Page 42 - Physical Principles of Sedimentary Basin Analysis

24 Properties of porous media

z
z′

x′

θ
x

Figure 2.15. The rotated block has a different permeability in the x -direction than in the z -direction

because of the layered structure.
The reference frame where the permeability becomes a diagonal matrix is called the
principal coordinate system, and the permeability values are the principal permeabilities.
The principal system is generally not aligned with the coordinate system we are working
in, where the x-axis is horizontal and the z-axis is vertical, because sedimentary layers may
be folded or tilted as shown in Figure 2.15. The diagonal permeability matrix (tensor) of
the tilted principal system must therefore be rotated to the coordinate system we are using.
The rotation of the Darcy flux, which is a vector, leads to the rotation of the permeability
tensor. The Darcy flux in the principal system is marked with a prime, and the rotation is
expressed as
1

v D = Rv =− RK ∇ (2.70)
D
μ
where R = R(θ) is the rotation matrix that rotates a vector an angle θ from the primed
to the unprimed system (see Figure 2.15). The primed gradient operator is for the primed
system. We can now insert the identity matrix written as R −1 R = I as follows:
1 −1 1

v D =− RK R R ∇ =− K∇ (2.71)
μ μ
where we identify

K = RK R −1 (2.72)
as the rotated permeability matrix (tensor) and ∇= R∇ as the rotated gradient operator

in the actual (unprimed) coordinate system.
Permeability is in general anisotropic and it is represented by a permeability matrix
(tensor)
⎛ ⎞
k xx k xy k xz
K = ⎝ k yx k yy k yz ⎠ . (2.73)
k zx k zy k zz