7252019 Analyzing Axial Stress and Deformation of Tubular

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Hindawi Publishing CorporationTe Scienti1047297c World JournalVolume 983090983088983089983091 Article ID 983093983094983093983096983097983089 983097 pageshttpdxdoiorg983089983088983089983089983093983093983090983088983089983091983093983094983093983096983097983089

Research Article Analyzing Axial Stress and Deformation of Tubular for Steam Injection Process inDeviated Wells Based on the Varied ( 1038389) Fields

Yunqiang Liu12 Jiuping Xu1 Shize Wang3 and Bin Qi3

983089 Uncertainty Decision-Making Laboratory Sichuan University Chengdu 983094983089983088983088983094983092 China983090 College of Economics amp Management Sichuan Agricultural University Chengdu 983094983089983089983089983091983088 China983091 Research School of Engineering echnology Te Southwest Petroleum and Gas Corp China Petroleum and Chemical Corp

Deyang 983094983089983096983088983088983088 China

Correspondence should be addressed to Jiuping Xu xujiupingscueducn

Received 983089983091 May 983090983088983089983091 Accepted 983090 August 983090983088983089983091

Academic Editors G Carbone S Park S orii and Q Yang

Copyright copy 983090983088983089983091 Yunqiang Liu et al Tis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Te axial stress and deormation o high temperature high pressure deviated gas wells are studied A new model is multiplenonlinear equation systems by comprehensive consideration o axial load o tubular string internal and external 1047298uid pressurenormal pressure between the tubular and well wall and riction and viscous riction o 1047298uid 1047298owing Te varied temperature and

pressure 1047297elds were researched by the coupled differential equations concerning mass momentum and energy equations insteado traditional methods Te axial load the normal pressure the riction and our deormation lengths o tubular string are gotten by means o the dimensionless iterative interpolation algorithm Te basic data o the X Well 983089983091983088983088 meters deep are used orcase history calculations Te results and some useul conclusions can provide technical reliability in the process o designing welltesting in oil or gas wells

1 Introduction

Te deviated wells had been wildly applicable or petroleumand natural gas industry Deviated wells have their distinctivecharacteristics which are distinguished rom that o otherwells (983089) High temperature high pressure the temperature

distribution and pressure on the tubing are signi1047297cantly different when outputs are varied (1047298ow velocity) but neitherhas a simple linear relationship because the 1047298uid density is not constant (983090) Deep well the sensibility o orce anddeormation in1047298uencing by the actors such as the temper-ature pressure density o 1047298uid viscous riction and 1047298uid

velocity and so orth will become high with the increase o tubing length Te completion test o a deep well is a new problem In the research o applied basic theory or deep welltesting tubular string mechanical analysis is very complexbut 1047298uid temperature and tubing pressure affect the orceo the tubular string heavily emperature pressure liquiddensity and 1047298uid velocity within tubing may change with o

the hole depth time and operations so that the axial orcechanges constantly A large compression load at low end caninduce the tubing plastic deormation and make the packerdamaged A large tension load at the top end may unpack thepacker or cause the tubing to break I the tubing ailed thewhole borehole can hardly maintain its integrity and saety

[983089] Tereore it is very important ordeviated wells to predictthe axial orces or the saety

Hammerlindl [983090] had made a great contribution abouttubular mechanics He had put orth the our effects betweenthe packer orces and length change o tubing temperatureeffect ballooning effect axial load effect and the helicalbuckling effect Tere is a large amount o papers to researchthe effect o buckling behavior Tereore it is considered thatin1047298exion is caused on its axial orce under certain conditionsby which colliding on partso the drill string with well bore isinduced When buckled o tubular beyond wellholersquos controlthe buckling con1047297guration which will be transormed at thestate o stabilization sinusoidal buckling and helical buckling

7252019 Analyzing Axial Stress and Deformation of Tubular

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983090 Te Scienti1047297c World Journal

with the increase o load Te problem o buckling o the tubewas 1047297rst studied and put into practice by Lubinski et al [983091]Tey had done the emulation experiment or the bucklingbehavior o tube in deviated wells and ound the computeormula on critical buckling load o tube in deviated wellsPaslay and Bogy [983092] ound that the number o sinusoids in

the buckling mode increases with the length o the tubeTe buckling behavior by inner and outer 1047298uid pressure o tubing was analyzed and the mathematical relation betweenpitch and axial pressures was deduced based on the principleo minimum potential energy (see Hammerlindl [983090]) Temptotic solution or sinusoidal buckling o an extremely longtube has been analyzed by Dawson and Paslay [983093] based on asinusoidal buckling mode o constant amplitude Numericalsolutions were also sought by Mitchell [983094] using the basicmechanics equations His solutions con1047297rm the thought thatunder a general loading the deormed shape o the tube is acombination o helices and sinusoids while helical deorma-tion occurs only under special values o the applied load Teormula about tubing orces had been put however which istoo simple or shallow wells to accommodate the complicatedstates o deep wells Up to now many researches are centeredon water injection tubular but not on steam injection Amongthem the values o temperature and pressure are consideredas constant or lineal unctions which will cause large errorson tubular deormation computing [983095]

In act the tubular string deormation includes trans- verse deormation and longitudinal deormation Because

the transverse length (its order o magnitude is 10minus3 m) ismuch and much smaller than the longitudinal length (its

order o magnitude is 103 m) we mainly consider the axial(longitudinal) deormation or the tubular string deormation

analysis in the paper In the paper the orce states o tubular in the process o steam injection are analyzed Te varied ( 1038389) 1047297elds are considered to compute the values o several deormations Te axial load and our deormationlengths o tubular string are obtained by the dimensionlessiterative interpolation algorithm Te basic data o the X Well(deviated well) 983089983091983088983088 meters deep in China are used or casehistory calculations Some useul suggestions are drawn

Tis paper is organized as ollows Section 983090 gives asystem model about tubular mechanics and deormationAnd the varied ( 1038389) 1047297elds were presented by model con-cerning mass momentum and energy balance Section 983091gives the parameters initial condition and algorithm or

solving model In Section 983092 we give an example rom adeviated well at 983089983091983088983088 meters o depth in China and the resultanalysis are made Section 983093 gives a conclusion

2 Model Building

983090983089 Basic Assumption Beore analyzing the orce on themicroelement some assumptions are introduced as ollows

(983089) the curvature o the hole o the considered modularsection is constant

Pi

P0

D

d

o

r

z

Tube

Casing

Packer

Packerfluid

F983145983143983157983154983141 983089 Te physical 1047297gure o orces analysis on tube

(983090) on the upper side or underside o the section whichis point o contact o the pipe and tube wall thecurvature is the same with the hole curvature

(983091) the radius o steam injection string in contrast tocurvature o borehole is insigni1047297cant

(983092) the string is at the state o linear elastic relationship

983090983090 Forces Analysis of ubular String Te orces o tubularstring are shown in Figure 983089 Consider the 1047298ow systemdepicted in Figure 983089 a constant cross-sectional 1047298ow area

1103925

inner diameter 907317 outer diameter material density 1packer 1047298uid density 2 and a total length Trough thistubing gas 1047298ows rom the bottom to the top with a mass 1047298ow rate Te distance coordinate in the 1047298ow direction alongthe tubing is denoted by Te cylindrical coordinate system origin o which is in wellhead and axis is down as theborehole axis is used

As shown rom Figure 983089 the tubular string is mainly actedupon by the ollowing orces at the process o steam injection

(983089) Initial Axial Force Te initial axial orce o tubularshould include the deadweight buoyant weight andinitial pull orce

(983090) Termal Stress On the process o steam injection thetemperature stress will act at the tubular with variedtemperature

(983091) Axial Force by the Varied Internal and External Pres-sure Tanks to the varied pressure with internal andexternal pressure the tubular will be acted by thebending orce piston orce and other axial orces

(983092) Friction Drag by Steam Injection On the process o steam injection the 1047298ow in tubular will produce

viscous 1047298ow which will cause the riction drag

7252019 Analyzing Axial Stress and Deformation of Tubular

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Te Scienti1047297c World Journal 983091

983090983091 Te Axial Load and Axial Stress of the ubular

983090983091983089 Initial Axial Load and Initial Axial Stress of Steam

Injection ubular

Initial Axial Load Te section to which the distance rom thewellhead is () was considered Te axial static load by thedeadweight o tubular is as ollows

1038389 = int1103925

1038389 cos907317 = 4 1 8520082 minus 9073172852009int1103925

1038389cos907317 (983089)

where 1038389 is the deadweight o tubular is the average unitlength weight o tubing is the length o tubular 1 is thedensity o tubular and is the inclination angle

Te axial static load by the buoyant weight is as ollows

9073171038389 = minus211039252 int1103925

1038389cos907317 = minus2983080

2 9830812 int1103925

1038389cos907317

(983090)

where 9073171038389 is the buoyant weight o tubular 2 is the density o packer 1047298uid

Te axial load by the steam injection pressure

1038389 = 1038389103838919073172

4 (983091)

where 103838910383891 represents the inner pressure at this section

Tereore summing (983089) (983090) and (983091) the axial orces inthe section are obtained as ollows

1038389 = 1038389 + 9073171038389 + 1038389 (983092)

Initial Axial Stress Te axial stress can be derived rom theollowing equation

1038389 = 41038389 10486162 minus 90731721048617 (983093)

983090983091983090 Axial Termal Stress of Steam Injection ubular In theprocess o steam injection the temperature o tubular willchange with time and depth which will make the tubulardeorm as ollows

1038389

= 104861610383891

minus 10383890

1048617 = Δ (983094)

where represents the steel elastic modulus o tubular isthe warm balloon coefficient o the tubular string and Δ isthe temperature change with beore and afer steam injection

983090983091983091 Axial Stress of Steam Injection ubular by the Changewith Pressure Te effect acting the tubular with pressurechange which is called ballooning effect normally

Ballooning Stress Analysis Te ballooning effect will beproduced rom pressure acted in inner and outer o the tubeGenerally there are two kinds o tubular in oil wells One isthe tubulars whose outerdiameteris 983096983096983097 mm inner diameter

Pz1

Pz2D

d

F983145983143983157983154983141 983090 Te radial and tangential stresses 1047297gure o tube

is 983095983094 mm and thickness o tubes is 983094983093 mm ((9073172) =171 gt 5) the other is the tubular whose outer diameter is983089983089983092983091 mm inner diameter is 983089983088983088983093 mm and thickness o tubesis 983094983097 mm ((9073172) = 137 gt 5) Neither is the thin-wallproblem Tereore it should be solvedby Lamersquos ormula [983096]

Te radial and tangential stresses in the thick-wall cylin-der can be shown as Figure 983090 Te two can be calculated asollows

1038389 = 9073172103838910383891 minus 21038389103838902 minus 9073172 minus 1048616103838910383891 minus 1038389103838901048617 29073172

10486162 minus 90731721048617 42

1038389 = 9073172103838910383891 minus 21038389103838902 minus 9073172 + 1048616103838910383891 minus 103838910383890104861729073172

10486162 minus 90731721048617 42 (983095)

where is radial stress is tangential stress (907317 le le ) isradial coordinate

103838910383891 is tube internal pressure at

point and

103838910383890 is tube external pressure at point

983090983091983092 Axial Stress of Steam Injection ubular by the FrictionLoss In act the 1047298ow in the tubular should be multi1047298ow Onthe process o steam injection the 1047298ow will be run and it willgive rise to riction effect to cause axial stress In our paperwe consider the 1047298ow gas-liquid mix 1047298ow and the liquid headloss is gotten by the Darcy-Weisbach ormula [983097] as ollows

ℎ = ( minus ) ]22907317 (983096)

where ℎ means heat loss o liquid 1047298ow is rictional headlosses coefficients and ] is the velocity o liquid 1047298ow

Te riction drag in tubular is 1038389 = ℎ9073172 ( isdensity o liquid 1047298ow) Te axial stress by 1047297ction drag can beobtained as ollows

1038389 = 41038389 10486162 minus 90731721048617 (983097)

983090983092 Analysis of Axial Deformation Based on the studiesand analyses mentioned above the axial deormation on thetubular is made up o the ollowing parts

7252019 Analyzing Axial Stress and Deformation of Tubular

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983092 Te Scienti1047297c World Journal

983090983092983089 Te Axial Deformation by the Axial Static Stress For themicroelement o the tubular 907317 the unit deormation by thestatic stress can be computed by generalized Hooke law

1 = 1 [1038389 minus 10486161038389 + 10383891048617] (983089983088)

where represents Poissonrsquos ratiosTe axial deormation at an element can be obtained

through integrating on the length o the element as ollows

Δ1 = int

minus1

1 [1038389 minus 10486161038389 + 10383891048617]907317 (983089983089)

Tereore the total axial deormation by the static stresscan be gotten accumulating each element as ollows

Δ1 = sum=1

Δ1 (983089983090)

983090983092983090 Te Axial Deformation with emperature Changed Forthe microelement o the tubular 907317 the unit deormation by the temperature change is as ollows

Δ2 = int

minus1

1038389 907317 = ΔΔ (983089983091)

Te same principle is that the total axial deormation by thevariedtemperature 1047297elds canbe gottenaccumulating eachelement as ollows

Δ2

=

sum=1Δ2

(983089983092)

983090983092983091 Te Axial Deformation with the Friction Drag For themicroelement o the tubular 907317 the unit deormation by theriction orce is as ollows

Δ3 = int

0

1038389 907317 = ]

29073172

10486162 minus 90731721048617 (983089983093)

983090983092983092 Te Axial Deformation with the ubular String BucklingResearchers in general call the buckling a bending effect Tetubular is reely suspended in the absence o 1047298uid inside asshown in Figure 983091(a) Because the orce

applied at the end

o the tubular which is large enough the tubular will buckleas shown in Figure 983091(b)

Lubinski et al [983091] had done many researches on thephenomenon From their work we canget the buckling effectDe1047297ne the virtual axial orce o tubing as ollows

= 1103925 104861610383891 minus 103838901048617 (983089983094)

where 10383891 is the pressure inside the tubular at the packerlength 10383890 is the pressure outside the tubular at the packerlength and 1103925 is the area corresponding to packer bore

By (983089983094) whether the tubular will buckle or not can be judged Te string will buckle i

is positive or remain

(a)

F

Neutral point

(b)

F983145983143983157983154983141 983091 Buckling o tubular

straight i is negative or zero Te axial deormation o thetubular string buckling is

Δ4 = minus211039252

1048616Δ10383891 minus Δ10383890104861728 (983089983095)

where means tubing-to-casing radialclearance is momento inertia o tubing cross-section with respect to its diameter( = (4 minus 9073174)64) Δ denotes change with beore and aferinjection and

is the unit weight o tubing as

Δ4 = sum=1

Δ4 (983089983096)

In addition the position o the neutral point is needed Telength () rom the packer to the point can be computed asollows

= (983089983097)

Generally the neutral point should be in tubular ( le )However at the multipackers it will occur that the neutral

point is outside the tubing between dual packers In thispaper we leave the latter phenomenon

o sum up the whole deormation length can be repre-sented as ollows

Δ = Δ1 + Δ2 + Δ3 + Δ4 (983090983088)

983090983093 Te Analysis of the Varied ( 1038389) Fields In the courseo dryness modeling we can 1047297nd that the numerical valueso deormation ((983089983088) (983089983091) and (983089983095)) were affected by thetemperature and pressure In act the two parameters variedaccording to the depth and time changing So the varied

(1038389) 1047297elds need to be researched Under the China Sinopec

7252019 Analyzing Axial Stress and Deformation of Tubular

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Te Scienti1047297c World Journal 983093

Group Hi-ech Project ldquoStress analysis and optimum designo well completionrdquo in 983090983088983088983097 [983094] undertaken by SichuanUniversity at early time Te varied ( 1038389) 1047297elds had beendeduced strictly based on the mass momentum and energy balance Te proo details can be shown in Xu et al [983089983089] Te

varied ( 1038389) 1047297elds is

9073171038389907317 = minus 104861611039251048617 + cos + (1103925) (907317907317)1 minus (1103925)

907317907317 = minus

]

907317907317 minus 9073171038389907317 minus

cos

minus ]34

+ 1048616 minus 1048617

1038389 104861601048617 = 10383890 1048616 01048617 = 0 907317104861601048617 = 9073170 1048616 01048617 = 0

(983090983089)

3 Numerical Implementation983091983089 Calculation of Some Parameters In this section we willgive the calculating method o some parameters

(983089) Each pointrsquos inclination

= minus1 + 1048616 minus minus11048617 ΔΔ (983090983090)

where represents segment point o calculation Δrepresents measurement depth o inclination angle and minus1 Δ is the step length o calculationransient heat transer unction [983089983090]

10486161048617 = 85209198316310486991128radic 10486161 minus 03radic 1048617 le 15104861604063 + 05 ln 1048617 1 + 06 gt 15

(983090983091)

(983090) Te density o wet steam Since the 1047298ow o the water vapor in is the gas-liquid two-phase 1047298ow there aremany researches about this problem [983089983091 983089983092] In thepaper we adopt the M-B model to calculate theaverage density o the mixture

(983091) Te heat transer coefficient to rom different posi-

tions o the axis o the wellbore to the second suraceTese resistances include the tubing wall possible insu-

lation around the tubing annular space (possibly 1047297lled witha gas or liquid but is sometimes vacuum) casing wall andcementing behind the casing as ollows

1

= 1ins

ln + 1ℎ + ℎ +

1cem

lncem (983090983092)

ins and cem are the heat conductivity o the heat insulatingmaterial and the cement sheath respectively ℎ and ℎ are thecoefficients o the convection heat transer and the radiationheat transer

983091983090 Initial Condition In order to solve model some de1047297niteconditions and initial conditions should be added Te initialconditions comprise the distribution o the pressure andtemperature at the well top In this paper we adopt the

value at the initial time by actual measurement Beore steaminjected the temperature o tubular just is initial temperature

o ormation (1038389 = 0 + cos is geothermal gradient)At the same time the pressure o inner tubular is assumed tobe equal to the outer tubular beore steam injected

983091983091 Steps of Algorithm o simpliy the calculation wedivided the wells into several short segments o the samelength Te length o a segment varies depending on varia-tions in wall thickness hole diameter 1047298uid density inside andoutside the pipe and wells geometry Te model begins withthe calculation at one particular position in the wells the topo the pipe

Step 983089 Set step length o depth In addition we denote

the relatively tolerant error by Te smaller ℎ is themore accurate the results are However it will lead to rapidincreasing calculating time In our paper we set ℎ = 1 (m)and = 5

Step 983090 Give the initial conditions

Step 983091 Compute each pointrsquos inclination

Step 983092 Compute the parameters under the initial conditionsor the last depth variables

Step 983093 Let = then we can get the by solving theollowing equation

= 22 + 1

1038389=0 = 0 + cos

1038389=1

= minus 12

907317907317

1038389 rarrinfin

= 0

(983090983093)

Let be the temperature at the injection time and

radial at the depth We apply the 1047297nite different methodto discretize the equations as ollows

+1 minus

= +1+1 minus 2

+1 + minus1+12 minus +1

+1 minus +1 (983090983094)

where is the interval o time and is the interval o radialrespectively It can be transormed into the standard orm asollows

minus + +1+1 + 2 +

+1 minus +1

minus1 = 2

(983090983095)

