frost heave pipeline

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Modelling of pipeline under differential frost heave

considering post-peak reduction of uplift

resistance in frozen soil

Bipul C. Hawlader, Vincent Morgan, and Jack I. Clark

Abstract: The interaction between buried chilled gas pipelines and the surrounding frozen soil subjected to differential

frost heave displacements has been investigated. A simplified semi-analytical solution has been developed considering

the post-peak reduction of uplift resistance in frozen soil as observed in laboratory tests. The nonlinear stressstrain be-

haviour of the pipeline at large strains has been incorporated in the analysis using an equivalent bending stiffness. The

predicted results agree well with our finite element analysis and also with numerical predictions available in the litera-

ture, hence the simple semi-analytical solution can be considered as an alternative to numerical techniques. A paramet-

ric study has been carried out to identify the influence of key factors that can modify the uplift resistance in frozen

soil. Among them, the residual uplift resistance has been found to be the important parameter for the development of

stresses and strains in the pipeline.

Key words: pipeline, frost heave, discontinuous permafrost, semi-analytical solution, uplift resistance, frozen soil.

Rsum : On a tudi linteraction entre les pipelines de gaz refroidi et le sol gel environnant assujettis des dpla-

cements en soulvement. On a dvelopp une solution semi-analytique simplifie en prenant en considration la rduc-

tion postpic de la rsistance au soulvement dans le sol gel telle quobserve dans les essais en laboratoire. Le

comportement contraintedformation non linaire du pipeline grandes dformations a t incorpor dans lanalyse en

utilisant une rigidit quivalente en flexion Les rsultats prdits concordent bien avec notre analyse en lments finis et

aussi avec les prdictions numriques disponibles dans la littrature; ainsi, on peut considrer la solution semi-anlytique

simple comme tant une alternative aux techniques numriques. On a ralis une tude paramtrique pour identifier

linfluence des facteurs cls qui peuvent modifier la rsistance au soulvement dans le sol gel. Parmi eux, on a trouv

que la rsistance rsiduelle au soulvement tait le paramtre important pour le dveloppement des contraintes et des

dformations dans le pipeline.

Mots cls : pipeline, soulvement d au gel, perglisol discontinu, solution semi-analytique, rsistance au soulvement,

sol gel.

[Traduit par la Rdaction] Hawlader et al. 293

Introduction

The growing interest in northern oil and gas resources de-mands an efficient design method for arctic pipelines. Pipe-lines in northern regions are often buried in continuous ordiscontinuous permafrost and traverse large distancesthrough a variety of soil types in terms of frost susceptibil-ity. The current design philosophy for natural gas pipelinesthrough permafrost is to chill the product to prevent thawingof frozen ground. This causes freezing of initially unfrozen

soil when the pipeline passes through unfrozen ground indiscontinuous permafrost, however, and results in frost heaveinduced vertical displacement. Uniform heave is not a major

issue for pipeline integrity as long as the minimum cover re-quirement is maintained; however, differential heave nearthe interface between two soils with different frost suscepti-bilities or frozen and unfrozen soil could be a critical issuein designing arctic pipelines. The maximum internal pres-sure usually dominates the design (wall thickness and grade)of conventional pipelines; however, differential frost heavemay be one of the major sources of load for buried chilledpipelines in discontinuous permafrost. The magnitude offrost heave induced stresses and strains in the pipeline de-

pends on the resistance offered by the surrounding soil. Forexample, a low uplift resistance near the interface could pro-duce a gradual movement of the pipe and therefore generateless frost heave induced stresses and strains (Nixon andHazen 1993). In a discontinuous permafrost section, the up-lift resistance of the frozen (nonheaving) side of the transi-tion is the main concern, as it dominates the design ofpipelines based on maximum stress and strain.

