CRREL REPORT 95-6

Applications of the

Clapeyron Equation to

Water and Ice in Porous Media

Patrick B. Black March 1995

WATER

ICE
Cover: Humpty Dympty falling off of the very high wall.

For conversion of SI units to non-SI units of measurement consult ASTM Standard

E380-93a, Standard Practice for Use of the International System of Units,

published by the American Society for Testing and Materials, 1916 Race St.,

Philadelphia, Pa. 19103.

Cover: Representation of the five solutions of the

Clapeyron equation.

Abstract

The equilibrium condition for water and ice in an air-free porous medium is

presented. The equation of state for this system is the Clapeyron equation. This

equation is presented in a general form that explicitly shows the pressuredifference dependence with temperature for water and ice. Five solution

scenarios are then discussed in terms of applicability to porous media.
Prepared for

OFFICE OF THE CHIEF OF ENGINEERS

Approved for public release; distribution is unlimited.

Patrick B. Black March 1995

Applications of the

Clapeyron Equation to

Water and Ice in Porous Media

CRREL Report 95-6

U.S. Army Corps

of Engineers

Cold Regions Research &

Engineering Laboratory
ii

PREFACE

This report was prepared by Patrick B. Black, Soil Physicist, Applied Research

Division, Research and Engineering Directorate, U.S. Army Cold Regions Research

and Engineering Laboratory. Funding was provided by the Office of the Chief of

Engineers under DA Project 4A762784AT42, Design, Construction and Operations

Technology for Cold Regions, Task CP, Work Unit C01, Freezing and Thawing of Soil

Water Systems. Samuel Colbeck, Edward Arons and David Cate provided technical

reviews of this report.

The contents of this report are not to be used for advertising or promotional

purposes. Citation of brand names does not constitute an official endorsement or

approval of the use of such commercial products.
Applications of the Clapeyron Equation to Water and

Ice in Porous Media

PATRICK B. BLACK

INTRODUCTION

Quantifying the relationship between water and

ice in a porous medium has been a subject of

interest for many years. Hudson (1906) first discussed possible applications of the Clapeyron equation to water and ice subjected to different pressure conditions for a given temperature change.

Much later, Edlefsen and Anderson (1943) extended Hudson’s analysis to the phase equilibrium condition of water and ice in soil. In both

cases, they approached the problem from an equilibrium thermodynamic assumption that included

a steady-state framework that did not include the

flow of water.

This report extends the traditional application

of the Clapeyron equation to include mass flow in

ice-containing porous media. It is not intended to

serve as an introduction to equilibrium thermodynamics for porous media. There are many other

references that discuss that topic, and some are

cited below. Instead, it is intended to explore the

use of phase equilibrium concepts for modeling

heat and mass transport in air-free, ice-containing

porous media.

CONDITIONS AT EQUILIBRIUM

The system of interest in the study of freezing in

a porous material is shown in Figure 1. It consists

of a porous material with water in two phases: ice

and liquid. Furthermore, it is assumed that any

liquid water in the system will always be located

between the surface of the porous matrix and the

ice. There is ample justification for the existence

and this configuration of the liquid film. One

group of theories attributes the film to an adsorption force emanating from the surface of the porous medium (Miller 1980), while another group

attributes the origin of the film to a liquid transition layer on the ice (Cahn et al. 1992). The result of

this behavior is that we can treat the porous medium as “inert” and need not include it as a separate chemical component.

The starting point for phase equilibrium thermodynamics is the Gibbs equation. This equation

expresses the chemical potential μ for a phase. This

results in two equations for a two-phase system

consisting of liquid water w and ice i:

 

μ

μ

w ww w

i ii i

d d

d d

=

=

Vp ST

Vp S

–

– (1)

in which V and S are molar volume and entropy,

and p and T are absolute pressure and temperature. The equilibrium condition (dμ = 0) for liquid

water and ice is obtained when the chemical potentials are equal (μw = μi). This condition results

in the Clapeyron equation:

 

V p Vp S S T H

T ww ii w i T wi

0

dd d d – – = ( ) = ∆ (2)

in which ∆H is the molar heat of fusion at T0 (273.15

K), the standard melting temperature of a planar

surface of ice.

  













































































Figure 1. Ice–water configuration surrounding a porous medium.
The temperature and pressure units are usually

changed to make the Clapeyron equation more

functional in experimental applications. First, temperature is expressed in terms of the Celsius scale:

 

θ = T T– 0 . (3)

Next, the pressures are expressed as gauge pressure by referencing to atmospheric pressure p0:

 

upp = – 0 . (4)

Finally, the specific gravity of ice γi (0.917) and the

volumetric latent heat of fusion h (Mg/m3) are

defined as

 

γ i

i

w

= V

V (5)

and

wi

w

h = H

V

∆ .

