Through-wall yield collapse pressure of casing based on unified strength theory

PETROLEUM EXPLORATION AND DEVELOPMENT Volume 43, Issue 3, June 2016 Online English edition of the Chinese language journal Cite this article as: PETROL. EXPLOR. DEVELOP., 2016, 43(3): 506–513.

RESEARCH PAPER

Through-wall yield collapse pressure of casing based on unified strength theory LIN Yuanhua1,*, DENG Kuanhai1, SUN Yongxing2, ZENG Dezhi3, XIA Tianguo4 1. State Key Laboratory of Oil and Gas Reservoir Geology and Exploitation (Southwest Petroleum University), Chengdu 610500, China; 2. CNPC Key Lab for Tubular Goods Engineering (Southwest Petroleum University), Chengdu 610500, China, 3. CCDC Drilling&Production Technology Research Institute 618300, Zhongshan Avenue South 2nd Section,Guanhan city,Sichuan Province. 4. Tarim Oil Field Company, CNPC, Korla, Xinjiang 841000, China.

Abstract: The unified algorithm of through-wall yield collapse pressure for casing with due consideration of strength differential (SD), yield-to-tensile strength ratio, material hardening and intermediate principal stress, which is suitable to calculate collapse strength of all casing has been obtained based on unified strength theory, and four classical through-wall yield collapse formulas of casing have been presented based on the L.Von Mises, TRESCA, GM and twin yield strength criterion. The calculated value is maximum based on the twin yield strength criterion, which can be used as upper limit of through-wall yield collapse pressure, and the calculated value is minimum based on the TRESCA strength criterion, which can be used as lower limit of through-wall yield collapse pressure in the design process. Numerical and experimental comparisons show that the equation proposed by this paper is much closer to the collapse testing values than that of other equations. Key words: unified strength theory; casing collapse; through-wall yield; collapse pressure; strength differential; intermediate principal stress

Introduction In recent years, with the deepening of oil and gas exploration and development, deep wells and ultra-deep wells[1–2] and wells encountering complex formations such as mudstone, shale and plastic rock salt stratum are increasing too, in these wells, casing is exposed in more complicated and harsh conditions, and subjected to higher external pressure in deep and super-deep wells in particular, so conventional API casing cannot meet the strength requirement of deep wells and ultra-deep wells, restricting the development of drilling technology for and drilling depth extension of deep wells[3]. In order to meet the strength requirement on casing in deep and ultra-deep wells, many kinds of heavy wall casings, non-API casings and high collapse casings have been developed successively[4–6], which have higher anti-collapsing capacity than conventional API casings[7]. Even so, there are still cases of failure of high strength casings in deep and ultra-deep wells reported in recent years[8]. Hence, the collapse strength prediction and design of casing are very important for ensuring the safety of oil and gas production. To predict and design the collapse strength of casing accurately, many researches on the casing collapse strength under uniform load[9–14] have been conducted in China and abroad, and some important results (such as building some classical

models ) have been achieved[12, 14]. But the design principles of most of those mechanical models are the initial yield of inner wall of casing under external pressure. In fact, the casing is not damaged completely and still has a significant amount of remaining anti-collapse capacity when the inner wall yields[15–16]. Many tubular products would be wasted in conventional wells and the selection of casing would be difficult if following this principle. In addition, most of the mechanical models[10–12] were built based on the classical Tresca yield criterion and von Mises yield criterion without due consideration of the effect of SD effect and intermediate principal stress on the yield collapse pressure of casing, which makes those mechanical models only applicable to materials with equal tensile and collapse strength[17]. Moreover, further study shows that yield collapse formula in ISO 10400 collapse model[14] is not a complete through-wall yield collapse formula, which underestimates the through-wall yield collapse pressure of casing, and makes the ISO 10400 model not suitable for predicting the collapse strength of casing of all sizes[8, 16]. Hence, a collapse formula considering SD effect, intermediate principal stress, material hardening and yield-to-tensile strength ratio, based on unified strength theory used widely in engineering field, which can predict through-wall yield collapse pressure of casing, has been presented in this paper[18–20].

