Assessment of structural integrity of subsea wellhead system: analytical and numerical study

Subsea wellhead systems exposed to severe fatigue loading are becoming increasingly a significant problem in offshore drilling operations due to their applications in wells with higher levels of pressure and temperature, situated at larger depths and in harsher environments. This has led to a substantial increase in the weight and size of offshore equipment, which, in combination with different loading conditions related to the environmental factors acting on the vessel and riser, has greatly increased the loads acting on subsea well systems. In particular, severe fatigue loading acting on the subsea wellhead system was detected. For this reason, a combined analytical and numerical study investigating the critical effect of crack depth on the overall structural integrity of subsea wellhead systems under cyclic loading was carried out. The study is based on a Linear Elastic Fracture Mechanics (LEFM) approach.


INTRODUCTION
owadays, offshore oil and gas exploration move into deeper waters and harsher environments. Technological advancements have enabled exploitation of previously inaccessible reservoirs, thanks to such important advancements as the improved ability to find offshore reservoirs and the capacity to drill deviated wells [1,2]. Simultaneously, new technology makes extraction of more hydrocarbons and gas from existing wells possible, and many of them are now more than twenty years old. At the same time, economic demands imply a prolonged use of the existing welds. Therefore, subsea wells that are used for longer periods are likely to experience dangerous levels of accumulated load cycles. As a result, regulatory bodies are seeking assurances that the well-system conditions and integrity are being N correctly supervised, particularly with regard to fatigue, fail-safe design as well as analytical and numerical methodologies for their prediction [1][2][3][4][5][6]. Circumferential girth welds are a critical location for the structural integrity of pipelines and risers, and are produced using mechanised welding processes. Welding involves the heating of metal to its melting temperature followed by rapid cooling. As the weld metal cools it contracts, and the cooling rate influences a type of microstructure [7][8][9][10], mechanical properties [11,12] and a level of residual stresses [13] in the welds. Besides, flaws are likely to occur during the welding process. Fracture mechanics-based structural integrity assessments, most commonly referred to as either engineering critical assessments (ECAs) or Fitness-For-Service (FFS) assessments have found widespread acceptance to deal with such problems over the years [1][2][3][4][5][6]. The nuclear and offshore oil and gas industries are the main drivers behind the development of formal FFS procedures. Structural integrity assessments [14,15] can be used at the design stage, to estimate the maximum flaw size that will not grow to intolerable levels during the life of the component, or to assess defects grown after some time in service. Such information about defect tolerance relies on the availability of representative and reliable experimental data, on which any defect-assessment calculation is based. This study aims at developing a fracture mechanics-based model to study and elucidate the effects linked to crack depth and shape as well as crack propagation in the subsea wellhead systems under cyclic loading conditions. In particular, this research focuses at simulations of crack propagation in a conductor/casing pipe girth welds and at understanding the impact of flaws on the drift-off/drive-off capacities and on the residual fatigue life of the wellhead system. In order to determine the maximum allowable flaw size and to simulate crack propagation under cyclic loading, the in-house software Zencrack was used. The validity of this software has been widely assessed in several research and industrial projects [16,17]. The wellhead system was calibrated to fit the S-N curve of Quality Category Q2 as described in BS 7910 Code [18], which is equivalent to BS 7608 Code design class E [19].

WELLHEAD SYSTEM
typical configuration of a wellhead system is shown in Fig. 1. In this study, a simple two-pipe wellhead system comprising a conductor pipe (diameter 36". thickness 1.5") and a surface casing pipe (diameter 22", thickness 1.0"), with a rigid lock wellhead, was considered. The girth weld for both the conductor and casing pipes to be investigated is class E type following BS7910 code. The effect of stress concentration or misalignment was not considered in this study. Both the conductor and the casing are considered fixed at the base, and a constant-amplitude cyclic bending moment was applied at the top end, with a crack plane situated half way along the length (Fig. 2). No presence of cement in the annular spacing between the conductor and the casing pipe or the surrounding soil was assumed. The crack geometries and the applied load, used in this study, are symmetric with respect to the YZ plane (Fig.  2). Hence, it was proposed to use half models for all analyses taking advantage of this symmetry. Material properties for X80 steel used for the casing and X65 steel for the conductor are summarised in Tab

