1. MODULE: MAR8097 – Dissertation
Dynamic behaviour of shallow water pipelines due to seabed liquefaction
Submitted by: Alejandro Marín Tamayo
Student No: 140047404
A Dissertation submitted
for the partial fulfilment of the Degree of Master of Science in
MSc Pipeline Engineering programme
School of Marine Science and Technology
Newcastle University
Supervisor name:
Dr. Longbin Tao
Deadline Date: [7 August 2015]
2. ii
To my father, for his tireless voice of encouragement and
for my mother, because once again made all of this possible
3. iii
Acknowledgments
The author of the dissertation acknowledges the discussion with Prof. Dr. Longbin Tao, who as
supervisor recommended and proposed key points to be addressed throughout the study. The
dissertation is a partial requisite for the qualification of MSc in Pipeline Engineering, which
academic programme was fully funded for the author by the Colombian governmental entity
COLFUTURO and by the Newcastle University by means of its NUIPS scholarship. The author is
grateful with Nikolas Georgiou, MSc student in Automation and Control and with Andrés Aldana,
PhD student of Nano Science for their valuable advices. Finally, the author acknowledges the
constant motivation given by the MSc Pipeline Engineering programme professors and by their
family.
4. iv
Abstract
Seabed is subjected to constant dynamic wave loading inherent to the metocean environment, and
extreme scenarios such as earthquakes, hurricanes and tsunamis, among other meteorological
features, may cause its liquefaction due to this cyclic wave loading. Conventional approaches on
defining pipeline stability do not describe properly the pipe-soil interaction once the seabed is
liquefied, therefore the need of enhanced and comprehensive approaches to do so is evident, since
liquefaction potential increases mainly in shallow waters, where sandy or non-cohesive soils
predominantly shape the seabed. In this way, the main purpose of this study is to assess the dynamic
pipeline behaviour once different lengths of seabed segments experiments liquefaction, where the
pipeline loses its support. In order to define the sensitiveness of the pipeline’s structural response to
seabed liquefaction, different water depths were assessed for the analysis as 25m water depth, 50m
water depth, 75m water depth and 100m water depth. Furthermore, four different pipeline diameters
were adopted (254mm, 406.4mm, 609.6mm and 914.4mm), in order to compare the structural
behaviour for different cross-sectional areas and D/t ratios. For the analysis, two seabed liquefied
behaviour were assumed, regarding solely the wave-induced pressure and regarding the dynamic
seabed response. After completing the set of analysis, it was found how for the adopted analysis
parameters and features, the dynamic behaviour of light pipelines (254mm and 406.4mm) are
sensitive to the seabed dynamic response once it is accounted, but for heavy pipelines (609.6mm
and 914.4mm) the dynamic behaviour is governed by the pipeline itself, where the seabed dynamic
response is irrelevant.
6. vi
List of Figures
Figure 1. Topographical features of sea floor....................................................................................3
Figure 2. Pore pressure increase area in a continental slope..............................................................6
Figure 3. Instability phases throughout time of a heavy pipeline over liquefied seabed...................7
Figure 4. Pipeline responses due to ground movement .....................................................................8
Figure 5. Upward water flow during liquefaction. After (Teh et al., 2006)....................................11
Figure 6. Results after laboratory tests on defining pipeline sinking ..............................................11
Figure 7. Excess of pore water pressure (u) and vertical effective stress (𝜎′𝑣)...............................12
Figure 8. Variables included in hoop stress calculation. After (Bai, 2005).....................................16
Figure 9. Components of axial or longitudinal stress. After (Bai, 2005).........................................16
Figure 10. Wave-like motion of the liquefied seabed. After (Foda and Hunt, 1993)......................22
Figure 11. Wave-induced stress over the seabed.............................................................................22
Figure 12. Different water depths analysed (T=5 s). .......................................................................27
Figure 13. Detail of modelling setting (T=5 s). ...............................................................................27
Figure 14. Schematic deflection of the pipeline based on a single supported beam behaviour. .....27
Figure 15. Different water depths analysed (T=10 s). .....................................................................28
Figure 16. Detail of modelling setting (T=10 s). .............................................................................28
Figure 17. Cross section of the model. ............................................................................................29
Figure 18. Detailed perspective of the half pipe..............................................................................29
Figure 19. Longitudinal view of the half-symmetric model............................................................30
Figure 20. Wave-induced pressure over the span. ...........................................................................30
Figure 21. Detail of amount of deformation of more than one diameter.........................................31
Figure 22. Concentration of equivalent stress where the support is lost. ........................................31
Figure 23. Maximum stress at the bottom of the pipe .....................................................................32
Figure 24. Localisation of maximum deformation at the centre of the liquefied segment..............32
Figure 25. Detail of amount of deformation of a few millimetres...................................................33
7. vii
Figure 26. Concentration of equivalent stress where the support is lost .........................................33
Figure 27. Maximum stress at the bottom of the pipe .....................................................................34
Figure 28. Detail of amount of deformation of more than one diameter.........................................34
Figure 29. Maximum stress at the top of the pipe............................................................................35
Figure 30. Localisation of maximum deformation at the centre of the liquefied segment..............35
Figure 31. Maximum stress at the bottom of the pipe .....................................................................36
Figure 32. Deformation (T=5 s, D=10”)..........................................................................................36
Figure 33. Deformation (T=5 s, D=16”)..........................................................................................36
Figure 34. Deformation (T=5 s, D=24”)..........................................................................................37
Figure 35. Deformation (T=5 s, D=36”)..........................................................................................37
Figure 36. 𝜎𝑒(T=5 s, D=10”)...........................................................................................................37
Figure 37. 𝜎𝑒(T=5 s, D=16”)...........................................................................................................37
Figure 38. 𝜎𝑒(T=5 s, D=24”)...........................................................................................................37
Figure 39. 𝜎𝑒(T=5 s, D=36”)...........................................................................................................37
Figure 40. Deformation (T=10s, D=10”).........................................................................................38
Figure 41. Deformation (T=10s, D=16”).........................................................................................38
Figure 42. Deformation (T=10s, D=24”).........................................................................................38
Figure 43. Deformation (T=10s, D=36”).........................................................................................38
Figure 44. 𝜎𝑒(T=10s, D=10”)..........................................................................................................38
Figure 45. 𝜎𝑒(T=10s, D=16”)..........................................................................................................38
Figure 46. 𝜎𝑒(T=10s, D=24”)..........................................................................................................39
Figure 47. 𝜎𝑒(T=10s, D=36”)..........................................................................................................39
Figure 48. Vertical Stress (T=5 s)....................................................................................................47
Figure 49. Shear Stress (T=5 s) .......................................................................................................47
Figure 50. Vertical Stress (T=10 s)..................................................................................................47
Figure 51. Horizontal Stress (T=5 s)................................................................................................47
Figure 52. Pore Pressure (T=5 s) .....................................................................................................47
9. ix
Figure 80. 𝜎𝑒(T=5s, D=16”, d=100m). ...........................................................................................56
Figure 81. 𝜎𝑒(T=5s, D=24”, 100m).................................................................................................56
Figure 82. 𝜎𝑒(T=5s, D=36”, 100m).................................................................................................56
Figure 83. 𝜎𝑒(T=10s, D=10”, d=25m). ...........................................................................................56
Figure 84. 𝜎𝑒(T=10s, D=16”, d=25m). ...........................................................................................56
Figure 85. 𝜎𝑒(T=10s, D=24”, d=25m). ...........................................................................................57
Figure 86. 𝜎𝑒(T=10s, D=36”, d=25m). ...........................................................................................57
Figure 87. 𝜎𝑒(T=10s, D=10”, d=100m). .........................................................................................57
Figure 88. 𝜎𝑒(T=10s, D=16”, d=100m). .........................................................................................57
Figure 89. 𝜎𝑒(T=10s, D=24”, d=100m). .........................................................................................57
Figure 90. 𝜎𝑒(T=10s, D=36”, d=100m). .........................................................................................57
Figure 91. Deformation (T=10s; D=10”, d=25m) ...........................................................................58
Figure 92. Deformation (T=10s; D=10”, d=50m) ...........................................................................58
Figure 93. Deformation (T=10s; D=10”, d=75m) ...........................................................................58
Figure 94. Deformation (T=10s; D=10”, d=100m) .........................................................................58
Figure 95. Deformation (T=10s; D=36”, d=25m) ...........................................................................58
Figure 96. Deformation (T=10s; D=36”, d=50m) ...........................................................................58
Figure 97. Deformation (T=10s; D=36”, d=75m) ...........................................................................59
Figure 98. Deformation (T=10s; D=36”, d=100m) .........................................................................59
Figure 99. Maximum equivalent stress variation with respect to water depth ................................61
Figure 100. Maximum equivalent stress variation with respect to water depth. .............................61
Figure 101. Maximum bending stress variation with respect to water depth ..................................62
Figure 102. Maximum bending stress variation with respect to water depth ..................................62
10. x
List of Tables
Table 1. Evaluated scenarios of pipeline structural response. .........................................................21
Table 2. Non-liquefied soil properties. ............................................................................................23
Table 3. Pipeline properties. ............................................................................................................23
Table 4. Pressures and design factors. .............................................................................................24
Table 5. Wave parameters for T=5 s................................................................................................24
Table 6. Wave parameters for T=10 s..............................................................................................25
Table 7. Wave-induced load over seabed, T=5 s.............................................................................26
Table 8. Wave-induced load over seabed, T=10 s...........................................................................26
Table 9. Seabed parameters for dynamic response analysis............................................................41
Table 10. Deformations for T=5 s, OD=10”, 25m water depth ......................................................50
Table 11. Selected vertical effective stress (red values) for second modelling stage, T=5 s...........50
Table 12. Deformations for T=10 s, OD=10”, 25m water depth ....................................................50
Table 13. Selected vertical effective stress (red values) for second modelling stage, T=10s..........51
11. 1 Introduction
1.1 Justification
Oil & Gas reservoirs are present extensively on the offshore continental margin, generating that an
accurate approach to its features, must be conducted with engineering knowledge for the adequate
fossil fuels extraction. However, since shallow water reservoirs are depleting (Randolph, 2011),
deeper exploration has being conducted in order to supply the increasing oil and gas demand. This
implies to long export pipelines and offshore installations to be founded on unfavourable
environmental conditions, such as changes on topography and seabed inclination, from the
platform`s location to the shore. These longer distances have risen the need of more rigorous and
exhaustive analysis of seabed behaviour, in order to develop accurate and economic feasible pipeline
and offshore installations design. Within these analysis, factors as geohazards associated to
metocean conditions (e.g. wave length, current velocity) like landslides on the continental slopes
and stress states` variations within the seabed leading to liquefaction, must be accounted.
Evidence of large seabed liquefaction areas are reported in Christian et al. (1997), where
identification of large zones exceeding 100m of submarine slope failures, due to seabed liquefaction
were exposed close to the Fraser River Delta, as well as those reported within the Yellow River
Delta by Jia et al. (2014). Therefore, large scale seabed failures due to earthquakes and wave induced
stresses causing seabed liquefaction are a reality, which must be addressed to guarantee subsea
pipelines integrity.
