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Cement failure probability analysis in water injection well
Journal of Petroleum Science and Engineering 107 (2013) 45–49
Contents lists available at SciVerse ScienceDirect
Journal of Petroleum Science and Engineering journal homepage: www.elsevier.com/locate/petrol
Cement failure probability analysis in water injection well Zhaoguang Yuan a,n, Paolo Gardoni b, Jerome Schubert a, Catalin Teodoriu c a b c
Department of Petroleum Engineering, Texas A&M University, TX, USA Department of Civil and Environmental Engineering, University of Illinois at Urbana-Champaign, IL, USA Department of Petroleum Engineering, TU Clausthal, Germany
art ic l e i nf o
a b s t r a c t
Article history: Received 1 November 2011 Accepted 26 April 2013 Available online 14 May 2013
In the petroleum industry, the salt water was produced along with oil and gas. Due to the environment issue, the oil companies choose, the cheap and convenient way, to inject salt water into the reservoir. Because of the cycle load that injects water periodically, the cement fatigue failure should be considered in cement design. In this paper, the cement compressive failure, shear failure, tensile failure and fatigue failure modes were considered with different bottomhole pressures. The uncertainties of casing mechanical properties, cement mechanical properties, formation mechanical properties and wellbore geometry were also considered in the study. Based on the analysis, it is shown that within the cement compressive strength requirement, the wellbore service life can be increased by enhancing cement plastic behavior. The elastic cement with higher plasticity showed better behavior than brittle cement though the brittle cement has higher compressive strength. & 2013 Elsevier B.V. All rights reserved.
Keywords: cement failure injection wells probability analysis random variables
1. Introduction The main objective of cementing operation is to provide zonal isolation of the formations which have been penetrated by the wellbore and provide support to the casing. Even if the initial cement was properly placed and initially provides a competent hydraulic seal, formation stresses and large changes in wellbore pressure and temperature from a variety of common well events could easily crack a cement sheath during service of the well. In water injection wells, water was injected into the reservoir formation through the well periodically. There were injection pressure load cycles during the service of the well. The load cycles can cause cement failure if the cement was not designed properly. The theory of thermo-poro-elasticity was used to predict the various modes of cement failure. The cement is assumed to behave as an elastic-brittle material (Thiercelin et al., 1998). A stressmodeling and risk analysis methodology was proposed using a complementary suite of software tools (Laidler et al., 2007). The cement mechanical properties were extensively studied under different conditions (Philippacopoulos and Berndt, 2001). However, there is seldom research on cement fatigue failure in the field of petroleum engineering. While, in the field of mechanical engineering, there are some researches on metal fatigue failure (Placido et al., 1997; Teodoriu et al., 2008; Wu et al., 2008). And some people have done research on cement and concrete fatigue failure in the area of civil n
Corresponding author. Tel.: +1 979 862 1195. E-mail address: [email protected] (Z. Yuan).
0920-4105/$ - see front matter & 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.petrol.2013.04.011
engineering. There are some experiments on the fatigue failure of portland cement concrete and paving concrete (Antrim, 1965; Joshi et al., 2004). In medical engineering, the probabilistic method was used to analyze the risk of failure of a cemented femoral component of a total hip replacement system and cement mantle of acetabular replacements (Nicolella et al., 2000; Nikolaus et al., 2007). In this study, four failure modes, compressive failure, shear failure, tensile failure and fatigue failure were considered. There are two main objectives of this study. The first objective is to quantify the probability of failure of the cement in terms of specific failure modes. Model parameters such as the applied loads and material properties are modeled as random variables to account for the uncertainty and variability. The second objective is to identify the variables that contribute most to the probability of failure. 2. Methods Two cement types were considered in this study. One is the unmodified Class G/40% silica flour (brittle cement) and the other is elastic cement. Risk of failure was calculated based on the performance function of the type: gðXÞ ¼ RðXÞ−SðXÞ
