CRREL REPORT 95-9

Initial Results From Small-Scale

Frost Heave Experiments in

a Centrifuge

Stephen A. Ketcham and Patrick B. Black May 1995
Abstract

Frost heave modeling is presented as a problem of small-scale experimental modeling. Scale factors applicable to frost heave model testing are reviewed, and initial data from frost heave experiments conducted as centrifuge model tests are presented. Ongoing improvements, modifications and

future model tests are discussed.

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

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

of Units, published by the American Society for Testing and Materials,

1916 Race St., Philadelphia, Pa. 19103.
CRREL Report 95-9

Initial Results From Small-Scale

Frost Heave Experiments in

a Centrifuge

Stephen A. Ketcham and Patrick B. Black May 1995

Prepared for

OFFICE OF THE CHIEF OF ENGINEERS

Approved for public release; distribution is unlimited.

US Army Corps


of Engineers

Cold Regions Research &


Engineering Laboratory
ii

PREFACE

This report was prepared by Stephen A. Ketcham, Research Civil Engineer, Civil and

Geotechnical Engineering Research Division, and Patrick B. Black, Research Physical Chemist, Applied Research Division, Research and Engineering Directorate, U.S. Army Cold

Regions Research and Engineering Laboratory, Hanover, New Hampshire.

Funding for this work was provided by the CRREL In-House Laboratory Independent

Research program and by the Office of the Chief of Engineers under DA Project

4A762784AT42, Design, Construction and Operations Technology for Cold Regions, Task CA,

Work Unit D02, Investigation and Design of Foundations in Cold Regions.

The authors thank Edwin Chamberlain and Sally Shoop for their technical reviews of

this report. They also thank James Morse, who provided valuable suggestions for the heater

design of the frost heave cell.

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

Citation of brand names does not constitute an official endorsement or approval of the use

of such commercial products.
iii

CONTENTS

Page

Preface ................................................................................................................................ ii

Introduction ........................................................................................................................ 1

Centrifuge modeling background ................................................................................... 1

Frost heaving modeling .................................................................................................... 3

Experimental technique .................................................................................................... 3

Results ................................................................................................................................ 5

Conclusions and future experiments .............................................................................. 7

Literature cited ................................................................................................................... 8

Appendix A: Some scale factors for conventional centrifuge modeling ................... 11

Appendix B: Scaling the Rigidice model ........................................................................ 15

Abstract ............................................................................................................................... 19

ILLUSTRATIONS

Figure

1. Frost heave scenarios ............................................................................................... 1

2. Sample containers and sample preloading equipment ...................................... 4

3. Sample containers in the centrifuge ...................................................................... 4

4. Sample used in the 8-g experiment ....................................................................... 5

5. Sample used in 32-g experiment ............................................................................ 6

6. Design of a simple frost heave cell for the small centrifuge.............................. 7

TABLES

Table

1. Scale factors for geotechnical centifuge modeling .............................................. 2

2. Centrifuge test conditions along with measured model and

calculated prototype results for New Hampshire silt ................................... 6
INTRODUCTION

Frost heave is the displacement of soil

caused by the formation of segregated

ice lenses in a freezing soil body. Although the process is complicated, a

number of researchers have developed

models to describe and analyze the mechanics of frost heave [see Kay and Perfect (1988) and Nakano (1990) for recent

reviews]. One of these is Miller’s (1978,

1980a) “Rigidice model.”

In the Rigidice model the soil body

is divided into three regions. The first region comprises already-frozen soil that might contain relic

ice lenses but definitely contains a growing ice

lens. The second region, which contains a growing ice lens at one end and unfrozen soil at the

other, is called the frozen fringe (Miller 1978,

1980b). The third comprises unfrozen soil and is

the source of mass and energy for the heaving

process. Miller (1978, 1980b) calls this three-region frost heave scenario “secondary heave.” In

a variation of secondary heave, which Miller designates “primary heave,” there is no frozen fringe

but rather an abrupt boundary between the growing ice lens and the unfrozen soil. Figure 1 illustrates primary and secondary heaving conditions.

Miller presented a mathematical description

of the Rigidice model in 1978. He followed with

finite-element (O’Neill and Miller 1985) and finite-difference implementations (Black and Miller

1985), comparisons to experimental data (Black

and Miller 1985) and a scaling analysis of the governing equations of the model (Black 1985, Miller

1990). Of particular interest for this paper is

Miller’s scaling analysis. This analysis and the

distinct role of gravitational body force in the heaving process led Miller to conclude that scale modeling of frost heave would best be conducted as

a body force analog and that a small-scale frost

heave test conducted in a centrifuge would be

appropriate for this purpose.

An objective of our ongoing work is to experimentally confirm Miller’s conclusion and the scale

factors inherent in his scaling analysis. This report presents a brief overview of frost heave centrifuge modeling from two perspectives: conventional geotechnical centrifuge modeling and scaling of the differential equations of the Rigidice

model. We describe two initial model experiments

in which silt specimens were partially frozen in

8-g and 32-g centrifugal fields. The experiments

were performed to provide a basis for equipment

development and further experiments. Plans for

the equipment and future tests are presented as

well.

CENTRIFUGE MODELING

BACKGROUND

Physical modeling is the testing of a small-scale

model so that the response of a prototype can be

inferred from the model response. A model is

a. Primary heave. b. Secondary heave.

Figure 1. Frost heave scenarios.

Initial Results From Small-Scale

Frost Heave Experiments in a Centrifuge

STEPHEN A. KETCHAM AND PATRICK B. BLACK

  























































































   


























































































said to be similar to the prototype when each significant engineering variable of the model is related by a proportionality or scale factor to the

corresponding variable of the prototype. Scale

factors are governed by the physics and model

laws of the problem. They are used to design the

model and to interpret the measured model response as the prototype response.

