Discontinuities and rock mass geometry

Int. d. Rock Mech. Min. Sci. & Geomech. Abstr. Vol. 16, pp. 339 to 362 Pergamon Press Ltd 1979. Printed in Great Britain

Discontinuities and Rock Mass GeomemJ J. A. HUDSON* S. D. PRIESTt~

Variation in discontinuity frequency as a function of scanline orientation in a plane is studied for rock masses containing sets of discontinuities. The spacing values between discontinuity intersection points that can occur along such scanlines are considered in order to develop a probability density distribution of block lengths. The ideas are extended to block area and volume distributions synthesized from the products of discontinuity spacirgt values along two and three axes respectively. Such distributions are also considered for rock masses where each discontinuity occurs with a random orientation. Histograms of discontinuity spacing and block area values compiled from measurements made on a variety of rock exposures from Pre-Cambrian to Jurassic are in general agreement with the theoretical distributions. The results are presented in dimensionless form and an explanation of the procedure for determining the scaling factors that allow computation of block area and volume distributions for a particular rock mass is included.

INTRODUCTION Initially, discontinuity frequency is discussed in terms This paper is concerned with rock mass geometry as a of borehole or scanline orientation. Consideration of function of both the orientation of discontinuities and the possible frequency distribution of spacings between the spacings between them. The research was stimu- discontinuities" then leads naturally to block length dislated within the context of civil and mining engineering tributions and a relation between discontinuity frein order to improve basic understanding of rock mass quency and the Rock Quality Designation (RQD). geometry and its characterization for analytical pur- When these ideas are considered along two orthogonal poses. It is well known that the mechanical behaviour axes, the concept of a block area distribution becomes of a rock mass can often be governed by the character- apparent; similarly, in three dimensions, a block istics of the discontinuities (a general term covering volume distribution can be generated. Such distribufaults, bedding planes, joints, fissures and micro-fis- tions are examined not only for discontinuities that sures) but a comprehensive statement about the discon- occur in sets of parallel planes but also for completely tinuities at a particular site can often be difficult to random geometries. On the experimental side, a variety produce due to the geometrical complexity of disconti- of rock exposures in the U K from Pre-Cambrian to nuity occurrence and the problems of reducing a large Jurassic was photographed in order to compare the quantity of associated meaurement data to a concise, theoretical discontinuity spacing and block area distriuseful form. The research presented here is aimed at the butions with those that occur in practice. The data are development of an approach to the fundamental geo- presented for individual locations and also in aggregate metry of discontinuity occurrence which will, in t u r n , for all locations, such aggregation being possible by lead to rational measurement techniques and character- normalizing the results and hence eliminating the scale. ization indices. It is emphasized, however, that the results in this paper refer to the geometry of rock BLOCK L E N G T H DISTRIBUTIONS masses: in order to predict the mechanical behaviour of The quantity of discontinuities present in a rock rock masses it is of course, also necessary to consider the mechanical properties of both the intact rock and mass is usually expressed by the discontinuity frethe discontinuities. quency which is defined as the mean number of discontinuities per unit length intersected along a borehole or * Directorate of Research Operations, Departments of Environscanline (measuring tape) set up on a rock exposure. ment and Transport, Westminster,London SWlP 3EB, U.K. ¢ Department of Mineral Resources Engineering Imperial College The way in which the discontinuity intersection points of Science and Technology,South Kensington, London SW7 2BP, are distributed along a line is, to some extent, taken U.K. ++Any correspondence concerning this paper should be addressed into account by the RQD value which is the percentage to Professor Priest as follows: Visiting Professor, Mining Engineer- of borehole core or scanline length that is composed of ing Universityof Wisconsin-Madison,Madison,WI 53706, U.S.A. intact rock lengths greater than 0.1 m. These two con339 R.M.M.S. 16/6--A

340

J. A. Hudson and S. D. Priest 0 iscontinuify

Scantine

I1!1 I I I U II II!11 I I i II

IIIII I lllL-"x, II ',1 I J. 4-1 I1 IIII II

I

I

I INiI

I

I

II

.,,,- ~

H

{b)

(a)

(C}

Fig. I. Discontinuity frequency along a line in a plane intersected by discontinuity sets.

cepts, the discontinuity frequency and the spacings between discontinuities will initially be discussed separately. Later, when considering the R Q D value along an arbitrary line through a rock mass and in extending the ideas to two and three dimensions, the concepts will be used together.

Discontinuity frequency Consider a single set of parallel, persistent and planar discontinuities producing, by their intersection with a plane, the linear features shown in Fig. la. The discontinuity frequency per unit length, 2, is measured in this plane along a line perpendicular to the discontinuity set by counting the number of discontinuities along a certain length of scanline, a method that can be used directly for field measurement [1]. If.. instead, the discontinuity frequency is determined along a line at angle 0 from the original scanline, then the new value, 20, will reduce from 2o = 2 when 0 = 0 ° to 2 0 = 0 , when 0 = 90 °. In fact, since any original spacing of x becomes x/cos& the mean spacing value, ~, is given by ~/cos0 and the frequency (being the reciprocal of the mean spacing) is given by 20 = 2 cos 0 where 0 is the acute angle between the scanline and the line perpendicular to the discontinuity set. If a general angle 0 is taken, then 20 = 2lcos 0l, the absolute value sign being necessary because the discontinuity frequency is always positive, and in this case is symmetrical about the 0 = 0 ° and 0 = 90 ° lines. Thus, the basic ~. value can readily be determined from a discontinuity frequency measurement made at an angle 0 to the pole of a discontinuity set. A similar ZO

approach to this was adopted by Terzaghi [2] in the analysis of joint survey errors. In the case where there are two discontinuity sets intersecting the plane with orthogonal traces and with fracture frequencies of 21 and 22 (Fig. lb), the value of 2o along an arbitrary line through the two sets at an acute angle 0 to the 2 t axis is the sum of the contributions from each set: 2 0 = ,Jq COS 0 + 2 2 s i n 0

The graph in Fig. 2a shows the variation of 2o as the scanline is rotated through two discontinuity sets with equal frequency, At = 22 = 10/m. The peak value of 20 = 14.14/m occurs for 0 = 45 °. The graph in Fig. 2b illustrates the variation of ~-0 when 2t = 5 and 22 = 15/m. As would be expected, 20 reaches a maximum further towards the 22 line in this case. It can readily be determined from equation (1) that the maximum value of ,i.o denoted by 2max occurs when the angle between the scanline and 21 axis is given by tan 0 = 22/21

-2 x ,4.ma

,~ + ,t~

18

16

16

12 10 h, = ~z: 101m 8 6

t~

h,:5/m

t~

2 o

2 I

I

I

I

i

15

30

~5

60

75

O(degrees} (a)

0

90

(3)

Thus, in the case of discontinuity sets with orthogonal traces in the plane, a knowledge of ,;q and 22 enables the direction and magnitude of the maximum discontinuity frequency in the plane to be determined from equations (2) and (3). Note, however, that the minimum value of 20 occurs along the ,i.t or 22 lines-whichever has the lowest value--and that this mini20

i8

(2)

and at this value of 0,

lB

12

(1)

I

I

I

t

I

15

30

~5

60

75

g0

O(de~rees) !b)

Fig. 2. Variation of discontinuity frequency as a scanline is rotated in a plane t h r o u g h two orthogonal discontinuity sets.

Discontinuities and Rock Mass Geometry mum is associated with a cusp on the ;to versus 0 curve, e.g. at 0 = 0 in Fig. 2b. If the two discontinuity sets intersect at angle ~b rather than 90 °, as in Fig lc, ).0 = )-licos 01 + ).21cos(0- 4~)1

341

Lower ht~misphere pro)~tion

(4) 062°

The maximum value of ;,o occurs between the )-~ and )-~ lines with a value derived from equation (4), .2

)-2 + 221).z cos ~ + 22

Area x

(5)

7t** W-

where 0 is given by tan 0 = ).2 sin ~/().i + ).e cos 0)

The variation in ).0 as a scanline is rotated through the traces of two discontinuity sets in a plane is shown in Fig. 3 for ).z = 10/m, )-2 = 5/m and ~b --- 30 °. The maximum value, obtained from equation (5) is 14.55/m and, from equation (6), this occurs in a direction 9.9 ° from the )-1 axis. The two abrupt changes in slope at 90 ° and 120 ° in Fig. 3 occur as the scanline becomes parallel to the )-1 and )-2 discontinuity sets respectively. The changes are abrupt because the contribution from a set does not asymptote to zero as the scanline becomes parallel to the set (d(cos O)/dO at 0 = 90 ° is not zero). The minimum ).0 value occurs along a scanline parallel to the set with the highest frequency, in this example parallel to the )-1 set. It can be shown from equation (6) that if )-1 = ).2, then the maximum value of 20 occurs when 0 = ~b/2: in other words, if the frequencies of two discontinuity sets are equal, the line of maximum discontinuity frequency bisects the perpendiculars to each set. The principle of adding the contribution from each set to establish the discontinuity frequency along a scanline can be extended to three dimensions and to any number of discontinuity sets: ).0 = ~ ).i cos 0i

(7)

i=1

where )-~ is the frequency of the ith set along the normal to the ith set and 0~ is the acute angle between the 16

12

.L to ~,

6

~z= S i r e

.L to )'L /

i"T,

E --

.