These substitutions result in the differential form

of the so-called generalized Clapeyron equation:

 

d d w d i

i

u

u h

T – γ = θ

0

. (6)

Relationships similar to the Clapeyron equation have a history of use in ice-bearing porous

media. In the second decade of this century,

Bouyoucos and McCool (1916) measured the freezing point depression of soil–water systems at various degrees of saturation as a method of inferring

the pressure of bound water. In the 1930s Schofield

(1935) also used the freezing point depression

approach to quantify the moisture tension of lowwater-content soils. This type of work was of questionable use since the results appeared to work

sometimes but fail many times. Edlefsen and

Anderson attempted to develop a comprehensive

theory of soil moisture in their classic monograph

in 1943 in which they developed and then discussed the Clapeyron equation for soil–water systems. This report will closely follow their discussion and add some interpretation. It is interesting

to note that they referred to eq 2 and 6 as the

Clausius–Clapeyron equation. This is not correct;

the Clausius–Clapeyron equation applies to liquid– vapor equilibrium, whereas the Clapeyron

equation refers to any equilibrium phase composition.

It should be noted that uw is the hydrostatic

pressure of pure liquid water. If there are any

impurities in solution, the water pressure must be

adjusted. This is conveniently performed by subtracting an osmotic pressure that is calculated for

a dilute solution by the van’t Hoff’s equation

(Castellan 1971). For the purpose of this report, it

is assumed that there are no impurities, so that an

osmotic pressure term is not required in eq 6.

Similarly the ice phase might also contain trapped

impurities that must be included in the expression

for the ice pressure. This case will also be excluded

by assuming the ice to be pure.

Inspection of eq 6 reveals that there are three

unknowns (uw, ui and θ) and one equation. Equilibrium is therefore described when eq 6 is integrated and two of the variables are given. Another

method to determine the number of intensive variables F required to describe the state of a system in

equilibrium is Gibb’s phase rule for a flat interface,

which states that (Castellan 1971)

 FCP = + – 2 (7)

in which C is the number of components and

P is the number of phases. There are no chemical

reactions in this example. This predicts that only

one intensive variable is required to fix the

equilibrium condition for the case of water and

two phases. But the system depicted in Figure 1

has an additional degree of freedom arising from

the curvature, which results in a pressure difference between the liquid water and the ice.

Thus, two intensive variables are required to describe a single-component, two-phase system

with curvature.

Five physically significant scenarios are applicable to eq 6, and each will give a different quantitative prediction of the pressure–temperature

relationship. Hudson (1906) and Edlefsen and

Anderson (1943) obtained four solution classes by

constraining one or both of the pressures or the

temperature. This approach reduces the number

of intensive variables required to describe the

equilibrium condition. The final case treats the

system in terms of the pressure difference between

the two phases. The particular choice will depend

on the prevailing environmental constraints on

the system under study.

APPLICATIONS OF THE CLAPEYRON

EQUATION

Figure 2 depicts the experimental setup of Miller

et al. (1960). In this experiment they were able to

2




 





  











uw

ui

Thermometer

Porous Plate

Rigid Piston

Ice

Frozen Soil

Water

subject the isothermal system to incremental

changes in ice or water pressures and monitor the

resulting change in temperature in the soil. In the

apparatus the liquid water in the soil is in hydraulic contact with the water in the lower horizontal

tube. The water pressure in that tube is therefore

the pressure of the water in the soil. Similarly the

pressure of the ice above the soil is the same as the

pressure of the ice in the soil, since it too is mutually connected.

Case 1: dθ = f(dui,duw); dui = duw

This is the standard textbook example for bulk

solutions. It was first developed in the 1850s by

Thompson for the freezing point depression of

bulk solutions subjected to equal pressure changes

on the ice and water phases. In this case, ui = uw in

Figure 2. The pressure–temperature behavior is

 

d fd d d θ= = = ( ) uu U i w (8)

 

d

d

i

i

U h

θ T

γ

γ =

0

1

– 1

 

= °

= °

–. /

–. /

13 49

0 074

MPa C

C MPa.

 or

This equation predicts a 0.074°C lowering of the

melting temperature for a 1-MPa increase in confining pressure on both the ice and the liquid water

phases. While this seems to be a very small change

in temperature, the inverse view is that there is a

very large change in pressure for a small change in

temperature.

