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Recoil analysis of deepwater drilling riser after emergency disconnection
Ocean Engineering 189 (2019) 106406
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Recoil analysis of deepwater drilling riser after emergency disconnection Yanbin Wang *, Deli Gao * MOE Key Laboratory of Petroleum Engineering, State Key Laboratory of Petroleum Resources and Prospecting, China University of Petroleum, Beijing, 102249, China
A R T I C L E I N F O
A B S T R A C T
Keywords: Deepwater drilling riser Emergency disconnection Recoil response Mechanical characteristics Influence factors Control measures
In this paper, the mechanical model and governing equations for the recoil response of deepwater drilling riser are established based on the mass-spring-damping system with three degrees of freedom. The direct acting tensioner (DAT) and axial elongation of riser before disconnection the are taken as the boundary condition and initial condition of the mechanical model, respectively. The model can consider the influence of frictional resistance induced by drilling mud discharge and elastic potential energy due to axial elongation on the recoil response. The clearance between the Low Marine Riser Package (LMRP) and the blowout preventer (BOP) after disconnection is figured out through solving the governing equations. Then, the parameter sensitivity analysis is given to present a comprehensive understanding of the recoil response. Analysis results show that the phase difference has a significant influence on the recoil response. The most favorable phase difference is 60� . The possibility of collision between the LMRP and the BOP increases with water depth and amplitude of the DAT. The top tensioner has an optimal heave motion frequency, which is the most advantageous for implementing riser emergency disconnection. Corresponding measures should be taken to prevent spring stiffness of the DAT from decreasing. Besides, riser with large diameter and wall thickness is conducive to a safe disconnection.
1. Introduction In deepwater drilling, emergency disconnection of riser between the LMRP and BOP is necessary under extreme marine environment. Before disconnection, elastic potential energy has been stored in riser due to the axial elongation in normal drilling. So, the riser will move upward like a spring after the disconnection, which is called recoil response. If the recoil control is improper, the drilling platform (Yang, 2013), the riser and the subsea wellhead (Miller et al., 1998; Gao and Wang, 2016) will be threatened. Therefore, the recoil response of emergency disconnection is an important issue for the safety of deepwater drilling. In the stipulation of the American Bureau of Shipping (ABS), the recoil analysis should cover the following issues (ABS, 2017): i) Establish the initial riser overpull tension and recoil settings, so that the LMRP will be able to clear the BOP stack for a specific wave condition. ii) Establish the riser tension and recoil and/or anti-recoil settings, so that the continuous riser movement will not cause the total collapse of the telescopic joint, slacked wire tensioners or com pressed direct-acting tensioners, negative tension along the riser string, or impact back to BOP stack for a specific wave condition.
iii) Establish a set of recoil control parameters for a range of envi ronmental conditions, if required. Therefore, a large number of scholars have done studies on the recoil analysis. Young et al. (1992a and 1992b) have described an analysis method to deal with the recoil problem and how to design an anti-recoil system. Puccio et al. (1998) have illustrated some significant variables affecting the velocity and displacement of the riser string and LMRP during an emergency disconnect sequence. Matthew et al. (2004) have proposed a recoil analysis of riser in a 4,308 ft water depth to determine the appropriate drilling riser configurations. Grytoyr et al. (2009, 2010 and 2011) have presented a methodology for the dynamic analysis of marine drilling riser disconnect and recoil using the finite element (FE) programs. Lang et al. (2009) have described the development of a disconnect and recoil analysis software tool. Some industrial standard (ISO/TR 13624-2, 2009) has also involved the riser recoil response which has provided reference and guidance of the recoil analysis. Bry nestad (2012) has investigated the drive-off and weak link of discon nection riser with pipe in pipe model. Ma et al. (2013) have presented the behavior of a 12,000 ft drilling riser during an emergency discon nection based on a 3-D nonlinear FE model. Grønevik (2013) has carried out recoil analyses using the computer program RIFLEX. Zhang (2014)
* Corresponding authors. E-mail addresses: [email protected] (Y. Wang), [email protected] (D. Gao). https://doi.org/10.1016/j.oceaneng.2019.106406 Received 18 January 2019; Received in revised form 15 July 2019; Accepted 1 September 2019 Available online 25 September 2019 0029-8018/© 2019 The Author(s). Published by Elsevier Ltd. This is an open (http://creativecommons.org/licenses/by-nc-nd/4.0/).
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Y. Wang and D. Gao
Ocean Engineering 189 (2019) 106406
have analyzed the main factors that affect the recoil response, and given simulations of a 1,500m riser with ANSYS software. Wang et al. (2018) have given recoil response simulation of 1,500m riser with ANSY S/AQWA and discussed the influence of mud discharge on the recoil characteristics. Due to the importance of the top tensioner in the riser recoil response, many scholars have studied the mechanical character istics of the tensioner. Dyngvold (2011) has proposed a mathematical model to analyze a wireline riser tensioner system and investigated the response of the system under different load-cases. Zhang et al. (2013a,b) have presented some simulations on the DAT based on the AME-Sim framework. Pestana et al. (2016) have studied the tensioner behavior during riser emergency disconnection using nonlinear parallel spring-damper scalar elements in Orcaflex software. Besides, some studies have been done with consideration of the influence of mud discharge on the recoil response. Li et al. (2015 and 2016) have given a numerical procedure to simulate the mud discharge and seawater refilling and given some discussion through a case study. Meng et al. (2018) have studied the recoil response of a deepwater drilling riser through a mass-spring-damping system with two degrees of freedom. Hock and Young (1993, 2000) have presented the design, installation, and testing procedures of an anti-recoil system in 6,000 ft water depth. He et al. (2016) have analyzed the composition of recoil control system and proposed a control method through taking the DAT as the research objective. At present, some progress has been made in the recoil research. However, published results under different methods and assumptions are very different, and the detailed solution process has not been re ported until now. Therefore, a new recoil analysis model has been established based on the previous study (Meng et al., 2018). The me chanical behavior and key influence factors of the recoil response have been obtained with consideration of the drilling mud discharge and heave motion of the DAT. This research can provide reference for the
Fig. 1. The mechanical model and the coordinate system of the recoil analysis.
Fig. 2. The WFCM analysis model for mud discharge. 2
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Ocean Engineering 189 (2019) 106406
Fig. 3. The working principle of the DAT.
optimal design of the DAT and emergency disconnection control. The remainder of the paper is organized as follows. The coordinate system and mechanical models (include the drilling mud discharge model, the DAT model, and the recoil response model) are introduced in Section 2. The detailed model solution process is presented in Section 3. The case study and parameters sensitivity analysis are shown in Section 4. The conclusions drawn and suggestions on the further study are given in Section 5.
system of X2 O2 Y2 stands for the clearance between the LMRP and the BOP during recoil, where the origin of the coordinate is selected as the joint point of the LMRP and the BOP. The coordinate system of X3 t represents the motion of the piston in DAT. L0 is the maximum clearance between LMRP and BOP. In the following sections, a prime stands for ∂ ðÞ=∂x and an overdot represents ∂ðÞ=∂t where t is time. Unless otherwise specified, all parameters in this paper are in the international system of units.
2. Mechanical models
2.1. Model of mud discharge
In normal operation, deepwater drilling riser will generate axial elongation due to the top tension, gravity and buoyancy. The elongation begins to spring back when the emergency disconnection occurs. Considering the heave movement of the top tensioner, the recoil response belongs to the forced vibration of an elastic body with infinite degrees of freedom under initial displacement. The following assump tions are accepted in order to establish the analysis model in this paper.
During the recoil, the drilling mud is discharged downward, and the riser is refilled by seawater, which makes the riser be subjected to the frictional resistance. Accurate calculation of the frictional resistance is the requirement for the recoil response. In the actual configuration, riser is often equipped with fill-up valves. Although the opening of one valve is usually small compared to the internal diameter of the riser. The number of valves and total volume flowing into the riser can guarantee that the mud is replaced by seawater at the same volumetric rate. This is a basic requirement to prevent the riser from being collapse. So, the Whole Fluid Column Model (WFCM), as shown in Fig. 2, is used to describe the mud discharge process (Grønevik, 2013; Meng et al., 2018). In this paper, the influence of drill pipe in the riser on the frictional resistance is considered, which is a significant different from the pre vious work. According to the Newton’s second law, the dynamic equilibrium equation of the drilling mud can be written as:
i) Since the riser recoil response is far stronger than its lateral characteristics, it is considered that only axial vibration exists during the recoil. ii) The influence of buoyancy modules on the riser stiffness is ignored. iii) The drill pipe and the riser are concentric during the recoil response. Accordingly, the mechanical model and the coordinate system of the recoil analysis are shown in Fig. 1. As shown in Fig. 1, the coordinate system of X1 O1 Y1 represents the riser axial elongation, where the origin of the coordinate is chosen at the connection point of the top tensioner and the riser. The coordinate
ðmw Lw þ mm Lm Þad ¼ Gw þ Gm þ Ftop
Fp
fr1
fr2
fd1
fd2
fp1
fp2
Fend (1) 3
Y. Wang and D. Gao
Ocean Engineering 189 (2019) 106406
where, mw is the mass of seawater in riser per length; mm is the mass of drilling mud in riser per length; Lw is the length of seawater in riser; Lm is the length of drilling mud in riser; ad is the acceleration of the whole fluid column; Gw is the weight of seawater in the riser; Gm is the weight of drilling mud in the riser; Ftop is the force on the top of seawater from the fill-up valves; Fp is the force on the bottom of drilling mud; Fend is the frontal force on the bottom of the drilling mud due to the “water hammer effect” based on Bernoulli’s equation. fr1 is the frictional resistance between the seawater and the riser inner wall; fr2 is the fric tional resistance between the drilling mud and the riser inner wall; fd1 is the frictional resistance between the seawater and the outer wall of the drill pipe; fd2 is the frictional resistance between the drilling mud and the outer wall of the drill pipe; fp1 is the frictional resistance between the seawater and the inner wall of the drill pipe; fp2 is the frictional resis tance between the drilling mud and the inner wall of the drill pipe. Therefore, the total frictional force on riser inner wall Ff can be written as: (2)
Ff ¼ fr1 þ fr2
The detailed derivation process of Eq. (2) is presented in Appendix A. 2.2. Model of the DAT The heave motion and stiffness of the top tensioner must be obtained to analyze the boundary conditions of the recoil response. In this paper, the DAT is taken as the research object, which working principle (Zhao et al., 2010) is shown in Fig. 3. The tension force is provided by the pressure difference in the cyl inder. The bottom of the rod end is connected to the tensioner ring, where the hydraulic pressure is provided by the high-pressure accu mulator. The accumulator can store and convert the energy produced by heave motion of the piston, which can maintain the stability of the tension force. The rear end is connected to the low-pressure NPV. The tension force generated by a single tensioner Ft can be calculated by: Ft ¼
phi V nhi �n Ar Vhi þ Ar xp
Vli
pli V nli �n Ap Ap xp
Fig. 4. The analysis model of spring-mass-damping system with three degrees of freedom.