I

~cantint or borehote axis

(6)

r

/ 75*

Fig. 4. Determination of the angles between the normats to discontinuity sets and a scanline or borehole through a rock mass.

scanline and the normal to the ith set, where there are n sets. Given the orientations of the scanline or borehole and the normal to each discontinuity set, the 0i values can readily be determined using stereographic projection methods as shown by the example in Fig. 4. The orientations used in the example are given in Table 1. The following brief explanation is included for those readers who are unfamiliar with stereographic projection techniques. The plunge is the angle, measured in a vertical plane, between a given Erie and the horizontal. Plunge is plotted on the projection by measuring from the perimeter of the net along the N - S or E - W diameter. Trend is the azimuth of the vertical plane that contains a given line. By convention, this azimuth is measured in the direction of plunge of the given line. Azimuth is plotted by counting in clockwise rotation from N around the perimeter of the net. Having plotted the borehole or scanline axis and the 21, 22 and )-3 axes as points on the projection, the angles between these axes are measured along great circles on the projection, resulting in the values shown in Table 1. Using this technique and equation (7), the discontinuity frequency along any line through any number of discontinuity sets can be found. Although only the discontinuity frequency associated with sets of parallel, persistent and planar discontinuities has been discussed so far, one important conclusion is evident: the discontinuity frequency along a line TABLE 1.

I I I I I 30

I 60

I 90

I 120

I 150

ANGLES

Type of line 1150

(degrees)

Fig. 3. Variation of discontinuity frequency as a scanline is rotated in a plane through two discontinuity sets inclined at 30 °.

16flo

S

Borehole or scanline 2z axis 2: axis )-3 axis

ASSOCIATED

WITH

THE EXAMPLE

IN FIG. 4

Trend (degrees)

Plunge (degrees)

Resultant Oi values (degrees)

168 210 062 297

75 35 80 26

44 20 74

342

J.A. Hudson and S. D. Priest

through a rock mass is a function of both the frequencies of the individual discontinuity sets and the orientation of the line relative to the sets. In the two-dimensional example in Fig. 3 for two discontinuity sets with frequencies of 10/m and 5/m intersecting at 30°, the discontinuity frequency along a line rotated through 180° varies from a maximum of 14.55/m to a minimum of 2.5/m and, as will be explained later, this corresponds to RQD values ranging from 57 to 97% respectively. Such variation obviously has important implications both for discontinuity measurement procedures and engineering interpretation: measurement procedures should, ideally, be directed towards determination of the basic ,~ values; and, in design, it may be useful to have a knowledge of the magnitudes and directions of the maximum and minimum discontinuity frequencies.

>,~

/Standarddeviation

~,fdMean i Spacing value Negative ex onenfial A

Scar~ine-...._

I.

S~ating value

Unordered spacings

Spacmg value

hiform B

Normal

Spacings occurring /in ¢yctes

ii ~.

i Oisconfinui ~y intersection points

Spacings between discontinuities T t + In the previous discussion on discontinuity freT quency, the governing equations are valid whether the discontinuities are clustered, evenly spaced or distributed in some other way along a line. In order to pro,r vide a complete description of the rock mass geometry, it is necessary, however, to consider the spacings between discontinuities in addition to the frequency values. In the previous section, the discontinuities were assumed to be parallel, persistent and planar; none of these assumptions is necessary in the following study of spacing values. Fig. 5. Examples of theoretical discontinuityspacing distributions. Consider a line through a rock mass intersecting discontinuities at a variety of points as illustrated in Fig. 5. It is convenient to study the pattern of spacing values small spacing values, one with equal numbers of spacusing their probability density distribution, which could ing values and one with values equally distributed follow one of the distribution functions illustrated in about a mean spacing value. These, in themselves, cover Fig. 5. The negative exponential distribution character- a wide range of possibilities and taken in combination izes spacings associated with randomly positioned dis- are likely to approximate any in-situ distribution. Incontinuity intersection points along the line, implying deed, since it is probable that a number of geological that the positions of discontinuities are mutually inde- and mechanical factors combine during the various pendent. The uniform distribution, in which all spacing phases of fracturing to produce the current discontivalues up to a limiting value have an equal probability nuity patterns in rock masses, it may be assumed that of occurrence, could occur as in lines A or B in Fig. 5. a current distribution of discontinuity spacings has In line A, the spacings are not ordered; in line B, there resulted from the superimposition of two or more comare repeated cycles of progressively fractured zones. ponent distributions. If this is the case, then the total Finally, in the normal distribution, the mean disconti- discontinuity frequency is the sum of the component nuity spacing is the most commonly occurring value. frequencies but the probability density of a given spacThe occurrence of one discontinuity is then related to ing value is not equal to the scaled sum of the contributhe position of adjacent discontinuities, reflecting the tions from the component distributions. This is because original formation process. For example, the formation the spacing values have been superimposed and not of a discontinuity at a given location within a rock simply added; small values have tended to survive but mass might relieve the stress for a distance of approxi- large values have tended to disappear. mately 1/2 (the mean spacing value) into the rock on Simulation techniques were used to study the results each side. At this distance, when the stress again of combining discontinuity spacing distributions subreaches the strength of the rock, another discontinuity ject to mutual interference. A diagrammatic represenis formed and the cycle recommences. Variability of tation of the combination of two spacing distributions stress and strength will cause variations in this distance is shown in Fig. 6 and the related Monte Carlo simulaand hence control the standard deviation of disconti- tion procedure is described in the Appendix. Spacing values are progressively selected from each of the comnuity spacing. These three distributions have been chosen to cover ponent distributions and, as the simulation proceeds, the three broad possibilities: a distribution with many the spacing values of the resultant distribution are

Discontinuities and Rock Mass Geometry

343

COHPONENT DISTRIBUTIONS RESULTANT DISTRIBUTION

;', Spacing value

),f =~,1-X2

Low spacings mutually additive

~ t

I

1 TT I

TTT

,

Spacing value

-

High spacings Lost by

mutuaL interference ). Spicing value

Fig. 6. C o m b i n a t i o n of t w o discontinuity spacing distributions.

2 = 1, the value of ;t for each component distribution represents its proportional contribution. The negative exponential curve for 2 = 1 plotted on each graph illustrates that seven of the nine histograms can be described as generally of negative exponential form; the two exceptions occur when there is an 80% contribution from a normal distribution. The convergence to negative exponential form occurs because of the tendency for large spacing values to be broken up randomly. Thus, in a rock mass that has suffered several stages of fracturing, equivalent to several of the mixes

generated from the mutual interference of the component distributions. The results of these simulations are shown in Fig. 7 for combinations of negative exponential, uniform and normal distributions taken in pairs with values of 2 ranging from 0.2 to 0.8.* Since each resultant distribution has a frequency of * O n these and s u b s e q u e n t histograms, the term 'frequency' refers to the n u m e r i c a l frequency within a given class interval divided by b o t h the m a g n i t u d e of the class interval and the total n u m b e r of values in the sample. This p r o c e d u r e m a k e s 'frequency" directly equivalent to probability density, f(x), f(a) a n d f(v), for a n y given histogram.

~o I\ O.B

iiii 0.6

1.0

Uniform X;, 0.2 Normal ),=0.8 o'= _I-,1.,g./d7

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[_

0

0.5

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~

lath diigrlm

~. 0.~ 0.2

1.5

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1.5

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0.~.

0.2

0.2

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i

i

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0.8

.8

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i

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0.8

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0.2

0.2

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I

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t

t

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2.5

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2'.0

i 0.S

i

i

1.0 1.5 Spacing vatue

~ 2.0

g 2.5

Fig. 7. T h e effect of c o m b i n i n g pairs of discontinuity spacing distributions.

