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Modeling of Frost Heave In Soils

A Report to the

Center for Transportation StudiesUniversity of Minnesota

From

Underground Space CenterUniversity of Minnesota790 CivMinE Building

500 Pillsbury DriveMinneapolis, MN 55455

Authors

Radoslaw L MichalowskiVaughn R. Voller

Principal Investigators

Raymond L SterlingRadoslaw L Michalowski

Vaughn R. Voller

October 1991

Table of Contents

Executive Summary i

Part I:

1.0 Introduction 12.0 Literature Overview 33.0 Philosophy of the Approach 74.0 Porosity Rate Function 105.0 Interpretation of the Stress State in Frozen Soil 156.0 Fundamental Equation 187.0 Simulations of Frost Heave Processes 218.0 Discussion of Results and Conclusions 279.0 Bibliography 3910.0 Notation 48

Part II:

1.0 Introduction 522.0 Metallurgical Solidification vs. Frost Heave in a Modelling Context 533.0 A Model of Freezing in Soils 574.0 References 62

Executive Summary

This report results from a research project undertaken over the period July 1, 1990 to

September 30, 1991. The research project itself resulted from the combination of two proposals

to the Center for Transportation Studies for research into the mechanics of frost heaving in soils.

Both proposals had a general objective of creating a better understanding of the behavior

of the soil-water-ice systems and frost susceptible soils at freezing temperatures and improving the

capabilities of the faculty, researchers and students in an area of study which is important for road

design in Minnesota.

One proposal had the following major objectives:

* deriving a basis for creating an improved engineering tool (numerical/analytical) for

predictions of frost heave in soils with the emphasis being on phenomenological modeling

of the frost heave process as initiated by Fremond and Blanchard at the Laboratory for

Roads and Bridges in Paris.

* assembling a laboratory for testing frost susceptible soils. No testing capabilities for

assessing the frost susceptibility of soils existed in Minnesota - despite the importance of

frost heave in roadway pavement design in Minnesota.

The second proposal was aimed at:

* stressing the multi-phase character of the phenomenon using a "local averaging technique"

to account for the micro-level processes. This approach was designed to determine the

applicability of Dr. Voller's previous research in metallurgical solidification phenomena to

the frost heave modeling process.

The combination of the two proposals retained the separate research approaches but

provided for comparisons of the approaches and their applicability to frost heaving modeling.

Early in the project period, some changes in project arrangements were necessitated by

the move of Dr. Michalowski to Johns Hopkins University. Without Dr. Michalowski's presence

in the Department, it was concluded that the investment in the laboratory testing apparatus would

not be justified at present. It was also necessary to write a subcontract with Dr. Michalowski for

him to be able to complete his work on the project while in Baltimore.

In Part I, Radoslaw Michalowski outlines the basic background of frost heave modelling

including a summary of the existing modelling approaches. The central element in this part of the

report is the extension and application of the Blanchard and Fremond frost heave model. Results

from these models are in good agreement with other modeling approaches and consistent with

current frost heave understanding.

In Part II, Vaughan Voller discusses various similarities between frost heave in soil and

solidification of metal alloys. Three components in the thermo-mechanical frost heave model are

identified (i) modelling phase change, (ii) modelling ground water flow and (iii) deformation

modelling. In the context of the first of these components a numerical technique, previously

developed for metallurgical phase change, is applied to a two dimensional case of ground freezing.

Results, obtained in a matter of seconds on a IBM PC, are in close agreement with existing frost

heave models. This part of the report concludes with a discussion of current and on-going work,

in particular the thesis work of the master student, Lingjun Hou, who will couple the numerical

techniques of Part II with the phenomenological model of Part I.

Part I

by

Radoslaw Michalowski

1. Introduction

Frost heave in soils has been a very active area of research in the past two decades. The first

very constructive findings, however, were reported by Taber as early as 1929. It was Taber who

gave unquestionable experimental evidence that frost heave in soils occurs due to mass transfer,

and not due to stationary water expansion upon freezing, as previously thought. It then became

evident that the description of the frost heaving process must involve elements beyond heat

transfer alone.

No comprehensive models exist, so far, which would be capable of simulating both

freezing and thawing processes in frost susceptible soils. While much research on soil freezing

has been reported in the literature, a relatively small number of papers are available on the topic

of soil thawing, and most of them consider the thawing process in a rather simplified fashion.

The research presented in this report also pertains to the freezing of soils, but some of the

elements, especially those related to stress considerations, may be extended later to thawing

processes. The clear split in the literature seems to be caused by the fact that thawing cannot

be simply considered as the reverse of the freezing process. The constitutive properties of the

skeleton become increasingly more important in the consolidation process related to the melting

of ice in originally frozen soil. While the importance of different components of the model may

change from freezing to thawing, it is believed that one consistent model can describe both

processes. Such comprehensive model is beyond the goals of the research presented here,

however, the consideration of soil thawing is a natural extension of this research.

The goal of the research presented here is to produce a constitutive model of frost

susceptible soil which would be capable of simulating the frost heave process. In short, this

model can be called a "frost heave model", however, from the standpoint of constitutive

modeling, frost heave is not a property of the material (susceptibility is), but is a process which

can occur when frost susceptible soil is placed under certain initial and boundary conditions.

The modeling effort presented here is very different from that described in the vast

number of papers on the subject. It is based on introducing a specific function describing the

volumetric changes in freezing soil. This idea is an extension of the proposal by Blanchard and

Fr1mond (1985). Since most research reported in the literature in the area of soil freezing is

concentrated on the micromechanic approach, and the phenomenological approach has not

2

received a very warm welcome, the philosophy behind this approach and justification for it is

presented in the third section of this report. A literature survey is presented in the next section.

Sections 4, 5, and 6 present the assumptions made, and fundamental equations. Section 7 shows

some numerical simulations of freezing processes under different boundary conditions for

different soils. The simulations were conducted using a computer program written exclusively

for the purpose of testing the theory presented in this report. Conclusions, discussion of results,

and bibliography are presented in the last two sections.

Thanks are due to the Underground Space Center of the University of Minnesota, where

this research was initiated while the author was a USC staff member, and gratitude is expressed

to both USC and the Center for Transportation Studies of the U of M for financially supporting

the research.

3

2. Literature Overview

This literature overview is not meant to describe all published efforts pertaining to modeling of

frost susceptible soils. It describes the major trends in frost heave modeling found in the

literature, and refers only to some representative papers. A wide bibliography of the subject can

be found in section 7.

Papers by Taber (1929, 1930) and Beskow (1935) present the first systematic research on

frost action in frost susceptible soils. Taber's experiments gave solid evidence that frost heave

occurs due to transfer of water into the freezing zone rather than due to water expansion during

freezing. Efforts to rigorously describe the process of freezing and the related frost heave,

however, did not start until about two decades ago. Several different approaches to modeling

frost heave can be distinguished.

The capillary theory of frost heaving (Penner, 1959; Everett, 1961) is based on the Laplace

surface tension equation: p - p' = 2o'"/r '~ (0' - interfacial ice water tension, r'" - radius of the

ice/water interface). Penner (1959), Everett(1961), and others defined p' - pW as heaving pressure.

The Laplace equation suggests then that under given stress in the soil skeleton, there is a critical

pore size (related to r? ) which leads to the growth of ice or frost damage. Williams (1967) used

the Clausius-Clapeyron equation to show that the capillary theory predicts the freezing point of

water, p' - p" being known from the Laplace equation. The simplicity of the capillary theory was

appealing, but experimental tests did not confirm its validity: heaving pressures during freezing

tests were found to be significantly larger than those predicted by the theory (Penner, 1973).

There was also evidence that ice lenses can form within frozen soil at some distance from the

freezing front (Hoekstra, 1966, 1969), which could not be explained by the capillary theory.

These shortcomings led to the concept of secondary heaving, primary heaving being assigned to the

capillary action. It also became evident that modeling frost heaving processes cannot be done

without rigorous description of the coupled heat and moisture flow within freezing soil, with

the appropriate simulation of latent heat effects (Harlan, 1973; Guymon and Luthin, 1974; Taylor

and Luthin, 1978).

The secondary heaving mechanism was proposed by Miller (1978) (see also Miller, 1980);

it can explain some of the phenomena which are not accounted for in the capillary mechanism,

namely the growth of ice lenses behind the freezing front. This secondary frost heave theory is

4

based on the regelation mechanism, causing the movement of particles imbedded in ice, under

a temperature gradient. A key experiment presenting the migration of particles in ice up the

temperature gradient was shown by R6mkens and Miller (1973). Pore ice and ice lenses are

treated in the secondary heave mechanism as one body. Due to kinematical constraints, the

mineral (solid) particles cannot move up the temperature gradient, thus it is ice which will move

down the temperature gradient, the relative particle-ice motion being the same. This mechanism

gives rise to secondary frost heave and its mathematical description is often called the rigid ice

model. Attempts at practical calculations of frost heave using the rigid ice model can be found

in papers by O'Neill and Miller (1980, 1895), Holden (1983), Holden, et al. (1985), and Black and

Miller (1985). All simulations involving the rigid ice model are restricted to one-dimensional

considerations.

Research on freezing of moist soils and their frost heaving lead to emphasizing the

importance of unfrozen water in frozen soils. The experiments reveal that, depending on the

type of soil, there may be a considerable amount of unfrozen water in soil at a few degrees

below 0°C (e.g., Williams, 1967; Tice et al., 1989). This water significantly influences the ability

of frozen soil to transport water (Burt and Williams, 1976), and is also necessary for the

regelation process to take place. The existence of the unfrozen water in frozen soil is crucial to

the secondary frost heave mechanism.

As the primary heave mechanism predicted a pressure at which frost heaving would

stop, the notion of a "shut-off' pressure was introduced. Laboratory experiments by Sutherland

and Guskin (1973) with constant-volume soil freezing confirmed the tendency of heaving

pressure to reach a limit and stop the heaving. However, experiments by Loch and Miller (1973)

showed that, given enough time, the process of heaving will continue (at a lower rate), and it

is now accepted that increased stress slows down the rate of heave but, with sufficient time,

heaving may continue almost indefinitely.

An approach to explaining and describing frost heave different from the primary or

secondary heave mechanism was that proposed by Takagi (1980), and is referred to as the

adsorption force theory. According to this theory, a suction potential is generated due to freezing

of the external layers of the unfrozen film of water surrounding solid particles in frozen or

partially frozen soil. This film, trying to recover its thickness (and equilibrium), generates a

suction force which moves the molecules of unfrozen water into the liquid film. The freezing

of water that generates suction is called segregation freezing (as opposed to freezing in situ). The

adsorption force theory gives a certain explanation for the occurrence of cryogenic suction at the

freezing front, but it never materialized as a technique for the calculation of frost heave.

Another approach, aimed at engineering applications, was proposed by Konrad and

Morgenstern (1980, 1981). This approach is based on the concept of a segregation potential,

defined as the ratio of the water flux into the freezing zone to the temperature gradient in the

frozen fringe (SP = V/(T/Az)). This model differs from all previous models in that it introduces

a material parameter (SP) solely for the purpose of constructing a mathematical description. In

this respect the approach of Konrad and Morgenstern relates to phenomenological modeling,

where global quantities (or quantities at the macroscopic level) are introduced a priori rather than

derived from the processes on the microscopic level. The model proposed by Konrad and

Morgenstern seems to be applicable particularly in steady-state processes of heaving (although

an attempt was also made at describing transient processes, Konrad and Morgenstern, 1982b).