Loading and Boundary Conditions
The applied load is a bending moment at the reference point (RP), shown in Fig. 2, which corresponds to a cyclic stress with a magnitude of 1 MPa (R=-1) on the outer surface of the conductor pipe. This is then scaled to get the required stress value, when needed. The initial study was based on the stress value of 80 MPa at the outer face of the conductor pipe. In the finite-element model, two loading surfaces are "constrained" together to the RP via kinematic coupling. The boundary conditions are described with the help of Fig. 2:  The bending moment was applied to RP (reference point) and via a kinematic coupling constraint to nodes on the remote end face at Z = L.  The conductor and casing were constrained to undergo zero displacements at Z = 0.

Boundary Conditions: Kinematic vs Distributing Coupling
A separate study was performed to check the adequacy of this coupling compared to the distributed boundary conditions. Kinematic coupling was enforced in a strict master-slave approach [20]. Degrees of freedom (DOFs) at the coupling nodes were eliminated, and the coupling nodes were constrained to move with the rigid-body motion of the reference node. Generally, kinematic coupling constraint does not allow relative motion among the constrained DOFs, while allowing relative motion of the unconstrained ones.Distributing coupling was enforced in an average sense [20]. DOFs at the coupling nodes were not eliminated. Rather, the constraint was enforced by distributing loads such that the resultants of the forces at the coupling nodes were equivalent to the forces and moments at the reference node, and force and moment equilibrium of the distributed loads about the reference point was maintained. A distributing coupling allows relative motion of the constrained and unconstrained DOFs. The effect of these two different coupling constraints should be evaluated during the crack propagation in the conductor and casing as depicted in Fig. 3. The kinematic coupling provides conditions that are more realistic for the case with parallel surfaces in the region, in which the bending moment is applied. For this reason this constraint was preferred to the distributing coupling, which allows more freedom to the displacements in the loaded section.

PRELIMINARY STUDY
LEFM-based methodology, which adopted a Paris-crack propagation law, was developed to calculate the crack growth rate and perform initial crack-size calibration exercise for the conductor and the casing considered separately. In order to achieve a close match between the residual life, based on the Paris law, and the fatigue life, based on the weld S-N curves (E class type [18]) in the conductor pipe, the following iterative procedure was implemented:  Determination of initial crack size for the calibration exercise.  Determination of crack-growth data for the appropriate material in a seawater environment (-1100 mV cathodic protection).  Analytical crack-growth analysis for different stress ranges.  Crack-growth analysis from the initial crack to the failure condition using Zencrack.  Comparison of preliminary numerical and analytical data.  Re-adjustment of the initial crack size depending upon the results of the analysis. The outcome of this activity was the determination of the Paris-law data and the initial crack size to be applied in subsequent crack modelling of the conductor/casing system. It is paramount to emphasise that FE simulations were based on linear elastic fracture mechanics with the values of stress intensity factor (SIF) K calculated employing the J contour integral [21][22]. SIFs were calculated at multiple points along the crack [16,17]. The failure condition was assumed to occur when the crack either grows through the wall thickness (and reaches the opposite surface) or reaches an unstable crack size (critical K) under the applied cyclic loading.

