Systems and methods for determining the moments and forces of two concentric pipes within a wellbore
Systems and methods for determining the bending moment and shear force of tubing and casing when the tubing buckles and contacts the casing.
1. A method for determining the moments and forces of two concentric pipes within a wellbore, comprising:
determining an external pipe displacement using a computer processor;
determining whether the external pipe contacts the wellbore based on the external pipe displacement;
determining a bending moment and a shear force of an internal pipe and the external pipe based on contact between the internal pipe and the external pipe and the external pipe displacement if the external pipe does not contact the wellbore;
determining whether contact forces between the internal pipe and the external pipe and between the external pipe and the wellbore are greater than or equal to zero if the external pipe contacts the wellbore;
determining the bending moment and the shear force of the internal pipe and the external pipe, using the computer processor, based on contact between the internal pipe and the external pipe and contact between the external pipe and the wellbore if the contact forces between the internal pipe and the external pipe and between the external pipe and the wellbore are greater than or equal to zero;
determining a displacement solution using a contact force between the internal pipe and the external pipe equal to zero if the contact forces between the internal pipe and the external pipe and between the external pipe and the wellbore are not greater than or equal to zero;
determining whether there is another displacement solution using a contact force between the external pipe and the wellbore equal to zero if the contact forces between the internal pipe and the external pipe and between the external pipe and wellbore are not greater than or equal to zero; and
determining the bending moment and the shear force of the internal pipe and the external pipe based on the displacement solution or the another displacement solution if the contact forces between the internal pipe and the external pipe and between the external pipe and the wellbore are not greater than or equal to zero.
2. The method of claim 1 , further comprising selecting the displacement solution to determine the bending moment and the shear force of the internal pipe and the external pipe if there is not another displacement solution.
3. The method of claim 1 , further comprising selecting the displacement solution to determine the bending moment and the shear force of the internal pipe and the external pipe if the displacement solution produces a total potential energy for a system represented by the internal pipe and the external pipe that is less than a total potential energy for the system produced by the another displacement solution.
4. The method of claim 1 , further comprising selecting the another displacement solution to determine the bending moment and the shear force of the internal pipe and the external pipe if the another displacement solution produces a total potential energy for a system represented by the internal pipe and the external pipe that is less than a total potential energy for the system produced by the displacement solution.
5. The method of claim 1 , further comprising performing a stress analysis of the internal pipe and the external pipe based on the bending moment and the shear force of the internal pipe and the external pipe.
6. The method of claim 1 , wherein
υ
=
r
c
PE
t
I
t
2
FE
t
I
t
+
P
(
E
c
I
c
-
E
t
I
t
)
is used to determine the casing displacement; r c is nominal radial clearance between the tubing and casing; P is axial compression in tubing; E t is Young's modulus of the tubing; I t is moment of inertia of the tubing; F is axial tension in casing; E c is Young's modulus of the casing and I c is moment of inertia of the casing.
7. The method of claim 1 , wherein
M
t
=
M
t
=
E
t
I
t
(
r
c
+
υ
)
β
2
M
c
=
r
c
P
2
E
c
I
c
2
P
(
E
c
I
c
-
E
t
I
t
)
+
4
FE
t
I
t
V
t
=
(
r
c
+
υ
)
β
E
t
I
t
β
2
-
P
V
c
=
F
-
PE
c
I
c
E
t
I
t
are used to determine the bending moment and the shear force of the internal pipe and the external pipe if the external pipe does not contact the wellbore; M t is bending moment of the tubing; E t is Young's modulus of the tubing; I t is moment of inertia of the tubing; r c is nominal radial clearance between the tubing and casing; (υ) is casing displacement; β is a possible displacement solution; M c is bending moment of the casing; E c is Young's modulus of the casing; I c is moment of inertia of the casing; F is axial tension in casing; V t is shear force in the tubing; P is axial compression in tubing; and V c is shear force in the casing.
