Keywords: Non-Darcy flow, Wellbore induced stresses, Analytical solution, Numerical models, Sensitivity analysis

Transcription

1 The effect of Non-Dacy Flo on Induced Stesses Aound a boe in an anisotopic in-situ stess field Amin Tohidi 1, Ahmad Fahimifa 1,*, Vamegh asouli 1 Depatment of civil and envionmental Engineeing, Amikabi Univesity of Technology Depatment of Petoleum Engineeing, Univesity of Noth Dakota, Gand Foks, ND, USA Abstact Afte dilling a ell, the stesses ill be alteed and the induced stesses pesent the ne state of stess. These induced stesses hich esult in geomechanical poblems. The studies indicate that the flo of hydocabon into the ellboe influences the induced stesses. Dacy equation has been used in the past; hoeve, the lamina flo assumption embedded in this equation cannot coectly model the flo of fluid hen a non- Dacy flo dominates nea a ellboe. Example of such situation includes the gas ells. In this study, analytical equations ae developed to incopoate the effect of non- Dacy flo on the induced stesses aound a ellboe. These equations developed based on Fochheime flo equation. Then the simplified solutions ee pesented by consideing the second tem of Fochheime flo equation. It as found that the diffeence beteen the esults fom Dacy and non- Dacy flo models is popotional to the dadon pessue. Futhe studies included numeical simulation of non-dacy fluid flo in a typical esevoi. Compaison of the esults ith the analytical models indicated that the magnitude of stesses in non-dacy flo is lage than of Dacy flo. Finally, sensitivity of the esevoi popeties on the induced stesses in the non-dacy flo egime as investigated. Keyods: Non-Dacy flo, boe induced stesses, Analytical solution, Numeical models, Sensitivity analysis 1. Intoduction Dilling opeation causes edistibute of stesses aound the ell. The edistibuted stesses, knon as induced stesses ae the tangential, adial and vetical stesses. As the ellboe all is a pincipal stess plane, it is most seveely affected by the pesence of induced stesses as a esult of dilling opeation. This ill esult in diffeent incidences; ith ellboe instability and sand poduction being only to examples of its many. The financial loss esulted fom these incidences ae significant and epoted to be ove six billion dollas in the case of ellboe instability [1], and tens of billion dollas due to sand poduction annually []. Estimation of dilling induced stesses is fundamental to seveal applications in oil and gas. These stesses ae the input to ellboe stability analysis, sanding pediction, hydaulic factuing design and detemining the optimum completion [3][4][5]. The impact of the flo of hydocabon hen entes into the ellboe on the induced stesses aound the ellboe have been epoted by eseaches[6][7][8][9]. Fo example the tangential stesses ee inceased due to stess concentation and fluid ithdaal that leads to compessive failue aound a cavity of a ell [6] o hen the poe pessue gadient is vey high, the effective adial stess may be tensile [8]. While the assumption of lamina flo nea ellboe fo oil esevois may be justified, this ill intoduce a significant eo hen applies to gas ells hee tubulent flo nea ellboe is dominant [1][11]. This implies that Dacy flo calculations cannot be used to modify the induced stesses aound the ellboe in case of a gas ell. This not only applies to ells poducing gas, but also to the ells hee the gas is injected fo enhanced oil ecovey (EO) o stoage puposes. The non-dacy fluid flo also exists in hydaulically factued fomations [1][13]In petoleum engineeing, the effect of non- * -Coesponding to: Ahmad Fahimifa, fahim@aut.ac.i

