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ACS Omega
ACS Omega
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ACS Omega
2470-1343
American Chemical Society

10.1021/acsomega.4c03836
Article
Quantitative Study on the Anticollapse Ability of Casing in Shale Gas Wells under Complex Load Conditions
Huang Jianbo †
Li Yiqiang †
Chen Rui †
Zhang Xiaojun ‡
Lian Wei §
Han Guangyao †
https://orcid.org/0009-0004-6145-4054
Li Jun *‡§
† Engineering Technology Research Institute of Xinjiang Oilfield Company, Karamay 834000, China
‡ China University of Petroleum (Beijing), Beijing 102249, China
§ China University of Petroleum (Beijing), Karamay Campus, Karamay 834000, China
* Email: lijun_paper806@163.com.
28 08 2024
10 09 2024
9 36 3785637868
22 04 2024
20 08 2024
05 08 2024
© 2024 The Authors. Published by American Chemical Society
2024
The Authors
https://creativecommons.org/licenses/by-nc-nd/4.0/ Permits non-commercial access and re-use, provided that author attribution and integrity are maintained; but does not permit creation of adaptations or other derivative works (https://creativecommons.org/licenses/by-nc-nd/4.0/).

Implementing novel technologies, including the “well factory” model and zipper fracturing techniques, has become prevalent in shale gas development. During completion operations such as lowering casing and multistage fracturing, the casing is subjected to many complex loads, reducing its strength and increasing the risk of casing deformation. By establishing a casing wear model and conducting multistage cyclic loading experiments and numerical simulations, we analyzed the change rule of casing anticollapse strength under complex loads, developed a calculation method for casing comprehensive anticollapse ability under complex loads, and applied the method to an illustrative calculation. The study shows that the wear effect during completion has a negligible impact on the strength of the casing. The casing anticollapse strength exhibits a linear decline in correlation with the number of cycles. The zipper fracturing operation resulted in a nonuniform distribution of geo-stress around the well, and the casing anticollapse strength demonstrated a nearly linear decline in correlation with the nonuniformity of geo-stress. In the presence of both internal and external effects, the casing anticollapse strength exhibited a decline exceeding 15%, thereby increasing the risk of casing deformation. This research method can provide computational guidance for preventing casing deformation in field fracturing construction.

National Natural Science Foundation of China 10.13039/501100001809 52204018 China University of Petroleum, Beijing 10.13039/501100002862 XQZX20220019 document-id-old-9ao4c03836
document-id-new-14ao4c03836
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pmc1 Introduction

At present, the global demand for oil and gas is increasing. The effective utilization of shale oil and gas is of paramount strategic importance in addressing the discrepancy between oil and gas supply and demand, ensuring national energy security, and facilitating the low-carbon transformation of the energy structure.1 With the breakthrough and large-scale application of horizontal well multistage fracturing technology since 2000, the development of the global shale oil and gas industry has entered the “fast lane”. With the continuous progress and maturity of development technology, the “well factory” model, three-dimensional development, and other technologies are gradually applied to shale gas reservoir development, expanding the effective fracture network from the local scale of a single well to the global scale of multiwell and even the entire gas field development, forming an efficient and economic development system, effectively improving the degree of shale gas reserves utilization, and improving shale gas development benefits.2,3 Due to the complexity of the development mode, new fracturing technologies such as zipper fracturing and cross-zipper fracturing have been developed based on conventional staged fracturing. This method optimizes the fracturing sequence and location and has achieved good application results in the Sichuan Basin, China.4−6 At the same time, shale gas wells’ horizontal sections have grown in length (to more than 1500 m) and the number of fracturing stages (to more than 30), which has improved fracture complexity and reservoir utilization.

In different stages of drilling, completion, and fracturing operations, the casing is subjected to various loads, mainly: (1) The long horizontal section of a shale gas well and the frequent adjustment of the trajectory, the casing is subjected to bending, stretching, and friction, resulting in casing wear during the running process, resulting in its strength reduction. (2) In the process of multistage fracturing, there are more fracturing stages and high pumping pressure, and the casing is subjected to multiple alternating internal pressure loads, which further reduces the strength of the casing. (3) In the “well factory” development mode, in the process of zipper fracturing technology, the stress interference between wells is serious. The distribution of in situ stress around wells is more complicated, which makes the mechanical environment around the casing more severe, and its strength is further affected. Under complex operating conditions such as completion and fracturing, casing deformation occurs frequently in shale gas wells, obstructing the bridge plug and causing other problems, resulting in lost and abandoned sections, which seriously affect the output of a single well of shale gas wells. Figure 1 illustrates the casing encounter in three fractured wells on a platform. Multiple casing deformations occurred in a single well, preventing the downhole tools from entering.

Figure 1 Casing deformation of fractured well.

