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A new analytical model of wellbore collapse pressure based on stability theory
Journal Pre-proof A new analytical model of wellbore collapse pressure based on stability theory Lisong Zhang, Menggang Jiang, Wenjie Li, Yinghui Bian PII:
S0920-4105(20)30027-9
DOI:
https://doi.org/10.1016/j.petrol.2020.106928
Reference:
PETROL 106928
To appear in:
Journal of Petroleum Science and Engineering
Received Date: 10 May 2018 Revised Date:
13 August 2019
Accepted Date: 6 January 2020
Please cite this article as: Zhang, L., Jiang, M., Li, W., Bian, Y., A new analytical model of wellbore collapse pressure based on stability theory, Journal of Petroleum Science and Engineering (2020), doi: https://doi.org/10.1016/j.petrol.2020.106928. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2020 Published by Elsevier B.V.
A new analytical model of wellbore collapse pressure based on stability theory Lisong Zhanga,*, Menggang Jianga, Wenjie Lia, Yinghui Bianb,**1 a
College of Pipeline and Civil Engineering, China University of Petroleum, Qingdao 266580, China
b
College of Electronic Engineering and Automation, Shandong University of Science and
Technology, Qingdao 266590, China Abstract: In theory, the wellbore collapse for the petroleum drilling belongs to the limit point instability, and should be treated as the stability problem, but instead of the strength problem. For this purpose, a new analytical model was established to determine the collapse pressure based on the stability theory. Using the elastoplastic mechanics, the stress and displacement around the wellbore were solved, and the equilibrium path (i.e., the relation between the inner pressure and the radial displacement) was proposed. According to the equilibrium path, it can be found that the wellbore collapse occurs at the end of the plastic softening stage and the beginning of the residual stage. Additionally, a comprehensive numerical validation was performed based on the numerical methods, to examine the proposed analytical model. The stress and displacement around the wellbore by the proposed method agree well with those from the numerical method, with a deviation of less than 8.9 %. Finally, the parametric analysis was performed to investigate the effects of the plastic hardening modulus, the plastic softening modulus, the yielding stress, the strength limit and the residual strength on the equilibrium path. The results show that the collapse pressure decreases as the plastic hardening modulus, the plastic softening modulus, the yield limit and the residual strength increasing. On the contrary, an increasing of the strength limit can increase the wellbore collapse pressure, which is different to the common understanding. Keywords: Wellbore stability; Stability theory; Equilibrium path; Limit point instability; Analytical model
1. Introduction 1
* Corresponding author. E-mail addresses: [email protected] (Lisong Zhang), [email protected] (Yinghui Bian).. 1
Wellbore collapse frequently occurs during the oil and gas drilling engineering, and therefore is treated as the most important problem for the drilling safety. Wellbore collapse not only causes the additional high drilling cost but also has a severe impact on the drilling schedule (Bradley, 1979; Aadnoy and Chenevert, 1987; Gentzis, 2009; Gentzis et al., 2009a; Gentzis et al., 2009b; Zhang et al., 2015a; Zhang et al., 2015b). At present, the rock strength criteria are widely used to address the wellbore collapse, and have become the main evaluation method for the wellbore collapse. Among these criteria, Mohr-Coulomb (Mohr, 1900), Hoek-Brown (Hoek and Brown, 1967, 1980, 1997; Priest, 2005), Drucker-Prager (Drucker and Prager; 1952) and Mogi-Coulomb (Mogi, 1967, 1971; Al-Ajmi and Zimmerman, 2005, 2006) criteria are most frequently used failure criteria for wellbore stability analysis. Currently, a series of achievements have been achieved for the wellbore stability prediction. Do et al. (2019) used the non-linear failure criterion proposed by Mroz and Maciejewski to predict the wellbore stability, after obtaining the stress state around the wellbore using a new closed-form solution. Li and Weijermars (2019) proposed a modified anisotropic Hoek-Brown failure criterion to predict the mud weight window for the transverse isotropic shale. Zhou et al. (2018) performed a wellbore stability analysis based on Mohr-Coulomb criterion, considering the effects of layer structures and anisotropic seepage. Alkamil et al. (2018) used Mohr-Coulomb, Mogi-Coulomb and Modified Lade criteria to determine the mud pressures for different wells in the Mishrif Formation. Abbasi et al. (2018) applied three different strength criteria, e.g., Mohr-Coulomb, Mogi-Coulomb and Modified Lade, to predict the drilling mud pressure in tight gas sand reservoir. The results showed that the Mogi-Coulomb criterion had a better prediction for the drilling mud pressure, referring to the measured values. Chen and Abousleiman (2017) established the strain hardening and/or softening plasticity models to analyze the wellbore stability. Meanwhile, the authors compared their results to the ones obtained by the elastic theory. Kanfar et al. (2015) introduced Mohr-Coulomb and modified Lade criteria to evaluate the time-dependent wellbore stability based
2
on the finite element method, considering the material anisotropy. Rahimi and Nygaard (2015) compared thirteen failure criteria used for predicting the wellbore collapse, and considered that Modified Lade, Modified Wiebols-Cook and Mogi-Coulomb had the better prediction than other strength criteria for the field case. Chabook et al. (2015) evaluated the role of different strength criteria in the wellbore stability, and recommended the Mogi-Coulomb criterion to calculate the wellbore collapse pressure. Shi et al. (2015) proposed the single-parameter parabolic criterion to predict the wellbore stability, to avoid the shortcoming of the Mohr-coulomb criterion (i.e., using its linear form to describe the nonlinear characteristic of the rock strength). Additionally, Song and Haimson (1997), Ewy (1999), Colmenares and Zoback (2002), Yi et al. (2006), Zhang and Zhu (2007), You (2009), Zhang et al. (2010), Cai (2010), Liu et al. (2012), Lee et al. (2012) and Maleki et al. (2014) also established the respective models to evaluate the wellbore stability based on the rock strength failure criterion. From the above analysis, it can be seen that many models were proposed to predict the wellbore collapse based on rock strength criteria. However, a problem was neglected for the models developed, i.e., rock strength failure at a point does not mean the wellbore collapse, which is a common knowledge in mechanics. In other words, the rock strength criterion can not explain the wellbore collapse or block slip from the wellbore. From this point of view, the strength theory can not illustrate the phenomenon of wellbore collapse. On the contrary, the stability theory can elaborate perfectly the immediate collapse of the structure. In fact, rock strength failure at a point and wellbore collapse belongs to different mechanics scopes. The former is the strength problem, and the latter is the instability problem. The plastic zone around the wellbore is very critical to evaluate the wellbore collapse. If the plastic zone is in stable equilibrium under the slightest disturbance, i.e., the plastic zone remains the initial equilibrium configuration under the slightest disturbed load, the plastic zone instability (i.e. wellbore collapse) does not occur. Conversely, the plastic zone instability occurs if the plastic
3
zone is in unstable equilibrium state under the slightest disturbance. The disturbance maybe results from the mud circulation or the bottomhole pressure fluctuation. Therefore, theoretically speaking, the wellbore collapse belongs to the stability problem of equilibrium, but instead of strength failure problem. Correspondingly, the wellbore collapse should be evaluated by the stability theory, but instead of the strength criteria. Furthermore, a larger plastic displacement certainly occurs before the wellbore instability. Therefore, the wellbore collapse satisfies the characteristic of the limit point instability, and can be treated as the limit point instability. Although important improvements have been achieved in the area of wellbore instability, the wellbore collapse mechanism has not been fully understood. In this paper, an analytical model was established based on the stability theory, to calculate the wellbore collapse pressure. In order to verify the proposed model, the analytical results of stresses and displacements in the plastic zone were compared with the finite element method.
2. Wellbore collapse mechanism 2.1 Equilibrium state In the stability theory, the equilibrium state can be divided into 3 categories: the stable equilibrium state, the unstable equilibrium state and the neutral equilibrium state. The stable equilibrium state is that structures can return to the initial equilibrium location after removing the slightest disturbance (seeing Fig.1 (a)). The unstable equilibrium state is that structures can not return to the initial equilibrium location after removing the slightest disturbance (seeing Fig.1 (b)). The neutral equilibrium state is that structures remain the equilibrium state at arbitrary location after removing the slightest disturbance (seeing Fig.1 (c)).
4
Fig.1. Different equilibrium states
2.2 Wellbore instability mechanism Generally speaking, a plastic zone around the wellbore has an important effect on wellbore instability (Yin and Chen, 2009; Yin and Di, 2014). If the plastic zone remains the stable equilibrium state, no wellbore instability occurs. On the contrary, if the plastic zone can not remain the stable equilibrium state, wellbore instability occurs. For the practical engineering, the wellbore collapse belongs to the limit point instability, i.e., the structure may collapse (destruction) immediately after reaching the limit point at which the maximum load appears. Correspondingly, the equilibrium path (i.e., the load-displacement curve) was shown in Fig. 2, where the segment OFG represents the stable equilibrium state, while the segment GHJ represents the unstable equilibrium state. For the wellbore collapse, the generalized force can be treated as a dimensionless form with respect to the inner pressure P0 , and the generalized displacement can be treated as a dimensionless form with respect to the radial displacement u .
Fig.2. The equilibrium path for limit point instability
3. Stress and displacement around the wellbore To derive analytically the elastoplastic solution of the stress and displacement, the wellbore model was divided into 3 parts, namely, the model I under the uniform stress field, the model II under the disturbed stress field and the model III under the non-uniform stress, as shown in Fig. 3.
5
Fig.3. Wellbore analysis model
To highlight the plastic behavior of rocks, the stress-strain relation was modeled a quad-linear model, as shown in Fig. 4, where segment OA represents the elastic stage with a elastic modulus of E , segment AB represents the hardening stage with the plastic hardening modulus of mE , segment
BC represents the softening stage with the plastic softening modulus of nE , segment CD represents the plastic residual stage. Meanwhile, σ s , σ b , σ t respectively represent the limit stresses of each stage, e.g., the yield limit σ s , the strength limit σ b and the residual stress σ t .
6
Fig.4. Stress-strain relation of the quad-linear model
3.1. Analytical solution in the elastic range For the model I, II and III, the stress and displacement solutions can be expressed as Eqs. (1)-(3) in the elastic range.
σ r′ = -σ h σ θ′ = -σ h U r′ = 0
(1)
a2 ′′ σ = r r 2 ∆P0 a2 ′′ = − ∆P0 σ θ 2 r 1 + µ a2 ′′ ∆P0 U r = − E r
(2)
7
σ − σ h a2 σ H − σ h a2 a2 1 − − 1 − 1 − 3 σ r′′′= − H cos2θ 2 r2 2 r 2 r2 2 4 σ ′′′ = − σ H − σ h 1 + a + σ H − σ h 1 + 3 a cos2θ θ 2 r2 2 r4 σ H − σ h a2 a2 ′′′ = 1 − 1 + 3 sin2θ τ rθ 2 2 2 r r 2 σ − σ h ) cos 2θ (1 + µ ) a2 9 (1 + µ ) a4 Ur′′′= − σ H − σ h (1 − µ ) r + (1 + µ ) a − ( H µ r + 6 + 4 − ( ) 2E r 5E r 2 r3 2 3 (σ H − σ h ) sin 2θ (1 + µ ) a2 (1 + µ ) a4 ′′′ U r = + 1 − + µ ( ) θ r 5E 2 r3 2
(3)
where σ r , σ θ and τ rθ are the radial, hoop and shear stresses, U r , Uθ are the radial and hoop displacements, r , θ are the radial and hoop coordinates, a is the borehole radius, P0 is the drilling fluid pressure, σ H , σ h are the maximum and minimum principle stresses, E is the elastic moulds, µ is the Poisson ratio. Using Eqs. (1)-(3), the total stresses and displacements can be given as:
σ r = σ r′ + σ r′′ + σ r′′′ σ = σ ′ + σ ′′ + σ ′′′ θ θ θ θ ′′′ τ = τ rθ rθ U = U ′ + U ′ + U ′′′ r r r r ′′′ Uθ = Uθ
(4)
In the actual engineering, the difference of the maximum and minimum horizontal principle stresses is limited. As a result, the shear stress is approaching to 0, i.e., τ rθ = 0 . Additionally, the hoop displacement has a smaller effect on the wellbore stability. Therefore, it can be neglected in this analysis. According to the analysis above, Eq. (4) can be further simplified as:
σ r = σ r′ + σ r′′ + σ r′′′ σ θ = σ θ′ + σ θ′′ + σ θ′′′ U r = U r′ + U r′ + U r′′′
(5)
To obtain the elastic limit pressure P0e , Tresca criterion was introduced to evaluate the initial yielding of rocks. Substituting Eq. (5) into the Tresca criterion, P0e can be determined: 8
P0e =
σ s + ( 3σ h − σ H )
(6)
2
3.2. Analytical solution in the hardening plastic zone Under P0 ≤ P0e , the plastic zone appears around the wellbore. In the plastic zone, the rock shows different mechanical behaviors at the different stages. Commonly, the rock has three plastic stages after yielding, namely, the hardening stage (segment AB), the softening stage (segment BC), and the residual stage (segment CD). For the plastic hardening stage, the plastic zone around the wellbore can be shown in Fig. 5, where ρ1 is the boundary of the elastic and plastic zones, b is the outer boundary of the elastic zone, much larger than the wellbore radius a .
