Environmental Engineering Reference
In-Depth Information
Box
9.6.1
Permeability in membranes and rocks
In Chapter 7, we introduced the permeability of a membrane in terms of the fl ux result-
ing from an applied concentration gradient:
P
(
)
j
=−
c
,
RP
−
c
L
where
c
R
and
c
P
are the concentrations of molecules on the retentate and permeate
side, respectively,
L
is the thickness, and
P
the permeability of the material.
A geological formation can be modeled as a membrane and we could use the
same law. In earth sciences, however, the starting point is hydrodynamics. In hydro-
dynamics, the main driving forces for fl ow are gravity and pressure differences.
Darcy's law relates the fl ux (in m
3
/s) to the pressure gradient:
k
k
(
)
j
=−
p
−
p
=−
∇
p
,
e
b
µ
L
µ
where
p
e
and
p
b
are the pressures at the end and beginning of the fl ow region, respec-
tively,
is the viscosity (in Pa s) and
k
is the
permeability of the medium (in m
2
). The viscosity appears in this equation because it
is derived from the Navier-Stokes equation, which in turn is derived from the conserva-
tion of momentum.
∇
p
is the gradient of the pressure. Here,
µ
of porosity for sandstones in the St. Peter sandstone of the Illinois Basin
[9.21]. A graphical representation of the variability in permeability is
shown in
Figure 9.6.2
[9.22]. Generally, as grain size increases in rocks,
the permeability increases by over eight orders of magnitude.
Fracture permeability
The data in
Figure 9.6.2
also show the effect of fractures on the perme-
ability of some rocks. Fractures can impart very high permeability to
otherwise low-permeability rocks. In fact, even a single fracture with a
small aperture can impart a fairly large overall permeability to a rock. It
has been shown that fractures provide permeability
k
(in m
2
) according
to the so-called Cubic Law, which follows from the fl ow of a fl uid between
two parallel plates:
3
Nb
k
=
,
12
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