Heat Flow Analysis for Unsaturated Soils - Unsaturated Soil Mechanics in Engineering Practice - page 492

Table 10.5 Specific Heat and Volumetric Heat
Capacity for Common Materials and Soils

the soils may be frozen for part of the year and then remain
unfrozen for the remainder of the year. The portion of the
soil profile that is frozen for part of the year and unfrozen
for the remainder of the year is referred to as the “active
zone.” The prediction of the depth of the active zone and
the time period over which the soil will remain frozen is
relevant to geotechnical engineering designs.
The thermal energy of a soil-water system becomes dis-
continuous when the temperature passes through the zero
isotherm. The discontinuity is the result of the latent heat of
water, as shown in Fig. 10.4. The thermal energy change is
essentially linear with respect to temperature when the tem-
perature is either above or below the freezing point. Heat is
either adsorbed or given off at the freezing point as a result
of the phase change in the water. The heat quantity associ-
ated with the phase change in water is called the latent heat
of fusion.
The freezing front in a soil is, in reality, more of a freezing
zone, as illustrated in Fig. 10.5. The freezing zone has a
particular thickness and the water within this zone goes from
being ice in the completely frozen portion to being water at
the completely unfrozen portion. Jumikis (1966), Dirksen and
Miller (1966), and Hoekstra (1966) viewed a freezing soil as
having three zones: a frozen zone, a freezing zone, and an
unfrozen soil. The water coefficient of permeability gradually
changes across the freezing zone. The unfrozen water content
varies across the freezing zone and, likewise, the coefficient
of water permeability changes across the freezing zone.
The freezing zone can also be viewed as a zone with a
water coefficient of permeability versus unfrozen water con-
tent function. High soil suction values are also generated in the
freezing zone. The soil suctions are related to the Clapeyron
equation (Newman, 1995). Consequently, the permeability in
the freezing zone can also be viewed as a relationship between
soil suction and water coefficient of permeability. While the
general mechanisms associated with the freezing of soils is
understood, there has not been general agreement on how
to predict the water coefficient of permeability in the freez-
ing zone (Newman, 1995). The water permeability function
forms an important part of mechanistic models that have been

Specific Volumetric

Heat, Heat

Density Capacity, C Capacity, ζ

(k/m 3 ) (J/m 3 / K) Substance (J/kg/K)

Air (20 ◦ C)

1.2 1000 1200

Water (20 ◦ C)

10 6

1000 4200 4 . 2 ×

Ice (0 ◦ C)

10 6

9200 2100 1 . 9 ×

10 6 Quartz 2660 800 2 ×

10 6

×

Mineral clay 2650 800 2

10 6 Soil organic matter 1300 2500 2 . 7 ×

Light soil with roots 1400 1300 500,000

10 6

Wet sand ( θ = 40%) 1600 1700 2 . 7 ×

Source: Data from de Vries (1963) and Rosenberg et al.
(1983).

is the ratio of the thermal conductivity of a material to the
product of specific heat and material density. The thermal
diffusivity term allows the mathematical equations of heat
flow to be written in a more convenient form for obtaining
closed-form solutions. When undertaking numerical model-
ing solutions in geotechnical engineering, it is usually better
to keep the heat flow and heat storage soil properties as
separate entities.
Diffusivity α is defined as the ratio of the thermal con-
ductivity to volumetric heat capacity:

λ
ζ

α = (10.9)

where:

λ = thermal conductivity of the soil, W/m/K , and

volumetric heat capacity, J/m 3 /K.

ζ =

Combining thermal conductivity and heat storage into a sin-
gle parameter is comparable to combining hydraulic conduc-
tivity and the coefficient of volume change into the coefficient
of consolidation when solving consolidation problems. Using
a single constant value for the material properties has distinct
advantages when performing long-hand calculations; how-
ever, when solving numerical models in geotechnical engi-
neering, it is generally advantageous to maintain independence
between the heat transfer and heat storage properties, particu-
larly when dealing with unsaturated soil conditions.

Unfrozen soil

L = latent heat

L

Freezing temperature

Frozen soil

10.3 THEORY OF FREEZING AND
THAWING SOILS

Thermal energy, J

The Arctic and Antarctic regions have soils that remain
frozen for the entire year. Moving away from the poles,

Figure 10.4 Thermal energy of soil-water system illustrating role
of latent heat of water.