Successful implementation of the segregation potential model in practical problems, where the

process has been simplified to be dependent on one space variable, can be found in the literature

(Konrad and Morgenstern, 1984; Nixon, 1987). It should be mentioned that the segregation

potential model introduced some controversy, especially in the scientific community.

A two-dimensional model for frost heaving of soils based on rigorous considerations of

coupled heat and moisture flow was developed by Guymon, Hromadka and Berg, and

summarized in their paper of 1984. The model presented by Guymon, et al., is one of a few

attempts at describing 2-dimensional freezing. This model, however, is not explicitly coupled

with the stress field in the freezing soil.

Another model for the frost heave problem was presented by Mu and Ladanyi, 1987.

Their model has a very clear structure, incorporates heat and moisture transfer, and is coupled

with the stress field. It is one of only two models that use a constitutive relation for the soil

matrix. Although this model is applicable to multi-dimensional freezing problems, only a 1-D

example was shown by the authors.

A model based on the continuum approach was suggested in 1985 by Blanchard and

Frmond. This model is based on the introduction of a specific function describing the growth

of pore spaces in freezing soil. While no micromechanical processes are considered, the material

functions introduced model the global (or the macro-scale) effect, i.e., the response of the soil-

water-ice mixture to freezing. This model rigorously considers coupled heat and mass flow, and

the stress state. The philosophy of this approach will be elaborated in section 3. The model

presented in this report falls into the same category as that of Blanchard and Fremond.

Most models were formulated as one-dimensional ones, and were not coupled with the

stress field (Gilpin, 1980; Hopke, 1980, and Konrad and Morgenstern, 1982b, investigated the

effect of overburden in a one-dimensional process, which is not the same as coupling of the heat

and mass flow equations to the equations describing the stress field). Their applications were

related to highway and airfield construction, where, indeed, a one-dimensional heat flow regime

seems to be a reasonable approximation. However, multi-dimensional formulation is important

when the interaction of soil with arbitrary structures is to be analyzed. These problems are

never one-dimensional, and the stress field is the important outcome that is expected from the

solution.

7

3. Philosophy of the Approach

It is the model of secondary heaving (or rigid ice model) which received the most attention in

the scientific community. However, the current stage of development of this model allows only

1-D freezing simulation. Like the model based on capillary action, the secondary frost heave

model is based on considering the micromechanical processes, and the heaving effect is obtained

as the integral of the growth of all ice lenses in the frozen column (1-D process). The term

"micromechanical" processes here pertains to processes taking place between the components of

the soil skeleton-ice-water mixture. Macromechanical (or global) effects are those seen as the

response of the whole mixture.

Constructing a model for predictions of frost heave based on the summation of the actual

micromechanical processes will, certainly, lead to realistic qualitative results. As to the

quantitative results, the micromechanical approach may not necessarily be the most reliable one.

This has certainly been true in the mechanics of solids, where phenomenological models based

on introducing material parameters on the macroscopic level were far more successful in

predicting such global quantities as displacements and limit loads than micromechanics-based

models. The simplest examples of constitutive (of phenomenological) models are Hook's model

of linear elasticity, and the model of perfect plasticity. As opposed to micromechanics-based

models they cannot explain why the deformation is elastic (or plastic), but they can be quite

accurate in determining the magnitudes of deformation or the stress state, given material

properties defined at the macroscopic level (Young's modulus, Poisson's ratio or the yield

condition and the flow rule). Micromechanical models can explain the nature of deformation,

but the quantitative predictions are usually orders of magnitude apart from the true result.

There is no reason to doubt that a phenomenological model formulated on the

macroscopic level can be constructed for frost susceptible soils so the increase of porosity due

to ice lens formation can be modelled. Macroscopic models require that some material functions

be defined at the macroscopic level (for the mixture rather than for all constituents separately).

The segregation potential introduced by Konrad and Morgenstern (1980) is an example of such

a function. The first macromechanical approach, consistently carried through, is that presented

by Blanchard and Frnmond (1985), where a specific function describing the porosity increase was

introduced. This effort is extended here to include more factors influencing growth of ice during

8

the freezing process of soils.

While the material functions formulated on the macroscopic level do not have to be

strictly derived from the microscopic processes, their mathematical form must reflect the

experimentally observed effects. For the model of frost susceptible soil this effect is the increase

in volume which will be modelled by a porosity rate function, n, as suggested earlier by

Fremond' (1987). Thus the ice growth will be described as the average increase in porosity

(volume) of the soil element, rather than as separate ice lens growth. The success of the

phenomenological model depends on how accurate the porosity growth is captured as a function

of factors affecting this growth.

The best constitutive models are those which can simulate the true behavior of the

material with the least parameters introduced. The porosity rate function introduced in the next

section has five parameters,as the true process of ice lens growth is affected by many factors.

Both the porosity rate function (t) and the unfrozen water content (w) are the constitutive

relations which are important to modeling of frost heave.

The frost heave itself is a process which occurs when the frost susceptible soil is placed

under certain conditions. Thus, frost heave can be calculated as the solution to a boundary value

problem in which the soil is subjected to specific initial and boundary conditions. In this sense,

frost heave itself is not a property of the material, but frost susceptibility (described by function

i) is.

To formulate and solve the problem of frost heave for specific thermal and hydraulic

conditions, constitutive functions in addition to i and w need to be adapted: the heat transfer

and mass flow laws, and the rule describing deformability (and/or yielding) of the soil skeleton

(a deformability law of the entire mixture also needs to be defined in multidimensional problems

with structure interaction). These equations along with the fundamental principles of energy,

mass, and momentum conservation form the set of equations to be solved for given boundary

conditions.

Related to the approach proposed is the interpretation of the freezing soil as a heat engine.

Heaving soil performs mechanical work against external loads (gravity and boundary forces),

and also performs work which is either dissipated (e.g., during plastic deformation of the

skeleton) or stored as elastic energy (in both ice and skeleton). Thus it can be argued that part

9

'Personal communication.

of the latent heat released when water freezes is transformed into mechanical work by means

of a specific heat engine of a very low efficiency. While the temperature gradients at the

macroscopic level are considered finite within the freezing soil, the specific conditions for the

heat engine must exist on the microscopic level.

As all functions are formulated on the macroscopic level, this approach does not utilize

the Clausius-Clapeyron equation which is used in the micromechanics-based approach (analysis

of a single ice lens formation). Specific equations are presented in the next three sections.

10

4. Porosity Rate Function

According to the approach taken, the formation of a single ice lens is not attempted here; instead

the average volume increase of the soil matrix-water-ice mixture is considered due to the ice

growth. This volume increase is modeled by a porosity rate function, n, which describes the

porosity increase of a soil element. Predictions of the location of a single ice lens based on an

approach involving description of processes between the components of the frozen soil is

influenced by quantities which vary within the soil. Consequently, such predictions give an

approximate location of ice lenses. However, if the increase of volume due to ice lens growth

is integrated over a larger soil element and averaged, this average growth is likely to be a more

accurate result. Introducing a function modeling this average growth is then a natural step in

constitutive modeling of a frost susceptible soil. Here, it is the porosity rate function.

The porosity rate function must account for characteristic features of the freezing process

of frost susceptible soils, as known from laboratory tests. In particular, it must be a function of

temperature, temperature gradient, porosity, and the stress state. The following form of the

porosity rate function is proposed

a r (1)

=i -a Te (1 - e- ) (1 - n ) e

where n is the porosity, n is the porosity rate (an/st), and a, •1, N, P3, and 4 are the material

parameters. The function is constructed in such a way that its first part, -aT eD'T, describes the

porosity rate, while the two expressions in parentheses and the last exponential function are

dimensionless factors, all ranging from 0 to 1.0, and thus reducing the porosity rate due to the

value of the thermal gradient, current porosity, and the stress state, respectively.

The dependence of the porosity rate on temperature is represented in eq. (1) by the

expression -aT ePT (temperature T in oC). Introducing parameters h, describing the maximum

possible porosity rate for a given soil, and T, describing the temperature (in oC) at which the

maximum porosity rate occurs, parameters a and P, can be calculated as P = -1/T,

11

and a = -(nJT,)/e, and eq. (1) can be rewritten as

T ar_ T 1r- _)02 (2)--- e .T(1-e )(1 n e(2)" T

The dependance of the porosity rate on temperature is shown graphically in Fig. 1(a) for

different parameters n, and T,. The hypothesis of a maximum porosity increase as a material

property comes from the fact that hydraulic conductivity decreases with temperature while

suction increases (both nonlinearly). It is then likely that the influx of water into a freezing soil

element has a maximum at some temperature below zero *C.

At 0°C no ice growth process takes place (h = 0), while the porosity rate increases

drastically with the decrease of temperature, reaches a maximum, and diminishes to quantities

near zero for temperatures well below zero. The rapid increase of porosity at temperatures

slightly below 0°C can be interpreted as primary frost heave, while its slow growth at lower

temperatures can be viewed as the secondary frost heave process.

Parameters hm and T, allow the simulation of cases known from laboratory experiments.

For example, silt exhibits a relatively high rate of heave (related to a high porosity rate) at

temperatures slightly below zero, but it diminishes quickly when the unfrozen water content

drops down and the hydraulic conductivity becomes very small, while clay, under the same

boundary conditions, heaves more slowly, but its rate of heave does not drop as rapidly with

a decrease of temperature. This part of the porosity rate function introduced here is the same

as suggested by Fremond1 (1987), and discussed with Duquennoi' (1989).

The second component of the porosity rate function, (1- e (aT/ra)), involves the

temperature gradient along the heat flow lines. Co-ordinate I coincides with the tangent to heat

flow lines and is directed from the warm side of the freezing front toward the cold side, hence

DT/al < 0. For a sufficiently large magnitude of thermal gradient, laT/Ia I, this function becomes

close to unity, Fig. l(b), and it drops to zero when the magnitude of the thermal gradient drops

to zero. Assuming that cryogenic suction for a given soil is a function of temperature alone, the

suction gradient must be zero when DT/l1 = 0. According to Darcy's law, no water influx due

to cryogenic suction is then possible when T = const. Water flow is possible through the frozen

soil at a constant temperature, however, due to an "externally applied" hydraulic gradient (Burt

and Williams, 1976). Hydraulic gradients due to variation of the water table in soils are likely

to be small when compared to cryogenic suction gradients in the freezing soil, and are not

12

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reflected in the porosity rate function. The cryogenic suction gradients are implicitly included

in parameters ., and T,.

The third component of the porosity rate function, (1 - n)a, makes the porosity rate a

function of the current value of porosity. Without getting into the details of a single ice lens

growth, the process can take place only if both a solid soil skeleton and water are present. An

increase in porosity at a given temperature below freezing causes an increase in the relative ice

content, 0" , while the relative unfrozen water content, 0("', and the solid content, 08", drop

down. This leads to a decrease in the hydraulic conductivity, and consequently, to a drop in the

water supply. When porosity becomes close to unity, the soil skeleton may be considered as

particles embedded in ice, and the increase of the ice volume due to the Darcian influx of water

is stopped, since the hydraulic flow paths in the soil-skeleton-ice mixture are discontinuous. At

the limit, n = 1, the ice fills the whole volume and no further increase is possible. The expression

(1 - n) in eq. (2) brings the value of ni to zero at n = 1 (see also Fig. 1(c)). Note that the non-

Darcian water and ice transport due to the regelation phenomenon (and related to the secondary

frost heaving) requires that the porosity be less than 1. The process cannot take place in the

absence of a skeleton and unfrozen water.