Seabed grain size distribution is given by wave and current velocity and a subsequent sediment
transportation capacity. Thus, in shallow water, due to relative higher velocity and wave influence
compared to deeper water, presence of coarser soil grains are predominantly found as seabed
sediments. These coarser soils are mainly composed by sands, ranging from coarse sands (0.42mm)
to fine sands (0.074mm). Moreover, these deposits can be found as dense or loose sands, what is
function of sedimentation features such as speed, quantity of sediments and environmental
characteristics such as seabed depth and stresses history. Due to the cohesionless nature of sands,
they are prone to liquefy once the excess of pore water pressure equals the effective stress of the
soil. The shallower the water depth, the higher the influence of wave-induced loads over the seabed.
These cyclic loads have the ability to increase the pore water pressure within the seabed, and
therefore have the potential to cause seabed liquefaction. Due to this, the scope of the study is
focused on shallow water depths where sandy seabed is predominant, where subsea pipelines are
therefore exposed to seabed liquefaction.
12. 2
There is still a deficiency of modelling and describing liquefied soil behaviour. However,
approaches to relate liquefaction onset and transient behaviour to the wave-induced pressure based
on linear wave theory have been done. Gao et al. (2011) established the seabed response in terms of
vertical stress, horizontal stress, shear stress and pore water pressure entirely as function of the
harmonic wave-load (
2𝜋𝑥
𝐿
−
2𝜋𝑡
𝑇
), and its repercussion at any depth by means of classic Boussinesq
principle; Wang et al. (2004) developed a numerical approach based on Biot’s consolidation theory
where interaction between soil skeleton and inter-granular water is regarded, but neglecting
acceleration components for simplification, justified on the pertinence of doing so once wave period
is long. Further developments based similarly in Biot´s consolidation theory such as the undertaken
by Zienkiewicz (1981), Ulker (2009) and Ulker (2012) have related fully dynamic, partially dynamic
and quasi-static formulations to account the seabed response to a wave-induced pressure, with
respect to the variability of both metocean features and seabed parameters.
Experimental studies as those conducted by Teh et al. (2003) have demonstrated that for subsea
pipelines design, current design methods and approaches fulfil sufficiently stability requirements
for a non-liquefied seabed, but are not adequate once the seabed experiences liquefaction. This, due
to the absence of liquefied seabed characterisation and a subsequent deficiency on pipe-liquefied
soil interaction prediction. In this way, the motivation of this study is to evaluate the influence of
sandy soils liquefaction over the dynamic response of subsea pipelines.
1.2 Background:
Even though the scope of the study corresponds to shallow water pipelines due to the predominance
of sandy soils in shallow water, which are prone to liquefaction, in the background will be also
mentioned the importance of liquefaction in continental slopes and landslides triggering, due to the
soil shear strength reduction once the soil liquefies.
According to Randolph (2011), all oceans` topography (i.e. bathymetry) can be classified into three
main geomorphological shapes: the continental margin, the continental rise and the abyssal plain.
The continental margin, can be subdivided into the continental shelf, the continental ridge and the
continental slope (Figure 1).
13. 3
Figure 1. Topographical features of sea floor, After Poulos (1988), referenced by (Randolph,
2011).
Continental ridge (break) and continental slope seabed is generally under several meters of water
column, 500m – 3000m, after Randolph (2011). Taking into account this feature, a complete soil
saturation (i.e. S=1) grade usually must be assumed, when conducting critical scenario assessments.
This assumption leads to liquefaction potential calculation for sandy soils.
Seabed is composed of two main sorts of sediments: terrigenous sediments (i.e. transported form
land) and pelagic sediments, which settle through the water column (Randolph, 2011).
Unconsolidated sediments generally exhibit low strength features. Even more, when these sediments
have some degree of inclination (i.e. slope), equilibrium conditions may be reduced once external
factors induce loads over the soil mass (e.g. earthquakes, waves). Therefore, since about 75% of
marine sediments are located within the continental margin (Randolph, 2011), it is crucial to
understand and predict accurately the unconsolidated soils` behaviour under external loads.
According to Randolph (2011), undrained conditions in soil behaviour can also be developed for
coarse grained soils (i.e. not only for fine grained soils) if the velocity rate of applied loads is
relatively high, e.g. under influence of earthquakes or wave loads. Regarding this, it is suggested
that consolidated undrained strength of soil deposits should be adopted when stress analysis is
conducted for calculating soil response to peak loads influence, as those generated by wave impacts.
Assessment of sands under cyclic loads usually considers the occurrence of liquefaction associated
to a critical excess pore pressure value, whilst clays behaviour analysis are tailored to define the
likelihood of excess pore pressure (Randolph, 2011).
When defining the right of way for an offshore pipeline or characterising the soil foundation of a
platform, a complete ground investigation must be conducted. Several laboratory tests are necessary
to define soil stress-strain thresholds. However, serviceability limits must be delimited regarding
the total shear strain when soil is subjected to cyclic loads. This means that it is not necessary that
14. 4
the onset of liquefaction has to be reached to represent the soil failure. Currently, 15% of total shear
deformation is an indicator of ´failure` (Randolph, 2011).
Although most of studies conducted have only involved one single layer on the evaluation of seabed
response to wave induced loading, the approximation developed by Ulker (2012) encompasses two
layer seabed to assess soil response in terms of liquefaction potential. The assumptions done by this
author varied both layer thickness (i.e. surface clay layer overlying sand layer) and wave period, in
order to establish a correlation to the potential liquefaction depth.
Assuming an almost saturated porous media (i.e. S≈1), the wave induced stress mentioned above
may develop an instantaneously reduction of the mean effective stress (Ulker, 2012). Thus,
instantaneous liquefaction may occur even though a low soil permeability is given Ulker (2012).
Furthermore, according to Ulker (2009) the induced cyclic wave loading and the consequent excess
in pore pressure lead to a downward and upward flow of the interstitial water. The subsequent effect
is a wave-induced liquefaction once the submerged unit weight of the soil is overcome by the
seepage force driven by the upward flow.
After the models and calculations conducted by Ulker (2012) on the two layer soil response analysis,
a relationship between the maximum liquefaction depth and the wave period (T) was found. Once
the wave period T increases, the liquefaction depth also increases. However, this liquefaction depth
decreases for a given increase of soil permeability (i.e. capability to dissipate excess on pore water
pressure). This results lead to the following hypothesis:
- Liquefaction depth may vary with a fluctuation of permeability for sandy soils.
- Long wave periods (T) increases liquefaction depths: Greater the wave length, greater the
period (T) increasing water column over a specific point on the seabed. This higher water
column increases stresses exerted over the seabed, rising excess pore water pressures for a
given time (t) and thus reducing the effective stress.
Wave induced repetition (i.e. cyclic harmonic oscillation) over the seabed may reach soil threshold
for liquefaction triggering. Once this threshold is overwhelmed, shear stress may be equal to soil
shear strength (i.e. zero effective stress due to liquefaction) and slip surfaces may develop for an
inclined seabed surface or may lose bearing capacity for both horizontal and inclined seabed
settings.
Randolph (2011) claims that the influence of surface wave action over seabed sediments varies
depending on the size and wave speed. For instance during a heavy storm, the influence may be
15. 5
around 200m depth. However, for normal conditions the influence may be reduced to 50m. The
same author also establish that pipelines and other Offshore systems could be affected regarding
this surface wave action, since liquefaction and seabed scouring can take place under generated
wave loading. Evidence of harmful submarine landslides and liquefaction triggered by hurricane
induced-waves is claimed by Hampton et al. (1996), referred to Hurricane Camile in 1969, which
caused the failure of three fixed rigs on the offshore area of the Mississippi Delta. Waves´ heights
up to 20m were found as responsible of generating soil foundation failure on depths greater than
100m.
As well as increasing the moment over potential failure surfaces, a stress field is induced over the
seabed due to the dynamic pressure exerted by wave propagation (Ulker, 2012). This dynamic
pressures produce excess on pore water pressures leading to effective stress reduction and potential
liquefaction. According to the latter statement, it is suggested that both global failure conditions (i.e.
development of a failure slip surface) and liquefaction can take place under cyclic wave-induced
stresses. However, liquefaction development will depend on soils` permeability and cyclic stress
duration.
Puzrin et al. (2004), conducted a detailed study about initiation and propagation of shear surfaces
on continental slopes, leading to large submarine landslides. The motivation of the study was based
on the evidence of wide shallow underwater landslides that took place in the continental slope of
the southern coast of Israel and Santa Barbara, California, associated to M7+ earthquakes. The
common features of both landslides was their low thickness (40m-45m thick), their length (4km-
10km) and the slope inclination (3°-6°). According to this author, even though earthquakes with
M7+ are able to trigger such sort of land movements, the likelihood of the development of
instantaneously shear (failure) surfaces is low.
Therefore, a detailed differentiation between progressive failure and catastrophic failure was shown
by the author in order to justify the triggering of the landslides above mentioned. Puzrin et al. (2004)
defines progressive failure as “… propagation of the shear band is stable in sense that it requires
work of the additional external forces”; on the other hand, catastrophic failure is defined as “…
propagation of the shear band is unstable, and takes place under existing forces”.
Once this terminology differentiation was done, Puzrin et al. (2004) recognised the probability of a
failure to become a catastrophic failure under particular conditions. One of this conditions (i.e.
geometrical slip surface feature) was found according to the energy balance criterion, where the
global failure of the slope initiates once the length of the shear band equals the catastrophic slip
16. 6
length. Given this, a strain softening of the soil within the shear band takes place, due to the pore
pressure development (Puzrin et al., 2004). A schematic geometry of the seabed increased pore
pressure zone is shown in Figure 2. Any linear weak band initiated within this area above the
maximum depth of slip surface development, found according to the energy balance criterion, may
become a shear band and its propagation can lead to a slope global failure (Puzrin et al., 2004).
One of the most important features of the calculations developed by Puzrin et al. (2004), relies on
the strain-softening behaviour under undrained loading process exhibited by the loose sand material
adopted for the given examples. This implies a gradual soil shear strength reduction once cyclic
induced stress over the seabed are imposed by storm waves, by increasing the excess of pore water
pressure leading to liquefaction and subsequent slip surfaces generation. After Puzrin et al. (2004)
it can be proposed a similar analysis taking into account the potential liquefaction depth regarding
wave cyclic loading proposed by Ulker (2009). This means an approximation to define the most
likely depth of slip surface regarding a transition from progressive failure to catastrophic failure,
after the accumulation of excess pore water pressure due to harmonic wave loading. After this, it
would be recognised a progressive (i.e. cyclic dependant) weakness of a potential shear surface
within the seabed, instead of sudden or immediate failure surface development.
Figure 2. Pore pressure increase area in a continental slope, after Puzrin et al. (2004).
Liquefaction occurrence is more likely to happen in soil depths where the effective shear stress is
low (i.e. near the surface). This is also shown by Ulker and Rahman (2009) once plotted pore water
pressure and effective stress variation thorough soil depth, where can be seen how the excess of pore
water pressure dissipates with an increase in seabed depth. Due to this, it is notorious how
liquefaction potential is critical and likely to happen at seabed surface and to certain depth depending
on soil permeability, density and on its ability to dissipate excess of pore water pressure.
In terms of the influence assessment of liquefied seabed over pipelines, according to Deng et al.
