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One of the major sources of induced load to pipelines is the differential heave near the interface between two types of soil with different frost heave susceptibilities or between frozen and unfrozen soils.
Several researchers performed Winkler models to predict the differential frost heave (e.g. Nixon et al. 1983; Rajani and Morgenstern 1992; 1993; 1994; Razaqpur and Wang 1996). For instance, Rajani and Morgenstern (1992) created a Winkler model, which assumed the ice rich permafrost as an elastic-plastic foundation. The Winkler model was applied to small-scale model steel pipelines embedded in polycrystalline ice and satisfactory comparisons were obtained between the observations and simulations (Rajani and Morgenstern 1993). The same Winkler model also simulated the differential frost heave observed at the Caen frost heave experiment (Rajani and Morgenstern 1994). Despite its simplicity, the developed Winkler model could predict the overall pipe defections and induced stresses of the Caen frost heave experiment. However, the Winkler models have limitations. Since the Winkler models used springs to account for the axial and radial restraints by surrounding soil, the soil pressure was only characterized in terms of the absolute pipe displacement. Consequently, the Winkler model neglected the impact of rigid body movements of the soil and the interaction through the soil from location to location.
Selvadurai et al. (1999) developed a more rigorous continuum modeling of soil-pipeline interaction due to differential frost heave. The continuum approach was established to model the interaction induced by the three-dimensional time-dependent growth of a frost bulb around the chilled pipeline, and the induced stress dependent expansion of the frost susceptible soil. The developed three-dimensional frost heave model could simulate the behavior of a buried pipeline in the Caen frost heave experiment. However, this three-dimensional approach modeled the pipe as one-dimensional beam elements that might influence the thermal analysis and the soil-pipeline interaction induced by axial, shear and flexural stiffness characteristics. Furthermore, the continuum model has not yet been applied to analyze large-diameter pipelines subjected to differential frost heave.
A two-dimensional numerical procedure using the SP porosity growth function was discussed in the previous chapter. The purpose of this chapter is to develop a three-dimensional frost heave model using the SP porosity growth function to predict the soil-pipeline interaction due to differential frost heave. Subsequently, the developed model is verified by the responses of a large-diameter pipeline due to differential frost heave using observations of the UAF frost heave experiment.
In the following sections, a description of the three-dimensional frost heave model is described first. The description of the UAF frost heave experiment at Fairbanks, Alaska follows. Verification of the 3-D frost heave model against the UAF frost heave experiment is presented. Finally, a qualitative analysis of abrupt pipeline uplift is conducted in response to fluctuation of pipeline temperature.
Three-dimensional frost heave modeling with SP porosity growth
The Segregation Potential (SP) concept is a macroscopic semi-empirical model based on laboratory observations and theoretical considerations such as the validity of the generalized Clausius-Clapeyron equation at the active ice lens. In frost-susceptible soils, the volume change occurs as a result of ice lens formation as in-situ pore water and migratory water freeze at the segregation freezing front. The stress-strain behavior of frozen soils depends mainly upon factors such as soil type, mineralogical composition, ice content, temperature, and strain rate. In freezing soils, stresses are applied slowly since the heave rates are usually small, for instance, on the order of tenths of millimeters per day in the UAF frost heave experiment (Kim et al. 2008). The frozen soils then deform in a ductile manner and the stress-strain relationship was approximated by a modified bilinear law as shown in Figure 5.2.
As described in previous chapters, the SP porosity growth function was modeled as porosity increment due to frost heave. The total porosity increment („nt) was clearly composed of two components: due to the in-situ freezing and the segregation freezing.
Defining the water content at time t as w(t), the porosity growth due to in-situ freezing („nin) at time t+„t can be expressed as:
where ±w(t) = volumetric fraction of water at time t.
The porosity growth due to segregation freezing („nsp) is calculated as:
where Œsp = the effective area of the segregation freezing temperature in the element; v = the rate of migrating water; Vsp = the effective volume of segregation freezing zone; SP = segregation potential; ³t = soil pressure acting on segregation freezing zone; and gradTsp = temperature gradient in the segregation freezing zone.
The gradTsp in eq. [6.0] is taken in direction 1 in Figure 6.1, which is the direction of heat flow and maximum temperature gradient direction, and is determined as:
where âˆ‚T/âˆ‚x, âˆ‚T/âˆ‚y, and âˆ‚T/âˆ‚z were temperature gradients in x, y and z principal directions of the global coordinating system, respectively.
