1. Chapter 2: Casing Design Calculations of Loads on a Casing

2. Casing Design Introduction
The casing design process involves three distinct operations:
The selection of the casing sizes and setting depths;
The definition of the operational scenarios which will result in burst, collapse and axial loads
The calculation of the magnitude of these loads and selection of an appropriate weight and grade of casing.

3. Calculation of Critical Loads Axial Load or Pipe Body Yield Strength
The axial load on the casing can be either tensile or compressive, depending on the operating conditions.

4. Calculation of Critical Loads Axial Load or Pipe Body Yield Strength
The force Ften tending to pull apart the pipe is resisted by the stregth of the pipe walls, which exert a counterforce F2.
Where syield is the minimum yield strength and As is the cross-sectional area of steel. Thus, the pipe-body strength:
Equation 2 is used to calculate the minimum force that would be expected to cause permanent deformation of the pipe. 1 2

5. Casing Design
Effects of Bending

6. Example 1
Compute the body-yield strength for 20’’, K-55 casing with a nominal wall thickness of 0.635’’ and a nominal weight per foot of 133 lbf/ft.

7. Example 1
Solution:
d = – 2(0.635) = 18.73’’

8. Calculation of Critical Loads Critical Burst Pressure
The casing will experience a net burst loading if the internal radial load exceeds
the external radial load.

9. Calculation of Critical Loads Critical Burst Pressure
Let q be small enough

10. Calculation of Critical Loads Critical Burst Pressure3
where ss is the nominal steel strength. Equation (3) is used only for thin- wall casing. In drilling application, it is suggested that one should use Barlow’s equation to calculate Pbr for thick-wall casing. 4
API recommends use of this equation with wall thickness rounded to the nearest 0.001’’ and the results rounded to the nearest 10 psi.

11. Calculation of Critical Loads Critical Burst Pressure
If casing is subjected to internal pressure higher than external, it is said that casing is exposed to burst pressure. Burst pressure conditions occur during well control operation or squeeze cementing.
Equation (4) is used to calculate the internal pressure at which the tangential stress at the inner wall of the pipe reaches the yield strength of the material. The factor represents the allowable manufactruing tolerance of -12.5% on wall thickness.
Because a burst pressure failure will not occur until after the stress exceeds the ultimate tensile strength, using a yield strength criterion as a measure of burst strength is an inherently conservative assumption.

12. Example 2
Compute the burst-pressure rating for 20’’, K-55 casing with a nominal wall thickness of 0.635’’ and a nominal weight per foot of 133 lbf/ft
Solution:
Rounded to the nearest 10 psi:

13. Calculation of Critical Loads Critical Collapse Pressure
The casing will experience a net collapse loading if the external radial load exceeds the internal radial load. The greatest collapse load on the casing will occur if the casing is evacuated (empty) for any reason.

14. Calculation of Critical Loads Critical Collapse Pressure
If external pressure exceeds internal pressure, the casing is subjected to collapse. Such conditions may exist during cementing operations or well evacuation. Collapse strength is primarily function of the material’s yield strength and its slenderness ratio, dn/t. Lame’s equations: Pe Pi ri ro r st sr dn t

15. Calculation of Critical Loads Critical Collapse Pressure5 6
pe, pi – external and internal pressure
sr, st – radial and tangential stresses

16. Calculation of Critical Loads Critical Collapse Pressure
Equations (5) and (6) are used under no axial tension or axial compression.
Data in Table 7.6 (Applied Drilling Engineering) apply only for zero axial tension and no pipe bending.
The maximal tangential stress, stmax, occurs at the internal surface of the pipe where r = ri.

17. Example 3
Consider a drillpipe of E-75 4 ½’’ outer diameter with a unit weight of 20 lb/ft inside a wellbore filled with 9.5 ppg mud. At a location of 3800 ft from the surface, pressure inside the pipe is 2000 psi, and pressure outside the pipe is 1700 psi. Determine the tangential and radial stresses at r = ro.