The recoil analysis model is shown in Fig. 4. The whole riser is dis cretized into three mass blocks with m1 , m2 and m3 . Kt is the spring stiffness of top tensioner. Kr is the axial stiffness of the riser, which derivation process is presented in Appendix C. F1 , F2 and F3 are the external force acting on the three mass blocks. c1 , c2 and c3 are the damping coefficients of the three mass blocks. x0 ðtÞ is the heave motion displacement of the piston. x1 , x2 and x3 are the vibration displacement of the three mass blocks during recoil. During actual recoil response process, the tensioner has damping effect to preventing the riser from rising too fast. However, it is difficult to accurately estimate the damping effect of tensioner. So, the damping effect of tensioner is neglected in this paper. The tensioner is idealized as a spring without damping. The top boundary condition of the recoil response is the ten sion force generated by the tension system. So, the recoil response re sults calculated by this assumption may be some faster than that the damping effect of tensioner is considered. Therefore, the control equation of the three mass blocks can be written as: 8 < m1 x€1 ¼ F1 Kt x1 c1 x_1 þ Kr ðx2 x1 Þ þ c2 ðx_2 x_1 Þ m x€ ¼ F2 Kr ðx2 x1 Þ c2 ðx_2 x_1 Þ þ c3 ðx_3 x_2 Þ þ Kr ðx3 x2 Þ : 2 2 m3 x€3 ¼ F3 Kr ðx3 x2 Þ c3 ðx_3 x_2 Þ
(3)
where, phi is the initial pressure of the rod end; pli is the initial pressure of the rear end; Vhi is the initial volume of the rod end; Vli is the initial volume of the rear end; n is the gas constan, and 1.3 is selected in this paper. Ar is the area of the rod end; Ap is the area of the rear end; xp is the piston displacement relative to the cylinder. The stiffness of a single tensioner can be obtained by taking the de rivative of the tension force with respect to the relative displacement, which is: Kt ¼ N
dFt dxp
(4)
where, N is the total number of the tensioner in the tension system. The detailed derivation process of Eq. (3) and Eq. (4) is shown in Appendix B.
(5) The mass of the three blocks can be calculated by: 8 < m1 ¼ ðms L þ mb Lb Þ=3 þ Mtensioner m ¼ ðms L þ mb Lb Þ=3 : 2 m3 ¼ ðms L þ mb Lb Þ=3 þ MLMRP
2.3. Model of recoil response In this paper, the riser is regarded as a spring-mass-damping system with three degrees of freedom. In addition, the following assumptions are accepted, which are:
(6)
where, ms is the mass of riser per length in seawater; L is the length of riser, which equals to the water depth in this paper. mb is the mass of buoyance module per length in seawater; Lb is the length of buoyance module; Mtensioner is the mass of top tension system, which is the total mass of rod and piston of all tensioners; MLMRP is the mass of the LMRP in seawater. The external force on the three mass blocks can be written as:
i) The buoyancy is uniform distributed along the riser axial direction. ii) The discrete riser segments have the same axial stiffness. iii) The frictional resistance is evenly distributed over the three mass blocks. 4
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Ocean Engineering 189 (2019) 106406
Table 1 The parameters of a deepwater drilling riser. Parameters
Value
L(m)
3000
Dro(mm)
533.4
Dri(mm)
482.6
Vli(m3)
2.25
Dso(mm) Dsi(mm)
Parameters
ε3
3.84e-4
Phi(Pa) Pli(Pa)
11.2e6 1.0e5
γ
0.35
Vhi(m3)
Value
Mtensioner(kg)
2.06e-4
ε2
6
Parameters
9.37e-5
ε1
127 108.62
N
Value
3
38842
MLMRP(kg) Lb(m)
1805.9
Ar(m2)
0.2048
2024.2 0.314
Ap(m2)
7850
ρw (kg/m3)
1025
1400
E(Pa) νm (m2/s)
2.1e11 1e-4
Sp(m)
4.42
νw (m2/s)
0.2463
Value
ρs (kg/m )
ρm (kg/m3)
103255
mb(m) ωt (rad/s)
9.38
Parameters
1.15e-6
* νm and νw are the viscosity of mud and sea water. Generally, the drilling mud is usually considered as a non-Newtonian fluid, and the viscosity coefficient is changed with its velocity. However, a constant viscosity is assumed as a simplification for the problem in this paper.
8 � < F1 ¼ m1 g þ Ff �3 þ fD1 F ¼ m2 g þ Ff �3 þ fD2 : 2 F3 ¼ m3 g þ Ff 3 þ fD3 þ Fend
� �K
(7)
That is, � � Kt þ Kr ζ2 m1 � � Kr � � 0
where, Fend ¼ 12ρm ðAin þ Aa Þv2 ; fDi ði ¼ 1; 2; 3Þ is the damping force on the mass blocks induced by seawater, which can be calculated by (Gobat, 2006; Meng et al., 2018): fDi ¼
π 2
ρw Dro Ct x_i jx_i j;
2 6 adj B ¼ 6 4
� ζ2 m2 Kr
Kt þ Kr
ζ 2 m3
Kr Kr
ζ2 m3 �
�
K 2r
ζ 2 m3 � 2 ζ m 1 Kr
Kr Kr ðKt þ Kr
K 2r
Kr Kt þ Kr
�
ζ2 m1
ζ2 m3 �
2
ðKt þ Kr
where, M ¼ 4 2
c1 þ c2 c2 4 c2 c3 c2 0 c3 2 Kt þ Kr Kr 4 Kt þ Kr Kr 0 Kr
½ F1
F2
3
T
m3
5 is the global mass matrix; C ¼
3 7 7 5
�
Kr Kt þ Kr ζ2 m1 � � ζ2 m1 Kt þ Kr ζ2 m2
1 ðiÞ AN ¼ pffiffiffiffiffiffiffiAðiÞ Mζi
0 c3 5 is the global damping matrix; K ¼ c3 3 0 Kr 5 is the global stiffness matrix. F ¼ Kr
(13)
K 2r
ði ¼ 1; 2; 3Þ
So, the modal matrix can be represented by: i h ð1Þ ð2Þ ð3Þ AN ¼ AN AN AN
(16)
(17)
Before riser disconnection, the three mass blocks have an initial displacement due to the axial elongation. Besides, the initial velocity of the three mass blocks is 0. So, the initial conditions of the recoil response are: � xð0Þ ¼ ½0 ΔL=2 ΔL�T (18) _ xð0Þ ¼ ½0 0 0�T
F3 � is the global force matrix.
This section presents the detailed solving process of Eq. (9). The eigenmatrix of Eq. (9) can be written as: ζ2 M
(12)
where, Мζ ¼ diag½Mζ1 Mζ2 Mζ3 � is the principal mass matrix. Kζ ¼ diag½Kζ1 Kζ2 Kζ3 � is the principal stiffness matrix. Thus, the normal modes of the system can be written as:
3. Model solution
B¼K
Kr
Due to the orthogonality of the mass matrix and stiffness matrix, one obtains: ( T Aζ МAζ ¼ Мζ (15) ATζ KAζ ¼ Kζ
3 m2
Kt þ Kr ζ 2 m 2 Kr
� � 0 � Kr �� ¼ 0 2 ζ m3 �
The three principal modes (Að1Þ , Að2Þ and Að3Þ ) can be figured out after substituting the three natural frequency into the adjoint matrix of the eigenmatrix. Then, the principal mode matrix can be written as: � � (14) Aζ ¼ Að1Þ Að2Þ Að3Þ
(9)
m1
Kr
K 2r
�
where Ct is the tangential drag coefficient, and 0.015 is selected in this paper according to the published literature (Gobat, 2006; Meng et al., 2018). Since the recoil velocity is time-varying, the drag force on riser also changes with time. In this paper, the frictional resistance in the following discrete time is calculated by substituting the velocity in the previous discrete time into Eq. (8). The linear damping model is adop ted, and the damping coefficients on the three mass blocks are consid ered to be the same. So, the damping coefficients of the three mass blocks can be figured out through dividing the damping force by the length of the riser and the velocity of the recoil. The matrix form of Eq. (5) is: Mx€ þ Cx_ þ Kx ¼ F
(11)
The three natural frequency ζ1 , ζ2 and ζ3 can be obtained after solving Eq. (12). The adjoint matrix of the eigenmatrix can be written as:
(8)
ði ¼ 1; 2; 3Þ
� ζ 2 M� ¼ 0
According to the mechanics of vibration, the decoupled differential equation can be obtained through substitute x ¼ AN xN into the initial conditions, which is:
(10)
If the natural frequency is ζ, one obtains: 5
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Ocean Engineering 189 (2019) 106406
ð1Þ
ð2Þ
ð3Þ
x ¼ AN xN1 þ AN xN2 þ AN xN3
(22)
4. Case study and analysis 4.1. Results of the top tensioner Taking an actual deepwater drilling as an example, the calculation data are shown in Table .1. In the case, it is assumed that there is no phase difference between the heave motion and the disconnection moment. Besides, the piston is in harmonic motion around its initial equilibrium position. Then, the variation of the tension force and the stiffness calculated are shown in Fig. 5 and Fig. 6, respectively. As shown in Fig. 5, the tension force varies approximately linearly with the piston displacement. When the piston displacement is negative (the riser moves downward), the tension force and stiffness provided by the tensioner is maximum. Because the tensioner needs to provide more force to “pull” the riser up. The stiffness of the tensioner decreases first and then increases with the piston displacement, as shown in Fig. 6. The difficulty of riser moving downward increases if the piston displacement is negative. Further study shows that the measures which can increase the stiff ness include: reducing the diameter of the piston rod (or increasing the diameter of the piston), increasing the initial pressure of the rear end (or increasing the initial pressure of the rod end), reducing the volume of the rod end. However, the frequency has almost no influence on the tension force and stiffness. This means that if the configuration of the tension system is given, the tension force and stiffness are constant when the piston moves to a certain position, which is not affected by the ve locity and acceleration of the piston passing through this point.
Fig. 5. The tension force vs piston displacement.
4.2. Results of the mud discharge In WFCM, the frictional resistance is needed firstly to calculate the whole column velocity, while the whole column velocity is prerequisite to calculate the frictional resistance. Therefore, in this paper, an initial timestep (usually less than 0.001s) is given firstly. During this timestep, the whole fluid column is discharged only under the force of gravity. The discharge velocity, acceleration and the length of the column can be figured at the end of the initial timestep. The results of the previous timesteps are imputed as the initial calculation conditions of the next timestep. Then the discharge parameters (velocity, acceleration, length of the column, and the frictional resistance) in each timestep can be calculated according to Eq. (A-1) to (A-3). The calculation is completed until the length of the column is 0. According to this analysis procedure, the calculation results of mud discharge are shown in Fig. 7. As shown in Fig. 7, the whole discharge process can be divided into two phases. When the column is beginning to discharge (the first phase), the frictional resistance generated by the drilling mud dominates. Because the column length is maximum, the acceleration is also the maximum as shown in Fig. 7-(b). As the discharge continues, the ve locity and frictional resistance increase, as shown in Fig. 7-(c) and Fig. 7(a), which makes the acceleration decrease as shown in Fig. 7-(b). However, the acceleration is always downward in the first phase, which results in the frictional resistance and velocity increase gradually. When the frictional resistance increases to a certain value, the discharge moves into the second phase. During this phase, the frictional resistance generated by seawater dominants. Due to the difference of seawater and drilling mud, the frictional resistance of the whole column decreases. However, the discharge velocity still increases slowly, as shown in Fig. 7-(c). During the whole process, the length of drilling mud approximately decreases linearly. In this example, the whole discharge time is 157.5s. The discharge time under different water depths is shown in Table 2. Then, curve fitting is needed to get the analytical expression of the
Fig. 6. The stiffness of tensioner vs piston displacement.
x€Ni þ 2δi ζi x_Ni þ ζ2i xNi ¼ QNi sinðωt tÞ
ði ¼ 1; 2; 3Þ
(19)
where, δi is the damping ratios of each modes; ωt is the heave motion frequency of the top tensioner; QNi ¼ ATNi Fi . Then, the response of the system in the normal coordinates can be written as: xNi ¼ BNi sinðωt t
ϕi Þ
ði ¼ 1; 2; 3Þ
(20)
where, ϕi is the phase difference between the heave motion and the riser disconnection. 8 QNi 1 > BNi ¼ 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > ζi > 2 2 > 1 λi þ ð2δi λi Þ2 > > > > < ωt λi ¼ > ζi > > > > > > > > : ϕi ¼ arctan 2δi λi 1 λ2i
(21)
Therefore, the recoil response the three mass blocks can be calcu lated by:
6
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Ocean Engineering 189 (2019) 106406
Fig. 7. Calculation results of mud discharge.