Analytical Evaluation of Stress Intensity Factor K I
An analytical expression was derived to estimate the stress intensity factor as a function of crack depth. This was subsequently used to estimate the initial crack size for the finite-element calibration analysis. A number of available publications [18,23] were considered to evaluate analytically the stress intensity factors at the deepest point of an elliptical crack located circumferentially on the inner wall of a cylinder as shown in Fig. 4. The procedure described in BS 7910 [18] using Eq. 1 for the evaluation of SIFs was found to give non-reliable results due to an error in the value of the term (Y) as prescribed in the equation M24 of the Annex M of BS 7910: This error was possibly due to a mistake in equation M24 for the contribution of stress due to the global bending moment. It was therefore decided to use the procedures described in the publication, "Stress Intensity Factor and Limit Load Handbook" edited by British Energy Generation Ltd [23]. The equation used to calculate the SIFs at the deepest point of the elliptical crack is: , , The sizes considered for the cylinder (conductor pipe) are:  outer radius ro = 18";  wall thickness (B) = 1.5";  internal radius ri = 16.5";  the dimensions of the planar circumferential elliptical crack on the inner wall of the cylinder are aand 2c, where a is the depth and 2c the length measured at the inner radius (Fig. 4). The values of the coefficient fbg used in the analytical evaluation of the SIFs and the calculated values of SIFs are given in Tab. 2. The term  bg in Eq. 2 is defined as the maximum outer-fibre bending stress. The SIF values have been determined for the following ratios a/B: 0.01 (~0.0); 0.2; 0.4; 0.6 and 0.8. The tabulated coefficients for Eq. 2 are provided in [23]. The (mode-I) stress intensity factor at the deepest point of the crack has been calculated for the following ratio between the crack depth a and half-arc length c: a/c = 1 This ratio is assumed to remain constant throughout the analytical calculation. This assumption was considered acceptable in the preliminary calibration study. Since in this study only bending stresses are taken into account, Eq. 2 can be simplified as follows: SIFs in Tab. 2 are given for unit outer fibre stress. The trend of the stress intensity factor I K is shown in Fig. 5.   Using the curve fitting techniques, a curve was fitted through the tabulated values for a/B, on the X-axis, and SIFs, related to the maximum fibre stress of 1 MPa, on Y-axis, as shown in Fig. 5. The polynomial equation representing this curve is: This equation was used to perform analytic crack-propagation calculations to estimate the initial crack size.

Finite-element strategy
After the preliminary analysis, a finite-element (FE) model was developed to evaluate the residual life of the pipe system under study with an initial semi-circular crack. The details of the numerical model used for the crack growth FE analyses are shown in Fig.6. The elements used were full-integration 20-node hexahedral elements. In the model with the crack, Zencrack introduced the crack details by replacement of a line of eight elements with crack blocks, resulting in the FE model of the cracked pipe. The mesh of the crack so created was a focussed mesh, consisting of quarter-point crack-tip elements along the crack front with r -1/2 singularity. The number of elements in the FE models with the crack and without it (conductor and casing separately considered) was never less than 135000. The node lying on the outer radius is referred to as external surface node, the node on the inner radius as internal surface node and the node on the wall thickness as thickness node (Fig. 6b).  In this section, the initial crack size and material coefficients of the Paris crack-growth law were evaluated and calibrated against the Quality Category 2 (Q2) S-N curve as given in BS 7910 Code.

Estimation of Initial Crack Size for S-N Relation
In order to evaluate the residual life, the Paris equation was used: where the coefficients A and m for the steel under analysis are: A = 5.21x10 -13 ; m = 3 for da/dN in mm/cycle and K in MPa mm 1/2 .

Minimum crack size
The size of the elliptical initial crack on the inner wall of the pipe was also estimated. It was of paramount importance that the estimated number of cycles for each crack size, at different stress range levels, was reasonably close to the fatigue cycles of the Q2 S-N curve. In BS 7910 Code, the quality categories refer to particular fatigue design requirements or to the actual fatigue strength of flaw-containing components. The quality categories are defined in terms of the ten S-N curves labelled Q1 to Q10. These are described by the following equation [18]: The value of such constant for the Quality Category 2 (Q2) for steel is: 1.04×10 12 cycles. At a stress range of 80 MPa, the estimated life is 2×10 6 for X80 steel. This reference stress and the number of fatigue life cycles were used in the first iterative calculations. The numerical integration of the Paris law (Eq. 5) was carried out using a fixed increment da. After each increment of crack advance an average number of life cycles was determined as The total life is therefore: The SIFs were calculated using the fitted curve described by Eq.  Fig. 7. The values 3.81 mm and 7.62 mm were chosen as they represent a/B=0.1 and 0.2. The initial crack size of 3.92 mm matches the S-N curve for all the stress ranges considered and, therefore, it was used for the preliminary finite-element analyses of crack growth. During this analytical study, the SIFs were constantly monitored and checked against the fracture toughness (KIC) value for X80 steel (225 MPam) to detect any failure due to fracture. The analytical solution showed that the stress intensity factor during the crack propagation did not exceed the KIC value; failure was achieved due to the break-through condition.

