9 3 1.0 4.0 0 100 0.08 0.072 1.561 1.633 0.9 19.7 20.6 4 2.0 1.0 0 100 0.08 0.592 0.532 1.124 15.4 13.0 28.4 5 2.0 2.0 0 100 0.08 0.323 1.136 1.459 7.6 27.9 35.5 6 2.0 4.0 0 100 0.08 0.148 1.470 1.618 3.6 36.7 40.3 7 4.0 1.0 0 100 0.08 1.009 0.084 1.093 55.9 4.2 60.1 8 4.0 2.0 0 100 0.08 0.525 0.672 1.197 26.6 32.7 59.3 9 4.0 4.0 0 100 0.08 0.280 1.234 1.514 13.0 60.7 73.7 10 2.0 2.0 0 150 0.08 0.388 .679 2.067 10.1 38.5 48.6 11 2.0 2.0 50 100 0.08 0.066 0.178 0.244 4.9 14.5 19.4 12 2.0 2.0 100 150 0.08 0.057 0.144 0.201 7.4 21.7 29.1 13 2.0 2.0 0 100 0.12 0.240 0.697 0.937 5.7 17.0 22.7 14 2.0 2.0 0 100 0.04 0.506 2.413 2.919 11.7 60.0 71.7 15 16.0 2.0 0 100 0.08 1.733 0.002 1.735 392.4 0.5 392.9


*Relative to immediately adjacent upland
**Expressed in millions of dollars


property values within project area













1.5 SU I= Z

Cnz 1.0 Jo



< 0.5 Z LW <0

00


20


30


40


50


60


YEARS INTO FUTURE a) Proportional Present Worth Storm Damage Reduction Benefit
Components vs. Years Into Future.


0


10


20


30


40


50


60


YEARS INTO FUTURE


b) Present Worth Recreational Benefits vs. Years Into Future.


Figure 13. Present Worth Storm Damage Reduction and Recreational Benefits. Hb = 2.0 ft, R = 2.0 miles, wo = 0.0, Aw = 100.0 ft., Interest Rate = 8%, Run No. 5.


26


10


I.


(Total

Adjacent Areas





~ /Project Area


- I I


V


40 30


20 10


-J

0 Wza

0

0)



U
Z Z


0-


Total

Adjacent Areas





Project Area
-














MI

owi

-Jo




0


40


YEARS INTO FUTURE


a) Proportional
Components


Present Worth Storm Damage Reduction Benefit vs. Years Into Future.


0


10


20 30


40


50


60


YEARS INTO FUTURE b) Present Worth Recreational Benefits vs. Years into Future.


Figure 14. Present Benefits. Hb = 1.0 Interest Rate = 8%,


Worth Storm Damage Reduction and Recreational ft, Z = 4.0 Miles, wo = 0.0, Aw = 100.0 ft, Run No. 7.


27


1.01


20 30


0.5



0


Total

---- .j-
Project Area




Adjacent Areas ( .L-4- -- -r -


0


10


50


60


-J
.4



00 a )




ccm


60 40 20


Total

Project Area


/

Adjacent Areas I 4 ---t----


0


L.


11









with Figure 13 favor Case B conditions and it is seen that the dominant benefits occur within the adjacent project areas. It is
also of interest to note that the benefits in the project adjacent areas lag those in the project area due to the time required for sediment transport to these adjacent areas. By contrast the longer project length and smaller wave height of Figure 14 favor Case A conditions and the benefits inside the project area dominate and commence quite early.

Table II summarizes results for all 15 runs conducted.


SUMMARY AND CONCLUSIONS

The methodology and results presented herein support the following statements.

Wider beaches seaward of structures perform as effective energy dissipators during storm conditions and, where the demand exists, also provide recreational benefits. These benefits can be enhanced through increasing beach widths by nourishment projects.

Beach nourishment projects conducted with good quality sand will evolve with erosion occurring within the project area and deposition in the project adjacent areas. Good quality sand will
remain within the active nearshore region and provide continuing storm damge reduction and recreational benefits.

A simple method is presented for quantifying the benefits in and adjacent to beach nourishment project areas. Considering limiting cases in which (a) all sand stays within the area placed, or (b) all sand placed spreads out rapidly demonstrates that the potential benefits are greater for the latter. Example
calculations for realistic cases demonstrate that the benefits for project adjacent areas can be substantial relative to those in project areas. The relative benefits in project adjacent areas increase with: short project length, large wave height, large sediment transport coefficient, low interest rate, and large additional beach width.