(2014) for a buried pipeline to floats once seabed is liquefied, two main features must be present: 1-
the constraint (i.e. normal stresses above the pipeline due to overburden pressure) exerted by the
17. 7
surrounding soil is not enough to maintain the pipeline buried; 2- the buoyancy of the pipeline is
greater than the total weight of the pipe itself and the product weight. Sumer et al. (1999) also claims
that one of the key issues on defining the stability of a pipeline laid on the seabed is to predict the
likelihood of sinking once the soil undergoes liquefaction. According to the experimental study
conducted by this author and the measures done, the sinking of a pipeline on a liquefied seabed takes
place before the excess in pore water pressure starts to decrease. This was evident as the pipeline
deformation and sinking took place while pore pressure build up was still measured.
According to the experimental studies undertaken by Teh et al. (2003), the instability phases of a
heavy pipeline, once seabed is liquefied, can be described as plotted in Figure 3. The mentioned
author claims that for time t1, the hydrodynamic wave induced pressure is not sufficient to move the
heavy pipeline, but is large enough to liquefy the seabed (i.e. the pipeline is stable); for times t2 and
t3 the pipeline starts to move and therefore sinking into the liquefied soil mass, up to a final position
for time t4.
Figure 3. Instability phases throughout time of a heavy pipeline over liquefied seabed. After
Teh et al. (2003)
On the completion of the referenced study, it was found that the extent of sinking of the pipeline is
function of its density and of the liquefied seabed parameters, regardless the wave conditions.
However, according to seabed response approaches as those conducted by Wang et al. (2004); Ulker
(2009); Ulker et al. (2009); Ulker (2012), it is common practice to relate the seabed liquefied
longitudinal length, equal to the wave length that induces harmonic pressure over it.
Once the pipeline response is desired to be calculated regarding pipe-soil interaction, Liu et
al.(2010) claims a deficiency of attention and rigorous analysis methods on defining pipeline
18. 8
response in terms of stresses, strains and deformations under soil deflections due to ground
movements. In this way, the author undertakes a Finite Element Analysis over an X65 buried
onshore pipeline subjected to large soil displacements. The utilised approaches consist on the
“modified Riks method”, associated to the iterative Newton-Raphson method and the “non-linear
stabilisation algorithm”, where damping is accounted for the convergence enhancement. After
modelling stages, total pipeline’s deformation, plastic strains as well as equivalent stresses were
found, as shown in Figure 4.
Figure 4. Pipeline responses due to ground movement, with a) equivalent stress, b)
equivalent plastic strain and c) total displacement. After Liu et al. (2010).
The mobilised soil exerting perpendicular load over the pipeline, was assumed to be a viscous soil
with elastic properties below than those for the stable soil. The latter can be appropriate to some
extent once the pipe-soil interaction is addressed on defining pipeline response due to near to solid
or viscous mobilised soils. However, since liquefied seabed soils exhibit a complex behaviour
induced by harmonic variations due to wave-induced pressures, it is necessary to account a time
dependant seabed response where the fluctuation of pore water pressure and soil shear strength
govern the pipeline structural dynamic response. Therefore, as previously mentioned, the objective
of this study is to assess the pipeline dynamic behaviour regarding a time dependant seabed response
in terms of stresses, developed by waves’ harmonic motion once the seabed is liquefied.
19. 9
1.3 Objectives
Regarding the aforementioned justifications and background, the proposed objectives for
developing the present study are mentioned below.
1.3.1 Main Objective
- Determine the influence of seabed liquefaction on the dynamic behaviour of shallow water
pipelines.
1.3.2 Specific Objectives
- Model different analysis scenarios, varying water depth, pipeline diameter as well as wave
parameters, in order to assess the sensitiveness of pipeline behaviour due to different
environmental settings;
- Estimate pipeline dynamic response in terms of equivalent stress, bending stress and
deformation, once a liquefied seabed segment is modelled as an incompressible fluid;
- Evaluate pipeline dynamic behaviour once seabed dynamic response is calculated, by
accounting the latter in terms of stresses;
- Compare obtained results from both previously mentioned modelling approaches.
1.4 Thesis outline
The developed study was assembled with a structure as following described: in chapter 1, a first
approach to the seabed liquefaction process and its influence over subsea infrastructure is presented,
by means of a literature review, with story cases and common approaches to understanding the
impact of liquefaction; also, the objectives of the study are specified, where these are subdivided
into the global main objective and specific and detailed objectives, which are necessary to fulfil the
proposed scope. In chapter 2, the fundamental theoretical framework is specified, where two main
topics are explained and reviewed, such as the principle of effective stress and how liquefaction
influences over this stress, and the mechanics of subsea pipelines.
Chapter 3 embraces the methodology and approaches undertaken in order to select, assume and
assign analysis inputs, as well as fundamental considerations adopted throughout the analysis
development. In chapter 4, all calculations, models and results are explained in detail and presented.
Since modelling stages encompassed two different approaches, the sub-chapter 4.1 covers the first
analysis approach where the pipeline structural behaviour is assessed regarding solely the wave-
induced pressure, whilst the sub-chapter 4.2 makes reference to the pipeline structural behaviour
regarding the dynamic soil response. Finally, chapter 5 includes conclusions and recommendations.
20. 10
2 Theoretical framework
Fundamental concepts which must be addressed on the development of the present study are
mentioned below. Since undertaken analysis embraces both soil mechanics concepts and mechanical
concepts in terms of stresses for the pipeline behaviour, this section is sub-divided in order to
mention relevant aspects of both topics separately.
2.1 Effective vertical stress principle and soil liquefaction
Undoubtedly the most relevant fundamental principle of soil mechanics is the effective vertical
stress principle proposed by Karl Terzaghi (Terzaghi, 1995), as the difference between the total
vertical stress and the pore water pressure, given as:
𝜎′ 𝑣 = 𝜎 − 𝑢 Equation 1
where the total vertical stress 𝜎 is given by the unit weight of the soil at a given depth z, as 𝜎 = 𝛾𝑧
and the pore water pressure 𝑢 is given by the unit weight of the water at the same given depth z,
as 𝑢 = 𝛾 𝑤 𝑧. This principle is a basic concept when describing volumetric changes and shear strength
features of saturated soils, since both are governed by the abovementioned effective stress
(Fredlund, 1993). Once an external excitation force is exerted over a cohesionless soil mass (e.g.
sands, non-cohesive silts), the static pore water pressure 𝑢 undertakes a transition to a flow state,
where an increase ∆𝑢 is developed (Terzaghi, 1995). For the developing of this excess of pore water
pressure ∆𝑢, the velocity of application of the external load must be faster than the ability of the soil
mass to drain or percolate the existing ground water, which is function of the soil permeability 𝑘 𝑠.
A positive increase in ∆𝑢 is related to an upward water seepage opposed to gravitational forces,
eventually equalising the value of the total vertical stress 𝜎 and thus leading the effective vertical
stress to a null value (i.e. 𝜎′ 𝑣 = 0). Once this occur, the friction between the seepage pressure and
the soil grains tends to lift the latter and liquefaction process takes place, as shown in the scheme of
upward flow lines in Figure 5, during soil liquefaction.
21. 11
Figure 5. Upward water flow during liquefaction. After Teh et al. (2006)
On the other hand, a negative variation in ∆𝑢 is related to a downward water seepage in the direction
of gravitational forces, which in turn increases the value of the effective vertical stress 𝜎′ 𝑣, avoiding
the development of liquefaction. To illustrate graphically the process of liquefaction under a cyclic
load (i.e. waves), two different plots of non-liquefaction (Figure 6a) and liquefaction due to an
excess of 𝜎′ 𝑣 (Figure 6b) were obtained after (Sumer et al., 1999) during the laboratory tests
conducted for the study of pipeline sinking due to seabed liquefaction, under different wave
parameters.
Figure 6. Results after laboratory tests on defining pipeline sinking, where a) soil does not
liquefy and b) where soil liquefies due to an excess of pore water pressure (Sumer et al.,
1999).
In this way, once the subsea pipeline behaviour due to seabed liquefaction is desired to be calculated,
a comprehensive understanding and characterisation of the liquefaction process, regarding the
aforementioned principle of effective stress, must be conducted. According to Teh et al. (2006),
both positive pore pressure and negative pore pressure (i.e. suction) may take place under cyclic
loading around a submarine structure (e.g. a pipeline). Also, the author claims that both sinking
velocity and depth are greater for a heavier pipe, whilst a lighter pipe tends to float once soil
liquefies. This author also states that during soil liquefaction, due to the depletion of effective stress
(i.e. ’v =0), the seabed loses its bearing capacity, leading to a consequent pipeline sinking. Having
22. 12
said this, Teh et al. (2006) states three different modes that governs the extent of pipeline sinking
once the seabed has experimented liquefaction, summarised as follows:
- Mode I: For a slow sinking light pipe, the gradient of the increasing pore pressure acts as
buoyancy force stopping the downward advance of it;
- Mode II: Due to the increase or recover of soil bearing capacity, once excess of pore water
pressure starts dissipating or when the pressure gradient is not sufficient;
- Mode III: For a fast sinking heavy pipe, it will continue to sink if whether the sinking
velocity is greater than the excess of pore pressure dissipation rate or the pressure gradient
is not enough to act as a buoyant force. Once it reaches a stable stratum, it may stop sinking.
Consequently to these sinking modes and to the sand liquefaction process based on the vertical stress
principle stated by Karl Terzaghi (Terzaghi, 1995), a need on defining the magnitude of the upward
water flow or buoyant force exerted by inter-granular water once liquefaction takes place (i.e. excess
of pore water pressure), is vital to predict and calculate pipeline response in terms of stresses and
deformation, regarding its extent of sinking. The latter, following the scheme proposed by Teh et
al. (2006) for the stresses state under a pipeline for a non-liquefied seabed (Figure 7a) and for a
liquefied one (Figure 7b).
Figure 7. Excess of pore water pressure (u) and vertical effective stress (𝝈′ 𝒗) under a) non-
liquefied conditions and b) liquefied conditions. After Teh et al. (2006).
Therefore, for the definition of seabed dynamic response due to wave cycling loads, in terms of
stresses and deformation due to soil liquefaction, the coupled model of soil skeleton-water flow was
the adopted approach. This seabed dynamic response approach was firstly introduced by (Biot,
1955) and (Biot, 1962) where invariants of strain components I1, I2 and I3, and fluid content 𝜉 –
governed by a linear variation of rate flow with the gradient of pressure proposed by Darcy (Ulker,
2012)– are related. With these strain-stress relations, accounting for pore water pressure 𝑝0 or 𝑝 𝑓, a
23. 13
simplified way for modelling nearly saturated soil mass to predict its response in presence of pore
fluid was possible, under plane strain conditions (i.e. 𝜀3 = 0).