2 A. Tohidi et al./scientia ianica, Tansaction A: Civil Engineeing Dacy flo on ell pefomance has been studied idely and the concept of skin effect has been intoduced to detemine the pessue dop [14][15][16]. This is hile most of the cuent solutions to estimate the stess aound the ellboe ae based on Dacy flo, i.e. lamina flo assumption nea ellboe [3] [7][14][17]. Thee ae only fe studies hich assumed non-dacy flo to developed sand poduction model fo high ate gas ells [6] o deived modified equations fo non-dacy effect on isotopic stess field [9]. This indicates the need fo developing solutions hich includes the non-dacy flo to coectly model the tubulent flo nea ellboe in case of, fo example, gas poduction o injection. This pape expands the application of the Fochheime equation to non-dacy flo. Numeical simulations also ee conducted to suppot the applicability of the developed models.. Coupling Fluid Flo and Induced Stesses Kisch s equations ae pehaps the most commonly used analytical solutions to estimate the stesses induced aound a ellboe. Kisch (1898) deived these equations fo a cicula hole in an infinite solid medium unde a unifom loads applied fa fom the hole [18]. Kisch s solution is based on assumptions such as a todimensional, elastic and non-poous medium. Futhe studies by Badley [19] based on Faihust s oks [] included the effect of mud pessue in calculating the dilling induced stesses. In this ok, hoeve, plane stain condition and omission the flo of fluid nea ellboe and its impact on induced stesses ee the to majo assumption. In 1986, Santaelli et al developed a nonlinea elastic elationship based on thei studies on shale fomations and concluded that maximum stess can take place nea the ellboe not at the ellboe all. Yet, this model also ignoes the nea ellboe fluid flo[1]. Pooelastic theoy poposed by Ye and Liu (199) addessed the issue of flo and defomation coupling in poous media []. They intoduced pooelasticity theoy in ode to study the effects of fluid flo on ellboe stability because this flo induces additional nomal stesses. Badfod and Cook (1994) developed a semi-analytical elastoplastic model [19]. This model applies to vetical ells subjected to isotopic in-situ stess and poe pessue fields that pedict the onset of sanding. Chen et al (3) developed a model that included the pooelastic, chemical, and themal effects [3]. Han and Dusseault (3) deived the effective stess distibutions aound a ellboe poducing oil ith poo-inelastic and diffeent bounday conditions[7]. All of the above models ae based on Dacy's fluid flo assumption. As stated ealie, only a fe studies consideed the effect of non-dacy flo in ock mechanics applications in paticula stess estimations. Wang et al (1991) have studied the effect of non-dacy flo on the stability of pefoation of gas ells using elastoplastic theoy. They have concluded that non-dacy flo ill significantly inceases the instability of pefoation tunnels [4]. Ong et al () have combined equilibium equations ith the Moh- Coulomb failue citeion and non-dacy fluid flo to study the onset of sanding in spheical o cylindical pefoation of high ate gas ells. They have infeed fom the esult of specific esevoi that the non-dacy flo educes the citical dadon pessue by half as compaed to the Dacy case [6]. They also have poposed the method of pedicting the onset of fomation solid poduction [5]. In this study based on the pooelastic theoy the effect of fluid flo (Dacy/non-Dacy) on induced stesses nea the ellboe ee investigated. 3. Developed Analytical Model Fochheime (191) established diffeential equation (1) to estimate the floing pessue nea ellboe as a function of ellboe adius [6]: dp f Q Q d k h 4 h (1)

