Engineering Rock Mechanics

Natural, pre-existing fradures 93

35-

e 30

8

* 25

x

p 20-

s

' 15-

10-

5-

0

• -^26

15

5

--

345670

Number of sides of polygon

Figure 7.4 Histogram of polygons in Fig. 7.3.

The mean number of sides of the polygons is given by

(3 x 38 + 4 x 26 + 5 x 15 + 6 x 5 + 8 x 1)

(38 + 26 + 15 + 5 + 1)

331

85

=-- - 3.9 sides.

The theoretical answer is exactly 4, found from the theory (Miles,
1964*) of 'Poisson flats', which is beyond the scope of this book.
However, for the average value to be equal to 4, there have to be
many triangles present, see Fig. 7.4. This provides an indication of
why there are many triangular areas bounded by fractures on a rock
exposure.
In a similar way, a histogram of the lengths along the scanline in
Fig. 7.3 is shown below in Fig. 7.5, together with the shape of the
negative exponential distribution. The intact rock lengths are relative to
a total scanline length in Fig. 7.3 of 23.4 units. Despite the small sample
size, the negative exponential trend is evident in this example.
Thus, the existence of many triangular areas bounded by fractures on
a rock exposure, or many small lengths along a scanline is a function
of the basic geometry and does not necessarily indicate anything about
the fracture genesis. Similarly, the negative exponential distribution
of lengths of scanline between fractures, or lengths of intact rock in
borehole core, occurs because the superimposition of a series of spacing
distributions (in this case, the successive fracturing events through
geological time) tends to a negative exponential distribution, whatever
the types of original spacing distributions 3.
Knowing that the probability density distribution of intact lengths,
x, can be well approximated by the negative exponential distribution,

2Miles R.E. (1964) Random polygons determined by random lines in a plane. Pruc.
Nafl. Acad. Sci. USA, 52,901-907.
3This was conjectured, and proved for certain cases, by Karlin and Taylor in their
1973 book, A First Course in Stochastic Processes, p. 221, providing a theoretical basis
for the existence of negative exponential distributions of spacings in rock masses. For
a more advanced treatment of the geometry and engineering assessment of fracture
occurrence, the reader is referred to the paper by Zhang Lianyang and Einstein H. H.
(2000), Estimating the Intensity of Discontinuities, Int. I. Rock Mech. Min. Sci., 37,819-837.