Table 2 The discharge time under different water depths. Water depth(m)
500
1000
1500
2000
2500
3000
Time of discharge(s)
28.0
54.0
79.9
105.8
131.6
157.5
Fig. 9. The clearance between LMRP and BOP.
increase also. Then, the external forces on the three mass blocks can be obtained through substituting Eq. (6), Eq. (8) and the fitted frictional resistance into Eq. (7). 4.3. Results of the clearance between LMRP and BOP
Fig. 8. The calculated frictional resistance and fitting curve on riser.
According to the mechanical model and solving approach proposed in this paper, calculation result of the clearance between the LMRP and the BOP is shown in Fig. 9. As shown in Fig. 9, the riser moves upward under the combined action of elastic potential energy and the frictional resistance after
frictional resistance, which is shown in Fig. 8. The sum of six sine expressions is used to fit the calculated frictional resistance on riser. More sinusoidal expressions will bring higher pre cision certainly, while the time and amount of computation needed 7
Y. Wang and D. Gao
Ocean Engineering 189 (2019) 106406
Fig. 10. The influence of phase difference on the clearance between LMRP and BOP. Table 3 The maximum and minimum clearance between the LMRP and the BOP. Phase difference
0�
30�
60�
90�
120�
150�
180�
210�
240�
270�
300�
330�
Δx3max (m)
7.25
10.69
15.12
12.79
12.10
10.25
7.22
9.81
12.11
12.90
12.12
9.94
4.8
13.8
1.4
26.4
26.3
26.1
24.8
13.3
11.5
9.9
8.2
6.5
Time (s)
Δx3min (m)
Time (s)
0.12 9.7
1.67 7.9
1.67 14.6
10.25 6.9
9.37
7.02
6.6
5.3
7.93 4.9
3.60 3.3
2.34 24.7
3.05 22.8
2.37 21.4
0.67 20.3
disconnection. As the recoil continues, the elastic potential energy is gradually released, the riser is compressed upward to the maximum position and then it starts to move downward. Due to the damping ef fect, the recoil response is in form of attenuation vibration, and will be weakened gradually. The maximum clearance between the LMRP and the BOP is 7.254m, which occurs at 4.8s after disconnection. The min imum clearance is 0.1172m occurs at 9.7s after disconnection, which means that there is possibility of LMRP colliding with BOP. The final clearance between them equals to the axial elongation of the riser in static. 4.4. Sensitive analysis The riser recoil response is a complex vibration process which can be affected by many factors. So, parameter sensitivity analysis is helpful for a comprehensive understanding of the mechanical phenomenon and can provide countermeasures for optimization of the anti-recoil system to ensure the safety of riser emergency disconnection. 4.4.1. Phase difference As mentioned above, the heave motion of the DAT is the boundary condition of the recoil response. In this paper, the heave motion is si nusoidal, which is shown in Fig. 5. However, the moment of
Fig. 11. The influence of water depth on the clearance between LMRP and BOP. 8
Y. Wang and D. Gao
Ocean Engineering 189 (2019) 106406
Fig. 13. The influence of frequency of the heave motion on the clearance be tween LMRP and BOP.
Fig. 12. The influence of amplitude of the heave motion on the clearance be tween the LMRP and BOP.
disconnection and heave motion does not take place at the same time. So, there is a phase difference between them. The clearance between the LMRP and the BOP is shown is Fig. 10 if the phase difference changes from 0� to 330� . As shown in Fig. 10, the phase difference has a significant influence on the recoil response. For deepwater drilling, it is indispensable to avoid the collision between the released LMRP and the BOP to ensure the safety of the subsea wellhead. Therefore, the minimum clearance between them is the most important issue. Table .3 presents the maximum and minimum clearance and the occurrence time under different phase difference. As shown in Table .3, when the phase difference is 60� , the riser has the maximum upward displacement which is 15.12m occurring at 1.4s after disconnection. When the phase difference is 180� , it has the min imum upward displacement which is 7.22m occurring at 24.8s after disconnection. When the phase difference is 90� , it has the maximum downward displacement which is 10.25m occurring at 6.9s after disconnection. When the phase difference is 60� , it has the minimum upward displacement which is 1.67m occurring at 14.6s after discon nection. Therefore, from the perspective of avoiding collision between the LMRP and the BOP, the most advantageous phase difference is 60� .
Fig. 14. The influence of the DAT stiffness on the clearance between the LMRP and the BOP.
4.4.2. Water depth Emergency disconnection operations can be implemented in different ocean environment. The recoil response of riser under different water depth when the phase difference is 0� is shown in Fig. 11. As shown in Fig. 11, with the increase of water depth, the upward maximum and minimum clearance between the LMRP and BOP gradu ally increase. Specifically, when the water depth is 500m, the vibration displacement of the LMRP is always above the BOP. The minimum clearance between them gradually decreases as the water depth in creases. When the water depth increases to 3000m, the LMRP will vibrate below the BOP. This means that the deeper the water depth is, the greater the possibility of collision between the LMRP and BOP is.
“peak value” is very sensitive to the amplitude of the heave motion. In this case, when the amplitude increases from 2.21m to 13.26m, the maximum clearance increases from 6.22m (4.8s after disconnection) to 18.69m (4.8s after disconnection), and the minimum clearance de creases from 1.24m (9.7s after disconnection) to 10.36m (9.7s after disconnection). Therefore, effective countermeasures should be taken to control the amplitude of heave motion to ensure the safety of the emergency disconnection. 4.4.4. Frequency of the heave motion The frequency is also a key parameter affecting the heave motion. When the frequency changes from π =20 to π =2, the clearance between the LMRP and BOP is shown in Fig. 13. As shown in Fig. 13, with the increase of frequency, the maximum clearance decreases first and then increases while the minimum clear ance increases and then decrease, which means there is an optimal frequency making the LMRP be most unlikely collision with the BOP. The optimal frequency is codetermined by the riser configuration and the parameters of the top tension system.
4.4.3. Amplitude of the heave motion After emergency disconnection, the riser top is still connected to the piston rod of the tensioner, as shown in Fig. 3. It is forced to vibrate under the heave motion of the piston rod. When the amplitude of the heave motion changes, the clearance between them is shown in Fig. 12. As can be seen from Fig. 12, with increase of the amplitude, both the maximum and the minimum clearance between them increase. But the moment when the “peak value” appears is not changed. However, the 9
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Fig. 15. The influence of riser geometric size on the clearance.
4.4.5. Stiffness of the DAT In the above analysis, the heave motion of the DAT is simplified as a spring and the maximum spring stiffness is chosen to analyze the recoil response. The clearance between the LMRP and the BOP is shown in Fig. 14 under different situations of spring stiffness. As shown in Fig. 14, when the DAT stiffness increases from K to 4K, the variation of the clearance changes slightly. However, when the stiffness decreases from K to K =2, the maximum clearance increases form 7.25m–16.84m and the minimum clearance decrease from 0.12m to 7.51m. So, strict control measures should be taken to ensure the stability of stiffness.
(2) The phase difference has a significant influence on the recoil response. The most favorable phase difference is 60� . The possi bility of collision between the LMRP and the BOP increases with water depth and amplitude of the DAT. The top tensioner has an optimal heave motion frequency, which is the most advantageous for implementing riser emergency disconnection. Corresponding measures should be taken to prevent spring stiffness of the DAT from decreasing. Besides, riser with large diameter and wall thickness is conducive to a safe disconnection. (3) This paper can provide reference for the study on riser recoil response. However, further study should be implemented to obtain a comprehensive understanding of this mechanical phe nomenon, which include: the fitting curve should be studied further to improve the accuracy of the frictional resistance during mud discharge, innovative mechanical model should be estab lished with consideration of the mass-spring-damping system with infinite degrees of freedom, the mechanical state of riser after disconnection should be studied through 3-D dynamic model with consideration of the wave-current effect to ensure no mechanical failure will occur. In addition, the safety of the moon pool is also needed to be assessed during the recoil response.
4.4.6. Riser geometric size The riser geometric size will not only change the mass of the three blocks but also affect the frictional resistance, which has a comprehen sive influence on the recoil response. Fig. 15 shows the response curve under the different conditions of riser outer diameter and wall thickness. As shown in Fig. 15, riser geometric size has a slight influence on the clearance between the LMRP and the BOP. Specifically, under the con dition of the same outer diameter, the greater the wall thickness is, the smaller the maximum clearance is. When the wall thickness is constant, the larger the outer diameter is, the smaller the maximum clearance is.
Acknowledgements
5. Conclusions and further study
The authors gratefully acknowledge the financial support from the Natural Science Foundation of China (Grant numbers: 51804321, 51821092, U1762214 and 51734010). This research is also supported by Science Foundation of China University of Petroleum, Beijing (No.00000),and also supported by other projects (Grant numbers: 2017ZX05005-005-007, 2017ZX05009-003, 2016YFC0303303, and 2018YFB0605502).
(1) In this paper, the mechanical model of mass-spring-damping system with three degrees of freedom has been established to analyze the riser recoil response. The model can consider the influence of the frictional resistance during mud discharge and the elastic potential energy stored in the riser. The clearance between the LMRP and BOP after disconnection has been figured out.
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Appendix D. Supplementary data Supplementary data to this article can be found online at https://doi.org/10.1016/j.oceaneng.2019.106406. Appendix A The appendix shows the solving process of the frictional resistance on the riser inner wall during mud discharge. The frictional resistance caused by drilling mud and seawater can be expressed as:
8 fr2 ¼ Δpr2 Aa < fr1 ¼ Δpr1 Aa ; fd1 ¼ Δpd1 Aa ; fd2 ¼ Δpd2 Aa : fp1 ¼ Δpp1 Ain ; fp2 ¼ Δpp2 Ain
(A-1)
where, Δpr1 is the pressure drop between seawater and riser inner wall; Δpr2 is the pressure drop between drilling mud and riser inner wall; Δpd1 is the pressure drop between seawater and drill pipe outer wall; Δpd2 is the pressure drop between drilling mud and drill pipe outer wall; Δpp1 is the pressure drop between seawater and drill pipe inner wall; Δpp2 is the pressure drop between drilling mud and drill pipe inner wall; Aa ¼ πðR2ri R2so Þ is the annular cross-sectional area between riser and drill pipe; Rri is the inner radius of riser; Rso is the outer radius of riser; Ain ¼ πR2si is the inner area of drill pipe; Rsi is the inner radius of drill pipe. According to the Darcy-Weisbach equation, the pressure drops can be calculated by: 8 > ρw v2 fwc1 Lw ρ v2 fmc1 Lm > > ; Δpr2 ¼ m > Δpr1 ¼ > 2ðRri Rso Þ 2ðRri Rso Þ > > > > < ρw v2 fwc2 Lw ρ v2 fmc2 Lm (A-2) Δpd1 ¼ ; Δpd2 ¼ m > 2ðRri Rso Þ 2ðRri Rso Þ > > > > > > ρ v2 fwc3 Lw ρ v2 fmc3 Lm > > : Δpp1 ¼ w ; Δpp2 ¼ w 4Rsi 4Rsi where, ρw is the density of seawater; ρm is the density of drilling mud; v is the discharge velocity of the whole column. fwc1 , fwc2 , fwc3 , fmc1 , fmc2 and fmc3 are dimensionless coefficients, which can be described by the Haaland formula as shown in Eq. (A-3). 8 � � � � > > 1 6:9 � ε1 �1:11 1 6:9 � ε1 �1:11 > > ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi p p ¼ 1:8 log ¼ 1:8 log þ þ ; > > Rew Rem 3:7 3:7 fwm1 > fwc1 > > > � � � � < 1 6:9 � ε2 �1:11 1 6:9 � ε2 �1:11 pffiffiffiffiffiffiffi ¼ 1:8 log pffiffiffiffiffiffiffiffi ¼ 1:8 log þ þ ; (A-3) > Rew Rem 3:7 3:7 f fwm2 > > wc2 > � � � > � � � > > 1 6:9 � ε3 �1:11 > pffi1ffiffiffiffiffiffi ¼ 1:8 log 6:9 þ ε3 1:11 ; > pffiffiffiffiffiffiffiffi ¼ 1:8 log þ > : fwc3 Rew Rem 3:7 3:7 fwm3 where, Rew and Rem are Reynolds number of seawater and drilling mud respectively; ε1 , ε2 and ε3 are the relative roughness of riser inner wall, riser outer wall and drill pipe inner wall. Considering that the riser outer wall is surrounded by seawater, the pressure difference on the top and bottom surfaces of the whole column can be calculated by: Fp ¼ ðρm
Ftop as:
(A-4)
ρw ÞgLm
In this paper, we assume that the mud is discharged directly through the LMRP. So, the frontal force (Rinaldi and Paȉ;doussis, 2010) can be written
1 Fend ¼ ρm ðAa þ Ain Þv2 2
(A-5)
Appendix B The appendix presents the detailed derivation process of Eq. (3) and Eq. (4).The analysis model of tension force and stiffness is shown in Fig. B-1. The following assumptions are accepted to analyze the tension force and stiffness of the DAT, which are: i) ii) iii) iv)
It is assumed that the gas in the tension system conforms to the rule of adiabatic state. The tension force is far greater than the weight of piston and piston rod. So, the weight of them is neglected. The liquid in the hydraulic cylinder is incompressible and the leakage of it is ignored. The damping effect of the piston is also neglected.