3 The effect of non-dacy Flo on induced stesses aound a ellboe The fist tem in this equation is the Dacy flo ith lamina flo assumption. The second tem accounts fo tubulent flo hee the Non-Dacy coefficient,β, is detemined empiically as a function of absolute pemeability. In equation (1), μ, Q, k, ρ,h and, ae viscosity, flo ate, pemeability, fluid density, thickness of poduction laye, and distance fom the ellboe, espectively. Afte solving diffeential equation (1), the bounday conditions need to be applied to convege the esults to the nea ellboe fluid flo hich is the objective of this study. Hee the bounday conditions ae established as: Pf P () P P (3) f f Which states that the fluid pessue at the ellboe all is P and it becomes equal to the esevoi pessue, P f at a lage distance aay fom the ellboe all ( =). Applying bounday conditions of equations () and (3) into equation (1) ill lead to equation (4). 1 1 P F o c h h e i m e flo a ln( ) b( ) P f ( ) f 1 1 a ln( ) b( ) P (4) In ode to detemine the nea ellboe induced stesses, e adopt the equations fo a homogenous medium subjected to anisotopic hoizontal stesses. These equations ae pesented as equation (A.1) in Appendix (A)[3]. Coupling equations (A.1) and (4) ill esult in equations (7) to (9) hich allos estimation of dilling induced adial, tangential and vetical stesses aound a ellboe dilled in a homogenous fomation subjected to anisotopic hoizontal stesses. ( F o c h h e i m e flo ) xy 4 a s i n a c o s θ 1 4 a. (ln( ) -.5) (.5 a b ) b ( - ) - a ( ln( ) ln( ( ).5 - b - P These equations ae based on the assumption that Fochheime equation (4) epesents the fluid flo nea ellboe. (7) In equation (4), coefficients a and b ae defined as: a = P P b( ) ln( ).. ln( ) (5) Q b (6) 4 h

4 A. Tohidi et al./scientia ianica, Tansaction A: Civil Engineeing ( F o c h h e i m e flo ) a a 1 3 cos 1 3 sin 4 xy 4 a ln( ).5 ln( ) b a 1 ( a b ) b b( ) a ln( ) ln( ).5 P ( 8 ) z ( F o c h h e i m e flo) 4 v a cos sin z xy 4 v aln( ) b (9) a ln( ) a b b Coefficients a1 and a ae defined as the equations (A.) in Appendix A. It is to note that the non-dacy paamete β is embedded into paamete b in equation 6. Detemining paamete is not staight foad and seveal expeimental coelations have been poposed to estimate this paamete [1]. In the next section, e discuss ho paamete can be discaded in the above equations consideing the second tem of equation (4) fo specific case of fluid flo in high pemeability gas ells [7]. 4. Induced Stesses aound a Gas Hee, equations (7) to (9) ae used to estimate the dilling induced stesses in geneal fom hee tubulent flo is dominant. The past eseach studies have indicated that the flo ate in gas ells is popotional to the squae oot of the hydaulic gadient [7]. Theefoe, the second tem of the Fochheime equation (equation (1)) as used to pesent the fluid flo in gas ells. By solving the second tem of the diffeential equation (1) and applying the bounday conditions (equations () and (3)) the poe pessue distibution as a function of adius fom the ell defined as equation (1): f P f ( Gas flo ) ( P P ) f ( ) ( P P ) ( ) 1P f 1P (1) Equation (1) only equies pessue and adius paametes and contay to equation (4) it does not include paamete β to calculate the poe pessue. On the othe hand, applying the bounday conditions implies that equation (11) is valid: ( P P ) Q h f ( ) 4 (11) Thus the left side of equation (11) as used instead of the ight side of this equation in equation (1). In ode to develop the stess equations coupled ith the non-dacy flo equation, e substituted equation (1) into equations (A.1). This esults in equations (1) to (14):