The fourth factor in the porosity rate function reflects the influence of the stress state in

the soil: exp(-ai/P)), where oa is the first invariant of the stress state in ice (acf = a 0 +

a22+ o'), and p3 is a material parameter. Note that the influence of the stress state isconsidered without introducing the notion of a frost heave shut-off pressure. The stresses infrozen soil are discussed in more detail in the next section. The decrease in porosity rate i withincreasing stress in ice is introduced here on the basis of a purely theoretical argument. In order

for the material to increase its porosity and cause frost heave, mechanical work has to be doneagainst the external forces (gravity and boundary forces), the dissipation of mechanical energy

will occur during irreversible deformation of some mixture components, and energy will be

stored during reversible deformation. This requires that part of the latent heat released when

water freezes is changed into mechanical work. Frost heave can then be interpreted as a result

of the action of a particular heat engine. Macroscopically, the temperature gradients are

considered finite, but, for the heat engine to have an efficiency larger than 0, a temperature

14

discontinuity must occur on the microscopic level2, otherwise all heat will be diffused. As long

as the conditions for the heat engine exist on the microscopic level, part of the latent heat (small,

however, due to very low efficiency) will be used to perform the mechanical work. The power

made available by the heat engine must then lead to heave rates where, the higher the

overburden pressure, the lower the heave rate. This relation is not linear due to dissipation and

energy storage terms in the energy balance, and also due to possible structural changes on the

microscopic level leading to changes in heat engine efficiency.

Porosity changes are described here as scalar i (eq. (2)). To capture the anisotropic

character of the void growth (due to ice lens formation), the porosity growth tensor is introduced

1 avk avn ki ( + )2 axI ax

ax3 ax2 3v3x 3x2 3

aV2 V2 v21I a2 3

av3 av3 av3

ax1 ax2 3X3

where v, is the velocity vector of the (incompressible) mineral skeleton and x, is the Cartesian

space co-ordinate. Assuming now that x, coincides with the heat flow lines across the freezing

soil element under consideration, it is hypothetically assumed that

id

S 0 0

0 01- ) 0

0 0 2

(4)

where 0.33 ~ < 1.0. The two limit values of 4 (0.33 and 1.0) describe the isotropic growth and

unidirectional growth of ice lenses (in the direction of heat flow lines), respectively.

2For example, efficiency of the Carnot heat engine for an ideal gas is represented as: p = 1 -Tj/T 2, where T, and T2 are temperatures (°K) of the two heat reservoirs, with T2 - T, being the

temperature discontinuity.

15

(3)

I

I

5. Interpretation of the Stress State in Frozen Soil

The influence of the stress state in soil on ice lens formation, simulated here as an increase in

porosity, is perhaps one of the more crucial elements in phenomenological modeling of frost

heave. Considering the stress state on the microscopic level, it is the stress in ice which is likely

to be the governing parameter (Williams and Wood, 1985).

There has been very little concern, so far, regarding the role of the stress state in freezing

soil, and its role in modeling frost susceptible soils. If considered at all, it is usually included

as "the influence of the overburden pressure", which, from the standpoint of constitutive

modeling, is not acceptable. The "overburden pressure" is a boundary condition (part of the

boundary value problem) and cannot influence the constitutive model of the material itself; it

is the stress state at the point of consideration which influences the behavior of the material at

that point. In general, there is an infinite number of stress boundary conditions causing the

same stress state at a given point. Successful use of the "overburden pressure" as a parameter

in some models comes from the fact that these models are restricted to one-dimensional

deformation processes (with fixed principal stress directions), where the normal stress

component in the direction of the overburden pressure throughout the soil is statically

determinate and equal to the overburden pressure plus the weight of the material above the

point of consideration.

The mechanism of the stress transfer through the soil is dependent on the microstructure

of the frozen soil. Miller (1978) adapted the concept of effective and neutral stress from soil

mechanics. This concept is also adapted here, and it is presented in terms of the full stress

tensors3

= °+ -( (5)

where -i is the total stress tensor in the mixture, if, is the effective stress tensor (stress in the

skeleton), and •,'"' is the neutral stress tensor (Fo, here, depicts the stress defined as the force

over the total cross-section of the mixture, and ol is the force over the cross-section of the

respective component - microstress). The neutral stress tensor is now decomposed into parts

3Subscript indices indicate space co-ordinates (vectors and tensors), and superscript indicesin parentheses denote the specific constituent of the mineral skeleton-water-ice mixture.

16

assigned to water and ice

S=" X u, + ( - X) (6)

where 8, is Kronecker's symbol, u is the pore water pressure, akC/ is the microstress tensor in

ice, and X is the partition factor (0 < X < 1). Note that such stress definition is based on the

parallel interaction of the skeleton and whatever fills the pores. It follows then, that under

a zero external load G, the buildup of microstress in ice, a5o, must be compensated for by the

stress in the skeleton, aOi, (assuming u unchanged). Thus the buildup of the compressive

normal stress increment in ice causes a tension increment in the skeleton. Williams and

Wood (1985) also gave a similar interpretation of the "heave pressure" transfer. This is possible

if the soil skeleton forms a continuous matrix and the matrix has enough strength to withstand

the tensile stresses. Ice inclusions, however, reduce the ability of the skeleton to resist tension,

and local slip between mineral particles (plastic flow) can be expected when the ice pockets

(lenses) increase their volume. The stress state in the clusters of the soil skeleton is expected to

be non-homogeneous and dependent on the geometrical location of these clusters with respect

to growing ice. The strength of the frozen silt or clay is, of course, much higher than that of the

unfrozen soil. However, this is the strength of the mineral-water-ice composite. The strength

of the skeleton matrix alone against expansion caused by the growing ice probably changes little

with freezing. The average stress state in the skeleton, o•~', over a representative frozen soil

element can presumably be described by a function similar to the yield functions of plastic

materials, and the level of tension is probably quite low (comparable to that following from the

Mohr-Coulomb yield condition in a tensile regime, for unfrozen silt and clay of comparable

liquid moisture content).

It is the first invariant of stress tensor a,/" which enters the porosity rate function. Based

on eq. (6), oag it can be calculated as

"-- x--(s)FY_ __ - -XuB u (7).I 1-x

It will be assumed that coefficient x depends on the unfrozen water concentration in the frozen

soil. The unfrozen water concentration is defined here as the ratio of the liquid water volume

to the volume of the pores

17

V(W) (8)V•,)

Assuming hypothetically that v = X, eq. (7) can be rewritten as

o -5vui (9)= l-v

In a one-dimensional freezing process in an ideal vertical column (with smooth boundaries), the

vertical normal stress component of the total stress, F, is statically determinate and equal to the

overburden pressure plus the weight of material above the point considered. The effective

stress, v'P), can be calculated from the strain introduced by the porosity changes in the elastic

range, or from the yield condition when the skeleton reaches the yield state. Suction u can be

back-calculated from the Darcy law since the continuous flow rate is uniquely related to the

porosity growth. The constitutive relations for deformation of the soil skeleton and the whole

mixture need to be known to calculate stress components other than the normal vertical one, and

aCo can be calculated iteratively from eq. (7) or eq. (9). The first stress state invariant in ice,

omm< = o /-0 + aOoG + o+ z, can now be determined and used in calculations of ii (eq. (2)). In

2- and 3-dimensional problems, all stress components of the total stress tensor and the effective

stress tensor need to be determined using the respective constitutive relations.

18

6. Fundamental Equation

The crucial part of the model capable of simulating frost heave introduced in this report is the

characteristic porosity rate function. In this respect, the model presented falls into the same

category as that of Blanchard and Fremond (1985), and is very different from most models

available in the literature.

The proposed model considers soil a porous medium, with pores filled with water, or

water and ice. Volumetric changes of the soil-water-ice mixture are assumed to occur due to

porosity increase which, in turn, is caused by the inflowing and freezing water driven by

cryogenic suction. The process of the formation of discrete ice lenses is then "smeared" over a

representative volume, and simulated by the porosity increase of the mixture. The porosity

increase is governed by the function characteristic to the specific soil (material function). This

function describes the effect on a macroscopic level and is not derived from the processes on

a microscopic level (processes between the components of the mixture). While such model

cannot explain physical phenomena leading to frost heave, it is expected to be a reliable

predictor of frost heave, provided the material parameters are derived from reliable experimental

tests.

In addition to the porosity rate function, hi, the relation representing the unfrozen water

content in the frozen soil, w, is important to the model presented. Based on experimental tests

(e.g., Burt and Williams, 1976, and Tice et al., 1989) the form of this function is assumed as

w = w* + (io - w')e "T (10)

where w is the unfrozen water content as a fraction of the dry weight, w° is the residual unfrozen

water content at some very low reference temperature, WI" is the minimum unfrozen water content

at 0°C, and a is the third material parameter.

For calculation purposes it is convenient to use the unfrozen water concentration, v,

defined in eq. (8). For the saturated soil considered here one can write

v Y"'1 - n (11)y(W) n

where f'' and j, are the specific weights of the soil skeleton and water, respectively. Note

that for unfrozen saturated soil v = 1, and the water content becomes uniquely determined by

19

the porosity (n)

W Y 7W n (12)y () 1-n

The skeleton, water and ice content in the frozen soil will be described by quantities 0',

0(' ), and 0' defined as

0( V(' V( 1) (13)ge) - 1 - n , = ) = vn, O _ = n(l-v) (13)V V V

where Vs ), w, and V( are the volumes of the soil skeleton, unfrozen water, and ice,

respectively, and V = Vs + V ' + V). Introducing the densities of the three components of the

mixture as p"', p(w), and pr), respectively, the density of the mixture can be written as

p = eO(S p(" + O()p( + ( p) = (1 - n)p() + vnp(w) + n(1 -v)p ) (14)

The heat capacity of the mixture per unit volume can be expressed as

C = (1 -n)p')C(S)+vnp(w)C(w)+n(1 -v)p('oC'O (15)

where C(', CC ), and C` are the mass heat capacities of the soil skeleton, water, and ice,

respectively. Due to changing proportions of the components in the freezing soil, the heat

capacity will vary.

The Fourier law of heat conduction is adopted

aTQ• -(T) (16)

ax.

where Q, is the heat flux vector, T is the temperature, and x, is the space co-ordinate. Thermal

conductivity X(T) is a function of temperature, and it pertains to the conductivity of the soil-

water-ice mixture.

Water transport is described by the Darcy law

Ahq. = - k(T) (17)

ax,

where q, is the flow rate vector, h is the hydraulic head, and k(T) is the hydraulic conductivity

20

(function of temperature).

The constitutive relations for the soil skeleton and the whole frozen composite are not

specified here in detail. It is only suggested that they may be described with hypoelastic-type

relations

a1) = B(T)" e() (18)Sij k1

where ec is the strain tensor and B(T) is the constitutive tensor.

Equations (2), (4), (10), and (16)-(18) describe the constitutive model of frost susceptible

soil. Frost heave, however, is a process dependent on the initial and boundary conditions, and

can be described only as a boundary value problem. Solution to the boundary value problem

requires that the unknown functions T, h, e'j, and oej, be found such that the fundamental

principles of energy, mass, and momentum conservation are satisfied. Using eq. (16) the energy

conservation principle can be written as

aT acpO (19)C - p~ - V( VT) = 0 (19)at at

where I is the latent heat of fusion of water per unit mass. It needs to be mentioned that eq. (19)

contains no term due to heat transport by water flow. It also neglects the terms due to

mechanical work. These terms are considered small compared to the latent heat released during

freezing. The mass conservation principle takes the form

(8) + WO(.. a 0+M (20)p t p t +pt - p V(kVh) = 0

at at at

or

( an () (nv) (21)(p" - p) (p -p) ~ n - p " V(kVh) = 0 (21)at at

Finally, the momentum conservation principle takes the form

a (s)

-p ̂gVVh+(p-p"')g O (22)

where gi is the gravity acceleration vector.