The objective of this study is to develop a semi-analyticalsolution considering the uplift resistance in frozen soilobserved in laboratory tests. The main attraction of this ap-proach is that it does not require complex numerical calcula-

Can. Geotech. J. 43 : 282293 (2006) doi:10.1139/T06-003 2006 NRC Canada

282

Received 15 March 2005. Accepted 16 December 2005.Published on the NRC Research Press Web site athttp://cgj.nrc.ca on 10 February 2006.

B.C. Hawlader,1 V. Morgan, and J.I. Clark. C-CORE,Captain Robert A Bartlett Building, Morrissey Road, St.Johns, NL A1B 3X5, Canada.

1Corresponding author (e-mail: [email protected]).


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tions. To verify the solution, the results have been comparedwith finite difference (Nixon 1994) and finite element re-sults. The effects and sensitivity to several soil parametersare also presented. Although the solution has been devel-oped for differential frost heave, it could also be used formodelling pipelines subjected to transverse soil movementswhere the post-peak reduction of soil resistance (softening)

is a concern.

Past works

The modelling of frost heave induced pipelinesoil inter-action is a complex, time-dependent thermo-hydro-mechanicalprocess. The complete coupled analysis of this process isrelatively difficult, and therefore in the state of practice twoindependent analyses are usually performed for frost heaveand mechanical response. Many useful concepts relating tofrost heave prediction and its implementation for structuralanalysis of pipelines have been proposed in the past. Con-sidering the pipeline as a passive structural member, Nixonet al. (1983) performed a series of finite element analyses

modelling the soil as an elastic and nonlinear viscous contin-uum. Selvadurai (1988) considered the soil purely as anelastic material. The uplift resistance in frozen soil has beenshown to be nonelastic at large displacement (Nixon 1998;Foriero and Ladanyi 1994), and therefore the application ofthis solution is limited. Shen and Ladanyi (1991) developeda combined numerical algorithm where the heat and mois-ture movements were simulated using finite difference, andthe mechanical behaviour was simulated using the finite ele-ment method. Considering elastic, plastic, and creep behav-iour of frozen soil, Selvadurai and his co-workers(Selvadurai and Shinde 1993; Selvadurai et al. 1999) pre-sented a more rigorous continuum modelling of pipesoil in-teraction. Although continuum modelling provides some

advantages, current state of practice uses a beam modelwhere the resistance offered by the surrounding soil is repre-sented by a series of Winkler springs. Several studies using aspring model for frozen soil are available in the literature(e.g., Ladanyi and Lemaire 1984; Rajani and Morgenstern1993; Nixon 1994; Razaqpur and Wang 1996). One of thekey parameters in this type of analysis is the accurate defini-tion of the resistance of the surrounding soil.

Nixon (1998) and Liu et al. (2004) performed a series ofmidscale laboratory tests to provide uplift resistance func-tions of frozen soils. It has been shown that the uplift resis-tance, as a function of pipe displacement, reaches a peakvalue and then reduces to a residual value at large displace-ment. Depending on test conditions, the post-peak resistancefalls to between 40% and 70% of the peak resistance at adisplacement of three times that at the peak resistance and issensitive to soil temperature, pipe diameter, burial depth, andloading rate. The formation of tension cracks in the frozensoil over and adjacent to the pipe can explain the reductionof this resistance (Nixon 1998; Liu et al. 2004). Foriero andLadanyi (1994) also performed uplift resistance tests infrozen soil using a 270 mm diameter pipe to understand themechanism of the full-scale test results from the Caen testfacility in France. A similarly shaped uplift resistance func-tion was obtained, although the reduction of resistance afterthe peak was relatively sharp in comparison to the results of

Nixon (1998). Although the mechanism could be differentdepending on the test conditions and soil type, the shape ofthe uplift resistance function is similar in all cases. As thetests progressed, the peak resistance developed and was fol-lowed by a post-peak reduction of resistance to a residualvalue at large displacement. Considering the post-peak re-duction of uplift resistance in frozen soil, Nixon (1994) de-

veloped a numerical algorithm for analyzing pipelines indiscontinuous permafrost. It was shown that the predictedbending strain is considerably reduced because of the post-peak reduction of uplift resistance.

