Modelling offshore pipelaying dynamics

The below model is a part of Ocean Intella software — open-source software for modelling offshore pipelaying.

UPDATED — April 9, 2023: PD controller enabled after t = 100 sec; removed restoring moment in pitch

Modern demand for operating oil and gas fields has boosted the number of subsea pipeline installation projects in deepwater. The S-lay pipelay method being fast and economical is dominating markets for pipeline installations (Fig. 1). In deepwater, conventional catenary equation for calculating pipeline tensions is not sufficient and pipelay dynamics must be assessed by other numerical methods for mitigating the risks during the installation procedure.

Fig. 1. The pipe configuration can be described by the radii of the overbend ro and the sagbend rs. These are represented by the inscribed circles. Note that rs > ro. It is desired that β → 90, [1]

In this article, I am heavily driven from and trying to recreate the results of pipelay dynamics obtained in [1, 2] for robotic pipe.

In [1] a model for the pipe is developed as part of a system joining a pipe and a vessel, that is suited for controller tasks while keeping the geometric configuration and the force balance of the pipeline. Instead of discretisation of catenary equations or finite element models which are disadvantageous due to their complexity, the standard robot model for a robot manipulator found in e.g. Spong and Vidyasagar [1989] or Sciavicco and Siciliano [2001] is utilised to model the pipeline. The vessel is included in the pipe model as the last link of the structure. A linked structure with many joints is termed hyper-redundant by Chirikjian and Burdick [1994]. Using a standard robot model formulation is advantageous since tools, developed for robot manipulators, for controller synthesis and stability analysis now can be applied directly.

Mathematical model

The pipeline is modelled as a series of connected links as illustrated in Fig 2. It moves in six degrees of freedom (DOF), and hence each pipe element must support longitudinal stretching, lateral bending and longitudinal rotation. Thus, each element has two rotational and a translational joint. In this paper and in [1] only planar motion of the pipeline in a vertical plane is considered. Hence the rotation is ignored, and the longitudinal bending is assumed to be small compared to the lateral bending, so it is also ignored. The model is based on the robot equation with minimal coordinates.

Fig. 2. The three first elements of the slender structure connected by rotational joints, [1]

The dynamics of the system is described by the second order ordinary differential equation (1):

Equation (1)

where

Vector of control inputs consists of the thruster forces and wave forces. Applied top tension is 14k ton-force for water depth of 700m to make the illustration vivid. For more details for the mathematical model and constants used in this article please consult [1,2].

Model implementation

Python programming language was used for the model implementation.

Python is an interpreted high-level general-purpose programming language. Its design philosophy emphasises code readability with its use of significant indentation. Its language constructs as well as its object-oriented approach aim to help programmers write clear, logical code for small and large-scale projects. Python is dynamically typed, and garbage collected. It supports multiple programming paradigms, including structured (particularly, procedural), object-oriented and functional programming. It is often described as a “batteries included” language due to its comprehensive standard library.

The modelling was conducted using SciPy package function odeint. The model consisted of 4 nodes and depicted in fig.3. Duration of simulation was 600 sec with dt=0.5 sec.

Fig.3 S-lay profile for pipelay dynamics modelling

For static solution, the robotic model dynamics (1) is reduced to

where the configuration of the pipe is shaped only by the bending stiffness, gravity and buoyancy (control forces from the stinger is not accounted in the model).

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The static solution angles obtained were as follows:

After solving statics, the dynamical solution was obtained by applying the main function for the pipe ‘manipulator’ as presented in fig. 4.

Fig.4 Main function for pipelay dynamics simulation.

Modelling results

The following diagrams depict coordinates alternations and horizontal tension during pipelay dynamics simulation.

Fig. 5. The state vector q, which are the joint angles.

Fig. 6. Dynamic horizontal tension

Conclusion

The overall results are mainly along the lines of those obtained in [1,2], considering slight differences in model settings. The dynamics of the pipeline is harsh in terms of variation of the angle vector q accounting specific constants used during modelling to make PD controller application more pronounced.

The motivation for this work was to obtain fully fledged pipelay dynamics model to apply in feasibility studies for offshore projects within the efforts at Gigala, [3]. The produced model was validated on [1].

To keep up with the project please visit https://gigala.io/

[1] Modelling and Control of Offshore Marine Pipeline during Pipelay Gullik A. Jensen et al, 2008

[2] Offshore Pipelaying Dynamics. Gullik Anthon Jensen. Doctoral theses at NTNU, 2010

[3] https://gigala.io/

More information on what the pipelay operation is can be found in [4]

[4] https://gigatskhondia.medium.com/pipelay-vessel-design-optimisation-using-genetic-algorithms-506aa04212f1

Code for the model can be found in [5]

[5] https://github.com/gigatskhondia/ocean_intella