Domestic and foreign scholars have carried out a great deal of research on the casing wear problem. Bradley et al. first put forward the problem of severe casing wear in directional Wells and carried out a series of casing wear experiments under different working conditions, finding that the rotation of drill string is the main factor leading to casing wear.7 Through experimental research and numerical simulation analysis, Williamson et al. proposed that a load of drill string joint and casing is the main factor in studying casing wear and gave the experimental relationship between contact pressure and casing wear rate.8 White and Dawson established a “wear efficiency” prediction model of casing metal loss based on the frictional functional loss and heat transfer principle.9 Shen et al. obtained an analytical solution for the stress distribution evolution of worn casing based on the boundary superposition principle, verified it by finite element numerical solution, and found that apparent stress concentration occurred in worn casing near the wear area.10 Gao et al. considered the influence of multiple wear depths when the contact pressure changed during casing wear and proposed the relationship between the wear depth and time. In addition, they demonstrate the impact of different drill string sizes on casing wear by analytical method and discuss three different types of casing wear: single-wear groove, sharp crescent groove, and blunt crescent groove.11−13 Kumar et al. proposed an “energy-wear” model to calculate casing wear in curved wells and discussed the effects of curvature and torque on casing wear volume, finding that hole curvature has a significant impact on casing wear.14 Yu et al. proposed a parallel calculation method based on finite element software and programming language to establish a numerical model of casing dynamic wear in directional Wells under the coupling of a nonuniform ground stress field. The research shows that casing wear depth is strongly correlated with the direction of ground stress, and the concepts of high-sensitivity zone and low-sensitivity zone of wear stress are proposed.15 Aichinger et al. confirmed Vavasseur’s view (that the drill string body and tool joint are in contact with the casing wall and both contribute to casing wear) through field data and multiarm diameter logging (MFCL) interpretation and developed a new method for predicting casing wear in three dimensions based on the MFCL interpretation data.16,17

Some scholars at home and abroad have conducted experimental research on the problem of casing strength change caused by multistage cyclic load. Placido et al. carried out an experimental study on casing deformation under temperature load and found that under alternating temperature conditions, casing would produce tensile load, which would increase with the increase of casing API grade, while casing strength would be weakened and deformation would occur.18 Terodoriu et al. carried out the failure experiment of casing cyclic axial pressure load and found that casing was more prone to deformation under cyclic load than under static load.19 Deng et al. carried out the test of casing anticollapse under nonuniform load, but the load application form is single, which is quite different from the actual underground situation.20,21 Gao et al. studied the influence of thermal stress and periodic changes of casing pressure on casing deformation during fracturing and concluded that periodic changes of casing pressure would cause cementing failure, cause the casing to bear local load, and increase the risk of deformation.22

The multistage fracturing of the shale gas well causes severe casing damage due to the change in the stress distribution characteristics around the well. Domestic and foreign scholars have done a great deal of research on this problem. Zoback and Daneshy, by studying the fracture propagation characteristics during fracturing, believe that asymmetric fractures will be formed after fracturing, resulting in uneven distribution of in situ stress around the well and resulting in casing bearing shear load.23,24 The study of Furui and Zhao et al. shows that fracturing will cause a region with high porosity near the wellbore, and the external compacted reservoir in this region will form an axial compression force, causing casing buckling and even collapse.25,26 Yu et al. carried out a numerical simulation study on casing deformation, and the results showed that the main factors leading to casing failure were the increase of nonuniformity of the geo-stress field during fracturing and the decrease of rock mechanical properties after the formation of pressure fracture network.27 Lin et al. and Lian et al. restored the fracturing area by inversion of microseismic monitoring event points and established a three-dimensional “casing–cement–formation” finite element model. The study showed that the in-suit stress field was redistributed during the multistage fracturing process, resulting in enhanced nonuniformity, and the asymmetry of the reconstruction area led to the casing being subjected to shear. Moreover, the interference between different stages of fracturing leads to repeated fracturing in local areas, the deterioration of rock performance is different, and finally causes the casing to appear “S” shaped bending deformation.28,29

Through comprehensive analysis, domestic and foreign scholars have made significant progress in casing deformation in shale gas wells. However, casing strength is reduced to different degrees due to varying load forms at different stages during the completion/fracturing process. Therefore, it is necessary to quantitatively study the casing anticollapse strength under various load conditions during completion/fracturing to clarify the casing residual strength and deformation mechanism during the completion/fracturing process of a platform well under the “well factory” mode.

This paper analyzed the variation law of casing anticollapse strength after casing wear, under multistage cyclic load, and under the zipper fracturing condition of the platform well by means of a mathematical model of casing wear, multistage cyclic load experiment, and numerical simulation. It also obtained the calculation method of casing comprehensive anticollapse strength and residual anticollapse strength under complex loads. Finally, the calculation and verification were carried out through field case wells.

