Page 43 - Physical Principles of Sedimentary Basin Analysis

2.10 Directional permeability 25

We will assume that there exists a coordinate system where the permeability tensor
becomes diagonal. The permeability tensor in any other reference frame with a different
orientation is obtained by rotation of the diagonal permeability tensor. The application of
the rotation matrix on a diagonal matrix leads to a symmetric matrix. The existence of a
diagonal permeability matrix therefore implies the symmetry property of the permeability
tensor. This means that k yx = k xy , k zx = k xz and k yz = k zy , and that there are only six
independent elements in the tensor (2.73).

2.10 Directional permeability
The Darcy flux v D is in the case of potential flow
1
v D =− K ∇ (2.74)
μ
but the direction of the Darcy flux (v D ) and the gradient of the potential (∇ ) are not
necessarily the same in the case of anisotropic permeability. We will now introduce the
directional permeability, which is the (scalar) permeability in the direction of the poten-
tial gradient. The directional permeability can be thought of as the permeability of a thin
cylindrical core taken in the wanted direction of the anisotropic rock. The gradient of the
potential can be written

∇ =|∇ | n (2.75)
where n is the unit vector in the direction of the gradient. The Darcy flux in the direction
of n is v n = n · v D , and the permeability in the same direction is therefore
v n T
k n = = n Kn. (2.76)
1
μ |∇ |
The directional permeability (2.76) becomes
2
2
k n = k x n + k y n + k z n 2 z (2.77)
x
y
in the principal system, where the permeability tensor is diagonal. The directional perme-
ability (2.77) can also be represented by an ellipsoid, and to see that we let n x = x/r,
√
2
2
2
2
n y = y/r and n z = z/r where x + y + z = r . In the case r = 1/ k n we get the
general equation for an ellipsoid
x 2 y 2 z 2
+ + = 1 (2.78)
a 2 b 2 c 2
√ √
with a = 1/ k x , b = 1/ k y and b = 1/ k z . The radius of the ellipsoid is therefore an
expression for directional permeability, and the semi-axis represents the principal perme-
ability. The Darcy flow v D is actually normal to the tangent plane of the ellipsoid at the
point (x, y, z), a property that is shown in Note 2.2 (see Figure 2.16).