Unpaginated      |      Vol_IIA-0154                                                                                                                  
EA-I. (Karl Terzaghi)

PHYSICS OF GROUND FROST

       

CONTENTS

	Page
Introduction	1
Soil Constituents	2
Soil Aggregate	3
Permeability and Degree of Saturation of Sandy and Silty Soils	8
Permeability and Degree of Saturation of Clay Strata	11
Thermal Properties of Soils	12
Geothermal Gradient	16
Relations between Surface and Ground Temperatures	18
Thickness and Continuity of Permafrost	23
Aggradation and Degradation of Permafrost	26
Ice Formation in Soils	31
Surface Movements Due to Freezing and Thawing	34
Strength of Thawed and Frozen Soils	36
Bibliography	40

        figures missing from copy 3

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copy 3.

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order in which they are clipped as this

follows correspondence instructions &

suggestions from published.

       

April 4, 1950

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EA-I. Terzaghi: Physics of Ground Frost

       

LIST OF FIGURES

		Page
Fig. 1	Diagram representing grain-size characteristics of soils	2-a
Fig. 2	Test arrangement for liquid limit determination	6-a
Fig. 3	Plasticity chart for classifying cohesive soils	7-a
Fig. 4	Plasticity characteristics of cohesive soils from different parts of Alaska. (Based on data furnished by St. Paul District, U.S. Army Engineers.)	7-b
Fig. 5	Degree of saturation of two originally saturated columns of sand, 2 1/2 years after drainage was started. (Tests by F. H. King, 1899.)	9-a
Fig. 6	Relation between effective grain size and degree of saturation of soils S r under field conditions. Boundaries of shaded area represent probabl y e upper ✓ and lower limits for S r	10-a
Fig. 7	Diagram representing the relation between porosity and thermal conductivity of frozen and unfrozen and unfrozen soils in a completely and partially saturated state. (Based on data furnished by the St. Paul District of the U.S. Army Engineers .) ✓	16-a
Fig. 8	Influence of thermal conductivity on increase of temperature with depth	18-a
Fig. 9	Influence of surface topography on geothermal gradient	18- a
Fig. 10	Diagram illustrating progress of change of ground temperatures due to sudden changes in surface temperature	21-a
Fig. 11	Diagrammatic presentation of diurnal changes of surface temperature	21-a
Fig. 12	Range of variations of ground temperature above the level of zero annual amplitude	25-a
Fig. 13	Diagram illustrating the influence of diffusivity on the maximum mean annual temperature at which permafrost can still persist	25-a

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EA-I. Terzaghi: Physics of Ground Frost

       

List of Figures -2-

		Page
Fig. 14	Diagram illustrating the cause of the degradation of permafrost beneath heated buildings	26-a
Fig. 15	Idealized diagram showing five stages in change of ground temperatures since the late Tertiary in the region located north of the present south boundary of the permafrost zone	26-a
Fig. 16	Distribution of ground temperatures in a region where two permafrost layers are separated by a layer of unfrozen ground	30-a
Fig. 17	Diagram illustrating formation of ice layers in (a) a closed and (b) an open system. (c) Shows method for transforming an open into a closed system by means of a layer of coarse sand which intercepts flow of capillary water toward zone of freezing	31-a

001      |      Vol_IIA-0157                                                                                                                  
EA-I. [Karl Terzaghi]

       

PHYSICS OF GROUND FROST

       

INTRODUCTION

        The term ground frost indicates the occurrence and the effects of freez–

ing temperatures below the ground surface. Even in arctic regions the top

layer of the ground is subject to alternate freezing and thawing. This top

layer is called the active layer . Below the active layer the ground tempera–

ture remains either above or below the freezing point throughout the year.

Those parts of the ground in which the temperature remains permanently below

the freezing point are known as “permanently frozen ground,” “ever-frozen

soil” , or, briefly, as “ permafrost ” ( q.v. ). This article deals with all those

physical properties of subsurface materials and physical processes which have

a direct or indirect bearing on the thickness of the active and the permafrost

zones and on the character of the ice segregation in those zones.

        The active layer and the permafrost zone may be located in rock, uncon–

solidated sediments or both. For a given surface-temperature regime the loca–

tion of the boundaries of the zone depends on the surface cover (bare ground,

grass cover, forest, etc.) and on the thermal properties of the subsurface

materials, rock or sediments. The thermal properties of rocks depend primarily

on the mineralogical composition of the rocks whereas those of unconsolidated

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EA-I. Terzaghi: Ground Frost

sediments depend also on grain size, porosity, degree of saturation, and

various other factors, to be described under the following subheadings. In

accordance with a widespread convention, unconsolidated sediments will be re–

ferred to briefly as soils . Information regarding the physical properties

of soils other than their thermal properties has been obtained chiefly by

research in soil mechanics (9).

       

Soil Constituents

        Significant properties of the soil constituents are their size, mineralogy–

cal composition, and shape.

        The grain - size characteristics of a soil are commonly represented by a —

Fig. 1 grain - size summation curve in a semilogarithmic grain - size diagram (Fig. 1). — —

The abscissas of the grain - size curve represent the logarithm of the gain —

size. The ordinates represent the percentage [ ?] P , by weight, of grains —

smaller than the size denoted by the abscissa. The more uniform the grain size

the steeper is the slope of the curve; a vertical line represents a perfectly

uniform powder.

        Experience has shown that the general character of a mixed-grained soil

is similar to that of a uniform one with a grain size D ₁₀ corresponding to

Fig. 1 P = 10 per cent in the grain - size diagram, Figure 1. The grain size D ₁₀ is — —

commonly referred to as effective size . The coarsest soil constituents which

may be encountered in sediments are designated as the sand and gravel frac–

tion , the finest ones as clay fraction, and the intermediate ones as silt fraction. Opinions regarding the boundaries of the grain-size range for these

fractions are not yet unanimous. The most satisfactory convention, known as

Fig. 1 the M.I.T. classification , is indicated in Figure 1, below the grain - size —

diagram.

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        If a mixed-grained soil is split up into grain - size fractions, it is —

always found that the mineralogical composition of the different fractions is

different. The sand fraction commonly consists of more or less equidimen–

sional grains of quartz, feldspar, calcite, and other rock-forming minerals.

The silt fraction commonly consists of a mixture of the same minerals and of

various micaceous minerals. The clay fraction is chiefly composed of a mixture

of micas and of clay minerals such as kaolinite, illite, or montmorillonite.

The equidimensional constituents of the sand fraction may be rounded, subangular,

or angular. Those of the silt fraction are commonly angular and the micas and

clay minerals occur in thin, flexible flanks or needles.

       

Soil Aggregate

        The term aggregate refers to the soil itself, in contrast to its constituent

parts. Soil aggregates consisting of identical constituents may differ in

porosity, relative density, water and air content, and consistency.

        The degree of porosity of a soil is expressed either by the porosity

n = (volume of voids)/(total volume of soil)

or by the void ratio

e = (volume of voids)/(volume of solid constituents)

        The water content w of a soil is commonly given in per cent of the dry

weight,

        w = 100 x (weight of water)/(weight of dried soil)

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If the soil contains both water and air, the degree of saturation is deter–

mined by the ratio

        S_r(%) = 100(volume occupied by water)/(total volume of voids)

        The mechanical properties of a soil depend on both the porosity and the

degree of saturation. For instance, the bearing capacity of a clean sand

increases rapidly with decreasing void ratio. Hence the description of a sand

is not complete unless it includes data concerning its relative density. If

e _o is the void ratio of the sand in its loosest state and e_min the correspond–

ing value for the densest state the relative density D_r of the sand at a void

ratio e is determined by the equation

D_r = (e_o – e)/(e_o – e_min)

The void ratio e _o of a clean sand may be as high as 0.8 and the void ratio

e_min as low as 0.4. Yet even a very dense clean sand has almost no cohesion

and the high bearing capacity of such a sand is exclusively due to inter–

locking and friction between grains.

