CALCULATION OF COLLAPSE PRESSURE IN SHALE GAS FORMATION AND THE INFLUENCE OF FORMATION ANISOTROPY L.Hu, J.Deng, F.Deng, H.Lin, C.Yan, Y.Li, H.Liu, W.Cao (Cina University of Petroleum) Sale gas formations feature anisotropy bot in mecanical properties and in strengt, so it is necessary to take into consideration te influence of te anisotropy on te calculation of collapse pressure wen drilling troug sale gas formations. Te sale gas formation is assumed to be a transversely isotropic material, and te modified Mor-Coulomb criterion is adopted to describe te failure of te sale gas formation wit consideration of te impact of te weak bedding plane. Ten, te influence of te elasticity modulus anisotropy and te Poisson s ratio anisotropy on collapse pressure is discussed. Te calculation result reveals tat te influence degree varies wit te deviation angle and te aimut angle of te wellbore. If te ratio of te oriontal elasticity modulus and te vertical elasticity modulus and tat of te vertical Poisson s ratio and te oriontal Poisson s ratio are smaller tan 2, te influence of te anisotropy is negligible since te degree of influence is smaller tan 7 percent compared wit te isotropic model. Te researc result can guide te calculation of collapse pressure wen drilling troug sale gas formations. Keywords: sale gas formation, transversely isotropic, collapse pressure, elasticity modulus, Poisson s ratio E-mail: dengfuceng128@163.com DOI: 10.5510/OGP20130400177 Introduction Collapse pressure is one of te most important parameters in drilling engineering, and it is te prerequisite of well structure design, selection of te proper drilling fluid density, te analysis of wellbore instability, etc. Te main object of te drilling work in oil and gas engineering is sedimentary rock, many of wic ave te caracteristic of anisotropy [1-4]. Researcers ave done a lot of work about te rock anisotropy. In 1987, B.S.Aadnoy [5] calculated te stress field around te wellbore in te transversely isotropic formation, based on te model of anisotropic elastic body establised by S.G.Leknitskii [6]. S.H.Ong [7] took into consideration te effect of te nonlinearity and te poroelasticity based on te Aadnoy B S s model. D.Gupta and M.Zaman [8] developed a program to calculate te fracture pressure and collapse pressure of anisotropic formation, and te result sows tat te fracture pressure and collapse pressure ave a closely relationsip wit te deviation angle of te wellbore and te formation, te anisotropic degree of te rock material, te in-situ stress state, etc. Te weak bedding plane (bedding plane) generally present in te non-isotropic sale gas formation, but previous researc about sale gas formation usually focused on te impact of te bedding plane and barely took into consideration bot of te anisotropy influence and te impact of te weak bedding plane [9-12]. Te frequent mecanical wellbore instability occurring in te sale gas formation indicates tat te calculation of collapse pressure of sale gas formation sould take into account bot of te anisotropy of te rock property and te weak bedding plane. Tis paper assumes te sale gas formation as a kind of transversely isotropic material, and calculates collapse pressure wit consideration of te impact of te weak bedding plane. 1. Te stress field distribution around te wellbore in a transversely isotropic formation Te analyed model and te in-situ stresses are sown in figure 1, and te formation is assumed to be oriontal and a transversely isotropic formation wit a vertical axis of symmetry. Tis paper focuses on te mecanical stability of te wellbore, so te interaction between te rock material and te drilling fluid is ignored. In order to calculate te stress field around te wellbore, te in-situ stress tensor sould be transformed from te coordinate system ( H,, ) to te coordinate system (x b, y b, b ) to gain te far field stress around te wellbore. Te transformation formula is given by eq.1. For transversely isotropic formation, te stiffness matrix can be expressed as eq.2. Te deducing process of te calculation of te Fig.1. Scematic sketc of te inclined boreole and te transversely isotropic formation RESERVOIR AND PETROLEUM ENGINEERING 49