The SP porosity growth function was obtained by adding eqs. [6.0] and [6.0]:
The total strain increment („¥€ ) was modeled to consist of two components: the modified bi-linear elastic strain increment („¥€ el) and the strain increment due to the SP porosity growth function („¥€ sp) as:
The strain increment due to the SP porosity growth function was modeled as anisotropic. The three-dimensional anisotropy could be written using orthogonal matrix as:
where º = dimensionless value between 1/3 and 1, representing conditions between isotropic and one-dimensional case, respectively. The direction 1 was specified as the major principal direction in eq. [6.0].
Component of the strain increment due to the three-dimensional SP porosity growth function in a global x-y-z coordinating system is obtained by the following transformation rule:
where ¦ = the angle axis x makes with the reflection of heat flow direction 1 on x-y plane; and ª = the angle axis z makes with heat flow direction 1 as defined in Figure 6.1.
The developed three-dimensional SP porosity growth function is applied to examine the soil-pipeline interaction induced by differential frost heave in a full-scale frost heave experiment.
UAF frost heave facility
The University of Alaska Fairbanks (UAF) and Hokkaido University, Japan, had conducted a full-scale field experiment to determine the differential heave of a 105m long pipe near frozen-unfrozen boundary from December 1999 to August 2003.
In this section, the UAF frost heave experiment is briefly described. Figure 6.2 shows the initial permafrost conditions along the pipeline indicating a rapid deepening of the permafrost table at 30m from the inlet riser. The pipeline crossed a boundary between permafrost and unfrozen ground. A 0.914m diameter, 105m long chilled pipe with X65 grade and 9mm wall thickness was used. The first 30m of the pipe were in a shallower supra-permafrost table area and the rest 75m were in unfrozen ground - a deeper supra-permafrost table area. The pipeline was backfilled with sand to the top of the pipeline in the shallower supra-permafrost area, and to the spring line of pipes in the deeper supra-permafrost table area, respectively. After the sand was bounded by water, the pipeline was covered with approximately 0.9m of in-situ crushed soil.
Figure 6.2 also shows the instrumentation and monitoring. The depth to the groundwater table was monitored by three open standpipes; Well #1, 2, and 3 were located approximately 104m, 113m, and 58m from the inlet riser, respectively.
Three thermal fences (TF) were designed to monitor changes in the thermal regime of the soil system. TFA and TFB were placed in the deeper supra-permafrost area and TFC in the shallower supra-permafrost area, respectively. Figures 6.3a and 6.3b show the location and spacing of the thermistors for TFB and TFC, which located at 36.5m and 13m from the inlet riser, respectively. TFB had three thermistor strings located between 1 and 3 m from the pipeline centerline with depths ranging from 0.09 to 7.76m (elevation 1.09 to -6.87 m) beneath the ground surface. Thermistor spacing ranged from 0.25 to 1.0m. TFC contained four thermistor strings with thermistor depths ranging from 0.04 to 7.00m (elevation 0.54 to -6.42m) beneath the ground surface. Thermistor spacing ranges from 0.5 to 1.0m. The reference elevation was defined as 1.00m. Pipeline temperatures were monitored by 9 thermistors placed along the outside of the pipeline.
Pipeline movement was monitored by 28 heave rods (HR) welded directly to the top of the pipe. In order to monitor heave of the soil directly beneath the pipeline, five heave gauges (HG) were installed. The gauges were located at 27.85, 30.96, 32.33, 37.04, and 68.85m, from the inlet riser, respectively. The heave gauges monitored movement of the first 1m of soil underneath the pipeline.
There were 11 stations along the pipeline in order to monitor the pipe strain. Forty weldable strain gauges (SG) were placed on the outside surface of the pipeline.
The results derived from the UAF frost heave experiment were fully documented by Bray (2003) and Huang et al. (2004) and will be presented, where appropriate, in the subsequent sections.
Modeling of the buried chilled gas pipeline problem
Figure 6.4a shows the pipeline movement profile for various days between day 4 and 1340. Day 4 was the first day of the survey on December 11, 1999 and the last day of the survey on August 8, 2003, respectively. Up to day 50, the pipeline experienced settlement. The initial settlement was possibly due to the thermal disturbance during excavation and the increased overburden pressure from the pipeline and berm. The shallower supra-permafrost area experienced approximately 0.025m initial settlement, but the deeper supra-permafrost area showed less settlement. Largest heave rate was observed between day 50 and 266. After day 266, lack of heave was observed at HR-26 and HR-27, which were at 74.04m and 89.275m from the inlet riser. One possible explanation was simply less frost heave susceptibility of the soil in that vicinity. Alternatively, groundwater condition could be locally different. Figure 6.4b shows the groundwater table fluctuations over a period of nearly two years because groundwater levels were measured starting from the second winter in 2000. Well #3 constantly showed a higher water table than otherwise. The lower groundwater might create the lack of heave at HR-26 and HR-27. The reduced heave also was observed at HR-28, which was at the end of the test section 105m. This behavior might be due to boundary effects. At the end of the test section, a vertical riser took the chilled air from the test section back to the refrigeration units via an above ground pipe. Since there was no chilling of the ground beyond the end of the pipeline test section, only a limited frost bulb and resulting heave would develop. After day 1060, the pipeline experienced a slow rate of settlement and had little heave. These phenomena continued until the end of the operation. The settlement occurred not only in the deeper supra-permafrost area but also in the shallower area. The reason of the settlement behavior was unknown, however, the settlement occurred right after a M7.9 earthquake in interior Alaska on November 3, 2002 (Huang et al. 2004).
