Page 44 - Physical Principles of Sedimentary Basin Analysis

26 Properties of porous media

z z
v D
c n
b y
r c
x a x
a

(a) (b)
√
Figure 2.16. (a) The scalar directional permeability k n gives the radius of the ellipsoid as r = 1/ k n .
(b) The unit vector n is in the direction of the gradient of the potential, and the Darcy flux vector v D
is normal to the tangent plane of the ellipsoid.

i j

(a) (b)
Figure 2.17. (a) The cell centers in a grid are connected by transmissibilities. (b) The transmissibility
connecting cell i and j.

The direction permeability can be used in numerical applications to compute the
transmissibilities, which is the conductivity of the bonds that connect the cells in a grid.
Figure 2.17a shows a grid where the cell centers are connected by bonds. The fluid flow in
Figure 2.17b from cell i to cell j is the transmissibility between the two cells multiplied
by the pressure difference. The flow rate (in units of volume fluid per time) from cell i
to cell j is
k n (p j − p i )
q = Av n = A (2.79)
μ l ij
where A is the area of the cross-section between the cells, k n is the directional perme-
ability between the cell centers, p j − p i is the pressure difference between the cells and
l ij is the distance between the cell centers. The directional permeability is made with
the unit vector in the direction of the pressure gradient, which is the direction of the
transmissibility. Most of the parameters in the flow rate (2.79) are given together as the
transmissibility

Ak n
T ij = . (2.80)
μl ij
The flow rate q from cell i to cell j is therefore the transmissibility T ij times the
pressure difference p j − p i between the cells. For a strongly anisotropic rock the
directional permeability will be quite nonuniform for connections pointing in different
directions.