5 The effect of non-dacy Flo on induced stesses aound a ellboe P P f 4 ( G as flo a ) 11 a cos sin xy 4 P ( Gas flo) sin xy 4 P P.. f (1) 4 a11 a 1 3 cos 4 P (13) z (G as flo ) P P f 4 v a cos sin xy P P f v z 1 1 (14) In ode to highlight the effect of non-dacy assumption, the above equations ae compaed ith the existing solutions based on Dacy assumption. Hee the compaison as pefomed ith the Dacy model pesented in efeence [3]. The esults sho that the diffeence beteen stesses in Dacy and non- Dacy flo egime is popotional to η (Pf- P). This means that in esevois ith high dadon pessue (Pf-P), the diffeence beteen Dacy and Non-Dacy stesses is moe significant. 5. Numeical Model In this section, the applicability of the above poposed analytical solutions is examined numeically. The data coesponding to a typical esevoi ee assumed fo this pupose as shon in Table 1. Abaqus softae as used fo numeical simulations. In numeical modeling thee steps ae folloed: defining the geomety and the bounday conditions; assigning mateial chaacteistics as ell as the loading conditions; and mesh geneation and the model analysis. In this study, axisymmetic simulation of a vetical ellboe as caied out. The poe pessue at the top suface as set to zeo, at the ell boe all as assumed to be P and at distances fa fom the ell as consideed equal to the esevoi pessue (Pf) acoss the poduction laye. The poous elastic mateial model as used in this simulation. At the bottom of the esevoi, displacements in the vetical diections ee estained and similaly the adial displacements at distances fa fom the ell ee estained. Fine mesh as used nea the ellboe because of the apid stess changes that ae expected nea the ellboe; and coase mesh as used in the egions fa fom the ellboe. Axisymmetic simulation as used in modeling ith 784 CAX8P type (eightnode biquadatic displacement, bilinea poe pessue, educe integation) elements as depicted in Figue 1. In addition, Non-Dacy flo as modeled by adding the velocitydependent tem to pemeability and defining β in Fochheime equation [8]. Figues to 5 sho the poe pessue, adial, tangential and vetical stesses coesponding to both Dacy and non-dacy flos. In all figues, the distance fom the ell is pesented fo a distance of 5 (i.e..6 m in this model) aay fom the ellboe all hee the distubed zone due to dilling is expected to occu [9].

6 A. Tohidi et al./scientia ianica, Tansaction A: Civil Engineeing Figues to 5 illustate the esults fo analytical and numeical solutions in to egimes of Dacy and non-dacy flos. As is obseved thee is the same tend beteen analytical and numeical esults in the ellboe vicinity. The aveage diffeence beteen analytical and numeical esults is less than 1.5 % nea the ellboe all in all cases. The diffeence may esult fom estimation of elastic input paametes fo numeical simulation o vaiation of pemeability ith velocity in numeical modeling. In Figue, the poe pessue at the ellboe all is equal to P and it appoaches Pf as moving ell fa fom the ellboe. The esults indicate that the poe pessue calculated fo non-dacy Fochheime flo (Eq 4) ae 7% lage than that of the Dacy flo (Heidaian et al). This numbe fom numeical simulations is 67% hich is compaable ith analytical esults. In analytical solution pesented by Fjae et al (199) (Appendix A, Figue A.) hen Pf is constant, the adial stess inceases fom P to σ H,, hich is a simila tend obseved in Figue 3. In addition, the esults of Figue 3 shos that the aveage adial stesses in non-dacy Fochheime flo egime (Eq 7) ae 7.3% lage than the Dacy s flo egime, espectively. Coespondingly, the adial stess in non-dacy flo is 7% lage than Dacy flo fom the esults of the numeical modeling. It should be mentioned that the amount of stess incease depends on the esevoi chaacteistics and it may vay fom case to case. Consideing Fjae et al (199) analytical solutions [3] hen Pf is constant (see Appendix A, Figue A.), the tangential stess aound the ellboe is at maximum, hoeve, it educes apidly moving aay fom the ellboe all and eaches the in-situ stess σ H.. Simila tend is obseved in Figue 4 fo the case of cuent study. Hee, the aveage tangential stess in the non-dacy Fochheime flo egime ae 1.5% geate than the Dacy flo egime based on the esults of analytical models. The esults of non-dacy flo in numeical simulations appea to be.16% lage than that of the Dacy flo. In Figue 5, the aveage vetical stess obtained using the analytical appoach in Non-Dacy Fochheime flo egime 5% geate than Dacy s flo egime, espectively. The esults of numeical simulations sho.5% lage value fo the vetical stess in non-dacy flo compaing to Dacy egime. In intepeting the above esults it is impotant to note the to implicit assumptions that ae embedded in the above pesented equations: these ae adial flo into the ellboe and homogeneity of the fomation as consideed in developing equations in Appendix A.1. In addition, the sensitivity of the esevoi popeties on the esults needs to be investigated as a continuation of this ok. 6. Sensitivity analysis of esevoi popeties In this section the sensitivity analysis of esevoi popeties on induced stesses aound a ellboe in non-dacy flo egime as pefomed. Fo this pupose, the effect of these paametes include dadon pessue, thickness of the pay zone and dainage adius on induced stesses ee investigated. Paametic analysis accoding to table as caied out to analytical equations (Eq 4 and Eqs 7 to 1) in the ay that one paamete as changed and othes as in table 1 ee kept unchanged. In figues 6 to 9 sho the effect of dadon pessue on adial, tangential and vetical stesses. As shon in numeical value and figue 6, the adial stesses nea the ellboe in non-dacy flo egime vey slightly deceased ith inceasing da don pessue that can be ignoed in pactical application.

