pressures than the lower, warmer pores (Fig. 7). That
Figure 7. Vertical gradient
is, the pore ice pressure increases upwards from the
of pore contents in a ver-
bottom of the frozen fringe, which causes effective
tical column of freezing
stress to decrease. Figure 8 depicts profiles of soil water
soil. (After Miller 1978.)
pressure, effective stress, and pore pressure in the frozen
Note that the top portion of
the soil particle has more
fringe just prior to ice lens initiation (after Miller 1978).
film water in contact with
Ice lens initiation occurs when the effective stress
R
ice than the bottom portion
reaches 0. At this condition, soil particles become incor-
and that the curvature of
porated into upward moving ice. Thus, the ice lens
the ice/water interface is
initiation condition is similar to that of Everett (1961).
greater at the top interface
than at the bottom inter-
However, the pore pressure that can be generated is
face.
greater than that proposed by Everett (1961) due to sur-
face effects of soil particles as well as temperature gra-
dients in the frozen fringe.
K(φ) = K(φa)
(68)
Ice movement within a soil is called regelation
(refreezing), and it involves the melting, transport
around soil grains in adsorbed films, and refreezing of
and
water. The heat released during the change of phase
from water to ice in freezing soil is far more significant
χ(φ) = χ(φa).
(69)
than the sensible heat transfer. Therefore, Miller ignored
sensible heat transfer during freezing, and accounted
Using laboratory data for unfrozen water content in a
only for heat transfer due to ice formation:
frozen soil, ϑw, as a function of temperature, along with
all of the relations defined by eq 62 through 69, and
q = -λ(φ) - ρi Lf vi (φ)
T
(71)
z
Terzaghi's equation for effective stress,
where q is the rate of heat flow in the soils, λ is the
σT = σ′ + u.
(70)
Equilibrium values of χ, φ, σ′, u, and K(φ) can be pre-
ric ice flux:
dicted as a function of temperature (or depth) in a freez-
vi(φ) = ϑi(φ)vI
(72)
ing soil.
Miller (1978) identified a downward force acting
where vI is the rate of frost heave. In the unfrozen zone,
on the granular skeleton in the freezing soil due to a
only heat conduction is considered:
vertical pressure gradient in the adsorbed film. This
T
q = -λ u .
pressure gradient results from the fact that the top, colder
(73)
z
pores have thinner films and therefore greater curva-
ture and greater differences between the ice and water
Equation 71 contains both heat and mass transfer
0
10
0
10
3
0.2
σ'
u
Level at which pore
uw
ui
ice first appears.
2
0.1
Z
(cm)
1
0
cm3/cm3
kPa
kPa
cm /s
0
0.2
0
200
0
0.2
0
200
8
6
10
10
φ
χ
K
Figure 8. Profiles of soil water pressure, effective stress, and pore pressure in the frozen fringe just prior to
ice lens initiation. (After Miller 1978.)
13
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