Bi-nonlinear vibration model of tubing string in oil & gas well and its experimental verification

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Bi-nonlinear Vibration Model of Tubing String in Oil&Gas Well and Its Experimental Verification Jun Liu , Xiaoqiang Guo , Guorong Wang , Qingyou Liu , Dake Fang , Liang Huang , Liangjie Mao PII: DOI: Reference:

S0307-904X(19)30588-8 https://doi.org/10.1016/j.apm.2019.09.057 APM 13061

To appear in:

Applied Mathematical Modelling

Received date: Revised date: Accepted date:

22 March 2019 26 September 2019 30 September 2019

Please cite this article as: Jun Liu , Xiaoqiang Guo , Guorong Wang , Qingyou Liu , Dake Fang , Liang Huang , Liangjie Mao , Bi-nonlinear Vibration Model of Tubing String in Oil&Gas Well and Its Experimental Verification, Applied Mathematical Modelling (2019), doi: https://doi.org/10.1016/j.apm.2019.09.057

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Highlights 

The finite element method and the contact theory are used to establish a bi-nonlinear vibration model of tubing string.



The longitudinal/lateral coupled vibration and the nonlinear contact of tubing-casing were considering in the model.



A similar experiment of tubing vibration is designed and completed to test the validity of bi-nonlinear vibration model.



The analysis shows that the model has good calculation accuracy and the vibration response law is basically consistent.

1

Bi-nonlinear Vibration Model of Tubing String in Oil&Gas Well and Its Experimental Verification Jun Liu1, Xiaoqiang Guo1, Guorong Wang1*, Qingyou Liu2, Dake Fang3, Liang Huang3, Liangjie Mao4 1

School of Mechatronic Engineering, Southwest Petroleum University, Chengdu, 610500, China. State Key Laboratory of Oil and Gas Reservoir Geology and Exploitation//Southwest Petroleum University-Chengdu University of Technology, Chengdu, Sichuan, 610500, China.. 3 Zhanjiang Branch, CNOOC (China) Co., Ltd., Zhanjiang, 524057, China. 4 School of Petroleum and Natural Gas Engineering, Southwest Petroleum University, Chengdu, 610500, China. 2

*

Corresponding author, Email: [email protected]

Abstract: Fluid-induced vibration (FIV) prediction is an important prerequisite work in wear and fatigue analysis of tubing string in oil&gas well. The finite element method, energy method and Hamiltonian principle are comprehensively used to establish a single nonlinear vibration model of pipe conveying fluid, taking into account the longitudinal/lateral coupled vibration. Based on the contact/impact theory of elastic/plastic body, the nonlinear contact-impact model of tubing-casing is established and introduced into the single nonlinear vibration model to form a bi-nonlinear vibration model of tubing string in oil&gas well. The bi-nonlinear model is numerically discretized by the finite element method, solved by Newmark   method, and verified preliminarily by a classical contact/impact example in literature in which the influence of inflow is not taken into account temporarily. A similar experiment of tubing vibration is designed and completed to further test the validity of the bi-nonlinear vibration model by comparing the frequency-domain and time-domain responses of the experiment with those from the model. The analysis shows that the bi-nonlinear model has good calculation accuracy and the vibration response law is basically consistent with the experimental results, which can provide an effective theoretical analysis tool for FIV behavior of tubing string in oil&gas well. Keywords: Pipe string mechanics, Tubing vibration, Fluid-induced vibration, Vibration experiment, Tubing fatigue.

1.Introduction During the production of oil&gas, due to the action of high-speed fluid in the tubing (Fig. 1), the tubing string is prone to longitudinal/lateral coupled nonlinear vibration [1], resulting in alternating load and dynamic stress in the tubing which can cause a series of problems, such as the fatigue damage of the tubing string, loosening of downhole tools matching the tubing and failure of connecting threads, etc. Moreover, this vibration may also causes buckling of the tubing due to excessive axial 2

force (Fig. 2(a)), and friction perforation (Fig. 2(c)) due to the increasing friction and wear between tubing and casing (Fig. 2(b)), resulting in serious downhole safety accidents and huge economic losses.

(a) (b) Fig. 1. Structural sketch of completion tubing string (a) tubing string (b) local contact between tubing string and casing. (a)

(b)

(c)

Fig. 2. Vibration-induced failure of completion tubing string (a) buckling deformation, (b) eccentric wear and (c) friction perforation.