7 The effect of non-dacy Flo on induced stesses aound a ellboe consideing figue 7, by inceasing the da don pessue, the tangential stess nea the ellboe inceases. Also in figue 8 the vetical stess deceases by inceasing the da don pessue. This means that hen da don pessue inceases the tangential stess inceases and vetical stess deceases leading to moe geomechanical poblems like sand poduction. Simila pocedue like above as done to find out the effect of thickness of the pay zone (h) on induced stesses aound the ellboe. thickness of the pay zone has a significant effect on adial stesses aound a ellboe. So that by inceasing the pay zone thickness fom 5m to m, adial stesses ee inceased apidly fom negative to positive value. The effect of the pay zone thickness on adial stess as shon in figue 9. Accoding to figue 1, hen thickness inceases fom 5 m to 5 m the tangential stess inceases apidly and then deceases. In figue 11 unlike the hoizontal stesses, by inceasing the pay zone thickness the vetical stess as deceased, so the potential of mechanical failue as inceased dominantly in thin pay zone. The dainage adius on stesses aound a boe hole has a mino effect. By inceasing the dainage adius, hoizontal stesses (adial and tangential) ee inceased and vetical stess as deceased. Hoeve, tangential stess is moe sensitive to the change of dainage adius than adial and vetical stesses. 7. Conclusions Accoding to this pape Dacy flo assumption as not appopiate in gas esevoi and in a situation like hydaulic factue, so to sets of stess equations ae developed that ill be useful in geomechanical poblem e.g. ellboe stability, sand poduction and completion design ith a non-dacy fluid flo assumption. Analytical solutions ee developed and compaed to numeical modelling fo a typical esevoi and the changes in poe pessue, adial, tangential and vetical stesses in non-dacy flo egime ee investigated. The esults indicated that the poe pessue and stesses in non-dacy flo egime have geate values than Dacy flo egime, hee the diffeence is a function of seveal paametes including the dadon pessue. Fom the pesented esults of this ok, it is concluded that using Dacy flo assumption fo stess calculations nea ellboe fo gas ells may esult in significant eo, poviding incoect stess data fo design puposes including ellboe stability analysis. Fo example, the induced stesses extacted fom equations 7 to 14 compaed ith appopiate failue citeia can be used in ellboe stability analysis. Also the sensitivity analysis of esevoi popeties on induced stesses aound a ell boe in non-dacy flo egime as done and conclude that effect of the thickness of the pay zone has a moe significant effect on induced stesses than da don pessue and dainage adius. efeences [1] Y. Kang, M. Yu, S. S. Z. Miska, N. Takach, and othes, boe Stability: A Citical evie and Intoduction to DEM, SPE Annu. Tech. Conf. Exhib., no. Octobe, pp. 4 7, 9. [] A. Acock et al., Pactical appoaches to sand management, Oil F. ev., vol. 16, no. 1, pp. 1 7, 4. [3] E. Fja,. M. Holt, A. M. aaen,. isnes, and P. Hosud, Petoleum elated ock mechanics, vol. 53. Elsevie, 8. [4]. isnes,. K. Batli, P. Hosud, and othes, Sand stesses aound a ellboe, Soc. Pet. Eng. J., vol., no. 6, pp , 198. [5] H. Agheshlui and S. K. Matthai, The

