Three-dimensional dynamic behaviour of flexible catenary risers with an internal slug flow

doi:10.1016/j.jfluidstructs.2021.103409

Journal of Fluids and Structures

Volume 107, November 2021, 103409

https://doi.org/10.1016/j.jfluidstructs.2021.103409 Get rights and content

Abstract

This work analyses the three-dimensional dynamic response of flexible catenary risers when subjected to the effects of an internal slug flow. The interaction between the flexible riser and the internal slug flow is a rather complex phenomenon, and this is because the forces due to the acceleration of the internal fluid influence the riser dynamics, which, in turn, influences the slug flow motion in its interior. In order to carry out this analysis, a new computational tool called SLUGFLEX has been developed, which is composed by two computational codes: the code that calculates the structural dynamic response of the flexible riser, and the other one that predicts the development of the internal slug flow. In the structural analysis, the spatial discretization of the riser is performed using spatial Euler beam finite elements, which suffer large displacements, although with small deformations. The tangent stiffness matrix and the internal force vector of the structure are calculated using the corotational approach. The internal slug flow, whose key feature is the intermittence of two different regions: a liquid slug and a large bubble of gas, is mathematically modelled by using a new one-dimensional lagrangian slug tracking method, which has been based on a model developed for straight pipes, and over which the effects of the curvature of the riser and its dynamics were added. The dynamic response of the riser is calculated by using the constant average acceleration method, in combination with the incremental iterative Newton–Raphson scheme for follower loads. The numerical results show good agreement when compared to the experimental ones obtained from the literature.

Introduction

Risers are cylindrical tubular structures of great importance in the oil and gas exploration and production operations in deep and ultra-deep water. The fluid inside the riser, which is transported from the well-head to the oil production platform, is a mixture of oil, gas, and water, and it is exposed to different conditions of pressure and temperature. This multiphase flow generates time-varying forces on the riser, thus inducing excitations that may cause large oscillations in the internal stresses of the riser, affecting its characteristics of natural vibration and reducing its lifetime due to fatigue.

In calculating the dynamic response of the riser, the biggest challenge is to mathematically model the fluid–structure interaction (FSI) that occurs between the multiphase fluid and the flexible riser. This interaction is based on the solution of two problems: the first one is predicting the development of an internal slug flow inside a moving riser, and the second one is modelling the forces generated by the internal slug flow on the riser.

The influence of the internal slug flow on the dynamic behaviour of risers has been studied by several authors, many of which used simplified mathematical models to represent the internal slug flow (see Ma and Srinil (2020) for a brief review), and also, without considering the influence of the riser movement on the development of the internal slug flow. One of the first works on this subject is the experimental and numerical study carried out by Patel and Seyed (1989), in which the authors represented the slug flow as a single-phase fluid with a density which varies sinusoidally along the riser. In addition, the authors conducted experiments using a catenary-shaped plastic tube, to examine the effects of forces arising due to the internal flow of air/water slugs. The obtained experimental results showed that the two-phase flow produces large oscillation amplitudes in the tube, mostly in areas of large curvature.

Valdivia (2008) also conducted an experimental and numerical study of the dynamic behaviour of risers with an internal two-phase liquid–gas flow. Using the proposed model by Patel and Seyed (1989) to represent the slug flow, the author obtained numerical results, which were compared with the experimental ones obtained by the author himself for a catenary-shaped silicone tube 18 metres in length in air.

Pollio and Mossa (2009) based on the slug flow model proposed by Patel and Seyed (1989), made comparisons between two simplified models of slug flow. The first model considers a constant slug wavelength along the riser, and the other model considers the slug flow wavelength as a function of the riser inclination, that is to say, with variable slug wavelength along the riser. The equations of motion of the riser were solved by using numerical integration in the time domain.

A numerical model to simulate the effect of the forces induced by the slug flow in catenary-shaped risers was proposed by Bordalo and Morooka (2018). According to the authors, the forces induced by the slug flow due to the gravity acceleration and due to the curvature of the flow path, are the ones that have a significant influence on the riser dynamics. Also, the authors proposed a simplified model to represent the internal slug flow, for which the liquid slugs were considered as steady regular periodic slugs.

Ortega (2015) developed a computational tool composed by two computational codes: one that predicts the development of slug flow in the flexible riser, and the other to calculate the two-dimensional dynamic response of risers. To predict the development of the slug flow, the author used a one-dimensional slug tracking model, which is based in the numerical model proposed by Nydal and Banerjee (1996).

Medina (2018) presented a computational scheme for coupled internal multiphase flow and structural dynamics of flexible pipes. To simulate the structural dynamic response of the pipe, the author used a three-dimensional model based on lumped mass formulation. The internal two-phase flow was simulated by using a computational code based on the slug tracking model. Both models: structural and flow model, were coupled by using a domain-decomposition method.

Meléndez and Avila (2019) carried out a parametric analysis of the influence of an internal slug flow on the dynamic response of flexible risers. The riser was discretized into several beam elements of equal length, and for the $2$ D formulation of the beam element, the corotational formulation was used. The mathematical model proposed by Patel and Seyed (1989) was used to represent the internal slug flow. The computational tool developed by the authors was used to simulate the dynamic behaviour of a reduced riser model, which was tested experimentally by Valdivia (2008). This work does not consider the FSI. The authors noted that the forces induced by the Coriolis acceleration have a negligible effect on the riser dynamics.