Spacing value

344

J.A. Hudson and S. D. Priest 10 , '"'.

for t=10/m

. ////

~

~

LIkxl

0.5

//

/

o

f

",,

h i

~

lencjth "

0.0 1

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flkxl/k . . _ ....... I

I

2

3

Normalized spacing vatue

-

-

" . . . . . .

kx

Fig. 8. Probability distributions and the cumulative length proportion for the negative exponential distribution of spacing values.

illustrated in Fig. 7, it is likely that the distribution of discontinuity spacing values along a line will be of negative exponential form. The probability density distribution of spacing values for a negative exponential distribution is given by f(x) = k

e -~x

where

kxf(x) dx to the scanline and hence the cumulative length proportion up to a spacing value l is given by

L(l) = For

the

negative

f2

kx f(x) dx

exponential

(8) distribution

L(I)--- 1 -e-X~(2/ + 1)

f(x) is the probability density. k is the mean discontinuity frequency. x is a spacing value. This function is shown for ~. = 1 in Fig. 7 and as the dotted line on the normalized graph in Fig. 8. The associated probability, F(x), that a given spacing value will be less than x is given by F(x)= 1 - e -~x which is plotted as the dashed line in Fig. 8.

where

f(x) = ke -~x, 19)

which is plotted as the solid line in Fig. 8. This Figure illustrates that, in this case, 50~o of the scanline consists of spacing values that are less than 1.68 times the mean spacing value, and 50~o of the spacing values are below 0.69 times the mean spacing value.

Relation with rock quality designation If, by regarding a scanline as analogous to a borehole, the Rock Quality Designation is defined as the percentage of scanline consisting of intact lengths over 0.1 m, the general relation between RQD and L(I) is

Cumulative length proportion Since the sampling method (i.e. the scanline) and the RQD = 100~1 - L(O.1)) resultant data (i.e. the spacing values) are both of linear form, it is possible to present the distribution of dis- and for the negative exponential distribution of spacing continuity spacings by means of a cumulative propor- values tion curve, which may have more direct geotechnical RQD = 100 e -°1~ (0.1~. + 1) (10) significance than the original distribution. The cumulative length proportion, L(I), is the proportion of scan- In Fig. 8, the value of unity on the kx axis could repline that consists of all spacing values up to a given resent a k value 10/m and an x value of 0.1 m (i.e. the value, l. This function can be determined once the prob- RQD threshold value). In this case, L(0.1) = 26~o and o/ Equation (10) and its ramifications ability density distribution has been identified for a par- the RQD = 74/0. ticular set of spacing values. The emphasis in the en- were discussed in an earlier paper by the authors [3]. The ideas presented in this section on discontinuity suing presentation is on the negative exponential distrispacing values and those in the previous section on bution because of the tendency for mixed distributions discontinuity frequency can now be combined. As a to converge to this form, but the concept can be applied scanline is rotated through discontinuity sets, the RQD to any distribution of spacing values. for any scanline position can be obtained from equaAlong a scanline containing, on average, k discontion (10) by inserting the appropriate value of ,i. (Note tinuity intersection points per unit length, the proporthat if the discontinuities do not follow a negative tion of spacing values lying in the range x to x + dx is exponential spacing value distribution, then an equivalf(x)dx where, as before, f(x) is the probability density ent expression to equation (10) must be determined.) In distribution of spacing values. Spacing values within the case where the traces of two discontinuity sets in a this range make a proportional length contribution of

Discontinuities and Rock Mass Geometry

345

l }~z II I1

I t Ill

I

i

ii

t

~t~

BIll I! 111 klP"

iI I lI I III

III I ~ t

~1111 III I I I I II

100

II

I

i

/ l Q O

ill

i

.

i

from X, discontinuities

i

80

6O ROD from k discontinuities-----7" ~ o

~0

o-

""ROD from both discontinuity sets

20

0

0

I

I

I

I

I

15

30

t,5

60

75

90

e (deqrees) Fig. 9. Variation of the Rock Quality Designation value as a scanline is rotated in a plane through two orthogonal discontinuity sets.

plane are orthogonal, equations (1) and (10) give RQD = 100e-°J;"(0.120 + 1)

(11)

where 2o = 2a cos 0 + 22 sin0 Equation (11) is plotted in Fig. 9 for a scanline rotating from 0 = 0 ° to 0 = 90 °. In this example, the RQDs perpendicular to the 21 and )+2 sets are 30% and 70% respectively. However, when the scanline makes an angle of 0 = 24.3 ° to the )+1 axis, the RQD reaches a minimum value of 25%. Clearly, as the 2o value rises and falls during scanline rotation, so the associated R Q D falls and rises following the relation in equation (11). Alternatively, if 21 = 22 = ll/rn for two orthogonal sets of traces, each equivalent to an RQD of 70~/~ the maximum value of 2o occurs for 0 = 45 ° where 2o = 15.56/m and the RQD is 54~o. BLOCK AREA DISTRIBUTIONS Having studied the discontinuity frequency and distribution of spacing values along a line, it is natural to extend the ideas to block area frequency and the distribution of block area values in a plane.

Area frequency In previous sections, discontinuity spacing values were considered in terms of a sample obtained from a scanline. The analogy in two dimensions is the sampling of areas bounded by discontinuities intersecting a scanplane, with the mean area frequency being the mean number of areas occurring per unit area of the scanplane. In a rock mass containing sets of discon* A tessellation is defined as the filling of a plane or volume by convex polygons or polyhedra that do not overlap.

tinuities, the mean area frequency determined from areas on a scanplane will be a function of the orientation of the scanplane relative to the discontinuity sets. This is directly analogous to the variation in discontinuity frequency along a rotating scanline and could be analysed in a similar way. Given that a specific number, n, of discontinuities intersect a unit area of a scanplane, there is a maximum and a minimum number of areas that can be bounded by the discontinuities. If none of the discontinuities intersect each other, e.g., they are parallel, the minimum number of areas, n + 1, is produced. Conversely, if all the discontinuities are mutually intersecting, the maximum number of areas, 1 + (n(n + 1)/2), is produced, an expression that can readily be established by induction. If n is 20, the minimum number of areas is 21 and the maximum is 211. Naturally, the number of areas produced by discontinuities in two orthogonal sets or occurring at random positions will be between these two extremes. It is, however, more instructive to consider such cases through the probability density distribution of areas because the mean area frequency, if required, can be obtained simply as the reciprocal of the mean area.

Area distributions for two discontinuity sets If the rock mass is divided into discrete blocks, a discontinuity spacing value can be regarded as a block dimension. The spacing value will represent a block edge length only if the scanline is parallel to one of the discontinuity sets; otherwise, the length refers to some traverse of a block, not necessarily through opposing faces. In Fig. 10a, a sample tessellation* of rectangles produced by two orthogonal discontinuity sets with negative exponential spacings is shown. Figure 10b illus-

346

J.A. Hudson and S. D. Priest I I I

I

III

I11

I 81

I I IIII

\ I

I ii':'

....... !! ,,!!!! II I [11 IH It llll

//

~j

! t

11 :!tit ..... ,

I I I Ill Ifln l I IIIl!

II I III i i ii

i,

X

/JP ~ t ~ l a

¢.//~

xy =a

//

I Ibl

(a)

tel

Fig. 10. Tessellations in a plane produced by two discontinuity sets.

trates the case where the two sets are not orthogonal. The )-t, set in Fig. 10b is the same as in Fig. 10a. The value 2~ in Fig. 10b is given by )-2 sin ~b since the )-~ set is inclined at angle 4~ to the )-~ set. However, the area of a given rectangle, R, in Fig. 10a is clearly the same as the corresponding parallelogram, P, in Fig. 10b. Thus, the distributions of areas are identical in these two cases and it is, therefore, sufficient to analyse the area distribution for two orthogonal sets to provide the area distribution for two discontinuity sets inclined at any angle ~b. Figure 10c illustrates how rectangles with equal areas are generated when the edge lengths, x and y, satisfy xy = a for a given value of area, a. 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(A ~ a), that an area, A, will be less than a given area, a, is found by integration of the product of the probability density function of x and the probability that y is less than a/x for all x values, where x and a/x are spacing values from each distribution along the orthogonal axes: P ( A ~< a) = ff(x)P(Y ~< y)dx

where (12)

For negative exponential spacing values, P(A ~< a) =

fo

2te-~x(1 - e-~2"/X)dx

where 2~ and 22 are the discontinuity frequencies along the x and y axes respectively. This reduces to P(a) = 1 - co KI(co)

where co = 2x//2~A2a and Kt(co) is a Bessel function of first order. The probability density distribution of areas, dP(a)/da, is given by f(a) = (co2/2a) Ko(cO) where Ko (co) is a Bessel function of zero order.