21

7. Simulations of Frost Heave Processes

Material parameters. Simulation of the freezing processes for specific soils requires that material

parameters be known from experimental tests. Reliable techniques of determining material

parameters for phenomenological models are those where the material sample tested can be

considered a material element. This can be done only if a homogeneous state is introduced to

the sample and the state parameters can be derived directly from the conservation principles.

The response of the material to independently controlled changes in the state parameters can be

monitored. Due to homogeneity of the state parameters the response of every material element

in the sample is the same, and the material properties can be calculated easily from the global

response of the sample.

Unfortunately, testing of the freezing processes (and heaving) requires that the sample

be subjected to a temperature gradient, thus non-homogeneous response of the material

elements throughout the sample must be expected. The contribution of the material in the

sample to the global effect (heave of the sample) varies throughout the sample. A feasible way

to obtain model parameters is then to simulate the freezing processes for which the lab tests

were done and try to match the lab and simulation results by adjusting the unknown

parameters. The number of lab tests (with different boundary conditions) must be at least equal

to the number of parameters to be evaluated.

Model parameters such as the unfrozen water content as a function of temperature,

hydraulic and thermal conductivity, initial porosity, and densities of the components of the

mixture are assumed to be known from independent tests. It is the parameters of the porosity

rate function that require some explanation as to their possible methods of estimation. It is

suggested that these parameters be derived from tests with "ramping temperatures". When the

rate of boundary temperature change is sufficiently slow, such tests allow a constant temperature

gradient to be maintained in the freezing sample, and, thus, the process in the freezing zone can

be considered as approximately stationary with respect to the co-ordinate system attached to the

moving freezing front (due to geometrical changes the process is not strictly stationary). The

22

heave rate is then expected to be nearly constant. The heave rate obtained from experiments

can now be described as

h fd dz = const (23)dt 0where h is the height of the sample, and n is expressed by eq. (2). Porosity rate i for material

under a negligible stress state is a function of parameters i., T., 12, and 4, thus a minimum of

four tests are needed, each with a different temperature gradient. Parameter 13 needs to be

evaluated from additional tests where the sample is loaded with an external load. While,

theoretically, five separate tests are needed to evaluate the five parameters, more tests are

desirable to obtain a reliable set of parameters. Determining the model parameters reduces then

to a "curve fitting" process with five free parameters. The tests used to determine the parameters

must not be used to verify the adequacy of the model.

Numerical simulations. A computer program was written for the purpose of performing

numerical simulations of a 1-D freezing process. The program considers the freezing soil an

assembly of "control volumes" (or elements) with the heat and mass flow occurring discretely

between these volumes. The temperature of the boundary volumes is adjusted to be equal to

the prescribed boundary temperatures. The diffusion process inside the "column" is numerically

generated through an explicit scheme (independently calculated fluxes between the elements).

The porosity growth was assumed to be anisotropic with t = 1 (eq. (4)). Hydraulic

conductivity undergoes large changes in quantity during freezing. While these changes easily

can be incorporated into the calculations, the experimental data for particular materials (for

which freezing was simulated) was not available. Therefore, only 2 values of hydraulic

conductivity k were used: one for unfrozen soil, and one for frozen soil. Consequently, the

suction in the soil was not accurately calculated and the calculations of the stress in ice,

according to eq. (9), were expected to be unreliable (the approximation X = v adds to this

inaccuracy). Therefore, to calculate the influence of the elevated stress in the soil the last

component of eq. (2) was replaced with

Sis the first invariant of the total stress tensor in the soil. The first invariant of the total

where o^ is the first invariant of the total stress tensor in the soil. The first invariant of the total

23

stress was then calculated assuming that the frozen soil is elastic

-a=Z +2 P 1 - -p (24)

where az is the normal vertical component in the total stress tensor (equal to overburden

pressure p; weight of sample neglected), and p is Poisson's ratio of the frozen composite.

No comprehensive experimental results were found in the literature to allow the porosity

rate parameters for soils to be evaluated with acceptable precision for practical purposes. The

parameters used in simulations presented in this section were adopted for the purpose of

presenting qualitative results. Comparison to lab test results available in the literature is shown

to present the potential of the model. Even though the simulations fit the lab data very well,

quantitatively these results cannot be considered a verification of the model.

Three one-dimensional freezing processes with different boundary conditions were

simulated (see Fig. 2):

1. A one-step freezing process, where the boundary temperatures are adjusted only once,

at the beginning of the process

2. A freezing process with ramping temperatures

3. A multi-step freezing process, where the temperature at the "warm" boundary is adjusted

three times during the process.

The one-step freezing process was simulated with parameters representing a freezing lab

test described by Konrad and Morgenstern in their 1980 paper (test NS-10). The initial height

of the sample was 18 cm, and its temperature +10C. The temperature of the top boundary was

dropped to -60C at t = 0, and kept at -60 C for ten days. Parameters used in the simulation are

shown in Table 1. Figs. 3, 4, and 5 represent the temperature, porosity, unfrozen water content,

ice content, suction, suction gradients, and heave rate.

The second process simulated is that for which the lab test was reported by Penner in

1986 (soil No. 1, run No. 1). The original height of the sample was 10 cm. Temperatures at the

cold and warm plates were ramped from -0.35 to -0.550 C, and from +0.55 to +0.35°C,

respectively, at 0.02°C/24h. It is assumed that a constant temperature gradient compatible with

the boundary temperatures exists in the sample at the beginning of the simulation. The

24

2 4 6 8 10

Time (days)

0 2 4 6 8 10

Time (days)

10

21

0

-2

-4

-6

-8

15

Time (days)

Fig. 2

Boundary conditions for simulated precesses; (a) stepfreezing, (b) ramped temperatures, (c) multi-stepfreezing.

25

0

CL

Q.E

I--

Dboattom.boundarv.

initial temperature: 1°Cinitial temp. gradient: 0

top boundary

,I I 1 I I I I 1 I

(a)

(b)

(C)

bottom boundary

initial temp. gradient: 90 C/m

_ I. top boundary

0C)0

o

E

0

a,0

I-

,,H.I--

.6

.4

.20

-.2-.4-.6-.8

4

2

0

-2

-4

-6

-8

-10

-------- bottom boundary-I------

initial temperature: 40Cinitial temp. gradient: 0

top boundary

_ I I , I

I

!A

Table 1. Data used in simulations of freezing processes.

Simulation I I m

Initial height of samplecm 18 10 15.2

Initial porosity 0.35 0.35 0.35

Heat conductivity Xfrozen soil, J/24h.m.°C 145,000 147,000 170,000

Heat conductivity Xunfrozen soil, J/24h.-m.C 175,000 181,000 190,000

Heat capacity of skeletonCs, J/kg-.C 900 900 900

Hydraulic conductivity kfrozen soil, m/24h 106 10" 10 5

Hydraulic conductivity kunfrozen soil, m/24h 10.2 0.1 0.1

h,, 1/24h 15 0.40 5001

Tm, oC -0.15 -0.15 -0.12

2~ 5 5 7

3, MPa 0.333

P4, m/°C 0.2 0.2 0.2

w 0.15 0.15 0.15

w* 0.02 0.02 0.02

a 0.9 0.9 0.9

p", kg/m 3 2700 2700 2700

Poisson's ratio p 0.3 -

'Very frost susceptible mixture of sand and ground chalk used by McCabe and Kettle (1985).

26

respective model parameters are given in Table 1, and the results are shown in Figs. 6, 7, and 8.

The third simulation reflects the results of the lab test given by McCabe and Kettle (1985).

The original height of the sample was 15.2 cm. The temperature of the sample before the test

was +4°C. At t = 0 the cold plate was adjusted to -7.50 C, and the warm plate to +30C, and,

thereafter, the warm plate temperature was dropped to +2°C at t = 8 days, and to +1°C at

t = 11.3 days. The results of the simulation are shown in Figs. 9-12.

27

8. Discussion of results and conclusions

Even though the model parameters used in simulations were not determined in laboratory tests,

the results allow one to indicate, in qualitative terms, the realistic character of the modelled

heave processes.

Figures 3 - 12 present the results for the three particular freezing processes for which the

boundary conditions are presented in Fig. 2: (a) one-step freezing, (b) freezing under "ramping"

temperatures, and (c) multi-step freezing. These figures represent the distribution of

temperature, unfrozen water content, porosity, volumetric ice content, and also the hydraulic

head (suction) and its gradient. All quantities are shown for a series of selected instants, and

the distribution along the sample is related to the current co-ordinate updated due to the heave

process (except Fig. 12). All results are presented in an "unsmoothened" fashion, thus some

discontinuities in derivatives of the distribution curves appear due to the division of the sample

into discrete volumes in the numerical technique used.

The results from a simulation of the one-step freezing process are shown in Figs. 3, 4, and

5. Under given boundary conditions, the heat flow process becomes almost steady after one day

(Fig. 3(a)). The departure from the steady-state process is caused by heaving. A slight change

in the temperature gradient at 0°C after one day is caused by the assumed abrupt change in

thermal conductivity (though small in quantity; see Table 1).

Appearance of the ice lenses is modelled here as regions of increased porosity, Fig. 3(b).

The quick advancement of the freezing zone in the first four hours of the process results in

predominantly freezing in situ where no significant increase in void volume is observed. Once

the speed of the 0°C isotherm becomes relatively small, the process of porosity growth becomes

very pronounced (Fig. 3(b)). The ice content increases significantly in the neighborhood of the

nearly-stationary freezing zone, which is indicated by the increasing "bulb" in ice content

distribution curves in Fig. 3(d). The peak in the ice content distribution can be interpreted as

the "final" ice lens formed during step freezing, while the increased ice content above that

indicates lower average ice concentration, e.g., thinner and widely spaced lenses. The constant

ice content distribution in the upper part of the sample indicates freezing in situ of that part.

The variation in the unfrozen water content is related to the temperature changes. The

unfrozen water content changes gradually as a function of temperature. The drop in unfrozen

28

(a)

C

CU0

0)E0)0J

-6 -4 -2 0

Temperature (oC)

(b).2.18

0

E.15

.05 o

.2 .4 .6

Porosity

(C)

ECO.0

0

NO0

o

E0C!,

0 .05 .1 .15 .20

Unfrozen water content(fraction of dry weight)

.2 (d)

.15 "C0

.1 o

0

.05 E0

00 .2 .4 .6

Ice content(fraction of total volume)

Fig. 3

Step freezing process; (a) temperature, (b) porosity,(c) unfrozen water content, (d) ice content.

29

v

.8

.2

.15

.1

.05

0

Pll

(b)

-15 -10 -5

hudraulic head (m)(suction)

.2

.15

.1

.05

0 -800 -600 -400 -200 0

suction gradient (m/m)

.2

.15

.1

.05

0

Fig. 4

Step freezing process; (a) hydraulic head, (b) hydraulichead gradient

(b)

0 2 4 6Time (days)

10

O 8EE 6

a 4

2

A

8 10 0 .2 .4 .6

Time (days)

Fig. 5

Frost heave during step freezing process.

30

(a)

cC

0

0

0

0E

0J

(a)25

20

15

10

5

0

E

C-0>I

.8 1||

water content with decreasing temperature (eq. (10)) affects the heat diffusion process through

the release of latent heat (this coupling leads to iterative numerical procedures for solving the

freezing process).

