Engineering Rock Mechanics

166 Anisotropy and inhomogeneity

and kp (parallel to bedding) we obtain

ka = kp. cos2 6, + k,,. sin2 6, k, = kp. cos2 6, + k,,. sin2 6,

which, in matrix form, are

cos2 6, sin2 6,

[:] = [cos26. sin26] * [:] '

Solving for k,, and kp as the principal values gives kp = 1.7 x lop6 m/s

and k,, = 3.008 x lo-* m/s.

Note that kp > k,, a feature which is common in such rocks. Defining

an anisotropy index in terms of the ratio of these principal values, we

have p = kp/kn and hence p = 56.5. The polar plot equations can be

used to produce a polar plot of the hydraulic conductivity in a section

perpendicular to the bedding, and this is shown below.

90.

270

This illustrates quite clearly the anisotropy of hydraulic conductivity

in this sandstone.

410.6 The diagram (see next page) shows a polar plot of the vari-

ation in hydraulic conductivity, K, of a sandstone, with the maximal

and minimal values occurring parallel and perpendicular to the

bedding.

Why is there a cusp in the (K,8) locus on the line representing the

direction normal to the bedding?

A1 0.6 The occurrence of cusps in rock mechanics loci is not unexpected.
In the fracture frequency formula, h, = E:=, hi1 COS~~I, used in A7.6,
there are cusps caused by the absolute value function, i.e. I cos 6, I.
However, that is not the case here.