2 Research Method

2.1 Establishment of Casing Wear Prediction Model

The “well factory” model of large-scale deployment of large offset horizontal wells with a complex borehole trajectory has become an essential part of shale gas development. For platform wells in the “well factory” mode, to avoid the collision between adjacent wells and maximize the development of shale gas, it is usually necessary to set the predeviation at the upper part of the reservoir segment in advance to meet the requirements of well spacing, as shown in Figure 2, where Dwell is the spacing of platform wells and DH is the spacing of each horizontal section of wells. Wells (A–J) on both sides of the platform have the largest borehole deviation distance and complex borehole trajectory, which causes serious casing wear problems and decreases casing strength.

Figure 2 Platform well deployment diagram in “well factory” mode.

Establishing the casing wear prediction model predicts the wear depth during casing running, and the change rule of the casing extrusion strength is calculated.

2.1.1 Casing Wear Model

The linear casing wear efficiency model (CWEM) established by White and Dawson was adopted, and its mathematical expression is shown in eq 1. According to this model, the main factors determining casing wear were wear coefficient, contact force, and relative motion path.1

where S is the casing wear area, m2; Wf is the wear coefficient of the wear efficiency model, 1/Pa; N is the contact force between the drill pipe joint and the casing surface, N; L is the relative motion distance of the pipe rod, m; and φ is the work function, N·m.

The reciprocating movement of the drill pipe has little influence, and casing wear is mainly caused by the rotation of the drill pipe. For the calculation expression of wear energy or work function, see eq 2.2

where ω is the rotary speed, r/min; Dtj is the outside diameter of the drill pipe joint, m; Ldr is the length of the drilling section, m; and Rop is the rate of penetration, m/h.

In the actual completion process, the main factor determining the casing wear efficiency is the contact pressure, which is determined by the contact force and contact width. Therefore, the mathematical expression of casing wear area is shown in eq 3.3

where fw is the coefficient of casing wear condition, m/Pa; w is half of the casing contact width, m; and Wnf is the wear coefficient of the nonlinear casing wear model, 1/Pa.

2.1.2 Casing Wear Depth Model

Wear on the outside of the casing can be regarded as the geometric cross-section formed by the intersection of casing and wellbore, as shown in Figure 3, where O1 is the center of the hole, R is the radius of the hole, O is the center of the casing, r is the casing radius, a is the center distance between O and O1, d is the maximum casing wear depth, and w is the half-width of the wear groove.

Figure 3 Casing wear diagram.

According to the geometric relationship in Figure 3, the maximum casing wear depth is4

The wear groove half-width is5

By integrating the casing crescent wear part, the casing wear area can be obtained as6

There is only one unknown a in eq 6, and the casing wear depth can be calculated through iteration. The iteration process is shown in Figure 4.

Figure 4 Casing wear depth prediction flowchart.

2.2 Microdeformation Experiment of Casing under Multistage Cyclic Load

In the process of multistage fracturing of shale gas wells, the casing is subjected to high internal pressure loads many times, and the internal pressure value is close to the limit of casing compressive strength, which may lead to microdeformation or strength reduction of the casing. The casing microdeformation experiment under a multistage cyclic internal pressure load is carried out using a full-size casing deformation monitoring test device under the combined wellbore condition of “casing–cement–formation.”

2.2.1 Experimental Apparatus

The experimental apparatus of this research is a full-size “casing–cement–formation” wellbore test system, as shown in Figure 5. It consists of a wellbore unit, a temperature and pressure control system, real-time stress–strain measurement, and a data collection and processing system. The wellbore unit consists of an inner casing (P110 steel grade, 139.7 mm outer diameter, 10.54 mm wall thickness) and an outer casing (N80 steel grade) that is filled with cement slurry to form a cement in the annulus between them. The temperature and pressure control systems include a temperature sensor and a heating rod, both fixed on the lower seal seat. The pressure control system consists of a pressure pump and a pressure control box connected to a pressure injection hole of the seal cover through a high-pressure pipeline. The temperature pressure control system simulates the downhole environment, with a temperature control range of 0–150 °C and a maximum pressure load of 120 MPa.

Figure 5 Schematic diagram and physical diagram of full-size “casing–cement ring–formation” wellbore simulation experimental apparatus (1. Casing; 2. Cement; 3. Formation; 4. Temperature sensor; 5. Heating rod; 6. Upper seal cover; 7. Lower seal seat; 8. Stress–strain gauge; 9. Signal collection editor; 10. Pressure pump; 11. Pressure control box; 12. Data processing system; 13. Temperature control box; 14. Outer sleeve, simulated formation).

2.2.2 Experimental Procedure

Given the research on casing strain characteristics under cyclic load, combined with the full-size wellbore assembly experimental apparatus, the following test steps are proposed, as shown in Figure 6, to achieve real-time and accurate acquisition of casing outer wall strain values under cyclic load.

Figure 6 Casing strain test procedure.

Experimental procedure mainly includes:① Paste strain gauge. First, sandpaper the outer wall of the casing to smooth it. Then, paste the first set of strain gauges in the middle of the casing and the second set of strain gauges near the casing, cover the polymer resin, and ensure that the wire is not directly exposed to the liquid.