        By contrast, all soils containing a high percentage of clay-size particles

Fig. 1 (see Fig. 1) possess cohesion. The cohesion is due to the great number of points

of contact between the soil particles per unit of volume of soil combined with

the physicochemical interaction between the water and the clay particles. If

the porosity of a clay is reduced its cohesion increases on account of the in–

crease of the number of points of intergranular contacts. If a lump of soft

clay dries out it becomes stiffer and stiffer and finally so hard that it can–

not be broken with the fingers, whereas a wet, clean, coarse sand remains

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practically conhesionless throughout the process of desiccation. Thus it is

evident that there is little resemblance between a sand and a clay, though

both are composed of individual mineral particles.

        In natural sand, strata are divided by the water table into a lower part

in which the sand is completely saturated and an upper one, in which parts of

the voids are filled with air. By contrast, clay soils are completely or

almost completely saturated from a depth of a few feet below the ground sur–

face downward, irrespective of the position of the water table. The clay may

be very soft, plastic, or stiff depending on the geologic history of the

deposit and its location with reference to the water table, but it does not

contain more than a trace of air. If the water content and the corresponding

porosity of a clay slurry is reduced by static pressure under conditions which

permit the escape of the excess water, the clay passes in succession from the

liquid into the plastic and finally into the solid state. Subsequent removal

of the load does not change the state. The water contents at which different

clays pass from one of these states into another are very different. There–

fore, the water contents at these transitions are used for identification and

comparison of different clays. However, the transition from one state to

another does not occur abruptly as soon as some critical water content is

reached. It occurs gradually over a fairly large range in the value of the

water content. For this reason every attempt to establish criteria for the

boundaries between the limits of consistency involves some arbitrary elements.

The method that has proved most suitable for engineering purposes was taken

over from agronomy. It is known as Atterberg’s method, and the water contents

that correspond to the boundaries between the states of consistency are called

the Atterberg limits .

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        The liquid limit L _W is the water content in per cent of the dry weight

Fig. 2 at which two sections of a pat of soil having the dimensions shown in Figure

2 barely touch each other but do not flow together when subjected in a cup

to the impact of sharp blows from below. The personal equation has an impor–

tant influence on the test results. In order to eliminate this factor, a

standardized mechanical device is used.

        The plastic limit P _W or lower limit of the plastic state is the water

content at which the soil begins to crumble when rolled out into thin threads.

        The range of water content within which a soil possesses plasticity is

known as the plastic range , and the numerical difference between the liquid

limit and the plastic limit is the plasticity index I _W . As the water content

w of a cohesive soil approaches the lower limit P _W of the plastic range, the

stiffness and degree of compaction of the soil increase. The ratio,

C_r = (L_w – w)/(L_w – P_w) = (L_w – w)/(I_w)

is called the relative consistency of the soil. It is analogous to the relative

density of cohesionless soils (see eq. 5).

        In accordance with their general character and outstanding physical proper–

ties, the cohesive soils can be divided into eight large groups: inorganic

clays of high, medium, or low plasticity; inorganic silty soils of high, medium,

or low compressibility; organic clays; and organic silts. This classification

is practically identical with the one used by foremen as a basis for their

entries into boring logs. However, even an experienced foreman or technician

cannot always distinguish between the various cohesive soils on the basis of

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their appearance alone, and the novice is likely to make serious errors. There–

fore, various attempts have been made to eliminate the danger of misjudgment.

As a result of these attempts, it has been found that the distinction between

members of the different groups it has been found that the distinction between

members of the different groups can be made far more reliably by means of

Fig. 3 the plasticity chart (see Fig. 3).

        In the plasticity chart, the ordinates represent the plasticity index I _W

and the abscissas the corresponding liquid limit L _W . The chart is divided

into six regions, three above line A and three below. The group to which a

given soil belongs is determined by the name of the region that contains the

point [ ?] representing the value of I _W and L _W for the soil. All points re–

presenting inorganic clays lie above line A , and all points for inorganic

silts lie below it. Therefore, if a soil is known to be inorganic, its group

affiliation can be ascertained on the basis of the value of I _W and L _W alone.

Points representing organic clays are commonly located within the same region

as those representing inorganic silts or high compressibility, and points

representing organic silts are located in the region assigned to inorganic

silts of medium compressibility.

        Experience has shown that the points which represent different samples

from the same soil stratum define a straight line that is roughly parallel to

line A . As the liquid limit of soils represented by such a line increases,

Fig. 4 the plasticity and the compressibility of the soils also increase. In Figure 4

each line represents several different clays from the same locality. If a

clay stratum was never located above the water table and if it has never carried

an overburden in excess of the present one its water content is likely to be

close to the liquid limit, L _W , up to a depth of one hundred feet or more

below the surface.

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Permeability and Degree of Saturation of Sandy and Silty Soils

        If the bedrock surface is covered with a stratum of gravel or sand, rain

and meltwater percolate through the voids of this stratum in a downward

direction and join the ground-water stream which flows above the bedrock sur–

face. Below the surface of the ground-water stream the voids of the soil are

completely filled with water. The rate at which the ground - water flows through —

the voids depends on the gradient of the surface of the ground-water stream and

the permeability of the soil. If i is the gradient (vertical drop of the

surface per unit of horizontal distance measured in the direction of the flow),

the quantity Q of water which flows through a section with area A , oriented

at right angles to the direction of the flow, is equal to

Q = Aki

wherein k (cm. per sec.) is known as the coefficient of permeability of the

ground. Equation ( 7 ) is valid for the flow of water at a gentle gradient through —

soils with an effective size D ₁₀ of less than about 1 mm. For loose sand the

value k is roughly equal to

k_o (cm. per sec.) = (100 to 150) x D2₁₀ —

wherein D ₁₀ is the effective size in centimeters. For a dense sand, with a

void ratio e , the coefficient of permeability is roughly equal to

k = 1.4k_De2

Between the surface of the groundwater stream and the surface of the ground, —

part of the void space of the soil is filled with air and the balance with

water. The water contained in the voids of the soil above the surface of the

ground water constitutes the soil moisture. By contrast to the ground water

which flows in an almost horizontal direction, the soil moisture is either sta–

tionary or it moves in a vertical direction.

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        The forces which retain the soil moisture in the voids of the soil are

identical with those which cause the water to rise in capillary tubes whose

lower ends are submerged. The height to which the water rises in such tubes

with reference to the outside water level is known as the height of capillary

rise, h _c . If D is the diameter of a capillary tube, the heigh t h _c in–

creases in direct proportion to 1/ D .

        The average width of the capillary channels in a soil with an effective

grain size D ₁₀ is roughly equal to D ₁₀ times the porosity n (eq. 1). Hence,

if the capillary openings in a soil had a uniform width, the water would rise in

the soil to the same height to which it rises in a capillary tube with a dia–

meter n × D ₁₀ . In reality the width of the voids in a soil ranges between a

minimum of almost zero and a maximum which is greater than n × D ₁₀ . The effect

Fig. 5 of this variation on the water content of soils is illustrated by Figure 5.

The figure shows the conditions of saturation in two originally saturated

columns of medium sand, two and one-half years after drainage of the columns

by discharge through their lower ends was started. The lowest section (I) of

the specimens was still completely saturated. In the middle section (II) part

of the void space was filled with air, but the water content of the sand formed

continuous threads. In the uppermost part (III) of the specimens the soil

moisture probably consisted of individual water particles surrounding the points

of contact between gains.

