GEOLOGICAL SURVEY CIRCULAR 632 -

Some Estimates of the 

Thermal Effects of a 

Heated Pipeline in Permafrost 

t 

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Some Estimates of the 

Thermal Effects of a 

Heated Pipeline in Permafrost 

By Arthur H. lachenbruch 

G E 0 L 0 G I C A L S U R V E Y C I R C U L A R 632 

Washington 1970 
United States Department of the Interior 

WALTER J. HICKEL, Secretary 

Geological Survey 

William T. Pecora, Director 

First printing 1970 

Second printing 1970 

Free on application to the U.S. Geological Survey, Washington, D.C. 20242 
FOREWORD 

One of the greatest mineral resource discoveries in the history of man is a colossal 

petroleum resource near Prudhoe Bay on the Arctic Slope of Alaska. Today, industry 

and government are engaged in an important dialogue as to how this resource may be 

developed and transported to market, and in what time frame, in the best interests of 

the Nation. The petroleum industry, by its own initiative, determined that a 800-mile 

pipeline, four feet in diameter, should be constructed reaching from Prudhoe Bay across 

all of Alaska to the ice-free south coast port of Valdez. 

Since the early years of this century, geologists and geophysicists of the U.S. Geological Survey have been investigating permafrost in Alaska. During the past six months, a 

concentrated effort has been made by our scientific and technical personnel to evaluate 

conditions of the terrain, in response to industry's request for a permit for a right-ofway and permit to construct the pipeline across Federal domain. 

A year or so ago the concept of the pipeline was being considered in terms of cold 

oil being transported. Subsequently, however, the Trans Alaska Pipeline System (TAPS) 

announced that a throughput of oil at a maximum capacity in the order of two million 

barrels per day, requiring several pumping stations, would result in maintaining a high 

temperature of the oil from the point of entry on the North Slope to its delivery point 

at Valdez. Estimates of the temperature of the flowing oil range between 150°F. to 

180°F. 

To bury a hot pipeline in hard rock or in coarse rocky terrain does not cause concern 

because the foundation is firm enough to support the pipeline without danger of disturbing it. But to bury a hot pipeline in permafrost ground composed of unconsolidated sediments with relatively high ice content could give us grave concern, not because of the 

melting by itself, but because of the consequences of the melting if safeguards were not 

designed into the engineering system. 

We have made detailed studies over the past few months in critical areas and have 

evaluated the possible effects through utilization of a computer program. We believe 

that rapid thawingofthepermafrostincriticalareas could have most significant effects 

on the local environment and upon the security of the pipeline itself. It is important, 

therefore, that these effects be anticipated and predicted as accurately as possible so 

that the problems can be accommodated by proper engineering design above or below 

ground or by selection of alternative routes. 

It is with these concerns in mind that the results of our preliminary studies are being 

made available at this time. This report presents an evaluation of the potential problems 

and effects of a buried hot oil pipeline in critical soil areas of the arctic and sub-arctic 

environment. 

W. T. Pecora 

Director, U.S. Geological Survey 

III 

Foreword-----------------------------

Abstract------------------------------

Introduction---------------------------

Temperature effects -------------------

Thawing in the first two decades-------

Thawing after very long times-equilibrium results--------------------

Effects of changes in pipeline ternperature--------------------------

Some subsurface manifestations of 

thawing---------------------------

Liquefaction-------------------------

Flow --------------------------- Buoyancy and related problems----

Convective thawing-------------------

Seismic liquefaction------------------

Differ entia! settlement ---------------

Some engineering considerations --

Ice wedges----------------------

CONTENTS 

Page 

III 

1 

2 

3 

4 

7 

8 

8 

8 

10 

12 

12 

13 

13 

13 

13 

Some subsurface manifestations of 

thawing--Continued. 

Differential settlement--Continued. 

Settlement in more homogeneous 

materials--------------------

Some surface effects-------------------

Thermal gradients and temperatures 

near the surface-------------------

Some hydrologic considerations -------

Snowmelt -----------------------

Evaporation---------------------

Water from permafrost-----------

Surface drainage-----------------

Convective heat transfer and the 

hydrologic balance------------

The construction disturbance----------

Concluding statement-------------------

Appendix A-numerical solutions --------

Appendix B-steady-state results--------

ILLUSTRATIONS 

Figure 1. Diagram illustration used in discussion of thermal effects of a hot pipeline 

Page 

15 

16 

17 

17 

18 

18 

18 

18 

19 

20 

20 

21 

21 

Page 

buried in permafrost ---------------------------------------------------------- 4 

2. Growth of the thawed cylinder around the pipe-------------------------------------- 5 

3. Vertical temperature profiles through the axis of the pipe --------------------------- 6 

4. Depth of thawing beneath the pipe ------------------------------------------------- 7 

5. Equilibrium temperature gradient at the surface for conductive 

transfer in homogeneous materials---------------------------------------------- 17 

6. Equilibrium position of the thawing interface--------------------------------------- 23 

v 
VI 

TABLES 

Page 

Table 1. Pa.rameters used in the thermal calculations--------------------------------------- 6 

2. Conditions for potential liquefaction----------------------------------------------- 9 

3. Rate of thawing beneath pipe as a function of time----------------------------------- 9 

4. Flow of viscous cylinder --------------------------------------------------------- 10 

5. Instability of a visco-plastic cylinder---------------------------------------------- 11 

6. Enhancement of heat flow by movement in direction of thermal gradient 

between surfaces separated by a distance, with a temperature difference of 80°C ---·-- 12 

7. Potential hydrologic effects of heat conducted to the surface from a buried pipe-------- 18 

8. Idealized relation between permeability and convective enhancement of 

surface heat .loss (D=8 ft) ----------------------------------------------------·-- 19 
Some Estimates of the Thermal Effects 

of a Heated Pipeline in Permafrost 

By Arthur H. Lachenbruch 

Abstract 

As one means of transporting crude oil from oil 

fields on the Alaskan Arctic coast to a year-round 

ice-free port, a large pipeline traversing most of the 

state of Alaska from north to south has been proposed. 

Plans call for a pipe 4 feet in diameter, which will be 

buried along most of the route in permafrost. According to preliminary estimates. the initial heat in the oil 

plus frictional heating in the pipe are expected to maintain oil temperatures in the neighborhood of 70° to 80°C 

(158° to 176°F) along the route when full production is 

achieved. Such an installation would thaw the surrounding permafrost. Where the ice content of permafrost 

is not high, and other conditions are favorable, thawing 

by the buried pipe might cause no special problems. 

Under adverse local conditions, however, this thawing 

could have significant effects on the environment, and 

possibly upon the security of the pipeline. It is important that any potential problem be identified prior 

to its occurrence so that it can be accommodated by 

proper pipeline design. Identifying a problem in advance depends upon an understanding of the conditions 

under which the problem will occur. For that reason 

much of this report is concerned with problems. If 

the pipeline system is properly designed, and if it is 

constructed and maintained in compliance with the design, they will not occur. Perhaps "proper design" in 

some areas will involve abandoning plans for burial 

or changing the route; in others it might involve burying the pipe and invoking special engineering designs 

or monitoring procedures. These are matters to be 

determined by much additional study and an intensive 

program of field and laboratory measurements of 

conditions along the route. 

In this report a few basic principles are applied to 

simplified models of permafrost regimes to identify 

some effects of a heated pipe, the conditions that control them, and the approximate ranges of physical 

properties for which these effects are likely to result 

in problems. The computations are approximate, and 

the problems discussed are only illustrative examples. 

Comprehensive discussions of these and re 1 ate d 

effects, taking account of physical and theoretical refinements, are beyond the scope of the report. Refined 

studies will probably be needed, however, to form an 

adequate basis for engineering design. 

It is difficult to summarize these effects briefly, 

but a few will be mentioned. The reader is urged to 

consult the full text for a more complete statement of 

the -conditions under which they are likely to occur. 

It should be emphasized that whether or· not such conditions exist is a matter yet to be determined b·r measurements on permafrost materials along the ·pipeline 

route. Such measurements are essential for predictions of the interaction between the pipeline and its 

environment. 

A 4-foot pipeline buried 6 feet in permaf-:-ost and 

heated to 80°C (176°F) will thaw a cylindrical region 

20 to 30 feet in d i am e t e r in a few years ir typical 

permafrost materials. At the end of the second decade 

of operation, typical thawing depths would be 40 to 50 

feet near the southern limit of permafrost a'ld 35 to 

40 feet in northern Alaskawherepermafrosti£' colder. 

Except for special materials near the northern end 

equilibrium conditions will not be reached and thawin~ will continue throughout the life of the pipeline, but 

at a progressively decreasing rate. If the thavred material or the water within it flows, these am':lunts of 

thawing can be increased several fold. If the pipeline 

temperature were only 30°C (instead of 80°C), t~e depth 

of thawing would probably be reduced by only 30 or 40 

percent. The principal effect of insulating the pipe 

would be to increase oil temperatures rather than to 

decrease thawing. 

If permafrost sediments have excess ice an1 a very 

low permeability when thawed, melting below the pipe 

could generate free water faster than it could filter to 

the surface. As a result the material in the thawed 

cylinder could persist as a semiliquid slurry. Where 
permeabilities are very low and excess ice contents 

are moderate, thawing rates could be sufficient to 

maintain this state for decades. 

If the strength of these slurries is less than 1 pound 

per square foot, they will flow with substantial velocities on such imperceptible slopes as are characteristic of "flat" basins. Theentirethawedcylinderwould 

tend to flow like a viscous river and seek a level. As 

an extreme example, if these slurries occured over 

distances of several miles on almost imperceptible 

slopes, the uphill end of the pipe could, in a few years, 

be lying at the bottom of a slumping trench tens of 

feet deep, while at the downhill end, millions of cubic 

feet of mud (containing the pipe) could be extruded out 

over the surface. Where the pipe settled to the bottom 

of the trench~ it would accelerate thawing and flow, and 

the process could be self-perpetuating. The pipeline 

could be jeopardized by loss o~ support in the trench 

and by displacement in the mudflow, and the disruption 

to the landscape could be substantial. Where such extreme conditions might occur, these problems could 

normally be anticipated by observations prior to or 

during construction. Less extreme conditions leading 

to partial liquefaction might be more d iff i c u 1 t to 

identify in advance. 

Where the pipe passed from strong material into a 

liquefied region, it would tend to float or sink, depending on the density of the slurry. It could be severly 

stressed by the resulting forces. 

Almost imperceptible systematic movements of the 

thawed material can accelerate the thawing process 

locally by as much as a thousand times. Hence if flow 

occurs the ultimate amount of thawing can be very great 

and difficult to predict. 

Seismic vibrations can cause loosely packed saturated sands and silts to liquefy. Hence where such material occupied the thawed zone around the pipe, the 

flow, buoyancy, and convective effects just discussed 

could also be caused by an earthquake. The southern 

part of the pipeline route lies in an active seismic 

zone. 

Differential settlement causing shearing stresses 

in a pipe can result from a variety of processes-the 

most conspicuous of which is probably the thawing of 

ice wedges. These massive vertical veins of ice form 

tight polygonal networks, commonly invisible from the 

surface and difficult to delineate with borings. They 

are widely distributed in northern Alaska. A pipeline 

crossing ice-wedge networks at random angles would 

thaw the wedges quickly and could thereby lose support 

over considerable spans. A statistical calculation suggests that in typical ice-wedge terrain, conditions which 

might exceed the design stress of the pipeline could 

occur on the average of once every mile. Most of these 

conditions could be anticipated by observations made 

during trenching. 

Settlement of the pipe due to thawing is a cumulative effect of all of the thawed material beneath it. 

2 

Only a negligibly small fraction of this material will 

be directly observed in separated bore holes. Rather 

small and subtle changes in porosities, noisture content, and other properties occurring over lateral distances of tens of feet could cause differential settlement resulting in excessive stress on the pipe. Such 

changes may be difficult or impossible to detect in 

advance by trenching and boring, even if holes were 

drilled every 1,000 feet along the route. 

Where the sediments are saturated or oversaturated, 

a trench one or more feet deep and tens of feet wide 

will probably develop over the pipeline in a few years; 

it will deepen and widen somewhat as time progresses. 

Where the trench is discontinuous it could create a 

series of ponds which could enlarge by thermal processes under certain conditions. Indiscriminate drainage of these ponds could create excessive stress on the 

pipe by removing buoyant forces that might be partially 

supporting it in these differentia 11 y settled zones. 

Where the trench above the pipeline is continuous, 

it could become a stream channel, altering drainage 

patterns and creating erosion problems a long the pipeline. 

Heat conducted from the pipe to the sur:"ace will have 

a significant effect on surface temperatures and plantroot temperatures over a band not more than about 60 

feet wide. Directly over the pipe snow will probably 

remain on the ground only after the hervier storms. 