ð1Þ
where RðXÞ is a random function describing the “resistance” or strength of the component or constituent, SðXÞ is the response of the structure, also a random variable, and X is the vector of
46
Z. Yuan et al. / Journal of Petroleum Science and Engineering 107 (2013) 45–49
Nomenclature c E Nf Jc kfe pe pw qw
cement cohesion, MPa Young's modulus, GPa load cycles, dimensionless specific heat of water at constant volume, J/kg K effective thermal conductivity of the formation, W/m K reservoir pressure, Pa wellbore pressure, Pa water injection rate, m3/s m
random variables. A negative or zero gðXÞrepresents a failure event. pf is the probability of failure defined as pf ¼ pðgðXÞ ≤0Þ
ð2Þ
r re srr sѲѲ srѲ ss st ω ρw X
radial co-ordinate, m boundary radius, m the radial stress, Pa hoop stress, Pa shear stress, Pa compressive strength, Pa tensile strength, MPa cement friction angle, radian density of injection fluid, kg/m3 vector of random variable
For elastic cement; g 3 ðNf ; s; ss Þ ¼ −LogðN f Þ−0:109
s þ 11:6 ss
ð12Þ
4. Tensile failure g 4 ðstensile ; st Þ ¼ −stensile þ st
ð13Þ
2.1. Solution to cement stresses Stresses for plane linear elasticity problems, with reference to a polar co-ordinate system Oðr; θÞ, are given in the following equation in terms of potential ΦðzÞ and Ψ ðzÞ of the complex variable z ¼ x þ iy: srr þ sθθ ¼ 2½ΦðzÞ þ ΦðzÞ
ð4Þ
z sθθ −srr þ 2isrθ ¼ 2 ½zΦ0 ðzÞ þ Ψ ðzÞ z
ð5Þ
3. Field applications The well is located in Cameron, Louisiana. As shown in Fig. 1, Well depth 1829 m; formation pore pressure gradient, 0.01074 MPa/m; overburden pressure gradient, 0.02262 MPa/m; formation horizontal principal stress, 30 MPa, 33 MPa; fracture
According to the boundary condition, the solution is expressed as Ψ ðkÞ ðzÞ ¼ DðkÞ þ
F ðkÞ GðkÞ þ 4 z2 z
ð6Þ
ΦðkÞ ðzÞ ¼ AðkÞ þ
Bð1Þ þ C ðkÞ z2 z2
ð7Þ
k ¼1 ,2, 3. 1 stands for casing; 2 stands for cement; 3 stands for formation. The partial differential equation (Fagley et al., 1982) used to describe the wellbore heat transfer is 1∂ ∂T ∂ ∂T q ρ C w ∂T rkf e þ kf e − w w r∂ ∂r ∂z ∂r ∂r 2πr qw pw −pe ∂T ¼ ðρCÞe ð8Þ þ ∂r 2πr 2 J c ln r e =r w From Eqs. (4)–(8), given bottomhole pressure, formation temperature, formation pressure and material properties, the radial stress srr , hoop stress sθθ and shear srθ can be solved. The solutions to A, B, C D, F, and G can be found (Atkinson and Eftaxiopoulos, 1996). 2.2. Cement failure modes 1. Compressive failure ð9Þ
g 1 ðs; ss Þ ¼ −s þ ss
2. Shear failure ð10Þ
g 2 ðτ; c; ss ; ωÞ ¼ −τ þ c þ ss tan ω
3. Fatigue failure For brittle cement; g 3 ðN f ; s; ss Þ ¼ −LogðN f Þ−0:082
s þ 8:48 ss
ð11Þ
Fig. 1. Wellbore geometry.
Z. Yuan et al. / Journal of Petroleum Science and Engineering 107 (2013) 45–49
gradient, 0.0185 MPa/m; casing inside radius, 11.22 cm; casing outside radius, 12.22 cm; and wellbore radius 15.56 cm. The confining pressure and compressive strengths relationship is expressed as follows: 2c cos ω 1 þ sin ω ð14Þ s ¼ sconf ining þ 1 þ sin ω 1−sin ω So, at the confining pressure of 19.64 MPa, compressive strength for brittle cement is 117.54 MPa, and compressive strength for elastic cement is 71.15 MPa. 3.1. Random variables and distribution Cement mechanical properties, casing mechanical properties, formation mechanical properties, wellbore geometry and the bottomhole pressure load are the random variables. Tables 1–3 show the random variables for the brittle cement. Tables 4–6 show the random variables for the elastic cement.
Fig. 2 shows that without cycle load, the cement tensile failure probability reaches one at the bottomhole pressure of 140 MPa. Table 1 Brittle cement random variables. Random variables
Mean
S.D.
Distribution type
Cement cohesion (c, MPa) Cement compressive strength (ss, MPa) Cement friction angle (ω, radian) Cement tensile strength ( st, MPa) Cement Young's modulus (E, GPa) Cement Poisson's ratio Formation Young's modulus (E, GPa) Formation Poisson's ratio Casing Young's modulus (E, GPa) Casing Poisson's ratio Casing ID (cm) Wellbore diameter (cm) Wellbore temperature (1C)
18.05 117.54 0.5044 5.136 9.93 0.2 7.2 0.25 204 0.3 22.44 31.12 54
0.903 11 0.001 0.0816 0.15 0.05 0.5 0.05 5 0.02 0.5 1.5 5
Lognormal Lognormal Lognormal Lognormal Lognormal Lognormal Lognormal Lognormal Lognormal Lognormal Lognormal Lognormal Lognormal
cycles cycles cycles cycles cycles cycles
(Nf) (Nf) (Nf) (Nf) (Nf) (Nf)
Bottomhole Bottomhole Bottomhole Bottomhole Bottomhole Bottomhole Bottomhole Bottomhole
pressure pressure pressure pressure pressure pressure pressure pressure
(MPa) (MPa) (MPa) (MPa) (MPa) (MPa) (MPa) (MPa)
Mean
S.D.
