Skip to main content
 Previous Next
  • Zoom In (+)
  • Zoom Out (-)
  • Rotate CW (r)
  • Rotate CCW (R)
  • Overview (h)
Physics of Ground Frost: Encyclopedia Arctica 2a: Permafrost-Engineering
Stefansson, Vilhjalmur, 1879-1962

Physics of Ground Frost

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
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

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

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

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

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.

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
<formula>n = (volume of voids)/(total volume of soil)</formula>
or by the void ratio
<formula>e = (volume of voids)/(volume of solid constituents)</formula>
The water content w of a soil is commonly given in per cent of the dry
weight,
<formula>w = 100 x (weight of water)/(weight of dried soil)</formula>

EA-I. Terzaghi: Ground Frost

If the soil contains both water and air, the degree of saturation is deter–
mined by the ratio
<formula>Sr(%) = 100(volume occupied by water)/(total volume of voids)</formula>
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
<formula>Dr = (eoe)/(eoemin)</formula>
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

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 .

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,
<formula>Cr = (Lww)/(LwPw) = (Lww)/(Iw)</formula>
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 Fig. 2

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.

EA-I. Terzaghi: Ground Frost

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
<formula>Q = Aki</formula>
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
<formula>ko (cm. per sec.) = (100 to 150) x D210</formula>
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
<formula>k = 1.4kDe2</formula>
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.

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 Fig 5

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

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.

EA-I. Terzaghi: Ground Frost

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
[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

EA-I. Terzaghi: Ground Frost

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 .

EA-I. Terzaghi: Ground Frost

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
<formula>ch = (0.19 + w)/(1 + w) cal.gm.-1(°C.)-1</formula>
If the soil freezes the heat capacity decreases to the value
<formula>c’h = (0.19 + 0.43w)/(1 + w)</formula>
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.

EA-I. Terzaghi: Ground Frost

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] <formula>d = k/(ch x wt)</formula>
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

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 Fig. 7

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
<formula>Qh = Aktig</formula>
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

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 Fig 98 Fig 109

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
<formula>t1 = z21/12d</formula>
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
<formula>t1 = 15202/(12 x 45 x 10-3) = 43 x 105 sec. = 50 days</formula>
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.

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
<formula>z1 = √(12dt1)</formula>
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

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
<formula>Ho (cm.) = √(12d x 12 x 3600) = 720√(d)</formula>
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
<formula>Ho = 720√(8 x 10-3) = 62 cm.</formula>
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.

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

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
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

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
<formula>Hp = 10/0.033 = 300 m. = 1000 ft.</formula>
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.

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.

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 Fig 1615

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.

EA-I. Terzaghi: Ground Frost

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
<formula>HD = 1.5/0.033 = 45 m. = 150 ft.</formula>
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

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

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.
Fig 1716

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
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.

EA-I. Terzaghi: Ground Frost

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

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

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
<formula>h = 0.1nH</formula>
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

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
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

EA-I. Terzaghi: Ground Frost

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
<formula>s = c + p tan(ɸ)</formula>
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 ) ,

EA-I. Terzaghi: Ground Frost

the unconfined compressive strength q u or the unit load under which a
cylindrical specimen fails [: ] is equal to
<formula>qu = 2c tan(45 + ɸ/2) </formula>
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

EA-I. Terzaghi: Ground Frost

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.

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] <formula>q’u = qi tan(45 + ɸ /2)</formula>
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.

EA-I. Terzaghi: Ground Frost

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.

HomePhysics of Ground Frost : Encyclopedia Arctica 2a: Permafrost-Engineering
 Text Only
 Text & Inline Image
 Text & Image Viewer
 Image Viewer Only