Basic Tubing Forces Model (TFM) Calculation

Transcription

1 CTS, L.C 9870 Pzs Lane Cnre, Texas phne: (409) fax: (409) asic Tubing rces Mdel (TM) Calculatin y Ken Newman & Kenneth halla January 13, 1999 Cntents Intrductin... Theretical Cncepts...3 Real versus ffective rce...4 Real rce versus Weight...6 Capstan r elt ffect...7 Sinusidal uckling Lad...7 Helical uckling Lad...8 Lckup...8 ffect f Curvature n Helical uckling Lad...9 Residual end...9 Stress Calculatins...10 Pressure Calculatins...10 Mdel quatins...10 asic quatin...10 ffect f luid lw...11 Helical uckling Lad...11 Wall Cntact rce...11 Helix Perid and Length Change..1 Calculatin xamples...1 Vertical Well xample...1 Inclined Well xample...14 Curved Well xample...15 Nmenclature...17 References...18 Summary The purpse f this dcument is t describe hw Orpheus calculates tubing frces. These calculatins are needed t determine the stresses in Ciled Tubing (CT) t ensure the safe wrking stresses are nt being exceeded. They als are needed t predict the behavir f the CT in a highly deviated well, t determine if the planned jb can be dne, r t determine if the jb being executed is prceeding as expected. Orpheus is a tubing frces mdel (TM) written specifically fr Ciled Tubing (CT). The basic TM calculatin is perfrmed by calculating the frces alng the length f a CT string at a specific depth in a well, as the string is being either run int the hle (RIH) r being pulled ut f the hle (POOH). This calculatin is perfrmed beginning at the dwnhle end f the CT string and calculating the frces n each segment f the string, prgressing up the string t the surface. Tech Nte CTS, L.C. 1

2 asic Tubing rces Mdel (TM) Calculatin Intrductin The basic TM calculatin is perfrmed by summing the frces n each segment as discussed abve. This is perfrmed with the end f the CT string at a specified depth. The basic calculatin is explained by using a simple example in which a CT segment is lcated in a straight, inclined sectin f a well withut fluids r pressures, shwn in igure 1. As is discussed later, the length f the segment culd vary frm a few feet t the entire length f the well depending n variatins in well gemetry and CT gemetry. R θ CT Segment N W S A Hle Wall IGUR 1 CT segment in a straight, inclined sectin f a well The vectr triangle in igure 1 shws hw a weight W S can be brken int tw cmpnent frces. A is the frce cmpnent in the axial directin (alng the axis f the hle). N is the frce cmpnent in the nrmal directin (nrmal r perpendicular t the axis f the hle). The equatins fr each f these cmpnents are: A = W S csθ Q 1 N = W S sinθ Q Tech Nte CTS, L.C.

3 asic Tubing rces Mdel (TM) Calculatin The frictin frce is calculated by multiplying the nrmal weight cmpnent by the frictin cefficient µ. = µ N Q 3 The real axial frce is fund by summing the weight cmpnent in the axial directin, A, with the frictin caused by the nrmal cmpnent f the weight, N. Nte that the axial cmpnent f the weight causes R t be in tensin, which is defined as a psitive frce. Hwever, the sign f the frictin frce depends n the directin f mtin. When RIH the frictin causes a cmpressive (negative) frce t be added t R. When POOH the frictin causes a tensile (psitive) frces t be added t R. = ± R A Q 4 This is the basic calculatin fr ne segment f the CT. Summing the results frm a series f segments up a CT string yields the axial frce n the CT in the well alng it s length. This frce versus length prfile is calculated by Orpheus when the run at depth functin is executed. The calculatin fr tripping the CT string in and ut f the hle simply repeats the run at depth TM calculatin abve fr many specified depths, stepping the CT string int and ut f the well. The amunt f data calculated during tripping is mre than can be easily displayed. Thus, nly the frces at surface, cmmnly knwn as the Weight, are displayed when the trip in and ut functin is used in Orpheus. ach f these values is the result f a TM calculatin alng the entire length f the string in the well. Theretical Cncepts The abve descriptin f a basic TM nly tk int cnsideratin mass and frictin, and ignred the mre cmplex issues f internal and external pressure, helical buckling, etc. In this sectin the mre difficult cncepts are discussed. Tech Nte CTS, L.C. 3

