Darcy's law
Water flow in porous media may be described by Darcy's law, eq 1. Cold temperatures
reduce water flows through porous media. As has been discussed, cold temperatures in-
crease the viscosity of water. Freezing reduces dramatically the permeability of soils and
ground. Geochemical solutions do not freeze uniformly at 0C. The equilibrium freezing
temperature of pore water is a function of pore geometry, freezing-point depressing effects
of solutes, and the charge behavior of the porous matrix (Everett 1961). The quantitative
description of these effects, which is discussed below, is an active research area.
Soil-water retention curve
The permeability of frozen porous media is affected by its liquid water content. Accord-
ingly, a discussion of the physics of unsaturated porous media is appropriate.
Matric potential
Two factors have been identified in determining the relationship between matric poten-
tial and water content in unsaturated soils. The first is the pressure difference between the
soil atmosphere and the liquid water bound to soil particles. The second factor is the work
required to expand the liquid water/air interface as soil-water content decreases. When a
stable interface is established, these two factors are balanced and the water content is sta-
ble. For a given matric potential, any environmental change that changes one of these bal-
anced factors relative to the other can be expected to cause the water content to change.
Effect of temperature
An obvious, and pertinent, example is temperature, which decreases the interfacial tension
of water against air. The effect of temperature on the soil-water retention curve may be esti-
mated via
∂θ
∂θ ∂Ψ
= -
(53)
∂T Ψ
∂Ψ t ∂T θ
where θ is the volumetric water content of the porous medium (dimensionless), and Ψ is
the matric potential of the water in the porous medium (Pa).
For a given pore radius, the relationship between capillary pressure and interfacial ten-
sion is direct:
(2γ wa cos φ)
Ψ=-
(54)
r
where r is pore radius (m) and φ is the contact angle of the air/water interface with the solid
(rad). The partial derivative in the denominator of eq 54 can be calculated with the formula
recommended by eq 9:
u-1
∂γ wa
T
[(uv + v)(T - T1 ) - uT1].
= γ 0 T1-2 1 -
(55)
T1
∂T
Accordingly, one would expect that the following relationship would hold exactly:
∂Ψ(t, θ)
∂T Ψ(t, θ) .
=
(56)
∂γ wa (t) γ wa (t)
∂T
In fact it does not, requiring the definition of an empirical variable [G(θ)] introduced by
Nimmo and Miller (1986) to account for the discrepancy:
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