Vibration of tubular structure caused by inside flow has attracted some researchers' attentions. In the early research, scholars mainly carried out preliminary study on the vibration of the pipe caused by the fluid in the pipe neglecting the external excitation [2, 3], and initially confirmed the phenomenon of pipe vibration induced by fluid in pipe without elaborating the interaction mechanism between fluid and pipe. Huangzhen [4] used hydrodynamic theory to study the vibration of tubing string induced by the flowing natural gas, and preliminarily revealed the interaction mechanism between the fluid and tubing string. On this basis, some scholars carried out theoretical model research on fluid-induced vibration of tubular structure. In the early theoretical model studies [5-8], the focus was put on seeking the analytical solution of pipe string vibration which was based on massive simplification and neglected the non-linear factors in the problem. Therefore, the calculation accuracy is difficult to meet the field requirements. Subsequently, some scholars [9-12] established the non-linear vibration model in the fluid-induced vibration analysis of the tubing string, taking into account the geometric non-linearity caused by large deformation. An in-house laboratory device for liquid–solid coupled flutter instability of salt cavern leaching tubing was developed to investigate these flow-induced vibration phenomena by Ge et al [13]. However, in these models only the vibration in one direction (longitudinal or lateral) was considered, neglecting the coupling vibration effect between different directions. The analysis results may be quite different from the actual pipe vibration because some scholars [14-16] found that the longitudinal/lateral coupled vibration effect has a significant impact on the vibration of slender structures. In the above work, the emphasis was put on the pipeline vibration induced by 3

internal fluid without considering the dynamic contact/collision between pipeline and surrounding structures. Whereas, the vibration of tubing string in oil&gas well belongs to typical pipe-in-pipe contact/collision problem, in which how to effectively describe the contact/collision action is a necessary work. Aimed at the static contact problems of slender structures, a few researchers [17-21] have tried to give the calculation methods of contact force between a beam and support structure, and the correctness of the methods were verified by experimental data. Moreover, the bracing effect of the outer pipe was taken into account by some researchers [5, 22-24] to analyze the static buckling deformation of a tubing string. About the dynamic contact/collision problem of slender structures, ANSYS finite element software was used by Yang et al [25] to investigate the impact force and friction force in the flow-induced vibration of completion tubing string in vertical well. The drill string vibration was investigated by Zhu et al [26] using a nonlinear dynamic model in which the restraint of the borehole wall is applied to the speed of the drill string, that is, when the drill string contacts the wellbore, its velocity was zero. Liu et al [27] used commercial software ABAQUS to analyze the dynamic behavior of a deepwater test tubing, taking into account the large deformation of riser-test pipes and the nonlinear contact between test tubing and riser. However, the longitudinal/lateral coupling effect of the test tubing was not taken into account. Up to now, due to the lack of effective flow-induced vibration (FIV) model and experimental analysis, the study on nonlinear FIV behavior of tubing string in oil&gas well is still very limited, and the understanding to the fatigue and wear mechanism of tubing string is far from enough. The purpose of this paper is to present a bi-nonlinear FIV model of tubing string in curved oil/gas well considering the nonlinear longitudinal/lateral coupling effect of the tubing string and the nonlinear contact/collision between tubing and casing, as well as provide an effective theoretical analysis tool for the FIV analysis of the tubing string. 4

2. Bi-nonlinear vibration model for tubing string in oil&gas well The tubing string in oil&gas well is a typical liquid-filled pipeline with large slenderness ratio, its longitudinal/lateral coupling vibration effect is obvious under external excitation [28]. Moreover, the tubing string may be in contact/collision with the casing during the FIV process. Therefore, in this section, firstly, a single nonlinear vibration model of pipe conveying fluid is presented, taking into account the longitudinal/lateral coupled vibration. Secondly, the nonlinear contact-impact model of tubing-casing is established and introduced into the single nonlinear vibration model to form a bi-nonlinear vibration model of tubing string in oil&gas well. Thirdly, the solution scheme of the bi-nonlinear model is presented. Finally, the validity of the model is preliminarily tested by an example in literature. 2.1 Longitudinal/lateral coupled nonlinear vibration model A vertically placed pipe conveying fluid and the corresponding coordinate system are shown in Fig. 3, in which the origin of the coordinate is located at the top of the pipe, the vertical downward direction and the horizontal right direction are respectively set as the positive direction of the x-axis, and the positive direction of the y-axis.

