Citation
Beach nourishment design

Material Information

Title:
Beach nourishment design probability of encountering rock with cores through a borrow pit : prepared for Office of Beaches and Coastal Systems, Florida Department of Environmental Protection ...
Series Title:
UFLCOEL-2000003
Creator:
Dean, Robert G ( Robert George ), 1930-
Florida -- Office of Beaches and Coastal Systems
Place of Publication:
Gainesville Fla
Publisher:
Coastal & Oceanographic Engineering Program, Dept. of Civil & Coastal Engineering
Publication Date:
Language:
English
Physical Description:
iii, 13 leaves : ill. ; 28 cm.

Subjects

Subjects / Keywords:
Sand -- Sampling -- Mathematical models ( lcsh )
Beach nourishment ( lcsh )
Genre:
government publication (state, provincial, terriorial, dependent) ( marcgt )
non-fiction ( marcgt )

Notes

General Note:
"March 24, 2000."
Statement of Responsibility:
prepared by Robert G. Dean.

Record Information

Source Institution:
University of Florida
Holding Location:
University of Florida
Rights Management:
All applicable rights reserved by the source institution and holding location.
Resource Identifier:
49506070 ( OCLC )

Full Text
UFL/COEL-2000/003

BEACH NOURISHMENT DESIGN: PROBABILITY OF ENCOUNTERING ROCK WITH CORES THROUGH A BORROW PIT
by
Robert G. Dean

March 24, 2000
Prepared for: Office of Beaches and Coastal Systems Florida Department of Environmental Protection Tallahassee, Florida




BEACH NOURISHMENT DESIGN:
PROBABILITY OF ENCOUNTERING ROCK WITH CORES THROUGH A BORROW PIT
March 2,4, 2000
Prepared for:
Office of Beaches and Coastal Systems
Florida Department of Environmental Protection
Tallahassee, Florida
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 O F FIGURES .......................................................... iii
LIST O F TABLES ........................................................... iii
1 INTRODUCTION ...................................................... 1
2 SITUATION CONSIDERED ............................................. 1
3 G EN ERA L ............................................................ 2
4 TERM INOLOGY ...................................................... 2
5 M ETHODOLOGY ..................................................... 2
5.1 Case (1) Uniform Distribution of Rock, Neglecting Possible Overlap of
Rock in Various Layers ............................................ 2
5.2 Case (2) Uniform Distribution of Rock, Accounting for Possible Overlap
of Rock in Various Layers .......................................... 3
5.3 Case (3) Variable Rock Concentration With Depth ..................... 5
6 EXAMPLE ILLUSTRATING APPLICATION OF THE RESULTS ............ 7
7 MOST PROBABLE NUMBER OF ROCKS ENCOUNTERED ............... 10
7.1 Background and Example ......................................... 10
8 SUMMARY AND CONCLUSIONS ...................................... 11




LIST OF FIGURES
FIGURE PAGE
1 Upper Four Layers of Borrow Pit, Each of One Rick Diameter Thickness ............ 1
2 Probability of Encountering Rock With N Cores Through M Layers of Rock
Thickness. C ase I ........................................................ 4
3 Comparison of Probability of Encountering Rock With and Without Sheltering. 10
Cores and 5 Layers ....................................................... 6
4 Distribution of Rock Concentration With Depth Into Borrow Pit ................... 8
5 Probability of Encountering Rock With 10 Cases in M Upper Layers, Each of
Thickness, D. Sheltering Included ........................................... 9
6 Most Probable Number of Rocks Encountered, Example Presented is for P, = 0.05 ... 12
LIST OF TABLES
TABLE PAGE
1 N otation to Be U sed ...................................................... 2
2 Results for Example Application ............................................ 7
3 Probabilities of Occurrence for Various Possibilities in Example of Four Flips of a
C oin ................................................................. 10




