modeled. The specimen was initially at 1C, then the

by the surface effects of the solid:

top temperature was gradually ramped down to 0.5C.

L = LB LΨ

(74)

puted as a geometric mean of the thermal conductivi-

where L is the chemical potential of the liquid in the

ties of soil solids, ice, and water during freezing. Fig-

film, LB is the chemical potential of the bulk liquid,

ures 9a and b show typical results. The model predicts

and LΨ is the depression in the chemical potential from

behavior of the soil column during freezing and the

values are reasonable.

bulk water caused by the presence of the solid surface.

Black and Miller (1985) applied a simplified ver-

Gilpin assumed that it had the form of a power law

sion of the rigid ice model to laboratory test results on

relationship:

a silt. They assumed that the liquid water content and

LΨ= *ay*α

(75)

the unsaturated hydraulic conductivities were simple

Brooks-and-Corey-type exponential functions of φ (e.g.,

where *y *is the distance from the soil particle surface

and *a *and α could be adjusted as needed.

Brooks and Corey 1964). Heave and frost penetration

rates and temperature gradients in the unfrozen soil were

The chemical potential of the bulk water is given

input variables, while temperature gradients in the fro-

by (an integrated form of eq 19)

zen soil and heaving pressure were outputs. The model

LB = Lo + *V*L (PL - *P*o ) - *S*L (TL - *T*o )

accurately predicted temperature gradients, but predict-

(76)

ed heaving pressures to be about half of those mea-

where Lo is the chemical potential at a reference con-

sured, indicating problems with the experimental pro-

cedure that were later found to exist.* The rigid ice

dition, (*P*o, *T*o). Substituting eq 75 into eq 74 and 74

model is now available in the form of a MathCad 5.0+

computer program (Black 1995).

that the temperatures of the bulk and film water are

The rigid ice model has now been developed into an

equal leads to the following expression for the varia-

engineering tool for prediction of heaving due to one-

tion of pressure in film water, with a distance from the

dimensional heat loss. Using the rigid ice model as a

solid surface (Fig. 10):

basis, Sheng (1994) developed a numerical model of

*a*

frost heave. This model, called PC-Heave, predicts

(*P*L - *P*o ) = *y * -α .

(77)

heave for stratified soils and unsaturated layers. Input

*V*L

variables for PC-Heave are the number of soil layers

and their thicknesses, the dry densities, water contents,

P

thermal and hydraulic conductivities and percentage

saturation of the soil layers, the boundary temperatures

at the top and the bottom, the depth of the groundwater

Bulk Water

table, and one unfrozen water content at a subfreezing

PL

temperature per FSL (a calibration factor). The model

predicts heave, location of ice lenses, frost penetration,

segregation temperature, and suction in the pore water

of the frozen fringe with time.

g

The modeling equations are the mass and heat bal-

y

ances at the base of the warmest ice lens and for the

frozen fringe, Darcy's Law, and the expression for the

Surface

pore water pressure in the frozen fringe that incorpo-

rates the generalized Clapeyron equation (Sheng 1994).

The model has been verified using both field and labo-

L

ratory soil freezing information.

Gilpin (1979, 1980) developed a model very similar

L

to the rigid ice model. He assumed that the chemical

potential of the water in the adsorbed film is lowered

* Personal communication, Dr. Patrick Black, CRREL, Hanover, New

Hampshire, 1999.