The inflow of water into the depresrion likely to 

develop over the pipe will probably more than supply 

the heat requirements for excess evaporation, and in 

general conditions will probably remain YTet. However, 

if the material overlying the pipe should be very permeable, thermal convection of water in the sediments 

could probably increase heat loss from the pipe to the 

surface one-hundred fold. Under these conditions evaporation would probably exceed the rate of local water 

supply and the region above the pipe could eventually 

become desiccated. Heat and moisture transfer above 

the pipe could have a significant effect on the formation 

of local ground fog. 

This study was not exhaustive. Poten-:ial problems 

certainly exist that have n::>t been considered, and some 

that have been considered may be shown rr:>t to exist by 

further studies. The report represents one perspective 

on an overall problem that transcends many disciplines 

and requires the perspectives of many hr an optimal 

solution. It is hoped that the report will provide one 

reference point for objective discussio"' between the 

people of many backgrounds who must communicate 

effectively on this issue. 

INTRODUCTION 

As one means of transporting crude oil from oil 

fields on the Alaskan Arctic coast to a year-round 

ice-free port, a large pipeline traversing most of the 

state of Alaska from north to south has b~en proposed. 

Plans call for a pipe 4 feet in diameter, which will be 
buried along most of the route in permafrost. According to preliminary estimates the initial heat in the oil 

plus frictional heating in the pipe are expected to maintain oil temperatures in the neighborhood of 70° to 80°C 

( 158° to 176°F) along the route when full production is 

achieved. Such an installation would thaw the surrounding permafrost. Under adverse local conditions this 

thawing could have significant effects on the environments, and possibly upon the security of the pipeline. 

Although adverse conditions may be present along only 

limited parts of the pipeline route, this report focuses 

on them and the problems they could create. It is important that these problems be anticipated as accurately as possible so that they may be avoided by appropriate engineering procedures or by the selection 

of alternative routes. 

The logical starting point for the study of thermal 

effects of a hot pipeline buried in permafrost is an 

estimate of the amount of thawing that will take place 

at various times after installation for the range of 

climatic conditions, permafrost properties, and pipeline temperatures that might be anticipated along the 

route. At the time of this writing no such study had 

been made, and hence no scientifically valid estimates 

of the thawing were available. Therefore as a starting 

point for the present study the theory of heat conduction has been applied in an attempt to provide such 

estimates; the results are described in the section 

that follows. Of the various quantities required for 

this analysis, the climatic conditions are reasonably 

well known and the range of possible permafrost properties can be assigned within rather specific limits 

from previous scientific studies. However, it was necessary to use a preliminary estimate of the temperature of the oil in the pipeline. This temperature is not 

yet known accurately, as it depends upon factors relating to production and transmission which are still 

under study. 

Once the thawing has been estimated, it is possible 

to apply a few basic principles to simplified models 

of permafrost regimes and thereby to identify some 

potential problems, the conditions that control them, 

and the approximate ranges of the physical properties 

for which the problems are likely to occur. In the section "Some subsurface manifestations of thawing" this 

procedure is used to discuss some conditions under 

which thawing might result in large stresses in the 

pipe or in mudflows on gentle slopes. The computations are approximate ones designed only to identify 

the order of magnitude of pertinent quantities. The 

specific problems discussed are only illustrative examples. Comprehensive discussions of these andrelated problems, taking account of stratification of the 

sediments and other important physical and theoretical refinements, are far beyond the scope of this report. Such studies will probably be needed, however, 

to form an ad e q u ate basis for engineering design. 

Where the ice content of permafrost is not high, 

and other conditions are favorable, thawing by a buried 

pipe might cause no special problems, and this could 

3 

very well be the common case. However, it is important that any potential problem be identified prior to 

its occurrence so that it can be accommodated by proper pipeline design. (Because our knowledge rr ay never 

be so complete as to achieve this ideal, it is desirable 

to develop monitoring procedures to detect a develop ... 

ing problem in time for remedial action.) Id~ntifying 

a problem in advance depends upon an understanding 

of the conditions under which the problem will occur. 

For that reason much of this report is concerned with 

problems. If the pipeline system is properly designed, 

and if it is constructed and maintained in co:-npliance 

with the design, they will not occur. Perhaps "proper 

design" in some areas will involve abandon~ng plans 

for burial or changing the route; in others it might 

involve burying the pipe and supporting it 1'? buried 

piling or exercising other engineering pre~autions. 

These are matters to be determined by much additional study and an intensive program of detsiled field 

and laboratory measurements of conditions along the 

route. 

Basic engineering problems relating to the security 

of the pipeline are of concern to ecologists to the extent that if they are not properly solved, trey might 

result in mudflows, oil spills, erosion, and so on. 

Apart from these major considerations there are other 

physical perturbations of the environment· that cannot be evaluated by ecologists until estimates of their 

magnitude are provided. These relate to such questions as "What will be the disturbancetothe temperatures at the surface and at plant-root level? Over how 

wide a band will it be appreciable? Will there be a 

winter snow cover over the pipeline? Will tre ground 

over the pipe be saturated or desiccated?" An attempt 

is made to provide approximate answers tC' some of 

these questions in the section "Some surface effects." 

To be useful it is n e cess a r y that this report be 

quantitative; it is based upon analyses and com':'lutations 

of many kinds. However, in order to keep it readable 

by nontechnical people with an interest in this general 

subject, no mathematical expressions are UE"ed in the 

text. Pertinent theoretical results are summarized in 

numerical tables or described verbally. Th?. mathematical foundation for the thermal calculations is presented in appendixes. 

The writer gratefully acknowledges the assistance 

of his colleagues, Patrick C. Doherty, who programmed 

the numerical solution for the heat conduction equation, 

and Reuben Kachadoorian, who helped gather geologic 

and engineering information essential for tr·is study. 

The writer is also grateful to them and to many other 

colleagues in the Geological Survey, in universities, 

and in industry for valuable discussions and for constructive comments on the draft manuscript. 

TEMPERATURE EFFECTS 

The purpose of this section is to obtain ro'.tgh estimates of the amount and rate of thawing and of the configuration of the thawing surface for the complete range 
of climatic conditions and sediment properties expected along the pipeline route. Figure 1 illustrates 

the notation used for the discussion. The pipe temperature (Tp) will be taken as 80°C, and the depth of burial 

of the pipe's axis (D) will be taken as 8 feet unless 

otherwise stated. Most of the results presented were 

obtained by the numerical solution of the differential 

equation of head conduction described in appendix A. 

The problem was programmed by Patrick C. Doherty, 

and it is documented in detail in a separate report. 1 

The numerical results were checked with analytical 

approximations, and separate analytical results were 

obtained for the equilibrium case (appendix B). 

'To 

/To 

and these values serve amply to bracket the possibilities. Vertical temperature profiles through the centerline of the pipe are illustrated for the arctic case in 

figure 3. In the solution leading to these results, provision has been made for the annual thawing and refreezing of near-surface materials wit'l the passage 

of the seasons (appendix A). The solid curves (fig. 2) 

denote conditions in the month of April and represent 

a more or less average annual condition. The dashed 

curves marked "(September)" represent conditions in 

the fall of the year at the time of maYimum annual 

thawing. They illustrate how the frozen c.ap can be expected to roll back each year to provirle continuity 

GROUND SURFACE 

To/ 

THAWED 

FROZEN To~ 

Figure 1.--Notation. Ta is the average annual temperature of the ground surface and of the undisturbed permafrost. Tp is the pipe te~perature.and Tf }s the freezing temperature (QOC). 

D is the depth of burial ot the pipe ax1s and R lS the dlstance of the thawing surface beneath the pipe axis. 

THAWING IN THE FIRST TWO DECADES 

Figure 2 illustrates the approximate positions of the 

thawing surface for various times after emplacement of 

an 80°C pipe in a typical permafrost medium. The 

curves on the right-hand side of the figure represent 

conditions near the southern limit of permafrost; for 

example, in the Copper River Basin, where the mean 

annual ground temperature (T a• fig. 1) is taken as 

-0.8°C. This condition will be termed the "interior 

case." The curves on the left denote conditions near 

the Arctic coast where the mean annual ground temperature is taken as -8. 9°C. This will be referred to 

as the "arctic case." The true range of ground temperatures is well known from many previous studies, 

1That report is released in open file as a 

Geological Survey Computer Contribution under 

the title "Hot Pipe" by Patrick C. Doherty 

and is available upon request from the Computer Center Division, U.S. Geological Survey 

Washington, D. C. 20242. 

4 

between the naturally thawed material in the active 

layer and the artifically thawed cylinder around the 

pipe. September curves are shown in fig"Jre 2 only for 

the 1- and 20-year cases, but the analogous curves 

for the other times are easily visualizec', as the depth 

of the shoulder is about the same for eacL The curves 

in figure 2 marked "t=oo (April)" represe'1tthe equilibrium configuration approached after a very long time 

(appendix B). They are discussed further under" Thawing after very long times-equilibrium r~sults." 

Figure 2 indicates the effect of the extreme range 

of climate on the thawing induced in a sirgle material. 

It is necessary to perform similar anal~rses on other 

materials to understand how the thawing: might be influenced by variations in physical prorerties of the 

permafrost along the route. The parame<:ers tabulated 

in table 1 represent the materials chosen for this purpose. The material designated "silt ( 17~ percent)" 

is the one represented in figures 2 and 3. In table 1, 

only four quantities are chosen independently; the 
others are derived from them. The four independently 

chosen parameters are actually severely constrained, 

as they must represent a combination that occurs typically in nature. Their selection was based upon extensive measurements made over several years in 

the vicinities of Barrow, Umiat, Fairbanks, andGlenallen, Alaska. For convenience the hypothetical materials used in the examples are designated by a sediment name, "clay," or "silt," with the percent moisture by dry weight in parentheses. Actual materials 

with these designations would typically have properties 

like the ones chosen, but other sediments can, of 

Approximate analytical formulations of th~ thawing 

problem suggest that moisture content by volume is 

the parameter of overriding importance in det,~rmining 

rate of thawing. This is confirmed by the n·1merical 

results pre sent e d in figure 4 for the dept'1 of thaw 

beneath the pipe axis (R) as a function of t;me. -The 

curves labeled "dry (0 percent)" represent a short 

computer run on material with the properties of silt 

( 17~ percent) with the moisture removed. Typical 

dry sediments tend to have high thermal diffusivities 

subject to rather little variation, and they are uninteresting insofar as thawing permafrost is ccncerned. 

~----ARCTIC CASE---------~;----- INTERIOR CASE------~ 

I-IORIZONTAL DISTANCE FROM CENTERLINE (FEET) 

-4~0 __________ -~3L0--------~-2~0~--------~-ILO--~==~;;~Or--=--~~~10~;::======2~0======~~~~~~------~40 

T~AWED 

20 

DEPT~ (FEET) 

Figure 2.-Growth of the thawed cylinder around a 4-foot-diameter pipe with its axis 8 feet 

beneath the surface and its temperature maintained at 80°C. Curves on the left represent 

conditions near the Arctic coast; those on the right represent conditions near the southern 

limit of permafrost. Dashed curves represent conditions at the conclusion of summer thawing 

(September) for 1 year and 20 years. Calculations were made by the method of appendix A for 

the medium "Silt 07t percent)" in table l. 

course, have similar thermal properties. The material 

designated "water/ ice" has properties almost identical 

to saturated peat, and insofar as thawing is concerned, 

similar to some very high ice content inorganic materials incorporated into permafrost in special ways. 

5 

The reasons for including these curves are to present 

an upper limit and to illustrate the great retarding 

effect of moderate amounts of moisture on thawing. 

There are two reasons for this effect, the most obvious being the great amount of heat required to change 
j..PIPE_.j 

ARCTIC CASE 

-8.9 !,-----~-----+------::~-----};~-----;!;;:-----~ 0 10 20 30 40 50 60 

Depth (feet) 

Figure 3.--Vertical temperature profiles through the axis of the pipeline (April). 

Table l.--Parameters used in the thermal aaZauZations 

[Parameters: Thermal conductivity, volumetric specific heat, moisture content, and wet 

density are independently chosen parameters; the others are derived] 

Silt (17 1/2 percent) Clay (65 percent) Water/ice 

Parameter 

Frozen Thawed Frozen Thawed Frozen Thawed 

Thermal conductivity 

(meal per em sec °C)----- 5.0 3.4 4.0 2.5 5.4 1.2 

Volumetric specific heat 

(cal per cm 3 °C)--------- .42 .57 .52 .83 .45 1.00 

Thermal diffusivity 

(cm2 per sec)------------ .012 .006 .008 .003 .012 .0012 

Latent heat per unit 

volume (cal per cm3 )----- 21.6 5.0.4 72 

Moisture content (percent 

wet weight)-------------- 15 39.5 100 

Moistur.e content (percent 

dry weight)-------------- 17.5 65 

Moisture content (percent 

by volume)--------------- 28.5 63 100 

Wet density (gm per cm 3 )--- 1.8 1.8 1.6 1.6 .9 1.0 

Wet density (lb per cu 

ft)---------------------- 112 112 100 100 56.2 62.4 

ice into water. Also important, however, is the anomalously high volumetric specific heat of water which 

demands large amounts of heat to raise the temperature 

of wet sediments after they are thawed. Both of these 

quantities, the latent heat and the volumetric specific 

heat, are best indicated by the moisture content by 

volume (see table 1 ). Moisture content by dry weight 

6 

is used to designate the materials, hov·<?.ver, as this 

parameter is more familiar to those concerned with 

the description of soils. 