FE Results: Conductor
The results obtained from the numerical analyses with the initial crack length equal to 3.92 mm (maximum crack depth) and the stress range of 80 MPa are shown in Fig. 8 and Fig.9.  The estimated residual life (i.e. the number of cycles for the crack to grow to the specific length) evaluated by Zencrack was 1.72×10 6 cycles whereas the analytical calculations gave a residual life of 2.03×10 6 cycles, with a difference of nearly 15%. This discrepancy between the analytical solution and the FE-based solution (with Zencrack) was likely due to the following factors:  r i /B = 11 is slightly out of the valid range of 10 (as given in the reference);  the analytical solution assumes that the shape of the crack remains constant, i.e the ratio a/c = 1, whereas in the FE simulation the crack shape would take a more natural shape, based on the energy release rate at each crack-front node. In order to get the residual life closer to the target value of 2.03×10 6 cycles it was decided to assess a more suitable initial crack size using Zencrack. The analytical study suggested an increase of about 17% in the crack dimensions to achieve the fatigue life of 1.72×10 6 cycles. Therefore, the size of the initial crack was modified to 4.74mm. Fig. 10 shows the results of the analytical calculations with initial crack size equal to 4.74 mm: with such an assumption, the analytical assessment matches exactly the Zencrack results calculated with an initial crack size equal to 3.92 mm. It was therefore decided to assume an initial crack size a = 3.1 mm (i.e.: 3.92 -0.82) in the FE analyses. Results of the numerical analysis for the crack size a = 3.1 mm (stress range 80 MPa) presented in Fig. 11 show that the residual life achieved for this crack size is very close to the target value of 2.03×10 6 cycles. This figure also presents a plot of the results of a Zencrack analysis, in which the crack shape was forced to maintain the rato a/c = 1 as the crack propagated through the wall thickness. This plot demonstrates that there is a tangible effect of the crack-front shape evolution.

FE Results: Casing
The same methodology described in Section 3.2 was used to evaluate the minimum size of an elliptical surface crack on the inner wall of the casing tube. Therefore, by means of the stress intensity factor KI, calculated analytically for different stress ranges and crack sizes, the number of fatigue cycles was evaluated and compared with the levels of the S-N curve of Quality Category 2 in Code BS7910. This iterative process was carried out until a close match was found. Using the K I values for different a/B ratios from the analytical solution, a polynomial curve K 1 =8.724(a/B) 3 -14.796(a/B) 2 + 12.583(a/B) + 0.4198 (9) was fitted.
The minimum value of a evaluated analytically was 3.18 mm, and the number of fatigue cycles evaluated under the reference stress range of 80 MPa was nearly equal to 2.03×10 6 . In order to achieve the fatigue life closer to 2.03×10 6 cycles, further iterations were carried out using Zencrack software to determine the minimum crack size. As described in     The crack propagation profiles showing the elliptical crack (symmetrical model) growing from its initial size a = 2.839 mm to the outer wall of the pipe are shown in Fig. 12.

FULL-SYSTEM STUDY
he undertaken preliminary crack-propagation analyses on the full model (combining the conductor and casing pipes), demonstrated that the crack in the conductor pipe grew much faster than that in the casing pipe. It was found that the inner-wall crack in the conductor pipe grew to become a through-wall-thickness crack and further, grew nearly 120 o wide in the cross-section of the conductor pipe before the inside-wall elliptical crack in the Casing pipe showed any substantial growth. A three-stage crack-propagation strategy was therefore adopted to evaluate the cumulative fatigue life of the system and of the individual pipes.

