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Discontinuity frequency and block volume distribution in rock masses
M. Stavropoulou n
Department of Dynamic, Tectonic and Applied Geology, Faculty of Geology and Geoenvironment, University of Athens, GR-15771 Athens, Greece
a r t i c l e i n f o
Article history:
Received 26 November 2012
Received in revised form
16 July 2013
Accepted 4 November 2013
Keywords:
Discontinuity
Scanline
Drill core
Joint spacing
Joint frequency
Block volume
a b s t r a c t
The discontinuity spacing density function is theoretically found by applying the rst principles of the
Maximum Entropy Theory. It is shown that this function is the negative exponential probability density
function. Then, the analytical relation between RQD and discontinuity frequency that may be derived,
provided that discontinuity sets follow the negative exponential model, is validated against simulation
data. It is also found that if the discontinuity spacings follow the negative exponential distribution, then
the number of fractures per length measured along scanlines or drilled cores follow quite well the two-
parameter Weibull distribution function. Subsequently, following the methodology proposed originally
by Hudson and Priest, the closed-form expression of block volume distribution in a rock mass transected
by three mutually orthogonal discontinuity sets is found. Also, the left-truncated block volume
proportion above a certain block volume size is found analytically. The theoretical results referring to
discontinuity frequency and block volume distributions are nally successfully validated against
measurements carried out on drill cores and exposed walls, in a dolomitic marble quarry. The
methodology presented herein can be applied to rock engineering applications that necessitate the
characterization of rock mass discontinuities and discontinuity spacings are reasonably well represented
by the negative exponential probability density function. The proposed method for the prediction of
marble block volume distribution was applied to data from a quarry from drill cores and scanlines on
exposed quarry walls.
& 2013 Published by Elsevier Ltd.
1. Introduction
There are a number of important practical rock engineering
applications in which the knowledge of the geometry of rock
discontinuities (or fractures) is of paramount signicance for the
correct design and construction of surface or underground works
in the rock mass. It is noted that fractures are discontinuities
because they form discontinuities in the mechanical continuum.
Throughout the paper, the word discontinuityis used as a general
term encompassing cracks, ssures, joints, shear fractures, slip
bands, bedding planes etc. penetrating the rock and characterized
by low cohesion, that may adversely affect the strength of the rockmass as well as the quality of an extracted block from a surface or
underground quarry. For example, the in situ distribution of rock
block sizes formed by the mutually intersecting discontinuities,
may be used for evaluating the production capability of a deposit
to be mined using the block caving or the sublevel caving mining
methods[1]and to assess the requirements and design of material
handling systems in the mine. Laubscher [2] has considered
natural rock fragmentation as a major factor affecting the design
process for block caving operations and stated that while all rock
masses will cave, the manner of their caving and the resultant
fragmentation size distribution need to be predicted if cave mining
is to be implemented successfully. In caving operations, fragmen-
tation has a bearing on drawpoint spacing, dilution entry into the
draw column, draw control, drawpoint productivity and secondary
blasting/breaking costs.
In addition, rock mass characterization for the production of
large prismatic or irregular blocks used as armourstone, i.e. blocks
weighing many tonnes are used for building cover layers to resist
wave action [3], constitutes the most important part of the
exploration. Also, the success of marble quarrying operations usingdiamond wire cutting, chain sawing or pre-splitting or combina-
tions of these techniques, depends on the yield of blocks of
orthogonal parallelepiped shape with volumes greater than 1 m3
or 2 m3 or more, depending on the market demands for this
specic marble. The same remark holds true also for monumental,
building or decorative natural stone quarries of other geological
origin, like for example limestones, sandstones, granites etc.
Further, surface or underground stability analyses employing
the rock block distribution as a factor for the quantitative descrip-
tion of the deterioration of intact rock properties [4,5]. For
example Barton [6] proposes the ratio RQD/Jn to represent the
relative block size, in the Q rock mass classication system, with
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International Journal ofRock Mechanics & Mining Sciences
1365-1609/$ - see front matter & 2013 Published by Elsevier Ltd.
http://dx.doi.org/10.1016/j.ijrmms.2013.11.003
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International Journal of Rock Mechanics & Mining Sciences 65 (2014) 62 74
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RQD denoting the Rock Quality Designation and Jn denoting the
number of joint sets transecting the rock mass.
Furthermore, intact rock properties and the discontinuity
structure of a rock mass are among the most important variables
inuencing blasting results. This inuence is considered to be a
composite intrinsic property of a rock mass and is referred to as
the blastability of a rock mass. It represents the ease with
which the in situ rock mass can be fragmented and displaced by
blasting[7].It could be also mentioned that the in situ block volume
distribution inuences the pressure transient behavior of wells
drilled in naturally fractured reservoirs[8].
The present research was stimulated by the relevant series of
milestone papers [912]. It aims at improving the approach of
prediction of joint spacings and frequencies, RQD and block size
distributions based on scanline measurements or borehole log-
gings. Regarding the prediction of block size distribution it is
concerned only with discontinuity sets occurring of parallel
persistent planes irrespectively of the size of joints.
Initially, in Section 2, discontinuity frequency is discussed in
terms of borehole or scanline orientation. Consideration of the
possible density function of spacings between discontinuities by
virtue of the maximum entropy method [13,14] is given in
Appendix A. By virtue of Monte-Carlo simulations it is demon-
strated that if the joint spacings are generated via a Poisson
process, then the measured joint number per xed length of a
scanline or borehole core follows the Weibull density function. The
relation between RQD and joint frequency proposed by Priest and
Hudson [9] assuming that the joint spacings follow the negative
exponential density function is then validated in the same section
against Monte-Carlo simulation data.