Figure 4 taken with figure 2 yields a rather complete description of the range of effec<:s of climate 

and properties on thawing for the first two decades 
-

Q) 

~ 

INTERIOR 

CD 70 

60 

INTERIOR 

ARCTIC 

INTERIOR 

ARCTIC 

INTERIOR 

ARCTIC 

or-~--~--~--~--L-~--~--~--~~~--

0 4 6 8 10 12 14 16 18 20 

TIME (years) 

Figure 4.--Depth of thawing beneath pipe axis 

(R) for media described in table 1. D=8 

feet. 

of operation of the pipeline. The case of water/ice 

gives a confident lower limit; most materials of interest will lie in the range for the silt and clay cases. 

Each of the curves for water/ice yield thawing distances (R) of about 50 percent of the corresponding 

values for silt, and those for clay are about 75 percent 

of the values for silt. A ro'..lgh indication of the complete configuration of the thawing surfaces for these 

materials can be obtained by applying scale reductions 

of those amounts to figure 2. (It should be mentioned 

that all of these numerical results may be subject to 

systematic adjustments (downward) on the order of 10 

percent as a result of a geometric simplification in 

the numerical solution (see appendix A). Such discrepancies are unimportant in the context of the present 

discussion.) 

One useful additional result can be derived from the 

calculations presented. The total amount of water 

thawed from permafrost after a given length of time 

is relatively independent of the moisture contentofthe 

material as long as at least m ode rat e amounts of 

moisture are present. The more ice-rich sediments 

thaw more slowly, but in these, a little thawing produces more water. This point will be elaborated in 

a later section. 

7 

THAWING AFTER VERY LONG TIMES-EQUILIBRIUM 

RESULTS 

If the pipe is maintained at a uniform terroerature 

for a very long time, heat will eventually leave the 

earth's surface as fast as it is being supplied by the 

pipe and the ground temperatures will remain steady. 

This equilibrium condition, which is easily ctescribed 

mathematically, has been the basis for previous discussions regarding thawing by the pipeline. ""be solution is based on the assumptions that the ground surface temperature is uniform, that the pipe terr1)erature 

eventually reaches a constant value, and tha': the materials are homogeneous. The solution prov~s that as 

long as these assumptions are satisfied, the f1.nal position of the thawing surface is a circle whose size and 

position is completely independent of the properties 

of the material; it depends only on the initial temperature conditions. 

A refinement of this solution (equations B-11 and 

B-12, appendix B) takes account of the chanr:ein conductivity on thawing, but even that result de}l~nds only 

upon a ratio of conductivities and not upon the absolute 

value of any property of the medium. The solution says 

nothing about the time required to approach this equilibrium configuration, and this time does de:->end upon 

properties, including in a very sensitive way the moisture content. It also depends upon the distanc~ the heat 

has to travel to achieve equilibrium. Figur~ 3 shows 

that immediately over the pipe, equilibrium is achieved 

in the first year. However, beneath the pir~, it will 

require on the order of a half century to approach 

equilibrium in typical materials in the arctic (except 

for the case of water/ice), and thousands of years to 

approach it in the interior (appendix B). It ''rould appear that during the life of the pipeline eqr1ilibrium 

conditions generally will not be reached, anct conductive thawing will continue, though at a progressively 

decreasing rate in typical materials. 

The curves labeled "t=co (April) "in figm·e 2 were 

computed from the refined solution in appenrlix B. The 

much greater equilibrium thawing for the interior case 

is typical, generally about 10 times as great as for 

the arctic case (appendix B). It is seen from figure 4, 

however, that during the first two decades results for 

the arctic and interior cases differ by only 10 or 20 

percent. For longer times the arctic and interior 

curves will diverge, and typically this divergence will 

occur earlier in the drier materials. (The case water/ 

ice is anomalous, as there the strong contrast in conductivity leads to an early conductive equilibrium state 

in the arctic.) 

Thus considerations of the equilibrium results alone 

can be very misleading. They suggest that thawing is 

very insensitive to material properties and that the 

quantity of overriding importance is climate, which 

causes variations in thaw by an order of magnitude 

along the pipeline route. Results for times of practical 
importance (fig. 4) indicate just the opposite; thawing 

is rather insensitive to climate, but sedimentproperties, particularly moisture content, are of overriding 

importance. 

EFFECTS OF CHANGES IN PIPELINE TEMPERATURE 

In addition to the sediment properties and permafrost temperature, the pipeline temperature is, of 

course, an important parameter in the thermal problem. It is worth considering the effects of lower pipe 

temperatures for two reasons: ( 1) The estimated value 

of 80°C may be subject to revision when detailed engineering studies are completed of heat balance along 

the pipeline, and (2) it is worsh considering whether 

sufficient advantage might accrue from lower pipe 

temperatures to justify de I i berate procedures to 

achieve them. 

The assumption that the pipe temperature will be 

80°C from the time the flow starts is unrealistic. The 

temperature wiii probably start out much lower and 

rise over a period of years as production increases 

and as heat loss to the warming permafrost decreases. 

For most purposes, however, quite satisfactory upper 

and lower limits to thawing under this condition can 

be' obtained by adjusting the time origin in figure 4. 

For example, if the pipe I in e temperature built up 

gradually to 80°C over a period of 4 years, then the 

thawing depths after 10 years would lie somewhere 

between the values given in figure 4 for 6 and 10 

years. 

It can be shown from an approximate an a I y tic a 1 

formulation of the thawing problem, that if the steady 

pipeline temperature were reduced frotn 80° to 30°C 

(that is, by 50°C), the thawing depths indicated in figure 4 would probably be reduced by only 30 or 40 percent in general. Hence substantial amounts of thawing 

can be expected promptly even for pipeline temperatures much lower than that assumed. 

Variation in the depth of burial of the pipe axis, say 

from 8 to 5 feet, is not expected to have a very large 

effect on the thaw depth (R of fig. 4). In general, however, deeper burial will result in somewhat greater 

thawing. 

If conventional insulation were placed around the 

pipe, its principal overall effect would be to increase 

oil temperatures rather than to decrease thawing. 

SOME SUBSURFACE MANIFESTATIONS OF 

THAWING 

LIQUEFACTION 

The most important difference between permafrost 

sediments and familiar unfrozen sediments stems from 

the fact that water generally has considerable strength 

when it is frozen (ice), and when it is thawed, it does 

not. In familiar stable sediments, gravity tends to pull 

the sediment grains together until they contact, and 

8 

hence the volume of water generally cannot exceed 

the volume of the voids left between contacting grains. 

No such :r,:ule applies to permafrost, as it is perfectly 

stable even ·when it contains large masses of p u r e 

water as ice. Excess ice (that volume, which after 

melting, cannot fit in the soil voids) can be incorporated into permafrost in various ways during its initial 

formation, and subsequently by the growth of ice 

wedges. When such permafrost thaws, it is no longer 

stable, and the mineral grains tend to settle and to 

exclude the excess water. Where this rettling is occurring the medium tends to behave as a dense fluid 

and has (virtually) no strength. Whether the I i quid 

state persists long enough to have adverse mechanical effects depends upon the rate of this settling out, 

or equivalently, the balance between the rate at which 

excess water is supplied by thawing and the rate at 

which water can get out of the thawed se<iiment. This, 

in turn, depends upon the properties of the sediment 

and the configuration of the impermeable boundary 

with the surrounding frozen material. 

Imagine the pipe to be embedded in sediments containing excess ice. As the thawed cylind~r (illustrated 

in fig. 2) grows, excess water will be generated at 

the advancing interface. If it can migrate to the ground 

surface as fast as it is generated, the grains can settle 

down to contact one another, and the thawed sediment 

will gain increasing strength by consolication ( excluding additional water). If the rate of mir,ration cannot 

keep pace with the rate of generation, excess water 

will persist in the thawed cylinder, part or all of which 

will tend to behave as a fluid. Predicting which type 

of behavior will occur is a very comple:~ problem requiring careful study of the particular materials involved by an experienced soils engineer. However, the 

simple model described in the following paragraph 

yields order-of-magnitude results of Eome interest. 

The excess water can migrate to the surface either 

by homogeneous flow through the mate-v-ial above it, 

or by channeled flow through or circumferentially 

around it. If the flow is not channeled and the material 

is homogeneous, a critical condition oc~urs when the 

vertical gradient of excess pore pressure is numericaiiy equal to the buoyant unit weight of the sediment 

(typically 0.5 to 1.0 cgs (centimeter-g:':"am-second)). 

Under this condition upward flow exerts sufficient 

force to support the weight of the sediment grains 

throughout the column, and the sediment can derive 

no strength from intergranular friction. This state 

will be referred to as the condition for potential liquefaction. Whether the sediment behaves more like a 

liquid or a plastic medium in this state depends upon 

the magnitude of any residual cohesive forces. If the 

flow is one-dimensional, the thawing rates required 

to maintain this critical condition can be calculated 

simply for any permeability and ice c'Jntent if it is 

assumed that Darcy's law and the conventional permeability concept apply for the process (table 2). 
Table 2.--Conditions for potentiat tiquefaation 

Permeability 

(ern per sec) 

Velocity of flow 

(ft per yr) 

10- 8 

----------- 0.005-0.01 

10- 7 ----------- .05-.1 

10- 6 

----------- .5-1 

10- 5 

----------- 5-10 

2 1/2 

Minimum 

0.2-0.4 

2-4 

20-40 

20Q-400 

The actual process by which excess water would 

exit from a deepening thaw zone is, of course, vastly 

more complicated than this idealized model and dependent upon small- and large-scale inhomogeneities 

in properties and their variation with time. If the excess ice were abundant and distributed homogeneously, after thawing the mineral grains would tend to be 

widely dispersed and the aggregate would be highly 

"permeable." In this case the exiting of water is most 

conveniently viewed in terms of settling velocities of 

the particles. As the grains settled closer together a 

more conventional view of permeability would apply, 

and this permeability would generally decrease as the 

mineral aggregate condensed. If the permeability decreased to the critical values given in table 2 before 

the sediment gained appreciable strength, further condensation would be arrested by flow forces and the 

fluid condition would persist (if homogeneous flow is 

assumed). Thus the permeabilities in table 2 refer to 

conditions of very low strength, perhaps at or above 

the liquid limit. If the excess ice were distributed in 

small discontinuous segregated masses, complex local 

processes would occur relating to absorption by overconsolidated clays or disaggregation of intervening 

soil masses. Typically these processes would have 

short time constants, and unless some form of channelized flow developed, a gross condition similar to 

the model could evolve. (In many materials, however, 

cohesive forces might remain intact, locally at least.) 

If the medium were composed of strata with contrasting permeabilities, conditions for potential liquefaction 

of the individual layers would be slightly more complicated, but the same principles would apply. A medium 

consisting of alternating strata of contrasting ice contents could be subject to transient liquefaction in icerich layers for a time depending upon the thickness of 

the strata, the consolidation coefficient (or permeability) and initial pore pressure of the overlying stratum, 

and other factors. Although these and many other complications may have to be pursued in connection with 

particular designs, they are clearly beyond the scope 

of the present discussion. The simple model is adequate to provide some insight into conditions under 

which a potential for liquefaction exists. In the interest of brevity the qualifications relating to its applicability will not be repeated in the discussions that 

follow. 

Table 3 gives the approximate thawing rate as a 

function of time for the materials described in table 

Excess water (percent by volume) 

5 10 20 50 

thawing rate for liquefaction (ft per yr) 

0.1-0.2 0.05-0.01 0.025-0.05 0.01-0.02 

1-2 . 5-l . 25-.5 .1-. 2 

10-20 5-10 2.5-5 1-2 

100-400 50-100 25-50 lC-20 

9 

1. For both arctic and interior cases, the thavring rates 

for all of them lie within about± 50 percent of the values 

tabulated. Thus if the permeability were 10-6 emf sec 

and the excess ice 10 percent by volume, thawing at 

the rate of 5 to 10 feet per year would supply water as 

fast as it could filter to the surface (table 2) if the flow 

were homogeneous (not channeled). Consolidation could 

not occur and the material would tend to persist as a 

dense fluid with little or no strength. It is s~en (table 

3) that thawing at the rate of 5 to 10 feet per year would 

not be expected after the first year, and liquefaction 

due to thawing for that material should not l'·~ a problem thereafter. 