        In nature such a complete state of drainage is never attained because

during every period of rain or melting snow additional water enters the soil

from above. On account of this periodic recharge, the degree of saturation of

a coarse-grained, homogenous soil located between the surface of the ground and

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EA-I. Terzaghi: Ground Frost

the water table is more or less uniform, but it varies with the seasons. Both

the average and the extreme degrees of saturation depend on the effective

grain size D ₁₀ and on the climatic conditions. With decreasing values of

Fig. 6 D ₁₀ the average degree of saturation increases as shown in Figure 6 by the

plain curve C . the probable range of seasonal variations is indicated by a

shaded area. The variations are greatest in soils with an effective size be–

tween about 0.1 and 0.02 mm. (fine sand and coarse silt). If D ₁₀ is greater

than about 1 mm. (coarse sand and gravel) the major part of the void space is

permanently occupied by air, and if it is smaller than about 0.005 mm. (fine

silt and clay) the soil remains completely saturated throughout the year, up

to within a few feet from the ground surface, as explained under the next sub–

heading.

        For a given average effective size of the subsoil and given conditions of

rainfall and evaporation the mean and the extreme degrees of saturation depend

to a large extent on the details of stratification. If, for instance, a stratum

of silty sand located above the water table contains a few thin layers of clean

coarse sand which communicate with a large body of clean sand, the presence of

these layers increases very considerably the degree of saturation of the

silty sand (8).

        On account of the influence of climatic factors and of the details of stra–

tification on the moisture content of sandy and silty soils located between the

water table and the ground surface, no rules of general validity can be estab–

lished regarding the relation between the effective grain size and the average

degree of saturation of sandy and silty soils located between the ground surface

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and the water table. Reasonably reliable estimates can be made only on the

basis of local experience obtained by determining the degree of saturation

of different types of soil at different depths and at different seasons of

the year. If the surface cover is changed, for instance, by stripping the

top layer and replacing it by a runway, both the average and the extreme

moisture content of the soil located below the affected area will change.

       

Permeability and Degree of Saturation of Clay Strata

        The effective size D ₁₀ of typical clays is smaller than 0.001 mm. or

10 −4 cm. Introducing this value into equation ( 8 ) , a value of k = (1 to 1.5)10 −6 —

cm. per second is obtained. Since part of the water contained in the voids of

a clay is in a semisolid state the real coefficient of permeability of clay is

even considerably smaller than 10 −6 cm. per second. Hence clay is almost

impermeable. On the other hand the height of capillary rise for clays is

so great that clay remains completely saturated throughout the year, even if

it is lifted, by geologic events, to a height of several hundred feet above

the surface of the body of water in which it was formed by sedimentation.

The capillary forces which retain the water in the voids of the clay produce

a certain amount of consolidation, but no air enters the voids of the clay ex–

cept in the top layer. In the top layer, which is subject to periodic desic–

cation and wetting and to seasonal variations of temperature, the clay [ ?]

crumbles and the seasonal variations of soil moisture in this layer are similar

to those in a layer of silty sand.

        If a saturated stratum of any kind freezes, it becomes perfectly imper–

meable. Hence, permafrost layers constitute impermeable horizons.

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Thermal Properties of Soils

        For a given surface-temperature regime the position of the boundaries

of the active and the permafrost layers with reference to the ground surface

depends on the thermal conductivity and the heat cap a city of the strata located

within and below the layers of ground frost. In the following text all the

numerical values referring to the thermal properties of soils are given in

centimeter-gram-second units.

        The thermal conductivity , k _t , in c.g.s. units, is the quantity of heat

which flows through a layer, 1 cm. thick, per unit of time and square centi–

meter of the layer, at a thermal gradient of 1°C. per centimeter. The heat

capacity , c _h , in c.g.s. units is equal to the quantity of heat per gram of

the weight of a body, required to raise the temperature of the body by 1°C.

        The essential relations between the thermal conductivity of a soil, k _t ,

cal. cm. −1 sec. −1 (°C.) −1 , the porosity, dry density in pounds per cubi c feet , — —

Fig. 7 and the degree of saturation are graphically represented in Figure 7. The

diagram is based on data obtained by Kersten (5) and various other sources

such as Birch (1). The abscissas represent the porosity n and the dry

weight w _d , respectively. The dry weight has been computed on the assumption —

that the average density of the solid [ ?] soil particles is 2.70. On this

assumption

The ordinates represent the thermal conductivity.

        As n approaches 100% the k _t value of a saturated unfrozen porous sub–

stance approaches the k _t value for water which is 1 × 10 −3 cgs, and that of —

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A frozen saturated porous substance the k _t value for ice which is approxi–

mately 5.3 × 10 −3 c . g . s. The k _t value of a very porous frozen material can —

hardly be smaller than that of ice, and the k _t value of a very porous

material with air-filled voids cannot be smaller than the k _t value of the

most effective insulating materials such as dry asbestos or cotton, which

ranges between 0.12 and 0.22 × 10 −3 c . g . s. On the other hand, as the porosity —

n of a soil approaches a value of zero, the k _t value of the soil must

approach the average k _t value of its mineral constituents. These relations

determine the position of the horizontal tangents or asymptotes of the curves

Fig. 7 shown in Figure 7. Since the thermal conductivity of ice is much higher than

that of water, the thermal [ ?] conductivity of frozen soils is higher than

that of unfrozen ones and the difference between the two values must increase

from zero for n = 0 to about 4.3 × 10 −3 c . g . s . for n = 100%. —

        The thermal conductivity of the soil-forming minerals ranges between a

maximum of more than 20 × 10 −3 c . g . s . for quartz and less than 3 × 10 −3 c.g.s. for micaceous minerals. As indicated in line missing

Figure 1, the coarsest grain-size fractions are dominated by quartz and the — p 14. orig.

finest ones by micaceous minerals, including the clay minerals. Therefore,

the curve S ₁₀₀ representing the relation between n and k _t for saturated

sand and is located high above the corresponding curve C ₁₀₀ for clay soils. —

        In the field, below the water table, all soils are completely saturated.

Fig. 7 For saturated, unfrozen sand, the k _t - n relation is indicated in Figure 7

by the curve S ₁₀₀ and for saturated frozen sand by S ' ₁₀₀ . The corresponding

curves for clay are marked C ₁₀₀ and C ' ₁₀₀ .

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        The factors which determine the degree of saturation of a soil located

above the water table have been discussed at the outset of this article. In

a general way the average degree of saturation of soils located above the

water table increases with decreasing effective grain size D ₁₀ as shown in

Fig. 6 Figure 6. With decreasing degree of saturation S _r the thermal conductivity

Fig. 7 k _t of a soil decreases. In Figure 7 the relation between k _t and the porosity

n of sand at a degree of saturation of 50%, 25%, 12.5%, and 0 % is indicated —

by the dotted lines S ₅₀ , S ₂₅ , S _12.5 , and S _o . It can be seen that the Question Terzaghi again - See letter of Feb 15, 1950.

effect of a reduction of the degree of saturation from 100% to 12.5% on k _t

is less important than that of a reduction from 12.5% to zero.

        The heat capacity, c _h , of soils, like their thermal conductivity, k _t ,

depends on the average mineral composition of the soil, the porosity, and the

[ ?] degree of saturation. However, the ratio between the extreme values of

the heat capacity of soils is very much smaller than the ratio between the ex–

treme thermal conductivities. The heat capacity of all the mineral constituents

of soils is close to 0.19, that of ice is about 0.43, and that of water 1.0.

The heat capacity c _h of a soil with a water content w in per cent of the — —

dry weight is roughly equal to

c_h = (0.19 + w)/(1 + w) cal.gm.-1(°C.)-1

If the soil freezes the heat capacity decreases to the value

c’_h = (0.19 + 0.43w)/(1 + w)

        — Both the thermal conducitivty conductivity , k _t , and the heat [ ?] capacity, c _h , depend —

to some extent on the temperature, but the effect of the temperature is not

important enough to require consideration in connection with permafrost problems.

