Page 45 - Physical Principles of Sedimentary Basin Analysis

2.11 Average permeability 27

Note 2.2 The Darcy flux vector v D is normal to the tangent plane of the ellipsoid, because
it is proportional to the gradient of the ellipsoid function

x 2 y 2 z 2
f (x, y, z) = + + . (2.81)
a 2 b 2 c 2
We have that ∇ f = (2xk x , 2yk y , 2zk z ) = 2r(k x n x , k y n y , k z n z ), which is parallel to v D .
Recall that ∇ is parallel to n and v D is therefore parallel to Kn = (k x n x , k y n y , k z n z ).

Exercise 2.16 The permeability in the principal system is

k x 0
K = . (2.82)
0 k z
2
(a) Show that the directional permeability is k n = k x cos θ + k z sin θ in the direction of
the unit vector n = (cos θ, sin θ).
(b) Let k z
k x and show that the directional permeability is reduced from its maximum
◦
at θ = 0 to approximately half at θ = 45 .
T
Exercise 2.17 Show how the directional permeability k n = n Kn for a general aniso-
tropic permeability tensor can be rewritten using a diagonal tensor.
Solution: Let R be the matrix that rotates the coordinate system into the principal
T
coordinate system, which makes D = RKR diagonal. We can then write
T T T T
k n = n R RKR Rn = m Dm (2.83)
where m = Rn is the unit vector in the principal coordinate system.

2.11 Average permeability

Sedimentary rocks are often layered due to the deposition of different lithologies. An
example of a block of sediments with a layered structure is shown in Figure 2.18.
We will need the average permeability of such a block, when the spatial resolution

w f

k i Δz i
k
H f
k 0

L w 0
(a) (b)
Figure 2.18. (a) A block of layered sedimentary rock, where layer i has thickness z i and permeability
k i . (b) A layer of rock with a vertical fracture zone.