Table 3.--Rate of thawing beneath pipe as a 

funation of time 

Time interval 

(years) 

Computed thawing rate (fig. 4) 

(ft per yr ±50 percent) 

0-1 ----------------- 10 

1-5 ----------------- 2 

5-10 ----------------- 1 

10-20 ----------------- .5 

Looking more systematically at tables 2 and 3, we 

see that for permeabilities greater than 10- 5 the formation of a liquid slurry poses no particular problem except possibly in the first year for materials 

that are more than half free water (ice). These materials, of course, would be more than half water even 

after the mineral grains had settled, and trey pose a 

different problem to discussed later. 

For permeabilities of 10- 6 , liquefaction can be a 

problem during the first year for moderate amounts 

of excess moisture, and during the first few yea r s 

for large amounts. Serious potential problems occur 

where the permeability is 10- 7 , as liquefac':ion could 

persist for decades with only moderate amounts of 

excess water, and for several years for small amounts. 

For permeabilities less than I0- 7 emf sec t;e thawed 

materials could remain in a fluid state for the life of 

the pipeline even in sediments with very small amounts 

of excess moisture. 

A material with a permeability of 10- 7 is very impermeable, but clays typically have permeabilities that 

are lower still. In addition to clays, materials referred 

to as loess, sandy silts, silty sands, silts, and sandy 
clays have been reported with permeabilities in the 

range from 10- 9 to I0- 7 (although typically such sediments have permeabilities that are much higher). Materials designated by the same descriptive words occur frequently along the pipeline route. It is important 

to measure their permeabilities and ice contents as 

an aid in anticipating their behavior. 

It has been pointed out that the calculations leading 

to the thawing rates in table 3 were based on the assumption that the pipe temperature was 80°C from the 

start of the operation of the pipeline. In rea 1 it y the 

pipe temperature will, of co u r s e, be lower at first 

and increase (presumably to that value) over a period 

of years as the production increases to capacity and 

the surrounding permafrost warms. If the pipe temperature increases to 80°C over a period of say 5 

years, the very high (and potentially most dangerous) 

initial thawing rates (- 10 ft per yr) might be avoided, 

but the values for subsequent times will generally be 

increased somewhat. It is important, therefore, that 

a slow increase of pipe temperature will not avoid 

problems relating to persistent liquefaction. 

It is seen that whether or not potential conditions 

for general liquefaction of the thawed cylinder occur 

depends upon the the presence of materials with very 

low permeability at or above the liquid limit (that is, 

in a condition of very small strength). However, in 

general as the water content of a sediment increases, 

its permeability increases and its strength decreases. 

The relationship greatly diminishes the probability of 

encountering such materials. Whether these conditions are possible along the pipeline route depends 

upon properties of materials yet to be determined in 

detail. However, if they occurred, substantial problems could result. 

The case of general liquefaction was considered as 

an example because it is the simplest to discuss. Innumerable examples involving partial liquefaction of 

the thawed cylinder can be imagined, and similar 

principles a p pI y. For example important problems 

relating to liquefaction can arise in a section that contains no excess ice on the average, if an isolated icerich layer is overlain by an impermeable one. The 

writer has not worked out the conditions for transient 

liquefaction for this case, but the problem is straightforward, and its solution should yield useful quantitative insight into conditions that are likely to occur. 

The liquefaction of an ice-rich layer beneath an impermeable relatively strong overburden would result 

in mechanical instability, and the slurry might migrate to the surface by channeled flow through or circumferentially around the denser consolidated core. 

In this case the core containing the hot pipe could settle downward and maintain undiminished thawing rates, 

and possibly perpetuate the process. If the trench were 

backfilled with gravel to provide a strong bed for the 

pipe in material s u b j e c t to liquefaction, the entire 

10 

gravel plug could settle in a similar manner if its effective density exceeded that of the slurry. In general, 

however, if some form of vertical channo.led flow develops, the effective permeability of the overburden 

will be greatly increased and the danger of liquefaction greatly diminished. 

Local liquefaction could also develop if the growing 

thaw zone intercepted an artesian aquifE:r confined by 

impermeable permafrost. The presence of such aquifers is sometimes indicated by the o c c u r r en c e of 

pingoes. 

Flow 

Recognizing that general liquefaction of the thawed 

cylinder could occur only under a limited set of conditions that may or may not occur, we shEll investigate 

some of its consequences, as the purpo~e of this report is to identify potential problems. lfmeasurements 

indicate that the conditions set forth ir the previous 

subsection are not approached, they cEn be taken as 

presumptive evidence that these problems will not occur, although further studies may be necessary. In 

general a potential for general liquefaction should be 

relatively easy to identify with appropriate measurements. 

The most obvious question concerning a thawed tube 

of liquefied sediments relates to the corditions under 

which it would flow on sloping surfacer. This can be 

investigated by applying the theory of laminar flow 

of viscous and visco-plastic fluids. Table 4 gives velocities of flow in an inclined tube of viscous material. 

The assumed viscosity of 10 3 poises is a value sometimes quoted for solifluction materials and probably 

represents a reasonable upper limit. Judging from a 

variety of experimental results, the visosity co u I d 

probably range downward to less than 1 poise. 

Table 4.--Flow of a viscous cylinder (viscosity=103 poise) 

Radius (ft) 

l --------

5 --------

25 --------

Slope (ft per rrile) 

l 5 25 50 

Average velocity (miles per day) 

0.02 

• 4 

10 

0.08 

2 

50 

0.4 0.8 

10 20 

(Turbulent?) 

It is seen that the flow velocity incr~ases sharply 

with the radius of the tube (table 4). But even for very 

thin cylinders and almost imperceptil·'~ gradients, 

material can be moved the total length of any naturally 

occurring continuous slopes in times 01. the order of 

a few days to a year. For example, the minimum slope 

over any few-mile section of the "flat" Copper River 
Basin is about 5 feet per mile. Viscous mud could flow 

on such slopes at a rate that would cross the entire 

basin in one month if the thawed cylinder had a radius 

of only 5 feet. (The conditions represented by the lower 

right-hand corner in table 4 probably correspond to 

a state of turbulent flow. Most of the entries in the 

table would represent turbulent conditions if the viscosity were 1 poise.) Hence if the material is t r u 1 y 

liquefied, viscosity is not likely to be a limiting factor in the flow. 

As the grains of a slurry settle down and begin to 

interact, the properties of the material pass from 

those of a liquid to those of a plastic medium. In this 

transition the material acquires a small strength, that 

is, a shearing stress which it can sustain without flowing. Even slurries which appear very much as liquids 

might have strengths from 0. 01 to 1 psf (p o u n d per 

square foot). Remou1ded clays at the liquid limit can 

have strengths from 10 to 100 psf. (Such strengths are 

determined from short-term laboratory tests, and the 

possibility must be considered that substantia 11 y 

smaller strengths might be applicable for stresses 

sustained over a period of years.) Such small strengths 

can keep the material stationary on gentle s 1 opes. 

However, as the radius of the thawed cylinder grows, 

the shearing stresses within it may eventually exceed 

the strength, and the material can become unstable 

and flow. Table 5 shows the radius of the cylinder reTable 5.--Instability of a visco-plastic 

cylinder 

Shear strength 

(psf) 

.01 ---------- .l ---------- l.O ---------- 10 ---------- 100 ----------

Slope (ft per mile) 

l 5 25 50 500 

Radius required for flow (ft) 

l 0. 2 0.04 0.02 0.002 

10 2 . 4 . 2 .02 

100 20 4 2 . 2 

200 40 20 2 

200 20 

quired to produce flow for various slopes and shear 

strengths. On a slope of 5 feet per mile material with 

a strength of 0.1 psf will flow when the radius exceeds 

2 feet. It can be shown that in a visco-plastic medium 

if the thawed radius is substantially greater than this 

mimimum, the flow velocity will approach the value 

for the corresponding viscous case (table 4). Thus in 

this case, if the radius grew to 5 feet the plastic material would flow at a rate approaching 2 miles per 

day (table 4) for a high viscosity, and for the lower 

viscosities it could be turbulent. From the section on 

temperature effects we know that radii on the order 

of 25 feet can be expected. Hence material with longterm strengths up to 1 psf can be expected to flow on 

slopes as small as 5 feet per mile. For material with 

strengths up to 10 psf flow could occur on slopes of 

11 

only 1 percent. Thus it would appear that flow could 

occur practically any place that such slurri~s occur, 

as imperceptible slopes such as these are alllostomnipresent. For slopes greater than 10 percent (500ft 

per mile), thawed cylinders composed of material as 

strong as 100 psf will probably flow. 

After an initial flow of the thawed cylinder, further 

flowage along the bottom of a trench would be approximated better by the case of an inclined layer with a 

free surface. The thickness r e qui red for flow of a 

plastic layer is half as great as the radius of the corresponding cylinder with the same slope, anc" the velocity of the layer in viscous flow is greater than that 

of the corresponding cylinder by a factor of 2 or 3. 

The pipe would, of course, have a retarding effect on 

these flows, but this effect can be shown to be unimportant. 

From figures 2 and 4 it is seen that a cylinder 20 

to 30 feet in diameter might be thawed in a few years. 

It is clear from tables 4 and 5 that if the cylinder contained a liquid slurry it would not reside where it was 

formed over such a period unless it were ccnfined by 

frequent dams of stronger material along the route. 

The material can be expected to "seek a level" within 

a matter of days to months, depending on the distance 

over which the liquid slurry extends and its viscosity. 

As an extreme example, iftherewerea S-mile stretch 

of liquefied material with a gradient of 25 feet per 

mile, the pipe could soon be lying at the bcr.tom of a 

50-foot slumping trench at one end, while some 10 

million cubic feet of mud (containing the pipe) could 

have been extruded over the surface near the other 

end. On the up h i 11 end the downward-settling pipe 

could continue to thaw to great depth. This_wculd favor 

liquefaction (tables 2 and 3) and further flew (tables 

4 and 5), and process could be self-perpetuating. It 

may be extremely difficult to stop slumping of large 

volumes from the walls into the widening trench. The 

slumped material would flow promptly to the downhill 

end where it would augment the mt~;d flowing onto the 

surface. The pipe could be jeopardized by loss of support in the trench and displacement in the mudflow, 

and the disruption to the landscape could be substantial. 

Because of the great radial distances to which the 

thawing extends (figs. 2 and 4), it may be d:.fficult to 

anchor structures to retard the flow. Sheet piling that 

was installed at regular intervals to darn the flow 

would have to be anchored in material not subject to 

liquefaction. If it were not, it might serve only to 

transfer the hydrostatic flow forces to the pipe. The 

cumulative effect of such structures over an elevation drop on the order of 100 feet might stress the 

pipe in an amount approaching its tensile strength. 

As in the previous discussion, the simplest and most 

adverse case, general liquefaction, was tal·en as an 

illustrative example. A variety of less ext':"eme and 
perhaps more probable cases can be imagined. Under 

static conditions, gravitational shearing stresses are 

greatest at the bottom of the thawed cylinder on a sloping surface. A liquefied layer there might flow laterally 

if it could exit through a channel to the surface downslope (perhaps at a thawed ice wedge or a river terrace). Although a very weak bonding oftheoverburden 

materials to the frozen wall would keep them from 

participating in this flow, loss of support could cause 

settlement which could break this bond and trigger 

more massive movement. 

This discussion has emphasized conditions relating 

to extremely small strengths because the forces that 

cause flow on gentle slopes are very small. On steeper 

slopes it will, of course, be necessary to analyze the 

stability of stronger materials. 

Buoyancy and related problems 

If the material surrounding the pipe were liquefied, 

the pipe would tend to sink or float depending upon 

whether its effective specific gravity were greater or 

less than that of the surrounding medium. The specific 

gravity of the oil-filled pipe will probably be from about 

1.15 to 1.2 while that ofthesurroundingmedium might 

vary from about 1.0 for highly ice-rich organic material to about 2. 0 for some inorganic slurries. To 

avoid vertical displacement of the pipe by positive or 

negative buoyant forces, it would be necessary either 

to support it with structures anchored in material with 

permanent strength, or to weight it in such a way that 

it were neutral. If it were weighted, a mismatch in 

the specific gravities of 0.1 would result in vertical 

forces along the pipe of about 80 pounds per running 

foot. If it were not weighted or supported, net forces 

on the pipe could range from about 150pounds per foot 

of pipe downward in purewaterwithorwithout organic 

material to about 600 pounds per running foot upward 

in inorganic slurries. To achieve neutrality in inorganic slurries by weighting with concrete, it would be 

necessary to use a volume of ballast comparable to 

the volume of the pipe. A further problem with weighting the pipe to match the specific gravity of the thawed 

medium is the uncertainty in the density of the sediment and the possibility that this densitywouldchange 

with time. Residual forces on the order of 100 pounds 

per running foot might be very difficult to avoid. Displacement by such forces would probably be suppressed 

by strengths on the order of 10 psf in the surrounding 

materials. 

