Casing Collapse Strength Analysis under Nonuniform Loading Using Experimental and Numerical Approach

College of Pipeline and Civil Engineering, China University of Petroleum (East China), Qingdao 266580, China CNPC Tubular Goods Research Institute, Xi’an 710077, Shaanxi, China Shanghai Key Laboratory for Digital Maintenance of Buildings and Infrastructure, Department of Civil Engineering, Shanghai Jiao Tong University, Shanghai 200240, China School of Electronic Engineering, Xi’an Shiyou University, Xi’an 710077, Shanxi, China


Introduction
Shale gas reservoirs have low porosity and low permeability. Generally, horizontal wells and large-scale multistage fracturing technology are used to improve the performance of reservoir production in an economical and effective way [1][2][3][4]. A shale gas well in Sichuan-Chongqing region has a drilling depth of 3500 m, a horizontal section length of more than 1500m, and a fracturing displacement of about 12 m 3 /min. e wellhead pressure of this well is about 80 MPa, and the production casing size is Φ139.7 × 12.7 mm/125 V, and the collapsing strength exceeds 160 MPa. In hydraulic fracturing operation, the bridge plug is often blocked in deflecting section or a horizontal section. e production casing is deformed, and the diameter is reduced by printing and calliper logs. Many studies [5][6][7][8][9] indicated that the rock formation is the reverse fault. When the fracturing fluid is injected into the formation, the magnitude of in situ stresses changes. In addition, as the horizontal in situ stress difference increases, the production casing cannot bear the nonuniform loading, resulting in the casing deformation [10][11][12][13]. It showed that the collapsing strength of casing under nonuniform loading is insufficient. Compared with that under uniform loading, the collapsing strength under nonuniform loading is reduced. e calculation formula for collapsing strength under uniform loading is provided by the American Petroleum Institute (API BUL 5C3) standard [12]. However, for tight reservoirs such as shale gas, the rock anisotropy and fracturing fluid injection lead to the complex in situ stress distribution [14][15][16].
e results show that the natural fracture length and dip angle have a great impact on the casing failure, which means that factors affecting collapsing strength of casing include nonuniform loading, the length of applied load, and loading angles. Cai et al. [17] carried out the test of casing collapsing strength under uniform loading by underwater strain and studied variation rules of casing strain with external loading. Pattillo et al. [18] analyzed the variation of casing collapse strength under the condition of a plane flattening test by the experiment and numerical simulation method.
It is found that the current research sets the shale gas reservoir and casing in full contact along the hole wellbore [19][20][21][22] and so that the partial contact is rarely studied, which is one of nonuniform loading distribution forms (in Figure 1, line AB is symmetry axis of the contact part between formation and casing). us, the variation rules of strain, deformation, and instability characteristics under nonuniform loading should be further studied. Lin et al. [23] studied casing deformation failure mechanism (yield strength 110ksi) under unidirectional loading through collapsing experiments. It was found that a point failure of the middle or outer wall of the casing rather than the yield point should be used as the basis of collapse failure. So, it is meaningful to determine casing collapse strength under nonuniform load by the experiential and numerical method.
According to the technical problem, this research carried out the casing deformation tests under different lengths, loading angles, and nonuniform loadings. e casing collapse failure and deformation characteristics for production casing (Φ139.7 × 12.7 mm/125 V) were analyzed in shale gas wells. In this paper, we firstly tested the influence of casing length and loading angle on casing collapse strength by the test method, and then we used the nonlinear buckling calculation method to carry out numerical experiments and got variation rule under different casing lengths, loading angles, and nonuniform loadings. is paper provided a method of analyzing casing deformation in actual service environment.

Experimental Research on Nonuniform Loading
2.1. Nonuniform Loading. In the condition of nonuniform loading, the load distribution is unevenly distributed along the casing outer wall. e stress distribution is expressed as where n is the loading unevenness coefficient and can be expressed as n � p 1 /p 2 . n � 1 corresponds to the uniform loading, and n ≠ 1 corresponds to nonuniform loading. e decrease in n increases the unevenness of loading distribution. p 1 is the horizontal minimum principal stress; p 2 is the horizontal maximum principal stress; and p(α) is the internal loading along the angle α.
Equations for uniform loading can be written as follows: ere are two special cases. One is the loading applied on the part of casing circumference, that is, the loading surface is an arc surface, as shown in Figure 2(a). e other one is uniaxial loading, that is, the loading surface is a line (see Figure 2(b)). When α � 180°, the casing is subjected to uniform loading, as shown in Figure 2(c).