7252019 Analyzing Axial Stress and Deformation of Tubular

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983094 Te Scienti1047297c World Journal

983137983138983148983141 983089 Parameters o pipes

Diameter (m) Tickness (m) Weight (Kg) Expansion Elastic (Gpa) Poissonrsquos ratios Using length (m)

983088983088983096983096983097 983088983088983089983090983097983093 983090983091983095983097 983088983088983088983088983088983089983089983093 983090983089983093 983088983091 983090983095983088

983088983088983096983096983097 983088983088983088983097983093983091 983089983096983090983096 983088983088983088983088983088983089983089983093 983090983089983093 983088983091 983089983090983088

983088983088983096983096983097 983088983088983088983095983091983092 983089983093983088983092 983088983088983088983088983088983089983089983093 983090983089983093 983088983091 983094983090983088

983088983088983096983096983097 983088983088983088983094983092983093 983089983091983093983096 983088983088983088983088983088983089983089983093 983090983089983093 983088983091 983090983097983088

983137983138983148983141 983090 Well parameters

Measured (m) Internal (m) External (m)

983091983091983094983095 983088983089983093983092983095983096 983088983089983095983095983096

983092983090983090983094 983088983089983093983090983093 983088983089983095983095983096

983089983091983088983088983088 983088983089983088983096983094983090 983088983089983090983095

983137983138983148983141 983091 Parameters o azimuth inclination and vertical depth

Number Measured

(m)Inclination

(∘)Azimuth

(∘)Vertical depth

(m)

983089 983089983091983093 983090983094983091 983090983092983089983088983089 983089983091983092983095983090

983090 983090983095983096 983089983090983091 983090983091983095983096983094 983090983095983095983097983089

983091 983091983094983092 983089983092983091 983090983089983091983096983094 983091983094983091983096983090

983092 983091983097983091 983090983089983095 983090983094983091983096 983091983097983090983093983091

983093 983092983090983090 983089983096983093 983092983092983093983094 983092983090983089983090983096

983094 983092983093983088 983088983096983090 983089983097983089983089983090 983092983092983097983094983090

983095 983092983096983094 983090983097983091 983090983094983097983088983095 983092983096983093983092983095

983096 983093983089983092 983089983088983091 983090983097983095983093983093 983093983089983091983096983091

983097 983093983092983091 983091983093983096 983091983090983092983093983089 983093983092983089983095983092

983089983088 983093983095983089 983090983097983096 983091983088983091983088983093 983093983095983088983092983091

983089983089 983094983088983088 983090983088983091 983090983088983092983095983092 983093983097983097983092983090

983089983090 983094983090983096 983090983091983092 983089983094983092983091983091 983094983090983095983090983096

983089983091 983094983094983088 983089983096983093 983089983097983093983090983096 983094983093983097983093983094

983089983092 983095983090983091 983091983089983092 983090983089983092983096983092 983095983090983089983095983088

983089983093 983095983096983090 983088983097983096 983090983089983094983092983096 983095983096983089983091983088

983089983094 983096983091983088 983090983089983093 983090983090983097983091983089 983096983090983097983089983090

983089983095 983096983094983088 983090983094983095 983090983092983092983088983091 983096983093983097983095983089

983089983096 983097983088983096 983092983096983093 983090983094983094983094983090 983097983088983092983088983096

983089983097 983097983090983096 983094983095983090 983090983093983096983095983096 983097983090983089983092983090

983090983088 983097983095983090 983090983088983091 983090983091983094983096983096 983097983095983089983095983089

983090983089 983089983088983090983093 983092983095983096 983090983091983097983090983095 983089983088983090983089983090983093

983090983090 983089983088983093983096 983092983088983089 983090983092983092983093983097 983089983088983093983093983093983096

983090983091 983089983088983096983097 983092983097983096 983090983090983096983090 983089983088983096983092983089983095

983090983092 983089983089983091983090 983091983095983093 983090983091983091983096983096 983089983089983090983097983090983096

983090983093 983089983089983095983092 983093983094983091 983090983091983093983089983092 983089983089983094983096983096983095

983090983094 983089983090983088983092 983092983090983091 983090983091983092983091983096 983089983090983088983088983097983097

983090983095 983089983090983091983093 983091983096983095 983090983091983092983097983097 983089983090983091983090983088983096

983090983096 983089983090983094983096 983092983097983095 983090983091983090983093983095 983089983090983094983091983092983093

983090983097 983089983091983088983088 983096983096983092 983090983091983091983090983096 983089983090983096983092983097983094

Ten the different method is used to discretize theboundary condition For = 1 we have

+12 minus 1 + 2

+11 = 2

(983090983096)

For = we have

+1 minus +1

minus1 = 0 (983090983097)

We can compute the symbolic solution o the temperatureo thestratumIn this step we willget thediscretedistributiono as the ollowing matrix

98313110486671048667104866710486671048667104866710486671048667104866710486671048667852059

11 2

1 sdot sdot sdot 1 sdot sdot sdot

12 2

2 sdot sdot sdot 2 sdot sdot sdot

sdot sdot sdot

1 2

sdot sdot sdot sdot sdot sdot

sdot sdot sdot

1 2

sdot sdot sdot sdot sdot sdot

98313310486691048669104866910486691048669104866910486691048669104866910486691048669852061

(983091983088)

where represents the injection time and represents theradial

Step 983094 Let the right parts o the coupled differential equa-tions be unctions where ( = 1 2) Ten we can obtain asystem o coupled unctions as ollows

1 = minus 104861611039251048617 + cos + (1103925) (907317907317)1 minus (1103925)

2 = minus ]

907317907317 minus 9073171038389907317 minus cos

minus ]34

+ 1048616 minus 1048617

(983091983089)

where at = 1

Step 983095 Assume that 1038389 are ( = 1 2) respectively Tenwe can obtain some basic parameters as ollows

= 10486161 21048617 = 1 + ℎ12 2 +

ℎ22 = 1 + ℎ12 2 +

ℎ22 907317 = 10486161 + ℎ1 2 + ℎ21048617

(983091983090)

7252019 Analyzing Axial Stress and Deformation of Tubular

httpslidepdfcomreaderfullanalyzing-axial-stress-and-deformation-of-tubular 79

Te Scienti1047297c World Journal 983095

983137983138983148983141 983092 Te results o the axial orce and various kinds o deormation lengths

Number Depth

(m)Axial orce

(N)

Displacement by temperaturechanged (m)

Displacement by pressure

changed (m)

Axial deormation(m)

Bucklingdeormation (m)

otal deormation(m)

983089 983089 983096983097983093983090983092983092983096 983088 983088 983088 983088 983088

983090 983089983088983088 983096983093983092983095983090983092983096 983088983089983090983088983089 983088983088983088983097983096983094 983088983088983090983092 983088 983088983089983093983092983092

983091 983090983088983088 983096983089983092983090983089983093983093 983088983090983091983094983090 983088983088983089983097983091983097983090 983088983088983093983090 983088 983088983091983088983095983090

983092 983091983088983088 983095983095983091983095983089983095983095 983088983091983092983096983091 983088983088983090983096983093983097983096 983088983088983096983090 983088 983088983092983093983097

983093 983092983088983088 983095983091983095983097983095983088 983088983092983093983094983092 983088983088983091983095983092983095983094 983088983089983089983093 minus983088983088983088983094 983088983094983088983090983097

983094 983093983088983088 983095983088983094983096983095983095983091 983088983093983094983088983094 983088983088983092983094983088983090983096 983088983089983093983090 minus983088983088983088983094 983088983095983093983090983091

983095 983094983088983088 983094983095983093983095983094983091983097 983088983094983094983088983095 983088983088983093983092983090983093983092 983088983089983097983090 minus983088983088983088983094 983088983097983088983088983094

983096 983095983088983088 983094983092983092983094983088983090983091 983088983095983093983094983097 983088983088983094983090983089983093983091 983088983090983091983093 minus983088983088983088983094 983089983088983092983096

983097 983096983088983088 983094983089983091983092983091983095983090 983088983096983092983097 983088983088983094983097983095983090983093 983088983090983096983091 minus983088983088983088983095 983089983089983097983092983094

983089983088 983097983088983088 983093983096983090983090983095983090983089 983088983097983091983095983089 983088983088983095983094983097983094983096 983088983091983091983093 minus983088983088983088983095 983089983091983092983090983090

983089983089 983089983088983088983088 983093983093983089983089983088983095983090 983089983088983090983089983090 983088983088983096983091983096983096983091 983088983091983097983089 minus983088983088983088983095 983089983092983096983097983094

983089983090 983089983089983088983088 983093983089983097983097983092983090983091 983089983089983088983089983092 983088983088983097983088983092983095983089 983088983092983093983090 minus983088983088983088983095 983089983094983091983094983095

983089983091 983089983090983088983088 983092983096983096983095983095983095983093 983089983089983095983095983093 983088983088983097983094983095983091983089 983088983093983089983095

minus983088983088983088983097 983089983095983096983090983090

983089983092 983089983091983088983088 983092983093983095983094983089983090983096 983089983090983092983097983094 983088983089983088983090983094983094983090 983088983093983096983092 minus983088983088983089 983089983097983090983094983089

Step 983096 Calculate the pressure and temperature at point( + 1)+1

=

+ ℎ 1048616 + 2 + 2 + 90731710486176 = 12 = 12

(983091983091)

Step 983097 Calculate the deormation Δ1 Δ2 and Δ4 by previous equations

Step 983089983088 Repeat the third step to the tenth step until tubularlength is calculated

Step 983089983089 Calculate the deormationΔ3 and total deormationlength as ollows

Δ = sum=1

Δ1 + sum=1

Δ2 + Δ3 + sum=1

Δ4 (983091983092)

4 Numerical Simulation

983092983089 Parameters o demonstrate the application o our the-

ory we study a pipe in X well which is in Sichuan ProvinceChina All the basic parameters are given as ollows deptho the well is 983089983091983088983088 m ground thermal conductivity parameteris 206 ground temperature is 16∘C ground temperaturegradient is 00218(∘Cm) roughness o the inner surace o the well is 0000015 and parameters o pipes inclined wellinclination azimuth and vertical depth are given in ables 983089983090 and 983091

983092983090 Main Results and Results Analysis Afer calculation weobtain a series o results o this well as able 983092 Te in1047298uenceo outputson the axial deormation o tubing was investigatedas shown by Figure 983092

25

2

15

1

05

0

T o t a l a x i a l d e

f o r m a t i o n

( m )

Depth (m)

1 1200800400

700000m3d

500000m3d

300000m3d

F983145983143983157983154983141 983092 Te total axial deormation under varied outputs

From the results as shown in Figure 983092 and able 983092 some

useul analysis can be drawn

(983089) Te amount o steam injected and inject pressureaffected the stretching orce with special severity

(983090) Te results were as ollows the length o tubulardeormation was risen with increased injected pres-sure or injected velocity

(983091) Te length o tubular deormation increases with theincreasing o outputs but more slowly

(983092) Te thermal stress is the main actor in1047298uencing thetubular deormation Tereore the temperature o steam injected should not be too high

7252019 Analyzing Axial Stress and Deformation of Tubular

httpslidepdfcomreaderfullanalyzing-axial-stress-and-deformation-of-tubular 89

983096 Te Scienti1047297c World Journal

(983093) Te lifing prestressed cementing technology hasimportant meanings to reduce the deormation o tubular

(983094) Te creeping displacement o downhole stings willproduce an upward contractility which causes packer

depressed or lapsed Tereore the effective measuresshould be adopted to control the companding o tubular

5 Conclusion

In this paper a total tubular deormation model aboutdeviated wells was given A coupled-system model o differ-ential equations concerningpressureand temperature in hightemperature-high pressure steam injection wells according tomass momentum andenergy balances which canreduce theerror o axial stress and axial deormation was given insteado theaveragevalueor simplelinear relationship in traditional

research Te basic data o the Well (high temperature andhigh pressure gas well) 983089983091983088983088 m deep in Sichuan China wereused or case history calculations Te results can providetechnical reliance or the process o designing well tests indeviated gas wells and dynamic analysis o production

Nomenclature

907317 Inner diameter (m)9073171038389 Microelement o the tubular Acceleration o gravity (ms2)ℎ Depth o top tubular located at the packer(m)

ime o down stroke (s) Dimensionless time (dimensionless)V Velocity o 1047298uid in tubing (ms)V Velocity o down stroke (ms) Distance coordinate in the 1047298ow direction

(m) along the tubing1103925 Constant cross-sectional 1047298ow area (m2)1103925 Effective area (m2)1103925 Area corresponding to packer bore (m2) Te Joule-Tomson coefficient(dimensionless) Heat capacity o 1047298uids (JKg sdotK)

Outer diameter (m)

Steel elastic modulus o tubular (Mpa) Axial orces in the section (N)1038389 Axial tensile strength (N) Friction orce (N) Piston orce or supporting packerrsquos pressure(N) Pumping orce (N) Length o tubular (m)9073171038389 Buoyant weight o tubular (Kg)1038389 Dead weight o tubular (Kg)10383890 Pressure outside the tubular (Mpa)10383891 Pressure inside the tubular at the packerlength (Mpa)

1038389 Pressure in tubing (Mpa) emperature in tubing (∘C) Initial temperature o ormation (∘C) Mass 1047298ow rate (Kgs) otal length (m)1 Material density (Kgm3)

2 Packer 1047298uid density (Kgm3

) Inclination angle (∘) Warm balloon coefficient o the tubularstring (dimensionless) Drop o any parameter1038389 Axial thermal stress () Density o 1047298uid in the tubing (Kgm3)Δ3 Te tubular string buckling axialdeormation (m)Δ otal axial deormation by variedtemperature 1047297elds (m)Δ907317 otal axial deormation by the variedpressure 1047297elds (m)

Δ1038389start Differential pressure at startup (Mpa)Δ1038389 Change in tubing pressure at the length(Mpa)Δ1038389 Change in annulus pressure at the length(Mpa)Δ1038389 Differential pressure rom top to bottom(Mpa)Δ emperature change with beore and aferwell shut-in (∘C)Δ Change in density o liquid in the tubing at

the length (Kgm3)Δ Change in density o liquid in the casing at

the length (Kgm3)

AcknowledgmentsTis research was supported by the Key Program o NSFC(Grant no 983095983088983096983091983089983088983088983093) and the Key Project o China Petroleumand Chemical Corporation (Grant no GJ-983095983091-983088983095983088983094)

References

[983089] D-L Gao and B-K Gao ldquoA method or calculating tubingbehavior in HPH wellsrdquo Journal of Petroleum Science and Engineering vol 983092983089 no 983089ndash983091 pp 983089983096983091ndash983089983096983096 983090983088983088983092

[983090] D J Hammerlindl ldquoMovement orces and stresses associatedwith combination tubing strings sealed in packersrdquo Journal of

Petroleum echnology vol 983090983097 pp 983089983097983093ndash983090983088983096 983089983097983095983095

[983091] A Lubinski W S Althouse and J L Logan ldquoHelical bucklingo tubular sealed in packersrdquo Journal of Petroleum echnology vol 983089983092 no 983094 pp 983094983093983093ndash983094983095983088 983089983097983094983090

[983092] P R Paslay and D B Bogy ldquoTe stability o a circular rodlaterally constrained to be in contact with an inclined circularcylinderrdquo Journal of Applied Mechanics vol 983091983089 pp 983094983088983093ndash983094983089983088983089983097983094983092

[983093] R Dawson and P R Paslay ldquoDrillpipe buckling in inclinedholesrdquo Journal of Petroleum echnology vol 983091983094 no 983089983088 pp 983089983095983091983092ndash983089983095983091983096 983089983097983096983092

[983094] R F MitchellldquoEffects o well deviationon helical bucklingrdquo SPEDrilling and Completion vol 983089983090 no 983089 pp 983094983091ndash983094983096 983089983097983097983095

7252019 Analyzing Axial Stress and Deformation of Tubular

httpslidepdfcomreaderfullanalyzing-axial-stress-and-deformation-of-tubular 99

Te Scienti1047297c World Journal 983097

[983095] P Ding and X Z Yan ldquoForce analysis o high pressure waterinjection stringrdquo Petroleum Dring echiques vol983091983094 no 983093p 983090983091983090983088983088983093

[983096] Z F Li ldquoCasing cementing with hal warm-up or thermalrecovery wellsrdquo Journal of Petroleum Science and Engineering vol 983094983089 no 983090ndash983092 pp 983097983092ndash983097983096 983090983088983088983096

[983097] A M Sun ldquoTe analysis and computing o water injectiontubularrdquo Drilling and Production echnology vol 983090983094 no 983091 pp983093983093ndash983093983095 983090983088983088983091 (Chinese)

[983089983088] J P Xu ldquoStress analysis and optimum design o well comple-tionrdquo echnical Report o Sinopec GJ-983095983091-983088983095983088983094 983090983088983088983097

[983089983089] J P Xu Y Q Liu S Z Wang and B Qi ldquoNumerical modellingo steam quality in deviated wells with variable ( P) 1047297eldsrdquoChemical Engineering Science vol 983096983092 pp 983090983092983090ndash983090983093983092 983090983088983089983090

[983089983090] A R Hasan and C S Kabir ldquowo-phase 1047298ow in vertical andinclined annulirdquo International Journal of Multiphase Flow vol983089983096 no 983090 pp 983090983095983097ndash983090983097983091 983089983097983097983090

[983089983091] HD Beggs and JR BrillldquoA studyo two-phase 1047298owin inclinedpipesrdquo Journal of Petroleum echnology vol 983090983093 no 983093 pp 983094983088983095ndash983094983089983095 983089983097983095983091 paper 983092983088983088983095-PA

[983089983092] H Mukherjee and J P Brill ldquoPressure drop correlations orinclined two-phase 1047298owrdquo Journal of Energy Resources echnol-ogy vol 983089983088983095 no 983092 pp 983093983092983097ndash983093983093983092 983089983097983096983093

7252019 Analyzing Axial Stress and Deformation of Tubular

httpslidepdfcomreaderfullanalyzing-axial-stress-and-deformation-of-tubular 39

Te Scienti1047297c World Journal 983091

983090983091 Te Axial Load and Axial Stress of the ubular

983090983091983089 Initial Axial Load and Initial Axial Stress of Steam

Injection ubular

Initial Axial Load Te section to which the distance rom thewellhead is () was considered Te axial static load by thedeadweight o tubular is as ollows

1038389 = int1103925

1038389 cos907317 = 4 1 8520082 minus 9073172852009int1103925

1038389cos907317 (983089)

where 1038389 is the deadweight o tubular is the average unitlength weight o tubing is the length o tubular 1 is thedensity o tubular and is the inclination angle

Te axial static load by the buoyant weight is as ollows

9073171038389 = minus211039252 int1103925

1038389cos907317 = minus2983080

2 9830812 int1103925

1038389cos907317

(983090)

where 9073171038389 is the buoyant weight o tubular 2 is the density o packer 1047298uid

