Page 249 - Geology and Geochemistry of Oil and Gas

216 MATHEMATICAL MODELING IN PETROLEUM GEOLOGY

or in terms of relative thicknesses,

o o o o o o 1=2
l ¼ ½ðh þ ah þ bh Þðh þ h =a þ h =bފ (11.11)
3
1
2
1
3
2
o
o
where h ¼ h i = P h i and the h is the relative frequency of a layer having the per-
i i
meability k i .
The last two equations show that the anisotropy coefficient is dependent on the
ratio of strata with a given thickness to the total thickness and on the empirical
factors. Inasmuch as the anisotropy coefficient is dimensionless, it is dependent on
the ratios of permeabilities and not on the permeability themselves.
The empirical factors a and b are constant for each particular formation or
a group of similar formations as a whole. For example, on assuming that
k 1ave ¼ 61 mD, k 2ave ¼ 5 mD, and k 3ave ¼ 0:5 mD, a ¼ 0:082 and b ¼ 0:0082, the
values of k i (i ¼ 1; 2; 3) and a and b average out the theoretical values of l. Meas-
urements of l, however, show that theoretically the thicknesses of the layer bands
have the major effect. Therefore, if only the h i for each stratigrafic–lithologic section
is known, one can calculate the anisotropy coefficients from Eq. 11.10 or 11.11.
This conclusion can be extended to multicomponent and multilayer media, as
shown by the following formulas:
o
o
o
o
o
o
o
o
l ¼ ½ðh þ ah þ bh þ þ oh Þðh þ h =a þ h =b þ þ h =oފ 1=2 (11.12)
1 2 3 n 1 2 3 n
or
1=2
n n
" #
X o X o
l ¼ ðah Þ ðh =a i Þ (11.12a)
i i
i¼1 i¼1
The median and the geometric-mean permeability is also used sometimes. From
Eq. 11.7, for a three-component multilayer medium, the geometric-mean permea-
bility (k geom ) is equal to
k geom ¼ ðk k k ? Þ 1=2 ¼ k 1 ½ðh 1 þ ah 2 þ bh 3 Þ=ðh 1 þ h 2 =a þ h 3 =bފ 1=2 (11.13)
or in terms of relative thicknesses,
o o o o o o 1=2
k geom ¼ k 1 ½ðh þ ah þ bh Þ=ðh þ h =a þ h =bފ (11.14)
1
2
3
3
1
2
The geometric-mean permeability differs from the anisotropy coefficient in being
dependent not only on the relative proportions of various strata and the empirical
coefficients, but also on the mean permeability (the base permeability). Thus, k geom
retains the dimensions of permeability.
In the case of a multicomponent multilayer medium, k geom is equal to
1=2
" , #
n n
X o X o
ða i h Þ ðh =a i Þ (11.15)
k geom ¼ k ave
i i
i¼1 i¼1
o
where k ave is the average (mean) permeability of the reservoir. The h in Eqs. 11.12,
i
11.12a, and 11.15 is the relative frequency of any empirical factor.