°[ t°a )1

P(a)=-am

log.

+ 1

fora ~ am

f ( a ) = - - I log eam am

a

where a m ---

4 = the maximum value of a. 212:

The term, ).t).z, which occurs in both solutions is the mean number of areas per unit area of the plane. In moving from the distribution of block lengths to the distribution of block areas, the probability density functions have become more complex: a negative exponential function has led to a Bessel function; and, in the case of the uniform distribution, a constant has led to a logarithm. In both cases, the highest probability is associated with the zero value, which is also a feature of the negative exponential distribution. Area distribution simulation

d

y = a/x.

Equation (13) obviously only applies for negative exponential spacing distributions. A uniform distribution of block edge lengths, for example, gives the following:

(13)

Solutions to equation (12), similar to those presented in the previous section, cannot readily be obtained for more complex edge length distributions such as the normal distribution. The Monte Carlo simulation procedure described in the Appendix was adopted to generate the probability functions for such intractable cases. Three tessellations were simulated for the cases of negative exponential, uniform and normal edge length distributions. The negative exponential and uniform cases were simulated in order to compare the simulation results with the theoretical solutions for these two distributions and thereby validate the simulation technique. The three probability density distributions of area produced by the simulation operation together with the theoretical solutions (shown by the dotted lines) are presented in Fig. 11. Clearly, the generation of 10,000 areas during simulation provides a close approximation to the probability density functions. The

347

Discontinuities and Rock Mass Geometry 2.~

Negative exponentiat )`1=1 )`2=1

2.0

1.6

7 g

1.2 fla)= (~ZaZ)ko (t~) 0.8

e~ = 2 X'/~'~-11)`2 a

O.t, i 0.5

1.2 -~ 1.0 \

=l

i 1.0 Normatised area, )'1 )`2a

I

I

1.5

2.0

2.5

Uniform )`1=1 )`2=1 ~

•

0.8

f(a)= a~- l°ge aa-~L

0.6 " -L 0.~,

~J"'-where i "~'-~l..~.~

am= )`~)` 1 2

,

0.2 0

- -===.==__~_ I

0.5

0

1.2

Normt

11)

)'1=1 ¢r1=0333 )'2=1 o'2=0.333

0.8

~

/

O.t,

j-

I

1.0 Nomalised area , )'1 12a

J

0.6

0.2

I

I-J-

I

I

I

1.5

FI

2'.0

2.5

I 1

0.5

_

I

1.0 Nomalised area , )'1 )'2 a

I

L--.

L_I___

I

I

1.5

2.0

--I

I

2.5

Fig. 11. Distributions of block areas generated by simulation of various edge length distributions.

values of 2a and 22 were taken as unity in the simulation because the probability density distribution of areas for any other values of 2 can be obtained from Fig. 11 simply by appropriately scaling the area axis.

Random tessellations

The previous solutions only apply for two sets of discontinuities, whether orthogonal or, with insertion of

348

J.A. Hudson and S. D. Priest 1,8 16 1., 12 f(a)

I/~~/ /

Random spacing. random orientation

Random spacing. orthogonatsets

1.0

i!;;ill:

0.8 0.6 0., 0.2

0'2

0., '

0.6 '

08 '

1.0 '

1.2 '

;,

,'.6

1.8 '

2.0 '

a

Fig. 12. Probability density distribution of areas for two types of r a n d o m tessellation.

the adjusted value, inclined at some angle. It is possible Rock Quality Designation. By direct extension, a to study the area distribution produced by a random cumulative area proportion, Ala), enables the proporgeometry where the discontinuities are distributed ran- tional contribution that areas below any value, a, make domly both in position and orientation throughout the to the total area of the plane, to be expressed. Such area rock mass, i.e., a completely random tessellation as proportion curves for areas produced by orthogonal illustrated in Fig. 12. Mathematical treatments of such sets with the three edge length distributions discussed random tessellations have been discussed by Goudsmit earlier and for the totally random geometry are shown 1/m. (The method of [4], Miles [5, 6] and Crain & Miles [7]. The area dis- in Fig. 13 for 21 = 2 2 = 2 = tributions for two sets of discontinuities and for a obtaining these curves is explained in the Appendix). totally random geometry are presented in Fig. 12. The The vertical axis of the graph in Fig. 13 is the percentcurve for the geometry with two discontinuity sets was age area of the plane, e.g., a rock surface, that consists plotted from equation (13); the curve for the random of block areas less than a specified area given by the geometry was obtained by Crain & Miles [7] who used value on the horizontal axis. The curves in Fig. 13 show that approximately 23% a Monte Carlo simulation technique involving the generation and mensuration by computer of nearly 200,000 of a rock face could be expected to consist of block polygons. Both plots have been normalized so that the value of unity on the area axis is the mean value. -- 1.0 ),1=1 )`2=1 ~ ~ The two curves in Fig. 12 are similar, indicating that the distribution of areas is not sensitive to discontinuity ~08 orientation assumptions. Also, in the rectangular tessellation, all areas have four sides; in the random geometry, the mean number of sides of all polygons is ......... ".... Randomspacing four 15, 6]. The mean area produced by two orthogonal ~" ntation sets is 1/2122; the mean area of the random polygons / /"'x~j J (extrapolated for 0.~, / / ~ ; ' ~ ~ ~1 )`Z a> 2) is 4/7Z2 2, where 2 is the discontinuity frequency along any line through the random tessellation [5,6]. The /~j ~ r m ~Brock edge tength mean perimeter in the orthogonal case is 2/2t + 2//22; ~02 y ~ Nega,ive i"dlsfributi°n' and it is 4/2 in the totally random case [5,6]. Cumulative area proportion In the one dimensional case discussed earlier, the idea of a cumulative length proportion was invoked both as a fundamental concept and as a link with the

0

1

2 3 Normatised area, ( )`1 )'2 a)

z~

5

Fig. 13. Cumulative block area proportion curves for two orthogonal discontinuity sets and for a totally r a n d o m discontinuity geometry.

Discontinuities and Rock Mass Geometry areas less than the mean area where the edge length spacings of two discontinuity sets follow a negative exponential distribution. This is equivalent to a two dimensional Rock Quality Designation of 77% (for ;.i = )~2 = 10/m and a threshold of 0.1 m x 0.I m). By comparison Fig. 8 shows that 26% of a total line is composed of individual lengths less than the mean length, assuming that the spacings are also negative exponentially distributed. Some discontinuities may not extend completely across the sample area; they may terminate either in intact rock or at another discontinuity. In the former case, certain discontinuities do not form convex block faces and therefore the applicability of the block area distributions described earlier will depend on the proportion of persistent discontinuities. In the latter case, if all discontinuities terminate at other discontinuities, the rock mass is indeed divided into discrete blocks and the distributions will apply. The extreme example of this latter case is that where no discontinuity extends beyond a block face, as is the case in the simulation. Thus, provided 21 and 22 have been correctly established, the assumptions necessary for the cumulative area distributions to apply are that the discontinuities are planar, are parallel within each set and that the sample plane is divided into discrete areas none of which contains a terminating discontinuity. BLOCK VOLUME DISTRIBUTIONS

Volume distributions for three discontinuity sets Three mutually orthogonal discontinuity sets will create a box tessellation in three dimensions. The probability density distribution of volumes can be determined by a direct extension of the methods described in the last section for the distribution of areas. This is illustrated in Fig. 14 for the simple case of a uniform distribution of edge length spacings. The results are P(v)=

(V/Vm){l + Iog,(Vm/V)+ ½(loge(Vm/V))2}

and f(v) =

~

(lOge(Vm/V)) 2

where v is the block volume and Vm = 8/~.12223, the maximum possible block volume when the edge lengths are uniformly distributed.

349

The block volume distributions produced by negative exponential and normally distributed edge length spacings present an intractable problem and have been determined by simulation. These distributions also apply for non-orthogonal sets provided that adjusted ~-i values are used, in the manner explained for area distributions.