Three-stage Analysis Strategy
The three stages of the used strategy are:  Stage 1: Application of the fatigue cyclic load (bending moment) to the combined model with the initial elliptical cracks both in the casing and conductor pipes; then the crack in the conductor pipe was allowed to grow until it broke through the wall thickness.  Stage2: Continuation of Stage-1, until the crack in the conductor stabilised.  Stage 3: Continuation until the inner-wall crack in the casing pipe became a through-wall-thickness crack.

Stage1
Crack-propagation analysis in Stage 1 was performed on the combined conductor/casing system described above with initial semi-circular cracks in both pipes (Fig. 13). The FE model and the location of the cracks are displayed in Fig. 14. It can be noticed that a negligible crack growth took place in the casing (Fig. 15b). The results, in terms of a-n and K-a relationships, for both the inner-surface crack-front node and the through-the-thickness crack-front node, at the end of Stage 1 (when the conductor pipes crack was about to break through the thickness) are presented in Figs. [16][17][18]. The respective number of fatigue cycles n was 2.6E+06.

Stage 2
The second stage of the analysis required an update of the crack shape in the conductor so that a through-thickness crack profile was created. The strategy to update a crack from a corner to through-the-thickness crack is described by Maligno et al. [16]. In Stage 2, the FE mesh of the conductor pipe, as obtained at the end of Stage-1 crack-growth analysis, was appropriately adjusted using a spline function for the new through-thickness crack front, as shown in Fig. 19. While the crack in the conductor grew during the simulation, a severely distorted mesh was generated. The consequence of this was that, to further extend the crack growth, some "manual" interventions were required to update the crack profile. In this study, different crack profiles were introduced to understand thoroughly the consequences of flaw extension in the conductor. The used through-the-thickness crack profile was determined analytically [16]. However, in order to investigate the effect of particularly large flaws in the structural integrity of wellhead systems (arising from a combination of variable loading conditions and aggressive environmental effects), two additional hypothetical shapes were investigated. The studied crack profiles are labelled as described in Tab. 7: Crack profiles Method of determination TW 1 analytical TW 2 analytical TW 3 analytical TW 4 hypothetical TW 5 hypothetical Table 7: Crack-profile labels and estimation technique.
Starting from the front of the initial through-the-thickness crack TW1 (Fig. 20a) the crack was extended in five steps up to a condition in which the remaining ligament of the conductor pipe was exposed only to compressive stresses [24,25] so that any crack opening was prevented. Throughout these analyses, the crack growth in the inner wall of the casing pipe was found to be negligible. For this reason and for sake of clarity only the conductor pipe is shown in the figures.
Step 1 During this step (TW1), the crack grew up to reaching a configuration, where its front extended circumferentially for nearly 120° (Fig. 20b). At this crack extension, further propagation was prevented from further development due to severe distortion of the FE mesh (formation of inside-out elements) near the crack front. The initial and final crack sizes and crack extensions are summarised in Tab. 8:  Step 2

Conductor Casing
Maintaining the same position and shape of the crack front as provided at the end of Step 1, the FE mesh near it was smoothed and the simulation analysis restarted allowing the crack to grow to position TW2 as depicted in Fig. 20c.
Step 3 The FE mesh at the crack front was modified as in Step 2, keeping the crack shape as calculated at the end of Step 2 to get further crack growth to position TW3 (Fig. 20d).

Steps 4 and 5
Although no crack propagation was detected at the end of Step 3, due to highly compressive stresses, two further positions, TW4 and TW5 (Figs. 20e and f, respectively), were considered to estimate the capability of the conductor to bear loads in presence of extended flaws.

