Qualitative Characterization of Granular Materials and Moisture Measurements - Broadband Reflectometry for Enhanced Diagnostics and Monitoring Applications - page 88

Topp's equation 3 is sufficiently general (in fact, it appears to be independent of
soil texture and structure, salinity and temperature) [52]. However, although (5.1)
has proved suitable for a wide range of homogeneous soils, it is inappropriate for
clayey soils [19] and for soils with high content of organic materials [56]. In these
latter cases, a third degree polynomial may still be used, but the coefficients should
be extrpolated through a fitting procedure [62].
Malicki et al. suggested that the influence of the soil matrix was remarkable and,
for this reason, they suggested including either dry bulk density or porosity [42].
However, the empirical functions used in current research for TDR data conversion
still remain anything but universal models [14].
A more accurate θ − ε app relationship can be obtained through volumetric mixing

models , which take into account the volume fraction and the dielectric constant of
each single constituent that is present in the MUT. The dielectric mixing models are,
therefore, particularly useful for the characterization of heterogeneous materials,
such as soils with low dry bulk density, a large amount of bound water, or a relatively
large permittivity of the solid phase.
In this approach, soils are usually considered as three-phased systems (made up
of soil particles, water and air) or as four-phased systems (considering that water
close to the soil particles behaves differently) [43].
The typical expression of a three-phased mixing model is as follows

ε m

= θε w − φ ) ε s +( φ − θ ) ε a +(

1 (5.2)

where the subscripts m, s, a, and w stand for measured, dry soil, air, and water,
respectively; is a parameter that depends on the spatial structure

of the mixture and its orientation with the respect to the outer electric field [49].
The four-phased systems take into account that the water that is close to the
soil particles has lower mobility than free water; hence, the thin layer of water that
surrounds soil particles has a lower dielectric permittivity than free water [20]. The
expression of a typical four-phase mixing model is the following [68]:

φ is the porosity; β

ε m =( − φ ) ε s +( φ − θ ) ε a +( θ − θ bw ) ε fw + θ bw ε bw 1 (5.3)

where the subscripts m, s, a, fw, and bw stand for measured, dry soil, air, free water
and bound water, respectively; is a parameter that takes into account the geometry

of the media particles with respect to the applied electromagnetic signal; and

α

θ bw is

the fraction of bound water, which can be obtained from

θ bw = n m δρ b S e (5.4)

3

Practically, for estimating water content from TDR measurements of the apparent dielec-
tric constant, the reciprocal of (5.1) is used:

θ = 4 . 3 × 10 − 6

app − 5 . 5 × 10 − 4 app + 2 . 92 × 10 − 2

ε app − 5 . 3 × 10 − 2

ε ε .