This situation might apply to porous media that

are purely colloidal, supersaturated with water, or

consisting of large grains with a low unfrozen

water content. When the system is supersaturated

with water or is essentially a colloidal suspension

of water, surface tension and curvature effects are

not important. The pressures of the liquid water

and ice are therefore essentially equal. This can

happen to the water in the surface soil that undergoes freezing to shallow depths so that there is no

confining pressure on the ice. This is also possible

in pure clay soils. When the porous medium is

composed of large grains and the unfrozen film is

thin, then the radius of curvature at the ice–water

interface is so small compared to the film thickness

that it is closely approximated as a flat surface with

equal liquid and ice pressures.

Case 2: dθ = 0 = f(dui,duw)

 

d

d

w

i i

u

u = 1γ

(9)

 

= 1 09 . .

This example happens under the unique condition that the curvature between the two phases

remains constant for a given temperature. While

this condition is of academic interest, it is of little

significance in environmentally important porous

media like soil–water systems. It just states that the

melting temperature remains constant if the change

in confining pressure on the water is 1.09 times

greater than the change in the ice pressure.

Once some ice is formed and then the volume of

liquid water and ice are fixed in Figure 2, the

Figure 2. Schematic representation of the Miller et al. (1960) experiment.

3
measured pressure changes with temperature

would follow eq 9. This might be approximated in

a soil system in which there is a volume of frozen

soil surrounded by rigid and impermeable walls.

Case 3: dθ = f(duw,dui); uw = constant

 

d – u h

T

i i

dθ

γ =

0

(10)

 

= °

= °

– .

– .

1 12

0 893

MPa/ C

C/MPa.

This expression also predicts a lowering of the

melting temperature of ice, but the change is greater

by an order of magnitude than predicted by eq 8.

On the other hand, there is an order-of-magnitude

smaller increase in pressure for an equal lowering

of temperature when compared to eq 6.

In this case the ice pressure is free to vary with

temperature, while the liquid water pressure in

Figure 2 remains constant. This condition was

used by Koopmans and Miller (1966) to infer the

ice pressure change with temperature in their tests,

which measured changes in unfrozen water content with changes in temperature. By applying eq

10 to their data, along with the assumption that the

geometry of the ice–water interface is the same as

an air–water interface at the same water content,

they obtained agreement with corresponding icefree data for unsaturated soil by suitable adjustment for ice–water and air–water surface tensions

(see eq 12).

Case 4: dθ = f(duw,dui); ui = constant

 

d

d

uw h

θ T =

0

(11)

 

=+ °

=+ °

1 22

0 819

.

.

MPa/ C

C/MPa.

The pressure–temperature behavior when the

ice pressure remains constant is of the same order

of change as when the water pressure remained

constant, but of opposite sign. This behavior was

recognized by Hudson (1906) and later used by

Schofield (1935) in his freezing point depression

measurements. The positive sign now means that

a decrease in liquid water pressure will result in a

lowering of the melting temperature.

This is a common form used by many modelers.

Unfortunately it is not as applicable as they infer.

Again, soil near the surface that lacks any overburden but has a water table beneath the surface is

described by eq 11. This indicates that the lower

the temperature, the greater the tension on the

liquid water.

Case 5: dθ ≠ 0 ≠ f(dui,duw)

Colbeck (1985) made direct observations of the

curved interface between the ice and the liquid

water in partially frozen porous media, just as

depicted in Figures 1 and 2. These observations,

along with his earlier images for partially frozen

unsaturated porous media (1982), also showed the

curvature to decrease with decreasing temperature. In terms of the Laplace surface tension equation, a decrease in the mean radius of curvature r

indicates an increase in the pressure difference

between the two phases:

 

u u

r

i w

iw – = 2σ (12)

in which σiw is the ice–water surface tension. The

earlier experiments also revealed that the curvature of the water–air interface indicated that the

liquid water pressure was always less than atmospheric, while the observed ice–air curvature indicated that the ice pressure was always greater than

atmospheric. It is difficult to determine which

constraint is required (uw = constant or ui = constant) to use the Clapeyron equation to describe

the behavior in these experiments. What is clear is

that the curvature, and therefore the pressure difference between the two phases, is a function of

temperature. To emphasize that this is a relationship for pressure difference between liquid water

and ice, it is convenient to introduce a variable that

is the difference:

 

φ = u u i w – . (13)

Substituting this expression into eq 6 results in

two possible expressions, one in which the water

pressure is an explicit variable and the other in

which the ice pressure is the explicit variable:

 

d dd i w

i φ γ γ = ( ) – – 1 θ

0

u

h

T (14)

 

d dd

i

φ i

γ = θ 



 



 1 1

0

– – u

h

T .