② Injection cement slurry. First, the cement slurry was prepared according to the formula of G grade cement +44% water +4% fluid loss reducer +0.5% defoamer mixed with API standard. The wellbore temperature was set at 90 °C. Finally, the cement slurry was injected into the annulus between the casing and the simulated formation.

③ Cement slurry is waiting to set. Adjust the temperature control system and keep it at a constant temperature of 90 °C for 48 h.

④ Assemble the “casing–cement ring–formation” wellbore unit. The sealing cap is attached to the casing, and the sealing ring is checked in advance to ensure the tightness of the entire wellbore unit.

⑤ Cyclic loading. The high-pressure pipeline connects the pressure pump with the internal pressure loading port of the wellbore. The upper and lower limits of the pressure and the loading rate are set. The ultimate load is maintained for 600s, and the number of cycles is set.

⑥ Casing strain is measured in real-time. The strain gauge data of the casing outer wall is transmitted to the computer software interface through a data wire to monitor and save the strain data in real time.

2.2.3 Experimental Scheme

According to literature and field data, fracturing pressure has a great influence on the casing deformation degree. Therefore, two experimental groups were set up to compare the casing strain conditions under standard internal pressure and high internal pressure. The experimental condition parameters were all based on the field data of fractured wells in the Xinjiang oilfield. The test scheme and parameters are shown in Table 1.

Table 1 Experimental Scheme and Parameters of Casing Deformation under Cyclic Internal Pressure Load

number	casing grade	casing shaft length, mm	casing standard/actual outside diameter, mm	casing wall thickness, mm	internal pressure load, MPa	wellbore temperature, °C	
1	P110	1000	139.7/138.26	10.54	0–70	90	
2	P110	1000	139.7/139.37	10.54	0–90	90	

2.2.4 Experimental Result

Table 2 shows the experimental results of casing deformation under different cyclic internal pressure loads. According to the average measured results, the effective strain values of the two experimental groups after the first internal pressure load were 0.157 and 0.194%, respectively.

Table 2 Experimental Results of Casing Deformation under Cyclic Internal Pressure Load

 	 	 	 	cycle number	 	 	
number	circumferential stress state of casing wall under ultimate load	relative strain of casing wall under single load, %	number of casing residual strain stages	stage 1	stage 2	residual strain (%) after 16 cycles	residual strain ratio	
1	compression	0.157	2	1–7	8–25	0.035	1	
2	tension	0.194	3	1–3	4–13	0.104	2.97	

The experimental results of the two groups show that the casing strain changes in stages with the number of cyclic loads. According to the transition of the strain change, the two groups have two stages, respectively, and the transition from the first stage to the second stage is different; the turning point of the experimental group with high cyclic internal pressure was only after three cycles. Further, the ratio of residual strain under high cyclic internal pressure to that under standard conditions was 2.97, indicating that high internal pressure had a more significant impact on casing deformation.

During the experiment, the measured casing strain after unloading was inconsistent with the casing strain at the loading point, indicating that the casing had plastic deformation during the loading process. The analysis suggests that this may be related to the interaction between the casing and the cement. In the actual underground environment, the cement may have plastic deformation under the action of internal pressure. Therefore, considering the interaction between the casing and cement ring interface, we measured that the strain at the unloading point on the outer wall of the casing cannot recover to the strain value at the loading point. The difference between the casing wall strain after unloading and the casing wall strain at the loading point is called residual strain, and its expression is shown in eq 7:7

where εR is the residual strain on the outer wall of the casing, %; εU is the circumferential strain of the casing wall at the loading point, %; and εL is the circumferential strain on the outer wall of the casing at the unloading point, %.

Figures 7 and 8 show the residual strain variation characteristics of the casing outer wall under standard internal pressure load (70 MPa) and high internal pressure load (90 MPa), respectively.

Figure 7 Residual strain on casing wall under standard internal pressure load (70 MPa).

Figure 8 Residual strain on casing wall under high internal pressure load (90 MPa).

As seen in Figure 7, the total residual strain on the outer wall of the casing gradually accumulates with the number of cycles, and different accumulation rates are shown. Within 25 cycles, the overall strain can be divided into two stages (stage 1: 1–7 times, stage 2: 7–25 times). In the first stage of cyclic loading, the micropores inside the casing and cement ring are compressed, and the residual strain of the casing increases gently with the number of cycles. In the second stage, after the eighth cycle, the casing residual strain suddenly increased, which may be caused by the sudden compression of the large cracks in the casing, and the cumulative growth rate of the casing residual strain after that was relatively large. In addition, the residual strain of the casing has a nearly linear relationship with the number of cyclic loads at each stage.

As shown in Figure 8, different from casing plastic deformation under low internal pressure cyclic load, plastic deformation of the casing outer wall under high internal pressure cyclic load appears in three significantly different stages. The transition from stage 1 to stage 2 occurs during the fourth cycle loading process, which is advanced substantially under the condition of low internal pressure (the eighth cycle), and the casing strain at the unloading point during the fourth cycle is significantly larger than that under the loading process, which means that the pores of the casing body are severely compressed and irrecoverable under the condition of high internal pressure. Under high internal pressure cyclic loading, the casing residual strain reached 0.0612% after four cycles of loading, which was much higher than that after 26 cycles of low internal pressure.