Fig. 3. Diagrammatic sketch of pipe conveying fluid

Considering the liquid-filled pipe as a uniform Rayleigh beam, the geometric relationship of displacement and deformation is given as follows, taking into account the longitudinal/lateral coupling effect. 2  u 1 1  u2  u 2    xx   ， yy  x 2  x  y     1  u 2  u1 ，      0 xz yz zz  xy 2  y y  

(1)

Where  ij (i, j  x, y, z) are strain components, ui i  1,2,3 are displacement components which can be further expressed as follows:

5

w  ( x, t ) u1 ( x, y, t )  u ( x, t )  y x   u2 ( x, t )  w( x, t )，u3 ( x, t )  0

(2)

Where u is longitudinal displacement, w is lateral displacement, t is time. The following equations can be obtained by substituting formula (2) into formula (1).

 u  2 w 1 w  y 2  ( )2  xx  x x 2 x   yy   xy  0 

(3)

The horizontal and vertical components of fluid velocity V in the pipeline can be expressed as Vx  V and Vy  w  V w . t

x

Therefore, the total kinetic energy T and potential energy U of the system including the pipe and the fluid can be respectively expressed as

1 L   u 2 w 2  2w 2 w 2  w  T     A ( )  ( )    I ( ) +0 A (  V )  V 2  dx 0 2   t t  tx x  t  U

Where

EA L  u 2 1 w 4 u w 2  EI ( )  ( ) + ( ) dx    0 2 4 x x x  2  x

L is the length of the pipeline,



L

0

(

2w 2 ) dx x 2

E is Young modulus,

(4a)

(4b) A is the

cross-sectional area,  is the fluid density, EI is the bending stiffness, u is the longitudinal displacement. The variational formulas of kinetic energy, potential energy and work done by external forces are given as follows:    u   u  2w 2w   w  w   A 2   2  ( )  2  I ( )  ( )          t2 t x t x  1 t2 L    t   t   t  t   T      dxdt t1 2 t1 0  w w w   w   0 A  2(  V ) (  V )   x t x   t  

    u   u   w 3 w    2         ( )   t2 t2 L  EA x  2 w   2 w   x   x   x     U     EI  2    2   dxdt t1 t1 0 2  w 2  u   u  w   w    x   x          2           x   x   x  x   x  

 W    f  x, t   u  p  x, t   w dx L

0

Where f  x, t  , p  x, t  is the longitudinal forces and lateral forces of pipeline. 6

(5a)

(5b)

(5c)

The following formula can be obtained by substituting the formulas (5) into Hamilton formula  t (T  U  W )dt  0 . t2

1

t2

  (T  U  W )dt t1

 2u  2u w  2 w )  EA ( )  EA ( )  f ( x, t )] udxdt t1 0 t 2 x 2 x x 2 t2 L 2w 4w 2w 2w 2w    [  A( 2 )   I ( 2 2 )   0 A( 2  2V V 2 2 ) t1 0 t  t x t t x x 2 2 2 4 3EA w 2  w u  w  u w  w  ( )  EA[( ) 2  ( 2 ) ]  EI ( 4 )  p ( x, t )] wdxdt 2 2 x x x x x x x 3 t2 w w w L EA w 3 L u w   (  I ( 2 ) 0L  0 A(V V 2 )0 ( ) 0  EA( )( ) 0L t1  t x t x 2 x x x 3 2 t w  w w u EA w 2 L 2  EI ( 3 ) 0L  EI ( 2 )( ) 0L  wdt   ( EA( )  ( ) ) 0  udt t 1 x x t x 2 x The boundary conditions of pipeline in oil&gas wells can be expressed as 

t2

L

[  A(

 2u (0, t )  2u ( L, t )  0, u ( L , t )  0, 0 2 x 2 x  2 w(0, t )  2 w( L, t ) w(0, t )  0,  0, w ( L , t )  0, 0 2 x 2 w The following equations can be obtained by formula (6a)

(6a)

u (0, t )  0,

(6b)

3w L w w EA w 3 L ) 0  0 A(V  V 2 ) 0L  ( ) 0 2 t1  t x t x 2 x u w 3w  2 w w  EA( )( ) 0L  EI ( 3 ) 0L  EI ( 2 )( ) 0L  wdt =0 x x x x t t2 u EA w 2 L t1 ( EA( x )  2 ( x ) ) 0  udt =0 t2

(  I (

(6c)

The differential equations of longitudinal and lateral vibrations of the pipeline can be extracted from equation (6a) ~ (6c).  2u  2u w  2 w )  EA ( )  EA ( )( )  f ( x, t )  0 t 2 x 2 x x 2

(7a)

2 2w 4w 2w 2w 2  w   A( 2 )   I ( 2 2 )   0 A( 2  2V V ) t  t x t t x x 2 3EA w 2  2 w u  2 w  2u w 4w  ( )  EA [( )  ( ) ]  EI ( )  p ( x, t )  0 2 x x 2 x x 2 x 2 x x 4

(7b)