The main purpose of this study is to predict and examine the soil-pipeline interaction due to differential frost heave traversing a different permafrost condition by applying a three-dimensional frost heave model developed in the study. Therefore, the area between the shallower and the deeper supra-permafrost area in the UAF frost heave experiment was chosen for the verification of the simulation. The dimensions (x, y, z) of the ground were modeled as 15m x 20m x 50m as shown in Figures 6.5a through 6.5c. Because of symmetry about the centre line, only half of the actual geometry was modeled. The pipe had been modeled using the one-dimensional beam element to reduce computational time (e.g. Selvadurai et al. 1999; Selvadurai and Shinde 1993). For the sake of more reasonable simulations, the pipe was model as an octagonal shape using solid mesh for this study.
The simulations were conducted with a mesh characterized by 11088 rectangular elements and 12913 nodes.
Boundary conditions and initial conditions:
Types of boundary conditions used for the simulation are illustrated in Figures 6.5a through 6.5c. Horizontal displacements are not allowed along vertical sides, while the bottom is fixed in both horizontal and vertical directions. The pipeline temperature data at 5 points, which were at 2m, 15m, 25m, 40m, and 50m from the inlet riser, were used for the simulation. Since the pipe temperature fluctuated with time, a step temperature was applied to the numerical simulation. The phases were divided into every 60days each. The average temperature during each phase was defined as input pipeline temperature. With increasing distance from the inlet riser, the pipeline temperature increased. For instance, the input pipeline temperature at 2m from the inlet riser was usually 2oC colder than that at 50m. The 50m modeled pipeline was divided into every 10m each. The input pipeline temperatures were applied to each section uniformly as shown in Figure 6.6. The air temperature was converted to the ground surface temperature by n-factor. Zero heat flux was applied at the vertical boundaries.
The initial ground temperature was created by the following procedure. First, a temperature of -0.1oC was applied to all nodes in the permafrost part, and 1oC was applied to the rest of the mesh nodes. The temperature of the bottom horizontal boundary (20m below the ground surface) was fixed at -0.1oC. Then, the simulation was executed without the pipe temperature input for 3years. Finally, -1oC was applied to an area of 1m wide and 1.8m deep at the center. It was assumed as the trench for the pipe was excavated during wintertime.
Figure 6.7a shows an example of foundation heave versus time for three heave gauges, HG-2, HG-4, and HG-5, respectively. The heave trend was essentially linear up to day 110, which suggests the simple linear volumetric pore water expansion. Initial stabilization period was observed after day 110. As the freezing front penetrated and passed through the anchor of the heave gauge, the heave gauges would no longer detect any movement between the plate and anchor. The first plateau occurred in response to latent heat release of pore water below groundwater table. Following day 130, HG-2 and HG-4 experienced abrupt jumps, which were caused by ice segregation, but HG-5 did not experience any additional jump in total heave amount (Huang et al. 2004). From the observation, the groundwater level would be determined as approximately -1.7m at HG-2 and HG-3 representing in the shallower supra-permafrost area, and approximately -2.2m at HG-5 representing in the deeper supra-permafrost area, respectively, in mid-April.
The groundwater table input for the three-dimensional simulation was created using the observation at Well #3 located approximately 58m from the inlet riser. In the early part of the summers, an abrupt change of groundwater level was observed. The abrupt change occurred due to the confinement by the frozen layer during winter. No monitor wells were installed closer than 58m from the inlet riser. However, temperature data of TFC indicated that the active layer completely froze in the shallower supra-permafrost area; suggesting the ground water within the shallower supra-permafrost area was also seasonally confined to summer months. Figure 6.7b shows the variation of the groundwater data input for the three-dimensional simulation. Measurement of the first year cycle was extrapolated using the data of the third year cycle and -1.7m elevation was applied to the stabilization period in between mid-December and mid-April of the first year cycle according to the heave gauge analysis above. The calculation of the segregation heave started when the segregation freezing zone reached the groundwater table.