Where the pipe emerged from strong material into 

liquid at a depth of say 6 feet beneath the surface, 

these buoyant or sinking forces would have to be balanced by shearing stresses in the pipe out to that distance (probably on the order of 100ft., at least) where 

it floated to the surface (for upward forces) or settled 

to the bottom (for downward forces). In such configura12 

tions the pipe would be subjected to se·rere bending 

moments which should be evaluated and compared to 

failure criteria. Upward forces might be released by 

tearing the pipe upward through the overburden at the 

point of support, but downward forces could not. However, the conditions for net sinking force2 would probably be limited to materials of restricted thickness 

and extent. (The thermal effects of dowrward settlement of a hot pipe are obvious.) 

CONVECTIVE THAWING 

The thermal calculations of the first E' e c t ion are 

based on the assumption that heat tram~fer would be 

only by conduction, the most inefficient ofheattransfer 

mechanisms. Although this is virtually tre only mechanism possible in frozen materials, heat can be transferred much more efficiently in thawed materials if 

they are mobilized or if their interstitial water is. In 

such cases the curves of figures 2 and 4 may have no 

resemblance to the actual amount of thawing; t h e i r 

main importance is that they indicate thrl: substantial 

amounts of thaw can be expected promptl~' even in immobile media. From figure 3 it is seen that substantial 

gradients occur in the thawed c y 1 in de r. When the 

thawed material moves, warm material is generally 

brought closer to the thawing surface and greatly accelerates thawing, while colder material is brought 

closer to the pipe and greatly accelerates heat extraction from it. Hence once the thawed material is 

mobilized, the thawing can be greatly accelerated and 

some of the effects described previously can be selfperpetuating. These convective effects have been estimated by applying the theory of heat conduction in a 

moving medium. Results presented in table 6 show 

the order of magnitude of the heat-flow enhancement 

by material flow between isothermal surfaces (for 

example, the pipe and the thawing interface). Fordistances of separation of several feet it is· unimportant 

Table 6.--Enhancement of heat flow by movement in direction of thermal gradient between surfaces separated by a distance, with 

a temperature difference of 80°C 

Velocity 

ft per yr ft per day 

l 0.003 

5 .015 

10 . 03 

20 .06 

50 .15 

100 . 3 

1,000 3 

10,000 30 

Convection+conduction 

Conduction 

Distance (feet) 

8 16 

l. 05 l.l 

1.3 1.6 

1.6 3. 2 

2. 3 4.6 

5.0 10 

10 20 

100 200 

1,000 2,000 
for velocities less than a few feet per year. However, 

for velocities of a few tenths of a foot per day, local 

thawing could be increased by an order of magnitude, 

and for 30 feet per day, by a factor of 1,000. A velocity of 30 feet per day is almost imperceptible; about 

the same as the speed of the end of the minute hand on 

a kitchen clock. 

The values shown in table 6 for movement of sediments plus their contained water are about the same 

as those for filtration of the water alone, provided the 

velocity is considered as the volume of water passing 

through a unit area in uni r_, time and not as the particle 

velocity. Tl;lis is the sense in which velocity was used 

in table 3; hence, it is seen from tables 2, 3, and 6 that 

heat transfer by excess water filtering from the thawing 

surface would be unimportant except for rare materials 

during the first year. However, the processes discussed in the section on flow could cause substantial 

convective thawing. 

SEISMIC LIQUEFACTION 

In the foregoing sections we have considered conditions under which the sediments had little or no 

strength because, after thawing, they were settling 

downward through water, or water was moving upward 

through them, and their grains were not firmly in 

contact. A similar situation can occur when a saturated granular sediment is subjected to vibrations 

which tend to compact it. While the grains are tumbling down into denser packing configurations, they lose 

contact and the aggregate liquefies. The water excluded 

from the diminishing pore space moves upward from 

the 1 i que fie d zones and can cause the process to 

spread. Loosely packed sands and silts in the thawed 

cylinder could be liquefied by an earthquake with consequences relating to flow, buoyancy, and convective 

thawing similar to those discussed in the preceding 

paragraphs. The southern part of the pipeline route 

lies in an active earthquake zone. 

DIFFERENTIAL SETTLEMENT 

Some engineering considerations 

It is well known from experience in the construction 

of roads, railroads, buildings, and other structures 

on permafrost that thawing can cause undulations of 

the surface resulting from differential settlement. For 

the most part this settlement results from reduction 

in volume of ice when it turns to water, or from loss 

by the sediments of water formerly contained in segregated ice masses or in the pore space of underconsolidated sediments: Such differential settlement can 

produce stresses in a pipeline just as it can in any 

other rigid structure the permafrost might have been 

supporting. Although analysis of these stresses is a 

matter for engineers, it is necessary to introduce some 

quantitative standard for pipe deformation to evaluate 

the possible impact of set tIe men t on the pipeline. 

An unsupported span of pipe will sag in th~ middle 

as _a .re~ult of stresses created by the pipe's weight; 

the larger the span, the greater the stresr and the 

sag. According to my understanding, the maximum 

stress for which the Alaskan pipeline is being designed 

(the "design stress") would be approached under operational conditions when such a free span is about 

60 feet long, and under these conditions the sag at the 

center of the oil-filled pipe would be about 3 inches. 

If the pipe were laid on the surface and a 60-foot section of the ground settled more than 3 inche2, conditions would be rather similar. From dimensional considerations it can be estimated that if the log_d on the 

pipe were four times its weight (as it might be if the 

pipe also supported its overburden in a fre~ span), 

similar stresses would develop in a 30-foot span and 

the sag w o u 1 d be about U2 inches. If the pipe were 

partially supported so that the load were onl~· 25 percent of the pipe weight, such stress would probably be 

approached in a 120-foot span with about M-foot displacement in the center. (It should be pointed out that 

the design stress probably includes a substan-:ial margin of safety insofar as failure is concerned.) 

In this discussion it should be borne in mind that 

settlement at the surface depends upon volume reduction throughout the entire thawed cylin~er. Typically 

less than 10 percent of the material in v o 1 v e d will 

ever be observed directly by boring and trenching. 

Settlement of the pipe depends primarily upcn behavior of thawed materials below pipe level, ofw"lich only 

a negligible fraction will pro b a b 1 y be dir~ctly observed. Properties below pipe level will probably be 

estimated by interpolation of information chained in 

separated bore holes. 

Ice wedges 

If the distribution of ice in permafrost an1 the details of the thermal disturbance are known in advance, 

settlement of the thawed materials can be predicted 

satisfactorily. The chief problem in determining the 

distribution of ice in advance (by borings) probably 

lies in detecting the presence of large masses of high 

ice-content material of limited lateral distribution 

such as ice wedges. The chief problem in predicting 

the thermal disturbance lies in accounting for convective thawing if these materials are mobilized. Very 

abrupt changes in ice content (and consequent severe 

differential settlement) might occur under a variety 

of conditions, but the most severe are probably those 

related to the thawing of ice wedges. 

It is well known that during the cold norther, winter, 

thermal contraction of the ground surface causes the 

sediments to crack in a quasi-polygonal pattern resembling desiccation cracks in mud. These cracks 

may extend to depths of tens of feet, and they are typically 30 to 100 feet apart. Spring melt water draining 

13 
into the cracks and freezing seals them to form a network of vertical ice veins in permafrost. Because of 

their weakness these veins are the sites of renewed 

cracking in succeeding winters. With the passage of 

centuries the veins grow into a honeycomb ofmassive 

ice wedges, sometimes 10 or more feet wide at the 

top. Because of the interaction of these wedges with 

surface processes, their traces at the surface are 

sometimes de 1 in eat e d by a conspicuous pattern of 

troughs or ridges or vegetation contrasts. Such patterns are called ice-wedgepolygons. Where ice-wedge 

polygons appear, the locations of many (though generally not all) ice wedges in an area can be determined 

by surface observations. Rather little is known of the 

distribution of ice wedges in permafrost areas where 

no surface polygons appear. However, judging from 

observations in random excavations in such areas and 

from the theory of ice-wedge formation, it is likely 

that they represent a rather general condition in permafrost sediment, becoming better developed northward. Their presence is revealed by surface markings 

only under special conditions favorable for it. 

Typically ice wedges underlie only 5 or 10 percent 

of the surface area. Hence on the average their presence would be revealed by exploratory drilling in only 

1 out of 10 or 20 borings. However, a pipeline crossing 

ice-wedge laden permafrost might encounter a wedge 

every SO or 100 feet on the average. As the wedges 

form closed figures (polygons) they generally could not 

be avoided by very local rerouting. 

Where the pipeline crosses an ice wedge, it is likely 

that much if not all of the wedge will be removed locally by thawing in a few years. Even if heat transfer 

were only by conduction, the thawing would be much 

faster than that indicated by the curves for water/ice 

(fig. 4) because the thermal regime of the surrounding 

material would dominate. However, with so much localized free water, convective heat transfer would 

probably dominate conduction from the outset. 

When ice beneath the pipe turns to water, its volume 

will reduce progressively, and the weight of the pipe 

and overlying materials will no longer be fully supported. As thawing of the ice wedge proceeds, the pipe 

and( or) the overlying material must settle downward 

into the water in a special way, or else they must sup-

- port the unbalanced load with their strength. Assume 

for the moment that the pipe and surrounding materials 

settle as a unit. Then the relative rates of downward 

movement of the pipe and the fluid level in the underlying ice-wedge trough determine the load on the pipe. 

This can be discussed in terms of a few simplified conditions. 

In the first condition, which will be called condition 

A, the water table is maintained near the surface by 

inward flow of water from surrounding materials to 

compensate for the volume reduction. In this case the 

17-l 

overburden materials (including the pipe) are partially 

supported by buoyant forces. However, an unbalanced 

load on the order of one to two times the weight of the 

pipe in a free span (depending upon derth of burial) 

will persist. Unless the surrounding sediments have 

appreciable long-term strength, strain will accumulate 

in the pipe which will ultimately bear thir load. 

, In the second condition (condition B), a void develops 

in the ice-wedge trough beneath the pipe. This could 

result from volume reduction of the ice wedge not compensated by inflow. of water, by drainage of the melt 

water into a seasonal contraction crack formed in the 

wedge, or by convective thawing of a drainage channel 

through the ice wedge in regions with locrl relief. Under this condition the unsupported 1 o ad on the pipe 

would typically be three or four times the weight of 

the pipe in a free span. 

A third condition (C) is possible where the pipe and 

overburden settle into the fluid in the trough and the 

surrounding materials are so impermerble that the 

unbalanced weight is supported by filtrati0n forces. In 

this case the material surrounding the wedge would 

probably be in a general state of liquefac':ion. 

Independently of the processes just discussed, material will slump into the trough from the walls, filling 

its bottom and widening its top (below th~ pipe). This 

process will not change the volume of water (condition 

A) or of the void (condition B) in the trough, but it will 

increase the length of the unsupported PDan of pipe 

crossing the trough. 

If the pipe and surrounding overburden material do 

not behave as a unit, thawed material will slump around 

the pipe into the ice-wedge trough and leave a depression on the surface across which the pipe would be exposed if the wedge were large enough. If the depression were full of water, the pipe would be partially 

supported by buoyancy, but if it were not, the pipe 

would be in a free span. The latter cas~ will be referred to as condition D and the former as condition E. 

A final condition (F) represents the case in which the 

wedge is so small that slumping of material around 

the pipe filis the trench without exposing the pipe. 

In summary, in conditions A, B, and C the ground 

surface remains more or less intact. In A the pipe is 

underlain by a water pocket and unbalanced forces on 

the pipe approach one to two times its weight. In B the 

pipe is underlain by a void, and the unbalanced forces 

on it approach three or four times its wej ght. In C the 

pipe overlies a slurry and conditions w o u 1 d be like 

those discussed in the section on liquefaction. In concitions D, E, and F a depression develops on the surface. In D the pipe is immersed in a pond of water, and 

the forces on it are only a small fraction of its weight. 

In E the pipe is exposed across a pit in a free span 

with the forces on it equal to its weight. In F it is 
buried in the bottom of a pit with no residual forces 

acting on it except the stresses resulting from its displacement. 

As a pipeline crosses a field of ice wedges, these 

six conditions might be expected with all possible intergradations depending upon the size of wedges, the 

properties of the materials, and details of local convective thawing and drainage processes. In many cases 

conditions A or B may precede conditions D, E, or F. 

