Geology Reference
In-Depth Information
length/time (Bear
1972
; Hubbert
1956
). For our purposes, it is more convenient to
express (
3.42
) in a form free of gravitational connotation that corresponds more
obviously to Ficks' first law, namely
q
¼
K
0
dp
dx
ð
3
:
43
Þ
where K
0
¼
K
=
qg is another conductivity constant expressing the rate of perme-
ation per unit pressure gradient. The relations (
3.42
) and (
3.43
) only apply for
relatively low fluid velocities (Hubbert
1956
).
The different roles of the solid and fluid phases are commonly distinguished,
under the assumption that they are independent of each other, by writing the
conductivity constant K
0
as the product of the fluidity 1
=
g of the fluid (g is the
dynamic viscosity) and a constant k characterizing the solid, called the perme-
ability or the intrinsic permeability. Thus we have
k
¼
K
0
g
¼
Kg
qg
ð
3
:
44
Þ
The permeability k has dimensions (length)
2
and hence the SI unit m
2
. Another
widely used unit is the darcy: 1 darcy & 10
-12
m
2
. The permeability k is thus a
geometric property of a rock and it can be measured in the laboratory or in the
field; see, for example, Brace (
1980
,
1984
). The ranges of typical values for
laboratory specimens of various types of rock are given in Fig.
3.4
. It should be
noted, however, that at pressures below a few megapascals in a gas, when the
mean free path becomes comparable to the dimensions of the connected pores or
cracks, the distribution of flow within the pores is no longer the same as at high
pressures and the values of K and K
0
will vary with pressure if Darcy's equation is
used (von Engelhardt
1960
, p. 127). Also the permeability is reduced, becoming a
function of fluid content, when the pores are not fully saturated with the fluid, as in
the case of soils partially saturated with water (Bear
1972
, Chap. 9).
Various models have been proposed as bases for calculating the permeability of
a porous solid (Bear
1972
, Chap. 5; Scheidegger
1960
, Chap. 6). In one widely
accepted and intuitively evocative model, the fluid movement is viewed as a flow
through an equivalent channel or parallel set of channels of mean length l
e
per
length l of porous solid traversed (Fig.
3.5
). In analogy with the Poiseuille formula
for flow in a pipe of diameter 4R
;
R
2
g
q
¼
1
2
dp
dx
;
one can then write for the flow in a porous solid
R
2
g
/
l
e
=ð
2
dp
dx
q
¼
C
ð
3
:
45
Þ
