Introduction

Hydraulic fracturing has been considered as one of the preferable technologies in petroleum industries for several decades for the purpose of hydrocarbon well stimulation to recover a huge volume of oil and gas. Because of this, the oilfield vigorously promotes the application of hydraulic fracturing technology to improve oil recovery. Further, the process of hydraulic fracturing, which is as a key stimulation technology with horizontal wells, has been widely used in order to improve the reservoir productivity (Davarpanah et al. 2019a, b, 2020; Sun et al. 2020). With the continuous application of fracturing technology, the downhole casing will bend laterally due to the influence of borehole configuration, and the problem of casing failure in oil field is becoming more and more serious. (Ghazvinian et al. 2014; Wang et al. 2015; Davarpanah et al. 2019a, b; Zhang et al. 2019).

Many effective casing design methods based on the design for vertical and directional wells have been developed and tried for bending borehole s in previous studies and applications (Yin and Gao 2014; Yu et al. 2016). Later on, these designs were extended to bending sections in horizontal wells. However, all previous studies have focused on analyzing a small section of casings in the buildup section; the stress and deformation of a single-point on the casing were studied separately (Akgun et al. 1994; Li et al. 2005; Zhuang et al. 2006).

In Quebec, Canada, 28 wells have experienced varying degrees of casing tensile failure after the completion of the large volume fracture (Zhang et al. 2020). The same thing happened in the Marcellus shale gas field in the USA. The casing problems are also serious in China. More than a dozen wells have experienced similar problems during fracturing operations, which have affected the development of reservoir reconstruction volume (Ye and Wang 2013; Tatomir et al. 2018). In the process of the down hole operation, the construction pipe string is subject to mechanical influences such as the gravity effect, temperature effect, swelling effect, bending effect and friction effect (Tang et al. 2013; Yu et al. 2014).

Tian et al. established a multifactor coupled casing tensile stress calculation and evaluation model (Tian et al. 2015). They claimed that the casing tensile failure of shale gas wells was a result of the coupling of multiple factors such as temperature effect, the casing bending, and axial pressure. Dai concluded that the casing deformation in the fractured formation was attributable mainly to the weakening of casing strength caused by the bending stress, strength fatigue and casing tensile failure and external load variations caused by the changes in nearby in situ stress field and local stress formed after failure of cement based on the characteristics of well testing and completion operation of shale wells in Sichuan (Dai 2015).

In recent years, there are many research work on the mechanism of casing tensile failure using finite element method; however, the numerical simulation of the bending sections was rarely done. In this paper, the stress of the whole string is studied by establishing the theoretical mechanical model of vertical section-bending sections-horizontal section, and it is concluded that the casing in the bending sections is most prone to the casing tensile failure (The casing tensile stress exceeds the yield strength, resulting in plastic deformation and local damage of casing). The finite element analysis method is used to study the casing tensile stress distribution in the bending sections during hydraulic fracturing. The effects of cement loss, the casing wall pressure, temperature change and Young’s modulus of elasticity of rock coupling cement on casing tensile stress distribution are obtained, which are analyzed and demonstrated by the actual borehole trajectory.

Mechanical model

Mechanical analysis of whole string

The whole string can be divided into three sections: the vertical section, the horizontal section and the bending sections for force analysis. In situ stress distribution of the whole string is shown in Fig. 1.