In the former case, if the water pressure is assumed

4
to be constant, eq 10 is obtained. Similarly, if ice

pressure is constant in the latter case, then eq 11 is

obtained.

There are at least three advantages to using eq

14 that are not realized by eq 10 and 11. First, eq 10

and 11 require that the pressure of one of the

phases must remain constant with change in temperature. The observations of Colbeck do not show

this to happen. Second, the pressure difference

form of eq 14 is a logical analog to the pressure

difference form used to model the water content of

ice-free unsaturated soil (Koopmans and Miller

1966). Third, as will be discussed, eq 14 gives a

simple explanation for the observed behavior of

liquid water flow into and out of the frozen fringe

during the formation of ice lenses.

DISCUSSION

The pressure–temperature behavior of liquid

water and ice is used to model material properties

of, and heat and mass transport through, porous

media. Traditional material properties such as

unfrozen water content and hydraulic conductivity are usually expressed as functions of temperature, while thermal conductivity and stress behavior are presented as functions of unfrozen water

content. These properties are then used in models

for mass and energy transport. It was shown above

that temperature alone is not always sufficient to

completely specify the conditions at equilibrium.

Instead of using temperature, it is more reasonable

to express the material properties as functions of φ.

Unfrozen water content can also be expressed as a

function of φ (Black and Tice 1989). Thus, all material properties for frozen soil are functions of the

pressure difference between the ice and water

phases φ.

For example, unfrozen water content data that

are collected as function of temperature are transformed into functions of φ by using eq 14, which

has two forms. The correct form depends on the

experimental method. In a constant-stress device

that has direct hydraulic connection to the atmosphere, the first form is used, with uw set to its

value of zero (Black and Miller 1990). When the

data are collected in an unconstrained manner,

then there is uncertainty as to the magnitude of

either phase pressure. In this case, a band of uncertainty is obtained by using both forms of eq 14,

with the explicit pressure set to zero (Black and

Tice 1989, Black 1991). Finally, modeling ground

freezing then proceeds in terms of φ, and the form

of eq 14 employed depends on the environmental

conditions.

An example is the modeling of mass transport

through a frozen fringe, shown in Figure 3. This is

the zone of partially frozen soil bounded by an ice

lens on the cold side and unfrozen soil on the

warm side. Black and Miller (1990) assumed that

uw at the base of the fringe was known (perhaps it

is zero), so that the former form of eq 14 was used

(Black and Miller 1990). They then calculated the

required liquid water pressure gradient at the base

of the fringe for given boundary conditions and

were able to start an iterative process to calculate

pressures of liquid water and ice throughout the

fringe.

Additional insight into the observed behavior

of ice and water flux in heaving soil is obtained

from eq 14. Forland and Ratkje (1980) contended

that the Clapeyron equation does not apply to

transport through the frozen fringe. They believed

that under the condition of restricted growth of an

ice lens (Radd and Oertle 1973), the Clapeyron

equation would predict a pressure gradient of

liquid water in reaction to a temperature gradient

across the fringe. This, they believed, was not

physically possible because it would induce a flux

of water. With no growth of a lens or penetration

of the fringe, there was no sink for a flux of water.

As a result they abandoned equilibrium thermodynamics and chose the route of irreversible thermodynamics.

It is not that the Clapeyron equation does not

   













































































Figure 3. Representation of the frozen fringe.

5
apply to the frozen fringe, it is that their interpretation does not apply. They implicitly assumed

that the ice pressure was constant, as in eq 11.

Therefore, a temperature gradient must be balanced by a pressure gradient in the liquid water.

The correct interpretation is given by eq 14. In this

case, there is balance between the liquid water and

ice pressures. A fixed temperature gradient results

in a fixed gradient of φ. Now, as a confining load on

top of the ice lens increases and its growth decreases, there is a corresponding decrease in the

flux of water to the ice lens. This results in a

decreased pressure gradient in the liquid water,

which is made up by an increased ice pressure

gradient to keep a constant gradient of φ for a

constant temperature gradient. In the extreme case

when there is no ice lens growth, the liquid water

pressure gradient is zero because the ice phase is

maintaining the necessary pressure gradient for

equilibrium.

Another observation from frost heave is the

discharge of liquid water out of the frozen fringe

into the unfrozen soil as both penetration of the

fringe and growth of the ice lens occur. This is

readily explained by the Clapeyron equation. These

conditions generally occur at the onset of heaving,

when the gradients are severe. The rapid ice penetration and slight growth of the ice lens results in

an overabundance of liquid water in the fringe due

to the density change from liquid to ice. The excess

liquid not required to feed ice lens growth results

in a liquid water pressure gradient directed towards the unfrozen soil and in the direction of

increasing temperature. Equilibrium is maintained

by a corresponding increase in the ice pressure

gradient. The resulting large ice pressure gradients also cause large gradients in neutral stress,

which result in rapid changes in the location for

optimum ice lens growth. This is shown by the

close spacing and small sizes of ice lenses under

these conditions (O’Neill and Miller 1985, Black

and Miller 1990).

