Stereological interpretation of joint trace data: Influence of joint shape and implications for geological surveys

blt, J. Rock Mech. Mm. Sci. & Geomech. Abstr. Vol. 17, pp. 305 to 316 ~' Pergamon Press Lid 1980. Printed in Great Britain

0020-7624/80 1201-0305502.00/0

Stereological Interpretation of Joint Trace Data: Influence of Joint Shape and Implications for Geological Surveys P. M. W A R B U R T O N *

The paper presents a new statistical model Jbr the ,qeometrical and spatial distributions of.joints, incorporating a joint shape based on the parallelo,qram. The model is linked with geological surveys by analytical predictions O[ trace patterns, coverinq area and line sampling, distributions ot trace lengths and spacings, and allowance fi)r truncation. Care is taken to express the equations in suitable forms lbr numerical evaluation. The predicted trace patterns are examined over the fidl ran qe o[ exposure orientations and are shown to be generally consistent with reported observations. Implications _.flor geological surveys are discussed in some detail, together with ways q[ obtaining the parameters of the model trom .tield data. The significance of the present theory is that it suggests a new approach to data collection aml interpretation and establishes a stereolo~tical [i'amework that can readily allow for fiaure model developments.

INTRODUCTION Discontinuities have a major influence on the stability of excavations in rock and are often incorporated, in some way, into models of mechanical response. Although most of these models are deterministic, many characteristics of discontinuities are best described statistically, and thcre are intrinsic advantages in mechanistic approaches that directly utilize the relevant statistical distributions. Estimates of probabilities of failure arise as a natural consequence of such formulations. Probabilistic models of this sort are still in their infancy, but there would appear to be certain directions in which developments might go. In circumstances where stability is controlled by geological structure, it is appropriate to set up models for the geometrical and spatial distributions of joints. These could be expected to lead to calculations of the likelihood of encountering certain unfavourable combinations of joints, and hence to estimates of probabilities of failure. The situation is considerably more complex when stress is important, because the models might then have to simulate not only deformations along individual discontinuities but also processes such as propagation of cracks and coalescence with other discontinuities. It is possible that

* Division of Applied Geomechanics, C.S.IR.O., P.O. Box 54, Mount Waverley, Victoria 3149, Australia. R xA ~l~

]7' 6

,~

special forms of statistical mechanics could be developed as part of the formulations. The starting point for these ventures is an acceptable model for the geometrical and spatial distributions of joints. Field evidence for such a model is necessarily indirect because of the almost invariable inaccessibility of the joints. The most complete data, in fact, comes from geological surveys of the traces observed where joints intersect large exposures. We shall refer to area or line sampling depending on whether the trace sample comprises all the traces in the area available or all the traces that intersect a suitable straight line drawn on the exposure. The process of extrapolation from the surface data of the traces to the hidden threedimensional structures of the joints then becomes an exercise in stereology.

A NEW STATISTICAL MODEL OF JOINTS A simple statistical model of joints was analysed in a previous paper by Warburton [1]. The joints were assumed to be parallel discs centred on random points whose volume density had a Poisson distribution, this being the mathematical description of randomness appropriate to the circumstances. Joint diameters were assigned an arbitrary distribution with no dependence

305

306

P.M. Warburton

on spatial location. The model was used to derive analytical predictions of trace patterns, and it was shown how its parameters could be obtained from field data. The assumption of random joint centres is worth retaining. In the previous model [1], and also in the model that we shall consider in this paper, the assumption leads to an exponential distribution of trace spacings along a sampling .line, in agreement with most reported field studies (e.g. Priest & Hudson [2], Call et al. [3], Baecher et al. [4]). The most restrictive assumption in the previous model was the circular shape of the joints. With circular joints it is impossible for trace patterns to vary with the component of exposure orientation in the planes of the joints. Yet just this sort of variation is likely to occur in practice. One only has to imagine, for example, a joint set in which the joints are elongated and have their major axes predominantly parallel. This is not an unreasonable possibility and the model should certainly be able to cope with it. There is little to guide one in the selection of a suitable non-circular joint shape for mathematical analysis. Virtually no direct experimental evidence on shape is available, and inferences from trace observations are sparse and equivocal. As a starting point it is assumed in this paper that the joints in a set can be represented by parallelograms of various sizes in which all similar sides are parallel. For simplicity it is also assumed, as in the previous model, that the joints are geometrically similar, which implies a constant ratio of longer to shorter sides for all parallelograms. It can be seen that the joints in the new model differ from each other only by scale and by simple translation in space with no rotation. Obviously the parallelogram model is highly idealized. Nevertheless it makes a good starting point because it opens the way to the analysis of more complex polygons, which could potentially provide approximations to a wide range of joint shapes. Later sections will show that, even in its present simple form, the model predicts trace patterns that are generally consistent with reported observations and suggests new interpretations of certain field data.

