Civil Engineering Reference
In-Depth Information
Using Equation (4.77), the flow resistivity
σ
in the
x
direction is
p
2
−
p
1
n(υπR
2
)
d
=
8
η
nπR
2
σ
=
(4.140)
cos
ϕ
where
υ
is the modulus of the average velocities
υ
1
and
υ
2
. From Equation (4.138),
σ
can be rewritten
8
η
σ
=
(4.141)
φR
2
cos
2
ϕ
The parameter
s
in Equation (4.128) is given by Equation (4.129). Eliminating
η/R
2
between Equations (4.16) and (4.141) we find
1
/
2
8
ωρ
o
σφ
cos
2
ϕ
s
=
(4.142)
The quantity 1
/
cos
2
ϕ
is the tortuosity of the material. The concept of tortuosity is
not recent, and it appears with different notation and meanings in previous works. For
Carman (1956) the tortuosity is related to
(
cos
ϕ)
−
1
and in the topic by Zwikker and
Kosten (1949), tortuosity is denoted by
k
s
and is called the structure form factor. In latter
works, tortuosity is denoted as
α
∞
1
cos
2
ϕ
α
∞
=
(4.143)
instead of
(
cos
ϕ)
−
2
,
σ
and
s
can be rewritten
Using
α
∞
8
ηα
∞
φR
2
σ
=
(4.144)
8
ωρ
o
α
∞
σφ
1
/
2
s
=
(4.145)
In the formalism of the Biot theory (Biot 1956) that will be presented in Chapter 6,
the tortuosity is related to an inertial coupling term
ρ
a
by
ρ
a
=
ρ
o
φ(α
∞
−
1
)
(4.146)
A method for measuring tortuosity exists, which, however, can only be used if the
frame does not conduct electricity. The porous material is saturated with a conduct-
ing fluid, and the resistivity of the saturated material is then measured, as indicated in
Figure 4.13.
Let
r
c
and
r
f
be the measured resistivities of the saturated material and the fluid,
respectively. It can easily be shown that the tortuosity
α
∞
is given by
cos
2
ϕ
=
φ
r
c
1
α
∞
=
(4.147)
r
f
which is independent of the shape of the cross-sections of the pores. As shown by Brown
(1980) the relation
α
∞
=
φr
c
/r
f
can be generalized to any porous medium. Another
