The Rigidice model of frost heave is a set of

1

ΓIW ≡

γ IW

γ IA

differential equations expressing mass, energy and

force balances in the frozen fringe for air-free, col-

and

(B3)

loid-free and solute-free soil (Miller 1978). A set

1

ΓAW ≡

γ AW .

of reduced or scaled variables for this model were

γ IA

later presented by Miller (1990). To avoid redun-

dancy, this later work did not contain derivations

To simplify, the remainder of this appendix will

but referred to the earlier classic works for ice-

address icewater systems. A similar set of equa-

free unsaturated soil (Miller and Miller 1955, Miller

tions for waterair and iceair systems are ob-

tained by direct analogy to the icewater equa-

pendix presents a brief derivation of the reduced

tion and are listed by Miller (1990).

variables presented by Miller for the liquid wa-

The distinguishing trait exhibited by water and

ter and solid ice phases. It also serves as an exam-

ice in a porous medium that does not exist in un-

ple of how the seemingly subjective process of

restricted bulk is the existence of curved inter-

scaling equations is performed. All scaled vari-

faces. The force balance across a curved interface

ables will be given upper-case letters to distin-

separating two phases in equilibrium is given by

guish them from the nonscaled dimensional vari-

the Laplace surface tension equation, which for

ables that are in lower case.

an icewater interface is

γ

(B4)

Air

where the absolute pressure is *p *(N/m2) and the

Water

mean radius of curvature is *r*. In the case of fro-

Ice

zen soil, this equation expresses the behavior of

water and ice contained within the pores. This

suggests a length scale on the order of the pore

γWA

size, so we introduce a microscopic length λ. Equa-

tion B4 contains three variables, *p*I, *p*W and *r, *and

one constant, γIW. Equation B3 shows how to re-

α

γIA

γIW

duce the surface tension. At this point we need

to decide how to reduce the remaining variables.

The simplest approach is to acknowledge that the

radius of curvature is a pore length dimension,

We start with Young's equation relating the

so it is reduced by dividing by λ. The reduced

contact angle α and the surface tensions for all

curvature is therefore

three phases of water:

1

γ IA = γ IW + γ WA cos α

(B1)

(B5)

λ IW

in which γ is the surface tension and the subscripts

Substituting B3 and B5 into the right hand side

of B4 determines the reduced pressure by dimen-

Figure B1 is a schematic of the equilibrium con-

sional equivalence:

figuration and force diagram about the point of

λ

mutual contact for all three interfaces. Since cos

(B6)

γ IA

α is already dimensionless, eq B1 can only be made

dimensionless by dividing through by one of the

giving the reduced form of the Laplace surface

surface tensions. A reasonable choice is γIA with

tension equation:

the largest magnitude, which results in the re-

ΓIW

duced Young's equation

.

(B7)

1 = ΓIW + ΓWA cos α

(B2)

At thermodynamic equilibrium the chemical

where the reduced surface tensions are

potentials of ice and water are equal and are ex-

14