8. The method of claim 1 , wherein
β
2
=
Pr
ic
2
-
Fr
oc
2
E
I
t
r
ic
2
+
EI
c
r
oc
2
r
ic
[
P
β
2
-
E
t
I
t
β
4
]
=
w
tc
r
oc
[
E
c
I
c
β
4
+
F
β
2
]
=
-
w
wc
+
w
tc
are used to determine the contact forces between the internal pipe and the external pipe and between the external pipe and the wellbore; P is axial compression in tubing; r ic is r oc −t c ; r oc is nominal radial clearance between the casing and exterior wellbore; t c is the thickness of the casing; F is axial tension in casing; E is Young's modulus; I t is moment of inertia of the tubing; I c is moment of inertia of the casing; E t is Young's modulus of the tubing; w tc is the contact force between the tubing and casing; E c is Young's modulus of the casing; β is a possible displacement solution; and w wc is the contact force between the wellbore and the casing.
9. The method of claim 1 , wherein
β
2
=
Pr
ic
2
-
Fr
oc
2
E
I
t
r
ic
2
+
EI
c
r
oc
2
is used to determine the bending moment and the shear force of the internal pipe and the external pipe if the contact forces between the internal pipe and the external pipe and between the external pipe and the wellbore are greater than or equal to zero; P is axial compression in tubing; r ic is r oc −t c ; r oc is nominal radial clearance between the casing and exterior wellbore; t c is the thickness of the casing; F is axial tension in casing; E is Young's modulus; I t is moment of inertia of the tubing; and I c is moment of inertia of the casing.
10. The method of claim 1 , wherein
w
tc
=
0
⇒
β
2
=
P
E
t
I
t
is used to determine the displacement solution; w tc is the contact force between the tubing and casing; P is axial compression in tubing; E t is Young's modulus of the tubing; and I t is moment of inertia of the tubing.
11. The method of claim 10 , wherein
w
wc
=
0
⇒
β
2
=
Pr
ic
-
Fr
oc
E
t
I
t
r
ic
+
E
c
I
c
r
oc
is used to determine the another displacement solution; w wc is the contact force between the wellbore and the casing; P is axial compression in tubing; r ic is r oc −t c ; r oc is nominal radial clearance between the casing and exterior wellbore; t c is the thickness of the casing; F is axial tension in casing; E t is Young's modulus of the tubing; I t is moment of inertia of the tubing; E c is Young's modulus of the casing; and I c is moment of inertia of the casing.
12. The method of claim 11 , wherein
w
tc
=
0
⇒
β
2
=
P
E
t
I
t
or
w
wc
=
0
⇒
β
2
=
Pr
ic
-
Fr
oc
E
t
I
t
r
ic
+
E
c
I
c
r
oc
is used to determine the bending moment and the shear force of the internal pipe and the external pipe if the contact forces between the internal pipe and the external pipe and between the external pipe and the wellbore are not greater than or equal to zero; w tc is the contact force between the tubing and casing; w wc is the contact force between the wellbore and the casing; P is axial compression in tubing; r ic is r oc −t c ; r oc is nominal radial clearance between the casing and exterior wellbore; t c is the thickness of the casing; F is axial tension in casing; E t is Young's modulus of the tubing; I t is moment of inertia of the tubing; E c is Young's modulus of the casing; and I c is moment of inertia of the casing.
13. The method of claim 3 , wherein
U= ½( E c I c r oc 2 +E t I t r ic 2 )β 4 +½( Fr oc 2 −Pr oc 2 )β 2
is used to determine the total potential energy for the system; E c is Young's modulus of the casing; I c is moment of inertia of the casing, r oc is nominal radial clearance between the casing and exterior wellbore; E t is Young's modulus of the tubing: I t is moment of inertia of the tubing; r ic is r oc −t c ; t c is the thickness of the casing; β is a possible displacement solution; F is axial tension in casing; and P is axial compression in tubing.