Te axial load by the steam injection pressure

1038389 = 1038389103838919073172

4 (983091)

where 103838910383891 represents the inner pressure at this section

Tereore summing (983089) (983090) and (983091) the axial orces inthe section are obtained as ollows

1038389 = 1038389 + 9073171038389 + 1038389 (983092)

Initial Axial Stress Te axial stress can be derived rom theollowing equation

1038389 = 41038389 10486162 minus 90731721048617 (983093)

983090983091983090 Axial Termal Stress of Steam Injection ubular In theprocess o steam injection the temperature o tubular willchange with time and depth which will make the tubulardeorm as ollows

1038389

= 104861610383891

minus 10383890

1048617 = Δ (983094)

where represents the steel elastic modulus o tubular isthe warm balloon coefficient o the tubular string and Δ isthe temperature change with beore and afer steam injection

983090983091983091 Axial Stress of Steam Injection ubular by the Changewith Pressure Te effect acting the tubular with pressurechange which is called ballooning effect normally

Ballooning Stress Analysis Te ballooning effect will beproduced rom pressure acted in inner and outer o the tubeGenerally there are two kinds o tubular in oil wells One isthe tubulars whose outerdiameteris 983096983096983097 mm inner diameter

Pz1

Pz2D

d

F983145983143983157983154983141 983090 Te radial and tangential stresses 1047297gure o tube

is 983095983094 mm and thickness o tubes is 983094983093 mm ((9073172) =171 gt 5) the other is the tubular whose outer diameter is983089983089983092983091 mm inner diameter is 983089983088983088983093 mm and thickness o tubesis 983094983097 mm ((9073172) = 137 gt 5) Neither is the thin-wallproblem Tereore it should be solvedby Lamersquos ormula [983096]

Te radial and tangential stresses in the thick-wall cylin-der can be shown as Figure 983090 Te two can be calculated asollows

1038389 = 9073172103838910383891 minus 21038389103838902 minus 9073172 minus 1048616103838910383891 minus 1038389103838901048617 29073172

10486162 minus 90731721048617 42

1038389 = 9073172103838910383891 minus 21038389103838902 minus 9073172 + 1048616103838910383891 minus 103838910383890104861729073172

10486162 minus 90731721048617 42 (983095)

where is radial stress is tangential stress (907317 le le ) isradial coordinate

103838910383891 is tube internal pressure at

point and

103838910383890 is tube external pressure at point

983090983091983092 Axial Stress of Steam Injection ubular by the FrictionLoss In act the 1047298ow in the tubular should be multi1047298ow Onthe process o steam injection the 1047298ow will be run and it willgive rise to riction effect to cause axial stress In our paperwe consider the 1047298ow gas-liquid mix 1047298ow and the liquid headloss is gotten by the Darcy-Weisbach ormula [983097] as ollows

ℎ = ( minus ) ]22907317 (983096)

where ℎ means heat loss o liquid 1047298ow is rictional headlosses coefficients and ] is the velocity o liquid 1047298ow

Te riction drag in tubular is 1038389 = ℎ9073172 ( isdensity o liquid 1047298ow) Te axial stress by 1047297ction drag can beobtained as ollows

1038389 = 41038389 10486162 minus 90731721048617 (983097)

983090983092 Analysis of Axial Deformation Based on the studiesand analyses mentioned above the axial deormation on thetubular is made up o the ollowing parts

7252019 Analyzing Axial Stress and Deformation of Tubular

httpslidepdfcomreaderfullanalyzing-axial-stress-and-deformation-of-tubular 49

983092 Te Scienti1047297c World Journal

983090983092983089 Te Axial Deformation by the Axial Static Stress For themicroelement o the tubular 907317 the unit deormation by thestatic stress can be computed by generalized Hooke law

1 = 1 [1038389 minus 10486161038389 + 10383891048617] (983089983088)

where represents Poissonrsquos ratiosTe axial deormation at an element can be obtained

through integrating on the length o the element as ollows

Δ1 = int

minus1

1 [1038389 minus 10486161038389 + 10383891048617]907317 (983089983089)

Tereore the total axial deormation by the static stresscan be gotten accumulating each element as ollows

Δ1 = sum=1

Δ1 (983089983090)

983090983092983090 Te Axial Deformation with emperature Changed Forthe microelement o the tubular 907317 the unit deormation by the temperature change is as ollows

Δ2 = int

minus1

1038389 907317 = ΔΔ (983089983091)

Te same principle is that the total axial deormation by thevariedtemperature 1047297elds canbe gottenaccumulating eachelement as ollows

Δ2

=

sum=1Δ2

(983089983092)

983090983092983091 Te Axial Deformation with the Friction Drag For themicroelement o the tubular 907317 the unit deormation by theriction orce is as ollows

Δ3 = int

0

1038389 907317 = ]

29073172

10486162 minus 90731721048617 (983089983093)

983090983092983092 Te Axial Deformation with the ubular String BucklingResearchers in general call the buckling a bending effect Tetubular is reely suspended in the absence o 1047298uid inside asshown in Figure 983091(a) Because the orce

applied at the end

o the tubular which is large enough the tubular will buckleas shown in Figure 983091(b)

Lubinski et al [983091] had done many researches on thephenomenon From their work we canget the buckling effectDe1047297ne the virtual axial orce o tubing as ollows

= 1103925 104861610383891 minus 103838901048617 (983089983094)

where 10383891 is the pressure inside the tubular at the packerlength 10383890 is the pressure outside the tubular at the packerlength and 1103925 is the area corresponding to packer bore

By (983089983094) whether the tubular will buckle or not can be judged Te string will buckle i

is positive or remain

(a)

F

Neutral point

(b)

F983145983143983157983154983141 983091 Buckling o tubular

straight i is negative or zero Te axial deormation o thetubular string buckling is

Δ4 = minus211039252

1048616Δ10383891 minus Δ10383890104861728 (983089983095)

where means tubing-to-casing radialclearance is momento inertia o tubing cross-section with respect to its diameter( = (4 minus 9073174)64) Δ denotes change with beore and aferinjection and

is the unit weight o tubing as

Δ4 = sum=1

Δ4 (983089983096)

In addition the position o the neutral point is needed Telength () rom the packer to the point can be computed asollows

= (983089983097)

Generally the neutral point should be in tubular ( le )However at the multipackers it will occur that the neutral

point is outside the tubing between dual packers In thispaper we leave the latter phenomenon

o sum up the whole deormation length can be repre-sented as ollows

Δ = Δ1 + Δ2 + Δ3 + Δ4 (983090983088)

983090983093 Te Analysis of the Varied ( 1038389) Fields In the courseo dryness modeling we can 1047297nd that the numerical valueso deormation ((983089983088) (983089983091) and (983089983095)) were affected by thetemperature and pressure In act the two parameters variedaccording to the depth and time changing So the varied

(1038389) 1047297elds need to be researched Under the China Sinopec

7252019 Analyzing Axial Stress and Deformation of Tubular

httpslidepdfcomreaderfullanalyzing-axial-stress-and-deformation-of-tubular 59

Te Scienti1047297c World Journal 983093

Group Hi-ech Project ldquoStress analysis and optimum designo well completionrdquo in 983090983088983088983097 [983094] undertaken by SichuanUniversity at early time Te varied ( 1038389) 1047297elds had beendeduced strictly based on the mass momentum and energy balance Te proo details can be shown in Xu et al [983089983089] Te

varied ( 1038389) 1047297elds is

9073171038389907317 = minus 104861611039251048617 + cos + (1103925) (907317907317)1 minus (1103925)

907317907317 = minus

]

907317907317 minus 9073171038389907317 minus

cos

minus ]34

+ 1048616 minus 1048617

1038389 104861601048617 = 10383890 1048616 01048617 = 0 907317104861601048617 = 9073170 1048616 01048617 = 0

(983090983089)

3 Numerical Implementation983091983089 Calculation of Some Parameters In this section we willgive the calculating method o some parameters

(983089) Each pointrsquos inclination

= minus1 + 1048616 minus minus11048617 ΔΔ (983090983090)

where represents segment point o calculation Δrepresents measurement depth o inclination angle and minus1 Δ is the step length o calculationransient heat transer unction [983089983090]

10486161048617 = 85209198316310486991128radic 10486161 minus 03radic 1048617 le 15104861604063 + 05 ln 1048617 1 + 06 gt 15

(983090983091)

(983090) Te density o wet steam Since the 1047298ow o the water vapor in is the gas-liquid two-phase 1047298ow there aremany researches about this problem [983089983091 983089983092] In thepaper we adopt the M-B model to calculate theaverage density o the mixture

(983091) Te heat transer coefficient to rom different posi-

tions o the axis o the wellbore to the second suraceTese resistances include the tubing wall possible insu-

lation around the tubing annular space (possibly 1047297lled witha gas or liquid but is sometimes vacuum) casing wall andcementing behind the casing as ollows

1

= 1ins

ln + 1ℎ + ℎ +

1cem

lncem (983090983092)

ins and cem are the heat conductivity o the heat insulatingmaterial and the cement sheath respectively ℎ and ℎ are thecoefficients o the convection heat transer and the radiationheat transer

983091983090 Initial Condition In order to solve model some de1047297niteconditions and initial conditions should be added Te initialconditions comprise the distribution o the pressure andtemperature at the well top In this paper we adopt the

value at the initial time by actual measurement Beore steaminjected the temperature o tubular just is initial temperature

o ormation (1038389 = 0 + cos is geothermal gradient)At the same time the pressure o inner tubular is assumed tobe equal to the outer tubular beore steam injected

983091983091 Steps of Algorithm o simpliy the calculation wedivided the wells into several short segments o the samelength Te length o a segment varies depending on varia-tions in wall thickness hole diameter 1047298uid density inside andoutside the pipe and wells geometry Te model begins withthe calculation at one particular position in the wells the topo the pipe

Step 983089 Set step length o depth In addition we denote

the relatively tolerant error by Te smaller ℎ is themore accurate the results are However it will lead to rapidincreasing calculating time In our paper we set ℎ = 1 (m)and = 5

Step 983090 Give the initial conditions

Step 983091 Compute each pointrsquos inclination

Step 983092 Compute the parameters under the initial conditionsor the last depth variables

Step 983093 Let = then we can get the by solving theollowing equation

= 22 + 1

1038389=0 = 0 + cos

1038389=1

= minus 12

907317907317

1038389 rarrinfin

= 0

(983090983093)

Let be the temperature at the injection time and

radial at the depth We apply the 1047297nite different methodto discretize the equations as ollows

+1 minus

= +1+1 minus 2

+1 + minus1+12 minus +1

+1 minus +1 (983090983094)

where is the interval o time and is the interval o radialrespectively It can be transormed into the standard orm asollows

minus + +1+1 + 2 +

+1 minus +1

minus1 = 2

(983090983095)

7252019 Analyzing Axial Stress and Deformation of Tubular

httpslidepdfcomreaderfullanalyzing-axial-stress-and-deformation-of-tubular 69

983094 Te Scienti1047297c World Journal

983137983138983148983141 983089 Parameters o pipes

Diameter (m) Tickness (m) Weight (Kg) Expansion Elastic (Gpa) Poissonrsquos ratios Using length (m)

983088983088983096983096983097 983088983088983089983090983097983093 983090983091983095983097 983088983088983088983088983088983089983089983093 983090983089983093 983088983091 983090983095983088

983088983088983096983096983097 983088983088983088983097983093983091 983089983096983090983096 983088983088983088983088983088983089983089983093 983090983089983093 983088983091 983089983090983088

983088983088983096983096983097 983088983088983088983095983091983092 983089983093983088983092 983088983088983088983088983088983089983089983093 983090983089983093 983088983091 983094983090983088

983088983088983096983096983097 983088983088983088983094983092983093 983089983091983093983096 983088983088983088983088983088983089983089983093 983090983089983093 983088983091 983090983097983088

983137983138983148983141 983090 Well parameters

Measured (m) Internal (m) External (m)

983091983091983094983095 983088983089983093983092983095983096 983088983089983095983095983096

983092983090983090983094 983088983089983093983090983093 983088983089983095983095983096

983089983091983088983088983088 983088983089983088983096983094983090 983088983089983090983095

983137983138983148983141 983091 Parameters o azimuth inclination and vertical depth

Number Measured

(m)Inclination

(∘)Azimuth

(∘)Vertical depth

(m)

983089 983089983091983093 983090983094983091 983090983092983089983088983089 983089983091983092983095983090

983090 983090983095983096 983089983090983091 983090983091983095983096983094 983090983095983095983097983089

983091 983091983094983092 983089983092983091 983090983089983091983096983094 983091983094983091983096983090

983092 983091983097983091 983090983089983095 983090983094983091983096 983091983097983090983093983091

983093 983092983090983090 983089983096983093 983092983092983093983094 983092983090983089983090983096

983094 983092983093983088 983088983096983090 983089983097983089983089983090 983092983092983097983094983090

983095 983092983096983094 983090983097983091 983090983094983097983088983095 983092983096983093983092983095

983096 983093983089983092 983089983088983091 983090983097983095983093983093 983093983089983091983096983091

983097 983093983092983091 983091983093983096 983091983090983092983093983089 983093983092983089983095983092

983089983088 983093983095983089 983090983097983096 983091983088983091983088983093 983093983095983088983092983091

983089983089 983094983088983088 983090983088983091 983090983088983092983095983092 983093983097983097983092983090

983089983090 983094983090983096 983090983091983092 983089983094983092983091983091 983094983090983095983090983096

983089983091 983094983094983088 983089983096983093 983089983097983093983090983096 983094983093983097983093983094

983089983092 983095983090983091 983091983089983092 983090983089983092983096983092 983095983090983089983095983088

983089983093 983095983096983090 983088983097983096 983090983089983094983092983096 983095983096983089983091983088

983089983094 983096983091983088 983090983089983093 983090983090983097983091983089 983096983090983097983089983090

983089983095 983096983094983088 983090983094983095 983090983092983092983088983091 983096983093983097983095983089

983089983096 983097983088983096 983092983096983093 983090983094983094983094983090 983097983088983092983088983096

983089983097 983097983090983096 983094983095983090 983090983093983096983095983096 983097983090983089983092983090

983090983088 983097983095983090 983090983088983091 983090983091983094983096983096 983097983095983089983095983089

983090983089 983089983088983090983093 983092983095983096 983090983091983097983090983095 983089983088983090983089983090983093

983090983090 983089983088983093983096 983092983088983089 983090983092983092983093983097 983089983088983093983093983093983096

983090983091 983089983088983096983097 983092983097983096 983090983090983096983090 983089983088983096983092983089983095

983090983092 983089983089983091983090 983091983095983093 983090983091983091983096983096 983089983089983090983097983090983096

983090983093 983089983089983095983092 983093983094983091 983090983091983093983089983092 983089983089983094983096983096983095

983090983094 983089983090983088983092 983092983090983091 983090983091983092983091983096 983089983090983088983088983097983097

983090983095 983089983090983091983093 983091983096983095 983090983091983092983097983097 983089983090983091983090983088983096

983090983096 983089983090983094983096 983092983097983095 983090983091983090983093983095 983089983090983094983091983092983093

983090983097 983089983091983088983088 983096983096983092 983090983091983091983090983096 983089983090983096983092983097983094

Ten the different method is used to discretize theboundary condition For = 1 we have

+12 minus 1 + 2

+11 = 2

(983090983096)

For = we have

+1 minus +1

minus1 = 0 (983090983097)

We can compute the symbolic solution o the temperatureo thestratumIn this step we willget thediscretedistributiono as the ollowing matrix

98313110486671048667104866710486671048667104866710486671048667104866710486671048667852059

11 2

1 sdot sdot sdot 1 sdot sdot sdot

12 2

2 sdot sdot sdot 2 sdot sdot sdot

sdot sdot sdot

1 2

sdot sdot sdot sdot sdot sdot

sdot sdot sdot

1 2

sdot sdot sdot sdot sdot sdot

98313310486691048669104866910486691048669104866910486691048669104866910486691048669852061

(983091983088)

where represents the injection time and represents theradial

Step 983094 Let the right parts o the coupled differential equa-tions be unctions where ( = 1 2) Ten we can obtain asystem o coupled unctions as ollows

1 = minus 104861611039251048617 + cos + (1103925) (907317907317)1 minus (1103925)

2 = minus ]

907317907317 minus 9073171038389907317 minus cos

minus ]34

+ 1048616 minus 1048617

(983091983089)

where at = 1

Step 983095 Assume that 1038389 are ( = 1 2) respectively Tenwe can obtain some basic parameters as ollows

= 10486161 21048617 = 1 + ℎ12 2 +

ℎ22 = 1 + ℎ12 2 +

ℎ22 907317 = 10486161 + ℎ1 2 + ℎ21048617

(983091983090)

7252019 Analyzing Axial Stress and Deformation of Tubular

httpslidepdfcomreaderfullanalyzing-axial-stress-and-deformation-of-tubular 79

Te Scienti1047297c World Journal 983095

983137983138983148983141 983092 Te results o the axial orce and various kinds o deormation lengths

Number Depth

(m)Axial orce

(N)

Displacement by temperaturechanged (m)

Displacement by pressure

changed (m)

Axial deormation(m)

Bucklingdeormation (m)

otal deormation(m)

983089 983089 983096983097983093983090983092983092983096 983088 983088 983088 983088 983088

983090 983089983088983088 983096983093983092983095983090983092983096 983088983089983090983088983089 983088983088983088983097983096983094 983088983088983090983092 983088 983088983089983093983092983092

983091 983090983088983088 983096983089983092983090983089983093983093 983088983090983091983094983090 983088983088983089983097983091983097983090 983088983088983093983090 983088 983088983091983088983095983090

983092 983091983088983088 983095983095983091983095983089983095983095 983088983091983092983096983091 983088983088983090983096983093983097983096 983088983088983096983090 983088 983088983092983093983097

983093 983092983088983088 983095983091983095983097983095983088 983088983092983093983094983092 983088983088983091983095983092983095983094 983088983089983089983093 minus983088983088983088983094 983088983094983088983090983097

983094 983093983088983088 983095983088983094983096983095983095983091 983088983093983094983088983094 983088983088983092983094983088983090983096 983088983089983093983090 minus983088983088983088983094 983088983095983093983090983091

983095 983094983088983088 983094983095983093983095983094983091983097 983088983094983094983088983095 983088983088983093983092983090983093983092 983088983089983097983090 minus983088983088983088983094 983088983097983088983088983094

983096 983095983088983088 983094983092983092983094983088983090983091 983088983095983093983094983097 983088983088983094983090983089983093983091 983088983090983091983093 minus983088983088983088983094 983089983088983092983096

983097 983096983088983088 983094983089983091983092983091983095983090 983088983096983092983097 983088983088983094983097983095983090983093 983088983090983096983091 minus983088983088983088983095 983089983089983097983092983094

983089983088 983097983088983088 983093983096983090983090983095983090983089 983088983097983091983095983089 983088983088983095983094983097983094983096 983088983091983091983093 minus983088983088983088983095 983089983091983092983090983090

983089983089 983089983088983088983088 983093983093983089983089983088983095983090 983089983088983090983089983090 983088983088983096983091983096983096983091 983088983091983097983089 minus983088983088983088983095 983089983092983096983097983094