28










Accounting methodologies for benefits of beach nourishment projects should be representative of modern understanding of
sediment transport processes and the equities of those residing along the shoreline and thus should recognize the benefits from sand transported from the project area and deposited in project adjacent areas.

Although the method presented here applies to the case of projects placed on long uninterrupted shorelines, similar procedures could be applied to situations where littoral controls exist, such as jetties at a channel entrance.


ACKNOWLEDGEMENTS

The work leading to this paper was carried out under funding
by the Office of Sea Grant and by the Division of Beaches and Shores of the Florida Department of Natural Resources. This support is gratefully acknowledged.


REFERENCES


Dean, R.G. (1988) "Sediment Interaction at Modified Coastal
Inlets: Processes and Policies", in Hydrodynamics and
Sediment Dynamics of Tidal Inlets, D. Aubrey, Editor, Woods Hole Oceanographic Institution, Woods Hole, Mass. (In Press).


Pelnard Considere, R. (1956) "Essai de Theorie de l'Evolution
des Formes de Rivate en Plages de Sable et de Galets", 4th Journees de l'Hydraulique, Les Energies de la Mar, Question
III, Rapport No. 1.


Shows, E.W. (1978) "Florida's Coastal Setback Line An Effort
to Regulate Beachfront Development", Vol. 4, Nos. 1/2,
Coastal Zone Management Journal, p. 151-164.


Tackney and Associates (1983) "Physical Monitoring Captiva Beach
Restoration Project", Final Report, August.


29




Full Text

PAGE 1

REPORT DOCUMENTATION PAGE 1. Report No. 2. 3. Recipient's Accession No. 4. Title and Subtitle 5. Report Date REALISTIC ECONOMIC BENEFITS FROM September 1988 BEACH NOURISHMENT 6. 7. Author(s) 8. Performing Organization Report No. Robert G. Dean UFL/COEL-88/009 9. Performing Organization Name and Address 10. project/Task/Work Unit No. Coastal and Oceanographic Engineering Dept. University of Florida 336 Weil Hal 11. Contract or Grant No. 336 Well Hall Gainesville, FL 32611 1. Te o 13. Type of Report 12. Sponsoring Organization Name and Address Miscellaneous 14. 15. Supplementary Notes 16. Abstract A method is presented and illustrated with examples to establish appropriate storm damage reduction and recreational benefits from beach nourishment projects. Unlike previous methods, benefits to project adjacent areas are recognized due to sand transport out of the project area and deposition on adjacent beaches. Assuming homogeniety along the shoreline, the character of storm damage reduction and recreational benefit relationships are such that sand transported from a project area and deposited on adjacent beaches always results in an increase rather than a reduction in benefits. A central element in calculating storm damage reduction benefits is the establishment of a proportional damage curve for upland structures as a function of beach width and storm return period. To illustrate the method, limiting cases are presented in which (A) all sediment remains within the area placed, and (B) all sediment spreads out immediately over a long segment of shoreline. Using Monte Carlo simulation to represent the random character of the storms, the method is applied to 15 realistic cases with varying project lengths, representative wave heights, added beach widths and interest rates. The present worth storm damage -continued 17. Originator's Key Words 18. Availability Statement Beach nourishment Damage reduction Nourishment benefits Recreational benefits Storm damage 19. U. S. Security Classif. of the Report 20. U. S. Security Classif. of This Page 21. No. of Pages 22. Price Unclassified Unclassified 29

PAGE 2

reduction and recreational benefits are calculated to demonstrate the effects of the various parameters. It is found that for short project lengths and relatively large wave heights, the benefits from project adjacent areas exceed those in the project area where the sand is placed. Although no littoral control structures, such as jetties are included in the present application, the method could be extended readily to include their effects.

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UFL/COEL-88/009 REALISTIC ECONOMIC BENEFITS FROM BEACH NOURISHMENT by Robert G. Dean

PAGE 4

TABLE OF CONTENTS PAGE LIST OF TABLES.............................................. 3 LIST OF FIGURES ............. ....... ....... ..... .............. .4 ABSTRACT ...................... .............................. .6 INTRODUCTION.................................... .......... ..7 CONCEPTS ... .............. ....... .......................... .7 METHODOLOGY ........ ..... .. .... .... .... ...................... 11 Shoreline Evolution Model.. ........ .................... 15 Storm Damage Reduction Benefits........................ 15 Idealized Cases.......................... ............... 20 Case (A). ... ......... ............. .............. ...... .22 Case (B)............ ..... ...... .................. .... .22 Case (A) ................... ....... ..... ................ 22 Case (B)................................................ 22 RESULTS .................................................... 24 SUMMARY AND CONCLUSIONS..... .............................. 28 ACKNOWLEDGEMENTS ....... ..................................... 29 REFERENCES.... ........... ....... ................ ....... ... 29 2