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Fig. B-1. The analysis model of tension force and stiffness.
When the piston has displacement xp relative to the cylinder, the volume of the rod end Vh is: (B-1)
Vh ¼ Vhi þ Ar xp The volume of the rear end Vl is: Vl ¼ Vli (
(B-2)
Ap x p
According to the first assumption, one obtains: �n phi V nhi ¼ ph Vhi þ Ar xp � n pli V nli ¼ pl Vli Ap xp
(B-3)
So, the tension force on the piston rod can be written as: Ft ¼ p h A r
(B-4)
p l Ap
Eq. (3) can be obtained through substituting Eq. (B-3) into Eq. (B-4). The stiffness of a single tensioner can be obtained through taking derivative of the tension force with respect to the relative displacement in Eq. (B4). Assume that all the tensioners are connected to the tensioner ring, the model, which is shown in Fig.B-2can be regarded as parallel springs with number of N.
Fig.B-2. The stiffness analysis model for parallel springs.2
Supposing that the stiffness of each tensioner spring is k1 k2 ; :::; kN and the elongation under the external force is x0 , then the external force can be written as: N X
F¼
(B-5)
ki x0 i¼1
Therefore, the total stiffness of the tension system is: N P
ki x0 X N Kt ¼ i¼1 ¼ ki x0 i¼1
(B-6)
Since the structure and parameters of each tensioner are the same, the spring stiffness of each tensioner is the same accordingly. So, the total stiffness of the tension system can be obtained as shown in Eq. (4) Appendix C This appendix shows the determination procedure of the riser axial stiffness. The axial stiffness of the riser can be calculated according to the work of the external force equaling to the elastic strain energy, which is: Z
L 0
N 2 ðxÞ 1 dx ¼ Ke ðΔLÞ2 2EA 2
(C-1)
where, ΔL is the total elongation of riser; NðxÞ is the axial tension force along riser; E is the elastic modulus of riser; A is the cross-sectional area of riser; 12
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Ke is the equivalent stiffness of the riser. The axial tension force can be calculated by: NðxÞ ¼ F0
(C-2)
ωx
Where, F0 is the top tension, which is 1.35 times the total weight of riser in sea water. ω ¼ ms g is the buoyancy. So, the axial elongation can be written as: Z L NðxÞ ΔL ¼ dx EA 0
ωb is the per length weight of riser in sea water; ωb
Then, the axial stiffness can be obtained after substituting Eq. (C-2) and Eq. (C-3) into Eq. (C-1), which is: � � � EA F 20 L þ ω2 L 3 F0 ω Ke ¼ ðF0 ωL=2Þ2
(C-3)
(C-4)
If γ represents the over-tension coefficient of riser, Eq. (C-4) can be written as: Ke ¼
γ2 þ γ þ 1=3 EA γ2 þ γ þ 1=4 L
(C-5)
In this paper, the mechanical model of stiffness is obtained through dividing the whole riser into 2 parallel segments, which is shown in Fig. C-1.
Fig. C-1. The analysis model for riser axial stiffness.3
Since the axial force is evenly distributed along riser, one obtains: N1 þ N3 N1 þ N2 N2 þ N3 ¼ þ 2Ke 2Kr 2Kr
(C-6)
where, N1 , N2 and N3 are the external force on the three mass blocks. According to Eq. (C-6), one obtains, Kr ¼
N1 þ 2N2 þ N3 Ke N1 þ N3
(C-7)
Due to F0 ¼ ð1 þ γÞωL, N1 ¼ ð1 þ γÞωL, N2 ¼ ð1=2 þ γÞωL and N3 ¼ γωL. Thus, the axial stiffness of each segment can be written as: Kr ¼
ð1 þ γÞ þ 2ð1=2 þ γÞ þ γ Ke ¼ 2Ke ð1 þ γÞ þ γ
(C-8)
References
Grønevik, A., 2013. Simulation of Drilling Riser Disconnection-Recoil Analysis. Master’s thesis of Norwegian University of Science and Technology. He, X.X., Zhang, F.F., Tian, X.J., Liu, X.Q., 2016. Modeling and simulation for recoil control system of deepwater drilling riser. China Pet. Mach. 44 (3), 63–67. Hock, C.J., Young, R.D., 1993. A deepwater riser emergency disconnect antirecoil system. J. Pet. Technol. 45 (08), 744–751. ISO/TR 13624-2:2009 (E), 2009. Petroleum and Natural Gas Industries-Drilling and Production Equipment- Part 2: Deepwater Drilling Riser Methodologies, Operations, and Integrity Technical Report. ISO. Lang, D.W., Real, J., Lane, M., 2009. Recent developments in drilling riser disconnect and recoil analysis for deepwater applications. In: International Conference on Ocean, Offshore and Arctic Engineering, Honolulu, Hawaii, USA, pp. OMAE2009–79427. Li, C.W., Fan, H.H., Wang, Z.M., Shen, W.G., Feng, X., Deng, S., 2015. Numerical simulation of mud discharge after drilling riser emergency disconnection. In: International Ocean and Polar Engineering Conference, Kona, Big Island. Hawaii, USA, ISOPE-I-15-736. Li, C.W., Fan, H.H., Wang, Z.M., Ji, R.Y., Ren, W.Y., Feng, X., 2016. Two methods for simulating mud discharge after emergency disconnection of a drilling riser. J. Nat. Gas Sci. Eng. 28, 142–152. Ma, P., Pyke, J., Vankadari, A., Whooley, A., 2013. Ensuring safe riser emergency disconnect in harsh environments: experience and design requirements. In:
American Bureau of Shipping, 2017. Guidance Notes on Drilling Riser Analysis. American Bureau of Shipping, Houston, USA. Brynestad, I.B., 2012. Disconnection of Workover Risers on Very Deep Water. Master’s thesis of Norwegian University of Science and Technology. Dyngvold, S.H.Ø., 2011. Development of Simulation Model for Virtual Testing and Design of a Riser Tensioner System. Master’s thesis of University of Agder. Gao, D.L., Wang, Y.B., 2016. Progress in tubular mechanics and design control techniques for deep-water drilling. Pet. Sci. Bull. 01, 61–80. Gobat, J.I., Grosenbaugh, M.A., 2006. Time-domain numerical simulation of ocean cable structures. Ocean Eng. 33, 1373–1400. Grytoyr, G., Rustad, A.M., Sodahl, N., Bunaes, P.C., 2009. Methodology for disconnection analysis of CWO risers in random seas. In: International Conference on Ocean, Offshore and Arctic Engineering, Honolulu, Hawaii, USA, pp. OMAE2009–79130. Grytoyr, G., 2010. Improving operating window for disconnect operations of CWO risers. In: International Conference on Ocean, Offshore and Arctic Engineering. Shanghai, China, OMAE2010-O2020. Grytoyr, G., Sharma, P., Vishnubotla, S., 2011. Marine drilling riser disconnect and recoil analysis. In: The AADE National Technical Conference and Exhibition. Houston, Texas, 2011, AADE-11-NTCE-80.
13
Y. Wang and D. Gao
Ocean Engineering 189 (2019) 106406 Wang, T., Wang, K., 2018. Analysis of resilient response of emergently disentangled drilling risers. China Offshore Oil Gas 30 (2), 120–125. Yang, H.L., 2013. Study on Disconnection Technique for Storm-Safe Drilling Riser. Master’s thesis of China University of Petroleum (East China). Young, R.D., Hock, C.J., Karlsen, G., Miller, J.E., 1992. Analysis and design of anti-recoil system for emergency disconnect of a deepwater riser: case study. In: Offshore Technology Conference. OTC, Houston, Texas, p. 6891. Young, R.D., Hock, C.J., Karlsen, G., Miller, J.E., 1992. Comparison of analysis and fullscale testing of anti-recoil system for emergency disconnect of deepwater riser. In: Offshore Technology Conference. OTC, Houston, Texas, p. 6892. Zhang, L., Chang, Y.J., Liu, X.Q., Liu, K., Chen, L.M., 2013. Recoil analysis for deepwater drilling riser after emergency disconnection with blowout preventer. Pet. Drill. Tech. 41 (3), 25–30. Zhang, L., 2014. Recoil Analysis for Deepwater Drilling Riser System. Master’s thesis of China University of Petroleum (East China). Zhang, L., Chang, Y.J., Liu, X.Q., Liu, K., Yang, H.L., 2013. AMESim-based modeling and simulation of platform riser tensioner. China Pet. Mach. 41 (12), 38–42. Zhao, J.T., Xue, Y., Pan, Y., Cheng, F., 2010. Study on riser tensioning system of floating drilling equipment. Shanghai Shipbuilding 4, 1–5.
International Offshore and Polar Engineering Anchorage. Alaska, USA, ISOPE-I-13299. Matthew, J.S., Momen, W., James, N.B., 2004. Riser recoil analysis at a harsh environment, deepwater site. In: International Conference on Offshore Mechanics and Arctic Engineering, Vancouver, British Columbia, Canada. OMAE2004-51628. Meng, S., Che, C.D., Zhang, W.J., 2018. Discharging flow effect on the recoil response of a deep-water drilling riser after an emergency disconnect. Ocean Eng. 151, 199–205. Miller, J.E., Stahl, M.J., Matice, C.J., 1998. Riser collapse pressures resulting from release of deepwater mud columns. In: Offshore Technology Conference. Houston. Texas, OTC-8853. Pestana, R.G., Roveri, F.E., Franciss, R., Ellwanger, G.B., 2016. Marine riser emergency disconnection analysis using scalar elements for tensioner modelling. Appl. Ocean Res. 59, 83–92. Puccio, W.F., Shaffer, V., Nuttall, R.V., 1998. Riser recoil during unscheduled lower marine riser package disconnects. In: IADC/SPE Drilling Conference. Dallas, Texas, IADC/SPE39296. Rinaldi, S., Paȉdoussis, M.P., 2010. Dynamics of a cantilevered pipe discharging fluid, fitted with a stabilizing end-piece. J. Fluids Struct. 26, 517–525. Stahl, M.J., Hock, C.J., 2000. Design of a riser recoil control system and validation through full-scale testing. In: SPE Annual Technical Conference and Exhibition. Dallas, Texas, SPE-62959.