InSection 3 these ideas are considered along three orthogo-
nal axes, and analytical formulae for the block volume distribu-
tion are presented. Such distributions are examined only for
discontinuities that occur in sets of parallel planes. On the
experimental side and in Section 4, the previous methods are
applied for an active open pit dolomitic marble quarry in order
to compare the theoretical discontinuity spacing and block
size distributions with those that occur in practice. Finally, the
main remarks that may be drawn from this work are outlined in
Section 5.
2. Discontinuity frequency along scanlines or boreholes and
its relation with RQD
As is mentioned in[9], Piteau[15]has used a scanline (measur-
ing tape) survey technique on rock faces and expressed disconti-
nuity intensity as the number of discontinuities per unit distance
normal to the strike of a set of sub-parallel discontinuities. It is
remarked here that the quantity of discontinuities present in arock mass is usually expressed by the discontinuity frequency
(or otherwise fracture frequency) denoted here by the symbol which is dened as the mean number of discontinuities per unit
length intersected along a borehole or scanline (measuring tape) set
up on a rock exposure. Furthermore, in the succeeding analysis the
symbolfwill denote the number of discontinuities per meter that
is measured at small regular intervals (i.e. sample support) along a
scanline or borehole core and it is assumed that all discontinuities
belonging to the same set or family are mutually parallel to
each other.
Discontinuities in rock are never uniformly distributed at all
orientations, but usually occur in sets. The mean true value of the
spacings of a given discontinuity set along a scanline or bore-
hole perpendicular to the discontinuities may be found by the
following formula
x 1N
N
i1xi 1
where N denotes the number of measured spacings along a
scanline and xi is the ith measurement of spacing including the
measurement resulting by adding the beginning and end values in
a core or interval [11]. For a sufciently large sample of spacing
measurements along a scanline of length L the following approx-imation holds true:
N=L 2The linear frequency value depends on the direction of the line
through the rock mass, and there is a maximum value in one
direction and a minimum value in another direction. Values of
linear frequency in different directions could be crucial in estimat-
ing both the true frequency perpendicular to the discontinuities, as
well as for the sizes of the formed rock blocks. Let be dened asthe solid acute angle subtended between the orientation of the
borehole or scanline, and the normal to the plane of the fractures.
Then, the apparent frequency along the scanline or boreholewill be less than that along the normal because the distance, x,
between two successive discontinuities intersected by the normalis increased toxx/cosalong the scanline where xdenotes theapparent or measured spacing. It may be easily inferred that the
apparent measured discontinuity frequency is given by
cos 3If the spacing between neighboring discontinuities of the same
set is considered to be a continuous Random Variable ( RV) and is
denoted by X, then its distribution function (known also as
cumulative function or cumulative distribution function, cdf) is
denoted as F(x)P{Xrx} and designates the probability that agiven spacing value X is less than x. For example the density
function (or probability density function, pdf)f(x) of spacing values
for a negative exponential distribution is given by
fx e x
4Then the corresponding cumulative function for a negative
exponential distribution is[11]
Fx Z x
0fd
Z x0
ed ex0 1e x 5
The negative exponential pdf of joint spacing values is a
frequently encountered function in fractured rock masses. The
reason for this, by recourse to the Maximum Entropyassumption
in heterogeneous rock masses, is demonstrated inAppendix A. The
analysis presented there shows that if the density function of
spacing values of a given joint set cannot be estimated from joint
spacing measurements for any reason, then it could be inferred
that the latter obey the negative exponential pdf based on the
hypothesis of maximum entropy (lack of information or uncer-tainty) of a joint system. Hence, the simple measurement of
number of joints per length along a scanline or borehole core of
sufcient total length gives the mean discontinuity frequency and the complete form of the distribution function. This hypoth-
esis is justied from many measurements of joint spacings in the
eld like in[11]and recently in [25]among others.
In the sequel we investigate which type of distribution function
is followed by the measured number of fractures per length f, that
is viewed also as a RV, provided that the joint spacings follow the
negative exponential distribution function. For this purpose we use
the Monte Carlo simulation method for producing a synthetic
sample of large size in the manner proposed by Hudson and Priest
[11]. That is to say, spacing values for any number of fracture sets
are progressively selected from each of the component distributions
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and as the simulation proceeds, the spacing values of the resultant
distribution are generated from the mutual interference of the
component distributions. All the component spacing distributions
are assumed to follow the negative exponential distribution func-
tion. In this simulation procedure the joint frequency values f, i.e.
the number of fractures per xed length along the synthetic
scanline or drilled core are also counted and stored as well. A
typical result of such a simulation in the form of a histogram of
number of joints per meter f (i.e. in this case we have chosen axed interval of length 1 m to count the number of fractures)
produced by the simulation of three joint sets with mean frequen-
cies 2, 2, and 0.4 m1 along a scanline of length 500 m is shown in
Fig. 1a. After quite many runs with different combinations of joint
sets from one to three and various associated mean joint frequen-
cies with this simulation procedure, it was found that the variance(var) of the f is approximately equal to the mean frequency value
which for the case of negative exponential distribution of spacings
is equal to the sum of the mean frequencies of the component
spacing distributions. Also, it was found that the Cumulative
Distribution Function (CDF) of the f values follows quite well the
Weibull distribution function that is given by the equation
Fwf 1ef=ab 6
where ina is the scale parameter and b the shape parameter of the
distribution. The mean and variance of the Weibull distribution
function are, respectively, given by
Ef 1b 1; varf 2f12b 121b1g 7
where E( ) denotes expectation, and ( ) is the Gamma function.Since there is no closed-form solution for the scale and shape
parameters of the two Eqs. (6) and (7) with the left-hand-sides of
these equations to be E(f) and var(f) (of course fordimensional homogeneity of both sides of the equation the value
of has to be multiplied by a constant of 1 m-1), the values of the
Weibull parameters are found by nonlinear regression analysis. At
a subsequent stage the mean and the variance predicted by the
Weibull distribution could be compared with the predicted values,
namely E(f) and var(f), respectively.Herein along with the Weibull distribution function we have
also tested the gamma distribution two-parameter function. The
gamma distribution models sum of exponentially distributed RVs
and is based on two parameters, namely the scale and shape
parameters, denoted here by the symbols ag, bg, respectively. Thechi-square and exponential distributions, which are children of
the gamma distribution, are one-parameter distributions that x
one of the two gamma parameters. The gamma pdfand cdf are
respectively
fgf 1
abg
gbgf
bg1
ef=ag;
Fgf bg;f=ag
bg 8
where ( ) denotes the incomplete Gamma function. The meanand variance of the gamma distribution function are, respectively,
Egf gbg; vargf a2gbg 9
The hypotheses that the observed frequencies of joint frequen-cies f in the various classes with those which would be given by
the Weibull and gamma density functions were tested by the
chi-square testthat is based on the frequencies in the various bins,
the Kolmogorov-Smirnov (KS) test and the regression coefcient.