Received date: 08 Apr. 2015; Revised date: 09 Mar. 2016. * Corresponding author. E-mail: [email protected] Foundation item: Supported by the National Natural Science Foundation of China (51274170). Copyright © 2016, Research Institute of Petroleum Exploration and Development, PetroChina. Published by Elsevier BV. All rights reserved.

LIN Yuanhua et al. / Petroleum Exploration and Development, 2016, 43(3): 506–513

1. 1.1.

Through-wall yield collapse formula for casing

Ri 2    Ri 2   1+ 2   1+    s 2  R  2 1  b   R 2  



Unified strength theory

Many existing strength criteria and some new strength criteria can be derived from the unified strength theory[18–19] which takes the effects of intermediate principal stress and SD effect of material on the material strength into account, and can be used to analyze plastic limit of various materials. The unified strength theory can be given as follows:

    3  2  1  f   1  1  b (b 2   3 )   s 1   1   3 f ( 1  b 2 )   3   s  2  1  1 b 1   t 0   1   c   b  1    s   t 0  b 1   t s

(1)

The stress components (σθ, σr, σz) of the casing increase with the increase of external pressure (po). The casing will be in the elastic limit state, when the inner wall yields initially. Assume that the casing is a long thick-walled cylinder. The long thick-walled cylinder is in the elastic state, when the external pressure (po) is lower. According to the Lame formula[17], the following stress components can be obtained:

(2)

The mechanical analysis of long thick-walled cylinder under external pressure belongs to the axisymmetric plane strain problem. Hence, according to the study on the plane strain problem[21–22], the plasticity coefficient m (m=2ν in the elastic region, m=1 in the plastic region) can be obtained. When the inner wall is in the elastic limit state (in the plastic region), by the Eq. (2), Eq. (3) can be obtained:

z 

m R 2p  r       2o o 2 2 Ro  Ri

The inner wall (R=Ri) of thick-walled cylinder yields firstly, when the external pressure po is equal to the elastic limit pressure (py). By the Eq. (4), the elastic limit pressure (py) can be obtained:

py 

1  b   Ro 2  Ri 2  s   2  b  Ro 2

When po>py, the plastic area will be formed near the inner wall of the thick-walled cylinder. Assume that Rc is the radius of interface between elastic region and plastic region. The plastic area will extend from inner surface to outer surface with the increase of external pressure, so the range of Ri  R  Rc is plastic area and the range of Rc  R  Ro is elastic area, as shown in Fig. 1. The interface between the elastic area and plastic area is a cylinder surface due to the axial symmetry of stress components (σθ and σr), and the interfacial pressure pi is applied to the interface. Therefore, the outer cylinder (elastic area) and inner cylinder (plastic area) can be also analyzed based on the principle of long thick-walled cylinder. Based on axial symmetry, the interfacial pressure (pi) and external pressure (po) act on inner wall and outer wall of outer cylinder respectively, and only interfacial pressure (pi) acts on outer wall of the inner cylinder. For the inner cylinder (plastic region Ri  R  Rc), according to the equilibrium equation of plastic mechanics, the Eq. (6) can be obtained: d r  r     0 (6) R dR By the Eq. (2) and Eq. (4), the relationship between stress component (  r    ) and yield stress (  s ) can be obtained, and the Eq. (7) can be given: 2 1  b   1 2 1  b  (7)  r    r +    2  b   2  b s Substituting Eq. (7) into Eq. (6) gives:

(3)

According to the stress state (σr  σz  σθ) of thick-walled cylinder, it can be known that the first, second and third principal stress (σ1, σ2 and σ3) are equal to σr, σz and σθ respectively. The second principal stress (σ2) is not more than  1   3  / 1    because the tensile strength-compressive strength ratio (α) is less than 1.0. By substituting the Eq. (2), Eq. (3) into Eq. (1), the Eq. (4) can be obtained:



(5)

1.3. Plastic limit analysis of casing under external pressure

1.2. Elastic limit analysis of casing under uniform external pressure

 Ro 2 po  Ri 2   r =  2 1  2  Ro  Ri 2  R    2 Ro po  Ri 2     =  2 1  2  Ro  Ri 2  R    m  z   r     2 

(4)

Ro 2 po  Ri 2  b  Ri 2  1 2   1  2   2 2  Ro  Ri  R  2 1  b   R 

Fig. 1.