Distribution type
Cement cohesion (c, MPa) Cement compressive strength (ss, MPa) Cement friction angle (ω, radian) Cement tensile strength (st, MPa) Cement Young's modulus (E, GPa) Cement Poisson's ratio Formation Young's modulus (E, GPa) Formation Poisson's ratio Casing Young's modulus (E, GPa) Casing Poisson's ratio Casing ID (cm) Wellbore diameter (cm) Wellbore temperature (1C)
10.6 71.15 0.356 4.558 6.53 0.3 7.2 0.3 204 0.3 22.44 31.12 54
0.53 6.84 0.001 0.5918 0.1 0.08 0.5 0.08 5 0.02 0.5 1.5 5
Lognormal Lognormal Lognormal Lognormal Lognormal Lognormal Lognormal Lognormal Lognormal Lognormal Lognormal Lognormal Lognormal
Load Load Load Load Load
cycles cycles cycles cycles cycles
(Nf) (Nf) (Nf) (Nf) (Nf)
Mean
S.D.
Distribution type
60,000 80,000 100,000 120,000 130,000
6000 8000 10,000 12,000 13,000
Lognormal Lognormal Lognormal Lognormal Lognormal
Table 6 Elastic cement random variables.
Bottomhole Bottomhole Bottomhole Bottomhole Bottomhole Bottomhole Bottomhole
pressure pressure pressure pressure pressure pressure pressure
(MPa) (MPa) (MPa) (MPa) (MPa) (MPa) (MPa)
Mean
S.D.
Distribution type
25.03 43.47 93 133 143 153 163
2 4 9 13 14 15 16
Lognormal Lognormal Lognormal Lognormal Lognormal Lognormal Lognormal
Mean
S.D.
Distribution type
3500 4000 4500 5000 5500 6000
350 400 450 500 550 600
Lognormal Lognormal Lognormal Lognormal Lognormal Lognormal
Table 3 Brittle cement random variables. Random variables
Random variables
While the tensile failure probability is high at the bottomhole pressure beyond 120 MPa, the compressive failure and shear failure probability is nearly zero. Fig. 3 shows that for different bottomhole pressure, the fatigue failure behavior is almost the same. As the load cycles increases from 3500 to 6000, the fatigue failure probability increases from zero to one. From Fig. 4, as the bottomhole pressure is lower than 127 MPa, the system failure behavior is almost the same as fatigue failure behavior in Fig. 3. However, when the bottomhole pressure increases to 127 MPa, the system failure probability is much higher than the fatigue failure probability. The reason is at this time the tensile failure probability is very high. During the normal operation, the bottomhole pressure is 25 MPa, so, the load cycles should be prevented beyond 3500 in case of fatigue failure.
Table 2 Brittle cement random variables.
Load Load Load Load Load Load
Table 4 Elastic cement random variables.
Table 5 Elastic cement random variables.
3.2. Results of brittle cement
Random variables
47
Mean
S.D.
Distribution type
3.3. Results of elastic cement
25.03 43.47 93 113 123 127 133 143
2 4 9 11 12 12 13 14
Lognormal Lognormal Lognormal Lognormal Lognormal Lognormal Lognormal Lognormal
The results of elastic cement show the same failure trend as the results of brittle cement. From Figs. 5 and 6, the cement tensile failure probability reaches one at the bottomhole pressure of 165 MPa. As the load cycles reach 80,000, the fatigue failure probability begins to increase above zero. Fig. 7 shows that the system failure probability is different from the fatigue failure probability because the tensile failure probability is very high at
Z. Yuan et al. / Journal of Petroleum Science and Engineering 107 (2013) 45–49
1.0
1.0
0.8
0.8 Failure Probability
Failure Probability
48
Compressive Failure
0.6
Tensile Failure Shear Failure
0.4
0.6
Tensile Failure Shear Failure
0.4 0.2
0.2 0.0
Compressive Failure
0.0
20
40
60 80 100 120 Bottom Hole Pressure, MPa
140
160
Fig. 2. The effect of bottomhole pressure on brittle cement failure probability.
20
40
60 80 100 120 140 Bottom Hole Pressure, MPa
160
180
Fig. 5. The effect of bottomhole pressure on elastic cement failure probability.
1.00
p=25.03 MPa p=43.47 MPa p=113 MPa p=127 MPa
0.40
p=43.47 MPa
0.60
p=143 MPa p=153 MPa
0.40 0.20
0
0 13
0,
00
00 0,
0 00 0,
12
0 11
00
00 0, 10
,0 90
00 ,0 80
00 ,0 60
00
0.00
0.20
0.00 3.000
p=25.03 MPa
,0
0.60
0.80
70
0.80
Fatigue Failure Probability
Fatigue Failure Probability
1.00
Load Cycles
4.000 5.000 Load Cycles
6.000
Fig. 6. The effect of load cycles on elastic cement fatigue failure probability.