Physical models of civil engineering structures

are often designed so that the model will be geometrically similar to the prototype, so that significant forces in the model are proportional to

the forces in the prototype, and so that the model’s

response to loads will be kinematically similar

to that of the prototype. Geometric similarity

means that all parts of the model have the same

shapes as the corresponding parts of the prototype, and kinematic similarity means that the

motions of the model are similar to the motions

of the prototype at corresponding times. When all

net forces are proportional, dynamic similarity is

said to exist (Langhaar 1951). In general, the mechanical response of a structural model to forces

is observed and interpreted as the prototype response.

Centrifuge modeling is a physical modeling

technique in which the weight stresses of a structure are simulated by the placement of a smallscale model in a centrifugal field. This technique

is of proven benefit for modeling soil structures,

because the form and magnitude of the soil response are often greatly dependent on weightgenerated effective stresses and because the loading can be dominated by weight loads. As described in Appendix A, when the prototype soil

is used in the model, model laws of soil mechanics, together with requirements for geometric and

kinematic similarity, lead to scale factors for centrifuge modeling. Several are shown in Table 1 as

ratios of the model quantity to the prototype quantity for a model that is 1/N the size of the prototype. Following Croce et al. (1985), a number of

these are derived in Appendix A.

Many of the scale factors listed in Table 1 are

basic structural modeling scale factors and can

be derived from the mechanical considerations

(Langhaar 1951). The scale factor for acceleration,

which is N, can be derived from the consideration that weight forces should be scaled as other

forces, and it implies the need for an increased

gravity field (Croce et al. 1985). The centrifuge is

used to approximate this condition by subjecting a model to the constant angular velocity Ω

according to

 

Ω = Ng

r

(1)

where g = acceleration due to Earth’s gravity

r = representative radius of the model

from the axis of rotation

rΩ2 = radial acceleration of the model at this

radius.

In this context, N can be considered as a nominal

gravity level:

 

N r

g = Ω2

(2)

and the model can be considered to be subjected

to an inertial field equivalent to N gravities, or N

g. As described by Langhaar (1951), if self-weight

stresses have a negligible influence on the prototype response, or if partial dynamic similarity is

acceptable to the modeler, the principles discussed

here for centrifuge modeling and the scaling relations of Table 1 can be applied to models tested

at 1 g. However, because self-weight stresses typically cannot be neglected for geotechnical prototypes, centrifuge modeling has become a conventional technique.

As indicated in Table 1, there are different time

scales for the forces of viscous, inertial and seepage phenomena. As a result, time scale conflicts

can occur for certain modeling problems, and dynamic similarity can be impossible to achieve.

Thus the experimenter must consider the limitations imposed by the model laws and scale factors when designing a model experiment. Appendix A presents derivations of these time scales.

Table 1. Scale factors for geotechnical centrifuge modeling. (After Scott and Morgan 1977, Croce et al. 1985,

Savidou 1988.)

Quantity Prototype Model

Linear dimension and displacement 1 1/N

Area 1 1/N2

Volume 1 1/N3

Mass density 1 1

Mass 1 1/N3

Acceleration 1 N

Stress 1 1

Force 1 1/N2

Strain 1 1

Energy density 1 1

Energy 1 1/N3

Temperature 1 1

Time (for viscous force similarity) 1 1

Time (for inertial force similarity) 1 1/N

Time (for seepage force similarity) 1 1/N2

2
The use of prototype soils in centrifuge models is an accepted practice and is an attempt to

achieve structural response similarity of the model

and prototype by ensuring that the constitutive

response of the model will be as close as possible

to the prototype material response. Langhaar

(1951) and Schmidt and Holsapple (1980) have

suggested that when a model is made of the material of the prototype, when there are no time

scale conflicts, and when the material response

is independent of model size, prototypes with

general constitutive behavior such as nonlinear

elasticity, plasticity and fracture can be modeled

by following the scaling relations of Table 1. The

use of prototype soils within typical scale models (e.g., 1/25 to 1/100 scale) does not generally

result in scale effects unless soils with particles

larger than sands are tested. It is common in centrifuge testing to ensure that the parts of models

are much larger than the soil particle sizes so that

particle size does not result in adverse scale effects. It is also common to test different scale

models of the same prototype to ensure that scale

effects are insignificant.

To correctly model heat transfer processes in

saturated ground, the model and prototype must

be thermally similar, in addition to being geometrically, kinematically and dynamically similar. Thermal similarity includes the similarity of internal

energy and energy density, as evidenced by similar

temperature changes and differences at homologous points. For centrifuge modeling, Savidou

(1988) has shown that the scale factor for temperature change is 1 and that the time scale factor for conductive and convective heat transfer

through a soil body is 1/N2, which is the same as

the time scale factor for seepage processes.

FROST HEAVE MODELING

When the differential equations of a particular mechanical problem or physical process are

known to the modeler, a scaling analysis can be

performed to determine techniques for appropriate small-scale modeling of the problem. Scaling

is a method of dissolving passive physical parameters into active variables in a differential

equation (Miller 1980a). Any solution to the scaled

equation is also a solution to any other distinct

physical system that can obtain the same scaled

variables and scaled boundary conditions from

its distinct physical parameters. The earliest scaling

analysis of soils containing water and its vapor

was conducted by Miller and Miller (1956). They

presented a set of scaled relationships describing the isothermal flow of liquids in the unsaturated soil, and they concluded from length, stress

and time scale factors that a centrifuge experiment would be appropriate for small-scale modeling of this flow (Miller and Miller 1955). Miller’s

1990 scaling analysis of the Rigidice equations

was an extension of these earlier investigations

of scaling laws for unsaturated porous flow. An

expansion of this analysis is presented in Appendix B.