Fig.5. Stress and displacement analysis model for the plastic hardening phase
In the plastic zone, the yielding surface can be expressed as Eq. (7) based on the Tresca criterion.
(
f1 = σ θ − σ r − σ s 1 + η1ε ip where η1 =
(
mE , ε p is the plastic strain, σ s 1+ η1ε ip (1 − m )σ s i
)
Meanwhile, the associated flow rule can be described as:
9
)
is the yield limit in the plastic flow.
(7)
dε ip =
∂f ∂σ i
dλ
(8)
where d λ is the plastic multiplier. Then, the plastic strain increment can be expressed as:
dε rp1 = − dλ1 p dε θ 1 = dλ1 p dε z1 = 0
(9)
Using Eq. (9), the physical equations can be solved:
σ r − µ (σ θ + σ z ) − λ1 (r ) ε r = E σ θ − µ (σ r + σ z ) + λ1 (r ) ε θ = E σ z − µ (σ r + σ θ ) ε z = E
(10)
where σ z is equal to µ (σ θ + σ r ) based on the plane strain state. The geometry and equilibrium equations can be expressed as:
dε θ ε θ − ε r =0 dr + r dσ r + σ r − σ θ = 0 dr r
(11)
According to the Tresca criterion, the radial and hoop stresses satisfy the following relation:
σ r − σ θ = σ s (1 + η1ε ip ) where ε i = Bλ1 (r ) , B = p
(12)
2 . 3
Eq. (12) can be further expressed as:
σ r − σ θ = σ s 1 + Bη1λ1 ( r )
(13)
Combining Eqs. (10), (11) and (13), the differential equation with respect to λ1 (r ) can be determined:
10
r
d ( λ1 ( r ) ) dr
+ 2λ1 ( r ) +
2 (1 − µ 2 ) σ s
(1 − µ )σ Bη 2
s
1 +E
=0
(14)
Substituting the boundary condition of λ1 (ρ1 ) = 0 into Eq. (14), λ1 (r ) can be solved:
ρ12 − 1 2 r
λ1 (r ) = K1
where K1 =
(15)
(1 − µ )σ . (1 − µ )σ Bη + E 2
s
2
s
Substituting λ1 (r ) into the equilibrium equation, the stresses in the hardening plastic zone can be expressed as:
1 σ s (1 − ξ1 ) ρ12 = − ln + + c2 r σ σ ξ r s 1 2 2 r σ s (1 − ξ1 ) ρ12 1 + c2 σ θ = −σ sξ1 (1 + ln r ) − 2 r2 σ 1z = µ −σ sξ1 (1 + 2 ln r ) + 2c2
(16)
where ξ1 = 1 − Bη1 K1 . Then, the volume strain can be solved: dU r1 U r1 1 d(rU r1 ) (1 − 2 µ )(1 + µ )(σ r + σ θ ) ε r + εθ + ε z = + = = dr r r dr E
(17)
Integrating Eq. (17) to lead to: U r1 =
(1 − 2 µ )(1 + µ ) r −σ ξ ln r + c + A1 [ s1 2] E
r
(18)
Eq. (18) has two unknown parameters, namely, c2 and A1 , which can be solved based on the following derivation. Using the stress boundary condition on r = a , i.e., σ r1 |r = a = −∆P0 , the constant c2 can be determined: c2 = σ sξ1 ln a −
σ s (1 − ξ1 ) ρ12 2 11
a2
+ ∆P0
(19)
Additionally, combining the stress and displacement continuity conditions at r = ρ1 , i.e.,
σ r1 |r = ρ = σ r |r = ρ and U r1 |r = ρ = U r |r = ρ , ∆P0 and A1 can be obtained: 1
1
1
1
∆P0 = σ sξ1 ln
A1 = −
ρ1 a
+
σ s (1 − ξ1 ) ρ12
σ 2 − 1 + s a 2
2
(1 + µ ) ρ 2 σ s − (1 − 2µ )(1 + µ ) ρ 2 −σ ξ ln ρ + c 1 1 [ s 1 1 2] E
2
E
(20)
(21)
Especially, the stress and displacement on the boundary of the elastic zone, i.e., σ r |r = ρ1 and U r |r = ρ1 , can be solved based on Eq. (5).
3.3. Analytical solution in the softening plastic zone After the rock enters into the plastic softening stage, the plastic zone near the wellbore can be shown in Fig. 6, where ρ 2 is the boundary of the plastic hardening stage and the plastic softening stage.
Fig.6. Stress and displacement analysis model for the plastic softening phase
At the plastic boundary of r = ρ 2 , the yield stress is equal to the strength limit, i.e., Eq. (22):
σ s 1 + Bη1λ1 ( r ) = σ b
12
(22)
Using Eq. (22), the plastic multiplier λ1 (r ) can be given:
λ1 (r ) |max =
σ b −σ s σ s Bη1
(23)
Substituting Eq. (23) into Eq. (15), the relation of the plastic radius ρ1 and ρ 2 for different plastic stages can be expressed as:
ρ12 σ b − σ s = +1 ρ 22 σ s Bη1 K1
(24)
In this stage, the radial stress σ r and the hoop stress σ θ also satisfy the Tresca criterion:
σ r − σ θ = σ b (1 − η 2ε ip ) where η2 =
(25)
nE σ −σs , ε ip = Bλ2 (r ) + b . σ sη1 (1 − n ) σ s
Then, Eq. (25) can be further rewritten as:
σ r − σ θ = σ b 1 − Bη 2λ2 ( r ) −
( σ b − σ s )η 2 σ sη1
(26)
Similar to the derivation of the plastic gardening stage, the differential equation with respect to
λ2 (r ) can be determined: r
d ( λ2 ( r ) ) dr
1 − µ 2 )σ b ( σ −σs =0 + 2λ2 ( r ) + 2 2 + b ( µ − 1) σ b Bη2 + E σ s Bη1
(27)
Substituting the boundary condition of λ2 (ρ 2 ) = 0 into Eq. (27), λ2 (r ) can be obtained:
ρ 22 − 1 2 r
λ2 (r ) = K 2
(1 − µ )σ = ( µ − 1)σ Bη 2
where K 2
2
b
(28)
σb −σs . σ s Bη1 2 −E b
+
Substituting λ2 (r ) into the equilibrium equation, the stresses in the plastic softening phase can be expressed as:
13
2 σ bξ3 ρ 22 + c3 σ r = −σ bξ 2 ln r − 2 r2 σ bξ3 ρ 22 2 + c3 σ θ = −σ bξ 2 (1 + ln r ) + 2 r2 σ z2 = µ −σ bξ 2 (1 + 2 ln r ) + 2c3 where ξ 2 = 1 + Bη2 K 2 −
(29)
σ b -σ s η 2 , ξ 3 = Bη 2 K 2 . σ s η1
Similar to Eq. (17), the radial displacement in the plastic softening stage can be yielded: r (1 − 2 µ )(1 + µ )( −σ bξ 2 ln r + c3 )
U r2 =
E
+
A2 r
(30)
where c3 and A2 are the unknown variables, and can be solved by the following derivation. Based on the stress condition on the boundary of r = a , i.e., σ r2 | r =a = − ∆P0 , the constant, c3 , can be given: c3 = σ bξ 2 ln a +
σ bξ3 ρ 22 a2
2
+ ∆P0
(31)
Additionally, the stress redistribution occurs in the hardening plastic zone, when the rock enters into the plastic softening stage. Therefore, the parameters, c2 and A1 , need to be resolved to ensure the stress and displacement accuracy. Combining the stress and displacement continuity conditions on r = ρ1 and r = ρ 2 , i.e., σ r1 |r = ρ1 = σ r |r = ρ1 , σ r2 |r = ρ 2 = σ r1 |r = ρ 2 , U r1 |r = ρ1 = U r |r = ρ1 and
U r2 |r = ρ 2 = U r1 |r = ρ 2 , the parameters, c2 , ∆P0 , A1 , A2 , can be determined accurately: c2 = σ sξ1 ln ρ1 −
∆P0 = σ bξ 2 ln
ρ2 a
A1 = − A2 = A1 +
+
σ s (1 − ξ1 ) σ s 2
+
2
ρ2 ρ σ (1 − ξ1 ) ρ12 σ s 1 − 2 + 1 − 22 - σ sξ1 ln 2 − s 2 a 2 ρ1 ρ2 2
σ bξ 3
(1 + µ ) ρ 2 σ s − (1 − 2µ )(1 + µ ) ρ 2 −σ ξ ln ρ + c 1 1 ( s 1 1 2) E
2
E
(1 − 2 µ )(1 + µ ) ρ 2 −σ ξ ln ρ + c − −σ ξ ln ρ + c 2 ( s 1 2 2) ( b 2 2 3 ) E
14
(32)
(33)
(34) (35)
3.4. Analytical solution in the residual plastic zone After the rock enters into the plastic residual stage, the plastic zone near the wellbore can be shown in Fig. 7, where ρ3 is the boundary of the plastic residual phase and the plastic softening phase.
