Figure.1 Casing-Cement Sheath-Formation System

Transcription

1 International Journal o Emerging Tecnology and Advanced Engineering Website: (ISSN , ISO 91:28 Certiied Journal, Volume 5, Issue 1, January 215) An Analytical Solution to Stress State o Casing-Cement Seat-Formation System wit te Consideration o its Initial Loaded State and Wellbore Temperature Variation Yucai Si 1, Ben Li 2, Boyun Guo 3, Zicuan Guan 4, Hui Li 5 1,4 Proessor & Qingdao, Sandong Prov., 26658, P. R. Cina 2,5 P.D Program & PO Box 4469 Laayette, LA 754, USA 3 Proessor & PO Box 4469 Laayette, LA 754, USA Abstract-- Stress state analysis to casing-cement seatormation system (CCSFS) is te oundation o te wellbore integrity design and examination. Many elastic mecanic models ave been establised to calculate te stresses and displacements o CCFS, but all analytical models and most inite element models took no account o te initial loaded state, and ignored te initial stresses and displacements on cement seat and ormation, and regarded te given outer casing diameter and te measured wellbore diameter as te initial inner and outer diameters o cement seat respectively. Now, a ew inite element models ave taken te initial stress o cement seat into account, but ignored te initial strains o cement seat and ormation. To accurately solve te stresses o CCSFS and better direct te wellbore integrity design and examination, a new analytical model to solve te stresses o CCSFS wit vertical wellbore and isotropic orizontal in-situ stress as been derived according to elastic mecanic teory, wic can take te initial loaded state and temperature variation into account. Combining wit an instance analysis, te stress distribution rules inluenced by its initial loaded state and inner casing pressure ave been discussed. Researc sowed tat weter considering te initial loaded state can cause signiicant dierences on te stress values and teir variation trends o CCSFS. Especially, te tangential stress in casing unit and te radial stress in cement seat ave te maximum dierences. Taking te initial loaded state into account is ully necessary and conducive to wellbore integrity design and examination. Keywords-- wellbore integrity; cement seat; initial loaded state; stress; analytical model I. INTRODUCTION In te wole lie o an oil and gas well, avoiding damages in casing, cement seat and ormation, keeping te integrity o casing-cement seat-ormation system (CCSFS) ave great signiicances to maintain regular production and extend te service lie. Solving te stresses and displacements o CCSFS is te basis to wellbore integrity design and examination. Many analysis models ave been establised to calculate te stresses and displacements o CCFS. 59 Fang Jun and Yue Boqian set up an elastic analytical model under anisotropic orizontal in-situ stress condition [1-2]. Li Jun and Cen Mian presented an elastic-plastic analytical solution under isotropic in-situ stress condition [3-4]. Yin Youquan and Cen Caowei provided an elastic solution and an elastic-plastic solution [5-8]. Yang Ziu and Meng Qingyuan presented a teoretical model o radial stresses on cementing aces in te process o expansive cement solidiication [9]. Haider [1] and Yong Li [11-12] sowed an analytical solution o stress distribution wit te consideration o wellbore temperature and pressure canges. A. Nabipour [13], Zaoguang Yuan [14], Z.Sen [15], Runar Nygaard [16] sowed te inite element models considering o temperature and pressure canges. A. Garnier, J. Saint-Marc and A.-P. Bois [18-22] introduced te cement seat integrity evaluation and design metods o Total Company, including analytical solutions and inite element solutions, considering te eect o wellbore temperature & pressure canges and te eect o initial state. Researces [9,16-22] ave sowed tat because te CCSFS orms under te downole temperature and pressure condition, its initial loaded state can aect te stress distribution and damage condition. However, all analytical models and most inite element models took no account o its initial loaded state, and ignored te initial stresses and strains on cement seat and ormation, and regarded te given outer diameter o casing and wellbore diameter as te initial inner and outer diameters o cement seat. Until now, only a ew inite element models [16, 17] considered te eect o initial stress o cement seat, but still ignored te initial displacements o cement seat and ormation. To accurately solve te stresses o CCSFS and better direct te wellbore integrity design and examination, a new analytical model to CCSFS wit vertical wellbore and isotropic orizontal in-situ stresses as been derived according to elastic mecanic teory, wic can take te initial loaded state and temperature variation into account.