As shown in eq. [6.0], the moisture migration depends not only on gradTsp but also on the stress field. The soil pressure acting on segregation freezing zone, which means ³t , was composed of two components: (1) induced soil stress due to frost heave at previous time step (³sp), and (2) overburden pressure of the soil above the segregation freezing zone (³ov). The overburden pressure was assumed to be a function of freezing depth, bulk and buoyant weights of soil, and groundwater table elevation. The overburden pressure was evaluated as following:
where W(t) = the depth of the groundwater table from the ground surface; Xs(t) = the depth of the segregation freezing zone from the ground surface; d(t) = dry soil unit weight; w = water unit weight; t(t) = bulk soil unit weight; and b(t) = buoyant soil unit weight.
Inline pipe pressure of 1.4MPa was applied based on the field data (Kim et al. 2005).
The simulation domain consisted of frost-susceptible and non-frost-susceptible materials. The initial permafrost conditions were reproduced in response to the permafrost condition. The fully-saturated part was modeled as frost susceptible, otherwise as non-frost susceptible.
Material properties of the UAF frost heave experiment were described in previous chapters. Initial dry densities of the Fairbanks silt and Lanzhou sand were calculated as 1308 and 1894kg/m3, respectively. The thermal material properties were summarized in Table 5.2.
The following temperature (T) dependent mechanical properties were used for the bi-linear elastic stress-strain relationships shown in Figure 5.2.
The temperature of the peak strength (³m) of the Fairbanks silt was determined as:
where ³m was in kPa. The peak strength for the unsaturated part was modeled as 50% of the fully saturated part.
The yield stress (³y) was estimated in terms of the initial dry density (²d = 1308kg/m3) as:
Temperature dependent Young's modulus (E) of the Fairbanks silt was determined as:
where E was in MPa.
When the Young's modulus of the frozen Fairbanks silt determined by eq. [6.0] was smaller than 11.2MPa, the value was assumed to be equal to 11.2MPa. Poisson's ratio was taken as € = 0.3 for both the frozen and unfrozen case.
The long-term yield strength and elastic modulus are temperature dependent and determined as:
where ³y was in kPa, and E was in MPa.
When the temperature was warmer than -0.1oC, the peak strength of the Lanzhou sand at -0.1oC was used. The Young's modulus of the unfrozen Lanzhou sand was assumed to be constant and equal to 20MPa. When the Young's modulus of the frozen Lanzhou sand determined by eq. [6.0] was smaller than 20MPa, the value was assumed to be equal to 20MPa.
The post-yield modulus of Lanzhou sand was modeled as zero. Poisson's ratio was taken as = 0.3 for both the frozen and unfrozen case.
The mechanical properties of steel (Kim 2003) were specified as:
The post-yield modulus of steel was modeled as zero. Poisson's ratio was taken as = 0.3.
Nixon (2003) reported that the relationship between SP and soil pressure was established for the undisturbed Fairbanks silt:
where b = 0.02596kPa-1; and ³t is in kPa.
Using the SP values in eq. [6.0], the author successfully simulated the pipeline movement in two-dimensional at TFA as compared to the field heave measurements of HR-25, which was located from 58m from the inlet riser (Kim et al. 2008). However, pipeline heave was non-uniform along the pipe axis as shown in Figure 6.4a. For instance, HR-22 shows the highest heave than any other location. It is common that considerable variation of field condition can exist even within a small geographical area, and the soils are often categorically classified as the same group. Although most soil profiles are layered and non-uniform, they are often presented by a uniform soil profile with average properties as shown in Figures 6.2a through 6.2c. In this simulation, SP0 was evaluated by using multiplication factor - while using a constant b value for the three-dimensional frost heave simulations.
Figure 6.8 summarizes the simulated freezing depth and temperature gradient of segregation freezing zone (gradTsp) at 1m from the centerline of the pipe. The simulated freezing depth agreed well with the observations in both -€ = 1 and 1.5 cases. The gradTsp is required by the SP porosity growth function. The gradTsp obtained from the simulations also agreed fairly well with the observed values in both cases. The reason for the agreements was that thermal processes were more dominated by the phase change process of the in-situ pore water rather than by that of the migrated water.
Figures 6.9a and 6.9b show the comparison of pipeline movement between observation and simulation at HR-16 and HR-22, which were at the location of TFB and the maximum heave observed, respectively. As expected, the simulation using - = 1 showed smaller heave than that calculated by - = 1.5 . Two unexpected settlements were observed: initial pipeline settlement up to day 90 and settlement with slow rate after day 1060 possibly due to the earthquake. The proposed model did not predict those two settlements. Although the proposed model had the shortcomings, the maximum pipeline movements observed on day 1060 were well reproduced. The maximum amount of the simulated heave using - = 1.5 was only 1.56% smaller than the observation at HR-16 and 2.14% smaller at HR-22 on day 1060, respectively. Therefore, the SP values were utilized with - = 1.5 in this study.