DEFINITION OF PARAMETERS OF JOINT SET The new model must be described mathematically. As mentioned above, the appropriate description of the random joint centres is a Poisson distribution for the volume density of the points. The Poisson distribution is defined by its mean N v , which represents the average number of joint centres per unit volume. Figure 1 shows the parameters used in the definition of joint shape. The two parallelograms are mirror images of each other, and the theory applies to either of them, though we shall normally choose the one on the left for illustration. To simplify the theory, the parameters are specified with reference to the trace directions that would be produced on an exposure. Thus the

__

po _ tt.t

,o

t ..s

/

.' S /kx-S~

Fig. 1. Definition of joint shape.

parameter x, which is a characteristic dimension (c.d.) defining joint size, is chosen to be the length of the sides that are not intersected by a line passing through the centre of the joint and parallel to the traces. The sides that are intersected by such a line are assigned length kx. As shown in Fig. 1, the angles fi and e are chosen so that they are associated with sides of length x and kx respectively. It should be clearly understood that the values of x, k, 3 and e for a given joint will not necessarily remain constant for different exposures, even though the joint itself is unchanged. A probability density (or frequency function) g(x) will be assumed for the characteristic dimensions x, which will be taken to be independent of spatial location. It can then readily be shown from the standard transformation formula of statistics that the joint sides of length kx have a probability density gk(kx) given by

1

(1)

gk(kx) = ~ ,q(x)

AREA SAMPLING: DERIVATION OF BASIC TRACE LENGTH EQUATIONS Consider an infinite plane section inside the rock. Every joint that intersects this plane creates a linear trace, which is in fact a chord of the joint, Let the parameters of the joint set be defined as in the previous section. The relative orientations of the Plane and the joint sides are then specified by the angles 6 and ~ and by the additional angle :( between the plane and the normals to the joints. For simplicity we shall initially concentrate on a typical joint with c,d. x. Let us examine under what h

normal to traces . . . . normal to joints

~ane

1/./

A

>,> ,>3 ,-" ,.]/) .,/~ p|o~ pwattet to Mnts 0

[}

Fig. 2. Construction used in area sampling derivation.

Stereological Interpretation of Joint Trace Data conditions the joint produces a trace with midpoint at a given point M on the plane section. Obviously one way in which this can occur is if the centre of the joint coincides with the point M, as in Fig. 2(a). In this position the joint is bisected by the line ABCD, which can be imagined to be fixed rigidly to the plane section at M. By visualizing different positions for the centre of the joint it is now readily seen that the joint must have its centre somewhere on the fixed line ABCD to produce a trace with midpoint at M. This argument can be extended to higher dimensions with the help of the construction shown in Fig. 2(b). In Fig. 2(b) the page represents the plane section, and the joint's normal is therefore inclined at an angle ~ to it. Since this angle is taken to lie in a vertical plane normal to the page, the joint would create a horizontal trace where it intersected the page. Let us now turn our attention to the box marked in Fig. 2(b). It will be noticed that the box extends equally on both sides of the page and intercepts unit area of it. Furthermore the shape of the box follows the contour of the line ABCD in Fig. 2(a). It is therefore possible to construct a line like ABCD through any point inside the box and know that the line will intersect the page at a point in the unit area intercepted by the box. In the light of the discussion above, the arbitrary point inside the box could represent the centre of our typical joint, and the point on the page would then coincide with the trace midpoint. The significance of the box now becomes apparent: each of our typical joints whose centre is in the box will produce a trace whose midpoint is in the unit area, and the number of such trace midpoints will be equal to the number of typical joints with centres in the box. Since the volume of the box is (k sin e + sin `5)x cos it contains on average a total of Nv(k sine + sin`5)x cos e centres of joints. A fraction g(x)dx of these have c.d. between x and × + dx and can be considered to be our typical joints (because dx is small). Consequently, average number of trace centres per unit area produced by joints with c.d. between x and x + dx = average number of such joints with centres in marked volume

- Nv(k sin • + sin `5)x cos c~g(x)dx.

N A

--

Nv(k sin • + sin ,5) cos ~m -7

[

x q(x)dx

(4)

d 0

Dividing (2) by (3), fraction of total traces produced by joints with c.d. between x and .x- + dx x

=

- , q ( x ) dx m

.[A(X) = X g(x) m

(6)

where the subscript A denotes area sampling. In the discussion that follows we shall classify joints that intersect the plane section into types A or B depending on whether they are intersected across opposite or adjoining sides. Of those joints with c.d. between x and x + dx that intersect the plane section (and hence have centres within a distance (k sin ¢ + sin 6)x 2 from it as measured in the planes of the joints), a fraction

2 dl (k sin e + sin `5).xhave centres at distances between l and t + dl from their intersections with the plane. It is clear from 15) that this elemental set of joints is responsible for a fraction 2

g(x) dx dl

m k s i n e + sin5 of the total traces. Two cases must now be considered depending on the value of/.

Case 1 1 is between zero and (k sin • - sin gi).v

2

The joints in the elemental set are of type A and produce traces with lengths between y and y + dy, where, from Fig. 3(a),

Sil

sin(6 + e)

y = - __

(7)

.,c

6

Consequently, fraction of total traces that have lengths

(3)

where the mean of x is, by definition, f

is worth noting that (5) shows that the probability density of traces produced by joints with c.d. x is given by

(2)

Integrating (2) over the range of all joint c.d. x and assuming a theoretically infinite upper limit, we find that the average total number of trace centres per unit area is given by

307

(5)

Equation (5) applies over the whole plane section. It

......~X _< C (o)

plane section

(b}

Fig. 3. Relationships between parameters for Ill type A and (b) type B joints.