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The knowledge of k _t and c _h forms the only reliable basis for a rational

explanation of frost phenomena such as permafrost and it also provided the

basis for estimating the rate of permafrost growth or degradation due to a

change in thermal regime of the subsoil.

        According to the laws of thermodynamics the rate at which the temperature

of a body with given dimensions and with a unit weight w _t (solid and water

combined) adapts itself to a change in the temperature of the surrounding medium

is determined by the ratio

d = k/(c_h x w_t)

known as diffusivity . In the c.g.s. system it has the dimension cm 2 . sec. −1 .

The role of diffusivity in thermodynamics corresponds to that of the coefficient

of consolidation c _v (cm 2 . sec. −1 ) in soil mechanics, which determines the rate

at which the water content of a saturated clay stratum adapts itself to an

increase of the load on the stratum. The numerical values of the diffusivity

in the c.g.s. system of different materials illustrate the range of this

value (see Table I).

Table I. Diffusivity of Various Materials . —

Material	Value, x 10 −3 c.g.s.	Material	Value, x 10 −3 c.g.s.
Copper	1 , 133	Dense saturated sand	8± —
Iron	173	Soft saturated clay	4±
Quartzite	45	Fresh snow	3.3
Granite	15	Dry soil	2.5±
Ice	11.2	Water	1.4

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        As indicated by this list the diffusivity of ice is very much higher

than that of water. Consequently the diffusivity of frozen soil is consi–

derably higher than that of the same soil in a thawed condition. On account

of this fact the average temperature of a body of saturated frozen soil in–

creases much more rapidly than that of a body of unfrozen soil with equal

dimensions, at equal difference between the initial temperature of the body

and that of the surrounding medium. However, if the temperature of the

medium surrounding the body of frozen soil is above the freezing point, in–

volving the thawing of the frozen soil, the increase of the temperature

of the frozen body is delayed by the latent heat of fusion of the ice. The

latent heat of fusion is the amount of heat, in thermal units per unit of

weight of water, which is consumed while the ice melts. In the c.g.s. system

it amounts to about 80 calories per gram of water.

       

Geothermal Gradient

        Volcanic phenomena and other geological evidence has led to the conclu–

sion that the temperature of the interior of the earth is high above the

melting point of rocks under moderate pressure. Even at the [ ?]

moderate depth of 20 miles below the surface the rocks have a temperature of

the order of magnitude of 500°C. Therefore, the interior of the earth represents

a heat reservoir. Since the temperature of the surface is relatively very

low there is a steady flow of heat from the interior of the earth toward the

surface, involving a continuous increase of the temperature with depth. The

temperature increase per unit of depth is called the geothermal gradient i _g

(°C. × cm. −1 in the c.g.s. system). If A is an area oriented at right angles

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to the direction of the flow of heat, the quantity of heat Q _h which flows

through this area per unit of time is equal to

        Q_h = Ak_ti_g —

wherein k _t is the thermal conductivity (Fig. 7). This equation corresponds

to equation ( 7 ) which determines the rate of flow of water through the voids of —

a fine-grained soil.

        Since k _t in equation ( 14 ) is different for different layers of the earth —

crust, the geothermal gradient varies along the lines of flow of heat. It

is greatest in strata with the lowest heat conductivity and vice versa as shown

in Figure 8. The figure represents a vertical section through the top layer

Fig. 8 of the ground in a fictitious region in which the temperature T _S (Fig. 8) of

the atmosphere is strictly constant and below the freezing point. On account

of the flow of heat from the interior of the earth of the temperature of the

ground increases with depth as indicated by the broken line a d . Since the

thermal conductivity of the successive soil strata is different, the geother–

mal gradient also varies. At any depth it is equal to the slope of the geo–

thermal line a d with reference to the vertical. To a depth H_p the tempera–

ture of the ground remains permanently below the freezing point. Hence to

depth H _p the ground contains permafrost.

        Experience as well as the theories of thermodynamics indicate that the

Fig. 9 geothermal gradient also depends on the surface topography, as shown in Figure 9.

Beneath mountain chains the lines of flow of heat diverge whereas they converge

toward the bottom of valleys. The lines of equal temperature intersect the lines

of flow of heat at right angles. (See, for instance, 4, pp. 200-205.) The

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isothermal grad eitn ient is equal to the increase of temperature between two adjacent

isothermal curves divided by the distance between these lines measured along a

Fig. 9 flow line. Hence, Figure 9 shows that the isothermal gradient beneath mountain

chains is subnormal and beneath valleys it is above normal.

        Finally, experience has shown that the rate of flow of heat, Q _t / A

(eq. 14) varies even below level ground. It may range between 2 × 10 −6 cal.sec. −1

cm. −2 in volcanic regions and less than 1 × 10 −6 cal.sec. −1 cm. −2 in nonvolcanic

ones. Hence the geothermal gradient is by no means a constant. During the

construction of the Mount Cenis tunnel in Switzerland it was found that the

temperature increased by 1°C. for about every 50 meters of depth. In some

regions of South Africa the geothermal gradient is even smaller than 1°C. per

100 meters. On the other hand, in coal-bearing formations, where the chemical

changes in t h he coal beds represent a supplementary source of heat, the tempera- —

ture may increase by 1°C. for every 12 meters of depth. At an average, the

temperature increases by 1°C. for every 30 meters of depth, corresponding to

an average thermal gradient of 0.033 (°C.) m. −1 .

       

Relations between Surface and Ground Temperatures

        Fig. 8 Figure 8 shows ground temperatures in a region of constant temperature. In

reality the temperature of the atmosphere undergoes diurnal, annual, and secular

variations. To demonstrate the fundamental relations which determine the effects

of such variations let us assume that the temperature of the atmosphere referred

to in Figure 8 is sudden t ly increased from T _o to T _o + Δ T _o . To simplify the —

investigation it is further assumed that the initial temperature of the ground

is uniform and equal to T _o .

        Fig. 10 The effects of the temperature increase are shown in Figure 10. As soon as

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the temperature of the atmosphere rises, the temperature of the ground starts

to increase and the heat penetrates the ground from the surface in a down–

ward direction.

        Methods for computing changes in the ground temperature due to sudden or

periodic changes of the surface temperature can be found in any textbook on

heat conduction. (See, for instance, 1a, pp. 43-51; 4, pp. 45-57.) By solving

the differential equation of the nonstationary flow of heat it is found that

the distribution of the temperature in homogeneous ground, at any time t ₁

after the surface temperature has been raised, is roughly parabolic as shown

Fig. 10 in Figure 10. The following discussion is based on the simplifying assumption

that the temperature distribution is strictly parabolic and that the tempera–

ture of t a he layer below point c is still unchanged. The error due to this

assumption does not exceed 4% which, from a practical point of view, is

negligible (10).

        According to the simplified theory the time t ₁ at which point c arrives

at depth z ₁ is equal to

t₁ = z2₁/12d —

wherein α is the heat diffusivity (eq. 13) (10). If the ground consists of

quartzite, α = 45 × 10 −3 cm. 2 sec. −1 , time t ₁ at which point c arrives at

a depth of z ₁ = 50 ft. = 1 , 520 cm. is —

t₁ = 15202/(12 x 45 x 10-3) = 43 x 105 sec. = 50 days

If the ground consists of granite, α = 15 × 10 −3 cm. 2 sec. −1 , t ₁ is equal to

150 days. For soft clay, α = 4 × 10 −3 cm. 2 sec. −1 , t ₁ = 562 days. These

figures show the decisive influence of the diffusivity [ ?] on the rate of pro–

pagation of temperature in the ground.