A typical result of such a t for the case at hand is shown inFig. 1a
and b, referring to the frequency histogram and the cumulative
distribution, respectively.
FromTable 1that displays the results of the chi-square test at
the signicance level of0.01, it may be observed that whileWeibull distribution is passing this criterion, the gamma distribu-
tion is not passing it. The p value is the probability, under
assumption of the null hypothesis, of observing the given statistic
or one more extreme.
The K
S test like the chi-square test is one of the agreementbetween an observed distribution and an assumed theoretical one,
but it is based on the cumulative distribution function rather than
on the frequencies in the separate classes (bins). As it may be
observed from Table 2, t h e KS goodness-of-t value of the
Weibull distribution is lower than that of the gamma function
but both of them are greater than the critical value for 0.01.
0 2 4 6 8 10 12 14 160
20
40
60
80
100
120joint frequencies histogram
Joint frequency (1/m)
Frequency
Data histogram
Weibull PDF
gamma PDF
0 2 4 6 8 10 12 140
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Joint frequency (1/m)
CDF
joint frequencies CDF
Weibull CDF
gamma CDF
Data CDF
Median Value
Fig. 1. Simulation results referring to (a) discontinuity frequency histogram and
best-tted Weibull and gamma density functions, and (b) cumulative distribu-
tion of observed discontinuity frequencies with Weibull and gamma best-tted
functions.
Table 1
Chi-squared goodness of t for simulated joint frequencies (critical value at
signicance level 0.01 is 2L23.2093).
Distribution Observed value p-Value Performance
Weibull 11.9492 0.71153 Accept
Gamma 246.1232 1 Reject
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In conclusion, after many simulation runs it may be inferred
that in all cases the Weibull distribution function displays a better
performance compared to the gamma distribution function.
Thecumulative length proportion,L(l), is the proportion of scanline
or borehole that consists of all spacing values up to a given value, l.
This function can be determined once the density function has been
identied for a particular set of spacing values. The emphasis in the
ensuing presentation is on the negative exponential distribution based
on the maximum entropy hypothesis, but the concept can be applied
to any distribution of spacing values. Along a scanline containing, on
average, discontinuity intersection points per unit length, theproportion of spacing values lying in the range x to xdx is f(x)dx[17,11]where, as before, f(x) is the density function of spacing values.
Spacing values within this range make a proportional length contribu-
tion of xf(x)dx to the scanline and hence the cumulative lengthproportion up to a spacing value is given by
Ll Z l
0
xfxdx 10
If we substitute the expression forf(x) as is given by Eq.(4)into
Eq.(10)and integrate we nd the result
L 1e1 11
If we are interested in nding the percentage cumulative
proportional length that refers to intact core lengths greater than
(commonly this is taken equal to 0.1 m) which essentially refers
to the theoretical denition of RQD as has been proposed by Deere[18]with l 0.1 m, then from Eq.(11)it is obtained
RQDn 1001L 100e1 12
where the asterisk as a superscript denotes that this is a theore-
tical estimation of RQD.
Priest and Hudson [9] have derived the same formula and
based on a thorough comparison of its predictions with eld
measurements with scanlines in several sites they have found that
the maximum error does not exceed 5%. Herein the validity of this
nding was checked against several synthetic joint spacing
distributions along a scanline produced by recourse to a Monte-
Carlo simulation program for a combination of three joint sets
intersecting a scanline and obeying the negative exponentialdensity distribution. In these simulations the RQD is measured
along segments of equal length along the scanline. This length was
always at least fty times the mean discontinuity spacing accord-
ing to the results presented in [9]. The simulation method is
similar to the method proposed in[11].A typical result of the RQD
with threshold value of 0.1 m calculated every 10 m of
such a simulation, for a scanline length of 250 m that intersects
three discontinuity sets which exhibit apparent frequencies
12 2 m 1; 3 1 m 1 is shown in Fig. 2a. The length ofthe sampling interval for RQD estimation was selected to be
always equal or larger than fty times the mean spacing of
discontinuities that is equal to 1=123. The relative errorof actual and theoretically estimated RQDn by formula (12),
denoted here by the symbol Re, is estimated with the following
expression
Re
RQDnRQDs
RQDs 100
13
Table 2
KolmogorovSmirnov goodness of t for simulated joint frequen-
cies (critical value at signicance level 0.01 isDL 0.07243).