  A(

2.2 Nonlinear contact/collision model of casing-tubing Under the action of fluid in the tubing, the tubing string will produce lateral and longitudinal vibrations. As the displacement of lateral vibration reaches the gap between the casing and the tubing, the tubing will contact and collide with the casing, and at the same time its contact and friction with the casing will occur in the 7

longitudinal direction, resulting in friction and wear on the surface of the tubing. Therefore, it is necessary for tubing fatigue and wear analysis to consider the tubing-casing contact/collision effect under longitudinal/lateral coupled vibration. (1) Collision force-deformation relationship between casing and tubing The deformation of tubing in wellbore and its contact with casing are shown in Fig. 4 in which R1 is the radius of casing, R2 is the radius of tubing, P is collision force on tubing. As the tubing collides with the casing, point A2 on the tubing deforms to point A1 on the casing. According to Fig. 4, the following geometric relationship can be obtained,

Ri2   Ri  zi   r 2 （i  1,2） 2

(8)

which can be further simplified as

r2 zi  2 Ri

（i  1,2）

(9)

Under the action of positive pressure, it is assumed that the relative deformation between the casing and tubing is  , the width of contact zone is 2b , and the radial displacements of the casing and tubing after contact are respectively 1 and  2 . From Fig. 4, the geometric relation shown below can be obtained by Xu et al [29].

  (2 +1 )+( z2 -z1 )=(2 +1 )+（

R1  R2 ）r 2 2 R1R2

(10)

Fig. 4. Schematic diagram of contact deformation between tubing and casing

If the width of the contact zone is much smaller than the radius of the tubing, each tubing string can be approximately considered as an elastic half-plane. Therefore, the formulas of 1 and  2 can be derived as follows: If the uniform pressure on the contact surface is q ( x) , the following relation expression can be obtained according to the symmetry



b

-b

q ( x)dx  p

Local axial deformation of tubing can be obtained from semi-infinite plane contact deformation formula by Xu et al [29]

8

(11)

d1  

r  x (1  1 ) 2(1  12 ) q( x)dx ln + q( x)dx  E1 R1  E1

(12)

r  x (1  1 )  2(1  12 )   q( x)  dx  q( x) ln  E1  R1  E1  Substituting the formula (11) into formula (12), the axial displacement of the tubing can be obtained as follow

 2(1  12 )  b 1 1    ln R1 ) p   b q( x) ln r  x dx-(  E1  2(1  1 ) 

(13a)

Similarly, the displacement of casing can be obtained as follow

2  

 2(1  22 )  b 1  ln R2 ) p   b q( x) ln r  x dx-(  E2  2(1  2 ) 

(13b)

where E1 and 1 are respectively the Yong’s modulus and Poisson’s ratio of tubing, E 2 and  2 are respectively the Yong’s modulus and Poisson’s ratio of casing.

Substituting the formulas (13a) and (13b) into the formula (10), the following formula can be obtained

2 1  12 1  22 b ( + )  q ( x) ln r  x dx   r 2  K b  E2 E1

(14)

where

=

R1  R2 2 R1R2

In formula (14), K is the sum of all the items unrelated to r . In order to eliminate these items, take the derivative of formula (14) with respect to r .

2 1  12 1  22 d b (  ) q( x) ln r  x dx  2  r  E1 E2 dr b

(15)

where the integral function becomes infinite at x  r , so the integral theorem of unbounded function must be used to perform the following operations d b d r  d b q( x) ln r  x dx  lim  q( x) ln r  x dx  lim  q ( x) ln r  x dx   0 dr  b  0 dr r  dr b (16) r  q ( x )dx b q ( x )dx  lim[   q(r   ) ln     q(r   ) ln  ] r  r  x  0  b rx

It’s worth noting that lim[q(r   ) ln   q (r   ) ln  ]  lim[  0

 0

q (r   )  q (r   )  2 ln  ]  q '(r )  0  0 2 9

(17)

Therefore, formula (16) can be further simplified as r  q ( x )dx b q ( x )dx b q ( x )dx d b q( x) ln r  x dx  lim      b  b r    b rx  0 dr rx rx

(18)

Substituting the formula (18) into formula (15) yields the following formula.

2 1  12 1  22 b q( x) (  ) dx  2  r  E1 E2  b r  x

(19)

It can be assumed that q  x  is proportional to the longitudinal coordinates of the semicircular arc with diameter 2b , i.e q( x) 

qmax b

b2  x2

(20)

Where qmax is the maximum value of q  x  which appears at the center of the contact surface and can be determined as follow by substituting the formula into the contact b

force formula p  b q( x)dx qmax 

2p b

(21)

It can be derived that



2(b 2  x 2 )  2 (b 2  x 2 )(b 2  r 2 ) b2  x2 x dx  b 2  x 2 ln  b 2  x 2  r sin 1 (22) rx rx b

Therefore  r  b2  x 2 b q( x)dx qmax b2  x 2   qmax  d x   lim b r  x b  0 b r  x r  r  x dx   b r   b

(23)

Using formula (23), formula (19) can be further written as

1  12 1  22 2qmax R  R2 (  )  2  1 E1 E2 b R1 R2 Substituting qmax 

(24)

2p , E1  E2  E and 1  2  0.3 into (24), b can be obtained in b

simplification form

b  1.522

P R1R2 E R2  R1

(25)

Substituting the formula (25) into formula (10), the formula (26) can be obtained through the necessary simplification process which can be found in Xu’s work [29].

