Site Investigation - Rock Mechanics Based on an Anisotropic Jointed Rock Model - page 392

According to Hudson & Priest (1979), spacing values depending on their formation
process should follow a negative exponential, uniform or normal distribution. A neg-
ative exponential spacing distribution characterizes a discontinuity set with randomly
appearing discontinuities. A uniform distribution may occur if the discontinuities are
either irregularly spaced or appear in repeated cycles of progressively fractured zones.
The normal distribution characterizes a regularly spaced discontinuity set. The occur-
rence of one discontinuity is then related to the position of adjacent discontinuities,
refl ecting the original formation process, as can be observed in basalt or bedded sand-
stone.
In nature, discontinuities may not be originated by one single formation process but
more likely by a combination of several processes. Numerical simulations of combi-
nations of exponential, uniform and normal distributions turned out to be similar to
negative exponential distributions (Hudson & Priest 1979). Thus, the frequency distri-
bution of spacing values of a discontinuity set tend to follow, in most cases, a negative
exponential distribution. This has been verifi ed by a large number of spacing mea-
surements (Baecher et al. 1977, Priest & Hudson 1976, Hudson & Priest 1979, Priest &
Hudson 1981, Einstein & Baecher 1983, Sen & Kazi 1984).
The PDF of a negative exponential distribution of spacing s is defi ned as

(13.14)

where both the mean value and the standard deviation are equal to the reciprocal

value of the parameter λ :

(13.15)

The and of negative exponential distributed spacing values can be calculated
from a frequency distribution of m spacing measurements si i as follows:

(13.16)

(13.17)

If is close to , then a spacing distribution can be adequately described by a nega-
tive exponential distribution.
More fl exible distributions, which include the exponential distribution as a special case,
such as the Weibull distribution and the gamma distribution, may be used in cases where
spacing distributions cannot be suffi ciently well described by a negative exponential dis-
tribution (Sen 1984, Rouleau & Gale 1985, Bardsley et al. 1990, Sen 1993). It was found
by Attewell & Farmer (1976) and Bridges (1979) that discontinuity spacing values of
stiff clay, clay shale and chalk tend to follow a log-normal distribution.