2.3 Distribution Characteristics of In Situ Stress around Well under Zipper Fracturing Condition

With the popularization of the “well factory” model of shale gas, zipper fracturing and cross-zipper fracturing technologies that optimize the fracturing sequence and location have emerged to improve fracture complexity and reservoir reconstruction volume. But at the same time, the stress interference between wells increases in the process of fracturing, which makes the stress environment around the wellbore more complicated and causes the change of casing anticollapse strength.

2.3.1 Establishment of Numerical Model of Zipper Fracturing

Taking the basic parameters of platform M in the Fengnan 4 well area of the Xinjiang oilfield, a three-dimensional numerical model of platform well zipper fracturing is established. The schematic diagram of zipping fracturing and the finite element numerical model are shown in Figure 9. The length, width, and height of the model are 1000 m, 500 m, and 500 m respectively, and three horizontal wells, H1, H2, and H3, are set. Each horizontal well had three stages of fracturing. The Cohesive unit is employed to simulate fracture expansion through the injection of fluids in the sequence Y1–Y9, as illustrated in Figure 9, with the objective of achieving zipper fracturing simulation calculations. Field platform well spacing is mainly 150 m/260 m, and 150 m is set in the model.

Figure 9 Diagram of zipper fracturing and finite element numerical model.

According to the field logging data, the average values of the maximum and minimum horizontal principal stress and vertical principal stress of the M platform reservoir are 50, 45, and 48 MPa, and the average values of the initial elastic modulus and Poisson’s ratio are 35 GPa and 0.25, respectively. In terms of setting boundary conditions, we will apply total displacement constraints on the far field boundary of the formation and the corresponding initial stress field conditions on the model using the predefined field function. Among them, the minimum horizontal principal stress (σmin) will be parallel to the wellbore direction, and the maximum horizontal principal stress (σmax) will be perpendicular to the wellbore direction. The analysis step types include Geostatic equilibrium (*Geostatic) and fluid–solid coupling (*Soil). The geostatic equilibrium analysis step performs the initial formation of skeleton stress and pore pressure balance before fracturing. The fluid–solid coupling analysis step represents each stage of fracturing. Regarding meshing, the formation is structured in axial and radial directions in a central form with mesh type C3D8P. The fracturing section realizes hydraulic fracture propagation through cohesive units with mesh type COH3D8P. Regarding load setting, a fracturing displacement of 14 m3/min was set in each fracturing stage to simulate the hydraulic fracture propagation process. Then, the characteristics of the in situ stress distribution around the well were analyzed.

2.3.2 Numerical Simulation Results

The intermediate H2 well (Figure 9) was selected as the research object for the finite element numerical model. A path along the H2 wellbore was chosen to extract the values of the ground stress at different fracturing moments and to analyze the influence law of the various stages of the platform well zipper fracturing on the ground stress around the H2 well. This is illustrated in Figure 10, in which the horizontal axis represents the distance along the horizontal direction of the H2 well and the vertical axis represents the value of the ground stress.

Figure 10 Distribution characteristics of in situ stress around well under zipper fracturing condition.

Following each segment’s fracturing, the geo-stress values in all directions around the H2 well will exhibit a discernible increase at the fracture extension. This increase in geo-stress values will be more pronounced in the vicinity of the well following the fracturing of the H2 well itself. The fracturing sequence of each well segment significantly influences the maximum horizontal, minimum horizontal, and vertical principal stresses around the H2 well, resulting in pronounced nonuniform characteristics. Furthermore, the fracturing process leads to the emergence of more pronounced superposition effects, whereby the geo-stresses in all directions exhibit a more pronounced tendency to superimpose.30

3 Results and Discussion

3.1 Variation of Anticollapse Strength of Worn Casing

3.1.1 Damage Factor and Residual Anticollapse Strength Model of Worn Casing

Casing wear that results in thinning of its wall thickness, or even wear through the casing, reduces casing strength and leads to failure of casing integrity. Based on Section 2.1.1, casing wear prediction model, and flowchart, the casing wear depth d value can be obtained. The casing damage factor εc is further defined, and its expression is shown in eq 8.8

where d is the casing wear depth, mm, and t is the casing wall thickness, mm.

Then, the expression of residual anticollapse strength of worn casing is shown in eq 9.9

where Pc is the anticollapse strength of completion casing after wear.

3.1.2 Case Calculation

Taking the M platform well in the Fengnan well area of the Xinjiang oilfield as an example, the casing damage factors and the change characteristics of the anticollapse strength of the three wells on the platform were calculated using the above casing wear model and the residual anticollapse strength calculation model. The calculation results are listed in Figure 11.