The developed three-dimensional heat transfer model was based on conduction only with isotropic thermal properties.
Temperature distribution and the freezing front penetration were compared in the deeper supra-permafrost area using TFB data. Figures 6.10 through 6.12 show the comparison between observed and simulated temperature contour profiles of TFB in mid-December for 3 years.
During the first year of operation, the soil mass beneath the pipe underwent a progressive cooling effect resulting in a uniform observed temperature near freezing temperature. After the first year of operation, no significant cooling of the soil mass beyond the frost bulb occurred. The observed temperature data revealed that the frost bulb grew beneath the pipe with cylindrical shape.
Although there were no direct temperature measurements deeper than -7.0m, the simulated results showed a similar trend with temperatures. The cylindrical frost bulb shape was reproduced according to the freezing front distribution. The simulated -1, -3, and -5oC isotherms were in agreement with the field observations as well. The simulated progressive cooling effect was verified by the distributions of the isotherms at 0.15oC above freezing (i.e. solid line in Figures. 6.10 through 6.12).
Next, temperature distributions were compared in shallower supra-permafrost area using TFC data. Figures 6.13 through 6.15 show the comparison between observed and simulated temperature contour profiles of TFC in the middle of December for 3 years. Compared with the rate of thermal influence of TFC, it was much greater than that of TFB because the latent heat effect was less significant in the initially frozen soil than in unfrozen soil. For instance, the isotherm propagation of "-0.3oC colder than freezing temperature" reached 6m from the center line in the second year cycle (i.e. solid lines in Figures 6.13 through 6.15). Furthermore, it could be seen that the permafrost region has cooled down and become more thermally stable with propagating freezing from the observed thawed layer. The overall distributions of each isotherm indicated by the simulation were consistent with the observation. Consequently, the initial permafrost condition and boundary conditions for the simulation were verified correctly.
Since the distance between the thermal fences were too large, detailed results in longitudinal (parallel to the longitudinal axis of the pipeline) thermal analysis were not given from observed data. Figure 6.16 shows only the simulated longitudinal temperature contour plots at the pipeline center line. In first year cycle, the freezing front did not reach the material boundary and the in-situ freezing zone, unfrozen soil widely distributed in between 25 and 30m from the inlet riser (Figure 6.16a). The in-situ freezing zone is in an unstable, quasi-steady thermal state that is very sensitive to changes in the surface conditions. As the existence of the chilled pipeline prevented warm up of the soil mass beneath the pipeline through operation, the in-situ freezing zone decreased in size toward one sided with increasing time.
A distinct vertical thermal boundary developed at 30m from the inlet riser. The vertical thermal boundary stabilized within first year cycle and hardly penetrated to the unfrozen soil after stabilization. Thermal regime and freezing front penetration were very similar through the deeper supra-permafrost area because of the effect of latent heat release.
Differential frost heave analysis
Deeper supra-permafrost area:
The analysis of the simulation is now extended to accommodate constitutive responses of the soils in the frost susceptible and non-frost susceptible areas and the flexural response of the buried pipeline.
Figure 6.17 shows the comparison between observed and simulated pipeline movement in the deeper supra-permafrost area. Differential frost heave was analyzed using HRs data. HR-14, HR-16, and HR-19 (i.e. located 32.9m, 35.945m, and 40.515m from the inlet riser, respectively) were chosen as representative points for pipeline movements in the deeper supra-permafrost area. The simulated results had some shortcomings. The simulation did not predict the initial settlement which occurred up to day 90. Also, following day 90 up to day 400, the heave rates calculated by the proposed model were lower than the observation. This is probably because of different frost heave susceptibility of soil layers.
Differential pipeline movements were observed after day 400. The main objective of this study is to examine pipeline movement due to differential frost heave. When segregation freezing zone reached the permafrost table between 25m and 30m from the inlet riser, the frost susceptible part would not experience any more volumetric expansion. Therefore, pipeline movements in the deeper supra-permafrost area were anchored by the frozen soil. As shown in Figure 6.16a, only little frost susceptible layer remained unfrozen in place between 25m and 30m from the inlet riser on day 381. Therefore, the proposed model could likely simulate the beginning time of the differential pipeline movement. The simulated heave rate agreed well with the observation at each HR after day 381. On day 1060, HR-14, HR-16, and HR-19 showed the maximum heaves of 0.136m, 0.156m and 0.172m, respectively. The maximum pipeline heaves were simulated to be 0.126m, 0.154m, and 0.170m at each HR location, which was in good agreement with the observations on day 1060. As mentioned above, the pipeline experienced a slow rate of settlement after day 1060 up to the end of operation possibly due to the earthquake, which the proposed model could not simulate.
