Engineering Rock Mechanics

1 18 Discontinuities

Figure 7.4 Quantifying discontinuity occurrence along a sampling line.

between fractures, denoted by xi in Fig. 7.4. When a sufficiently large
sample of these individual spacing values (preferably more than 200
individual measurements) is plotted in histogram form, a negative
exponential distribution is often evident, as illustrated in Fig. 7.5. It should
be noted that the general trend of this histogram is for there to be many
small spacing values and few very large spacing values in the distribution.
We utilize histograms when there is a finite number of values (Le. the
discrete case) and the data can be assigned to chosen class intervals. On
approaching the limiting case of an infinite number of spacing values and
infinitely small class intervals, the histogram tends to a continuous curve,
which can be expressed as the probability density function.

f(x) =

Note that the mean of this distribution is l/& and the standard deviation

is also l/A. It is a one-parameter distribution, with the mean and the

standard deviation being equal.

This negative exponential distribution is the spacing distribution

associated with the Poisson process of random events. However, it must be

emphasized that we are not treating the occurence of discontinuities as

random events, rather we are using the equation because we expect it to

apply and field data support its use. In some statistical cases, such as repeated

sampling to determine the mean of a population with any distribution, there

is convergence of the results to the well-known normal distribution: this is

called the central limit theorem. The negative exponential distribution is an

analogue to the normal distribution, except that it is the distribution to

which the spacing values converge when successive spacing distributions

of any type are superimposed on the sampling line. In other words, the

occurrence of the negative exponential distribution is expected as the result

of a suite of superimposed geological events, each of which produces

fracturing of a given distribution. It should be noted that fracturing is

deterministic in the sense that it occurs as the result of direct mechanical

causes, but that in aggregate a probabilistic model is mathematically

convenient to use, as will be explained in the next sub-section.

7.2.2 The Rock Quality Designation, RQD

As is evident in Fig. 7.5, a natural clustering of discontinuities occurs through

this genetic process of superimposed fracture phases, each of which could

have a different spacing distribution. An important feature for engineering