Fig. 1
figure 1

Schematic diagram of in situ stress full string

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Vertical section

During casing running, the casing joints will have some contact and friction with wellbore wall, but the contact has no fixed point as the pipes are run down and lasts for a short time. The friction loss is small, and the casing will not be damaged. Because of this, in the force analysis of casing, only the role of buoyed weight (the weight of casing in drilling fluid, referred to as buoyed weight) is considered. At this time, the friction resistance of casing is zero, and the axial force increment is only the buoyed weight of casing, that is: (Karolczuk and Macha 2008):

$$\Delta T = qL$$
(1)

Bending sections

Taking the micro-element for force analysis, the total lateral force N of casing micro-element is (Xiao 2019):

$$N = \sqrt {N_{{\text{v}}}^{2} + N_{{\text{h}}}^{2} } = \sqrt {\left( {1/\mu \sin \frac{\gamma }{2} - \cos \frac{\gamma }{2}\left( {G\sin \frac{\gamma }{2} + G_{{\text{v}}} \cos \frac{\gamma }{2} + 2T_{{\text{A}}} \sin \frac{\gamma }{2}\cos \frac{\gamma }{2}} \right)} \right)^{2} + \left( {qL\cos \gamma_{{{\text{gb}}}} } \right)^{2} }$$
(2)

When Nv > 0, that is:

$$N_{{\text{v}}} = \frac{1}{{\mu \sin \frac{\gamma }{2} - \cos \frac{\gamma }{2}}}\left( {G\sin \frac{\gamma }{2} + G_{{\text{v}}} \cos \frac{\gamma }{2} + 2T_{{\text{A}}} \sin \frac{\gamma }{2}\cos \frac{\gamma }{2}} \right)$$
(3)

The total lateral force on the casing microelement is as follows:

$$N = \sqrt {N_{{\text{v}}}^{2} N_{h}^{2} } = \sqrt {\left( {1/\mu \sin \frac{\gamma }{2} - \cos \frac{\gamma }{2}\left( {G\sin \frac{\gamma }{2} + G_{{\text{v}}} \cos \frac{\gamma }{2} + 2T_{{\text{A}}} \sin \frac{\gamma }{2}\cos \frac{\gamma }{2}} \right)} \right)^{2} + \left( {qL\cos \gamma_{{{\text{gb}}}} } \right)^{2} }$$
(4)

When Nv < 0, that is:

$$N_{v} = \frac{1}{{\mu \sin \frac{\gamma }{2} + \cos \frac{\gamma }{2}}}\left( {G\sin \frac{\gamma }{2} + G_{{\text{v}}} \cos \frac{\gamma }{2} + 2T_{{\text{A}}} \sin \frac{\gamma }{2}\cos \frac{\gamma }{2}} \right)$$
(5)

The total lateral force on the casing microelement is as follows:

$$N = \sqrt {N_{{\text{v}}}^{2} N_{{\text{h}}}^{2} } = \sqrt {\left( {1/\mu \sin \frac{\gamma }{2} + \cos \frac{\gamma }{2}\left( {G\sin \frac{\gamma }{2} + G_{{\text{v}}} \cos \frac{\gamma }{2} + 2T_{{\text{A}}} \sin \frac{\gamma }{2}\cos \frac{\gamma }{2}} \right)} \right)^{2} + \left( {qL\cos \gamma_{{{\text{gb}}}} } \right)^{2} }$$
(6)

The total friction of casing microelement is as follows:

$$F_{{\text{D}}} = \mu \sqrt {N_{{\text{v}}}^{2} + N_{{\text{h}}}^{2} }$$
(7)

Horizontal section

When the casing in the horizontal section is attached to the lower side of the sidewall, it can be considered that the sidewall completely bears the buoyancy of casing. The friction resistance of horizontal casing is as follows (Jia 2020):

$$F_{{\text{d}}} = \mu N = \mu qL\sin \alpha$$
(8)

The axial increment is:

$$\Delta T = T_{{\text{B}}} - T_{{\text{A}}} = F_{{\text{d}}} - qL\cos \alpha = qL(\mu \sin \alpha - \cos \alpha )$$
(9)

From the force analysis of the vertical section, the horizontal section and the bending sections, when the casing goes into the bending sections of horizontal well, the friction resistance of casing the bending sections of horizontal well is the highest due to the interaction of various factors, such as borehole bending and the casing dead weight. It shows that the section where the casing tensile failure occurs most easily in the whole string is located in the bending sections.

