Geoscience Reference
In-Depth Information
Fig. 3.5
Effective permeability and upscaled block permeability (
a
) Real rock medium has some (unknown)
effective permeability. (
b
) Modelled rock medium has an estimated block permeability with the same average flow
as the real thing
Full permeability tensor
é
k
k
k
ù
xx
xy
xz
ê
ú
Coordinates
=
k
eff
k
k
k
ê
ú
Permeable medium
yx
yy
yz
x
ê
ú
k
k
k
y
ë
û
zx
zy
zz
z
Diagonal permeability tensor
k
0
0
é
ù
xx
ê
ú
k
eff
=
0
k
0
ê
ú
yy
ê
ú
0
0
k
ë
û
zz
Fig. 3.6
The permeability tensor
Note that in petrophysical analysis, the term
'effective porosity' refers to the porosity of move-
able fluids excluding micro-porosity and chemi-
cally bound water, while total porosity
encompasses all pore types. Although effective
porosity and effective permeability both represent
properties relevant to, and controlling,
macroscopic flow, they are defined on different
bases. Effective permeability is essentially a
larger scale property requiring statistical homoge-
neity in the medium, whereas effective porosity is
essentially a pore-scale physical attribute. Of
course, if both properties are estimated at, or
rescaled to, the same appropriate volume, they
may correspond and are correctly used together
in flow modelling. They should not, however, be
automatically associated. For example, in an
upscaled heterogeneous volume there could be
effective porosity elements (e.g. vuggy pores)
which do not contribute to the flow and therefore
do not influence the effective permeability.
In general, k
b
is a tensor property (Fig.
3.6
)
where, for example, k
xy
represents flow in the
x direction due to a pressure gradient in the
y direction. In practice k
b
is commonly assumed
to be a diagonal tensor where off-diagonal terms
are neglected. A further simplification in many
reservoir modelling studies is the assumption
that k
h
¼
k
zz.
The calculation or estimation of k
b
is depen-
dent on the boundary conditions (Fig.
3.7
). Note
that
k
xx
¼
k
yy
and that k
v
¼
the assumption of a no-flow or sealed