308

P.M. Warburton

between y and y + dy and that are produced by type A joints 1 k sin • - sin 6 dx = m k sin • + sin 6 xg(x) ~y dy 1 k sin • - sin 6 [ sin• ]2 [ y sin• ] m k sin • + sin 6 sin~ q- E)J YO[ sin(3-+- e);d dy (8)

right side of (12) is integrated separately. Taking the second term first, we reverse the order of integration and perform the resulting integration with respect to y. In essence this yields

f;( d.v

g(x) dx (y sin ~)/sin(6 + c)]

~

oc

Case 2

=

[(Y sin ~)/sin(6 + ~)]

l is between (k sin e - sin 6)x 2

and

(k sin • + sin 6) x 2

(Ysin,e)/sin(tS+.)l

The joints in the elemental set are of type B and produce traces with lengths between y and y + dy, where, from Fig. 3(b), sin 6 sin • k sin • + sin 6 sin(6 + •) y 2 x - I

= m k s i n e + s i n 6 9(x) dx dl dy 2 1 sin 6 sin • O(x) dx dy m k sin• + sin 6 sin(6 + •)

Integrating (10) over the range of all possible type B joint c.d. x that could yield trace lengths between y and y + dy, we find that fraction of total traces that have lengths between y and y + dy and that are produced by type B joints 2 1 sin 6 sin• fro 9(x)dxdy m k sin• + sin 6 sin(~5 + •) Ir sin,)/s~n(~+,)l

(11) The fraction of the total traces with lengths between y and y + dy is obtained by adding (8) and (11). It is immediately apparent from the resulting expression that the probability density (or frequency function) of trace lengths y is given by 1 ksin,-sin6[ sin_E ]2 [- y s i n , ] ha(y) = m k sine + sin 6 Lsin(6 + •)_~ YgLsin(~-e)-J 2 1 sin 6 sin • m k sin e + sin 6 sin(6 + e)

x

9(x) dx

fr~

(12)

[(y sin ~)/sin(/t + ~)1

where the subscript A again denotes area sampling. Before leaving this section we shall obtain a convenient expression for the numerical evaluation of the integral

f ~ ha(y) dy, which will be needed later. Initially each term on the

Y O(x)dx (13)

si--nE

[- y s i n , ~ sin(6+E) ygLsin-~- ) dy= Sin, × ft ®

x sin(6 + E) #(x) dx

(Y sin E)/sin(6+ ~)l

(14)

sine

Finally equations (13) and (14) together enable us to obtain from (12) the result ha(y) dy = m

(10)

dy

,./Y

The integral of the first term can be expressed in a similar form by a simple change of variable. Thus

(9)

Consequently, fraction of total traces that have lengths between y and y + dy and that are produced by type B joints with c.d. between x and x + dx

+

p {[x sin(6 + ~)l/sin (}

9(x) dx |

×

x-

(r sin~)/sin(,4+,)]

2Y sin ~ s i n e ] k s i n , + s i n 6 - : -d- ~x" +s ,-v/g(x) OJ n ( o

(15)

LINE SAMPLING: DERIVATION OF BASIC TRACE LENGTH EQUATIONS We shall start with the same geometry as before for the plane section and the joint set. The additional Feature Is the sampling line. which will be taken to be an infinite straight line lying in the plane section and making an angle fl with the normals to the traces. Every joint that intersects the sampling line at a point also intersects the plane section along a line, which is in fact a chord of the joint. It is the set of such chords that constitutes the population of traces in line sampling. As before, we shall initially concentrate on a typical j.omt with c.d. x Let us examine under what conditions the joint passes through a given point P on the sampiing line. Obviously one way in which this can occur is if the centre of the joint coincides with the point P. as in Fig. 4(a). In this position the joint occupies the area ABCD. which can be imagined to be fixed rigidly to the plane section. By visualizing different positions for the centre of the joint it is now readily seen that the joint must have its centre somewhere in the fixed area ABCD to intersect the sampling line at P. The extension of this argument to higher dimensions is made with the help of the construction shown in Fig. 4(b). The parallelepiped intercepts unit length of the sampling line and has ABCD as its section parallel to the joints. Figure 4(b) is interpreted similarly to Fig. 2(b), and the parallelepiped shown in the former is

Stereological Interpretation of Joint Trace Data line~

sampling

lnormat to traces

line, a fraction

normal ,o joints

0 sampling line \..\

--

plane

~,

309

2w dl kX 2 sin(a + ~}

1c

have centres at distances between I and l + dl from their intersections with the plane section. It is clear from (19) that this elemental set of joints is responsible for a fraction

,, to ,race,

sectionAlX( S L x

2 wg(x) d x dl

B

y~ k sin(fi + 6.)

plane p~allel to joints

of the total traces intersecting the sampling line. Two cases must now be considered depending on the value of 1. Case 1

(a}

/ is between zero and

F i g 4. C o n s t r u c t i o n u s e d in line s a m p l i n g d e r i v a t i o n .