 507 

Mechanical model of casing.

LIN Yuanhua et al. / Petroleum Exploration and Development, 2016, 43(3): 506–513

d r A r +B s  0 dR R where A

2 1  b   1  2  b

B

2 1  b    2  b

Substituting boundary conditions (R=Ri, σr=0) into Eq. (8) gives:

 R r  s     1  Ri 

2 1 1 b 

  2b



s  1

(9)

Substituting Eq. (9) into Eq. (7) gives: 21 1 b    2  2b  b   s  R    2b  s  21  b      s     2  b    1  Ri    1   2  b    (10) For the outer cylinder (elastic region Rc  R  Ro), based on the Lame formula, the stress components can be given:  Rc 2 pi  Ro 2 po  pi  po  Rc 2 Ro 2  2  =  r Ro 2  Rc 2 Ro  Rc 2 R 2   Rc 2 pi  Ro 2 po  pi  po  Rc 2 Ro 2 (11)  2   = Ro 2  Rc 2 Ro  Rc 2 R 2    m     / 2 r   z  When R=Rc, from the Eq. (11), the stress components on the inner wall of outer cylinder can be obtained:   =  p i  r  Rc 2 +Ro 2  pi  2 Ro 2 po    = Ro 2  Rc 2   R 2 p  Ro 2 po  z  m c 2i Ro  Rc 2 

(12)

Similarly, the intermediate principal stress (  z  m / 2       , where m = 1) can be obtained in the elastic limit state of outer cylinder. By comparing the stress components in Eq. (12), the relationships between stress components can be obtained:  r   z   (13) Based on the relationships between stress components in Eq. (13), substituting Eq. (12) into Eq. (1) gives:

pi 

  b  2  Ro 2 po  1  b   Ro 2  Rc 2  s Ro 2 1  b     Rc 2 1  b  b   

(14)

According to continuity of the radial stress, the radial stress of elastic region is equal to plastic region at the interface (R=Rc). As a result, from the Eq. (9), Eq. (11) and Eq. (14), Eq. (15) can be obtained: CE (15) po   F where

 R  C  r  s  c    1  Ri 

2 1 1 b 

  2 b



s  1

1  b   Ro 2  Rc 2  s Ro 2 1  b     Rc 2 1  b  b      b  2  Ro 2 F 2 Ro 1  b     Rc 2 1  b  b   

(8)

E

The plastic area increases with the increase of external pressure. The whole wall of thick-walled cylinder is in plastic state, when the radius of interface (Rc) is equal to the outer radius (Ro) of thick-walled cylinder. According to the Eq. (15), the through-wall yield collapse pressure of casing can be obtained: 21b1    s  D     2b pty  1 (16)    1  D   2   

The Eq. (16) gives adequate consideration to the impact of SD effect and intermediate principal stress on through-wall yield collapse pressure of casing. Hence, the through-wall yield collapse pressure of different casing (API casing, high collapse casing and non-API casing) can be obtained by changing the value of α and b, so the Eq. (16) not only solves the problem that the ISO collapse model isn't suitable for predicting the through-wall yield collapse pressure but also improves the prediction accuracy with due consideration of SD effect and intermediate principal stress. In addition, the research results can improve strength design of casing of all sizes, and provide a theoretical basis for the development and improvement of ISO 10400 collapse model. 2. Analysis and discussion of through-wall yield collapse pressure 2.1.

Analysis of through-wall yield collapse pressure

2.1.1. Effect of intermediate principal stress on through-wall yield collapse pressure

In order to analyze the effect of intermediate principal stress on through-wall yield collapse pressure of casing, it is assumed that the tensile strength is equal to compressive strength of casing material (α=1). By calculating the limit of the Eq. (16), the Eq. (17) can be obtained: 21  b D (17) pty  s ln 2b D  2 The influence rule of the intermediate principal stress on through-wall yield collapse pressure of casing can be obtained according to the Eq. (17). The frequently used P110 casing was chosen to analyze the effect of the intermediate principal stress on through-wall yield collapse pressure. The ratio of through-wall yield collapse pressure to yield stress of P110 casing at different influence coefficients of 0, 0.25, 0.50, 0.75 which reflects the effect of intermediate principal stress on the material failure and different radius-thickness ratio (5< D/δ=2Ro/(RoRi)<30) were calculated using Eq. (17), as shown in Fig. 2. The collapse ratio is the ratio of through-wall yield collapse pressure to yield strength of casing in Fig. 2. Fig. 2 shows that the collapse ratio of casing increases with

