Tubing String Analisis

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Completion Systems Tubing String Analysis

The information and results computed by this analysis are based on generally accepted data and physical relationships described withinpapers presented to the Society of Petroleum Engineers. Due to evolving well conditions and other unknown information, WeatherfordInternational makes no warranty, expressed or implied, concerning results obtained from the use of this analysis.

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Technical ManualFor


Stacked Packers(option to 3 packers)

and

Tapered String

(option to 3 sections per string)

©Weatherford International, Inc.November 21, 1999





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Introduction

In order for a completion system to successfully perform, it must be properly selectedand installed. There are many forces acting upon the tubing string and packer duringcompletion, production and workover operations. To ensure successful operation,service, and retrieval of the string, well conditions and resulting forces and/or tubinglength changes must be understood and evaluated properly prior to selecting or performing certain operations. It should also be understood that the results of thisanalysis are time dependent and may not occur simultaneously, even though theanalysis reports values in that manner.

Information presented in this manual serves to present a compilation of data fromgenerally accepted technical literature as it applies to combination completion stringsand stacked packers. The founding work in this field is the technical paper authoredby Lubinski, Althouse, and Logan titled Helical Buckling of Tubing Sealed in Packers(J. Pet. Tech.; June 1962 665-670; Trans. AIME, 225) as well as Developments inPetroleum Engineering, Vol. One; Collected Works of Arthur Lubinski (Gulf Publ.1987). Both works cover single string performance in great detail.

This analysis includes performance of tapered or combinations strings. Thegoverning technical paper is Movement, Forces, and Stresses Associated WithCombination Tubing Strings Sealed in Packers by D.J. Hammerlindl (SPE 5143,1974). Hammerlindl generalizes the work of Lubinski to include strings of tubing (fromsurface to packer or packer to packer) made up from different tubing sizes or gradesand/or within different casing inside diameters.

In conjunction with this analysis is a Microsoft Excel spreadsheet that performscalculations on a completion design with conditions ranging from a single string of tubing in a single packer to a combination completion of three stacked packers eachwith a three section tapered string of tubing. Throughout this manual, reference to thesoftware, the assumptions made, and results will be discussed.





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Input Data

Certain data must be known in order to complete the analysis. This data is bestpresented on the Input Data page of the software. A wellbore schematic including allpossible equipment for the software analysis is shown on the following page. For a

single string, only the Upper Packer and Top Tubing Section would be used. For asimple tapered string analysis, the Upper Packer and two or three tubing sectionswould be used.

Software Note:Input data format for the spreadsheet defines a tubing section by change in tubingdimension, tubing yield strength, or casing inside diameter. Conventional taperedstring analysis defines a change in section as a change in tubing or annular fluid.Software inputs call the same fluid for all tubing above the designated packer. Fluidscan change between packers.

Tubing for the entire well must be of the same basic material (e.g. alloy steel, CRA,etc.) but can be of different size and strength. Tubing must have identical Young’s

Modulus, Co-efficient of Thermal Expansion, and Poisson’s ratio.

Two conditions are required for a length and force change to be computed. Theseconditions are commonly known as initial and final . Initial is defined as that wellborecondition when the tubing is run and the individual packer set. It is assumed thepacker is run in a wellbore and the packer set with zero differential across the packer element. Final is defined as that condition of temperature and pressure that exists inthe well at some period after the packer has been set. Both conditions areconsidered to be quasi-static, which is they are considered as “snapshots” in time of wellbore conditions. When the “snapshot” is taken, well conditions are assumed to bestatic and in equilibrium.

Input Values Typical to ALL Packers and ALL Tubulars:Initial Surface Temperature:Temperature just below surface where the value remains stable over time (does notchange with outdoor ambient). In the case of a low fluid level well, temperature of thewellbore fluid should be used if the level is near surface and ambient air temperatureshould be used if the level is low on the string. In the case of multiple packers, usewellbore fluid temperature nearest surface.

Initial Bottom Hole Temperature:Temperature of the wellbore fluid at the packer when the packer is set. In the case of multiple packers, use wellbore fluid temperature at the lowest packer to be set.Temperature will be modified (using a calculated temperature gradient and assuminglinear temperature change along the wellbore) for upper packers.

Final Surface Temperature:Temperature of the wellbore fluid at surface when the operation under considerationis complete. This may be produced or injected fluid temperature. The value shouldreflect the temperature of the tubulars at surface.





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Final Bottom Hole Temperature:Temperature of the wellbore fluid at the bottom packer when the operation under consideration is complete.

Depth of BHT (MD):Measured depth at which both of the bottom hole temperatures were taken.

Depth of BHT (TVD): True vertical depth at which both bottom hole temperatures were taken.

Software Note:The spreadsheet calculates a temperature gradient based on the TVD to bottomhole. This gradient is used to calculate the temperature at each section of tubing andat each packer based on the TVD of each. Using MD would result in an erroneous

temperature gradient calculation for highly deviated wells.

Initial Tubing Fluid:

Density of the fluid in the tubing when the packer was run in pounds per gallon.

Initial Tubing Fluid Level:If the packer is set in a low fluid level well, hydrostatic pressure is affected.

Initial Casing Fluid:Density of the fluid in the casing when the packer was run in pounds per gallon. Thisis often the same as the fluid in the tubing, however packer fluid could be circulatedinto the annulus prior to setting the packer.

Initial Casing Fluid Level:If the packer is set in a low fluid level well, hydrostatic pressure and (possibly)temperature are affected. To balance tubing fluid of different density, fluid level in the

casing may be at a different level (as opposed to applying pressure to tubing or annulus to balance).

Software Note:Tubing and casing fluid density and fluid level are used to calculate hydrostaticpressure conditions at each tubing section and at the packer to obtain the totalpressure when added to applied pressure. Fluid levels are also used to obtain thestring weight in fluid.

Young’s Modulus:The assumption is made that the tubulars under analysis are operating within their elastic range (stress has not exceeded yield strength). Commonly known as theModulus of Elasticity, it is defined as the proportional constant between stress and

strain. It is also assumed that tubulars are homogeneous in material properties; thatis, physical properties are identical in along all three principle axes. Most metal tubingbehave in this manner, however fiberglass or fiber-wound tubing does not.For steel tubing, E = 30,000,000 PSI (or 3.00E+07)





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Co-Efficient of Thermal Expansion:When temperature changes, homogeneous tubing tends to change in length. If tubing is heated it will grow longer and when cooled it will shorten. This co-efficientdefines the linear relationship between change in average tubing temperaturechange and length change.