18. Example 3
E-75 4 ½’’ and 20 lb/ft drillpipe has an inner diameter of 3.64 in. Considering “r” is equal to ro = 4.5’’

19. Calculation of Critical Loads
Critical Collapse Pressure – Collapse pressure Regimes
Yield Range
Elastic Range
Transition Range
Stress
Strain
Plastic range

20. Calculation of Critical Loads
Critical Collapse Pressure – Collapse pressure Regimes
Primary collapse loads are generated by the hydrostatic head of the fluid column outside the casing string. These fluids are usually drilling fluids and sometimes cement slurry.
Casing is also subjected to sever collapse pressure when drilling through troublesome formations such as: plastic clays and salts.
Strength of the casing under external pressure depends in: length, diameter, wall thickness of the casing and the physical properties of the casing materials (yield point, elastic limit, poisson’s ration, etc.).

21. Calculation of Critical Loads
Critical Collapse Pressure – Collapse pressure Regimes
Casing having a low dn/t ratio and low strength, reaches the critical collapse value as soon as the material begins to yield under the action of external pressure.
Casings exhibit ideally plastic collapse behavior and the failure due to external pressure occurs in the so-called yield range.
Casing with high dn/t ratio and high strength, collapses below the yield strength of the material. In this case, failure is caused by purely elastic deformation of the casing. The collapse behavior is known as failure in the elastic range.

22. Calculation of Critical Loads
Critical Collapse Pressure – Yield Strength Collapse
The collapse strength criteria consist of four collapse regimes determined by yield strength and dn/t. Each criterion is discussed next in order of increasing dn/t.
Yield strength collapse: Yield strength collapse is based on yield at the inner wall. This criterion does not represent a “collapse” pressure at all. For thick wall pipes (dn/t < 15), the tangential stress exceeds the yield strength of the material before a collapse instability failure occurs.
Assumed that the pipe is subjected only to an external pressure pe. From eq. (6), the absolute value of tangential stress st is always greatest at the inner wall of the pipe. Hence, the yield strength collapse occurs at the inner wall: r = ri then equation (6) becomes:

23. Calculation of Critical Loads
Critical Collapse Pressure – Yield Strength Collapse 7 8 9
Rearrange equation (8) gives equation (9) to calculate the critical pressure for yield strength collapse, Pcr

24. Calculation of Critical Loads
Critical Collapse Pressure – Yield Strength Collapse
Plastic collapse:
Plastic collapse is based on empirical data from 2,488 tests of K-55, N-80 and P-110 seamless casing. No analytic expression has been derived that accurately models collapse behavior in this regime. The minimum collapse pressure for the plastic range of collapse is calculated by equation (10). 10

25. Calculation of Critical Loads
Critical Collapse Pressure – Transition Collapse
Transition Collapse:
Transition collapse is obtained by a numerical curve fitting between the plastic and elastic regimes. The minimum collapse pressure for the plastic-to-elastic transition zone is calculated by equation (11) 11

26. Calculation of Critical Loads
Critical Collapse Pressure – Elastic Collapse
Elastic Collapse:
Elastic collapse is based on theoretical elastic instability failure; this criterion is independent of yield strength and applicable to thin-wall pipe (dn/t > 25). The minimum collapse pressure for the elastic range of collapse is calculated by using equation (12)
Most oilfield tubulars experience collapse in the plastic and transition regimes. 12

27. Calculation of Critical Loads Critical Collapse Pressure

28. Calculation of Critical Loads Critical Collapse Pressure

29. Calculation of Critical Loads Critical Collapse Pressure
Apply only when axial stress is zero and no internal pressure

30. Casing Design
Effects of Bending

31. Example 4
Compute the collapse pressure rating for 20’’, K-55 casing with a nominal wall thickness of 0.635’’ and a nominal weight per foot of 133 lbf/ft.

32. Example 4
Solution:
dn/t = 20/0.635 = 31.49
This is the transition collapse

33. Example

34. Critical Collapse Pressure Combined Stress Effects
All the pipe strength equations previously given are based on a zero axial stress state. This idealized situation never occurs in oilfield applications because pipe in a wellbore is always subjected to combined loading conditions.
The fundamental basis of casing design is that if stresses in the pipe wall exceed the yield strength of the material, a failure condition exists.
Hence the yield strength is a measure of the maximum allowable stress. To evaluate the pipe strength under combined loading conditions, the uniaxial yield strength is compared to the yielding condition.