Fig.7. Stress and displacement analysis model for the plastic residual phase
At the plastic boundary of r = ρ3 , the yield stress is equal to the residual stress, i.e.,
σ b 1 − Bη 2λ2 ( r ) −
( σ b − σ s )η 2 = σ σ sη1
(36)
t
According to Eq. (36), the plastic multiplier λ2 (r ) can be yielded:
λ2 ( r ) |max =
σ b 1 −
(σ b − σ s )η 2 − σ σ sη1 σ b Bη2
t
(37)
Substituting Eq. (37) into Eq. (28), the relation of the radius ρ 2 and ρ3 of the plastic zones can be expressed as:
ρ = ρ 2 2 2 3
σ b 1 −
( σ b − σ s )η 2 − σ σ sη1 σ b Bη2 K 2
15
t
+1
(38)
In the residual plastic zone, the radial stress σ r and the hoop stress σ θ still satisfy the Tresca criterion:
σ r − σθ = σ t
(39)
Substituting Eq. (39) into the equilibrium equation, the stresses in the residual plastic zone can be expressed as: σ 3 = −σ lnr + c t 4 r 3 3 σ θ = σ t + σ r = −σ t ( lnr + 1) + c 4 3 3 3 σ z = µ (σ θ + σ r ) = − µσ t ( 2lnr + 1) + 2 µ c 4
(40)
Meanwhile, the radial displacement in the residual zone can be expressed as: U r3 =
(1 − 2µ )(1 + µ ) r −σ ln r + c + A3 ( t 4) E
r
(41)
In Eq. (41), the parameters, c4 , A3 , are unknown. According to the stress boundary condition of σ r3 |r =a = −∆P0 , the parameter, c4 , can be obtained: c4 = σ t ln a + ∆P0
(42)
Similarly, the stress redistribution occurs in the hardening and softening plastic zones, when the rock enters into the residual plastic stage. Therefore, the parameters, c2 , c3 , A1 and A2 , need to be resolved to ensure the stress and displacement accuracy. Combining the stress and displacement continuity conditions at r = ρ1 , r = ρ 2 and r = ρ3 , i.e., σ r1 |r = ρ1 = σ r |r = ρ1 ,
σ r2 |r = ρ = σ r1 |r = ρ , σ r3 |r = ρ = σ r2 |r = ρ , U r1 |r = ρ = U r |r = ρ , U r2 |r = ρ = U r1 |r = ρ 2
2
3
3
1
1
2
2
and U r3 |r = ρ 2 = U r3 |r = ρ 2 ,
the unknown parameters, c2 , c3 , ∆P0 , A1 , A2 , A3 , can be obtained: c2 = σ sξ1 ln ρ1 −
c3 = σ bξ 2 ln ρ 2 +
σ bξ 3 2
σ s (1 − ξ1 ) σ s
− σ sξ1 ln
2
+
2
ρ 2 σ s (1 − ξ1 ) ρ12 σ s − 1 − 2 + 2 ρ1 ρ2 2 16
(43)
(44)
ρ3 σ bξ3 ρ22 ρ2 σ s (1 − ξ1 ) ρ12 σ s ∆P0 = σ t ln − σ bξ 2 ln − − 1 − σ sξ1 ln − 1 − 2 + a 2 ρ32 2 ρ2 ρ1 ρ2 2 ρ3
(1 + µ ) ρ 2 σ s − (1 − 2µ )(1 + µ ) ρ 2 −σ ξ ln ρ + c 1 1 ( s 1 1 2)
(46)
(1 − 2µ )(1 + µ ) ρ 2 −σ ξ ln ρ + c − −σ ξ ln ρ + c 2 ( s 1 2 2) ( b 2 2 3 )
(47)
(1 − 2µ )(1 + µ ) ρ 2 −σ ξ ln ρ + c − −σ lnρ + c 3 ( b 2 3 3) ( t 3 4 )
(48)
A1 = − A2 = A1 +
(45)
A3 = A2 +
E
2
E
E
E
After calculating the stress and displacement of the plastic zone under the uniform stress field, the total stress and displacement under the non-uniform stress field can be calculated approximately, seeing "Appendix".
4. Wellbore collapse analysis 4.1. Equilibrium path The equilibrium path is the most important step for analyzing the wellbore collapse problem using the stability theory. Considering that P0 and U r are the control variables for the wellbore stability, the equilibrium path depends on the inner pressure P0 and the radial displacement U r . Note that, the radial displacement U r is a smallness item and is not convenient to be used in the equilibrium path. Therefore, the equilibrium path is expressed as a dimensionless form with respect to P0 and U r . The dimensionless process of P0 and U r is herein expressed as:
2 (σ h − P0 ) 2 (σ H − σ h ) − (5 − 2 cos 2θ ) U r = σs 5 (1 + µ ) σ s 2 P0 P = σ s + 4σ h + (σ H − σ h ) cos 2θ
(49)
where p is the dimensionless force, U r is the dimensionless displacement. Using Eq. (5), the equilibrium path in the elastic range can be determined: Ur =
2σ h − σ s + 4σ h + (σ H − σ h ) cos 2θ p 2 (σ H − σ h ) − (5 − 2 cos 2θ ) σs 5 (1 + µ ) σ s 17
(50)
Using Eqs. (18) and (20), the equilibrium path in the plastic hardening zone can be concluded as:
ρ12 ρ1 ρ12 2 (σ H − σ h ) U 1 2 µ ξ ( 2 ln 1) (5 − 2 cos 2θ ) = − + − + − ) 1 2 r ( 2 a a a 5 1 + µ σ ( ) s 2 ρ ρ 2σ h − σ s ξ1 (1 + 2 ln 1 ) + (1 − ξ1 ) 12 a a P = σ s + 4σ h + (σ H − σ h ) cos 2θ
(51)
Using Eqs. (30) and (33), the equilibrium path in the plastic softening zone can be derived as:
2 U = 2 (1 − 2 µ ) σ b ξ ln a − σ b ξ − ξ ( 1 − ln ρ 2 ) − 1 − ξ1 ρ1 2 3 1 r 2 2 ρ 22 ρ 2 2σ s ρ1 σs 2 (σ H − σ h ) σ ρ2 + 22 2 (1 − µ ) + (1 − 2 µ ) b ξ3 − (5 − 2 cos 2θ ) a σ s 5 (1 + µ ) σ s σb ρ2 σ b ρ22 ρ2 ρ12 2σ h − σ s 2 ξ 2 ln + ξ3 (1 − 2 ) + ξ1 (1 − 2 ln ) + (1 − ξ1 ) 2 σs ρ1 ρ2 a σs a P = σ s + 4σ h + (σ H − σ h ) cos 2θ
(52)
Using Eqs. (41) and (45), the equilibrium path in the plastic residual zone can be given as: 2 2 U = 2 (1 − 2 µ ) σ t ln a + σ b ξ ln ρ3 + σ b ξ ( ρ 2 − 1) − ξ ( 1 − ln ρ 2 ) − 1 − ξ1 ρ1 1 r σ s ρ3 σ s 2 ρ 2 2σ s 3 ρ32 ρ1 2 2 ρ 22 ρ 2 2 (σ H − σ h ) + 2 (1 − µ ) 12 − (5 − 2 cos 2θ ) a 5 (1 + µ ) σ s σ ρ σ ρ σ ρ2 ρ ρ2 2σ h − σ s 2 t ln 3 − 2 b ξ 2 ln 3 − b ξ3 ( 22 − 1) + ξ1 (1 − 2 ln 2 ) + (1 − ξ1 ) 12 a σs ρ2 σ s ρ3 ρ1 ρ2 σs P = σ s + 4σ h + (σ H − σ h ) cos 2θ
(53)
According to Eqs. (50), (51), (52) and (53), the equilibrium path for the wellbore collapse problem can be determined. To show the solution process of the equilibrium path more clearly, a flowchart was depicted in Fig. 8. Especially, the selection of the independent variable was highlighted in this analysis. Generally speaking, the critical instability state can be achieved by changing the generalized force. That is to say, the wellbore collapse can occur through decreasing the inner pressure of the wellbore. 18
In such case, the inner pressure should be treated as the independent variable, and the other parameters can be treated as the dependent variable. However, a problem appears under this case, i.e., the plastic zone boundary of ρ1 , ρ 2 and ρ3 are difficult to be solved based on Eqs. (20), (30) and (45), which directly increases the solution difficulty for U r . In view of this, ρ1 was selected as the independent variable, mainly because: (1) the other parameters can be obtained by changing
ρ1 ; (2) the wellbore collapse can be achieved by increasing the plastic zone boundary of ρ1 .
Fig. 8. Calculation flowchart of the equilibrium path.
4.2. Collapse pressure analysis Using the condition that the first derivative of Eq. (52) is equal to 0, the critical radius where p reaches the maximum value p max can be solved as Eq. (54). In fact, the critical radius ρ cr is the boundary of the softening zone and the residual zone.
ρcr = a
ξ 2 (σ b σ s − ξ1 ) ξ3 (1 − ξ1 ) 19
(54)
According to the limit point instability, the wellbore collapse occurs immediately once
p = p max . Therefore, the collapse pressure Pci of the wellbore can be solved by substituting Eq. (54) into Eq. (52): Pcr =
σb 2
(ξ 2 ln
σ σ σ −ξ ξ2 + ξ3 − ξ 2 + 1) − s ξ1 ln b s 1 2 1 − ξ1 ξ3
(55)
Eq. (55) is the collapse pressure obtained by the stability theory, which is obviously different from the one by the strength criterion. Additionally, the wellbore collapse occurs at the end of the softening phase or the beginning of the residual phase, which is different to the conclusion of the rock strength criterion.
5. Results and discussions TS-1 well was drilled as a vertical well in Xinjiang oil field. Due to the tectonic stress, the wellbore collapse occurs frequently in this region. In such case, it is very important to predict accurately the collapse pressure. In this analysis, a plane strain model was used, and the basic parameters were listed as follows: the depth H = 2044 m, the borehole radius a = 157.05 mm, the horizontal maximum in-situ stress σ H = 43.33 MPa, the horizontal minimum in-situ stress
σ h =41.90 MPa. Additionally, the mechanical parameters of the rock were concluded as follows: the elastic modules E =33.9 GPa, the Poisson ratio µ = 0.33, the yield stress σ s = 70 MPa, the strength limit σ b = 90 MPa, the residual stress σ t = 35 MPa, m = 0.20, n = 1.20.
5.1. Validation of the proposed method Based on the previous parameters, the equilibrium path of arbitrary point near the wellbore for TS-1 well can be determined by calculating the dimensionless parameters of U r and p . To exhibit the wellbore instability, 7 track points were set to determine the detailed instability location, with the angles of θ = 0°, 15°,
, 90° at the inner surface of the wellbore. Then, the equilibrium
paths of 7 track points were solved and shown in Fig. 9. 20
Fig. 9. The equilibrium paths for different track points.
As seen from Fig. 9, the critical pressures for 7 track points are different, due to the non-uniform in-situ stress field. However, the difference of the critical pressures is limited, mainly because the difference of the horizontal maximum and minimum stresses is small. The wellbore instability occurs firstly at θ = 90 °, because the critical pressure pcr is highest in this location. Substituting pcr into Eq. (55) or (49), the collapse pressure Pci of TS-1 well can be calculated, equaling to 19.56 MPa. Meanwhile, the critical pressure is lowest at θ = 0 °, which indicates that the wellbore collapse is difficult to occur at this location. Specially, in order to further verify the accuracy of the proposed method, the stresses and displacements from the proposed method were compared to the ones from the numerical method (see Fig.10).
21
(a) Radial displacements (r = a)
(b) Hoop displacements (r = a)
(c) Radial stresses (r = a)
(d) Hoop stresses (r = a)
(e) Radial displacements (r = 2a)
(f) Hoop displacements (r = 2a)
22
(g) Radial stresses (r = 2a)
(h) Hoop stresses (r = 2a)
Fig. 10. Result comparison of the proposed method and numerical method.
As seen from Fig. 10, the analytical results agree well with the results from the numerical method at r=a and 2a, regardless of the stress or displacement results. The deviations between the analytical and the numerical results are within 1.6 % ~ 8.9 %, mainly because of different assumptions made in the analytical and numerical methods. According to analysis above, the proposed analytical model can calculate accurately the stress and displacement around the wellbore. Meanwhile, the finite element method shows a better agreement to the analytical method for the stress and displacement results, mainly because of the same mechanics assumption. A rigid symmetry can be observed for the results from the analytical and finite element methods in Fig. 11, regardless of the stress or displacement.
5.2. Parametric analysis As seen from Eqs. (51), (52), (53), (54), the plastic hardening modulus, the plastic softening modulus, the yielding stress, the strength limit and the residual strength have important effects on the equilibrium path and the wellbore collapse pressure. Therefore, the parametric analysis is performed to investigate the effects of these parameters on the equilibrium path.
5.2.1 Effect of the plastic hardening modulus To investigate the effect of the plastic hardening modulus on the equilibrium path, the plastic hardening modulus varies at a range of 0.10 E ~0.30 E . 23
Fig. 11. The effect of the plastic hardening modulus on the equilibrium path.
