Geoscience Reference
In-Depth Information
For the relative permeability this results again in (8.36) with the value of the exponent
n
=
2
+
2
/
b
(8.50)
The original version of the series-parallel model, both in its finite difference forms and
in the integral forms (8.47) and (8.48), has been tested with experimental data (see Childs
and Collis-George, 1950; Marshall, 1958; Millington and Quirk, 1964; Nielsen
et al
., 1960;
Jackson
et al
., 1965). Although there is a wide variation for different soils, it appears to
produce reasonable results for unstructured soils (without macropores). However, it also
tends to overestimate the relative permeability somewhat under drier conditions.
Several shortcomings of the model are obvious from the assumptions invoked in its
derivation. These are that: (i) the sizes of pores in sequence are independent of each other;
(ii) the flow rate in a pore sequence is controlled by the smallest diameter; and (iii) the pores
in sequence are lined up perfectly, and they are straight without tortuosity. Assumptions
(i) and (ii) will not cause any overestimate in the calculated result. As regards (i), this can
be seen by considering that in the parallel models the sizes of the pores in sequence are
assumed to be totally dependent on each other, since each flow channel is assumed to have a
uniform cross section over its whole length; parallel models, without tortuosity correction,
severely overestimate the permeabilty. Thus the assumption of any partial correlation would
result in a further overestimate. Similarly, as regards (ii), inclusion of the larger pore size
in the sequence into the expression for the fluid velocity would also result in a larger rate
of flow. This means that the overestimate is not the result of assumptions (i) and (ii) but
mainly of assumption (iii).
To compensate for these shortcomings, in several studies use was made of the concept
of tortuosity. Although the concept is intuitively clear, there has been no unanimity in
defining it conceptually or mathematically. Clearly, the drier the soil is, the less perfect the
remaining pores with water line up, and the more tortuous the flow paths become. For this
reason, a common way of implementing the tortuosity effect has consisted of assuming
that it is directly proportional with some power of the size of the largest pores, that contain
water, and thus of the water content of the soil (see Millington and Quirk, 1964; Mualem,
1976), say
S
e
, where
c
is an empirical constant. As noted, this assumption was already
used earlier in several of the parallel models (Burdine, 1953; Brooks and Corey, 1966).
Subsequently, however, it was observed (Brutsaert, 2000) that this assumption is incapable
of producing agreement with experimental data. Rather, it was found necessary to assume
that the tortuosity of a flow path through any given pore depends on the characteristic
spatial scale of that specific pore, and not just on the scale of the largest pores. Actually, this
assumption had already been used by Fatt and Dykstra (1951), in their parallel model, with
the physical justification that liquid flowing through smaller pores travels a more tortuous
path; accordingly, they assumed that the tortuosity is inversely proportional to a power of
the pore size, say
r
c
, in which
c
is another constant to be determined experimentally.
It is straightforward to incorporate this assumption into the series-parallel model pre-
sented above, to adjust it for tortuosity and possibly other factors that may not be fully
taken into account. Thus, in Equation (8.46) the power of
z
and
y
should be taken as (2
+
c
)
instead of 2, so that instead of (8.48) one obtains
S
e
k
=
S
r
)]
2
x
) [
H
(
x
)]
−
2
−
c
dx
(2 Ge)[(2
σ/γ
)
θ
0
(1
−
(
S
e
−
(8.51)
0