(k sine - sin #)x 2 analogous to the box shown in the latter. It is only by having its centre in the parallelepiped that our typical joint can intersect the given unit length of sampling line. Consequently, by analogy with the reasoning that led to (2), average number of intersections per unit length produced by joints with c.d. between x and x + dx = average number of such joints with centres in marked parallelepiped -- N v k x 2 sin(~ + e)cos:~cosfl #(x)dx.

(16t

Integrating t161 over the range of all joint c.d, x and assuming a theoretically intinite upper limit, we find that the average total number of intersections per unit length is given by Nc = N v k sin(?5 + e)cosc~cosflll2

(17i

where the second moment of v about the origin is, by definition, H2 -=

x2c/(x) dx

(18}

Dividing (16) by (17), fraction of total intersections produced by joints with c.d. between x and x + dx X2

,q{xl d x (19) He Equation 119) applies over the whole sampling line. It is worth noting that (19) shows that the probability density of mtersections produced by.joints with c.d. x is given by -

,

1 k sin e - sin 6 dx = #~ -ksin(6 -+ 6.)-xyg(x) dv dy l k s i n e - sin 6 sin2e , [ vsin6. ] = 11~ k - -sin3(6 + ~iY-g[ sin(,S + ~jj dy (21) Case 2

1 is between {ksine-sin6)x 2

(201

Ire

where the subscript L denotes line sampling. In the discussion that follows we shall classify joints that intersect the sampling line into types A or B depending on whether the plane section intersects them across opposite or adjoining sides. It can be seen from Fig. 5 that, of those joints with c.d. between x and x + dx that intersect the sampling

and

(ksin6. + s i n f ) x 2

The joints in the elemental set are of type B and produce traces with lengths between v and v + dy, where y is obtained from Fig. 3(b) and is given by (9}. Since Fig. 5 yields an identical equation for w, it follows that fraction of total traces that have lengths between v and y + dy and that are produced by' type B joints with c.d. between x and x + dx 2

1

dl[

= 5'2 k Sin(; + 6.) s o ( ' ) d\- dy[ dy 2

X2

l},(,c) = -- ,q(x)

The joints in the elemental set are of type A and produce traces with lengths between v and y + dy, where y is obtained from Fig. 3(a) and is given by (7). Since Fig. 5 yields an identical equation for w, it follows that fraction of total traces that have lengths between y and y + dy and that are produced by type A joints

sin 6 sin e

- H2 7 ksin2(c~ + 6-)

vg(x) dx dv

122)

Integrating (22) over the range of all possible type B joint c.d. x that could yield trace lengths between y and y + dy, we find that fraction of total traces that have lengths between y and y + dy and that are produced by type B joints 2 sin ,6 s i n e f ~ /t~ k sin2(6 + e) y

J,[(y 'qn~) ,'.stnl5+~t] .q~x) dx dv • .s"

"

(23)

310

P . M . Warburton

The fraction of the total traces with lengths between y and y + dy is obtained by adding (21) and (23). It is immediately apparent from the resulting expression that the probability density (or frequency function) of trace lengths y is given by 1 ksinE - sin6

hL(y) = #'2

k

sin2~ y2g[- ysinE ] sina(6 + e) I_sin~ +-Ej]

2 sin6sinE t"~ + ~ k sinZ(6 +~) y Jt (y sin~)/sin(6+ c)] O(x) dx (24) where the subscript L again denotes line sampling. Before leaving this section we shall obtain a convenient expression for the numerical evaluation of the integral

r~ hL(y) dy, which will be needed later. The method is similar to that used for the analogous area sampling integral in the previous section. Corresponding to (13) and (14) we have the equations

f;f" dy

yg(x) dx

[(y sin O/sin(~ + c)]

=

9(x) dx (Y sin ~)/sin(6 +E)]

= 2

y dy dY

[(Ysin~)/sin(6+O] I_

Hi(Y) =-

£

hi(y) dy

as

f;

Hi(Y) = 1 -

of the traces can be recognized, the values of Ni that are actually observed are given by

_ sin(~ + E) f f

x2 sin2(6 + e) g(x) dx (26)

(Y sina)/sin(6+~)]

sin 2 E

Equations (25) and (26) together enable us to obtain from (24) the result =

[12

(29)

fffhi(y)dy

z [- ysin~ q .

hL(y) dy

hi(y) dy

The form of (29) facilitates numerical evaluation via (15) or (27). In practice the sample distributions of trace lengths that are actually observed are both truncated and censored. Truncation results from the inability to recognize traces shorter than a certain threshold length y, and censoring arises from the impossibility of measuring the full lengths of traces that extend out of the field of view. Analysis of censoring was discussed briefly in a previous paper by Warburton [1] and reference was made to forthcoming publications that obtain appropriate correction formulae by taking proper account of the point process of the trace centres. It is therefore sufficient to derive only the uncensored distributions of trace lengths in this paper. Physically this is equivalent to assuming a large exposure dimension in the direction of the traces. Truncation is readily allowed for. Firstly, because only a fraction

Y gLsin(~¥dJ °y Sln E

(28)