stress field around te wellbore is very complicated, so in tis paper te autor just directly cites te result of previous researcers. According to te work of researcers suc as [5-8], te stress field around te wellbore provided is given as eq.3, and te detail process can be seen in literature ([7]. xo, xy, o x, o =,,, = ij yx o y o y o x, o y, o, o = L H L (1a) (1b) Were: L is te coordinate system transformation; = ab - as; a s is te angle from te nort direction counterclockwise to te direction of te maximum oriontal principle stress, ; a b is te angle from te nort direction counterclockwise to te projection of te axis of te wellbore on te oriontal plane, ; is te deviation angle of te wellbore,. 1 E E E 1 E E E 1 A E E Ev 1 1 2v (2) E E E 1 1 2v E E E 2(1 v ) E Were: E, are te elasticity modulus and te Poisson s ratio normal to te isotropic plane, tat is, at te vertical direction respectively; E, are te elasticity modulus and te Poisson s ratio in te isotropic plane, tat is, in te oriontal plane respectively; T cos cos cos sin sin L sin cos 0 sin cos sin cos cos 2 2 2Re x x 1 1 1 2 2 2 2Re y y 1 1 2 2 2Re xy xy 1 1 1 22 2 2Re x x 33 3 2Re y y 3 3 1 a31 x, a32 y, a33 (3a) 2 2 2Re1 1 1 2 x 2 2, 2Re y 1 1 2 2 (3b) Were: x, y,, xy, x, y are te stress components around te wellbore, MPa; x, y,, xy, x, y are te stress components of te far field stress of te wellbore, MPa; Te oter components are te stress caused by te anisotropy of te formation and te wellbore pressure p W, and determined by te location of te researced point, te stiffness matrix of te formation, and te wellbore pressure; Te detail calculation process can be found in te literature [7]. In te analysis of te mecanical wellbore stability, te principle stresses are often used and gained from te stress components of te researced point. According to elastic mecanics, te principle stresses are te eigenvalues of te stress tensor wic is sown in eq.4 and te direction cosine of te principle stress is te eigenvectors of te stress tensor. So, if te in-situ stresses, te elasticity modulus and te Poisson s ratio, te parameter of te wellbore and te wellbore pressure are known, te stress field around te wellbore can be calculated wit te metod discussed above. = ij x xy x yx y y x y In order to illustrate te influence of te anisotropy on te stress distribution around te wellbore, te maximum principle stress, te minimum principle stress of te anisotropic formation and te isotropic formation and te percent ratio error between tem are sown in figures 2 and 3 respectively wit te parameters tabulated in Table 1 In te table, R is te radius of te wellbore and te oters are te same as mentioned above. Te percent ratio error is defined in eq.5. Te results reveals tat te anisotropy of te formation as a great impact on te stress distribution around te wellbore, and tat Isotropic formation Parameters used in te example Table 1 Transversely isotropic formation P W, MPa 32, MPa 66 P W, MPa 32, MPa 66 a b, 0 E, GPa 12 a b, 0 E, GPa 36 a s, 45 E, GPa 12 a s, 45 E, GPa 12, 45 0.25, 45 0.125 H, MPa 55 0.25 H, MPa 55 0.25, MPa 46.2 R, m 0.108, MPa 46.2 R, m 0.108 (4) 50 RESERVOIR AND PETROLEUM ENGINEERING
and te transversely isotropic formation and teir percent ratio error Fig.3. Te minimum principle stress of te isotropic formation and te transversely isotropic formation and teir percent ratio error it as a greater influence on te maximum principle stress tan on te minimum principle stress, wic indicates tat te anisotropy sould be taken into consideration wen calculating te wellbore stress distribution. V V PRE= Anis Iso 100 (5) VIso Were, PRE is te percent ratio error of a certain parameter, %; V Anis, V Iso are te value of te anisotropy model and isotropy model respectively. of te transversely isotropic formation Te rock mecanical property is anoter important factor for te formation stability. To determine te proper criteria of te rock failure in sale gas formation, te compressive strengt was measured on specimens cored at different angles wit respect to te normal of te sale bedding plane under different confining pressures. Te suitable failure criteria for laminated formation is te continuously variable coesion strengt criterion developed by McLamore [3-14], wic is expressed in eq.6. 