Distribution Observed value
Gamma 0.1556
Weibull 0.11138
0 50 100 150 200 25086
88
90
92
94
96
98
Comparison of RQD values
Distance along scanline [m]
RQD[%]
Exact
Estimated
0 50 100 150 200 250-4
-3
-2
-1
0
1
2
3
4
Distance along scanline [m]
Relativeerror[%]
0 0.5 1 1.5 2 2.5 3 3.5 40
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Absolute error [%]
CDF
maximum absolute error for 95% of measurements
Fig. 2. (a) Comparison between measured from synthetic experiments and
theoretical RQD; (b) distribution of relative error along the scanline and
(c) maximum absolute error including the 95% of the measurements.
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wherein RQDsdenotes the simulated RQD that represents here the
measured RQD.
For this simulation run, the distribution of the relative error
along the scanline is displayed in Fig. 2b. Also, the maximum
absolute value of the relative error corresponding to 95% of
measurements that is 3.5%, is also shown in Fig. 2c. Based on
several such simulations it was found that the formula (12)
predicts in a sufciently accurate manner the RQD of joints
following the negative exponential distribution, thus giving addi-tional validity to the observations and conclusions made in [9].
Hence, since RQD measurement on rock cores is a time consuming
process prone to measurement errors, the application of relation-
ship (12) can give good results by simply measuring the mean
number of discontinuitiesfencountered in segments along thescanline or borehole of equal length, provided that this length is
sufciently larger than the mean discontinuity spacing.
3. Block volume distribution
It is natural to extend the above ideas regarding spacing
distribution along lines to block volume frequency and thedistribution of block volume values in a 3D domain. Herein we
follow the methodology proposed originally by Hudson and Priest
[11]. These authors did not eventually nd the block volume
distribution in a closed form but rather they have presented
results based on the Monte Carlo simulation method.
In order to achieve the above aim we start here with the
simplest model proposed in [11] namely that of the block areas
distribution produced by two discontinuity sets having the same
strike in a plane perpendicular to their strike which intersect at an
acute angle in this plane as is shown in Fig. 3. The sketch at theright-hand side in the same gure illustrates how parallelograms
with equal areas are generated when the edge lengths, x and y,
satisfy xya for a given value of area, a. In the case of two non-orthogonally intersected discontinuity sets as shown in Fig. 3,
yy/sin, where y is the perpendicular distance between con-secutive joints of system 2 corresponding to the true frequency 2.The probability distribution of areas can be found by integrating
the product of the edge length densities for x and y along the
appropriate equal area hyperbolae. Alternatively, the probability, P
{Ara}, that an area, A, isolated between mutually intersecting
cracks will be less than a given area, a, is found by integration of
the product of the density function ofx and the probability that y
is less than a/x for allx values, where x and a/x are spacing values
from each distribution along the orthogonal axes [16]
PfArag Z
fxPfYrygdx 14
where yra/x. Substituting the result that may be found from
Eq.(5)
PfYryg Fy 1e2 sin y 1e2 sin a=x 15into the integral(14)
PfArag Z 1
0
1e1x1e 2 sin a=xdx 16
we may nd
Fa PfArag 1 ffiffiffi
ap
K1 ffiffiffi
ap 17
whereK1denotes the modied Bessel function of the 2nd kind and
of 1st order, and we have dened the following variable
2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi12 sin
q 18
The term 12sin appearing in the denition of above,indicates the mean number of block areas formed between
mutually intersection fractures per unit area of the plane. By
differentiation Eq. (16) w.r.t. a we nd the following density
function for the areas
fa dFada
2
2K0
ffiffiffia
p 19
where K0 denotes the modied Bessel function of 0th order. Thesame results represented by Eq's.(17)(19) have been also reached
by Hudson and Priest[11]for 901.Along the same line of reasoning, we may extend the above
results, to the case of distribution of volumes produced by three
mutually intersecting discontinuity sets in the three-dimensional
space. In this case we assume that the strike of the third joint set
makes an anglewith the common strike of the former two sets.This means that the block volumes are parallelepipeds formed by
six parallelograms. Hence, the probability P(Vrv) that a volume,
V, will be less than a given volume, v, is found by integration of the
product of the density function of areas given by Eq. (19) and
the probability thatzis less than v/a for allzvalues, where v/a are
the apparent spacing values along the Oz-axis that is orthogonal to
Ox and Oy-axes.
PfVrvg Z
faPfZrv=agda 20
in which zrv/a and
PfZrzg Fz 1e3 sin z 1e3 sin v=a 21Combining Eqs.(19)(21), the following integral representation
for the block volume distribution function is derived:
Fv P Vrvf g 12
Z 10
2K0 ffiffiffi
ap1e3 sin v=ada 22
The analytical evaluation of the above integral in terms of
known functions may be done by recourse to known properties of
Fig. 3. Block area distribution in a plane produced by two persistent discontinuity sets.
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Bessel functions[19]as follows:
Fv 1
k0
13k23 sin vk 1AklnvBk22k 3k2k12
23
where in we have dened the following parameters:
Ak 22k22 ln3 sin 4k14 ln 4 ln 2;Bk 4k2k12 ln3 sin 2
4 ln3 sin k24 ln k2k12 1k221k14k124 ln3 sin k14 ln 2k28 ln 8k1ln 2 8k1ln ln3 sin ln4 ln 2 ln3 sin 8 ln 2 ln8 ln 2 24
and(x) denoted the digamma function that is dened as
x dlnxdx
1x
dxdx
25
Also,(n)(x) stands for the nth polygamma function, which isthe nth derivative of the digamma function [20]:
nx dn
dxnx d
n 1
dxn 1ln x 26
By differentiation Eq. (23) w.r.t. the volume, v, we nd the
following density function for the volumes
fv 1
k0
13k23 sin vk 1v 1k1fAklnvBkgAk22k 3k2k12
27The above innite series converges rapidly and a nite number
of terms n may be retained for accurate results, i.e. it may be
approximated with the expression
fv n
k0
13k4v123 sin sin k 1v 1k1fAklnvBkgAk22k 3k2k12
28It is recommended though that several values of the number of
terms n in the series expression above should be used to
determine the convergent value of frequency for a given
123sin sin. The term 123sinsin appearing in theexpression for the cumulative volume distribution in Eq. (28)
above, indicates the mean number of volumes per unit volume
of the 3D space occupied by the rock mass and gives the scale of
the block volume distribution. To illustrate this, in Fig. 4 the
frequencies of block volumes f(v) as they given by Eq. (28) have
been plotted for three different values of the product 123at hand, assuming mutually orthogonal discontinuity sets (i.e.