For steel tubing, α = 0.0000069 inches per inch-degree F (or 6.9E-06)

Poisson’s Ratio:When tubulars with homogeneous material properties remain in the elastic range,there exists a proportionality between lateral and axial strain that was demonstratedby Poisson. That proportionality is defined as Poisson’s Ratio.

For steel tubing, µ = 0.30 (and is dimensionless).

Tubing Pressure Initial:Pressure applied to the tubing at surface under initial conditions. Pressure may beapplied to balance wellbore fluid or to set a packer.

Casing Pressure Initial:

Pressure applied to the annulus at surface under initial conditions.

Wireline Tool Diameter to Pass:When tubulars are subjected to helical buckling, it is often difficult to pass wireline or other service tools through the helix. Diameter of future logging or perforating tools isoften known prior to running the completion. Since most tubulars experience somedegree of helical buckling, there is a calculation that determines the maximum lengthof (solid) tool of this given diameter that can easily pass through the helix.

Number of Packers:Used for the software, a number between one and three must be entered. If anumber less than one or greater than three is entered, erroneous output will result.

Input Values Typical to EACH Packer:Depth (MD):Measured depth at which this packer was set. Should be identical to the MD of tubingfor this packer.

Depth (TVD):True vertical depth at which this packer was set. Should be identical to the TVD of tubing for this packer.

Packer Type:While listed as “Packer Type” this value reflects the type of attachment between theupper tubing string (seals) and the packer. Three types are permitted in this analysis:

Free: Seal assembly has no mechanical means of applying load to the packer. Theseal assembly, and thus the bottom of the tubing string, is free to move axially withinthe packer bore. This type of packer cannot sustain a tubing to packer load other than seal friction.Landed: Seal assembly has a locator that allows tubing weight to be “set down” onthe packer but is free to move in the upward direction. Compressive load can passfrom the string to the packer, but tensile load cannot. The string is free to movedownward in the packer until the locator “lands” on the packer. At this point, attempts





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to apply further downward motion result in application of compressive force to thepacker. Upward motion is permitted without restriction once the string is picked up off bottom. Anchored: Seal assembly has a device to fix the bottom of the tubing string to thepacker. There is no motion of the tubing permitted. Changes result in the application

of tensile or compressive forces to the bottom of the packer.

Packer Seal Bore or Valve Diameter:Honed bore inside the packer where the seal assembly seals. When the sealassembly is run inside the packer, pressure acts on the bottom of the string at theseal bore diameter. On a mechanical type tool, bypass valve area is to entered here.

Slack-off or Pickup Force:When the packer is set, tubing weight can either be slacked-off or picked-up from thepacker (if the packer type allows). Following sign convention, weight slacked-off is apositive slackoff force and weight picked-up is a negative force.

Tubing Fluid Final:

Density of the fluid inside the tubing (pounds per gallon) is entered. This is mostimportant when the tubing fluid is a gas.

Casing Fluid Final:Density of the fluid in the annular area between the tubing OD and the casing ID isentered. Remember that in the case of multiple packers, this may be produced fluidthat may be a gas.

Tubing Pressure Final:Surface pressure applied to, or induced within the tubing should be entered. If youwere reading a gage on the tubing at surface, this would be the gage reading.

Casing Pressure Final:Surface pressure applied to the annulus in the case of the upper packer. For subsequent packers, the value would be that pressure that would be measured on agage at the top of that section’s annular area (just below the next higher packer).

Number of Tubing Sections:Three tubing sections are possible for each packer. Enter the number of sections of tubing in your analysis. Conditional branches in the software complete the analysisbased on your entry.

Input Values Typical to EACH Tubing Section:Enter data for each individual tubing section under analysis. The last section of tubing under analysis should have a MD and TVD matching that of the packer.

Tubing OD:Outside diameter of the tubing.

Tubing ID:Inside diameter of the tubing





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Tubing Weight:Weight (pounds per foot) of the tubing in this section, including couplings.

Tubing Yield Strength:Mechanical properties of the tubing specify minimum yield strength. Yield strength is

defined as that point at (or near) which stress is no longer proportional to strain, thematerial is no longer elastic, and any further load results in permanent deformation of the tube. For API tubulars, this value is designated as grade; for example, N-80tubing has a yield strength of 80,000 PSI, while P-110 tubing has a yield strength of 110,000 PSI.

Measured Depth to Bottom of Section: Actual length of tubing used to make up this section.

TVD to Bottom of Section:When run in the well, the bottom of this section resides at the entered true verticaldepth.

Casing ID:Inside diameter of the casing within which the tubing resides.





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Variables & Constants: Aa = Annular area outside tubing and inside casing; in

2

Ai = Area of the inside diameter of the tubing; in2

Ao = Area of the outside diameter of the tubing; in2

Ap = Area of the packer seal bore diameter or valve diameter; in2

As = Tubing cross sectional area; in2

Ats = Area from the packer seal bore to the tubing inside diameter; in2

ATS = Area from the packer seal bore to the tubing outside diameter; in2

CFLi = Initial casing fluid level; ftdT = Change in average tubing temperature; FE = Modulus of elasticity (Young’s Modulus); psiF = Generic force; lbF’a = Actual force acting on the string as a result of pressure; lbFa = Actual force acting on the string as a result of pressure and tubing below; lbFa* = Resultant actual force on the tubing (considers Fp); lbFb = Bending force in tubing; lbFb* = Resultant bending force on the tubing (considers Fp); lbFf = Fictitious force; lb

Ff * = Resultant fictitious force (considers Fp); lbFp = Restraining force of the packer on the bottom of the tubing; lbFso = Slackoff force; lbFtj = Force in the top joint of tubing; lbHcf = Final hydrostatic pressure in the casing; psiHci = Initial hydrostatic pressure in the casing; psiHtf

= Final hydrostatic pressure in the tubing; psi

Hti = Initial hydrostatic pressure in the tubing; psi

I = Moment of Inertia; in4

IDc = Inside diameter of casing; inIDt = Tubing inside diameter; inK = Constant in slackoff force equationL = Generic length; ft or in

