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

(A1)

where *k*a, *k*g, *k*i and *k*w are the thermal conductivities of air, soil solids, ice and water; *x*a, *x*g, *x*w and *x*w 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.

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

(A2)

(1- ε)

(A3)

(kw / *k*i )ε γ (1- ε)

=

(A4)

where γ = 0.9825 is a temperature correction for *k*g (Lachenbruch et al. 1982) and *k*w/*k*i = 1.34/5.45 =

0.2459.

The volumetric specific heat for the system may be expressed as follows, for the thawed and frozen

states

(A5)

(A6)

where *C*su and *C*sf are the volumetric specific heats of unfrozen and frozen solids, and *C*w and *C*i 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

(A7)

0.4296 + 0.5708ε

= *C*21 =

.

(A8)

0.4202 + 0.0388ε

The density ratio is

ρu

1 - 0.6154ε

= ρ21 =

.

(A9)

ρf

1 - 0.650ε

23

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