Engineering Rock Mechanics

142 Permeability

z 42

Cross

Cross permeability flows

Shear stresses

Figure 9.2 Equivalence of the shear stresses and the cross-permeabilities

(Fig. 9.2). If this assumption is not made, there are nine independent
components (for example, as in the Cosserat continuum which does
allow for internal moments). In the case of permeability, the equivalent
assumption is that the resistances to cross-flow from the x to z axis and
from the z to x axis are equal (Fig. 9.2). In intact rock, this assumption
seems reasonable, but in a fractured rock mass, the assumption is more
questionable.
Thus, the tensor quantities of stress, strain and permeability are
mathematically identical (see Table 9.1): the same mathematics repres-
ents the different physical quantities. As a consequence, we can use
Mohr's circle for permeability in the same way as for stress and strain.
The units of permeability, k, are metres squared (L2). For the case
when the fluid is water, the term 'hydraulic conductivity', K, is used;
this has units of metre/second (LT-I). The relation between the two is
k = (p/yf)K where p is the water viscosity (L-'h4T-') and yf is the unit
weight of water (L-*MT2).
We distinguish between the permeabilities of the intact rock and the
fractured rock mass by using the terms 'primary permeability' and 'sec-
ondary permeability', respectively. It is reasonable to assume that the
intact rock is continuous and that the primary permeability can be rep-
resented by a tensor, with the three orthogonal principal permeabilities.
However, for secondary permeability, it should be remembered that the
fractures tend to occur in sets and the directions of maximal and min-

Table 9.1 The general and principal components of stress, strain and permeability

stress

Strain

Permeability

Components of the stress, strain

and permeability matrices with

reference to known x, y and z axes

...... py Y 52

symm. ...........

symm.

p. symm. .....$ ":I

........ kzz

The principal components

of stress, strain and permeability

0

symm. ........

0

.......

symm."" ...