2 International Journal o Emerging Tecnology and Advanced Engineering Website: (ISSN , ISO 91:28 Certiied Journal, Volume 5, Issue 1, January 215) Te stress distribution rules inluenced by its initial loaded state ave been discussed. way to do tis is simply to download te template, and replace(copy-paste) te content wit your own material. II. BASIC MODEL OF CCSFS As sown in Figure 1, te CCSFS is equivalent to an assembling cylinder wit tree kinds o material. Te inner unit is casing (Young's modulus & Poisson s ratio is & respectively), te middle unit is cement seat (Young's modulus & Poisson s ratio is & respectively), and te outer unit is ormation (Young's modulus & Poisson s ratio is & respectively). Te external loads on te assembling cylinder are regarded as known parameters, including te inner casing pressure and te outer isotropic orizontal in-situ stress. In te wole service lie o an oil &gas well, te inner casing pressure is dierent during well completion, testing and production, but te orizontal in-situ stress is regarded uncanged in most cases. Except external loads above, te termal stress due to wellbore temperature variation is taken into account. However, considering tat te CCSFS always undergoes downole temperature and pressure variations, tere is no initial state witout any external loads. Tat is to say, te inner and outer radii o cement seat are always unknown parameters during well completion, testing and production. According to elastic mecanic teory, solving te stresses o CCSFS wit vertical wellbore and isotropic orizontal in-situ stress is an axisymmetric plane strain problem. Tis document is template. We ask tat autors ollow some simple guidelines. In essence, we ask you to make your paper look exactly like tis document. Te easiest way to do tis is simply to download te template, and replace (copy-paste) te content wit your own material. III. SOLUTIONS OF KEY PARAMETERS Tree basic states wit or witout external loads are discussed ere, wic are named as initial loaded state, reloaded state and virtual complete unloaded state. To solve te problem, usually adopt te ollowing basic assumptions [1-4]. 1) Te CCSFS is always in elastic deormation state. Tus, te plastic deormation can be ignored, and elastic mecanics teory can be used. 2) Cement seat cements well and contacts continuously wit casing and ormation. Tat is to say, under te action o external loads, inner and outer cement seat radii are always equal to outer casing radius and wellbore radius, respectively; te radial stresses on inner and outer cement seat walls are equal to tose on outer casing wall and wellbore wall, respectively. Figure.1 Casing-Cement Seat-Formation System 3.1 Initial loaded state Here, te initial loaded state reers to a loaded state ater cementing operation tat slurry as completely solidiied under te given downole temperatures and pressures, and mecanical parameters o cement seat ave turned into stable values. Obviously, initial loaded state as initial stresses and displacements anywere. Te known key parameters only include te inner casing pressure P mi (drilling luid pressure in casing) and te isotropic orizontal in-situ stress σ. Te unknown key parameters include inner cement seat radius r cai (outer casing radius), outer cement seat radius r cbi (wellbore radius), as well as te initial radial stresses P cai on inner cement seat wall (outer casing wall) and P cbi on outer cement seat wall(wellbore wall). 3.2 Virtual complete unloaded state As sown in Figure 2, te virtual complete unloaded state reers to an unloaded state tat te assembling cylinder is split into tree separate cylinder units, and all loads and elastic deormations are released witout any temperature variations. 6

3 International Journal o Emerging Tecnology and Advanced Engineering Website: (ISSN , ISO 91:28 Certiied Journal, Volume 5, Issue 1, January 215) Figure 2 Completed Unloaded State Considering tat te mecanical parameters o casing, cement seat and ormation are dierent, it can be inerred tat under complete unloaded state, te inner cement seat radius is likely not equal to te outer casing radius, and te outer cement seat radius is not likely equal to te wellbore radius. However, all analytical models and most inite element models ave taken no account o te inluence o te initial loaded state, and ignored te initial stresses and displacements, and regarded te given outer casing diameter and wellbore diameter as te initial inner and outer diameters o cement seat under unloaded state respectively. Obviously, it is inconsistent wit te act tat CCSFS always undergoes te downole temperature and pressures, and inevitably