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        The depth z ₁ within which the temperature of the ground has perceptibly

increased during a given time t ₁ is

z₁ = √(12dt₁)

        If, at some time t ₁ , the temperature of the atmosphere is sudden t ly —

reduced from T _o + Δ T _o to T _o − Δ T ₁ , the cooling, like the preceding heating,

proceeds from the surface of the ground in a downward direction. In the

earlier stages of the process of cooling the distribution of the temperature

will resemble that indicated by the broken line def . The position of this

curve, with reference to the vertical line a f (initial temperature line), shown

that the temperature of a layer located between the elevations of points a ₁ and

f will be higher than the temperature in the ground immediately above and below

the layer.

        In connection with periodic changes of the surface temperature, such as those

associated with the sequence of day and night, one-half of the difference be–

tween the minimum and the maximum temperature is called the amplitude of the

temperature wave and the lapse of time between two successive temperature

maxima is the period of the wave. The theory of periodic flow of heat shows

that every temperature wave on the surface produces a similar temperature

variation in the ground. However, with increasing depth, the amplitude of the

temperature wave decreases and the time lag between the temperature maximum

at depth with reference to that of the surface increases. The level below which

the amplitude of the temperature variations becomes imperceptible is the level

of zero amplitude .

        The greater the period of the surface wave the greater is the depth [ ?] to

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Fig. 11 which the temperature of the ground varies perceptibly. Figure 11 shows the

diurnal variations of the surface temperature. The period [ ?] of these

variations is 24 hours. For 12 hours the temperature is above and for 12

hours below the average for the day. A rough estimate of the position of

the level of zero amplitude of the temperature variations produced by the

periodic change of the surface temperature can be made on the assumption

that the change of the surface temperature occurs abruptly as indicated in

Figure 11 by a broken line. On this assumption the time t ₁ = 12 hours

corresponds to t ₁ in equation ( 16 ) . According to this equation the depth —

H _o to which a sudden rise of the surface temperature increases the ground

temperature in 12 hours is

        H_o (cm.) = √(12d x 12 x 3600) = 720√(d)

wherein α is the average diffusivity of the ground in cm. 2 sec. −1 . As soon

as the cooling has proceeded to depth H _o the temperature again becomes

higher than the average. Therefore H _o represents the depth to which the

diurnal variations of the surface temperature produces perceptible changes

of the subsurface temperature. In order to compute this depth for wet sand,

we introduce into the preceding equation α = 8 × 10 −3 cm. 2 sec. −1 and obtain

H_o = 720√(8 x 10-3) = 62 cm.

or about 2 ft. hence the diurnal temperature variations do not extend beyond a

depth of a few feet. For the seasonal variations, with a period of 365 days,

we obtain values of H _o of the order of magnitude of 40 ft. Finally, the

effects of secular cold waves like the wave which produced the last period of

glaciations extend to a depth of many thousand feet.

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        In connection with permafrost, the most important cold waves are the

annual ones, caused by the alternation between summer and winter. Their

effects on ground frost conditions are illustrated by Figures 12a and b.

        Fig. 12a and 12b Figure 12a represents the temperature conditions in the subsoil of a

region with a temperate climate and Figure 12b refers to a similar region

with an arctic climate. The mean annual temperature a o in the temperate

region is above the freezing point and that of the arctic region is well

below this point. In both regions the level of zero annual amplitude is

assumed to be located at the same depth. Within this depth the temperature

conditions change continuously. On account of the diurnal variations of

the surface temperature and the time lag between the maximum surface tempera–

ture and the corresponding subsurface temperature, every temperature line

representing the ground temperatures at a given moment can have one or more

points of inflection. However, in both regions all the temperature lines are

located within a roughly triangular space b c d . The two sides [ ?]

b d and c d of this triangle determine the maximum range of the ground

temperature for every depth.

        The two diagrams lead to the following conclusion regarding the ground

frost conditions. In the temperate region the ground will freeze-in every

winter to a depth equal to or somewhat smaller than H _a . This is the depth

of the active layer . Below the depth H _a the soil moisture is permanently

liquid. In the arctic region the frost penetrates to depth H _a + H _p . To

depth H _a it thaws every summer, whereas, between depth H _a and H _a + H _p it

is permanently frozen. In exceptionally mild winters after a hot summer, the

frost may not penetrate to the full depth H _a . In such winters the frozen

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ground in the active zone will be separated from the permafrost by a layer

of unfrozen ground or talik . Wherever the layer of permafrost is in a state

of progressive degradation this condition is normal.

        Fig. 12b Figure 12b also discloses the existence of a simple relation between

the thickness of the active and the permafrost zone. The thickness H _a of

the active zone is equal to the ordinate of point e at which the freezing

line O f intersects the lateral boundaries b c d of the temperature tri–

Fig. 12a angle. In a temperate zone (Fig. 12a) point e is located on the boundary

c d , representing the minimum ground temperatures. If the mean annual surface

temperature O a decrease at unaltered range of annual temperature variations,

point e moves down, until the freezing line O f becomes tangent to the

line b d , representing the maximum ground temperatures. A further decrease

of the surface temperature is associated with the formation of permafrost, but,

at the same time, point e moves up, involving a decrease of the depth of

the active zone. Hence, in a general way, an increase of the thickness of

the permafrost zone must be associated with a decrease of that of the active

layer.

       

Thickness and Continuity of Permafrost

        In the light of what is known about permafrost, it can be taken for granted

that permafrost is a product of existing climatic conditions and not a relic

of the last ice age. If this statement is correct, the thickness of the sheet

of permafrost should everywhere be equal to the mean annual surface temperature

in °C. below zero divided by the average geothermal gradient i _g (increase of

the temperature per unit of depth) for the region.

        Along the coast of the Arctic Ocean Sea (north coast of Siberia, Victoria and

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Southampton Island in Canada) the mean annual surface temperature is some–

what below −10°C. and the average geothermal gradient i _g is 0.033 m, −1 x °C.

Hence, along the Arctic Ocean Sea the depth H _p of the permafrost zone should —

be about

        H_p = 10/0.033 = 300 m. = 1000 ft.

According to Muller (6, Fig. 3), the permafrost layer has, at the north coast

of Siberia, a depth of 200 to 400 meters. The southern boundary of the sheet

of permafrost follows, also in accordance with present climatic conditions,

the 0°C. isotherm for the mean annual temperature. In the Amur region in

Manchuria, both the 0°C. and the southern boundary of the permafrost sheet

Fig. 12 come close to latitude 50° N. Figure 12 led to the conclusion that the

decrease of the thickness of the permafrost must be associated with an

increase of the thickness of the active layer. As a matter of fact, the

thickness of the active layer increases from 0.2-1.6 meters at the Arctic

Ocean Sea to 0.7-4.0 meters at the southern boundary of the permafrost territory. —

        In those parts of northern Siberia where the thickness of the permafrost

is close to its maximum, gaps in the permafrost have been found only beneath

big river valleys such as the valley of the Yenisei at Ust-Port (6, Fig. 14).

These major gaps are probably due to the fact that the big rivers of Siberia

are flowing throughout the year. They constitute large veins of heat supply.

Over the area covered by the rivers and the [ ?] perennial part of the ground–

water stream associated with the rivers, the mean annual surface temperature

is well above the freezing point whereas on both sides of the valley it may

be −10°C.

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        Along the southern margin of the permafrost territory the permafrost

becomes discontinuous. The permafrost zone surrounds patches of talik or

unfrozen ground and vice versa. Most of these gaps in the continuity of the

permafrost are probably due to variations in the average heat diffusivity of

Fig. 13a 13b the top layer of the ground in horizontal directions, as shown in Figures 13a

and b. In each figure T _o represents the mean annual temperature. (Ex–

perience has shown that the mean annual temperature near the surface is some–

what lower than the temperature T _o obtained by extrapolation from the line

dg representing the geothermal gradient. This is probably due to the heat

which is consumed in the top layer by the evaporation of soil moisture.

