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The purpose of the above discussion was the calculation of the temperature
dependent volume fraction of bound water, θ bw (T). Wang and Schmugge [99]
show that the quantity of bound water in soil increases with the increase of clay
fraction in it. This is due to large specific area of clay surface compared to other
soil fractions. The bound water volume fraction of the soil is the product of solid
phase surface, S, temperature dependent thickness of bound water layer, x(T ), and
bulk density, ρ, as follows:
( T ) x( T ) S ρ
θbw = (55)
Having determined the dependence of bound water volume fraction of soil,
θ bw (T ), on temperature using equation (55), it is now possible to find the overall
temperature dependence of bulk soil dielectric permittivity, ε b , by application of
dielectric mixing models. Among many dielectric mixing models describing soil
as the mixture of solids, liquid, gas and also bound water phase there are two the
most used: model α (Birtchak et al. [9], Dirksen and Dasberg [27], Roth et al.
[77]) and model de Loor (de Loor [25], de Loor [26], Dirksen and Dasberg [27],
Dobson and Ulaby [28]).
Model α treats the soil as the mixture of volume fractions of four phases
described in the mathematical form:
α
α
s
α
a
α α
( θ −θ
) ε θ ε
ε = ( 1−φ)
ε + ( φ −θ
) ε +
+
(56)
b
where: ε b , ε s , ε a , ε fw , ε bw are respective values of bulk soil, solid, gas, liquid and
bound water phases dielectric permittivity, and θ, φ and θ bw are volume fractions
of liquid, pore and bound water respectively in the soil, α is a curve-fitting
parameter that can be interpreted as a measure of geometry of the medium with
relation to the applied electric field (Roth et al. [77]). Roth et al. [77] found that
an average value of α = 0.5 for three phase model while for four phase model
α = 0.65.
Four-phase soil model of de Loor is based on Maxwell equations and it
describes the homogenous mixture of substances with different dielectric
permittivity with the host solid phase medium. It can be described in following
form:
bw
( ε − ε ) + 2θ
( ε − ε ) + 2θ
( ε − ε )
3ε
s + 2θ
fw fw s bw bw s a a s
ε b =
(57)
⎛ ⎞ ⎛ ⎞ ⎛ ⎞
⎜
ε s
+ − ⎟
ε s ε s
3 θ fw 1 + θ
⎜ −1
⎟ +
⎜ −1
⎟
bw θ a
⎝ ε fw ⎠ ⎝ ε bw ⎠ ⎝ ε a ⎠
fw
bw
bw
65