A fully dynamic approach, where inertial terms correspondent to both acceleration of soil skeleton
(𝑢̈) and the acceleration of pore water pressure relative to soil skeleton (𝑤̈ ), due to soil permeability
of analysed sands, was adopted. Thus, based on the dynamic response formulation further developed
by Zienkiewicz (1981) and extended by Ulker et al. (2009) for different seabed characteristics, the
global equilibrium for a unit volume of soil mass is given as:
𝜎𝑖𝑗,𝑗 + 𝜌𝑔𝑖 = 𝜌𝑢̈ 𝑖 + 𝜌 𝑓 𝑤̅̈ 𝑖 Equation 2
As mentioned before, since the dynamic response of a saturated porous media is governed by fluid
flow and soil skeleton deformation interaction, Darcy’s law governs flow and its fluid equilibrium
phase, given as:
−𝑝,𝑖 + 𝜌 𝑓 𝑔𝑖 = 𝜌 𝑓 𝑢̈ 𝑖 +
𝜌 𝑓 𝑤̅̈ 𝑖
𝜂
+
𝜌 𝑓 𝑔𝑖
𝑘𝑖
𝑤̅̇ 𝑖 Equation 3
By establishing a continuity condition due to mass balance:
𝑢̈ 𝑖,𝑖 + 𝑤̅̇ 𝑖,𝑖 = −
𝜂
𝐾𝑓
𝑝̇ Equation 4
Where:
𝜎𝑖𝑗: total stress;
p: pore water pressure;
𝑢̈ 𝑖: soil skeleton acceleration;
𝑤̅̇ 𝑖: average relative water velocity;
𝑤̅̈ 𝑖: average relative water acceleration;
𝑔𝑖: component of gravitational acceleration;
𝑘𝑖: component of permeability;
𝜂: porosity;
24. 14
𝐾𝑓: Bulk modulus of pore water;
𝜌: total density;
𝜌 𝑓: density of pore water.
Where 𝐾𝑓 is related to the degree of saturation (S) as:
𝐾𝑓 =
𝜌 𝑤 𝑔𝑑𝐾 𝑤
𝜌 𝑤 𝑔𝑑 + 𝐾 𝑤(1 − 𝑆) Equation 5
With 𝐾 𝑤 as the bulk density of seawater and d as the water depth.
The previous mentioned effective stress principle, is written as:
𝜎′𝑖𝑗 = 𝜎𝑖𝑗 − 𝛿𝑖𝑗 𝑝 Equation 6
With:
𝜎′𝑖𝑗: effective stress;
𝜎𝑖𝑗: total stress;
𝛿𝑖𝑗: Kronecker delta;
p: pore water pressure.
Furthermore, by means of Lame’s parameters 𝜆 and G, the stress-strain relations are taken into
account as linear-elastic under a plane strain condition, as:
𝜎′𝑖𝑗 = 𝜆𝜀 𝑘𝑘 𝛿𝑖𝑗 + 2𝐺𝜀𝑖𝑗 Equation 7
With 𝜀 𝑘𝑘 and 𝜀𝑖𝑗 as volumetric and deviatoric components of strain. In this way, by means of
equations Equation 2Equation 3Equation 4Equation 6Equation 7, a coupled flow-deformation
response is given to characterise seabed behaviour under dynamic wave loads. By re-writing
Equation 4 as:
𝐾𝑓
𝜂
(𝑢𝑖,𝑖 + 𝑤̅ 𝑖,𝑖)𝑖
= −𝑝,𝑖 Equation 8
25. 15
By substituting Equation 8 in Equation 3 and by means of the effective stress relation of Equation
6, two final coupled equations results as follows:
𝐾𝑓
𝜂
(𝑢𝑖,𝑖 + 𝑤̅ 𝑖,𝑖)𝑖
= 𝜌 𝑓 𝑢̈ 𝑖 +
𝜌 𝑓 𝑤̅̈ 𝑖
𝜂
+
𝜌 𝑓 𝑔𝑖
𝑘𝑖
𝑤̅̇ 𝑖 Equation 9
𝜎′𝑖𝑗,𝑗 +
𝐾𝑓
𝜂
(𝑢𝑖,𝑖 + 𝑤̅ 𝑖,𝑖)𝑖
= 𝜌𝑢̈ 𝑖 + 𝜌 𝑓 𝑤̅̈ 𝑖 Equation 10
Having established the above coupled equations, the seabed dynamic response will have a form
𝑢(𝑥, 𝑧, 𝑡) = 𝑈(𝑧)𝑒 𝑖(𝑘𝑥−𝜔𝑡)
, described by the harmonic complex form related in section 4.2 by
equationsEquation 26,Equation 27,Equation 28 andEquation 29, where are solved in order to predict
responses in terms of stresses.
2.2 Pipeline structural behaviour
The pipeline strength is given by the steel yielding strength (i.e. the maximum stress at which linear
elastic deformations take place) and by its ultimate tensile strength. Beyond these thresholds, the
pipeline will deform plastically and fail, respectively. A subsea pipeline is under different sort of
loads generated by both the external environment such as external pressure, seabed friction, tidal
and wave currents, and by operational conditions such as internal pressure of the transported
product, its temperature and the subsequent thermal stresses regarding the temperature differential
between the latter and the environment temperature.
Therefore, stresses definition and quantification are necessary to guarantee an adequate material
grade selection in order to withstand the operational loads throughout the pipeline’s design life. To
do so, the hoop or circumferential stress generated by the internal pressure must be calculated
according to Barlow´s formula as:
𝜎ℎ =
(𝑝𝑖 − 𝑝0)𝐷
2𝑡
Equation 11
Where 𝑝𝑖 corresponds to the internal pressure, 𝑝0 to the external pressure, D corresponds to the
pipeline’s external diameter and t represents its thickness, as shown in Figure 8.
26. 16
Figure 8. Variables included in hoop stress calculation. After (Bai, 2005).
Due to the aforementioned external and internal factors exerting loads and stresses over the pipe,
additional longitudinal or axial stresses must be calculated in order to define completely the stresses
state at which the pipeline may be subjected. These additional stresses corresponds to the end cap
stress, the axial component of the abovementioned hoop stress, thermal stress due to the thermal
delta between the seawater temperature and the transported product temperature, and bending
stresses. These stresses are schematically represented in Figure 9.
Figure 9. Components of axial or longitudinal stress. After (Bai, 2005).
Where the end cap stress is calculated as the difference of the internal force 𝐹𝑖 due to the internal
pressure and the external force 𝐹𝑜 due to the water column pressure, over the cross sectional area of
the pipe A, given as:
𝜎 𝑎𝑒 =
𝐹𝑖 − 𝐹𝑜
𝐴
Equation 12
27. 17
The thermal stress, is function of the steel Young’s Modulus E, the steel coefficient of thermal
expansion 𝛼, and the temperature difference between the internal temperature and the external
temperature Δ𝑇, as:
𝜎 𝑎𝑇 = 𝐸𝛼Δ𝑇 Equation 13
The axial component of the hoop stress, is governed by the steel Poisson’s ratio 𝜈 = 0.3, given as:
𝜎 𝑎ℎ = 0.3𝜎ℎ Equation 14
Finally, the bending stress is given by the general bending moment formulation for the elastic range,
as function of the bending moment Mb, the moment of inertia I, the pipeline´s radius y, the Young’s
Modulus E and the radius of curvature R, as follows:
𝑀 𝑏
𝐼
=
𝜎𝑏
𝑦
=
𝐸
𝑅 Equation 15
Where the minimum bending radius can be estimated as:
𝑅 =
𝐸𝐷
2𝜎 𝑦 𝐷𝐹 Equation 16
With 𝜎 𝑦 as the minimum specified yielding stress (i.e. SMYS) and DF as a design factor usually
assumed as 0.85, according to Mousselli (1981). Once the totality of the axial or longitudinal stress
are computed, the total longitudinal stress 𝜎𝐿 can be calculated as:
𝜎𝐿 = 𝜎 𝑎𝑒 + 𝜎 𝑎𝑇 + 𝜎 𝑎ℎ + 𝜎𝑏 Equation 17
In this way, following the von Mises yield criterion, the equivalent stress that takes into account the
interaction of the previously mentioned stresses can be calculated as:
𝜎𝑒 = √𝜎ℎ
2
− (𝜎ℎ 𝜎𝐿) + 𝜎𝐿
2
Equation 18
Knowing the equivalent stress resulting after the combination of stresses acting over the subsea
pipeline, it is possible to know if whether the yield strength of the particular grade of steel selected
28. 18
will be able to withstand all stresses throughout the operational life of the pipeline. However, a
fundamental feature of a subsea pipeline is its ability to withstand the external pressure exerted by
the water column without collapsing or buckling, due to its characteristic thin-walled property.
Therefore, the critical collapse pressure must be defined as follows:
𝑃𝑐𝑟 =
2𝐸( 𝑡
𝐷⁄ )
3
(1 − 𝜈2)
Equation 19
Where E is the material Young´s Modulus, t is the pipe wall thickness, D is the external diameter,
and 𝜈 is the Poisson’s ratio. Since the greater the compressive stresses over the pipe walls, the greater
the likelihood of buckling from occurring, the critical bending moment of the pipeline must be
additionally estimated, given as:
𝑀𝑐𝑟 =
0.99𝐸𝐷𝑡2
2(1 − 𝜈2)
Equation 20
Once both critical collapse pressure 𝑃𝑐𝑟 and the critical bending moment 𝑀𝑐𝑟 are known, the
maximum allowable bending moment Mall can be obtained by means of the combined bending and
collapse formulation, as follows:
𝑀 𝑎𝑙𝑙
𝑀𝑐𝑟
+
𝑃
𝑃𝑐𝑟
=
1
𝑆𝐹 Equation 21
Where P is the external pressure due to the water column and SF is the safety factor with a value of
one (1.0), in order to define the ultimate bending moment at which the pipeline may buckle. Thus,
the bending stress 𝜎𝑏 generated once the allowable bending (i.e. ultimate bending moment since
SF=1.0) takes place, is given by:
𝑀 𝑎𝑙𝑙
𝐼
=
𝜎𝑏
𝑦
=
𝐸
𝑅 Equation 22
Having established this, pipeline’s ultimate strengths in terms of equivalent stress and buckling
resistance (i.e. due to bending moments and collapse pressure) are defined.
29. 19
3 Methodology
The pipeline dynamic behaviour assessment included in this document is subdivided into two stages,
in order to compare the difference in pipeline responses in terms of deformation, bending stress and
equivalent stress once the liquefied seabed behaviour is assumed as: a) an incompressible and fully
saturated fluidised media where the pipeline has already sunk, and b) a fully saturated porous media
where its dynamic response is calculated regarding soil skeleton and fluid flow interaction. For the
first approach, an incompressible fluid behaviour governing the liquefied seabed stress is assumed,
which can be described by Bernoulli’s Principle in terms of the developed pore water pressure within
the studied thickness for an infinitesimal element of area dA, due to the wave-induced pressure. For
the second approach, the seabed dynamic response due to a cyclic wave-induced pressure was
calculated according to the soil deformation-water flow coupled methodology proposed originally
by (Biot, 1955) and (Biot, 1962), and further developed by (Zienkiewicz, 1981) and (Ulker et al.,
2009).
For both cases, a dynamic analysis by means of the Finite Element Method was conducted to
compute the pipeline response as function of time, where the mechanical analysis software ANSYS
was utilised. The modelling stage was conducted by allocating a free span where the liquefied
seabed segment was supposed to be, and in order to model the pipe-liquefied soil interaction, stresses
were assigned to the external surface (i.e. contact stresses) of the pipe a) regarding the wave-induced
pressure, based on linear wave theory as function of time, for the case where the liquefied soil was
assumed as an incompressible fluidised media, and b) with respect to the calculated dynamic
response in terms of vertical effective stress, horizontal effective stress, shear stress and pore water
pressure.