1 3 2( tan( )) 0 3 2 tan( ) tan ( ) 1 (6a) m o 0=A 1B1[cos 2( 1)] (0 1) n o (6b) 0=A 2 B2[cos 2( 1)] ( 1 90 ) Were, 1, 3 are te maximum and te minimum principle stress respectively, MPa; A 1, B 1, A 2, B 2, m, n are te constants tat describe te beave of te 0 over te range 0 o 1 and 1 90 o ; is te internal friction angle; According to te experiment result, te constant parameters in eq.6 can be gained wit regression metod, and te regression result is: A1 = -18.2, A2 = -15.8, B1 = -9.88, B2 = -7.94, 1 = 47, = 33.79, m = 4, n = 4. Te measured data and te regression surface are sown in figure 4, from wic it can be concluded tat wen te angle between te axial stress and te isotropic plane is between 50 and 60, te compressive strengt is very low. In tis paper, te angle witin tis range is called te weak plane angle. 3. Collapse pressure calculation Wen te pressure in te wellbore, tat is te wellbore pressure, is too low, te wellbore wall may collapse, and te critical wellbore pressure is called collapse pressure. In tis paper, collapse pressure is calculated using te criteria described in eq.6. Wat sould be noticed is tat te stresses in eq.6 are effective stress, so te effective stress around te wellbore sould be calculated at first. Te effective stress can be gained by Biot s effective stress teory: ij ij Ppij Were `ij is te effective stress component around te wellbore; is te Biot s coefficient; P p is te pore (6) RESERVOIR AND PETROLEUM ENGINEERING 51
Te compressive strengt, MPa Te confining pressure, MPa Te operation angle, MPa te samples of te sale formation in Sicuan Basin, Cina. pressure, and ij is te Kronecker symbol. Te analytic solution of collapse pressure is very difficult to get, so numerical calculation metod is applied to determine collapse pressure. A brief describe of te process is as following: 1) Get constant parameters suc as te in-situ stress, te pore pressure, te deviation angle and te aimut angle of te wellbore, te parameters in te failure criteria eq.5, etc by experiments or referencing oter researc result; 2) Assume a very small wellbore pressure (small enoug to make sure te wellbore will collapse) and ten calculate te effective principle stress around te wellbore; 3) Use te criteria in eq.5 to ceck if te wellbore will collapse or not; 4) If te well will collapse, ten increase te wellbore pressure by a proper value; if te well will not collapse, ten te current wellbore pressure could be regarded as collapse pressure and te calculation is accomplised; 5) Calculate te effective principle stress around te wellbore and go to te step 3. and te Poisson s ratio on collapse pressure Te focus of tis paper is to analye te influence of te anisotropy on collapse pressure, so some of te parameters sould be fixed and some sould be variable. Te parameters used for analysis are tabulated in table 2. 4.1 Te influence of te elasticity modulus on collapse pressure To analye te influence of te elasticity modulus on collapse pressure, te elasticity modulus in te isotropic plane is te only variable and te value of te oters is fixed as in te table 2. Te anisotropic coefficient of te elasticity modulus is defined as te ratio of te elasticity modulus in te isotropic plane and tat normal to te isotropic plan, tat is, K E = E /E. Te variation of collapse pressure is calculated wit different value of K E. Figure 5 illustrates collapse pressure of anisotropy model and te isotropy model and te percent ratio error versus different deviation angle and aimut angle of te wellbore wen K E is 3.4. Form figure 5, it is clear tat te influence degree of te elasticity modulus on collapse pressure varies wit te aimut angle and te deviation angle. Compared wit isotropy model, wen te wellbore aimut is equal to te direction of te maximum oriontal stress and te deviation angle is close to te weak plane angle, te influence of te elasticity modulus reaces te peak and collapse pressure of te anisotropy model is 15 percent greater tan tat of te isotropy model. It 52 RESERVOIR AND PETROLEUM ENGINEERING