4 asic Tubing rces Mdel (TM) Calculatin Real versus ffective rce There is ften cnfusin abut the difference between the real axial frce ( R ) and the effective frce ( ), smetimes called the fictitius frce. Suri Suryanarayana f Mbil has clarified this situatin. This clarificatin is dcumented belw using a simple example. Imagine a clsed ended pipe suspended in a well as shwn belw: A P i P x ξ i ξ IGUR Clsed ended pipe suspended in a well Let us cnsider nly the lwer sectin f this pipe frm sme pint A dwnward. The variables used in this discussin are defined in the nmenclature at the end f this dcument. The axial frce cmpnents acting belw pint A are: 1. weight f the pipe acting dwnward = W S X. upward frce n the end f the pipe due t the external pressure = P A 3. dwnward frce n the end f the pipe due t the internal pressure = P i A i Summing these frces t btain the real axial frce at A yields: R = W X + P A P A S i i Q 5 P i and P can be calculated as fllws: P i = P ia + Xξ i Q 6 P = P A + Xξ Q 7 Defining the buyant weight per ft as: W = W S +ξ A ξ A i i Q 8 Tech Nte CTS, L.C. 4

5 asic Tubing rces Mdel (TM) Calculatin Substituting quatins 6, 7, and 8 int quatin 5 and arranging terms yields: R = W X + P A P A ia i A Q 9 Anther frce, the effective frce is nw defined as: = P A + P R ia i A A Q 10 Nte that this is a definitin, nt a real physical frce. The effective frce is the real frce withut the effects f pressure included. This frce turns ut t be much mre cnvenient t wrk with fr several reasns. Nte that the effective frce at pint A can nw be written by cmbining quatins 9 and 10 as: = W X Q 11 This is a much simpler equatin t wrk with in a tubing frces mdel than quatin 9. Als, as shwn in the Tech Nte The ffective rce, the buckling characteristics f a pipe depend upn the effective frce, nt the real frce. The physical significance is that buyancy, which is independent f depth, affects buckling; hwever, pressure, which is dependent n depth, des nt affect buckling. The nly significant quantities that depend upn the real frce are the stresses and strains. Thus, the Orpheus tubing frces mdel wrks in effective frce. The effective frce is cnverted t real frce nly fr stress calculatins and utput purpses. One questin that is ften asked: Des the bttm hle pressure r the well head pressure try t frce the CT ut f the hle? The same questin can be asked anther way: When pushing pipe in against pressure, Des the wellhead pressure multiplied by the crss sectinal area frce need t be snubbed against r des the pressure at the bttm end f the pipe need t be snubbed against? T answer this questin assume pint A in the abve analysis is at the surface. The real frce in quatin 5 is a functin f the bttm f the pipe. Hwever, the real frce in quatin 9 is a functin f the pressure at the tp f the pipe! When the buyant weight is being used t calculate the weight f the pipe, as is usually the case, the wellhead pressure shuld be used t calculate the snubbing frce. urthermre, since the TM calculatin is perfrmed segmentally frm the bttm f the CT t the surface, a bundary cnditin r starting cnditin is required at the bttm f the CT. Cnsider igure : Tech Nte CTS, L.C. 5

6 asic Tubing rces Mdel (TM) Calculatin 1. If the end f the CT is clsed, the real frce R (x=0) = P i A i -P A. rm the definitin f the effective frce, (x=0) = R (x=0) - P i A i +P A. Substituting fr the real frce yields, (x=0) = 0.. If the end f the CT is pen, the real frce R (x=0) = P (A i -A ). Nw frm the definitin f the effective frce and the fact that P =P i fr an pen tube gives, (x=0) = 0. Real rce versus Weight The real frce is calculated by rewriting quatin 10 t be: R = + P A P ia Cnsider the diagram shwn in igure 3. i A A Q 1 Reel ack Tensin Guide Arch P i A i Inner rame Outer rame Weight Sensr Hinge Stripper P i A i P A IGUR 3 rces at surface The real frce just belw the stripper can be calculated using quatin 1. Recall that it is this real frce that must be used fr stress calculatins. The stripper causes the effective frce t change such that the effective frce Tech Nte CTS, L.C. 6

