discontinuity frecuency and block volume distribution in rock masses

8/13/2019 Discontinuity Frecuency and Block Volume Distribution in Rock Masses

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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

Contents lists available atScienceDirect

journal homepage: www.elsevier.com/locate/ijrmms

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

n Tel.:30 210 727 4778; fax:30 210 727 4096.E-mail address: [email protected]

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

M. Stavropoulou / International Journal of Rock Mechanics & Mining Sciences 65 (2014) 62 74 63


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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

M. Stavropoulou / International Journal of Rock Mechanics & Mining Sciences 65 (2014) 62 7464


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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.



8/13

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).



9/13

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


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


Spacing [m]

Frequency

Data histogram


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


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.



10/13

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



11/13

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


Spacing [m]

Frequen

cy

Data histogram


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



12/13

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).



13/13

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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discontinuity frecuency and block volume distribution in rock masses

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