To take into account the influence of wave parameters variability on seabed response and therefore
on pipeline dynamic behaviour, two wave periods were assumed for the conducted analysis. The
first wave period corresponds to a five seconds period (i.e. T=5 s) and the second, to a ten seconds
period (i.e. T=10 s). For both cases, an initial deep water wave length (L0) and wave height (H0)
were assumed, and by means of the dispersion relationship, the resultant shallow water length (L)
and wave height (H) were calculated by keeping fixed the assumed wave periods. This, since the
scope of the present study addresses solely to shallow water pipelines’ environment. In this way,
four different scenarios correspondent to shallow water depths were adopted, as: 25m water depth,
50m water depth, 75m water depth and 100m water depth. Thus, for each water depth scenario
30. 20
corresponds a particular segment length of liquefied seabed equal to the wave length regarding each
wave period and the correspondent water depth (i.e. a 40m segment of liquefied seabed corresponds
to a 40m wave length for T=5s; a 120m segment of liquefied seabed corresponds to a 120m wave
length for T=10s and 25m of water depth), as summarised in Table 5 and Table 6.
Seabed liquefaction potential is function of a combination of several metocean parameters such as
wave length, wave height, water depth, wave period, wave frequency and of different soil
parameters such as permeability, degree of saturation, bulk density, porosity, relative density, grain
size, seabed thickness and stresses history. Therefore, an adequate combination of the
aforementioned aspects must take place simultaneously in order to seabed liquefaction to occur. In
this way, to guarantee a liquefied seabed scenario for the proposed dynamic assessment, a previously
liquefied state was assumed in the onset of each analysis, triggered by an external cyclic load capable
of generating excess of pore water pressure that exceeds the effective vertical stress of the soil, such
as earthquakes, hurricanes or greater wave periods than assumed for the analyses. In this way, the
influence of the supposed T=5 s and T=10 s wave periods on the seabed liquefaction process, for
both analysis approaches (i.e. regarding solely wave-induced water pressure and regarding the
dynamic seabed response analysis) was analysed by assuming that liquefaction has already taken
place. Hence, its further development or its dissipation may depend on the interaction of both
assumed wave periods and the seabed features, and their ability to allow inter-granular water
drainage or to increase the excess of pore water pressure.
Furthermore, for each water depth scenario four different pipeline diameters were adopted and
different pipe wall thicknesses were assigned to each diameter. The selected diameters were 254mm
(10 inches), 406.4mm (16 inches), 609.6mm (24”) and 914.4mm (36 inches), whilst the selected
wall thicknesses were 11mm, 19mm, 20mm and 31.7mm respectively. The main purpose of varying
pipelines’ diameters and pipes’ wall thicknesses was to find a relationship between pipeline dynamic
behaviour once its geometric features were varied in terms of diameter to thickness ratio (i.e. D/t).
For each analysed scenario, external and internal pressures were taken into account as related in
Table 4. Steel properties assumed for the conducted analysis were not varied, and no plastic
behaviour was taken into account, which implies that an isotropic elastic behaviour of the pipeline
governs the dynamic responses during the analysis.
31. 21
4 Analysis and results
As mentioned earlier, this chapter is subdivided into two sections. The first section (numeral 4.1)
considers the pipeline dynamic motions and stresses solely as function of the wave induced pressure
over the seabed, once the latter has experimented liquefaction. The second section (numeral 4.2)
focuses on pipeline dynamic behaviour as function of the seabed dynamic response as a porous
media interacting with inter-granular water flow, where the conducted modelling takes into account
how stresses variation throughout the analysed wave periods affects the mentioned pipeline
behaviour, once the seabed has been liquefied. For both sections, also water depth and pipeline
diameter were varied in order to appreciate how sensitive is the structural dynamic response of the
pipeline due to soil liquefaction, regarding different environment settings.
Summarising, the analysed scenarios are shown in Table 1. It is important to mention that since two
different wave periods were taken into account for modelling, the following scenarios embrace both
wave parameters.
Table 1. Evaluated scenarios of pipeline structural response.
Accordingly, aforementioned scenarios are described below and the calculation procedures shown.
4.1 Pipeline structural behaviour regarding solely wave induced stress over seabed
Once the excess of pore water pressure due to a cyclic external excitation source (e.g. induced
vibrations, earthquakes, waves) equals or exceed the effective vertical stress of the soil, a fluidisation
state takes place where the bearing capacity and shear strength reduce to zero and the soil behaviour
can be described as a liquid, with a similar harmonic behaviour than the above wave motion, as
suggested by Foda and Hunt (1993) in Figure 10. Due to this liquid behaviour of the liquefied soil,
an overburden pressure q acting over a surficial area A of the seabed, will have a constant value q at
a depth z keeping fixed the area A due to water incompressibility.
Pipeline structural behaviour
regarding only wave induced
stress over seabed
254
(10)
406.4
(16)
609.6
(24)
914.4
(36)
254
(10)
406.4
(16)
609.6
(24)
914.4
(36)
254
(10)
406.4
(16)
609.6
(24)
914.4
(36)
254
(10)
406.4
(16)
609.6
(24)
914.4
(36)
Pipeline structural behaviour
regarding dynamic seabed
response
254
(10)
406.4
(16)
609.6
(24)
914.4
(36)
254
(10)
406.4
(16)
609.6
(24)
914.4
(36)
254
(10)
406.4
(16)
609.6
(24)
914.4
(36)
254
(10)
406.4
(16)
609.6
(24)
914.4
(36)
Water depth
Scenario
Outer Diameter mm (in) Outer Diameter mm (in) Outer Diameter mm (in) Outer Diameter mm (in)
25m 50m 75m 100m
32. 22
Figure 10. Wave-like motion of the liquefied seabed. After Foda and Hunt (1993)
In terms of oceanic environment and based on linear wave theory, the harmonic wave motion
transmits a stress over the seabed described as:
𝑝(𝑥, 𝑡) =
𝜌 𝑤 𝑔𝐻
2cosh(𝑘𝑑)
𝑒 𝑖(𝑘𝑥−𝜔𝑡)
This wave-transmitted stress to the seabed has also a consequent harmonic behaviour regarding the
wave number (k), the angular frequency (𝜔) and its length (x), where the stress related to the crest
of the wave is compressive (+) whilst the trough exerts a tensile stress (-), as shown schematically
in Figure 11.
Figure 11. Wave-induced stress over the seabed.
33. 23
According to the above mentioned, it is possible to describe the behaviour of a liquefied seabed
under the wave-induced stress on a simplified way, as a liquid mass under an harmonic pressure
related to the wave parameters. Adopting a fully saturated (i.e. S=1) and an almost incompressible
(i.e. 𝜈 → 0.5) condition of the liquefied seabed, it is appropriate to assume that the stresses
throughout the liquid-like thickness have a value of
𝜌 𝑤 𝑔𝐻
2cosh(𝑘𝑑)
𝑒 𝑖(𝑘𝑥−𝜔𝑡)
. Furthermore, the null values
of the vertical effective stress (i.e. 𝜎′ 𝑧𝑧 = 0) and of the shear strength (i.e. 𝜏 𝑥𝑧 = 0) once the soil
reaches the liquefaction state, lead to a loss of bearing capacity of the seabed. Therefore, if a steel
pipeline is laid on the seabed and the latter liquefies, it will deflect and may sink due to its own
weight, where the extent of deflection and sinking is function of its moment of inertia (I), the density
of the steel (𝜌𝑠) and the length of the liquefied seabed segment, which can be described as a ‘span’
due to the loss of support (i.e. bearing capacity).
During modelling, the non-liquefied seabed was assumed to be an elastic homogeneous solid, with
density and elastic properties as shown in Table 2.
Table 2. Non-liquefied soil properties.
Parameter Units Value
Seabed thickness, h m 30
Density of soil, s T/m³ 2
Elasticity Modulus, E kPa 14000
Shear Modulus, G kPa 4698
Poisson´s ratio, - 0.35
On the other hand, pipeline´s properties are specified in Table 3, whilst assumed internal and
external pressures are included in Table 4, as well as the design factor related to these pressures for
each water depth scenario.
Table 3. Pipeline properties.
Grade
SMYS
(MPa)
UTS
(MPa)
Poisson’s
ratio
𝜌𝑠
(kg/m³)
Wall Thickness (mm)
OD 254
(10")
OD 406.4
(16")
OD 609.6
(24")
OD 914.4
(36")
API 5L X65 464 563.8 0.3 6800 11 19 20 31.7
34. 24
Table 4. Pressures and design factors.
Pipeline
Diameter
Pipeline
Pressures
Water Depth
25m 50m 75m 100m
OD 254
(10")
Pi (MPa) 23.5 23.5 23.5 23.5
Po (MPa) 0.25 0.5 0.75 1.00
Design Factor 0.58 0.57 0.57 0.56
OD 406.4
(16")
Pi (MPa) 20 20 20 20
Po (MPa) 0.25 0.5 0.75 1.00
Design Factor 0.46 0.45 0.44 0.44
OD 609.6
(24")
Pi (MPa) 16.5 16.5 16.5 16.5
Po (MPa) 0.25 0.5 0.75 1.00
Design Factor 0.53 0.53 0.52 0.51
OD 914.4
(36")
Pi (MPa) 23 23 23 23
Po (MPa) 0.25 0.5 0.75 1.00
Design Factor 0.71 0.70 0.69 0.68
Additionally, wave parameters were calculated in order to define completely the modelling inputs
for the aforementioned analysis scenarios in Table 1. These wave parameters are shown in Table 5
and Table 6, regarding the different adopted water depth scenarios.
Table 5. Wave parameters for T=5 s.
Deep water wave
parameters
Units
Water
depth
25m
Water
depth
50m
Water
depth
75m
Water depth
100m
Value Value Value Value
Wave period, T s 5 5 5 5
Wave height, H0 m 3 3 3 3
Water depth, d m 500 500 500 500
Angular frequency, 1/s 1.257 1.257 1.257 1.257
Wave number, k 1/m 0.161 0.161 0.161 0.161
Wave length, L0 m 39.033 39.033 39.033 39.033
Shallow water wave
parameters
Units Value Value Value Value
Wave period, T s 5 5 5 5
Water depth, d m 25 50 75 100
Angular frequency, 1/s 1.257 1.257 1.257 1.257
Wave number, k 1/m 0.080 0.057 0.046 0.040
Wave length, L m 37.645 38.766 38.958 39.007
Wave height, H m 3.049 3.010 3.003 3.001
35. 25
Table 6. Wave parameters for T=10 s.
