Refined analytical models for pipe-lay on elasto-plastic seabed
Introduction
Pipe–soil interaction is of great importance during pipeline installation, as it influences the pipeline shape and internal force distribution, as well as the pipeline embedment and stability during subsequent operations. Analytical solutions are complicated by the need to consider non-linear, plastic response of the seabed, and to allow for the effects of cyclic remoulding of the soil due to the dynamic pipe motions in the touchdown zone [1], [2]. The earliest solutions for pipe-lay assumed the seabed as rigid, which greatly simplified the pipe–soil interaction problem [3], [4]. Such solutions exaggerate the maximum pipe–soil contact force and the curvature at the point of contact. Later, solutions based on linear elastic seabed response were developed [5], [6], [7]. These solutions, in particular the closed form solution and analysis framework of Lenci and Callegari [5], were relatively simple and easy to implement, and represented a significant advance. From these solutions it is possible to obtain reasonably accurate estimates of the pipe shape and distribution of pipe–soil contact force through the touchdown zone (TDZ). Preliminary estimates of pipe embedment could be made by adopting values of secant stiffness that reflected the non-linear response of seabed sediments, although the solutions could not capture the pipeline profile in the seabed beyond the TDZ.
As the exploitation of oil and gas has moved from shallow to deep water, where the seabed typically comprises soft, fine-grained sediments, the importance of capturing the plastic response of the seabed has increased. Palmer [8] suggested an analytical pipe–soil interaction model for pipelines on rigid-plastic seabeds. Although certain simplifications were adopted in the model, including uniformly distributed soil resistance and limited consideration of the overall pipeline shape, it represented an important step towards improved analytical modelling of pipe–soil interaction in the TDZ. Wang et al. [9] and Yuan et al. [10] combined the models of Lenci and Callegari [5] and Palmer [8] to establish an analytical solution for the complete pipeline for a rigid-plastic seabed where the pipe-soil resistance was assumed to increase proportionally with pipeline embedment. This model explored the effects of cyclic remoulding of soil in the TDZ, as suggested by Westgate et al. [1], by considering softened soil resistance profiles. The main limitation was that, in order to simplify the solution process, the rebound compliance of the seabed was neglected. This resulted in a discontinuity in the soil resistance, as detailed later. The solution presented here has improved the analytical model of Yuan et al. [10] by considering the rebound stiffness of the seabed, thus preserving continuity of soil resistance. It has also considered a different assumption for the axial force in the pipe segments close to the seabed, as discussed in the following section.
Section snippets
Analytical solutions
The analytical boundary layer solution of Lenci and Callegari [5], on which the present work is based, adopted an assumption of constant axial tension in the segments close to the seabed, with the tension fixed at the value (T) at the transition to the catenary section (point P1 in Fig. 1). The assumption is reasonable, since the angle of the pipeline to the seabed is relatively small there, but it leads eventually to a tension in the near-horizontal pipeline along the seabed that is
Solution
No explicit solution can be obtained for Eq. (36). In the present study, we have adopted a numerical method, similar to that of Yuan et al. [10]. For a constant axial force of H, the solution is shown schematically in Fig. 5. The value of H estimated from Eq. (35) is taken as the initial value. Meanwhile, a two dimensional (X2, X3) space is established with X2lower < X2 < X2upper, X3lower < X3 < X3upper, where the subscripts “lower” and “upper” represent the lower and upper bound respectively. Then, the
Soil properties
In moderate to deep water, the shear strength profile may be approximate as increasing proportionally with depth with a mudline intercept of zero. The strength gradient will typically lie in the range 1–2 kPa/m for normally or lightly overconsolidated fine-grained sediments. Where a surface crust exists, which is common [15], [16], much higher strength gradients pertain in the zone of interest for pipeline penetration, as illustrated in Fig. 6. Cyclic pipeline motions during laying will remould
Validation and comparison
In order to illustrate the above model with some examples, a typical pipeline is selected with elastic modulus E = 210 GPa, outer and inner diameters of the pipeline d1 = 0.6 m and d2 = 0.55 m. The density of steel pipe is ρs = 7.85 × 103 kg/m3, inclination slope at P1 is φ0 = 80°, the density and depth of sea water are ρw = 1.03 × 103 kg/m3 and Y0 = 1000 m, and the intact shear strength gradient is ksu = 12 kPa/m. Two sets of comparison examples with St = 1 and 8 are made to validate the present models and quantify the
Influence of soil sensitivity
The influence of the seabed stiffness is explored for an initial strength gradient of ksu = 12 kPa/m, and sensitivities ranging from 1 to 8, so that the operative strength gradient ranges from ksu = 12 kPa/m down to ksu = 1.5 kPa/m. The rebound stiffness is maintained at kr = 100kp = 375ksu. The results are shown in Fig. 10. The maximum embedment increases by a factor of 5.5, from 0.12 m to 0.66 m, as the operative strength gradient is reduced from 12 to 1.5 kPa/m. Correspondingly, the maximum contact force R
Influence of rebound stiffness
The effect of the rebound stiffness is explored in Fig. 11, for six different rebound stiffnesses of: kr = 10kp, kr = 20kp, kr = 40kp, kr = 60kp, kr = 80kp and kr = 100kp. The operative soil strength gradient was taken as ksu = 1.5 kPa/m, so kp = 5.64 kPa. The maximum embedment of the pipe exceeds the final embedment in accordance with the rebound stiffness. Also, the greater the rebound stiffness ratio, kr/kp, the greater is both the maximum and the final embedment. For rebound stiffness ratios of 60 or more,
Conclusions
This paper has presented an analytical model for pipe-lay on an elasto-plastic seabed. The analysis improved previous analytical work of Yuan et al. [10] by considering rebound of the pipe beyond the point of maximum contact force (and embedment). In addition, an alternative assumption of constant tension was adopted in solving the differential equations within the boundary layer, touchdown and rebound segments. The constant tension was taken as H, the horizontal component of tension in the
Acknowledgement
The authors would like to acknowledge the support of the Grant No. 2014T70574 from Postdoctoral Science Foundation of China, China, and the Grant No. 51409228 from the National Natural Science Foundation of China.
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