As shown in Fig. 11, with the increasing of the plastic hardening modulus, the dimensionless critical pressure (i.e., the peak at the equilibrium path) decreases obviously. In other words, the wellbore collapse pressure decreases as the plastic hardening modulus increasing, referring to the relation of p and P0 in Eq. (49). This means that the wellbore collapse occurs more difficultly with the increasing of the plastic hardening modulus. Note that, there has no peak point at the equilibrium path for the smaller plastic hardening modulus, which means that the wellbore collapse pressure is higher for m= 0.10 and 0.15. That is to say, the wellbore is difficult to maintain stable when the plastic hardening modulus is lower in the practice engineering. Meanwhile, the plastic hardening modulus is only related to the plastic hardening stage, and has no effect on the elastic stage. Therefore, the dimensionless displacement and pressure is approximately same at the elastic stage, even if the plastic hardening modulus changes. Overall, an increasing of the plastic hardening modulus is beneficial for maintaining the wellbore stability, because it decreases the collapse pressure.
5.2.2 Effect of the plastic softening modulus To investigate the effect of the plastic softening modulus on the equilibrium path, the plastic 24
softening modulus changes from 1.0 E to 1.4 E .
Fig. 12. The effect of the plastic softening modulus on the equilibrium path.
As shown from Fig. 12, the collapse pressure (i.e., the peak on the equilibrium path) decreases obviously, as the plastic softening modulus increasing. Note that, there has no peak point to appear on the equilibrium path at n= 1.0, meaning that the wellbore collapse pressure is very high. In such case, the wellbore stability is difficult to maintain in the practice engineering. Meanwhile, the plastic softening modulus has no effect on the elastic and plastic hardening stages. As a result, the equilibrium paths are close to each other in the elastic and plastic hardening stages. Overall, a higher plastic softening modulus is good for maintaining the wellbore stability, because it decreases the wellbore collapse pressure.
5.2.3 Effect of the yielding stress To investigate the effect of the yielding stress on the equilibrium path, the yielding stress varies at a range of 60 MPa ~ 80 MPa.
25
Fig. 13. The effect of the yielding stress on the equilibrium path.
As seen from Fig. 13, with the increasing of the yielding stress, the dimensionless critical pressure (i.e., the peak on the equilibrium path) decreases obviously. According to the physical meaning of the dimensionless pressure, it can be determined that the wellbore collapse pressure decreases as the yielding stress increasing. Additionally, the dimensionless critical displacement decreases as the yielding stress increasing. This is mainly because the larger yielding stress decreases the plastic deformation capacity in the hardening stage, when the strength limit keeps constant. Overall, an increasing of the yielding stress can avoid effectively the wellbore collapse based on the stability theory, because it decreases the wellbore collapse pressure.
5.2.4 Effect of the strength limit To investigate the effect of the strength limit on the equilibrium path, the strength limit varies at a range of 80 MPa ~ 100 MPa.
26
Fig. 14. The effect of the strength limit on the equilibrium path.
As seen from Fig. 14, the dimensionless critical pressure increases significantly, as the strength limit increasing. Correspondingly, the wellbore collapse pressure increases with the increasing of the strength limit. This is because the larger strength limit extends the plastic deformation capacity of the hardening stage, and further increases wellbore collapse pressure. In the previous strength models, the effect of the strength limit on the wellbore collapse can not be reflected, because the strength limit can not be contained in the strength criterion. From the point of view, it is an advantage to analyze the wellbore collapse pressure using the stability theory. Overall, the larger strength limit is adverse for maintaining the wellbore stability, because it increases the wellbore collapse pressure. This conclusion is different to the common understanding.
5.2.5 Effect of the residual strength To investigate the effect of the residual strength on the equilibrium path, the residual strength varies at a range of 25 MPa ~ 45 MPa.
27
Fig. 15. The effect of the residual strength on the equilibrium path.
As shown from Fig. 15, with the increasing of the residual strength, the dimensionless critical pressure decreases obviously, i.e., the wellbore collapse pressure decreases obviously. Especially, the trends of the equilibrium path are similar each other, mainly because the residual strength has no effect on the elastic, plastic hardening and plastic softening stages. Additionally, the stability theory is more suitable than the strength criterion to analyze the wellbore stability, because the residual strength can be reflected in this analysis. Overall, a higher residual strength is very important for maintaining the wellbore stability, because it decreases the wellbore collapse pressure.
6. Conclusions (1) In essence, the welbore collapse belongs to the stability problem, but instead of the strength problem. Using the stability theory, an analytical model was established to determine the wellbore collapse pressure. In this model, the radial displacement under the non-uniform in-situ stress field can be solved based on the elastoplastic mechanics, and then the equilibrium path can be obtained with respect to the radial displacement and the inner pressure. (2) The proposed method agrees well with the numerical method. The deviation of less than
28
8.9 % can be observed between the proposed method and the numerical method, which means that the proposed method can satisfy the practice engineering. (3) Using the stability theory, the effects of the parameters on the wellbore collapse pressure can be investigated, including the plastic hardening modulus, the plastic softening modulus, the yielding stress, the strength limit and the residual strength, while the effects mentioned above can not be discussed using the strength criterion. This can be treated an obvious advantage on the stability method over the strength criterion. (4) The parametric analysis shows that: 1) the higher plastic hardening modulus, plastic softening modulus, yield limit and residual strength are beneficial for the wellbore stability, because of decreasing the wellbore collapse pressure; 2) the higher strength limit can increase the wellbore collapse pressure, which is adverse for the wellbore stability. This conclusion is different from the previous understanding.
Acknowledgment The authors are very much indebted to the Projects Supported by PetroChina Innovation Foundation (2018D-5007-0309), Focus on Research and Development Plan in Shandong Province (2019GGX103007), and the Fundamental Research Funds for the Central Universities (19CX02034A) for the financial support.
Appendix In the plastic range, the total stress and displacement under the non-uniform stress field can be assumed as a sum of the plastic ones under the uniform stress field and the elastic ones under the non-uniform stress field (i.e., Model III). By adding Eq. (3) to Eqs. (16) and (18), the total stress and displacement in the plastic hardening zone can be expressed as:
29
1 σs σH −σ h r σ s (1 − ξ1 ) ρ12 a2 σ H −σ h a2 a2 = − + − + − − − − − − σ σ ξ ln ( 1) σ (1 ) (1 )(1 3 )cos2θ r s 1 h 2 2 2 2 ρ 2 r 2 2 r 2 r r 1 1 σ (1 − ξ1 ) ρ12 σ σ −σh r a2 σ −σ h a4 ( 2 + 1) + s − σ h − H (1 + 2 ) + H (1 + 3 4 )cos2θ σ θ = −σ sξ1 (1 + ln ) − s ρ1 2 r 2 2 r 2 r 2 σ 1z = −2 µ (σ sξ1 ln r − σ h ) + (σ H − σ h ) µ (2 a 2 cos2θ − 1) r ρ1 (56) 2 µ µ µ 1 − 2 1 + 1 + σ ξ σ ( )( ) r ( ) ρ r s 1 s 1 U 1 = (1 − 2 ln ) − 1 + (1 − 2 µ ) ξ1 r ρ1 2 E E r 2 (σ H − σ h )(1 + µ ) r (1 − µ + a 2 ) − (σ H − σ h )(1 + µ ) r (1 + 6 + 4µ a 2 − 9 a 4 ) cos 2θ − 2E 1+ µ r2 10 E 1+ µ r2 r4 1 3 (σ H − σ h ) sin 2θ (1 + µ ) a 2 (1 + µ ) a 4 + 1 − + r µ ( ) Uθ = 5E 2 r3 r 2
By adding Eq. (3) to Eqs. (29) and (30), the total stress and displacement in the plastic softening zone can be given: 2 r σ bξ3 ρ22 ρ2 σ s (1 − ξ1 ) ρ12 σ s = − ln + (1 − ) − ln − (1 − )+ σ σ ξ σ ξ r b 2 s 1 2 2 2 r 2 2 ρ ρ ρ 2 1 2 2 2 2 σ −σh σ −σh a a a −σh − H (1 − 2 ) − H (1 − 2 )(1 − 3 2 )cos2θ r r r 2 2 2 2 σξ ρ ρ σ s (1 − ξ1 ) ρ2 σ r (1 − 12 ) + s σ θ = −σ bξ2 (1 + ln ) + b 3 (1 + 22 ) − σ sξ1 ln 2 − ρ2 ρ1 ρ2 2 r 2 2 σ −σh a2 σ −σ h a4 −σh − H (1 + 2 ) + H (1 + 3 4 )cos2θ r 2 2 r ρ ρ2 r σ z2 = µ −σ bξ2 (1 + 2ln ) + σ bξ3 − 2σ sξ1 ln 2 − σ s (1 − ξ1 ) (1 − 12 ) + σ s − 2σ h ρ2 ρ1 ρ2 2 a + (σ H − σ h ) µ (2 2 cos2θ − 1) r 1 − 2µ )(1 + µ ) ρ σ (1 − ξ1 ) ρ2 σ r σξ U r2 = ( r −σ bξ2 ln + b 3 − σ sξ1 ln 2 − s (1 − 12 ) + s ρ2 ρ1 ρ2 E 2 2 2 2 2 2 (1 − µ )(1 + µ ) ρ1 σ s (1 − 2µ )(1 + µ ) ρ2 σ bξ3 ρ − − ( − σ sξ1 ln 2 ) E r 2 E r 2 ρ1 (σ H − σ h )(1 + µ ) r (1 − µ + a 2 ) − (σ H − σ h )(1 + µ ) r (1 + 6 + 4µ a 2 − 9 a 4 )cos 2θ − 2E 1+ µ r 2 10 E 1+ µ r2 r4 3 (σ H − σ h ) sin 2θ (1 + µ ) a 2 (1 + µ ) a 4 Uθ2 = + 1 − r µ ( ) + 5E 2 r 2 r3
(57)
Similarly, by adding Eq. (3) to Eqs. (40) and (41), the total stress and displacement in the
30
plastic softening zone can be given: 2 2 σ3 =−σ ln r −σ ξ ln ρ3 − σbξ3 ( ρ2 −1) −σ ξ ln ρ2 − σs (1−ξ1 ) (1− ρ1 ) + σs t b 2 s 1 r 2 ρ3 ρ2 2 ρ32 ρ1 ρ22 2 σH −σh a2 σH −σh a2 a2 σ − − (1 − ) − (1 − )(1 − 3 )cos2θ h 2 r2 2 r2 r2 2 2 σ3 =−σ (ln r +1) −σ ξ ln ρ3 − σbξ3 ( ρ2 −1) −σ ξ ln ρ2 − σs (1−ξ1 ) (1− ρ1 ) + σs t b 2 s 1 θ ρ3 ρ2 2 ρ32 ρ1 ρ22 2 2 σH −σh a2 σH −σh a4 σ − − (1 + ) + (1 + 3 )cos2θ h 2 4 2 2 r r σt 3 σb ρ3 ρ22 r ρ2 ρ12 ρ12 3 3 σz = µ (σθ +σr ) =−µσs (2ln +1) + ξ3 (2ln + 2 −1) +ξ1(2ln + 2 −1) − 2 ρ3 σs ρ2 ρ3 ρ1 ρ2 ρ2 σs a2 + (σH −σh )µ(2 2 cos2θ −1) r (1−2µ)(1+ µ) σs r 2σt ln r + σb ξ (2ln ρ3 + ρ22 −1) +ξ (2ln ρ2 + ρ12 −1) − ρ12 − 2(1− µ)(1+ µ) ρ12 σs Ur3 =− 1 3 2 σs ρ3 σs E E r 2 ρ1 ρ22 ρ22 ρ2 ρ32 2 2 4 − (σH −σh )(1+ µ) r(1− µ + a ) − (σH −σh )(1+ µ) r(1+ 6 + 4µ a −9 a )cos2θ 2E 1+ µ r2 10E 1+ µ r2 r4 (58) 3 3(σH −σh ) sin2θ (1+ µ) a2 (1+ µ) a4 r + (1− µ) + Uθ = 5E r 2 r3 2
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34
Fig.1. Different equilibrium states
Highlights:
► Wellbore collapse is a stability problem, but instead of strength failure problem. ► An analytical model of wellbore stability was derived based on stability theory. ► The results of proposed model agree well with those of the numerical model. ► New model can perform the parametric study that can not be analyzed by previous model. ► An important conclusion on strength limit is different to the previous understanding.
