# US 7,412,368 B2 · Methods and computer-readable media for determining design parameters to prevent tubing buckling in deviated wellbores

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- Number: US 7,412,368 B2
- Document ID: US7412368B2
- Status: Granted
- Kind: B2
- Application number: 11/274,637
- Filed: 2005-11-15
- Published: 2006-05-18
- Granted: 2008-08-12
- Earliest claimed: 2004-11-15
- Assignees: Landmark Graphics Corporation
- Inventors: Robert F. Mitchell
- Web page: https://patentgrep.com/patent/US7412368B2

## CPC

- E21B 17/00: Drilling rods or pipes; Flexible drill strings; Kellies; Drill collars; Sucker rods; {Cables;} Casings; Tubings
- E21B 41/00: Equipment or details not covered by groups E21B15/00 - E21B40/00
- E21B 43/10: Setting of casings, screens, liners {or the like} in wells
- E21B: EARTH OR ROCK DRILLING; OBTAINING OIL, GAS, WATER, SOLUBLE OR MELTABLE MATERIALS OR A SLURRY OF MINERALS FROM WELLS

## Abstract

Methods and computer-readable media are provided for determining design parameters for oil well casing and tubing to prevent buckling in deviated wellbores. Well parameter data including tubing size, tubing weight, well depth, and well geometry is obtained and may be utilized to calculate parameters for predicting the movement of tubing near a packer or centralizer in the deviated wellbore based on the received well parameter data, predicting a total bending moment near the packer or centralizer, predicting a maximum bending stress near the packer or centralizer based on the total bending moment, and predicting the minimum axial force necessary to initiate buckling due to friction, and predicting the onset of buckling for the connection of tubing of different sizes. After the parameters have been calculated, they may be utilized in the design of the oil well casing and tubing to prevent buckling in the deviated wellbore.

## Claims

### Claim 1

A method of determining design parameters for oil well casing and tubing to prevent buckling in a deviated wellbore, comprising:
receiving well parameter data comprising at least one of tubing size, tubing weight, well depth, and well geometry;
calculating a first parameter used in predicting movement of the tubing near at least one boundary condition in the deviated wellbore based on the received well parameter data, wherein calculating a first parameter used in predicting movement of the tubing near at least one boundary condition in the deviated wellbore based on the received well parameter data comprises calculating a parameter used in predicting the movement of the tubing near a packer in the deviated wellbore using the formula:
θ
⁢
⁢
(
ξ
)
=
ln
[
cosh
⁢
⁢
(
2
2
⁢
Δ
⁢
⁢
ξ
+
ϕ
s
)
cosh
⁢
⁢
(
ϕ
s
)
]
where θ(ξ) is a buckling parameter for a beam-column solution for tubing located near the packer in the deviated wellbore;
Δξ is the change in dimensionless length associated with the tubing where ξ is given by the relationship:
ξ
=
s
⁢
P
EI
where s is the measured depth of the tubing;
P is the axial buckling force of the tubing; and
EI is the bending stiffness of the tubing; and
φs is a numerical constant;
calculating a second parameter used in predicting a total bending moment near the at least one boundary condition based on the received well parameter data;
calculating a third parameter used in predicting a maximum bending stress near the at least one boundary condition in the deviated wellbore based on the total bending moment; and
calculating a fourth parameter used in predicting a minimum axial force necessary to initiate buckling based on the received well parameter data, wherein the first, second, third, and fourth parameters are utilized in a design of the oil well casing and tubing to prevent buckling in the deviated wellbore.

### Claim 2 (depends on 1)

The method of claim 1 further comprising:
calculating a fifth parameter used in predicting an onset of buckling for a connection of tubing of different sizes based on the received well parameter data, wherein the fifth parameter is utilized in the design of the oil well casing and tubing to prevent buckling in the deviated wellbore.