Because the hi(y) are normalized, (28) can be rewritten

sin2 ¢

and f~

cumulative distribution Hi(Y), which is defined by

(¥ sin,)/sin(6+e)]

x x2

y2 sin 6 sin • -I ~lo(x)dx k

(27)

N °Bs = Ni

f;

hdy) dy

(30)

t

N °Bs represents either the average number of observed trace centres per unit area or the average number of observed intersections per unit length, depending on whether area or line sampling is being considered. Furthermore, the probability densities of observed trace lengths, h°BS(y), correspond to the portion of h~(y) in the range y >~ y,, so that when normalized,

O, y < Y t hOBS(y) =

hi(y)

y >! y,

(31)

fy

~ hi(y) dy t

DERIVATION O F CUMULATIVE DISTRIBUTIONS AND CORRECTIONS FOR TRUNCATION

The theory in this section is common to both area and line sampling. This wilt be indicated by the use of a single subscript i, which can be replaced by either A or L for area or line sampling respectively. The probability densities of trace lengths y have been derived in the two previous sections and are given by (12) and (24) for area and line sampling respectively. Associated with each probability density hi(y) is a

The associated cumulative distributions are therefore given by

t H°ns(Y) =

fr 1

oo

~

O, Y < Y t hi(Y) dy , Y >1 y~

(32)

f~ h~(y)dy t

Equations (30), (31) and (32) are intentionally expressed

Stereological Interpretation of Joint Trace Data in forms that facilitate numerical evaluation via (15) or (27). In comparing line and area sampling, it is interesting to note that the ratio of probability densities of trace lengths is directly proportional to trace length y. This applies to both 'true' and observed probability densities. It is a reflection of the fact that, for this particular model, the probability of an area trace intersecting the sampling line is directly proportional to the length of the trace.

Furthermore, from the well-known theory of the distribution of interval sizes in a Poisson process, the distances s between adjacent observed intersections on the sampling line are distributed exponentially and have the probability density (or frequency function) , . NL oBs )-~ .[(s,

TRACE

N °~s)

SPACINGS

e N~'"~(NORS)~ = r!

e ,~'~'"~.~

(34)

EXAMINATION OF S O M E PREDICTIONS FROM MODEL

Consider again the infinite volume elements shown in section by the strips marked 'd(area)' in Fig. 5. In particular, consider the points within the elements that are the centres of joints with c.d. between x and x + dx. By a postulate of the model, the volume density of these points has a Poisson distribution. Consequently the joints centred on the points create sampling line intersections whose line density also has a Poisson distribulion, the mean of which represents the average number of such intersections per unit length. The set of intersections defined above will be recognized from two sections back. It is aptly called the elemental set, because other sets of interest, such as those associated with NL and N TM, can be constructed from it by appropriate integrations. Because the distribution of intersection density resulting from such a construction is compounded from elemental Poisson distributions, it is itself Poisson, and its mean represents the average integrated number of intersections per unit length. Suppose, for example, that we are interested in all the intersections that are actually observed on the sampling line, allowing for the effect of truncation at the threshold trace length y,. The average number of such intersections per unit length has been derived two sections back as N °Ss, but because the intersection density has a Poisson distribution, the probability of there being r of the intersections in a given unit length is P(r:

N°RS

The mean of s is (N °~s) ~, which represents the average observed interval length.

LINE S A M P L I N G : DERIVATION OF

311

It is convenient to adopt a standard analytical form for the probability density ,q(xl. The problem of determining g(x) is then reduced to the problem of estimating the limited number of parameters that define the analytical function. Another advantage is that further calculations involving .q(x) are usually simplified if an analytical form is available. Following Baecher et al. [4], Barton [53 and Warburton [l], we assume that the joint c.d..,~ are distributed lognormally, q(x) can then be expressed in the form ,q(x) -

.

a\," 27zx

exp -- 2 \

o-

]

(35)

where I~ and a are the parameters of the distribution. (It and a are, in fact, the mean and standard deviation of the normally distributed random variable In x.) By substituting (35) into (1) it can readily be shown that the lengths kx of the other pair of joint sides are also distributed lognormally, but with parameters (I~ ~- In k) and O.

Various consequences of this assumption will be examined shortly over the full range of exposure orientations. One result, however, can be shown to be reasonable merely by inspection of the relewmt equations without numerical computation. Consider (12) and (24). These equations are simplest when 6 = 0, which is the condition for the exposure being parallel to a side of the .joints. We then havc 1

hA(y) = m " vfl(v) '

(33)

(36)

and

where r = 0, 1,2 . . . . .