Like the previous scaling investigations,

Miller’s Rigidice analysis was based on scaling

the effects of surface tension and viscous flow,

but it also incorporated heat transfer, phase change

of the pore water, and ice lens initiation and

growth. The analysis yielded reduced equations

that are consistent with the techniques of conventional centrifuge modeling and the scale factors in Table 1. It provided a complete theoretical basis for small-scale frost heave modeling in

a geotechnical centrifuge. Miller (1990) emphasized that the ratio of model time to prototype

time for the freezing process in a centrifuge model

would be 1/N2, and he cited Pokrovsky and

Fyodorov (1969) as the first to suggest the use of

a geotechnical centrifuge for modeling frozen and

freezing ground structures. On the basis of Miller’s

earlier unpublished discussions on model laws

for soil freezing, Black (1985) presented a set of

scaling relationships for frost heave centrifuge

modeling, recommended a simple model study

of frost penetration and heave, and demonstrated

that a great time savings would be obtained over

a full-scale experiment.

EXPERIMENTAL TECHNIQUE

Our preliminary soil freezing experiments were

conducted on an International Equipment Company Model PR-2 centrifuge. The device is a small

refrigerated centrifuge manufactured for applications in biology, bacteriology, physiology and

biochemistry. It has a nominal operating radius

of 0.2 m for the rotor and cups used in our test

configuration. During the experiments the centrifuge was located in a coldroom with an ambient

temperature of approximately 4.5°C.

Two insulated sample containers were made

to allow simultaneous testing of saturated soil

samples. In each container a Plexiglas cylinder

enclosed the soil sample. The cylinder was mount3
der of the assembled sample container. The nominal sample dimensions were 10 mm in height and

34 mm in diameter. We prepared each sample at

room temperature by mixing 12.5 g of the New

Hampshire silt with water to form a slurry with

an 80% water content. The slurry was poured into

the Plexiglas cylinder, preloaded and consolidated

with a vertical stress of 14.1 kPa. Free water expelled at the sample surface was removed. The

dry mass density of the consolidated samples was

approximately 2040 kg/m3. The sample containers and sample preloading equipment are shown

in Figure 2.

Following the sample preparation, the insulated sample containers were taken to the

coldroom, placed in the centrifuge cups, and

immediately tested. Figure 3 shows the sample

containers in the centrifuge. The radius from

the centrifuge axis to the sample bottom was

0.196 m. To achieve nominal gravity levels of

8 and 32 g at desired reference radii in the tests

(Cooke 1991), the centrifuge was operated at

rotational velocities close to 193 and 387 rpm,

respectively. The effect of Earth’s gravity on the

equilibrium orientation of the centrifuge cups was

considered in our calculation of these velocities.

The experiments were conducted with minimal control of the temperature of the sample

and with minimal measurement of the sample

response. Our hope for these preliminary experiments was simply that we could freeze the

samples from the top down. We operated the

centrifuge with the centrifuge chamber closed

to the coldroom atmosphere, and we maintained the chamber temperature close to our

desired temperature of –2°C by manually turning on and off the refrigeration system of the

ed on a porous plastic base to allow water flow to

the sample from an exterior annular reservoir. Except for the upper end of the Plexiglas cylinder,

the cylinder and reservoir were surrounded by

insulation. The upper end of the cylinder was left

open to the air. The insulated sample containers

were made to fit into cylindrical cups that were

supplied with the centrifuge. These cups have

inside dimensions of 98 mm in diameter and 120

mm in height.

The test samples comprised a saturated New

Hampshire silt compacted in the Plexiglas cylinFigure 3. Sample containers in the centrifuge.

Figure 2. Sample containers and sample preloading equipment.

4
Figure 4. Sample used in the 8-g experiment.

centrifuge. Manual operation was necessary because of the crude response of the centrifuge’s

refrigeration control system. The duration of the

experiments was approximately five hours. A thermocouple located at the top of the closed centrifuge chamber provided the only temperature

measurement for the tests. There was no measurement of the temperature of the sample. In addition there was no measurement of the sample

heave as a function of time.

RESULTS

The experimental data consist of measurements

of the temperature in the centrifuge chamber, measurements of the height of the samples at the beginning and end of the tests, and observations

and photographs of the samples following the

tests. Table 2 presents the measurements for the

8-g and 32-g tests along with the corresponding

1-g prototype values calculated using either the

Table 1 scale factors or the scaling process described in Appendix B. The temperature records

reveal that the chamber temperature varied by

±0.25°C from the desired temperature of –2°C.

Figures 4 and 5 show the 8-g and 32-g samples,

respectively, before and after their extrusion from

the cylinder and after crudely cutting them to

examine the interior ice lens configuration. The

air temperature surrounding the samples during

cutting was 4.5°C. As a result the figures show

the samples in progressive stages of melting.

In the five-hour test period, only the upper 5–

8 mm of the 8-g and 32-g samples froze. This was

a desired result, as it confirms that the freezing

occurred from the top down. This was important

since we had no other means of monitoring the

progress of the freezing front. That the 8-g and

32-g samples would freeze to about the same depth

a. Prior to extrusion from the cylinder.

c. After cutting to examine the interior ice lens

configuration.

b. After extrusion from the cylinder.

5
Table 2. Centrifuge test conditions along with measured model and calculated

prototype results for New Hampshire silt.

1/8-scale experiment 1/32-scale experiment

Quantity Model Prototype Model Prototype

Acceleration (g) 8 1 32 1

Dry bulk density (kg/m3) 2040 2040 2040 2040

Sample height (mm) 10 80* 10 320*

Sample diameter (mm) 34 272* 34 1088*

Upper surface temperature (°C) –2 –2† –2 –2†

Duration of test (hr) 5 320** 5 5120**

Frost penetration (mm) 5–8 40–64* 5–8 160–256*

Heave (mm) 1–2 8–16* 3 96*

*Table 1 or eq B18

†Table 1 or eq B10

**Table 1 or eq B24 with B18

a. Prior to extrusion from the cylinder. b. After extrusion from the cylinder.

c. After cutting to examine the interior ice lens configuration.