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Fig. 2. Relationship between intermediate principal stress and collapse ratio.

increase of the influence coefficient (b), which is in agreement with the research results of literature [13] basically, and the influence rule of the value of influence coefficient (b) on collapse ratio of casing at different radius-thickness ratios (5< D/δ<30) is basically the same. It can be concluded that the intermediate principal stress has a great impact on throughwall yield collapse pressure and can’t be ignored. The potential strength of casing material can be given full play by considering of the intermediate principal stress rationally in the process of casing strength design. 2.1.2. Effect of SD effect on through-wall yield collapse pressure

To analyze the impact of SD effect on through-wall yield collapse pressure of casing, it is assumed that the influence coefficient (b) is equal to 1, by calculating the Eq. (16), the Eq. (18) can be obtained: 41    s  D   3   pty  1     1  D   2   

It can be seen from Fig. 3 that the collapse ratio of casing decreases with the increase of tensile strength-compressive strength ratio (SD effect). The greater the strength differential between tensile strength and compressive strength, the greater the collapse ratio will be. The impact extent of SD effect on collapse ratio decreases gradually with the increase of radius-thickness ratio. However, in general, the SD effect has a great impact on the through-wall yield collapse pressure. Hence, the potential strength of casing material can be given full play by considering the SD effect rationally in the process of casing strength design, so the collapse property of casing will be improved to some extent. 2.1.3. Through-wall yield collapse pressure under different yield criteria

Many kinds of important strength criteria can be derived from the unified strength theory under the extreme case that the tensile strength is equal to compressive strength, such as VON MISES, TRESCA, GM and twin shear stress yield criterion[2425]. Hence, many kinds of through-wall yield collapse formulas of casing can be obtained by changing the value of α and b in Eq. (16). 2.1.3.1.

VON MISES yield criterion

VON MISES yield criterion can be derived from the unified strength theory, when α is equal to 1 and the b is equal to 1/(1+√3). By calculating the limit of the Eq. (16), the throughwall yield collapse formula of casing can be obtained based on the VON MISES yield criterion: 21b1    D s  Ro    2b   lim pyL  lim 1     1.155s ln D   2  1  1  1  R  i  

(18)

(19)

The influence rule of SD effect on through-wall yield collapse pressure of casing can be figured out from the Eq. (18). It has been found that the tensile strength-compressive strength ratio of casing material is about 0.8 generally[23]. Hence, the collapse ratio of P110 casing at different tensile strength-compressive strength ratios of 0.7, 0.8, 0.9, 1.0 and different radius-thickness ratios (5
2.1.3.2.

TRESCA yield criterion

TRESCA yield criterion can be derived from the unified strength theory, when α and b are equal to 1 and 0. By calculating the limit of the Eq. (16), the through-wall yield collapse formula of casing can be obtained based on the TRESCA yield criterion: 21b1    D s  Ro    2b   lim pyT  lim 1     s ln D   2  1  1  1  R  i  

2.1.3.3.

(20)

GM yield criterion

GM yield criterion can be derived from the unified strength theory, when α and b are equal to 1 and 2/5 respectively. By calculating the limit of the Eq. (16), the through-wall yield collapse formula of casing can be obtained based on the GM yield criterion:

Fig. 3.

21b1    D s  Ro    2b   lim pyG  lim 1     1.167s ln D   2  1  1  1  R  i  

Relationship between SD effect and collapse ratio.

 509 

(21)

LIN Yuanhua et al. / Petroleum Exploration and Development, 2016, 43(3): 506–513

2.1.3.4.