### Claim 3 (depends on 2)

The method of claim 2, wherein calculating a fifth parameter used in predicting an onset of buckling for a connection of tubing of different sizes based on the received well parameter data comprises using the formula:
ⅆ
θ
⁢
⁢
(
s
)
ⅆ
s
=
±
2
⁢
α
j
⁢
r
i
⁢
sd
⁡
(
λs
-
2
⁢
⁢
π
α
b
⁢
λ
,
k
)
2
⁢
r
j
⁢
1
+
∑
+
(
1
-
∑
)
⁢
sd
2
⁡
(
λ
⁢
⁢
s
-
2
⁢
⁢
π
α
b
⁢
λ
,
k
)
where
ⅆ
θ
⁢
⁢
(
s
)
ⅆ
s
is a buckling parameter for a beam-column solution to predict the onset of buckling for a connection of a first tubing and a second tubing;
ri is the radial clearance of the first tubing;
rj is the radial clearance of the second tubing, wherein ri<rj;
∑
=
r
j
2
-
r
i
2
r
j
2
k
=
1
-
∑
1
+
∑
α
j
=
P
E
j
⁢
I
j
,
where P is the buckling force associated with the connection of the first tubing and the second tubing and EjIj is the bending stiffness of the second tubing;
λ
=
2
2
⁢
α
j
⁢
1
+
Σ
;
α
b
=
F
E
b
⁢
I
b
,
where the subscript b refers to the properties of the beam-column solution, where F is the axial buckling force associated with the connection of the first tubing and the second tubing, and EbIb is the bending stiffness; and
s
∈
(
0
,
2
⁢
⁢
π
α
b
)
,
where sd(*,k) is a Jacobi elliptic function with parameter k.

### Claim 4 (depends on 1)

The method of claim 1, wherein calculating a second parameter used in predicting a total bending moment near at least one boundary condition in the deviated wellbore based on the received well parameter data comprises calculating a parameter used in predicting a total bending moment of the tubing near at least one of a packer and a centralizer in the deviated wellbore using the formula:
M
=
Fr
⁢
(
ⅆ
2
⁢
u
1
ⅆ
ξ
2
)
2
+
(
ⅆ
2
⁢
u
2
ⅆ
ξ
2
)
2
where ξ is a dimensionless length;
M is the total bending moment in a beam-column solution for the packer or centralizer in the deviated wellbore;
F is the bending stiffness of the tubing;
r is the radial clearance of the tubing in the packer or centralizer; and
u1 and u2 are measures of the lateral displacement of the tubing in the deviated wellbore.

### Claim 5 (depends on 4)

The method of claim 4, wherein calculating a third parameter used in predicting a maximum bending stress near the at least one boundary condition based on the total bending moment comprises calculating a parameter used in predicting a maximum bending stress near the at least one of a packer and a centralizer in the deviated wellbore using the formula:
σ
b
=
M
⁢
⁢
d
o
2
⁢
I
where σb the maximum bending stress;
do is the outside diameter of the tubing; and
I is the moment of inertia of the tubing.

### Claim 6 (depends on 1)

The method of claim 1, wherein calculating a fourth parameter used in predicting a minimum axial force necessary to initiate buckling when the tubing is constrained by friction forces based on the received well parameter data comprises using the formula:
F
=
GJ
r
p
2
+
4
⁢
EIw
c
r
c
where F is the minimum axial force necessary to initiate buckling in the tubing when the tubing is rolling in the deviated wellbore;
G is the shear modulus of the tubing;
J is the polar moment of inertia of the tubing;
rp is the radius of the tubing;
EI is the bending stiffness of the tubing;
wc is the contact load between the deviated wellbore and the tubing; and
rc is the radial clearance of the tubing.

### Claim 7

A computer-readable storage medium having computer-executable instructions, which when executed by a computer cause the computer to perform a method of determining design parameters for oil well casing and tubing to prevent buckling in a deviated wellbore, the method comprising:
receiving well parameter data comprising at least one of tubing size, tubing weight, well depth, and well geometry;
calculating a first parameter used in predicting movement of the tubing near at least one boundary condition in the deviated wellbore based on the received well parameter data, wherein calculating a first parameter used in predicting movement of the tubing near at least one boundary condition in the deviated wellbore based on the received well parameter data comprises calculating a parameter used in predicting the movement of the tubing near a packer in the deviated wellbore using the formula:
θ
⁢
⁢
(
ξ
)
=
ln
⁢
[
cosh
⁢
⁢
(
2
2
⁢
Δ
⁢
⁢
ξ
+
ϕ
s
)
cosh
⁢
⁢
(
ϕ
s
)
]
where θ(ξ) is a buckling parameter for a beam-column solution for tubing located near the packer in the deviated wellbore;
Δξ is the change in dimensionless length associated with the tubing where ξ is given by the relationship:
ξ
=
s
⁢
P
EI
where s is the measured depth of the tubing;
P is the axial buckling force of the tubing; and
EI is the bending stiffness of the tubing; and
φs is a numerical constant;
calculating a second parameter used in predicting a total bending moment near the at least one boundary condition based on the received well parameter data;
calculating a third parameter used in predicting a maximum bending stress near the at least one boundary condition in the deviated wellbore based on the total bending moment; and
calculating a fourth parameter used in predicting a minimum axial force necessary to initiate buckling based on the received well parameter data, wherein the first, second, third, and fourth parameters are utilized in a design of the oil well casing and tubing to prevent buckling in the deviated wellbore.