Deep water wave
parameters
Units
Water
depth
25m
Water
depth
50m
Water
depth
75m
Water
depth
100m
Value Value Value Value
Wave period, T s 10 10 10 10
Wave height, H0 m 4 4 4 4
Water depth, d m 500 500 500 500
Angular frequency, 1/s 0.628 0.628 0.628 0.628
Wave number, k 1/m 0.040 0.040 0.040 0.040
Wave length, L0 m 156.131 156.131 156.131 156.131
Shallow water wave
parameters
Units Value Value Value Value
Wave period, T s 10 10 10 10
Water depth, d m 25 50 75 100
Angular frequency, 1/s 0.628 0.628 0.628 0.628
Wave number, k 1/m 0.040 0.028 0.023 0.020
Wave length, L m 119.107 138.845 146.750 150.581
Wave height, H m 3.900 4.048 4.083 4.065
For the previous calculations, the wave length for deep water L0 was calculated as function of the
wave period as,
𝐿0 =
𝑔𝑇2
2𝜋
Equation 23
Once this variable was known, the wave length for shallow water L was calculated in function of it
as,
𝐿 = 𝐿0tanh (
2𝜋𝑑
𝐿
) Equation 24
Finally, the height H for the shallow water was found as function of the previous calculated wave
length L and the assumed wave height H0 for deep water, as follows:
36. 26
𝐻 = 𝐻0 {[1 +
4𝜋𝑑
𝐿⁄
𝑠𝑖𝑛ℎ(4𝜋𝑑
𝐿⁄ )
] 𝑡𝑎𝑛ℎ (
2𝜋𝑑
𝐿
)}
−1
2⁄
Equation 25
Therefore, as of included parameters in Table 5 and Table 6 for the evaluated wave periods, the
wave-induced pressure 𝑝(𝑥, 𝑡) was calculated for each water depth, as shown in Table 7 and Table
8.
Table 7. Wave-induced load over seabed, T=5 s.
Water Depth
(m)
Wave load (Pa)
t=0 t=1 t=2 t=3 t=4 t=5
25 -3922.84 -759.20 3453.62 2893.66 -1665.25 -3922.84
50 -1014.34 1013.08 1640.46 0.78 -1639.98 -1014.34
75 -211.41 777.98 692.23 -350.16 -908.64 -211.41
100 3.08 507.41 310.52 -315.50 -505.51 3.08
Table 8. Wave-induced load over seabed, T=10 s.
Water
Depth
(m)
Wave load (Pa)
t=0 t=1 t=2 t=3 t=4 t=5 t=6 t=7 t=8 t=9 t=10
25 819.79 -6589.78 -11482.27 -11988.93 -7916.23 -819.79 6589.78 11482.27 11988.93 7916.23 819.79
50 -6343.14 -8951.11 -8140.06 -4219.78 1312.31 6343.14 8951.11 8140.06 4219.78 -1312.31 -6343.14
75 -6612.48 -6374.09 -3701.02 385.72 4325.12 6612.48 6374.09 3701.02 -385.72 -4325.12 -6612.48
100 -5230.45 -3858.21 -1012.27 2220.33 4604.83 5230.45 3858.21 1012.27 -2220.33 -4604.83 -5230.45
For an enhanced understanding of the proposed analysis scenarios and proposed settings, a
schematic illustration of the four depths’ scenarios assessed is shown in Figure 12, for a wave period
T=5s, whilst in Figure 13, a detailed plot of the pipeline and liquefied seabed is presented. It is
important to mention that for modelling, the length of the right and left segments of the pipeline
which are not over the liquefied soil, have the same length that the latter, in order to reproduce the
scheme of as single supported beam shown in Figure 14.
37. 27
Figure 12. Different water depths analysed (T=5 s).
Figure 13. Detail of modelling setting (T=5 s).
Figure 14. Schematic deflection of the pipeline based on a single supported beam behaviour.
Similarly, in Figure 15 a schematic illustration of the four depths’ scenarios assessed is shown for a
wave period T=10s and in Figure 16, a detailed plot of the pipeline and liquefied seabed is presented.
38. 28
The same consideration was adopted in order to reproduce the scheme of as single supported beam
shown in Figure 14.
Figure 15. Different water depths analysed (T=10 s).
Figure 16. Detail of modelling setting (T=10 s).
Having defined the analysis scenarios and all inputs such as geometries, loads and analysis times
(i.e. equal than adopted wave periods), modelling results are shown below.
In order to illustrate the procedures adopted on modelling the pipeline dynamic response due to the
seabed liquefaction and under the wave-induced pressure, from Figure 17 to Figure 31 relevant
obtained results from the structural analysis software ANSYS are shown. Since several models were
run in order to satisfy the adopted water depths (i.e. 25m, 50m, 75m and 100m), the assumed wave
39. 29
periods (i.e. T=5 s and T=10 s) as well as the selected pipeline diameters (i.e. 10”, 16”, 24” and
36”), in the aforementioned figures (Figure 17 to Figure 31) only the obtained results in terms of
deformation and stresses for 10” and 36” pipeline diameters, for 25m water depth and wave period
T=5 s are presented. Remaining analyses for the additional scenarios were conducted by means of
analogue procedures. Even though totality of the ANSYS outputs are not presented, summary tables
and plots with all the analysis results are exposed in order to undertake an adequate interpretation
and discussion of the latter.
In Figure 17 and Figure 18, a detail of the general cross section of the models adopted is shown. It
can be seen how in order to optimise processing time, a symmetry condition was adopted and only
half of the cross section of both seabed and pipeline were taken into account.
Figure 17. Cross section of the model.
Figure 18. Detailed perspective of the half pipe.
40. 30
In a similar way, the symmetry condition was accounted for the total longitudinal dimension of the
models, as shown in Figure 19. Therefore, the modelled geometry corresponds to a quarter of the
overall geometry (i.e. half cross section, half longitudinal dimension), optimising time of
processing. It is important to mention that boundary conditions in terms of degree of freedom were
cautiously assigned to this symmetry regions, in order to maintain the continuity of the modelled
geometries.
Figure 19. Longitudinal view of the half-symmetric model.
Since the main assumption of the analysis is that liquefaction has already taken place and the
induced-wave pressure is exerted over this incompressible liquid-like soil, the harmonic stress
transmitted to this fluidised mass and therefore to the deformed pipeline was assigned to the pipeline
surface, as shown in Figure 20.
Figure 20. Wave-induced pressure over the span.
Consequently, once the 5 seconds water period exerts a pressure over the 40m segment long of
liquefied seabed (stresses correspondent to Table 7), a 10” diameter pipeline with a 11mm wall
thickness may deflect in the centre of the generated ‘span’ or section without support, a total length
of 0.54m, correspondent to a deflection of more than one time its own diameter (i.e.
OD=254mm=0.254m).
41. 31
Figure 21. Detail of amount of deformation of more than one diameter in the centre of the
liquefied section (OD=10”, T=5 s, 25m water depth).
In terms of stress concentration, the maximum equivalent stress develops in the liquefied and non-
liquefied soil transition, with a value of 289.51MPa for the 11mm wall thickness and 10” diameter
pipeline as shown in Figure 22. In Figure 23, a detail of the concentration of the maximum
equivalent stress on the bottom of the pipeline is presented. This location may occur due to the high
compressive stress developed in the transition from non-liquefied seabed to the liquefied section
once the pipeline lost support and under the cyclic load, where buckling and collapse potential may
also increase.
Figure 22. Concentration of equivalent stress where the support is lost (OD=10”, T=5 s, 25m
water depth).
42. 32
Figure 23. Maximum stress at the bottom of the pipe at the non-liquefied and liquefied
transition (OD=10”, T=5 s, 25m water depth).
The main purpose of displaying the results of the smallest and the biggest pipelines’ diameters
analysed, was to compare how the moment of inertia (i.e. function of the diameter and the wall
thickness) plays an important role in terms of total deformation once the seabed is assumed as
liquefied, keeping fixed all the other parameters such as wave-induced pressure and water depth. As
shown in Figure 24 and Figure 25, the amount of total deformation at the centre of the liquefied
section is significantly smaller for a 36” pipeline if compared to the deformation of the 10” pipeline
in Figure 21.
Figure 24. Localisation of maximum deformation at the centre of the liquefied segment
(OD=36”, T=5 s, 25m water depth).
43. 33
Figure 25. Detail of amount of deformation of a few millimetres in the centre of the liquefied
section (OD=36”, T=5 s, 25m water depth).
On the other hand, the concentration of the maximum stress takes place once more in the liquefied
and non-liquefied soil transition (Figure 26). In terms of stress magnitude, it is evident how a higher
value of the equivalent stress develops for the 36” pipeline (403.5MPa) if compared to the developed
stress for the 10” pipeline. However, the design factor for the latter is less than the design factor for
the 36” pipeline (i.e. 0.58 for the OD=10” and 0.71 for the OD=36”) justifying this magnitude
difference.
Figure 26. Concentration of equivalent stress where the support is lost (OD=36”, T=5 s, 25m
water depth).
Similarly than for the 10” pipeline, the stress accumulation for the 36” pipeline takes place in the
bottom of it (Figure 27).
44. 34
Figure 27. Maximum stress at the bottom of the pipe at the non-liquefied and liquefied
transition (OD=36”, T=5 s, 25m water depth).
For wave period T=10, a similar procedure was conducted for all assumed scenarios. In order to
compare the behaviour of same pipelines described above for the wave period T=5 s (i.e. OD=10”
and OD=36”), ANSYS outputs for maximum total deformations and equivalent stress are presented
below.
In accordance to the wave induced pressures calculated for T=10 s (Table 8), the 10” diameter
pipeline will deflect a considerable magnitude of 2.52m in the centre of a 120m liquefied seabed
segment, with respect to its undeformed vertical alignment, as shown in Figure 28.
Figure 28. Detail of amount of deformation of more than one diameter in the centre of the
liquefied section (OD=10”, T=10 s, 25m water depth).
Different from the 40m liquefied segment correspondent to a wave period T=5 s previously
analysed, the maximum equivalent stress for a wave period T=10 s develops over the top of the pipe,
as plotted in Figure 29. It can be also seen how the maximum equivalent exceeds the material
yielding strength (i.e. 𝜎𝑒 = 505.61MPa > 464MPa), indicating that once the pipeline is under a cyclic
45. 35
wave load throughout a 120m of unsupported segment, it will work within the material plastic
region.
Figure 29. Maximum stress at the top of the pipe at the non-liquefied and liquefied transition
(OD=10”, T=10 s, 25m water depth).
In the same way, the 36” pipeline was assessed under the effect of a T=10 s wave-induced pressure
for the same scenario as for the previously analysed 10” pipeline (i.e. 120m of liquefied seabed,
25m of water depth). Conversely to the T=5 s case, for this case the 36” pipeline deflects
significantly at the centre of the liquefied segment (i.e. 2.41m), as shown in Figure 30. This may
indicate that for the unsupported span of 120m, the moment of inertia of the heavy thick wall does
not longer influence the amount of deformation, since the maximum displacement of the 10”
pipeline for the same scenario (i.e. 25m water depth, wave period T=10) is slightly different once
compared to the aforementioned deformation (2.52m of maximum deformation for the 10”
pipeline).
Figure 30. Localisation of maximum deformation at the centre of the liquefied segment
(OD=36”, T=10 s, 25m water depth).
46. 36
In terms of stress concentration, the 36” pipeline exposes an excess of equivalent stress over the
ultimate tensile strength of the API 5L X65 steel, with a maximum stress of 859.33MPa, as evident
in Figure 31. However, it is important to remember that due to the scope of the study, an elastic
isotropic behaviour of the steel was taken into account, what means that no effects of strain
hardening and plastic deformation, and thus their influence in the dynamic response of the pipeline,
were regarded.
Figure 31. Maximum stress at the bottom of the pipe at the non-liquefied-liquefied transition
(OD=36”, T=10 s, 25m water depth).
In order to compare how the pipeline dynamic response varies regarding the adopted scenarios and
conditions, a set of plots for both wave period T=5 s and T=10 s are exposed below.
Figure 32. Deformation (T=5 s, D=10”). Figure 33. Deformation (T=5 s, D=16”).
