Improved Energy Method on Helical Buckling of Tubing in Vertical and Inclined Wellbore Casing and tubing buckling Lubinski's Error in old energy method Onset and Post buckling behavior of PIP Helical Buckling Updated equations of critical helical buckling forces Helical Buckling Zone - HBZ Confirmation using ABAQUS FEA Model

1. Onset and Post Buckling of Pipe-in-Pipe Helical Buckling
- Using Improved Energy Method
OMAE2018-77032 June 17 - 22, 2018, Madrid, SPAIN
Date: 21 June2018 Prepared for: OMAE 2018
Prepared by: Lixin Gong, PhD

2. 2an employee-owned company
Purpose
• The purpose of this paper is to present theoretical solutions
based on an improved energy method for predicting the
helical buckling (HB) behavior of pipes in vertical, inclined,
and horizontal wells.
2

3. 3an employee-owned company
Outline
3
• Introduction of Improved Energy Method
• Helical buckling of tubing in a vertical well
• Helical buckling of tubing in an inclined well
 Helical Buckling Zone (HBZ)
 Critical helical buckling force
• Verification of critical helical buckling force by ABAQUS FEA
modelling
• Conclusions

4. 4an employee-owned company
Helical Buckling of the Down-Hole Tubular String
• Helical buckling of the down-hole tubular string is always a
concern during drilling and casing/tubing running operations.
4
(courtesy of Gao and Huang 2015: “A review of down-hole tubular string buckling in well engineering” )

5. 5an employee-owned company
Energy Method for PIP Helical Buckling
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• Vertical Well: (Lubinski, et al [1962]) the well-known force-
pitch equation for helical buckling: (However, the deductive
procedure is not correct.)
• Inclined well: (Paslay and Bogy [1964])
• Horizontal well: (Chen et al [1990])
2
2
8
p
EI
F


r
EIw
Fo
sin8

r
EIw
Fo
8


6. 6an employee-owned company
Energy Method for HB
6
• The energy method is based on the law of the Conservation of
Energy:
• Where:
 𝛿𝛱 = Total energy change
 WF = External work by compressive force F
 Ub = Bending strain energy
 Vw = Potential energy of the gravitational force
 VN= Potential energy of the distributed contact normal force
• In the preceding studies, the potential energy term of the
distributed contact normal force (VN) is missing.
𝛿𝛱 = 𝑊 𝐹 − 𝑈𝑏 − 𝑉𝑤 − 𝑉𝑁 = 0
𝛿𝛱 = 𝑊 𝐹 − 𝑈𝑏 − 𝑉𝑤 = 0

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Potential Energy of the Distributed Contact
Force
• The potential energy of the distributed contact force:
• The potential energy of the distributed contact force is corresponding
to the helix configuration relative to a straight line configuration along
the side of the wellbore wall, similar to the potential energy of the
gravitational force.
   
p
N dsNr
p
L
V
0
coscos1 
Straight line configuration along the side
of the wellbore wall
Helix configuration

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Helical Buckling in a Vertical Well

9. 9an employee-owned company
HB in a Vertical Well
• Helix geometry:
2
2
222
2
4
4
4
p
r
rp
r
C





 2
2
222
2
4
4
4
p
EIr
rp
EIr
EICM






𝑁 𝛾 = −𝑀
4𝜋2
𝑝2
cos 𝜃 cos 𝛾 𝑖 − 𝑀
4𝜋2
𝑝2
cos 𝜃 sin 𝛾 𝑗
𝑁 = 𝑀
4𝜋2
𝑝2
cos 𝜃 ≈ 𝐸𝐼𝑟
2𝜋
𝑝
4

10. 10an employee-owned company
HB in a Vertical Well
• The strain energy of the bending of a helix (Lubinski 1962):
• The external work by compression force F during buckling (Lubinski
1962):
• The potential energy of the distributed contact force:
• The law of the Conservation of Energy:
  4
24
2222
24
8
4
8
p
LEIr
rp
LEIr
Ub





 

sin
p
ML
Ub 




sin
cos1
sin2
p
FrL
p
FrL
WF 


   
p
N dsNr
p
L
V
0
coscos1  

sin
p
ML
VN 
0 NwbF VVUW 2
2
8
p
EI
F


EI
rF
N
4
2


11. 11an employee-owned company
Helical Buckling in an Inclined Well

12. 12an employee-owned company
HB in an Inclined Well
• For a fully developed helical buckling in an inclined well with an inclination
angle of , the helix’s distributed contact normal force will become a
function of the hoop angle  (Belayneh, 2006 ):
• To satisfy the equilibrium equation of the forces in the direction normal to
the inclination well:
• A general form of the distributed contact force:
     cossin
4
2
Bw
EI
rF
N 
  0sincos
0
  wpdzN
P
2B
   cossin2sin wAwN 
EI
rF
AwNmean
4
sin
2
 
sin4
2
2
EIw
rF
A   0N

13. 13an employee-owned company
HB in an Inclined Well
• The potential energy of the distributed contact force:
• The potential energy of the gravitational force :
• The law of the Conservation of Energy:
• At the onset of the helical buckling:
     
