This example illustrates the use of the coupled temperature–pore
pressure displacement modeling capability in
Abaqus/Standard
to model the effects of thaw settlement that commonly occurs in relatively warm
pipelines buried in permafrost.
Thaw settlement generally occurs in shallow waters and at shore crossings
where soil ice-bonded permafrost underlies the pipeline (Xu et al., 2009). Due
to the heat released from the relatively warm pipeline, the surrounding
permafrost may gradually thaw over years of operation and create a permafrost
thaw bulb. The load-carrying capacity of the soil is generally believed to be
reduced as a result of the thaw formation, which could in turn (as a result of
the weight of the soil above it) lead to excessive deformation and stress in
the pipeline leading to its eventual damage.
Two primary mechanisms control the behavior of the problem: heat transfer
from the warm pipeline to the surrounding permafrost leading to thawing of the
permafrost and diffusion of the pore fluid and associated consolidation of the
thawed soil mass surrounding the pipeline. The analysis is carried out
utilizing two approaches. The first approach is a sequential one, in which the
permafrost melting process is modeled as a pure heat transfer analysis in
Abaqus,
and the resulting temperature is used to drive a soils consolidation analysis.
The second approach is fully coupled, in which the heat transfer and soils
consolidation problems are solved in a fully coupled manner in a single
analysis. This approach uses the coupled temperature–pore pressure elements
that solve for the temperature field as a nodal degree of freedom in addition
to the displacement and the pore pressure fields. The results obtained using
these two approaches are compared, and the advantages of using a fully coupled
approach are discussed. The model presented here has been adapted from Xu et
al. (2009), where the first approach alone was used to investigate this
problem.
Geometry
Figure 1
shows the geometry of the model consisting of two outer sections of sand
separated by an inner section of clay, which approximates the nonuniform soil
properties that are often encountered in practice along a pipeline. The
geometry assumes symmetry about the right vertical plane in the figure, and the
dimensions of the soil mass are as shown. The pipe diameter and thickness are
assumed to be 0.4 m (15.76 in) and 0.096 m (3.8 in), respectively, and it is
assumed to be buried 2.5 m below the soil surface.
Materials
The material properties assumed for this analysis are shown in
Table 1,
Table 2,
Table 3,
and
Table 4
for clay, sand, steel, and water (pore fluid), respectively. The symbols
,
,
,
,
,
,
,
,
and
represent the Young's modulus, Poison's ratio, thermal conductivity, density,
specific heat, latent heat, thermal expansion coefficient, permeability, and
void ratio, respectively. The elastic and the pore fluid flow properties of the
soil are assumed to depend on whether the soil is frozen (more precisely,
whether the water in the soil is frozen) or thawed. The soil is assumed to be
significantly stiff and impermeable when frozen compared to the corresponding
properties when thawed. The properties in the frozen state and the thawed state
are distinguished from each other using the symbols
F and T,
respectively, next to a property symbol. This approach approximately captures
the effects of phase change of the pore fluid on the mechanical properties of
the soil and can be improved upon with a more detailed modeling of the physics
of the phase change process. In the heat transfer part of the sequential
approach, the thermal properties are assumed to be weighted averages of the
properties of the individual constituents (clay or sand, and water). The latent
heat of fusion from ice to water is utilized to define the phase change of the
soil. The pipe is assumed to be made out of steel, with the mechanical behavior
defined using a linear elastic model along with thermal expansion. Inelastic
response of the pipe material, such as plasticity and creep, can be easily
included in the model if needed. Both the soil and the pore fluid are also
assumed to undergo thermal expansion associated with the temperature variation.
Thermal expansion is an important part of the overall physics of the problem,
with the differential expansion of the pore fluid relative to pores in the soil
often determining the initial rise in the pore pressure.