7 asic Tubing rces Mdel (TM) Calculatin abve the stripper is reduced by the amunt f the wellhead pressure times the crss sectinal area, plus r minus the stripper frictin depending upn the directin f mvement. This can be written as: AbveStripper = elwstripper whp ± P A Stripper Q 13 Since P A is zer abve the stripper, the real frce abve the stripper is the effective frce plus the internal pressure times internal area. Thus: P A R AbveStripper = elwstripper whp + i i ± P A Stripper Q 14 It is this real frce abve the stripper which must be used in the stress calculatin. (It shuld be remembered that the real axial frces calculated in the Hercules mdule d nt take these additinal factrs int cnsideratin.) Als, this frce is different frm the frce measured by the weight indicatr, typically knwn as the weight. The weight is affected by the frces acting abve the injectr and thus the weight is: Weight = R AbveStripper P A RT i i Q 15 where RT is the reel back tensin. Capstan r elt ffect Assume that a sectin f CT is in tensin when it passes arund a curve in a well. The tensin causes the CT t be pulled against the inside f the curve. The greater the tensin, the greater the radial lad pushing the CT against the casing. This radial lad causes the frictin with the casing t increase. This increased frictin is knwn as the capstan effect r belt effect. The same argument can be made if the CT is in cmpressin. Nw the CT is pushed against the utside f the curve in the well. Again, additinal frictin frces are generated which must be cnsidered in a tubing frces calculatin. Thus, any curvature in a well, either in the inclinatin r the azimuth directins, causes additinal frictin which adversely affects the mvement f the CT int and ut f a well. Later we will see that there are cases where curvature is beneficial. Sinusidal uckling Lad Imagine a straight CT string is being pushed int a straight hrizntal casing. As the length f CT pushed int the casing increases the frce required t push it increases. This frce is equal t the ttal weight f the CT string in the casing multiplied by the frictin cefficient. As the length increases the weight increases and thus the frce required t push it increases. r the initial distance the CT remains straight, lying nicely in the trugh frmed by the bttm f the casing. Tech Nte CTS, L.C. 7

8 asic Tubing rces Mdel (TM) Calculatin Once the frce required t push the CT reaches a certain amunt (lad), the CT will begin t snake in a sinusidal fashin back and frth acrss the bttm f the casing. This certain amunt is referred t as the sinusidal buckling lad r smetimes the snake buckling lad. In drill pipe TMs this is ften referred t as the critical buckling lad. Hwever, there is nthing critical abut this type f buckling. It des nt prevent the CT frm mving further int the well. The perid f the sine wave is very large (usually 30 t 100 ft), and f curse its amplitude is n greater than the ID f the casing. Thus the bending that is ccurring is trivial. Orpheus des nt even calculate when the sinusidal buckling lad is reached, since it has n impact n the tubing frces calculatin. Helical uckling Lad Lckup Cntinuing t push the CT int the casing cntinues t increase the frce required t push the CT. The first prtin f the CT will still be lying straight in the casing. The secnd prtin, which has an axial lad greater than the sinusidal buckling lad, will be lying in a sine wave in the bttm f the casing. Again, a certain lad is reached at which the CT begins t frm a helix inside f the casing. This lad is referred t as the helical buckling lad. Again, this lad isn t critical. The perid f the helix cntinues t be large, and n significant bending stresses ccur in the CT material. Hwever, at this pint the tubing frces calculatin changes. Helical buckling itself des nt prevent the CT frm ging further int the well. Hwever, as the helix is pushed int the casing there are additinal wall cntact frces due t the helix. These wall cntact frces increase the frictin with the wall f the casing. The additinal wall cntact frces and thus additinal frictin frces increase as the axial lad applied t the CT increases. Nw the CT has 3 distinct sectins. irst there is a straight sectin up t the pint where the sinusidal buckling lad is reached. This is fllwed by a sectin which is buckled int a sine wave, until the helical buckling lad is reached. inally there is a sectin f the CT which is buckled int a helix. It is nly this third, helical sectin in which the additinal wall cntact frces are being generated. These wall cntact frces increase faster than the rate f increase f the axial lad and eventually a vicius circle is created in which the additinal axial frce required t vercme frictin increases faster than the applied axial lad. This pint is referred t as helical lckup. It is nt pssible t push the CT further int the casing nce helical lckup is reached, n matter hw much axial lad is applied. The riginal lckup calculatin used in Orpheus (Lckup 1.0) did nt take int cnsideratin helical buckling, but rather was based n the yield strength f the CT material. Lckup was determined t ccur when any additinal frce n end wuld cause the CT stresses t exceed the yield limit f the pipe. Usually this required setting dwn enugh weight with the injectr t yield the CT at the surface, which is unrealistic. Tech Nte CTS, L.C. 8