 =1.82 10

P (1  ln b) E

(26)

(2) Collision force and friction force of casing-tubing There may be multiple contact/collision locations between casing and tubing. Because of the complexity of the problem, the contact/collision analysis method of casing/tubing at one point (Fig. 5) is studied first, and then the method is extended to contact/collision analysis of the whole tubing string.

Fig. 5. Schematic diagram of contact/collision between tubing/casing

As shown in Fig. 5, the interaction between the casing and the tubing is simulated by a spring-damper system. Therefore, the impact force exerted by the casing on the tubing can be determined from the calculation formula (25) given in the previous section. The formula for calculating the damping force can be given as follow shown in reference [20].

f c =  cY  t 

(27)

 3  Where Y  t  is the longitudinal displacement of the contact point, c   ap  is  2 

damping coefficient where a is constant with value 0.2~0.3 for steel material by Hunt [30]. Obviously, the impact damping force is also a nonlinear function of deformation. Thus, the formula for calculating the contact force between tubing and casing after collision can be obtained as follow 3 f t  [ P  aPY  t ] 2

(28)

The contact force calculation model is introduced into tubing vibration model presented in section 2.1. Spring-dampers are set on each side of each tubing element. At some point in time in the nonlinear dynamic analysis, as the transverse displacement of the tubing is larger than the casing radius, the tubing will be in contact with the casing. At the same time, the relative deformation between tubing and casing is obtained from the transverse displacement. The collision force on tubing can be determined by formula (25). Using formula (27), the collision damping force on the tubing can be determined by the lateral displacement. Finally, the 11

contact force and friction force acting on the tubing in the next time step can be determined by formula (28) and the following formula.

fF   ft

(29)

where  is the friction coefficient between tubing string and casing, which is generally taken as 0.3 in monograph by Wen et al [31]. 2.3 Solution scheme of bi-nonlinear model (1) Displacement function The linear Lagrange function and the cubic Hermite function are used to represent the longitudinal and lateral displacements u and w of the tubing string. T u  ψ d  T  w  φ d

(30)

where d T  [u1

w1

dw1 dx

u2

 3x 2 2 x3 φ  0 1  2  3 l l  T

w2

dw2 ] dx

2 x 2 x3 x  2 l l

ψ T  [1 

x 0 0 l

3x 2 2 x 3 0  3 l2 l

x 0 0] l

x 2 x3    2 l l 

where l is the element length. (2) Discrete equations By substituting the displacement function (30) into the energy function, the standard forms of strain energy function U and kinetic energy function T expressed by the nodal displacement vector can be obtained. L EA  L EAψψT dx   φφT w  w T φφT dx    0 4 1 T  0 U d  L L 2  2 EAψw T φφT dx   EIφφT dx d 0 0  

  Ad T ψψ T ddx    Ad T φφT ddx    Id T φφT ddx  1 0 0 o T  L L T T T T 2   Ad φφ ddx  2  AVφw φ ddx  L  AV 2dT φφT ddx 0 0 0 0  0 0 L

L

L

The above formula can be further written as abbreviately

12

(31)

1 U  d T kd 2 1 1 1 T  d T md + cd  d T kd 2 2 2

(32)

where the element stiffness matrix k , mass matrix d and damping matrix c can be expanded as follows

k  k  k  k  k T  k +k 1 2 3 3 4 5  l l  T T T m  0  Aψψ dx  0 (  Aφφ + Iφ ' φ ' )dx  L c  2  0 AVφwT φ T dx  0

(33)

where k 1   EAψ ' ψ 'T dx

l 1 k 2   2  EAφ ' φ 'T ww T φ ' φ 'T dx 0 4

1 l k 3    EAψ ' ψ 'T φ ' φ 'T dx 2 0

k 4   EIφ '' φ ''T dx

l

0

l

0

(34)

L

k 5    0 AV 2φφT dx 0

After assembling the structural element matrix, the discrete dynamic equation of the tubing string can be obtained according to the variational principle.