Lwt = Length of wireline tool to pass through tubing; inMDt = Measured depth to the bottom of a tubing section; ftn = Neutral point location; ftODt = Tubing outside Diameter; inODwt = Outside diameter of wireline tool; inPbf = Pitch of helix under final conditions; inPbi = Pitch of helix under initial conditions; inPcf = Final applied pressure in the casing; psiPCF = Final total pressure in the casing at depth; psiPci = Initial applied pressure in the casing; psiPCI = Initial total pressure in the casing at depth; psiPtf = Final applied pressure in the tubing; psiPTF = Final total pressure in the tubing at depth; psiP

ti = Initial applied pressure in the tubing; psi

PTI = Initial total pressure in the tubing at depth; psir = Radial clearance from tubing OD to casing ID; inR = Ratio of tubing OD to ID; [dimensionless]

T ≅ Generic reference to terms of an equation [see individual sections of this doc]T AVGi = Initial average tubing temperature; FT AVGf = Final average tubing temperature; FTBHi = Initial bottom hole temperature; F





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TBHf = Final bottom hole temperature; FTBOTi = Initial temperature at the bottom of a section; FTBOTf = Final temperature at the bottom of a section; FTSi = Initial surface temperature; FTSf = Final surface temperature; F

TTOPi = Initial temperature at the top of a section; FTTOPf = Final temperature at the top of a section; FTFLi = Initial tubing fluid level; ftTVDBHT = True vertical depth at which BHT was measured; ftW = Generic weight; lbWair = Weight of the string in air; lbWliq = Weight of the string in liquid; lbwi = Wieght of the fluid inside the tubing; lb/inwo = Weight of the fluid in the annulus displaced by the tubing OD; lb/inws = Weight of the tubing string in air; lb/inwtbg = Weight of the tubing; lb/ft

∆Fa = Change in actual force; lb

∆L1 = Length change due to piston effect; in

∆L2 = Length change in a partially buckled section; in∆L’2 = Length change in a completely buckled section; in

∆L3 = Length change due to ballooning effect; in

∆L4 = Length change due to temperature effect; in

∆L5 = Length change due to slackoff force; in

∆Pc = Change in applied casing pressure; psi

∆PC = Change in casing pressure at depth; psi

∆Pp = Pressure differential across packer elements; psi

∆Pt = Change in applied tubing pressure; psi

∆PT = Change in tubing pressure at depth; psi

∇Ti = Initial temperature gradient; F/ft

∇Tf = Final temperature gradient; F/ft

ϕf = Angle of helix under final conditions; deg

ϕi = Angle of helix under initial conditions; deg

µ = Poisson’s ratio [0.30 for steel]; [dimensionless]

ρti = Initial density of fluid in tubing; lb/gal

ρci = Initial density of fluid in casing; lb/gal

σaf = Axial stress in the tubing under final conditions; psi

σai = Axial stress in the tubing under initial conditions; psi

σbf = Bending stress under final conditions; psi

σbi = Bending stress under initial conditions; psi

σif = Corkscrew stress on the inner fiber under final conditions; psi

σii = Corkscrew stress on the inner fiber under initial conditions; psi

σof = Corkscrew stress on the outer fiber under final conditions; psi

σoi = Corkscrew stress on the outer fiber under initial conditions; psi

σp = Tubing stress from pressure; psiσtj = Stress in the top joint; psi

∆ρc = Change in casing fluid density; lb/gal

∆ρt = Change in tubing fluid density; lb/gal





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Calculations:

There are a number of basic calculations whose results are used repeatedlythroughout this analysis. Those basic calculations are presented below:

Basic:Moment of Inertia:Strength and stress calculations require the calculation of a tubing section propertycalled Moment of Inertia. Whenever bending forces are present in a body (such asthe forces resulting from helical buckling), Moment of Inertia is used to define thetubing section property over which the force is dispersed. The engineering termdefines Moment of Inertia as:

∫ = dA y I 2

where y is the distance from the neutral axis to tubing cross section carrying the loadand dA is an integral cross section of area.

For round tubing with a concentric inside diameter and center of the tubing being the

neutral axis, the following formula applies:

( )44

64 t t IDOD I −=

π

Cross Sectional Area:End area of the tubing cross section:

( )22

4 t t s IDOD A −=

π

Length:Tubing Length is entered as the MD to the top and bottom of a particular section. Todetermine the length of the section, the difference in MD is taken then multiplied by

12 to convert feet to inches. Length in inches is used in subsequent calculations for continuity of units.

( ) ( ) 121−−= nt nt MD MD L

Annular Area:Cross sectional area between the tubing outside diameter and the casing insidediameter:

( )22

4 t ca OD ID A −=

π

Radial Clearance:Radial distance from the outside diameter of the tubing to the inside diameter of the

casing:( )

2

t c OD IDr

−=





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Outside Area:Total end area of the tubing string, using the outside diameter:

( )2

4 t o OD A

π=

Inside Area:Total end area of the tubing string, using the inside diameter:

( )2

4 t i ID A

π=

Packer Seal Bore Area:Cross sectional area of the complete packer seal bore:

( )2

4 s p ID A

π=

Seal Bore to Tubing ID Area:

Internal tubing pressure acts on an area from the seal bore inside diameter to theinside diameter of the tubing; one of the elements of the hydraulic piston force.

( )22

4 t sts ID ID A −=

π

Seal Bore to Tubing OD Area: Annular casing pressure acts on an area from the seal bore inside diameter to theoutside diameter of the tubing; one of the elements of the hydraulic piston force.

( )22

4 t sTS OD ID A −=

π

True Vertical Depth to Section:

To calculate accurate temperature and hydrostatic pressure gradients, the truevertical location of each section must be defined. The assumption is made that theTVD of the top of the first section of tubing is zero feet. The TVD of the bottom of thatsection is an input to the software.

Temperature Calculations:Temperature Gradient:The analysis calculates the initial and final temperature in each section of tubing andat each packer. The assumption is made that temperature increases (or decreases)linearly with depth. To establish temperature we must establish the gradient indegrees F per foot of TVD. Remember to use TVD to calculate gradient andtemperature at each section. If the well is highly deviated, error becomes significant.Initial Gradient:

BHT

Si BHi

iTVD

T T T −=∇





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Final Gradient:

BHT

Sf BHf

f TVD

T T T

−=∇

Initial Temperature at the Top of a Section:

SURFiitopTOPi T T TVDT +∇×=

Initial Temperature at the Bottom of a Section:

( ) SURFiibot BOTi T T TVDT +∇×=

Average Initial Tubing Temperature:Tubing length or force change is based on the change in average tubingtemperature. This value calculates the average temperature in the tubing section; theaverage of top and bottom temperatures.