S0920-4105(20)30027-9
DOI:
https://doi.org/10.1016/j.petrol.2020.106928
Reference:
PETROL 106928
To appear in:
Journal of Petroleum Science and Engineering
Received Date: 10 May 2018 Revised Date:
13 August 2019
Accepted Date: 6 January 2020
Please cite this article as: Zhang, L., Jiang, M., Li, W., Bian, Y., A new analytical model of wellbore collapse pressure based on stability theory, Journal of Petroleum Science and Engineering (2020), doi: https://doi.org/10.1016/j.petrol.2020.106928. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2020 Published by Elsevier B.V.
A new analytical model of wellbore collapse pressure based on stability theory Lisong Zhanga,*, Menggang Jianga, Wenjie Lia, Yinghui Bianb,**1 a
College of Pipeline and Civil Engineering, China University of Petroleum, Qingdao 266580, China
b
College of Electronic Engineering and Automation, Shandong University of Science and
Technology, Qingdao 266590, China Abstract: In theory, the wellbore collapse for the petroleum drilling belongs to the limit point instability, and should be treated as the stability problem, but instead of the strength problem. For this purpose, a new analytical model was established to determine the collapse pressure based on the stability theory. Using the elastoplastic mechanics, the stress and displacement around the wellbore were solved, and the equilibrium path (i.e., the relation between the inner pressure and the radial displacement) was proposed. According to the equilibrium path, it can be found that the wellbore collapse occurs at the end of the plastic softening stage and the beginning of the residual stage. Additionally, a comprehensive numerical validation was performed based on the numerical methods, to examine the proposed analytical model. The stress and displacement around the wellbore by the proposed method agree well with those from the numerical method, with a deviation of less than 8.9 %. Finally, the parametric analysis was performed to investigate the effects of the plastic hardening modulus, the plastic softening modulus, the yielding stress, the strength limit and the residual strength on the equilibrium path. The results show that the collapse pressure decreases as the plastic hardening modulus, the plastic softening modulus, the yield limit and the residual strength increasing. On the contrary, an increasing of the strength limit can increase the wellbore collapse pressure, which is different to the common understanding. Keywords: Wellbore stability; Stability theory; Equilibrium path; Limit point instability; Analytical model
1. Introduction 1
* Corresponding author. E-mail addresses: [email protected] (Lisong Zhang), [email protected] (Yinghui Bian).. 1
Wellbore collapse frequently occurs during the oil and gas drilling engineering, and therefore is treated as the most important problem for the drilling safety. Wellbore collapse not only causes the additional high drilling cost but also has a severe impact on the drilling schedule (Bradley, 1979; Aadnoy and Chenevert, 1987; Gentzis, 2009; Gentzis et al., 2009a; Gentzis et al., 2009b; Zhang et al., 2015a; Zhang et al., 2015b). At present, the rock strength criteria are widely used to address the wellbore collapse, and have become the main evaluation method for the wellbore collapse. Among these criteria, Mohr-Coulomb (Mohr, 1900), Hoek-Brown (Hoek and Brown, 1967, 1980, 1997; Priest, 2005), Drucker-Prager (Drucker and Prager; 1952) and Mogi-Coulomb (Mogi, 1967, 1971; Al-Ajmi and Zimmerman, 2005, 2006) criteria are most frequently used failure criteria for wellbore stability analysis. Currently, a series of achievements have been achieved for the wellbore stability prediction. Do et al. (2019) used the non-linear failure criterion proposed by Mroz and Maciejewski to predict the wellbore stability, after obtaining the stress state around the wellbore using a new closed-form solution. Li and Weijermars (2019) proposed a modified anisotropic Hoek-Brown failure criterion to predict the mud weight window for the transverse isotropic shale. Zhou et al. (2018) performed a wellbore stability analysis based on Mohr-Coulomb criterion, considering the effects of layer structures and anisotropic seepage. Alkamil et al. (2018) used Mohr-Coulomb, Mogi-Coulomb and Modified Lade criteria to determine the mud pressures for different wells in the Mishrif Formation. Abbasi et al. (2018) applied three different strength criteria, e.g., Mohr-Coulomb, Mogi-Coulomb and Modified Lade, to predict the drilling mud pressure in tight gas sand reservoir. The results showed that the Mogi-Coulomb criterion had a better prediction for the drilling mud pressure, referring to the measured values. Chen and Abousleiman (2017) established the strain hardening and/or softening plasticity models to analyze the wellbore stability. Meanwhile, the authors compared their results to the ones obtained by the elastic theory. Kanfar et al. (2015) introduced Mohr-Coulomb and modified Lade criteria to evaluate the time-dependent wellbore stability based
2
on the finite element method, considering the material anisotropy. Rahimi and Nygaard (2015) compared thirteen failure criteria used for predicting the wellbore collapse, and considered that Modified Lade, Modified Wiebols-Cook and Mogi-Coulomb had the better prediction than other strength criteria for the field case. Chabook et al. (2015) evaluated the role of different strength criteria in the wellbore stability, and recommended the Mogi-Coulomb criterion to calculate the wellbore collapse pressure. Shi et al. (2015) proposed the single-parameter parabolic criterion to predict the wellbore stability, to avoid the shortcoming of the Mohr-coulomb criterion (i.e., using its linear form to describe the nonlinear characteristic of the rock strength). Additionally, Song and Haimson (1997), Ewy (1999), Colmenares and Zoback (2002), Yi et al. (2006), Zhang and Zhu (2007), You (2009), Zhang et al. (2010), Cai (2010), Liu et al. (2012), Lee et al. (2012) and Maleki et al. (2014) also established the respective models to evaluate the wellbore stability based on the rock strength failure criterion. From the above analysis, it can be seen that many models were proposed to predict the wellbore collapse based on rock strength criteria. However, a problem was neglected for the models developed, i.e., rock strength failure at a point does not mean the wellbore collapse, which is a common knowledge in mechanics. In other words, the rock strength criterion can not explain the wellbore collapse or block slip from the wellbore. From this point of view, the strength theory can not illustrate the phenomenon of wellbore collapse. On the contrary, the stability theory can elaborate perfectly the immediate collapse of the structure. In fact, rock strength failure at a point and wellbore collapse belongs to different mechanics scopes. The former is the strength problem, and the latter is the instability problem. The plastic zone around the wellbore is very critical to evaluate the wellbore collapse. If the plastic zone is in stable equilibrium under the slightest disturbance, i.e., the plastic zone remains the initial equilibrium configuration under the slightest disturbed load, the plastic zone instability (i.e. wellbore collapse) does not occur. Conversely, the plastic zone instability occurs if the plastic
3
zone is in unstable equilibrium state under the slightest disturbance. The disturbance maybe results from the mud circulation or the bottomhole pressure fluctuation. Therefore, theoretically speaking, the wellbore collapse belongs to the stability problem of equilibrium, but instead of strength failure problem. Correspondingly, the wellbore collapse should be evaluated by the stability theory, but instead of the strength criteria. Furthermore, a larger plastic displacement certainly occurs before the wellbore instability. Therefore, the wellbore collapse satisfies the characteristic of the limit point instability, and can be treated as the limit point instability. Although important improvements have been achieved in the area of wellbore instability, the wellbore collapse mechanism has not been fully understood. In this paper, an analytical model was established based on the stability theory, to calculate the wellbore collapse pressure. In order to verify the proposed model, the analytical results of stresses and displacements in the plastic zone were compared with the finite element method.
2. Wellbore collapse mechanism 2.1 Equilibrium state In the stability theory, the equilibrium state can be divided into 3 categories: the stable equilibrium state, the unstable equilibrium state and the neutral equilibrium state. The stable equilibrium state is that structures can return to the initial equilibrium location after removing the slightest disturbance (seeing Fig.1 (a)). The unstable equilibrium state is that structures can not return to the initial equilibrium location after removing the slightest disturbance (seeing Fig.1 (b)). The neutral equilibrium state is that structures remain the equilibrium state at arbitrary location after removing the slightest disturbance (seeing Fig.1 (c)).
4
Fig.1. Different equilibrium states
2.2 Wellbore instability mechanism Generally speaking, a plastic zone around the wellbore has an important effect on wellbore instability (Yin and Chen, 2009; Yin and Di, 2014). If the plastic zone remains the stable equilibrium state, no wellbore instability occurs. On the contrary, if the plastic zone can not remain the stable equilibrium state, wellbore instability occurs. For the practical engineering, the wellbore collapse belongs to the limit point instability, i.e., the structure may collapse (destruction) immediately after reaching the limit point at which the maximum load appears. Correspondingly, the equilibrium path (i.e., the load-displacement curve) was shown in Fig. 2, where the segment OFG represents the stable equilibrium state, while the segment GHJ represents the unstable equilibrium state. For the wellbore collapse, the generalized force can be treated as a dimensionless form with respect to the inner pressure P0 , and the generalized displacement can be treated as a dimensionless form with respect to the radial displacement u .
Fig.2. The equilibrium path for limit point instability
3. Stress and displacement around the wellbore To derive analytically the elastoplastic solution of the stress and displacement, the wellbore model was divided into 3 parts, namely, the model I under the uniform stress field, the model II under the disturbed stress field and the model III under the non-uniform stress, as shown in Fig. 3.
5
Fig.3. Wellbore analysis model
To highlight the plastic behavior of rocks, the stress-strain relation was modeled a quad-linear model, as shown in Fig. 4, where segment OA represents the elastic stage with a elastic modulus of E , segment AB represents the hardening stage with the plastic hardening modulus of mE , segment
BC represents the softening stage with the plastic softening modulus of nE , segment CD represents the plastic residual stage. Meanwhile, σ s , σ b , σ t respectively represent the limit stresses of each stage, e.g., the yield limit σ s , the strength limit σ b and the residual stress σ t .
6
Fig.4. Stress-strain relation of the quad-linear model
3.1. Analytical solution in the elastic range For the model I, II and III, the stress and displacement solutions can be expressed as Eqs. (1)-(3) in the elastic range.
σ r′ = -σ h σ θ′ = -σ h U r′ = 0
(1)
a2 ′′ σ = r r 2 ∆P0 a2 ′′ = − ∆P0 σ θ 2 r 1 + µ a2 ′′ ∆P0 U r = − E r
(2)
7
σ − σ h a2 σ H − σ h a2 a2 1 − − 1 − 1 − 3 σ r′′′= − H cos2θ 2 r2 2 r 2 r2 2 4 σ ′′′ = − σ H − σ h 1 + a + σ H − σ h 1 + 3 a cos2θ θ 2 r2 2 r4 σ H − σ h a2 a2 ′′′ = 1 − 1 + 3 sin2θ τ rθ 2 2 2 r r 2 σ − σ h ) cos 2θ (1 + µ ) a2 9 (1 + µ ) a4 Ur′′′= − σ H − σ h (1 − µ ) r + (1 + µ ) a − ( H µ r + 6 + 4 − ( ) 2E r 5E r 2 r3 2 3 (σ H − σ h ) sin 2θ (1 + µ ) a2 (1 + µ ) a4 ′′′ U r = + 1 − + µ ( ) θ r 5E 2 r3 2
(3)
where σ r , σ θ and τ rθ are the radial, hoop and shear stresses, U r , Uθ are the radial and hoop displacements, r , θ are the radial and hoop coordinates, a is the borehole radius, P0 is the drilling fluid pressure, σ H , σ h are the maximum and minimum principle stresses, E is the elastic moulds, µ is the Poisson ratio. Using Eqs. (1)-(3), the total stresses and displacements can be given as:
σ r = σ r′ + σ r′′ + σ r′′′ σ = σ ′ + σ ′′ + σ ′′′ θ θ θ θ ′′′ τ = τ rθ rθ U = U ′ + U ′ + U ′′′ r r r r ′′′ Uθ = Uθ
(4)
In the actual engineering, the difference of the maximum and minimum horizontal principle stresses is limited. As a result, the shear stress is approaching to 0, i.e., τ rθ = 0 . Additionally, the hoop displacement has a smaller effect on the wellbore stability. Therefore, it can be neglected in this analysis. According to the analysis above, Eq. (4) can be further simplified as:
σ r = σ r′ + σ r′′ + σ r′′′ σ θ = σ θ′ + σ θ′′ + σ θ′′′ U r = U r′ + U r′ + U r′′′
(5)
To obtain the elastic limit pressure P0e , Tresca criterion was introduced to evaluate the initial yielding of rocks. Substituting Eq. (5) into the Tresca criterion, P0e can be determined: 8
P0e =
σ s + ( 3σ h − σ H )
(6)
2
3.2. Analytical solution in the hardening plastic zone Under P0 ≤ P0e , the plastic zone appears around the wellbore. In the plastic zone, the rock shows different mechanical behaviors at the different stages. Commonly, the rock has three plastic stages after yielding, namely, the hardening stage (segment AB), the softening stage (segment BC), and the residual stage (segment CD). For the plastic hardening stage, the plastic zone around the wellbore can be shown in Fig. 5, where ρ1 is the boundary of the elastic and plastic zones, b is the outer boundary of the elastic zone, much larger than the wellbore radius a .