M(t )D  C(t )D  K(t )D  F(t )

(35)

where M(t ) , C(t ) and K(t ) are respectively the overall mass matrix, damping matrix and stiffness matrix, F(t ) is the load column vector, including the impact force of gas on the string, the contact force and friction force of the tubing and casing. (3) Newmark -  solution scheme Newmark- step-by-step integration method which avoids the application of

superposition and has good adaptability to nonlinearity, is adopted to solve equations (35) in which the displacement and velocity corresponding to each time step are defined as follows:

{u}t t  {u}t  [(1   ){u}t  {u}t t ]t

(36a)

1 {u}t t  {u}t  {u}t  [(   ){u}t   {u}t t ]t 2 2

(36b)

where  and  are respectively the parameters adjusted according to the accuracy and stability requirements of integration.   0.5 and   0.25 means that the constant 13

average acceleration method is chosen, in which it is assumed that the velocity is constant from moment t to moment t +t , namely 1 ({u}t  {u}t t ) . 2

The following formulas can be obtained according to formulas (36).

{u}t t 

1 1 1 ({u}t t  {u}t )  {u}t  (  1){u}t 2 t t 2

(37a)

{u}t t 

   ({u}t t  {u}t )  (1  ){u}t  (1  )t{u}t t  2

(37b)

Using formulas (36) and (37), the vibration equilibrium equations at t +t moment can be written as follows

[M]ut t  [C]ut t  [K]ut t   f t t [K]ut t   f 

t t

where

 f t t

(38a) (38b)

is the load column vector at a given time.

[K ]  [K ]  { f }  { f }t t  [M](  [C](

1  [M ]  [C] 2 t t

1 1 1 {u}t  {u}t  (  1){u}t ) 2 t t 2

   {u}t  (  1){u}t  (  1)t{u}t ) t  2

In order to determine the displacement, velocity and acceleration at t +t moment, the procedure shown in Fig. 6 is used to solve equation (35).

Fig. 6. Solution flow chart

2.4 Preliminary verification of bi-nonlinear model Literature data presented by Shen et al [20] are used to preliminarily verify the validity of the bi-nonlinear vibration model. The calculation model parameters and structure in the literature are shown in Table. 1 and Fig. 7 respectively. The pipe is empty and it’s upper and lower ends are fixed, and spring-damper contact elements are set in the middle part. In the reference, the contact force in the middle of the pipeline was analyzed by finite element method and experiment. In this paper, the contact force of the same example presented in the literature is investigated, using the proposed bi-nonlinear vibration model. The simulation time is 14

0.06 seconds and 16 elements are used for discretization both in literature and this paper. The difference between them is that in the literature the longitudinal/lateral coupling vibration effect of the pipe are not taken into account in the finite element analysis. Table.1 Model parameter

Fig. 7. Contact/collision diagram between pipeline and support.

Figure. 8 shows the time history response curves of the contact force obtained from single nonlinear model, bi-nonlinear model and experimental result in literature. It can be found from the figure that the result of the bi-nonlinear model presented in this paper is closer to the experimental result than that of the single-nonlinear model in the literature, both in terms of the amplitude and the change law. Moreover, the bi-nonlinearity model can better reflect the high-frequency response characteristics of the system than the single-nonlinearity model. Fig. 8. Result comparison between the bi-nonlinear model and experiment in literature.

3. Experimental verification In section 2.4, the validity of the bi-nonlinear model is tested preliminarily, but the effect of internal flow is not taken into account, and the analysis object is confined to straight pipe. In this section, based on the similarity principle and taking the actual parameters of a typical gas well in the South China Sea as the object, a FIV experiment of completion tubing in gas well is designed and carried out to further test the validity of the bi-nonlinear vibration, taking into the effect of inflow and wellbore structure. 3.1 Experimental design (1) Simulated experimental parameters Three criterions should be satisfied for the similarity experiment of pipe string vibration: geometrical similarity, motion similarity and dynamic similarity [32, 33]. In this paper, the basic sizes of tubing and casing (inner diameter, outer diameter, tube length, etc.) in the simulation experiment are determined by using geometrical similarity. Since the size difference between the length direction and radial direction are great, the uniform scale ratio is not adopted. The similarity ratios in radial and longitudinal directions are respectively set as 5.0 and 438.0. 15

According to the work presented by Huang [34], the material density and elastic modulus of the experimental pipe string and their counterparts of the actual pipe string should satisfy the following formulas.

p

m

Ep

 

1 5

(39)

Em where  p and Ep are respectively material density and elastic modulus of actual tubing,  m and Em are respectively material density and elastic modulus of experimental pipe,  is similarity ratio. Substituting the actual material parameters p =7850kg / m3 and

Ep =210GPa into

equation (39), the following result is obtained. Em

m



1 1 Ep  kg  5.35 103 GPa   3  5 p m 

(40)