te percent ratio error versus different deviation angle and aimut angle wen K E Te parameters used for analysis Te well vertical dept, H 3200 m Te pore pressure, Pp 46.3 MPa Te direction of te maximum oriontal principle stress, s N45 W Te maximum oriontal principle stress, H 55 MPa Te minimum oriontal principle stress, 46.2 MPa Te vertical principle stress, 66 MPa Te elasticity modulus normal to te isotropic plane, E 12GPa Te elasticity modulus in 36 GPa te isotropic plane, E (variable) Te Poisson s ratio normal to te isotropic plane, 0.25 Te Poisson s ratio in te isotropic plane, 0.125 (variable) Te radius of te wellbore, R 0.108 m Te Biot s coefficient, 0.7 indicates tat if te anisotropy of te formation is ignored, collapse pressure will be underestimated by 15 percent, wic may lead to serious wellbore collapsing problems in te drilling operation. Figure 6 sows te relationsip of te elasticity modulus and te percent ratio error at te worst situation. It is obvious tat te anisotropic coefficient of te elasticity modulus and te percent ratio error as a linear relationsip and tat te percent ratio error increase wit te increasing of te elasticity modulus. 4.2 Te influence of te Poisson s ratio on collapse pressure To analye te influence of te Poisson s ratio on collapse pressure, te Poisson s ratio in te Percent ratio error 18.0 16.0 14.0 12.0 10.0 8.0 6.0 4.0 2.0 0.0 isotropic plane is te only variable and te value of te oters is fixed as in te table 2. Te anisotropic coefficient of te Poisson s ratio is defined as te ratio of te Poisson s ratio in te isotropic plane and tat normal to te isotropic plan, tat is, K = /. Te variation of collapse pressure is calculated wit different value of K. Figure 7 illustrates te collapse pressure of anisotropy model and te isotropy model and te percent ratio error versus different deviation angle and aimut angle of te wellbore wen K is 2.4. Form figure 7, it is obvious tat te degree of te impact of te Poisson s ratio on collapse pressure also varies wit te aimut angle and te deviation angle. Compared wit isotropy model, wen te wellbore aimut is equal to te direction of te maximum oriontal stress and te deviation angle is close to te weak plane angle, te influence of te Poisson s ratio reaces maximum value and collapse pressure of te anisotropy model is 4.2 percent greater tan tat of te isotropy model. Figure 8 sows te relationsip of te Poisson s ratio and te percent ratio error of te worst situation. It is obvious tat wen te anisotropic coefficient of Poisson s ratio increases te percent ratio error of collapse pressure will increase. However, compared wit 1 1.5 2 2.5 3 3.5 Te anisotropic coefficient of te elasticity modulus KE modulus on collapse pressure RESERVOIR AND PETROLEUM ENGINEERING 53
Percent ratio error 4.5 4 3.5 3 2.5 2 1.5 1 1 1.2 1.4 1.6 1.8 2 2.2 2.4 Te anisotropic coefficient of te Poisson s ratio Kv Fig.8. Te influence of te Poisson s ratio on collapse pressure different value of K E and K and te result is sown in figure 9. Te figure reveals tat te impact of te elasticity modulus on collapse pressure is muc iger tan tat of te Poisson s ratio and tat in tis situation te isotropy model will underestimate collapse pressure wic may lead to te improper use of drilling fluid wit low density. Figure 9 also indicates tat te biggest underestimate of collapse pressure is close to 15 percent and tat if te anisotropy degree of te sale gas formation is witin a low level (te anisotropic coefficient of te elasticity modulus and te Poisson s ratio is less tan 2), te percent ratio error of te isotropy model and te transversely model is smaller tan 7 percent. te anisotropy of elasticity modulus, te anisotropy of te Poisson s ratio as a muc less impact on collapse pressure. 