Figure 5. Sample used in 32-g experiment.

6
Figure 6. Design of a simple frost heave cell for the small centrifuge. The

diagram show the cross section of the inside of the centrifuge chamber during

operation; the centrifuge rotor arm is not shown.

was also an expected result, since an inertial field

should not directly affect the heat transfer. From

the sample height measurements we estimated

the average heave of each of the two 8-g samples

to be approximately 1–2 mm and the average

heave of each of the two 32-g samples to be approximately 3 mm.

The 8-g test samples froze under a stress field

that is approximately similar to the stress field of

an 8-cm column of soil at 1 g, and the 32-g test samples

froze under a stress field approximately similar to

a 32-cm column of soil at 1 g. The differences in the

configurations of the interior lenses of the 8-g samples

and the 32-g samples were the most interesting result of the experiments. In the 8-g samples the ice

lenses did not form in distinct horizontal layers.

The ice clearly formed thin (< 0.5 mm) lenses that

separated the soil particles, but the lenses appeared

randomly oriented. In the 32-g samples, however,

a horizontal bias to the lens orientation was evident. These observations are consistent with an

explanation that might be used for similar observations in the upper 4 cm and the upper 16 cm of a

silt column in 1-g freezing tests: that a randomly

oriented lens configuration can develop in shallow soils that do not have a significant vertical

overburden stress, and a horizontally oriented lens

pattern can develop in deeper soils that have a

greater overburden stress.

The results of the experiments are crude, yet

they provide two pieces of information for the

design of equipment for further experiments. First,

the indication that the freezing occurred from the

top down in the samples suggests that we can

proceed with a one-dimensional cylindrical sample configuration in our small centrifuge tests,

and second, the apparent consistency of the lens

configurations of the 8-g and 32-g samples with

corresponding prototype behavior suggests that

with control of the experiments the applicability

of the centrifuge for frost heave modeling can be

investigated.

CONCLUSIONS AND

FUTURE EXPERIMENTS

A design of a simple frost heave cell for the

small centrifuge is illustrated schematically in

Figure 6. The cell will be instrumented with lower

and upper heaters and with temperature and displacement measurement sensors. The lower heater

will be a thin heating pad mounted onto the bottom of the centrifuge cup, and the upper heater

will be a cartridge heater mounted in a radial-fin

heat sink. A fan will be installed above the heater

to circulate air past the heat sink fins to the upper sample surface. The upper heater will not be

7
in contact with the sample and thus will not load

the sample. Both heaters will be controlled using

feedback controllers. An electrical slip ring assembly will be installed on the centrifuge axis to

allow power and instrumentation signals to be

passed to and from the cell while the centrifuge

is operating. The centrifuge will be operated in a

coldroom at a temperature below freezing (e.g.,

–5°C), and the heaters will be controlled to maintain upper and lower sample surfaces at desired

temperatures. Thin sectioning and image analyzing techniques will be used to examine the lens

ice distribution, orientation and thickness.

The modeling at different scales or acceleration fields will be used to examine the validity of

our frost heave modeling. Specifically a series of

different scale models of the same prototype will

be tested, and the model displacement vs. time

records will be interpreted to see if each of the

prototype heave vs. time predictions are essentially the same. If the one-dimensional heave process can be modeled, then other tests will be conducted regarding heave forces and the interaction

of heaving soil with foundation and other structural members.

Additional analysis will compare experimental measurements to predictions using a recent

version of the Rigidice model (Black 1994). While

the exact location and thickness of individual ice

lenses in the model and prototype will most likely

not be predictable, the total amount of heave, penetration and fluxes should. This will be the topic

of future discussions.

LITERATURE CITED

Black, P.B. (1985) The RIGIDICE model of frost

heave and its input functions. Ph.D. Thesis, Cornell

University.

Black, P.B. (1992) Soil water: liquid, vapor and

ice. Encyclopedia of Earth System Science, 4: 259–

269.

Black, P.B. (1994) Heave 4.0, A finite difference

implementation of the RIGIDICE model. Unpublished.

Black, P.B. and Miller, R.D. (1985) A continuum

approach to modeling frost heaving. Freezing and

Thawing of Soil Water Systems (D.M. Anderson and

P.J. Williams, Ed.). American Society of Civil Engineers, Technical Council on Cold Regions Engineering, p. 36–45.

Cooke, B. (1991) Selection of operative centrifuge

radius to minimize stress error in calculations.

Canadian Geotechnical Journal, 28: 160–161.

Croce, P., V. Pane, D. Znidarcic, H.-Y. Ko, H.W.

Olsen and R.L. Schiffman (1985) Evaluation of

consolidation theories by centrifuge modelling.

Application of Centrifuge Modelling to Geotechnical

Design, p. 381–401 (W.H. Craig, Ed.). Rotterdam:

Balkema.

Kay, B.D. and E. Perfect (1988) State of the art:

Heat and mass transfer in freezing soils. Ground

Freezing 88, Proceedings of the Fifth International

Symposium on Ground Freezing (R.H. Jones and J.T.

Holden, Ed.). Rotterdam: Balkema.

Langhaar, H.L. (1951) Dimensional Analysis and

Theory of Models. New York: John Wiley and Son.

Loch, J.P.G. (1978) Thermodynamic equilibrium

between ice and water in porous media. Soil Science, 126(2): 77–80.

Miller, R.D. (1978) Frost heaving in non-colloidal soils. Proceedings of the Third International Conference on Permafrost, vol. 1, p. 708–713.