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Contents lists available at ScienceDirect
Ocean Engineering journal homepage: www.elsevier.com/locate/oceaneng
Recoil analysis of deepwater drilling riser after emergency disconnection Yanbin Wang *, Deli Gao * MOE Key Laboratory of Petroleum Engineering, State Key Laboratory of Petroleum Resources and Prospecting, China University of Petroleum, Beijing, 102249, China
A R T I C L E I N F O
A B S T R A C T
Keywords: Deepwater drilling riser Emergency disconnection Recoil response Mechanical characteristics Influence factors Control measures
In this paper, the mechanical model and governing equations for the recoil response of deepwater drilling riser are established based on the mass-spring-damping system with three degrees of freedom. The direct acting tensioner (DAT) and axial elongation of riser before disconnection the are taken as the boundary condition and initial condition of the mechanical model, respectively. The model can consider the influence of frictional resistance induced by drilling mud discharge and elastic potential energy due to axial elongation on the recoil response. The clearance between the Low Marine Riser Package (LMRP) and the blowout preventer (BOP) after disconnection is figured out through solving the governing equations. Then, the parameter sensitivity analysis is given to present a comprehensive understanding of the recoil response. Analysis results show that the phase difference has a significant influence on the recoil response. The most favorable phase difference is 60� . The possibility of collision between the LMRP and the BOP increases with water depth and amplitude of the DAT. The top tensioner has an optimal heave motion frequency, which is the most advantageous for implementing riser emergency disconnection. Corresponding measures should be taken to prevent spring stiffness of the DAT from decreasing. Besides, riser with large diameter and wall thickness is conducive to a safe disconnection.
1. Introduction In deepwater drilling, emergency disconnection of riser between the LMRP and BOP is necessary under extreme marine environment. Before disconnection, elastic potential energy has been stored in riser due to the axial elongation in normal drilling. So, the riser will move upward like a spring after the disconnection, which is called recoil response. If the recoil control is improper, the drilling platform (Yang, 2013), the riser and the subsea wellhead (Miller et al., 1998; Gao and Wang, 2016) will be threatened. Therefore, the recoil response of emergency disconnection is an important issue for the safety of deepwater drilling. In the stipulation of the American Bureau of Shipping (ABS), the recoil analysis should cover the following issues (ABS, 2017): i) Establish the initial riser overpull tension and recoil settings, so that the LMRP will be able to clear the BOP stack for a specific wave condition. ii) Establish the riser tension and recoil and/or anti-recoil settings, so that the continuous riser movement will not cause the total collapse of the telescopic joint, slacked wire tensioners or com pressed direct-acting tensioners, negative tension along the riser string, or impact back to BOP stack for a specific wave condition.
iii) Establish a set of recoil control parameters for a range of envi ronmental conditions, if required. Therefore, a large number of scholars have done studies on the recoil analysis. Young et al. (1992a and 1992b) have described an analysis method to deal with the recoil problem and how to design an anti-recoil system. Puccio et al. (1998) have illustrated some significant variables affecting the velocity and displacement of the riser string and LMRP during an emergency disconnect sequence. Matthew et al. (2004) have proposed a recoil analysis of riser in a 4,308 ft water depth to determine the appropriate drilling riser configurations. Grytoyr et al. (2009, 2010 and 2011) have presented a methodology for the dynamic analysis of marine drilling riser disconnect and recoil using the finite element (FE) programs. Lang et al. (2009) have described the development of a disconnect and recoil analysis software tool. Some industrial standard (ISO/TR 13624-2, 2009) has also involved the riser recoil response which has provided reference and guidance of the recoil analysis. Bry nestad (2012) has investigated the drive-off and weak link of discon nection riser with pipe in pipe model. Ma et al. (2013) have presented the behavior of a 12,000 ft drilling riser during an emergency discon nection based on a 3-D nonlinear FE model. Grønevik (2013) has carried out recoil analyses using the computer program RIFLEX. Zhang (2014)
* Corresponding authors. E-mail addresses: [email protected] (Y. Wang), [email protected] (D. Gao). https://doi.org/10.1016/j.oceaneng.2019.106406 Received 18 January 2019; Received in revised form 15 July 2019; Accepted 1 September 2019 Available online 25 September 2019 0029-8018/© 2019 The Author(s). Published by Elsevier Ltd. This is an open (http://creativecommons.org/licenses/by-nc-nd/4.0/).
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have analyzed the main factors that affect the recoil response, and given simulations of a 1,500m riser with ANSYS software. Wang et al. (2018) have given recoil response simulation of 1,500m riser with ANSY S/AQWA and discussed the influence of mud discharge on the recoil characteristics. Due to the importance of the top tensioner in the riser recoil response, many scholars have studied the mechanical character istics of the tensioner. Dyngvold (2011) has proposed a mathematical model to analyze a wireline riser tensioner system and investigated the response of the system under different load-cases. Zhang et al. (2013a,b) have presented some simulations on the DAT based on the AME-Sim framework. Pestana et al. (2016) have studied the tensioner behavior during riser emergency disconnection using nonlinear parallel spring-damper scalar elements in Orcaflex software. Besides, some studies have been done with consideration of the influence of mud discharge on the recoil response. Li et al. (2015 and 2016) have given a numerical procedure to simulate the mud discharge and seawater refilling and given some discussion through a case study. Meng et al. (2018) have studied the recoil response of a deepwater drilling riser through a mass-spring-damping system with two degrees of freedom. Hock and Young (1993, 2000) have presented the design, installation, and testing procedures of an anti-recoil system in 6,000 ft water depth. He et al. (2016) have analyzed the composition of recoil control system and proposed a control method through taking the DAT as the research objective. At present, some progress has been made in the recoil research. However, published results under different methods and assumptions are very different, and the detailed solution process has not been re ported until now. Therefore, a new recoil analysis model has been established based on the previous study (Meng et al., 2018). The me chanical behavior and key influence factors of the recoil response have been obtained with consideration of the drilling mud discharge and heave motion of the DAT. This research can provide reference for the
Fig. 1. The mechanical model and the coordinate system of the recoil analysis.
Fig. 2. The WFCM analysis model for mud discharge. 2
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Fig. 3. The working principle of the DAT.
optimal design of the DAT and emergency disconnection control. The remainder of the paper is organized as follows. The coordinate system and mechanical models (include the drilling mud discharge model, the DAT model, and the recoil response model) are introduced in Section 2. The detailed model solution process is presented in Section 3. The case study and parameters sensitivity analysis are shown in Section 4. The conclusions drawn and suggestions on the further study are given in Section 5.
system of X2 O2 Y2 stands for the clearance between the LMRP and the BOP during recoil, where the origin of the coordinate is selected as the joint point of the LMRP and the BOP. The coordinate system of X3 t represents the motion of the piston in DAT. L0 is the maximum clearance between LMRP and BOP. In the following sections, a prime stands for ∂ ðÞ=∂x and an overdot represents ∂ðÞ=∂t where t is time. Unless otherwise specified, all parameters in this paper are in the international system of units.
2. Mechanical models
2.1. Model of mud discharge
In normal operation, deepwater drilling riser will generate axial elongation due to the top tension, gravity and buoyancy. The elongation begins to spring back when the emergency disconnection occurs. Considering the heave movement of the top tensioner, the recoil response belongs to the forced vibration of an elastic body with infinite degrees of freedom under initial displacement. The following assump tions are accepted in order to establish the analysis model in this paper.
During the recoil, the drilling mud is discharged downward, and the riser is refilled by seawater, which makes the riser be subjected to the frictional resistance. Accurate calculation of the frictional resistance is the requirement for the recoil response. In the actual configuration, riser is often equipped with fill-up valves. Although the opening of one valve is usually small compared to the internal diameter of the riser. The number of valves and total volume flowing into the riser can guarantee that the mud is replaced by seawater at the same volumetric rate. This is a basic requirement to prevent the riser from being collapse. So, the Whole Fluid Column Model (WFCM), as shown in Fig. 2, is used to describe the mud discharge process (Grønevik, 2013; Meng et al., 2018). In this paper, the influence of drill pipe in the riser on the frictional resistance is considered, which is a significant different from the pre vious work. According to the Newton’s second law, the dynamic equilibrium equation of the drilling mud can be written as:
i) Since the riser recoil response is far stronger than its lateral characteristics, it is considered that only axial vibration exists during the recoil. ii) The influence of buoyancy modules on the riser stiffness is ignored. iii) The drill pipe and the riser are concentric during the recoil response. Accordingly, the mechanical model and the coordinate system of the recoil analysis are shown in Fig. 1. As shown in Fig. 1, the coordinate system of X1 O1 Y1 represents the riser axial elongation, where the origin of the coordinate is chosen at the connection point of the top tensioner and the riser. The coordinate
ðmw Lw þ mm Lm Þad ¼ Gw þ Gm þ Ftop
Fp
fr1
fr2
fd1
fd2
fp1
fp2
Fend (1) 3
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Ocean Engineering 189 (2019) 106406
where, mw is the mass of seawater in riser per length; mm is the mass of drilling mud in riser per length; Lw is the length of seawater in riser; Lm is the length of drilling mud in riser; ad is the acceleration of the whole fluid column; Gw is the weight of seawater in the riser; Gm is the weight of drilling mud in the riser; Ftop is the force on the top of seawater from the fill-up valves; Fp is the force on the bottom of drilling mud; Fend is the frontal force on the bottom of the drilling mud due to the “water hammer effect” based on Bernoulli’s equation. fr1 is the frictional resistance between the seawater and the riser inner wall; fr2 is the fric tional resistance between the drilling mud and the riser inner wall; fd1 is the frictional resistance between the seawater and the outer wall of the drill pipe; fd2 is the frictional resistance between the drilling mud and the outer wall of the drill pipe; fp1 is the frictional resistance between the seawater and the inner wall of the drill pipe; fp2 is the frictional resis tance between the drilling mud and the inner wall of the drill pipe. Therefore, the total frictional force on riser inner wall Ff can be written as: (2)
Ff ¼ fr1 þ fr2
The detailed derivation process of Eq. (2) is presented in Appendix A. 2.2. Model of the DAT The heave motion and stiffness of the top tensioner must be obtained to analyze the boundary conditions of the recoil response. In this paper, the DAT is taken as the research object, which working principle (Zhao et al., 2010) is shown in Fig. 3. The tension force is provided by the pressure difference in the cyl inder. The bottom of the rod end is connected to the tensioner ring, where the hydraulic pressure is provided by the high-pressure accu mulator. The accumulator can store and convert the energy produced by heave motion of the piston, which can maintain the stability of the tension force. The rear end is connected to the low-pressure NPV. The tension force generated by a single tensioner Ft can be calculated by: Ft ¼
phi V nhi �n Ar Vhi þ Ar xp
Vli
pli V nli �n Ap Ap xp
Fig. 4. The analysis model of spring-mass-damping system with three degrees of freedom.