For pipelines in northern regions, transitions from frozento unfrozen frost-susceptible ground can occur several timesper kilometre (Greenslade and Nixon 2000). In view of un -certainties in the modelling of frost heave and its effects inthe analysis of pipelines in discontinuous permafrost, a sim-ple but sufficiently accurate method of prediction, such as ananalytical or semi-analytical solution, would be an efficientmethod for analyzing pipelines operating under these condi-tions. Ladanyi and Lemaire (1984) developed a simplifiedanalytical solution considering both soil and pipeline as lin-

ear elastic materials. Rajani and Morgenstern (1993) devel-oped an enhanced analytical solution for an elastic pipelineby idealizing the behaviour of frozen soil as an elastic per-fectly plastic material. Major limitations of these solutionsare that the uplift resistance in frozen soil has not been mod-elled as observed in the laboratory (Foriero and Ladanyi1994; Nixon 1998; Liu et al. 2004), and the nonlinearstressstrain behaviour of the pipeline was not considered.

Soilpipeline interaction

The soilpipeline interaction considered in this study islimited to vertical ground movement perpendicular to thepipeline axis. The stress and strain developed in the pipe de-

pend on soil characteristics, pipeline geometry and materialproperties, and the amount of frost heave experienced. Fig-ure 1 schematically presents an overview of the model usedin this study. The free-field frost heave (wf), where the frostheave is not influenced by the restraint of the pipeline infrozen ground, can be calculated using available frost heavemodels (Konrad and Morgenstern 1984; Nixon 1986; Shahand Razaqpur 1993; Konrad and Shen 1996; Hawlader et al.2004). In the same way as described by Rajani andMorgenstern (1992, 1993), it is assumed that the displace-ment profile of the pipeline due to differential heave followsa double-curvature profile. Also, the process of frost heavein frost-susceptible soil has been decoupled from the me-chanical modelling of the pipeline in non-frost-susceptiblesoil (Rajani and Morgenstern 1993). The action due to frostheave is transferred to the pipeline by applying an end dis-placement (w0) at the interface equal to half of the free-fieldfrost heave (wf/2), and the analysis has been carried out onlyfor the portion of the pipeline located in frozen soil. The au-thors are aware of the fact that the end displacement dependson the coupled process of frost heave and pipelinesoil inter-action. The decoupling technique greatly simplifies the anal-ysis, however. Figure 1b shows the end displacements fortwo freezing periods in the Caen large-scale test (Carlson1994), where an 18 m long pipeline was buried with one halfin a frost-susceptible silt and the other half in a non-frost-

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284 Can. Geotech. J. Vol. 43, 2006

susceptible sand, and the effects of differential frost heavewere investigated. In this figure, w frepresents the differencebetween the vertical displacement of the two ends of thepipe, and w0 represents the difference between the verticaldisplacement at the sandsilt interface and the end of thepipe in the non-frost-susceptible sand layer. As shown, mea-sured w0 is slightly less than wf/2. Therefore, the analysespresented in this study using w0 = wf/2 will tend to beslightly conservative for this case.

Modelling of soil resistance

The uplift resistance in frozen soil as a function of dis-placement has been divided into three segments (Fig. 1c).The formulation has been performed for three levels of enddisplacement (small, intermediate, and large) to describe thesequence of deformation during the process of heaving.