983089983090 983089983089983088983088 983093983089983097983097983092983090983091 983089983089983088983089983092 983088983088983097983088983092983095983089 983088983092983093983090 minus983088983088983088983095 983089983094983091983094983095

983089983091 983089983090983088983088 983092983096983096983095983095983095983093 983089983089983095983095983093 983088983088983097983094983095983091983089 983088983093983089983095

minus983088983088983088983097 983089983095983096983090983090

983089983092 983089983091983088983088 983092983093983095983094983089983090983096 983089983090983092983097983094 983088983089983088983090983094983094983090 983088983093983096983092 minus983088983088983089 983089983097983090983094983089

Step 983096 Calculate the pressure and temperature at point( + 1)+1

=

+ ℎ 1048616 + 2 + 2 + 90731710486176 = 12 = 12

(983091983091)

Step 983097 Calculate the deormation Δ1 Δ2 and Δ4 by previous equations

Step 983089983088 Repeat the third step to the tenth step until tubularlength is calculated

Step 983089983089 Calculate the deormationΔ3 and total deormationlength as ollows

Δ = sum=1

Δ1 + sum=1

Δ2 + Δ3 + sum=1

Δ4 (983091983092)

4 Numerical Simulation

983092983089 Parameters o demonstrate the application o our the-

ory we study a pipe in X well which is in Sichuan ProvinceChina All the basic parameters are given as ollows deptho the well is 983089983091983088983088 m ground thermal conductivity parameteris 206 ground temperature is 16∘C ground temperaturegradient is 00218(∘Cm) roughness o the inner surace o the well is 0000015 and parameters o pipes inclined wellinclination azimuth and vertical depth are given in ables 983089983090 and 983091

983092983090 Main Results and Results Analysis Afer calculation weobtain a series o results o this well as able 983092 Te in1047298uenceo outputson the axial deormation o tubing was investigatedas shown by Figure 983092

25

2

15

1

05

0

T o t a l a x i a l d e

f o r m a t i o n

( m )

Depth (m)

1 1200800400

700000m3d

500000m3d

300000m3d

F983145983143983157983154983141 983092 Te total axial deormation under varied outputs

From the results as shown in Figure 983092 and able 983092 some

useul analysis can be drawn

(983089) Te amount o steam injected and inject pressureaffected the stretching orce with special severity

(983090) Te results were as ollows the length o tubulardeormation was risen with increased injected pres-sure or injected velocity

(983091) Te length o tubular deormation increases with theincreasing o outputs but more slowly

(983092) Te thermal stress is the main actor in1047298uencing thetubular deormation Tereore the temperature o steam injected should not be too high

7252019 Analyzing Axial Stress and Deformation of Tubular

httpslidepdfcomreaderfullanalyzing-axial-stress-and-deformation-of-tubular 89

983096 Te Scienti1047297c World Journal

(983093) Te lifing prestressed cementing technology hasimportant meanings to reduce the deormation o tubular

(983094) Te creeping displacement o downhole stings willproduce an upward contractility which causes packer

depressed or lapsed Tereore the effective measuresshould be adopted to control the companding o tubular

5 Conclusion

In this paper a total tubular deormation model aboutdeviated wells was given A coupled-system model o differ-ential equations concerningpressureand temperature in hightemperature-high pressure steam injection wells according tomass momentum andenergy balances which canreduce theerror o axial stress and axial deormation was given insteado theaveragevalueor simplelinear relationship in traditional

research Te basic data o the Well (high temperature andhigh pressure gas well) 983089983091983088983088 m deep in Sichuan China wereused or case history calculations Te results can providetechnical reliance or the process o designing well tests indeviated gas wells and dynamic analysis o production

Nomenclature

907317 Inner diameter (m)9073171038389 Microelement o the tubular Acceleration o gravity (ms2)ℎ Depth o top tubular located at the packer(m)

ime o down stroke (s) Dimensionless time (dimensionless)V Velocity o 1047298uid in tubing (ms)V Velocity o down stroke (ms) Distance coordinate in the 1047298ow direction

(m) along the tubing1103925 Constant cross-sectional 1047298ow area (m2)1103925 Effective area (m2)1103925 Area corresponding to packer bore (m2) Te Joule-Tomson coefficient(dimensionless) Heat capacity o 1047298uids (JKg sdotK)

Outer diameter (m)

Steel elastic modulus o tubular (Mpa) Axial orces in the section (N)1038389 Axial tensile strength (N) Friction orce (N) Piston orce or supporting packerrsquos pressure(N) Pumping orce (N) Length o tubular (m)9073171038389 Buoyant weight o tubular (Kg)1038389 Dead weight o tubular (Kg)10383890 Pressure outside the tubular (Mpa)10383891 Pressure inside the tubular at the packerlength (Mpa)

1038389 Pressure in tubing (Mpa) emperature in tubing (∘C) Initial temperature o ormation (∘C) Mass 1047298ow rate (Kgs) otal length (m)1 Material density (Kgm3)

2 Packer 1047298uid density (Kgm3

) Inclination angle (∘) Warm balloon coefficient o the tubularstring (dimensionless) Drop o any parameter1038389 Axial thermal stress () Density o 1047298uid in the tubing (Kgm3)Δ3 Te tubular string buckling axialdeormation (m)Δ otal axial deormation by variedtemperature 1047297elds (m)Δ907317 otal axial deormation by the variedpressure 1047297elds (m)

Δ1038389start Differential pressure at startup (Mpa)Δ1038389 Change in tubing pressure at the length(Mpa)Δ1038389 Change in annulus pressure at the length(Mpa)Δ1038389 Differential pressure rom top to bottom(Mpa)Δ emperature change with beore and aferwell shut-in (∘C)Δ Change in density o liquid in the tubing at

the length (Kgm3)Δ Change in density o liquid in the casing at

the length (Kgm3)

AcknowledgmentsTis research was supported by the Key Program o NSFC(Grant no 983095983088983096983091983089983088983088983093) and the Key Project o China Petroleumand Chemical Corporation (Grant no GJ-983095983091-983088983095983088983094)

References

[983089] D-L Gao and B-K Gao ldquoA method or calculating tubingbehavior in HPH wellsrdquo Journal of Petroleum Science and Engineering vol 983092983089 no 983089ndash983091 pp 983089983096983091ndash983089983096983096 983090983088983088983092

[983090] D J Hammerlindl ldquoMovement orces and stresses associatedwith combination tubing strings sealed in packersrdquo Journal of

Petroleum echnology vol 983090983097 pp 983089983097983093ndash983090983088983096 983089983097983095983095

[983091] A Lubinski W S Althouse and J L Logan ldquoHelical bucklingo tubular sealed in packersrdquo Journal of Petroleum echnology vol 983089983092 no 983094 pp 983094983093983093ndash983094983095983088 983089983097983094983090

[983092] P R Paslay and D B Bogy ldquoTe stability o a circular rodlaterally constrained to be in contact with an inclined circularcylinderrdquo Journal of Applied Mechanics vol 983091983089 pp 983094983088983093ndash983094983089983088983089983097983094983092

[983093] R Dawson and P R Paslay ldquoDrillpipe buckling in inclinedholesrdquo Journal of Petroleum echnology vol 983091983094 no 983089983088 pp 983089983095983091983092ndash983089983095983091983096 983089983097983096983092

[983094] R F MitchellldquoEffects o well deviationon helical bucklingrdquo SPEDrilling and Completion vol 983089983090 no 983089 pp 983094983091ndash983094983096 983089983097983097983095

7252019 Analyzing Axial Stress and Deformation of Tubular

httpslidepdfcomreaderfullanalyzing-axial-stress-and-deformation-of-tubular 99

Te Scienti1047297c World Journal 983097

[983095] P Ding and X Z Yan ldquoForce analysis o high pressure waterinjection stringrdquo Petroleum Dring echiques vol983091983094 no 983093p 983090983091983090983088983088983093

[983096] Z F Li ldquoCasing cementing with hal warm-up or thermalrecovery wellsrdquo Journal of Petroleum Science and Engineering vol 983094983089 no 983090ndash983092 pp 983097983092ndash983097983096 983090983088983088983096

[983097] A M Sun ldquoTe analysis and computing o water injectiontubularrdquo Drilling and Production echnology vol 983090983094 no 983091 pp983093983093ndash983093983095 983090983088983088983091 (Chinese)

[983089983088] J P Xu ldquoStress analysis and optimum design o well comple-tionrdquo echnical Report o Sinopec GJ-983095983091-983088983095983088983094 983090983088983088983097

[983089983089] J P Xu Y Q Liu S Z Wang and B Qi ldquoNumerical modellingo steam quality in deviated wells with variable ( P) 1047297eldsrdquoChemical Engineering Science vol 983096983092 pp 983090983092983090ndash983090983093983092 983090983088983089983090

[983089983090] A R Hasan and C S Kabir ldquowo-phase 1047298ow in vertical andinclined annulirdquo International Journal of Multiphase Flow vol983089983096 no 983090 pp 983090983095983097ndash983090983097983091 983089983097983097983090

[983089983091] HD Beggs and JR BrillldquoA studyo two-phase 1047298owin inclinedpipesrdquo Journal of Petroleum echnology vol 983090983093 no 983093 pp 983094983088983095ndash983094983089983095 983089983097983095983091 paper 983092983088983088983095-PA

[983089983092] H Mukherjee and J P Brill ldquoPressure drop correlations orinclined two-phase 1047298owrdquo Journal of Energy Resources echnol-ogy vol 983089983088983095 no 983092 pp 983093983092983097ndash983093983093983092 983089983097983096983093

7252019 Analyzing Axial Stress and Deformation of Tubular

httpslidepdfcomreaderfullanalyzing-axial-stress-and-deformation-of-tubular 49

983092 Te Scienti1047297c World Journal

983090983092983089 Te Axial Deformation by the Axial Static Stress For themicroelement o the tubular 907317 the unit deormation by thestatic stress can be computed by generalized Hooke law

1 = 1 [1038389 minus 10486161038389 + 10383891048617] (983089983088)

where represents Poissonrsquos ratiosTe axial deormation at an element can be obtained

through integrating on the length o the element as ollows

Δ1 = int

minus1

1 [1038389 minus 10486161038389 + 10383891048617]907317 (983089983089)

Tereore the total axial deormation by the static stresscan be gotten accumulating each element as ollows

Δ1 = sum=1

Δ1 (983089983090)

983090983092983090 Te Axial Deformation with emperature Changed Forthe microelement o the tubular 907317 the unit deormation by the temperature change is as ollows

Δ2 = int

minus1

1038389 907317 = ΔΔ (983089983091)

Te same principle is that the total axial deormation by thevariedtemperature 1047297elds canbe gottenaccumulating eachelement as ollows

Δ2

=

sum=1Δ2

(983089983092)

983090983092983091 Te Axial Deformation with the Friction Drag For themicroelement o the tubular 907317 the unit deormation by theriction orce is as ollows

Δ3 = int

0

1038389 907317 = ]

29073172

10486162 minus 90731721048617 (983089983093)

983090983092983092 Te Axial Deformation with the ubular String BucklingResearchers in general call the buckling a bending effect Tetubular is reely suspended in the absence o 1047298uid inside asshown in Figure 983091(a) Because the orce

applied at the end

o the tubular which is large enough the tubular will buckleas shown in Figure 983091(b)

Lubinski et al [983091] had done many researches on thephenomenon From their work we canget the buckling effectDe1047297ne the virtual axial orce o tubing as ollows

= 1103925 104861610383891 minus 103838901048617 (983089983094)

where 10383891 is the pressure inside the tubular at the packerlength 10383890 is the pressure outside the tubular at the packerlength and 1103925 is the area corresponding to packer bore

By (983089983094) whether the tubular will buckle or not can be judged Te string will buckle i

is positive or remain

(a)

F

Neutral point

(b)

F983145983143983157983154983141 983091 Buckling o tubular

straight i is negative or zero Te axial deormation o thetubular string buckling is

Δ4 = minus211039252

1048616Δ10383891 minus Δ10383890104861728 (983089983095)

where means tubing-to-casing radialclearance is momento inertia o tubing cross-section with respect to its diameter( = (4 minus 9073174)64) Δ denotes change with beore and aferinjection and

is the unit weight o tubing as

Δ4 = sum=1

Δ4 (983089983096)

In addition the position o the neutral point is needed Telength () rom the packer to the point can be computed asollows

= (983089983097)

Generally the neutral point should be in tubular ( le )However at the multipackers it will occur that the neutral

point is outside the tubing between dual packers In thispaper we leave the latter phenomenon

o sum up the whole deormation length can be repre-sented as ollows

Δ = Δ1 + Δ2 + Δ3 + Δ4 (983090983088)

983090983093 Te Analysis of the Varied ( 1038389) Fields In the courseo dryness modeling we can 1047297nd that the numerical valueso deormation ((983089983088) (983089983091) and (983089983095)) were affected by thetemperature and pressure In act the two parameters variedaccording to the depth and time changing So the varied

(1038389) 1047297elds need to be researched Under the China Sinopec

7252019 Analyzing Axial Stress and Deformation of Tubular

httpslidepdfcomreaderfullanalyzing-axial-stress-and-deformation-of-tubular 59

Te Scienti1047297c World Journal 983093

Group Hi-ech Project ldquoStress analysis and optimum designo well completionrdquo in 983090983088983088983097 [983094] undertaken by SichuanUniversity at early time Te varied ( 1038389) 1047297elds had beendeduced strictly based on the mass momentum and energy balance Te proo details can be shown in Xu et al [983089983089] Te

varied ( 1038389) 1047297elds is

9073171038389907317 = minus 104861611039251048617 + cos + (1103925) (907317907317)1 minus (1103925)

907317907317 = minus

]

907317907317 minus 9073171038389907317 minus

cos

minus ]34

+ 1048616 minus 1048617

1038389 104861601048617 = 10383890 1048616 01048617 = 0 907317104861601048617 = 9073170 1048616 01048617 = 0

(983090983089)

3 Numerical Implementation983091983089 Calculation of Some Parameters In this section we willgive the calculating method o some parameters

(983089) Each pointrsquos inclination

= minus1 + 1048616 minus minus11048617 ΔΔ (983090983090)

where represents segment point o calculation Δrepresents measurement depth o inclination angle and minus1 Δ is the step length o calculationransient heat transer unction [983089983090]

10486161048617 = 85209198316310486991128radic 10486161 minus 03radic 1048617 le 15104861604063 + 05 ln 1048617 1 + 06 gt 15

(983090983091)

(983090) Te density o wet steam Since the 1047298ow o the water vapor in is the gas-liquid two-phase 1047298ow there aremany researches about this problem [983089983091 983089983092] In thepaper we adopt the M-B model to calculate theaverage density o the mixture

(983091) Te heat transer coefficient to rom different posi-

tions o the axis o the wellbore to the second suraceTese resistances include the tubing wall possible insu-

lation around the tubing annular space (possibly 1047297lled witha gas or liquid but is sometimes vacuum) casing wall andcementing behind the casing as ollows

1

= 1ins

ln + 1ℎ + ℎ +

1cem

lncem (983090983092)

ins and cem are the heat conductivity o the heat insulatingmaterial and the cement sheath respectively ℎ and ℎ are thecoefficients o the convection heat transer and the radiationheat transer

983091983090 Initial Condition In order to solve model some de1047297niteconditions and initial conditions should be added Te initialconditions comprise the distribution o the pressure andtemperature at the well top In this paper we adopt the

value at the initial time by actual measurement Beore steaminjected the temperature o tubular just is initial temperature

o ormation (1038389 = 0 + cos is geothermal gradient)At the same time the pressure o inner tubular is assumed tobe equal to the outer tubular beore steam injected

983091983091 Steps of Algorithm o simpliy the calculation wedivided the wells into several short segments o the samelength Te length o a segment varies depending on varia-tions in wall thickness hole diameter 1047298uid density inside andoutside the pipe and wells geometry Te model begins withthe calculation at one particular position in the wells the topo the pipe

Step 983089 Set step length o depth In addition we denote

the relatively tolerant error by Te smaller ℎ is themore accurate the results are However it will lead to rapidincreasing calculating time In our paper we set ℎ = 1 (m)and = 5

Step 983090 Give the initial conditions

Step 983091 Compute each pointrsquos inclination

Step 983092 Compute the parameters under the initial conditionsor the last depth variables

Step 983093 Let = then we can get the by solving theollowing equation

= 22 + 1

1038389=0 = 0 + cos

1038389=1

= minus 12

907317907317

1038389 rarrinfin

= 0

(983090983093)

Let be the temperature at the injection time and

radial at the depth We apply the 1047297nite different methodto discretize the equations as ollows

+1 minus

= +1+1 minus 2

+1 + minus1+12 minus +1

+1 minus +1 (983090983094)

where is the interval o time and is the interval o radialrespectively It can be transormed into the standard orm asollows

minus + +1+1 + 2 +

+1 minus +1

minus1 = 2

(983090983095)

7252019 Analyzing Axial Stress and Deformation of Tubular

httpslidepdfcomreaderfullanalyzing-axial-stress-and-deformation-of-tubular 69

983094 Te Scienti1047297c World Journal

983137983138983148983141 983089 Parameters o pipes

Diameter (m) Tickness (m) Weight (Kg) Expansion Elastic (Gpa) Poissonrsquos ratios Using length (m)

983088983088983096983096983097 983088983088983089983090983097983093 983090983091983095983097 983088983088983088983088983088983089983089983093 983090983089983093 983088983091 983090983095983088

983088983088983096983096983097 983088983088983088983097983093983091 983089983096983090983096 983088983088983088983088983088983089983089983093 983090983089983093 983088983091 983089983090983088

983088983088983096983096983097 983088983088983088983095983091983092 983089983093983088983092 983088983088983088983088983088983089983089983093 983090983089983093 983088983091 983094983090983088

983088983088983096983096983097 983088983088983088983094983092983093 983089983091983093983096 983088983088983088983088983088983089983089983093 983090983089983093 983088983091 983090983097983088

983137983138983148983141 983090 Well parameters

Measured (m) Internal (m) External (m)

983091983091983094983095 983088983089983093983092983095983096 983088983089983095983095983096

983092983090983090983094 983088983089983093983090983093 983088983089983095983095983096

983089983091983088983088983088 983088983089983088983096983094983090 983088983089983090983095

983137983138983148983141 983091 Parameters o azimuth inclination and vertical depth

Number Measured

(m)Inclination

(∘)Azimuth

(∘)Vertical depth

(m)

983089 983089983091983093 983090983094983091 983090983092983089983088983089 983089983091983092983095983090

983090 983090983095983096 983089983090983091 983090983091983095983096983094 983090983095983095983097983089

983091 983091983094983092 983089983092983091 983090983089983091983096983094 983091983094983091983096983090

983092 983091983097983091 983090983089983095 983090983094983091983096 983091983097983090983093983091

983093 983092983090983090 983089983096983093 983092983092983093983094 983092983090983089983090983096

983094 983092983093983088 983088983096983090 983089983097983089983089983090 983092983092983097983094983090

983095 983092983096983094 983090983097983091 983090983094983097983088983095 983092983096983093983092983095

983096 983093983089983092 983089983088983091 983090983097983095983093983093 983093983089983091983096983091