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LIST OF TABLES TABLE PAGE I. PRESENT WORTH DAMAGE FUNCTION, PWDF(w,-) VERSUS BEACH WIDTH, w, FOR ALL FUTURE DAMAGE .................... 20 II. PRESENT WORTH STORM DAMAGE AND RECREATIONAL BENEFITS FOR VARIOUS WAVE AND PROJECT CONDITIONS .................. 25 3

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LIST OF FIGURES FIGURE PAGE 1. Structural Damages Due to Hurricane Eloise (1975) and Example of Reduced Damages by Beach Nourishment Advancing the Shoreline Seaward by Fifty Feet ........................................ .9 2. Damage Reduction Per Structure Resulting from a One Foot Wide Additional Beach, as a Function of Structure Location Relative to Control Line. Based on Hurricane Eloise Data........................... 10 3. Hypothetical Usage and Recreational Benefits of Sandy Beaches....................................... 12 4. Annual Recreational Benefits Per Additional Foot of Beach Width as a Function of Initial Beach Width, Per Foot of Beach Length. Developed from Figure 3b........................................... 13 5. Schematic of Erosion of Nourished Area and Deposition in Project Adjacent Areas........................... 14 6. Example Solution of Evolution of Initially Rectangular Beach Planform. Pelnard Considere Method. Wave Height, Hb = 2.0 ft, Initial Nourished Beach Width = 100 ft, Fill Length, X = 4 miles, t = time............................... 16 7. Hypothetical Proportional Storm Damage, D, as a Function of Storm Return Period, TR, and Beach Width, w.................. .............. ............ 17 8. Flow Chart of Methodology........................... 18 9. Expected Damage D(w) Due to a Single Storm as a Function of Beach Width, w.......................... 21 10. Interpretation of Damage Reduction Benefits if Sand Remains Where Placed (Case A) and if Sand Spreads Out Immediately (Case B).................... 21 11. Ratio of Storm Damage Benefits, RSD, vs Initial Beach Width, wo, and Additional Beach Width, Aw. RSD is Ratio of Storm Damage Benefits for Sand Which Spreads Out Immediately to Those for Which Sand Remains Where Placed................................ 23 4

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PAGE 12. Ratio of Recreational Benefits, RRB, vs Initial Beach Width, wo, and Additional Beach Width, Aw. RRB is Ratio of Recreational Benefits for Sand Which Spreads Out Immediately to Those for Which Sand Remains Where Placed........................... 23 13. Present Worth Storm Damage Reduction and Recreational Benefits. Hb = 2.0 ft, A = 2.0 miles, wo = 0.0, Aw = 100.0 ft., Interest Rate = 8%, Run No. 5....... 26 14. Present Worth Storm Damage Reduction and Recreational Benefits. Hb = 1.0 ft, £ = 4.0 Miles, wo = 0.0, Aw = 100.0 ft, Interest Rate = 8%, Run No. 7........ 26 5

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REALISTIC ECONOMIC BENEFITS FROM BEACH NOURISHMENT by Robert G. Dean ABSTRACT A method is presented and illustrated with examples to establish appropriate storm damage reduction and recreational benefits from beach nourishment projects. Unlike previous methods, benefits to project adjacent areas are recognized due to sand transport out of the project area and deposition on adjacent beaches. Assuming homogeniety along the shoreline, the character of storm damage reduction and recreational benefit relationships are such that sand transported from a project area and deposited on adjacent beaches always results in an increase rather than a reduction in benefits. A central element in calculating storm damage reduction benefits is the establishment of a proportional damage curve for upland structures as a function of beach width and storm return period. To illustrate the method, limiting cases are presented in which (A) all sediment remains within the area placed, and (B) all sediment spreads out immediately over a long segment of shoreline. Using Monte Carlo simulation to represent the random character of the storms, the method is applied to 15 realistic cases with varying project lengths, representative wave heights, added beach widths and interest rates. The present worth storm damage reduction and recreational benefits are calculated to demonstrate the effects of the various parameters. It is found that for short project lengths and relatively large wave heights, the benefits from project adjacent areas exceed those in the project area where the sand is placed. Although no littoral control structures, such as jetties are included in the present application, the method could be extended readily to include their effects. 6