Ma and Srinil (2020) investigated numerically slug liquid–gas flow-induced vibrations of a long inclined curved flexible risers. The slug flow is assumed fully developed and modelled as a sequence of slug units travelling through the flexible riser. In addition, the slug flow is assumed to be stable, which implies that the slug length stays constant as each slug travels along the riser. This study is limited to a two-dimensional slug flow induced vibrations problem.

In the present work, the simulation of the interaction between the flexible riser and the two-phase slug flow, transported inside, is carried out in a simultaneous way by using a new computational tool called SLUGFLEX. To predict the development of the internal fluid inside the flexible riser, a new one-dimensional slug tracking model was developed, which is based on the numerical model proposed by Rodrigues (2009). Unlike the latter, which was developed for two-phase flows inside straight pipes without movement, this new model considers the influence of the flexible riser on the internal fluid: curvature and the effects due to the riser dynamics. In the structural analysis, to describe the kinematics of the three-dimensional beam element, the corotational formulation has been used to beam elements subject to large displacements and large rotations, but with small deformations.

This research paper is structured as follows: Section 2 presents the mathematical modelling and the numerical methods for predicting the movement of the internal flow through the flexible riser. In Section 3, the equation of motion for flexible risers under the influence of an internal slug flow is obtained. The FSI between the riser and the internal fluid is explained in Section 4, and Section 5 shows the numerical results obtained by using the new computational tool SLUGFLEX. The paper ends with conclusions in Section 6. The corotational approach, used for the three-dimensional formulation of beams, is not explained in this work because it is well-documented in Crisfield (1990) and Crisfield (1997).

Section snippets

Mathematical modelling of the internal fluid

The two-phase slug flow is mathematically modelled by using a one-dimensional lagrangian slug tracking model. In the modelling, the unit cell of slug flow is divided into two regions: the liquid slug and a region of stratified flow, as shown in Fig. 1.

In the analysis, the following assumptions are made:

1.
the aeration of the liquid slugs has been neglected;
2.
the gas phase behaves as an ideal gas;
3.
the film thickness of a unit cell remains unchanged;
4.
there are no temperature variations;
5.
there is no mass

Structural analysis

The dynamic response of the flexible riser, the focus of this study, is calculated from its static equilibrium configuration. The equations of motion for a riser element are obtained through Hamilton’s principle, which states that $δ \int_{t_{1}}^{t_{2}} (K - E + W) d t = 0$ where $K$ and $E$ are the kinetic energy and the internal energy of the riser element, respectively, and $W$ is the work done by external forces. Applying the variational operator $δ$ in Eq. (38), and after applying the finite element method (FEM) for the

Fluid-structure interaction

The computational tool SLUGFLEX, developed in this work, is composed of two computational codes:

1.
Code for calculating the three-dimensional dynamic response of the riser. This code also computes the force vectors due to the internal slug flow and the sea current.
2.
Code for predicting the internal two-phase flow development inside the riser. This code considers the influence of the riser dynamics on the two-phase flow.

Both computational codes interact by sending each other information. Fig. 11

Results

To ensure the correct operation of the SLUGFLEX tool, the latter has been validated by comparing the numerical results with the experimental ones obtained by Valdivia (2008), whose experiment consisted of a silicone catenary pipe under the effect of a liquid–gas two-phase flow. After the validation, the numerical results obtained for a catenary riser with real dimensions are shown.

Conclusions

The computational tool SLUGFLEX, developed and implemented in this work, calculates the three-dimensional dynamic response of flexible catenary risers with an internal two-phase slug flow. SLUGFLEX has been developed on the basis of two computational codes: code of structural analysis and the code for predicting the internal two-phase development inside the riser. SLUGFLEX considers fluid–structure interaction; on one hand, the influence of the slug flow on the dynamic behaviour of the riser,

CRediT authorship contribution statement

Joseph Arthur Meléndez Vásquez: Conception and design of study, Acquisition of data, Analysis and/or interpretation of data, Writing – original draft, Writing – review & editing. Juan Pablo Julca Avila: Conception and design of study, Acquisition of data, Writing – review & editing.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

Approval of the version of the manuscript to be published: J. A. Meléndez Vásquez, J. P. Julca Avila.

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Citation Excerpt :
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Three-dimensional dynamic behaviour of flexible catenary risers with an internal slug flow

Abstract

Introduction

Section snippets

Mathematical modelling of the internal fluid

Structural analysis

Fluid-structure interaction

Results

Conclusions

CRediT authorship contribution statement

Declaration of Competing Interest

Acknowledgements

J. Pet. Sci. Eng.

Appl. Ocean Res.

Comput. Methods Appl. Mech. Engrg.

Chem. Eng. Sci.

Eng. Struct.

Adv. Heat Transf.

Subsea Pipelines and Risers

Finite Element Procedures

Nonlinear Finite Element Analysis of Solids and Structures

Planar dynamics of inclined curved flexible riser carrying slug liquid-gas flows

J. Fluids Struct.