14. A non-transitory program carrier device tangibly carrying computer executable instructions for determining the moments and forces of two concentric pipes within a wellbore, the instructions being executable to implement:
determining an external pipe displacement;
determining whether the external pipe contacts the wellbore based on the external pipe displacement;
determining a bending moment and a shear force of an internal pipe and the external pipe based on contact between the internal pipe and the external pipe and the external pipe displacement if the external pipe does not contact the wellbore;
determining whether contact forces between the internal pipe and the external pipe and between the external pipe and the wellbore are greater than or equal to zero if the external pipe contacts the wellbore;
determining the bending moment and the shear force of the internal pipe and the external pipe based on contact between the internal pipe and the external pipe and contact between the external pipe and the wellbore if the contact forces between the internal pipe and the external pipe and between the external pipe and the wellbore are greater than or equal to zero;
determining a displacement solution using a contact force between the internal pipe and the external pipe equal to zero if the contact forces between the internal pipe and the external pipe and between the external pipe and the wellbore are not greater than or equal to zero;
determining whether there is another displacement solution using a contact force between the external pipe and the wellbore equal to zero if the contact forces between the internal pipe and the external pipe and between the external pipe and wellbore are not greater than or equal to zero; and
determining the bending moment and the shear force of the internal pipe and the external pipe based on the displacement solution or the another displacement solution if the contact forces between the internal pipe and the external pipe and between the external pipe and the wellbore are not greater than or equal to zero.
15. The program carrier device of claim 14 , further comprising selecting the displacement solution to determine the bending moment and the shear force of the internal pipe and the external pipe if there is not another displacement solution.
16. The program carrier device of claim 14 , further comprising selecting the displacement solution to determine the bending moment and the shear force of the internal pipe and the external pipe if the displacement solution produces a total potential energy for a system represented by the internal pipe and the external pipe that is less than a total potential energy for the system produced by the another displacement solution.
17. The program carrier device of claim 14 , further comprising selecting the another displacement solution to determine the bending moment and the shear force of the internal pipe and the external pipe if the another displacement solution produces a total potential energy for a system represented by the internal pipe and the external pipe that is less than a total potential energy for the system produced by the displacement solution.
18. The program carrier device of claim 14 , further comprising performing a stress analysis of the internal pipe and the external pipe based on the bending moment and the shear force of the internal pipe and the external pipe.
19. The program carrier device of claim 14 , wherein
υ
=
r
c
PE
t
I
t
2
FE
t
I
t
+
P
(
E
c
I
c
-
E
t
I
t
)
is used to determine the casing displacement; r c is nominal radial clearance between the tubing and casing; P is axial compression in tubing; E t is Young's modulus of the tubing; I t is moment of inertia of the tubing; F is axial tension in casing; E c is Young's modulus of the casing and I c is moment of inertia of the casing.
20. The program carrier device of claim 14 , wherein
M
t
=
M
t
=
E
t
I
t
(
r
c
+
υ
)
β
2
M
c
=
r
c
P
2
E
c
I
c
2
P
(
E
c
I
c
-
E
t
I
t
)
+
4
FE
t
I
t
V
t
=
(
r
c
+
υ
)
β
E
t
I
t
β
2
-
P
V
c
=
F
-
PE
c
I
c
E
t
I
t
are used to determine the bending moment and the shear force of the internal pipe and the external pipe if the external pipe does not contact the wellbore; M t is bending moment of the tubing; E t is Young's modulus of the tubing; I t is moment of inertia of the tubing; r c is nominal radial clearance between the tubing and casing; (υ) is casing displacement; β is a possible displacement solution; M c is bending moment of the casing; F c is Young's modulus of the casing; I c is moment of inertia of the casing; F is axial tension in casing; V t is shear force in the tubing; P is axial compression in tubing; and V c is shear force in the casing.
