Geoscience Reference
In-Depth Information
(
y
+
δ
y
/
2) that are in contact with the pores with size between (
z
−
δ
z
/
2) and (
z
+
δ
z
/
2),
is
Ge
g
ν
∂
h
∂
[
θ
0
(1
−
S
r
)]
2
s
e
(
y
)
s
e
(
z
)
y
2
−
δ
y
δ
z
x
n
in which the subscript n indicates the direction normal to the section. The pores with a
size between (
y
2) of the first surface are in contact with pores of all
possible sizes of the second surface. The discharge rate through these pores is
−
δ
y
/
2) and (
y
+
δ
y
/
[
θ
0
(1
−
S
r
)]
2
s
e
(
y
)
y
0
s
e
(
z
)
dz
δ
y
s
e
(
z
)
z
2
dz
δ
y
+
s
e
(
y
)
y
2
R
y
Ge
g
ν
∂
h
∂
x
n
−
where
R
is the size of the largest pores that are still available for flow at the given degree
of saturation. The first term gives the flow rate from the pores of the first surface with size
between (
y
−
δ
y
/
2) and (
y
+
δ
y
/
2) into all the pores of the second surface that are smaller
than
y
; the second term gives the flow rate into the pores of the second surface that are
larger than
y
. Integration over
y
yields finally the total discharge per unit cross-sectional
area of porous medium. Hence the intrinsic permeability, defined in Equation (8.23), can
be written as
k
=
Ge [
θ
0
(1
−
S
r
)]
2
R
0
s
e
(
y
)
y
0
R
s
e
(
y
)
y
2
R
y
s
e
(
z
)
dzdy
s
e
(
z
)
z
2
dzdy
+
0
(8.46)
where
y
and
z
are dummy variables representing
R
. This result can be applied to fully
saturated media simply by putting
R
=∞
. It can also be expressed directly in terms of
the soil water characteristic function, by means of (8.5), Laplace's equation for capillary
rise,
k
=
Ge [(2
σ/γ
)
θ
0
(1
−
S
r
)]
2
S
e
0
dy dx
x
S
e
[
H
(
x
)]
−
2
S
e
x
[
H
(
y
)]
−
2
dydx
+
0
0
(8.47)
where now
x
and
y
are the dummy variables representing
S
e
. One can show by integration
by parts that the first double integral on the right is identical with the second; thus (8.47)
can be expressed in a more condensed form as
S
e
k
=
(2 Ge) [(2
σ/γ
)
θ
0
(1
−
S
r
)]
2
(
S
e
−
x
) [
H
(
x
)]
−
2
dx
(8.48)
0
Equations (8.47) and (8.48) can now be applied immediately with suitable expressions
for
S
e
(
R
)or
S
e
(
H
) (see Brutsaert, 1968a). Although they can be applied to fully saturated
media to obtain
k
0
by putting
R
=∞
and
S
e
=
1.0, respectively, they have been applied
mostly to obtain the relative permeability
κ
r
=
k
/
k
0
.
Example 8.6. Calculation with the power function
As before, the integration is especially simple with Equation (8.14) and it produces
Ge[(2
σ/γ
)
θ
0
(1
−
S
r
)
b
]
2
(
b
+
k
=
S
2
+
2
/
b
e
(8.49)
1) (
b
+
2)
H
b