### Claim 8 (depends on 7)

The computer-readable storage medium of claim 7 further comprising:
calculating a fifth parameter used in predicting an onset of buckling for a connection of tubing of different sizes based on the received wherein the fifth parameter is utilized in the design of the oil well casing and tubing to prevent buckling in the deviated wellbore.

### Claim 9 (depends on 8)

The computer-readable storage medium of claim 8, wherein calculating a fifth parameter used in predicting an onset of buckling for a connection of tubing of different sizes based on the received well parameter data comprises using the formula:
ⅆ
θ
⁡
(
s
)
ⅆ
s
=
±
2
⁢
α
j
⁢
r
i
⁢
sd
⁡
(
λ
⁢
⁢
s
-
2
⁢
⁢
π
α
b
⁢
λ
,
k
)
2
⁢
r
j
⁢
1
+
∑
+
(
1
-
∑
)
⁢
sd
2
⁡
(
λ
⁢
⁢
s
-
2
⁢
⁢
π
α
b
⁢
λ
,
k
)
where
ⅆ
θ
⁡
(
s
)
ⅆ
s
is a buckling parameter for a beam-column solution to predict the onset of buckling for a connection of a first tubing and a second tubing;
ri is the radial clearance of the first tubing;
rj is the radial clearance of the second tubing, wherein ri<rj;
∑
=
r
j
2
-
r
i
2
r
j
2
k
=
1
-
∑
1
+
∑
α
j
=
P
E
j
⁢
I
j
,
where P is the buckling force associated with the connection of the first tubing and the second tubing and EjEj is the bending stiffness of the second tubing;
λ
=
2
2
⁢
α
j
⁢
1
+
Σ
;
α
b
=
F
E
b
⁢
I
b
,
where the subscript b refers to the properties of the beam-column solution, where F is the axial buckling force associated with the connection of the first tubing and the second tubing, and EbIb is the bending stiffness; and
s
∈
(
0
,
2
⁢
π
α
b
)
,
where sd(*,k) is a Jacobi elliptic function with parameter k.

### Claim 10 (depends on 7)

The computer-readable storage medium of claim 7, wherein calculating a second parameter used in predicting a total bending moment near at least one boundary condition in the deviated wellbore based on the received well parameter data comprises calculating a parameter used in predicting a total bending moment of the tubing near at least one of a packer and a centralizer in the deviated wellbore using the formula:
M
=
Fr
⁢
(
ⅆ
2
⁢
u
1
ⅆ
ξ
2
)
2
+
(
ⅆ
2
⁢
u
2
ⅆ
ξ
2
)
2
where ξ is a dimensionless length;
where M is the total bending moment in a beam-column solution for the packer or centralizer in the deviated wellbore;
F is the axial buckling force of the tubing;
r is the radial clearance of the tubing in the packer or centralizer; and
u1 and u2 are measures of the lateral displacement of the tubing in the deviated wellbore.

### Claim 11 (depends on 10)

The computer-readable storage medium of claim 10, wherein calculating a third parameter used in predicting a maximum bending stress near the at least one boundary condition based on the total bending moment comprises calculating a parameter used in predicting a maximum bending stress near the at least one of a packer and a centralizer in the deviated wellbore using the formula:
σ
b
=
Md
o
2
⁢
I
where σb the maximum bending stress;
do is the outside diameter of the tubing; and
I is the moment of inertia of the tubing.