Conditions A and B are potentially the most dangerous, 

not only because they result in the greatest loading of 

the pipe, but also because they may occur with little 

or no surface expression. The other conditions can be 

detected by surface observations and remedial action 

can then be taken. 

In considering what happens when an ice wedge thaws 

from beneath the pipe, it is important to keep in mind 

that the wedges are randomly oriented, and the pipe 

may cross them at any angle. (It has been explained 

that where ice wedges are well developed, local rerouting to avoid individual wedges is not feasible.) The 

length of the unsupported pipe span over an ice-wedge 

trough increases as the width of the trough increases 

and as the angle of intersection between the pipe and 

the iee wedge decreases (that is, the intersection of 

wider troughs at smaller angles creates larger unsupported spans). A simp 1 e statistical calculation 

shows that where individual wedges are about 100 feet 

long and the width of the thawed and slumped region 

beneath the pipe is 5 feet, the pipe centerline will be 

unsupported for distances greater than 60 feet, about 

once in every 50 crossings, or about once every mile 

for a spacing of 100 feet. Under comparable conditions 

unsupported c en t e r 1 in e spans greater than 30 feet 

could be expected once in every 13 crossings, or about 

four times per mile if these 5-foot wide slumped zones 

occur at intervals of 100 feet. If the slumped region 

were 10 feet wide, these frequencies would he approximately doubled. It is possible that the spans greater 

than 60 feet could exceed the design stress of the pipeline under conditions A, B, and E and that those greater 

than 30 feet could do so under condition B. 

In some respects a worse threat is posed by i c e 

wedges which pass close to the trench but are not exposed by it. As the thawed zone is typically more than 

10 times as large as the trench, the thawing of unexposed wedges in ice-wedge terrains is probable. Their 

presence might not be recognized until after they had 

caused disruptive effects. Perhaps the greatest displacement of the pipe would be caused by wedges which 

run close to the trench and parallel with it and those 

that pass under the trench, but whose tops are below 

trenching depths. Massive extinct wedges with their 

tops many feet beneath the surface have been observed 

in several areas. 

15 

Settlement in more homogeneous materials 

Although the most conspicuous differential settlement usually results from the thawing of ice wedges, 

substantial amounts can be caused by more s·1btle lateral changes in properties. To illustrate this we shall 

first consider some simplified relations between the 

water content and absolute amounts of settlement to 

be expected in homogeneous sediments. 

It has been pointed out in an earlier section that for 

a given climatic regime (for example, arctic or interior) the rate and amount of water melted f1·om permafrost is rather independent of moisture content of 

the materials (unless they should be anomalously dry); 

the wetter the material, the more slowly it tb1ws, but 

the more water a little thawing will produce. For the 

thermal models presented in the section on temperature effects, it can be shown that roughly 750 cu ft 

(cubic feet) of ice is melted per foot of pipeline for 

the interior cases in the first two decades, and about 

450 cu ft for the arctic cases in the same period. 

Roughly one-third of these amounts would te melted 

by the end of the first 2 years and about two-thirds by 

the end of the first decade. If the sediments v·<?.re sufficiently undersaturated (at least by 10 per,.~ent) and 

homogeneous that they did not expand appredably on 

freezing, the melting of this ice would not result in 

appreciable settlement. If, however, the s e dim en t s 

were saturated when they were frozen, they must have 

expanded by at least 10 percent of the volume of water 

present during the process. They would, therefore, reduce in volume by that amount when they thawed and 

returned to their original state of consolidati':ln. Thus 

for such sediments the thawed cylinder would undergo 

a volume reduction in two decades of about 75 cu ft 

(per foot of pipeline) in the interior, and 45 cu ft in 

the arctic. This volume reduction would have to be expressed by settlement of the surface, and this settlement could occur only over the distance bet·•reen the 

frozen shoulders (see September curves, fig. 2). For 

"silt" (fig. 2) this distance is roughly 30 feet in the 

arctic and 50 feet in the interior. Hence for both cases 

the surface settlement is about the same, I~ feet. The 

corresponding value for the clay cases is about 2 feet 

and for water/ice about 3 feet, because the width of 

the band of settlement for them is narrower. After 2 

years the volume reduction is aboutone-thirdas great 

and the shoulders are closer together, but not proportionately so, because equilibrium is achieved faster 

near the surface than at depth. Hence the settlement 

after 2 years is roughly one-half as great as it is after 

20 years. Thus for typical saturated sediments a widening trench can be expected to develop over the pipeline, deepening from about a foot after a few years to 

about twice that after two decades. If the sediments 

were oversaturated and 20 percent of the ice were excess, the values would be approximately tripled if the 
excess water could filter to the surface; 3 feet of settlement in a few years and 6 feet in two decades. (If 

the excess water could not filter to the surface, liquefaction would occur.) If the sediments were sufficiently 

undersaturated, settlement at the surface w o u 1 d be 

negligible. If the pipe tempe rat u r e were increased 

gradually during the first few years, the settlement 

would develop more slowly during that period. 

In this discussion we are interested in the settlement that might occur below the pipe. Inspection of 

figures 2 and 4 suggests that typically it will be well 

over 50 percent of the surface settlement in homogeneous materials. If we now imagine the pipeline to 

be passing over materials that are vertically homogeneous, but whose degree of saturation or moisture content changes along the route, it is seen that differential 

settlements on the order of 1 foot might not be uncommon, and that they could occur even in the absence 

of oversaturated sections. They could result from 

changes in moisture content of only a few percent (on 

the order of 10 percent of the water present) or they 

could occur with no lateral change in moisture content 

at all if the porosity changed by only a few percent. 

It is even possible for the greater settlement to occur 

over the dryer of two materials (where the dryer material has a lower porosity and greater degree of saturation). Where oversaturation (excess water) occurs 

locally, several feet of differential settlement are possible at pipe level. 

The homogeneous case discussed above is the one 

that generally results in the least settlement. Where 

the sediments are not homogeneous vertically, they 

may contain several percent of segregated ice even 

though they may be undersaturated on the average. 

Reabsorption or exclusion of the water formed by melting of this ice might cause settlement of a foot or more 

in inhomogeneous undersaturated sections. 

One would hope that these subtle changes would be 

so gradual as not to result in abrupt vertical displacements of the pipe, but there is no assurance that this 

will always be so. Nor is there any assurance that 

lateral changes which could cause significant differential settlement would be revealed, even after extensive exploratory drilling and the completion of trenching for the pipe. The most troublesome changes would 

occur over lateral distances of a few tens of feet, too 

small to be detected by most exploratory drilling. They 

would occur at depths below the pipe level; generally 

too deep to be detected in trenching. Perhaps the most 

obvious place where sharp lateral changes could occur would be where the pipeline passed from bedrock 

to saturated sediments. There the problems could be 

anticipated by surface observations and local drilling. 

Although lateral contrasts undetected by drilling or 

trenching might be rare, they might occur, for example, at the banks of ancient river meanders or lakes, 

now buried in the sedimentary section. 

16 

It is convenient to discuss these differential settlements in terms of conditions similar t') A, B, and C 

used in the discussion of ice wedges. In t'le first (condition A') the material initially forming the bed for the 

pipe settles downward and loses contact with the pipe, 

but the water table is at the surface. In this case the 

load is equal to the buoyant weight ofthe pipe plus the 

overburden material (one or two times, the weight of 

the pipe in a free span). In the second (B') the water 

has drained off, and a void exists beneath the pipe. 

Here the load may be t h r e e or four t:1mes the free 

weight of the pipe. In the third (C') the surrounding 

material flows or falls into the void beneath the pipe 

to give it partial or complete support. This is the most 

favorable case, and perhaps it would be t'le most common one. It is possible, however, that ifthe pipe were 

laid in a gravel bed, the development ofthe third case 

(C') could be inhibited. 

It is seen that for differential settlement on the order 

of 1 foot over a 60-foot span, the design stress could 

be equaled or exceeded under condition A'. If the material at and above pipe level were well drained (condition B'), the design stress could be exceeded by 200 

or 300 percent unless the metal yielded to relieve local stress concentrations. (Much more extreme examples involving longer spans and several feet of settlement in oversaturated sediments are easily imagined.) It is inter-esting to note that if a drainage trench 

were dug to remove excess water around the pipeline, 

conditions could change from A' to B'. The loss of 

buoyant support for the pipe and overlying sediments 

could double the unsupported load on the pipe. Hence 

decisions relating to field maintenance problems might 

have significant engineering consequences. 

It is important_ to emphasize here as elsewhere that 

the discussion is based upon approximate_calculations 

and simplified physical models. However, the orders 

of magnitude of the quantities derived suggest potential problems that deserve further attention, including 

a more refined treatment. 

SOME SURFACE EFFECTS 

Most of the processes discussed in the preceding 

section on subsurface effects, are mecranical manifestations of thawing around and beneath the pipe. Although they can have profound effects or the surface, 

their controlling mechanisms are generally processes 

at depth. Although the classification is eomewhat arbitrary, there is another group of effects for w h i c h 

the controlling mechanism operates primarily at or 

near the ground surface, and some of these are the 

subject of this section. These effects depend more or 

less directly upon the amount of excess reatdelivered 

to the surface from the buried pipe. Th~re seems to 

be two main ways that this heat can be transferred to 

the surface: ( 1) by thermal conduction through the 

overlying sediments, and (2) by thermal convection of 
water driven through the sediments by heat extracted 

from the pipe. The subsection that follows considers 

the near-surface temperature effects resulting from 

thermal conduction. It is followed (in the subsection 

on hydrologic effects) by a discussion of heat transferred to the surface by both convection and conduction. 

It should be anticipated here that,under certain circumstances to be discussed, the convective process 

can dominate the surface thermal regime. 

THERMAL GRADIENTS AND TEMPERATURES 

NEAR THE SURFACE 

To investigate the temperature disturbance near the 

surface by conduction from the pipe, it is convenient 

to use the steady-state solution (appendix B) because 

it is applicable after the first year directly over the 

pipe. At lateral distances of a few tens of feet, the 

steady-state solution will overestimate the heat losses 

somewhat, but the discrepancy is unimportant for the 

present purposes. 

The excess thermal gradients produced by the heated 

pipe at the ground surface are shown a•..,proximately 

as a function of distance from the centerline in figure 

5 for a pipe with its axis 5 feet beneath the surface 

(0=5 ft) and one with its axis 8 feet beneath the surface 

(D=8 ft). These can be viewed as average annual excess values if the heat transfer is primarily by conduction, and the material is more or less homogeneous. The thermal gradient anomaly at the surface im-

~ 1-

z 

w 

0 

<[ 

a:: 

(!) 

w 

(.) 

Li: 

a:: 

::::> 

CfJ 

I 

w 

a:: 

::::> 

1-

<[ 

a:: 

w 

Q_ ~ 

:2>-

w __j 

1-Z 

wo 

(.):2 

30 Li: a:: a::o 

::::>LL 

CfJ 

a:: 

s 

20 

\ D=5feet 

\) 

\ 

\ 

10~, 

30 20 10 0 10 2.0 30 

HORIZONTAL DISTANCE FROM CENTERLINE (feet) 

Figure 5.--Equilibrium temperature gradient at 

the surface for conductive transfer in homogeneous materials over a pipeline maintained at 80°C. D is depth of burial of pipe 

axis. Curves also give approximate form of 

surface temperature when local meteorological conditions are uniform. 

17 

mediately over the deeper pipe is about 10°C per ft, 

while that for the shallow one is more than twice as 

great. However, the anomaly from the shalk•ver pipe 

falls off sharply, and at distances from the c~nterline 

exceeding 10 feet, both cases produce about the same 

effect (fig. 5). Thus the average annual terrperature 

of plant roots 1 foot beneath the surface at the centerline would be elevated above the surface terrperature 

some 22°C by a pipe at a depth of 5 feet and l0°C by 

one at 8 feet. Ten feet from the centerline both cases 

would elevate plant-root temperatures at the 1-foot 

level by 3~t to 4°C, still an appreciable anount. At 

distances greater than 30 feet, the effect would amount 

to only a fraction of a degree; on the same order as 

natural random variations on a microenvironmental 

scale. 

Some interest is attached to the elevation of the surface temperature in the vicinity of the pipe. (For example, the total temperature anomaly at roo~ level is 

the one caused by the anomalous gradient, just discussed, plus the temperature anomaly at the surface.) 