9 asic Tubing rces Mdel (TM) Calculatin The new lckup calculatin (Lckup.0) is a mre sphisticated mdel which apprximates the depth/frce cmbinatin at which the wall cntact frces resulting frm helical buckling begin t verwhelm the applied axial lad. Specifically, lckup is nw defined t ccur when a large increase in set dwn weight causes nly a very small increase in frce at the end f the tl (dwnhle frce). igure 4 shws the relatinship between the dwnhle frce and the set dwn weight at a specific depth. Althugh set dwn weight and dwnhle frce are treated as psitive quantities in the graph and in this discussin, in reality they tend t be cmpressive frces and hence negative.) D wnh le rce Set Dwn Weight IGUR 4 Dwnhle frce vs. set dwn weight In igure 4, d is the change in dwnhle frce, and dw is the crrespnding change in set dwn weight. The weight transfer is the slpe, d/dw. If the weight transfer is less than an arbitrarily designated percentage, then the CT is cnsidered t be lcked up. ffect f Curvature n Helical uckling Lad The abve thery applies t straight CT in a straight hle. Nw let s cnsider what happens if the hle is nt straight. Imagine the CT lying in a curved casing. The axial lad applied t the CT causes it t seat itself in the trugh frmed by the casing. As the axial lad increases, the radial lad pushing the CT int the seat increases. Thus, the axial lad required t cause the CT t pp ut f the seat and frm a helix is much greater than the helical buckling lad fr a straight hle. Increasing the helical buckling lad delays the nset f helical buckling, and thus delays the nset f lckup. Thus it culd be argued that curvature in the well is beneficial. Hwever, the belt effect caused by the curvature increases the frictin. In mst cases CT can be pushed further int a straight hle than int a curved hle. Residual end The abve thery applies t straight CT. Hwever, the bending that ccurs t the CT at the reel and at the guide arch causes residual stresses in the CT material, which causes the CT t be bent when nt in tensin. This residual bend causes lckup t ccur mre quickly. Tech Nte CTS, L.C. 9

10 asic Tubing rces Mdel (TM) Calculatin Orpheus handles residual bend by assuming that the CT behaves as thugh it were straight. Hwever, the frictin cefficient fr RIH is increased t accunt fr residual bend. The typical frictin cefficient f 0. is increased t 0.3, fr RIH nly. The increase f cefficient f frictin fr running in hle accunts fr the additinal wall cntact frces due t residual bend. Stress Calculatins Orpheus has tw stress calculatins. y default, Orpheus uses the vn Mises stress calculatin. This calculatin cmbines the axial stress due t the frce n the tubing with the hp stress caused by external and internal pressure and the radial stress caused by internal pressure, t calculate the cmbined stress at the inside surface f the CT. Nte that fr the stress calculatin the real frce, R, must be used. R is calculated frm using quatin 10 slved fr R. r mre details n hw the vn Mises stress is calculated see the Hercules mdule f Cerberus. CTS recmmends that the vn Mises stress be used. If the user chses t turn vn Mises stress calculatin ff, Orpheus nly calculates the axial stress. This is simply the real axial frce in the CT divided by the crss-sectinal area. The axial stress is prvided nly t allw the user t see this majr cmpnent f the ttal stress by itself. The vn Mises stress can be turned ff under Optins, Preferences. There is ne additinal cmpnent f stress which the user can chse t include in either the vn Mises r the axial stress calculatin. This is the additinal axial stress due t the helical buckling. y default this stress is nt included because it is a very lcalized stress and des nt tend t cause CT failures. This stress cmpnent can be included under the Optins, Preferences menu. Pressure Calculatins The pressure values used in Orpheus are calculated frm user inputs fr fluid density and flw rate. Currently, the hydraulics mdel nly supprts single-phase liquids; hwever, in the future this mdel will be extended t include multi-phase fluids and gases. In the case f flwing liquids, the pressures calculated include the frictinal pressure lss cmpnent resulting frm cntact with the pipe and casing walls. Mdel quatins asic quatin As was described previusly, the Orpheus mdel begins a calculatin fr the CT string at ne psitin in the well: the bttm end f the string. The effective frce calculatin is perfrmed fr each successive segment f the CT up t the surface. Nte that the string segment discussed here has nthing t d with the string segments in String Manager. The length f a segment varies depending n variatins in wall thickness, hle diameter, Tech Nte CTS, L.C. 10