983097 983093983092983091 983091983093983096 983091983090983092983093983089 983093983092983089983095983092

983089983088 983093983095983089 983090983097983096 983091983088983091983088983093 983093983095983088983092983091

983089983089 983094983088983088 983090983088983091 983090983088983092983095983092 983093983097983097983092983090

983089983090 983094983090983096 983090983091983092 983089983094983092983091983091 983094983090983095983090983096

983089983091 983094983094983088 983089983096983093 983089983097983093983090983096 983094983093983097983093983094

983089983092 983095983090983091 983091983089983092 983090983089983092983096983092 983095983090983089983095983088

983089983093 983095983096983090 983088983097983096 983090983089983094983092983096 983095983096983089983091983088

983089983094 983096983091983088 983090983089983093 983090983090983097983091983089 983096983090983097983089983090

983089983095 983096983094983088 983090983094983095 983090983092983092983088983091 983096983093983097983095983089

983089983096 983097983088983096 983092983096983093 983090983094983094983094983090 983097983088983092983088983096

983089983097 983097983090983096 983094983095983090 983090983093983096983095983096 983097983090983089983092983090

983090983088 983097983095983090 983090983088983091 983090983091983094983096983096 983097983095983089983095983089

983090983089 983089983088983090983093 983092983095983096 983090983091983097983090983095 983089983088983090983089983090983093

983090983090 983089983088983093983096 983092983088983089 983090983092983092983093983097 983089983088983093983093983093983096

983090983091 983089983088983096983097 983092983097983096 983090983090983096983090 983089983088983096983092983089983095

983090983092 983089983089983091983090 983091983095983093 983090983091983091983096983096 983089983089983090983097983090983096

983090983093 983089983089983095983092 983093983094983091 983090983091983093983089983092 983089983089983094983096983096983095

983090983094 983089983090983088983092 983092983090983091 983090983091983092983091983096 983089983090983088983088983097983097

983090983095 983089983090983091983093 983091983096983095 983090983091983092983097983097 983089983090983091983090983088983096

983090983096 983089983090983094983096 983092983097983095 983090983091983090983093983095 983089983090983094983091983092983093

983090983097 983089983091983088983088 983096983096983092 983090983091983091983090983096 983089983090983096983092983097983094

Ten the different method is used to discretize theboundary condition For = 1 we have

+12 minus 1 + 2

+11 = 2

(983090983096)

For = we have

+1 minus +1

minus1 = 0 (983090983097)

We can compute the symbolic solution o the temperatureo thestratumIn this step we willget thediscretedistributiono as the ollowing matrix

98313110486671048667104866710486671048667104866710486671048667104866710486671048667852059

11 2

1 sdot sdot sdot 1 sdot sdot sdot

12 2

2 sdot sdot sdot 2 sdot sdot sdot

sdot sdot sdot

1 2

sdot sdot sdot sdot sdot sdot

sdot sdot sdot

1 2

sdot sdot sdot sdot sdot sdot

98313310486691048669104866910486691048669104866910486691048669104866910486691048669852061

(983091983088)

where represents the injection time and represents theradial

Step 983094 Let the right parts o the coupled differential equa-tions be unctions where ( = 1 2) Ten we can obtain asystem o coupled unctions as ollows

1 = minus 104861611039251048617 + cos + (1103925) (907317907317)1 minus (1103925)

2 = minus ]

907317907317 minus 9073171038389907317 minus cos

minus ]34

+ 1048616 minus 1048617

(983091983089)

where at = 1

Step 983095 Assume that 1038389 are ( = 1 2) respectively Tenwe can obtain some basic parameters as ollows

= 10486161 21048617 = 1 + ℎ12 2 +

ℎ22 = 1 + ℎ12 2 +

ℎ22 907317 = 10486161 + ℎ1 2 + ℎ21048617

(983091983090)

7252019 Analyzing Axial Stress and Deformation of Tubular

httpslidepdfcomreaderfullanalyzing-axial-stress-and-deformation-of-tubular 79

Te Scienti1047297c World Journal 983095

983137983138983148983141 983092 Te results o the axial orce and various kinds o deormation lengths

Number Depth

(m)Axial orce

(N)

Displacement by temperaturechanged (m)

Displacement by pressure

changed (m)

Axial deormation(m)

Bucklingdeormation (m)

otal deormation(m)

983089 983089 983096983097983093983090983092983092983096 983088 983088 983088 983088 983088

983090 983089983088983088 983096983093983092983095983090983092983096 983088983089983090983088983089 983088983088983088983097983096983094 983088983088983090983092 983088 983088983089983093983092983092

983091 983090983088983088 983096983089983092983090983089983093983093 983088983090983091983094983090 983088983088983089983097983091983097983090 983088983088983093983090 983088 983088983091983088983095983090

983092 983091983088983088 983095983095983091983095983089983095983095 983088983091983092983096983091 983088983088983090983096983093983097983096 983088983088983096983090 983088 983088983092983093983097

983093 983092983088983088 983095983091983095983097983095983088 983088983092983093983094983092 983088983088983091983095983092983095983094 983088983089983089983093 minus983088983088983088983094 983088983094983088983090983097

983094 983093983088983088 983095983088983094983096983095983095983091 983088983093983094983088983094 983088983088983092983094983088983090983096 983088983089983093983090 minus983088983088983088983094 983088983095983093983090983091

983095 983094983088983088 983094983095983093983095983094983091983097 983088983094983094983088983095 983088983088983093983092983090983093983092 983088983089983097983090 minus983088983088983088983094 983088983097983088983088983094

983096 983095983088983088 983094983092983092983094983088983090983091 983088983095983093983094983097 983088983088983094983090983089983093983091 983088983090983091983093 minus983088983088983088983094 983089983088983092983096

983097 983096983088983088 983094983089983091983092983091983095983090 983088983096983092983097 983088983088983094983097983095983090983093 983088983090983096983091 minus983088983088983088983095 983089983089983097983092983094

983089983088 983097983088983088 983093983096983090983090983095983090983089 983088983097983091983095983089 983088983088983095983094983097983094983096 983088983091983091983093 minus983088983088983088983095 983089983091983092983090983090

983089983089 983089983088983088983088 983093983093983089983089983088983095983090 983089983088983090983089983090 983088983088983096983091983096983096983091 983088983091983097983089 minus983088983088983088983095 983089983092983096983097983094

983089983090 983089983089983088983088 983093983089983097983097983092983090983091 983089983089983088983089983092 983088983088983097983088983092983095983089 983088983092983093983090 minus983088983088983088983095 983089983094983091983094983095

983089983091 983089983090983088983088 983092983096983096983095983095983095983093 983089983089983095983095983093 983088983088983097983094983095983091983089 983088983093983089983095

minus983088983088983088983097 983089983095983096983090983090

983089983092 983089983091983088983088 983092983093983095983094983089983090983096 983089983090983092983097983094 983088983089983088983090983094983094983090 983088983093983096983092 minus983088983088983089 983089983097983090983094983089

Step 983096 Calculate the pressure and temperature at point( + 1)+1

=

+ ℎ 1048616 + 2 + 2 + 90731710486176 = 12 = 12

(983091983091)

Step 983097 Calculate the deormation Δ1 Δ2 and Δ4 by previous equations

Step 983089983088 Repeat the third step to the tenth step until tubularlength is calculated

Step 983089983089 Calculate the deormationΔ3 and total deormationlength as ollows

Δ = sum=1

Δ1 + sum=1

Δ2 + Δ3 + sum=1

Δ4 (983091983092)

4 Numerical Simulation

983092983089 Parameters o demonstrate the application o our the-

ory we study a pipe in X well which is in Sichuan ProvinceChina All the basic parameters are given as ollows deptho the well is 983089983091983088983088 m ground thermal conductivity parameteris 206 ground temperature is 16∘C ground temperaturegradient is 00218(∘Cm) roughness o the inner surace o the well is 0000015 and parameters o pipes inclined wellinclination azimuth and vertical depth are given in ables 983089983090 and 983091

983092983090 Main Results and Results Analysis Afer calculation weobtain a series o results o this well as able 983092 Te in1047298uenceo outputson the axial deormation o tubing was investigatedas shown by Figure 983092

25

2

15

1

05

0

T o t a l a x i a l d e

f o r m a t i o n

( m )

Depth (m)

1 1200800400

700000m3d

500000m3d

300000m3d

F983145983143983157983154983141 983092 Te total axial deormation under varied outputs

From the results as shown in Figure 983092 and able 983092 some

useul analysis can be drawn

(983089) Te amount o steam injected and inject pressureaffected the stretching orce with special severity

(983090) Te results were as ollows the length o tubulardeormation was risen with increased injected pres-sure or injected velocity

(983091) Te length o tubular deormation increases with theincreasing o outputs but more slowly

(983092) Te thermal stress is the main actor in1047298uencing thetubular deormation Tereore the temperature o steam injected should not be too high

7252019 Analyzing Axial Stress and Deformation of Tubular

httpslidepdfcomreaderfullanalyzing-axial-stress-and-deformation-of-tubular 89

983096 Te Scienti1047297c World Journal

(983093) Te lifing prestressed cementing technology hasimportant meanings to reduce the deormation o tubular

(983094) Te creeping displacement o downhole stings willproduce an upward contractility which causes packer

depressed or lapsed Tereore the effective measuresshould be adopted to control the companding o tubular

5 Conclusion

In this paper a total tubular deormation model aboutdeviated wells was given A coupled-system model o differ-ential equations concerningpressureand temperature in hightemperature-high pressure steam injection wells according tomass momentum andenergy balances which canreduce theerror o axial stress and axial deormation was given insteado theaveragevalueor simplelinear relationship in traditional

research Te basic data o the Well (high temperature andhigh pressure gas well) 983089983091983088983088 m deep in Sichuan China wereused or case history calculations Te results can providetechnical reliance or the process o designing well tests indeviated gas wells and dynamic analysis o production

Nomenclature

907317 Inner diameter (m)9073171038389 Microelement o the tubular Acceleration o gravity (ms2)ℎ Depth o top tubular located at the packer(m)

ime o down stroke (s) Dimensionless time (dimensionless)V Velocity o 1047298uid in tubing (ms)V Velocity o down stroke (ms) Distance coordinate in the 1047298ow direction

(m) along the tubing1103925 Constant cross-sectional 1047298ow area (m2)1103925 Effective area (m2)1103925 Area corresponding to packer bore (m2) Te Joule-Tomson coefficient(dimensionless) Heat capacity o 1047298uids (JKg sdotK)

Outer diameter (m)

Steel elastic modulus o tubular (Mpa) Axial orces in the section (N)1038389 Axial tensile strength (N) Friction orce (N) Piston orce or supporting packerrsquos pressure(N) Pumping orce (N) Length o tubular (m)9073171038389 Buoyant weight o tubular (Kg)1038389 Dead weight o tubular (Kg)10383890 Pressure outside the tubular (Mpa)10383891 Pressure inside the tubular at the packerlength (Mpa)

1038389 Pressure in tubing (Mpa) emperature in tubing (∘C) Initial temperature o ormation (∘C) Mass 1047298ow rate (Kgs) otal length (m)1 Material density (Kgm3)

2 Packer 1047298uid density (Kgm3

) Inclination angle (∘) Warm balloon coefficient o the tubularstring (dimensionless) Drop o any parameter1038389 Axial thermal stress () Density o 1047298uid in the tubing (Kgm3)Δ3 Te tubular string buckling axialdeormation (m)Δ otal axial deormation by variedtemperature 1047297elds (m)Δ907317 otal axial deormation by the variedpressure 1047297elds (m)

Δ1038389start Differential pressure at startup (Mpa)Δ1038389 Change in tubing pressure at the length(Mpa)Δ1038389 Change in annulus pressure at the length(Mpa)Δ1038389 Differential pressure rom top to bottom(Mpa)Δ emperature change with beore and aferwell shut-in (∘C)Δ Change in density o liquid in the tubing at

the length (Kgm3)Δ Change in density o liquid in the casing at

the length (Kgm3)

AcknowledgmentsTis research was supported by the Key Program o NSFC(Grant no 983095983088983096983091983089983088983088983093) and the Key Project o China Petroleumand Chemical Corporation (Grant no GJ-983095983091-983088983095983088983094)

References

[983089] D-L Gao and B-K Gao ldquoA method or calculating tubingbehavior in HPH wellsrdquo Journal of Petroleum Science and Engineering vol 983092983089 no 983089ndash983091 pp 983089983096983091ndash983089983096983096 983090983088983088983092

[983090] D J Hammerlindl ldquoMovement orces and stresses associatedwith combination tubing strings sealed in packersrdquo Journal of

Petroleum echnology vol 983090983097 pp 983089983097983093ndash983090983088983096 983089983097983095983095

[983091] A Lubinski W S Althouse and J L Logan ldquoHelical bucklingo tubular sealed in packersrdquo Journal of Petroleum echnology vol 983089983092 no 983094 pp 983094983093983093ndash983094983095983088 983089983097983094983090

[983092] P R Paslay and D B Bogy ldquoTe stability o a circular rodlaterally constrained to be in contact with an inclined circularcylinderrdquo Journal of Applied Mechanics vol 983091983089 pp 983094983088983093ndash983094983089983088983089983097983094983092

[983093] R Dawson and P R Paslay ldquoDrillpipe buckling in inclinedholesrdquo Journal of Petroleum echnology vol 983091983094 no 983089983088 pp 983089983095983091983092ndash983089983095983091983096 983089983097983096983092

[983094] R F MitchellldquoEffects o well deviationon helical bucklingrdquo SPEDrilling and Completion vol 983089983090 no 983089 pp 983094983091ndash983094983096 983089983097983097983095

7252019 Analyzing Axial Stress and Deformation of Tubular

httpslidepdfcomreaderfullanalyzing-axial-stress-and-deformation-of-tubular 99

Te Scienti1047297c World Journal 983097

[983095] P Ding and X Z Yan ldquoForce analysis o high pressure waterinjection stringrdquo Petroleum Dring echiques vol983091983094 no 983093p 983090983091983090983088983088983093

[983096] Z F Li ldquoCasing cementing with hal warm-up or thermalrecovery wellsrdquo Journal of Petroleum Science and Engineering vol 983094983089 no 983090ndash983092 pp 983097983092ndash983097983096 983090983088983088983096

[983097] A M Sun ldquoTe analysis and computing o water injectiontubularrdquo Drilling and Production echnology vol 983090983094 no 983091 pp983093983093ndash983093983095 983090983088983088983091 (Chinese)

[983089983088] J P Xu ldquoStress analysis and optimum design o well comple-tionrdquo echnical Report o Sinopec GJ-983095983091-983088983095983088983094 983090983088983088983097

[983089983089] J P Xu Y Q Liu S Z Wang and B Qi ldquoNumerical modellingo steam quality in deviated wells with variable ( P) 1047297eldsrdquoChemical Engineering Science vol 983096983092 pp 983090983092983090ndash983090983093983092 983090983088983089983090

[983089983090] A R Hasan and C S Kabir ldquowo-phase 1047298ow in vertical andinclined annulirdquo International Journal of Multiphase Flow vol983089983096 no 983090 pp 983090983095983097ndash983090983097983091 983089983097983097983090

[983089983091] HD Beggs and JR BrillldquoA studyo two-phase 1047298owin inclinedpipesrdquo Journal of Petroleum echnology vol 983090983093 no 983093 pp 983094983088983095ndash983094983089983095 983089983097983095983091 paper 983092983088983088983095-PA

[983089983092] H Mukherjee and J P Brill ldquoPressure drop correlations orinclined two-phase 1047298owrdquo Journal of Energy Resources echnol-ogy vol 983089983088983095 no 983092 pp 983093983092983097ndash983093983093983092 983089983097983096983093

7252019 Analyzing Axial Stress and Deformation of Tubular

httpslidepdfcomreaderfullanalyzing-axial-stress-and-deformation-of-tubular 59

Te Scienti1047297c World Journal 983093

Group Hi-ech Project ldquoStress analysis and optimum designo well completionrdquo in 983090983088983088983097 [983094] undertaken by SichuanUniversity at early time Te varied ( 1038389) 1047297elds had beendeduced strictly based on the mass momentum and energy balance Te proo details can be shown in Xu et al [983089983089] Te

varied ( 1038389) 1047297elds is

9073171038389907317 = minus 104861611039251048617 + cos + (1103925) (907317907317)1 minus (1103925)

907317907317 = minus

]

907317907317 minus 9073171038389907317 minus

cos

minus ]34

+ 1048616 minus 1048617

1038389 104861601048617 = 10383890 1048616 01048617 = 0 907317104861601048617 = 9073170 1048616 01048617 = 0

(983090983089)

3 Numerical Implementation983091983089 Calculation of Some Parameters In this section we willgive the calculating method o some parameters

(983089) Each pointrsquos inclination

= minus1 + 1048616 minus minus11048617 ΔΔ (983090983090)

where represents segment point o calculation Δrepresents measurement depth o inclination angle and minus1 Δ is the step length o calculationransient heat transer unction [983089983090]

10486161048617 = 85209198316310486991128radic 10486161 minus 03radic 1048617 le 15104861604063 + 05 ln 1048617 1 + 06 gt 15

(983090983091)

(983090) Te density o wet steam Since the 1047298ow o the water vapor in is the gas-liquid two-phase 1047298ow there aremany researches about this problem [983089983091 983089983092] In thepaper we adopt the M-B model to calculate theaverage density o the mixture

(983091) Te heat transer coefficient to rom different posi-

tions o the axis o the wellbore to the second suraceTese resistances include the tubing wall possible insu-

lation around the tubing annular space (possibly 1047297lled witha gas or liquid but is sometimes vacuum) casing wall andcementing behind the casing as ollows

1

= 1ins

ln + 1ℎ + ℎ +

1cem

lncem (983090983092)

ins and cem are the heat conductivity o the heat insulatingmaterial and the cement sheath respectively ℎ and ℎ are thecoefficients o the convection heat transer and the radiationheat transer

983091983090 Initial Condition In order to solve model some de1047297niteconditions and initial conditions should be added Te initialconditions comprise the distribution o the pressure andtemperature at the well top In this paper we adopt the

value at the initial time by actual measurement Beore steaminjected the temperature o tubular just is initial temperature

o ormation (1038389 = 0 + cos is geothermal gradient)At the same time the pressure o inner tubular is assumed tobe equal to the outer tubular beore steam injected

983091983091 Steps of Algorithm o simpliy the calculation wedivided the wells into several short segments o the samelength Te length o a segment varies depending on varia-tions in wall thickness hole diameter 1047298uid density inside andoutside the pipe and wells geometry Te model begins withthe calculation at one particular position in the wells the topo the pipe

Step 983089 Set step length o depth In addition we denote

the relatively tolerant error by Te smaller ℎ is themore accurate the results are However it will lead to rapidincreasing calculating time In our paper we set ℎ = 1 (m)and = 5

Step 983090 Give the initial conditions

Step 983091 Compute each pointrsquos inclination

Step 983092 Compute the parameters under the initial conditionsor the last depth variables

