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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 24, 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. O. Box 116580
Gainesville, Florida 32611-6580
TABLE OF CONTENTS
LIST OF FIGURES .......................................................... iii
LIST OF TABLES ......................................................... iii
1 INTRODUCTION .................................................. 1
2 SITUATION CONSIDERED ............ ................................ 1
3 GENERAL ..... ...................................................... 2
4 TERMINOLOGY ..................................................... 2
5 METHODOLOGY ..................................................... 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. Case 1 .... ................................................... 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 Pv = 0.05 ... 12
LIST OF TABLES
TABLE PAGE
1 Notation to Be Used ........................... ... ................ ........ 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
1 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, Py.
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 -Sand
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, P,,
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 (
NR
p 4 3 (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 m" Layer
PNOE:N:m Probability of Not Encountering Rock With N Cores in m" Layer = 1 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:I,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 = ( -PA)N (3)
The probability that rock will be encountered by N cores through the m sh layer is
PE:N:m (1-PA)N (4)
The probability that rock will be encountered by N cores through M thicknesses of rock is
PE:N:1,M -(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
PE:1:2=PA(I -PA) (6)
Thus the probability that rock will not be encountered in either the upper layer or second layer by
one core is
PNOE:1:1,2 [-PA] 1 -PA(1 -PA)] (7)
The probability that rock will be encountered in the third layer by one core is
PE:1:3 PA1-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
0
0)
Cu
4-
.-
0
4--
0
L0
.0
2 3 4 5 6 7 8 101
2 3 4 5 678102
2 3 4 5 6 7 8 103
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
i -i i-i .' -
-----
:: : : : :Pv : : : : = 0.01
. . . . .. . ...... .. PV = 0.02 -
: : : : ,: : : : : : : .-.--- PV = 0.05
,. ,, .------- PV=0.10
I
i i : : : : ./ : : ,: : : : : :
/
. . . :. : .
'. .
i -,
,-
.
., -. . i .
' . .... . .
,r : : : : : .J : : : : : : '
s ... ... . .- IJ : : : '
u.u
100
We can now generalize to the mh layer with the probability of an encounter in the mth layer by one
core
m-i
PE:l: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-l
PE:1:m PA ( -P) = PA -PA) (11)
j=1
as before. The probability of non-encounter with N cores in the mh 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:,M PNOE:N:m (13)
m=l
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-PAI 1 _PAO _PA 11 _PAO _PA )2]
0
0
0
Ow
4-
0
CU
L..
0-
O
2D
0-
0
0Q
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
Density,
0.07
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
.. ..... .. ..
- /.
S. -- Without Sheltering
-.y' ......... -W ith Sheltering
-
-h
)0
m-1
PE:I:m=PA(m) I (1-PA(i))
j=1
(14)
and the probability that rock will not be encountered in the mh layer with N cores is
PNOE:N: (= (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 NOE:N:m
m=l
(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 VgocK (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
o -10 -10 ---
-J
-o
ca
-30
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
- o^.
0- s:
-o
Rock Volumetric Density, Pv
Depth Into Borrow Pit
Fig4 Mar. 15, 2000 9:09:43 AM
0
O
-o
Cn
a)
Co
Co
-0 -------------------..................-----...........
-30
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.
Fig Mar. 15, 2000 9:01:21 AM
Fig Mar 15 200 9:12 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 non-
occurrence is q (= l-p), the probabilities of k occurrences in N trials is given in terms of the
binomial coefficient, (-), where
k
N 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 = E k qN (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
xa k k qN-k (19)
k=o k
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
MN MN IIkiT\
Xavg = E k M k MN-k (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
0
w
CO
-o
I..
0
9)
Ia)
C
0
C
w
a)
0
cc
0
or
I-
U)
o
E
U)
-o
rCu
(0
0
L..
0
4J
c,)
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.
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