The model presented does not require calculations of hydraulic head (suction) to simulate

heave. This is because the porosity rate function becomes independent of the suction if the

influence of the stress state in ice is neglected. It is possible, however, to back-calculate suction

assuming that the water transfer is governed by Darcy's law. Hydraulic conductivity in frozen

soil is very much dependent on temperature (especially in the region of intensive porosity

increase), and its exact variation is seldom tested for practical purposes. Back calculations of

suction based on approximately estimated hydraulic conductivity are not very accurate, but they

are presented here so some qualitative conclusions can be drawn. The two values of hydraulic

conductivity (for frozen and unfrozen soil) used in calculations are given in Table 1. The level

of the hydraulic head at the bottom of the sample was assumed to be zero. The hydraulic head

and its gradient for the simulated step freezing process are presented in Fig. 4. The distribution

of the cryogenic suction changes during the process, and it reaches its maximum before a big

concentration of increased porosity is formed. As expected, the maximum of the suction

gradient is related to the maximum heave rate, and the suction gradient decrease coincides with

a decrease in the heave rate. Some oscillations in the magnitude of the cryogenic suction and

suction gradient were observed during the process (see, for example, the sequence of results for

2, 5, and 10 days). These oscillations seem to be of a numeric nature and become smaller with

the decrease in time step and element size.

The heave of the sample as a function of time is shown in Fig. 5. During the step

freezing process the heave rate (derivative of the heave curve) tapers down once the freezing

zone becomes almost stationary. Such behavior of the simulated sample is to be expected, since

it is postulated in eq. (2) that the porosity rate decreases with an increase in porosity itself. The

heave rate reaches its maximum somewhere at the beginning of the process, but not at the very

start of freezing. The temperature gradient is the highest in the frozen part of the sample at the

beginning of step freezing. The porosity rate increases with an increase in |3T/Az . Heave of

the sample, however, is the integral of the volume increase over the frozen part of the sample,

and this frozen part is small at first, when the magnitude of the temperature gradient is high.

Consequently, the sample (during step freezing) reaches the maximum of the heave rate not at

the very start of the process, but shortly after, see Fig. 5(b) (some fluctuations in the simulated

31

heave rate were also observed, due to discretization of the sample for numerical calculations).

Discrete points in Fig. 5(a) represent the heave measured by Konrad and

Morgenstern (1980). The height of the sample and the boundary conditions used in the

simulation were identical to those reported by Konrad and Morgenstern, and the other

parameters (Table 1) were selected in such a way so that the simulated heave would match that

measured by Konrad and Morgenstern quantitatively. While such simulation cannot be

considered a verification of the model, the numerical results indicate that the model produces

very reasonable qualitative results. The four curves for p > 0 in Fig. 5(a) represent heaving for

overburden pressures of 30, 100, 200, and 300 kPa, respectively. As mentioned earlier, the stress

state in ice, which is likely to be a governing factor in decreasing the heave rate, cannot be

calculated accurately. Therefore, the decrease of the porosity rate due to an elevated stress level

was expressed as a function of the first invariant of the total stress state (in the entire frozen

composite). This hypothesis requires more study, though the reduction in frost heave in Fig. 5(a)

due to overburden seems to be reasonable.

Results from simulations of soil freezing with ramped temperatures are shown in Figs. 6,

7, and 8. A constant temperature gradient throughout the sample was assumed at the beginning

of the process. As heat conductivity is assumed different for soil above 0°C and below, this

gradient becomes discontinuous at the 0°C isotherm once the process advances, Fig. 6(a) (this

discontinuity is relatively small due to the small change in conductivity, Table. 1). The porosity

increase is quite different from that during step freezing. Porosity now grows gradually

(Fig. 6(b)) due to the slow movement of the freezing zone. No regions of high void

concentration are created. If the process was continued sufficiently long, a nearly constant

porosity would be left some distance behind the freezing zone. This constant distribution of the

average porosity could be interpreted as the rhythmic formation of lenses that are identical in

thickness and evenly spaced.

The unfrozen water content drops down rapidly from the initial 0.199 to 0.15 at 0°C, and

further drops down almost linearly due to a small change in temperature (it is, however, an

exponential function, eq. (10)). According to the model presented here, the rapid drop from

0.199 to 0.15 occurs at 0°C. The inclination of the distribution lines in this range in Fig. 6(c) is

caused by discretizing the sample for calculations. As no discontinuities were allowed within

a discrete volume, this otherwise abrupt drop is plotted between the centers of two adjacent

discrete volumes, giving the impression of a continual decrease.

32

.1 Ec.

.08 .

.06 00

.04 Eo0

.02

n

0 .2 .4 .6

Temperature (oC)

4 (b)

.2 .3 .4Porosity

(d).12

.1

.08

.06

.04

.02

00 .05 .1 .15 .2

Unfrozen water content

(fraction of dry weight)

.1.

.1

.08

.06

.04

.02

0Iv

.5 .6 n

0 .1 .2 .3 .4

Ice content(fraction of total volume)

Fig. 6

Freezing by temperature ramping; (a) temperature,(b) porosity, (c) unfrozen water content, (d) ice content

33

(a)

c0

0

4-9

E

Co

-.6 -.4 -.2

(C) .12

.1

.08

.06

.04

.02

0

4 ̂

(a)

C0

o- -

- -a)E

CD

-3 -2 -1

Hydraulic head (m)

Ik\.12 \U/

.1

.08

.06

.04

.02

n

-100 -50

.12

.1

.08

.06

.04

.02

0

Suction gradient (m/m)

Fig. 7

Freezing by temperature ramping; (a) hydraulic head,(b) hydraulic head gradient

EE

a)

2LL

12

10

8

6

4

2

0 2 4 6 8 10 12

Time (days)

Fig. 8

Frost heave during freezing process with temperatureramping.

34

Oh

Ice content, Fig. 6(d), in the area behind the freezing front still increases after 11 days.

After sufficiently extended "ramping", the ice content could be expected to be nearly constant

in the upper part of the sample. While the calculations of the suction are not considered

accurate quantitatively, Fig. 7(a) indicates that it varies little, and these variations are probably

due to numerical instabilities. The suction gradient (which drives the water into the freezing

part of the sample) decreases gradually toward the cold boundary (Fig. 7(b)), and not as rapidly

as in the step freezing, since the drop in temperature toward the cold boundary is now much

smaller.

Heaving of the sample during the process with ramped temperatures occurs with a nearly

constant heave rate, Fig. 8. The simulated heave is shown along with the heave curve reported

by Penner (1986) for an experiment with similar boundary conditions (the actual heat flow in

Penner's experiment was from the top down). The material parameters were selected in such

a way so that the quantitative results matched Penner's laboratory test. As in the previous

instances, the simulation cannot be considered a verification of the model, but its qualitative

result is quite realistic.

The multi-step freezing simulation, according to Fig. 2(c), is shown in Figs. 9 - 12. The

periodic porosity increase of the freezing soil is now clearly related to temperature jumps in the

warm end of the sample. The porosity distribution has characteristic maxima. The most upper

peak in porosity distribution is formed as in the single-step freezing. The next peak is caused

by the rapid movement of the 0°C isotherm from its nearly steady position at 8 days, to its new

position, which is associated with the change in the warm boundary temperature. This is later

repeated at 12.3 days. These peaks are related to the material and not to a steady location in

space, therefore they move up while the heave process takes place. Fig. 10(a) shows the same

porosity increase, but drawn at the original positions of the material elements. The unfrozen

water content is shown in Fig. 9(c), and the ice content in Fig. 9(d). As with the porosity, the

ice content distribution has maxima characteristic to a multi-step freezing process (see also

Fig. 10(b)).

The hydraulic head and its gradient have a clear tendency to increase at the beginning

of each initiation of a porosity peak, and to decrease with elapsing time until the subsequent step

in warm boundary temperature. This decrease causes the heave rate to drop after the initial

increase at the beginning of each step in boundary temperature, which is shown in Fig. 12.

35

.2 (b)

E.15

S-0

0.1

E.05 0

C)

A

-8 -6 -4 -2 0 2 4T

Temperature (oC)

0 .05 .15 .1 .2

Unfrozen water content

(fraction of dry weight)

0 .2 .4

Porosity

.2 (d)

.15

.1

.05

0

.15

.15

.1

.05

nv.6 .8 n

0 .2 .4 .6 .8

Ice content(fraction of total volume)

.2

.15

.1

.05

0

Fig. 9Multi-step freezing process; (a) temperature, (b)porosity, (c) unfrozen water content, (d) ice content.

36

(a)

C

0c0

E00)

(c)

el

(a)

c0

0

-

EO0

0)0

0 .2 .4

Porosity.6 .8 n

(b).2

.15

.05

O1

.9

.05 o0)0

initial height of sample

.05

7.99- 1.

11.29days 15

I I i i i

0 .2 .4 .6 .8Ice content

(fraction of total volume)

.2

.15

.1

.05

0

Fig. 10

Multi-step freezing; porosity (a), and ice content (b)related to the original position of the material elements.

37

n

(a)

0

E0aQ0I3

.5 -.4 -.3 -.2 -.1

Hydraulic head (m)

Multi-step freezing;gradient.

I.C

50

40

30

20

10

0

.2 (b)

.15

.1

.05

.2

.15

.1

.05

-40 -30 -20 -10 0

Hydraulic gradient (m/m)

Fig. 11

(a) hydraulic head, (b) hydraulic head

5 10

Time (days)

15

Fig. 12Frost heave during multi-step freezing process.

38

Conclusions.

1. As the model is not derived from micromechanical processes, it is not targeted at

explaining the frost heave phenomenon, but at qualitative and quantitative predictions

of frost heave in saturated soils.

2. The hypothetical dependencies of the void increase due to temperature, temperature

gradient, current magnitude of porosity, and the stress state need experimental

verification. While available in the literature raw data do not allow verification of the

model, the simulations for different freezing processes indicate very realistic

(qualitatively) behavior of soil under freezing conditions.

39

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40

December 1986.

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41

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42

Harlan, R.L., Analysis of coupled heat-fluid transport in partially frozen soil, Water ResourcesResearch, 9(1973), 1314-1323.

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Hoekstra, P. and R D. Miller, On the mobility of water molecules in the transition layer betweenice and a solid surface, J. Coll. Interf. Sci., 25 (1967), 166-173; see also Report 153, U.S. Army ColdRegion Research and Engng. Laboratory, Hanover, N.H., 1965.

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Holden, J.T., Approximate solutions for Miller's theory of secondary heave, Fourth Int. Conf.Permafrost, Fairbanks, Alaska, 1983, 498-503.

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Holden, J.T., D. Piper and R.H. Jones, Some developments of a rigid-ice model of frost heave,4th Int. Symp. Ground Freezing, Sapporo 1985, 93-99.

Hopke, S., A model for frost heave including overburden, Cold Regions Science and Technology,Vol. 3 (1980), 111-127.

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Horiguchi, K., Effect of the rate of heat removal on the rate of frost heaving, Proceedings of theFirst International Symposium on Ground Freezing, Bochum, March 8-10, 1978, Ed. H. L.Jessberger, Elsevier, 1979, 63-71.

Horiguchi, K. and R.D. Miller, Hydraulic conductivilty functions of frozen materials, Fourth Int.Conf. Permafrost, Fairbanks 1983, 504-508.

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43

Jame, Y.W. and D.I. Norum, Heat and mass transfer in a freezing unsaturated porous medium,Water Resources Research, 16 (1980), 811-819.

Jackson, K.A. and B. Chalmers, Freezing of liquids in porous media with special reference tofrost heave soils, J. Appl. Physics, 29 (1958), 1178-1181.