Initial conditions
The initial temperature is assumed to be −5°C everywhere in the model. The
initial void ratio, pore pressure, and saturation are assumed to be 1.8, 0.0,
and 1.0, respectively, everywhere in the model. Thus, the pore fluid flow is
assumed to occur under fully saturated conditions. Initial effective geostatic
stress in the vertical direction is defined as a function of the depth below
the soil surface. The stress in the horizontal direction is assumed to be a
fraction of the stress in the vertical direction.
Boundary conditions and loading
All the outer surfaces of the model (except the top surface) are assumed to
be restrained from moving in a perpendicular direction. Following Xu et al.
(2009), the model assumes the flow of heat and fluid across all the outer
surfaces (except the top surface) to be zero. The top surface allows free
drainage (zero pore pressure). The axial displacement at both ends of the pipe
is restrained. This simple example model does not include gravity in the
loading. In general, gravity should be taken into account for more realistic
consolidation analyses.
Abaqus modeling approaches and simulation techniques
In the consolidation part of the analysis in the sequential approach, the
soil is modeled using pore pressure elements with displacement and pore
pressure degrees of freedom, while the pipe is modeled using solid continuum
elements using displacement degrees of freedom only. In the coupled analysis,
the soil is modeled using the coupled elements with displacement, pore
pressure, and temperature degrees of freedom, while the pipe is modeled using
coupled elements with displacement and temperature degrees of freedom only. The
soil and the pipe share nodes at the interface. An alternate approach would
model this interface utilizing contact constraints; however, this approach has
not been used because contact is not supported with the coupled
temperature–pore pressure elements.
Analysis types
The sequential approach utilizes a heat-transfer analysis first to determine
the temperature profile. This temperature profile is subsequently used to drive
the mechanical part of the simulation that does not model heat transfer. The
latter consists of two steps: a geostatic analysis, followed by a soils
consolidation analysis. On the other hand, the coupled approach solves the
thermal and mechanical problem in a fully coupled manner and consists of two
steps: a geostatic analysis, followed by a soils consolidation analysis.
Mesh design
The heat transfer part of the sequential approach uses first-order brick (DC3D8) elements, while the consolidation part of the sequential
approach uses first-order reduced-integration (C3D8RP and C3D8R) elements for the soil and pipe regions, respectively. The
coupled analysis uses first-order reduced-integration fully coupled (C3D8RPT and C3D8RT) elements for the soil and pipe regions, respectively.
Loads
Inside the pipe the fluid is assumed to be flowing at 20°C. A constant
convection coefficient on the inner pipe wall is assumed. The top surface of
the soil is subjected to a sinusoidal temperature variation ranging between
−5°C and 5°C. The above boundary condition approximately represents the season
variation of temperature at the soil surface. The weight of the pipe is
considered as a load and is modeled as a body force in the vertical direction.
While the physics of this problem will likely span two or more decades, the
present analysis is carried out for a period of one year in order to present
the basic framework for an analysis of this type.
Solution controls
The simulations use solution controls to specify a nondefault initial value
of the time average pore fluid flux. The default choice may not be appropriate
in situations such as those encountered in the present problem where the fluid
velocities are, in relative terms, lower compared to typical flux values
encountered for other fields (such as displacements or rotations). Without the
above specification, the increments would be treated as linear from the
viewpoint of the continuity equation. In other words, without using solution
controls to specify a nondefault initial value of the time average pore fluid
flux, the pore fluid part of the incrementation will be treated as linear.
Consequently, the continuity equation would be assumed to have been satisfied
at the first iteration itself, without performing any further iterations to
compute corrections to pore pressure.
Results and discussion
Figure 2
shows a contour of the temperature field around the buried pipe after a period
of one year. The temperature profile can be used to determine the thaw bulb
around the pipe. The results are obtained from the heat transfer part of the
sequential approach; the coupled analysis produces similar results. The thaw
bulb represents the region where the permafrost has thawed and the temperature
is above freezing.
