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    Physics of Ground Frost

    Encyclopedia Arctica 2a: Permafrost-Engineering

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




    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

    are clipped together with explan–

    atory notes & fastened as bach 8

    copy 3.

            Please leave these figures in

    order in which they are clipped as this

    follows correspondence instructions &

    suggestions from published.


    April 4, 1950

    Unpaginated      |      Vol_IIA-0155                                                                                                                  
    EA-I. Terzaghi: Physics of Ground Frost



    Fig. 1 Diagram representing grain-size characteristics of

    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.)
    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.)
    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
    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 .)
    Fig. 8 Influence of thermal conductivity on increase of

    temperature with depth
    Fig. 9 Influence of surface topography on geothermal

    18- a
    Fig. 10 Diagram illustrating progress of change of ground

    temperatures due to sudden changes in surface

    Fig. 11 Diagrammatic presentation of diurnal changes of

    surface temperature
    Fig. 12 Range of variations of ground temperature above the

    level of zero annual amplitude
    Fig. 13 Diagram illustrating the influence of diffusivity on

    the maximum mean annual temperature at which

    permafrost can still persist

    Unpaginated      |      Vol_IIA-0156                                                                                                                  
    EA-I. Terzaghi: Physics of Ground Frost


    List of Figures -2-

    Fig. 14 Diagram illustrating the cause of the degradation of

    permafrost beneath heated buildings
    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
    Fig. 16 Distribution of ground temperatures in a region

    where two permafrost layers are separated by a

    layer of unfrozen ground
    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

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





            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


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    EA-I. Terzaghi: Ground Frost

            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


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

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    EA-I. Terzaghi: Ground Frost

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

    mined by the ratio

            Sr(%) = 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 emin the correspond–

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

    ratio e is determined by the equation

    Dr = (eoe)/(eoemin)

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

    emin 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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    EA-I. Terzaghi: Ground Frost

    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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    EA-I. Terzaghi: Ground Frost

            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,

    Cr = (Lww)/(LwPw) = (Lww)/(Iw)

    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

    006a      |      Vol_IIA-0163                                                                                                                  

    Fig. 2

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    EA-I. Terzaghi: Ground Frost

    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

    ko (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 = 1.4kDe2

    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.

    009      |      Vol_IIA-0166                                                                                                                  
    EA-I. Terzaghi: Ground Frost

            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

    009a      |      Vol_IIA-0167                                                                                                                  

    Fig 5

    010      |      Vol_IIA-0168                                                                                                                  
    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–


            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

    011      |      Vol_IIA-0169                                                                                                                  
    EA-I. Terzaghi: Ground Frost

    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


    [Math Formula]

    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

    ch = (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

    [Math Formula]

    d = k/(ch x wt)

    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
    Iron 173 Soft saturated clay
    Quartzite 45 Fresh snow 3.3
    Granite 15 Dry soil 2.5±
    Ice 11.2 Water 1.4

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    EA-I. Terzaghi: Ground Frost

            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

    016a      |      Vol_IIA-0175                                                                                                                  

    Fig. 7

    017      |      Vol_IIA-0176                                                                                                                  
    EA-I. Terzaghi: Ground Frost

    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

            Qh = Aktig

    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 Hp 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

    018      |      Vol_IIA-0177                                                                                                                  
    EA-I. Terzaghi: Ground Frost

    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

    018a      |      Vol_IIA-0178                                                                                                                  

    Fig 98

    Fig 109

    019      |      Vol_IIA-0179                                                                                                                  
    EA-I. Terzaghi: Ground Frost

    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.

    020      |      Vol_IIA-0180                                                                                                                  
    EA-I. Terzaghi: Ground Frost

            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

    021      |      Vol_IIA-0181                                                                                                                  
    EA-I. Terzaghi: Ground Frost

    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

            Ho (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

    Ho = 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.

    022      |      Vol_IIA-0182                                                                                                                  
    EA-I. Terzaghi: Ground Frost

            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

    023      |      Vol_IIA-0183                                                                                                                  
    EA-I. Terzaghi: Ground Frost

    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



    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

    024      |      Vol_IIA-0184                                                                                                                  
    EA-I. Terzaghi: Ground Frost

    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

            Hp = 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.

    025      |      Vol_IIA-0185                                                                                                                  
    EA-I. Terzaghi: Ground Frost

            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.

    026      |      Vol_IIA-0186                                                                                                                  
    EA-I. Terzaghi: Ground Frost

    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

    026a      |      Vol_IIA-0187                                                                                                                  

    Fig 1615

    027      |      Vol_IIA-0188                                                                                                                  
    EA-I. Terzaghi: Ground Frost

    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.

    028      |      Vol_IIA-0189                                                                                                                  
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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

    HD = 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

    029      |      Vol_IIA-0190                                                                                                                  
    EA-I. Terzaghi: Ground Frost

    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

    030      |      Vol_IIA-0191                                                                                                                  
    EA-I. Terzaghi: Ground Frost

    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.

    030a      |      Vol_IIA-0192                                                                                                                  

    Fig 1716

    031      |      Vol_IIA-0193                                                                                                                  
    EA-I. Terzaghi: Ground Frost


    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


            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.

    032      |      Vol_IIA-0194                                                                                                                  
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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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    EA-I. Terzaghi: Ground Frost

    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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    EA-I. Terzaghi: Ground Frost

    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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    EA-I. Terzaghi: Ground Frost

    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


            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

            qu = 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,

    Consistency q u value,

    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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    EA-I. Terzaghi: Ground Frost

            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

    [Math Formula]

    q’u = qi 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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    EA-I. Terzaghi: Ground Frost


    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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