Piston Force
Piston force arises from the hydrostatic pressure acting on the internal and external shoulders of the casing string (Fig. 2.28). For a given casing size, the external piston forces acting on the casing collars cancel each other leaving only the internal piston forces. Assuming that the bottom section has a larger internal diameter than the upper section, the piston force. Fap, on the casing can be
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GRADE 1 |
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GRADE 2 |
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GRAOE 3 |
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! Aup |
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Fig. 2.28: Diagrammatic presentation of the piston effect arising from a change in internal diameter. |
expressed as:
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(2.181) |
where:
Pdлл. = GP:D&a, — internal pressure at depth D±a.- psi.
D&as = depth of the change in cross-section, .4,. of the pipe. ft.
Aupi = internal area of the upper section of the pipe at depth D±a,- in.2
Aiowi = internal area of the lower section of the pipe at depth in.2
In this case the piston force is a compressive force.
Consider the string of 9| in., N-80 casing subjected to the conditions in Table
2.6. Is it likely to buckle?
The top of the cement, Djoc, >s at 9,100 ft. From the earlier discussion, an unstable equilibrium is given by Eq. 2.179:
<T( + <rT aa <
where:
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Table 2.6: EXAMPLE 2-13: Lengths and downhole conditions for N-80 casing string.
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_ (D — x)Tn — Aa(GPox + GPcm(D — x)) — AtGPiD + Fap ~ A,
Similarly:
<?t + <rT _ (Ajpi — A0pa)
2 “
Consider first the radial and tangential stresses along the length of casing. A0 = 72.760 sq. in.
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Table 2.7: EXAMPLE 2-13: Radial and tangential stress calculations.
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Next, consider the axial stresses. First, the buoyancy force, Fbu, at the shoe is given by:
Fbu = 72.76 x 0.78 [(10,000 — 9,100)+ 0.702 x 9,100]
-55.88 x 0.702 x 105 = 123,605 lbf
For the remaining axial stresses it is more convenient to use a table (Table 2.8): Fa = Wn(D — x)
Fbu = A0{GVoDTOC + GPcm(D — Dtoc)) ~ A, GVtD Fap PD^Asi. AUpi Aiowl )
|
Depth |
Weight |
At |
Fa |
Fbu |
Fap |
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|
(ft) |
(lb/ft) |
(sq. in.) |
(lbf) |
(lbf) |
(lbf) |
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10,000 |
58.4 |
16.879 |
0 |
123.605 |
0 |
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9,100 |
58.4 |
16.879 |
58.4(900) |
123.605 |
0 |
|
|
7,500 |
58.4 |
16.879 |
58.4 (2.500) |
123.605 |
0 |
|
|
7,500 |
47 |
13.572 |
58.4(900) |
123.605 |
7,500 x |
0.702 |
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x (59.19 — |
55.88) |
|||||
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2,000 |
47 |
13.572 |
58.4 (2,500) |
123,605 |
7500 x |
0.702 |
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+ 47 (5.500) |
x (59.19 — |
— 5.88) |
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2,000 |
58.4 |
16.879 |
58.4 (2,500) |
123,605 |
7,500 x |
0.702 |
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+ 47 (5.500) |
x (59.19 — |
55.88) |
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+ 2,000 x |
0.702 |
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(55.88 — |
59.19) |
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0 |
58.4 |
16.879 |
58.4 ((2,500) |
123,603 |
7.500 x |
0.702 |
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+ 2000 |
x (59.19- |
55.88) |
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+ 47 (5,500) |
+ 2.000 x |
0.702 |
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x (55.88 — |
59.19) |
One can combine all the available information into a final table (Table 2.9). Clearly at no time does the condition for instability, cra < (crt + яу)/2, occur in the above example. The neutral point is at the casing shoe.
Example 2-13 is a simplified example because temperature and pressure have been assumed constant.