/2).In direct analogy to the cumulative length proportion, L, the
cumulative volume proportion V(v) is found from the following
formula:
Vv 1Vtot
n
i1fvi iv
v
2
29
where v is the class interval of the volume frequency distribution
histogram,f(vi) is the numerical frequency of volume values in the ith
class interval of the frequency distribution, andVtotis the total volume
of rock mass. Following the above method, the cumulative volume
proportion curves illustrated inFig. 5were produced. The vertical axis
on the graph gives the percentage volume of rock mass that consists of
block volumes less than a volume specied by the value on the
horizontal axis. For example, taking values 1231/m, thepercentage of a rock mass that consists of blocks with a volume less
than 1 m3 is approximately 23% for negative exponential distributions
of discontinuity spacings. In marble quarries (a) the volume proportion
of blocks larger than a given volume as well as (b) the maximum block
volume that could be extracted is also of special interest. From
the same graph it may be seen that the maximum block size
for 12310 m3 is around 7 m3; whereas for the case of12316 m3 it is approximately equal to 5 m3.
It is a usual practice of a decorative stone quarry that only the
block volume distribution of extracted marble blocks above a
certain volume is assessed (for example vLZ1 m3), since the
blocks with lower volume from this threshold are dumped
as waste material without further measuring their volumes. Inorder to compare this experimental block volume distribution
with the theoretical one derived previously, we should be able to
derive the left-truncated block volume proportion above a certain
block volume size. Hence, based on basic statistical principles,
given a point of truncation, vL, the left-truncated cumulative
volume distribution V(v) denoted as VLTN(v) can be stated as
follows
VLTNv 0 vr0Vv vLVvL
1 VvL
( ; vZ0 30
Fig. 6 displays the truncated distributions of block volumes
above v1 m3
for the three mean block volumes at hand.
Fig. 4. Probability density functions of block volumes for three mean block
volumes 123at hand.
Fig. 5. Cumulative distributions with blocks smaller than given volume for three
mean block volumes 123at hand.
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4. Case study of a white dolomitic marble quarry
4.1. Basic structural features of the quarry and the method of
excavation of blocks
Understanding a quarry in terms of its potential for production
of decorative or building stones in the form of orthogonal blocks,
presents a special challenge for the mining engineer and the
engineering geologist. Unlike blasting in aggregates and miningoperations, optimization of the extraction process in a marble
quarry, for example, has a focus on the potential for production of
large orthogonal parallelepiped blocks with volume greater than
1 or 2 m3 that are free of crack-like defects.
An actual quarry is considered here that is located in a
dolomitic marble formation. The quarry has the form of an open
surface excavation with vertical benches of 6 m height. There were
identied three joint sets transecting the marble, namely the
grain, the head-grain that has the same strike and it is almost
orthogonal to the former set, and a secondary set with a strike
orthogonal to the common strike of the other two families of
joints. All the three sets were created during the uplift of the
Fig. 6. Left-truncated cumulative distributions above vL1 m3 with blocks smallerthan given volume for the three cases at hand.
Fig. 7. Lower hemisphere equal area stereographic projection of joint sets; (a) poles concentration, and (b) great circles of the three principal joint systems.
Fig. 8. Isometric view of the model of a jointed marble bench with excavated panel
with dimensions 10 10 6 m3. Diamond wire cutting planes oriented N201W(direction of head-grain planes) and orthogonal to it. Oy-axis points to the North.
The trace of the grain and head-grain are indicated in the ZOX plane, whereas the
trace of the secondary plane is indicated in the XOY plane.
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dolomitic marble layer of a thickness of 200300 m up to the
surface. The marble has been initially failed in shear along the
weakest bedding planes; at the same time the head-grain planes
as well as the secondary joints have been formed in order to
accommodate the large shear displacements along the master
sliding grain planes. The orientations of the poles of these three
sets processed with specialized software[21]are illustrated in the
lower-hemisphere stereographic projection diagram of Fig. 7. In
the same gure the great circles of these sets are also displayed.
The marble is excavated by using diamond wire cuts at three
mutually perpendicular planes as usual, i.e. one horizontal in a
rst place 10 10 m2, and then two vertical cuts with dimensions10 (horizontal) 6 (vertical) m2 oriented along the head-grainsand secondary joints, respectively. As was mentioned before the
bench height is 6 m according to the usual quarrying practice. The
initial dimensions of the panel are 10 10 6 m3. Subsequentlyfour vertical cuts are made along the grain planes at 22.5 m apart
in order to produce orthogonal parallelepiped sub-panels that areeasy to be tilted by the excavator. Such a typical panel of marble
with volume 10 10 6 m3 inside the quarry constructed byvirtue of a distinct element code [22] is shown in Fig. 8; since
this gure is for illustrating the directions of cuts with respect to
the joint orientations we have assumed a uniform distribution of
spacings of all three joint sets.
In order to facilitate the measurements of the spacings between
adjacent joints as well as the frequencies of the same set on each
photo of a box with cores, the joint traces appearing on the marble
cores have been properly identied and then marked carefully
with different colors as is shown in Fig. 9, namely: (a) grain planes
were marked with green color; (b) head-grain planes were marked
with red color, and (c) secondary planes were marked with
blue color.