Volume distribution simulation Distributions of block volumes produced by numerical simulation for various edge length distributions are shown in Fig. 15. The theoretical solution for the uniform distribution included in Fig. 15 demonstrates that the simulation of 10,000 blocks provides a good approximation to the appropriate probability density function. Using these volume frequency distributions and following the method described in the Appendix, the cumulative volume proportion curves illustrated in -Fig. 16 were produced. The vertical axis on the graphs gives the percentage volume of rock mass that consists of block volumes less than a volume specified by the value on the horizontal axis. For example, taking values 21 = ;t2 = 23 = 10/m, the percentage of a rock mass that consists of blocks with a volume less than 0.001 m 3 (i.e., the R Q D threshold of (0.1 m) 3) is 26% for negative exponential distributions of discontinuity spacings. This is equivalent to a three dimensional RQD of 74%.

The random division of space Miles [8] has studied the mathematical aspects of the random division of space. He states that "the fundamental and natural property without which little progress seems possible is that of stochastic homogeneity" and that "technically, a stochastic system is homogeneous if it is stochastically invariant under arbitrary translation". Whilst such tessellations can be anisotropic, having different 2 values in different directions, homogeneity is essential for the distributions to be valid. In other words, the theory can only be applied to a domain of a rock mass where the 2 values in a given direction are invariant, although 2 values measured within this domain will of course, exhibit sampling variability. Miles [8] also points to the necessity for consideration of edge effects in simulation procedures

Lengths in "~x direction Ill

f(x) fly) Lengths in

y direction

2

x2

~1

X2X3 ( t o g e ~ 16 t~ am= "7"7", ~2

. -

}2

•m- ~1 )'2;~3 Fig. 14. Derivation of the probability density distribution of block volumes for uniformly distributed edge lengths.

350

J.A. Hudson and S. D. Priest 3.6

Negative exponential ),1=1 ~,2=1 ~,3=I

3.2

2.8

2./*

2.0

~

1.6

f i t curve through histogram mid-points

_•Best ½

1.2

O.B

0./*

I

0

I

O.5

0

1.0

I~5

2.0

2.S

Normalised volume , t 1~,2),3 v

ZO 1.a

Uniform

~1=1 ),~=I ),3=1

16

12 10

=~I.0

0B

u~

Nomat 11=1 k2=1 13=1 o"1=0333 o ' ~

0.6

/where

O.t,

vm=

8

~o~ ,.,0.2

0.2

L__,.__

0

i

0.5 •

I

i

10 15 Normatisedvolume, tlk213v

L

i

2.0

2.5

i

05

10

15

20

25

Normahsedvotume , klk2k3v

Fig. 15. Distributions of block volumes generated by simulation of various edge length distributions.

of bounded domains but the reader will have noticed that the simulation of an implicit tessellation as described in the Appendix is automatically free from edge effects. Another important point mentioned by Miles [8] is that the magnitude of the mean block volume is not significant for theoretical considerations since the mean value only indicates "the scale of the volume distribution. The point also holds for one and two dimensional studies so, for this reason, scale in both the theory and experimental results has been eliminated by normaliz-

ing all values with respect to the mean value. When this is done for two sets of data in histogram form, the histograms may be added directly. In rock mechanics engineering practice, however, the value of the mean discontinuity spacing, block area or block volume may, of course, be absolutely vital to the stability of a given structure. Miles [-8] considered the box tessellation, which is produced by three sets of mutually orthogonal discontinuities with spacings following the negative exponential distribution and with parameters )-1, )-2 and ~-3-

Discontinuities and Rock Mass Geometry -~

1.0

~1--1 )

'

2

pared to the box tessellation, where the equivalent mean volume is 1/23.

~

0.8

] / / / ~

"~

~

0

J

0

~

i

1

I

Analogy with the particle size distribution for soils A block size distribution for rocks is analogous to the particle size distribution for soils, the former being inferred from measurements made along scanlines on rock exposures and the latter being determined directly by sieving. The particle size distribution for a soil could, however, be inferred from the measurement of grain boundaries along travelling microscope scanlines. To compare the cumulative volume distributions of soils and rocks, both are plotted in Fig. 17 with the sieve size related to mean particle volume using Marschner's empirical method [9]. The soil grading curves were taken from CP 2001:1957 [10]. In soil mechanics terms, the block volume curves in Fig. 17 show that normally distributed spacing values produce a relatively 'poorly graded' material. Clearly, this reflects the tendency for blocks with normally distributed edge lengths to have similar volumes. Conversely, blocks with widely differing edge lengths have widely differing volumes as in the case of the negative exponential distribution of edge lengths. However, comparison between the soil and rock curves shows that, when compared to soil, all the rock mass block curves represent 'poorly graded material' almost completely concentrated within two orders of magnitude of block volume 10 -4 to 10 - 2 m 3 for 21 = 2 2 = ;t3 = lO/m. In this context, the difference between soils and rocks is that degradation and sorting effects of transported particles can cause a relatively large proportion of the volume of a well graded soil deposit to be composed of particles of microscopic dimensions mixed with sand, gravel and pebbles. In the case of rock, the block size distribution is generated by the fracturing of a continuum without subsequent sorting and mixing of size fractions. The absence of these processes prevents very small rock blocks from forming a significant proportion

L Block s:lge length

~,Uniform Negafive

I

[- disfribufions

I

2 3 ~ Normatised volume, (Xl),2A3v)

I

I

f

5

6

7

Fig. 16. Cumulative block volume proportion curves for three orthogonal discontinuity sets.

In particular, he notes that E(vk) = (k!)3/(;q;t223)k for k - 1,2, 3 . . . where E(vk) is the expected value of vk. This gives the mean block volume as 1/Z1~2).3 and 222 the variance of block volumes as 7/;tl;t2).3. Miles [81 extends this work to cover the tessellation produced by a homogeneous, isotropic, Poisson, plane process, i.e., the polyhedra produced by random planes in space. From this work, it is possible to establish the probability of a given number of planes intersecting a given volume of space. This has clear implications for rock mechanics studies, e.g., the number of discontinuities intersected during construction of an underground excavation. Miles shows that the mean block volume is given by F-.,(V)= 6/n). 3 where 2 is the negative exponential parameter along any line through the tessellation. This illustrates the polyhedral influence in the random tessellation as corn-

~.~

Soil particle size (ram) 0.1 ,

0.01 ,

100

351

80

1.0 ,

10 ,

orma[

°~

0

10-7

10-6

10-5

10-4.

10-3

Approximate parfide volume(ram3)

10-2

10-1

100

101

102

Btock volume (m3}

Fig. 17. Grading curves for soil particlesand rock blocks.

103

104`

352

J.A. Hudson and S. D. Priest

of the rock mass volume in a statistically homogeneous domain. EXPERIMENTAL M E T H O D AND DATA The theoretical ideas presented in the first part of this paper indicate that mixing distributions of discontinuity spacing values results in convergence towards a negative exponential form of frequency versus spacing value curve. The fact that this distribution of spacing values occurs when the discontinuity intersection points are randomly positioned along a scanline (i.e., the points are a Poisson process) reflects the effectively random result when several independent fracture phases cumulate in the rock mass. However, because the discontinuity spacing distribution is the key to obtaining quantitative results from the theory, field measurements were required in order to verify the validity of the negative exponential distribution for characterizing discontinuity spacings. Similarly, it was necessary to measure the distribution of areas that can occur on a plane traversing a rock mass in order to provide a comparison with the theoretical area distribution curves presented in Fig. 12.

Site locations A variety of sites in the UK was chosen where rocks ranging in age from Pre-Cambrian to Jurassic were examined. The site locations are listed in Table 2. At each location, a reasonably planar unweathered rock face exhibiting clearly defined discontinuities was selected and photographed with the film plane parallel to the rock face. Scales were included in the field of view for subsequent determination of block dimensions from the photographs.

Measurement of discontinuity spacing and block area values Scanlines were extablished on 275 x 350 mm prints of the rock faces which are shown in Figs 18-27. The scanline technique involves the sampling and measurement of only those discontinuities that intersect a measuring tape set up on the rock face (or, in this context, a photograph of the rock face), thereby automati-

Fig. 18. Location 1: Jurassic limestone.

Fig. 20. Location 3: Carboniferouslimestone.

Fig. 19. Location 2: Carboniferoussandstone.

Fig. 21. Location 4: Carboniferouslimestone.

Discontinuities and Rock Mass Geometry

Fig. 25. Location 8: Ordovician mudstone.

Fig. 22. Location 5: Devonian limestone.

Fig. 23. Location 6: Silurian sandstone.

Fig. 24. Location 7: Ordovician dolerite.

Fig. 26. Location 9: Cambrian sandstone.

Fig. 27. Location 10: Pre-Cambrian diorite.