as initial stresses and displacements. Under complete unloaded state, te known key parameters only include inner casing radius r si and outer casing radius r so. Te unknown key parameters include inner cement seat radius r ca and outer cement seat radius r cb, wellbore radius r w. Due to te outer ormation radius r is always ar greater tan te wellbore radius r w, it can be regarded as known parameter. How to solve te unknown parameters depends on te initial loaded state. Tat is to say, eiter te initial stresses or displacements o cement seat must be known in advance. However, all initial stresses and displacements cannot be measured directly and accurately until now. Tree metods ave been presented to estimate te initial radial stresses on cementing aces. All te metods assume tat te initial radial stresses on two cementing aces reac to te same value, but dierent metod provides dierent values. Te irst assumes tat te initial radial stresses are equal to te annulus luid pressure wen cement solidiying. Te second assumes tat te initial radial stresses are equal to te annulus luid pressure wile solidiying minus te ormation pore pressure. Te tird assumes tat te initial radial stresses are equal to zero. For unexpansive cement slurry, te irst gives te upper limit value; te tird gives te lower limit value; te second gives a medium value and is commonly used. 61 Assuming tat tere is no temperature cange wen te initial loaded state turns into te complete unloaded state, set te initial radial stresses on cementing aces equal to te same value P ci and regard it as known parameter. Te inner cement seat radius r ca and te outer cement seat radius r cb under complete unloaded state are inversely calculated according to te ormula o tick wall cylinder displacement. Usually, wellbore diameter is measured in open ole state beore running casing. Wen measuring wellbore diameter, set te drilling luid pressure on wellbore wall as P m and te isotropic orizontal in-situ stress as σ, te measured wellbore radius as r w. Te wellbore radius r w under complete unloaded state is inversely calculated according to te ormula o tick wall cylinder displacement. 2(1 s ) rsi pmi (1 ) s (1 2 s ) rso r si p ci 1 Es ( rso rsi ) rca rso (1 c)(1 2 c) p ci 1 Ec 2 (1 ) pci 2(1 ) 1 E rcb r w (1 c)(1 2 c) p ci 1 Ec (1) r w w 2 (1 ) pm 2(1 ) 1 r E 3.3 Reloaded state Here, te reloaded state reers to a loaded state ater cementing and in te process o well testing and regular production tat bears inner casing pressure on inner casing wall and orizontal in-situ stress on outer boundary o ormation, and termal stress due to temperature variation. According to te continuous contact assumption, te known key parameters under reloaded state only include te inner casing pressure P i and te isotropic orizontal insitu stress σ. Te unknown key parameters include inner cement seat radius r ca (outer casing radius), outer cement seat radius r cb (wellbore radius), as well as te radial stress P ca on inner cement seat wall (outer casing wall) and P cb on outer cement seat wall (wellbore wall). (2)

4 International Journal o Emerging Tecnology and Advanced Engineering Website: (ISSN , ISO 91:28 Certiied Journal, Volume 5, Issue 1, January 215) In addition, consider te inluence o termal stress due to temperature variation, and set te termal expansive coeicient o casing unit, cement seat unit and ormation unit is α s, α c and α respectively. Under initial loaded state, set te initial temperature o eac unit equal to te same value T i. Under reloaded state, set te average temperature o casing unit, cement seat unit and ormation unit as T i, T c and T respectively. Ater solving te key geometric parameters under complete unloaded state, to calculate te radial stress P ca on inner cement seat wall and P cb on outer cement seat wall is te key to solve te stress distributions o CCSFS. According to te ormula o tick wall cylinder displacement and te continuous contact assumption on cementing aces, a dual equations at a ti me is derived at last, in wic te unknown parameters P ca and P cb are te radial stresses on cementing aces. c p c p b p b c p c p b b 11 ca 12 cb 1 i 1 21 ca 22 cb 2 2 Were, (1 s )[(1 2 s ) rso rsi] rso Es (1 c)[(1 2 c) rca rcb ] rca c11 rso rsi Ec ( rcb rca ) 2 Es (1 c ) rcb rca c12 Ec ( rcb rca ) 2 E (1 c ) rca r cb c21 Ec ( rcb rca ) E (1 c)[(1 2 c) rcb rca ] rcb c22 (1 ) rw Ec ( rcb rca ) 2(1 s ) rsirso b1 rso rsi 2 b2 2(1 ) rwo b1 Es ( rca rso) (1 c) c( Tc Ti ) rca (1 s ) s( Ts Ti ) rso b2 E ( rw rcb ) (1 ) ( T Ti ) rw (1 c) c( Tc Ti ) r cb (4) According to Eq.(3) and Eq.