Geology Reference
In-Depth Information
to which q is related; and l
e
=ðÞ
enters twice in order to take into account, first, that
the local velocity in the channel is increased by the ratio l
e
=ðÞ
relative to that in a
channel parallel to the direction of bulk flow and, second, that the pressure gra-
dient along the equivalent channel is less than in the direction of bulk flow
(Sullivan and Hertel
1942
). The derivation of Chapman (
1981
, Chap. 3), which
leads to l
e
=ð
3
in the above equation appears to be in error because of using
/ l
e
=ðÞ
instead / of for the cross-sectional area factor since this factor relates to
flow in the direction for which q is defined, not the local flow direction. From
(
3.45
), one therefore obtains the following expression for the permeability,
k
¼
C/R
2
l
e
=ð
2
ð
3
:
46
Þ
The quantity l
e
=ð
2
is often called the tortuosity and designated by T, although
this name and symbol are variously used also for l
e
=ðÞ
and l
=
l
ð
2
in the literature.
The relative length l
e
=ðÞ
will depend on the porosity and can be expected to
increase as the porosity decreases and the paths followed by the fluid become more
tortuous, but it is difficult to obtain a direct measure of it. It is therefore common to
approach its evaluation indirectly by invoking an analogy with electrical con-
ductivity and introducing the ''formation resistivity factor'' F, which is the ratio of
the electrical resistivity of the saturated porous body to that of the pure fluid, it
being assumed that the fluid is electrically conducting and the solid parts of the
body are not, and that the electric current and the fluid flow will follow identical
paths (Archie
1942
). From Ohm's law, the electrical equivalent of Poiseuille's
law, we have
j
¼
j
dV
dx
¼
j
dV
dx
¼
1
dV
dx
j
f
F
j
f
ð
3
:
47
Þ
j
f
where j is the macroscopic current density, dV
=
dx the macroscopic voltage gra-
dient, j the conductivity of the saturated porous body, and j
f
the conductivity of
the fluid. Then, by applying to the analogous model of an equivalent electrical
conduit similar arguments to those used to generalize Poiseuille's law to (
3.45
), we
obtain 1
=
F
¼
/
=
l
e
=ð
2
which contains a factor/
=
l
e
=ðÞ
relating to the cross-sec-
tional area of the conduit and a factor 1
=
l
e
=ðÞ
relating to its length. This rela-
tionship can then be used in (
3.46
) to give
k
¼
CR
2
F
ð
3
:
48
Þ
The same result can be obtained by noting, in comparing (
3.45
) and (
3.47
) for
the equivalent channel model, that the conductivity j
f
in the electrical case is the
analog of CR
2
=
g in the fluid flow case. If we now insert in (
3.48
) the Archie
empirical relation F
¼
/
m
;
we obtain
k
¼
C/
m
R
2
ð
3
:
49
Þ