Step 983093 Let = then we can get the by solving theollowing equation

= 22 + 1

1038389=0 = 0 + cos

1038389=1

= minus 12

907317907317

1038389 rarrinfin

= 0

(983090983093)

Let be the temperature at the injection time and

radial at the depth We apply the 1047297nite different methodto discretize the equations as ollows

+1 minus

= +1+1 minus 2

+1 + minus1+12 minus +1

+1 minus +1 (983090983094)

where is the interval o time and is the interval o radialrespectively It can be transormed into the standard orm asollows

minus + +1+1 + 2 +

+1 minus +1

minus1 = 2

(983090983095)

7252019 Analyzing Axial Stress and Deformation of Tubular

httpslidepdfcomreaderfullanalyzing-axial-stress-and-deformation-of-tubular 69

983094 Te Scienti1047297c World Journal

983137983138983148983141 983089 Parameters o pipes

Diameter (m) Tickness (m) Weight (Kg) Expansion Elastic (Gpa) Poissonrsquos ratios Using length (m)

983088983088983096983096983097 983088983088983089983090983097983093 983090983091983095983097 983088983088983088983088983088983089983089983093 983090983089983093 983088983091 983090983095983088

983088983088983096983096983097 983088983088983088983097983093983091 983089983096983090983096 983088983088983088983088983088983089983089983093 983090983089983093 983088983091 983089983090983088

983088983088983096983096983097 983088983088983088983095983091983092 983089983093983088983092 983088983088983088983088983088983089983089983093 983090983089983093 983088983091 983094983090983088

983088983088983096983096983097 983088983088983088983094983092983093 983089983091983093983096 983088983088983088983088983088983089983089983093 983090983089983093 983088983091 983090983097983088

983137983138983148983141 983090 Well parameters

Measured (m) Internal (m) External (m)

983091983091983094983095 983088983089983093983092983095983096 983088983089983095983095983096

983092983090983090983094 983088983089983093983090983093 983088983089983095983095983096

983089983091983088983088983088 983088983089983088983096983094983090 983088983089983090983095

983137983138983148983141 983091 Parameters o azimuth inclination and vertical depth

Number Measured

(m)Inclination

(∘)Azimuth

(∘)Vertical depth

(m)

983089 983089983091983093 983090983094983091 983090983092983089983088983089 983089983091983092983095983090

983090 983090983095983096 983089983090983091 983090983091983095983096983094 983090983095983095983097983089

983091 983091983094983092 983089983092983091 983090983089983091983096983094 983091983094983091983096983090

983092 983091983097983091 983090983089983095 983090983094983091983096 983091983097983090983093983091

983093 983092983090983090 983089983096983093 983092983092983093983094 983092983090983089983090983096

983094 983092983093983088 983088983096983090 983089983097983089983089983090 983092983092983097983094983090

983095 983092983096983094 983090983097983091 983090983094983097983088983095 983092983096983093983092983095

983096 983093983089983092 983089983088983091 983090983097983095983093983093 983093983089983091983096983091

983097 983093983092983091 983091983093983096 983091983090983092983093983089 983093983092983089983095983092

983089983088 983093983095983089 983090983097983096 983091983088983091983088983093 983093983095983088983092983091

983089983089 983094983088983088 983090983088983091 983090983088983092983095983092 983093983097983097983092983090

983089983090 983094983090983096 983090983091983092 983089983094983092983091983091 983094983090983095983090983096

983089983091 983094983094983088 983089983096983093 983089983097983093983090983096 983094983093983097983093983094

983089983092 983095983090983091 983091983089983092 983090983089983092983096983092 983095983090983089983095983088

983089983093 983095983096983090 983088983097983096 983090983089983094983092983096 983095983096983089983091983088

983089983094 983096983091983088 983090983089983093 983090983090983097983091983089 983096983090983097983089983090

983089983095 983096983094983088 983090983094983095 983090983092983092983088983091 983096983093983097983095983089

983089983096 983097983088983096 983092983096983093 983090983094983094983094983090 983097983088983092983088983096

983089983097 983097983090983096 983094983095983090 983090983093983096983095983096 983097983090983089983092983090

983090983088 983097983095983090 983090983088983091 983090983091983094983096983096 983097983095983089983095983089

983090983089 983089983088983090983093 983092983095983096 983090983091983097983090983095 983089983088983090983089983090983093

983090983090 983089983088983093983096 983092983088983089 983090983092983092983093983097 983089983088983093983093983093983096

983090983091 983089983088983096983097 983092983097983096 983090983090983096983090 983089983088983096983092983089983095

983090983092 983089983089983091983090 983091983095983093 983090983091983091983096983096 983089983089983090983097983090983096

983090983093 983089983089983095983092 983093983094983091 983090983091983093983089983092 983089983089983094983096983096983095

983090983094 983089983090983088983092 983092983090983091 983090983091983092983091983096 983089983090983088983088983097983097

983090983095 983089983090983091983093 983091983096983095 983090983091983092983097983097 983089983090983091983090983088983096

983090983096 983089983090983094983096 983092983097983095 983090983091983090983093983095 983089983090983094983091983092983093

983090983097 983089983091983088983088 983096983096983092 983090983091983091983090983096 983089983090983096983092983097983094

Ten the different method is used to discretize theboundary condition For = 1 we have

+12 minus 1 + 2

+11 = 2

(983090983096)

For = we have

+1 minus +1

minus1 = 0 (983090983097)

We can compute the symbolic solution o the temperatureo thestratumIn this step we willget thediscretedistributiono as the ollowing matrix

98313110486671048667104866710486671048667104866710486671048667104866710486671048667852059

11 2

1 sdot sdot sdot 1 sdot sdot sdot

12 2

2 sdot sdot sdot 2 sdot sdot sdot

sdot sdot sdot

1 2

sdot sdot sdot sdot sdot sdot

sdot sdot sdot

1 2

sdot sdot sdot sdot sdot sdot

98313310486691048669104866910486691048669104866910486691048669104866910486691048669852061

(983091983088)

where represents the injection time and represents theradial

Step 983094 Let the right parts o the coupled differential equa-tions be unctions where ( = 1 2) Ten we can obtain asystem o coupled unctions as ollows

1 = minus 104861611039251048617 + cos + (1103925) (907317907317)1 minus (1103925)

2 = minus ]

907317907317 minus 9073171038389907317 minus cos

minus ]34

+ 1048616 minus 1048617

(983091983089)

where at = 1

Step 983095 Assume that 1038389 are ( = 1 2) respectively Tenwe can obtain some basic parameters as ollows

= 10486161 21048617 = 1 + ℎ12 2 +

ℎ22 = 1 + ℎ12 2 +

ℎ22 907317 = 10486161 + ℎ1 2 + ℎ21048617

(983091983090)

7252019 Analyzing Axial Stress and Deformation of Tubular

httpslidepdfcomreaderfullanalyzing-axial-stress-and-deformation-of-tubular 79

Te Scienti1047297c World Journal 983095

983137983138983148983141 983092 Te results o the axial orce and various kinds o deormation lengths

Number Depth

(m)Axial orce

(N)

Displacement by temperaturechanged (m)

Displacement by pressure

changed (m)

Axial deormation(m)

Bucklingdeormation (m)

otal deormation(m)

983089 983089 983096983097983093983090983092983092983096 983088 983088 983088 983088 983088

983090 983089983088983088 983096983093983092983095983090983092983096 983088983089983090983088983089 983088983088983088983097983096983094 983088983088983090983092 983088 983088983089983093983092983092

983091 983090983088983088 983096983089983092983090983089983093983093 983088983090983091983094983090 983088983088983089983097983091983097983090 983088983088983093983090 983088 983088983091983088983095983090

983092 983091983088983088 983095983095983091983095983089983095983095 983088983091983092983096983091 983088983088983090983096983093983097983096 983088983088983096983090 983088 983088983092983093983097

983093 983092983088983088 983095983091983095983097983095983088 983088983092983093983094983092 983088983088983091983095983092983095983094 983088983089983089983093 minus983088983088983088983094 983088983094983088983090983097

983094 983093983088983088 983095983088983094983096983095983095983091 983088983093983094983088983094 983088983088983092983094983088983090983096 983088983089983093983090 minus983088983088983088983094 983088983095983093983090983091

983095 983094983088983088 983094983095983093983095983094983091983097 983088983094983094983088983095 983088983088983093983092983090983093983092 983088983089983097983090 minus983088983088983088983094 983088983097983088983088983094

983096 983095983088983088 983094983092983092983094983088983090983091 983088983095983093983094983097 983088983088983094983090983089983093983091 983088983090983091983093 minus983088983088983088983094 983089983088983092983096

983097 983096983088983088 983094983089983091983092983091983095983090 983088983096983092983097 983088983088983094983097983095983090983093 983088983090983096983091 minus983088983088983088983095 983089983089983097983092983094

983089983088 983097983088983088 983093983096983090983090983095983090983089 983088983097983091983095983089 983088983088983095983094983097983094983096 983088983091983091983093 minus983088983088983088983095 983089983091983092983090983090

983089983089 983089983088983088983088 983093983093983089983089983088983095983090 983089983088983090983089983090 983088983088983096983091983096983096983091 983088983091983097983089 minus983088983088983088983095 983089983092983096983097983094

983089983090 983089983089983088983088 983093983089983097983097983092983090983091 983089983089983088983089983092 983088983088983097983088983092983095983089 983088983092983093983090 minus983088983088983088983095 983089983094983091983094983095

983089983091 983089983090983088983088 983092983096983096983095983095983095983093 983089983089983095983095983093 983088983088983097983094983095983091983089 983088983093983089983095

minus983088983088983088983097 983089983095983096983090983090

983089983092 983089983091983088983088 983092983093983095983094983089983090983096 983089983090983092983097983094 983088983089983088983090983094983094983090 983088983093983096983092 minus983088983088983089 983089983097983090983094983089

Step 983096 Calculate the pressure and temperature at point( + 1)+1

=

+ ℎ 1048616 + 2 + 2 + 90731710486176 = 12 = 12

(983091983091)

Step 983097 Calculate the deormation Δ1 Δ2 and Δ4 by previous equations

Step 983089983088 Repeat the third step to the tenth step until tubularlength is calculated

Step 983089983089 Calculate the deormationΔ3 and total deormationlength as ollows

Δ = sum=1

Δ1 + sum=1

Δ2 + Δ3 + sum=1

Δ4 (983091983092)

4 Numerical Simulation

983092983089 Parameters o demonstrate the application o our the-

ory we study a pipe in X well which is in Sichuan ProvinceChina All the basic parameters are given as ollows deptho the well is 983089983091983088983088 m ground thermal conductivity parameteris 206 ground temperature is 16∘C ground temperaturegradient is 00218(∘Cm) roughness o the inner surace o the well is 0000015 and parameters o pipes inclined wellinclination azimuth and vertical depth are given in ables 983089983090 and 983091

983092983090 Main Results and Results Analysis Afer calculation weobtain a series o results o this well as able 983092 Te in1047298uenceo outputson the axial deormation o tubing was investigatedas shown by Figure 983092

25

2

15

1

05

0

T o t a l a x i a l d e

f o r m a t i o n

( m )

Depth (m)

1 1200800400

700000m3d

500000m3d

300000m3d

F983145983143983157983154983141 983092 Te total axial deormation under varied outputs

From the results as shown in Figure 983092 and able 983092 some

useul analysis can be drawn

(983089) Te amount o steam injected and inject pressureaffected the stretching orce with special severity

(983090) Te results were as ollows the length o tubulardeormation was risen with increased injected pres-sure or injected velocity

(983091) Te length o tubular deormation increases with theincreasing o outputs but more slowly

(983092) Te thermal stress is the main actor in1047298uencing thetubular deormation Tereore the temperature o steam injected should not be too high

7252019 Analyzing Axial Stress and Deformation of Tubular

httpslidepdfcomreaderfullanalyzing-axial-stress-and-deformation-of-tubular 89

983096 Te Scienti1047297c World Journal

(983093) Te lifing prestressed cementing technology hasimportant meanings to reduce the deormation o tubular

(983094) Te creeping displacement o downhole stings willproduce an upward contractility which causes packer

depressed or lapsed Tereore the effective measuresshould be adopted to control the companding o tubular

5 Conclusion

In this paper a total tubular deormation model aboutdeviated wells was given A coupled-system model o differ-ential equations concerningpressureand temperature in hightemperature-high pressure steam injection wells according tomass momentum andenergy balances which canreduce theerror o axial stress and axial deormation was given insteado theaveragevalueor simplelinear relationship in traditional

research Te basic data o the Well (high temperature andhigh pressure gas well) 983089983091983088983088 m deep in Sichuan China wereused or case history calculations Te results can providetechnical reliance or the process o designing well tests indeviated gas wells and dynamic analysis o production

Nomenclature

907317 Inner diameter (m)9073171038389 Microelement o the tubular Acceleration o gravity (ms2)ℎ Depth o top tubular located at the packer(m)

ime o down stroke (s) Dimensionless time (dimensionless)V Velocity o 1047298uid in tubing (ms)V Velocity o down stroke (ms) Distance coordinate in the 1047298ow direction

(m) along the tubing1103925 Constant cross-sectional 1047298ow area (m2)1103925 Effective area (m2)1103925 Area corresponding to packer bore (m2) Te Joule-Tomson coefficient(dimensionless) Heat capacity o 1047298uids (JKg sdotK)

Outer diameter (m)

Steel elastic modulus o tubular (Mpa) Axial orces in the section (N)1038389 Axial tensile strength (N) Friction orce (N) Piston orce or supporting packerrsquos pressure(N) Pumping orce (N) Length o tubular (m)9073171038389 Buoyant weight o tubular (Kg)1038389 Dead weight o tubular (Kg)10383890 Pressure outside the tubular (Mpa)10383891 Pressure inside the tubular at the packerlength (Mpa)

1038389 Pressure in tubing (Mpa) emperature in tubing (∘C) Initial temperature o ormation (∘C) Mass 1047298ow rate (Kgs) otal length (m)1 Material density (Kgm3)

2 Packer 1047298uid density (Kgm3

) Inclination angle (∘) Warm balloon coefficient o the tubularstring (dimensionless) Drop o any parameter1038389 Axial thermal stress () Density o 1047298uid in the tubing (Kgm3)Δ3 Te tubular string buckling axialdeormation (m)Δ otal axial deormation by variedtemperature 1047297elds (m)Δ907317 otal axial deormation by the variedpressure 1047297elds (m)

Δ1038389start Differential pressure at startup (Mpa)Δ1038389 Change in tubing pressure at the length(Mpa)Δ1038389 Change in annulus pressure at the length(Mpa)Δ1038389 Differential pressure rom top to bottom(Mpa)Δ emperature change with beore and aferwell shut-in (∘C)Δ Change in density o liquid in the tubing at

the length (Kgm3)Δ Change in density o liquid in the casing at

the length (Kgm3)

AcknowledgmentsTis research was supported by the Key Program o NSFC(Grant no 983095983088983096983091983089983088983088983093) and the Key Project o China Petroleumand Chemical Corporation (Grant no GJ-983095983091-983088983095983088983094)

References

[983089] D-L Gao and B-K Gao ldquoA method or calculating tubingbehavior in HPH wellsrdquo Journal of Petroleum Science and Engineering vol 983092983089 no 983089ndash983091 pp 983089983096983091ndash983089983096983096 983090983088983088983092

[983090] D J Hammerlindl ldquoMovement orces and stresses associatedwith combination tubing strings sealed in packersrdquo Journal of

Petroleum echnology vol 983090983097 pp 983089983097983093ndash983090983088983096 983089983097983095983095

[983091] A Lubinski W S Althouse and J L Logan ldquoHelical bucklingo tubular sealed in packersrdquo Journal of Petroleum echnology vol 983089983092 no 983094 pp 983094983093983093ndash983094983095983088 983089983097983094983090

[983092] P R Paslay and D B Bogy ldquoTe stability o a circular rodlaterally constrained to be in contact with an inclined circularcylinderrdquo Journal of Applied Mechanics vol 983091983089 pp 983094983088983093ndash983094983089983088983089983097983094983092

[983093] R Dawson and P R Paslay ldquoDrillpipe buckling in inclinedholesrdquo Journal of Petroleum echnology vol 983091983094 no 983089983088 pp 983089983095983091983092ndash983089983095983091983096 983089983097983096983092

[983094] R F MitchellldquoEffects o well deviationon helical bucklingrdquo SPEDrilling and Completion vol 983089983090 no 983089 pp 983094983091ndash983094983096 983089983097983097983095

7252019 Analyzing Axial Stress and Deformation of Tubular

httpslidepdfcomreaderfullanalyzing-axial-stress-and-deformation-of-tubular 99

Te Scienti1047297c World Journal 983097

[983095] P Ding and X Z Yan ldquoForce analysis o high pressure waterinjection stringrdquo Petroleum Dring echiques vol983091983094 no 983093p 983090983091983090983088983088983093

[983096] Z F Li ldquoCasing cementing with hal warm-up or thermalrecovery wellsrdquo Journal of Petroleum Science and Engineering vol 983094983089 no 983090ndash983092 pp 983097983092ndash983097983096 983090983088983088983096

[983097] A M Sun ldquoTe analysis and computing o water injectiontubularrdquo Drilling and Production echnology vol 983090983094 no 983091 pp983093983093ndash983093983095 983090983088983088983091 (Chinese)

[983089983088] J P Xu ldquoStress analysis and optimum design o well comple-tionrdquo echnical Report o Sinopec GJ-983095983091-983088983095983088983094 983090983088983088983097

[983089983089] J P Xu Y Q Liu S Z Wang and B Qi ldquoNumerical modellingo steam quality in deviated wells with variable ( P) 1047297eldsrdquoChemical Engineering Science vol 983096983092 pp 983090983092983090ndash983090983093983092 983090983088983089983090

[983089983090] A R Hasan and C S Kabir ldquowo-phase 1047298ow in vertical andinclined annulirdquo International Journal of Multiphase Flow vol983089983096 no 983090 pp 983090983095983097ndash983090983097983091 983089983097983097983090

[983089983091] HD Beggs and JR BrillldquoA studyo two-phase 1047298owin inclinedpipesrdquo Journal of Petroleum echnology vol 983090983093 no 983093 pp 983094983088983095ndash983094983089983095 983089983097983095983091 paper 983092983088983088983095-PA

[983089983092] H Mukherjee and J P Brill ldquoPressure drop correlations orinclined two-phase 1047298owrdquo Journal of Energy Resources echnol-ogy vol 983089983088983095 no 983092 pp 983093983092983097ndash983093983093983092 983089983097983096983093

7252019 Analyzing Axial Stress and Deformation of Tubular

httpslidepdfcomreaderfullanalyzing-axial-stress-and-deformation-of-tubular 69

983094 Te Scienti1047297c World Journal

983137983138983148983141 983089 Parameters o pipes

Diameter (m) Tickness (m) Weight (Kg) Expansion Elastic (Gpa) Poissonrsquos ratios Using length (m)

983088983088983096983096983097 983088983088983089983090983097983093 983090983091983095983097 983088983088983088983088983088983089983089983093 983090983089983093 983088983091 983090983095983088

983088983088983096983096983097 983088983088983088983097983093983091 983089983096983090983096 983088983088983088983088983088983089983089983093 983090983089983093 983088983091 983089983090983088