BEACH NOURISHMENT DESIGN:
PROBABILITY OF ENCOUNTERING ROCK WITH CORES THROUGH A BORROW PIT
I INTRODUCTION
An emerging concern in beach nourishment projects is the presence of rocks in a borrow pit. A mixture of sand and a relatively small percentage of rocks placed as nourishment will, after profile equilibration and/or storm activity, leave a concentration of rocks on the beach surface which is undesirable from both aesthetics and safety considerations. If substantial quantities of rock are present in the borrow area, it may be warranted to employ specialized equipment which separates the sand and rock at either the source or the point of delivery. However, use of this specialized equipment is expensive. A second difficulty which is addressed in this report is, given a volumetric percentage of rock in the borrow pit, the probability of identifying the presence of that rock by standard coring operations.
2 SITUATION CONSIDERED
Consider the case in which core samples are to be taken through a potential borrow pit to establish the quality of the material. The question considered here is the probability of encountering rock with N cores through M layers of rock diameter thickness, D. The rocks are considered to be spherical in shape and of uniform diameter, D, and to be uniformly dispersed horizontally within a layer of rock diameter with a volumetric concentration, Pv.
Three cases will be examined: (1) Uniform distribution of rock neglecting possible overlap of rock by upper layers, (2) Same as (1), except the possible overlap with overlying layers of rock is taken into consideration, and (3) A distribution of rock concentration with depth into the borrow area. Figure 1 illustrates the general situation.
Water S a n di
Rocks D
Figure 1. Upper Four Layers of Borrow Pit, Each of One Rock Diameter
Thickness.




3 GENERAL
Considerations common to all three cases are discussed first. Within each layer of rock diameter thickness, uniform rock size is considered with diameter, D. With a volumetric density, Pv, the number of spherical rocks, NR, in a layer of one rock diameter thickness is ADPV
NR-=__D3 (1)
6
in which A is the plan area of the borrow pit under consideration such that the product, A D, is the total volume of one layer. The plan area of rock in one rock thickness, which is relevant to the probability of one or more cores encountering rock, is the product of the number of rocks and the plan area of each rock. Thus the proportion of the plan area occupied by rock, PA, is
NR tD (
PA 4 _=3 p (2)
A A 2
4 TERMINOLOGY
The terminology in Table 1 will be utilized.
Table 1
Notation to Be Used
Notation Significance
PE:N:m Probability of Encountering Rock With N Cores in mt' Layer PNOE:N:m Probability of Not Encountering Rock With N Cores in m Layer = I- PE:N:m PE:N:I,M Probability of Encountering Rock With N Cores in Upper M Layers PNOE:N:I,M Probability of Not Encountering Rock With N Cores in Upper M Layers = 1 PNOE:N:1,M
5 METHODOLOGY
5.1 Case (1) Uniform Distribution of Rock, Neglecting Possible Overlap of Rock in
Various Layers
The probability that one core passing through one rock thickness of the borrow pit will encounter rock is PA. The probability that N cores through one rock thickness will not encounter rock in the mth layer is




PNOE:N:m c ( e -PtA) N (3)
The probability that rock will be encountered by N cores through the m ss layer is PE:N:m = I (1 -PA )N (4)
The probability that rock will be encountered by N cores through M thicknesses of rock is
PE:N:,M 1 -(1 -PA)MN (5)
Figure 2 presents the probability of encountering rock versus the product of the number of cores and the number of layers penetrated, each of one rock diameter thickness. Four values of rock volumetric density, P,, are considered in Figure 2.
5.2 Case (2) Uniform Distribution of Rock, Accounting for Possible Overlap of Rock in
Various Layers
In this treatment, we allow for the probability that a rock in a particular lower layer may be wholly or partially below and thus "blocked" or "occluded" from coring by an overlying rock. Thus, in determining the probability that rock will be encountered, it is necessary to reduce the probability due to this blocking by the overlying rock. Commencing with the upper layer, the probability that rock will be encountered by one core is the same as for Case 1 since there are no overlying layers. However, in the second layer of thickness, D, the probability of encountering rock by one core is P E::2=P3A(I -PA) (6)
Thus the probability that rock will not be encountered in either the upper layer or second layer by one core is
P NOE::I,2 =[ 1 -P A] [11 -PA(1 -PA (7)
The probability that rock will be encountered in the third layer by one core is PE:I:3 =PA(1-PA )2 (8)
with the term (1-PA)2 representing the blocking by the two overlying layers. The probability that rock will not be encountered with one core through 3 layers of rock thickness, PNOE:1:1,3 is




0
CY)
-
.-
0
U
4-
0
-.Q
0
0-

2 3 4 5 6 78101

2 3 4 5678102

2 3 4 5678103

Product of Number of Cores and Number of Rock Layers, MN
Figure 2. Probability of Encountering Rock With N Cores
Through M Layers of Rock Thickness. Case 1.