11 asic Tubing rces Mdel (TM) Calculatin fluid density inside and utside the CT and well gemetry. The maximum segment length can be set by the user, but usually it defaults t 100 ft. The basic differential equatin 1 which is integrated ver the segment is: d ds = W csθ ± µ d ds N Q 16 where d ds N dγ dθ = sinθ + + W sinθ ds ds Q 17 This equatin is similar t quatin 4 except that it includes the additinal frictin due t the capstan effect. Nte that if the curvature terms dγ/ds and dθ/ds are zer and the internal and external pressures are zer (n fluids), quatin 1 with 13 included becmes the same as quatin 4. ffect f luid lw The flw f fluid in the CT and in the annulus arund the CT prduces tw types f frces which must be accunted fr in the equatin f axial equilibrium, i.e. quatin 16. irst, there is a lss in the nrmal cmpnent f fluid pressure due t frictinal cntact between the fluid and the CT surface. Secnd, there is an additinal tangential cmpnent caused by the shear stresses (r viscus drag) n the CT due t the fluid flw. As a result, it has been shwn 4 that the fllwing term, which accunts fr bth f the fluid flw effects mentined here, must be added t quatin 16: d ds l π r r c = rc r ( r τ r τ ) c c Q 18 Helical uckling Lad The primary equatin fr the helical buckling lad, ignring the effect f frictin n the helical buckling lad, is H I dγ dθ 4 H sinθ + H + W sin rc ds ds = θ Q 19 Mbil has prvided CTS with prprietary mdificatins t this equatin which accunt fr the effect f frictin n the helical buckling lad. Anther prprietary equatin frm Mbil is used fr the helical buckling lad when the inclinatin is less than 15 degrees. Wall Cntact rce quatin 17 is used by Orpheus t calculate the nrmal frce per unit length that the CT makes with the hle wall due t weight and the curvature effect. If the CT is helically buckled, an additinal wall cntact frce Tech Nte CTS, L.C. 11

12 asic Tubing rces Mdel (TM) Calculatin must be added t quatin 17 t accunt fr the additinal wall cntact frce caused by the helix. This additinal wall cntact frce due t the helix is given by the fllwing equatin 3 : ds NH = rc 4I Q 0 Helix Perid and Length Change The ttal wall cntact frce per unit length is fund by summing quatin 17 and 0. Orpheus utputs a curve shwing these values. The perid f the helix is calculated using the fllwing equatin 4 : λ = π I Orpheus uses this equatin t calculate and utput the perid length. Q 1 The helical shape f the CT requires that the CT be lnger than the sectin f the well it is in. In mst cases the difference in length between the CT and the well sectin is quite small. Orpheus calculates this length difference using the fllwing equatin derived frm gemetry: πr L = L c λ Q Calculatin xamples Vertical Well xample The fllwing examples shw calculatins fr the real and effective frce. Cnsider a CT string f uter diameter 1.5 and thickness 0.109, the CT string is hanging in a vertical well f 10,000 ft. Let the fluid density in the CT and the annulus between the CT and cmpletin be 8.5 lb/gal (i.e. water f density lb/ft 3 ). Let the CT be clsed at the dwn hle end. The weight per unit length f the CT is 1.63 lb/ft. Tech Nte CTS, L.C. 1