1

hL(y) = 7-ye#(y)

(37)

/~2

~ d [area;

secfion (× dt

-el (areal SSgO

Fig. 5. Section through parallelepiped in Fig. 4(b) taken parallel to joints

For most of the sensible forms that 9(x) might take, both (361 and (37) would yield skew unimodal curves passing through the origin. In practice it is found [4] that trace length distributions with these characteristics are well fitted by tognormal models. Although this experimental observation restricts the choice of functions for 9(x), our assumption of a lognormal ,q(x) is still acceptable, because further simplification of (36i and (37) is then possible, resulting in exact tognormal forms for both ha(3, ) and hr,(y) when 6 = 0. In fact if ,q(xt is given by (35), a little algebra on (36) and (371 shows that

312

P.M. Warburton

tion chosen for the longer sides and the corresponding derived distribution for the shorter sides. Equations (12) and (24) have been used to compute the predictions for area and line sampling respectively. Consequently the plots in Fig. 8 show only the 'true" trace length distributions without truncation, but this has the advantage that the behaviour of the short traces is not masked. frQces The 6 orientations in Fig. 8 are selected to give a good indication of the varying trace length distributions that would be expected in a full 360 ° rotation of Fig. 6. Archetypal joints with ~ = 0. exposures about any axis not parallel to the joints. The sequence shown would actually occur twice during a ha(y) is lognormal with parameters (# + tr 2) and tr and full 360 ° rotation, because exposures 180 ° apart in orientation yield identical results. Care is taken to inthat hL(y) is lognormal with parameters (/a + 2e 2) and clude all cases in which the exposure is parallel to a O'. It is worth pointing out in passing that (36) and (37) side or a diagonal of the joints, because these cases give are independent of k and ~, so that, for instance, the rise to special curve shapes. If the exposure is parallel archetypal joints shown in Fig. 6 will all produce ident- to a side, all the joints producing traces are of type A ical trace length distributions. They will not, however, and the second term on the right side of (12) and (24) produce identical trace spacing distributions in line vanishes. Conversely, if the exposure is parallel to a sampling, because (34), with substitutions from (30) and diagonal, all the joints producing traces are of type B (17), shows that f(s; N °as) is a function of both k and E and the first term on the right side of (12) and (24) vanishes. At orientations between 2 adjacent special when ~ = 0. Before proceeding, we note from a standard result for cases there is a transition region in which both types the lognormal distribution that the rth moment of x of joint participate and both terms contribute to the expressions for ha(y) and hL(Y). Thus Fig. 8(a) is associabout the origin is given by ated with pure type A joints. There is then a transition #, = e~,,+~r~2~. (38) (Figs 8(b) and 8(c)), in which both types A and B parIn particular m and g~, both of which occur in the ticipate, to pure type B (Fig. 8d}. This is followed by derivation of the theory, are obtained from (38) by put- further transitions to pure type A (Fig. 8e), pure type B (Fig. 8f), and back to the ortginal pure type A (Fig. 8a) ting r = 1 and r = 2 respectively. Figure 8 illustrates the influence of joint shape by for a repeat of the sequence. A number of geological surveys (summarized in showing predicted probability densities of trace lengths Baecher et al. [4]) indicate that trace length distribufor a range of differently oriented exposures. The joint tions are usually well fitted by either lognormal or set is identical in each case. The archetypal joint is exponential models. One would expect the predicted shown in Fig. 7, together with the lognormal distribuprobability densities in Figure 8 to be consistent with this experimental observation. Certainly the trace length distributions from pure type A joints--as in Figs 8(a) and 8(e)--are exactly lognormal for both area and line sampling. As explained in the discussion following (36) and (37), lognormality of such distributions is ensured by the adoption of a lognormal form for g(x). In the case of pure type B joints, Figs 8(d) and 8(t) show that the trace length distributions superficially >i--resemble exponential forms for area sampling and lognormal forms for line sampling. In order to examine these resemblances more easily we take the exposure orientation in Fig. 8(d) as an example and use (29) to compute (1 - HA(Y)) and HL(Y). The results are plotted in Figs 9 and 10 respectively. Scales on the axes are such that an exponential HA(Y) and a lognormal HL(Y) g would produce a linear plot in each figure, and in addition the straight line in Fig. 9 would pass through the point (0, 1). Clearly neither of the curves in Figs 9 and 10 satisfies the linearity condition along its entire length. In prac-O 10 2O 3O ~ SO 60 tice, however, the ends of a 'true' trace length distribuSlOE LENGTH -, tion do not contribute to the distribution that is Fig. 7. Archetypal joint with lognormal probability densities of side actually observed. In the case of truncation, for lengths.

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Fig. 11. Mean lengths of traces and joint sides. bination of types A and B. It may be concluded that the predicted trace length distributions have shapes that are generally consistent with reported observations. We have seen that the predicted trace length distributions vary with exposure orientation about any axis not parallel to the joints. Figure 8 shows not only that the effects are different for area and line sampling but also that the variations are by no means uniform over the 360 ° range of orientations. This can be demonstrated quantitatively by various statistical measures. We shall c~ take the mean as an example. The means of ha(y) and hL(Y) can be obtained from 1;o -J200 the defining integrals by techniques similar to those TRACE LENGTH '~ used in the derivations of (15) and (27). For comparison Fig. 10. Plot of HL(Y) for exposure orientation in Fig. 8(d). it is convenient to have in addition the means of g(x) and gk(kX)~ the latter being obtained with the help of (1). example, (31) and (32) show that h°aS(y) and H°as(Y) The results are given in Table 1, where the means of all are unaffected by the behaviour of the traces with four distributions are expressed in terms of the lengths below the truncation threshold Yr. In fact the moments of an arbitrary g(x) and also in terms of/~ and very long traces are also virtually excluded because of for a lognormal 9(x). the low likelihood of observing them in a typical Figure 11 shows plots of the expressions in Table 1 limited sample size. If the ends of the 'true' trace length for the joint set represented in Fig. 7. The means of the curves in Figs 9 and 10 are omitted, it is seen that in four distributions are plotted against exposure orientaeach case the remaining portion can be satisfactorily tion measured, like 6 and e, in the planes of the joints. represented by part of an appropriate straight line from It is only necessary to show a range of 180 ° because, as an exponential or lognormal trace length distribution explained earlier, this cycle would be repeated in a respectively. Similar considerations apply to the other pair of further 180° rotation. It can be seen from Fig. 11 that, despite the considertrace length distributions produced by pure type B joints (Fig. 8f) and to distributions produced by a corn- able difference between the maximum and minimum of TABLE l. RELATIONSHIP BETWEEN MEANS OF JOINT SIDE AND 'TRUE' TRACE LENGTH DISTRIBUTIONS Mean Distribution