Fig2 Mar. 20, 2000 1:55:59 PM

0.9 0.8 0.7 0.6 0.5
0.4 0.3
0.2 0.1

... .. . . . P . . .0 .
. . .... .. . . .
. . . :: : : : '-- - - - P V = 0 .0 5
. 0 .
. . ,.," .. .PV 0 1
... . .. .
... I .... .
...* o,. . ... .
* : : : : : : :

u.U
-i &




We can now generalize to the mt layer with the probability of an encounter in the m th layer by one core
m-i
PE:I:m=PA(m) II (1 PA(J)) (10)
j=1
where the symbol "II" denotes "product" and Eq. (10) has been written for the case of a variable probability of encounter with depth layer, j. In the case of uniform rock distribution with depth, Eq.
(10) reduces to
m-i
PE:I:m = PA I (1 -PA) = PA(1 PA) (1) j=i
as before. The probability of non-encounter with N cores in the mt layer is
PNOE:N,m = (1 PE:i:m)N (12)
The probability that N cores through M layers of rock thickness will encounter rock, P E:N:,M is
M
PE:N:I,M = -11 PNOE:N:m (13)
m=i
Figure 3 compares the results for various values of Pv and for N = 10 and M = 5 for Case 1 which does not account for blocking and the present case (Case 2) which does account for blocking by the overlying layers. It is seen that the differences are small as might be expected from intuition based on the small proportions of rock density considered.
5.3 Case (3) Variable Rock Concentration With Depth
Rock concentrations usually increase with depth into the borrow pit. This is a simple extension of Case 2 by generalizing such that Pv ( or PA) is a function of the layer number, m, i.e., PA(m). In this case, the probability of encountering rock in the mt layer with one core has been given by Eq. (10) repeated here as

PNOE:1:1,3 = [ I -PA] [11 _PA(O -PA [11 _PA(O -PA)2]




0
a)
0
w
4
0
CU
0

1.0 0.9 0.8 0.7 0.6 0.5
0.4 0.3
0.2 0.1 0.0 0.c

0.06 -0.07 Density, Pv

0.08 0.09 0.10

Figure 3. Comparison of Probability of Encountering Rock With and Without Sheltering. 10 Cores and 5 Layers.
Fig3 Mar. 21, 2000 10:14:03AM

0.04 0.05 Volumetric

0.01 0.02 0.03
Rock

* .. .Without Sheltering
-....... ...... With Sheltering
. . . . . . . . . . . - - - - - - - - - - - - - -

)0




m-I
PE:l:m=PA(m) II (1 -PA())
j=1

(14)

and the probability that rock will not be encountered in the mt layer with N cores is

PNOE:N:m = (1 PE:I:m)N

Thus the probability that rock will be encountered by N cores through the upper M layers is

M
PE:N:I,M = 1 H PNOE:N:m
m=

(15)

(16)

Figure 4 shows a reasonable distribution of rock with depth. In the upper 30 rock thicknesses, the rock density varies from zero in the upper three layers to 0.05 in the lower layers. Applying Eq. (16), the probability that rock will be encountered within the upper M layers with ten cores is shown in Figure 5.
6 EXAMPLE ILLUSTRATING APPLICATION OF THE RESULTS
Consider the case in which rocks in a borrow pit are uniformly dispersed, the borrow pit plan dimensions are 3,000 ft by 3,000 ft and the rocks are of uniform diameter = 0.5 ft. On the basis of a core spacing of approximately 1,000 ft, we will assume that 10 cores have been taken. Four rock percentages by volume will be considered: Pv = 0.01, 0.02, 0.05 and 0.10. Calculate the number of rocks that would be within a uniform borrow area thickness which is defined by a 0.5 rock encounter probability.
Referring to Figure 2, the associated product of number of cores (N) and rock thicknesses
(M) are presented in Column 2 of Table 2.
Table 2
Results for Example Application
Pv MN@50% M VgOC (yd3) No. of Rocks
0.01 45 4.5 7,000 3,094,000
0.02 22 2.2 7,333 3,025,000
0.05 8 0.8 6,667 2,750,000
0.10 4.2 0.4 6,667 2,750,000