2 BOTiTOPi AVGi

T T T

+=

Final Tubing Temperature at the Top of a Section:

Sf f topTOPf T T TVDT +∇×=

Final Temperature at the Bottom of a Section:

Sf f bot BOTf T T TVDT +∇×=

Average Final Tubing Temperature:

2

BOTf TOPf

AVGf

T T T

+=

Change in Average Tubing Temperature:

( ) AVGi AVGf T T dT −=

Length Change Due to Temperature Effect (dL4):

LdT L α=∆ 4

Software Note:Length change is calculated for each individual tubing section based on input data,whether it is used or not. There exists a conditional branch statement that looks atthe number of tubing sections for a particular packer. If there is one tubing section,

length change for that packer is the length change for the first section alone. If twosections are chosen, length change for that packer is the sum of the first and second

tubing sections. For three tubing sections, the result is the sum of all sections.





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Pressure Calculations:Many of the calculations in this analysis depend on the “state of pressure” throughoutthe completion. Total pressure is defined as applied (pressure that can be measuredby a gage installed at the top of a fluid column) and hydrostatic (pressure that isinduced by the weight of a column of fluid at a depth).

Hydrostatic Pressure -- Initial Conditions:Under initial conditions, fluid may not completely fill the wellbore. To account for lowfluid level, use the input value of Initial Tubing Fluid Level and Initial Casing Fluid Level .

( )( )( )ititi TFLTVD H −= ρ052.0 for tubing

and

( )( )( )icici CFLTVD H −= ρ052.0 for casing

Hydrostatic Pressure in Tubing – Final Conditions:Under final conditions, fluid is assumed to fill the wellbore. General form of theequation for hydrostatic pressure is given as:

( ) ( )TVD H tf tf ρ052.0=

Software Note:The spreadsheet provides for different tubing fluids in each of the three tubing strings(between packers). The assumption also holds that the wellbore is completely full of fluid. The following calculations are made in the spreadsheet:For the upper or only packer:

( ) ( )111 052.0 TVD H tf tf ρ=For the middle or second packer:

( ) ( ) 11222 052.0 tf tf tf H TVDTVD H +−= ρ

For the bottom packer:

( ) ( ) 22333 052.0 tf tf tf H TVDTVD H +−= ρ

Hydrostatic Pressure in Casing – Final Conditions:Under final conditions, fluid is assumed to completely fill the wellbore. However,when multiple packers are set, it is assumed that none of them leak and that the actof setting an upper packer isolates the second tubing string from hydrostatic pressurein the upper string’s annular area. If a second packer is set, it is assumed thathydrostatic pressure in the annulus just below that packer is zero as the upper packer’s element system isolates the lower annular area from fluid in the upper annular area. This assumption is likely to be in error if the analysis is made for conditions existing very shortly after the upper packer is set.The equation for the top packer is:

( ) ( )111 052.0 BOT cf cf TVD H ρ=For the middle or second packer:

( ) ( )1222 052.0 BOT BOT cf cf TVDTVD H −= ρ

For the Bottom Packer:

( ) ( )2333 052.0 BOT BOT cf cf TVDTVD H −= ρ





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Software Note:The spreadsheet calculates hydrostatic pressure at the bottom of each tubing sectionas well as at each packer. Viewing the detailed calculation section (not available at alluser levels) gives a matrix of tubing pressure differentials at the bottom of each

tubing section.

Caution:

The assumption that the contribution of initial annular hydrostatic pressure at the topof a section is zero over time may not be valid for all situations when multiple packersare installed.

Total Pressure Inside Tubing – Initial:Total pressure is defined at applied plus hydrostatic pressure for any point in thestring(s) for section designated as n.

( ) ( ) ( )ntintinTI P H P +=

Total Pressure Inside Casing – Initial:

( ) ( ) ( )ncincinCI P H P +=

Total Pressure Inside Tubing – Final:

( ) ( ) ( )ntf ntf nTF P H P +=

Total Pressure Inside Casing – Final:

( ) ( ) ( )ncf ncf nCF P H P +=

Pressure Differential Across the Packer:Pressure differential across the packer is defined as the difference in pressure acrossthe packer’s sealing system to the casing. This should not be confused with

differential across the tubing just above the packer. In the case of a single packer,differential would simply be the difference between total pressure in the tubing andtotal pressure in the casing at the packer. In the case of multiple packers, differentialwould be the difference between total casing pressure at the lower end of the upper annulus and total casing pressure at the upper end of the lower annulus. Picturing aconventional packer with rubber elastomer sealing system, pressure differentialwould be the difference in pressure between the two sides of the set element. Prior tosetting the packer, this value would be zero.

For a single packer:

CF TF p P P P −=∆For a system with 2 packers installed, the upper packer dP would be:

( ) ( ) ( )121 CF cf p

P P P −=∆For a system with 3 packers installed, the upper packer would be defined by theequation above, the lower packer by the equation for a single packer, and the middlepacker by the equation:

( ) ( ) ( )232 CF cf p P P P −=∆





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Software Note:The spreadsheet calculates pressure differential across the packer for threeconditions: one packer, two packers, and three packers installed. Output data for differential pressure refers to the input value for the number of packers chosen, and

enters the appropriate pressure differential value.

Change in Tubing Pressure at Surface:The difference between input values of initial and final applied tubing pressure:

titf t P P P −=∆

Change in Casing Pressure at Surface:The difference between input values of initial and final applied casing pressure:

cicf c P P P −=∆

Change in Tubing Pressure at Packer:The difference in absolute tubing pressure at the packer between initial and finalconditions:

TI TF T P P P −=∆

Change in Casing Pressure at Surface:The difference in absolute casing pressure at the packer between initial and finalconditions:

CI CF C P P P −=∆





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Helical Buckling Calculations:Consider a string of tubing, freely suspended in the absence of any fluid inside thecasing. Now consider an upward force F applied at the lower end of the tubing. Thisforce compresses the string and if the compression is large enough (which is almostalways the case under actual well conditions) the lower portion of the string will

buckle into a helix. This compressive force decreases with distance from the packer and becomes zero at the neutral point. Above the neutral point, the string is intension and remains straight.

Formulae to calculate length change due to helical buckling differ dependent onwhether the section under analysis is completely buckled or partially buckled. Toproperly apply the equations, the neutral point of the string must be located. Locatingthe neutral point is accomplished by the general equation:

W

F n =

Where n is the location of the neutral point upwards from the packer, F is resultantforce to the tubing and W is the weight per unit length, of the string.

