Geoscience Reference
In-Depth Information
The geometric average is often proposed as a
useful or more correct average to use for more
variable rock systems. Indeed for flow in a
correlated random 2D permeability field with a
log-normal distribution and a low variance the
effective permeability is equal to the geometric
mean:
where p
1 corresponds to the harmonic
mean, p ~ 0 to the geometric mean and p
¼
¼
1
0 is invalid and the
geometric mean is calculated using Eq. (
3.6
)).
For a specific case with some arbitrary hetero-
geneity structure, a value for p can be found (e.g.
by finding a p value which gives best fit to results
of numerical simulations). This can be a very
useful form of the permeability average. For
example, after some detailed work on estimating
the permeability of a particular reservoir unit or
facies (based on a key well or near-well model)
one can derive plausible values for p for general
application in the full field reservoir model (e.g.
Ringrose et al.
2005
). In general, p for k
h
will be
positive and p for k
v
will be negative.
Note that for the general case, when applying
averages to numerical models with varying cell
sizes, we use volume weighted averages. Thus,
the most general form of the permeability esti-
mate using averages is:
to the arithmetic mean (p
¼
"
#
exp
X
n
k
geometric
¼
ln
k
i
=
n
ð
3
:
6
Þ
1
This can be adapted for 3D as long as account
is also taken for the variance of the distribution.
Gutjahr et al. (
1978
) showed that for a log-
normally distributed permeability field in 3D:
2
k
eff
¼
k
geometric
1
þ ˃
=
6
ð
3
:
7
Þ
2
is the variance of ln(k).
Thus in 3D, the theoretical effective perme-
ability is slightly higher than the geometric aver-
age, or indeed significantly higher if the variance
is large.
An important condition for k
eff
where
˃
k
p
dV
Z
1
=p
Z
k
estimate
¼
dV
h
1
<
p
<
1
j
k
geometric
is
, of the permeability
variation must be significantly smaller than the
size of the averaging volume, L. That is:
that correlation length,
ʻ
ð
3
:
9
Þ
where p is estimated or postulated.
ʻ
x
ʻ
y
ʻ
z
L
x
L
y
L
z
3.2.5 Numerical Estimation of Block
Permeability
This relates to the condition of statistical
homogeneity. In practice, we have found that
ʻ
needs to be at least 5 times smaller than L for
k
b
!
For the general case, where an average perme-
ability cannot be assumed,
a priori,
numerical
methods must be used to calculate the block
permeability (k
b
). This subject has occupied
many minds in the fields of petroleum and
groundwater engineering and there is a large
literature on this subject. The numerical methods
used are based on the assumptions of conserva-
tion of mass and energy, and generally assume
steady-state conditions. The founding father of
the subject in the petroleum field is arguably
Muskat (
1937
), while Matheron (
1967
) founded
much of the theory related to estimation of flow
properties. De Marsilly (
1986
) gives an excellent
foundation from a groundwater perspective and
Renard and de Marsily (
1997
) give a more recent
review on
k
geometric
for a log-lognormal permeability
field. This implies that the assumption (some-
times made) that k
geometric
is the 'right' average
for a heterogenous reservoir interval is not gen-
erally true. Neither does the existence of a log-
normal permeability distribution imply that the
geometric average is the right average. This is
evident in the case of a perfectly layered system
with permeability values drawn from a log nor-
mal distribution - in such a case k
eff
¼
k
arithmetic
.
Averages between the outer-bound limits to k
eff
can be generalised in terms of the power average
(Kendall and Staurt
1977
; Journel et al.
1986
):
h
i
1
=p
X
k
i
=
k
power
¼
n
ð
3
:
8
Þ
the
calculation
of
equivalent
