APPENDIX A: SOIL PROPERTIES AND RATIOS
The thermal conductivity of a mixture such as a soil can be estimated by using the weighted geometric
mean (Lachenbruch et al. 1982, Gold and Lachenbruch 1973, Lunardini 1981a). This can be written for a
general soil as
()
(kw )xw (ka )xa (ki )xi
xg
k = kg
(A1)
where ka, kg, ki and kw are the thermal conductivities of air, soil solids, ice and water; xa, xg, xw and xw are
the volumetric fractions of air, soil solids, ice and water. The geometric mean is usually better than the as-
sumption of parallel geometry (weighted arithmetic mean), which is often used for simplicity.
Saturated soil
Many assumptions can be made concerning the soil saturation and porosity but simple approximations
will be used here. If the soil is always saturated, has a constant void ratio ε, and all of the water freezes,
( )( )
1- ε
ε
ku = kw γkg
(A2)
(1- ε)
kf = kiε (kg )
(A3)
(kw / ki )ε γ (1- ε)
ku / kf = k21
=
(A4)
where γ = 0.9825 is a temperature correction for kg (Lachenbruch et al. 1982) and kw/ki = 1.34/5.45 =
0.2459.
The volumetric specific heat for the system may be expressed as follows, for the thawed and frozen
states
Cu = Csu (1 xw) + Cw xw
(A5)
Cf = Csf (1 xi) + Ci xi
(A6)
where Csu and Csf are the volumetric specific heats of unfrozen and frozen solids, and Cw and Ci are the
volumetric specific heats of water and ice.
It is fortunate that the volumetric specific heats of soil solids and ice are all about the same. For exam-
ple, the specific heat of organic solids is 0.461 cal/cm3 C, for mineral solids it is 0.420, and for ice it is
0.459 (Lunardini 1981a). If one assumes that the values for the solids, except for ice, change little through
the phase change then
Cf = 0.4202 + 0.0388ε
(A7)
0.4296 + 0.5708ε
Cu
= C21 =
.
(A8)
0.4202 + 0.0388ε
Cf
The density ratio is
ρu
1 - 0.6154ε
= ρ21 =
.
(A9)
ρf
1 - 0.650ε
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