In helical buckling analyses, F is replaced by a value commonly known as thefictitious force since part of it does not appear to exist. The proof of this theory iscovered in depth in the Appendix of the Lubinski paper. Further explanation isavailable in the following references:Buckling of Tubing in Pumping Wells, Its Effects and Means for Controlling It ;Lubinski and Blenkarn; Trans. AIME (1957) 210; pp. 73-88 AndVarious Methods of High Pressure Testing Oil Country Tubular Material ; Texter; Pet.Eng. (March 1953) 25; No. 3; B-45

The fictitious force Ff is defined as the area of the packer seal bore multiplied by thedifference in pressure inside the packer and outside the packer. Thus:

( )C T p f

P P A F

−=and can exist under either initial or final conditions. The basic fictitious force remainsconstant regardless of the number of tubing sections between packers.

Software Note:The spreadsheet calculates a matrix for Ff independent of the number of tubingsections chosen; that is, an Ff for each tubing section (row 185). A conditional branchis applied to determine the actual Ff dependent on the number of tubing sectionschosen for analysis (which tubing section is sealed in the packer; row 189). Note thatthe formula for Ff is independent of tubing size, and does not depend on tubingdimensions. The force does depend on the values of absolute pressure at the bottomof a particular tubing section, and the conditional branch is used to apply the correctvalues for these two pressures.

Fictitious force at any point in the string can be calculated by subtracting the weightof the string (in fluid) below the point of interest, from the actual fictitious force and isdefined by the general equation:

∑=

+−=n

i

i f fn LW F F 1

1





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What we can see by this equation is that when the weight of the string in fluidbecomes greater than the fictitious force at the packer, the fictitious force at that pointin the string becomes negative. Above this point, we would not expect to see helicalbuckling.

Software Note:The spreadsheet calculates the fictitious force at each tubing section according to theformula above and places results in row 191. In the detailed calculation section,determination of neutral point can be verified by comparing the transition (whichsection) at which the fictitious force changes from positive to negative. The neutralpoint should reside in the last section (from the bottom up) that has a positivefictitious force. Strings that have positive fictitious force in all sections are likely tohave the neutral point above the string.

Weight of the String Per Unit Length:Lubinski proves that in the presence of fluids, the weight per unit length of the stringmust be considered as:

oi s

wwwW −+=where:Weight of the tubing in air, in pounds per inch

12

tbg

s

ww =

Weight of the fluid inside the tubing (initial or final), in pounds per inch:

=231

it i

Aw ρ

Weight of the fluid in the annulus (initial or final), in pounds per inch:

=231

oco

Aw ρ

Note that wo does not use the volume of casing fluid outside the tubing, but uses thevolume of casing fluid displaced when tubing is inside. This term accounts for thebuoyant weight of the tubing string.

Software Note:Under initial conditions, low fluid level may result in the string weight in the well beingequal to the string weight in air. The spreadsheet accounts for low fluid level in thecalculation of string weight per unit length in the same manner as calculation of hydrostatic pressure. Values can be found in the detailed calculations section, rows107 for initial and 111 for final conditions.





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Neutral Point:The general formula for and discussion of neutral is presented at the beginning of this section. We now have all the intermediate values necessary to begin calculationfor a multi-section tubing string. Determination of the neutral begins with theassumption that n is in the lowest section. We then apply the following formula to the

first (lowest) section of tubing:

( )oi s

f

www

F n

−+=

*

Comparing the value of n calculated above to the length of the lowest section, if n isgreater than the length of that section, then the neutral point is above that section.We also know that the lowest section is completely buckled (this fact becomesimportant when applying the formula for helical buckling).

We now apply the following formula to the second section of tubing from the bottomof this string:

( )( )

( ) bottom

ond oi s

bottomoi s f

L

www

www L F n +

−+

−+−=

sec

*

The numerator is a specific form of the general equation for the fictitious force at thebottom of the second string. We also know that the neutral point is above the bottomtubing section, so we add the length of the bottom section (Lbottom) to that portion of the string in the second section that remains buckled (the fraction in the equationabove). Again, if n is greater than the length of the bottom and second sections, weknow that the neutral point is above the second section. We also know that sectiontwo is completely buckled.

Moving up the string, we apply the equation one more time as follows:

( )( ) ( )( )

( ) bottomond

topoi s

bottomoi sond oi s f

L Lwww

www Lwww L F n ++

−+−+−−+−

= secsec

*

If n is greater than the length of all three sections of tubing, the neutral point is abovethe string and all sections are completely buckled.

Software Note:Detailed calculation section row 195 gives results for n assuming that section is thelowest section and is partially buckled. This value is valid for the chosen lowestsection of tubing only and is designated as preliminary n value.

A matrix follows with calculations for each packer with three sections of tubing (row197), two sections of tubing (row 198), and one section of tubing (row 199). Thismatrix contains calculations using the generic formulae above.

Another matrix follows that performs the length comparison calculation to determine

which section in the string contains the neutral point. The conditional branch looks atthe lowest section and if n is less than the length of that section, the branch returnsthe number of that section (3 being the lowest section). The next cell contains thecalculation of n for that section and divides by 12 (to convert inches to feet) for thelocation of the neutral point above the packer. If the neutral point is above the string,the matrix returns a neutral point location of “above” with location in section “0”.





Page 19

Length Change Due to Helical Buckling in a Partially Buckled Section:If the neutral point is determined to be within the tubing section, the formula for lengthchange due to helical buckling is:

EIW

F r L

f

8

22

2 −=∆

Length Change Due to Helical Buckling in a Completely Buckled Section:The following formula is used:

−

−=∆

f f

f

F

LW

F

LW

EIW

F r L 2

8'

22

2

Software Note:In the detailed calculation section, the spreadsheet calculates a matrix of values for dL2 assuming each section meets the criteria for a partially buckled section. If this isnot the case, the partially buckled result is the first term of the completely buckled

equation. Using a conditional branch to verify that the actual fictitious force is greater than zero, partially buckled results are placed in row 223.

Assuming each section meets the criteria for a completely buckled section, thesecond term of that equation is calculated in row 226. The two terms are multipliedtogether and placed in row 228.

Another matrix of values applies a conditional branch to present the total lengthchange in that section, returning a zero value if the neutral point is in the sectionbelow, completely buckled value if the neutral point is in the section above and thepartially buckled value if the neutral point is within the section.

Results for each section are summed for each packer based on a conditional branch

that keys on the number of tubing sections chosen for that packer.