983088983088983096983096983097 983088983088983088983095983091983092 983089983093983088983092 983088983088983088983088983088983089983089983093 983090983089983093 983088983091 983094983090983088

983088983088983096983096983097 983088983088983088983094983092983093 983089983091983093983096 983088983088983088983088983088983089983089983093 983090983089983093 983088983091 983090983097983088

983137983138983148983141 983090 Well parameters

Measured (m) Internal (m) External (m)

983091983091983094983095 983088983089983093983092983095983096 983088983089983095983095983096

983092983090983090983094 983088983089983093983090983093 983088983089983095983095983096

983089983091983088983088983088 983088983089983088983096983094983090 983088983089983090983095

983137983138983148983141 983091 Parameters o azimuth inclination and vertical depth

Number Measured

(m)Inclination

(∘)Azimuth

(∘)Vertical depth

(m)

983089 983089983091983093 983090983094983091 983090983092983089983088983089 983089983091983092983095983090

983090 983090983095983096 983089983090983091 983090983091983095983096983094 983090983095983095983097983089

983091 983091983094983092 983089983092983091 983090983089983091983096983094 983091983094983091983096983090

983092 983091983097983091 983090983089983095 983090983094983091983096 983091983097983090983093983091

983093 983092983090983090 983089983096983093 983092983092983093983094 983092983090983089983090983096

983094 983092983093983088 983088983096983090 983089983097983089983089983090 983092983092983097983094983090

983095 983092983096983094 983090983097983091 983090983094983097983088983095 983092983096983093983092983095

983096 983093983089983092 983089983088983091 983090983097983095983093983093 983093983089983091983096983091

983097 983093983092983091 983091983093983096 983091983090983092983093983089 983093983092983089983095983092

983089983088 983093983095983089 983090983097983096 983091983088983091983088983093 983093983095983088983092983091

983089983089 983094983088983088 983090983088983091 983090983088983092983095983092 983093983097983097983092983090

983089983090 983094983090983096 983090983091983092 983089983094983092983091983091 983094983090983095983090983096

983089983091 983094983094983088 983089983096983093 983089983097983093983090983096 983094983093983097983093983094

983089983092 983095983090983091 983091983089983092 983090983089983092983096983092 983095983090983089983095983088

983089983093 983095983096983090 983088983097983096 983090983089983094983092983096 983095983096983089983091983088

983089983094 983096983091983088 983090983089983093 983090983090983097983091983089 983096983090983097983089983090

983089983095 983096983094983088 983090983094983095 983090983092983092983088983091 983096983093983097983095983089

983089983096 983097983088983096 983092983096983093 983090983094983094983094983090 983097983088983092983088983096

983089983097 983097983090983096 983094983095983090 983090983093983096983095983096 983097983090983089983092983090

983090983088 983097983095983090 983090983088983091 983090983091983094983096983096 983097983095983089983095983089

983090983089 983089983088983090983093 983092983095983096 983090983091983097983090983095 983089983088983090983089983090983093

983090983090 983089983088983093983096 983092983088983089 983090983092983092983093983097 983089983088983093983093983093983096

983090983091 983089983088983096983097 983092983097983096 983090983090983096983090 983089983088983096983092983089983095

983090983092 983089983089983091983090 983091983095983093 983090983091983091983096983096 983089983089983090983097983090983096

983090983093 983089983089983095983092 983093983094983091 983090983091983093983089983092 983089983089983094983096983096983095

983090983094 983089983090983088983092 983092983090983091 983090983091983092983091983096 983089983090983088983088983097983097

983090983095 983089983090983091983093 983091983096983095 983090983091983092983097983097 983089983090983091983090983088983096

983090983096 983089983090983094983096 983092983097983095 983090983091983090983093983095 983089983090983094983091983092983093

983090983097 983089983091983088983088 983096983096983092 983090983091983091983090983096 983089983090983096983092983097983094

Ten the different method is used to discretize theboundary condition For = 1 we have

+12 minus 1 + 2

+11 = 2

(983090983096)

For = we have

+1 minus +1

minus1 = 0 (983090983097)

We can compute the symbolic solution o the temperatureo thestratumIn this step we willget thediscretedistributiono as the ollowing matrix

98313110486671048667104866710486671048667104866710486671048667104866710486671048667852059

11 2

1 sdot sdot sdot 1 sdot sdot sdot

12 2

2 sdot sdot sdot 2 sdot sdot sdot

sdot sdot sdot

1 2

sdot sdot sdot sdot sdot sdot

sdot sdot sdot

1 2

sdot sdot sdot sdot sdot sdot

98313310486691048669104866910486691048669104866910486691048669104866910486691048669852061

(983091983088)

where represents the injection time and represents theradial

Step 983094 Let the right parts o the coupled differential equa-tions be unctions where ( = 1 2) Ten we can obtain asystem o coupled unctions as ollows

1 = minus 104861611039251048617 + cos + (1103925) (907317907317)1 minus (1103925)

2 = minus ]

907317907317 minus 9073171038389907317 minus cos

minus ]34

+ 1048616 minus 1048617

(983091983089)

where at = 1

Step 983095 Assume that 1038389 are ( = 1 2) respectively Tenwe can obtain some basic parameters as ollows

= 10486161 21048617 = 1 + ℎ12 2 +

ℎ22 = 1 + ℎ12 2 +

ℎ22 907317 = 10486161 + ℎ1 2 + ℎ21048617

(983091983090)

7252019 Analyzing Axial Stress and Deformation of Tubular

httpslidepdfcomreaderfullanalyzing-axial-stress-and-deformation-of-tubular 79

Te Scienti1047297c World Journal 983095

983137983138983148983141 983092 Te results o the axial orce and various kinds o deormation lengths

Number Depth

(m)Axial orce

(N)

Displacement by temperaturechanged (m)

Displacement by pressure

changed (m)

Axial deormation(m)

Bucklingdeormation (m)

otal deormation(m)

983089 983089 983096983097983093983090983092983092983096 983088 983088 983088 983088 983088

983090 983089983088983088 983096983093983092983095983090983092983096 983088983089983090983088983089 983088983088983088983097983096983094 983088983088983090983092 983088 983088983089983093983092983092

983091 983090983088983088 983096983089983092983090983089983093983093 983088983090983091983094983090 983088983088983089983097983091983097983090 983088983088983093983090 983088 983088983091983088983095983090

983092 983091983088983088 983095983095983091983095983089983095983095 983088983091983092983096983091 983088983088983090983096983093983097983096 983088983088983096983090 983088 983088983092983093983097

983093 983092983088983088 983095983091983095983097983095983088 983088983092983093983094983092 983088983088983091983095983092983095983094 983088983089983089983093 minus983088983088983088983094 983088983094983088983090983097

983094 983093983088983088 983095983088983094983096983095983095983091 983088983093983094983088983094 983088983088983092983094983088983090983096 983088983089983093983090 minus983088983088983088983094 983088983095983093983090983091

983095 983094983088983088 983094983095983093983095983094983091983097 983088983094983094983088983095 983088983088983093983092983090983093983092 983088983089983097983090 minus983088983088983088983094 983088983097983088983088983094

983096 983095983088983088 983094983092983092983094983088983090983091 983088983095983093983094983097 983088983088983094983090983089983093983091 983088983090983091983093 minus983088983088983088983094 983089983088983092983096

983097 983096983088983088 983094983089983091983092983091983095983090 983088983096983092983097 983088983088983094983097983095983090983093 983088983090983096983091 minus983088983088983088983095 983089983089983097983092983094

983089983088 983097983088983088 983093983096983090983090983095983090983089 983088983097983091983095983089 983088983088983095983094983097983094983096 983088983091983091983093 minus983088983088983088983095 983089983091983092983090983090

983089983089 983089983088983088983088 983093983093983089983089983088983095983090 983089983088983090983089983090 983088983088983096983091983096983096983091 983088983091983097983089 minus983088983088983088983095 983089983092983096983097983094

983089983090 983089983089983088983088 983093983089983097983097983092983090983091 983089983089983088983089983092 983088983088983097983088983092983095983089 983088983092983093983090 minus983088983088983088983095 983089983094983091983094983095

983089983091 983089983090983088983088 983092983096983096983095983095983095983093 983089983089983095983095983093 983088983088983097983094983095983091983089 983088983093983089983095

minus983088983088983088983097 983089983095983096983090983090

983089983092 983089983091983088983088 983092983093983095983094983089983090983096 983089983090983092983097983094 983088983089983088983090983094983094983090 983088983093983096983092 minus983088983088983089 983089983097983090983094983089

Step 983096 Calculate the pressure and temperature at point( + 1)+1

=

+ ℎ 1048616 + 2 + 2 + 90731710486176 = 12 = 12

(983091983091)

Step 983097 Calculate the deormation Δ1 Δ2 and Δ4 by previous equations

Step 983089983088 Repeat the third step to the tenth step until tubularlength is calculated

Step 983089983089 Calculate the deormationΔ3 and total deormationlength as ollows

Δ = sum=1

Δ1 + sum=1

Δ2 + Δ3 + sum=1

Δ4 (983091983092)

4 Numerical Simulation

983092983089 Parameters o demonstrate the application o our the-

ory we study a pipe in X well which is in Sichuan ProvinceChina All the basic parameters are given as ollows deptho the well is 983089983091983088983088 m ground thermal conductivity parameteris 206 ground temperature is 16∘C ground temperaturegradient is 00218(∘Cm) roughness o the inner surace o the well is 0000015 and parameters o pipes inclined wellinclination azimuth and vertical depth are given in ables 983089983090 and 983091

983092983090 Main Results and Results Analysis Afer calculation weobtain a series o results o this well as able 983092 Te in1047298uenceo outputson the axial deormation o tubing was investigatedas shown by Figure 983092

25

2

15

1

05

0

T o t a l a x i a l d e

f o r m a t i o n

( m )

Depth (m)

1 1200800400

700000m3d

500000m3d

300000m3d

F983145983143983157983154983141 983092 Te total axial deormation under varied outputs

From the results as shown in Figure 983092 and able 983092 some

useul analysis can be drawn

(983089) Te amount o steam injected and inject pressureaffected the stretching orce with special severity

(983090) Te results were as ollows the length o tubulardeormation was risen with increased injected pres-sure or injected velocity

(983091) Te length o tubular deormation increases with theincreasing o outputs but more slowly

(983092) Te thermal stress is the main actor in1047298uencing thetubular deormation Tereore the temperature o steam injected should not be too high

7252019 Analyzing Axial Stress and Deformation of Tubular

httpslidepdfcomreaderfullanalyzing-axial-stress-and-deformation-of-tubular 89

983096 Te Scienti1047297c World Journal

(983093) Te lifing prestressed cementing technology hasimportant meanings to reduce the deormation o tubular

(983094) Te creeping displacement o downhole stings willproduce an upward contractility which causes packer

depressed or lapsed Tereore the effective measuresshould be adopted to control the companding o tubular

5 Conclusion

In this paper a total tubular deormation model aboutdeviated wells was given A coupled-system model o differ-ential equations concerningpressureand temperature in hightemperature-high pressure steam injection wells according tomass momentum andenergy balances which canreduce theerror o axial stress and axial deormation was given insteado theaveragevalueor simplelinear relationship in traditional

research Te basic data o the Well (high temperature andhigh pressure gas well) 983089983091983088983088 m deep in Sichuan China wereused or case history calculations Te results can providetechnical reliance or the process o designing well tests indeviated gas wells and dynamic analysis o production

Nomenclature

907317 Inner diameter (m)9073171038389 Microelement o the tubular Acceleration o gravity (ms2)ℎ Depth o top tubular located at the packer(m)

ime o down stroke (s) Dimensionless time (dimensionless)V Velocity o 1047298uid in tubing (ms)V Velocity o down stroke (ms) Distance coordinate in the 1047298ow direction

(m) along the tubing1103925 Constant cross-sectional 1047298ow area (m2)1103925 Effective area (m2)1103925 Area corresponding to packer bore (m2) Te Joule-Tomson coefficient(dimensionless) Heat capacity o 1047298uids (JKg sdotK)

Outer diameter (m)

Steel elastic modulus o tubular (Mpa) Axial orces in the section (N)1038389 Axial tensile strength (N) Friction orce (N) Piston orce or supporting packerrsquos pressure(N) Pumping orce (N) Length o tubular (m)9073171038389 Buoyant weight o tubular (Kg)1038389 Dead weight o tubular (Kg)10383890 Pressure outside the tubular (Mpa)10383891 Pressure inside the tubular at the packerlength (Mpa)

1038389 Pressure in tubing (Mpa) emperature in tubing (∘C) Initial temperature o ormation (∘C) Mass 1047298ow rate (Kgs) otal length (m)1 Material density (Kgm3)

2 Packer 1047298uid density (Kgm3

) Inclination angle (∘) Warm balloon coefficient o the tubularstring (dimensionless) Drop o any parameter1038389 Axial thermal stress () Density o 1047298uid in the tubing (Kgm3)Δ3 Te tubular string buckling axialdeormation (m)Δ otal axial deormation by variedtemperature 1047297elds (m)Δ907317 otal axial deormation by the variedpressure 1047297elds (m)

Δ1038389start Differential pressure at startup (Mpa)Δ1038389 Change in tubing pressure at the length(Mpa)Δ1038389 Change in annulus pressure at the length(Mpa)Δ1038389 Differential pressure rom top to bottom(Mpa)Δ emperature change with beore and aferwell shut-in (∘C)Δ Change in density o liquid in the tubing at

the length (Kgm3)Δ Change in density o liquid in the casing at

the length (Kgm3)

AcknowledgmentsTis research was supported by the Key Program o NSFC(Grant no 983095983088983096983091983089983088983088983093) and the Key Project o China Petroleumand Chemical Corporation (Grant no GJ-983095983091-983088983095983088983094)

References

[983089] D-L Gao and B-K Gao ldquoA method or calculating tubingbehavior in HPH wellsrdquo Journal of Petroleum Science and Engineering vol 983092983089 no 983089ndash983091 pp 983089983096983091ndash983089983096983096 983090983088983088983092

[983090] D J Hammerlindl ldquoMovement orces and stresses associatedwith combination tubing strings sealed in packersrdquo Journal of

Petroleum echnology vol 983090983097 pp 983089983097983093ndash983090983088983096 983089983097983095983095

[983091] A Lubinski W S Althouse and J L Logan ldquoHelical bucklingo tubular sealed in packersrdquo Journal of Petroleum echnology vol 983089983092 no 983094 pp 983094983093983093ndash983094983095983088 983089983097983094983090

[983092] P R Paslay and D B Bogy ldquoTe stability o a circular rodlaterally constrained to be in contact with an inclined circularcylinderrdquo Journal of Applied Mechanics vol 983091983089 pp 983094983088983093ndash983094983089983088983089983097983094983092

[983093] R Dawson and P R Paslay ldquoDrillpipe buckling in inclinedholesrdquo Journal of Petroleum echnology vol 983091983094 no 983089983088 pp 983089983095983091983092ndash983089983095983091983096 983089983097983096983092

[983094] R F MitchellldquoEffects o well deviationon helical bucklingrdquo SPEDrilling and Completion vol 983089983090 no 983089 pp 983094983091ndash983094983096 983089983097983097983095

7252019 Analyzing Axial Stress and Deformation of Tubular

httpslidepdfcomreaderfullanalyzing-axial-stress-and-deformation-of-tubular 99

Te Scienti1047297c World Journal 983097

[983095] P Ding and X Z Yan ldquoForce analysis o high pressure waterinjection stringrdquo Petroleum Dring echiques vol983091983094 no 983093p 983090983091983090983088983088983093

[983096] Z F Li ldquoCasing cementing with hal warm-up or thermalrecovery wellsrdquo Journal of Petroleum Science and Engineering vol 983094983089 no 983090ndash983092 pp 983097983092ndash983097983096 983090983088983088983096

[983097] A M Sun ldquoTe analysis and computing o water injectiontubularrdquo Drilling and Production echnology vol 983090983094 no 983091 pp983093983093ndash983093983095 983090983088983088983091 (Chinese)

[983089983088] J P Xu ldquoStress analysis and optimum design o well comple-tionrdquo echnical Report o Sinopec GJ-983095983091-983088983095983088983094 983090983088983088983097

[983089983089] J P Xu Y Q Liu S Z Wang and B Qi ldquoNumerical modellingo steam quality in deviated wells with variable ( P) 1047297eldsrdquoChemical Engineering Science vol 983096983092 pp 983090983092983090ndash983090983093983092 983090983088983089983090

[983089983090] A R Hasan and C S Kabir ldquowo-phase 1047298ow in vertical andinclined annulirdquo International Journal of Multiphase Flow vol983089983096 no 983090 pp 983090983095983097ndash983090983097983091 983089983097983097983090

[983089983091] HD Beggs and JR BrillldquoA studyo two-phase 1047298owin inclinedpipesrdquo Journal of Petroleum echnology vol 983090983093 no 983093 pp 983094983088983095ndash983094983089983095 983089983097983095983091 paper 983092983088983088983095-PA

[983089983092] H Mukherjee and J P Brill ldquoPressure drop correlations orinclined two-phase 1047298owrdquo Journal of Energy Resources echnol-ogy vol 983089983088983095 no 983092 pp 983093983092983097ndash983093983093983092 983089983097983096983093

7252019 Analyzing Axial Stress and Deformation of Tubular

httpslidepdfcomreaderfullanalyzing-axial-stress-and-deformation-of-tubular 79

Te Scienti1047297c World Journal 983095

983137983138983148983141 983092 Te results o the axial orce and various kinds o deormation lengths

Number Depth

(m)Axial orce

(N)

Displacement by temperaturechanged (m)

Displacement by pressure

changed (m)

Axial deormation(m)

Bucklingdeormation (m)

otal deormation(m)

983089 983089 983096983097983093983090983092983092983096 983088 983088 983088 983088 983088

983090 983089983088983088 983096983093983092983095983090983092983096 983088983089983090983088983089 983088983088983088983097983096983094 983088983088983090983092 983088 983088983089983093983092983092

983091 983090983088983088 983096983089983092983090983089983093983093 983088983090983091983094983090 983088983088983089983097983091983097983090 983088983088983093983090 983088 983088983091983088983095983090

983092 983091983088983088 983095983095983091983095983089983095983095 983088983091983092983096983091 983088983088983090983096983093983097983096 983088983088983096983090 983088 983088983092983093983097

983093 983092983088983088 983095983091983095983097983095983088 983088983092983093983094983092 983088983088983091983095983092983095983094 983088983089983089983093 minus983088983088983088983094 983088983094983088983090983097

983094 983093983088983088 983095983088983094983096983095983095983091 983088983093983094983088983094 983088983088983092983094983088983090983096 983088983089983093983090 minus983088983088983088983094 983088983095983093983090983091

983095 983094983088983088 983094983095983093983095983094983091983097 983088983094983094983088983095 983088983088983093983092983090983093983092 983088983089983097983090 minus983088983088983088983094 983088983097983088983088983094

983096 983095983088983088 983094983092983092983094983088983090983091 983088983095983093983094983097 983088983088983094983090983089983093983091 983088983090983091983093 minus983088983088983088983094 983089983088983092983096