49. 39
Figure 46. 𝝈 𝒆(T=10s, D=24”). Figure 47. 𝝈 𝒆(T=10s, D=36”).
From the above presented plots, the following observations can be done:
Related to deformation behaviour for the analysis corresponding to the wave period T=5 s, it can be
seen that in general, the higher deformation corresponds to the smallest pipeline diameter (i.e.
OD=254mm) and the smallest deformation was plotted for the larger diameter (i.e. OD=914.4mm).
Therefore, an influence of the moment of inertia regarding the “span” length can be associated to
the restricted deformation for the largest pipeline diameter.
Independently for each pipe diameter, the largest deformation for the OD=254mm pipeline was
registered for the 75m depth scenario, and the smallest deformation for the 100m depth scenario.
Conversely, for the OD=406.4mm pipeline the largest deformations were computed for the 75m and
100m water depth scenarios, whilst the smallest for the 25m one. For the OD=609.6mm pipeline,
the largest deformation was for the 25m water depth scenario and the smallest for the 50m water
depth scenario, in contrast to the computed deformations for the OD=914.4mm, where the highest
correspond to the 25m water depth scenario and the 100m one, whilst the lower was registered for
the 75m water depth scenario.
The above mentioned shows how once maximum deformations, for a 40m of liquefied seabed and
for different pipe diameters are compared between each other, the bigger the diameter and the
thicker the wall thickness, the higher the moment of inertia I. The latter minimise the amount of
deformation. Contrariwise, the smaller the pipe diameter and the thinner the pipe wall, the larger the
deformation.
50. 40
On the other hand, once the different assessed scenarios (i.e. different water depths) are compared
independently for each pipe diameter, no path was found in terms of maximum deformations once
the water depth is varied.
In terms of equivalent stress, it was found how its magnitude varied inversely to the amount of
deformation. Thus, the lowest equivalent stress was computed for the 254mm pipeline, followed by
the 406.4mm pipeline. However, the following greater stress was found for the largest pipe diameter
(914.4mm), and the highest stress level was computed for the 609.6mm pipe. The latter even
exceeded the ultimate tensile strength of the X65 steel. This switch between the 609.6mm pipeline
and the 914.4mm pipe can be justified by a higher D/t ratio of the 609.6mm pipeline (i.e. 30.5),
whilst the 914.4mm pipeline has a D/t ratio of 28.8.
Opposite to the deformation behaviour, in terms of stress response a common behaviour with the
maximum developed stress for the 25m water depth scenario was found. However, for the 609.6mm
pipeline, after t=2 the dynamic response for the mentioned scenario fell below the stresses reported
for the other scenarios; for the 406.4mm pipeline, the stress regarding this water depth was recorded
as well as the lowest. Once again, the D/t ratio may have implications in this behaviour since the
406.4mm pipeline has the lowest ratio with a value of 21.4.
For deformations recorded after the conducted analysis regarding the T=10 s wave period, two
aspects can be subtracted from plots: As for the T=5 s wave period deformations, the smaller the
pipe diameter, the larger the total deformation. Conversely to the conducted analysis for T=5 s, for
the T=10 s wave period’s scenarios a pattern with the largest deformations for the 100m water depth
scenario was found, except for the 254mm pipeline. In the same way, for totality of the analysed
pipeline diameters, a pattern with the smallest deformations were recorded for the 25m water depth
scenario.
In terms of stress responses, a generalised behaviour for all the assessed pipe diameters was found
as the larger the deformation, the higher the generated stress. This is opposed to particular
behaviours for the T=5 s wave period analysis, where the higher the deformation, the lower the
stress recorded. This may depend on the fact that almost all the structural responses for a wave
period T=5 s (i.e. 40m liquefied segment), are within the elastic range of the adopted steel. However,
for the wave period T=10 s, an also generalised behaviour of stress levels above the steel yield
strength and the ultimate tensile stress is evident. Furthermore, it can be noticed how a gradual
increase in the equivalent stress was reached once the pipe diameter was increased, as the response
51. 41
plots for the 914.4mm pipeline are well above the yielding strength and the ultimate tensile strength
limits (marked as the horizontal red lines), if compared to the response of the 254mm pipeline.
4.2 Pipeline structural behaviour regarding dynamic seabed response
In this section, all material’s properties, scenarios and analysis considerations mentioned in the
previous section (4.1) were accounted similarly on the describing of the pipeline structural
behaviour regarding the dynamic seabed response. Nonetheless, once the seabed dynamic response
was evaluated, new inputs for the pipeline modelling by means of ANSYS were available.
Therefore, the basis and procedures of analysis are essentially the same for both sections (i.e. 4.1
and 4.2) and will not be described again in this section, but an explanation of how the above
mentioned inputs resultant of the dynamic seabed response analysis were taken into account during
modelling, are provided.
Based on the analysis approach proposed by (Ulker and Rahman, 2009), the dynamic response of
the seabed due to a cyclic wave loading was calculated. Wave periods of five seconds (i.e. T=5 s)
and ten seconds (i.e. T=10 s) were accounted as inputs for the mentioned analyses. Since the
employed approach of calculation is based on water flow-soil skeleton coupled equations, both
elastic properties and flow parameters of the porous seabed assumed for calculations are shown
below in Table 9. Also, wave parameters are described for the different water depths adopted.
Table 9. Seabed parameters for dynamic response analysis.
Parameter Units Value
Seabed thickness, h m 30
Permeability, kz m/s 1.00E-04
Elasticity Modulus, E kPa 14000
Shear Modulus, G kPa 4698
Poisson´s ratio, - 0.49
Bulk modulus of water, Kw MPa 2000
Bulk modulus of seabed, K MPa 233.33
Saturation, S - 1
Porosity, n - 0.33
Density of soil, s T/m³ 2
Density of seawater, w T/m³ 1
Where the bulk modulus was calculated as 𝐾 =
𝐸
3(1−2𝜈)
and the shear modulus as 𝐺 =
𝐸
2(1+𝜈)
.
52. 42
From Table 9, it can be seen that assumed soil parameters were intentionally selected to reflect an
already liquefied state, represented by a Poisson’s ratio tending to an uncompressible behaviour (i.e.
𝜈 → 0.5) and a fully saturated state (i.e. S=1.0). This in accordance with the accounted premise
where the seabed has been previously liquefied and its dynamic response in terms of cyclic stresses
(i.e. vertical stress, normal stress, shear stress and pore water pressure) is calculated for the assumed
wave periods. For the non-liquefied seabed segments (i.e. first third and last third of the analysed
pipeline where it is still supported), the elastic properties are same as defined in Table 2.
On the other hand, wave parameters are summarised in Table 5 and Table 6, regarding the different
water depth scenarios (i.e. 25m, 50m, 75m and 100m) and the assessed wave periods (i.e. T=5s and
T=10s) respectively.
Having defined seabed elastic parameters, hydraulic parameters and wave parameters, by means of
the coupled equations mentioned in numeral 2 a system of complex harmonic governing equations,
function of soil displacements Ux, Uz and fluid displacements 𝑤̅ 𝑥, 𝑤̅ 𝑧 arises as follows:
(
𝜌 𝑓ℎ2
𝜔2
− 𝑘2
ℎ2 𝑘 𝑓
𝑛
𝐾 +
𝑘 𝑓
𝑛
) 𝑈 𝑥
+ ℎ
(
𝜌 𝑓ℎ𝜔2
𝑛
+
𝑖𝜔𝜌 𝑓ℎ𝑔
𝑘 𝑥
− 𝑘2
ℎ
𝑘 𝑓
𝑛
𝐾 +
𝑘 𝑓
𝑛 )
𝑊̅𝑥
+ 𝑖𝑘ℎ
𝑘 𝑓
𝑛
𝐾 +
𝑘 𝑓
𝑛
(
𝑑𝑈𝑧
𝑑𝑧̅
+
𝑑𝑊̅𝑧
𝑑𝑧̅
) = 0
Equation 26
𝑘 𝑓
𝑛
𝐾 +
𝑘 𝑓
𝑛
(
𝑑2
𝑈𝑧
𝑑𝑧̅2
+
𝑑2
𝑊̅𝑧
𝑑𝑧̅2
) +
𝑖𝑘ℎ
𝑘 𝑓
𝑛
𝐾 +
𝑘 𝑓
𝑛
∗
𝑑𝑈 𝑥
𝑑𝑧̅
+
𝑖𝑘ℎ
𝑘 𝑓
𝑛
𝐾 +
𝑘 𝑓
𝑛
∗
𝑑𝑊̅𝑥
𝑑𝑧̅
+
𝜌 𝑓ℎ2
𝜔2
𝐾 +
𝑘 𝑓
𝑛
∗ 𝑈𝑧
+ ℎ2
(
𝜌 𝑓 𝜔2
𝑛
+
𝑖𝜔𝜌 𝑓 𝑔
𝑘 𝑧
𝐾 +
𝑘 𝑓
𝑛 )
𝑊̅𝑧 = 0
Equation 27
53. 43
ℎ2
(
𝜌𝜔2
𝑛
𝐾 +
𝑘 𝑓
𝑛
− 𝑘2
) 𝑈 𝑥 + 𝑖𝑘ℎ (
𝑘 𝑓
𝑛
+ 𝜆 + 𝐺
𝐾 +
𝑘 𝑓
𝑛
)
𝑑𝑈𝑧
𝑑𝑧̅
+ ℎ2
(
𝜌 𝑓 𝜔2
− 𝑘2 𝑘 𝑓
𝑛
𝐾 +
𝑘 𝑓
𝑛
) 𝑊̅𝑥 +
𝑖𝑘ℎ
𝑘 𝑓
𝑛
𝐾 +
𝑘 𝑓
𝑛
∗
𝑑𝑊̅𝑧
𝑑𝑧̅
+
𝐺
𝐾 +
𝑘 𝑓
𝑛
∗
𝑑2
𝑈 𝑥
𝑑𝑧̅2
= 0
Equation 28
𝑑2
𝑈𝑧
𝑑𝑧̅2
+
𝑖𝑘ℎ (
𝑘 𝑓
𝑛
+ 𝜆 + 𝐺)
𝐾 +
𝑘 𝑓
𝑛
∗
𝑑𝑈 𝑥
𝑑𝑧̅
+
𝑘 𝑓
𝑛
𝐾 +
𝑘 𝑓
𝑛
∗
𝑑2
𝑊̅𝑧
𝑑𝑧̅2
+
𝑖𝑘ℎ
𝑘 𝑓
𝑛
𝐾 +
𝑘 𝑓
𝑛
∗
𝑑𝑊̅𝑥
𝑑𝑧̅
+ ℎ2
(
𝜌𝜔2
− 𝑘2
𝐺
𝐾 +
𝑘 𝑓
𝑛
) 𝑈𝑧
+
𝜌 𝑓ℎ2
𝜔2
𝐾 +
𝑘 𝑓
𝑛
∗ 𝑊̅𝑧 = 0
Equation 29
Having established this simultaneous equations derived from the interaction between soil skeleton
and water flow, non-dimensional parameters must be taken into account in order to solve the
simultaneous equation system. These parameters are:
𝜅 =
𝑘 𝑓
𝑛
𝐾+
𝑘 𝑓
𝑛
as the ratio of the fluid bulk modulus and the soil bulk modulus;
𝜅1 =
𝜆
𝐾+
𝑘 𝑓
𝑛
as the ratio of the volumetric term of Lamé constants and the system bulk modulus;
𝜅2 =
𝐺
𝐾+
𝑘 𝑓
𝑛
as the ratio of the soil shear modulus to the system bulk modulus;
m=kh, with k as the wave number and h as the total seabed thickness;
𝛽 =
𝜌 𝑓
𝜌
as the ratio between the fluid density and the soil density;
Π1𝑥 =
𝑘 𝑥 𝑉𝑐
2
𝑔𝛽𝜔ℎ2
as a ratio between the time for the pore fluid in x direction and the time for wave to
travel;
54. 44
Π1𝑧 =
𝑘 𝑧 𝑉𝑐
2
𝑔𝛽𝜔ℎ2
as previous but in z direction;
𝑉𝑐
2
=
𝐾+
𝑘 𝑓
𝑛
𝜌
as the ratio of the bulk modulus of the system and the soil density (i.e. compression wave
speed).