Fig. 3. The effect of load cycles of brittle cement fatigue failure probability.
1.00
0.80
0.60 p=25.03 MPa p=43.47 MPa
0.80
0.60
0.40
p=25.03 MPa p=43.47 MPa p=143 MPa
0.20
p=153 MPa
p=113 MPa p=127 MPa
0 00 0, 13
0,
00
0
0 00 0,
0 00 0,
00 ,0 90
00
00 ,0 80
11
6.000
10
4.000 5.000 Load Cycles
,0
60
0.00 3.000
,0
00
0.00
70
0.20
12
0.40
System Failure Probability
System Failure Probability
1.00
Load Cycles Fig. 7. The effect of load cycles on elastic cement system failure probability.
Fig. 4. The effect of load cycles on brittle cement system failure probability.
4. Fatigue failure sensitivity measure the bottomhole pressure of 153 MPa. In the normal operation, the bottomhole pressure is 25 MPa, so, the load cycles should be prevented beyond 80,000 in case of fatigue failure.
For brittle cement, at the bottomhole pressure of p ¼113 MPa, load cycles Nf ¼ 4000, the sensitivities of the reliability index are
Z. Yuan et al. / Journal of Petroleum Science and Engineering 107 (2013) 45–49
Table 7 Sensitivities of the reliability index with respect to distribution parameters (brittle cement). Variable
Mean
Standard deviation
Load cycles (Nf) Compressive stress (s) Compressive strength (ss)
−2.12032e−003 −4.13361e−009 1.32974e−009
−3.85465e−003 −1.18237e−011 −4.10930e−011
Table 8 Sensitivities of the reliability index with respect to distribution parameters (elastic cement). Variable
Mean
Standard deviation
Load cycles (Nf) Compressive stress (s) Compressive strength (ss)
−9.66630e−005 −1.19195e−008 8.38311e−009
−2.84587e−004 −5.69055e−011 −1.10395e−009
49
sensitivity measure, the compressive strength has little influence on the cement capacity because the value of compressive strength sensitivity is very small. 4. Within the cement compressive strength requirement, the wellbore service life can be increased by enhancing cement plastic behavior. The elastic cement shows better behavior than brittle cement though brittle cement has much higher compressive strength.
Acknowledgments This research was funded by Research Chevron Center for Well Construction and Production of Crisman Institute for Petroleum in Texas A&M Petroleum Engineering Department. We gratefully acknowledge these supports.
References shown in Table 7. For elastic cement, at the bottomhole pressure of p ¼143 MPa, load cycles Nf ¼80,000, the sensitivities of the reliability index are shown in Table 8. For both cement systems, the cement compressive strength tends to increase the cement capacity, however, the compressive strength has little influence on the cement capacity because the value of compressive strength sensitivity is very small. 5. Conclusions 1. For water injection wells, this paper proposed a probabilistic method to predict the wellbore service life and failure probability based on casing mechanical properties, cement mechanical properties, formation mechanical properties and wellbore geometry. Fatigue failure is studied together with the other three cement failure modes: compressive failure, tensile failure and shear failure 2. Both cement systems show the same failure characteristic. Without cycle load, comparing to compressive failure and shear failure, tensile failure is the most likely occurring failure mode. At the same bottomhole pressure, the elastic cement can hold more load cycle than the brittle cement. During the normal operation, the load cycles should be prevented beyond 80,000 in case of fatigue failure for elastic cement and 3500 for brittle cement. 3. It is little effect to improve the cement fatigue behavior by increasing the cement compressive strength. In the fatigue failure
Antrim, J.D., 1965. A Study of The Mechanism of Fatigue in Cement Paste and Plain Concrete. Purdue University, Research report no. 6. Atkinson, C., Eftaxiopoulos, D.A., 1996. A plane model for the stress field around an inclined, cased and cemented wellbore. Int. J. Numer. Anal. Methods Geomech. 20, 549–569. Fagley, John, Fogler, H. Scott, Davenport, C. Brent, 1982. An improved simulation for interpreting temperature logs in water injection wells. SPE J. 22 (5), 709–718. Joao, C.R. Placido, Ademar, P. Jr., Luiz, A.B.F. Paulo, Ilson P. Pasqualino. 1997. Stress analysis of casing string submitted to cyclic steam injection, SPE Paper 38978, Fifth Latin American and Caribbean Petroleum Engineering Conference and Exhibition, Rio de Janeiro, Brazil, 30 August–3 September. Joshi, Y.P., Dhang, N., Pandey, B.B., 2004. Probability analysis of fatigue of paving concrete. J. Inst. Eng., 85 (11), 163–168. Laidler, A., Taoutaou, S., Johnson, C.R., 2007. A risk analysis approach using stress analysis models to design for cement sheath integrity in a multilateral well, in: IPTC Paper 11059, International Petroleum Technology Conference, Dubai, U.A. E. 4–6 December. Nicolella, Daniel P., Thacker, Ben H., Hamid Katoozian., Davy, Dwight T., 2000. Probabilistic risk analysis of a cemented hip implant. J. Orthop. Res. Nikolaus, P. Zant, Charles, K.Y. Wong, Tong, Jie, 2007. Fatigue failure in the cement mantle of a simplified acetabular replacement model. Int. J. Fatigue 29, 1245–1252. Philippacopoulos, A.J., Berndt, M.L., 2001. Mechanical property issues for geothermal well cements. Geothermal Resour. Counc. Trans. 25, 119–224. Teodoriu, C., Ulmanu, V., Badicioiu, M. 2008. Casing fatigue life prediction using local stress concept: theoretical and experimental results, in: SPE Paper 110785, SPE Western Regional and Pacific Section AAPG Joint Meeting, Bakersfield, California, USA, 31 March–2 April. Thiercelin, M., Dargaud, B., Baret, J.F., Rodriguez, W.J., 1998. Cement design based on cement mechanical response, in: Proceedings of SPE 52890, SPE Annual Technical Conference and Exhibition, SanAntonio, Texas, USA, 5–8 October. Wu, Jiang, Knauss, Martin E., Kritzler, Torsten, 2008. Casing failure in cyclic steam injection wells, in: IADC/SPE Paper 114231, IADC/SPE Asia Pacific Drilling Technology Conference and Exhibition, Jakarta, Indonesia, 25–27 August.
