Page 65 - Well Logging and Formation Evaluation

Full Interpretation 55

in the direction of the bedding plane. Small, impermeable streaks will
have only very little effect on the average.
The geometric average is given by:

k i *
k geom = exp ( Âlog ( ) h i  ) (3.4.2)
h i
In effect, one takes the average of the logarithms of the individual k’s,
then takes the exponential at the end. This average is appropriate to use
if the flow in the reservoir is partially in the direction of the bedding plane
and partly normal to it. Impermeable streaks will have some influence but
not completely kill off the zonal average.
The harmonic average is given by:

k harm = ( Â( h k i ) Â ) (3.4.3)
1
i
h i
In effect, one takes the average of the inverse of the individual k’s, then
inverts the result at the end. This average is appropriate to use if the flow
in the reservoir is normal to the direction of the bedding plane. Imper-
meable streaks will completely dominate the zonal average.
Depending on which method is used, the petrophysicist can get widely
different results. Typically the arithmetic average will be at least 10
times higher than the harmonic, with the geometric lying somewhere in
between.
Note that in horizontal wells there is an additional effect due to the fact
that k v / k h on the microscopic scale is usually less than 1. The effect of
this may be estimated as follows. Let a= k v / k h , where k v = permeability
of the vertical well and k h = that of the horizontal. It may be shown that
the average permeability (k av ) seen by the wellbore, which will be par-
tially influenced by k v and partly by k h , is given by:
2p
k av = ( k h 2* p )* Ú 0 sqrt (cos 2 ( ) + *sinq a 2 ( )) qd = k h * (1 + ) 2
q
a
The result, for various values of a, is shown in Figure 3.4.1.
The parameter of k v / k h will generally be assumed over an entire reser-
voir within a dynamic model. Typical values are between 0.1 and 0.3.
From Figure 3.4.1, it may be seen that the permeabilities, as determined
from a poroperm relationship, need to be adjusted in a horizontal well,
even if the formation appears homogeneous throughout.
When giving zonal averages, it is usual to also include the product
k*h, where h is the thickness of the zone, since it is this which can be