8 A. Tohidi et al./scientia ianica, Tansaction A: Civil Engineeing Impact of Layeing and Heteogeneity on Stesses aound Boeholes, in ECMO XIV-15th Euopean Confeence on the Mathematics of Oil ecovey, 16. [6] S. Ong, B. Atlas,. amos, A. Exploation, P. Technology, and Z. Zheng, SPE 5871 Sand Poduction Pediction in High ate, Pefoated and Open-hole Gas s, Poduction,. [7] G. Han and M. B. Dusseault, Desciption of fluid flo aound a ellboe ith stess-dependent poosity and pemeability, J. Pet. Sci. Eng., vol. 4, no. 1, pp. 1 16, 3. [8] G. Chen, Y. Jin, L. Li, and B. Zhou, Analysis of Vetical Collapse Volume in adial Poous Media Flos of a High Pessue Gas, vol. i, 16. [9] A. H. Hassani, M. Veyskaami, A. M. Al-ajmi, and M. Masihi, ock Mechanics & Mining Sciences A modi fi ed method fo pedicting the stesses aound poducing boeholes in an isotopic in-situ stess fi eld, Int. J. ock Mech. Min. Sci., vol. 96, no. Octobe 16, pp , 17. [1] D. Li, T. W. Engle, and othes, Liteatue evie on Coelations of the Non-Dacy Coefficient, SPE Pemian Basin Oil Gas ecove. Conf., no., pp. 1 8, 1. [11]. Al-Hussainy, H. J. amey J, P. B. Cafod, and othes, The flo of eal gases though poous media, J. Pet. Technol., vol. 18, no. 5, pp , [1] J. G. Wang, A. Kabi, and Z. Chen, Effects of non-dacy flo on the pefomance of coal seam gas ells, no. Januay, 1. [13] J. L. Miskimins, H. D. J. Lopez,. D. Baee, and othes, Non-Dacy flo in hydaulic factues: does it eally matte?, in SPE Annual Technical Confeence and Exhibition, 5. [14] E. Detounay and a..-d. Cheng, Pooelastic esponse of a boehole in a non-hydostatic stess field, Int. J. ock Mech. Min. Sci. Geomech. Abst., vol. 5, no. 3, pp , [15] A. M. Amao, Mathematical model fo Dacy Fochheime flo ith applications to ell pefomance analysis, Texas Tech Univesity, 7. [16] P. Kundu, V. Kuma, and I. M. Misha, Expeimental and numeical investigation of fluid flo hydodynamics in poous media: Chaacteization of Dacy and non- Dacy flo egimes, Pode Technol., 16. [17] A. H.-D. Cheng, E. Detounay, and Y. Abousleiman, Pooelasticity, vol. 7. Spinge, 16. [18] S. P. Timoshenko and J. M. Gee, Theoy of elastic stability. Couie Copoation, 9. [19] W. B. Badley, Failue of Inclined Boeholes, J. Enegy esou. Technol., vol. 11, no. 4, p. 3, [] C. Faihust, Methods of detemining in-situ ock stesses at geat depths: Tech, ep. TI-68, Mo. ive Div. Cops Eng., Omaha, Neb, [1] F. J. Santaelli, E. T. Bon, and V. Mauy, Analysis of boehole stesses using pessue-dependent, linea elasticity, in Intenational Jounal of ock Mechanics and Mining Sciences & Geomechanics Abstacts, 1986, vol. 3, no. 6, pp [] P. Li, K. Wang, and D. Lu, Analytical Solution of Plane-Stain Pooelasticity due to Suface Loading ithin a Finite ectangula Domain, no. 14, pp. 1 8, 15. [3] J. E. Olson and S. Sinivasan, The impact of shale popeties on ellboe stability, 5. [4] Z. Wang, J. M. Peden, E. S. H. Damasena, and S. P. A. S, The Pediction of Opeating Conditions To Constain Sand Poduction Fom a Gas, [5] highate gas ell (patent).pdf..