According to formula (40) and referring to the material manual, it can be found that the PE pipe material with Yong’s modulus Em =6.0GPa and density  m =1200kg / m3 can meet the similar experimental requirement。 According to the actual condition of an oil&gas field in the western South China Sea (yield 1.0-2.0 million square /day, the formation pressure 46.7 MPa, the formation temperature 150℃, ground temperature 25℃ and standard atmospheric pressure 0.1 MPa), the similar fluid velocity in pipe can be determined by the state equation and the conversion formula as following

V

PV PV 1 1  2 2 T1 T2

(41)

Q 24  60  60  S

(42)

Similar and actual parameters in the experiment are shown in table 2. Table 2. Simulation experimental parameters of tubing string

(2) Experimental system The experimental system is mainly composed of gas injection subsystem, string subsystem and connection subsystem (Fig. 9), specifically including gas source 16

(screw air compressor, gas storage tank), pipeline, solenoid valve, gas flowmeter, completion string model, casing model, fixing device, axial force device (weight, pulley), etc. Fig. 9. Flow chart of experimental system design

Fig. 10. Schematic diagram of horizontal well test bench

(a) (b) Fig. 11. Experimental system (a) experimental bench and (b) airflow Generator.

Strain gages are used to collect the strain characteristics at different locations of the pipe. As shown in Fig. 12, eight acquisition points are arranged around the pipe (CF1, CF2, IL1, IL2), totaling 32 points. Two strain gauges are arranged vertically and horizontally at each point to eliminate the errors induced by temperature and initial deformation during the experiment. Nodes 1 and 8 are 0.15m away from both ends of the pipe, and eight acquisition points are evenly arranged along the length direction of the pipe with distance 1.0 m. Fig. 12. Strain gauge pasting scheme

(3) Experimental procedure According to the experiment characteristics, the following steps are taken to complete the experiment. ① Turn on the air compressor and make it work for 4 minutes to ensure that the gas storage tank is filled with gas and reaches the required pressure (4MPa). ② Open the valve of the gas storage tank, and control the high-pressure gas to flow out of the gas storage tank through the solenoid valve. The gas flowmeter is used to record the gas flow rate of the pipe string, and the air compressor is used to keep the gas flow rate unchanged during the pipe vibration. ③ The axial strain of the pipe string is measured by strain gauges at different positions of the pipe and are recorded in the computer by dynamic strain gauge. ④ Repeat steps ① ~ ③ to measure a number of groups of data. Through the selection, analysis and conversion of the measured data, the vibration response (dynamic displacement and vibration frequency) of the pipe is obtained.

17

3.2 Comparative analysis between experimental and bi-nonlinear model Using the same parameters as the experiment, the bi-nonlinear vibration model presented in section 2 is used to analyze the nonlinear dynamic response of the tubing string in curved well. The calculation model is shown in Fig. 13, in which the pipe string is divided into 300 elements. The total simulation time is 70s and the step length is 0.0001s. Other parameters used in simulation experiments shown in table 2. Vibration responses at 8 measuring locations (which are the same positions as the places in the experiment where the sensors are installed) are extracted. Fig. 13. Calculation sketch in bi-nonlinear model (a) Measuring point 2

(b) Measuring point 3

(c) Measuring point 6 (d) Measuring point 7 Fig. 14. Lateral displacement time-history response of tubing at different positions in horizontal well.

Fig. 14 shows that the amplitude of the bi-nonlinear vibration model is close to that of the experimental measurement. Both of them indicate that the displacement response is transient before 50 seconds and stable after 50 seconds. The high frequency components of the experimental results are relatively much, which is mainly due to the interference of some field environmental factors. (a) Measuring point 2

(b) Measuring point 3

(c) Measuring point 6

(d) Measuring point 7

Fig. 15. Lateral displacement amplitude-frequency response of tubing at different positions in horizontal well.

Fig. 15 shows that the amplitude-frequency response of the model is basically consistent with the experimental results. Both of them show that the maximum response frequency is about 1.5HZ, and the vibration energy of the pipe string in the range of 0-2.0Hz is relatively large.