353

354

J.A. Hudson and S. D. Priest

cally generating data on the spacings between the discontinuity intersection points. A suitable sample area was delineated on each photograph and scanlines were positioned in order to provide data representative of the full sample area. From the theoretical ideas on discontinuity frequency discussed at the very beginning of this paper, it is clear that if there are two orthogonal discontinuity sets, a corresponding pair of orthogonal scanlines should be established, each perpendicular to a set, and only those discontinuities perpendicular to the scanline should be counted in each case. If there are two nonorthogonal discontinuity sets, two scanlines should again be established, each perpendicular to a set with only those discontinuities perpendicular to the scanlines being counted in each case. If there are three discontinuity sets, the same procedure should be adopted with three scanlines, and so on. Examination of the photographs, indicated, however, that considerable subjectivity could be involved in deciding to which discontinuity set, if any, an intermediate discontinuity belonged. In addition, for engineering purposes, this direct procedure for determining the frequency values could become too complicated for use as a general measurement technique. Thus to avoid these problems, scanline positions were established using the following two methods: 1. Orthogonal pairs of scanlines were established without reference to discontinuity orientation. 2. Pairs of scanlines were established where possible parallel to two major discontinuity sets. Neither of these methods allows measurement of the 2 values directly but both have their advantages. The first method has the advantage of objectivity and the second method has the advantage of removing one discontinuity set from the measurements along each scanline. It should be noted, however, that in general N scanlines are required for a complete analysis of N discontinuity sets (a full description of discontinuity scanline surveys will be the subject of a future paper). Also, if the discontinuities do not occur in sets but each with a random orientation, then, allowing for sampling error, the same 2 value will be obtained regardless of the scanline direction. Hence comparison of the discontinuity frequency measured along the scanlines used in both of the above methods provided some indication of the extent to which the discontinuities are actually orientated in sets. In both methods, the total scanline length in each direction was at least 50 times the mean discontinuity spacing. From theoretical studies [3], this gives a value of mean discontinuity frequency within + 20% of the true value at the 86~o confidence level. The discontinuity spacing values were measured directly from the scanlines drawn on the photographs and then converted to real lengths using a scaling factor determined from the metre scale included in the photographs. Each photograph was then cut along all discontinuity traces within the sample area. The resulting pieces, i.e., block

photographs, were weighed to +0.000l g. Using the appropriate scaling factor, the real areas of the blocks wer.e determined. On average, about 500 block photographs were processed in this way for each sample area. A calibration test carried out on 10 x 10 mm squares of photographic paper showed that the weighing accuracy and consistency of paper weight gave results accurate to within +lVo for this area. (The majority of pieces had an area of between 20 mm 2 and 200 mm2). The hisl~ograms of block lengths and block areas were plotted in dimensionless form by scaling the data by the mean value, thus producing a mean value of unity for each data set. Block length and area results

Table 2 gives both the actual values of mean and standard deviation and the normalized standard deviation of the spacing values measured in the four scanline directions at each location. For example, at Location 1, Jurassic limestone, the numbers (1) and (2) indicate the line of results in the table for the first and second of the pair of orthogonal scanlines. The numbers (3) and (4) indicate the results for the scanlines parallel to discontinuity sets. This layout applies for each location. One of the theoretical characteristics of the negative exponential distribution is that the mean and the standard deviation are equal. In Table 2, 35 out of the 40 scanline results, have a standard deviation within _+20~o of the mean value. The histograms of normalized spacing values at each location are shown in Fig. 28. The spacing values measured in each scanline direction were normalized by their individual means and then the four sets of results were added to produce the composite histogram for each location. The frequency axis represents the number of readings in each class interval divided by both the class interval and the total number of readings, thus making the values compatible with the probability density. The curve superimposed on each histogram is the negative exponential function for ,i-- 1. Clearly, there is a general tendency for the data to follow a negative exponential form. Similar histograms for normalized area values are shown in Fig. 29. The two superimposed curves on each histogram are the probability density distributions for area previously presented in Fig. 12. Again, there is a definite tendency for the data to follow the theoretical curves. Since the normalizing procedure has eliminated scale, it is possible to aggregate all the data for block lengths into a single histogram and all the block areas into a single histogram. An interpretation of this operation is that, when the data for each location are scaled by their own mean, they may be considered as a sample from a single rock mass which has a mean block length (or area) value of unity. Aggregating the data in this way increases the sample size and thereby improves the precision. The resulting histograms, one for block lengths and one for block areas, are shown in Fig. 30. It can be seen that the effect of having effectively ten times as much data 12489 measurements of block length and

355

Discontinuities and Rock Mass Geometry MEANANDSTANDARDDEVIATIONOFDISCONTINUITYSPACINGVALUESMEASUREDALONGSCANLINESAND TI-IEDISCONTINUITYFREQUENCIESDETERMINEDFROMTHESPACINGANDBLOCKAREAMEASUREMENTS

TABLE 2.

Location number, rock type 1 Jurassic limestone 2 Carboniferous sandstone 3 Carboniferous limestone 4 Carboniferous limestone 5 Devonian limestone 6 Silurian sandstone 7 Ordovician dolerite 8 Ordovician mudstone 9 Cambrian sandstone 10 Pre-Cambrian diorite

Scanlines mutually orthogonal Mean SD SD (m) (m) Mean (1)0.016 (2)0.022 0.041 0.041 0.116 0.058 0.083 0.111 0.046 0.033 0.298 0.332 0.054 0.054 0.176 0.233 0.044 0.061 0.060 0.040

0.015 0.021 0.030 0.033 0.095 0.050 0.085 0.104 0.044 0.026 0.289 0.301 0.047 0.057 0.165 0.215 0.040 0.038 0.063 0.034

5014 measurements of block area) has smoothed the histograms and made the agreement with theory much better. There is a tendency for the data in histogram form to show lower frequencies than might be expected in the first few class intervals. There are three possible explanations for this phenomenon. Firstly, the resolution of the experimental method may not have been sufficient to detect very small values so that a significant number may have been omitted (note that both the length and area theoretical distributions have a mode for the zero value). Secondly, very small pieces of rock may have weathered out, resulting in distortion of the original fracturing pattern. Thirdly, the real length and area distributions may not follow the predicted distributions in the lower class intervals. Further research is required to establish which of these explanations is valid. From an engineering point of view, it is of considerable interest to known whether the distribution of block areas could have been adequately predicted from scanline measurements. The theoretical block area distribution has been presented as a single dimensionless curve so that the determination of discontinuity frequency from scanlines is necessary to provide scale at a particular location. A comparison between the scaling factor determined from scanlines and that determined from measurement of block areas provides an indication, therefore, of how accurately the block area distribution could have been predicted from the scanline measurements. One method of determining the scaling factor from a set of scanlines is to compute the weighted mean discontinuity frequency along all scanlines at a particular location. The corresponding scaling factor for block areas is the square root of the mean number of block areas per unit area. These scaling facR,M M,S. I 0 6

B

Scanlines parallel to main sets 2 Mean SD SD from (m) (m) Mean scanlines

0.90 (3)0.017 0.98 (4)0.021 0.72 0.037 0.80 0.034 0.82 0.108 0.87 0.093 1.02 0.109 0.94 0.157 0.96 0.045 0.79 0.033 0.97 0.313 0.91 0.314 0.88 0.057 1.06 0.064 0.93 0.280 0.92 0.252 0.91 0.089 o.62 0.043 Io5 0.043 0.84 0.043

0.016 0.020 0.026 0.031 0.108 0.094 0.072 0.154 0.045 0.028 0.333 0.317 0.060 0.073 0.268 0.208 0.091 0.035 0.041 0.044

0.94 0.96 0.70 0.91 1.00 1.01 0.66 0.98 1.00 0.85 1.06 1.01 1.06 1.14 0.96 0.83 1.02 0.81 0.95 1.02

~. from areas

53.08

42.43

26.25

20.15

10.66

9.22

8.70

6.87

24.88

24.48

3.22

2.68

17.49

19.74

4.24

4.89

18.25

12.84

21.96

19.91

tors are compared at the right hand side of Table 2. Since there is reasonable agreement between the two columns of scaling factors, the distributions of block area could have been predicted sufficiently well for practical purposes from the scanline data using either of the theoretical curves presented in Fig. 12 and applying the appropriate scaling factor. For example, assume that a set of scanline measurements taken in various directions on a rock face has given a mean discontinuity frequency of say, 3.27/m. The mean area frequency is, therefore, (3.27)2 = 10.69/m 2 giving a mean block area of 0.094 m 2. The value of unity on the horizontal axis in Fig. 12 would therefore be scaled to 0.094 m 2 and a suitable class interval, Aa, could then be selected. The block area histogram can now be constructed by reading values of f(a) from either of the theoretical curves at the mid-point of each class interval. The percentage of block areas lying within a given class interval, Aa, is given by 100 f(a)Aa. It has been demonstrated that the distribution of block areas generated by discontinuities following a negative exponential spacing distribution along two orthogonal axes is very similar to actual block area distributions. It is likely, therefore, that the simulated block volume distribution plotted in dimensionless form in Fig. 15 for negative exponentionally distributed edge lengths will provide a good representation of the block volume distribution in rock masses. The block volume distribution for a given site may therefore be scaled off the curve fitted to the simulation results in Fig. 15 in a similar way to that described for block areas, faking in this case the mean block volume as the reciprocal of the overall mean discontinuity frequency cubed (1/23). The value of ). in this case should be determined from scanlines taken in various directions on the rock mass, ideally from three orthogonal rock faces.

