Geoscience Reference
In-Depth Information
where
u
concept is widely used, and abused, and requires
some care in its use and application. It is also
rather fundamental - if it was a simple thing to
estimate the correctly upscaled permeability for a
reservoir unit, there would be little value in res-
ervoir modelling (apart from simple volume
estimates).
The upscaled (or block) permeability, k
b
,
is defined as the permeability of an homoge-
neous block, which under the same pressure
boundary conditions will give the same average
flows as the heterogeneous region the block is
representing (Fig.
3.5
). The upscaled block per-
meability could be estimated, given a fine set of
values in a permeability field or model, or it
could be measured at the larger scale (e.g. in a
well test or core analysis), in which case the fine-
scale permeabilities need not be known.
The
effective permeability
is defined strictly in
terms of effective medium theory and is an
intrinsic large-scale property which is indepen-
dent of the boundary conditions. The main theo-
retical conditions for estimation of the effective
permeability, k
eff
, are:
That the flow is linear and steady state;
That the medium is statistically homogeneous
at the large scale.
When the upscaled domain is large enough,
such that these conditions are nearly satisfied,
then k
b
approaches k
eff
. The term
equivalent
permeability
, is also used (Renard and de Marsily
1997
) and refers to a general large-scale
permeability which can be applied to a wide
range of boundary conditions, to some extent
encompassing both k
b
and k
eff
. These terms are
often confused or misused, and in this treatment
we will refer to the permeability upscaled from
a model as the
block permeability
,k
b
, and use
effective permeability
as the ideal upscaled per-
meability we would generally wish to estimate
if we could satisfy the necessary conditions.
In reservoir modelling we are usually estimating
k
b
in practice, because we rarely fully satisfy
the demands of effective medium theory. How-
ever, k
eff
is an important concept with many
constraints that we try to satisfy when estimating
the upscaled (block) permeability.
¼
intrinsic fluid velocity
k
¼
intrinsic permeability
ʼ
¼
fluid viscosity
P
¼
applied pressure gradient
∇
ˁ
gz
¼
pressure gradient due to gravity
∇
P (grad P) is the pressure gradient, which can
be solved in a cartesian coordinate system as:
dP
dx
þ
dP
dy
þ
dP
dz
P
¼
ð
3
:
5
Þ
∇
The pressure gradient due to gravity is then
ˁ
z. For a homogeneous, uniform medium k
has a single value, which represents the
medium's ability to permit flow (independent of
the fluid type). For the general case of a hetero-
geneous rock medium, k is a tensor property.
g
∇
Exercise 3.3
Dimensions of permeability
What are the dimensions of perme-
ability? Do a dimensional analysis for
Darcy's Law.
For the volumetric flux equation
Q
¼
KA
ð
ʔ
H
=
L
Þ
The dimensions are
L
3
T
1
¼
L
2
LT
1
Therefore the SI unit for K is:
ms
1
Do the same for Darcy's Law:
u
¼
=ð :
∇
k
ð
P
þ ˁ
gz
Þ
The dimensions are
½
¼½
=
½
ð
Þ:
½
Therefore the SI unit for k is
____
3.2.2 Upscaled Permeability
In general terms,
upscaled permeability
refers to
the permeability of a larger volume given some
fine scale observations or measurements. The