The recoil analysis model is shown in Fig. 4. The whole riser is dis cretized into three mass blocks with m1 , m2 and m3 . Kt is the spring stiffness of top tensioner. Kr is the axial stiffness of the riser, which derivation process is presented in Appendix C. F1 , F2 and F3 are the external force acting on the three mass blocks. c1 , c2 and c3 are the damping coefficients of the three mass blocks. x0 ðtÞ is the heave motion displacement of the piston. x1 , x2 and x3 are the vibration displacement of the three mass blocks during recoil. During actual recoil response process, the tensioner has damping effect to preventing the riser from rising too fast. However, it is difficult to accurately estimate the damping effect of tensioner. So, the damping effect of tensioner is neglected in this paper. The tensioner is idealized as a spring without damping. The top boundary condition of the recoil response is the ten sion force generated by the tension system. So, the recoil response re sults calculated by this assumption may be some faster than that the damping effect of tensioner is considered. Therefore, the control equation of the three mass blocks can be written as: 8 < m1 x€1 ¼ F1 Kt x1 c1 x_1 þ Kr ðx2 x1 Þ þ c2 ðx_2 x_1 Þ m x€ ¼ F2 Kr ðx2 x1 Þ c2 ðx_2 x_1 Þ þ c3 ðx_3 x_2 Þ þ Kr ðx3 x2 Þ : 2 2 m3 x€3 ¼ F3 Kr ðx3 x2 Þ c3 ðx_3 x_2 Þ
(3)
where, phi is the initial pressure of the rod end; pli is the initial pressure of the rear end; Vhi is the initial volume of the rod end; Vli is the initial volume of the rear end; n is the gas constan, and 1.3 is selected in this paper. Ar is the area of the rod end; Ap is the area of the rear end; xp is the piston displacement relative to the cylinder. The stiffness of a single tensioner can be obtained by taking the de rivative of the tension force with respect to the relative displacement, which is: Kt ¼ N
dFt dxp
(4)
where, N is the total number of the tensioner in the tension system. The detailed derivation process of Eq. (3) and Eq. (4) is shown in Appendix B.
(5) The mass of the three blocks can be calculated by: 8 < m1 ¼ ðms L þ mb Lb Þ=3 þ Mtensioner m ¼ ðms L þ mb Lb Þ=3 : 2 m3 ¼ ðms L þ mb Lb Þ=3 þ MLMRP
2.3. Model of recoil response In this paper, the riser is regarded as a spring-mass-damping system with three degrees of freedom. In addition, the following assumptions are accepted, which are:
(6)
where, ms is the mass of riser per length in seawater; L is the length of riser, which equals to the water depth in this paper. mb is the mass of buoyance module per length in seawater; Lb is the length of buoyance module; Mtensioner is the mass of top tension system, which is the total mass of rod and piston of all tensioners; MLMRP is the mass of the LMRP in seawater. The external force on the three mass blocks can be written as:
i) The buoyancy is uniform distributed along the riser axial direction. ii) The discrete riser segments have the same axial stiffness. iii) The frictional resistance is evenly distributed over the three mass blocks. 4
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Table 1 The parameters of a deepwater drilling riser. Parameters
Value
L(m)
3000
Dro(mm)
533.4
Dri(mm)
482.6
Vli(m3)
2.25
Dso(mm) Dsi(mm)
Parameters
ε3
3.84e-4
Phi(Pa) Pli(Pa)
11.2e6 1.0e5
γ
0.35
Vhi(m3)
Value
Mtensioner(kg)
2.06e-4
ε2
6
Parameters
9.37e-5
ε1
127 108.62
N
Value
3
38842
MLMRP(kg) Lb(m)
1805.9
Ar(m2)
0.2048
2024.2 0.314
Ap(m2)
7850
ρw (kg/m3)
1025
1400
E(Pa) νm (m2/s)
2.1e11 1e-4
Sp(m)
4.42
νw (m2/s)
0.2463
Value
ρs (kg/m )
ρm (kg/m3)
103255
mb(m) ωt (rad/s)
9.38
Parameters
1.15e-6
* νm and νw are the viscosity of mud and sea water. Generally, the drilling mud is usually considered as a non-Newtonian fluid, and the viscosity coefficient is changed with its velocity. However, a constant viscosity is assumed as a simplification for the problem in this paper.
8 � < F1 ¼ m1 g þ Ff �3 þ fD1 F ¼ m2 g þ Ff �3 þ fD2 : 2 F3 ¼ m3 g þ Ff 3 þ fD3 þ Fend
� �K
(7)
That is, � � Kt þ Kr ζ2 m1 � � Kr � � 0
where, Fend ¼ 12ρm ðAin þ Aa Þv2 ; fDi ði ¼ 1; 2; 3Þ is the damping force on the mass blocks induced by seawater, which can be calculated by (Gobat, 2006; Meng et al., 2018): fDi ¼
π 2
ρw Dro Ct x_i jx_i j;
2 6 adj B ¼ 6 4
� ζ2 m2 Kr
Kt þ Kr
ζ 2 m3
Kr Kr
ζ2 m3 �
�
K 2r
ζ 2 m3 � 2 ζ m 1 Kr
Kr Kr ðKt þ Kr
K 2r
Kr Kt þ Kr
�
ζ2 m1
ζ2 m3 �
2
ðKt þ Kr
where, M ¼ 4 2
c1 þ c2 c2 4 c2 c3 c2 0 c3 2 Kt þ Kr Kr 4 Kt þ Kr Kr 0 Kr
½ F1
F2
3
T
m3
5 is the global mass matrix; C ¼
3 7 7 5
�
Kr Kt þ Kr ζ2 m1 � � ζ2 m1 Kt þ Kr ζ2 m2
1 ðiÞ AN ¼ pffiffiffiffiffiffiffiAðiÞ Mζi
0 c3 5 is the global damping matrix; K ¼ c3 3 0 Kr 5 is the global stiffness matrix. F ¼ Kr
(13)
K 2r
ði ¼ 1; 2; 3Þ
So, the modal matrix can be represented by: i h ð1Þ ð2Þ ð3Þ AN ¼ AN AN AN
(16)
(17)
Before riser disconnection, the three mass blocks have an initial displacement due to the axial elongation. Besides, the initial velocity of the three mass blocks is 0. So, the initial conditions of the recoil response are: � xð0Þ ¼ ½0 ΔL=2 ΔL�T (18) _ xð0Þ ¼ ½0 0 0�T
F3 � is the global force matrix.
This section presents the detailed solving process of Eq. (9). The eigenmatrix of Eq. (9) can be written as: ζ2 M
(12)
where, Мζ ¼ diag½Mζ1 Mζ2 Mζ3 � is the principal mass matrix. Kζ ¼ diag½Kζ1 Kζ2 Kζ3 � is the principal stiffness matrix. Thus, the normal modes of the system can be written as:
3. Model solution
B¼K
Kr
Due to the orthogonality of the mass matrix and stiffness matrix, one obtains: ( T Aζ МAζ ¼ Мζ (15) ATζ KAζ ¼ Kζ
3 m2
Kt þ Kr ζ 2 m 2 Kr
� � 0 � Kr �� ¼ 0 2 ζ m3 �
The three principal modes (Að1Þ , Að2Þ and Að3Þ ) can be figured out after substituting the three natural frequency into the adjoint matrix of the eigenmatrix. Then, the principal mode matrix can be written as: � � (14) Aζ ¼ Að1Þ Að2Þ Að3Þ
(9)
m1
Kr
K 2r
�
where Ct is the tangential drag coefficient, and 0.015 is selected in this paper according to the published literature (Gobat, 2006; Meng et al., 2018). Since the recoil velocity is time-varying, the drag force on riser also changes with time. In this paper, the frictional resistance in the following discrete time is calculated by substituting the velocity in the previous discrete time into Eq. (8). The linear damping model is adop ted, and the damping coefficients on the three mass blocks are consid ered to be the same. So, the damping coefficients of the three mass blocks can be figured out through dividing the damping force by the length of the riser and the velocity of the recoil. The matrix form of Eq. (5) is: Mx€ þ Cx_ þ Kx ¼ F
(11)
The three natural frequency ζ1 , ζ2 and ζ3 can be obtained after solving Eq. (12). The adjoint matrix of the eigenmatrix can be written as:
(8)
ði ¼ 1; 2; 3Þ
� ζ 2 M� ¼ 0
According to the mechanics of vibration, the decoupled differential equation can be obtained through substitute x ¼ AN xN into the initial conditions, which is:
(10)
If the natural frequency is ζ, one obtains: 5
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Ocean Engineering 189 (2019) 106406
ð1Þ
ð2Þ
ð3Þ
x ¼ AN xN1 þ AN xN2 þ AN xN3
(22)
4. Case study and analysis 4.1. Results of the top tensioner Taking an actual deepwater drilling as an example, the calculation data are shown in Table .1. In the case, it is assumed that there is no phase difference between the heave motion and the disconnection moment. Besides, the piston is in harmonic motion around its initial equilibrium position. Then, the variation of the tension force and the stiffness calculated are shown in Fig. 5 and Fig. 6, respectively. As shown in Fig. 5, the tension force varies approximately linearly with the piston displacement. When the piston displacement is negative (the riser moves downward), the tension force and stiffness provided by the tensioner is maximum. Because the tensioner needs to provide more force to “pull” the riser up. The stiffness of the tensioner decreases first and then increases with the piston displacement, as shown in Fig. 6. The difficulty of riser moving downward increases if the piston displacement is negative. Further study shows that the measures which can increase the stiff ness include: reducing the diameter of the piston rod (or increasing the diameter of the piston), increasing the initial pressure of the rear end (or increasing the initial pressure of the rod end), reducing the volume of the rod end. However, the frequency has almost no influence on the tension force and stiffness. This means that if the configuration of the tension system is given, the tension force and stiffness are constant when the piston moves to a certain position, which is not affected by the ve locity and acceleration of the piston passing through this point.
Fig. 5. The tension force vs piston displacement.
4.2. Results of the mud discharge In WFCM, the frictional resistance is needed firstly to calculate the whole column velocity, while the whole column velocity is prerequisite to calculate the frictional resistance. Therefore, in this paper, an initial timestep (usually less than 0.001s) is given firstly. During this timestep, the whole fluid column is discharged only under the force of gravity. The discharge velocity, acceleration and the length of the column can be figured at the end of the initial timestep. The results of the previous timesteps are imputed as the initial calculation conditions of the next timestep. Then the discharge parameters (velocity, acceleration, length of the column, and the frictional resistance) in each timestep can be calculated according to Eq. (A-1) to (A-3). The calculation is completed until the length of the column is 0. According to this analysis procedure, the calculation results of mud discharge are shown in Fig. 7. As shown in Fig. 7, the whole discharge process can be divided into two phases. When the column is beginning to discharge (the first phase), the frictional resistance generated by the drilling mud dominates. Because the column length is maximum, the acceleration is also the maximum as shown in Fig. 7-(b). As the discharge continues, the ve locity and frictional resistance increase, as shown in Fig. 7-(c) and Fig. 7(a), which makes the acceleration decrease as shown in Fig. 7-(b). However, the acceleration is always downward in the first phase, which results in the frictional resistance and velocity increase gradually. When the frictional resistance increases to a certain value, the discharge moves into the second phase. During this phase, the frictional resistance generated by seawater dominants. Due to the difference of seawater and drilling mud, the frictional resistance of the whole column decreases. However, the discharge velocity still increases slowly, as shown in Fig. 7-(c). During the whole process, the length of drilling mud approximately decreases linearly. In this example, the whole discharge time is 157.5s. The discharge time under different water depths is shown in Table 2. Then, curve fitting is needed to get the analytical expression of the
Fig. 6. The stiffness of tensioner vs piston displacement.
x€Ni þ 2δi ζi x_Ni þ ζ2i xNi ¼ QNi sinðωt tÞ
ði ¼ 1; 2; 3Þ
(19)
where, δi is the damping ratios of each modes; ωt is the heave motion frequency of the top tensioner; QNi ¼ ATNi Fi . Then, the response of the system in the normal coordinates can be written as: xNi ¼ BNi sinðωt t
ϕi Þ
ði ¼ 1; 2; 3Þ
(20)
where, ϕi is the phase difference between the heave motion and the riser disconnection. 8 QNi 1 > BNi ¼ 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > ζi > 2 2 > 1 λi þ ð2δi λi Þ2 > > > > < ωt λi ¼ > ζi > > > > > > > > : ϕi ¼ arctan 2δi λi 1 λ2i
(21)
Therefore, the recoil response the three mass blocks can be calcu lated by:
6
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Ocean Engineering 189 (2019) 106406
Fig. 7. Calculation results of mud discharge.
