Engineering Rock Mechanics

Questions and answers: anisotropy and inhomogeneity 171

structural domains, it should be utilized where possible, and is the basis

of the next question.

470.10 The subiect of geostatistics deals with the variation of prop-

erty values in space, and so anisotropy and inhomogeneity can

be quantitativety studied. The basic device of geostatistics is the

semi-variogram, defined as

where y(h) is the semi-variogram statistic for samples distance h
apart, n is the number of sample pairs, p(x) is the rock property
value at location x, and p(x + h) is the value at location x + h.
Using this statistic, a graph can be constructed of y(h) versus h
to indicate the variation in rock property values as a function of
distance between the observations.

I/ /

h h h h

In the diagram above, there are four examples of such semi-vari-
ogram curves for a rock mass. In each case, the length of the h axis
represents 50 m.
(a) Explain what type of variation in rock property values, or
inhomogeneity, each type of semi-variogram represents.
(b) For each type of semi-variogram behaviour, how far away
from a borehole would you feel confident in extrapolating results
obtained from a borehole core?

A10.70 (a) If the rock mass were homogeneous, p(x) and p(x + h)
would always have the same value for all values of x and h, and the
semi-variogram statistic y(h) would be zero for all values of x and h.
However, given the form of the equation for y(h), the statistic will take
on positive values when there are differences between the rock property
values at different sample locations, i.e. when there are differences
between p(x), the rock property value at location x, and p(x + h), the
value at location x + h. Thus, we might expect that as h is increased,
i.e. the distance between the samples is increased, that y(h) will increase
because the rock properties will have changed by a greater amount over
larger h distances.
The magnitude of y(h) and the way in which y(h) increases is thus
an indication of the type of heterogeneity present in the rock mass.

' Borehole information represents specific information from within the rock mass; we

have to interpolate between the boreholes to estimate the properties between them.