(4), te radial stresses on cementing aces are solved. p p ca cb c22( b1 pi b1 ) c12 ( b2 b2 ) c c c c c11 ( b2 b2 ) c21( b1 pi b1 ) c c c c (3) (5) Ater te radial stresses on cementing aces are solved, te stresses and displacement at anywere can be calculated according to te ormula o te stress and displacement o tick wall cylinder. IV. COMPUTATIONAL ANALYSIS OF A REAL EXAMPLE To veriy and empasize te importance and necessity to take te initial loaded state into account, wellbore temperature variation is ignored ere. A typical analytical model [6,7] witout te consideration o initial loaded state, a typical inite element model[16] wit te consideration o initial stress ave been selected and compared to te new analytical model. 4.1 Given Computational Conditions Tis living example is selected rom an existing literature [16]. All geometrical parameters and mecanic parameters are sown in Table 1. In addition, te ormation pore pressure is 2.MPa; te isotropic orizontal in-situ stress is 4.MPa; te normal drilling luid pressure wile drilling is 22.MPa; te annulus luid pressure wile cement solidiying is 3.MPa. Table 1 Basic Parameters o a Real Example Parameters Casing Cement Seat Formation Inner Radius (m) Outer Radius (m) / Young s Modulus (GPa) Poisson s Ratio Compare to te Existing Models Set te initial radial stresses on cementing aces equal to 1.MPa and equal to tat o te existing literature. Set te inner casing pressure as 4.MPa, calculate te stresses on cementing aces by te new analytical model and te existing analytical model. Ten extract te computational results rom te existing inite element model. A wole contrastive analysis is presented in Figure 3 and Figure 4. It is sown tat: 1) Weter considering te inluence o initial loaded state, te values and variation trends o te stress distributions solved by te new analytical model are dierent rom tose o te existing typical analytical model. Especially, te maximum dierences lie on te radial stress o cement seat unit and te tangential stress o casing unit. 62

5 Tangential stress(mpa) Radial stress(mpa) Tangential stress σ θ (MPa) International Journal o Emerging Tecnology and Advanced Engineering Website: (ISSN , ISO 91:28 Certiied Journal, Volume 5, Issue 1, January 215) 2) Te values and variation trends o te radial and tangential stresses solved by te new analytical model ave obvious dierences rom tose o te existing typical inite element model wic considers te initial stresses and ignores te initial displacements. In general, te radial and tangential stresses solved by tese two models ave te maximum dierences in casing unit; te radial stresses are quite close and te tangential stresses are close in cement seat unit and ormation unit. Te contrastive analysis as sown tat wen solving te stresses and displacements o CCSFS, it is ully necessary to take te inluence o te initial loaded state into account. I not, te computational results may departure rom actual values severely. 4.3 Analysis o te Inluence Due to Initial Loaded State. Set inner casing pressure equal to 4.MPa, select te initial radial stresses on cementing aces equal to 5.MPa, 1.MPa, 2.MPa respectively, solve te stresses o CCSFS under dierent initial radial stresses by te new analytical model. A wole contrastive analysis is presented in Figure 5 and Figure 6. Under te same inner casing pressure, it is sown tat: 1) Te radial stress (compressive stress) at anywere increase wile increasing te initial radial stresses on cementing aces. 2) Wen increasing te initial radial stresses on cementing aces, te tangential stress (not always compressive stresses) ave dierent variation trends in dierent units. Te absolute value o tangential stress (tensile stress) in casing unit decrease; te tangential stress (compressive stress) in cement seat unit increase, and te tangential stress (compressive stress) in ormation unit decrease. 3) Te radial stress in cement seat unit and te tangential stress in casing unit bear te maximum inluence due to te initial radial stress Analytical model [6,7] New analytical model Finite element model [16] Distance rom wellbore centre(m) Figure.4 Comparison o Tangential Stress =5MPa =1MPa =2MPa Distance rom wellbore centre(m) Figure.5 Inluence on Radial Stress due to Initial Loaded State =5MPa =1MPa =2MPa Distance rom wellbore centre(m) Figure.3 Comparison o Radial Stress 63 Figure.6 Inluence on Tangential Stress due to Initial Loaded State 4.4 Analysis o te Inluence Due to Inner Casing Pressure. Set te initial radial stress on cementing aces equal to 1.MPa, select inner casing pressure equal to 2.MPa, 4.MPa, 6.MPa respectively, solve te stress o CCSFS under dierent inner casing pressure by te new analytical model.