Although the proposed model showed some limitations, it should be emphasized that the proposed model could simulate the differential pipeline movements in the deeper supra-permafrost area.
Shallower supra-permafrost area:
Figure 6.18a shows the comparison between observed and simulated pipe movement in the shallower supra-permafrost area. HR-1 and HR-2 (i.e. located 8.53m, and 14.63m, respectively) were chosen as representative points for pipeline movement in the shallower supra-permafrost area. The HRs in the shallower supra-permafrost area experienced initial settlement similar to those HRs in the deeper supra-permafrost area. The initial settlement was approximately 0.025m followed by a period of very slow heave up to day 510. In between day 510 and day 542, an abrupt upheaval was observed in the shallower supra-permafrost area as shown in Figure 6.17b. After the abrupt upheaval, the pipeline movement followed hardly upward trend which was only less than 0.005m of heave up to the earthquake. After the earthquake, no additional heave had occurred and was followed by settlement. The simulated heave at TFC (i.e. located 13m) was only less than 0.002m through operation, and did not predict the subtle variations of the observed heaves.
Qualitative analysis of abrupt pipeline upheaval movement in shallower supra-permafrost area
Although pipeline movement is of importance, pipeline designers have a greater need to know what the bending moment will be. Six order polynomial fitting analyses were performed from the pipeline movement results. Then, bending moment due to differential frost heave could be determined from the second derivative of the fitted pipeline profiles.
Figure 6.19a shows the comparison between observed and simulated profiles of pipeline movement on day 521, which was before the abrupt upheaval in the shallower permafrost area. The simulated result agreed well with the observation. The bending moment profiles by the fitting were in agreement with the profile determined from the strain gauge (SG) data at 9 locations (i.e. 18.53, 22.1, 24, 26.24, 30.68, 32.16, 33.51, 36.8, 42.75m from the inlet riser) as shown in Figure 6.19b. Figures 6.20a and 6.20b shows the comparison between observed and simulated profiles of pipeline movement and bending moment, respectively, on day 548, which was after the abrupt upheaval movement. The bending moment profiles from observations show that the pipeline experienced relaxation in the shallower supra-permafrost area due to the abrupt upheaval, and the simulated bending moment in the shallower supra-permafrost area was approximately three times larger than the observations.
The most likely explanation for the abrupt pipeline movement in the shallower permafrost area is uplift buckling. Palmer and Williams (2003) developed a simple model to evaluate the abrupt upheaval movement on pipeline. The uplift buckling is caused by the high axial stresses in the pipe resulting from the large temperature difference between installation and operation temperatures, coupled with inadequate soil resistance to resist the tendency for the pipe to buckle upwards. The mechanism for abrupt upheaval movement was modeled as a combination of longitudinal compressive stress and overbend irregularities in the profile. In the model, the pipeline was assumed as a thin-walled cylindrical shell and to remain elastic. The induced longitudinal stress has two components: inline pressure and thermal expansion. Following the customary sign convention in this study, compression is considered as positive.
Consider an element of pipeline in an arbitrary profile defined by a vertical distance y (y: measured positive upwards from a datum), which is a function of longitudinal distance z. In Figure 6.21, p is the longitudinal stress, S is the shear force, q is the external vertical force per unit length, and M is the bending moment. The vertical force and moment equilibrium of the element is described as:
and therefore, differentiating eq. [6.0] and eliminating „S/„z,
if the pipe remains elastic,
where flexural rigidity for a thin-walled elastic cylinder (F) with elastic modulus (E) is given by,
where ‘ = mean diameter (twice the mean radius, measured from the centre to half way through the wall); and Š = wall thickness.
In eqs. [6.0] and [6.0], the first term on the right is a curvature term, the product of the longitudinal stress and the curvature „2y/„z2, which is positive in concave and negative in convex, respectively. The pipeline tends to push upwards due to frost heave, and therefore requires a positive value of q to hold it down. The less obvious second term is proportional to changes in shear force and vanishes when the curvature is uniform.
When frost heave lifts the pipeline level, the deflection profile from the initial is idealized as an arc of a circle with uniform overbend curvature k (so that „2y/„z2 is -k and the overbend radius is 1/k).