At small end displacements (0 < w0 we) the upward de-formation of soil above the pipeline is essentially elastic,where we is the displacement at which the peak uplift resis-tance (Fe) is developed. The uplift resistance in this range ofdisplacement has been defined using the line OA in Fig. 1c,and the segment of the pipe where the surrounding soil be-haves elastically has been defined as region A (Fig. 1a). Atintermediate end displacements (we < w0 wp) the uplift re-sistance in a segment of the pipe near the interface for a

length x0, where the vertical displacement of the pipe isgreater thanwe, decreases from the peak value. This segmenthas been defined as region B. Here, wp is the displacementat which the uplift resistance reduces to the residual value(Fp). The uplift resistance in region B has been defined byusing the linear line AB shown in Fig. 1c. At large end dis-placements (w0 > wp) a constant residual uplift resistance(Fp) is developed on a segment of the pipe near the interfacewhich has been defined as region C. The uplift resistance inthis region has been defined using the linear line BC inFig. 1c. The horizontal distance from the interface to thepoint that separates regions B and C is denoted by x(Fig. 1a).

Mathematical formulation

As three types of uplift resistance are possible dependingon the end displacement, the mathematical formulation ofthese combinations is described separately in the followingsections. Note that the displacement (w), slope (), bendingmoment (M), and shear force (S) are always a function ofdistance (x) along the pipe.

Small end displacement (0 < w0 we)The soil is elastic at this level of end displacement and

Fig. 1. Modelling of differential frost heave. (a) Response of pipeline at large end displacement (at intermediate end displacement, x =

0; at small end displacement, x0 = x = 0). (b) Caen full-scale test (data from Carlson 1994). (c) Uplift resistance function.


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only region A is formed. Both x0 and x(Fig. 1a) are equal tozero, and the origin is at the interface. The governing differ-ential equation and its solution can be written as (Hetnyi1946)

[1] EI w

xk w

d

d

As A

4

4 0+ =

[2] wA = exp(x)(C1 cos x + C2 sin x)

where EI is the bending stiffness of the pipeline (in which Eis the elastic modulus and I is the moment of inertia), wA isthe displacement in region A, C1 and C2 are constants, and

is the reciprocal of characteristic length (= k EIs/ 44 , where

ks represents the stiffness of the soil spring). The expres-

sions for slope (), bending moment (M), and shear force (S)can be obtained by successive differentiation of eq. [2] andare listed in Appendix A (eqs. [A1][A3]).

Imposing boundary conditions at x = 0 (M = 0 and w =w0), it can be shown that C1 = w0 and C2 = 0, which have

been used to calculate the displacement, slope, bending mo-ment, and shear force using eqs. [2] and [A1][A3].

Intermediate end displacement (we < w0 wp)At this level of end displacement the uplift resistance for a

length x0 near the interface is governed by the post-peakdegradation line AB (Fig. 1c), and the resistance for the restof the length is still governed by elastic line OA (Fig. 1 c).Therefore, two segments of uplift resistance function OAand AB define the response of the pipeline in regions A andB, respectively (Fig. 1a). Note that x =0 at this stage. Formathematical convenience the origin is considered at thepoint that separates regions A and B. The governing differ-ential equation for region B can be written as

[3] EI w

xk w k w w

d

d

Bs B s B e

4

4 0+ = ( )

with s p p

s p e

=

k w F

k w w( )

where wB is the displacement in region B. The factor (>1)defines the rate of degradation of uplift resistance after thepeak: the higher the value of, the faster the rate of degra-dation. Note that equals 0 and 1 represent elastic andelastic perfectly plastic conditions, respectively. Equation[3] can be rewritten as

[4] d

d

B sB

s e4

4

10

w

x

k

EIw

k w

EI

+

=

( )

The solution of this differential equation is

[5] wB = we/( 1) + C3 cos x + C4 sin x

+ C5 cosh x + C6 sinh x

where = k EIs ( )/14 , and C3C6 are unknown con-

stants. Again, the expressions for slope, bending moment,and shear force can be obtained by successive differentiationof eq. [5] and are listed in Appendix A (eqs. [A4][A6]).