Jackson, K.A., D.R. Uhlmann and B. Chalmers, Frost heave in soils, J. Appl. Physics, 37 (1966),848-852.

Kaplar, C.W., New experiments to simplify frost susceptibility testing of soils, Highway ResearchRecord, Vol. 215 (1968), 48-59.

Kay, B.D. and Groenevelt, P.H., On the interaction of water and heat transport in frozen andunfrozen soils: I. Basic theory; Vapor phase, Soil Sci. Soc. Amer. Proc., 38(1974), 395-404.

Kay, B.D., M. Fukuda, H. Izuta and M.I. Sheppard, The importance of water migration in themeasurement of the thermal conductivity of unsaturated frozen soils, Cold Reg. Sci. Techn., 5(1981), 95-106.

Kinoshita, S., Effects of initial soil-water conditions on frost heaving characteristics, EngineeringGeology, 13 (1979), 41-52.

Konrad, J.M. and N.R. Morgenstern, A mechanistic theory of ice lens formation in fine-grainedsoils, Canadian Geotechnical Journal, 17 (1980), 473-486.

Konrad, J.M. and N.R Morgenstern, The segregation potential of a frozen soil, Canadian Geot.J., 18 (1981), 482-491.

Konrad, J.M. and N.R. Morgenstern, Predictions of frost heave in the laboratory during transientfreezing, Canadian Geot. J., 19 (1982), 250-259.

Konrad, J.M. and N.R. Morgenstern, Effects of applied pressure on freezing soils, Canadian Geot.J., 19 (1982b), 494-505.

Konrad, J.M. and N.R. Morgenstern, Frost heave prediction of chilled pipelines buried inunfrozen soils, Canadian Geot., 21 (1984), 100-115.

Konrad, J.M., Influence of freezing mode on frost heave characteristics, Cold Regions Science andTechnology, 15(1988), 161-175.

Koopmans, R.W.R. and Miller, R.D., Soil freezing and soil water characteristic curves, Soil Sci.Soc. Amer., 30(1966), 680-685.

Li, Y., R.L. Michalowski, B. Dasgupta and R.L. Sterling, Preliminary results from simulation ofretaining wall displacement by frost action, 5th Int. Conf. Cold Regions Engineering, ed. R.L.Michalowski, 1989, 389-396.

Loch, J.P.G., State-of-the-art report - Frost Action in Soils, Engineering Geology, 18 (1981),

44

213-224.

Loch, J.P.G., Thermodynamic equilibrium between ice and water in porous media, Soil Science,126 (1978), 77-80.

Loch, J.P.G. and B.D. Kay, Water redistribution in partially frozen, unsaturated soil under severaltemperature gradients and overburden loads, Soil Sci. Soc. Am. J., 42 (1978), 400-406.

Mageau, D.W. and N.R. Morgenstern, Observations of moisture migration in frozen soils,Canadian Geotechnical Journal, 17 (1980), 54-60.

McCaba, E.Y. and Kettle, RJ., Thermal aspects of frost action, 4th Int. Symp. on Ground Frezing,Sapporo 1985, 47-54.

McRoberts, E.C. and J.F. Nixon, Some geotechnical observations on the role of surcharge pressurein soil freezing, Proc. 1st Conf. Soil-Water Problems in Cold Regions, Calgary, Canada, 42-57.

Miller, R.D., Soil freezing in relation to pore water pressure and temperature, 2nd Int. Conf.Permafrost, Yakutsk 1973, 344-352.

Miller, RD., Freezing and heaving of saturated and unsaturated soils, Highway Research Record,393 (1972), 1-11.

Miller, R.D., Frost heaving in non-colloidal soils, Proc. Third Int. Conf. Permafrost, Edmonton,Alberta, 1978, 707-713.

Miller, R.D., Lens initiation in secondary heaving, Proc. Int. Symp. Frost Action in Soils, Lulea1977, 2, 68-74.

Miller, R.D., The adsorbed film controversy, Cold Reg. Sci. Techn., 3 (1980), 83-86.

Miller, RD. and E.E. Koslow, Computation of rate of heave versus load under quasi-steady state,Cold Regions Science and Technology, Vol. 3 (1980), 143-252.

Miller, RD., Freezing phenomena in soils, in: Applications of Soil Physics, ed. D. Hillel,Academic Press, 1980.

Miller, R.D., Thermally induced regelation: a qualitative discussion, 4th Int. Conf. Permafrost,1983, 61-63.

Milly, P.C.D., Moisture and heat transport in hysteretic, inhomogeneous porous media: a matrichead-based formulation and a numerical model, Water Res. Research, 18 (1982), 489-498.

Mostaghimi, J. and E. Pfender, Measurement of thermal conductivities of soils, Wairme- undStoffiibertragung, 13 (1980), 3-9.

Mu, S. and B. Ladanyi, Modelling of coupled heat, moisture and stress field in freezing soil, ColdReg. Sci. Techn., 14 (1987), 237-246.

45

Nixon, J.F., Thermally induced heave beneath chilled pipelines in frozen ground, Canadian Geot.J., 24 (1987), 260-266.

O'Neill, K. and Miller, RD., Numerical solutions for a rigid ice model of secondary frost heave,2nd Int. Symp. Ground Freezing, Trondheim 1980, 656-669.

O'Neill, K., The physics of mathematical frost heave models: a review, Cold Reg. Sci. Techn., 6(1983), 275-291.

O'Neill, K. and R.D. Miller, Exploration of a rigid ice model of frost heave, Water ResourcesResearch, 21 (1985), 281-296.

Palmer, A.C., Ice lensing, thermal diffusion and water migration in freezing soil, Journal ofGlaciology, 6(1967), 681-694.

Penner, E., Soil moisture tension and ice segregation, Highway Research Board Bulletin, 168(1957), 50-64.

Penner, E. and W. Gold, Transfer of heaving forces by adfreezing to columns and foundationwalls in frost-susceplible soils, Canadian Geotechnical Journal, Vol. 8 (1971), 514-526.

Penner, E., Frost heaving pressures in particulate materials, Proc. Symp. Frost Action on Roads,Oslo, 1973, Vol. 1, 279-285.

Penner, E., Ice lensing in layered soils, Canadian Geotechnical Journal, 23 (1986), 334-340.

Penner, E., Frost heaving in Soils, Proc. Int. Conf. on Prmafrost, 11-15 Nov., 1963, Lafayette,Indiana, 197-202; Nat. Res. Council, Washington, D.C., 1963.

Penner, E., Heaving pressures in soil during unidirectional freezing, Canadian GeotechnicalJournal, Vol. 4 (1967), 398-408.

Penner, E., Particle size as a basis for predicting frost action in soils, Soils and Foundations, 8(1968), 21-29.

Penner, E., Frost heaving forces in Leda clay, Canadian Geotechnical Journal, Vol. 7 (1970), 8-16.

Penner, E., Heave and heaving pressures in frozen soils: Discussion, Canadian GeotechnicalJournal, 8 (1971), 499-501.

Penner, E., Aspects of ice lens growth in soils, Cold Reg. Sci. Techn., 13 (1986), 91-100.

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46

Penner, E. and T. Walton, Effects of temperature and pressure on frost heaving, Proceedings ofthe First International Symposium on Ground Freezing, Bochum, March 8-10, 1978, Ed. H. L.Jessberger, Elsevier, 1979, 29-39.

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Perfect, E. and P.J. Williams, Thermally induced water migration in frozen soils, Cold RegionsScience and Technology, 3 (1980), 101-109.

Philip, J.R. and D.A. Vries, Moisture movement in porous materials under temperature gradients,American Geoph. Union, Trans., 38 (1957), 222-232.

Radd, F.J. and D.H. Oertle, Experimental pressure studies of frost heave mechanisms and thegrowth-fusion behavior of ice, 2nd Int. Conf. on Permafrost, 13-28 July, 1973, Yakutsk, U.S.S.R.,North American Contribution, 409-419; Nat. Acad. Sci., Washington, D.C., 1973.

Radhakrishna, H.S., K.C. Lau and A.M. Crawford, Coupled heat and moisture flow throughsoils, J. Geot. Engng., 110 (1984), 1766-1784.

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Romkens, M.J.M. and R.D. Miller, Migration of mineral particles in ice with a temperaturegradient, J. Coll. Interf. Sci., 42 (1973), 103-111.Sophocleous, M, Analysis of water and heat flow in unsatureted-saturated porous media, WaterRes. Research, 15 (1979), 1195-1206.

Sheppard, M.I., Kay, B.D. and Loch, J.P.G., Development and testing of a computer model forheat and mass flow in freezing soils, 3rd Int. Conf.Permafrost, Edmonton 1978, vol. 1, 76-81.

Sutherland, H.B. and P.N. Gaskin, Pore water and heaving pressures developed in partiallyfrozen soils, 2nd Int. Conf. on Permafrost, 13-28 July, 1973, Yakutsk, U.S.S.R., North AmericanContribution, 377-384; Nat. Acad. Sci., Washington, D.C., 1973.

Svec, O.J., A new concept of frost-heave characteristics of soils, Cold Regions Science andTechnology, 16(1989), 271-279.

Takagi, S., Fundamentals of the theory of frost-heaving, Proc. Int. Conf. on Permafrost, 11-15Nov., 1963, Lafayette, Indiana, 203-207; Nat. Acad. Sci., Nat. Res. Counc., Washington, D.C., 1963.

Takagi, S., An analysis of ice lens formation, Water Resources Research, 6 (1970), 736-749.

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Taber, S., Frost heaving, Journal of Geology, Vol. 37 (1929), 428-461.

47

Taber, S., The mechanics of frost heaving, Journal of Geology, Vol. 38 (1930), 303-317.

Takagi, S., The adsorption force theory of frost heaving, Cold Region Science and Technology,Vol. 3 (1980), 57-81; see also Report 140, U.S. Army Cold Regions Research and Engng.Laboratory, Hanover, N.H., 1965.

Taylor, G.S. and J.S. Luthin, Numeric results of coupled heat-mass flow during freezing andthawing, 2nd Conf. Soil Water Problems in Cold Regions, Edmonton 1976, 155-172.

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Tice, A.R., Black, P.B. and Berg, R.L., Unfrozen water contents of undisturbed and remouldedalaskan silt, Cold Regions Science and Technology, 17(1989), 103-111.

Vignes, M. and K.D. Dijkema, A model for the freezing of water in a dispersed medium, J.Colloid Interf. Sci., 49 (1974), 166-172.

Williams, P.J., Properties and behaviour of freezing soils, Norwegian Geotechnical Institute, Publ.No. 72, 1976, Oslo.

Williams, P.J. and Wood, J.A., Internal stresses in frozen soil, Canadian Geotech. J., 22(1985), 413-416.

Williams, P.J. and Smith, M.W., The Frozen Earth. Fundamentals of Geocryology, CambridgeUniv. Press, 1989.

Wood, J.A. and Williams, P.J., Stress distribution in frost heaving soil, 4th InternationalSymposium on Ground Freezing, Sapporo 1985, 165-171.

Yong, R.N. and J.C. Osler, Heave and heaving pressures in frozen soils, Canadian GeotechnicalJournal, Vol. 8 (1971), 272-282.