Fig.5. Stress and displacement analysis model for the plastic hardening phase
In the plastic zone, the yielding surface can be expressed as Eq. (7) based on the Tresca criterion.
(
f1 = σ θ − σ r − σ s 1 + η1ε ip where η1 =
(
mE , ε p is the plastic strain, σ s 1+ η1ε ip (1 − m )σ s i
)
Meanwhile, the associated flow rule can be described as:
9
)
is the yield limit in the plastic flow.
(7)
dε ip =
∂f ∂σ i
dλ
(8)
where d λ is the plastic multiplier. Then, the plastic strain increment can be expressed as:
dε rp1 = − dλ1 p dε θ 1 = dλ1 p dε z1 = 0
(9)
Using Eq. (9), the physical equations can be solved:
σ r − µ (σ θ + σ z ) − λ1 (r ) ε r = E σ θ − µ (σ r + σ z ) + λ1 (r ) ε θ = E σ z − µ (σ r + σ θ ) ε z = E
(10)
where σ z is equal to µ (σ θ + σ r ) based on the plane strain state. The geometry and equilibrium equations can be expressed as:
dε θ ε θ − ε r =0 dr + r dσ r + σ r − σ θ = 0 dr r
(11)
According to the Tresca criterion, the radial and hoop stresses satisfy the following relation:
σ r − σ θ = σ s (1 + η1ε ip ) where ε i = Bλ1 (r ) , B = p
(12)
2 . 3
Eq. (12) can be further expressed as:
σ r − σ θ = σ s 1 + Bη1λ1 ( r )
(13)
Combining Eqs. (10), (11) and (13), the differential equation with respect to λ1 (r ) can be determined:
10
r
d ( λ1 ( r ) ) dr
+ 2λ1 ( r ) +
2 (1 − µ 2 ) σ s
(1 − µ )σ Bη 2
s
1 +E
=0
(14)
Substituting the boundary condition of λ1 (ρ1 ) = 0 into Eq. (14), λ1 (r ) can be solved:
ρ12 − 1 2 r
λ1 (r ) = K1
where K1 =
(15)
(1 − µ )σ . (1 − µ )σ Bη + E 2
s
2
s
Substituting λ1 (r ) into the equilibrium equation, the stresses in the hardening plastic zone can be expressed as:
1 σ s (1 − ξ1 ) ρ12 = − ln + + c2 r σ σ ξ r s 1 2 2 r σ s (1 − ξ1 ) ρ12 1 + c2 σ θ = −σ sξ1 (1 + ln r ) − 2 r2 σ 1z = µ −σ sξ1 (1 + 2 ln r ) + 2c2
(16)
where ξ1 = 1 − Bη1 K1 . Then, the volume strain can be solved: dU r1 U r1 1 d(rU r1 ) (1 − 2 µ )(1 + µ )(σ r + σ θ ) ε r + εθ + ε z = + = = dr r r dr E
(17)
Integrating Eq. (17) to lead to: U r1 =
(1 − 2 µ )(1 + µ ) r −σ ξ ln r + c + A1 [ s1 2] E
r
(18)
Eq. (18) has two unknown parameters, namely, c2 and A1 , which can be solved based on the following derivation. Using the stress boundary condition on r = a , i.e., σ r1 |r = a = −∆P0 , the constant c2 can be determined: c2 = σ sξ1 ln a −
σ s (1 − ξ1 ) ρ12 2 11
a2
+ ∆P0
(19)
Additionally, combining the stress and displacement continuity conditions at r = ρ1 , i.e.,
σ r1 |r = ρ = σ r |r = ρ and U r1 |r = ρ = U r |r = ρ , ∆P0 and A1 can be obtained: 1
1
1
1
∆P0 = σ sξ1 ln
A1 = −
ρ1 a
+
σ s (1 − ξ1 ) ρ12
σ 2 − 1 + s a 2
2
(1 + µ ) ρ 2 σ s − (1 − 2µ )(1 + µ ) ρ 2 −σ ξ ln ρ + c 1 1 [ s 1 1 2] E
2
E
(20)
(21)
Especially, the stress and displacement on the boundary of the elastic zone, i.e., σ r |r = ρ1 and U r |r = ρ1 , can be solved based on Eq. (5).
3.3. Analytical solution in the softening plastic zone After the rock enters into the plastic softening stage, the plastic zone near the wellbore can be shown in Fig. 6, where ρ 2 is the boundary of the plastic hardening stage and the plastic softening stage.
Fig.6. Stress and displacement analysis model for the plastic softening phase
At the plastic boundary of r = ρ 2 , the yield stress is equal to the strength limit, i.e., Eq. (22):
σ s 1 + Bη1λ1 ( r ) = σ b
12
(22)
Using Eq. (22), the plastic multiplier λ1 (r ) can be given:
λ1 (r ) |max =
σ b −σ s σ s Bη1
(23)
Substituting Eq. (23) into Eq. (15), the relation of the plastic radius ρ1 and ρ 2 for different plastic stages can be expressed as:
ρ12 σ b − σ s = +1 ρ 22 σ s Bη1 K1
(24)
In this stage, the radial stress σ r and the hoop stress σ θ also satisfy the Tresca criterion:
σ r − σ θ = σ b (1 − η 2ε ip ) where η2 =
(25)
nE σ −σs , ε ip = Bλ2 (r ) + b . σ sη1 (1 − n ) σ s
Then, Eq. (25) can be further rewritten as:
σ r − σ θ = σ b 1 − Bη 2λ2 ( r ) −
( σ b − σ s )η 2 σ sη1
(26)
Similar to the derivation of the plastic gardening stage, the differential equation with respect to
λ2 (r ) can be determined: r
d ( λ2 ( r ) ) dr
1 − µ 2 )σ b ( σ −σs =0 + 2λ2 ( r ) + 2 2 + b ( µ − 1) σ b Bη2 + E σ s Bη1
(27)
Substituting the boundary condition of λ2 (ρ 2 ) = 0 into Eq. (27), λ2 (r ) can be obtained:
ρ 22 − 1 2 r
λ2 (r ) = K 2
(1 − µ )σ = ( µ − 1)σ Bη 2
where K 2
2
b
(28)
σb −σs . σ s Bη1 2 −E b
+
Substituting λ2 (r ) into the equilibrium equation, the stresses in the plastic softening phase can be expressed as:
13
2 σ bξ3 ρ 22 + c3 σ r = −σ bξ 2 ln r − 2 r2 σ bξ3 ρ 22 2 + c3 σ θ = −σ bξ 2 (1 + ln r ) + 2 r2 σ z2 = µ −σ bξ 2 (1 + 2 ln r ) + 2c3 where ξ 2 = 1 + Bη2 K 2 −
(29)
σ b -σ s η 2 , ξ 3 = Bη 2 K 2 . σ s η1
Similar to Eq. (17), the radial displacement in the plastic softening stage can be yielded: r (1 − 2 µ )(1 + µ )( −σ bξ 2 ln r + c3 )
U r2 =
E
+
A2 r
(30)
where c3 and A2 are the unknown variables, and can be solved by the following derivation. Based on the stress condition on the boundary of r = a , i.e., σ r2 | r =a = − ∆P0 , the constant, c3 , can be given: c3 = σ bξ 2 ln a +
σ bξ3 ρ 22 a2
2
+ ∆P0
(31)
Additionally, the stress redistribution occurs in the hardening plastic zone, when the rock enters into the plastic softening stage. Therefore, the parameters, c2 and A1 , need to be resolved to ensure the stress and displacement accuracy. Combining the stress and displacement continuity conditions on r = ρ1 and r = ρ 2 , i.e., σ r1 |r = ρ1 = σ r |r = ρ1 , σ r2 |r = ρ 2 = σ r1 |r = ρ 2 , U r1 |r = ρ1 = U r |r = ρ1 and
U r2 |r = ρ 2 = U r1 |r = ρ 2 , the parameters, c2 , ∆P0 , A1 , A2 , can be determined accurately: c2 = σ sξ1 ln ρ1 −
∆P0 = σ bξ 2 ln
ρ2 a
A1 = − A2 = A1 +
+
σ s (1 − ξ1 ) σ s 2
+
2
ρ2 ρ σ (1 − ξ1 ) ρ12 σ s 1 − 2 + 1 − 22 - σ sξ1 ln 2 − s 2 a 2 ρ1 ρ2 2
σ bξ 3
(1 + µ ) ρ 2 σ s − (1 − 2µ )(1 + µ ) ρ 2 −σ ξ ln ρ + c 1 1 ( s 1 1 2) E
2
E
(1 − 2 µ )(1 + µ ) ρ 2 −σ ξ ln ρ + c − −σ ξ ln ρ + c 2 ( s 1 2 2) ( b 2 2 3 ) E
14
(32)
(33)
(34) (35)
3.4. Analytical solution in the residual plastic zone After the rock enters into the plastic residual stage, the plastic zone near the wellbore can be shown in Fig. 7, where ρ3 is the boundary of the plastic residual phase and the plastic softening phase.
