Geoscience Reference
In-Depth Information
of the hydraulic radius concept; it was proposed originally by Kozeny (1927) for saturated
materials and can be defined as the ratio of pore volume to particle area. Irmay (1954)
extended this to unsaturated porous media and derived (8.36) with
n
=
3.
Parallel models
In this approach the pore system is assumed to be equivalent with a bundle of uniform
capillary tubes of many different sizes. The distribution of the pore sizes is derived from
the soil water characteristic, i.e.
S
e
=
S
e
(
H
), through Equation (8.5), i.e.
H
=
H
(
R
), as
explained after Equation (8.6). The true mean velocity in each pore can be described by a
Hagen-Poiseuille type equation for creeping flow, namely (8.24) with a value of Ge around
(1/8).
Because
s
e
(
R
)
=
θ
0
(1
−
S
r
)(
dS
e
/
dR
)
δ
R
=
θ
0
(1
−
S
r
)
s
e
(
R
)
δ
R
is the portion of the pore volume occupied by
“active” pores, with radius between (
R
−
δ
R
/
2) and (
R
+
δ
R
/
2), where
δ
R
denotes a very
small increment of
R
. It follows that [
θ
0
(1
−
S
r
)
s
e
(
R
)
δ
R
] is also the area, per unit bulk
cross-sectional area of porous material, occupied by openings whose sizes are between
(
R
−
δ
R
/
2) and (
R
+
δ
R
/
2). The flow rate through this elemental area is, by virtue of
(8.24),
=
dS
e
(
R
)
/
dR
represents the pore size density,
δθ
(
R
)
=
(
d
θ/
dR
)
δ
R
Ge
g
ν
∂
h
∂
x
n
[
θ
0
(1
−
S
r
)]
s
e
(
R
)
R
2
−
δ
R
(8.42)
in which the subscript n refers to the direction normal to the area under consideration.
With
s
e
(
R
)
dR
=
dS
e
and with Laplace's equation (8.5), i.e.
R
=
2
σ/
(
γ
H
), one obtains the
intrinsic permeabilty, by integration over all pores filled with water,
S
e
k
=
[
H
(
x
)
]
−
2
dx
(2
σ/γ
)
2
Ge[
θ
0
(1
−
S
r
)]
(8.43)
0
where
x
is the dummy variable representing
S
e
.
Purcell (1949) and Gates and Tempelaar-Lietz (1950) were among the first to apply
this approach, and came up with expressions similar to Equation (8.43). However, because
(8.43) tended to yield values considerably larger than available experimental data, several
subsequent authors included a tortuosity factor in the formulation to account for the limita-
tions inherent in this model of straight parallel tubes. The tortuosity concept had originally
been introduced by Carman (1937; 1956) as an improvement on the uniform hydraulic
radius model of Kozeny (1927), and it can be expressed as
T
=
(
L
e
/
L
)
2
, in which
L
e
is the
actual or microscopic path length of the fluid particles in the pores, and
L
is their apparent
or macroscopic path length along the Darcy streamlines. In several studies this tortuosity
was assumed to depend on the water content, i.e.
S
e
; in this case the relative permeability
κ
r
=
k
/
k
0
(
=
k
/
k
0
) can be written as
⎡
⎣
β
⎤
⎦
⎡
⎣
β
0
⎤
⎦
S
e
1
[
H
(
x
)]
−
2
dx
[
H
(
x
)]
−
2
dx
κ
r
=
(8.44)
0
0
where the variable
β
=
β
(
S
e
) is related to the tortuosity and
β
0
is its value at
S
e
=
1
.
0.
Burdine (1953) proposed on the basis of his experimental data that (
β/β
0
)
=
S
e
.