However, in the following discussion this difference will be disregarded.)

At points O ₁ and O ₂ , located at the margin of the permafrost territory,

the maximum, minimum, and mean annual temperatures are assumed to be the

same, but the average diffusivity of the ground is different.

        According to equation ( 16 ) , the depth H _o at which the level of zero annual

amplitude is located increases in direct proportion to the square root of the

diffusivity, α (eq. 13). At point O ₁ , located above strata with low diffusivity,

the line b d , representing the maximum ground temperatures, intersects the

freezing line O ₁ g , whereas in the region represented by Figure 13b, the

maximum ground temperatures (line b d ) are above freezing. Hence, beneath

point O ₁ there will be a layer of permafrost whereas beneath point O ₂ no

permafrost can exist. Yet the surface-temperature conditions at the two

points are perfectly identical.

        Gaps in the permafrost layer can also be produced by installing and

continuously operating heating systems in buildings located above permafrost.

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If a heating system in a building maintains a constant temperature, T ₁

Fig. 14 (Fig. 14) the temperature conditions in the subsoil will gradually approach

a stationary state represented by a temperature line similar to m n r . The

position of this line can be computed on the basis of the laws of the sta–

tionary flow of heat through solids. If this line, like m n r in Figure 14,

does not intersect the freezing line O f , the heat emanating from the

building will gradually thaw a hole through the layer of permafrost. Other–

wise the permafrost located below the building will merely shrink, whereby

the base of the permafrost layer will rise and its surface will descend.

       

Aggradation and Degradation of Permafrost

        On the basis of paleontological evidence it can be considered certain

that the present state of permafrost in arctic and subarctic regions was pre–

ceded by a state in which no permafrost and, probably, not even seasonal frost

occurred. Hence, there is no difference between permafrost and seasonal

frost except in the time which elapses between the beginning and the end

of the period of the existence of ground temperatures below the freezing point.

        The successive stages in the formation and subsequent degradation of

Fig. 15 permafrost are illustrated by Figure 15. In all stages the slope of the

temperature line T , representing the increase of the temperature with depth

remains practically unaltered whereas the abscissa of the upper end of the

line, equal to the mean annual surface temperature, varies.

        Stage (a) corresponds to tropical climatic conditions similar to those

which prevailed, for instance, in Greenland and in Spitsbergen during the Ter–

tiary period. Following stage (a) the temperature decreased. As soon as the

minimum surface temperature reached the freezing point the formation of an

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active layer started (stage b). The thickness of the active layer became a

maximum in stage (c) when the curve of maximum temperatures touched the ver–

tical line representing zero °C. A further decrease of the mean annual tempera–

ture led to the formation of permafrost, stage (d), associated with a decrease

of the thickness of the active layer. This condition prevailed in arctic

regions up to the present day, with no change other than occasional increase

or decrease of the thickness of the permafrost layer, associated with a de–

crease or increase of that of the active layer, due to changes of the mean

annual temperature, for instance, from −105°C. (stage d) to −5°C. (stage e). —

        A departure from this normal course of events took place only in those

regions in which the Pleistocene ice sheet advanced over the terrain under–

laid by permafrost. To realize the peculiar temperature conditions which

prevailed during the period of glaciations, the icecap of Greenland may be

considered. Over large areas the thickness of the icecap is about 8,000 ft.

or 2,400 m. The mean annual surface temperature is probably −40°C. Since the

thermal conductivity of ice at low temperatures is roughly equal to that of

granite and other crystalline rocks, the geothermal gradient in the ice

ought to be about equal to that in rock formations, 0.033°C. per meter of

depth. Since the ice sheet is 2,400 meters thick, the temperature at its

base would be 80°C. above that of the surface or equal to +40°C. This is

obviously impossible. Hence, we are compelled to assume that the lower part

of the inland ice of Greenland is in a state of continuous melting, which keeps

the temperature at the base of the ice sheet at the freezing point by heat

absorption. On account of the weight of the ice the freezing point at the base

of an ice sheet with a thickness of 2,400 meters would be about −2°C.

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        The quantity of heat which flows from the interior of the earth toward

the surface is about 1.3 × 10 −6 cal.cm. −2 sec. −1 or about 40 calories per

square centimeter per year. The heat which is absorbed during the melting

of ice is about 70 cal. per cubic centimeter of ice. Hence, the amount of

melting required to absorb the heat which comes from the interior of the

earth is about half a centimeter per year. During the Pleistocene periods of

glaciations the average thickness of the ice sheet was of the order of magni–

tude of 6,000 ft. or 1,800 meters. At the bottom of an ice sheet with

such a thickness the freezing point is at about −1.5°C. Assuming an average

geothermal gradient of 0.033 m. −1 × °C., the corresponding depth of perma–

frost cannot exceed

H_D = 1.5/0.033 = 45 m. = 150 ft.

which is very small compared to the depth of the permafrost in the non–

glaciated parts of the Pleistocene arctic zone. As a consequence, in formerly

glaciated regions, such as Canada and the major part of Alaska, the advance

of the Pleistocene ice sheet was associated with a temporary degradation or

even a disappearance of the permafrost and not with an aggradation.

        In accordance with the laws of heat conduction the change of the thickness

of the layer of permafrost lags behind the change of the mean annual tempera–

ture and the lag increases approximately with the square of the thickness of

the layer. The lag due to the time required for the temperature of the

ground to adapt itself to a change in the surface temperature must be added

to the lag due to the heat of fusion of the ice. On account of the heat which

is liberated at freezing and absorbed during thawing (about 80 cal. per gram

of water), the change of the temperature in permafrost proceeds as if each

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body of permafrost were surrounded by a skin with a very low heat conductivity.

The insulating effect of the heat of fusion will be realized if the following

in considered. Suppose the porosity of a stratum of completely saturated

and frozen silt stratum is 50%. It takes less than 2 cal. per cubic centi–

meter to increase the temperature of the silt-water system from −1°C. to

+1°C., but the thawing of ice contained in the voids of the silt requires an

additional inflow of heat of about 40 cal. per cubic centimeter.

        Along the boundaries of the permafrost sheet, where the thickness of

the sheet does not exceed a hundred feet, the lag may be less than a century,

whereas, along the coast of the Arctic Ocean Sea , where the permafrost penetrates —

to depths up to a thousand feet, the lag may amount to several thousands of

years. Nevertheless, one can hardly expect any permafrost remnants of the

ice age because even a lag of several thousand years is short compared to

the period of roughly twenty-five thousand years which has elapsed since the

retreat of the last ice sheet.

        On account of the time lag between a change of the mean annual surface

temperature and the corresponding change of the ground temperature, it is

possible that the geothermal gradient between the ground surface and the

center of the permafrost stratum is temporarily reversed, involving a

decrease of the temperature with increasing depth below the level of zero

annual amplitude. Such a case has been reported by Sumgin (6, Fig. 7, p. 15).

        The most conspicuous consequence of the time lag between the surface

and the ground temperature is the occasional occurrence of one or several

layers of unfrozen ground or talik between layers of permafrost (6, p. 10).

To demonstrate the prerequisites for the formation of layered talik, two layers

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Fig. 16 of permafrost (Fig. 16) will be considered, separated by a layer of talik.

The surface of the lower permafrost layer is located far below the bottom

of the present active zone. This fact indicates that the lower layer re–

presents the result of the progressive degradation of a very much thicker

layer. Progressive degradation of a permafrost layer can be started only

by an increase of the mean annual surface temperature. The presence of the

upper permafrost layer indicates that the increase of the temperature was

followed by a decrease. Since cooling, like heating, proceeds from the sur–

face in a downward direction, the secular cold wave started a new permafrost

layer above the original, degrading one. Since the degrading layer is trapped

between two unfrozen layers, its degradation inevitably continues as long

as the talik layer exists, while the upper permafrost stratum aggrades. Hence,

the presence of a layer of talik between two layers of permafrost is a

transitory phenomenon caused by important secular variations of the mean

annual surface temperature.

