Geoscience Reference
In-Depth Information
of the wetting fluid with respect to the solid is
{
[(
∂
w
/∂
t
)
/
(
n
0
S
)]
−
(
∂
u
/∂
t
)
}
and an anal-
ogous expression for the nonwetting fluid, so that the respective Darcy fluxes are given
by [(
∂
w
/∂
t
)
−
n
0
S
(
∂
u
/∂
t
)] and [(
∂
v
/∂
t)
−
n
0
(1
−
S
)(
∂
u
/∂
t
)].
When there are two fluids within the same pores, it is clear that the relative motion
between the fluids may give rise to an additional “head” loss. This means that Darcy's law,
which expresses a linear relationship between flux and the gradient of pressure and body
force, can be written in a general form as follows:
k
w
1
∂
t
+
k
wa
1
∂
t
h
w
=
μ
w
n
0
S
γ
w
n
0
S
∂
w
−
∂
u
1
n
0
S
∂
w
1
−
S
)
∂
v
−∇
−
∂
t
∂
t
n
0
(1
k
a
∂
t
+
k
wa
∂
t
(8.72)
h
a
=
μ
a
n
0
(1
−
S
)
γ
a
n
0
(1
−
S
)
∂
v
1
∂
t
−
∂
u
1
n
0
(1
−
S
)
∂
v
1
n
0
S
∂
w
1
−∇
∂
t
−
where
k
is the (intrinsic) permeability and
μ
is the Newtonian viscosity. (Note that in what
follows in this section for convenience of notation the prime symbol is omitted from the
permeability terms
k
w
,
k
a
,
and
k
wa
.) Recall that the total pressure is the sum of the initial
and the incremental pressures, or
p
w
=
p
wi
+
p
w
. Since the initial pressure is hydrostatic,
so that
∇
z
+
(
∇
p
wi
/γ
w
)
=
0, it follows that
∇
h
w
γ
w
∇
p
w
1
1
γ
w
∇
p
w
≡∇
z
+
=
(8.73)
where
z
is the vertical coordinate and
γ
w
=
ρ
w
g
the specific weight; the quantity
h
w
is
defined in Equation (8.21). The subscript w refers to the wetting fluid; the same quantities,
but with the subscript a, refer to the nonwetting phase. The cross-permeability term
k
wa
arises
from the relative motion between the two fluids. For most practical problems, the effect
of this relative motion is probably negligible. It is conceivable, however, that it becomes
important under conditions of counterflow; this would be the case, for example, of water
infiltration into a soil profile in which the displaced air is being prevented from escaping
downward so that it bubbles upward while the wetting front moves down. The possibility
of momentum exchange as a result of this relative motion has been considered already by
Yuster (1951) and Scott and Rose (1953). Mainly because it is practically impossible to
determine experimentally at present and probably small, it is omitted in what follows. Then
Equations (8.72) become
∂
w
∂
t
−
n
0
S
∂
u
k
w
μ
w
∇
p
w
∂
t
=−
(8.74)
∂
∂
t
−
n
0
(1
−
S
)
∂
v
u
∂
t
=−
k
a
μ
a
∇
p
a
The equations presented so far form a complete system. They can be combined to
eliminate certain less useful variables and to leave only those pertinent to most practical
problems. One way of accomplishing this is to consider the solid displacements and the
fluid pressures. Together with the porosity and the degree of saturation, these are seven
variables:
u
,
p
w
,
p
a
,
n
0
,
and
S
. If it is assumed that the porous matrix is homogeneous
and inert, so that
μ
and
λ
are constant and independent of
S
and thus of space and time (this
would not be the case, for example, in a clay-water-air system), substitution of (8.67) or
(8.70) into the well-known equilibrium equations (for incompressible grains) yields
2
u
+∇
[(
μ
+
λ
)
e
−
χ
p
w
−
(1
−
χ
)
p
a
]
=
0
μ
∇
(8.75)