The force per unit length available to hold the pipeline down is the sum of the pipeline weight per unit length (·) and the uplift resistance per unit length provided by the overburden. The uplift resistance per unit length (¨) is calculated as:
where ˜ = the thickness of the overburden (measured from the top of the pipeline to the ground surface); and f = uplift resistance coefficient determined experimentally.
Assembling the results from eqs. [6.0], [6.0], and [6.0], the pipe becomes unstable when
which can be rewritten as:
The non-dimensional term ·/(p‘2t /4) in eq. [6.0] has a simple physical interpretation as the relative density of the pipeline, relative to the soil it is buried in. The last term highlights the importance of the ratio of overburden cover thickness to pipeline diameter.
As an example, observations at 24m from the inlet riser are presented between Day 518 and Day 553, which the abrupt upheaval occurred during the period. The 5-week history of pipeline temperature and inline pressure is shown in Figure 6.22a. Pipeline temperature suddenly increased from approximately -10oC to 6oC from Day 531 to Day 537. The axial stress increased in response to the pipeline temperature fluctuation during the phase as shown in Figure 6.22b. As the test was operated at a relatively constant pressure of approximately 1.4MPa during the phase, it is safe to say that the compressive axial stress was induced mainly due to thermal expansion.
For the evaluation of the abrupt uplift in the shallower permafrost area, the input values for eq. [6.0] was determined as: t = 18kN/m3, p = 10MPa, · = 2.0kN/m, ˜ = 0.9m, ‘ = 0.905m, = 0.3, and f = 0.5 (Palmer and William 2003). The calculated overbend curvature at which the pipeline becomes unstable is 0.0093m-1. On day 543, the observed maximum overbend curvature was induced around 35m from the inlet riser as approximately 0.000465m-1, which is one-magnitude smaller than the calculated value. Furthermore, as shown in Figure 6.18b, little upheaval occurred around 35m from the inlet riser between day 521 and day 548. However, with the location close to the inlet riser, larger upheaval was observed than otherwise. The maximum movement was approximately 0.025m around 15m from the inlet riser.
Figure 6.23 shows the schematic of the abrupt upheaval: as the longitude compressive stress is induced, the pipe moves inward against the longitudinal resistance of the surrounding soil, and then the upheaval grows in the shallower supra-permafrost area. Since the first station of pipeline movement (HR-1) was located at 8.53m from the inlet riser, there were no direct measurements around the inlet pipeline riser. As shown in Figure 6.18a, the pipeline experienced non-uniform settlement initially, for instance, 0.02m at HR-1 and 0.01m at HR-2, respectively. Over the length between the inlet riser and HR-1, the calculated curvature is 0.0093m-1 (Figure 6.19b) at which the pipeline becomes unstable using the input values above. This corresponds to a 0.083m high "hill" profile. Any overbending becomes more sharply curved than that will become unstable.
Figures 6.24a and 6.24b shows the comparison between observed and simulated profiles of pipeline movement and bending moment, respectively, on day 1060, which was two days before the M7.9 earthquake in interior Alaska on November 3, 2002. The simulated heave profile agreed well with the observation in the deeper supra-permafrost area. Eventually, without considering the stress relaxation, the simulation overestimated the bending moment by about 60% in the shallower supra-permafrost area. The results suggested that the abrupt upheaval regarding of the UAF frost heave experiment was on the safe side in estimation of pipeline bending. However, it is wrong to conclude that uplift buckling would always yield conservative results. For instance, uplift of 1.1m or more was observed at one location (kilometer post 5.2) of the Norman Wells oil pipeline (Nixon and Burgess 1999). The uplift event was extremely dramatic such as that the pipeline exposed above the surrounding ground surface elevation.
When the pipeline temperature fluctuates through the operation of arctic pipelines, compressive longitudinal stress will likely be induced in the pipelines. Even though many numerical simulations have been done to predict the vertical pipeline movement due to differential frost heave, it is undoubtedly critical for arctic pipeline designers to evaluate the effect of the longitudinal stress induced by temperature fluctuation on differential pipeline movement.
A three-dimensional frost heave model applying the SP porosity growth function was developed to simulate the differential pipeline movement at a transition zone between a pre-frozen soil and an unfrozen frost susceptible soil. The developed three-dimensional frost heave model was verified by the UAF full-scale frost heave experiment using a large diameter pipe.
The developed three-dimensional frost heave model had limitations and shortcomings. However, overall simulated results agreed well with the trends presented by the full-scale experiment.
Significant findings from this chapter are:
Temperature distributions were simulated, and were in a good agreement with the observation in both the pre-frozen soil and the unfrozen frost susceptible soil with the effect of latent heat release.