Numerical Results: Stage 2
Similarly to Stage 1, throughout Stage 2 propagation of the crack on the inner wall of the casing pipe was found to be negligible. Fig. 21 shows the variation of stress intensity factor K with growth of the through-the-thickness crack along the circumference of the conductor pipe. Apparently, K increased as the crack grew but once the crack front partially crossed the compressive zone at the pipe's cross-section the stress intensity started dropping. This phenomenon was further studied and it is explained in Analytical Study: Stage 2.  It is understandable that when the crack front approached some 90° on the circumference the crack front started changing its shape and elongated. The external node was likely to enter the compressive zone [24,25], and the crack growth slowed down, whereas the internal node remained in the tensile-stress zone and continued to propagate (Fig. 22). The number of cycles was monitored for the whole set of crack configurations; the respective results are summarised in Tab. 9.

Crack Shape
Cycles Corner Crack 2.600E+06 (to reach external surface) TW 1 1.910E+05 (to reach configuration TW 2) TW 2 1.450E+03 (to reach configuration TW 3) TW 3 No growth TW 4 No growth TW 5 No growth Total number of cycles 2.7925E+06 It is interesting to observe that at crack configuration TW 3 the crack propagation can be considered stabilised and no subsequent crack propagation was detected. It is paramount to notice that, while the crack advanced, an inversion of the trend for the stress intensity factor occurred. The consequence of this particular behaviour can be attributed to the position of the crack in relation to the neutral axis, as also described in [24,25]. In fact, the crack, at its initial stage, was entirely in tension (Fig. 22a), but when it advanced, the neutral axis changed its initial position; as soon as the crack position was between 85° (external surface) and 90° (internal surface), the crack started to be exposed to compressive stresses (Fig. 22b). In particular, from Fig. 22b it is possible to deduce that for the crack positions close to 110° the K value underwent an abrupt reduction. Therefore, the flaw in proximity of the external surface was totally under compressive stress and no crack propagation could be detected (according to the Linear Elastic Fracture Mechanics criteria). Analytical Study: Stage 2 In order to further investigate inversion of the trend of the stress-intensity-factor, an analytical approach was carried out.
The overall procedure for the analytical study was based on the following steps: 1) Evaluation of an area of the cracked conductor at different crack extensions; 2) Evaluation of the cracked conductor's moment of inertia; 3) Determination of the position of the neutral axis (NA) for the cracked conductor; 4) Estimation of the moment of inertia of the combined system (cracked conductor/casing pipes); 5) Determination of the position of the NA of the combined system; 6) Assessment of the updated distance from the crack; 7) Stress assessment in the conductor at the crack level; 8) Evaluation of stress intensity factors for different crack extensions. The Roark's Formulas [26] were used to calculate the second moment of inertia of the cracked conductor. The area and distances from the centroid to extremities can be calculated using the following formulae for a sector of the hollow circle ( Fig. 23): All the terms present in the previous expressions are explained in Fig. 23. Figure 23: Sector of hollow circle [26].
The moment of inertia about the central axis X for different values of angle  is given by the following expression: The analytically determined stress intensity factors (related to the bending stress of 1 MPa applied on the external surface of the conductor) were plotted versus the crack position . The respective curve in Fig. 24 shows a very close agreement between the values of the inversion angles, calculated with FE analyses (nearly 85 o ) and analytical analysis (nearly 82 o ).

Stage 3
The crack-propagation analysis at Stage-3 was carried out when the crack in the conductor pipe was prevented to grow further because its crack front reached the compression zone. The FE mesh of the model at the end of Stage 2 (after crack in the conductor pipe reached TW5) was modified as illustrated in Fig. 25, and the crack-propagation analysis started until the crack propagating in the casing pipe through its thickness became a through-the-thickness crack. In this case, the residual ligament in the conductor was deemed to remain of constant size throughout the simulation.  In Fig. 26 the profile of the inside-wall crack growing through the thickness in the casing pipe (while no further crack growth happens in the Conductor pipe) is presented.

Stage 3: Numerical results
The number of cycles necessary for the thickness node to reach the external surface of the casing and the effect of crack depth on the stress intensity factor are presented respectively in Figs. 27 and 28.  In Fig. 28, it is interesting to notice that the magnitude of K values is now comparable to the calculated values of K for the previously investigated configurations. This behaviour clearly demonstrates that the casing bears most of the bending load.