6 Tangential stress(mpa) Radial stress(mpa) International Journal o Emerging Tecnology and Advanced Engineering Website: (ISSN , ISO 91:28 Certiied Journal, Volume 5, Issue 1, January 215) A wole contrastive analysis is presented in Figure 7 and Figure 8. Under te same initial radial stress, it is sown tat: 1) Wen increasing inner casing pressure, te radial stress (compressive stress) at anywere increase, and te tangential stress (not always compressive stress) decrease. I te tangential stress in casing unit are tensile stress, teir absolute values increase. 2) Te stresses in casing unit bear te maximum inluence due to te inner casing pressure; te stress in ormation unit bear te minimum inluence due to te inner casing pressure =2MPa =4MPa =6MPa Distance rom wellbore centre(m) Figure.7 Inluence on Radial Stress due to Inner Casing Pressure =2MPa =4MPa =6MPa Distance rom wellbore centre(m) Figure.8 Inluence on Tangential Stress due to Inner Casing Pressure V. CONCLUSIONS AND SUGGESTIONS Based on teoretical derivation and contrastive analysis o a living example, some conclusions ave drawn as ollows. 1) Casing-cement seat-ormation system as initial loaded state wit initial stresses and initial displacements. I te system is unloaded completely, te inner and outer radii o cement seat are not equal to te given outer casing radius and wellbore radius respectively. 2) According to elastic mecanic teory, a new analytical model to solve te stresses o casing-cement seatormation system wit isotropic orizontal in-situ stress is derived, wic as taken te initial loaded state (including initial stresses and displacements) and wellbore temperature variation into account. 3) Contrastive analysis wit living example is sown tat weter considering te inluence o initial state may cause magniicent dierences to te values and variation trends o stresses o te system. Taking te initial state into account is ully necessary. 4) Wen increasing te initial radial stresses on cementing aces, te radial stresses(compressive stresses) at anywere increase; te tangential stresses(not always compressive stresses) ave dierent variation trends in dierent units; te radial stress in cement seat unit and te tangential stress in casing unit bear te maximum inluence due to te initial radial stresses.. 5) Wen increasing te inner casing pressure, te radial stresses(compressive stresses) at anywere increase; te tangential stresses(not always compressive stresses) decrease; te stresses in casing unit bear te maximum inluence due to te inner casing pressure; te stresses in ormation unit bear te minimum inluence due to te inner casing pressure. 6) Until now, because te initial radial stresses on cementing aces cannot be obtained easily and accurately, it is necessary to continue to explore te initial state o cement seat and estimate te initial stresses on cementing aces reasonably. Funding 1) Cina National Key Basic Researc Development Plan (973 Plan- 21CB22676), Cina Department o Science & Tecnology (DST); 2) Cina National Hig Tecnology Researc and Development Program (863 Program- 212AA9151), Cina Department o Science & Tecnology (DST); 3) United States Department o Energy s National Energy Tecnology Laboratory (NETL), Federal Grant ID Number: DE FE

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