PAGE 9

INTRODUCTION Policies and methodologies should evolve continuously to remain consistent with modern understanding of coastal processes and the true equities of those residing along the shoreline. Several changes have occurred in the last few decades that argue for an examination and modifications of present economic analysis procedures relating to beach nourishment: (1) It is now clear that on a long, uninterrupted shoreline, good quality sand placed in a beach nourishment project will eventually be transported out of the region placed, but will remain within the active nearshore system, (2) Sand transported from a project area and deposited on project adjacent areas provides not only continuing damage reduction and recreational benefits, but provides enhanced benefits, and (3) With increasing concern over the use of "hard structures" as a means of shoreline control, beach nourishment will play an increasing future role. This paper considers the economic consequences of sand eroded from a beach nourishment project area and deposited on project adjacent areas. Realistic damage reduction relationships and recreational benefits for a widened beach are utilized to demonstrate that this evolution process actually results in a net increase in project benefits. Benefits from simple limiting cases are examined in which (1) the sand remains in the area placed, and (2) the sand spreads out immediately. A direct procedure is presented to account for total present worth project benefits. The procedure utilizes Monte Carlo simulation to faithfully represent the probability of storm occurrences. Although the methodology presented here is not applicable to shorelines which include features which would cause longshore sediment transport interruptions, the concepts could be extended readily for such cases. CONCEPTS There are two simple concepts which are critical to the methodology presented here: 7

PAGE 10

(1) Good quality sand placed in a beach nourishment project will be eroded from the area placed but will remain indefinitely in the active nearshore region, and (2) The greatest storm damage and recreational benefits are generally realized for the initially narrower beaches. The first concept will be considered as valid without much discussion. Although "good quality sand" is a matter of degree, here it refers to sand that is greater than 0.14 mm or so in diameter and that is coarser than or as coarse as the material originally present on the beach. For those nourishment materials in which the above is not the case, this paper refers to that sand fraction which is compatible. Monitoring results from a number of beach nourishment projects have demonstrated the first concept, for example at Port Canaveral, FL (Dean, 1988) and Captiva Island, FL (Tackney and Associates, 1983). The second concept is illustrated by Figure la which represents a survey (by Shows, 1978) of the structural damage caused by Hurricane Eloise (1975) in Bay County, FL as a function of proximity of the structures relative to a jurisdictional control line which is generally parallel to the shoreline. Of particular significance in Figure la is the steeply sloped portion of the damage curve near its seaward end and the relatively mild slope near its landward end. It is instructive to consider the effect of a beach nourishment project which displaces the beach seaward by a certain amount such as 50 ft as shown in Figure lb. It is seen that due to the slope characteristics discussed above, the greatest damage reductions occur for those structures which initially have very little beach in front of them. Figure 2 presents the damage reduction per structure associated with an additional one foot of beach width. For the narrower initial beach widths, the reduction is approximately $3,000 per structure whereas for greater initial beach widths, the damage reduction per structure is less than $500. In summary, the damage reduction benefits are greater for beaches which are initially much narrower. 8

PAGE 11

0 200 o0 WL" 160 < 120<80CL MO L U3 ) 0 II 40 C, -150 -100 -50 0 50 100 150 Seaward 1 Landward DISTANCE FROM CONTROL LINE (ft) a) Damage to Structures in Relation to their Location with Control Line (Resulting from Study of 540 Structures in Bay County after Hurricane Eloise, by Shows, 1978). 0 -200 W o 160 U 120 wLU .Reduced Damages <'O 80 cDamage Curve W M Shifted by 50 ft > !40 40 0 00 150 100 50 0 50 100 150 Seaward Landward DISTANCE FROM CONTROL LINE (ft) b) Damage Reduction Due to Beach Nourishment Advancing the Profile Fifty Feet Seaward. Figure 1. Structural Damages Due to Hurricane Eloise (1975) and Example of Reduced Damages by Beach Nourishment Advancing the Shoreline Seaward by Fifty Feet. 9

PAGE 12

4 W Z 3 0F-~f W 2 = I0 150 100 50 0 50 100 150 Seaward I Landward DISTANCE FROM CONTROL LINE (ft) Figure 2. Damage Reduction Per Structure Resulting from a One Foot Wide Additional Beach, as a Function of Structure Location Relative to Control Line. Based on Hurricane Eloise Data. 10