Table 2 The discharge time under different water depths. Water depth(m)
500
1000
1500
2000
2500
3000
Time of discharge(s)
28.0
54.0
79.9
105.8
131.6
157.5
Fig. 9. The clearance between LMRP and BOP.
increase also. Then, the external forces on the three mass blocks can be obtained through substituting Eq. (6), Eq. (8) and the fitted frictional resistance into Eq. (7). 4.3. Results of the clearance between LMRP and BOP
Fig. 8. The calculated frictional resistance and fitting curve on riser.
According to the mechanical model and solving approach proposed in this paper, calculation result of the clearance between the LMRP and the BOP is shown in Fig. 9. As shown in Fig. 9, the riser moves upward under the combined action of elastic potential energy and the frictional resistance after
frictional resistance, which is shown in Fig. 8. The sum of six sine expressions is used to fit the calculated frictional resistance on riser. More sinusoidal expressions will bring higher pre cision certainly, while the time and amount of computation needed 7
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Ocean Engineering 189 (2019) 106406
Fig. 10. The influence of phase difference on the clearance between LMRP and BOP. Table 3 The maximum and minimum clearance between the LMRP and the BOP. Phase difference
0�
30�
60�
90�
120�
150�
180�
210�
240�
270�
300�
330�
Δx3max (m)
7.25
10.69
15.12
12.79
12.10
10.25
7.22
9.81
12.11
12.90
12.12
9.94
4.8
13.8
1.4
26.4
26.3
26.1
24.8
13.3
11.5
9.9
8.2
6.5
Time (s)
Δx3min (m)
Time (s)
0.12 9.7
1.67 7.9
1.67 14.6
10.25 6.9
9.37
7.02
6.6
5.3
7.93 4.9
3.60 3.3
2.34 24.7
3.05 22.8
2.37 21.4
0.67 20.3
disconnection. As the recoil continues, the elastic potential energy is gradually released, the riser is compressed upward to the maximum position and then it starts to move downward. Due to the damping ef fect, the recoil response is in form of attenuation vibration, and will be weakened gradually. The maximum clearance between the LMRP and the BOP is 7.254m, which occurs at 4.8s after disconnection. The min imum clearance is 0.1172m occurs at 9.7s after disconnection, which means that there is possibility of LMRP colliding with BOP. The final clearance between them equals to the axial elongation of the riser in static. 4.4. Sensitive analysis The riser recoil response is a complex vibration process which can be affected by many factors. So, parameter sensitivity analysis is helpful for a comprehensive understanding of the mechanical phenomenon and can provide countermeasures for optimization of the anti-recoil system to ensure the safety of riser emergency disconnection. 4.4.1. Phase difference As mentioned above, the heave motion of the DAT is the boundary condition of the recoil response. In this paper, the heave motion is si nusoidal, which is shown in Fig. 5. However, the moment of
Fig. 11. The influence of water depth on the clearance between LMRP and BOP. 8
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Ocean Engineering 189 (2019) 106406
Fig. 13. The influence of frequency of the heave motion on the clearance be tween LMRP and BOP.
Fig. 12. The influence of amplitude of the heave motion on the clearance be tween the LMRP and BOP.
disconnection and heave motion does not take place at the same time. So, there is a phase difference between them. The clearance between the LMRP and the BOP is shown is Fig. 10 if the phase difference changes from 0� to 330� . As shown in Fig. 10, the phase difference has a significant influence on the recoil response. For deepwater drilling, it is indispensable to avoid the collision between the released LMRP and the BOP to ensure the safety of the subsea wellhead. Therefore, the minimum clearance between them is the most important issue. Table .3 presents the maximum and minimum clearance and the occurrence time under different phase difference. As shown in Table .3, when the phase difference is 60� , the riser has the maximum upward displacement which is 15.12m occurring at 1.4s after disconnection. When the phase difference is 180� , it has the min imum upward displacement which is 7.22m occurring at 24.8s after disconnection. When the phase difference is 90� , it has the maximum downward displacement which is 10.25m occurring at 6.9s after disconnection. When the phase difference is 60� , it has the minimum upward displacement which is 1.67m occurring at 14.6s after discon nection. Therefore, from the perspective of avoiding collision between the LMRP and the BOP, the most advantageous phase difference is 60� .
Fig. 14. The influence of the DAT stiffness on the clearance between the LMRP and the BOP.
4.4.2. Water depth Emergency disconnection operations can be implemented in different ocean environment. The recoil response of riser under different water depth when the phase difference is 0� is shown in Fig. 11. As shown in Fig. 11, with the increase of water depth, the upward maximum and minimum clearance between the LMRP and BOP gradu ally increase. Specifically, when the water depth is 500m, the vibration displacement of the LMRP is always above the BOP. The minimum clearance between them gradually decreases as the water depth in creases. When the water depth increases to 3000m, the LMRP will vibrate below the BOP. This means that the deeper the water depth is, the greater the possibility of collision between the LMRP and BOP is.
“peak value” is very sensitive to the amplitude of the heave motion. In this case, when the amplitude increases from 2.21m to 13.26m, the maximum clearance increases from 6.22m (4.8s after disconnection) to 18.69m (4.8s after disconnection), and the minimum clearance de creases from 1.24m (9.7s after disconnection) to 10.36m (9.7s after disconnection). Therefore, effective countermeasures should be taken to control the amplitude of heave motion to ensure the safety of the emergency disconnection. 4.4.4. Frequency of the heave motion The frequency is also a key parameter affecting the heave motion. When the frequency changes from π =20 to π =2, the clearance between the LMRP and BOP is shown in Fig. 13. As shown in Fig. 13, with the increase of frequency, the maximum clearance decreases first and then increases while the minimum clear ance increases and then decrease, which means there is an optimal frequency making the LMRP be most unlikely collision with the BOP. The optimal frequency is codetermined by the riser configuration and the parameters of the top tension system.
4.4.3. Amplitude of the heave motion After emergency disconnection, the riser top is still connected to the piston rod of the tensioner, as shown in Fig. 3. It is forced to vibrate under the heave motion of the piston rod. When the amplitude of the heave motion changes, the clearance between them is shown in Fig. 12. As can be seen from Fig. 12, with increase of the amplitude, both the maximum and the minimum clearance between them increase. But the moment when the “peak value” appears is not changed. However, the 9
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Ocean Engineering 189 (2019) 106406
Fig. 15. The influence of riser geometric size on the clearance.
4.4.5. Stiffness of the DAT In the above analysis, the heave motion of the DAT is simplified as a spring and the maximum spring stiffness is chosen to analyze the recoil response. The clearance between the LMRP and the BOP is shown in Fig. 14 under different situations of spring stiffness. As shown in Fig. 14, when the DAT stiffness increases from K to 4K, the variation of the clearance changes slightly. However, when the stiffness decreases from K to K =2, the maximum clearance increases form 7.25m–16.84m and the minimum clearance decrease from 0.12m to 7.51m. So, strict control measures should be taken to ensure the stability of stiffness.
(2) The phase difference has a significant influence on the recoil response. The most favorable phase difference is 60� . The possi bility of collision between the LMRP and the BOP increases with water depth and amplitude of the DAT. The top tensioner has an optimal heave motion frequency, which is the most advantageous for implementing riser emergency disconnection. Corresponding measures should be taken to prevent spring stiffness of the DAT from decreasing. Besides, riser with large diameter and wall thickness is conducive to a safe disconnection. (3) This paper can provide reference for the study on riser recoil response. However, further study should be implemented to obtain a comprehensive understanding of this mechanical phe nomenon, which include: the fitting curve should be studied further to improve the accuracy of the frictional resistance during mud discharge, innovative mechanical model should be estab lished with consideration of the mass-spring-damping system with infinite degrees of freedom, the mechanical state of riser after disconnection should be studied through 3-D dynamic model with consideration of the wave-current effect to ensure no mechanical failure will occur. In addition, the safety of the moon pool is also needed to be assessed during the recoil response.
4.4.6. Riser geometric size The riser geometric size will not only change the mass of the three blocks but also affect the frictional resistance, which has a comprehen sive influence on the recoil response. Fig. 15 shows the response curve under the different conditions of riser outer diameter and wall thickness. As shown in Fig. 15, riser geometric size has a slight influence on the clearance between the LMRP and the BOP. Specifically, under the con dition of the same outer diameter, the greater the wall thickness is, the smaller the maximum clearance is. When the wall thickness is constant, the larger the outer diameter is, the smaller the maximum clearance is.
Acknowledgements
5. Conclusions and further study
The authors gratefully acknowledge the financial support from the Natural Science Foundation of China (Grant numbers: 51804321, 51821092, U1762214 and 51734010). This research is also supported by Science Foundation of China University of Petroleum, Beijing (No.00000),and also supported by other projects (Grant numbers: 2017ZX05005-005-007, 2017ZX05009-003, 2016YFC0303303, and 2018YFB0605502).
(1) In this paper, the mechanical model of mass-spring-damping system with three degrees of freedom has been established to analyze the riser recoil response. The model can consider the influence of the frictional resistance during mud discharge and the elastic potential energy stored in the riser. The clearance between the LMRP and BOP after disconnection has been figured out.
10
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Appendix D. Supplementary data Supplementary data to this article can be found online at https://doi.org/10.1016/j.oceaneng.2019.106406. Appendix A The appendix shows the solving process of the frictional resistance on the riser inner wall during mud discharge. The frictional resistance caused by drilling mud and seawater can be expressed as:
8 fr2 ¼ Δpr2 Aa < fr1 ¼ Δpr1 Aa ; fd1 ¼ Δpd1 Aa ; fd2 ¼ Δpd2 Aa : fp1 ¼ Δpp1 Ain ; fp2 ¼ Δpp2 Ain
(A-1)
where, Δpr1 is the pressure drop between seawater and riser inner wall; Δpr2 is the pressure drop between drilling mud and riser inner wall; Δpd1 is the pressure drop between seawater and drill pipe outer wall; Δpd2 is the pressure drop between drilling mud and drill pipe outer wall; Δpp1 is the pressure drop between seawater and drill pipe inner wall; Δpp2 is the pressure drop between drilling mud and drill pipe inner wall; Aa ¼ πðR2ri R2so Þ is the annular cross-sectional area between riser and drill pipe; Rri is the inner radius of riser; Rso is the outer radius of riser; Ain ¼ πR2si is the inner area of drill pipe; Rsi is the inner radius of drill pipe. According to the Darcy-Weisbach equation, the pressure drops can be calculated by: 8 > ρw v2 fwc1 Lw ρ v2 fmc1 Lm > > ; Δpr2 ¼ m > Δpr1 ¼ > 2ðRri Rso Þ 2ðRri Rso Þ > > > > < ρw v2 fwc2 Lw ρ v2 fmc2 Lm (A-2) Δpd1 ¼ ; Δpd2 ¼ m > 2ðRri Rso Þ 2ðRri Rso Þ > > > > > > ρ v2 fwc3 Lw ρ v2 fmc3 Lm > > : Δpp1 ¼ w ; Δpp2 ¼ w 4Rsi 4Rsi where, ρw is the density of seawater; ρm is the density of drilling mud; v is the discharge velocity of the whole column. fwc1 , fwc2 , fwc3 , fmc1 , fmc2 and fmc3 are dimensionless coefficients, which can be described by the Haaland formula as shown in Eq. (A-3). 8 � � � � > > 1 6:9 � ε1 �1:11 1 6:9 � ε1 �1:11 > > ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi p p ¼ 1:8 log ¼ 1:8 log þ þ ; > > Rew Rem 3:7 3:7 fwm1 > fwc1 > > > � � � � < 1 6:9 � ε2 �1:11 1 6:9 � ε2 �1:11 pffiffiffiffiffiffiffi ¼ 1:8 log pffiffiffiffiffiffiffiffi ¼ 1:8 log þ þ ; (A-3) > Rew Rem 3:7 3:7 f fwm2 > > wc2 > � � � > � � � > > 1 6:9 � ε3 �1:11 > pffi1ffiffiffiffiffiffi ¼ 1:8 log 6:9 þ ε3 1:11 ; > pffiffiffiffiffiffiffiffi ¼ 1:8 log þ > : fwc3 Rew Rem 3:7 3:7 fwm3 where, Rew and Rem are Reynolds number of seawater and drilling mud respectively; ε1 , ε2 and ε3 are the relative roughness of riser inner wall, riser outer wall and drill pipe inner wall. Considering that the riser outer wall is surrounded by seawater, the pressure difference on the top and bottom surfaces of the whole column can be calculated by: Fp ¼ ðρm
Ftop as:
(A-4)
ρw ÞgLm
In this paper, we assume that the mud is discharged directly through the LMRP. So, the frontal force (Rinaldi and Paȉ;doussis, 2010) can be written
1 Fend ¼ ρm ðAa þ Ain Þv2 2
(A-5)
Appendix B The appendix presents the detailed derivation process of Eq. (3) and Eq. (4).The analysis model of tension force and stiffness is shown in Fig. B-1. The following assumptions are accepted to analyze the tension force and stiffness of the DAT, which are: i) ii) iii) iv)
It is assumed that the gas in the tension system conforms to the rule of adiabatic state. The tension force is far greater than the weight of piston and piston rod. So, the weight of them is neglected. The liquid in the hydraulic cylinder is incompressible and the leakage of it is ignored. The damping effect of the piston is also neglected.