Pitch of Helix under Initial Conditions:General formula for pitch of the buckled helix is:

128

÷= so

bi F

EI P π

When Fso is less than or equal to zero, there is no helix. Division by 12 converts frominches to feet for convention.

Pitch of Helix under Final Conditions:General formula for pitch of the buckled helix is:

12

8* ÷= f

bf F

EI

P π

When Ff * is less than or equal to zero, there is no helix. F f * is the fictitious forceadded to the packer restraining force.





Page 20

Helix Angle under Initial Conditions:Helix angle is calculated from helix pitch by the formula:

( )

−= −

bi

t c

i

P

OD IDTAN

24

21 πφ

Helix Angle under Final Conditions:

( )

−= −

bf

t c

f P

OD IDTAN

2421 πφ

Software Note:Helix pitch is calculated for each tubing section in rows 277 and 278 verifying that F af

is greater than or equal to zero in row 278. Rows 281 and 282 choose helix pitch asthe minimum value within the number of tubing sections chosen. Row 281 verifiesthat Fso is greater than or equal to zero.

Helix angle is then calculated for each tubing section and the minimum chosen withinthe range of chosen tubing sections.

String Weight Calculations:String weight is the value that would be read on a scale attached to the top of thetubing string. There are two common references to string weight: weight in air andweight in liquid.

String weight in air is the weight of the tubing string if it were suspended in a wellborewith no fluid inside. Calculation of string weight in air is simply:String Weight in Air:

LwW sair =since tubing weight is input in pounds per foot.

String Weight in Liquid:Weight in liquid is the measured weight of the tubing string if it were suspended in awellbore that was partially or completely filled with liquid. There are two commonmethods of calculating this value.

The first (and the one not used in the software) is to assume density of steel is 65pounds per gallon. Divide the string weight in air by 65 to get the number of gallonsof casing fluid displaced. Since we know casing fluid density (in pounds per gallon)we can multiply the number of “gallons of steel” by the casing fluid density to get thebuoyant force. Subtract the buoyant force from the string weight in air to get the stringweight in liquid.

The second (used in the software) considers the density of the fluid inside the tubing.Theory is that fluid inside the tubing affects the hook load. Consider the case of 7-5/8” tubing inside 9-5/8” casing; leave the 7-5/8” casing empty (filled with air) with aplug on bottom and run in the hole. There will be a depth at which the 7-5/8” isweightless (floats) even though the weight of the tubing displaces only a smallamount of casing fluid. This is how casing float equipment works.





Page 21

The software returns a value of:

( )oi sliq www LW −+=

Software Note:Input initial weight of the string in liquid takes into account the condition of low fluidlevel (if entered as such). There is a conditional branch that looks at fluid level todetermine how much, if any of the section is in fluid of initial density.

The output data section returns weight of the string in fluid under final conditions.Weight in liquid under initial conditions does not appear.

Actual Forces Acting on the Tubing:There exists an actual force on the steel and elastomer cross section of the tubing atthe packer. This actual force is given by the following:

CSGo pTBGi pa P A A P A A F −−−=which may be either positive or negative.

The actual force at any point in the string may be determined by subtracting theweight of the tubing (in air) below the point on interest from the actual force shownabove. In combination strings, if the tubing ID or OD changes, a concentrated force isintroduced at the transition point due to fluid pressure. This concentrated force isadded to the actual force at the bottom of the string to obtain the actual force at thebottom of the section.

( ) ( ) 1121121 ' CSGooTBGiia P A A P A A F −−−=

For a three section tubing string with tubing dimension changes at the transitions andabsolute pressure differential across the tubing wall, the following:

( ) ( )32211 '' s saaaa Lw Lw F F F F −−++=The above formula shows the actual force at the upper (section 1 to section 2)transition by summing the concentrated force at that transition, the concentratedforce at the second transition (section 2 to section 3) and the actual force wheresection 3 is sealed in the packer, and subtracts the weight in air of tubing sectionstwo and three.

Author ’s Note:

The equation for actual force (previous page) at any point in the string appliesconcentrated forces due to total pressure acting on differences in end area. Whenconsidering a point in the string off bottom, we sum the forces and subtract theweight of the string in air. Since the string is actually in fluid, it might be more prudent to subtract the weight of the string in fluid using the (ws+wi-wo) weight per unit length

times length of the section.





Page 22

Software Note:Software calculates a matrix of values for Fa’ assuming the transition betweensections 1 and 2, and 2 and 3 are intermediate points in the string; that is, the bottomof section one and/or the bottom of section two does not terminate in a packer. For

those transition areas, change in tubing inside area (row 141) and change in tubingoutside area (row 142) are calculated. Total pressure in the annulus is multiplied bychange in tubing outside area (row 149-final, row 163-initial) and change in totaltubing pressure is multiplied by the change in tubing inside area (row 150-final, row164-initial). The two values are summed in rows 145 (final) and 165 (initial) to obtainF’ for that section.

A second matrix is calculated under the assumption that each section is the tubingtermination into the packer. The equation for Fa (as given above) is made using theappropriate tubing diameter in conjunction with packer seal bore diameter.

A third matrix is calculates Fa at transitions using the general form of the equationand assumes the three possible cases of one, two and three tubing sections.

All above conditions are completed for initial and final conditions.

Length Change Due to Piston Effect:Piston effect, or Hooke’s Law effect usually results in shortening of the tubing stringdue to hydraulic forces acting on the string. These forces result from differences intotal pressure and/or differences in area upon which total pressure acts. Piston forcecalculations use the actual force on the tubing calculated in the previous section.

To obtain length change, change in actual force at each transition must be calculatedby the simple formula:

aiaf a F F F −=∆for each section. Change in force is transformed in change in length using Hooke’sLaw and the assumption that tubing material remains elastic under the results of thisanalysis.

a

s

F EA

L L ∆=∆ 1

Length changes for each section are simply summed to obtain total length change for the string.

Software Note: A final matrix of force change (dFa) is calculated for each section under theassumption for all tubing string conditions (1,2, or 3 strings) in rows 180-182. Lengthchange is calculated in another matrix for all conditions in rows 215-217. Row 218contains a conditional branch statement that looks at the number of tubing sectionschosen and presents the total length change for the entire string by summing lengthchanges in the appropriate row.





Page 23

Ballooning Effect:Changes in pressure result in changes in radial force on the tubing. With increase ininternal tubing pressure, tubing commonly increases in diameter and decreases inlength (all other things being equal and the tubing not restrained). Since the tubingincreases in diameter, the effect has been named ballooning.