0
-20
ca C)
0 -1 0 ----------U) 'C
C)
-
E 00
z 0.00 0.01 0.02 0.03 0.04 0.05
Rock Volumetric Density, Pv Figure 4. Distribution of Rock Concentration With Depth Into Borrow Pit Fig4 Mar. 15, 2000 9:09:43 AM




0
(U)
. . . . . . . . . . . . . - - . . . . . . . . . . . . . . . . . . . . . . . . .
-1

C
Cn
C')
0
z 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Probability of Encountering Rock
Figure 5. Probability of Encountering Rock With 10 Cores in M
Upper Layers, Each of Thickness, D. Sheltering Included.
Fig5 Mar. 15, 2000 9:01:21 AM




It is interesting that for the fixed encounter probability of this problem of 50%, the volume and numbers of rock incorporated into the borrow area so defined are nearly independent of the proportion of the rock, Pv, present. In fact, it is believed that the volumes and numbers of rock are independent of Pv, and the variations in Table 2 are due to approximations made including reading values from Figure 2.
7 MOST PROBABLE NUMBER OF ROCKS ENCOUNTERED
7.1 Background and Example
In assessing the probability of encountering rocks through coring operations, it is useful to quantify the expectations that rocks will be encountered. To establish a correspondence, it may be useful to first refer to a coin flipping exercise in which the probability of encountering a head or tail in each flip is 0.5. It can be shown that for an exercise in which, say four flips are made, that the probability of exactly no heads occurring is 1/16. The probability of exactly two heads occurring is 3/8. The probability for each of the possible combinations is presented in Table 3.
Table 3
Probabilities of Occurrence for Various Possibilities in Example of Four Flips of a Coin
Occurrence Probability
No Heads, Four Tails 1/16
One Head, Three Tails 1/4
Two Heads, Two Heads 3/8
Three Heads, One Tail 1/4
Four Heads, No Tails 1/16
In general, if the probability of an occurrence in one event is p and the probability of a nonoccurrence is q (= 1-p), the probabilities of k occurrences in N trials is given in terms of the binomial coefficient, ( N ), where
k
N N! (17)
k (N k)!k!




where the "!" indicates factorial. The probability of a particular combination is (--)pkq N-k. It can
k
also be shown that the most probable number of encounters, Xavg is given by
Xavg = k p kqN-k (18)
For the coin flipping example, the most probable number of heads (or tails) is two, a not surprising result. This methodology can also be applied effectively to the case of rocks present in a borrow pit medium.
Suppose that the areal probability of a rock is PA (= p) as before. The most probable number of rock encounters by N cores is
x k )p k qN-k (19)
Xavg
The most probable number of rocks encountered by N cores penetrating M layers for a uniform distribution with depth, not including sheltering by the overlying layers, is
X MN MN k MN\
Xavg = E k pkqMNk (20)
Figure 6 presents the most probable number of encounters as a function of MN for P, = 0.05. Since PA (= P) = 1.5 Pv = 0.075 for this example, the results in Figure 6 demonstrate that
Xavg = p(NM) (21)
8 SUMMARY AND CONCLUSIONS
Methodology is developed at the conceptual level which relates the rock density within a potential borrow pit to the probability of encountering rock with N cores penetrating through M rock layers. Cases of uniformly distributed rocks and rock density varying with depth into the borrow pit are examined. The methods are presented for spherical rocks of a single diameter but could be generalized for more realist site-specific conditions. An example illustrates the volume and number of rocks that would be dredged if the borrow pit depth is defined by an encounter probability of 50%. The most probable number of rocks encountered by N cores penetrating M layers of a homogeneous




C
0
w
4
0
9)
a)
-0
a,)

U
-o
E
Cu
.0

0 10 20 30 40 50 60 70 80 90 Product of Number of Cores and Number of Rock Layers, MN Figure 6. Most Probable Number of Rocks Encountered, Example Presented is for Pv = 0.05.

100

Fig6 Mar. 21, 2000 10:09:27 AM




rock field is presented. A relationship is presented quantifying the most probable number of rocks encountered by N cores passing through M rock thickness layers. The utility of this expression is in judging whether with the number of cores and rock layers, rocks should have been encountered.
This methods developed here could be extended both to represent more fully the actual borrow pit conditions and in providing useful tools for interpreting coring results in situations where the unknown rock concentration varies with depth into the borrow pit.