4.2. Experimental results on discontinuity spacing
distributions and frequencies
Twenty vertical boreholes from the drilling campaign have
been logged for Fracture Frequency (FF) in units of (1/m) by
counting the number of joints per meter f, i.e. the support inthe original data is equal to 1 m. Support is a geostatistical term
indicating the size of a sample. The twenty vertical bore-
holes penetrating the marble were drilled mainly along EW and
NS directions. This approximates a direction perpendicular and
parallel to the strike of the bedding or grain planes that is almost
coincident with the strike of the head-grains (e.g.Fig. 7). Since the
secondary joints are steeply dipping, the joint frequency observed
along the drilled cores is mainly due to the grain and head-grain
Fig. 9. Method of marking the joints along the drilled core and apparent spacingmeasurements s1, s2, etc.
Table 3
Estimated apparent and true frequencies from spacing measurements on the drill
cores and on an exposed wall of the quarry.
Method of
measurement
Joint set Dip
angle
[deg]
Number of
measurements
Apparent
true
frequency,
[1/m]
Mean true
frequency,
[1/m]
Drill-core Grain 40 733 1.5 2
Drill-core Head-grain 70 618 1.3 3.7
Drill-core Secondary 85 41 0.14 1.66
Exposed
quarry wall
Secondary 85 354 1.71
Fig. 10. Distribution of the four marble qualities expressed as Fracture Frequency (1/m) along the vertical boreholes inside the planned quarry limits (see color bar for the four marble
qualities).
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joints (i.e. Table 3). For this reason in order to validate the joint
frequency distribution inferred from the drill cores, it was neces-
sary at a later stage to map the secondary planes along a
horizontal scanline oriented perpendicularly with the mean strike
of these planes. The location of the boreholes and the measured
FF's along them at every 1 m apart are shown in Fig. 10.
Joint apparent spacing data for the three principal joint sets
obtained from the drill core inspection are presented below in the
form of frequency histograms and cumulative distribution plots.
Three distribution functions have been best-tted on each set of
data namely, the one-parameter negative exponential distribution
function, the Weibull, as well as, the gamma two-parameter
distribution functions.Fig. 11a shows the grain spacings histogram
deduced from the spacing measurements along the vertical drill
cores and the best-tted density functions at hand. Fig. 11b
illustrates the cumulative distributions of measurements and of
0 1 2 3 4 5 60
100
200
300
400
500
600
700
800
joint spacings histogram
Spacing [m]
Frequency
Data histogram
Negative exponential PDF
gamma PDF
Weibull PDF
0 1 2 3 4 5 60
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
joint spacing [m]
CDF
joint spacings CDF
neg exponential CDF
gamma CDF
Weibull CDF
Data CDF
Median Value
0 1 2 3 4 5 6 7 80
100
200
300
400
500
600
joint spacings histogram
Spacing [m]
Frequency
0 1 2 3 4 5 6 7 80
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
joint spacing [m]
CDF
joint spacings CDF
neg exponential CDF
gamma CDF
Weibull CDF
Data CDFMedian Value
0 5 10 15 20 250
1
2
3
4
5
joint spacings histogram
Spacing [m]
Frequency
Data histogram
Negative exponential PDF
gamma PDF
Weibull PDF
0 5 10 15 20 250
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
joint spacing [m]
CDF
joint spacings CDF
neg exponential CDF
gamma CDF
Weibull CDF
Data CDF
Median Value
Data histogram
Negative exponential PDF
gamma PDF
Weibull PDF
Fig. 11. Histograms of measured apparent joint spacings measured along vertical drill cores and best-tted distribution functions for each joint set, i.e. (a) histogram of
grains, (b) cumulative curve of grains, (c) histogram of head-grains, (d) cumulative curve of head-grains, (e) histogram of secondary joints, and (f) cumulative curve of
secondary joints.
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the three theoretical cdf's for the same joint set. In a similar
fashion, Fig. 11c and d illustrates the results for the head-grains
andFig. 11e and f for the secondary joints occurring in the quarry.
Table 3 presents the main results of the mean apparent (mea-
sured) and true (corrected) frequencies of the three main joint sets
in marble.
Table 4 presents the results of the chi-square goodness-of-t
test of the three distribution functions on each set of joint spacing
data at the signicance level of0.01. The following observa-tions could be done from these results, namely: (a) The Weibulland gamma two-parameter distribution functions always exhibit
better performance compared to the one-parameter negative
exponential distribution function, with the only exception for the
case of secondary joint system in which the negative exponential
has better performance. This was expected since the negative
exponential function has only one parameter while the other two
have two parameters. (b) In only one case of the grain system the
negative exponential distribution displays a higher observed value
than the critical value of the test. (c) The Weibull distribution
function always displays a lower observed value and larger
correlation coefcient than the other two distribution functions.
In order to have a better picture of the performance of the
considered three distribution functions against the drill core data,
Table 5illustrates the results pertaining to the KS test at the same
signicance level. The main results of these KS tests may be
summarized as follows: (i) The observed values of all theoretical
functions are larger than the critical value for the grain and head-
grain joint sets, while the Weibull displays the lower observed
value. (ii) Regarding the secondary joint set all the distribution
functions display a lower observed value compared to the critical
one, with the negative exponential exhibiting the better perfor-
mance compared to the other two.