Figure 11 Variation law of casing damage factor and anticollapse strength in platform well.

According to the calculation results in Figure 11, it can be seen that the casing damage factor increases with an increase in well depth. The casing bending degree increases at the build-up section, resulting in a sudden rise in the damage factor. Due to the effect of gravity in the horizontal section, the contact area between the string and the bottom end of the well increases, resulting in increased casing wear degree and further increased damage factor. However, overall, the range of damage factor after casing running is generally less than 0.01. Therefore, in the process of casing running, the impact on the casing anticollapse strength is small, and the drop is less than 1 MPa.

3.2 Analysis of Casing Anticollapse Strength under Multistage Cyclic Load

3.2.1 Dynamic Diameter-to-Thickness Ratio of Casing Strength Calculation Theory

The experimental results of full-size wellbore deformation under multistage cyclic load show that the casing has experienced multistage fracturing pressure loads, resulting in irrecoverable residual strain, so it is necessary to correct the casing size according to the fracturing load. The correction equation is shown in eq 10.10

where D is the casing outer diameter after fracturing, mm; D0 is the standard casing outer diameter before fracturing, mm; and εR is the residual casing strain after fracturing.

Since the plastic cumulative deformation of casing under cyclic internal pressure load is very small, it is assumed that the casing deformation conforms to Hooke’s law; the casing volume remains unchanged. Then, the casing wall thickness after multistage cyclic internal pressure loading can be expressed in eq 11.11

where ΔD0 is the standard casing wall thickness before fracturing, mm, and ΔD is the casing wall thickness after fracturing, mm.

After the casing deformation, the anticollapse strength changes compared with the original state.31 According to the modified anticollapse strength equation, the casing anticollapse strength expression under different fracturing stages, that is, under different cyclic internal pressure loading times, can be calculated as shown in eq 12.12

where pc is the correction formula of casing anticollapse strength, MPa; pe is the critical external pressure for elastic instability, MPa; p is the equivalent pressure MPa; α is the casing defect coefficient; and σy is the minimum yield strength of an ideal circular tube, MPa.

3.2.2 Variation of Casing Anticollapse Strength under Multistage Cyclic Load

According to the modified eq 12 of casing anticollapse strength and the circumferential residual strain curve of casing obtained by multistage cyclic loading experiment, the casing anticollapse strength under different fracturing stages under cyclic loading of standard internal pressure and high internal pressure (P110 casing, outer diameter 139.7 mm, wall thickness 10.54 mm) was obtained, as shown in Figure 12.

Figure 12 Change curve of casing anticollapse strength under multistage fracturing.

According to the calculation results of casing anticollapse strength, the casing anticollapse strength does not decrease significantly at the initial stage of fracturing under standard internal pressure, and the cyclic loading of internal pressure has little effect on casing strength. After the eighth fracturing, the casing anticollapse strength decreases obviously; that is, the casing deformation risk increases after the eighth fracturing. Under high internal pressure, the effect of the fracturing stage on the casing anticollapse strength is more prominent. After the fourth fracturing, the casing anticollapse strength decreases to 82.17 MPa, and the casing anticollapse strength further decreases with the progress of fracturing. Therefore, casing deformation is more likely to occur under a high internal pressure.

3.2.3 Change Function of Casing Anticollapse Strength under Multistage Cyclic Load

Based on on-site fracturing design data, most wells are constructed with a fracturing pressure of 70 MPa. Therefore, the relationship between casing anticollapse strength and fracturing stage (number of cycles) can be obtained by fitting the curve of casing anticollapse strength under standard internal pressure cyclic load. The fitting function curve is shown in Figure 13.

Figure 13 Fitting function diagram of casing anticollapse strength under standard internal pressure.

According to the fitting function, the relationship between casing anticollapse strength and fracturing stage is obtained as shown in eq 13.13

where x is the fracturing stage; Px is the casing anticollapse strength after x times of fracturing, MPa; and a and b are fitting coefficients of casing anticollapse strength.

3.3 Analysis of Variation of Casing Anticollapse Strength under Nonuniform Stress Distribution

Based on the numerical simulation results of zipper fracturing in the platform well, the nonuniform degree of ground stress around the well increases, which causes the change of external extrusion load on the casing and the change of its anticollapse strength. This section provides the variation rule of casing anticollapse strength by simulating different nonuniform ground stress conditions.

3.3.1 Simulation of Nonuniform Ground Stress

To study the influence of nonuniform ground stress on casing compressive strength, it is defined that ground stress nonuniformity k = maximum principal stress/minimum principal stress. In horizontal wells, the maximum principal stress and minimum principal stress correspond to the maximum horizontal principal stress and vertical principal stress, respectively. The greater the k value, the more significant the difference between the maximum and minimum principal stresses; that is, the greater the uneven distribution of ground stresses.