Fig.7. Stress and displacement analysis model for the plastic residual phase
At the plastic boundary of r = ρ3 , the yield stress is equal to the residual stress, i.e.,
σ b 1 − Bη 2λ2 ( r ) −
( σ b − σ s )η 2 = σ σ sη1
(36)
t
According to Eq. (36), the plastic multiplier λ2 (r ) can be yielded:
λ2 ( r ) |max =
σ b 1 −
(σ b − σ s )η 2 − σ σ sη1 σ b Bη2
t
(37)
Substituting Eq. (37) into Eq. (28), the relation of the radius ρ 2 and ρ3 of the plastic zones can be expressed as:
ρ = ρ 2 2 2 3
σ b 1 −
( σ b − σ s )η 2 − σ σ sη1 σ b Bη2 K 2
15
t
+1
(38)
In the residual plastic zone, the radial stress σ r and the hoop stress σ θ still satisfy the Tresca criterion:
σ r − σθ = σ t
(39)
Substituting Eq. (39) into the equilibrium equation, the stresses in the residual plastic zone can be expressed as: σ 3 = −σ lnr + c t 4 r 3 3 σ θ = σ t + σ r = −σ t ( lnr + 1) + c 4 3 3 3 σ z = µ (σ θ + σ r ) = − µσ t ( 2lnr + 1) + 2 µ c 4
(40)
Meanwhile, the radial displacement in the residual zone can be expressed as: U r3 =
(1 − 2µ )(1 + µ ) r −σ ln r + c + A3 ( t 4) E
r
(41)
In Eq. (41), the parameters, c4 , A3 , are unknown. According to the stress boundary condition of σ r3 |r =a = −∆P0 , the parameter, c4 , can be obtained: c4 = σ t ln a + ∆P0
(42)
Similarly, the stress redistribution occurs in the hardening and softening plastic zones, when the rock enters into the residual plastic stage. Therefore, the parameters, c2 , c3 , A1 and A2 , need to be resolved to ensure the stress and displacement accuracy. Combining the stress and displacement continuity conditions at r = ρ1 , r = ρ 2 and r = ρ3 , i.e., σ r1 |r = ρ1 = σ r |r = ρ1 ,
σ r2 |r = ρ = σ r1 |r = ρ , σ r3 |r = ρ = σ r2 |r = ρ , U r1 |r = ρ = U r |r = ρ , U r2 |r = ρ = U r1 |r = ρ 2
2
3
3
1
1
2
2
and U r3 |r = ρ 2 = U r3 |r = ρ 2 ,
the unknown parameters, c2 , c3 , ∆P0 , A1 , A2 , A3 , can be obtained: c2 = σ sξ1 ln ρ1 −
c3 = σ bξ 2 ln ρ 2 +
σ bξ 3 2
σ s (1 − ξ1 ) σ s
− σ sξ1 ln
2
+
2
ρ 2 σ s (1 − ξ1 ) ρ12 σ s − 1 − 2 + 2 ρ1 ρ2 2 16
(43)
(44)
ρ3 σ bξ3 ρ22 ρ2 σ s (1 − ξ1 ) ρ12 σ s ∆P0 = σ t ln − σ bξ 2 ln − − 1 − σ sξ1 ln − 1 − 2 + a 2 ρ32 2 ρ2 ρ1 ρ2 2 ρ3
(1 + µ ) ρ 2 σ s − (1 − 2µ )(1 + µ ) ρ 2 −σ ξ ln ρ + c 1 1 ( s 1 1 2)
(46)
(1 − 2µ )(1 + µ ) ρ 2 −σ ξ ln ρ + c − −σ ξ ln ρ + c 2 ( s 1 2 2) ( b 2 2 3 )
(47)
(1 − 2µ )(1 + µ ) ρ 2 −σ ξ ln ρ + c − −σ lnρ + c 3 ( b 2 3 3) ( t 3 4 )
(48)
A1 = − A2 = A1 +
(45)
A3 = A2 +
E
2
E
E
E
After calculating the stress and displacement of the plastic zone under the uniform stress field, the total stress and displacement under the non-uniform stress field can be calculated approximately, seeing "Appendix".
4. Wellbore collapse analysis 4.1. Equilibrium path The equilibrium path is the most important step for analyzing the wellbore collapse problem using the stability theory. Considering that P0 and U r are the control variables for the wellbore stability, the equilibrium path depends on the inner pressure P0 and the radial displacement U r . Note that, the radial displacement U r is a smallness item and is not convenient to be used in the equilibrium path. Therefore, the equilibrium path is expressed as a dimensionless form with respect to P0 and U r . The dimensionless process of P0 and U r is herein expressed as:
2 (σ h − P0 ) 2 (σ H − σ h ) − (5 − 2 cos 2θ ) U r = σs 5 (1 + µ ) σ s 2 P0 P = σ s + 4σ h + (σ H − σ h ) cos 2θ
(49)
where p is the dimensionless force, U r is the dimensionless displacement. Using Eq. (5), the equilibrium path in the elastic range can be determined: Ur =
2σ h − σ s + 4σ h + (σ H − σ h ) cos 2θ p 2 (σ H − σ h ) − (5 − 2 cos 2θ ) σs 5 (1 + µ ) σ s 17
(50)
Using Eqs. (18) and (20), the equilibrium path in the plastic hardening zone can be concluded as:
ρ12 ρ1 ρ12 2 (σ H − σ h ) U 1 2 µ ξ ( 2 ln 1) (5 − 2 cos 2θ ) = − + − + − ) 1 2 r ( 2 a a a 5 1 + µ σ ( ) s 2 ρ ρ 2σ h − σ s ξ1 (1 + 2 ln 1 ) + (1 − ξ1 ) 12 a a P = σ s + 4σ h + (σ H − σ h ) cos 2θ
(51)
Using Eqs. (30) and (33), the equilibrium path in the plastic softening zone can be derived as:
2 U = 2 (1 − 2 µ ) σ b ξ ln a − σ b ξ − ξ ( 1 − ln ρ 2 ) − 1 − ξ1 ρ1 2 3 1 r 2 2 ρ 22 ρ 2 2σ s ρ1 σs 2 (σ H − σ h ) σ ρ2 + 22 2 (1 − µ ) + (1 − 2 µ ) b ξ3 − (5 − 2 cos 2θ ) a σ s 5 (1 + µ ) σ s σb ρ2 σ b ρ22 ρ2 ρ12 2σ h − σ s 2 ξ 2 ln + ξ3 (1 − 2 ) + ξ1 (1 − 2 ln ) + (1 − ξ1 ) 2 σs ρ1 ρ2 a σs a P = σ s + 4σ h + (σ H − σ h ) cos 2θ
(52)
Using Eqs. (41) and (45), the equilibrium path in the plastic residual zone can be given as: 2 2 U = 2 (1 − 2 µ ) σ t ln a + σ b ξ ln ρ3 + σ b ξ ( ρ 2 − 1) − ξ ( 1 − ln ρ 2 ) − 1 − ξ1 ρ1 1 r σ s ρ3 σ s 2 ρ 2 2σ s 3 ρ32 ρ1 2 2 ρ 22 ρ 2 2 (σ H − σ h ) + 2 (1 − µ ) 12 − (5 − 2 cos 2θ ) a 5 (1 + µ ) σ s σ ρ σ ρ σ ρ2 ρ ρ2 2σ h − σ s 2 t ln 3 − 2 b ξ 2 ln 3 − b ξ3 ( 22 − 1) + ξ1 (1 − 2 ln 2 ) + (1 − ξ1 ) 12 a σs ρ2 σ s ρ3 ρ1 ρ2 σs P = σ s + 4σ h + (σ H − σ h ) cos 2θ
(53)
According to Eqs. (50), (51), (52) and (53), the equilibrium path for the wellbore collapse problem can be determined. To show the solution process of the equilibrium path more clearly, a flowchart was depicted in Fig. 8. Especially, the selection of the independent variable was highlighted in this analysis. Generally speaking, the critical instability state can be achieved by changing the generalized force. That is to say, the wellbore collapse can occur through decreasing the inner pressure of the wellbore. 18
In such case, the inner pressure should be treated as the independent variable, and the other parameters can be treated as the dependent variable. However, a problem appears under this case, i.e., the plastic zone boundary of ρ1 , ρ 2 and ρ3 are difficult to be solved based on Eqs. (20), (30) and (45), which directly increases the solution difficulty for U r . In view of this, ρ1 was selected as the independent variable, mainly because: (1) the other parameters can be obtained by changing
ρ1 ; (2) the wellbore collapse can be achieved by increasing the plastic zone boundary of ρ1 .
Fig. 8. Calculation flowchart of the equilibrium path.
4.2. Collapse pressure analysis Using the condition that the first derivative of Eq. (52) is equal to 0, the critical radius where p reaches the maximum value p max can be solved as Eq. (54). In fact, the critical radius ρ cr is the boundary of the softening zone and the residual zone.
ρcr = a
ξ 2 (σ b σ s − ξ1 ) ξ3 (1 − ξ1 ) 19
(54)
According to the limit point instability, the wellbore collapse occurs immediately once
p = p max . Therefore, the collapse pressure Pci of the wellbore can be solved by substituting Eq. (54) into Eq. (52): Pcr =
σb 2
(ξ 2 ln
σ σ σ −ξ ξ2 + ξ3 − ξ 2 + 1) − s ξ1 ln b s 1 2 1 − ξ1 ξ3
(55)
Eq. (55) is the collapse pressure obtained by the stability theory, which is obviously different from the one by the strength criterion. Additionally, the wellbore collapse occurs at the end of the softening phase or the beginning of the residual phase, which is different to the conclusion of the rock strength criterion.
5. Results and discussions TS-1 well was drilled as a vertical well in Xinjiang oil field. Due to the tectonic stress, the wellbore collapse occurs frequently in this region. In such case, it is very important to predict accurately the collapse pressure. In this analysis, a plane strain model was used, and the basic parameters were listed as follows: the depth H = 2044 m, the borehole radius a = 157.05 mm, the horizontal maximum in-situ stress σ H = 43.33 MPa, the horizontal minimum in-situ stress
σ h =41.90 MPa. Additionally, the mechanical parameters of the rock were concluded as follows: the elastic modules E =33.9 GPa, the Poisson ratio µ = 0.33, the yield stress σ s = 70 MPa, the strength limit σ b = 90 MPa, the residual stress σ t = 35 MPa, m = 0.20, n = 1.20.
5.1. Validation of the proposed method Based on the previous parameters, the equilibrium path of arbitrary point near the wellbore for TS-1 well can be determined by calculating the dimensionless parameters of U r and p . To exhibit the wellbore instability, 7 track points were set to determine the detailed instability location, with the angles of θ = 0°, 15°,
, 90° at the inner surface of the wellbore. Then, the equilibrium
paths of 7 track points were solved and shown in Fig. 9. 20
Fig. 9. The equilibrium paths for different track points.
As seen from Fig. 9, the critical pressures for 7 track points are different, due to the non-uniform in-situ stress field. However, the difference of the critical pressures is limited, mainly because the difference of the horizontal maximum and minimum stresses is small. The wellbore instability occurs firstly at θ = 90 °, because the critical pressure pcr is highest in this location. Substituting pcr into Eq. (55) or (49), the collapse pressure Pci of TS-1 well can be calculated, equaling to 19.56 MPa. Meanwhile, the critical pressure is lowest at θ = 0 °, which indicates that the wellbore collapse is difficult to occur at this location. Specially, in order to further verify the accuracy of the proposed method, the stresses and displacements from the proposed method were compared to the ones from the numerical method (see Fig.10).
21
(a) Radial displacements (r = a)
(b) Hoop displacements (r = a)
(c) Radial stresses (r = a)
(d) Hoop stresses (r = a)
(e) Radial displacements (r = 2a)
(f) Hoop displacements (r = 2a)
22
(g) Radial stresses (r = 2a)
(h) Hoop stresses (r = 2a)
Fig. 10. Result comparison of the proposed method and numerical method.
As seen from Fig. 10, the analytical results agree well with the results from the numerical method at r=a and 2a, regardless of the stress or displacement results. The deviations between the analytical and the numerical results are within 1.6 % ~ 8.9 %, mainly because of different assumptions made in the analytical and numerical methods. According to analysis above, the proposed analytical model can calculate accurately the stress and displacement around the wellbore. Meanwhile, the finite element method shows a better agreement to the analytical method for the stress and displacement results, mainly because of the same mechanics assumption. A rigid symmetry can be observed for the results from the analytical and finite element methods in Fig. 11, regardless of the stress or displacement.
5.2. Parametric analysis As seen from Eqs. (51), (52), (53), (54), the plastic hardening modulus, the plastic softening modulus, the yielding stress, the strength limit and the residual strength have important effects on the equilibrium path and the wellbore collapse pressure. Therefore, the parametric analysis is performed to investigate the effects of these parameters on the equilibrium path.
5.2.1 Effect of the plastic hardening modulus To investigate the effect of the plastic hardening modulus on the equilibrium path, the plastic hardening modulus varies at a range of 0.10 E ~0.30 E . 23
Fig. 11. The effect of the plastic hardening modulus on the equilibrium path.
As shown in Fig. 11, with the increasing of the plastic hardening modulus, the dimensionless critical pressure (i.e., the peak at the equilibrium path) decreases obviously. In other words, the wellbore collapse pressure decreases as the plastic hardening modulus increasing, referring to the relation of p and P0 in Eq. (49). This means that the wellbore collapse occurs more difficultly with the increasing of the plastic hardening modulus. Note that, there has no peak point at the equilibrium path for the smaller plastic hardening modulus, which means that the wellbore collapse pressure is higher for m= 0.10 and 0.15. That is to say, the wellbore is difficult to maintain stable when the plastic hardening modulus is lower in the practice engineering. Meanwhile, the plastic hardening modulus is only related to the plastic hardening stage, and has no effect on the elastic stage. Therefore, the dimensionless displacement and pressure is approximately same at the elastic stage, even if the plastic hardening modulus changes. Overall, an increasing of the plastic hardening modulus is beneficial for maintaining the wellbore stability, because it decreases the collapse pressure.
