Permeability and Seepage Flow - Rock Mechanics Based on an Anisotropic Jointed Rock Model - page 137

Thus, the mean velocity of turbulent fl ow is no longer proportional to the hydraulic

gradient I.
If 0.0168 < k/D h 0.032, the transition from laminar to turbulent irrotational fl ow

according to Rißler (1977) takes place at the following critical Reynolds number, de-
pending on k/D h (Fig. 6.8, left):

≤

(6.24)

If Re C is exceeded, Equation (6.23) is recommended to be used to describe turbulent
fl ow in this range of relative roughness (Wittke 1990).

If k/D h > 0.032, the following fl ow equation based on relationships proposed by Louis
(1967) and Rißler (1977) can be used to describe turbulent rotational fl ow in disconti-
nuities (Fig 6.8, right):

(6.25)

Equations (6.24) and (6.25) indicate that the transition to turbulent fl ow with increasing
relative roughness is displaced towards smaller critical Reynolds numbers.
In Fig. 6.9, as an example, the mean fl ow velocity in a discontinuity is represented as
a function of the hydraulic gradient, the relative roughness (k/D h = 0, k/D h = 0.02,
k/D h = 0.2) and the mean aperture ( = 0.1 mm, = 0.5 mm, = 1.0 mm).
It can be seen that irrotational fl ow in a hydraulically smooth discontinuity with
a mean aperture of 0.5 mm remains laminar under hydraulic gradients up to 17

(Fig. 6.9, upper). Rotational fl ow in a discontinuity with relative roughness of 0.2
and a mean aperture of

≤

0.5 mm remains laminar under hydraulic gradients up to
9. The transition to turbulent fl ow at small hydraulic gradients takes place only in
discontinuities with large mean apertures of some 1 mm (Fig. 6.9, lower).
Thus, turbulent fl ow conditions arise only at comparatively large discontinuity aper-
tures or at high hydraulic gradients, and the assumption of laminar fl ow is reasonable
for most practical problems in rock engineering (Wittke 1990).

≤