Page 46 - Physical Principles of Sedimentary Basin Analysis

28 Properties of porous media

is the size of the block. The average permeability k x is first found in the x-direction,
(parallel to the bedding), where it is the permeability that yields the average Darcy flux.
The average Darcy flux v x is obtained from the total rate of fluid passing through the
block
z i k i
v x z i =− (2.84)
μ L
i i
where k i is the permeability (in the x-direction) of the ith layer, z i is the layer thickness,
L is the width of the block and is the potential difference across the block. Since the
average Darcy flux v x is related to the average permeability k x , we get
k x 1
v x =− where k x = k i z i (2.85)
μ L H
i

where H = z i is the height of the block. The average permeability k x along the
i
bedding is the arithmetic average of the bedding permeability.
Next we find the average permeability normal to the bedding. We note that the Darcy
flow normal to the bedding is the same through each layer. The average Darcy flux normal
to the bedding is therefore

k i i
v z =− (2.86)
μ z i
where i is the potential difference across the ith layer. The potential difference across
the entire block in the z-direction is
z i

= i =−v z μ . (2.87)
k i
i i
The average permeability in the z-direction is the permeability that gives v z from Darcy’s

law when there is a potential difference applied across a block with height H = i z i .
We then have
k z 1 1 z i
v z =− where = . (2.88)
μ H k z H k i
i
We see that the average permeability normal to the bedding is the harmonic aver-
age of the bedding permeabilities (in the z-direction). In case the bedding perme-
ability is a continuous function of the z-coordinate k = k(z), instead of a discrete
bedding permeability, we can express the average permeabilities (2.85) and (2.88)as
integrals
1 1 1 dz
k x = k(z) dz and = . (2.89)
H k z H k(z)
It is also possible to go further and find the average permeability of more general structures
than those that are layered, but the problem is then that the flow field inside the block is no
longer 1D.