EFFECT OF LOADING
n order to achieve a full understanding of the mechanical behaviour of the Subsea Wellhead System in presence of flaws and to develop the most effective damage-tolerant design, five representative scenarios were investigated under variable loading conditions. In particular, the maximum stresses in the range from 80 MPa to 200 MPa were applied. These further analyses gave the possibility to understand the effect of different bending moments on damage tolerance capability of five full-system configurations. The examined five scenarios are presented in Tab. 10.

Bending moment and fatigue life cycles assessment
The fatigue life, related to the five scenarios, was calculated for stress of 80 MPa evaluated on the outside surface of the uncracked conductor pipe. A similar exercise was also carried out for stress of 200 MPa. The fatigue cycles for each of the scenarios and their combinations are shown in Tab. 11 and plotted in Fig. 29. As expected, the maximum fatigue life is achieved for Scenario 2. In this case, the conductor pipe remained intact and a relatively reduced load was borne by the casing pipe. A more realistic fatigue life for the system was obtained when considering a combination of scenarios 1, 3 and 5. In Scenario 1, the inner wall crack in the conductor pipe grew into a through-the-thickness crack first and then, as I shown in Scenario 3, this crack propagated circumferentially in the conductor until it stabilised thanks to the compressive stress state established in the remaining ligament of the pipe due to the shifting neutral axis. At that stage, the inner-wall crack in the casing pipe started growing until it became a through crack, as shown in Scenario 5.

Scenario
Scenario description Scenario presentation   Once the casing-pipe crack became a through crack, the entire pressure barrier provided by the casing pipe disappeared.
In the graph (Fig. 29), each line represents the fatigue life of the pipe-in-pipe system for a particular damaging condition for different cyclic-loading combinations. It can be clearly seen that the majority of the fatigue life is borne by the conductor pipe (Scenario 1), whereas the fatigue life represented by Scenario 2 is rather unlikely. All the other combinations are between Scenarios 1 and 2. Of paramount importance is the magnitude of the bending stress, which may lead to reaching the critical stress intensity factor. Based on fracture mechanics, the fatigue life of a component is reached when the value of the stress intensity factor exceeds the material's critical fracture strength K c causing the component failure. Stress intensity factors K on the inner surface as the through crack propagates around the conductor (Scenario 3), are plotted as a function of loading cycles in Fig 30. The lowest curve corresponds to the load level applied in the analysis, and shows that the K value does not exceed K c (which is 7190 MPa mm 1/2 ). The K-N response was studied for the levels of bending moment multiplied by factors of 2.5, 3, 4 and 5 to generate additional curves in Fig. 30. The scale factor 2.5 gives a peak K close to Kc. Higher scaling values give K values above Kc. Only for comparison purposes, the analyses exceeding Kc value were not prevented to complete the full crack growth simulation, albeit the critical stress intensity value was already reached.

CONCLUSION
n analytical and numerical study based on the LEFM theory was carried out for the girth weld (although the weld was not explicitly modelled) of the subsea wellhead system. Analytical studies were performed either to establish necessary parameters (e.g. initial crack dimensions) or to validate the finite-element results. The combined study allowed a thorough assessment of the overall behaviour of the wellhead system under cyclic loading conditions for different flaw extensions and various stress ranges. It was demonstrated that most of the load was borne by the conductor pipe even for relatively extended flaws while the crack in the casing pipe underwent a virtually negligible crack growth in the fully coupled system with a kinematic constraint. Furthermore, this study allowed the evaluation of an adequate modelling strategy (e.g. kinematic coupling), which will help pipeline engineers to assess accurately structures with flaws and, in particular, compound systems. The effect of welding residual stress [27,28] is also important for subsea wellhead systems, especially for low-toughness steels, where a high tensile residual stress can provide a significant part of the crack driving force. Therefore, although this study proposed an efficient approach to the structural integrity assessment of subsea wellhead systems, the future work will contemplate the direct modelling of girth welds and the effect of residual stresses, induced during the welding process, on crack initiation and propagation.