CONCLUSIONS

The water and ice contained within a porous

medium do not necessarily exhibit the same behavior as the same water and ice in bulk. The most

important difference is that the geometric restrictions imposed by porous media lead to curved

interfaces separating the ice and water phases. The

presence of a curved interface indicates that the

pressures of the two phases are different. How

they differ depends on the environmental conditions that are imposed on the system. The discussion presented in this report should help interpret

and model the behavior of water and ice in freezing porous media.

LITERATURE CITED

Black, P.B. (1991) Interpreting unconfined unfrozen water content. In Ground Freezing 91, Proceedings of the Sixth International Symposium on Ground

Freezing, Beijing.

Black, P.B. and A.R. Tice (1989) Comparison of

soil freezing and soil water curve data for Windsor

sandy loam. Water Resources Research,25(10): 2205–

2210.

Black, P.B. and R.D. Miller (1990) Hydraulic conductivity and unfrozen water content of air-free

frozen silt. Water Resources Research, 25(2): 323–

329.

Castellan, G.W. (1971) Physical Chemistry. Second

Edition. Reading, Massachusetts: Addison–Wesley

Publishing Company.

Colbeck, S.C. (1982) Configuration of ice in frozen

media. Soil Science, 133(2): 116–123.

Colbeck, S.C. (1985) A technique for observing

freezing fronts. Soil Science, 139(1): 13–20.

Bouyoucos, G.J. and M.M. McCool (1916) The

freezing-point depression method as a new means

of measuring the concentration of the soil solution

directly in the soil. Michigan Agricultural Experiment Station Technical Bulletin, 24: 592–631.

J.W. Cahn, J.G. Dash and H. Fu (1992) Theory of

ice premelting in monosized powders. Journal of

Crystal Growth, 123: 101.

Edlefsen, N.E. and A.B.C. Anderson (1943) Thermodynamics of soil moisture. Hilgardia, 15(2).

Forland, T. and S.K. Ratkje (1980) On the theory

of frost heave. Frost i Jord, 21: 45–48.

Hudson, C.S. (1906) The freezing of pure liquids

and solutions under various kinds of positive and

negative pressure and the similarity between osmotic and negative pressure. Physiological Reviews,

22(5): 257–264.

Koopmans, R.W.R. and R.D. Miller (1966) Soil

freezing and soil water characteristic curves. Soil

Science Society of America Proceedings, 30: 680–685.

Miller, R.D. (1980) The adsorbed film controversy. Cold Regions Science and Technology, 3: 83–86.

Miller, R.D., J.H. Baker and J.H. Kolaian (1960)

Particle size, overburden pressure, pore water pres6
sure and freezing temperature of ice lenses in soil.

Proceedings, Seventh International Congress of Soil

Science, Madison, Wisconsin, p. 122–129.

O’Neill, K. and R.D. Miller (1985) Exploration of

the rigid ice model of frost heave. Water Resources

Research, 21(3): 281–296.

Radd, F.J. and D.H. Oertle (1973) Experimental

pressure studies of frost heave mechanism and the

growth-fusion behavior of ice. Second International

Permafrost Conference, Washington, D.C., p. 377–

383.

Schofield, R.K. (1935) The pf of water in soil.

Transactions of the Third International Congress on

Soil Science, Oxford, Great Britain, 2: 37–48.

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Prescribed by ANSI Std. Z39-18

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March 1995

Applications of the Clapeyron Equation to Water and

Ice in Porous Media

Patrick B. Black

U.S. Army Cold Regions Research and Engineering Laboratory

72 Lyme Road

Hanover, N.H. 03755-1290

CRREL Report 95-6

Approved for public release; distribution is unlimited.

Available from NTIS, Springfield, Virginia 22161

The equilibrium condition for water and ice in an air-free porous medium is presented. The equation of state for this

system is the Clapeyron equation. This equation is presented in a general form that explicitly shows the pressuredifference dependence with temperature for water and ice. Five solution scenarios are then discussed in terms of

applicability to porous media.

Office of the Chief of Engineers

Washington, D.C. 20314-1000

13

UNCLASSIFIED UNCLASSIFIED UNCLASSIFIED UL

Clapeyron equation Porous media

Frozen soil Water

Ice

PR: 4A762784AT42

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