Permeability and Seepage Flow - Rock Mechanics Based on an Anisotropic Jointed Rock Model - page 131

The factor of proportionality k IR between v s and I in (6.4) is referred to as the intact
rock's “coeffi cient of water permeability” or simply “permeability coeffi cient”.
Equation (6.4) can be generalized to the three-dimensional case as:

(6.8)

where {v s } and {I} are the vectors of seepage velocity and hydraulic gradient

(6.8a)

(6.8b)

and is the Nabla operator:

Equation (6.8) is valid if the intact rock's permeability is isotropic. Because of the irreg-
ular network of fl ow paths (Fig. 6.3) the permeability of most intact rocks, if relevant,
can be considered as isotropic. Otherwise k IR must be replaced by a tensor (Section
6.4.2).

6.3 Discontinuities

6.3.1 Laminar Flow
Flow is referred to as “laminar” if the streamlines run parallel to each other, i.e. if there
are no noticeable interactions between the water particles. Because of the low velocities
of water particles, in most cases fl ow of water in discontinuities can be considered as
laminar.
In the following, the laminar fl ow of water in a persistent, planar and open discontinu-
ity D with constant aperture 2a i is considered. The discontinuity-oriented coordinate
system (x',y',z') is selected in such a way that the coordinate axes x' and y' lie on the
middle plane of the discontinuity and the z' axis is oriented perpendicular to the same
(Fig. 6.5). For a two-dimensional laminar fl ow, which is of particular signifi cance in
rock mechanics, the following parabolic velocity distributions v x' and v y' can be derived
(Louis 1967, Wittke 1970, Wittke 1990):