983097 983096983088983088 983094983089983091983092983091983095983090 983088983096983092983097 983088983088983094983097983095983090983093 983088983090983096983091 minus983088983088983088983095 983089983089983097983092983094

983089983088 983097983088983088 983093983096983090983090983095983090983089 983088983097983091983095983089 983088983088983095983094983097983094983096 983088983091983091983093 minus983088983088983088983095 983089983091983092983090983090

983089983089 983089983088983088983088 983093983093983089983089983088983095983090 983089983088983090983089983090 983088983088983096983091983096983096983091 983088983091983097983089 minus983088983088983088983095 983089983092983096983097983094

983089983090 983089983089983088983088 983093983089983097983097983092983090983091 983089983089983088983089983092 983088983088983097983088983092983095983089 983088983092983093983090 minus983088983088983088983095 983089983094983091983094983095

983089983091 983089983090983088983088 983092983096983096983095983095983095983093 983089983089983095983095983093 983088983088983097983094983095983091983089 983088983093983089983095

minus983088983088983088983097 983089983095983096983090983090

983089983092 983089983091983088983088 983092983093983095983094983089983090983096 983089983090983092983097983094 983088983089983088983090983094983094983090 983088983093983096983092 minus983088983088983089 983089983097983090983094983089

Step 983096 Calculate the pressure and temperature at point( + 1)+1

=

+ ℎ 1048616 + 2 + 2 + 90731710486176 = 12 = 12

(983091983091)

Step 983097 Calculate the deormation Δ1 Δ2 and Δ4 by previous equations

Step 983089983088 Repeat the third step to the tenth step until tubularlength is calculated

Step 983089983089 Calculate the deormationΔ3 and total deormationlength as ollows

Δ = sum=1

Δ1 + sum=1

Δ2 + Δ3 + sum=1

Δ4 (983091983092)

4 Numerical Simulation

983092983089 Parameters o demonstrate the application o our the-

ory we study a pipe in X well which is in Sichuan ProvinceChina All the basic parameters are given as ollows deptho the well is 983089983091983088983088 m ground thermal conductivity parameteris 206 ground temperature is 16∘C ground temperaturegradient is 00218(∘Cm) roughness o the inner surace o the well is 0000015 and parameters o pipes inclined wellinclination azimuth and vertical depth are given in ables 983089983090 and 983091

983092983090 Main Results and Results Analysis Afer calculation weobtain a series o results o this well as able 983092 Te in1047298uenceo outputson the axial deormation o tubing was investigatedas shown by Figure 983092

25

2

15

1

05

0

T o t a l a x i a l d e

f o r m a t i o n

( m )

Depth (m)

1 1200800400

700000m3d

500000m3d

300000m3d

F983145983143983157983154983141 983092 Te total axial deormation under varied outputs

From the results as shown in Figure 983092 and able 983092 some

useul analysis can be drawn

(983089) Te amount o steam injected and inject pressureaffected the stretching orce with special severity

(983090) Te results were as ollows the length o tubulardeormation was risen with increased injected pres-sure or injected velocity

(983091) Te length o tubular deormation increases with theincreasing o outputs but more slowly

(983092) Te thermal stress is the main actor in1047298uencing thetubular deormation Tereore the temperature o steam injected should not be too high

7252019 Analyzing Axial Stress and Deformation of Tubular

httpslidepdfcomreaderfullanalyzing-axial-stress-and-deformation-of-tubular 89

983096 Te Scienti1047297c World Journal

(983093) Te lifing prestressed cementing technology hasimportant meanings to reduce the deormation o tubular

(983094) Te creeping displacement o downhole stings willproduce an upward contractility which causes packer

depressed or lapsed Tereore the effective measuresshould be adopted to control the companding o tubular

5 Conclusion

In this paper a total tubular deormation model aboutdeviated wells was given A coupled-system model o differ-ential equations concerningpressureand temperature in hightemperature-high pressure steam injection wells according tomass momentum andenergy balances which canreduce theerror o axial stress and axial deormation was given insteado theaveragevalueor simplelinear relationship in traditional

research Te basic data o the Well (high temperature andhigh pressure gas well) 983089983091983088983088 m deep in Sichuan China wereused or case history calculations Te results can providetechnical reliance or the process o designing well tests indeviated gas wells and dynamic analysis o production

Nomenclature

907317 Inner diameter (m)9073171038389 Microelement o the tubular Acceleration o gravity (ms2)ℎ Depth o top tubular located at the packer(m)

ime o down stroke (s) Dimensionless time (dimensionless)V Velocity o 1047298uid in tubing (ms)V Velocity o down stroke (ms) Distance coordinate in the 1047298ow direction

(m) along the tubing1103925 Constant cross-sectional 1047298ow area (m2)1103925 Effective area (m2)1103925 Area corresponding to packer bore (m2) Te Joule-Tomson coefficient(dimensionless) Heat capacity o 1047298uids (JKg sdotK)

Outer diameter (m)

Steel elastic modulus o tubular (Mpa) Axial orces in the section (N)1038389 Axial tensile strength (N) Friction orce (N) Piston orce or supporting packerrsquos pressure(N) Pumping orce (N) Length o tubular (m)9073171038389 Buoyant weight o tubular (Kg)1038389 Dead weight o tubular (Kg)10383890 Pressure outside the tubular (Mpa)10383891 Pressure inside the tubular at the packerlength (Mpa)

1038389 Pressure in tubing (Mpa) emperature in tubing (∘C) Initial temperature o ormation (∘C) Mass 1047298ow rate (Kgs) otal length (m)1 Material density (Kgm3)

2 Packer 1047298uid density (Kgm3

) Inclination angle (∘) Warm balloon coefficient o the tubularstring (dimensionless) Drop o any parameter1038389 Axial thermal stress () Density o 1047298uid in the tubing (Kgm3)Δ3 Te tubular string buckling axialdeormation (m)Δ otal axial deormation by variedtemperature 1047297elds (m)Δ907317 otal axial deormation by the variedpressure 1047297elds (m)

Δ1038389start Differential pressure at startup (Mpa)Δ1038389 Change in tubing pressure at the length(Mpa)Δ1038389 Change in annulus pressure at the length(Mpa)Δ1038389 Differential pressure rom top to bottom(Mpa)Δ emperature change with beore and aferwell shut-in (∘C)Δ Change in density o liquid in the tubing at

the length (Kgm3)Δ Change in density o liquid in the casing at

the length (Kgm3)

AcknowledgmentsTis research was supported by the Key Program o NSFC(Grant no 983095983088983096983091983089983088983088983093) and the Key Project o China Petroleumand Chemical Corporation (Grant no GJ-983095983091-983088983095983088983094)

References

[983089] D-L Gao and B-K Gao ldquoA method or calculating tubingbehavior in HPH wellsrdquo Journal of Petroleum Science and Engineering vol 983092983089 no 983089ndash983091 pp 983089983096983091ndash983089983096983096 983090983088983088983092

[983090] D J Hammerlindl ldquoMovement orces and stresses associatedwith combination tubing strings sealed in packersrdquo Journal of

Petroleum echnology vol 983090983097 pp 983089983097983093ndash983090983088983096 983089983097983095983095

[983091] A Lubinski W S Althouse and J L Logan ldquoHelical bucklingo tubular sealed in packersrdquo Journal of Petroleum echnology vol 983089983092 no 983094 pp 983094983093983093ndash983094983095983088 983089983097983094983090

[983092] P R Paslay and D B Bogy ldquoTe stability o a circular rodlaterally constrained to be in contact with an inclined circularcylinderrdquo Journal of Applied Mechanics vol 983091983089 pp 983094983088983093ndash983094983089983088983089983097983094983092

[983093] R Dawson and P R Paslay ldquoDrillpipe buckling in inclinedholesrdquo Journal of Petroleum echnology vol 983091983094 no 983089983088 pp 983089983095983091983092ndash983089983095983091983096 983089983097983096983092

[983094] R F MitchellldquoEffects o well deviationon helical bucklingrdquo SPEDrilling and Completion vol 983089983090 no 983089 pp 983094983091ndash983094983096 983089983097983097983095

7252019 Analyzing Axial Stress and Deformation of Tubular

httpslidepdfcomreaderfullanalyzing-axial-stress-and-deformation-of-tubular 99

Te Scienti1047297c World Journal 983097

[983095] P Ding and X Z Yan ldquoForce analysis o high pressure waterinjection stringrdquo Petroleum Dring echiques vol983091983094 no 983093p 983090983091983090983088983088983093

[983096] Z F Li ldquoCasing cementing with hal warm-up or thermalrecovery wellsrdquo Journal of Petroleum Science and Engineering vol 983094983089 no 983090ndash983092 pp 983097983092ndash983097983096 983090983088983088983096

[983097] A M Sun ldquoTe analysis and computing o water injectiontubularrdquo Drilling and Production echnology vol 983090983094 no 983091 pp983093983093ndash983093983095 983090983088983088983091 (Chinese)

[983089983088] J P Xu ldquoStress analysis and optimum design o well comple-tionrdquo echnical Report o Sinopec GJ-983095983091-983088983095983088983094 983090983088983088983097

[983089983089] J P Xu Y Q Liu S Z Wang and B Qi ldquoNumerical modellingo steam quality in deviated wells with variable ( P) 1047297eldsrdquoChemical Engineering Science vol 983096983092 pp 983090983092983090ndash983090983093983092 983090983088983089983090

[983089983090] A R Hasan and C S Kabir ldquowo-phase 1047298ow in vertical andinclined annulirdquo International Journal of Multiphase Flow vol983089983096 no 983090 pp 983090983095983097ndash983090983097983091 983089983097983097983090

[983089983091] HD Beggs and JR BrillldquoA studyo two-phase 1047298owin inclinedpipesrdquo Journal of Petroleum echnology vol 983090983093 no 983093 pp 983094983088983095ndash983094983089983095 983089983097983095983091 paper 983092983088983088983095-PA

[983089983092] H Mukherjee and J P Brill ldquoPressure drop correlations orinclined two-phase 1047298owrdquo Journal of Energy Resources echnol-ogy vol 983089983088983095 no 983092 pp 983093983092983097ndash983093983093983092 983089983097983096983093

7252019 Analyzing Axial Stress and Deformation of Tubular

httpslidepdfcomreaderfullanalyzing-axial-stress-and-deformation-of-tubular 89

983096 Te Scienti1047297c World Journal

(983093) Te lifing prestressed cementing technology hasimportant meanings to reduce the deormation o tubular

(983094) Te creeping displacement o downhole stings willproduce an upward contractility which causes packer

depressed or lapsed Tereore the effective measuresshould be adopted to control the companding o tubular

5 Conclusion

In this paper a total tubular deormation model aboutdeviated wells was given A coupled-system model o differ-ential equations concerningpressureand temperature in hightemperature-high pressure steam injection wells according tomass momentum andenergy balances which canreduce theerror o axial stress and axial deormation was given insteado theaveragevalueor simplelinear relationship in traditional

research Te basic data o the Well (high temperature andhigh pressure gas well) 983089983091983088983088 m deep in Sichuan China wereused or case history calculations Te results can providetechnical reliance or the process o designing well tests indeviated gas wells and dynamic analysis o production

Nomenclature

907317 Inner diameter (m)9073171038389 Microelement o the tubular Acceleration o gravity (ms2)ℎ Depth o top tubular located at the packer(m)

ime o down stroke (s) Dimensionless time (dimensionless)V Velocity o 1047298uid in tubing (ms)V Velocity o down stroke (ms) Distance coordinate in the 1047298ow direction

(m) along the tubing1103925 Constant cross-sectional 1047298ow area (m2)1103925 Effective area (m2)1103925 Area corresponding to packer bore (m2) Te Joule-Tomson coefficient(dimensionless) Heat capacity o 1047298uids (JKg sdotK)

Outer diameter (m)

Steel elastic modulus o tubular (Mpa) Axial orces in the section (N)1038389 Axial tensile strength (N) Friction orce (N) Piston orce or supporting packerrsquos pressure(N) Pumping orce (N) Length o tubular (m)9073171038389 Buoyant weight o tubular (Kg)1038389 Dead weight o tubular (Kg)10383890 Pressure outside the tubular (Mpa)10383891 Pressure inside the tubular at the packerlength (Mpa)

1038389 Pressure in tubing (Mpa) emperature in tubing (∘C) Initial temperature o ormation (∘C) Mass 1047298ow rate (Kgs) otal length (m)1 Material density (Kgm3)

2 Packer 1047298uid density (Kgm3

) Inclination angle (∘) Warm balloon coefficient o the tubularstring (dimensionless) Drop o any parameter1038389 Axial thermal stress () Density o 1047298uid in the tubing (Kgm3)Δ3 Te tubular string buckling axialdeormation (m)Δ otal axial deormation by variedtemperature 1047297elds (m)Δ907317 otal axial deormation by the variedpressure 1047297elds (m)

Δ1038389start Differential pressure at startup (Mpa)Δ1038389 Change in tubing pressure at the length(Mpa)Δ1038389 Change in annulus pressure at the length(Mpa)Δ1038389 Differential pressure rom top to bottom(Mpa)Δ emperature change with beore and aferwell shut-in (∘C)Δ Change in density o liquid in the tubing at

the length (Kgm3)Δ Change in density o liquid in the casing at

the length (Kgm3)

AcknowledgmentsTis research was supported by the Key Program o NSFC(Grant no 983095983088983096983091983089983088983088983093) and the Key Project o China Petroleumand Chemical Corporation (Grant no GJ-983095983091-983088983095983088983094)

References

[983089] D-L Gao and B-K Gao ldquoA method or calculating tubingbehavior in HPH wellsrdquo Journal of Petroleum Science and Engineering vol 983092983089 no 983089ndash983091 pp 983089983096983091ndash983089983096983096 983090983088983088983092

[983090] D J Hammerlindl ldquoMovement orces and stresses associatedwith combination tubing strings sealed in packersrdquo Journal of

Petroleum echnology vol 983090983097 pp 983089983097983093ndash983090983088983096 983089983097983095983095

[983091] A Lubinski W S Althouse and J L Logan ldquoHelical bucklingo tubular sealed in packersrdquo Journal of Petroleum echnology vol 983089983092 no 983094 pp 983094983093983093ndash983094983095983088 983089983097983094983090

[983092] P R Paslay and D B Bogy ldquoTe stability o a circular rodlaterally constrained to be in contact with an inclined circularcylinderrdquo Journal of Applied Mechanics vol 983091983089 pp 983094983088983093ndash983094983089983088983089983097983094983092

[983093] R Dawson and P R Paslay ldquoDrillpipe buckling in inclinedholesrdquo Journal of Petroleum echnology vol 983091983094 no 983089983088 pp 983089983095983091983092ndash983089983095983091983096 983089983097983096983092

[983094] R F MitchellldquoEffects o well deviationon helical bucklingrdquo SPEDrilling and Completion vol 983089983090 no 983089 pp 983094983091ndash983094983096 983089983097983097983095

7252019 Analyzing Axial Stress and Deformation of Tubular

httpslidepdfcomreaderfullanalyzing-axial-stress-and-deformation-of-tubular 99

Te Scienti1047297c World Journal 983097

[983095] P Ding and X Z Yan ldquoForce analysis o high pressure waterinjection stringrdquo Petroleum Dring echiques vol983091983094 no 983093p 983090983091983090983088983088983093

[983096] Z F Li ldquoCasing cementing with hal warm-up or thermalrecovery wellsrdquo Journal of Petroleum Science and Engineering vol 983094983089 no 983090ndash983092 pp 983097983092ndash983097983096 983090983088983088983096

[983097] A M Sun ldquoTe analysis and computing o water injectiontubularrdquo Drilling and Production echnology vol 983090983094 no 983091 pp983093983093ndash983093983095 983090983088983088983091 (Chinese)

[983089983088] J P Xu ldquoStress analysis and optimum design o well comple-tionrdquo echnical Report o Sinopec GJ-983095983091-983088983095983088983094 983090983088983088983097

[983089983089] J P Xu Y Q Liu S Z Wang and B Qi ldquoNumerical modellingo steam quality in deviated wells with variable ( P) 1047297eldsrdquoChemical Engineering Science vol 983096983092 pp 983090983092983090ndash983090983093983092 983090983088983089983090

[983089983090] A R Hasan and C S Kabir ldquowo-phase 1047298ow in vertical andinclined annulirdquo International Journal of Multiphase Flow vol983089983096 no 983090 pp 983090983095983097ndash983090983097983091 983089983097983097983090

[983089983091] HD Beggs and JR BrillldquoA studyo two-phase 1047298owin inclinedpipesrdquo Journal of Petroleum echnology vol 983090983093 no 983093 pp 983094983088983095ndash983094983089983095 983089983097983095983091 paper 983092983088983088983095-PA

[983089983092] H Mukherjee and J P Brill ldquoPressure drop correlations orinclined two-phase 1047298owrdquo Journal of Energy Resources echnol-ogy vol 983089983088983095 no 983092 pp 983093983092983097ndash983093983093983092 983089983097983096983093

7252019 Analyzing Axial Stress and Deformation of Tubular

httpslidepdfcomreaderfullanalyzing-axial-stress-and-deformation-of-tubular 99

Te Scienti1047297c World Journal 983097

[983095] P Ding and X Z Yan ldquoForce analysis o high pressure waterinjection stringrdquo Petroleum Dring echiques vol983091983094 no 983093p 983090983091983090983088983088983093

[983096] Z F Li ldquoCasing cementing with hal warm-up or thermalrecovery wellsrdquo Journal of Petroleum Science and Engineering vol 983094983089 no 983090ndash983092 pp 983097983092ndash983097983096 983090983088983088983096

[983097] A M Sun ldquoTe analysis and computing o water injectiontubularrdquo Drilling and Production echnology vol 983090983094 no 983091 pp983093983093ndash983093983095 983090983088983088983091 (Chinese)

[983089983088] J P Xu ldquoStress analysis and optimum design o well comple-tionrdquo echnical Report o Sinopec GJ-983095983091-983088983095983088983094 983090983088983088983097

[983089983089] J P Xu Y Q Liu S Z Wang and B Qi ldquoNumerical modellingo steam quality in deviated wells with variable ( P) 1047297eldsrdquoChemical Engineering Science vol 983096983092 pp 983090983092983090ndash983090983093983092 983090983088983089983090

[983089983090] A R Hasan and C S Kabir ldquowo-phase 1047298ow in vertical andinclined annulirdquo International Journal of Multiphase Flow vol983089983096 no 983090 pp 983090983095983097ndash983090983097983091 983089983097983097983090

[983089983091] HD Beggs and JR BrillldquoA studyo two-phase 1047298owin inclinedpipesrdquo Journal of Petroleum echnology vol 983090983093 no 983093 pp 983094983088983095ndash983094983089983095 983089983097983095983091 paper 983092983088983088983095-PA

[983089983092] H Mukherjee and J P Brill ldquoPressure drop correlations orinclined two-phase 1047298owrdquo Journal of Energy Resources echnol-ogy vol 983089983088983095 no 983092 pp 983093983092983097ndash983093983093983092 983089983097983096983093