This problem has no simple solution, as the surface 

temperature is a consequence of complex processes 

of heat and mass transfer which are imperfectly known 

in detail. The temperature anomaly at the surface can 

be expected to be extremely variable with wind, moisture, and radiation conditions. However, when these 

conditions are uniform in the vi c in it y, the surface 

temperature will be approximately proportio"'lal to the 

temperature gradient at the solid surface (by Newton's 

low of cooling). Hence to a ro;.~gh approxirr ation the 

form of the temperature disturbance at the s·trface at 

any time would be given by the curves in figure 5; that 

is, we might view figure 5 as graphs of surface temperature versus distance from the centerline if we 

leave the scale on the ordinate axis (temperature) unspecified. At the centerline the surface temperature 

would probably be elevated about twice as much for 

the 5-foot burial as for the 8-foot burial, and in both 

cases the temperature elevation would probably be 

insignificant at distances greater than 30 feet. Combining results of the last two paragraphs, plant-root 

temperatures probably would not be affected significantly by conducted heat beyond a 60-foot syrath centered over the pipeline. 

SOME HYDROLOGIC CONSIDERATIONS 

Heat plays a key role in hydrologic processes in 

cold climates, and the excess heat supplied to the environment near the pipeline can have some conspicuous effects. Table 7 indicates some possible thermal 

effects on surface water, ice, and snow for an assumed 

thermal conductivity (2 x 10- 3 cal/ em sec °C) and a 

gradient of 10°C per ft. This represents conditions at 

the centerline for D=8 feet; conditions elsew'lere can 

be scaled directly from fig u r e 5. Variations in the 

conductivity could cause variations in the values in 

table 7 by a factor of 2, and heat transfer by moisture 
Table 7.--Potential hydrologic effects of heat conducted to the surface from a buried pipe 

[Values are for centerline over a pipe with i-ts axis buried 8 ft; other values can be scaled 

from figure 5. Effective width of the layer is 25 ft for D=8 ft or 32 ft for D=5 ft] 

Layer thickness per year Layer thickno.ss per day 

Process 

Water at 0°C to vapor--------------------

Ice at 0°C to water at 0°C---------------

Water at 0°C to water at 20°C------------

Snow (density=0.25 gm per em 3 ) to 

water at 0°C---------------------------

migration could change the values much more. These 

qualifications will not be repeated in the discussion 

that follows. 

The total rate of heat loss at the surface per unit 

of pipe length for 0•8 is about the same as if the centerline flux obtained over a band about 25 feet wide, 

and no loss occurred beyond that (equation B-6 ). Hence, 

for the purpose of computing volumes of materials involved, the processes of table 7 can be viewed as occurring over a 25-foot-wide strip. The comparable 

values for 0=5 can be computed by imagining the 0=8 

centerline flux to be distributed over a 32-foot band. 

Snowmelt 

It is seen from table 7 that heat from the pipe is 

capable of melting 34 feet of partially compacted snow 

in a year, or over 1 inch per day (centerline, 0=8). 

Indeed, whenever snow is on the ground, its insulating 

effect will cause rapid thawing (from below) of any 

crust that might have frozen in the ground, and thereafter all of the available heat will go in to melting the 

snow. At the centerline snow will lie on the ground for 

only a short while even after the heaviest snows. (Ten 

feet from the centerline it will remain about three 

times as long, while at the centerline for 0=5 it will 

remain about half as long, fig. 5.) It is important that 

the snow cannot melt from below unless the ground 

upon which it rests is first thawed; hence, water from 

the melting snow will flow directly into the thawed cy J.-

inder around the pipe below, if material there is not 

already saturated. South of the Brooks Range on the 

order of one-fourth to one-half foot of water is precipitated annually as snow, and it is likely that significant additional amounts of water may be supplied 

by snow drifting onto the bare ground above the pipe. 

North of the Brooks Range the precipitation is much 

less, but drifting is much more important, and it is 

possible that more water would be supplied by snowmelt there than farther south. 

Evaporation 

Whereas the pipe can melt tens of feet of snow annually, all of the conducted heat delivered to the surface 

em in. ft em in. 

35 14 l.l 0.1 0.04 

260 100 8.5 • 7 . 3 

1,040 410 34 3. 0 l.l 

1,040 410 34 3.0 l.l 

(centerline, 0=8) could be consumed by th?. evaporation 

of only about 1 foot of water; that is, if excess evaporation over the pipe amounted to only 0.1 em per day, 

the conducted heat would not even raise the surface 

temperature (table 7). Over much of the terrain traversed by the pipeline the natural evaporation rate is 

on the order of 1 foot per year. Ooublin7, this v a 1 u e 

very locally is not likely to produce a gro~s disruptive 

effect-perhaps ground fogs will be freq'Ient over the 

pipe, particularly after a snowfall. 

Most of the heat conducted from the pine will probably be consumed by evaporation if water is available. 

If it is not, surface temperatures will rise sharply. 

Water from permafrost 

If the sediments do not contain excess ice, in general 

no water will be delivered to the surface as a result 

of conductive thawing of permafrost. If 20 percent of 

the water were excess, it would appear at the rate of 

perhaps 20 or 30 cu ft per year (per foct of pipeline) 

in the first year. This would fall off rapidly in succeeding years to a few cubic feet per year. Hence, after 

the first year or two, water thawed from permafrost 

may be enough to provide an inch or so annually over 

a band a few tens of feet wide. This water probably 

will not play a significant role in the surface hydrologic 

regime or in hydraulic erosional processes; its cumulative contribution over 10 or 20 miles of pipe 1 in e 

would be equivalent to the output of one household water 

faucet. 

If permafrost near the surface contained abundant 

excess ice, artificial disturbances could trigger selfperpetuating processes of thawing from solar radiation 

and convection. These co u 1 d contribute significant 

amounts of surface water locally. 

Surface drainage 

In the section on temperature effects (fig. 2),it was 

shown that any frozen cap that might form over the 

thawed cylinder in winter would roll back in the spring 

to provide continuity with the naturally thawed active 

layer. In the section on differential settlement it was 

18 
pointed out that this region above the pipe is likely to 

be the site of a depression. Typically, the surface has 

excess W3ter in spring and early summer in permafrost regions, and it is likely that this water would tend 

to drain into the region around the pipe from the surrounding area. It is seen that an amount of water comparable to the excess annual evaporation (by conduction) could be supplied if an inch or so were drained 

from the surrounding terrain for distances on the order 

of 100 feet. If a depression occurred over the pipe and 

it were filled with permeable gravel, this drainage 

would not be impeded. 

The depression, likely to occur over the pipeline in 

regions where the sediments are saturated, could have 

a profound effect on the surface drainage. If the depression were discontinuous, it could result in a series 

of ponds which can collect solar radiation and grow 

rapidly by thawing their banks in regions where the 

near-surface permafrost is ice rich. (It has been 

pointed out in the section on differential settlement that 

indiscriminate drainage of such ponds could result in 

severe stresses on the pipeline.) 

If the depression over the pipe were relatively continuous, it could act as a focus for local surface drainage and intercept preexisting streams and create a 

drainage system where none existed before. This could 

affect the hydrologic regime over a substantial area, 

and it could cause problems of erosion along the pipeline. 

Convective heat transfer and the 

hydrologic balance 

It has been pointed out that conductive heat from the 

pipeline is not likely to have a major disruptive local 

effect on the hydrologic balance. It will result in excess annual evaporation equivalent to a layer ofwater 

on the order of 1 foot thick over a band effectively 25 

or 30 feet wide. This is not different in order of magnitude from the natural annual evaporation over much 

of the route. It is likely that the deficiency will be more 

than compensated by the combined effects oflocal winter snowmelt that cannot run off, and by the influx of 

spring runoff into the depression likely to oc-:-ur over 

much of the pipeline. If sediments in the thawed cylinder around the pipe are not saturated initirlly, it is 

likely that they will become saturated along much of 

it if the heat transfer to the surface is primarily by 

conduction. 

We now consider the possible effects of heat transfer by convection arising from the potential irstability 

of the water column heated from below ln the sediments 

above the pipe. Density changes in the heated column 

will give rise to hydraulic gradients on the order of 2 

percent. They must be balanced by dynamic forces 

arising from upward flow of warm interst.itial water 

immediately above the pipe, and a downwarc counter 

flow of denser water at some distance from the pipe. 

The downward flow would normally represent a return 

of the upwelling water after it had given ur its heat 

at the surface. The permeability of the materials determines the velocity of this flow, and the velocity and 

thermal gradient determine the am o u n t of heat delivered to the surface by the convecting system. Thus, 

using Darcy's law for fluid flow and the heat-cC'nduction 

equation, we can attempt an estimate of the vertical 

velocity of water movement and the convective heat 

flow for any permeability. A rigorous formulation of 

this co:tvection problem is extremely complex and unwarranted by the objectives of the present study. It is 

possible that the numerical results presentee in table 

8 do not even represent the correct order of magnitude, 

but they are suggestive and of some interest. Tl'~ fourth 

column (table 8) shows the enhancement of centerline 

heat flux resulting from convection. For permeabilities 

of IO-'+ or less the convective contribution is unimportant. For perrneabilities of 10- 3 the contril'utions of 

convection and conduction at the centerline rre cornparable, and the effect on the surface regime becomes 

significant. However, the effective width of the convective contribution is about 4 feet, but that of the conduction is about 25 feet. Henc~ total loss by cC'nduction 

still tends to dominate loss by convection as is seen 

in column 5 (table 8). For permeabilities greater than 

Table a.--Idealized ~elation between permeability and aonveative enhanaement of 

surfaae heat loss (D=B ftJ 

Sediment Vertical velocit:t 

permeability (ft per yr) (ft per hr) 

(em per sec) 

10- 5 

--------- 0. 2 

10-.. --------- 2. 0 

10- 3 

--------- 20 

10- 2 

--------- 200 . 023 

10- 1 

--------- 2,000 . 23 

10- 0 --------- 20,000 2. 3 

Convection+conduction 

Conduction 

Centerline Total heat 

flux loss 

enhancement enhancement 

l. 008 l. 00 

l. 08 l. 01 

2. 0 1.2 

16 3.4 

160 27 

1,600 260 

19 

Vaporization rate for total 

heat loss 

(cu ft per yr per ft of pipe) 

28 

28 

32 

94 

730 

7070 
w- 3 , however, both centerline flux and total heat loss 

are dominated by convection; for permeable gravels 

(·vl0- 0 ) the total heat loss is enhanced by two orders 

of magnitude (column 5, table 8). 

Although the conductive centerline flux for 0=5 is 

more than twice as great as that for D=8, the net convective heat loss is about the same in each case because it is insensitive to depth of burial. 

The last column of table 8 shows the amount of water 

that would have to be vaporized annually to consume 

all of the heat that could be delivered to the surface 

from the convecting system. It is unlikely that the local 

hydrologic system would generally be able to supply 

the 7,000 cu ft or so ofwaterper foot of pipe required 

annually by an 80°C pipe in permeable gravel (this is 

equivalent to about 1 ft of evaporation per day over a 

20-ft band). Hence, con d i t ion s of desiccation would 

probably develop in most places where the pipe is embedded in very permeable materials. Until they did, 

however, the ground surface would remain unfrozen in 

winter, snow would melt almost as soon as it landed, 

and the pipeline might be shrouded in mist. 

The remarkable variation of heat loss with permeability suggested by this calculation could offer an opportunity to control excess moisture or desiccation by 

controlling the permeability of the material above and 

adjacent to the pipe. To do this effectively it would not 

be enough to control permeability of backfill in the 4-

foot-wide trench, as the convective system could not 

develop fully in such a restricted zone. 

This convective effect might be utilized also to reduce the temp~rature of the oil in the pipe 1 in e and 

thereby reduce some of the adverse effects of thawing 

permafrost. If the entire pipe were laid in a trench 

whose width exceeded its depth and the trench were 

backfilled with permeable gravel, surely the oil could 

not maintain such high temperatures. If the pipe temperature were reduced from 80°C to 25°C, the hydraulic gradients would be reduced by a factor of 10 and 

thermal gradients by a factor of about 3. The convective heat loss would then be reduced by a factor of 

about 30. However, the total heat loss would still exceed, by an order of magnitude, the conductive loss 

from an 80°C pipeline, and further cooling might be 

expected. Even in this case, however, water sup p 1 y 

may be a problem, as approximately 10 times the natural eva p or at ion over a 25-foot band would be required. The ultimate amount of cooling possible by this 

process would depend upon heat generation by viscous 

forces in the oil. However, temper at u r e s probably 

could not go below 15° to 20°C where the convective 

loss in permeable gravels would probably be of the 

same order as the conductive loss from an 80°C pipe. 

At such temperatures the water for coo 1 in g could 

probably be supplied by the natural regime. There 

are many problems with this scheme related to the 

availability of gravel and water, and the fact that significant perm afro s t thawing would occur even for 

20 

greatly reduced pipe temperatures. The system is 

stable in the sense that as the pipe tempe~ature tends 

to rise, the convective heat loss will rise sharply to 

counteract it. It is u n stab 1 e in the sense that, if increasing heat loss outstrips the local su~ply ofwater, 

desiccation will commence, and pipet em perature s 

would rise sharply. 