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Stereological Interpretation of Joint Trace Data each trace length mean, there are extensive ranges of exposure orientation over which the means show little variation. This could well explain why Bridges [6] found different trace length distributions and means on differently oriented exposures, whereas Robertson [7] and Barton [8] observed them to be approximately similar. In each of these reports the number of different orientations was strictly limited and the authors could not have inferred the variation over a full 360 °. Naturally a detailed numerical comparison with field data would require the equations for the observed trace length distributions from two sections back, and calculation of the means would involve h°BS(y) and h°BS(y) rather than ha(y ) and hdy). Equation (34) shows that the model of joints always predicts exponentially distributed trace spacings in line sampling. This is consistent with most reported field studies (e.g. Priest & Hudson [2], Call et al. [3], Baecher et al. [4]). As explained in the previous section, the exponential distributions have means of (ND 1 or (~ToBs~ L depending on whether 'true' or observed spacings are being considered. Since an exponential distribution is completely defined by its mean, it follows that the influence of various parameters on the trace spacing distributions can be predicted entirely from the reciprocals of the expressions for NL and N °'Ss in (17) and (30) respectively. To conform with the previous discussion in this section, we shall examine the influence of exposure and sampling line orientations on the trace spacing distributions from a given joint set. Equation (17) shows that (NL) 1 is inversely proportional to the factor cosc~ cosfl, which is simply the cosine of the angle 7 between the sampling line and the normals to the joints. Changes in (NL)-1 are determined solely by this angle, because the rest of the expression for (ND 1 is invariant for the given joint set. It follows from (30) that (NLO B S ) - 1 too must be inversely proportional to cosT. Equation (30) shows that (NL . o n s)- 1 is also affected by changes in hL(y) and hence by the component of exposure orientation in the planes of the joints, though the effect is clearly slight if Yt is small. IMPLICATIONS FOR GEOLOGICAL SURVEYS Special sorts of geological surveys will be needed to obtain the parameters of the statistical model of joints. Although the discussion in this section is restricted to the present model, the approach to data collection and interpretation would be similar for more sophisticated models developed within the same general stereological framework. The approach here is seen as supplementing the 'Suggested Methods' published by the ISRM Commission on Standardization of Laboratory and Field Tests [9]. For the present model to be applicable, the joints in the chosen set should be approximately flat and parallel to each other. Their shapes will almost never be known experimentally and will usually have to be inferred

315

from other evidence, such as the process of formation and the influence of other joint sets created earlier in the geological history. There is a pressing need for experimental work on joint shape. The chosen joint set must be surveyed on its own, preferably by area and line sampling on a number of differently oriented exposures. Surveys on different exposures must be kept separate. Each exposure should be plane, with sufficient length in the direction of the traces to minimize censoring. All orientations should be recorded, and it is generally good experimental practice to avoid a large angle fl between a sampling line and the normals to the traces. Apart from their orientations, however, it is important that sampling lines should be truly random to be consistent with the assumptions of the theory. The truncation threshold Yt, marking the lower limit of trace length measurement, should be set at a definite value for a given exposure to provide a unique lower limit for the relevant integrals in (30), (31) and (32). Furthermore the value of Yt should always be as low as practicable. This is because determination of g(x) is based entirely on the shapes of the trace distributions, and the more completely the latter are known, the less ambiguity there is in determining the parameters of the model. For the same reason, Yt should be small even if one intends using the model only to obtain the distributions of the large joints. It is certainly fallacious to make inferences about the large joints purely from the long traces, because the theory demonstrates that large joints produce a range of trace lengths, including short ones (except in the rare circumstances where an exposure is parallel to a side of the joints). In order to obtain as much information as possible from the data, an attempt should be made to measure the parameter N °Bs from the area sample on each exposure. It will be recalled that N °~s, the average number of observed trace centres per unit area, is analogous to the line sampling parameter N °Bs, which was defined to be the average number of observed trace intersections per unit length of sampling line. N °Rs would normally be available from routine measurements, possibly as the reciprocal, which represents the average observed spacing, Direct measurement of N °us is much less common and would often be hindered in practice by censored trace length data (though even then forthcoming publications referred to in [l] will permit indirect estimation of N°BS). Before determining the parameters of the model it is desirable to check that the trace spacings in each line sample are distributed exponentially, as predicted by the theory. It has been pointed out earlier that this prediction is consistent with most reported field studies, though Priest & Hudson [2] found that a sample size of at least 200 was generally needed for the exponential form to be clearly recognized from a histogram. A preliminary test that may be useful is to compare the sample mean and standard deviation, which would theoretically be equal for an exponential distribution. In practice [2], for sample sizes over 200, agreement