PAGE 13

The same concepts demonstrated above for damage reduction benefits apply for recreational benefits. Figure 3 presents the hypothetical usage and associated recreational benefits for beaches of varying widths. The number of people using the beach will increase with beach width; however, the rate of increase decreases for the greater widths. The results in Figure 3b are based on a visitation value of $6.00 per visitor per day and a plan area visitation requirement of 200 square feet. The annual recreational benefits associated with an additional foot of beach width versus initial beach width, based on Figure 3, are presented in Figure 4. As before, it is seen that the greatest benefits occur for the initially narrower beaches. Referring to Figure 5, the significance of greater benefits for initially narrower beaches is that as a beach nourishment project evolves with the beach fronting the project area narrowing and the project adjacent beaches widening, benefits are lost in the initially wider project area. This loss of benefits is small compared to the gain of relatively large benefits in the initially narrow project adjacent areas. Assuming that the value of the upland structures protected by the project and the initial beach widths in project adjacent areas are uniform along the beach, there is always a net gain in storm reduction benefits as a result of project evolution. Similarly with respect to recreational benefits, assuming that the need for and access to recreational beaches are uniform, etc., the net effect of project evolution is a gain in recreational benefits. METHODOLOGY The methodology will be described and illustrated for idealized cases of no project evolution and rapid project evolution and general cases of benefits due to project evolution over realistic time frames. 11

PAGE 14

C/ 300 O -Weekend 200 -Visitation w m 100z 0 0 50 100 150 BEACH WIDTH, w(ft) a) Number of Days Per Year that Full Beach Width Is Used Z 500O (n 400L) C300WooSu. 2004| 100z 0 0 50 100 150 BEACH WIDTH, w(ft) b) Annual Recreational Benefits vs Beach Width Per Foot of Beach Length Figure 3. Hypothetical Usage and Recreational Benefits of Sandy Beaches. 12

PAGE 15

' 10 cn tD C, LU z -J 2 5 O cc -j z z 0 S I I I 0 II I I I I -I-I < 0 50 100 150 INITIAL BEACH WIDTH (ft) Figure 4. Annual Recreational Benefits Per Additional Foot of Beach Width as a Function of Initial Beach Width, Per Foot of Beach Length. Developed from Figure 3b. 13

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SBenefits Gained I I I \ X Benefits Lost I I I I I Benefits Gained a) Initial Nourished Planform b) Eroded Nourished Planform with Material Deposited on Adjacent Beaches Figure 5. Schematic of Erosion of Nourished Area and Deposition in Project Adjacent Areas. 14

PAGE 17

Shoreline Evolution Model The shoreline evolution model adopted here will be that due to Pelnard-Considere for an initially rectangular planform as presented in Figure 6. The factor G is the so-called "longshore diffusivity" and for small angles of wave incidence is K H 5/2 S K Hb G = 8(s-1)(l-p)(h*+ B) in which K is the sediment transport factor usually taken as 0.77, Hb is the representative breaking wave height, g is gravity, K is the spilling breaker ratio (on the order of 0.8), s is the ratio of sediment specific gravity to that of the water in which transport is occurring, p is the in situ porosity and (h*+ B) is the vertical extent of beach profile response. Storm Damage Reduction Benefits Development of storm damage reduction benefits commences with the establishment of the relationship of a proportional storm damage factor, D, as a function of beach width fronting the structure, w, and storm return period, TR. Figure 7 presents one example of such a relationship which has been used in the state of Florida Beach Management Plan. Development of this relationship is by no means trivial and should be based on an analysis of the expected damage to a range of representative structures as well as calibration with available storm results if such data are available. The proportional storm damage factor, D, depends on the foundation and structural and elevation characteristics of the buildings as well as the beach morphology, and presence and integrity of coastal protection structures, etc. With the availability of D(w,TR) it is possible to predict the present worth damage reduction benefits PWDRB(N) during N years by the general procedure described in Figure 8 and the following equation 15

PAGE 18

DISTANCE FROM ORIGINAL SHORELINE, y (ft) 100!00. < *-Original Nourished Beach Planform iI /: \ --Planform After 3 Months S -80 \ 10 Months S60 --7 Years 0 -30 Years -40 ., _--s-130 Years Pre-Nourished S ---~-y -20 --~ Shoreline 6 4 2 0 2 4 6 8 ALONGSHORE DISTANCE, x (miles) Figure 6. Example Solution of Evolution of Initially Rectangular Beach Planform. Pelnard Considere Method. Wave Height, Hb = 2.0 ft, Initial Nourished Beach Width = 100 ft, Fill Length, k = 4 miles, t = time.

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1.0 0.8O o 0 0.6I / IY LL. 4 S0.42 0.20.0 1 10 100 1000 RETURN PERIOD, TR (Years) Figure 7. Hypothetical Proportional Storm Damage, D, as a Function of Storm Return Period, TR, and Beach Width, w.

