Geoscience Reference
In-Depth Information
j
+
1
j
+
1
j
+
1
j
+
1
∆
∆
t
z
j
TT
z
−
TT
z
−
( )
=
/
1
/
CT T
j
+
1
j
+
1
j
λ
j
+
1
2
i
−
1
i
−
λ
+
j
i
i
+
1
(9.13)
/
/
s
i
i
i
−
1
2
i
+
1
2
∆
∆
i
u
where the superscript
j
denotes the time level, the subscript
i
is the node number,
Δz
u
=
z
i+1
-
z
i
and
Δz
l
=
z
i
-
z
i+1
. As the coeficients
C
s
and
λ
s
are not affected by the
soil temperature itself, Eq. (
9.13
) is a linear equation, which can be solved efi-
ciently.
Both volumetric heat capacity and thermal conductivity depend on the soil com-
ponents quartz, clay mineral, organic matter, water and air. The volumetric soil heat
capacity
C
s
can be calculated as weighted mean of the heat capacities of each com-
ponent:
n
∑
1
C
=
f C
(9.14)
s
i
i
i
=
where
f
is the volume fraction (m
3
m
-3
),
C
is the volumetric heat capacity (J m
-3
K
-1
)
and
n is
the number of soil components.
Table 9.2
gives values of
C
for the different
soil components.
Table 9.2
also lists thermal conductivity values, which are largest for sand and
clay, an order smaller for organic matter and water, and again an order smaller for
dry air. Because the thermal conductivity of air is much smaller than that of water or
solid matter, a high air content (or low water content) corresponds to a low thermal
conductivity.
The components that affect
λ
s
are the same as those affecting
C
s
. However, the
variation in
λ
s
is much larger. In the range of soil wetness normally experienced in
the ield,
C
s
may undergo a threefold or fourfold change, whereas the corresponding
change in
λ
s
may be hundredfold or more. As discussed in
Chapter 2
, thermal conduc-
tivity is sensitive to the sizes, shapes and spatial arrangements of the solid particles.
In the case of dry soil, the addition of a small amount of water increases the con-
tact area between soil particles considerably, and therefore the thermal conductivity
increases rapidly (see
Fig. 2.21
). At larger water contents this increase becomes less
Farouki (
1986
) gives an overview of various methods to calculate the thermal con-
ductivity as function of soil moisture content. SWAP employs the method of De Vries
(
1963
), which compares well to laboratory measurements (Ochsner et al.,
2001
). In
this method the soil is considered a continuous liquid or gaseous phase in which soil
and respectively gas or liquid 'particles' are dispersed. In the case of a 'wet' soil (
θ
>
θ
wet
) liquid water is assumed to be the continuous phase and the thermal conductivity
is given by:
kf
λ
+
kf
+ ++
++++
λ
kf
λ λ
kf
λ
q
qq
cc c
o
oo
w
a
aa
(9.15)
λ
=
s
kf
kf
kf
θ
kf
q
q
c
c
o
o
a
a
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