Contents lists available at SciVerse ScienceDirect
Journal of Petroleum Science and Engineering journal homepage: www.elsevier.com/locate/petrol
Cement failure probability analysis in water injection well Zhaoguang Yuan a,n, Paolo Gardoni b, Jerome Schubert a, Catalin Teodoriu c a b c
Department of Petroleum Engineering, Texas A&M University, TX, USA Department of Civil and Environmental Engineering, University of Illinois at Urbana-Champaign, IL, USA Department of Petroleum Engineering, TU Clausthal, Germany
art ic l e i nf o
a b s t r a c t
Article history: Received 1 November 2011 Accepted 26 April 2013 Available online 14 May 2013
In the petroleum industry, the salt water was produced along with oil and gas. Due to the environment issue, the oil companies choose, the cheap and convenient way, to inject salt water into the reservoir. Because of the cycle load that injects water periodically, the cement fatigue failure should be considered in cement design. In this paper, the cement compressive failure, shear failure, tensile failure and fatigue failure modes were considered with different bottomhole pressures. The uncertainties of casing mechanical properties, cement mechanical properties, formation mechanical properties and wellbore geometry were also considered in the study. Based on the analysis, it is shown that within the cement compressive strength requirement, the wellbore service life can be increased by enhancing cement plastic behavior. The elastic cement with higher plasticity showed better behavior than brittle cement though the brittle cement has higher compressive strength. & 2013 Elsevier B.V. All rights reserved.
Keywords: cement failure injection wells probability analysis random variables
1. Introduction The main objective of cementing operation is to provide zonal isolation of the formations which have been penetrated by the wellbore and provide support to the casing. Even if the initial cement was properly placed and initially provides a competent hydraulic seal, formation stresses and large changes in wellbore pressure and temperature from a variety of common well events could easily crack a cement sheath during service of the well. In water injection wells, water was injected into the reservoir formation through the well periodically. There were injection pressure load cycles during the service of the well. The load cycles can cause cement failure if the cement was not designed properly. The theory of thermo-poro-elasticity was used to predict the various modes of cement failure. The cement is assumed to behave as an elastic-brittle material (Thiercelin et al., 1998). A stressmodeling and risk analysis methodology was proposed using a complementary suite of software tools (Laidler et al., 2007). The cement mechanical properties were extensively studied under different conditions (Philippacopoulos and Berndt, 2001). However, there is seldom research on cement fatigue failure in the field of petroleum engineering. While, in the field of mechanical engineering, there are some researches on metal fatigue failure (Placido et al., 1997; Teodoriu et al., 2008; Wu et al., 2008). And some people have done research on cement and concrete fatigue failure in the area of civil n
Corresponding author. Tel.: +1 979 862 1195. E-mail address: [email protected] (Z. Yuan).