5.2.2 Effect of the plastic softening modulus To investigate the effect of the plastic softening modulus on the equilibrium path, the plastic 24
softening modulus changes from 1.0 E to 1.4 E .
Fig. 12. The effect of the plastic softening modulus on the equilibrium path.
As shown from Fig. 12, the collapse pressure (i.e., the peak on the equilibrium path) decreases obviously, as the plastic softening modulus increasing. Note that, there has no peak point to appear on the equilibrium path at n= 1.0, meaning that the wellbore collapse pressure is very high. In such case, the wellbore stability is difficult to maintain in the practice engineering. Meanwhile, the plastic softening modulus has no effect on the elastic and plastic hardening stages. As a result, the equilibrium paths are close to each other in the elastic and plastic hardening stages. Overall, a higher plastic softening modulus is good for maintaining the wellbore stability, because it decreases the wellbore collapse pressure.
5.2.3 Effect of the yielding stress To investigate the effect of the yielding stress on the equilibrium path, the yielding stress varies at a range of 60 MPa ~ 80 MPa.
25
Fig. 13. The effect of the yielding stress on the equilibrium path.
As seen from Fig. 13, with the increasing of the yielding stress, the dimensionless critical pressure (i.e., the peak on the equilibrium path) decreases obviously. According to the physical meaning of the dimensionless pressure, it can be determined that the wellbore collapse pressure decreases as the yielding stress increasing. Additionally, the dimensionless critical displacement decreases as the yielding stress increasing. This is mainly because the larger yielding stress decreases the plastic deformation capacity in the hardening stage, when the strength limit keeps constant. Overall, an increasing of the yielding stress can avoid effectively the wellbore collapse based on the stability theory, because it decreases the wellbore collapse pressure.
5.2.4 Effect of the strength limit To investigate the effect of the strength limit on the equilibrium path, the strength limit varies at a range of 80 MPa ~ 100 MPa.
26
Fig. 14. The effect of the strength limit on the equilibrium path.
As seen from Fig. 14, the dimensionless critical pressure increases significantly, as the strength limit increasing. Correspondingly, the wellbore collapse pressure increases with the increasing of the strength limit. This is because the larger strength limit extends the plastic deformation capacity of the hardening stage, and further increases wellbore collapse pressure. In the previous strength models, the effect of the strength limit on the wellbore collapse can not be reflected, because the strength limit can not be contained in the strength criterion. From the point of view, it is an advantage to analyze the wellbore collapse pressure using the stability theory. Overall, the larger strength limit is adverse for maintaining the wellbore stability, because it increases the wellbore collapse pressure. This conclusion is different to the common understanding.
5.2.5 Effect of the residual strength To investigate the effect of the residual strength on the equilibrium path, the residual strength varies at a range of 25 MPa ~ 45 MPa.
27
Fig. 15. The effect of the residual strength on the equilibrium path.
As shown from Fig. 15, with the increasing of the residual strength, the dimensionless critical pressure decreases obviously, i.e., the wellbore collapse pressure decreases obviously. Especially, the trends of the equilibrium path are similar each other, mainly because the residual strength has no effect on the elastic, plastic hardening and plastic softening stages. Additionally, the stability theory is more suitable than the strength criterion to analyze the wellbore stability, because the residual strength can be reflected in this analysis. Overall, a higher residual strength is very important for maintaining the wellbore stability, because it decreases the wellbore collapse pressure.
6. Conclusions (1) In essence, the welbore collapse belongs to the stability problem, but instead of the strength problem. Using the stability theory, an analytical model was established to determine the wellbore collapse pressure. In this model, the radial displacement under the non-uniform in-situ stress field can be solved based on the elastoplastic mechanics, and then the equilibrium path can be obtained with respect to the radial displacement and the inner pressure. (2) The proposed method agrees well with the numerical method. The deviation of less than
28
8.9 % can be observed between the proposed method and the numerical method, which means that the proposed method can satisfy the practice engineering. (3) Using the stability theory, the effects of the parameters on the wellbore collapse pressure can be investigated, including the plastic hardening modulus, the plastic softening modulus, the yielding stress, the strength limit and the residual strength, while the effects mentioned above can not be discussed using the strength criterion. This can be treated an obvious advantage on the stability method over the strength criterion. (4) The parametric analysis shows that: 1) the higher plastic hardening modulus, plastic softening modulus, yield limit and residual strength are beneficial for the wellbore stability, because of decreasing the wellbore collapse pressure; 2) the higher strength limit can increase the wellbore collapse pressure, which is adverse for the wellbore stability. This conclusion is different from the previous understanding.
Acknowledgment The authors are very much indebted to the Projects Supported by PetroChina Innovation Foundation (2018D-5007-0309), Focus on Research and Development Plan in Shandong Province (2019GGX103007), and the Fundamental Research Funds for the Central Universities (19CX02034A) for the financial support.
Appendix In the plastic range, the total stress and displacement under the non-uniform stress field can be assumed as a sum of the plastic ones under the uniform stress field and the elastic ones under the non-uniform stress field (i.e., Model III). By adding Eq. (3) to Eqs. (16) and (18), the total stress and displacement in the plastic hardening zone can be expressed as:
29
1 σs σH −σ h r σ s (1 − ξ1 ) ρ12 a2 σ H −σ h a2 a2 = − + − + − − − − − − σ σ ξ ln ( 1) σ (1 ) (1 )(1 3 )cos2θ r s 1 h 2 2 2 2 ρ 2 r 2 2 r 2 r r 1 1 σ (1 − ξ1 ) ρ12 σ σ −σh r a2 σ −σ h a4 ( 2 + 1) + s − σ h − H (1 + 2 ) + H (1 + 3 4 )cos2θ σ θ = −σ sξ1 (1 + ln ) − s ρ1 2 r 2 2 r 2 r 2 σ 1z = −2 µ (σ sξ1 ln r − σ h ) + (σ H − σ h ) µ (2 a 2 cos2θ − 1) r ρ1 (56) 2 µ µ µ 1 − 2 1 + 1 + σ ξ σ ( )( ) r ( ) ρ r s 1 s 1 U 1 = (1 − 2 ln ) − 1 + (1 − 2 µ ) ξ1 r ρ1 2 E E r 2 (σ H − σ h )(1 + µ ) r (1 − µ + a 2 ) − (σ H − σ h )(1 + µ ) r (1 + 6 + 4µ a 2 − 9 a 4 ) cos 2θ − 2E 1+ µ r2 10 E 1+ µ r2 r4 1 3 (σ H − σ h ) sin 2θ (1 + µ ) a 2 (1 + µ ) a 4 + 1 − + r µ ( ) Uθ = 5E 2 r3 r 2
By adding Eq. (3) to Eqs. (29) and (30), the total stress and displacement in the plastic softening zone can be given: 2 r σ bξ3 ρ22 ρ2 σ s (1 − ξ1 ) ρ12 σ s = − ln + (1 − ) − ln − (1 − )+ σ σ ξ σ ξ r b 2 s 1 2 2 2 r 2 2 ρ ρ ρ 2 1 2 2 2 2 σ −σh σ −σh a a a −σh − H (1 − 2 ) − H (1 − 2 )(1 − 3 2 )cos2θ r r r 2 2 2 2 σξ ρ ρ σ s (1 − ξ1 ) ρ2 σ r (1 − 12 ) + s σ θ = −σ bξ2 (1 + ln ) + b 3 (1 + 22 ) − σ sξ1 ln 2 − ρ2 ρ1 ρ2 2 r 2 2 σ −σh a2 σ −σ h a4 −σh − H (1 + 2 ) + H (1 + 3 4 )cos2θ r 2 2 r ρ ρ2 r σ z2 = µ −σ bξ2 (1 + 2ln ) + σ bξ3 − 2σ sξ1 ln 2 − σ s (1 − ξ1 ) (1 − 12 ) + σ s − 2σ h ρ2 ρ1 ρ2 2 a + (σ H − σ h ) µ (2 2 cos2θ − 1) r 1 − 2µ )(1 + µ ) ρ σ (1 − ξ1 ) ρ2 σ r σξ U r2 = ( r −σ bξ2 ln + b 3 − σ sξ1 ln 2 − s (1 − 12 ) + s ρ2 ρ1 ρ2 E 2 2 2 2 2 2 (1 − µ )(1 + µ ) ρ1 σ s (1 − 2µ )(1 + µ ) ρ2 σ bξ3 ρ − − ( − σ sξ1 ln 2 ) E r 2 E r 2 ρ1 (σ H − σ h )(1 + µ ) r (1 − µ + a 2 ) − (σ H − σ h )(1 + µ ) r (1 + 6 + 4µ a 2 − 9 a 4 )cos 2θ − 2E 1+ µ r 2 10 E 1+ µ r2 r4 3 (σ H − σ h ) sin 2θ (1 + µ ) a 2 (1 + µ ) a 4 Uθ2 = + 1 − r µ ( ) + 5E 2 r 2 r3
(57)
Similarly, by adding Eq. (3) to Eqs. (40) and (41), the total stress and displacement in the
30
plastic softening zone can be given: 2 2 σ3 =−σ ln r −σ ξ ln ρ3 − σbξ3 ( ρ2 −1) −σ ξ ln ρ2 − σs (1−ξ1 ) (1− ρ1 ) + σs t b 2 s 1 r 2 ρ3 ρ2 2 ρ32 ρ1 ρ22 2 σH −σh a2 σH −σh a2 a2 σ − − (1 − ) − (1 − )(1 − 3 )cos2θ h 2 r2 2 r2 r2 2 2 σ3 =−σ (ln r +1) −σ ξ ln ρ3 − σbξ3 ( ρ2 −1) −σ ξ ln ρ2 − σs (1−ξ1 ) (1− ρ1 ) + σs t b 2 s 1 θ ρ3 ρ2 2 ρ32 ρ1 ρ22 2 2 σH −σh a2 σH −σh a4 σ − − (1 + ) + (1 + 3 )cos2θ h 2 4 2 2 r r σt 3 σb ρ3 ρ22 r ρ2 ρ12 ρ12 3 3 σz = µ (σθ +σr ) =−µσs (2ln +1) + ξ3 (2ln + 2 −1) +ξ1(2ln + 2 −1) − 2 ρ3 σs ρ2 ρ3 ρ1 ρ2 ρ2 σs a2 + (σH −σh )µ(2 2 cos2θ −1) r (1−2µ)(1+ µ) σs r 2σt ln r + σb ξ (2ln ρ3 + ρ22 −1) +ξ (2ln ρ2 + ρ12 −1) − ρ12 − 2(1− µ)(1+ µ) ρ12 σs Ur3 =− 1 3 2 σs ρ3 σs E E r 2 ρ1 ρ22 ρ22 ρ2 ρ32 2 2 4 − (σH −σh )(1+ µ) r(1− µ + a ) − (σH −σh )(1+ µ) r(1+ 6 + 4µ a −9 a )cos2θ 2E 1+ µ r2 10E 1+ µ r2 r4 (58) 3 3(σH −σh ) sin2θ (1+ µ) a2 (1+ µ) a4 r + (1− µ) + Uθ = 5E r 2 r3 2
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34
Fig.1. Different equilibrium states
Highlights:
► Wellbore collapse is a stability problem, but instead of strength failure problem. ► An analytical model of wellbore stability was derived based on stability theory. ► The results of proposed model agree well with those of the numerical model. ► New model can perform the parametric study that can not be analyzed by previous model. ► An important conclusion on strength limit is different to the previous understanding.