P
N dzNr
p
L
V
0
coscos1   rLw
ArLw
VN 

sin
2
sin

 rLwVw sin
0 NwbF VVUW
22
4
sin
4 













 p
r
Aw
p
EIF
2A
22
2
sin
4 













 p
r
w
p
EIFcr

14. 14an employee-owned company
HB in an Inclined Well
• For a “long” pipe (i.e. L po), the minimum value of F to initiate helical
buckling can be determined by taking derivative of Fcr w.r.t. pitch p:
• For a “short” pipe (i.e. L po), the minimum value of F to initiate helical
buckling can be determined approximately by:
• For a horizontal well:  = /2 and sin = 1
0


p
Fcr
r
EIw
Fo
sin8

41
sin
8









w
EIr
po
22
2
sin
4 













 L
r
w
L
EIFcr
r
EIw
Fo
8

41
8







w
EIr
po 

15. 15an employee-owned company
HB in an Inclined Well – Pipe’s Properties
• The pipe’s properties used in the following helical buckling examples:
 Production Tubing 5.5": OD = 5.5 in, ID = 4.548 in & wall thickness = 0.476 in
 Self weight of tubing: w=26 lbf/ft
 radial clearance between tubing and hole: r = 0.1875 ft

16. 16an employee-owned company
HB in an Inclined Well – Helical Buckling
Zone (HBZ)
• The green shaded area in-between the two curves is defined as the helical
buckling zone (HBZ):
 The curve of the onset of the helical buckling (i.e. onset curve)
 The curve of the fully developed helical buckling (i.e. Lubinski curve)

17. 17an employee-owned company
Critical Helical Buckling Force
• The log-log plots of the helical pitch vs.
the critical helical buckling force for
wells of various inclination angles
 The Lubinski curve depicts force-pitch relation
for a fully developed helical buckling. For a
“long pipe”, the minimum value of F to initiate
helical buckling and its corresponding helical
pitch are on the Lubinski curve.
 For a “short” pipe, the minimum value of F to
initiate helical buckling and its corresponding
helical pitch are on the onset curves.
 As the helical pitch decreases, the helical
buckling onset curves will converge to the
lower bound of the onset curve, which is half
of the Lubinski Eqn.

18. 18an employee-owned company
ABAQUS FEA Modeling
• Finite element ABAQUS models have been utilized to verify the critical
helical buckling force predicted by the onset Equation.
• FEA beam models using the structural elements:
 Properties of the pipe in the models
 Pipe modeled as PIPE31 elements
 ITT31 elements for contact between pipe and borehole
 No friction between pipe and borehole
 Displacement control
 Periodic boundary conditions at the two ends
• The ABAQUS results show remarkable agreement with the onset
Equation.
Tubing resting at bottom of casing Tubing’s helical buckling

19. 19an employee-owned company
Conclusions
19
• The energy method is improved by adding the term of the potential
energy of the distributed contact normal force.
• For inclined wells,  from 0 to 90 deg, equations are provided to
determine the critical forces required to initiate the helical buckling for
both “long” and “short” pipes. In addition, the post buckling behavior is
also described, and a new concept of HBZ for “short” pipes is introduced
based on the force-pitch plots as an area in-between the helical buckling’s
onset curve and the classical Lubinski curve.
• The lower bound of the onset curve, which is half of the Lubinski Eqn., is
appropriate and conservative to evaluate the critical helical buckling force
for field operations and experiments on the PIP’s helical buckling
response.
2
4 






p
EIFcr


20. 20an employee-owned company
Questions
20

21. 21an employee-owned company
Thank You!

22. 22an employee-owned company
Assumptions of Helical Buckling Study
• The subsequent HB study is based upon the following assumptions:
1) The pipe’s cross-section is uniform. The pipe-to-wellbore’s radial clearance, r, is
uniform.
2) The axial force in the pipe is uniform.
3) The ratio of the radial clearance to the pipe’s helix pitch is very small, i.e. r/p<<1.
4) Small helical deformation is within the pipe material’s linear elastic range.
5) During helical buckling, the pipe is in full contact with the wellbore’s inner surface.
6) Axial work and axial strain energy on the pipe are considered secondary and neglected.
7) The torque induced by the helical buckling is considered to be very small and is
negligible.
8) For situations with inclined and horizontal wells, the helical buckling mode is assumed
to be a helix with uniform bending moment (or curvature) to achieve analytical
solutions. Otherwise, the problems can only be solved numerically.
9) The friction between the pipe and the wellbore is not considered.

23. 23an employee-owned company
Buckling of Tubing in a Horizontal Well
• The axial compression load is
normalized by “sqrt(EIw/r)”.
 The normalized load for sinusoidal
buckling is 2.
 The normalized load for helical
buckling is approx. 2.828.
• The tubing’s axial response deviates
from the trivial solution when the
sinusoidal buckling starts.
• Once the helical buckling initiates,
there is a instability in the tubing
system. The axial load drop to
critical helical buckling load, and
then starts to increase again.

24. 24an employee-owned company
Buckling of Tubing in a Horizontal Well
• Lateral displacement is magnified 100 times
X-Z Plane View Y-Z Plane ViewX-Y Plane View