4.3 Te combined effect of te elasticity modulus and te Poisson s ratio on collapse pressure In te most case, anisotropy of te elasticity modulus and tat of te Poisson s ratio present are concurrent in te sale gas formation. So it is necessary to discuss te combined effect of tem. According to te calculation result above, it is found tat wen te wellbore aimut is equal to te direction of te maximum oriontal stress and te deviation angle is close to te weak plane angle, te influence of te elasticity modulus or te Poisson s ratio reaces maximum value. So, in tis section, te wellbore aimut is assumed to be equal to te direction of te maximum oriontal stress and te deviation angle is assumed to be 65 and te oter parameters are constants. Te percent ratio error of te isotropy model compared wit te transversely model is calculated wit Fig.9. Te combined effect of te elasticity modulus and te Poisson s ratio on collapse pressure 54 RESERVOIR AND PETROLEUM ENGINEERING
1. Te anisotropy of te sale gas formation as a great impact on te stress distribution around te wellbore, so te anisotropy sould be taken into consideration wen calculating te wellbore stress distribution. 2. Te degree of te influence of te elasticity modulus or te Poisson s ratio on collapse pressure varies wit te aimut angle and te deviation angle. Compared wit isotropy model, wen te wellbore aimut is equal to te direction of te maximum oriontal stress and te deviation angle is close to te weak plane angle, te influence of te elasticity modulus or te Poisson s ratio reaces te peak. Te impact of te elasticity modulus on collapse pressure is muc iger tan tat of te Poisson s ratio. Collapse pressure of te anisotropy model is 15 percent greater tan tat of te isotropy model wen te anisotropy coefficient of elasticity modulus is 3.4 and tat value is 4.2 percent wen te anisotropy coefficient of Poisson s ratio is 2.4. Te percent ratio error increases wit te increasing of anisotropic coefficient of te elasticity modulus or te Poisson s ratio. 3. If te anisotropy degree of te sale gas formation is witin a low level (bot of te anisotropic coefficient of te elasticity modulus and te Poisson s ratio are less tan 2), te percent ratio error of te isotropy model and te transversely model is smaller tan 7 percent wic may could be ignored. References 1. L.Xianui. Rock mecanical properties. Beijing: Coal industry publiser, 1983. 2. A.Ou. Tectonic Stress Field. Beijing: Seismological Press. 1992. 3. L.Dongyan, Z.Kesan. A study of strengt anisotropy of rock mass containing intermittent joints //Cinese Journal of Rock Mecanics and Engineering. -1998. -Vol.17(4). -P.366 371 4. G.Worotnicki. CSIRO Triaxial stress measurement cell, in compreensive rock engineering /edited by J.A.Houdson. Oxford: Pergamon, 1993. 5. B.S.Aadnoy. Continuum mecanics analysis of te stability of boreoles in anisotropic rock formations. Norway: Norwegian Institute of Tecnology, University of Trondeim, 1987. 6. S.G.Leknitskii. Teory of elasticity of an anisotropic body. M.: Mir Publisers, 1981. 7. S.H.Ong. Boreole stability. Norman OK: University of Oklaoma, 1994. 8. D.Gupta, M.Zaman. Stability of boreoles in a geologic medium including te effects of anisotropy //Applied Matematics and Mecanics. -1999. -Vol.20. -No8. -P.837-866. 9. D.Okland, J.M.Cook. Bedding -related boreole instability in ig -angle wells. //SPE 47285. -1988 10. J.Yan, C.Mian, L.Gongui et al. Wellbore stability analysis of inclined wells in weak plane formations //Journal of te University of Petroleum (Cina). Natural Science Edition. -1999. -Vol.23. -No4. -P.33-35. 11. L.Xiangjun, C.Yijian, X.Yong. Effect of weak plane dip angle and dip aimut angle on wellbore stability //Journal of Soutwest Petroleum Institute. -2001. -Vol.23. -No6. -P.12-13. 12. M.E.Cenevert, C.Gatlin. Mecanical anisotropies of laminated sedimentary //SPE 890, Annual Fall Meeting of SPE in Houston, 1964. 13. J.C.Jaeger, N.G.W.Cook, R.Zimmerman. Fundamentals of rock mecanics. Oxford: Blackwell, 2007. 14. R.T.McLamore. A strengt criterion for anisotropic rocks based upon experimental observations //SPE 1721. -1967. RESERVOIR AND PETROLEUM ENGINEERING 55
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