ABAQUS finite element software was used to establish a casing-cement-formation assembly model to simulate the casing under nonuniform ground stress. The model is shown in Figure 14. The model size is 3 m × 3 m, casing outer diameter is 139.7 mm, wall thickness is 10.54 mm, elastic modulus is 210 GPa, and Poisson’s ratio is 0.3. The elastic modulus of cement is 7 GPa, and Poisson’s ratio is 0.2. The formation elastic modulus is 35 GPa, and Poisson’s ratio is 0.25. The variation of casing anticollapse strength is simulated when the ground stress nonuniformity k is 1, 1.1, 1.2, 1.3, 1.4, 1.5, 1.6, 1.7, 1.8, and 1.9, respectively.

Figure 14 Numerical model of nonuniform ground stress.

When the ground stress load is applied, the X stress and Y stress are applied in proportion to each nonuniformity degree until the maximum stress of the casing reaches the yield limit. The corresponding applied load, that is, the anticollapse strength of the casing under the corresponding nonuniformity condition, is recorded.

3.3.2 Variation of Casing Anticollapse Strength under Nonuniform Ground Stress

The variation law of casing anticollapse strength under different ground stress nonuniformity was obtained through the simulation of nonuniform ground stress conditions, as shown in Figure 15.

Figure 15 Variation of casing anticollapse strength and its decline under nonuniform ground stress.

With the increase of ground stress heterogeneity, the casing anticollapse strength decreases continuously, consistent with the findings of Zhang et al.32 When the ground stress heterogeneity increases from 1 to 1.9, the casing compressive strength decreases from 100.2 to 76.76 MPa, a 22.39% decrease.

3.3.3 Function of Casing Anticollapse Strength under Nonuniform Ground Stress

According to the calculation results of casing anticollapse strength under different ground stress heterogeneity, the variation function of casing anticollapse strength under nonuniform stress conditions can be fitted, as shown in eq 14.14

where P0 and Pk are the rated anticollapse strength and the casing anticollapse strength under nonuniform ground stress, respectively, and k is the nonuniformity of ground stress.

3.4 Calculation of Comprehensive Anticollapse Ability of Casing under Complex Load Conditions

According to the research results in Section 2, the casing compressive strength functions under casing running, multistage cyclic load, and nonuniform ground stress are obtained, respectively. In this section, the coefficient of casing anticollapse ability is defined for each stage, and the comprehensive coefficient of casing anticollapse ability is further obtained under complex load conditions during the completion/fracturing process.

3.4.1 Calculation of Casing Anticollapse Ability during Well Completion

fc is defined as the coefficient of anticollapse ability of casing in the completion process, and its expression is shown in eq 15.15

where P0 and Pc are the rated anticollapse strength and anticollapse strength of casing after running, respectively.

Based on the calculation results of anticollapse strength of field fractured wells after completion in Section 2.1, the coefficient of anticollapse ability of casing of each well is obtained, as shown in Figure 16. It can be seen that well completion has little effect on casing anticollapse ability, and the coefficient fc of anticollapse ability is greater than 0.99. Therefore, according to the worst case, an fc of 0.99 is taken.

Figure 16 Coefficient of casing anticollapse ability after completion.

3.4.2 Calculation of Casing Anticollapse Ability under Multistage Cyclic Load

Considering that the multistage fracturing of shale gas wells is mostly 20–30 stages, or even more than 30 stages, eq 13 can be expressed as16

Based on the above equation, fx is defined as the coefficient of anticollapse ability of casing under multistage cyclic load, and the expression is shown in eq 17.17

The coefficient of casing anticollapse ability under different cycles (x ≥ 8) was calculated, as shown in Figure 17.

Figure 17 Coefficient of casing anticollapse ability under multistage cyclic load.

3.4.3 Calculation of Casing Anticollapse Ability under Zipper Fracturing Condition

Based on eq 14, fk is defined as the coefficient of anticollapse ability of casing under nonuniform distribution of ground stress, and its expression is shown in eq 18.18

where Pk is the casing anticollapse strength under nonuniform ground stress condition.

The coefficient of casing anticollapse ability under different ground stress nonuniformity (k = 1, 1.1, 1.2, 1.3, 1.4, 1.5, 1.6, 1.7, 1.8, 1.9) was calculated, as shown in Figure 18.

Figure 18 Coefficient of anticollapse ability of casing behind casing under nonuniform ground stress.

3.4.4 Calculation of Casing Anticollapse Ability under Complex Load Conditions

Considering the influence of casing running, multistage cyclic load, and nonuniform distribution of ground stress caused by zipper fracturing on casing anticollapse strength comprehensively, f is defined as the coefficient of casing comprehensive anticollapse ability under complex load conditions in completion/fracturing process, and its expression is shown in eq 19.19

Further, the residual casing anticollapse strength under complex load conditions during completion/fracturing is obtained, and the expression is shown in eq 20.20

where PL is the residual casing anticollapse strength under complex load condition.