35. Critical Collapse Pressure Combined Stress Effects
The most widely accepted yielding criterion is based on the maximum distortion energy theory, which is known as the Huber-Von-Mises Theory.
This theory states that if the triaxial stress exceeds the yield strength, a yield failure is indicated.
Note that the triaxial stress is not a true stress. It is a theoretical value that allows a generalized three-dimensional stress state to be compared with a uniaxial failure criterion (the yield strength).

36. Critical Collapse Pressure
Combined Stress Effects – Von Mises Equivalent 13
Where
sY – minimum yield stress, psi
sVME – triaxial stress, psi
VME: Von Mises Equivalent
sz, st, sr – axial tress, tangential stress, and radial stress, psi

37. Critical Collapse Pressure
Combined Stress Effects – Von Mises Equivalent
Setting the triaxial stress equal to the yield strength and solving equation (13) give the results:
Equation (14) is for the ellipse of plasticity. Combining Eq. (14) and eq. (6) together and let r = ri, will give the combinations of internal pressure, external pressure and axial stress that will result in a yield strength mode of failure.
< 0 for collapse and > for burst 14

38. Critical Collapse Pressure
Combined Stress Effects – Von Mises Equivalent

39. Critical Collapse Pressure Combined Stress Effects
As axial tension increases, the critical burst-pressure increases and the critical collapse-pressure decreases.
In contrast, as the axial compression increases, the critical burst-pressure decreases and the critical collapse-pressure increases.

40. Example 5
Compute the nominal collapse pressure rating for 5.5’’, N-80 casing with a nominal wall thickness of 0.476’’ and a nominal weight per foot of 26 lbf/ft. In addition, determine the collapse pressure for in-service conditions in which the pipe is subjected to a 40,000 psi axial tension stress and a 10,000 psi internal pressure. Assume a yield strength mode of failure.

41. Example 5
For collapse pressure rating, r = ri then eq. (6) becomes

42. Example 5
From eq. (14) with we have 14

43. For in-service conditions of sz = 40,000 psi and pi = 10,000 psi
Example 5
For in-service conditions of sz = 40,000 psi and pi = 10,000 psi
Solving eq. (14) gives 14

44. Example 5

45. Casing Design
Effects of Bending
In directional wells, the effect of wellbore curvature and vertical deviation angle on axial stress in the casing and couplings must be considered in the casing design.
When a casing is forced to bend, the axial tension on the convex side of the bend can increase greatly.
On the other hand, in relatively straight sections of hole with a significant vertical deviation angle, the axial stress caused by the weight of the pipe is reduced.

46. Casing Design
Effects of Bending
The maximum increase in axial stress, sb, on the convex side of the pipe is given by Crandall and Dahl (1995)
sb = ± 0.5EdnK.
In oil field units, where dogleg severity, K, is expressed as the change in angle in degrees per 100 ft; E (lbf/in2) = stress/strain is elastic modulus.
sb = ± 218dnK.
In terms of an equivalent axial force
Fab = ± 218dnKAs.
In terms of weight per foot of pipe
Fab = ± 64dnKwp.

47. Casing Design
Effects of Bending
Example: Determine the maximum axial stress for in, 39-lbf/ft, N-80 casing if the casing is subjected to a 400,000 lbf axial tension load in a portion of a directional wellbore having a dogleg severity of 40/100ft. Assuming uniform contact between the casing and the borehole wall.
Note that, the definition for the pipe body yield strength:

48. Casing Design
Effects of Bending

49. Casing Design
Effects of Bending
Solution: Nominal API pipe body yield strength for this casing is 895,000 lbf, and the ID is The cross sectional area of steel in the pipe body is
A = p/4( – ) = in2.
The axial stress without bending: 400,000/ = 35,740 psi
The additional axial stress on the convex side of the pipe due to the bending
sb = ± 218dnK = 218(7.625)(4) = 6,649 psi
The total axial stress = 35, ,649 = 42,389 psi