9 m The effect of non-dacy Flo on induced stesses aound a ellboe boe CL 1 m [6] S. Imay, On the theoetical deivation of Dacy and Fochheime fomulas, Eos, Tans. Am. Geophys. Union, vol. 39, no. 4, pp. 7 77, [7] H. Al-Atta and S. Al-Zuhai, A geneal appoach fo deliveability calculations of gas ells, J. Pet. Sci. Eng., vol. 67, no. 3 4, pp , 9. [8] D. Systèmes, Abaqus analysis uses manual, Simulia Cop. Povid. I, USA, 7. [9] Z. Zamanipou, S. Z. Miska, P.. Haihaan, and othes, Effect of Tansient Suge Pessue on Stess Distibution aound Diectional boes, in IADC/SPE Dilling Confeence and Exhibition, 16. [3] M. Heidaian, H. Jalalifa, M. Schaffie, S. Jafai, and othes, Ne Analytical Model fo Pedicting the Unstable Zone Aound the Boehole, SPE J., vol. 19, no. 6, pp , 14. [31] S. K. Al-Shaaibi, A. M. Al-Ajmi, and Y. Al-Wahaibi, Thee dimensional modeling fo pedicting sand poduction, J. Pet. Sci. Eng., vol. 19, pp , 13. Table 1- esevoi chaacteistics fo lamina and tubulent flos 57 MPa σ v σ H =σ h P P f 8.5 MPa 5 MPa MPa 1 m.1 m η.5 ϑ.46 β.5 ρ Q h Kg m m3 s m Pay Zone Fig. 1 Axisymmetic mesh used fo the model 4 m

10 adial Stess (MPa) Vetical Stess (MPa) Poe Pessue (MPa) Tangential Stess (MPa) A. Tohidi et al./scientia ianica, Tansaction A: Civil Engineeing boe 3 boe (m) Analytical Dacy Flo (Heidaian et al) Analytical Gas flo (Eq 1) Analytical Focheime Flo (Eq 4) Numeical Dacy Flo Numeical Focheime Flo Fig. Poe Pessue vaiation aound the ellboe in Dacy and non-dacy flo egimes (m) Analytical Dacy Flo (Heidaian et al) Analytical Gas Flo ( Eq 13) Analytical Fochheime Flo ( Eq 8) Numeical Dacy Flo Numeical Fochheime Flo Fig. 4 Tangential stesses aound the ellboe in Dacy and non-dacy flo egimes ell boe boe (m) Analytical Dacy Flo(Heidaian et al) Analytical Gas Flo (Eq 1) Analytical Fochheime Flo (Eq 7) Numeical Dacy Flo Numeical Fochheime Flo Fig. 3. adial stesses aound the ellboe in Dacy and non-dacy flo egimes (m) Analytical Dacy Flo(Heidaian et al) Analytical Gas Flo (Eq 14) Analytical Fochheime Flo (Eq 9) Numeical Dacy Flo Numeical Fochheime Flo Fig. 5 Vetical stesses aound the ellboe in Dacy and non-dacy flo egimes

11 Vetical stess (MPa) adial stess (MPa) Tangential stess (MPa) The effect of non-dacy Flo on induced stesses aound a ellboe Table -Paametic analysis of esevoi popeties paametes Da don pessue (Pf-P) Thickness of The pay zone (h) Dainage adius () MPa 5 MPa 1 MPa 15 MPa 5 m 5 m 5 m m 5 m 5 m 75 m 1 m boe Fig. 7 The effect of da don pessue on tangential stesses in non-dacy flo egime 65 P-P= Mpa P-P=1 Mpa (m) P-P=5 Mpa P-P=15 Mpa 15 boe (m) P-P= Mpa P-P=5 Mpa P-P=1 Mpa boe Fig. 6 The effect of da don pessue on adial stesses in non-dacy flo egime (m) P-P= Mpa P-P=1 Mpa P-P=5 Mpa P-P=15 Mpa Fig. 8 The effect of da don pessue on vetical stesses in non-dacy flo egime