356

J.A. Hudson and St D. Priest

Location 1 .~urasslc llmeSl~le

!.0

Locahon 2 Carbomfero~s sandstone

1.0

N = 257 0.8

N = Z61

'"-L

08

0.6

06

0.4

02

0.2

o.o 0.0

'

o'e

O.Z~

'

1.2

'

16

z'.0

0.0 00

i

i

i

i

i

0.4

OB

12

16

20

Spaci~ vatue

SDgting value

12

1.2 Location 5: Devonian

10

limestone

Lo(ation 6

10

Si[urlan sandstone

N = 230 08

08

0.6

~" ==

0.6

0.&

02

02

ool 00

i

O~

i

r

I 2

08

(

1.6

I

00

20

O0

i

i

0.~

OB

Spacincj value

Spacing

i

1.2

16

20

value

1Z

Locat$on 9 Cambrian sandstone

10

N = Z99

O~

02

O0

O0

,

L

~

i

j

0/~

08

12

16

20

Spacmg vaiue

Fig. 28. Histograms of normalized discontinuity spacing values for

Discontinuities and Rock Mass Geometry

357

1.2

1.2

Location 3: Cenboniferous Limestone

1.0

Location 4: C~rboniferous (imestone

10

N = 2L.5

N = 226 08

08

0.6

06

O~

0/*

02

0.2

O0

i

O0

• O~

i 06

~ 1.2

i 16

P 2.0

"%.

0.0 0.0

"-4-. i

,

0.g

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Spacin9 value

,

i

1.2

1.6

2.0

Spacing value

12

1.2

Locafion 7: Ordovician dolerite

1.0

~

.

.

~

Locittion e Ordovtcian mudstone

10

N=2S0

N= 236 08

0.8

== o6

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02

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0'~*

1.2

01.8

1'6

2'.o

12 Locafion 10: Pre-t'.ambrian diorite

N= 21,1 0.8 -=, 06

0.~.

0.2

o o oo

o'~

o'o

O0

o'~

o'.e Spacir~ vaLue

Spacing value

1.0

0.0

11z

1'6

Spacing value

each location, including the negative expon¢ntial curve for 2 -- 1.

zi0

112

116

2'.0

358

J.A. Hudson and S. D. Priest

16

T~

Locahon 1

1.2

i

Jurasstc limestone

i 10 i

Carbon(fecous sandsfone

LOCation 2

I\

N=672

N=S20

i\

...-Random spacang, random orientation

s

c ® 0@

s-Random spacing, orthogonal se~

0.6

06

I

0.6,

,i I 0.2

02~ I

0.0

b

o12

0.0

*

06,

Q6

i

i

L

0.8

10

12

Area

1 6"-

,

,

16,

16

,

i

18

20

ooJ

00

i

i

i

,

02

0 t,

06

08

l

,

10

I'2

16.

l

16

J

l

18

20

1:B

210

Area va(ue

value

2.000

. . . . .

16,

1.2

Location 6 : Silurian N=513

Location S: Oevonien limestone

~\i

N=6,35

1.2

l'\l

sandstone

'01

10

\ 08 \

\

06

0.6

0.6, O2

0.2 i

0.0

i

00

02

06,

~

L

i

i

06

08

10

12

i

16,

i

1.6

i i

i

18

Z0

0'00.0

012

016,

016

018

110

1:2

11.

116

Area value

Area value

16

16

LSt,oo,

12

c.0 .......

°s,ooe

10

oe

(16

O~

02

O0

00

L

i

J

b

i

i

~

=

,

,

02

0~,

06

08

10

12

1~.

16

18

20

Area value

Fig. 29. Histograms of normalized block area values for each

Discontinuities and Rock Mass Geometry 1.6

359

16 I~+

12

Location 3 Carboniferous hmestone

Location L. Carboniferous timesfone

12

N = 337

N= 281

10

10

08

== 0e

! 06

0.6

0~. ~

02

=

02

0.0

00

0'.2

0'"

016

018

,

1.0

112

,

I"

116

oo

,

1.8 210

O0

,

02

,

0L

,

06

.

0.8

1.868

16

1."

1,2

Location ? : Ordovicmn doterite

.

12

.

.

1..

.

16

18

20

1'8

210

1666,

Location 8

Ordov¢ian mudstone

N = 502

1.0

\

f, 0.8 g

~

0 . 8

&

0.6

06

0..

O.g

0.2

0.2

;2

0.0

o', o'~

o'8

,'o ;.2 Areavalue

,i.

,16

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1'.8

2'o

1682

16 16

- -If ~

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N= 570

0.0

.

10

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.

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Areo vILu!

1 Location10: Pre-[amDriandiorite N= 571

1.2 10 ~ O.B u0.6

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000.0 o',2

o:,

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Area vatue

location with superimposed theoretical distributions.

1',

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~,

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0'.6

;0 1'2 Areavalue

,i,

;6

360

J.A. Hudson and S. D. Priest 12

10

Locations l t o 1 0

aggregated data

N= 2t,89 08 w

= 0.6 ~u 0.~

0.2

0.0 0,0

J 0.2

~ 0.6

~ 0.6

~ 0.8

1=0

' 1.2

1' 4-

' 1.6

L 18

~ 20

Spacing vatue (a)

1.8

l t,

12

Locations 1 folO:

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.

data

= 10

~, 0.8

0.6

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0.2

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02

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Area vatue (b)

Fig. 30. H i s t o g r a m s of n o r m a l i z e d b l o c k length a n d b l o c k area d a t a a g g r e g a t e d from all locations.

CONCLUSIONS It has been shown from theoretical considerations that, where discontinuities occur in sets, the discontinuity frequency along a scanline through a rock mass is a function of scanline orientation. It follows that there will be a maximum frequency in one direction and a minimum in another. The spacings between the discontinuities along a scanline were also studied leading to the conclusion, from both theory and simulation results, that the spacings between discontinuities are likely to follow a negative exponential distribution. The cumulative length proportion (the proportion of scanline that consists of intact lengths below a certain value) can be quantified once the discontinuity spacing distribution is known. This provides a relation between the

Rock Quality Designation (RQD) and the discontinuity frequency since the RQD can be found from the cumulative length proportion at a spacing of 0.1 m, the RQD threshold. Since RQD and discontinuity frequency are related, RQD is also a function of scanline (or borehole) orientation in the rock mass, attaining a maximum in one direction and a minimum in another. For orthogonal sets of discontinuities with negative exponentional spacings, the frequency distribution of block areas on a plane is a Bessel function. Simulation techniques, which were verified by the Bessel function solution, were developed for generating the area frequency distribution for any discontinuity spacing function. The Bessel function solution for orthogonal sets and the corresponding area histogram for a completely