### Claim 12 (depends on 7)

The computer-readable storage medium of claim 7, wherein calculating a fourth parameter used in predicting a minimum axial force necessary to initiate buckling when the tubing is constrained by friction forces based on the received well parameter data comprises using the formula:
F
=
GJ
r
p
2
+
4
⁢
EIw
c
r
c
where F is the minimum axial force necessary to initiate buckling in the tubing when the tubing is rolling in the deviated wellbore;
G is the shear modulus of the tubing;
J is the polar moment of inertia of the tubing;
rp is the radius of the tubing;
EI is the bending stiffness of the tubing;
wc is the contact load between the deviated wellbore and the tubing; and
rc is the radial clearance of the tubing.

### Claim 13

A method of determining design parameters for oil well casing and tubing to prevent buckling in a deviated wellbore, comprising:
receiving well parameter data comprising at least one of tubing size, tubing weight, well depth, and well geometry;
calculating a first parameter used in predicting movement of the tubing near at least one boundary condition in the deviated wellbore based on the received well parameter data, wherein calculating a first parameter used in predicting movement of the tubing near at least one boundary condition in the deviated wellbore based on the received well parameter data comprises calculating a parameter used in predicting the movement of the tubing near a centralizer in the deviated wellbore using the formula:
θ
⁡
(
ξ
)
=
ln
[
cosh
(
2
2
⁢
⁢
Δ
⁢
⁢
ξ
+
ϕ
s
)
cosh
⁢
⁢
(
ϕ
s
)
]
where θ(ξ) is a buckling parameter for a beam-column solution for tubing located near the centralizer in the deviated wellbore;
Δξ is the change in dimensionless length associated with the tubing where ξ is given by the relationship:
ξ
=
s
⁢
P
EI
where s is the measured depth of the tubing;
P is the axial buckling force of the tubing;
EI is the bending stiffness of the tubing; and
φc is a numerical constant;
calculating a second parameter used in predicting a total bending moment near the at least one boundary condition based on the received well parameter data;
calculating a third parameter used in predicting a maximum bending stress near the at least one boundary condition in the deviated wellbore based on the total bending moment;
calculating a fourth parameter used in predicting a minimum axial force necessary to initiate buckling based on the received well parameter data; and
calculating a fifth parameter used in predicting an onset of buckling for the connection of tubing of different sizes based on the received well parameter data, wherein the at least one boundary condition comprises at least one of a centralizer installed in the deviated wellbore to concentrically position the oil well casing and a packer installed in the deviated wellbore to hold the tubing and wherein the first, second, third, fourth parameters, and fifth parameters are utilized in a design of the oil well casing and tubing to prevent buckling in the deviated wellbore.

### Claim 14 (depends on 13)

The method of claim 13, wherein calculating a second parameter used in predicting a total bending moment near at least one boundary condition in the deviated wellbore based on the received well parameter data comprises calculating a parameter used in predicting a total bending moment near at least one of a packer and a centralizer in the deviated wellbore using the formula:
M
=
Fr
⁢
(
ⅆ
θ
ⅆ
ξ
)
4
+
(
ⅆ
2
⁢
θ
ⅆ
2
⁢
ξ
)
2
where ξ is a dimensionless length;
M is the total bending moment in a full contact solution for the packer or centralizer in the deviated wellbore;
F is the axial buckling force of the tubing;
r is the radial clearance of the tubing in the packer or centralizer; and
θ is the angle between a tubing center location and an x coordinate on a coordinate axis from the tubing center location to a point tangent to the wall of the deviated wellbore, wherein x=dθ/dξ.

### Claim 15 (depends on 13)

The method of claim 13, wherein calculating a fourth parameter used in predicting a minimum axial force necessary to initiate buckling when the tubing is constrained by friction forces based on the received well parameter data comprises using the formula:
F
=
4
⁢
EIw
c
r
c
where F is the minimum axial force necessary to initiate buckling in the tubing when the tubing is rotating in the deviated wellbore;
EI is the bending stiffness of the tubing;
rp is the radius of the tubing;
rc is the radial clearance of the tubing; and
wc is the contact load between the deviated wellbore and the tubing, wherein wc is given by the relationship:
w
c
=
(
w
bp
⁢
n
z
-
F
⁢
⁢
κ
)
2
+
(
w
bp
⁢
b
z
)
2
1
+
μ
2
where wbp is the buoyant weight of the tubing;
nz is the vertical component of the normal to the trajectory of the deviated wellbore;
bz is the vertical component to the binormal to the trajectory of the deviated wellbore;
κ is the curvature of the deviated wellbore; and
μ is the dynamic coefficient of friction with respect to the tubing in the deviated wellbore.