0920-4105/$ - see front matter & 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.petrol.2013.04.011
engineering. There are some experiments on the fatigue failure of portland cement concrete and paving concrete (Antrim, 1965; Joshi et al., 2004). In medical engineering, the probabilistic method was used to analyze the risk of failure of a cemented femoral component of a total hip replacement system and cement mantle of acetabular replacements (Nicolella et al., 2000; Nikolaus et al., 2007). In this study, four failure modes, compressive failure, shear failure, tensile failure and fatigue failure were considered. There are two main objectives of this study. The first objective is to quantify the probability of failure of the cement in terms of specific failure modes. Model parameters such as the applied loads and material properties are modeled as random variables to account for the uncertainty and variability. The second objective is to identify the variables that contribute most to the probability of failure. 2. Methods Two cement types were considered in this study. One is the unmodified Class G/40% silica flour (brittle cement) and the other is elastic cement. Risk of failure was calculated based on the performance function of the type: gðXÞ ¼ RðXÞ−SðXÞ
ð1Þ
where RðXÞ is a random function describing the “resistance” or strength of the component or constituent, SðXÞ is the response of the structure, also a random variable, and X is the vector of
46
Z. Yuan et al. / Journal of Petroleum Science and Engineering 107 (2013) 45–49
Nomenclature c E Nf Jc kfe pe pw qw
cement cohesion, MPa Young's modulus, GPa load cycles, dimensionless specific heat of water at constant volume, J/kg K effective thermal conductivity of the formation, W/m K reservoir pressure, Pa wellbore pressure, Pa water injection rate, m3/s m
random variables. A negative or zero gðXÞrepresents a failure event. pf is the probability of failure defined as pf ¼ pðgðXÞ ≤0Þ
ð2Þ
r re srr sѲѲ srѲ ss st ω ρw X
radial co-ordinate, m boundary radius, m the radial stress, Pa hoop stress, Pa shear stress, Pa compressive strength, Pa tensile strength, MPa cement friction angle, radian density of injection fluid, kg/m3 vector of random variable
For elastic cement; g 3 ðNf ; s; ss Þ ¼ −LogðN f Þ−0:109
s þ 11:6 ss
ð12Þ
4. Tensile failure g 4 ðstensile ; st Þ ¼ −stensile þ st
ð13Þ
2.1. Solution to cement stresses Stresses for plane linear elasticity problems, with reference to a polar co-ordinate system Oðr; θÞ, are given in the following equation in terms of potential ΦðzÞ and Ψ ðzÞ of the complex variable z ¼ x þ iy: srr þ sθθ ¼ 2½ΦðzÞ þ ΦðzÞ
ð4Þ
z sθθ −srr þ 2isrθ ¼ 2 ½zΦ0 ðzÞ þ Ψ ðzÞ z
ð5Þ
3. Field applications The well is located in Cameron, Louisiana. As shown in Fig. 1, Well depth 1829 m; formation pore pressure gradient, 0.01074 MPa/m; overburden pressure gradient, 0.02262 MPa/m; formation horizontal principal stress, 30 MPa, 33 MPa; fracture
According to the boundary condition, the solution is expressed as Ψ ðkÞ ðzÞ ¼ DðkÞ þ
F ðkÞ GðkÞ þ 4 z2 z
ð6Þ
ΦðkÞ ðzÞ ¼ AðkÞ þ
Bð1Þ þ C ðkÞ z2 z2
ð7Þ
k ¼1 ,2, 3. 1 stands for casing; 2 stands for cement; 3 stands for formation. The partial differential equation (Fagley et al., 1982) used to describe the wellbore heat transfer is 1∂ ∂T ∂ ∂T q ρ C w ∂T rkf e þ kf e − w w r∂ ∂r ∂z ∂r ∂r 2πr qw pw −pe ∂T ¼ ðρCÞe ð8Þ þ ∂r 2πr 2 J c ln r e =r w From Eqs. (4)–(8), given bottomhole pressure, formation temperature, formation pressure and material properties, the radial stress srr , hoop stress sθθ and shear srθ can be solved. The solutions to A, B, C D, F, and G can be found (Atkinson and Eftaxiopoulos, 1996). 2.2. Cement failure modes 1. Compressive failure ð9Þ
g 1 ðs; ss Þ ¼ −s þ ss
2. Shear failure ð10Þ
g 2 ðτ; c; ss ; ωÞ ¼ −τ þ c þ ss tan ω
3. Fatigue failure For brittle cement; g 3 ðN f ; s; ss Þ ¼ −LogðN f Þ−0:082
s þ 8:48 ss
ð11Þ
Fig. 1. Wellbore geometry.
Z. Yuan et al. / Journal of Petroleum Science and Engineering 107 (2013) 45–49
gradient, 0.0185 MPa/m; casing inside radius, 11.22 cm; casing outside radius, 12.22 cm; and wellbore radius 15.56 cm. The confining pressure and compressive strengths relationship is expressed as follows: 2c cos ω 1 þ sin ω ð14Þ s ¼ sconf ining þ 1 þ sin ω 1−sin ω So, at the confining pressure of 19.64 MPa, compressive strength for brittle cement is 117.54 MPa, and compressive strength for elastic cement is 71.15 MPa. 3.1. Random variables and distribution Cement mechanical properties, casing mechanical properties, formation mechanical properties, wellbore geometry and the bottomhole pressure load are the random variables. Tables 1–3 show the random variables for the brittle cement. Tables 4–6 show the random variables for the elastic cement.