4 Case Calculation

According to the proposed calculation method for casing anticollapse ability under complex load conditions in the completion/fracturing process, the casing anticollapse ability at the blocked location of the pumping bridge plug is calculated, and the cause of deformation is determined based on the parameters of fractured well H2 (the middle well of the M platform) in the Fengnan 4 well area, Xinjiang.

4.1 Case Well Basic Data

4.1.1 H2 Well Completion Basic Data and Wellbore Structure

The H2 well belongs to one of the horizontal wells on the M platform. Its complete drilling depth is 4178 m, and the horizontal section length is 1436 m. After the beginning of the directional section, due to the build-up rate reaching 11.5°/30 m, the borehole deviation is not ideal, so the drill bit has been lifted and lowered many times to adjust the trajectory, as shown in Figure 19. The parameters of the casing are shown in Table 3. The anticollapse strength is determined according to the API casing strength data.

Figure 19 Wellbore structure of the H2 well.

Table 3 Casing Parameter

casing specification (mm)	joint	grade	wall thickness (mm)	inside diameter (mm)	anticollapse strength (MPa)	
Φ139.7	ladder joint	P110	10.54	118.62	100.2	

4.1.2 Casing Deformation Characteristics

The M platform adopts the zipper fracturing process as a whole. During the staged fracturing process of the H2 well, the pumping bridge plug encountered resistance several times, especially at 3511–3515, 3997–4001, and 4016–4024 m after multistage pumping measures failed to pump successfully. By analyzing the three-dimensional imaging map of the multiarm well diameter measurement, as shown in Figure 20, in which the more significant the color difference, the more serious the casing deformation, it can be judged that the casing of the H2 well is mainly extrusion deformation. According to the segmented cluster design, the bridge plugs’ blocked positions correspond to the 2nd, 4th, and 14th fracturing stages, respectively.

Figure 20 Location of casing deformation points in H2 well and the interpretation results of multiarm well diameter.

4.2 Calculation Result

Based on the analysis of casing anticollapse ability during the completion of Section 3, fw = 0.99 is taken.

According to the numerical simulation results of zipper fracturing of the platform well, under zipper fracturing conditions, the distribution of ground stress around well H2 presents a strong nonuniformity, with an increase of ground stress of 5–20% and an increase of in situ stress nonuniformity of 5–15%. k = 1.106, 1.123, and 1.173, respectively.

Using eqs 17 and 18, the coefficient of casing anticollapse ability is calculated, respectively, when the casing deformation position of the H2 well is under the action of multistage cyclic load and the in-suit stress is not evenly distributed due to zipper fracturing. Further, the coefficient of casing comprehensive anticollapse ability is calculated under complex completion/fracturing load conditions by using eq 19. The calculation results are listed in Table 4.

Table 4 Calculation Result of Casing Anticollapse Ability at Casing Deformation in H2 Well

casing deformation position	4016–4024 m	3997–4001 m	3511–3515 m	
coefficient of casing anticollapse ability under multistage loads fx	0.948	0.941	0.904	
coefficient of casing anticollapse ability under nonuniform stress fk	0.942	0.935	0.919	
coefficient of casing comprehensive anticollapse ability f	0.893	0.879	0.830	

The calculations’ results demonstrate that, by combining the casing under completion, multistage cyclic loading, and zipper fracturing action, the casing anticollapse ability coefficients at each location are significantly lower than the casing design safety coefficient values. Consequently, the casing is at a higher risk of extrusion deformation under complex working conditions.

5 Conclusions

This paper conducted a quantitative study on casing anticollapse ability under complex loads during completion/fracturing through the casing wear mathematical model, multistage cyclic load experiment, and numerical simulation. The main conclusions are as follows:(1) the wear in the completion process has little effect on the casing strength.

(2) The casing accumulates microdeformation plastically under the action of a multistage cyclic load. It has a prominent stage, and the casing anticollapse strength gradually decreases. The high internal pressure has a greater effect on the casing anticollapse strength.

(3) Under the zipper fracturing condition, due to the interwell severe stress interference, the nonuniform degree of in-suit stress around the well increases continuously, which leads to the uneven stress of casing and the continuous reduction of the anticollapse strength.

(4) In the presence of both internal and external effects, the casing anticollapse strength exhibited a decline exceeding 15%, and the risk of casing extrusion deformation is increasing.

Data Availability Statement

Data is contained within the article.

Author Contributions

J.H.: Field data analysis; Y.L.: experiment and data processing; R.C.: numerical simulation and data analysis writing; X.Z.: data curation, writing (original draft preparation); W.L.: reviewing and editing; G.H.: reviewing and editing; J.L.: reviewing and editing.

The authors declare no competing financial interest.

Acknowledgments

This work was supported by National Natural Science Foundation of China “Study on quantitative calculation model of fault slip induced by multistage fracturing and casing deformation control method in shale gas well” (no. 52204018) and Research Foundation of China University of Petroleum-Beijing at Karamay “Wellbore Integrity Analysis and Optimization System Construction of Shale Oil and Gas Wells During Life Cycle” (no. XQZX20220019).
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