Discontinuities and Rock Mass Geometry

361

random geometry are very similar, indicating that the Brown and R. Coleman of Imperial College, Professor S. L. Crouch area frequency distribution is insensitive to disconti- of the University of Minnesota and Dr F. Garwood formerly of TRRL for extremely helpful discussions. The views expressed in this nuity orientation assumptions. Cumulative area pro- paper are those of the authors and not necessarily those of their portion curves, analogous to cumulative length propor- employers. tion curves, were also developed. Received 22 May 1979. The concepts were extended to three dimensions, leading to frequency distribution curves for block volumes and a cumulative volume proportion curve. The latter is basically a block size distribution curve directly equivalent to the particle size distribution curve REFERENCES for soils. In soil mechanics terms, all rock mass block 1. Price N. J. Fault and Joint Development in Brittle and Semi-Brittle size distribution curves are likely to be poorly graded, Rock, p. 176, Pergamon Press, London (1966). lying almost completely within two orders of magni2. Terzaghi R. D. Sources of errors in joint surveys. Geotechnique tude of block volume. 15, 287-304 (1965). 3. Priest S. D. & Hudson J. A. Discontinuity spacings in rock. Int. Ten rock exposures ranging through the geological J. Rock Mech. Min. Sci. & Geomech. Abstr. 13, 135-148 (1976). spectrum were studied in order to provide experimental 4. Goudsmit S. Random distribution of lines in a plane. Ret,. rood. block length and block area distribution data. In all Phys. 17, 321-322 (1945). 5. Miles R. E. Random polygons determined by random lines in a cases, the histograms of measured discontinuity spacplane. Proc. hath. Acad. Sci. U.S.A. 52, 901-907 (1964). ings were of negative exponential form. Also, in all 6. Miles R. E. The various aggregates of random polygons detercases, the histograms of measured block areas were in mined by random lines in a plane. Adv. Math. 10, 256-290 (1973). 7. Crain K. & Miles R. E. Monte Carlo estimates of the distribureasonable agreement with the area size distribution for tions of the random polygons determined by random lines in a random tessellations. It was found that the actual area plane. Statist. Comput. Siraul. 4, 293-325 (1976). distributions could have been adequately predicted 8. Miles R. E. The random division of space. Suppl. Adv. appl. Prob. 243-266 (1972). from discontinuity frequency measurements made 9. Mafschner A. W. A method for the size analysis of sand on a along scanlines for most locations. The methods for number frequency basis. J. sedim. Petrol. 23, 49-59 (1953). determining the scaling factors necessary to produce 10. Site investigations, British Standard Code of Practice CP2001, B.S.I., London (1957). area size distributions and block size distributions at a specific site were explained. The work presented in this paper has been aimed at providing an introduction to the concept of block length, area and volume distributions for rock masses. APPENDIX--SIMULATION TECHNIQUES Valid assessment of the engineering properties of a rock mass is hindered by the high cost of large scale field Monte Carlo simulation techniques were used to study the comtests and the questionable relevance of small scale bination of negative exponential, uniform and normal spacing distrilaboratory tests. In view of this, the characterization of butions. The simulation was carried out using a 500 step program on rock mass discontinuities and the evaluation of their a Hewlett Packard Model 9810A calculator, and the techniques are described below. relation with engineering performance is one of the keys to rational rock mechanics design practice. Some Generation of random variables Negative exponential distribution. Random spacing values, R~i, form of statistical approach must be invoked when rock from a negative exponential distribution with a parameter },E (the masses are analysed partly because of their inherent mean discontinuity frequency) were generated as follows: stochastic nature and partly because complete informa- Rr.i = ['-log~(l - R)]/2E where i = I. 2. . . . (}.eL - I), (2~L), where L tion concerning their geometry can never be obtained. is the scanline length and R is a random value from a uniform distribution over (0,1) generated within the calculator. The authors feel that the ability to summarize block Uniform distribution. Random spacing values, Rui, from a uniform lengths, areas and volumes by statistical distribution distribution with a parameter }-t were generated as follows: functions will prove to be of assistance in the character- R~,~ = 2R,'}.~ where i = 1,2....(,;.~L - II. (}.~L), where L is the scanline length and R is a random variable from a uniform distribution ization of rock mass geometry. Further research should over (0,I). Note R . . . . = 2/}.. be concentrated on the incorporation of the mechanical Normal distribution. Random variables, Z, from a normal distribuproperties of the block elements and the discontinuities tion with a mean of zero and a standard deviation of unity were generated by taking triplets of R values, R~, R 2, R3, where R is a into the model in order to allow complete characteriza- random value from a uniform distribution over (0,1). A pair of values tion of a rock mass in terms of its mechanical proper- of Z was calculated as follows: ties. Z1 = - (2 log~ R 1)1/2 cos(2nR2). Z2 = - ( 2 log, R1)1/2 sin(2nR2). Acknowledgements--The authors are particularly indebted to Dr L. J. Griffin and Mr M. P. O'Reilly (Heads of Structures Department and Tunnels Division, Transport and Road Research Laboratory) for their continued support of this work. This paper includes research results which were obtained when the authors were stationed at TRRL and these elements of the research are published by permission of the Director, TRRL. The authors are also grateful to Drs E. T.

if

* Handbook of Mathematical Functions, Applied Mathematics Series, US Department of Commerce, 1964, page 953.

where i = 1,2. . . . (2sL - 1), (2NL), and L is the scanline length. It was

One of these values of Z, selected at random, was then used to calculate RN~, a random value from a normal distribution with a mean of 1/2r~ and a standard deviation of 1/32 N using the criterion* if

R 3 < 0.5, RNi = (Z1/32r~) + (1/2N), discard Z2. Ra > 0.5,

RN~ = (Z2/3),r~) + (lfi;,N), discard ZI

362

J. A. H u d s o n

a n d S. D, P r i e s t

TABLE At. COMPUTATION OF CUMULATIVE LENGTH, AREAAND VOLUMEPROPORTIONS Mean discontinuity frequency of parent spacing distribution

Total length, area or volume

21

N,,'21

Total length, area or volume up to and including nth class interval

Cumulative length, area or volume proportion up to and including a value In& ) n

Length

t&,NI Y fix0 [ l i & )

-

i&/2)]

i=1

Area

2t, 22

• ftx~) [i&

N/~122

i=1

-

(&,"2)]

(.Lt.L2[N) 3" f(x,) [(iA~) - (&,/2}] i=l a

Volume

2t,22,2a

N/~t2223

(~.122.~a/N) ~ f(x,) ['(i&,) - {A~/2)] i=t

therefore possible to generate the following random variables: Negative exponential . . . . . . . . . . . .

Re~, parameter ~-E.

Uniform . . . . . . . . . . . . . . . . . . .

Ru~, parameter 2u

Normal . . . . . . . . . . . . . . . . . . . .

RN~, parameter ";'N

Combination of spacing distributions In order to combine a pair of distributions, it was necessary to specify the following: i. Total scanline length to be simulated, 2. The value of 2 for each of the component distributions. 3. The nature of the component spacing distributions. The simulation procedure involved selecting random variables from the component distributions, progressively combining them on a single scanline, and then re-computing the spacing values for the resultant distribution. This process is illustrated diagrammatically in Fig. 6. The process continued until a scanline length, L, of 10,000 had been simulated. As each spacing value for the resultant distribution was computed, it was allocated to one of the 24 spacing classes shown in Fig. 7. All combinations were designed to give a resultant 2 of I; thus, for a scanline length of 10,000 a total of 10,000 spacing values was generated.

Block size distributions Monte Carlo simulation techniques were also used to model the distributions of lengths, areas and volumes---in one, two and three dimensions--assuming uniform, normal and negative exponential distributions of discontinuity spacing values along lines through the simulated rock mass. The methods of generating random variables from negative exponential, uniform and normal distributions are described above. Area and volume distributions were synthesized by multiplication of pairs and triplets of random variables from each of these three spacing distributions. For each simulation, it was necessary to specify the following information:

1. The number of dimensions required. 2. The number, N, of units of length, area or volume to be generated. In all cases, N was set at 10,000. 3. The nature of the parent spacing distribution, i.e., whether negative exponential, uniform or normal. 4. The value of 2 for the parent distribution in each of the orthogonal directions as appropriate. For the one-dimensional simulation, values of discontinuity spacing were generated and allocated to one of 24 spaci~ag classes. The results of these simulations were used to check the program and are not presented in this paper. For the two-dimensional simulation, 10,000 values of area were generated and allocated to one of 24 area classes shown in Fig. 11. Similarly, for three dimensions, 10,t300 values of volume were generated and allocated to one of 24 volume classes shown in Fig. 15. The class interval for each histogram was chosen on the basis of the expected mean value for length, area and volume.

Computation of cumulative length, area and volume proportions Expressions have been presented in this p a ~ r that give the cumulative length proportion, L0), Ithe proportion of scanline that consists of spacing values up to a given value, l) as a function of ;. for uniform and negative exponential distributions of spacing values. Cumulative length (area or volume) proportions must, however, be computed from frequency distributions of length (area or volume), generated using simulation techniques where an analytical solution is not available. The terms used in the computation of these cumulative proportions are given below and the associated expressions are presented in Table A1. N = total number of length (area or volumet values generated in the simulation. Ax = class interval of frequency distribution histogram. f(x0 = numerical frequency of length {area or volume) values in the ith class interval of the frequency distribution. )q, 5.2, 23 = Mean discontinuity frequencies of the parent spacing distribution.