316

P.M. Warburton

within 20~ indicates that the measured distribution may be exponential and that it is worth carrying out more detailed tests for goodness of fit. If a particular joint set produces trace spacing distributions that are not exponential, the model in its present form is inapplicable, though Baczynski's zonal approach [10] suggests a possible adaptation. As explained in the previous section, it is convenient to adopt a standard analytical form for g(x). We may, for example, assume #(x)_to be lognormal as in (35). The unknown parameters /~ and tr can then be estimated from one of the sample distributions of trace lengths. Details of the estimation will not be given here because the topic has already been covered in a previous paper by Warburton [1]. After mentioning some problems of robustness, the discussion in [1] described an estimation procedure based on the minimum X2 method and went on to demonstrate its application with field data from a mine. The same procedure is applicable here, with (32) being used to compute the theoretical distribution expected from the model. Once the parameters of the model are known, predictions of the various observed trace distributions can be computed from (31), (32), (33) and (34) for comparison with the field data. In addition the measurements of N °Bs and N °Bs on each exposure can be checked for consistency with the model. The theoretical relationship between the two parameters is obtained from (30), with substitutions from (17), (3) and (30), which gives the result NOBS = NOBS k sin(fi + ~) k sin E + sin ~i

x cos fl #'2 m

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t

The theory indicates some circumstances in which determination of the model's parameters is especially simple. For example, if there is evidence that an exposure is parallel to a side of the joints, then we know from the discussion following (36) and (37) that the 'true' trace length distributions are independent of k and E, which define the shape of the archetypal joint when 6 = 0. It follows from (31) and (32) that this independence also holds good for the observed trace length distributions, which are used in the estimation of # and a. Hence # and a can be estimated without knowledge

of the parallelogram shape. Furthermore the discussion following (35) shows that a similar estimation made from a second exposure parallel to the adjoining side of the joints would be expected to yield (# + In k) and a. This would permit determination of k, because # would already be known from the first exposure. CONCLUSION The new statistical model of joints incorporates an alternative joint shape based o n the parallelogram and predicts trace patterns that are generally consistent with reported observations. Special sorts of geological surveys are now needed to obtain the parameters of the model and to test its predictions in detail, The significance of the present theory is that it suggests a new approach to data collection and interpretation and establishes a stereological framework that can readily allow for future model developments. Received 20 Auoust 1979; in revised form 12 May 1980.

REFERENCES 1. Warburton P. M A stereological interpretation of joint trace data. Int. J. Rock Mech & Min. Sci. & Geomech. Abstr. 17, 181-190 (1980). 2. Priest S. D. & Hudson J. A. Discontinuity spacings in rock. Int. J. Rock Mech. & Min. Sci. 13, 135-148 (1976). 3. Call R. D., Savely J. P. & Nicholas D. E. Estimation of joint set characteristics from surface mapping data. Proc. 17th U.S. Syrup. on Rock Mechanics, Utah, pp. 2B2-1-2B2-9 (1976). 4. Baecher G. B., Lanney N. A. & Einstein H. H Statistical description of rock properties and sampling. Proc. 18th U.S. Syrup. on Rock Mechanics, Colorado, pp. 5C1-1-5CI-8 (1977). 5. Barton C. M. Analysis of joint traces. Proc. 19th U.S. Syrup. on Rock Mechanics, Nevada, pp. 38--41a (1978). 6. Bridges M. C. Presentation of fracture data for rock mechanics. 2nd Aust.-N.Z. Conf. on Geomechanics, Brisbane, Inst. of Engrs Nat. Conf. Publn No. 75/4, pp. 144-148 (I975). 7. Robertson A. MaeG. The interpretation of geological factors for use in slope theory, in Planning Open Pit Mines. Proc. Sth Aft. Inst. of Minino and Metall. Syrup. on the Theoretical Background to the Planning of Open Pit Mines with Special Reference to Slope Stability, Sess. 4, pp. 55-71 (1970). 8. Barton C. M. A geotechnical analysis of rock structure and fabric in the CSA Mine, Cobar, New South Wales, CSIRO Aust.. Div. of Applied Geomechanics Technical Paper No. 24, 30 pp. (1977). 9. ISRM Commission on Standardization of Laboratory and Field Tests. Suggested methods for the quantitative description of discontinuities in rock masses. Int. J. Rock Mech. Min. Sci.& Geomech. Abstr. 15, 319-368 (1978). 10. Baczynski N. R. A three-dimensional model for the spatial distribution of extension fractures within the dolomitic shales at the Mount Isa Mine, North-western Queensland. Report to Dept Min. and Metall., Univ. of Melbourne (1978).