PAGE 20

FLOW CHART OF METHODOLOGY Damage Reduction Benefits (a) In Project Area (b) In Project Adjacent Areas Initial Nourished Beach anform Evoluio Planform Evolution Planform (Pelnard-Considered) (Rectangular) (Pelnard-Considered) Recreational Benefits (1) (2) (a) In Project Area (b) In Project Adjacent Areas (3) --------------------------------------------------------Damage Reduction Benefits (a) Random Storm Magnitudes in Accordance C with Actual Distribution x x SY (b) Damage Reduction vs. Storm Return Period -and Beach Width ... (c) Present Worth Damage Reduction Benefits Recreational Benefits ) (2) (a) Recreational Benefits vs. Beach Width y erf[A(l + 2)] + erf [A(1 -2)] (b) Present Worth Recreational Benefits 2 -e .cp A4Gt Figure 8. Flow Chart of Methodology.

PAGE 21

PWDRB(N) = N I ( n V(x,n)[D(w(x,n)TR(n))-D(w ,TR(n))]dx n=1 (1+I) Project Area (1) N 1 + I n f V(x,n)[D(w(x,n),TR(n))-D(wo,TR(n))]dx n=1 (1+I) Project Adjacent Areas in which I is the interest rate and V(x,n) represents the structure value at a location, x, at a time n years into the future. The two integrals differ only in their respective intervals of integration and are written separately here to illustrate the contributions from the two areas. Eq. (1) accomplishes the objective of providing methodology for quantifying storm damage reduction. However, it is instructive to develop concepts further. Referring to Figure 7 which presents the proportional storm damage factor, D, the expected damage by a single storm D(w) as a function of beach width, w, is -1 D(w) = f D(w,TR)p(D)dD (2) 0 in which p is the probability density function and is related to the cumulative probability distribution P by dP p(D) -d (3) and noting that TR = -(4) Eq. (2) simplifies to -1 dP D(w) = f D(w,TR) -dD = f D(w,TR)dP (5) 0 0 19

PAGE 22

Figure 9 presents D as a function of beach width as developed from Eq. (5). It is noted that this distribution is qualitatively similar to damages experienced in Hurricane Eloise (Figure 1) which was approximately a 70 year storm. Assuming that the value of the upland structures remain constant with time and that damaged structures are rebuilt to the same standards (both considerable assumptions), the present worth damage factor, PWDF(w) as a function of beach width for N years into the future is N -PWDF(w,N) = I D(w) = [1 -D(w) (6) n=1 (1+I)n (1+1) and again, I is the interest. The bracketed factor in Eq. (6) approaches unity with large N. Table I presents values of PWDF(w,-) for several beach widths interest rates. It is noted that the present worth damage factor can range as high as 1.31 for the case of zero beach width and an interest rate of 8%. TABLE I PRESENT WORTH DAMAGE FUNCTION, PWDF(w,-) VERSUS BEACH WIDTH, w, FOR ALL FUTURE DAMAGE Present Worth Damage Function, PWDF(w,m) Interest For Beach Width, w Rate 0 ft 50 ft 100 ft 150 ft 6% 1.75 0.67 0.47 0.35 8% 1.31 0.50 0.35 0.26 12% 0.88 0.33 0.23 0.18 Idealized Cases In contrasting project benefits realized within the project area to those outside the project area, it is instructive to consider two simple cases: 20

PAGE 23

S0.10 LU-0 o( 0 .05 0.1 uO \ SI I I 0 50 100 150 BEACH WIDTH, w(ft) Figure 9. Expected Damage D(w) Due to a Single Storm as a Function of Beach Width, w. 0.1 U.IO WDamage Reduction if .-Sand Remains Where Placed (Case A) Sr--Aw --Additional Damage Reduction W 0 Due to Sand Spreading [L IM Out Immediatel 0L 0 w 50 wo+Aw 100 150 BEACH WIDTH, w(ft) Figure 10. Interpretation of Damage Reduction Benefits if Sand Remains Where Placed (Case A) and if Sand Spreads Out Immediately (Case B). 21

PAGE 24

Case (A). All sediment remains within the area placed, and Case (B). The sediment placed spreads out immediately over a long segment of shoreline. Case (A). The expected storm damage reduction benefits due to a single storm are (SDRB)A = [D(wo + Aw) -D(wo)]a (7) Case (B). Denoting the (long) distance over which the sediment has been distributed as V' and the associated additional width as Aw', we have (SDRB)B = -()w Aw'' (8) o and since sediment is conserved Aw2 = Aw'a', (SDRB)B = -()w Awk (9) O The ratio, RSD, of storm damage reduction benefits for the case of sand spreading out immediately to the case in which sand remains where placed is /D -(w)w Aw aw w RSD = -(10) [D(wo + Aw) -D(wo)] It is noted that the ratio RSD is always greater than unity. As shown in Figure 10, the interpretation is simple with the numerator representing the tangent of the damage curve at wo and the denominator the secant slope between wo and Wo + Aw. Due to the character of the curve, the ratio will always exceed unity. Figure 11 presents the ratio RSD vs wo for several values of Aw. 22