Fig. 2 shows that without cycle load, the cement tensile failure probability reaches one at the bottomhole pressure of 140 MPa. Table 1 Brittle cement random variables. Random variables
Mean
S.D.
Distribution type
Cement cohesion (c, MPa) Cement compressive strength (ss, MPa) Cement friction angle (ω, radian) Cement tensile strength ( st, MPa) Cement Young's modulus (E, GPa) Cement Poisson's ratio Formation Young's modulus (E, GPa) Formation Poisson's ratio Casing Young's modulus (E, GPa) Casing Poisson's ratio Casing ID (cm) Wellbore diameter (cm) Wellbore temperature (1C)
18.05 117.54 0.5044 5.136 9.93 0.2 7.2 0.25 204 0.3 22.44 31.12 54
0.903 11 0.001 0.0816 0.15 0.05 0.5 0.05 5 0.02 0.5 1.5 5
Lognormal Lognormal Lognormal Lognormal Lognormal Lognormal Lognormal Lognormal Lognormal Lognormal Lognormal Lognormal Lognormal
cycles cycles cycles cycles cycles cycles
(Nf) (Nf) (Nf) (Nf) (Nf) (Nf)
Bottomhole Bottomhole Bottomhole Bottomhole Bottomhole Bottomhole Bottomhole Bottomhole
pressure pressure pressure pressure pressure pressure pressure pressure
(MPa) (MPa) (MPa) (MPa) (MPa) (MPa) (MPa) (MPa)
Mean
S.D.
Distribution type
Cement cohesion (c, MPa) Cement compressive strength (ss, MPa) Cement friction angle (ω, radian) Cement tensile strength (st, MPa) Cement Young's modulus (E, GPa) Cement Poisson's ratio Formation Young's modulus (E, GPa) Formation Poisson's ratio Casing Young's modulus (E, GPa) Casing Poisson's ratio Casing ID (cm) Wellbore diameter (cm) Wellbore temperature (1C)
10.6 71.15 0.356 4.558 6.53 0.3 7.2 0.3 204 0.3 22.44 31.12 54
0.53 6.84 0.001 0.5918 0.1 0.08 0.5 0.08 5 0.02 0.5 1.5 5
Lognormal Lognormal Lognormal Lognormal Lognormal Lognormal Lognormal Lognormal Lognormal Lognormal Lognormal Lognormal Lognormal
Load Load Load Load Load
cycles cycles cycles cycles cycles
(Nf) (Nf) (Nf) (Nf) (Nf)
Mean
S.D.
Distribution type
60,000 80,000 100,000 120,000 130,000
6000 8000 10,000 12,000 13,000
Lognormal Lognormal Lognormal Lognormal Lognormal
Table 6 Elastic cement random variables.
Bottomhole Bottomhole Bottomhole Bottomhole Bottomhole Bottomhole Bottomhole
pressure pressure pressure pressure pressure pressure pressure
(MPa) (MPa) (MPa) (MPa) (MPa) (MPa) (MPa)
Mean
S.D.
Distribution type
25.03 43.47 93 133 143 153 163
2 4 9 13 14 15 16
Lognormal Lognormal Lognormal Lognormal Lognormal Lognormal Lognormal
Mean
S.D.
Distribution type
3500 4000 4500 5000 5500 6000
350 400 450 500 550 600
Lognormal Lognormal Lognormal Lognormal Lognormal Lognormal
Table 3 Brittle cement random variables. Random variables
Random variables
While the tensile failure probability is high at the bottomhole pressure beyond 120 MPa, the compressive failure and shear failure probability is nearly zero. Fig. 3 shows that for different bottomhole pressure, the fatigue failure behavior is almost the same. As the load cycles increases from 3500 to 6000, the fatigue failure probability increases from zero to one. From Fig. 4, as the bottomhole pressure is lower than 127 MPa, the system failure behavior is almost the same as fatigue failure behavior in Fig. 3. However, when the bottomhole pressure increases to 127 MPa, the system failure probability is much higher than the fatigue failure probability. The reason is at this time the tensile failure probability is very high. During the normal operation, the bottomhole pressure is 25 MPa, so, the load cycles should be prevented beyond 3500 in case of fatigue failure.
Table 2 Brittle cement random variables.
Load Load Load Load Load Load
Table 4 Elastic cement random variables.
Table 5 Elastic cement random variables.
3.2. Results of brittle cement
Random variables
47
Mean
S.D.
Distribution type
3.3. Results of elastic cement
25.03 43.47 93 113 123 127 133 143
2 4 9 11 12 12 13 14
Lognormal Lognormal Lognormal Lognormal Lognormal Lognormal Lognormal Lognormal
The results of elastic cement show the same failure trend as the results of brittle cement. From Figs. 5 and 6, the cement tensile failure probability reaches one at the bottomhole pressure of 165 MPa. As the load cycles reach 80,000, the fatigue failure probability begins to increase above zero. Fig. 7 shows that the system failure probability is different from the fatigue failure probability because the tensile failure probability is very high at
Z. Yuan et al. / Journal of Petroleum Science and Engineering 107 (2013) 45–49
1.0
1.0
0.8
0.8 Failure Probability
Failure Probability
48
Compressive Failure
0.6
Tensile Failure Shear Failure
0.4
0.6
Tensile Failure Shear Failure
0.4 0.2
0.2 0.0
Compressive Failure
0.0
20
40
60 80 100 120 Bottom Hole Pressure, MPa
140
160
Fig. 2. The effect of bottomhole pressure on brittle cement failure probability.
