Page 47 - Physical Principles of Sedimentary Basin Analysis

2.11 Average permeability 29

Exercise 2.19 shows when a vertical fracture dominates the average permeability of a
layer. This example is in 2D, while it might be that fracture flow in 3D is more similar to
channel flow in thread-like structures in the fracture zone.
Exercise 2.21 studies the anisotropy of a formation made of two layers of different
thickness and different permeabilities. The anisotropy ratio of the formation is the ratio
of the average parallel and the average normal permeabilities. The exercise shows that the
anisotropy ratio is a function of the thickness ratio of the two layers and the ratio of their
permeabilities (as we would expect).

Exercise 2.18 Show that the average permeability of a thick layer of sediments is
n(z 2 − z 1 ) k 1 k 2
k z = · (2.90)
z 0 k 1 − k 2
n
where the sediment permeability is a function of the porosity, k(φ) = k 0 · (φ/φ 0 ) , and
the porosity is a function of the depth, φ(z) = φ 0 exp(−z/z 0 ). The layer is in the depth
interval from z 1 to z 2 , and the permeabilities at z 1 and z 2 are k 1 and k 2 , respectively.
Exercise 2.19 Let us assume that a layer of rock with later extent w 0 has a vertical fracture
zone of width w f where w o
w f (see Figure 2.18b). The rock has the permeability k 0
and the fracture zone has the permeability k f .
(a) Show that the average vertical permeability of the layer can be approximated as

w f
k av ≈ k 0 + k f . (2.91)
w 0
(b) Show that permeability of the layer can be characterized as either rock-dominated or
fracture-dominated as follows:

k 0 , k f w f
k 0 w 0 , rock-dominated
k av = (2.92)
k f (w f /w 0 ), k f w f
k 0 w 0 , fracture-dominated.
The fracture zone is negligible with respect to vertical fluid flow in the rock-dominated
regime. But the layer has an average permeability that is much larger than the rock per-
meability in the fracture-dominated regime, and most of the vertical fluid flow, therefore,
takes place in the fracture zone.
(c) Assume that the fracture zone is a parallel plate fracture, which has the permeability
2
k f = w /12. What is the condition for a fracture-dominated fracture in terms of the frac-
f
ture width w f ?
(d) How wide must the fracture be for the layer to be fracture-dominated when
2
k 0 = 10 −18 m and w 0 = 100 m?
Solution:
(a) The average permeability is the linear average in this case
k 0 w 0 + k f w f
w f
k av = ≈ k 0 + k f (2.93)
w 0 + w f w 0
when w 0
w f .