Geology Reference
In-Depth Information
where d is the grain size (Bear
1972
, p. 133; Hubbert
1956
; Rumer
1969
).
Comparing this expression with (
3.49
) indicates that the hydraulic radius in
granular media is approximately proportional to the grain size since the factor C/
m
does not vary markedly; putting C
¼
0
:
4
;
/
¼
0
:
4 and m
¼
1
:
3 shows that the
expression corresponds to taking the hydraulic radius to be approximately d
=
14
:
It is possible that in general the permeability can be expressed in the form
k
¼
Cf
ðÞ
R
2
ð
3
:
50
Þ
where f
ðÞ
is a function of / that becomes more sensitive to variation in / as the
value of / decreases, especially when connectivity begins to decrease significantly
(Bernabé et al.
1982
). The fall in permeability toward zero as connectivity is lost is
related to similar connectivity effects in electrical conductivity in rocks (see, for
example, Waff
1974
for the case of partial melting). This limit is analogous to the
percolation limit in resistor networks (Chelidze
1982
; Kirkpatrick
1973
) or to the
critical point in phase transitions. It has been treated by the theory of renormal-
ization groups (Allègre et al.
1982
; Madden
1983
).
The approach to fluid permeation through Darcy's law with a constant per-
meability does not apply when local flow velocities become high enough for
inertial or turbulent effects to become significant (Scheidegger
1960
, Chap. 7).
Limitations due to molecular effects in gasses at low pressures have already been
mentioned but complications can also arise from ionic effects in the flow of
electrolyte solutions, especially if clays are present (Scheidegger
1960
, Chap. 7).
Finally, external stress applied to the porous medium influences its permeability
but the effect depends on both confining pressure and pore pressure and can be
related
to
an
effective
pressure
(see
Walsh
1981
for
the
case
of fracture
permeability).
For nonsteady-state flow in the Darcy-law regime, the governing equation,
analogous to Fick's second law and the heat flow equation, is
or
ot
¼
D
o
2
p
o
p
ot
¼
o
D
o
p
ox
op
ð
3
:
51
Þ
ox
ox
2
where the second equation applies when D is independent of p. In analogy with the
thermal diffusivity, the fluid diffusivity D has the form
D
¼
K
0
b
0
¼
k
ð
3
:
52
Þ
gb
0
where b
0
is the fluid storage capacity per unit volume of the porous body, given by
b
0
¼
/b
þ
b
e
1
þ
/
ð
Þ
b
s
ð
3
:
53
Þ
b being the compressibility of the fluid, b
s
the compressibility of the solid parts of
the body, and b
e
the macroscopic compressibility of the porous body with zero or
constant pore pressure (Brace et al.
1968
); for a summary of the elasticity of