Experimental Equipment.
e experimental equipment is composite loading tester of 600t. According to the different experimental conditions, loading has been applied until the casing failure, and the stress-strain curve and loading value are collected and recorded. Six strain gauges are installed every 180°on the casing circumference and every 50 or 100 mm in the axial direction, as shown in Figure 3. e sample installation is shown in Figure 4. e sample size is Φ139.70 × 12.70 mm/125 V. It is assumed that the casing is an isotropic homogeneous elastomer, ignoring the effects of residual stress, ovality, wall thickness unevenness, and so on. e mechanical and chemical properties are identical during test analysis.

Experimental Scheme.
In order to study the casing deformation rule under different lengths, different loading angles, and nonuniform loading, the experimental scheme is shown in Table 1.
For the experiment, the loading is uniformly increased, and the loading rate is 1 kN/s.

Impact of Sample Length on Collapsing Strength.
Tooling length is 1D, and the lengths of samples1-1, 1-2, and 1-3 are 1D, 2D, and 3D, respectively. When the loading angle is 0, the plane loading plane fails. is can be called a plane flattening test. e shape and tested results of sample 1-1 before and after tests are shown in Figure 5. e test results for three group samples are shown in Figure 6. e sample experienced three stages of elasticity, yield, and instability in the failure process. When loading is 330 kN and the diameter variation rate is 3.0%, the yield occurs for sample 1-1, as shown in Figure 6. e diameter variation rate (the ratio of diameter variation value to the diameter for casing) of the casing does not increase linearly, as the load increases. For sample 1-2, yield occurs when loading is 670 kN, and the diameter variation rate is 3.1%. For sample 1-3, yield occurs when loading is 780 kN, and the diameter variation rate is 3.3%. e yield loading increases with the increase in sample length; under the same loading, the diameter variation rate decreases with the increase in sample length. When the sample length is more than 3D, the increasing trend of yield loading decreases. When the loading area is identical, the collapsing strength of casing increases with the increase in the sample length.

Impact of Loading Angle on Collapsing Strength.
e tooling length is 2D, the sample length is 7D, and the loading angles are 0°, 15°, 30°, and 45°, respectively. ese are called the curve flattening tests. For sample 2-2, the tooling shape and sample shape after testing and test results are shown in Figures 7 and 8. e test results of sample 2-2 are shown in Figure 9.
As shown in Figure 9, the sample undergoes three stages of elasticity, yield, and instability. e yield load increases with the increase in α, but the diameter variation rate decreases; under the same loading, the greater α, the less of diameter variation rate; the larger of the loading area, the stronger of casing collapsing strength. Under nonuniform external loading, the casing collapse strength increases with the increase in α.

