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Chapter 6 Modeling of Coupled Water and Heat Transfer in Freezing and Thawing Soil
described using the modified Richards equation (e.g., Hansson et al., 2004):
∂h
∂T
∂h
∂T
[ ( h)
+ K ( h)
+ K ( h)
+ K ( θ ) + K ( θ ) ] S
∂θ
( h) pi
∂θi
( T ) ∂
∂t
+ p t = z K
w ∂ ∂ Lh ∂z
Lh
LT ∂z
vh ∂z
vT ∂z
u −
where θu [L 3 L -3 ] is the volumetric unfrozen water content (=θ+θv), θ is the
volumetric liquid water content, θv is the volumetric vapor content, θi is the
volumetric ice content, t is time, z is soil depth, pw and pi is the density of liquid
and ice water, respectively, h is the pressure head, T is the temperature, and S is
a sink/source term usually accounting for root water uptake.
In Eq. 1, the first five terms on the right-hand side represent liquid flows due
to gradient in pressure head (KLh, [L T -1 ]), gravity, and temperature (KLT, [L 2 T -1
K -1 ]), and vapor flows due to gradient in pressure head (Kvh) and temperature
gradients (KvT), respectively. Eq. 1 is nonlinear, namely dependencies of the
water content and the hydraulic conductivity on the gradient of pressure head
and temperature, i.e. θ(h), θi(T), KLh(h), and KLT(h). The KLh is described by the
following expressions (van Genuchten, 1980):
θ ( h)
−θ
r 1
Se
( h)
= =
n
θ −θ
[ 1+
αh
]
s
K ( h)
−
Lh
r
m
l
1/
m m 2
= Ks
× Se
( 1−
( 1 Se
) )
[3]
where Se is the effective saturation, θs and θr are the saturated and residual
water contents, respectively; the symbols α [L -1 ], n, and m=1-1/n are empirical
shape parameters, and the inverse of α is often referred to as the air entry value
or bubbling pressure; Ks is the saturated hydraulic conductivity [L T -1 ], and L is a
pore connectivity parameter, which normally a value of 0.5 is given.
The hydraulic conductivity of frozen soil is significantly reduced by ice
lenses, blocking parts of the pore space. To account for this blocking effect, the
hydraulic conductivity in our study is reduced by an impedance factor, Ω (Lundin,
1990), which is multiplied by Q. Namely, the Ω reduces the hydraulic conductivity
of the partially frozen soil, KfLh, as follows:
KfLh =10 -ΩQ KLh [4]
the parameter Q is the ratio of the ice content to the total water content,
which accounts for the more significant blocking effects with the increase in ice
[1]
[2]
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