13 asic Tubing rces Mdel (TM) Calculatin 5,000 ft ρ ρ i x R =? =? Therefre the real frce at the end f the CT is R =P i A i -P A, the internal and external pressures at the CT end are gverned by the hydrstatic pressure in the CT and annulus arund the CT at the depth in questin. R (x=0) = ρ g h A - ρ i g h A i = 1, lbf rm the definitin f the effective frce, quatin 1, = 0. The buyed weight, per unit length, f the CT = lbf/ft, frm quatin 8. rm quatin 11, the effective frce at the surface ( x=5,000 ft) = W X = 5000 W = 7,064 lbf Cnsider tw cases: Case 1. Let the WHP, the circulating pressure, the stripper frictin and the reel back tensin be equal t zer. Then the real frce at the surface = 5,000 W, frm quatin 8, thus the surface weight as the CT is run in and pulled ut f the well is a linear functin f the amunt L f the CT run int hle and equal t L. Case. If the WHP = 5,000 psi, the stripper frictin frce = 300 lbf, the reelback tensin is equal t 500 lbf while running in hle and 800 lbf while pulling ut f hle, then frm quatins 14 and 15, the variatin f surface weight as the CT is run in hle (RIH) and pulled ut f hle (POOH) is: W RIH = W L - WHP A + Stripper rictin rce - RT RIH = lbf at 5,000 ft. W POOH = W L - WHP A - Stripper rictin rce - RT POOH = -871 lbf at 5,000 ft. Tech Nte CTS, L.C. 13

14 asic Tubing rces Mdel (TM) Calculatin As the CT is being snubbed against the WHP frce, the effect f WHP is t decrease the surface weight by a cnstant amunt. Similarly the effect f the reelback tensin is t decrease the surface weight. Inclined Well xample Cnsider the same CT gemetry as in the example fr a straight well. The CT is nw run in and pulled ut f a well that is inclined at an angle, θ = 30 degrees. Again, let the CT be buyed by fluid f density 8.5 lb/gal in the CT and in the annulus arund the CT and cmpletin. Let the cefficient f frictin between the CT and cmpletin be 0.5 fr RIH. 5,000 ft 30 deg s rce equilibrium (r quatin 16 with θ = cnstant and γ = 0) gives: d ds = W csθ ± µ W sinθ where s is the measured depth alng the well. This equatin can be integrated t give the effective frce as a functin f measured depth, nting that the effective frce at the dwnhle end f the CT is zer (if the CT is pen r clse ended). Thus, the effective frce distributin in the CT string is () s = ( W csθ ± µ W sinθ )s The effective frce at surface, given the CT is run in hle t a measured depth, L equal t 5,000 ft is: () l = ( W cs θ µ W sinθ ) L = L = lbf Tech Nte CTS, L.C. 14

15 asic Tubing rces Mdel (TM) Calculatin S it can be seen by changing the well gemetry frm a vertical t an inclined well, the surface weight has decreased, while RIH, due t frictinal resistance. Nw again if the WHP = 5,000 psi, the stripper frictin frce = 300 lbf, the reelback tensin is equal t 500 lbf while running in hle, then the variatin f surface weight as the CT is run in hle (RIH) is: W RIH = ( W csθ µ W sinθ ) RIH + Stripper rictin rce RT = 1.047L 9, L WHP A RIH Again, the surface weight is a linear functin f the amunt f CT run in hle, but is ffset by the well head pressure, stripper frictin and reelback tensin. urthermre, if the stresses in the CT string are t be examined, the effective frce needs t be cnverted back t the real frce. r instance, if we wish t determine the true frce and stress in the CT string at a measured depth f 1,000 ft frm the surface, while the string is being run in hle at a measured depth f 5,000 ft. Then the true frce = effective frce at 1,000 ft + internal pressure f the CT at 1,000 ft * A I - external pressure f the CT at 1,000 ft* A. The effective frce at a measured depth f 1,000 ft is: ( W cs θ µ W sinθ ) 4,000 = 4, lbf RIH 060 Nte we measure frm the bttm f the CT. The internal and external pressure f the CT is ρ i g 1,000 Cs θ and ρ g 1,000 Cs θ, nte 1,000 Cs θ is the true vertical depth f the pint at a measured depth f 1,000 ft. Thus, the true frce at a depth f 1,000 ft is 3,990 lbf and hence the axial stress in the CT at this pint is: 3,990/ (A i - A ) = 8,38.4 psi. Curved Well xample Cnsider nw a curved well with a cnstant radius f curvature (equal t R, r cnstant curvature, κ = 1/R). It can be shwn that fr a well with a cnstant curvature, the differential equatin gverning the effective frce, i.e quatin 16 becmes: d ds dθ = W csθ ± µ + W sinθ ds, by definitin dθ ds 1 = = cnstant R Tech Nte CTS, L.C. 15