48

10. Notation

a - material parameter in the unfrozen water content function

B,, - constitutive tensor for deformation of the soil-water-ice mixture

Bij constitutive tensor for deformation of the skeleton

C - heat capacity of soil-water-ice mixture per unit volume

C• - heat capacity of ice per unit mass

C( - heat capacity of mineral skeleton per unit mass

C - heat capacity of water per unit mass

h - hydraulic head

k - hydraulic conductivity

n - porosity

n - porosity rate

n, - maximum porosity rate

Q - heat flux vector

q - hydraulic flow rate vector

T - temperature (0C)

Tm - temperature at which the maximum porosity rate occurs

t - time

u - pore water pressure

V - volume of a soil element

V) - volume of ice in the representative soil element

VS - volume of mineral skeleton in the representative soil element

V•w - volume of water in the representative soil element

VP - volume of pores

w - unfrozen water content in frozen soil (fraction of dry skeleton weight)

w - residual unfrozen water content at some low reference temperature

(for practical purposes can be assumed 0)

w - minimum unfrozen water content at 0°C

x - space co-ordinate

49

a

i -04

OP)

oo"'x

vV

O'"'xp(i)p"'

aiiij(s)

(i")

aqu>

50

- material parameter in the porosity rate function

- material parameters in porosity rate function

- specific weight of mineral skeleton

- specific weight of water

- stress partition factor

- strain tensor (skeleton)

- unfrozen water concentration (VJV)

coefficient describing anisotropic growth of porosity

- volumetric ice fraction in frozen soil

- volumetric soil skeleton fraction in frozen soil

- volumetric water fraction in frozen soil

- thermal conductivity

- ice density

skeleton density

- water density

- total stress tensor

- stress tensor in skeleton

- neutral stress tensor

- stress tensor in ice (microstress)

Part II

by

Vaughn Voller

51

1. Introduction

The objective of the second part of the project was to explore the commonality between

metallurgical phase change systems and frost heave with the aim of:

1. Transferring the applicable computational and modelling approaches for metallurgical phase

change systems to frost heave.

2. Developing and extending the research capabilities of the principal investigator to an area of

interest to the CTS. This talent development component also included a masters student

(Lingjun Hou) who would develop his thesis topic in the area of frost heave.

3. Provide an interface with the first part of the project. In particular lay the ground work for

further activity that would include the coupling of the phenomenological model developed

by Michalowski in part one to the numerical methods developed in part two.

The two major components to be reported here are:

1. A discussion of the relationship between frost heave and metallurgical phase change

modelling, and

2. An illustration of the application of a numerical phase change models (originally developed

for metallurgical phase change systems) to the case of frost heave.

52

2. Metallurgical Solidification vs. Frost Heave in a Modelling Context

In the first part of this report Radoslaw Michalowski provided an overview of frost heave

phenomena and the modelling approaches that can be used in its analysis. This overview will provide

an excellent source for information on frost heave in discussing the commonality between

metallurgical phase change and frost heave. Information on metallurgical phase change systems can

be found in the texts by Flemings (1974) and Kurtz and Fisher (1989).

Although frost heave and metallurgical solidification are both thermo-mechanical systems at

the process level there is little in common between them. When one looks at the underlying

phenomena however, commonalities do emerge. This is particularly useful in a modelling context in

that an appropriate process model can be developed on piecing together phenomena models. In

other words, if well defined modelling techniques exist for modelling metallurgical phase change

phenomena it may be possible to use them to model related frost heave phenomena.

Common phenomena found in both metallurgical phase change and frost heave are outlined

below. In addition, commonalities in the modelling approach are noted.

Phase Change. Clearly both frost heave and solidification involve a liquid to solid phase change. In

this respect there is an additional common feature; in both systems, due to impurities (or alloying

elements) and undercooling effects, the solid to liquid phase change does not occur at a unique

temperature. In metals, solidification can occur over a reasonably wide range of temperatures. Such

a range is referred to as a "mushy" region and is characterized by a local liquid fraction temperature

relationship. The form of this relationship for a aluminum copper alloy is given by the curve in

Figure la. This curve has been derived on considering the effect of local solute redistribution

(microsegregation) (Flemings (1974) and Kurtz and Fisher (1989)). In frost heave systems, the

temperature range is referred to as the "frozen fringe" (Miller (1980) and O'Neill and Miller (1985)).

This region can be characterized by pore unfrozen water content and its nature is determined from

experiment (O'Neill and Miller (1985)). A typical shape of this curve (following Blanchard and

Fremond (1985)) is given in Figure lb. Although the metal and frost heave curves have different

shapes, the important point is that a freezing range characterized by a liquid fraction temperature

relationship exists in each case. Hence, if a robust numerical treatment of a liquid fraction

temperature range phase change can be found it can be effectively used for both metal phase change

and frost heave phase change. The demonstration of such a technique will be made at a later point

in this report.

53

Mushy Region. The natures of the metal mushy region and the frost heave frozen fringe are

schematically illustrated in Figure 2. In metal systems the characteristic size of the mushy region is

defined by the size of the secondary dendritic arm spacings (Flemings (1974)) of the order of 10's of

microns. The frozen fringe is characterized by the soil particle size. In modelling both the metal

solidification and the frost heave systems, it is true to say that events occurring in the mushy region

(frozen fringe) are occurring at the microscopic scale. These microscopic events, however, can not

be neglected at the macroscopic process level. This is a point we will return to below.

Fluid Flow. The fluid flow in both metal solidification and frost heave takes a significant role. In

metal solidification, the shrinkage in the mushy region results in fluid flow. This fluid flow plays an

important part in the heat and mass transfer of the solidification process (Flemings (1974)). A

solidification fluid flow is also a fundamental part of frost heave. In order to maintain equilibrium

in the mushy region, a suction is created in the absorbed film and new water is brought up into the

frozen fringe. This so called "primary heave" is one of the prime causes of frost heave 1. A detailed

discussion is given in Michalowski (1991)). The fluid flow in both frost heave and solidification (see

Voller et al (1989)) can be adequately described by Darcy type equations. Hence, fluid flow

modelling approaches developed for one system will be suitable for the other.

Deformation. Both frost heave and metallurgical phase change can be classed as thermo-mechanical

processes. The analysis requires the coupling of heat and mass transfer modelling with the modelling

of the deformation and stress. In looking at the deformation modelling of the two systems, however,

there is a fundamental difference in that frost heave involves expansion and metal solidification

involves contraction (shrinkage). There are distinct similarities in the modelling approach of the

deformation processes, however, in particular the coupling of the heat and mass transfer models to

suitable constitutive equations. In part one of this report Michalowski (1991) has outlined suitable

constitutive equations for frost heave systems. An equivalent discussion for metals solidification

systems can be found in Dantzig (1989).

Micro vs. Macro Modelling and the Engineering Modelling Approach. In closing our discussion of

the similarities between solidification and frost heave systems,we will compare and contrast the

modelling approaches used. Many of the key features of metallurgical solidification (frost heave)

occur in the mushy region (frozen fringe). An appropriate model at the macroscopic scale will in

1The other cause of frost heave ("Secondary Heave") is the so called"regulation", Miller (1980), the movement of soil particles through the ice.

54

general have a characteristic length which is too large to fully resolve events at the microscopic mushy

region (frozen fringe) scale. This mismatch between micro and macro scales is not only found in frost

heave and metal solidification it is common in the modelling of many engineering systems.

Modelling of solidification has taken two distinct paths. On one path, modelling has been

carried out in the mushy region, at the microscopic scale, concentrated on the investigation of

microstructure and micro segregation phenomena. In these models, macroscopic phenomena, eg. heat

and fluid flow, are only treated approximately, if at all, and the emphasis is on a fundamental

scientific understanding of the solidification phenomena. On the other path, modelling has been

carried out at the macroscopic level centered on the heat transfer aspects and the engineering

performance of the process as a whole. In these models, microscopic effects are included by the

definition of appropriate constitutive models (e.g. a liquid fraction temperature relationship, see

Figure la.). The different types of micro and macro models have been extensively reviewed by

Rappaz (1989). This review paper clearly indicates the success of both modelling approaches but

notes that further advances can only be made if coupling between the micro and macro scales is

improved. Recent approaches that attempt to do this include (i) the so called micro-macro models

(Rappaz and Stefanesq (1988)) which combine a macro and micro models acting on two time scales

and (ii) the use of Representative Elementary Volume (REV) averaging technique which allows for

a more comprehensive treatment of the microscopic behavior in macroscopic models, Beckerman

(1990).

A similar distinction in modelling paths can be found in the frost heave literature. On one

hand, are what Michalowski (1989) refers to as the micromechanical models of frost heave. These

microscopic models (by far the majority) emphasis the fundamental physical nature of the frost heave

process. Although these models provide insight into the possible micromechanisms they are, by and

large, unable to provide a comprehensive treatment of frost heave and structural interactions at the

macro process scale. On the other hand, are the phenomenological models (e.g. Blanchard and

Fremond (1985)). These continuum models use constitutive equations to capture the microscopic

behavior and in every respect are similar in scope and purpose to the macro solidification models.

In view of this similarity, it is valid to ask the question; can the micro-macro metal solidification

modelling approach (Rappaz and Stefanesq (1988)) be adopted for frost heave? A drawback in using

such an approach is that there is still controversy (e.g. see the paper by Takagi (1980) and the

subsequent discussion by Miller (1980)) in determining the exact nature of microscopic frost heave

55

phenomena. In the metals literature the microscopic phenomena is reasonably well understood and

consistent.

To summarize, a thermo-mechanical model, can be used to describe both frost heave and

metallurgical solidification. Common components in theses models include:

1. A treatment of a non-linear phase change, i.e., a non unique phased change temperature with

a non-linear liquid fraction temperature relationship (Figure 1.)

2. A multi phase region mushy (=solid + liquid metal) or fringe (=water + ice + soil), in which

the microscopic phenomena are key to the macroscopic process behavior.

3. Fluid flow driven by suction in the mushy region or frozen fringe.

4. Deformation driven by the expansion of pores (frost heave) or the contraction of the

solidifying metal.

In arriving at a full and comprehensive model all of the above features need to be carefully

coupled. An example of such a coupling can be found in the first part of this project. In arriving

at the final best model, however, there is some merit in treating each of the above components in

turn. This was the approach followed in the second part of this project with the aim of identifying

metallurgical modelling approaches that could be easily and efficiently applied in the modelling of a

component of the frost heave problem. In the next section, a recently developed metallurgical phase

change technique (Voller (1990)) and Voller and Swaminathan (1991) will be successfully applied to

a realistic frost heave phase change.

56

3. A Model of Freezing in Soils

3.1 The Test Problem

The test problem is taken from the work of Li et al. (1985). The geometry for this test

problem is given in Figure 3. The boundary conditions are given in Table 1 (note the cyclic boundary

condition over a 120 day period). In modelling this problem, we will only consider the heat transfer

aspects. Flow due to suction in the frozen fringe and the subsequent heave and deformation will be

neglected. With this assumption the governing heat transfer equation can be written as, Crank

(1982).

pH = V.(KV)(1)at

where p is density H is the enthalpy, K is conductivity and T is temperature. In a case of constant

specific heat, Cp, and a zero reference temperature the enthalpy can be defined as:

H = CT+.fL

where L is the latent heat and f is the local liquid fraction (see Figure Ib). In this work the

following f vs T curve equation will be assumed:

1 if >0

f/ = (2)

(T+2) if -2T<016

and the following thermal properties will be used where a is the thermal diffusivity.

Note that the latent heat is based on assuming a porosity of 0.4 in the soil.