Miller, E.E. (1980a) Similitude and scaling of soilwater phenomena. In Chapter 12, Applications of

Soil Physics (D. Hillel, Ed.). New York: Academic

Press.

Miller, R.D. (1980b) Freezing phenomena in soils.

In Chapter 11, Applications of Soil Physics (D. Hillel,

Ed.). New York: Academic Press.

Miller, R.D. (1990) Scaling of freezing phenomena in soils. In Chapter 1, Scaling in Soil Physics:

Principles and Applications (D. Hillel and D.E. Elrick,

Ed.). Madison, Wisconsin: Soil Science Society of

America.

Miller, E.E. and R.D. Miller (1955) Theory of capillary flow: I. Practical implications. Soil Science

Society of America Proceedings, 19: 267–271.

Miller, E.E. and R.D. Miller (1956) Physical theory

for capillary flow phenomena. Journal of Applied

Physics, 27: 324–332.

Mitchell, J.K. (1976) Fundamentals of Soil Behavior. New York: John Wiley and Son.

Nakano, Y. (1990) Quasi-steady problems in freezing soils: 1. Analysis on the steady growth of an

ice layer. Cold Regions Science and Technology, 17:

207–226.

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

of a rigid ice model of frost heave. Water Resources

Research, 21(3): 281–296.

Pokrovsky, G.I. and I.S. Fyodorov (1969) Centrifuge Model Testing in the Construction Industry. English translation from Russian by Building Research Establishment Library Translation Service.

Savidou, C. (1988) Centrifuge modelling of heat

8
transfer in soil. Centrifuge 88 (J.-F. Corté, Ed.), p.

583–591. Rotterdam: Balkema.

Schmidt, R.M. and K.A. Holsapple (1980). Theory

and experiments on centrifuge cratering. Journal

of Geophysical Research, 85(B1): 235–252.

Scott, R.F. and N.R. Morgan (1977) Feasibility and

desirability of constructing a very large centrifuge for geotechnical studies. Report to the National Science Foundation. Grant No. ENG 76-

18871. California Institute of Technology.

Snyder, V.A. and R.D. Miller (1985) Tensile

strength of unsaturated soils. Soil Science Society

of America Journal, 49: 58–65.

9
APPENDIX A: SOME SCALE FACTORS FOR

CONVENTIONAL CENTRIFUGE MODELING

PHYSICAL MODELING CONCEPTS

Following the presentation of Langhaar (1951),

consider a physical variable of a prototype fp(xp,

yp, zp, tp) and the corresponding model quantity

fm(xm, ym, zm, tm), where x, y and z are coordinate measures of a point within the structure, t is

time, and the subscripts p and m refer to the prototype and model, respectively. The function fm

is said to be similar to the function fp if the ratio

fm/fp is constant for homologous points and homologous times. The constant ratio f = fm/fp is

called the scale factor for the function f. Scale factors can be derived from model laws, i.e. ratios

of physical laws for the model and prototype.

When similarity is achieved, a model’s response

is interpreted using the scale factors to infer the

response of the corresponding prototype.

The ideal for physical modeling is complete

similarity, which includes geometric, kinematic,

thermal and dynamic similarity. Complete similarity is usually not achieved for a model test.

Scale factor conflicts and scale effects, i.e., disturbing influences that are associated with the

small scale of the model, can limit similarity to a

partial similarity.

When a model is made of the material of the

prototype, and when there are no scale factor conflicts or disturbing scale effects, then the model

material will have the same constitutive behavior as the prototype material, and prototypes with

general constitutive behavior such as nonlinear

elasticity and plasticity can be modeled.

CROCE’S DERIVATION OF

SCALE FACTORS FOR CONVENTIONAL

GEOTECHNICAL MODELING USING

SCALAR MODEL LAWS

The objective of conventional small-scale

geotechnical modeling is to achieve a model response that is similar to the mechanical behavior

of the prototype, and the approach is to use the

prototype soil in a geometrically similar model.

Thus it is important to ensure that the effective

stresses in the model are the same as in the prototype at homologous points and times. The scale

factors for these conditions can be derived using

scalar model laws. Closely following the analysis of Croce et al. (1985), this approach is presented

here. Two assumptions are made: that soil can

be treated as a continuum and that soil properties are not affected by a change in acceleration.

Dynamic similarity requires that all forces (and

all kinds of forces) have the same scale factor. The

forces of interest include the weight force Fw, the

external force Fe, the viscous force Fv, the inertia

force Fi and the seepage force Fs.

Scale factors for length l, mass m and time t

are

 

K l

l l m

p

= , (A1)

 

K m

m m m

p

= , (A2)

and

 

K t

t t m

p

= . (A3)

Thus the scale factors for area A and volume V

are

 

K K A

A A l m

p

= = 2 (A4)

and

 

K K V

V V l m

p

= = 3 . (A5)

Since the same material is used in the model and

prototype at the same mass density, the scale factor

for mass is equal to the scale factor for volume:

 

K K m

m m l m

p

= = 3 . (A6)

For identical effective stresses in the model and

prototype,

 

K KK u

u ′ === ′

′ σ σ === σ

σ

σ

σ u m

p

m

p

m

p

1 (A7)

where σ′, σ and u are, respectively, the effective

stress, the total stress and the excess pore water

pressure, including the capillary pressure caused

by surface tension. Considering eq A4 and A7 and

the definition of stress, it follows that the scale

factor for all forces F should be

11
 

K K F

F F l

2 m

p

= = . (A8)

From the definition of weight force, the weight

force model law and scale factor KFw is

 

K

F

F

m a

m a

F K w

w

m g

p g w l m

p

m

p

== = 2 (A9)

where ag is the acceleration due to gravity. Considering eq A6, A9 and the relation 

K KK Fw F l = = 2

from eq A8,

 

a

a K

g

g l

m

p

= 1 . (A10)

Equation A10 implies that geotechnical modeling requires a technique that elevates the gravitational acceleration of the model.