        Fig. 16 Figure 16 shows the temperature conditions which prevail in a two–

layer sheet of permafrost. As time goes on, the temperature line a f g h afhg —

moves into the ultimate position a e g , provided the mean annual surface —

temperature retains its low value long enough. The last stage of the transi–

tion may be associated with a downward movement of the base of the permafrost

sheet as indicated in the figure, but this movement cannot possibly start

before the layer of talik is completely frozen.

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Ice Formation in Solis

        During the last decades the formation of ice in the voids of soils has

received increasing attention on account of the detrimental effects of ice

formation on highways and airports. An annotated bibliography on frost

action in soils, covering 283 items, has been published by the Highway

Research Board (7). The frost investigations have led to the following

conclusions.

        If the temperature of a mass of clean sand or gravel in a moist or

saturated state is lowered below the freezing point, the water contained in

the voids of the mass freezes in situ . The freezing is associated with a

volume expansion of the water by almost ten percent. However, this expansion

does not necessarily lead to a ten percent increase in the volume of the

voids of a saturated sand because part of the liquid water may be expelled

while freezing proceeds.

        On the other hand, if a mass of saturated, fine-grained soil such as

silt or clay is exposed to freezing temperatures, the major part of the

frozen water accumulates in the form of layers of crystal-clear ice oriented

parallel to the surface exposed to the freezing temperature. As a consequence

the frozen soil consists of a series of layers of soil separated from each

other by layers of clear ice. The formation of the ice layers requires that

water be pulled through the voids of the soil toward the seat of the ice–

layer formation. The water which enters the layers may come out of the

soil which freezes or it may be drawn out of an aquifer located below the zone

Fig. 17 of freezing. These possibilities are illustrated by Figure 17.

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        Fig. 17. Figure 17. represents three cylind ir ri cal specimens of a fine saturated —

silt. Specimen (A) rests on a solid base, whereas the lower ends of

specimens (B) and (C) are immersed in water. The temperature of the upper

end of each specimen is kept below the freezing point. In (A) the water

that enters the ice layers is drawn out of the lower part of the specimen.

As a consequence, the lower part consolidates in the same manner as if the

water were pulled by capillarity toward a surface of evaporation at the

upper end. The growth of the ice layers probably continues until the water

content of the lower part is reduced to the shrinkage limit. Since all the

water entering the ice layers comes from within the specimen, the sample ; is —

referred to as a closed system . The volume increase associated with the

freezing of a closed system does not exceed the volume increase of the line missing

water contained in the system. It ranges between about 3 and 5 per cent of

the total volume.

        In (B) the water required for the initial growth of the ice layers

is also drawn out of the specimen, whereupon the lower part of the sample

consolidates. However, as the consolidation progresses, more and more water

is drawn from the pool of free water located below the specimen. Finally,

both the rate of flow toward the zone of freezing and the water content of the

unfrozen zone through which the water percolates become constant. Such a

sample constitutes an open system . The total thickness of the ice layers

contained in such a system can increase to tens of feet. This is demonstrated

by the pingo phenomenon. Pingos can assume a height up to 300 feet and con–

sist of clear ice, covered with a thin layer of soil (6, p. 59 and Fig. 28).

        The open system represented by sample (B) can be transformed into a

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closed system by inserting a layer of coarse-grained material between the

zone of freezing temperature and the water table, as shown by (C). Since

the water cannot rise by capillarity through the coarse layer, the upper

part of the sample represented in (C) constitutes a closed system. The

lower part of the system is subject to drainage by frost action.

        Opinions regarding the molecular mechanics of the formation of the ice

layers and the intensity of the forces involved are still controversial.

Nevertheless, the conditions for the formation of the layers and the means

for preventing it are already known.

        Ice layers develop only in fine-grained soils. However, the critical

grain size marking the boundary between soils that are subject to ice-layer

formation and those that are not depends on the uniformity of the soil. In

perfectly uniform soils, ice layers do not develop unless the grains are

smaller than 0.01 mm. Fairly uniform soils must contain at least 10% of

grains smaller than 0.02 mm. The formation of ice layers in mixed-grained

soils requires, as a rule, that grains with a size less than 0.02 mm.

constitute at least 3% of the total aggregate. On soils with less than 1%

of grains smaller than 0.02 mm., ice layers are not formed under any condi–

tions which may be encountered in the field.

        The mechanical causes of the flow of water toward a zone of freezing

are identical with those compelling the water to flow from a ground-water

reservoir through the voids of a soil toward a surface of [ ?] evaporation.

Abundant growth of the ice layers, such as the growth of pingos, can take place

only if the [ ?] vertical distance between the base of the ice layer and the

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water table is smaller than the height to which the water can rise from the

water table by capillarity. If the base of the ice layer is located above

this height, or if the capillary communication between the water table and

the ice layer is intercepted, for instance, by a layer of gravel (Fig. 17,

system (C)), the sum of the quantity of ice and water in the stratum subject

to freezing remains unchanged and the frost produces only a redistribution of

the water content of the stratum.

        If thick layers or large pockets of ice are encountered in sand or gravel

strata, it is almost certain that the ice was formed by the freezing of a

pool, or moved to its present location and subsequently buried. On the other

hand, pockets of clear ice in silt strata may either [ ?] be the remnants of

bodies of ice formed in the open or else they may have developed in the silt,

Fig. 17 beneath the surface, like the layers shown in Figure 17. In the second case

their formation must have been associated with an intense consolidation of

the surrounding material, because at least part of the water which entered

the pocket was withdrawn from the silt.

       

Surface Movements Due to Freezing and Thawing

        As the ice crystals grow in the voids of a soil they ca ac t lik e jacks which — —

push the soil grains apart and thus increase the void space. If the freezing

occurs in the voids of a coarse-grained soil, such as sand or gravel, or in a

closed system, the corresponding rise of the ground surface does not exceed

h = 0.1nH

wherein n is the average porosity and H the thickness of the layer subject

to freezing. On the other hand, if the freezing takes place in an open system

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susceptible to the formation of ice layers, h can assume values which are

far in excess of the thickness H of the stratum in which the formation

of ice layers started. This is demonstrated by the occurrence of pingos

and other frost mounds. The corresponding rise of the ground surface is

known as frost heave. If a structure is located above the seat of ice

formation, it goes up together with the surface of the ground.

        Conclusive information regarding the maximum load which can be lifted

by a growing ice layer is not yet available. However, the occurrence of

pingos with a height of 100 meters indicates that this load is greater than

10 tons per square foot. The greatest pressure which can possibly be exerted

by freezing water is about 2,000 tons per square foot, because under higher

pressures the water crystallizes into ice which does not involve a volume

increase.

        The thawing of the ice in the ground is associated with a volume

contraction. The corresponding settlement of the ground surface depends

on the distribution of the ice throughout the ground. If the ice was formed

in the voids of the ground without preceding migration of water toward the

centers of freezing, such as the ice in a sand or gravel stratum, the settle–

ment due to thawing cannot exceed the value h , equation ( 18 ) . It may even —

be considerably smaller. Reliable information concerning the settlement

associated with the thawing of frozen sand and gravel is not yet available.

        On the other hand, if the ice occurs in the form of pockets or lenses

of clear ice, the thawing of the ground involves a settlement which can be

almost equal to the total thickness of the bodies of ice, because the roof

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of the cavities produced by the melting of the ice gradually collapses due

to stoping. If a structure is located above the seat of thawing, such as

Fig. 14 the b bu ilding shown in Figure 14, the resulting settlement can be catastrophic.

