Engineering Rock Mechanics

Ramifications for analysis 169

The semi-variogram illustrated in the top right of Fig. 10.3 is derived from

the equation

where y(h) is the semi-variogram statistic for samples a distance

h apart,

is the number of sample pairs,

is the rock property at location x, and

is the rock property at location x + h.

n

p(x)

p(x + h)

Each statistic, y(h), refers to the overall variation for samples taken at a

distance h from each other. In Fig. 10.3, this statistic is plotted against h to

indicate the rock property variation as a function of distance between the

observations. Naturally, when h tends to zero, we might expect y(h) to tend

to zero (although this is not always the case because of measurement

inaccuracies and sudden differences in the rock property, the so-called

'nugget effect'). More interestingly, ash increases, so y(h) will increase until

it reaches a constant value-indicating no correlation between the data

points making up each pair. This occurs at a distance h = u, which is the

range of influence of a sample, and is at the value of y(h) = C, termed the

sill of the semi-variogram.

Although the techniques of geostatistics and Geographical Information

Systems have not been fully exploited in rock mechanics, it is clear that the

approach does take the location of the sample into account and does

provide a method for quantifylng inhomogeneity. The concept of the range

of influence is important in establishing the distance to which one can

extrapolate borehole information. Also, one can examine anisotropy by

constructing semi-variograms in different directions.

Figure 10.4 shows simulated discontinuity patterns for both statistically

homogeneous and statistically inhomogeneous cases. These patterns illus-

trate the need to account for inhomogeneity in order to develop a correct

understanding of rock mass variability or rock mass structure at any site.

10.4 Ramifications for analysis

The overall validity of models has been discussed in terms of the CHILE
and DIANE assumptions and, at this stage, it will be helpful to consider
anisotropy and inhomogeneity in the modelling procedures. The models
are either solutions for continuous materials or solutions for discontinuous
materials, and in a few cases a combination of the two.
In the first case, with 'classical' solutions we have little room for manoeu-
vre. For example, as illustrated in Fig. 10.5, from Daemen's work, a set of
'laminations' (i.e. one set of parallel, planar and persistent discontinuities)
has been included. Daemen has assumed, by applying the Mohr-Coulomb
failure criterion for potential slip along the discontinuities, that the presence
of the discontinuities does not affect the fundamentaI stress distribution
around the tunnel, but does affect the strength of the material. This
approach provides a useful indication of the likely areas subject to failure
under these circumstances, and hence also provides guidance on support