57

L = 60.00 (KJIkg)

a = 0.11 (m2/day) Silt= 0.10 (m2/day) Sand

C, = 1.00 (KJlkg- C)

3.2 The governing equation

Since f is a function of temperature, T equation (1) is non-linear. In nearly all the frost heave

models to date (see references in Michalowski (1991)) the above equation is solved using the so

called apparent heat capacity approach. In essence on noting that:

aH dH aT (3)9t dT St

equation (1) becomes

C aT= V.(KV7) (4)9at

where C = dH/dT is an enhanced specific heat. On the surface, equation (4) looks attractive since

it has the same form as the linear heat conduction equation and the basic numerical (finite element

of finite difference) techniques can be employed. In practice, however, a number of complications

arise. In particular, sharp changes in the f vs T relationship will result in unstable schemes. As a

result jumps in the f vs T curve have to be artificially smoothed out. Further, even after this

smoothing small and hence inefficient time steps are required for stable solutions. A discussion of

apparent heat capacity methods as used in the metallurgical industry can be found in a review by

Voller et al (1990).

An alternative to the enhanced specific heat approach is to use a so called source-based

method (Voller (1989) Voller and Swaminathan (1990)). With this approach phase change problems

with arbitrary liquid fraction temperature curves (e.g. Figure 1.) can be efficiently resolved. This

approach has been successfully used on a wide range of metallurgical systems (Swaminathan and

Voller (1991)). The aim of the current work is to apply it to the frost heave freezing problem.

Substitution of equation (2) in equation (1) with rearrangement results in:

aT = V.(aVT) - L -- (5)at (t

58

where L* = L/Cp. The essential feature of this equation is that the non-linearity associated with the

phase change has been isolated in a source term.

3.3 The Numerical Solution

The numerical solution of equation (5) will follow that proposed by Voller and Swaminathan

(1990). The reader is referred to this source for additional information. Following a suitable spatial

discretization of the domain of interest (i.e. finite elements (Zienkiewicz (1977), or control volumes

Patankar (1980), etc.) and a backward Euler time discretization with a lumped capacitance

formulation the generic discretization equation corresponding to the governing equation (5) subject

to appropriate boundary and initial conditions can be conveniently written down in point-form as:

V V (6

aT = an, Tnb T+ + L * P-nb At AtP

where the a's are coefficients of the discretization equation which depend on the thermal properties

and the chosen form of the numerical scheme, Vp is a volume associated with node P and AT is

the time step. The superscript [] refers to the old time value and the subscript []p and []nb refer to

the node P and the neighboring nodes respectively. In essence, equation (6) provides a relationship

between the temperature value at node P and the neighboring nodes nb see Figure 4. In the case

of bilinear square finite elements with side Ax the coefficients assuming constant thermal

diffusivity are:

a = 8+ a a = a, V= Ax2At 3 nb 3 P

The system of equations resulting form the assembly of the point equation (6) will be non-linear since

the liquid fraction field is a strong function of temperature. On using a truncated Taylor series

expansion for fpm+1, we have:

59

,+1 dF + +1 - 1 n] (7)

where F is the given liquid fraction temperature relationship the term [dF/dT] is evaluated at fpm and

the superscript ]m refers to the iteration level. Using this linearization, the term:

(8)S = VLf

can be conveniently expressed as:

where:

S = Sc + S, T..t

Sc= -VL f -vF [(f)]

SdFS = - V L* -d

dT

(9)

(10)

The iterative scheme now reads:

(11)* anb• *+ I V; V= E anb I+ p+ pO + Sc

nb At At9

Based on equation (11), the solution procedure in a single time step is as follows:

(1) At the start of the time step, the iterative procedure is initiated on setting f" = fo.

(2) The coefficients and the values of Sp and Sc are evaluated and the linearized system (in the

form Ax=b )is solved to give the temperature field, Tm+1. (A simple iterative point by point

60

(ap -SP) rI

scheme is often sufficient to solve the Ax=b).

(3) In general, this temperature field will not be consistent with the current liquid fraction field.

Consequently, the liquid fraction field is corrected via equation (7).

(4) In practice, the liquid fraction update, equation (7) is applied at every node point followed

by the over/under shoot correction.

+1 = 0 if f.+ < 0(12)

F1 = 1 if fp+1 >1

Equation (12) eliminates the need to check the necessity of an update at each and every node

and efficiently accounts for the jumping of the latent heat peak. It also provides a mechanism

by which the nodal volume can start or finish a phase change in a single time step.

(5) The procedure outlined in steps (2) to (4) is continued to convergence.

Some general comments on the proposed method point:

(1) The optimum performance is obtained without the need for any under-relaxation.

(2) The jump discontinuities at the top of the f=F(T) curve (equation (2) is dealt with on setting

[dF/dT] to an arbitrarily large value (e.g. 1010). This is equivalent to the introduction of a

finite freezing interval. However, the scheme is not dependent in any way on the size of the

interval.

(3) In order to ensure a smooth transition from the fully liquid to a phase change state, the

proposed liquid fraction update, equation (11), is only used for nodes for which fpm+1 < 1.

At nodes where fpm+1 = 1, the liquid fraction update of Brent et al. (1986) is used i.e.

1 t1= I b + a [ - F- (f,,M)] (13)a,

(4) In many respects, the proposed scheme has similarities to the apparent heat capacity method.

In particular, the addition of the Sp term to the diagonal of the coefficient matrix can be

thought of as an apparent capacity. As with the conventional apparent capacity methods, this

will introduce some control in the iterative solution in that it ties down the predicted

61

temperatures at nodes where the phase change is occurring. The inclusion of the liquid

fraction update will mean that remedial actions and averaging schemes are not required at

jumps on the F(T) curve.

3.4 Solution

The test problem outlines in section 3.1 has been solved using the approach outlined above.

Approximately 1000 square bilinear finite elements of size 0.25 m with a time step of 2.5 days (note

that this time step is 8 times larger than that allowed by an explicit scheme) was used in this solution.

The solution time to reach 120 days was on the order of 10 minutes on an IBM 386 clone with math

co-processor. Results in terms of temperature contours at 120 days are given in Figure 4. Note the

trapped "frozen sandwich" caused by the subsequent thawing of the soil. These results also compare

well with the results obtained by Li et al. using the Blanchard and Fremond model on a main frame

computer see Figure 5.

The significance of the above result is not so much in the numbers but in the method. In

essence a metallurgical phase change approach has been successfully applied to a component of the

frost heave problem. Although the freezing is only one component of the frost heave problem it is

not an insignificant problem and much effort has been used up on it in the past. The efficient

solution presented here (10 minutes of CPU on a 386 is insignificant when compared with similar or

longer main frame times) is only achieved through the use of a methodology developed in the

metallurgical literature - a method that has been under development for almost five years (the

original source based method dates back to Voller and Prakash (1987)). It is hoped that in follow

up studies more robust and efficient metallurgical methodologies will be found to be useful in the

analysis of frost heave.

Conclusions and Further Work

The major aim of this part of the project was to explore and exploit the commonality between

metallurgical solidification and frost heave. In the first part of this report a number of common

phenomena and modelling issues were outlines and discussed. This commonality was emphasized in

the second part of the project on successfully applying a numerical method developed in the

metallurgical literature to the freezing component of the frost heave system.

In follow up studies it will be hoped to couple the freezing model to the phenomenological

62

heave and deformation model developed in part one. We feel that this approach is promising

because the analysis of real engineering frost heave problems (e.g. the interaction of frost heave with

engineering structures) becomes possible.

63

(a) 1.

C00.

4-,

O0co0

L

0J0

0.

(b) 1.00

00. 00-

O-

©0.60-LL

0.,40"

CT-J

0.20

0.00- ", . .. I.. ... .. . i f. . . , ,. .. .. .I-4.00 -3.00 -2.00 -1.00 0.00 1.00 2.00

Temperature C

Figure 1 Liquid Fraction Temperature Relationships in (a) metal [Al - 4.5% Cu] and (b) poresin frozen soil

64

%

Columnar Dendrite

(a)

Equiaxed Dendrite

(b)

Figure 2 Mush Region / Frozen Fringe in (a) Metal [Al - 4.5% Cu] and (b) Freezing Soil

65

(a)

Figure 3 Test Problem Geometry

Figure 3 Test Problem Geometry

Figure 4 Predicted Temperature (OC) Isotherms After 120 Days

Figure 5 Predicted Temperature (oC) Isotherms After 120 Days as Predicted

by Li et al. (1989).

66

Table 1. Boundary Conditions

67

4.0 References

Blanchard, D. and M. Fremond, "Soil Frost Heaving and Thaw Settlement, 4th Int. Symp. Ground

Freezing, Sapporo (1985), 209-216.

Crank, J. "Free and Moving Boundary Problems", Clarendon Press, Oxford 1984).

Dantzig, J.A., 'Thermal Stress Development in Casting Processes", Metallurgical Science and

Technology, 7, (1989), 133-178.

Flemings, M.C., "Solidification Processing", McGraw Hill, New York, (1974).

Kurz, W. and D.J. Fisher, "Fundamentals of Solidification", Trans Tech Publications,

Aedermannsdorf, Switzerland (1989).

Li, Y., R.L. Michalowski, B. Dasgupta and R.L. Sterling, "Preliminary Results from Simulation of

Retaining Wall Displacement by Frost Action," Proceedings, 5th Int. ASCE Specialty Conference

on Cold Regions Engineering, Feb. 6-8, 1989, St. Paul, MN. ASCE, N. Y., pp. 389-396.

Michalowski, R.L. "Modelling of Frost Heave in Soils" Part 1. Report to CTS. (1991).

Miller, R.D., "The Absorbed Film Controversy" Cold Regions Science and Technology", 3 (1980) 83-

86.

Miller, R.D., "Freezing Phenomena in Soils" Applications of Soil Physics Academic Press, (1980).

Ni, J. and C. Beckermann, "A volume averaged two-phase model for transport phenomena during

solidification, "Metall. Trans. B, 22B, (1991), 349-361. Transfer Conference, Seattle, June 1990).

O'Neill, K and R.D. Miller, "Exploration of a Rigid Ice Model of Frost Heave" Water Resources

Research, 21, (1985) 281-296.

Patankar, S.V., "Numerical Heat Transfer and Fluid Flow", McGraw Hill, New York (1980).

Rappaz, M. and D.M. Stefanesq, "Modelling of Microstructural Evolution", Computer Applications

in Metals Casting, Metals Handbook, 15 (1988), 883-891.

Rappaz, M., "Modelling of Microstructure Formation in Solidification Processes," Int. Mat. Review,

34 (1989), 93-123).

Swaminathan, C.R. and V.R. Voller, "A general enthalpy method for modeling solidification

processes" submitted to Metall. Trans. B, (1991).

Takagi, S., "The Adsorption Force Theory of Frost Heaving", Cold Regions Science and Technology,

3 (1980), 57-81.

Voller, V.R., and C. Prakash, "A fixed grid numerical modelling methodology for convection-diffusion

mushy region phase change", Int. J. Heat and Mass Transfer, 30, (1987), 1709-1719.

68

Voller, V.R., A.D.Brent and C. Prakash, "The Modelling of Heat, Mass and Solute Transport in

Solidification Systems", Int. J. Heat Mass Transf., 32 (1989), 1719-1731.

Voller, V.R., "Fast Implicit Finite Difference Method for the Analysis of Phase Change Problems",

Numerical Heat Transfer B. 17B, (1990), 155-169.

Voller, V.R. and C.R. Swaminathan, "A general source based method for solidification phase

change", Numerical Heat Transfer B. 19B, (1990) 175-190.

Voller, V.R., C.R. Swaminatnan and B.G. Thomas, "Fixed grid techniques for phase change

problems: A review," Int. J. Numer. Methods Eng. 30 (1990), 875-898.

Zienkiewicz, O.C., "The Finite Element Method", McGraw Hill (1977).

69

TE 210 .M525 1991Michalowski, Radoslaw L.Modeling of frost heave insoils