### Claim 16

A computer-readable storage medium having computer-executable instructions, which when executed by a computer cause the computer to perform a method of determining design parameters for oil well casing and tubing to prevent buckling in a deviated wellbore, the method comprising:
receiving well parameter data comprising at least one of tubing size, tubing weight, well depth, and well geometry;
calculating a first parameter used in predicting movement of the tubing near at least one boundary condition in the deviated wellbore based on the received well parameter data, wherein calculating a first parameter used in predicting movement of the tubing near at least one boundary condition in the deviated wellbore based on the received well parameter data comprises calculating a parameter used in predicting the movement of the tubing near a centralizer in the deviated wellbore using the formula:
θ
⁡
(
ξ
)
=
ln
[
cosh
(
2
2
⁢
⁢
Δ
⁢
⁢
ξ
+
ϕ
c
)
cosh
⁢
⁢
(
ϕ
c
)
]
where θ(ξ) is a buckling parameter for a beam-column solution for tubing located near the centralizer in the deviated wellbore;
Δξ is the change in dimensionless length associated with the tubing where ξ is given by the relationship:
ξ
=
s
⁢
P
EI
where s is the measured depth of the tubing;
P is the axial buckling force of the tubing; and
EI is the bending stiffness of the tubing; and
φc is a numerical constant;
calculating a second parameter used in predicting a total bending moment near the at least one boundary condition based on the received well parameter data;
calculating a third parameter used in predicting a maximum bending stress near the at least one boundary condition in the deviated wellbore based on the total bending moment; and
calculating a fourth parameter used in predicting a minimum axial force necessary to initiate buckling based on the received well parameter data, wherein the first, second, third, and fourth parameters are utilized in a design of the oil well casing and tubing to prevent buckling in the deviated wellbore.

### Claim 17 (depends on 16)

The computer-readable storage medium of claim 16, wherein calculating a second parameter used in predicting a total bending moment near at least one boundary condition in the deviated wellbore based on the received well parameter data comprises calculating a parameter used in predicting a total bending moment near at least one of a packer and a centralizer in the deviated wellbore using the formula:
M
=
Fr
⁢
(
ⅆ
θ
ⅆ
ξ
)
4
+
(
ⅆ
2
⁢
θ
ⅆ
2
⁢
ξ
)
2
where ξ is a dimensionless length;
M is the total bending moment in a full contact solution for the packer or centralizer in the deviated wellbore;
F is the bending stiffness of the tubing;
r is the radial clearance of the tubing in the packer or centralizer; and
θ is the angle between a tubing center location and an x coordinate on a coordinate axis from the tubing center location to a point tangent to the wall of the deviated wellbore, wherein x=dθ/dξ.

### Claim 18 (depends on 16)

The computer-readable storage medium of claim 16, wherein calculating a fourth parameter used in predicting a minimum axial force necessary to initiate buckling when the tubing is constrained by friction forces based on the received well parameter data comprises using the formula:
F
=
4
⁢
EIw
c
r
c
where F is the minimum axial force necessary to initiate buckling in the tubing when the tubing is rotating in the deviated wellbore;
EI is the bending stiffness of the tubing;
rp is the radius of the tubing;
rc is the radial clearance of the tubing; and
wc is the contact load between the deviated wellbore and the tubing,
wherein wc is given by the relationship:
w
c
=
(
w
bp
⁢
n
z
-
F
⁢
⁢
κ
)
2
+
(
w
bp
⁢
b
z
)
2
1
+
μ
2
where wbp is the buoyant weight of the tubing;
nz is the vertical component of the normal to the trajectory of the deviated wellbore;
bz is the vertical component to the binormal to the trajectory of the deviated wellbore;
κ is the curvature of the deviated wellbore; and
μ is the dynamic coefficient of friction with respect to the tubing in the deviated wellbore.