16 The abve equatin assume n buckling ccurs. Thus, fr running in hle: asic Tubing rces Mdel (TM) Calculatin d ds = W csκ s µ + W sinκ s R This can be integrated t give, assuming that the CT cntacts the bttm side f the well: = µ 1+ µ µκs [( 1 µ ) sinκ s µ csκ s] e WR WR + 1+ µ The slutin beys the cnditin that = 0 at s = 0. Let the radius f curvature R = 10,000 ft S=L=5,000 ft R=10,000 ft Let s examine, the effective frce, real frce, surface weight and stresses f CT with 1.5 OD and (W = 1.63 lb/ft) thickness being run in hle t a measured depth f 5,000 ft. r a measured depth f 5,000 ft the true vertical depth is 5,000 Cs(8.65) = 4,387.8 ft. r a measured depth f 1,000 ft the true vertical depth is 1,000 Cs(5.73) = 995 ft. (The angles are btained frm the radius and measured depth being the arc f the well path). irstly, calculating the effective frce at the surface, given 5,000 ft f CT has been RIH gives: Tech Nte CTS, L.C. 16

17 asic Tubing rces Mdel (TM) Calculatin = + ( 0.5)( 1.43)( 10,000) 1+ ( 0.5) ( 1.43)( 10,000) 1 ( 0.5) 1+ ( 0.5) = 6, lbf [( ) sin( 8.65) ( 0.5) cs( 8.65) ] e Nw again if the WHP = 5,000 psi, the stripper frictin frce = 300 lbf, the reelback tensin is equal t 500 lbf while running in hle and 800 lbf while pulling ut f hle, then the variatin f surface weight as the CT is run in hle (RIH) and pulled ut f hle (POOH) is: W RIH = 6, WHP A + Stripper rictin rce RTRIH = 8.69 lbf Again, lets calculate the effective frce at 1,000 ft, the crrespnding true frce and the stress at 1,000 ft: (MD = 1,000 ft) = 5,050 lbf R = 4,840 lbf Thus the axial stress at 1,000 ft in the CT = 10,168 psi Nmenclature A i = crss sectinal area f the inside f the pipe - in A = crss sectinal area f the utside f the pipe - in A H N NH R = Yung s Mdulus - psi = axial frce cmpnent - lbs = effective frce - lbs = helical buckling frce - lbs = nrmal frce cmpnent - lbs = nrmal frce cmpnent due t helix - lbs = real frce - lbs I = mment f inertia - in 4 L = length f well segment - ft L = amunt CT segment is lnger than well segment due t helix - ft P ia P A = internal pressure at pint A - psi = external pressure at pint A - psi Tech Nte CTS, L.C. 17

18 asic Tubing rces Mdel (TM) Calculatin P i P r c s W W S X ξ I ξ γ θ λ τ c τ = internal pressure at the end (pint ) - psi = external pressure at the end (pint ) - psi = radial clearance between CT and hle wall - in = axis alng length f CT = buyant weight f the pipe - lb/ft = weight f the steel pipe - lb/ft = length f pipe belw pint A - ft = density f the fluid inside the pipe - psi/ft = density f the fluid utside the pipe - psi/ft = azimuth angle - degrees = azimuth angle - degrees = perid f helix - ft = shear stress term n the uter radius f cmpletin = shear stress term n the uter surface f the CT References 1. halla, K., Implementing Residual end in a Tubing rces Mdel, SP 8303, 69 th ATC, New Orleans, September Chen Y.C. Pst uckling ehavir f a Circular Rd Cnstrained within an Inclined Hle, Master s Thesis, Rice University, (September 1987). 3. Lubinski, A. Althuse, W., Lgan, J., Helical uckling f Tubing Sealed in Packers, Jurnal f Petrleum Technlgy, June 1956, , AIM, halla, K., Waltn, I.C., The ffect f luid lw n Ciled Tubing Reach, SP 36464, 71 st ATC, Denver, Octber CTS Tech Nte CTS, L.C. 18