(a) Measuring point 1

(b) Measuring point 2

18

(c) Measuring point 3

(d) Measuring point 4

(e) Measuring point 5

(f) Measuring point 6

(g) Measuring point 7

(h) Measuring point 8

Fig. 16. Longitudinal vibration displacement at measuring points of pipe string

Because of the randomness of contact collision in the simulation experiment, It is difficult to determine the position of contact collision in advance, as well as the installation position of sensor. Due to the small gap between tubing and casing in the simulation experiment, it is very difficult to install longitudinal displacement sensor. Therefore, the time history curve of longitudinal displacement is not shown. Here, the longitudinal displacement (Fig. 16) and contact force (Fig. 17) calculated by the bi-nonlinear vibration model are presented. Fig. 16 shows that there is a transient response in the first two seconds. The main reason is the sudden action of gravity. After 25 seconds, the vibration of the pipe string tends to be stable. The longitudinal vibration amplitude of the pipe string near the inclined section is the largest, whereas the amplitudes near the tubing hanger and the bottom packer are the smallest. Fig. 17 shows that at measuring points 1 and 8, the tubing is not in contact with the casing because of the restrictions of tubing hanger and packer. The collision forces at measuring points 2, 6 and 7 which locate near the oblique section are larger and have higher response frequency. The contact force at points 3 and 4 changes slightly, due to the fact that these measuring points locate in the stable inclined section. Under the action of gravity, the tubing in the stable inclined section almost lies on the casing, and the vibration is suppressed. Figs 16 and 17 show that the response of longitudinal displacement and contact force obtained by the model is basically consistent with common knowledge, which further shows the effectiveness of the bi-nonlinear model.

(a) Measuring point 1

(b) Measuring point 2

(c) Measuring point 3

(d) Measuring point 4

19

(e) Measuring point 5

(f) Measuring point 6

(g) Measuring point 7

(h) Measuring point 8

Fig. 17. Contact/collision force between tubing and casing at measuring points of pipe string

4. Conclusion Using the finite element method, energy method and Hamiltonian principle, a bi-nonlinear FIV model of tubing string in oil&gas well is established and verified by a similar experiment. The following conclusions can be drawn. (1) Fatigue and wear phenomenon caused by the vibration of tubing string are becoming more and more prominent in some complex oil&gas wells, such as high production wells and multi-section well. The main reason is that high-speed fluid flows through curved tubing string, and the FIV mechanism will become more prominent. However, the vibration of tubing string is a typical "tube in tube" problem, and it is challenging to perform the theoretical analysis, experimental analysis and field testing. In addition to some necessary simplifications, the bi-nonlinear model presented in this paper takes into account the effects of nonlinear factors such as longitudinal/lateral coupling vibration, contact/collision, and well structure, therefore it is especially suitable for analyzing FIV behavior of tubing string in curved well. (2) Because of the complexity of the problem, it is difficult to validate the bi-nonlinear FIV model using field testing measure since the safety, economy and reliability of the method are difficult to be guaranteed due to the high yield often accompanied by high temperature and high pressure. In this paper, the validity of the bi-nonlinear vibration model is preliminarily validated by a typical example as well as the experimental results in the literature. Then, taking a gas well in the South China Sea as an example, a similar experiment of tubing string in three-section well was carried out. The comparative analysis of displacements, contact forces and amplitude-frequency responses show that the results of the bi-linear model presented in this paper are close to the experimental results and are basically in line with common sense. The model can provide an effective analysis tool for FIV, fatigue and wear analysis of tubing string.

Acknowledgments This work has been supported in part by National Natural Science Foundation of China (Grant No. 51875489), Major Projects of CNOOC (China) Co., Ltd. (Grant No. 20

CNOOC-KJ 135ZDXM24LTDZJ03) and Open Fund of Key Laboratory of Oil & Gas Equipment, Ministry of Education (Southwest Petroleum University) (Grant No. OGE201701-01).

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Table.1 Model parameter Parameter

Val

Parameter

Valu

ue

e 0.99

Pipe length (m)

06 9.52

Inside diameter (mm)

5 15.3

Outside diameter (mm)

(Hz)

Time step length (s)

0.06 0.00 01

Element number

16

2.95

Tubing density (kg/m3)

84

Clearance (mm)

7740 0.12

75

External excitation amplitude (N) External excitation frequency

Computing time (s)

7

Table 2. Simulation experimental parameters of tubing string Parameters

Actual value

Simulatio n value

Parameters

Actual value

Simulation value

Outside diameter of tubing (mm)

114.3

23.0

Inner diameter of tubing (mm)

100.3

20.0

Length of tubing (m)

3500

7.3

Velocity of gas (m/s)

6.0

6.0

Density of Gas (kg/m3)

600.0

600.0

Dynamic viscosity (μPa.s)

11.0

11.0

Outer diameter of casing (mm)

177.8

35.0

Inner diameter of casing (mm)

152.5

30.0

Elastic modulus of tubing (GPa)

210.0

6.0

Density of tubing (kg/m3)

7850

1200

24

Figure 1

Figure 2

25

Figure 3

26

Figure 4

27

Figure 5

28

Figure 6

29

Figure 7

30

Figure 8

Figure 9

31

Figure 10

Figure 11

32

Figure 12

33

Figure 13

34

Figure 14

35

Figure 15

36

Figure 16

37

Figure 17

38