Figure 2
shows the mesh for the whole model along with the contour plot of the fluid
velocities. The zero pore pressure boundary condition on the top surface allows
the seepage of pore fluid across this surface. Clay is significantly less
permeable compared to sand. The effect can be observed in
Figure 3:
the flow across the top surface in the clay section is less compared to the
flow in the sand section.
Figure 4
shows the temperature history at a point (node number 1591) in the soil in the
vicinity of the pipe. The two curves were obtained from the uncoupled
(sequential) and coupled analysis. The temperature results are, in relative
terms, close to each other before thawing. However, upon thawing the results
from the coupled analysis predict a higher temperature. The latter may be
explained by recalling that unlike the coupled analysis, the heat transfer part
of the uncoupled (sequential) analysis does not account for heat transfer
associated with pore fluid flow. The results from both the sequential and
coupled analyses predict that the temperature does not increase significantly
for a certain period of time after it reaches 0°C. This behavior is due to
phase change of the pore fluid as a result of thawing; all the available heat
is used up as the latent heat of fusion.
Figure 5
shows the normalized (by the maximum vertical displacement from the uncoupled
analysis) vertical displacement as a function of length along the span of the
pipe for both the uncoupled (sequential) and coupled analyses. The uncoupled
analysis predicts a higher peak displacement compared to the coupled analysis.
This can be attributed to the difference in temperatures at points in the
vicinity of the pipe as predicted by the two analyses and the associated
effects on the elastic stiffness of the surrounding soil mass. It is
particularly interesting to note that a fully coupled analysis can be used as a
basis for a less conservative design as the predicted displacements and
associated stresses are lower.
Xu, J., B.
Abdalla, A. Eltaher, and
P. Jukes,
“Permafrost Thawing-Pipeline
Interaction Advanced Finite Element Model,” Proceedings of the
ASME 2009 28th International Conference on
Ocean, Offshore, and Arctic Engineering,
OMAE2009–79554, 2009.
Tables
Table 1. Properties for clay (F-frozen state, T-thawed state).
Property
Value
(F)
1.03421 × 1011 Pa
(T)
6.89476 × 105 Pa
0.30
2.0 J/sec m°C
1602.0 kg/m3
1381.0 J/kg °C
9.67 × 10−6 /°C
(F)
1.0 × 10–14 m/sec
at
(T)
1.0 × 10−8 m/sec
at
(T)
2.0 × 10−8 m/sec
Table 2. Properties for sand (F-frozen state, T-thawed state).
Property
Value
(F)
1.03421 × 1011 Pa
(T)
1.03421 × 107 Pa
0.30
2.0 J/sec m°C
1602.0 kg/m3
1381.0 J/kg °C
5.60 × 10−6 /°C
(F)
1.0 × 10–13 m/sec
at
(T)
1.0 × 10−7 m/sec
at
(T)
2.0 × 10−7 m/sec
Table 3. Properties for steel.
Property
Value
2.0 × 1011 Pa
0.30
52.0 J/sec m°C
7832.0 kg/m3
434.0 J/kg °C
12.0 × 10−6 /°C
Table 4. Properties for water (pore fluid).
Property
Value
0.58 J/sec m°C
1000.0 kg/m3
4186.0 J/kg °C
(at −5°C)
51.0 × 10−6 /°C
(at 0°C)
51.0 × 10−6 /°C
(at 1°C)
–16.67 × 10−6 /°C
(at 4°C)
0.0
(at 10°C)
29.33 × 10−6 /°C
Figures
Figure 1. Pipeline buried in permafrost (half-symmetry model). Figure 2. Thaw bulb around the relatively warm pipe after one year. Figure 3. The meshed buried pipeline model showing the contour of fluid
velocities. The relatively impermeable clay region exhibits low fluid
velocities (blue-colored contour in top center section). Figure 4. Temperature history in the soil near the pipe. Figure 5. Normalized vertical displacement along the length of the pipe.