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Ocean Engineering 189 (2019) 106406
Fig. B-1. The analysis model of tension force and stiffness.
When the piston has displacement xp relative to the cylinder, the volume of the rod end Vh is: (B-1)
Vh ¼ Vhi þ Ar xp The volume of the rear end Vl is: Vl ¼ Vli (
(B-2)
Ap x p
According to the first assumption, one obtains: �n phi V nhi ¼ ph Vhi þ Ar xp � n pli V nli ¼ pl Vli Ap xp
(B-3)
So, the tension force on the piston rod can be written as: Ft ¼ p h A r
(B-4)
p l Ap
Eq. (3) can be obtained through substituting Eq. (B-3) into Eq. (B-4). The stiffness of a single tensioner can be obtained through taking derivative of the tension force with respect to the relative displacement in Eq. (B4). Assume that all the tensioners are connected to the tensioner ring, the model, which is shown in Fig.B-2can be regarded as parallel springs with number of N.
Fig.B-2. The stiffness analysis model for parallel springs.2
Supposing that the stiffness of each tensioner spring is k1 k2 ; :::; kN and the elongation under the external force is x0 , then the external force can be written as: N X
F¼
(B-5)
ki x0 i¼1
Therefore, the total stiffness of the tension system is: N P
ki x0 X N Kt ¼ i¼1 ¼ ki x0 i¼1
(B-6)
Since the structure and parameters of each tensioner are the same, the spring stiffness of each tensioner is the same accordingly. So, the total stiffness of the tension system can be obtained as shown in Eq. (4) Appendix C This appendix shows the determination procedure of the riser axial stiffness. The axial stiffness of the riser can be calculated according to the work of the external force equaling to the elastic strain energy, which is: Z
L 0
N 2 ðxÞ 1 dx ¼ Ke ðΔLÞ2 2EA 2
(C-1)
where, ΔL is the total elongation of riser; NðxÞ is the axial tension force along riser; E is the elastic modulus of riser; A is the cross-sectional area of riser; 12
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Ocean Engineering 189 (2019) 106406
Ke is the equivalent stiffness of the riser. The axial tension force can be calculated by: NðxÞ ¼ F0
(C-2)
ωx
Where, F0 is the top tension, which is 1.35 times the total weight of riser in sea water. ω ¼ ms g is the buoyancy. So, the axial elongation can be written as: Z L NðxÞ ΔL ¼ dx EA 0
ωb is the per length weight of riser in sea water; ωb
Then, the axial stiffness can be obtained after substituting Eq. (C-2) and Eq. (C-3) into Eq. (C-1), which is: � � � EA F 20 L þ ω2 L 3 F0 ω Ke ¼ ðF0 ωL=2Þ2
(C-3)
(C-4)
If γ represents the over-tension coefficient of riser, Eq. (C-4) can be written as: Ke ¼
γ2 þ γ þ 1=3 EA γ2 þ γ þ 1=4 L
(C-5)
In this paper, the mechanical model of stiffness is obtained through dividing the whole riser into 2 parallel segments, which is shown in Fig. C-1.
Fig. C-1. The analysis model for riser axial stiffness.3
Since the axial force is evenly distributed along riser, one obtains: N1 þ N3 N1 þ N2 N2 þ N3 ¼ þ 2Ke 2Kr 2Kr
(C-6)
where, N1 , N2 and N3 are the external force on the three mass blocks. According to Eq. (C-6), one obtains, Kr ¼
N1 þ 2N2 þ N3 Ke N1 þ N3
(C-7)
Due to F0 ¼ ð1 þ γÞωL, N1 ¼ ð1 þ γÞωL, N2 ¼ ð1=2 þ γÞωL and N3 ¼ γωL. Thus, the axial stiffness of each segment can be written as: Kr ¼
ð1 þ γÞ þ 2ð1=2 þ γÞ þ γ Ke ¼ 2Ke ð1 þ γÞ þ γ
(C-8)
References
Grønevik, A., 2013. Simulation of Drilling Riser Disconnection-Recoil Analysis. Master’s thesis of Norwegian University of Science and Technology. He, X.X., Zhang, F.F., Tian, X.J., Liu, X.Q., 2016. Modeling and simulation for recoil control system of deepwater drilling riser. China Pet. Mach. 44 (3), 63–67. Hock, C.J., Young, R.D., 1993. A deepwater riser emergency disconnect antirecoil system. J. Pet. Technol. 45 (08), 744–751. ISO/TR 13624-2:2009 (E), 2009. Petroleum and Natural Gas Industries-Drilling and Production Equipment- Part 2: Deepwater Drilling Riser Methodologies, Operations, and Integrity Technical Report. ISO. Lang, D.W., Real, J., Lane, M., 2009. Recent developments in drilling riser disconnect and recoil analysis for deepwater applications. In: International Conference on Ocean, Offshore and Arctic Engineering, Honolulu, Hawaii, USA, pp. OMAE2009–79427. Li, C.W., Fan, H.H., Wang, Z.M., Shen, W.G., Feng, X., Deng, S., 2015. Numerical simulation of mud discharge after drilling riser emergency disconnection. In: International Ocean and Polar Engineering Conference, Kona, Big Island. Hawaii, USA, ISOPE-I-15-736. Li, C.W., Fan, H.H., Wang, Z.M., Ji, R.Y., Ren, W.Y., Feng, X., 2016. Two methods for simulating mud discharge after emergency disconnection of a drilling riser. J. Nat. Gas Sci. Eng. 28, 142–152. Ma, P., Pyke, J., Vankadari, A., Whooley, A., 2013. Ensuring safe riser emergency disconnect in harsh environments: experience and design requirements. In:
American Bureau of Shipping, 2017. Guidance Notes on Drilling Riser Analysis. American Bureau of Shipping, Houston, USA. Brynestad, I.B., 2012. Disconnection of Workover Risers on Very Deep Water. Master’s thesis of Norwegian University of Science and Technology. Dyngvold, S.H.Ø., 2011. Development of Simulation Model for Virtual Testing and Design of a Riser Tensioner System. Master’s thesis of University of Agder. Gao, D.L., Wang, Y.B., 2016. Progress in tubular mechanics and design control techniques for deep-water drilling. Pet. Sci. Bull. 01, 61–80. Gobat, J.I., Grosenbaugh, M.A., 2006. Time-domain numerical simulation of ocean cable structures. Ocean Eng. 33, 1373–1400. Grytoyr, G., Rustad, A.M., Sodahl, N., Bunaes, P.C., 2009. Methodology for disconnection analysis of CWO risers in random seas. In: International Conference on Ocean, Offshore and Arctic Engineering, Honolulu, Hawaii, USA, pp. OMAE2009–79130. Grytoyr, G., 2010. Improving operating window for disconnect operations of CWO risers. In: International Conference on Ocean, Offshore and Arctic Engineering. Shanghai, China, OMAE2010-O2020. Grytoyr, G., Sharma, P., Vishnubotla, S., 2011. Marine drilling riser disconnect and recoil analysis. In: The AADE National Technical Conference and Exhibition. Houston, Texas, 2011, AADE-11-NTCE-80.
13
Y. Wang and D. Gao
Ocean Engineering 189 (2019) 106406 Wang, T., Wang, K., 2018. Analysis of resilient response of emergently disentangled drilling risers. China Offshore Oil Gas 30 (2), 120–125. Yang, H.L., 2013. Study on Disconnection Technique for Storm-Safe Drilling Riser. Master’s thesis of China University of Petroleum (East China). Young, R.D., Hock, C.J., Karlsen, G., Miller, J.E., 1992. Analysis and design of anti-recoil system for emergency disconnect of a deepwater riser: case study. In: Offshore Technology Conference. OTC, Houston, Texas, p. 6891. Young, R.D., Hock, C.J., Karlsen, G., Miller, J.E., 1992. Comparison of analysis and fullscale testing of anti-recoil system for emergency disconnect of deepwater riser. In: Offshore Technology Conference. OTC, Houston, Texas, p. 6892. Zhang, L., Chang, Y.J., Liu, X.Q., Liu, K., Chen, L.M., 2013. Recoil analysis for deepwater drilling riser after emergency disconnection with blowout preventer. Pet. Drill. Tech. 41 (3), 25–30. Zhang, L., 2014. Recoil Analysis for Deepwater Drilling Riser System. Master’s thesis of China University of Petroleum (East China). Zhang, L., Chang, Y.J., Liu, X.Q., Liu, K., Yang, H.L., 2013. AMESim-based modeling and simulation of platform riser tensioner. China Pet. Mach. 41 (12), 38–42. Zhao, J.T., Xue, Y., Pan, Y., Cheng, F., 2010. Study on riser tensioning system of floating drilling equipment. Shanghai Shipbuilding 4, 1–5.
International Offshore and Polar Engineering Anchorage. Alaska, USA, ISOPE-I-13299. Matthew, J.S., Momen, W., James, N.B., 2004. Riser recoil analysis at a harsh environment, deepwater site. In: International Conference on Offshore Mechanics and Arctic Engineering, Vancouver, British Columbia, Canada. OMAE2004-51628. Meng, S., Che, C.D., Zhang, W.J., 2018. Discharging flow effect on the recoil response of a deep-water drilling riser after an emergency disconnect. Ocean Eng. 151, 199–205. Miller, J.E., Stahl, M.J., Matice, C.J., 1998. Riser collapse pressures resulting from release of deepwater mud columns. In: Offshore Technology Conference. Houston. Texas, OTC-8853. Pestana, R.G., Roveri, F.E., Franciss, R., Ellwanger, G.B., 2016. Marine riser emergency disconnection analysis using scalar elements for tensioner modelling. Appl. Ocean Res. 59, 83–92. Puccio, W.F., Shaffer, V., Nuttall, R.V., 1998. Riser recoil during unscheduled lower marine riser package disconnects. In: IADC/SPE Drilling Conference. Dallas, Texas, IADC/SPE39296. Rinaldi, S., Paȉdoussis, M.P., 2010. Dynamics of a cantilevered pipe discharging fluid, fitted with a stabilizing end-piece. J. Fluids Struct. 26, 517–525. Stahl, M.J., Hock, C.J., 2000. Design of a riser recoil control system and validation through full-scale testing. In: SPE Annual Technical Conference and Exhibition. Dallas, Texas, SPE-62959.
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