Numerical Model Validation.
According to the experiment results, the finite element simulation method was used to analyze collapse strength changes for different casing samples lengths from 1D to 7D. Simulation parameters of casing are shown in Table 2. We assumed that casing is an ideal cylinder, and uneven wall thickness, the roundness of outer diameter, and ao on are not considered in the model. e radial dimension of the casing is much smaller than the axial length of the casing, which is simplified as a plane strain problem. Solid four nodes and 183 elements can be used for mesh generation, stresses, and deformation calculation. For sample 1-1, the finite element calculation results are shown in Figure 10, and the comparison of the experiment results and numerical simulation is shown in Figure 11.
As shown in Figures 10 and 11, the variation rate of casing diameter increases with the increase in loading. e maximum stress is located in the upper and lower of casing inner walls. e segment (0A) of Figure 11 shows that the loading and diameter variation rate increases linearly, which belongs to the inelastic stage. e segment (AB) of Figure 11 shows that when the diameter variation rate is greater than 3%, the relationship between loading and diameter variation rate is a nonlinear increase. is belongs to the yield stage. e segment (BC) of Figure 11 shows that the diameter variation rate increases rapidly when it is greater than 10%. However, this value is not sensitive to loading in the stage of instability. When the diameter variation rate continues to increase, the loading increases again, as the casing inner wall of the left and right sides begin to plying-up. is belongs to the strength strengthening stage. e experimental results are in good agreements with the results of finite element analysis.
For sample 2-3, the experimental and numerical simulation results are shown in Figure 12.
As shown in Figure 12, the numerical simulation results are in good agreement with the experimental ones. e diameter variation rate increases as the loading increases.
e maximum stress appears on the upper and lower parts of the casing inner wall. In the initial loading stage, the diameter variation rate increases linearly with the increase in loading, and the sample is in the elastic stage. When the diameter variation rate is greater than 3%, with the increase in loading, the diameter variation rate increases nonlinearly, and the sample is in the yield stage. When the diameter variation rate is greater than 10%, the diameter variation rate increases rapidly. However, the loading increase rate is not significant, which is in the instability stage. When the diameter variation rate continues to increase, the casing inner wall begins to plying-up, and loading has an increasing trend again. is is in the stage of plastic strengthening.

e Influence of Different Sample Lengths on the Casing
Collapse Strength. According to the experimental conditions of the first group, numerical simulation results can be obtained by changing the sample length, as shown in Figure 13. e diameter variation rate decreases at the contact surface. e casing deformation becomes smaller at the end of the sample with the increase in the sample length. However, when the sample is shorter, the end and contact surface of the sample have greater deformation. e relationships between the diameter variation rate and the loading for the flattening test of different length samples are shown in Figure 14. e tooling length remains unchanged. As the length of the sample increases, the casing collapse strength (loading capacity) increases. When the sample length is less than or equal to 3D, the casing collapse strength (loading capacity) will be significantly reduced. When the sample length is more than 3D but less than 6D, casing collapse strength (loading capacity) gradually increases, and the increase rate gradually decreases. When the sample length is more than or equal to 6D, the impact of sample length on casing collapse strength (loading capacity) can be ignored. e relationship between diameter variation rate and critical load for different length samples using the flattening test is shown in Figure 15. As the length of the sample increases, the critical loading and the diameter variation rate of instability increase, and the relationship between them is approximately proportional. It shows that the end effect is 1D testing 2D testing 3D testing  Mathematical Problems in Engineering 5 smaller when the sample is shorter. As the length of the sample increases, the end effect on the instability part is enhanced. From the above results, it can give a suggestion that the spacing between fracturing points should be more than 6D in shale multistage hydraulic fracturing.

e Influence of Different Loading Angles on the Casing
Collapse Strength. In order to eliminate the end effect, the ratio of sample length to tooling length has been set as 3.5 : 1. e finite element simulation method is used to analyze the influence of different loading angles on the casing collapse strength. e relationships between the diameter variation rate and loading at different loading angles are shown in Figure 16. As the loading angle increases, the casing collapse strength (loading capacity) increases. e increase rate of loading capacity accelerates as the loading angle increases.
is indicates that the casing collapse strength is enhanced. e results demonstrate that the collapse strength reaches the maximum under the uniform loading cases, that is, α is 180°. In the nonuniform loading cases, the casing collapse strength is reduced. In particular, when α is 0°, the collapse strength reduces the minimum value. Consequently, the casing is more likely to deform.
With the increase in loading angle, the critical loading increases, but the diameter variation rate decreases under the condition of instability, and the correlation between buckling loads and diameter variation rates are roughly inversely proportional, as shown in Figure 17. In the initial state, the loading surface fits the pipe body closely for different loading angles, which has certain constraints on the pipe body. However, there is no constraint on the plane load. For the same loading, the casing deformation becomes smaller as the loading angle increases. e greater the  loading angle, the greater the restriction on the casing. In addition, the casing deformation is smaller as the loading angle increases. ere are two main reasons for the increase in collapse pressure with the increase in loading angle. (1) Under the same condition of casing stress, the increase in angle will cause the increase in contact area so that the total load level will increase. It can be seen from Figure 17 law that the load variation is not linear with different load angles. is means that there are other reasons for the increase in load. (2) e increase in the angle will change the stress state of the dangerous point in the casing. With the increase in the angle, the stress distribution in some areas of the casing is similar to that of the hydrostatic pressure. It means that the deviation stress of dangerous point in casing will decrease, so the collapse strength will be enhanced based on Mises strength theory.
e curve in Figure 16 shows the strengthening phenomenon at the later stage, which is due to the increase        in contact surface between the sample and tooling after casing deformation.