21. The program carrier device of claim 14 , wherein
β
2
=
Pr
ic
2
-
Fr
oc
2
E
I
t
r
ic
2
+
EI
c
r
oc
2
r
ic
[
P
β
2
-
E
t
I
t
β
4
]
=
w
tc
r
oc
[
E
c
I
c
β
4
+
F
β
2
]
=
-
w
wc
+
w
tc
are used to determine the contact forces between the internal pipe and the external pipe and between the external pipe and the wellbore; P is axial compression in tubing; r ic is r oc −t c ; r oc is nominal radial clearance between the casing and exterior wellbore; t c is the thickness of the casing; F is axial tension in casing; E is Young's modulus; I t is moment of inertia of the tubing; I c is moment of inertia of the casing; E t is Young's modulus of the tubing; w tc is the contact force between the tubing and casing; E c is Young's modulus of the casing; β is a possible displacement solution; and w wc is the contact force between the wellbore and the casing.
22. The program carrier device of claim 14 , wherein
β
2
=
Pr
ic
2
-
Fr
oc
2
E
I
t
r
ic
2
+
EI
c
r
oc
2
is used to determine the bending moment and the shear force of the internal pipe and the external pipe if the contact forces between the internal pipe and the external pipe and between the external pipe and the wellbore are greater than or equal to zero; P is axial compression in tubing; r ic is r oc −t c ; r oc is nominal radial clearance between the casing and exterior wellbore; t c is the thickness of the casing; F is axial tension in casing; E is Young's modulus; I t is moment of inertia of the tubing; and I c is moment of inertia of the casing.
23. The program carrier device of claim 14 , wherein
w
tc
=
0
⇒
β
2
=
P
E
t
I
t
is used to determine the displacement solution; w tc is the contact force between the tubing and casing; P is axial compression in tubing; E t is Young's modulus of the tubing; and I t is moment of inertia of the tubing.
24. The program carrier device of claim 19 , wherein
w
wc
=
0
⇒
β
2
=
Pr
ic
-
Fr
oc
E
t
I
t
r
ic
+
E
c
I
c
r
oc
is used to determine the another displacement solution; w wc is the contact force between the wellbore and the casing; P is axial compression in tubing; r ic is r oc −t c ; r oc is nominal radial clearance between the casing and exterior wellbore; t c is the thickness of the casing; F is axial tension in casing; E t is Young's modulus of the tubing; I t is moment of inertia of the tubing; E c is Young's modulus of the casing; and I c is moment of inertia of the casing.
25. The program carrier device of claim 20 , wherein
w
tc
=
0
⇒
β
2
=
P
E
t
I
t
or
w
wc
=
0
⇒
β
2
=
Pr
ic
-
Fr
oc
E
t
I
t
r
ic
+
E
c
I
c
r
oc
is used to determine the bending moment and the shear force of the internal pipe and the external pipe if the contact forces between the internal pipe and the external pipe and between the external pipe and the wellbore are not greater than or equal to zero; w tc is the contact three between the tubing and casing; w wc is the contact force between the wellbore and the casing; P is axial compression in tubing; r ic is r oc −t c ; r oc is nominal radial clearance between the casing and exterior wellbore; t c is the thickness of the casing; F is axial tension in casing; E t is Young's modulus of the tubing; I t is moment of inertia of the tubing; E c is Young's modulus of the casing; and I c is moment of inertia of the casing.
26. The program carrier device of claim 16 , wherein
U= ½( E c I c r oc 2 +E t I t r ic 2 )β 4 +½( Fr oc 2 −Pr oc 2 )β 2
is used to determine the total potential energy for the system; E c is Young's modulus of the casing; I c is moment of inertia of the casing, r oc is nominal radial clearance between the casing and exterior wellbore; E t is Young's modulus of the tubing; I t is moment of inertia of the tubing; r ic is r oc −t c ; t c is the thickness of the casing; β is a possible displacement solution; F is axial tension in casing; and P is axial compression in tubing.
27. A method for determining the moments and forces of two concentric pipes within a wellbore, comprising:
determining an external pipe displacement using a computer processor;
determining whether the external pipe contacts the wellbore based on the external pipe displacement; and
determining a bending moment and a shear force of an internal pipe and the external pipe, using the computer processor, based on at least one of contact between the internal pipe and the external pipe and contact between the external pipe and the wellbore.
