ing variables are known for a given soil. As a first

This relationship introduces the soil function *k*w,

approximation, the amount of residual unfrozen

the hydraulic conductivity. Each soil will have its

water is assumed to be equal to the lower limit of

unique hydraulic conductivity function. It is not

freezing for the soil. In some situations, this might

a specific property of heaving soils, but is rather a

represent all the required information for the mod-

general material property of all frozen soils, heav-

eler. More often, the heaving pressure, and lens

ing or non-heaving. It must be known in order to

thickness and spacing are also required.

calculate the heaving process.

To date, most efforts to calculate this material

property are by inference. It is inferred from ther-

mal analysis (van Loon et al. 1988), back calcula-

tions (Ratkje et al. 1982 ) and unsubstantiated in-

ference to non-frozen soil (Guymon and Luthin

To calculate the heaving pressure and lens lo-

1974). Black and Miller (1990) were able to directly

cation within the fringe, profiles of temperature

measure the change in hydraulic conductivity in

and pressures must be calculated throughout the

air-free, lens-free and solute-free frozen soil as a

fringe. To perform such calculations, a method of

function of unfrozen water content. Their analy-

executing mass and energy balances within the

sis found that if the measured hydraulic conduc-

fringe must be obtained. This is accomplished with

tivity was expressed a function of the difference

the second physical approximation. It states that

between the ice and water pressures

the instantaneous fluxes of matter and energy at

the beginning and the end of the lensing cycle are

φ = *u*i - *u*w

(6)

equal to the averaged values of the fluxes during

the lensing cycle.

then the analogous expression given by Brooks

Another way of stating the second approxima-

and Corey (1964) for partially saturated and ice-

tion is that any instantaneous fluctuations in mag-

free soil could be transformed to

nitude of the mass and energy fluxes during the

α

φb

lensing cycle are negligible. This means that the

+ *W*d

(7)

magnitude of the penetration rate, heave rate and

φ

temperature gradient within the unfrozen soil are

and

invariant with time and space. In finite difference

β

φb

form, the local mass balance within the fringe is

.

(8)

φ

(qw )n+1 = (qw )n + *v*b

- *Y*(vb + *v*i )[Wn - *W*n+1] ,

The flux of thermal energy through the fringe

(3)

is assumed to follow the Fourier 's law

and thermal balance is

=- h.

(9)

(qh )n+1 = (qh )n + *h *[ (Wn - *W*n+1) vb

This relationship introduces another soil function,

]

+ (qw )n - (qw )n+1 .

(4)

Just as the hydraulic conductivity, this material

property is also a function of the pressure differ-

The remaining information required to com-

ence between the ice and water pressures. One

plete all calculations are statements for water and

standard expression is the geometric mean formu-

ice pressures, temperature and a criterion for lens

lation of Farouki (1981)

initiation.

(kh )I*I*(φ) .

(10)

The flux of water through the fringe is assumed

to obey Darcy's law

The equilibrium condition between the ice and

= *f*w - w .

water pressures and temperature is given by the

(5)

Clapeyron equation

3

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