Yield Range
Collapse strength of the composite pipe is defined with reference to a state in which the tangential stress of the inner or outer pipe attains the value of its yield
strength. According to the theory of distortional energy, the yield strength of the inner or outer pipe can be expressed as follows:
|
(4.95) |
2^1 = J — <ytl Y + (<7tl — <T2l )2 + (cr21 — 0>, )2
|
(4.96) |
2cts/2 = /{°n ~ <A2)2 + (^2 — ^2)2 + K-2 — °V2)2
|
0.5 ~ |
|
0 — J |
|
E • 100 N/mm cm |
|
°2 1.0 |
|
CEMENT INNER CASING |
|
M |
|
Inner Casing Outer Casing |
|
O. D. 9 5/8’ lb/ft 43.5 |
|
13 3/8′ 68.0 |
|
E — 1000 N/mm2 cm E — 104 N/mm2 cm E • 1<? N/mm* |
Fig. 4.19: Radial stress in 13| — 9|-in. composite casing as a function of modulus of elasticity and Poisson’s ratio of cement sheath. (After El-Sayed, 1985; courtesy of ITE-TU Clausthal.)
Defining cryi and oy,2 as the yield strengths of the inner and outer pipes with a permanent deformation of 0.2%, and substituting the values of oy, oy, Ecm = 5,691 — f 376 oym — 1.19 anc^ E ~ x Ю5 N/mm2, the yield strength of the individual pipe can be obtained in metric units as follows (El — Sayed, 1985):
|
‘(Po, — Ph)r^’ |
2 + |
L |
, РоЛ ‘ |
|
(И — rf ) V Oi 4 / J |
r2 ___ — p2 . о 1 ‘ *i |
г2 — г* 01 *1 . |
о Н 1————————————————- 1—————————- 1———————- 1 1 1 Г—I 1 ►
О 10 20 30 40 50 60 70 80 90
а, N/mm 2
cm
Fig. 4.20: Collapse resistance of the composite pipe as a function of compressive strength and Poisson’s ratio of cement. (After El-Sayed, 1985; courtesy of ITE — TU Clausthal.)
and
|
Р.2Г |
|
Po 2ri2 |
|
‘2 ‘2 |
|
cr y2 — 3 |
|
(4.98) |
|
+ |
|
+ |
|
(Po2 ~ Pi2) r, |
|
(r2 — r2 ) V 02 ‘2 / |
where:
<7cm — compressive strength of cement, N/mm2.
Using Eqs. 4.93, 4.94, 4.97 and 4.98, the stress distribution in the composite pipe and its collapse resistance were computed by El-Sayed (1985) (see Figs. 4.18 through 4.20). From the figures the following observations can be made:
1. Maximum stress occurs in the outer pipe.
2. Minimum stress occurs in the cement sheath.
3. Stress on the outer pipe increases with increasing Ecm. i. e.. the collapse resistance increases.
4. Stress on the inner pipe decreases with increasing Ecm. i. e.. the collapse resistance decreases.
|
о, N/mm2 cm ’ Fig. 4.21: Collapse resistance of the composite pipe as a function of compressive strength of cement. (After El-Sayed. 1985: courtesy of ITE-TU Clausthal.) |
This behavior is explained by the fact that at low values of Ecm, the tangential stress in the outer pipe exceeds its yield strength and results in collapse. At high values of Ecm the composite pipe starts to collapse at the inner pipe. This suggests that cement with a high modulus of elasticity does not necessarily increase the collapse resistance of the composite pipe. Collapse resistance in the yield range (Fig. 4.21) displays similar behavior to that observed in the elastic range.
Test results obtained on two sets of composite pipes (131 — 9|-in. and 7 — 5-in.) by Marx and El-Sayed (1984) show behavior (Fig. 4.22) similar to that predicted by their theoretical model. The pipe failure observed for all specimens was, however, in the plastic range (Fig. 4.23). Collapse failure in the plastic range can be explained as follows. As the external and internal pressures increase, the cement sheath experiences a confining pressure, which results both in an increase in compressive strength and the modulus of elasticity of cement and a corresponding decrease in Poisson’s ratio. With further increases in the external pressure, the modulus of elasticity of the cement decreases and Poisson s ratio increases. As the changes in the modulus of elasticity, Poisson s ratio and external pressure (increasing) continue, the composite pipe reaches a stage where the tangential stress exceeds the value of the yield strength of any one of the pipes. Consequently, the composite pipe starts to yield and finally collapses. The effect of the combined loads improves the collapse resistance (Fig. 4.22), thereby improving the behavior of the cement sheath.
|
О in N/mm2 cm |
|
2 |
|
c in N/mm |
|
cm |
|
(a) |
(b)
Fig. 4.22: Collapse resistance of the outer and inner pipe as a function of compressive strength of cement; (a) 13| — 9|-in. and (b) 7 — 5-in. (After El — Sayed, 1985; courtesy of ITE-TU Clausthal.)
(b)
Fig. 4.23: Collapse failure of the composite pipe as a function of external pressure; (a) 131 — 9|-in. and (b) 7 — 5-in. (After El-Sayed, 1985; courtesy of ITE-TU Clausthal.)
On the basis of the results obtained from the theoretical model and the laboratory experiments, Marx and El-Sayed (1985) suggested the following formula for calculating the collapse resistance of composite casing:
|
2.05 |
|
— 0.028 |
|
(4.99) |
|
(d0/t)C! |
|
Pccp = Pct + Pc2 + oc |
|
where: {d0/t)cs Pccp Pc2 Pc2 G Г Sr.- |
ratio of outside diameter of the cement sheath to its thickness.
overall collapse resistance of the composite (pipe) body, psi.
collapse resistance of the inside pipe. psi.
collapse resistance of the outside pipe. psi.
collapse stress of the cement sheath
under the external pressure pCl. psi.
|
-X- |
+ 2 pCl
|
= |
1 — sin t compressive strength of the cement, angle of internal friction calculated from Mohr’s circle.
The compressive strength of cement and the angle of internal friction for the collapse resistance of the composite pipe can be computed from Eq. 4.99. The equation also shows that the collapse resistance of the composite pipe is the sum of the collapse resistance of the individual pipes. Inasmuch as the collapse resistance of the cement sheath cannot be predicted as a single pipe, Marx and El-Sayed (1985) suggested the following simplified equation:
|
(4.100) |
Pccp = Kt (Pd + Pc2)
where:
KT — reinforcement factor.
The value of KT lies between 1.17 and 2.03.