PAGE 25

5.0 M 5--Aw=150 ft CC -4.0 OI .Aw=100 ft r1 3.0 LL W 0 m 2.0 -1.0 ----------------------------0 0 50 100 150 INITIAL BEACH WIDTH, wo (ft) Figure 11. Ratio of Storm Damage Benefits, RSD, vs Initial Beach Width, Wo, and Additional Beach Width, Aw. RSD is Ratio of Storm Damage Benefits for Sand Which Spreads Out Immediately to Those for Which Sand Remains Where Placed. 3.0 30 --Aw=150 ft z 00 < 2.0 Aw=100 ft Ra mU .\ Aw=50 ft WLL z u W 1.0-------------O0 n0 50 100 150 INITIAL BEACH WIDTH, wo (ft) Figure 12. Ratio of Recreational Benefits, RRB, vs Initial Beach Width, Wo, and Additional Beach Width, Aw. RRB is Ratio of Recreational Benefits for Sand Which Spreads Out Immediately to Those for Which Sand Remains Where Placed. 23

PAGE 26

The same general discussion presented above applies to recreational benefits relationships. The ratio of benefits RRB for a project that spreads out immediately to one that remains in place is aR aw w RRB = (11) [R(w + Aw) -R(w )] and this ratio will always exceed unity by the same argument as for the damage reduction benefits. For the recreational benefits shown in Figure 3, values of the ratio, RRB, are presented in Figure 12. RESULTS Prior to presenting results for the general case, in which the beach planform evolves with time, it is worthwhile to consider the variables which will tend to favor Case A (sand remains in place) or Case B (sand spreads out immediately). Case A conditions would tend to dominate for: Large Beach Fill Lengths, z Low Wave Height, Hb Small Transport Coefficient, K Small Additional Beach Widths, Aw High Interest Rates, I and vice versa for Case B. The methodology described in the previous section was incorporated into a computer program which was "exercised" for the variable values shown in Table II. Results will be presented in two different forms. A schematic of the methodology has been presented as Figure 8. Figures 13 and 14 present variations of storm damage and recreational benefits with time for Runs 5 and 7, respectively. The relatively large wave height and short beach fill associated 24

PAGE 27

TABLE II. PRESENT WORTH STORM DAMAGE AND RECREATIONAL BENEFITS FOR VARIOUS WAVE AND PROJECT CONDITIONS Characteristics Storm Damage Reduction* Recreational Benefits** In In Initial Added In Project In Project Project Wave Beach Beach Interest Project Adjacent Total Project Adjacent Total Run Length, Height Width Width Rate Area Area Area Area i, (miles) Hb (ft) wo (ft) Aw (ft) I 1 1.0 1.0 0 100 0.08 0.368 1.037 1.405 4.4 12.7 17.1 2 1.0 2.0 0 100 0.08 0.176 1.434 1.610 2.1 17.8 19.9 3 1.0 4.0 0 100 0.08 0.072 1.561 1.633 0.9 19.7 20.6 4 2.0 1.0 0 100 0.08 0.592 0.532 1.124 15.4 13.0 28.4 5 2.0 2.0 0 100 0.08 0.323 1.136 1.459 7.6 27.9 35.5 6 2.0 4.0 0 100 0.08 0.148 1.470 1.618 3.6 36.7 40.3 7 4.0 1.0 0 100 0.08 !.009 0.084 1.093 55.9 4.2 60.1 0 8 4.0 2.0 0 100 0.08 0.525 0.672 1.197 26.6 32.7 59.3 9 4.0 4.0 0 100 0.08 0.280 1.234 1.514 13.0 60.7 73.7 10 2.0 2.0 0 150 0.08 0.388 .679 2.067 10.1 38.5 48.6 11 2.0 2.0 50 100 0.08 0.066 0.178 0.244 4.9 14.5 19.4 12 2.0 2.0 100 150 0.08 0.057 0.144 0.201 7.4 21.7 29.1 13 2,0 2.0 0 100 0.12 0.240 0.697 0.937 5.7 17.0 22.7 14 2.0 2.0 0 100 0.04 0.506 2.413 2.919 11.7 60.0 71.7 15 16.0 2.0 0 100 0.08 1.733 0.002 1.735 392.4 0.5 392.9 *Relative to immediately adjacent upland property values within project area **Expressed in millions of dollars

