Geoscience Reference
In-Depth Information
for confined aquifer flow in terms of the physical properties of the aquifer and the
fluid. Subsequently, however, the exact derivation of Equation (8.86) remained the
subject of controversy, until Verruijt (1969) showed how it can be reconciled with
Biot's (1941; 1955) analysis.
In addition to the limitations inherent in Equation (8.84), the Terzaghi-Jacob equa-
tion (8.86) is also restricted by the assumption of a constant hydraulic conductivity.
As shown earlier in this chapter, the permeability
k
is dependent on the porosity
n
0
;
for instance, in Equations (8.48) and (8.51)
k
is proportional to
n
0
. Thus, in deriving
(8.86) from (8.85), it is assumed that
n
0
and
ρ
w
=
ρ
w
(
p
w
) are constant, even though
they are in fact unknown dependent variables. In other words, it is assumed that the
fluid and the porous matrix are compressible on the left side of (8.85) but not on the
right. This assumption may have its limitations whenever
∇
p
w
is not small. Still,
in spite of this inconsistency, for most problems the exact formulation of (8.86) in
terms of physical properties of the porous material and the fluid is probably not very
crucial, since (
S
s
/
k
) is usually determined from field experiments. Consequently, as
irrotational or unidirectional displacement and the constancy of
k
and
ρ
w
may be
difficult to justify, the main problem is not how to express
S
s
in terms of
n
0
,β
w
,μ,λ
,
etc., at the micro- or Darcy scale, but rather whether the heat flow equation (8.86) is
adequate to solve the practical problem at hand at the larger scale of the aquifer.
(iii)
Flow of two immiscible fluids in an incompressible porous material. In this special
case, the solid phase cannot move, so that the displacement
u
and the rate of displace-
ment (
∂
u
/∂
t
) of the solid aggregate are equal to zero; therefore Equations (8.76) can
be written as:
n
0
∂
ρ
w
k
w
μ
w
p
w
∂
t
(
ρ
w
S
)
=∇·
∇
(8.87)
and
n
0
∂
∂
t
ρ
a
k
a
μ
a
∇
p
a
(
ρ
a
(1
−
S
))
=∇·
(8.88)
These equations are equivalent to those first proposed by Muskat and Meres (1936)
but taking account of the solubility of the non-wetting fluid in the wetting fluid.
Equations (8.88) have been used to study the effect of soil air movement on the infiltration
of water (Le Van Phuc and Morel-Seytoux, 1972; Morel-Seytoux, 1973).
Whenever the effect of the non-wetting fluid is negligible, owing to small viscosity
μ
a
and small pressure changes
∇
p
a
, only the wetting fluid is of interest; if the density of this
fluid
ρ
w
is constant, (8.87) reduces to
n
0
∂
S
(
k
∇
p
w
)
∂
t
=∇·
/γ
w
(8.89)
which is equivalent to Equation (8.55), first derived by Richards (1931) for soil water
movement.
REFERENCES
Arbhabhirama, A. and Dinoy, A. A. (1973). Friction factor and Reynolds number in porous media flow.
J. Hydraul. Div., Proc. ASCE
,
99
, 901-911.