The governing differential equation and its solution for re-gion A are the same as eqs. [1] and [2], respectively. Notethat eq. [2] also represents the elastic behaviour for smallend displacements, but the boundary conditions for obtain-ing the constants C1 and C2 are different. The followingboundary conditions should be used to obtain the constants(C1C6 and x0) for intermediate end displacement.

Boundary conditionsAt x = x0, the displacement and bending moment are

equal to w0 and zero, respectively. Therefore, using eqs. [5]and [A5] we have

[6] C3 cos x0 C4 sin x0 + C5 cosh x0

C6 sinh x0 = w0 + we/(1 )

[7] C3 cos x0 + C4 sin x0 + C5 cosh x0

C6 sinh x0 = 0

Considering the displacement and slope compatibility andthe moment and shear force equilibrium at x= 0, the follow-

ing relationships can be obtained:

[8] C3 + C5 C1 = we/(1 )

[9] C4 + C6 = [C1 + C2] where/ =

[10] C3 + C5 = 2 2C2

[11] C4 + C6 = 23[C1 + C2]

Lastly, the rate of shear force variation, EI(d4wA/dx4), at

x= 0 is equal to the peak soil resistance ( Fe), which yields

[12] C1 = Fe/44EI

Equations [8][12] have been used to obtain the unknown

constants (C1C6 and x0). A simple solution procedure hasbeen described in Appendix B. Once the values of these con-stants are known, the variation of displacement, slope, bend-ing moment, and shear force can be calculated usingeqs. [2], [5], and [A1][A6].

Large end displacement (w0 > wp)When the end displacement (w0) is greater than wp, the

constant residual uplift resistance (Fp) will be developed ona portion of the pipe near the interface (region C in Fig. 1 a)where the displacement of the pipe exceeds wp. Next to thisregion, where the displacement of the pipe is more than webut less than wp, there will be another region B where thepost-peak degradation line AB (Fig. 1c) represents the uplift

resistance. The uplift resistance on the rest of the pipe (re-gion A) is still governed by elastic soil deformation. Formathematical convenience, the origin is considered at thepoint that separates regions B and C.

The governing differential equation and its solution for re-gion C ( x x 0) can be written as

[13] EI w

xF

d

d

Cp

4

4 =

[14] wF x

EIC

xC

xC x CC

p= + + + +

4

7

3

8

2

9 1024 6 2

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where wC is the displacement in region C. Once again, theexpressions for slope, bending moment, and shear force canbe obtained by successive differentiation of eq. [14] and arelisted in Appendix A (eqs. [A7][A9]).

The displacements for region A (x> x0) and region B (0 wp)

Inserting (C3 + C5) from eq. [25] into eq. [17] gives

[C1] C w F

EI10

1=

e p

4( )

Using eqs. [19] and [25], we have

[C2] CF

EI

C3 4

822 2

= p

and

[C3] C FEI

C5 4

822 2

= +p

As the equations obtained from the boundary conditions(eqs. [15][26]) are not linear, the following trail and errorapproach has been used to solve these equations for un-knowns:(1) Set the values of C4, C6, x, and x0. The first trial starts

with their values in the previous step.(2) Calculate C7 (eq. [20]), C8 (eq. [16]), C9 (eq. [18]), C10

(eq. [C1]), C3 (eq. [C2]), and C5 (eq. [C3]).(3) Check whether it satisfies eq. [15]. If not, increase or

decrease the value ofxdepending on a negative or posi-tive value of the left-hand side of eq. [15]. Repeat steps2 and 3 until it becomes within the limit of tolerance.Using the calculated values of C3 and C5 and replacingC1 using eq. [26], calculate the values ofC2, C4, and C6using eqs. [22][24].

(4) Check whether it satisfies eq. [21]. If not, update thevalue ofx0 and repeat steps 4 and 5 until it satisfies thedesired level of tolerance.

(5) Update the values ofC4 and C6 as the averages of theirassumed and calculated values, and repeat the calcula-tion until the difference between the calculated values ofunknowns in two successive trials is within the limit oftolerance.

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