The scale factor for an external force KFe must

satisfy eq A8, and only technical constraints should

limit this.

A viscous force acting on a small area A can be

defined as

 

F v

n v s A d

d = μ (A11)

where μs is the viscosity of the soil skeleton and

dv/dn is the velocity gradient. The model law

and scale factor for the viscous force KFv can then

be written

 

K

F

F

v

v

n

n

A

A F

v

v

s

s

m

p

p

m

m

p v

m

p

m

p

d

d

d

d = = μμ . (A12)

From the definition of velocity it follows that

 

K

K

v

v

l

t

m

p

= . (A13)

From eq A1, A4 and A13 and the fact that viscosity is independent of gravity, KFv

 can be written

 

K K

K F

l

2

t v = . (A14)

Because KFv must equal K K F l

2 = , then

Kt = 1 . (A15)

That is, the time scale factor is 1 if the viscous

force is to be scaled like the weight force.

An inertia force can be written as

Fi = ma (A16)

where a is the acceleration. The model law and

scale factor KFi for the inertia force is

 

K

F

F

m

m

a

a

F

i

i

m

p

m

p i

m

p

= = . (A17)

From the definition of acceleration it follows that

 

K

K

a

a

l

t

2

m

p

= . (A18)

From eq A6 and A18, eq A17 becomes

 

K K

K F

l

4

t i 2 = . (A19)

Because KFi must equal K K F l

2 = , then

Kt = Kl . (A20)

That is, the time scale factor is equal to the length

scale factor if the inertia force is to be scaled like

the weight force.

A seepage force can be written as

Fs = iWF (A21)

where WF is the weight of the fluid phase and i is

the hydraulic gradient defined by

 

i v

k = (A22)

in which v is the superficial velocity of the fluid

and k is Darcy’s coefficient of permeability. From

eq A21 and A22 the scale factor KFs for the seepage force can be written as

 

K

F

F

v

v

k

k

W

W F

s

s

m

p

p

m

F

F s

m

p

m

p

= = . (A23)

Equation A8 implies that

 

W

W

K K F

F

F l m 2

p

= = . (A24)

Then, from eq A13 and A24, eq A23 becomes

 

K K

K

k

k F

l

3

t

p

m s = . (A25)

A general expression for the coefficient of permeability is (Mitchell 1976)

12
scale factor squared if the seepage force is to be

scaled like the weight force.

CENTRIFUGE TECHNIQUE

The centrifuge is used to approximate the condition of eq A10 by subjecting a model to the constant angular velocity Ω such that the following

approximate relation holds:

 

a

a K

r

a

g

g l g

m

p p

≈ = 1 Ω2

(A30)

where agp is the acceleration due to Earth’s gravity, r is a representative radius of the model from

the axis of rotation (Cooke 1991), and rΩ2 is the

radial centripetal acceleration of the model at this

radius. In the centrifuge modeling literature, N

or n is often used as the symbol for the inverse of

the length scale factor, i.e., 1/Kl.

 

k

a

= κ

ρ

μ

w g

w

(A26)

where ρw and μw are, respectively, the density

and the viscosity of the fluid phase, and κ is defined as the absolute permeability, which depends

only on the geometry of the soil skeleton. Because

ρw, μw and κ do not depend on gravity, it follows

that

 

k

k

a

a K

m

p

g

g l

m

p

= = 1 . (A27)

Thus eq A25 becomes

 

K K

K F

l

4

t s = . (A28)

Because KFs must equal K K F l

2 = , then

 

K K t l

2 = . (A29)

That is, the time scale factor is equal to the length

13
The Rigidice model of frost heave is a set of

differential equations expressing mass, energy and

force balances in the frozen fringe for air-free, colloid-free and solute-free soil (Miller 1978). A set

of reduced or scaled variables for this model were

later presented by Miller (1990). To avoid redundancy, this later work did not contain derivations

but referred to the earlier classic works for icefree unsaturated soil (Miller and Miller 1955, Miller

1980a). For completeness of this paper, this appendix presents a brief derivation of the reduced

variables presented by Miller for the liquid water and solid ice phases. It also serves as an example of how the seemingly subjective process of

scaling equations is performed. All scaled variables will be given upper-case letters to distinguish them from the nonscaled dimensional variables that are in lower case.

APPENDIX B: SCALING THE RIGIDICE MODEL

 

ΓIW

IA

IW

1 ≡ γ γ

and (B3)

 

ΓAW

IA

AW

1 ≡ γ γ .

To simplify, the remainder of this appendix will

address ice–water systems. A similar set of equations for water–air and ice–air systems are obtained by direct analogy to the ice–water equation and are listed by Miller (1990).

The distinguishing trait exhibited by water and

ice in a porous medium that does not exist in unrestricted bulk is the existence of curved interfaces. The force balance across a curved interface

separating two phases in equilibrium is given by

the Laplace surface tension equation, which for

an ice–water interface is

 

p p r I

IW

IW

W − = 2 γ (B4)

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

mean radius of curvature is r. In the case of frozen soil, this equation expresses the behavior of

water and ice contained within the pores. This

suggests a length scale on the order of the pore

size, so we introduce a microscopic length λ. Equation B4 contains three variables, pI, pW and r, and

one constant, γIW. Equation B3 shows how to reduce the surface tension. At this point we need

to decide how to reduce the remaining variables.