28. The method of claim 27 , wherein determining the bending moment and the shear force of the internal pipe and the external pipe is based on contact between the internal pipe and the external pipe and the external pipe displacement if the external pipe does not contact the wellbore.
29. The method of claim 27 , wherein determining the bending moment and the shear force of the internal pipe and the external pipe is based on contact between the internal pipe and the external pipe and contact between the external pipe and the wellbore if the contact forces between the internal pipe and the external pipe and between the external pipe and the wellbore are greater than or equal to zero.
30. The method claim 27 , wherein determining the bending moment and the shear force of the internal pipe and the external pipe is based on a displacement solution or another displacement solution if the contact forces between the internal pipe and the external pipe and between the external pipe and the wellbore are not greater than or equal to zero.
31. The method of claim 30 , wherein the displacement solution is determined using a contact force between the internal pipe and the external pipe equal to zero.
32. The method of claim 30 , wherein the another displacement solution is determined using a contact force between the external pipe and wellbore equal to zero.
33. The method of claim 30 , wherein the displacement solution is used to determine the bending moment and the shear force of the internal pipe and the external pipe if there is not another displacement solution.
34. The method of claim 30 , further comprising selecting the displacement solution to determine the bending moment and the shear force of the internal pipe and the external pipe if the displacement solution produces a total potential energy for a system represented by the internal pipe and the external pipe that is less than a total potential energy for the system produced by the another displacement solution.
35. The method of claim 30 , further comprising selecting the another displacement solution to determine the bending moment and the shear force of the internal pipe and the external pipe if the another displacement solution produces a total potential energy for a system represented by the internal pipe and the external pipe that is less than a total potential energy for the system produced by the displacement solution.
36. A non-transitory program carrier device tangibly carrying computer executable instructions for determining the moments and forces of two concentric pipes within a wellbore, the instructions being executable to implement:
determining an external pipe displacement;
determining whether the external pipe contacts the wellbore based on the external pipe displacement; and
determining a bending moment and a shear force of an internal pipe and the external pipe based on at least one of contact between the internal pipe and the external pipe and contact between the external pipe and the wellbore.
37. The program carrier device of claim 36 , wherein determining the bending moment and the shear force of the internal pipe and the external pipe is based on contact between the internal pipe and the external pipe and the external pipe displacement if the external pipe does not contact the wellbore.
38. The program carrier device of claim 36 , wherein determining the bending moment and the shear force of the internal pipe and the external pipe is based on contact between the internal pipe and the external pipe and contact between the external pipe and the wellbore if the contact forces between the internal pipe and the external pipe and between the external pipe and the wellbore are greater than or equal to zero.
39. The program carrier device claim 36 , wherein determining the bending moment and the shear force of the internal pipe and the external pipe is based on a displacement solution or another displacement solution if the contact forces between the internal pipe and the external pipe and between the external pipe and the wellbore are not greater than or equal to zero.
40. The program carrier device of claim 39 , wherein the displacement solution is determined using a contact force between the internal pipe and the external pipe equal to zero.
41. The program carrier device of claim 39 , wherein the another displacement solution is determined using a contact force between the external pipe and wellbore equal to zero.
42. The program carrier device of claim 39 , wherein the displacement solution is used to determine the bending moment and the shear force of the internal pipe and the external pipe if there is not another displacement solution.
43. The program carrier device of claim 39 , further comprising selecting the displacement solution to determine the bending moment and the shear force of the internal pipe and the external pipe if the displacement solution produces a total potential energy for a system represented by the internal pipe and the external pipe that is less than a total potential energy for the system produced by the another displacement solution.
44. The program carrier device of claim 39 , further comprising selecting the another displacement solution to determine the bending moment and the shear force of the internal pipe and the external pipe if the another displacement solution produces a total potential energy for a system represented by the internal pipe and the external pipe that is less than a total potential energy for the system produced by the displacement solution.