Ψ(t2 , θ)
-1
Ψ(t1 , θ)
G(θ) =
(where t2 > t1).
(57)
γ wa (t2 )
-1
γ wa (t1)
Soil freezing characteristic curve
Liquid water (albeit generally very small amounts) exists in frozen soil at temperatures
significantly below 0C. For water-saturated frozen soils, an approximation of the liquid-
water content can be calculated from the soil-water retention curve, if known. The physics
of the two situations are similar: in unsaturated, unfrozen soil, the liquid-water content is
determined by the balancing of the pressure difference across the interface between the soil
water and the soil atmosphere with the work necessary to deform this interface against the
interfacial tension of water against air. The validity of this hypothetical mechanism for con-
trolling the soil-water retention curve is supported by the observation that, for a given
matric potential, the quantity of water held by the soil decreases as the temperature
increases; that is, as interfacial tension decreases with increasing temperatures, less work is
required to deform the air/water interface.
In frozen, water-saturated soils, there is a water/ice interface between the liquid water
surrounding the soil particles and the ice. This situation is qualitatively similar to the water/
air interface that exists in unsaturated soils. A schematic displaying these parallel physical
systems is presented in Figure 15. It is expected then that, for a single soil sample of unal-
tered internal structure, liquid water contents in the frozen, unsaturated state should be a
function of the pressure gradient across the water/ice interface. This function should be
similar to the empirical equation that describes the soil-water retention curve.
Pw
Pa
Pw
Pi
Solid Mineral Particle
Liquid Water
Water Ice
Air
Figure 15. The similarity of capillary water of unsaturated, unfrozen soils and
unfrozen water in saturated, frozen soils.
Generalized Clapeyron equation
Here we follow closely the lucid, complete presentation of the chemicalthermodynam-
ic theory of frozen porous media presented by Brun et al. (1977). Consider three phases:
solid (s), liquid (l), and gas (g), which are in thermal, but not necessarily in hydrostatic,
classic equalities: the GibbsDuhem equations, and the Kelvin equation.
GibbsDuhem equations. For each phase, the GibbsDuhem relation holds:
0 = Sm dT - Vm dpg + ∑ xBd B
g
g
g
g
B
0 = Sm dT - Vm dpl + ∑ xBd B
l
l
l
l
(58)
B
0 = Sm dT - Vm dps + ∑ xBd s
s
s
s
B
B
21
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