In order to check the validity of the estimated mean frequency
of the secondary joints from drill cores, additional measurements
of spacings of joints from this set have been carried out along a
horizontal scanline of 200 m length on an exposed vertical wall of
the quarry oriented in an orthogonal direction with the strike of
the almost vertical secondary joints. The results of this additional
survey are presented in the form of a frequency histogram and a
cumulative distribution inFig.12a and b, respectively; whereas the
mean joint frequency (in this case the apparent joint frequency isidentical with the true) is shown inTable 3. From this table it may
be observed that the secondary joints frequency is almost the
same for the two sampling methods. Again for this set of data the
three theoretical distribution functions have been best-tted and
displayed in the two graphs ofFig. 12a and b. The goodness-of-t
test results referring to the chi-squared and KS tests are also
displayed in Tables 4and5, respectively. FromTable 4it may be
seen that the negative exponential function exhibits the worst
performance, while the Weibull distribution has lower observed
value than the gamma distribution function. Also fromTable 5 it
could be observed that all the three distribution functions display
larger observed value compared to the KS critical value, and the
Weibull function exhibits the lowest observed value hence better
matches the data.
According to the results presented inSection 2, an indirect way
to check the validity of the exponential density hypothesis for the
joint spacings is by the virtue of the experimental distribution of
number of joints measured alongxed length intervals on the drill
cores, instead of the time consuming measurement of individual
joint spacings. For the case of simply measuring the number of
joints of all sets occurring every one meter of drill core extracted
from vertical boreholes, labeled as FF and corresponding to the
experimental f values, Fig. 13a to b presents the all 868 data
Table 4
Chi-squared goodness-of-t for measured apparent joint spacings at signicance level 0.01 and correlation coefcient.
Joint set Distribution function Critical value at 0.01 Observed value Correlation coef cient
Grain Negative exponential 27.6882 43.2453 0.98585
Weibull 26.217 4.2194 0.99409
Gamma 26.217 10.5802 0.99066
Head-grain Negative exponential 30.5779 15.5321 0.98905
Weibull 29.1412 2.0986 0.99403
Gamma 29.1412 6.018 0.99138
Secondary (drill core) Negative exponential 44.3141 3.0168 0.99548
Weibull 42.9798 3.3869 0.99552
Gamma 42.9798 3.086 0.99548
Secondary (exposed vertical wall) Negative exponential 30.5779 135.0054 0.95131
Weibull 29.1412 3.7125 0.98871
Gamma 29.1412 12.5709 0.97697
Table 5KolmogorovSmirnov goodness-of-t for measured apparent joint spacings at signicance level 0.01.
Joint set Distribution function Critical value at 0.01 Observed value
Grain Negative exponential 0.059876 0.13642
Weibull 0.059876 0.070794
Gamma 0.059876 0.094703
Head-grain Negative exponential 0.065184 0.10992
Weibull 0.065184 0.065866
Gamma 0.065184 0.086522
Secondary (drill core) Negative exponential 0.24904 0.079552
Weibull 0.24904 0.082099
Gamma 0.24904 0.08007
Secondary (exposed vertical wall) Negative exponential 0.085993 0.27336
Weibull 0.085993 0.092263
Gamma 0.085993 0.13967
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measured on borehole cores lying inside the planned nal excava-
tion boundaries along with the best-tted Weibull cdf. In order to
compare the performance of the Weibull distribution function
with another candidate function, we have plotted in Fig. 13a and b
the pdf and cdf curves of the gamma distribution function.
The goodness-of-t results performed by virtue of the chi-
squared and KS tests are illustrated in Tables 6 and 7, respec-
tively. As it may be observed from Table 6 both distribution
functions are passing the criterion with the Weibull functionbetter matching the data since it exhibits a lower observed value.
From Table 7 it may be observed that both functions display a
higher observed value than the critical one prescribed by the KS
goodness-of-t test. However, also in this case the observed value
of the Weibull distribution is lower than that of the gamma
function.
4.3. Block volume distribution
Having estimated the mean true joint frequencies of the three
main joint sets in the quarry and having measured the volumes of
the extracted marble blocks above a certain volume size at the
quarry, a comparison could be made between the theoretical
model expressed by Eq. (30) and the actual block volume
measurements performed after tilting of a large number of sub-
panels. It is noted that according to the practice followed in this
particular quarry, there are measured volumes of blocks of volume
larger than 1 m3, which means a left-truncated distribution of
0 2 4 6 8 100
50
100
150
200
250
300
350
400
450
joint spacings histogram
Spacing [m]
Frequen
cy
Data histogram
Negative exponential PDF
gamma PDF
Weibull PDF
0 2 4 6 8 100
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
joint spacing [m]
CDF
joint spacings CDF
neg exponential CDF
gamma CDF
Weibull CDF
Data CDF
Median Value
Fig.12. Histogram (a) and cumulative distribution (b) of measured secondary joint
spacings on an exposed quarry wall aligned perpendicular to the strike of
secondary joints and best-tted distribution functions.
0 5 10 15 20 25 30 350
20
40
60
80
100
120
140
160
180
joint frequencies histogram
Joint frequency [1/m]
Frequency
Data histogram
Weibull Distribution PDF
gamma PDF
0 5 10 15 20 25 300
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Joint frequency [1/m]
CDF
joint frequencies CDF
Weibull CDF
gamma CDF
Data CDF
Median Value
Fig. 13. Nonlinear regression of measured total number of joints per meter along
drill cores with the Weibull and gamma distribution functions; (a) experimental
and theoretical histograms and (b) experimental and theoretical cumulative
distributions.
Table 6
Chi-squared goodness of t for measured joint frequencies along the boreholes
inside the nal excavation boundaries (critical value at signicance level 0.01 is2L
42.9798).
Distribution Observed value
Weibull 2.5565
Gamma 2.9936
Table 7
KolmogorovSmirnov goodness of t for counted joint frequencies on the drill
cores inside nal excavation boundaries (critical value at signicance level 0.01is DL0.058577).