PAGE 28

C) l 1.5 au Total 0y Adjacent Areas C Zz 1.0---I --O 1 OM uJ / .Project Area O. 0. I--\-----i---0 0 10 20 30 40 50 60 YEARS INTO FUTURE a) Proportional Present Worth Storm Damage Reduction Benefit Components vs. Years Into Future. -. <: 40 -' Total 00 .U 0o 30 -Adjacent Areas 5o ... ---""" 0-) 0 O 20 0 / / Project Area m 10 -I -0 10 20 30 40 50 60 aYEARS INTO FUTURE b) Present Worth Recreational Benefits vs. Years Into Future. Figure 13. Present Worth Storm Damage Reduction and Recreational Benefits. Hb = 2.0 ft, a = 2.0 miles, wo = 0.0, Aw = 100.0 ft., Interest Rate = 8%, Run No. 5. 26

PAGE 29

S1.5 I: z Ow Total Z 1.0 --J .*0 1Project Area 0 • ~ o.s/ SCLI o'S 0~0 B= / Adjacent Areas < 0 V _--4----T a 0 10 20 30 40 50 60 YEARS INTO FUTURE a) Proportional Present Worth Storm Damage Reduction Benefit Components vs. Years Into Future. -J 60 Total SJ 40-n Project Area cc 40 -" § ./ &=5 20 :: Adjacent Areas !E 0 I---4---t -u O 0 10 20 30 40 50 60 m YEARS INTO FUTURE Q. b) Present Worth Recreational Benefits vs. Years Into Future. Figure 14. Present Worth Storm Damage Reduction and Recreational Benefits. Hb = 1.0 ft, z = 4.0 Miles, wo = 0.0, Aw = 100.0 ft, Interest Rate = 8%, Run No. 7. 27

PAGE 30

with Figure 13 favor Case B conditions and it is seen that the dominant benefits occur within the adjacent project areas. It is also of interest to note that the benefits in the project adjacent areas lag those in the project area due to the time required for sediment transport to these adjacent areas. By contrast the longer project length and smaller wave height of Figure 14 favor Case A conditions and the benefits inside the project area dominate and commence quite early. Table II summarizes results for all 15 runs conducted. SUMMARY AND CONCLUSIONS The methodology and results presented herein support the following statements. Wider beaches seaward of structures perform as effective energy dissipators during storm conditions and, where the demand exists, also provide recreational benefits. These benefits can be enhanced through increasing beach widths by nourishment projects. Beach nourishment projects conducted with good quality sand will evolve with erosion occurring within the project area and deposition in the project adjacent areas. Good quality sand will remain within the active nearshore region and provide continuing storm damge reduction and recreational benefits. A simple method is presented for quantifying the benefits in and adjacent to beach nourishment project areas. Considering limiting cases in which (a) all sand stays within the area placed, or (b) all sand placed spreads out rapidly demonstrates that the potential benefits are greater for the latter. Example calculations for realistic cases demonstrate that the benefits for project adjacent areas can be substantial relative to those in project areas. The relative benefits in project adjacent areas increase with: short project length, large wave height, large sediment transport coefficient, low interest rate, and large additional beach width. 28

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Accounting methodologies for benefits of beach nourishment projects should be representative of modern understanding of sediment transport processes and the equities of those residing along the shoreline and thus should recognize the benefits from sand transported from the project area and deposited in project adjacent areas. Although the method presented here applies to the case of projects placed on long uninterrupted shorelines, similar procedures could be applied to situations where littoral controls exist, such as jetties at a channel entrance. ACKNOWLEDGEMENTS The work leading to this paper was carried out under funding by the Office of Sea Grant and by the Division of Beaches and Shores of the Florida Department of Natural Resources. This support is gratefully acknowledged. REFERENCES Dean, R.G. (1988) "Sediment Interaction at Modified Coastal Inlets: Processes and Policies", in Hydrodynamics and Sediment Dynamics of Tidal Inlets, D. Aubrey, Editor, Woods Hole Oceanographic Institution, Woods Hole, Mass. (In Press). Pelnard Considere, R. (1956) "Essai de Theorie de 1'Evolution des Formes de Rivate en Plages de Sable et de Galets", 4th Journees de l'Hydraulique, Les Energies de la Mar, Question III, Rapport No. 1. Shows, E.W. (1978) "Florida's Coastal Setback Line -An Effort to Regulate Beachfront Development", Vol. 4, Nos. 1/2, Coastal Zone Management Journal, p. 151-164. Tackney and Associates (1983) "Physical Monitoring Captiva Beach Restoration Project", Final Report, August. 29