Engineering Rock Mechanics

94 Fractures and hemispherical projection

6-

O 5-

1 3-

p 4-

2-

1-

oi

91

0.0-1.0 1.0-2.0 2.0-3.0 3.0-4.0

Length, x

+ Frbability density,

I b

Length, x

Figure 7.5 Distribution of lengths between fracture intersections along the scanline in

Fig. 7.3.

f(x) = Ach where h is the mean frequency, the mean and standard

deviation values for the intact rock lengths can be established. The

definition of the mean, i, using probability theory for continuous dis-

tributions is X = [Sxf(x)dx]/[J f(x)dx]. In the case of the negative

exponential distribution with f(x) = he-Ax, with the limits of integra-

tion being taken as zero and infinity, and knowing that the integral

1: f(x)dr = 1 (because the total probability is equal to unity), we find

that X = $rxAe-Axdx = l/h.

The variance, u2, of a probability distribution is defined as

$(x -X)2f(x)dx, so for the case of a negative exponential distribution we

have

02 = (x - (l/h))2Ae-A"&

00

which reduces to 1/X2.

Thus, the standard deviation4, (T, is also l/h. This means that for the

negative exponential distribution the mean and standard deviation are

Note that the symbols used here for the mean and standard deviation of a population

are f and u. Thus, the symbol (J is used in this book for both a stress component and

the standard deviation of a statistical population. Because u is used here in a specific

statistical context, there should be no confusion -we will not be discussing the standard

deviation of stress components.