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Stress Distribution in a Composite Pipe

Previously, it was shown that the casing suffers an axial stress during heating and cooling operations. In practice, however, all three principal stresses, radial stress, tangential stress and axial stress, are present (Fig. 4.28). A reasonably accurate description of the behavior of these stresses in the elastic range can be provided by assuming that the casing, the cement sheath, and the formation form a rotationally symmetric composite pipe, subjected to an internal pressure, external pressure, and a quasi-steady-state temperature distribution. Figure 4.29 presents the different elements of the composite pipe under internal and external pressures and temperatures.

<т«(г) = — —— |дГ(г) — — r AT (r) dr|

According to Szabo (Goetzen, 1986), if the ratio of the length to external diameter is comparatively large and axial displacement of the pipe is prevented, the radial and the tangential stresses of the pipe body can be expressed by the following relationships:

г г

Fig. 4.29: Rotationally symmetric composite pipe under pressure and temper­

ature.

(4.102)

where AT{r) is the change in temperature with respect to r. For a quasi-steady — state temperature distribution. AT(r) can be expressed as:

AT(r) = where:

(4.103)

ATj In(r0/r) — АГ0 ln(r,-/r) ln(r0/r,)

r, = internal radius of the pipe body. in.

r0 = external radius of the pipe body. in.

C5 and С’б = constants obtained by substituting the boundary conditions: o> (r,) = pi and <7r (r0) = p0 (4.104)

Pi = internal pressure, psi.

Po = external pressure, psi.

T = coefficient of thermal expansion, in./in. °C.

A Tj — A T0 r2 , ^

4 ln(r0/r,) 2 (;}

According to Szabo (Goetzen. 1986). the change in temperature at any radius can be found as follows:

Substituting Eqs. 4.104 and 4.105 in Eqs. 4.101 and 4.102. the solution for radial and tangential stresses is obtained:

<Tr(r

T E (AT;■ — AT0

— AT(r)

2(1 — v) f r‘0 — r,

(AT? i’j — AT? r0) (ro — rf)

(4.106)

(Po l’l — Рг Г?) (Pi — Pa) (ГГГсА 2

(rl ~ rf) (rf — rf) V Г

and,

2

T E f AT, — AT0

(АТ>г,-АТ}г,) (АТ,-AT,

(rf-rf) ln(r0/r,)

ЛРоГ1-РгГ2) (p,—p0) /r, roy

TOC o "1-5" h z (r02-r?) K-r?) I r j ]

In the Eqs. 4.106 and 4.107, p, and p0 are negative.

Inasmuch as the pipe is prevented from axial movement (t. = 0):

<ra(r) = V {crr(r) + <T((r)} — T E AT (r) + crTfS (4.108)

where <7rej is the residual axial stress present in the material prior to heating of

the pipe body.

From classical distortion energy theory (Goetzen, 1986). the equivalent stress can be calculated as follows:

Ue = /d i(<Tr ~ CT‘)2 + (°7 ~ a*)2 + “ (7′-)2} (4.109)

For an elastic composite pipe as shown in Fig. 4.29, the radial, tangential and axial stresses can be determined by using Eqs. 4.106. 4.107. 4.108, and 4.109, pro­vided that the influences of the boundary layers for each element j are neglected. The radial interlayer stresses between the elements are not known; however, they can be expressed in terms of the internal and external pressures of the individual elements as follows:

с

г

— 30 -25 — -20

— 15 -10

— 5 —

‘Ч

Z1

С1

C1 :

7й x 9.19 mm

———

: ores — 256 N/mm2

—-

°res " 0

P. —

100 bar

P —

77 bar

0

AT

300 — 250 ■ 200 — 150 — 100 50

<J

а

-800 — -700 — -600 — -500 — -400 — — 300 —

t

-250 — 200 —

— 150 —

— 100 — — 50 —

0

У

800

700

600

500

400

300

CASING

CEMENT

(Po)j = <rT (r0)j

(Pi)j = °v(r.)j

(Po)j = {Pi)j +1

where:

Fig. 4.30: Stress distribution in a single-casing completion with packed-off annulus ( AT in °C and о in N/mm2). (After Goetzen, 1987; courtesy of ITE — TU Clausthal.)

(Po)n — (Po) formation pressure (p,)i = (p,) inside annulus pressure for 1 < j < n — 1

..n Pt

Po

Pjtj+i

system element.

inner pressure of element j.

outer pressure of element j.

< 10-3 N/r

Similarly, the radial displacement is given by Szabo (Goetzen, 1986):

ur (ro)j — ur (ri)j+1

ur(r0)j = {т0£«(т0)Ь

Ur(r,-)j + l = {г,£((г,)}_, + 1

{Aur)Jtj+l = {ur (r„)j — ur (r,)}+1} for element j < 1 < n — 1.

Using the above relationships for radial displacement, the pressure between two

Cl :

7" x 9.19mm

res

« 256 N/mm2

C2:

10 3/4" x 10.16mm

о

-0

res

P.

— 100 bar

Fig. 4.31: Stress distribution in a double-casing completion with packed-off annulus ( AT in °C and a in N/mm2). (After Goetzen. 1987; courtesy of ITE — TU Clausthal.)

с

I

-30 — -25 — — 20

— 15 —

— 10 — -5 —

adjacent elements at any radius r is obtained:

= (Ацг);,;+1 [1 Л + Gj+1

;,;+1 r I Г ( 1 _ C2 J+1

ri, j+i I ^j+i A <~’j+1

(4110)

Wl + G-

where:

G=— (4.111)

In order to solve the analytical equations for radial, tangential and axial stresses, extensive calculation is involved. In a recent study, Goetzen (1986) developed a computer program based on an iterative solution and presented numerous data for radial, tangential, axial and equivalent stresses for different steam stimulation situations. Some of these results are presented in Figs. 4.30 and 4.31 and are based on the following wellbore situations:

Cl: 7" x 9.19mm °res *°

/

Pj — 100 bar p — 88 bar

t — 2 days t — 50 days

at

-250 —

-200 —

— 150 —

— 100 —

-50 — 0

°a -800 — 700 -600 -500 -400 -300

адшД

У

800 — 700 — 600 — 500 — 400 — 300

CASING

F ■ ^casing

■^cement T •

casing

"^cement

^casing

^cement

CEMENT

Fig. 4.32: Stress distribution in a single-casing completion and the casing ex­posed directly to the injected steam ( AT in °C and a in N/mm2). (After Goetzen. 1987; courtesy of ITE-TU Clausthal.)

1. Steam is injected through 3|-in. bare tubing and the casing temperature is calculated based on the model proposed by Willhite (1966).

2. Casing temperature is varied from 68 °F (20 °C) to 590°F (310 °C) and the annular pressure is kept constant at 1.294 psi (88 bar).

completion with residual stress ares — 0, the equivalent stress. crre}. is 88,200 psi (600 N/mm2). When the residual stress, <rres, is increased to 37,632 psi (256 N/mm2), the equivalent stress reduces to 51,450 psi (350 N/mm2). For the double casing completion in Fig. 4.31 when ares — 0, the equivalent stress in the internal pipe is only 58,800 psi (400 N/mm2). Contrary to what was observed in the first case, an increase in the residual stress does not lead to an appreciable decrease in the equivalent stress.

Figure 4.32 illustrates the effect of direct contact of steam with the casing. It shows that the stresses in the casing are much greater than those with a packed-off annulus as in Fig. 4.30.

From this study, it is evident that the residual axial stresses in the casing and the type of completion are the major factors controlling the casing stresses during the heating cycle.

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