The simplest approach is to acknowledge that the

radius of curvature is a pore length dimension,

so it is reduced by dividing by λ. The reduced

curvature is therefore

 

R r IW IW ≡ 1

λ . (B5)

Substituting B3 and B5 into the right hand side

of B4 determines the reduced pressure by dimensional equivalence:

 

P p ≡ λ

γ IA

(B6)

giving the reduced form of the Laplace surface

tension equation:

 

P P

R I W

IW

IW

− = 2

Γ . (B7)

At thermodynamic equilibrium the chemical

potentials of ice and water are equal and are exWe start with Young’s equation relating the

contact angle α and the surface tensions for all

three phases of water:

 

γγγ α IA IW WA = + cos (B1)

in which γ is the surface tension and the subscripts

I, W and A are for ice, water and air, respectively.

Figure B1 is a schematic of the equilibrium configuration and force diagram about the point of

mutual contact for all three interfaces. Since cos

α is already dimensionless, eq B1 can only be made

dimensionless by dividing through by one of the

surface tensions. A reasonable choice is γIA with

the largest magnitude, which results in the reduced Young’s equation

 

1 = + Γ Γ IW WA cosα (B2)

where the reduced surface tensions are

Air

Water

Ice

α

γWA

γIW γIA

Figure B1. Equilibrium

configuration.

14
pressed by the differential form of the Gibb’s relationship. It has been shown (Loch 1978, Black

1992) that the Gibb’s relationship for monocomponent and multicomponent systems contained

within a porous medium that subjects the water

to a general body force can be expressed in a simplified Clapeyron equation. When pressures are

given in standard gauge pressures u, the Clapeyron equation is written as

 

u

u

Y

h

W

I

IW

IW

o

− = θθ (B8)

in which hIW is the volumetric latent heat of fusion (J/m3),YIW is the specific gravity of ice and

θ (°C) is the elevation of the melting point of ice

from the standard melting point of ice θo. For

normal water, θ will be negative.

The gauge pressures will scale the same as the

absolute pressures:

 

U u ≡ λ

γ IA

. (B9)

The specific gravity of ice is already dimensionless, so the remaining constants θo and hIW must

be used to reduced the only variable: temperature θ. Earlier scaling attempts by Miller (Black

1985) developed a reduced temperature that included the latent heat term, but for the benefit of

clarity, Miller (1990) suggested a different reduced

temperature:

 

Θ ≡ 1

θ

θ

o

 . (B10)

Substituting these scaling parameters for pressure and temperature into eq B8 results in an additional reduced term called the reduced latent

heat:

 

H h IW

IA

≡ IW

λ

γ . (B11)

These scaling relationships give a reduced

Clapeyron equation:

 

U U

Y W H

IW

IW

I −=Θ. (B12)

The Laplace and Clapeyron equations show

that the pressures, temperature and curvature are

interrelated. When the porous medium is geometrically complicated, as in the case of soil, it is

helpful to simplify the pressures–curvature behavior by expressing the Laplace equation with

empirically determined material properties. In our

case the necessary material properties are the volume fractions for water W(uW, uI), ice I(uW, uI)

and grains G. These volume fractions are related

by

 

G Wu u Iu u + ( ) WI WI , ,1 + ( ) = . (B13)

The complete scaled form is simply

 

G WU U IU U + ( ) WI WI , ,1 + ( ) = . (B14)

The criterion for lens initiation is obtained from

the geotechnical concept of effective stress, or the

particle-to-particle contact stress. When this stress

is zero, the particles no longer remain in contact

and an ice lens is able to form. The Snyder–Terzaghi

equation partitions the total confining stress σT

(N/m2) between the effective σe (N/m2) and neutral or pore σn (N/m2) stresses at locations near the

initiation of an ice lens:

 

σ σσ T en = + . (B15)

Figure B2 shows the partitioning of these stresses.

The effective and neutral stresses in this relationship are different from the traditional stresses in

the Terzaghi equation because these include

Snyder’s flawed solid theory (Snyder and Miller

1985). These stresses, like the pressure, take place

at the pore scale, so they will scale just as pressures do:

 

∑ = λ

γ

σ

IA

. (B16)

The flow of water through the frozen fringe is

assumed to obey Darcy’s law:

 

v kuuf u W WW W W = ( ) , I ( ) − ∇ (B17)

in which vW is the volumetric flux of water

(m3/m2s), kW(uW, uI) is the capillary conductivity

(m4/Ns), fW is the body force per unit volume

(N/m3) and ∇ is the gradient. To reduce Darcy’s

law we begin by examining the last term. This is

the gradient of water pressure across the region

of interest. If we are looking at flow through an

individual pore, then this length would appear

to be λ. This would be correct for this special and

very limited case. In general, far more complex

flow patterns through a myriad of pore geometries containing different amounts of ice and

σn σe

σ T

Figure B2. Stress partitioning of ice, water and air.

15
water are of concern. At this scale we refer to the

porous medium behaving as a continuum displaying uniform material properties, and we are interested in the flow into and then out of a volume

of material. This suggests that a different length

scale is required to describe the gradient, since λ

is applicable only at the pore scale. The additional

macroscopic length scale is ζ. Using this new

length scale to reduce the gradient operator results in the reduced body force:

 

F f W

AW

≡ W

ζλ

γ . (B18)

The proportionality factor in Darcy’s equation

is the source of much confusion. In eq B17 it is

expressed as capillary conductivity kW. Soil scientists tend to use hydraulic conductivity K (m/

s), and geologists and petroleum engineers use

intrinsic permeability k (m2). These three parameters are related by

 

k K

g

k

W = = ρ η (B19)

in which ρ (kg/m3) is the density of water, g is

the acceleration due to gravity (or other body

force) and η (N/m2s) is the viscosity of water.

They are all functions of water and ice content,

which are functions of the water and ice pressures.

The process of scaling the capillary conductivity consists of first realizing that it is already

inversely proportional to viscosity (i.e., eq B19),

so it must therefore be scaled by multiplying by

the viscosity. It then becomes the intrinsic permeability, which is in dimensions of length squared.

