The fact that the effective permeability
has dimensions of area raises the question whether
has an interpretation as a length
scale.
The traditional answer to this question is provided
by hydraulic radius theory which uses the approximate
result (5.61) for the capillary tube model
to postulate more generally the relation
![]() |
(5.98) |
where is the hydraulic
radius.
This generalization has been modified by incorporating the
formation factor to write [327, 112]
![]() |
(5.99) |
where the length scale is still given
by the hydraulic radius, and the geometrical tortuosity
was replaced by the electrical tortuosity defined as
.
Because the length scale is still given by the hydraulic
radius this theory is still faced with the objection that
the hydraulic radius
contains contributions from
the dead ends which do not contribute to the transport.
An alternative was proposed in [318, 43].
It postulates where
is a length scale
related to the breakthrough pressure in mercury injection
experiments.
The length scale
is well defined for network models
with a broad distribution of cylindrical pores.
A dynamical interpretation of
was proposed in
[319, 320, 328] as
![]() |
(5.100) |
where is the unknown exact solution of the
microscopic dielectric problem.
This “electrical length” is expected to measure,
somehow, the “dynamically connected pore size
[319, 328, 4].
The interpretation of
within local porosity
theory is obtained by eliminating
between
the result (5.48) for the conductivity, and
equation (5.95) for the permeability.
This yields in general
![]() |
(5.101) |
where and
are
the local electrical conductivity and the local permeability.
Thus
involves macroscopic geometrical information
through
and
and microscopic dynamical and
geometrical information through the local transport coefficients.
If one assumes the hydraulic radius expressions
and
locally
and the expression
valid for large measurment cells, then it follows that
becomes the
local hydraulic radius [170].
This expression is no longer proportional to the total
internal surface but only to the average local internal
surface, and thus the argument against hydraulic radius
theories no longer apply.