Twin shear stress yield criterion

Twin shear stress yield criterion can be derived from the unified strength theory, when α and b are both equal to 1. By calculating the limit of the Eq. (16), the through-wall yield collapse formula of casing can be obtained based on the twin shear stress yield criterion: 21b1    Ro s  Ro    2b   lim pyS  lim 1 (22)     1.333s ln R  1  1  1  R i  i    In order to analyze the effect of different yield criteria above on through-wall yield collapse pressure, the throughwall yield collapse pressure of N80 and J55 casing under different radius-thickness ratios have been calculated by using through-wall yield collapse formulas (the Eq. (19), Eq. (20), Eq. (21) and Eq. (22)) obtained based on different yield criteria, as shown in Fig. 4 and Fig. 5. Fig. 4 and Fig. 5 show that through-wall yield collapse pressure of casing decreases rapidly with the increase of radius-thickness ratio. The calculation value of the Eq. (22) derived from the unified strength theory based on twin shear stress yield criterion is maximum, and the calculation value of the Eq. (20) derived from the unified strength theory based on TRESCA yield criterion is minimum. The calculation values of the Eq. (19) and Eq. (21) derived from the unified strength theory based on the other two yield criteria are basically the same. The maximum calculation value obtained by the Eq.

Fig. 4.

Fig. 5.

Through-wall yield collapse pressure of N80 casing.

Through-wall yield collapse pressure of J55 casing.

(22) is about 30 percent higher than the minimum calculation value obtained by the Eq. (20), which indicates that the yield criteria have a significant impact on through-wall yield collapse pressure of casing. Hence, under the extreme case (the tensile strength is equal to compressive strength), the calculation value of the Eq. (22) can be used as upper limit of through-wall yield collapse pressure, and the calculated value of the Eq. (20) can be used as lower limit of through-wall yield collapse pressure in the design process. The calculation value of the Eq. (19) derived from VON MISES yield criterion and Eq. (21) derived from GM yield criterion can be used as average of through-wall yield collapse pressure. 2.2.

Discussion on through-wall yield collapse formula

The accuracy and reliability of the through-wall yield collapse pressure formula based on the yield strength (σs) is not enough and needs to be further improved since the real casing will undergo hardening stage and plastic flow (large plastic deformation) stage from inner wall yield to through-wall yield. As a result, in plastic flow stage, the flow stress (σf) is used to replace yield strength (σs) of casing for calculating the through-wall yield collapse pressure. Klever’s research results[26] also demonstrated that the flow stress (σf) considering the impact of material hardening on the plastic collapse pressure of casing ranged from the yield strength to tensile strength. In addition, for the different metal pipes, many researchers[2729] have presented an empirical formula of flow stress, as follows:  f   s  68.95 (23) To give full consideration of the effect of material hardening on through-wall yield collapse pressure, the through-wall yield collapse formula of casing is presented based on the study[30] on the effect of yield-to-tensile strength ratio (σs/σt) on the through-wall yield collapse pressure. 21b1       f  D     2b   f  s , 0.8  s  1 1  1  D   2   t     pty   21b1       2  b     D     f 1 f  s  68.95, 0  s  0.8  1  D   2  t    (24) The Eq. (24) can be used to calculate the through-wall yield collapse pressure of casing (especially high collapse casing) accurately, which will improve the casing design of deep and ultra-deep wells to a great extent on the guarantee of material safety, and provide important reference and new idea for strength design of casing. Comparing the inner-wall yield collapse pressure with through-wall yield collapse pressure, it can be seen that the inner wall yields only and the local plastic deformation occurs, when the external pressure applied to casing is equal to inner-wall yield collapse pressure, which is defined as in-