Formulae for the calculation of length change due to ballooning are quitecumbersome. Performing some intermediate calculations makes calculation easier and more accurate.

Change in Tubing Fluid Density: Calculations require change in fluid density in units of PSI/inch of length. Thegeneral form of the equation for this intermediate value is:

( )( )

12

052.0 tintf

nt

ρρρ

−=∆

Change in Casing Fluid Density:

Formula is similar to change in tubing fluid density above:

( )( )

12

052.0 cincf

nc

ρρρ

−=∆

Dimensionless Tubing Constant:Calculations use a dimensionless tubing ratio of OD/ID as follows:

( )nt

nt

ID

OD R

)(=

Ballooning Effect Equations:There are two terms to the ballooning equation that are separated for this analysis for reasons that will be made clear later in this document.

First Term (density change effect):

( )( ) ( )

−

∆−∆

−=

12

22

1 R

R

E

LT

ncnt

n

ρρµ

Second Term (pressure change effect):

( )( ) ( )

−

∆−∆

−=

1

22

2

2 R

P R P

E

LT

ncnt

n

µ

The total effect of ballooning is the sum of terms one and two:

( ) ( ) ( )nnn T T L 213 +=∆yielding the length change due to ballooning in a particular tubing section n. For thecomplete string, technical papers advise summing values for each section.





Page 24

Author’s Note:Following the procedure given in Hammerlindl’s paper, to sum individual sections for the total ballooning effect is not correct. Consider the case of a single string of tubing 9000 feet long under specified initial and final conditions. Perform an analysis on anidentical string of tubing as three sections of tubing, all identical in length (3000 feet)

and in dimension with identical fluids.

The second term of the ballooning equation can be summed as it deals with length tothe first power; that is, (1/3) + (1/3) + (1/3) = 1. The first term deals with length to thesecond power cannot be summed; for example (1/3)

2 + (1/3)

2 + (1/3)

2 = 1/3, not 1.

The sum of ballooning effects in multiple sections will never agree with the effect inone continuous section if Hammerlindl’s procedure is applied.

It should be noted that the software uses Hammerlindl’s procedure as it is the published and accepted (?) method of calculation.

Software Note:Preliminary values can be found in matrix format for change in tubing fluid density

(row 240), change in casing fluid density (row 241), change in surface tubingpressure (row 242), change in surface casing pressure (row 243), and thedimensionless tubing constant (row 244). Values for the first and second terms of theballooning equation can be found in rows 245 and 246 respectively. Thesecalculations are separated to allow evaluation of the effects of fluid density andpressure change on tubing length change.

There is a conditional branch statement applied in row 249 that sums the ballooningeffect terms using the chosen number of tubing sections, according to Hammerlindl’s

procedure.

Slackoff Force:Slackoff force is applied to the string from surface and can be either positive (weightslacked off) or negative (weight picked up). Complete discussion of slackoff force isgiven in SPE paper #26511 with the results given here.

There is an intermediate calculation for slackoff force reaching the packer as:

n

n

n EI

r K

4

µ=

calculated for each tubing section.

Slackoff Force Reaching the Packer:

The value of K is used in the following relationship for slackoff force reaching thepacker:

( )

( )

( )

( )

= so

n s

n

n

n s

soP F w

K TANH

K

w F

5.05.0





Page 25

To determine slackoff for the entire tubing string, the value can be calculated for eachsection, then summed using a weighted average.

Length Change Due to Slackoff:Slackoff adds length to the string so a positive value of slackoff results in a positive

length change. There are two terms to the slackoff length change equation, one termcalculating pure elastic length change according to Hooke’s Law and a second termto account for buckling inside the casing.Hooke’s Law Term:

( )

( ) E A

L F T

n s

n so

so =1

Buckling Term:

( )

−+=

)()(

22

)(

28

noi sn

son

sowww EI

F r T

Total slackoff is the sum of both terms. For multiple sections, the Hooke’s Law term issummed and the buckling term added one time using a weighted average.

Author’s Note: Although these formulae are published and generally accepted, there are some basic questions that must be asked prior to their use on critical applications. The formulafor slackoff force reaching the packer is independent of length. The implication is that slacking off weight 10,000 feet or one inch would yield identical force reaching the packer.

Software Note:Since Fso reaching the packer is independent of length, values for K and F so arecalculated for each section in rows 252 and 253, based on tubing and casing

properties. The buckling term was also calculated for each section in row 255.

Weighted average of slackoff force and the buckling term are calculated for threetubing sections by:

321

332211

L L L

L F L F L F F so so so

so ++++

=

and for two sections by:

21

2211

L L

L F L F F so so so +

+=

Conversion of Length Change to Force Change: Assuming we are operating in the elastic range of the tubulars, Hooke’s Law statesthat we can convert the previously calculated length changes into force changes inthe string. To accomplish this, we first need to normalize section properties over thestring.





Page 26

Conversion from length to force involves the application of the formula:

∆=∆ L

E A L F s

Since tubing sections have (potentially) different length and cross sectional area, and

length changes are calculated fro an entire string, weighted average of tubingproperties was assumed according to the formula:For three sections:

( )2321

332211

L L L

A L A L A L

L

A s s s s

++++

=

and for two sections:

( )221

2211

L L

A L A L

L

A s s s

++

=

To convert length change to force change, multiply the normalized section propertyfactor above by the length change and modulus as follows:

normalized

s

L

A E L F

∆=∆ −− 4141

Software Note:Results of normalized section properties and length to force conversions can be

found in rows 262 to 265.

Force on Packer:The bottom of the tubing string exerts a force on the packer depending on thedirection of the force and the type of packer (seal assembly) used.

Packers that permit free motion (type 1) sustain no tubing to packer force (other thantheoretical seal friction). Free motion tubing is free to move over the complete lengthchange calculated on the output data sheet.

Packers that permit limited motion (type 2) or landed packers are capable of sustaining a compressive (or positive) packer to tubing force. Resultant tensile in thecalculation output data sheet is shown as zero tubing to packer load and in effectinvolves some upward seal movement.

Packers that permit no motion (type 3) or anchored packers are capable of sustainingtensile or compressive loads applied by the tubing, and permit zero seal movement.Care must be taken when using packers with shear release anchor seal assembliesto assure a net tensile load will not release the seals. It may also be of interest toevaluate top joint tension under final conditions when such a seal assembly is

sheared for release.