To complete the scaling we must decide on the

correct length scale to use. Again relying on the

continuum approximation, the conductivity

should not depend on the size of the sample once

it attains the minimum size required for the continuum approximation. This requires the use of

the microscopic length scale λ to scale the conductivity. Miller and Miller (1955) deduced the

same length scale choice by considering the

Navier–Stokes equation applied to flow through

the pore. The reduced capillary conductivity is

then

 

K k W W ≡ η

λ2 . (B20)

Substituting eq B20, B19, B6 and the scaling factor ζ to reduce the gradient operator into eq B17

gives the scaled form of Darcy’s equation:

 

V K UU F U W WIW W W = ( ) , ( ) − ∇ (B21)

in which the reduced volumetric flux of water

VW is, by balancing constants,

 

V v W

IA

≡ W

ζη

λγ . (B22)

The continuity equation for pore constituents

is

 

∇• + ( ) v Yv W IW I

= − ∂

∂ [ ] ( ) + ( ) t

Wu u Y Iu u W I IW W I , , (B23)

in which vI (m3/m2s) is the volumetric flux of

ice and t is time. The operator ∇• is the divergence, which simply scales by multiplying by ζ.

The volumetric flux of ice must scale like the

volumetric flux of water to maintain dimensional

equality on the left side of eq B23. Since the volumetric water and ice contents are already dimensionless, the reduced time T is

 

T t ≡ λγ

ζ η

IA

2 . (B24)

The conservation of thermal energy is

 

∇• + ( ) = − ∂

∂

v hv [ ( ) t H IW W H W I Cuu, θ

θ + h Wu u IW W I ( ) , ] (B25)

in which vH (W/m2) is the macroscopic flux of sensible heat and CH(uW, ur) (J/m3 K) is the volumetric heat capacity excluding heat resulting from

phase change. The volumetric heat capacity is assumed to be a material property that is constant

in the continuum assumption for a given state of

ice and water contents. Starting with the left side

of eq B25 the reduced macroscopic flux of sensible heat VH is obtained

 

V v

IA

H H ≡ ζη

γ 2 . (B26)

The reduced volumetric heat capacity CH must

therefore be

 

CU U Cu u HWI

IA

HWI o ( ) , , ≡ ( ) λθ

γ . (B27)

The macroscopic flux of sensible heat is given

by Fourier’s law:

16
 

V kuu H HWI = − ( ) , ∇θ (B28)

in which kH (W/mK) is the thermal conductivity. This conductivity is also a material property

that is assumed to be constant in the continuum

sense for a given state of water and ice contents.

The reduced thermal conductivity is then

 

KUU kuu

IA

HWI o HWI ( ) , , ≡ ( ) ηθ

γ 2 . (B29)

The most important characteristic of the Rigidice

model that sets it apart from others is the concept

of thermally induced regelation. This results in the

inclusion of an extra mass flux resulting from the

bulk movement of ice in the frozen fringe. The velocity of ice movement in the fringe is the heave

velocity vR (m/s) relative to the stationary grains

of the porous media below the fringe. The volumetric flux of ice vI (m3/m2s) resulting from this

regelation is

 

v Iu u v I WIR = ( ) , . (B30)

From eq B22 we know how volumetric fluxes scale,

so the reduced regelation velocity is therefore

 

VR

IA

≡ ζη

λγ . (B31)

This completes the scaling of the more important variables and parameters involved in ground

freezing. A list of all scaled variables applicable

to ground freezing, as well as all water phases,

was presented by Miller (1990).

An example of how heaving experiments should

be interpreted by reduced variables seems in order. Given two experiments using the same soil

but subjected to different body forces, eq B18, B10

and B24 show how they must relate. Using the

convention of calling the laboratory reference

experiment the prototype p and the experiment

subjected to the different body force m for model,

the reduced body force FW must be equal for both:

 

ζ λ

γ

ζ λ

γ

p p

IA p m mIA

f f = m . (B32)

If we say that fm = 100 fp, as might be expected in

a centrifuge, and we use the same soil in both

experiments, then the total lengths of the two experiments are related by ζm = 0.01 ζp. Since the

reduced temperatures Θ must be equal at corresponding locations, the actual temperatures θ will

therefore be the same in both the prototype and

the model. This leads to the most important experimental consequence of centrifuge modeling.

Since the reduced times T must be equal, the actual times are related by tm = tp/1002. Scaled experiments are finished in a very timely manner

compared to similar laboratory experiments.

A final note is on the meaning of the microscopic length λ. If the porous medium consisted

of identical pores, then this length would just be

the pore size. But in real porous media there is a

distribution of sizes and geometries of pores. The

microscopic length in these cases represents some

averaged value of the pores. It is more like the

size of the smallest representative elementary

volume (REV) that begins to describe the porous

media as a continuum of uniform material properties. The difficulty is in actually measuring this

parameter without completely destroying the

original material.

17
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Initial Results From Small-Scale Frost Heave Experiments in a Centrifuge

Stephen A. Ketcham and Patrick B. Black

U.S. Army Cold Regions Research and Engineering Laboratory

72 Lyme Road CRREL Report 95-9

Hanover, New Hampshire 03755-1290

Office of the Chief of Engineers

Washington, D.C. 20314-1000

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23

Frost heave Modeling

UNCLASSIFIED UNCLASSIFIED UNCLASSIFIED UL

Frost heave modeling is presented as a problem of small-scale experimental modeling. Scale factors applicable to

frost heave model testing are reviewed, and initial data from frost heave experiments conducted as centrifuge

model tests are presented. Ongoing improvements, modifications and future model tests are discussed.

ILIR and

PE: 6.27.84A

PR: 4A762784AT42

TA: CA

WU: D02