        The rapid melting of permafrost can be produced by the heating of a

buildings, but it can also be produced by a radical change in the surface

cover, such as deforestation, or the removal of a layer of peat involving

an increase of the heat conductivity of the top layer. If the ground be–

neath the affected area contains large pockets or thick layers of ice, the

melting of the ice causes the formation of troughs and sinkholes without any

outlet similar to the sinkholes in limestone terrances, produced by the col–

lapse of the roof of solution channels. Therefore, the resulting irregular

surface topography has been given the name thermokarst (6, p. 83).

       

Strength of Thawed and Frozen Soils

        Under the influence of concentrated loads , both thawed and frozen soils —

commonly fail by shear. According to Coulomb’s classical concept the resistance

s per unit of area against failure by shear along a section through any

material is

s = c + p tan(ɸ)

wherein D is the unit pressure on the surface of sliding, ø the angle of

internal friction , and c the cohesion (shearing resistance for p = O). The

validity of this equation is subject to various limitations (see, for instance,

9, pp. 78-93), but in connection with the following discussions these limitations

can be disregarded.

        If the shearing resistance of a material is determined by equation ( 19 ) , —

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the unconfined compressive strength q _u or the unit load under which a

cylindrical specimen fails [ ?] is equal to

        q_u = 2c tan(45 + ɸ/2)

        For coarse-grained soils like clean sand or gravel in a dry or

completely saturated state, c = O and q _u = O. In other words, these soils

owe their capacity to sustain concentrated loads exclusively to internal fric–

tion. Their angle of internal friction lies between about 35° and 45° and

its value depends on the relative density D _r (eq. 5). Fine and very fine

sand, in a moist state, has a slight cohesion but the cohesion disappears as

soon as the sand is submerged.

        The compressive strength q _u of a saturated silt or clay depends on

its relative consistency C _r (eq. 6), which, in turn, depends on its geologic

history and the physical and chemical properties of the clay constituents. The

q _u value determines the consistency of the clay, which is commonly designated

by one of the following terms:

Consistency	q u value, kg./sq.cm.	Consistency	q u value, kg./sq.cm.
Very soft	< 0.25	Stiff	1.0-2.0 —
Soft	0.25-0. 0 5	Very stiff	2.0-4.0 —
Medium	0.5-1.0	Extremely stiff	> 4.0 — see p. 38 orig

        If a soil freezes, the free water contained in the voids of the soil

freezes whereupon the ice interconnects the soil particles. Therefore, the

strength of the soil increases. The unconfined compressive strength q ' _u

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of the frozen soil depends on the unconfined compressive strength q _i of

the ice, on the degree of saturation S _r (eq. 4), and the angle of internal

friction ø (eq. 20).

        The unconfined compressive strength q _i of ice depends on the

temperature of the ice, the structure of the ice, and the rate of loading.

According to Brown, the q _i of ice depends on the temperature of the ice, delete - repeated

the structure of the ice, and the rate of loading. . According to Brown, the

q _i value of river ice increased from 21.0 kg. per square centimeter at −2.2°C.

to 62.0 kg. per square centimeter at −16°C. (2, p. 449). E. Bucher obtained

for ice produced by the freezing of saturated snow at −3.5°C. an average

value of 34 kg. per square centimeter (3, p. 449). Russian investigators

reported that the q _i value for the top layer increased from 20.7 to 38.4

kg. per square centimeter, whereas that of the middle part of the layer in–

creased from 35.8 to 76.0 kg. per square centimeter. They also reported that

the compressive strength decreased with increasing rate of loading. By in–

creasing the rate of loading from 20 to 50 kg. per square centimeter a minute,

the q _i value was reduced from 60 to 24 kg. per square centimeter (6, p. 36).

        Another important mechanical property of the ice is its capacity to

“creep.” If a block of ice is permanently kept under a load which is con–

siderably smaller than the failure load ( q _i × loaded area), the block

gradually flattens out. At a load of less than about 2 kg. per square centi–

meter, this slow deformation or creep is imperceptible. However, under loads

of more than 2 kg. per square centimeter, the rate of creep rapidly increases

with increasing load (3, p. 137). The capacity of the ice to creep under rela–

tively very low deviator stresses is responsible for the movement of glaciers.

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        Since frozen soil owes its cohesion chiefly or entirely to that of the

ice, the behavior of a frozen soil under stress must have at least some

features in common with the behavior of pure ice under similar stress

conditions. Since the angle of internal friction of ice is equal to zero,

equation ( 20 ) requires that the cohesion c of ice is equal to q _i /2. —

        If the degree of saturation of soil S _r (eq. 4) is smaller than 100%,

the freezing of the soil moisture imparts to the soil the character of a mild

sandstone. The grains of this sandstone-like material are interconnected by

minute patches of ice. On the other hand, if a saturated soil freezes, it

turns into a block of ice, reinforced by a skeleton of solid soil particles.

The strength, q _ú , of such a soil should roughly be equal to

q’_u = q_i tan(45 + ɸ /2)

wherein q _i is the unconfined compressive strength of the ice and [ ?]

ø the angle of internal friction of the soil.

        The q _ú values reported by Russian investigators range between 22 kg. per

square centimeter at −0.5°C. and 30 kg. per square centimeter at −2°C. for

saturated sand, and between 5 kg. per square centimeter at −0.5°C. and 23

kg. per square centimeter at −2°C. for saturated soil (6, p. 40). German in–

vestigators obtained at −15°C.: 37 kg. per square centimeter for o r ganic clay, — —

72 kg. per square centimeter for inorganic clay, 90 kg. per square centimeter

for sandy clay , and 138 kg. per square centimeter for clean sand. Conclusive —

information regarding the relation between temperature, relative density , and —

q _ú value for the principal types of soils in a saturated state and regarding

the creep of frozen soil under moderate loads is not yet available.

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---

BIBLIOGRAPHY

1. Birch, Francis, Schairer, J.F., and Spicer, H.C., eds. Handbook

of Physical Constants . N.Y., 1942. Geol.Soc.Amer. Spec.

Pap . no.36.

1a. Carslaw, H.S., and Jaeger, J.C. Conduction of Heat in Solids.

N.Y., Oxford, 1947.

2. Dorsey, N.E. Properties of Ordinary Water Substance in all its

Phases . N.Y., Reinhold, 1940. American Chemical Society.

Monograph Series (no.81)

3. Haefeli, R. “Schneemechanik,” Geologie der Schweiz-Geotechnische

Serie-Hydrologie. Lieferung 3: Der Schnee und seine

Metamorphose . 1938.

4. Ingersoll, L.R., Zobel, O.J., and Ingersoll, A.C. Heat Conduction.

N.Y., McGraw-Hill, 1948.

5. Kersten, M.S. Determination of Thermal Properties of Soils for

Investigation of Airfield Construction in Arctic and Subarctic

Regions . St. Paul, Minn., Corps of Engineers, St. Paul

District, Dept. of the Army, 1948.

6. Muller, S.W. Permafrost . Ann Arbor, Mich., Edwards, 1947.

7. National Research Council. Highway Research Board. Bibliography on

Frost Action in Soils. Wash.,D.C., The Council, 1948.

Bibliography no.3.

8. ✓ Ter a z aghi, Karl. “Soil moisture and capillary phenomena in soils,”

National Research Council. Committee on Physics of the Earth.

✓ Physics of the Earth . Vol.9. Hydrology . N.Y., McGraw-Hill,

1942, pp.331-63.

9. ---, and Peck, R.B. Soil Mechanics in Engineering Practice . N.Y.,

Wiley, 1948.

10. ----. Publication scheduled for 1949.

       

Karl Terzaghi

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