The developed frost heave model was modeled as that volumetric expansion due to frost heave only occurs in fully saturated part. After segregation freezing zone reached the permafrost table between 25m and 30m from the inlet riser, differential pipeline movement started. The simulated results showed good agreement with the observation.
In between day 510 and day 542, approximately 0.02m abrupt upheaval was observed in the shallower supra-permafrost area. Approximately 0.02m abrupt upheaval was observed in shallower-supra permafrost area. The abrupt upheaval event was postulated by a combination of longitudinal compressive stress induced by pipe temperature fluctuation.
Suggestions for further improvement of the developed three-dimensional frost heave model are:
Further calibration and sensitivity studies will be exercised using other field-scale frost heave experiments.
The simulation will be extended to include thaw weakening and settlement.
The simulation will adapt a combination of longitudinal stress and vertical pipeline bending stress due to differential frost heave in the profile to simulate the abrupt upheaval event.
Figure 6.1 3-D coordinate system of an anisotropic heave element.
Figure 6.2 Initial permafrost condition, instrumentation, and monitoring of the UAF frost heave experiment.
Figure 6.3 Cross section of (a) TFB and (c) TFC showing the placement of thermistor beads and the generalized backfill materials.
Figure 6.4 Observations (Huang et al. 2004) of (a) pipeline movement profile along the length of the pipeline; and (b) groundwater table elevations monitored at the test facility.
Figure 6.5 The finite element discretion of (a) geometry; (b) cross section at rapid deepening area (at 30m from the inlet riser); and (c) cross section in the deeper supra-permafrost area (beyond 30m from the inlet riser).
Figure 6.6 Input pipe temperatures (a) from 0 to 10m; (b) from 10 to 20m; (c) from 20 to 30m; (d) from 30 to 40m; and (e) from 40 to 50m.
Figure 6.7 (a) Heave gauge data showing the foundation heave within the first 1m of native silt below the bottom of the pipeline (Huang et al, 2004); and (b) variation of the input groundwater table elevation.
Figure 6.8 Comparison between the observed (Bray 2003) and the simulated results of (a) freezing depth and (b) temperature gradient of frozen fringe at 1m from the center at TFB.
Figure 6.9 Comparison between the observed (Huang et al., 2004) and the simulated pipe displacement using different SP values for UAF frost heave experiment: (a) at 35.945m (TFB); and (b) at 46.615m from the inlet riser.
Figure 6.10 Temperature distribution at TFB in early December of the first year cycle with comparison between (a) the observed (Bray, 2003) and (b) the simulated results.
Figure 6.11 Temperature distribution at TFB in early December of the second year cycle with comparison between (a) the observed (Bray, 2003) and (b) the simulated results.
Figure 6.12 Temperature distribution at TFB in early December of the third year cycle with comparison between (a) the observed (Bray, 2003) and (b) the simulated results.
Figure 6.13 Temperature distribution at TFC in early December of the first year cycle with comparison between (a) the observed (Bray, 2003) and (b) the simulated results.
Figure 6.14 Temperature distribution at TFC in early December of the second year cycle with comparison between (a) the observed (Bray, 2003) and (b) the simulated results.
Figure 6.15 Temperature distribution at TFC in early December of the third year cycle with comparison between (a) the observed (Bray, 2003) and (b) the simulated results.
Figure 6.16 Distribution of simulated temperature and induced stress at the center line in early December of (a) the first year cycle, (b) the second year cycle, and (c) the third year cycle.
Figure 6.17 Comparison between the observed (Huang et al., 2004) and the simulated pipe displacement in the deeper supra-permafrost area.
Figure 6.18 (a) Comparison between the observed (Huang et al., 2004) and the simulated pipe displacement in the deeper supra-permafrost area; and (b) pipeline movement profile during the abrupt upheaval in the shallower supra-permafrost area.
Figure 6.19 Comparison between observed (Huang et al., 2004) and simulated distribution of (a) pipe movement; and (b) bending moment along the pipeline on day 521, which is before the abrupt upheaval event.
Figure 6.20 Comparison between observed (Huang et al., 2004) and simulated distribution of (a) pipe movement; and (b) bending moment along the pipeline on day 534, which is after the abrupt upheaval event.
Figure 6.21 Pipeline element (modified from Palmer and Williams, 2003).
Figure 6.22 History of (a) the observed pipeline temperature and inline pressure; and (b) axial stress at 24m from the inlet riser during the abrupt upheaval event.
Figure 6.23 Schematic drawing of abrupt movement of pipeline (not to scale).
Figure 6.24 Comparison between observed (Huang et al., 2004) and simulated distribution of (a) pipe movement; and (b) bending moment along the pipeline on day 1060, which is two days before the earthquake.