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ner-wall yield collapse design. However, the casing is not collapsed, on the contrary, it still has a significant amount of remaining anti-collapse capacity, when the inner wall yields. The whole wall (from inner wall to outer wall) of casing yields and it might be damaged in theory, when the external pressure is equal to through-wall yield collapse pressure, which is defined as through-wall yield collapse design. The yield collapse design method (inner-wall yield and through-wall yield) is suitable for casing (especially high collapse casing) only subjected to uniform external pressure under conventional environment without the effect of acid, temperature and temperature difference. However, the yield failure criterion is not suitable for the acid environment where the environment fracture is the main failure mode. Hence, for the acid environment, the casing strength not only need meet the requirement of strength design criterion, but also the requirement of fracture mechanics design. For the high temperature environment, the effect of extra thermal stress produced by temperature and temperature difference on collapse pressure of casing should be given a consideration. Finally, it should be noted that the through-wall yield collapse formula is suitable for the condition of uniform external pressure only. However, for the non-uniform pressure, the non-uniform pressure should be simplified and related assumptions should be adopted before the through-wall yield collapse formula of casing is derived and solved by using unified strength theory and elastic/plastic limit analysis method. It is necessary to study the through-wall yield collapse pressure under non-uniform pressure by adopting the unified strength theory and related analysis method because the non-uniform pressure has a great impact on the collapse strength of casing.

3.

Numerical and experimental comparisons

It can be known that the calculation value of the Eq. (22) derived from unified strength theory based on the twin shear stress yield criterion can be used as upper limit of throughwall yield collapse pressure according to the analysis. On the contrary, the predicted value of API and ISO 10400 collapse model are too conservative, especially the API model. Hence, the Eq. (22) is used to replace the through-wall yield collapse formula in ISO 10400 collapse model in this paper, which forms the new collapse model used to calculate the collapse strength of casing. The calculation results of the new collapse model have been compared with the calculation results of the API model[12], ISO model[14] and the nine groups of experimental values from the literature[31], as shown in Fig. 6 and Table 1. Fig. 6 and Table 1 show that the calculation results of this new model are much closer to the test data than that of API and ISO model, and the error is less than 10%. The calculation results of API and ISO model are less than the test data, especially the API model, and the error is larger than 10%. It can be conclude that the through-wall yield collapse formula pre-

Fig. 6. Comparison of collapse strength calculation results of three models with experimental values. Table 1.

Comparison of collapse strength calculation results

with experimental values No. D/ Grade 77 13.03 Q125 62 17.57 A110 144 21.21 Q125 139 25.49 A95 140 26.81 P110 115 27.08 N80 136 28.41 P110 146 30.30 N80 148 31.41 N80

ISO model ptest/pISO Error/% 1.337 33.7 1.148 14.8 1.304 30.4 1.253 25.3 1.269 26.9 1.266 26.6 1.280 28.0 1.325 32.5 1.340 34.0

New model API model ptest/pty Error/% ptest/pAPI Error/% 1.037 3.7 1.445 44.5 1.015 1.5 1.211 21.1 1.058 5.8 1.228 22.8 1.010 1.0 1.191 19.1 1.034 3.4 1.384 38.4 1.032 3.2 1.389 38.9 1.045 4.5 1.401 40.1 1.087 8.7 1.479 47.9 1.098 9.8 1.492 49.2

sented by this paper is accurate and reliable and can provide important reference for the prediction and design of casing collapse strength.

4.

Conclusions

The through-wall yield collapse formula of casing presented in this paper with due consideration of SD effect, intermediate principal stress effect, material hardening and yield-to-tensile strength ratio in its calculation, can be used to calculate the through-wall yield collapse pressure of casing with SD effect and intermediate principal stress effect, not only solving the problem that the ISO 10400 collapse model can’t predict the through-wall yield collapse pressure of casing accurately and reasonably, but also improving the prediction accuracy of collapse strength of casing of all sizes. The influence rules of SD effect and intermediate principal stress effect on through-wall yield collapse pressure of casing have been analyzed. The through-wall yield collapse formulas based on four kinds of classic yield criteria have been presented, and the calculation value of collapse formula based on twin shear stress yield criterion is maximum, which can be used as the upper limit of through-wall yield collapse pressure, and the calculation value of collapse formula based on TRESCA yield criterion is minimum, which can be used as the lower limit of through-wall yield collapse pressure in the design process, and the calculation value of collapse formulas based on VON MISES yield criterion and GM yield criterion

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LIN Yuanhua et al. / Petroleum Exploration and Development, 2016, 43(3): 506–513

are basically the same and can be used as average of throughwall yield collapse pressure. The accuracy of this calculation model presented in this paper has been validated by the experimental data.

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