Distribution Observed Value
Gamma 0.17712
Weibull 0.18906
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measured volumes. Fig. 14 shows the comparison of the actual
left-truncated marble block distribution with the predicted dis-
tribution given by the analytical Eqs. (28)(30) with a mean joint
frequency of the three main joint sets (i.e. the sum of the mean
frequencies 123 of the joint sets) of 7.4 1/m displayed inTable 3and based on exploratory borehole data.
The comparison shown in Fig. 14 is very good given the
inherent assumptions of the theory and the complexity of the
natural rock fragmentation conditions. The maximum measure
block volume is 6.5 m3 whereas the theoretical model predicts
slightly larger maximum block volume of around 7.5 m3. This can
be explained by the fact that at the quarry the original block
volume distribution is inevitably affected by the vertical and
horizontal diamond wire sawing cuts. It may be also noticed that
the theoretical curve is shifted to the left relatively to the
experimental one. This can be explained by the inherent assump-
tion of the model that the measured discontinuities along the drill
cores are persistent whereas in reality a percentage of them
correspond to intermittent joints.
5. Concluding remarks
The work presented above was stimulated by the relevant
series of milestone papers [912]. It aims at improving the
approach of prediction of joint spacings and number of joints
per length, RQD and block size distributions based on scanline
measurements or measurements on drill cores at the exploratory
phase. Regarding the prediction of block size distribution it is
concerned only with discontinuity sets occurring of parallel
persistent planes irrespectively of the size of joints. Finally, theresults found here are validated against measurements of joints on
drill cores taken from a dolomitic marble quarry. The following
conclusions may be drawn from this study:
The joint density function found in a theoretical manner by
applying the maximum entropy theory is the one-parameter
negative exponential function. It is noted that a more rened
and elaborate model of joint spacings such that presented in
Appendix A with appropriate mechanical constraints could be
created but this is out of the scope of this paper.
The relation between RQD and joint frequency found by Priest
and Hudson[9] has been validated against simulation data.
Aiming at inferring the mean joint frequency from measure-
ments of number of joints per meter along drill cores instead from
the more cumbersome joint spacing measurements, it has been
found that if the joint spacings follow the negative exponential
distribution, then the measured number of joints per length of
drilled core follows the Weibull density function with scale and
shape parameters related only to the mean frequency of joints.
Following the methodology proposed originally by Hudson and
Priest[10]the closed-form expression of block volume distribution
has been found.
Also, the distribution of block volumes has been found analy-
tically. Furthermore, the left-truncated block volumes distributionhas been found in analytical form.
The theoretical results are validated against experimental data
collected at a dolomitic marble quarry referring to joint spacings
and frequencies sampling, as well as marble block volumes. Joint
spacings data have been best-tted by three pdfs namely the one-
parameter negative exponential, and the two-parameters gamma
and Weibull pdfs. As was expected in most of the cases it was
found that the Weibull and gamma pdfs t better with the data,
however the much simpler negative exponential pdf was found to
describe adequately the experimental data.
Appendix A. Maximum entropy theory applied inheterogeneous rock masses
Methods from Information Theory [23] and entropy theory
[13,14] are employed in order to derive the form of density
function for the joint spacings in the case of a lack of previous
information. This lack of information pertains to: (a) the inuence
of random factors on stress distribution inside the heterogeneous
rock mass that cause high variability of the local stresses from the
values which are calculated using the averaged constants of the
elastic or plastic solutions, (b) the heterogeneity of rock strength,
and (c) the type, succession and intensity of previous tectonic
episodes responsible for the current state of fracturing of a
rock mass.
The density function is derived here from a condition ofmaximum likelihood of a given state of rock mass fracturing,
which corresponds to a maximum entropy of the blocky and
fractured rock mass. The condition for the maximum entropy of
this process could be written in the following manner
Z 1
0fxlnfxdx-max A:1
However, the density function must satisfy two constraints.
Since Fx Rx0 fd, and F(1)1, the rst constraint is thefollowing well known integral equation
Z 10
fxdx 1 A:2
The second constraint may be derived by an energy balance
applied to a given volume of the rock V of the internal specic
volume energy absorbed by the rock denoted by wV (units of
energy divided by the volume of strained rock) and the assump-
tion that all the volume energy is converted into surface energy of
cracks wA (units of energy divided by area). Then we assume that
the expected value or mean of joint spacings is proportional to the
ratio of specic energies as follows
Z 10
xfxdx kwAwV
A:3
wherek is a proportionality constant.
The Lagrangian of the system [24] as usual is the sum of the
objective function we want to maximize (i.e. Eq. (A.1)), plus the
Fig. 14. Comparison of the left-truncated actual block volume distribution (circles)
with the analytical distribution function given by Eq. (30)(line).
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constraints i.e (A.2) a n d (A.3) each multiplied by a Lagrange
multiplier, i.e.
f;x;1;2 Z 1
0
fxlnfxdx
1 1Z 1
0
fxdx
2 kwAwV
Z 1
0
xfxdx
A:4
where {1, 2} denote the Lagrange multipliers. From Eq. (A.4) it
may be observed that the Lagrangian Z is a function of fourvariables. Maximizing the Lagrangian, one may obtain the follow-
ing result
Z
f 0 3 fx e 1 1e2x; A:5
Finally, inserting the above expression (A.5)for the estimated
frequency into Eq. (A.2) one of the Lagrange multipliers may be
eliminated as follows:Z 10
fxdx 1 3 2 e 1 1 A:6
Hence,
f x 2e2x A:7
and it may be noted that the mean frequency of joints isessentially a Lagrange multiplier. Finally, combining the above
Eq.(A.7)into(A.3)the frequency parameter2 may be expressedas follows
2
Z 10
xe2xdx kwAwV
3 21
k
wVwA
A:8
The above formula means that according to the supposed
simple model the mean frequency of joints is proportional to the
ratio of specic volume to specic fracture surface energies that
are responsible for the rock fracturing.
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