After adopting these parameters, a linear system of equations results which can be expressed in form
of the matrix below:
[
𝛽Π2 − 𝑚2
𝜅 𝑖𝑚𝜅
𝜕
𝜕𝑧̅
(
𝛽Π2
𝑛
+
𝑖
Π1𝑥
− 𝑚2
𝜅) 𝑖𝑚𝜅
𝜕
𝜕𝑧̅
𝑖𝑚𝜅
𝜕
𝜕𝑧̅
(𝛽Π2 + 𝜅
𝜕2
𝜕𝑧̅2
) 𝑖𝑚𝜅
𝜕
𝜕𝑧̅
(
𝛽Π2
𝑛
+
𝑖
Π1𝑧
+ 𝜅
𝜕2
𝜕𝑧̅2
)
(Π2 − 𝑚2
+ 𝜅2
𝜕2
𝜕𝑧̅2
) 𝑖𝑚(𝜅 + 𝜅1 + 𝜅2)
𝜕
𝜕𝑧̅
(𝛽Π2 − 𝑚2
𝜅) 𝑖𝑚𝜅
𝜕
𝜕𝑧̅
𝑖𝑚(𝜅 + 𝜅1 + 𝜅2)
𝜕
𝜕𝑧̅
(Π2 − 𝑚2
𝜅2 +
𝜕2
𝜕𝑧̅2
) 𝑖𝑚𝜅
𝜕
𝜕𝑧̅
(𝛽Π2 + 𝜅
𝜕2
𝜕𝑧̅2
)
]
{
𝑈𝑥
𝑈𝑧
𝑊̅𝑥
𝑊̅𝑧}
= 0
Once this matrix is solved by finding its determinant (i.e. 𝑑𝑒𝑡[𝑀] = 0), a characteristic equation
results as follows:
𝛼1
𝜕6
𝜕𝑧̅6 + 𝛼2
𝜕4
𝜕𝑧̅4 + 𝛼3
𝜕2
𝜕𝑧̅2 + 𝛼4 = 0 Equation 30
The six roots of the previous equation are the eigenvalues 𝜂𝑖(𝑖 = 1, 2, 3, 4, 5, 6) of the eigenvectors,
𝑉𝑖 = {
1
𝑏𝑖
𝑐𝑖
𝑑𝑖
}
where bi, ci and di coefficients can be found as [𝑀]{𝑉𝑖} = 0, leading to a system of six values for
each coefficient: 𝑏𝑖(𝑖 = 1, 2, 3, 4, 5, 6), 𝑐𝑖(𝑖 = 1, 2, 3, 4, 5, 6) and 𝑑𝑖(𝑖 = 1, 2, 3, 4, 5, 6).
Remaining 𝑎𝑗 coefficients can be obtained from the matrix of nodal coordinates of the second order
Langranian Triangular Element, 𝜉𝑖,𝑗(𝑖, 𝑗 = 1, 2, 3, 4, 5, 6) which are written in terms of the
previously calculated eigenvalues 𝜂𝑖, as follows:
𝜉1𝑗 = 1;
𝜉2𝑗 = 𝑏𝑗;
𝜉3𝑗 = 𝑑𝑗;
55. 45
𝜉4𝑗 =
𝐾
ℎ
𝑏𝑗 𝜂 𝑗 + 𝑖𝑘𝜆;
𝜉5𝑗 = 𝐺 (
𝜂 𝑗
ℎ
+ 𝑖𝑘𝑏𝑗) ;
𝜉6𝑗 = 𝑖𝑘(1 + 𝑐𝑗) +
𝜂 𝑗
ℎ
(𝑏𝑗 + 𝑑𝑗)
In this way, the matrix of nodal coordinates can be written as:
[
𝜉11 𝜉21 𝜉31 𝜉41 𝜉51 𝜉61
𝜉12 𝜉22 𝜉32 𝜉42 𝜉52 𝜉62
𝜉13 𝜉23 𝜉33 𝜉43 𝜉53 𝜉63
𝜉14 𝜉24 𝜉34 𝜉44 𝜉54 𝜉64
𝜉15 𝜉25 𝜉35 𝜉45 𝜉55 𝜉65
𝜉16 𝜉26 𝜉36 𝜉46 𝜉56 𝜉66]
Where values of i and j vary from 1 to 6 according to the eigenvalues and constants previously
calculated. To find the solution of the resultant linear system, boundary conditions must be set as:
- At the surface of the seabed (i.e. mudline) where 𝑧 = 0, both vertical effective stress 𝜎′ 𝑧𝑧
and shear stress 𝜏 𝑥𝑧 are equal to zero (i.e. 𝜎′ 𝑧𝑧 = 𝜏 𝑥𝑧 = 0) and the pore pressure takes a
value of 𝑝 =
𝜌 𝑤 𝑔𝐻
2cosh(𝑘𝑑)
𝑒 𝑖(𝑘𝑥−𝜔𝑡)
;
- At the bottom of the seabed (i.e. contact with impermeable rock) where 𝑧 = −ℎ, three of the
four displacements cannot take place (i.e. 𝑊̅𝑧 = 𝑈 𝑥 = 𝑈𝑧 = 0).
Once this, 𝑎𝑗 coefficients were obtained and the dynamic seabed response in terms of effective
vertical stress 𝜎′ 𝑧𝑧, horizontal effective stress 𝜎′ 𝑥𝑥, shear stress 𝜏 𝑥𝑧 and pore water pressure 𝑝 was
determined for the four different assumed water depths, by means of the following equations:
𝜎′ 𝑧𝑧 = [∑ (𝑖𝑘𝜆 + 𝑏𝑗 𝐾
𝜂 𝑗
ℎ
)
6
𝑗=1
𝑎𝑗 𝑒 𝜂 𝑗
𝑧
ℎ] 𝑒 𝑖(𝑘𝑥−𝜔𝑡)
Equation 31
𝜎′ 𝑥𝑥 = [∑ (𝑖𝑘𝐾 + 𝑏𝑗 𝜆
𝜂 𝑗
ℎ
)
6
𝑗=1
𝑎𝑗 𝑒 𝜂 𝑗
𝑧
ℎ] 𝑒 𝑖(𝑘𝑥−𝜔𝑡)
Equation 32
56. 46
𝜏′ 𝑥𝑧 = [∑ (𝑖𝑘𝑏𝑗 +
𝜂 𝑗
ℎ
)
6
𝑗=1
𝑎𝑗 𝑒 𝜂 𝑗
𝑧
ℎ] 𝑒 𝑖(𝑘𝑥−𝜔𝑡)
Equation 33
𝑝 = −
𝐾𝑓
𝑛
{∑ [𝑖𝑘(1 + 𝑐𝑗) +
𝜂 𝑗
ℎ
∗ (𝑏𝑗 + 𝑑𝑗)] 𝑎𝑗 𝑒 𝜂 𝑗
𝑧
ℎ
6
𝑗=1
} 𝑒 𝑖(𝑘𝑥−𝜔𝑡)
Equation 34
Since previous equations for stresses calculation are function of seabed depth (z) and time (t),
several calculations were conducted in order to compute each stress variation throughout the wave
periods assumed (i.e. T=5 s and T=10 s) and for a seabed depth of 10 meters. Therefore, different
sets of results were obtained summarised as follows:
- Vertical effective stress 𝜎′ 𝑧𝑧: 6 curves for wave period T=5 s (i.e. t=0 s to t=5 s) varying
throughout a depth of 10m, for each different water depth assessed (i.e. 25m, 50m, 75m and
100m);
- Horizontal effective stress 𝜎′ 𝑥𝑥: 6 curves for wave period T=5 s (i.e. t=0 s to t=5 s) varying
throughout a depth of 10m, for each different water depth assessed (i.e. 25m, 50m, 75m and
100m);
- Shear stress 𝜏 𝑥𝑧: 6 curves for wave period T=5 s (i.e. t=0 s to t=5 s) varying throughout a
depth of 10m, for each different water depth assessed (i.e. 25m, 50m, 75m and 100m);
- Pore water pressure p: 6 curves for wave period T=5 s (i.e. t=0 s to t=5 s) varying throughout
a depth of 10m, for each different water depth assessed (i.e. 25m, 50m, 75m and 100m).
In the same way, for T=10 s results are:
- Vertical effective stress 𝜎′ 𝑧𝑧: 11 curves for wave period T=10 s (i.e. t=0 s to t=10) varying
throughout a depth of 10m, for each different water depth assessed (i.e. 25m, 50m, 75m and
100m);
- Horizontal effective stress 𝜎′ 𝑥𝑥: 11 curves for wave period T=10 s (i.e. t=0 s to t=10 s)
varying throughout a depth of 10m, for each different water depth assessed (i.e. 25m, 50m,
75m and 100m);
- Shear stress 𝜏 𝑥𝑧: 11 curves for wave period T=10 s (i.e. t=0 s to t=10 s) varying throughout
a depth of 10m, for each different water depth assessed (i.e. 25m, 50m, 75m and 100m).
- Pore water pressure p: 11 curves for wave period T=10 s (i.e. t=0 s to t=10 s) varying
throughout a depth of 10m, for each different water depth assessed (i.e. 25m, 50m, 75m and
100m).
57. 47
In consequence, the variance of stresses throughout the wave periods and for a seabed depth of 10m
are shown below. However, since several plots contain the results of the seabed dynamic response,
only the correspondent for a water depth of 50m and for both wave periods are shown below as
illustrative examples. Remaining plots are included in the Appendix.
Figure 48. Vertical Stress (T=5 s)
Figure 49. Shear Stress (T=5 s)
Figure 50. Vertical Stress (T=10 s)
Figure 51. Horizontal Stress (T=5 s)
Figure 52. Pore Pressure (T=5 s)
Figure 53. Horizontal Stress (T=10 s)
![MODULE: MAR8097 – Dissertation
Dynamic behaviour of shallow water pipelines due to seabed liquefaction
Submitted by: Alejandro Marín Tamayo
Student No: 140047404
A Dissertation submitted
for the partial fulfilment of the Degree of Master of Science in
MSc Pipeline Engineering programme
School of Marine Science and Technology
Newcastle University
Supervisor name:
Dr. Longbin Tao
Deadline Date: [7 August 2015]](https://image.slidesharecdn.com/3850295b-5760-449c-aaa3-8cbab4b6b695-160129185519/85/Dissertation-Alejandro-Marin-T-1-320.jpg)