12 Tangential stess(gpa) adial stess (GPa) Vetical stess (GPa) A. Tohidi et al./scientia ianica, Tansaction A: Civil Engineeing boe 6 boe (m) h=5 h=5 h=5 h= Fig. 9 The effect of pay zone thickness on adial stesses in non-dacy flo egime boe (m) h=5 h=5 h=5 h= Fig. 1 The effect of pay zone thickness on tangential stesses in non-dacy flo egime Fig. 11 The effect of pay zone thickness on vetical stesses in non-dacy flo egime Appendix A Dilling induced stesses aound a deviated ellboe in a homogenous media subjected to anisotopic hoizontal stesses ae detemined as [3]: a a cos sin xy 4 P P ( ) d f P ( ) d f (m) h=5 h=5 h=5 h=, (A. 1)

13 The effect of non-dacy Flo on induced stesses aound a ellboe a cos 4 4 xy sin 4 P z a z Pf ( ) d Pf P ( ) d, (A.) 4 a cos sin xy P f 4 P ( ) d f In equations A.1 to A.3: f (A.3) σ x σ y σ x - σ y a 1, a 1 - v,η α ( 1 - v ) ΔP ( ) P ( ) - P, f f f σ I σ I σ I σ, x xx H xy h xz v σ I σ I σ I σ, y yx H yy h yz v σ I σ I σ I σ, z zx H zy h zz v τ I I σ I I σ I I σ, xy xx yx H xy yy h xz yz v hee I cos a cos i ; I - sin a; I cos a sin i, I xx yx zx xy sin a cos i ; I cos a; I sin a sin i, yy I - sin i ; I ; I cos i, xz yz zz zy In equations A.1 to A., σh and σh ae the maximum and minimum hoizontal stesses, is the adius of the ellboe, is the dainage adius, Pf is the poe pessue in the distance fom the ellboe cente, Pf is the esevoi pessue, v is fomation s Poisson s atio and α is Biot coefficient. As depicted In Figue (A.1), the angles a and i ae used fo tansfomation of stesses fom a vetical to a deviated ellboe[3].

14 A. Tohidi et al./scientia ianica, Tansaction A: Civil Engineeing head of enhanced oil ecovey Institute of Amikabi Univesity of technology. His eseach inteest about ock mechanics. Fig A.1. Angles a and i ae used fo tansfomation of stesses fom a vetical to a deviated ellboe [31]. Vamegh asouli is the Depatment Chai and continental esouces distinguished Pofesso at the Depatment of Petoleum Engineeing, Univesity of Noth Dakota, USA. Vamegh has been a consulting Enginee to Schlumbege in the past and is no seving as an instucto fo thei NExT (Netok of Excellence in Taining) pogam, deliveing industy shot couses oldide in diffeent subjects including Petoleum Geomechanics, Dilling Eng, Hydaulic Factuing, Poe pessue estimation and Sand Contol. His eseach inteest about geomechanic and dilling. Fo solving equations A.1 to A.3 simplified assumptions have been made by eseaches; the gaphical pesentation of equations A.1 to A.3, hen p is shon in figue (A.)[3]. Fig A.. Analytical solution of equations A.1 fo Pf = constant (Elastic stess solution). Biogaphies Amin Tohidi is a PhD candidate at Amikabi Univesity of technology. He eceived B.S. and M.S. degees fom Shaif Univesity of Technology in Civil and Envionmental depatment. His eseach inteest is the petoleumelated ock mechanics. Ahmad Fahimifa is a Pofesso of Depatment of civil and envionmental engineeing, Amikabi Univesity of Technology. He is also