20
40
60 80 100 120 140 Bottom Hole Pressure, MPa
160
180
Fig. 5. The effect of bottomhole pressure on elastic cement failure probability.
1.00
p=25.03 MPa p=43.47 MPa p=113 MPa p=127 MPa
0.40
p=43.47 MPa
0.60
p=143 MPa p=153 MPa
0.40 0.20
0
0 13
0,
00
00 0,
0 00 0,
12
0 11
00
00 0, 10
,0 90
00 ,0 80
00 ,0 60
00
0.00
0.20
0.00 3.000
p=25.03 MPa
,0
0.60
0.80
70
0.80
Fatigue Failure Probability
Fatigue Failure Probability
1.00
Load Cycles
4.000 5.000 Load Cycles
6.000
Fig. 6. The effect of load cycles on elastic cement fatigue failure probability.
Fig. 3. The effect of load cycles of brittle cement fatigue failure probability.
1.00
0.80
0.60 p=25.03 MPa p=43.47 MPa
0.80
0.60
0.40
p=25.03 MPa p=43.47 MPa p=143 MPa
0.20
p=153 MPa
p=113 MPa p=127 MPa
0 00 0, 13
0,
00
0
0 00 0,
0 00 0,
00 ,0 90
00
00 ,0 80
11
6.000
10
4.000 5.000 Load Cycles
,0
60
0.00 3.000
,0
00
0.00
70
0.20
12
0.40
System Failure Probability
System Failure Probability
1.00
Load Cycles Fig. 7. The effect of load cycles on elastic cement system failure probability.
Fig. 4. The effect of load cycles on brittle cement system failure probability.
4. Fatigue failure sensitivity measure the bottomhole pressure of 153 MPa. In the normal operation, the bottomhole pressure is 25 MPa, so, the load cycles should be prevented beyond 80,000 in case of fatigue failure.
For brittle cement, at the bottomhole pressure of p ¼113 MPa, load cycles Nf ¼ 4000, the sensitivities of the reliability index are
Z. Yuan et al. / Journal of Petroleum Science and Engineering 107 (2013) 45–49
Table 7 Sensitivities of the reliability index with respect to distribution parameters (brittle cement). Variable
Mean
Standard deviation
Load cycles (Nf) Compressive stress (s) Compressive strength (ss)
−2.12032e−003 −4.13361e−009 1.32974e−009
−3.85465e−003 −1.18237e−011 −4.10930e−011
Table 8 Sensitivities of the reliability index with respect to distribution parameters (elastic cement). Variable
Mean
Standard deviation
Load cycles (Nf) Compressive stress (s) Compressive strength (ss)
−9.66630e−005 −1.19195e−008 8.38311e−009
−2.84587e−004 −5.69055e−011 −1.10395e−009
49
sensitivity measure, the compressive strength has little influence on the cement capacity because the value of compressive strength sensitivity is very small. 4. Within the cement compressive strength requirement, the wellbore service life can be increased by enhancing cement plastic behavior. The elastic cement shows better behavior than brittle cement though brittle cement has much higher compressive strength.
Acknowledgments This research was funded by Research Chevron Center for Well Construction and Production of Crisman Institute for Petroleum in Texas A&M Petroleum Engineering Department. We gratefully acknowledge these supports.
References shown in Table 7. For elastic cement, at the bottomhole pressure of p ¼143 MPa, load cycles Nf ¼80,000, the sensitivities of the reliability index are shown in Table 8. For both cement systems, the cement compressive strength tends to increase the cement capacity, however, the compressive strength has little influence on the cement capacity because the value of compressive strength sensitivity is very small. 5. Conclusions 1. For water injection wells, this paper proposed a probabilistic method to predict the wellbore service life and failure probability based on casing mechanical properties, cement mechanical properties, formation mechanical properties and wellbore geometry. Fatigue failure is studied together with the other three cement failure modes: compressive failure, tensile failure and shear failure 2. Both cement systems show the same failure characteristic. Without cycle load, comparing to compressive failure and shear failure, tensile failure is the most likely occurring failure mode. At the same bottomhole pressure, the elastic cement can hold more load cycle than the brittle cement. During the normal operation, the load cycles should be prevented beyond 80,000 in case of fatigue failure for elastic cement and 3500 for brittle cement. 3. It is little effect to improve the cement fatigue behavior by increasing the cement compressive strength. In the fatigue failure
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