Geoscience Reference
In-Depth Information
5.
The fluid compressibility
β
is defined in Equation (8.80) in terms of partial time
derivatives.
6.
The terms (
∂
u
/∂
t
)
·∇
(
ρ
w
S
) and (
∂
u
/∂
t
)
·∇
[
ρ
a
(1
−
S
)] are negligible.
Equations (8.82) are the main result of this section. In principle it should be possible to
solve them for appropriate boundary conditions, provided the values of the parameters are
known. However, for many common situations they are more general than necessary, and
they can be simplified considerably.
Some special cases
(i)
Flow of one partially saturating, elastic fluid in an elastic porous material. Whenever
the air pressure can be assumed to be constant, the second equation of (8.82) becomes
irrelevant; the first equation of (8.82) can then be written as
k
w
ρ
w
μ
w
∇
p
w
ρ
w
S
α
∂
∂
t
(
χ
w
p
w
)
+
ρ
w
Sn
0
β
w
∂
p
w
∂
t
+
n
0
ρ
w
∂
S
∂
t
=∇·
(8.84)
This equation was examined in Brutsaert and El-Kadi (1984) to study the relative
effects of partial saturation and compressibility on the flow in unconfined systems.
It may be noted that in the groundwater literature, various derivations have been
presented; these have yielded results somewhat different from (8.84), mainly in the
first term on the left. Some reasons for the discrepancies between these other equations
and (8.84) stem from the neglect of the relative velocity in Darcy's law (8.72) and
of the equation of continuity of the solid (8.63). The latter assumption is especially
serious, since (8.63) involves the compression of the solid, which in turn gives rise to
the compressibility
α
. Other differences result from the use of the total pressure
p
w
,
rather than
p
w
, as is done here, and also from the neglect of
χ
w
.
(ii)
Flow of one elastic fluid in an elastic porous material. This case is the one to which
the theory of Biot (1941; 1955) is applicable. Because the pores are filled with one
fluid only, one has
S
=
1
.
0
,
and
χ
w
=
1
.
0; this reduces Equation (8.84) immediately
to
ρ
w
k
μ
w
∇
p
w
n
0
β
w
)
∂
p
w
∂
t
ρ
w
(
α
+
=∇·
(8.85)
in which the symbol
k
w
has been replaced by the more common
k
for the permeabil-
ity. If now also the hydraulic conductivity
k
=
(
ρ
w
gk
/μ
w
) is assumed to be constant,
(8.85) assumes the well-known linear form:
S
s
∂
p
w
∂
t
2
p
w
=
k
∇
(8.86)
where
S
s
=
ρ
w
g
[
n
0
β
w
+
(2
μ
+
λ
)
−
1
]. This form is the same as that of the equa-
tions describing heat conduction and diffusion (cf. also Equations (5.88) and (5.92)).
Equation (8.86), but with various expressions for
S
s
, has been applied widely in the
description of soil consolidation and of flow in confined aquifers. It was proposed
for one-dimensional consolidation by Terzaghi (1925), who later (1943) extended it
to three dimensions. Independently, Theis (1935) adopted the heat flow equation to
analyze horizontal unsteady flow in an elastic artesian aquifer, but he justified it only
on the basis of heuristic arguments concerning the analogy between Fourier's law
and Darcy's law. But it was Jacob (1940; 1950) who derived this heat flow equation