Page 27

Software Note: A matrix is used to evaluate tubing to packer forces with conditional branches toverify packer type and load carrying capability. The following chart is an example of the formulae and conditions applied to determine tubing to packer force.

Packer Type Initial FinalFree (type 1) 0 0

Landed (type 2) ΣF1-5* Fso*

Anchored (type 3) ΣF1-5 Fso

* Where summation of forces 1-5 and Fso must be greater than zero, otherwise forceon tubing to packer is zero

Top Joint Tension:Based on published and accepted formulae, tensile force in the top joint follows thegeneric formula:

pa stj F F W F −−=For this calculation, Fp has been modified to include the full value of slackoff force.

Even though only a portion of the slackoff force reaches the packer, all of the slackoff force is applied to the top joint. Normally, Fp would be the amount of tubing to packer force.

Software Note:Since tubing to packer forces are tabulated on the output data sheet, some of thecalculations for top joint tension are done there as well; specifically the modificationto the application of slackoff weight. Conditional branches are employed to assurethat Fp does not include a tensile force sustained by a landed seal assembly. Sincethere was doubt regarding the calculation method and conditional branch, top jointtension is calculated twice in the program.

Author’s Note:

Top joint tension formulae require the use of string weight in air less the packer totubing force, less the actual force from pressure. Since strings are seldom evaluated in air, we may want to consider the weight of the string in liquid. Using the formula as presented yields a somewhat conservative method of evaluating top joint tension.

Stress in the Top Joint:Stress in the top joint is considered to be the result of top joint tension alone. Thisvalue is represented as:

stj

tj

tj A

F =σ

and is calculated using both initial and final top joint tension.

Author’s Note:If performing a workover operation where casing pressure may exceed tubing pressure by a considerable amount, and you are concerned about top joint collapse, API Specification 5C3 has a formula for the calculation of tubular collapse in the

presence of tensile load.





Page 28

Normal Axial Stress Calculations:There exists a normal axial stress in the tubing due to the actual (axial) force Faf inconjunction with tubing to packer forces Fp acting on the tubing cross sectional area.To calculate this stress, the resultant actual tubing force Fa* is calculated for eachtubing section as follows:

paa F F F +=*

The resultant actual force is calculated for initial and final conditions (using the Fa

calculated previously) and Fp based on packer type and summation of forces at thepacker (using slackoff weight at the packer). The reason slackoff weight at the packer is used (instead of full slackoff weight) is that the result of the normal axial stresscalculation is used as a component in the corkscrew stress formula. Since corkscrewstress is greatest where helical buckling is greatest (at the packer), this value is judged to be most representative.

Having calculated the resultant axial force, we can calculate normal axial stress ineach section by:

s

ai

ai A

F *

=σ

And

s

af

af A

F *

=σ

Bending Stress Calculations:Corkscrew stress calculations are made following the work in SPE #9265. Written inslightly different form than Lubinski, Durham represents the bending force under initial conditions as:

( ) ( )CI oTI iaibi P A P A F F −+= And under final conditions as

( ) ( )CF oTF iaf bf P A P A F F −+=These values require consideration of packer restraint forces for landed andanchored tubing situations. If either F*bi or F*bf return a values less than or equal tozero, their value is set at zero. Bending exists only if the bending force is greater thanzero.

Bending stress under initial conditions is calculated using the bending forces above:

I

rF OD bit bi

4

*

=σ

And under final conditions as:

I

rF OD bf t

bf 4

*

=σ

Now that we have axial and bending stress values, common practice is to apply themaximum distortion-energy theory for calculating tri-axial stress in the tubulars. Thefollowing stress formulae are derived in the appendix of Lubinski’s paper and used inthe works of Hammerlindl. Equations are presented in their general form for corkscrew stress at the inner and outer fiber of the tubing.





Page 29

Outer Fiber Stress General Formula:

2

2

22

2 113

±+

−−

+

−−

= ba

cT C T

o R

P R P

R

P P σσσ

Inner Fiber Stress General Formula:

2

2

22

2

2

11

)(3

±+

−−

+

−−

= R R

P R P

R

P P R ba

cT C T i

σσσ

Where initial and final values are substituted in the formula for initial and final

conditions. Note that the formula includes a ± σb term. Stress is calculated onceadding bending stress and once subtracting bending stress. The maximum value isreported as the final stress.

Axial stress tends to be uniform over the cross-section, bending stress tends to behigher at the outer wall and stress due to pressure greater at the inner wall.

If both stresses remain less than yield strength of the tubing, theory states that thetubing will not be permanently corkscrewed.

Software Note:The spreadsheet f a* values for each section for both initial and final conditions usingthe actual tubing to packer force permitted by the selected packer type. Axial stress iscalculated using the appropriate tubing cross sectional area.

Corkscrew tri-axial stress calculates the first term and second term of the equationshown above, then combines them to calculate stress adding and subtracting thebending stress term. Maximum value of tri-axial stress is determined based onbending stress and the number of sections chosen using conditional branchstatements.

Longest Wireline Tool To Pass:In sections where net tubing force is in tension, there is no helix and therefore no limiton the length of wireline tool that will pass. Where tubing force is compressive, thereis assumed to be a helix that prevents and infinite length tool to be passed throughthe tubing.

The first portion of this analysis establishes tubing force:

( )C oT ia P A P A F F −+= *

substituting in the formula:

( )

−

−=

2

4t c

wt t wt

OD ID F

OD ID EI L using initial and final conditions.




The information and results computed by this analysis are based on generally accepted data and physical relationships described withinpapers presented to the Society of Petroleum Engineers. Due to evolving well conditions and other unknown information, WeatherfordInternational makes no warranty expressed or implied concerning results obtained from the use of this analysis

Maximum Stress on String:To evaluate state of stress in the tubing, it is prudent to review all stress valuescalculated to determine the cause of the highest stress in the string. Valuescompared are:

Tubing Stress from Pressure:( )

( )t t

t C T p

IDOD

OD P P

−−

=875.0

σ

Along with tubing stress above the packer (σa), top joint initial stress (σtji), top joint

final stress (σtjf ), corkscrew outer fiber (σof ) and corkscrew inner fiber (σif ). Maximumstress is chosen to output.

Tubing Margin:Safety margin compares the highest stress value reported above to the minimumyield stress in the string. This method tends to be somewhat conservative when thereis a considerable difference in tubing materials throughout the string (such as P-110

top joint and J-55 lower in the string).

Tubing String Analisis

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