THE CONSTRUCTION DISTURBANCE 

Among the most conspicuous surface effects relating 

to the pipeline could be the artificial modifications resulting from vehicular activity, excavatio"", grading, 

artificial drainage, and clearing of timber. These can 

have significant effects on the insulating a"'d radiationco 11 e c tin g properties of the s u r face and on heattransfer processes relating to evaporaticn and movement of water. These, in turn, can have immediate and 

long-term effects on the thermal regime of the active 

layer and permafrost, and on the total environment. 

These effects are mentioned because some could be 

important. They are not elaborated because they are 

only indirectly related to the thermal effe~ts of a pipeline, and further discussion is beyond the scope of this 

report. 

CONCLUDING STATEMENT 

This report has outlined some potent~.al effects of 

a heated pipeline buried in permafrost. Effects selected for discussion were some that might be related 

to problems of disruption of the environment or to security of the pipeline. An attempt has been made to 

apply fundamental physical concepts to describe these 

effects in sufficient detail to identify th<:! controlling 

parameters. These are the qua n tit i e s that must be 

measured to ascertain whether the effects described 

mjght result in potential problems. An attempt has also 

been made to indicate the quantitative relation between 

these parameters and the effects they con~rol, to serve 

as an aid in anticipating the problem aft~r the measurements are made. 

No attempt has been made to elaborate the many detailed environmental and engineering consequences 

that might arise from each of the effects and the i r 

various combinations. Those consequeno~s discussed 

were generally in the nature of illustrativ~ examples. 

To the extent that these effects exist, t'leir detailed 

consequences are best considered by specialists: engineers, ecologists, hydrologists, geologif'ts, and those 

concerned wlth matters of administration and control. 

No attempt has been made to suggest engineering solutions to the problems raised. Nor is there any intention to imply that these problems have not already 

been considered, or indeed solved, by engineers concerned with them. 

This study was not exhaustive. Potent~al problems 

certainly exist that have not been considered, and some 

that have been considered me:.y be shown not to exist by 

further studies. The report represents one perspective 

on an overall problem that transcends many disciplines 
and requires the perspectives of many for an optimal 

solution. It is hoped that the report will provide one 

reference point for objective discussion between the 

people of many backgrounds who must communicate 

effectively on this issue. 

APPENDIX A--NUMERICAL SOLUTIONS 

The numerical results presented in the temperature 

effects section were ob~ained on the U.S. Geological 

Survey IBM 360/65 computer. The program was written by Patrick C. Doherty at the U.S. Geological Survey 

Computation Center. It is documented in detail by him 

in a separate report. The p•Jrpose ,of this appendix is 

to describe what has been done. 

The differential e q u at i on of heat conduction was 

solved numerically for the two-dimensional case of a 

pipe embedded in a semi-infinite homogeneous medium 

with a mean annual temperature Ta· From time t=O 

onward the pipe temperature (Tp) is assumed to be 

80°C, and the surface temperature fluctuates sinusoidally with a period of 1 year, an amplitude of 20°C, 

and a mean value ofT a· The initialtemperature chosen 

for the medium is the quasi-steady-state periodic solution consistent with this surface temperature. Two 

different values of T a were selected; -0. 8°C termed 

the 11 interior case, 11 and -8. 9°C term e d the 11 arctic 

case. 11 The effects of latent heat absorption and liberation on thawing and freezing at the 0°C isotherm were 

accounted for. Also accounted for were the differences 

in thermal conductivity and vo~umetric specific heat 

between the frozen and thawed states. The geometric 

consequences of volume change of water on freezing 

were neglected as these are effects of higher order. 

The thermal behavior of the permafrost in the presence of the hot pipe was s i m u 1 a ted over a 20-year 

period for three media described in the text (see table 

1 ). 

The principal geometric simplification of the problem resulted from representing the pipe as a prism 

with a square cross section 4 feet on a side. This permitted a substantial economy in the computations. It 

can be shown that this might result in an overestimate 

of the thawing distances by 10 percent or so, for early 

times. However, for longer times truncation errors in 

the time steps probably generated a small compensating error. For the purposes of the present study, 

discrepancies of t h i s magnitude are not considered 

important; arbitrary decisions in the choice of thermal 

parameters could result in larger ones. 

The problem solved is stated analytically below. The 

following notation is used (see also fig. 1): 

T =temperature 

t=time 

a =thermal diffusivity 

p=density 

21 

P =percent moisture by wet weight 

K=thermal conductivity 

x ,y=cartesian coordinates 

w=angular velocity for a period of 1 y~ar 

A=amplitude of surface temperature variation 

y* =bottom of problem space 

x"'=edge of problem space 

X(t), Y(t)=coordinates of O"C isotherm 

L=latent heat of melting ice, per unit mass 

The subscript 11 111 refers to the thawed state, and the 

sttbscript 11 211 refers to the frozen state. 

Differential equations: 

a 2T1 + a 2T1 _ 1 aT1 

ax2 ay2 - al a=r-

~ + a2T2 

ax 2 ay 2 

Initial condition: 

y ~ 0 

T > 0°C 

y ~ 0 

T < 0°C 

(x, y) not 

in pipe 

(x, y) not 

in pipe 

-yJw ~ T=T + A e 2a 2 sin (wt - y V~) , t=O a 2a2 

fixed boundary conditions: 

T=Ta + A sin w t , y=O 

T=Ta , y=y* 

aT=O , x-0, x* ax 

T=T , (x, y) on pipe surface 

p 

moving boundary conditions: 

Tz=T2=0, x=X, y=Y 

K aTz K aT 2 _L P aY z ay- - 2 ay-- P at 

x=X, y=Y 

APPENDIX B--STEADY-STATE RESULTf 

The equilibrium solution for an isothermal pipe at 

temperature Tp embedded in a semi-infinite homogeneous medium with ambient and surface terrnerature 

T a mc-.y be obtained directly from well-known results 

in potential theory. The problem may be stated formally as follows: (Notation is the same as in appendix 

A and fig. 1). 
Differential equation: 

~ + ~=0 ' y + 0 

ax 2 ay 2 

Boundary condition: 

T=Ta , y=O, (x, y) -+ oo 

T=T , on circle on radius a 

p with center at x=O, y=D 

(B-1) 

(B-2a) 

(B-2b) 

The solution to this problem shows that the isotherms 

for any temperature (T) between Tp and Ta are circles with their centers on the axis of symmetry (y 

axis). Their depths beneath the surface (d) and radii 

(r) are given by 

l/2 

d=(D 2 -a 2 ) coth [T - Ta cosh- 1 ~~ (B-3) T - T a p a 

r; [D 2-a 2 /

12 

roth2 ~p-_ T ~a cosh - 1 ~ -j'tB-4) 

The temperature T can be written as a function of d and 

r as follows 

T - T a 

T-=---T 

p a cosh- 1 ~ (B-5) 

An additional result, useful fo·r computing the rate of 

heat loss from the pipe to the surface in equilibrium 

is 

Q 2TIK (T _ T 

cosh- 1 ~ P a) (B-6) 

a 

where Q is the total heat lost from the pipe per unit 

time per unit length. 

Numerical results from (B-3) and (B-4) are presented in figure 6, for a pipe radius (a) of 2 feet and 

for burial depths (D) of 5 feet and 8 feet. We are interested in the parameters ( d, r) for the freezing isotherm and, therefore, T is replaced by T f (that is, 

0°C). The quanity 

will be ·referred to as the "temperature ratio." It is 

the ratio of the amount by which the freezing point 

exceeds the ambient temperature to the amount by 

which the pipe temperature exceeds the ambient temperature. For temperature ratios less than about 0.2 

equations (B-3) and B-4) can be replaced by the approximations 

TE - T 

d :::: r 3.75 a D=8 (B-7) Tf - T ' a 

TE - T 

2.92 a D=5 (B-8) Tf - T ' a 

22 

where d and r are given in feet and the te'llperature is 

in °C. If the pipe temperature T p is f0°C we have 

for the temperature ratio in the arctic case: 

0 - (-8.9) 

80 - (-8.9) 

and the interior case 

0.1 

0 - (-0.8) = 0.01 

80 - c-o.8) 

(B-9) 

(B-10) 

Combining the last four equations we se~ that for 0=8, 

the equilibrium configuration of the fre~zing surface 

in the arctic case is a circle whose center is about 

37.5 feet beneath the surfaceandwhoser~diusis about 

37.5 feet; that is, it is almost tangent to the surface 

and represents maximum thawing to a depth of about 

75 feet. Correspondingly for the interbr case (0=8) 

the radius is about 375 feet and maximum thawing is 

to about 750 feet. 

In the idealized problem,d is always greater than r 

as long as T a < T f• but their difference, representing 

the thickness of a frozen surface crust, is generally 

very small, for example, 0.08 feet and 0.8 feet, respectively, for the interior and arctic cases (0 • 8). 

Values for d and r can be read directly from the abscissa of figure 6 for any temperature ratio entered 

on the first ordinate scale (on the right). The second 

ordinate scale permits a direct reading of the parameters ( d, r) for any mean annual temoerature (T a) 

under the condition that the pipe temperature is 80°C. 

The third ordinate scale permits a dire~t determination of d and r for different pipe temperatures in the 

arctic case (T a = -8. 9°C), and the fourth ordinate scale 

yields the same in form at ion for the interior case 

(T a= -0.8°C). Thus, if the pipe temperature is reduced 

from 80°C to 20°C, the total equilibrium thaw depth 

( d + r) will be about 190 feet in the interior but on 1 y 

about 25 feet in the arctic case for 0 = 8. 

It is seen that as the temperature rat;') approaches 

zero (that is, as the mean annual temperature approaches the freezing pQint) the equilibrium thawing 

depths approach infinity. Physically this means that 

eventually the heated pipe would remove all . of the 

permafrost in this case; clearly this wo·.lld take a long 

time. Actually for equilibrium values of d and r of a 

few hundred feet, thousands of years would be required 

to approach that condition at depth, anc the solutions 

have little meaning in an engineering context. Above 

the pipe, however, equilibrium is approached rather 

quickly in all cases, and the solutions are useful for 

estimating conductive heat losses t':l the surface. 

These solutions are adeqiJate for order-ofmagnitude steady-state calculations. They depart from 

reality chiefly by not accounting for th~ fact that the 
oc oc oc 

0 0 

(.) 

0 

0 

co 

II 

~ 

... 

.g 

~ 

0.1 

(.) 

0 ~,~ I I co (.) '+-c.. 

0 0 1-1-

I ~ 

co 

~ I 

II 

.g 

... ~ 

.... c.. .g 1- c.. 

1-

= cu 

VI 

0 cu u VI 

0 

... (.) 

0 (_) -;:::: :;::: cu (_) 

=E. =.4. 

2 5 10 20 50 100 200 500 1000 

r and d, feet 

Figure 6.--Equilibriurn position o"f the thawing interface for a 4-foot-diarneter pipe with its 

axis at depth D (8 or 5 ft) in homogeneous materials. First ordinate scale on the right 

represents temperature ratio. Other ordinate scales are adjusted to give corresponding ternperature ratios for fixed values of the pipe temperature (Tp) or ambient temperature (Ta). 

For refined calculations reduce Tp by the factor K1 /K2 (see appendix B). 

conductivity of the thawed material (K 1) is usually less 

than the conductivity of the frozen material (K2 ). The 

effect of this contrast in conductivity can be accounted 

for by replacing T and Tp in foregoing equations by T' 

and Tp defined by 2 

T'=.!S...!_ K2 (T - Tf) + Tf T > Tf (B-lla) 

= T Ts_ Tf (B-llb) 

and 

T'•!S_!_ (T p Kz p - Tf) + Tf (B-12) 

2

The writer is indebted to Mr. William Joyner for a suggestion leading to this solution. 

For purposes of calculating d and r from figure 6 this 

is equivalent to reducing the pipe temperature in °C 

by the fraction K 1 /K2. If the true pipe tem·...,erature 

is 80°C, it is seen from table 1 that the effective pipe 

temperatures to be used in figure 6 for silt, clay, and 

water/ice are respectively 54.4°C, and 50.0°C, and 

17. 8°C. These lead to equilibrium thawing depths ( d + 

r) in the arctic cases of 52 feet; 50 feet, and 24 feet, 

respectively (fig. 6). Corresponding valueP for the 

interior case are 520 feet for silt, 480 feet for clay, 

and 185 feet for water/ice. It can be estimated from 

figure 4 and scaling laws for heat conduction that in 

the arctic case equilibrium might be appror~hed for 

water/ice in two decades, but for more typical materials it will take on the order of a half century. For 

the interior cases equilibrium would not be approached 

for tens of centuries. 

fr U.S. GOVERNMENT PRINTING OFFICE: 1970 0- 383-781 

23