e Influence of Load Unevenness Coefficient on Casing
Collapse Strength. Using a nonlinear buckling calculation method, the casing collapse strength is defined as the critical pressure of casing flattening or instability. When loading unevenness coefficient is 0.5 (n � p 1 /p 2 � 0.5), numerical simulation results are shown in Figure 18.
As shown in Figure 18, when the loading condition (n � p 1 /p 2 � 0.5) is applied, the stress of the casing inner wall in the short axis direction and the casing outer wall in the long axis direction is higher. Stress level and diameter variation rate increase as the loading increases. When the diameter variation rate reaches 2.0%, the casing failure occurs.
e pipe body of the long axis direction shows flattening deformation. On the other hand, the radial expansion deformation occurs in the short axis direction-the diameter variation rate of radial direction increases as the     Figure 18: e finite element stress cloud diagram: (a) diameter variation rate is 0.7903%; (b) diameter variation rate is 1.944%; (c) diameter variation rate is 2.061%; (d) diameter variation rate is 2.145%.
loading unevenness coefficient decreases. When the casing loading capacity is reached, the collapse failure occurs. e relationship between load unevenness coefficient and the ratio of the casing collapse strength of uniform load and nonuniform load is shown in Figure 19. e uniform loading turns to nonuniform loading, and the casing collapse strength decreases rapidly. When the load unevenness coefficient n is 0.8, the casing collapse strength is about 60% for collapse strength of uniform load (n � 1). e load unevenness coefficient continues to decrease, and the reduction rate of casing collapse strength slows down. When the load unevenness coefficient n is 0, the casing collapse strength is about 28% for the casing collapse strength of uniform load (n � 1). e model of load unevenness coefficient and the casing collapse strength is y � 1.728n 3 − 1.634n 2 + 0.603n + 0.270, where y is the ratio of casing collapse strength for load unevenness coefficient n ∈ [0,1] separately and p n is the casing collapse strength for load unevenness coefficient n. e relationship of load unevenness coefficient, diameter variation rate, and casing collapse strength is shown in Figure 20. As the load unevenness coefficient increases, the casing flexural strength of the casing decreases. When the casing failure occurs, the diameter variation rate increases. e relationship between casing collapse strength and diameter variation rate is roughly inversely proportional. When the load unevenness coefficient is small, the casing is more prone to failure and deformation. As the load unevenness coefficient increases, the casing is more difficult to deform and the load capacity is stronger.

Conclusions
For shale gas production casing deformation, it is suggested to consider nonuniform loading changes caused by fracturing for casing strength design. Based on the experimental and numerical results, the variation law and extremum for calculating the casing collapse strength for different loading unevenness coefficients are obtained.
(1) When the sample length is less than or equal to 3D, the casing collapse strength is significantly reduced.  When the sample length is more than 3D but less than 6D, the casing collapse strength gradually increases as the sample length increases, but the increase rate is gradually slow down. When the sample length is more than or equal to 6D, the effect of length on the loading capacity can be ignored.
(2) As the loading angle increases, the casing collapse strength and the increase rates of casing collapse strength increase. When the loading angle increases from 0 to 90, the critical load value increases from 1600 kN to 4000 kN. (3) As the load unevenness coefficient decreases, the casing collapse strength reduces. When the load unevenness coefficient n is 0.8, the casing collapse strength reduces to 60%, and when the load unevenness coefficient n is 0, the casing collapse strength reduces to 28%.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that there are no conflicts of interest.