Collapse strength of casing in the bending sections

The casing tensile failure is closely related to its yield strength, collapse strength and other parameters; it is necessary to understand the casing strength data under the API standard as shown in Table 1.

Table 1 Casing size, grade and strength based on the API standard
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In this work, P110 grade casing was selected in the analysis. The collapse strength of the casing is based on the API standard. The collapse strength is 76.5 MPa, the tensile strength is 862 MPa, and the yield strength is 758 MPa. The casing tensile stress exceeds the yield strength, and the casing is ruptured. However, the P110 casing under the API standard cannot be directly used to analyze the casing tensile failure in the bending sections under the action of in situ stress during the fracturing process. Because of this, it is necessary to conduct in-depth research in combination with the fracturing conditions.

The radial stress, circumferential stress and axial stress of bending casing are as follows (Huang and Gao 2015):

$$\sigma_{\theta } = - \frac{{D^{2} }}{{D^{2} - d^{2} }}\left[ {1 + \frac{{d^{2} }}{{4r^{2} }}} \right]P_{{\text{o}}}$$
(10)
$$\sigma_{r} = - \frac{{D^{2} }}{{D^{2} - d^{2} }}\left[ {1 - \frac{{d^{2} }}{{4r^{2} }}} \right]P_{{\text{o}}}$$
(11)
$$\sigma_{{\text{z}}} = \frac{{F_{{{\text{ax}}}} }}{A} + Erk$$
(12)

When the borehole curvature exceeds 2°/30 m, the casing is bending and greater tensile stress is induced outside of the casing produces. The Von Mises yield criterion of casing is as follows (Lin et al. 2014):

$$\left( { \sigma_{{{\text{r}} - }} \sigma_{\theta } } \right)^{2} + \left( { \sigma_{\theta - } \sigma_{{\text{z}}} } \right)^{2} + \left( { \sigma_{{{\text{z}} - }} \sigma_{{\text{r}}} } \right)^{2} > 2\sigma_{{\text{s}}}^{2}$$
(13)

When the casing equivalent stress (Under Von Mises yield criterion, the value of yield criterion) in the bending section reaches the minimum yield strength of casing, the casing will be crushed to (Huang and Gao 2015):

$$2\sigma_{{\text{s}}}^{2} = \left( {\sigma_{r} - \sigma_{\theta } } \right)^{2} + \left( {\sigma_{\theta }^{2} + \sigma_{{\text{r}}}^{2} } \right) - \sigma_{{\text{z}}} \left( {\sigma_{\theta } + \sigma_{{\text{r}}} } \right) + 2\sigma_{{\text{z}}}^{2}$$
(14)

Replace Eq. (10), Eq. (11) and Eq. (12) into Eq. (14) to obtain:

$$\sigma_{{\text{s}}} = \sqrt {\frac{6}{{r^{4} }}\left[ {\frac{{D^{2} d^{2} P_{{\text{o}}} }}{{D^{2} - d^{2} }}} \right]^{2} + 2\left[ {\sigma_{z} + \frac{{D^{2} P_{{\text{o}}} }}{{D^{2} - d^{2} }}} \right]^{2} }$$
(15)

By sorting out the above equations, the following equations can be obtained (Deng et al. 2015):

$$P_{{\text{b}}} = P_{{\text{t}}} \left[ {\sqrt {1 - \frac{3}{4}\left( {\frac{{\sigma_{{\text{z}}} }}{{\sigma_{{\text{s}}} }}} \right)^{2} - \frac{1}{2}\frac{{\sigma_{{\text{z}}} }}{{\sigma_{{\text{s}}} }}} } \right]$$
(16)

According to the equation of casing collapse strength in curved section, the casing collapse strength corresponding to different borehole curvature is calculated, and the relationship between wellbore curvature and casing collapse strength is analyzed. The casing wall thickness is 7.72 mm, 10.54 mm and 12.7 mm, respectively. The calculation results are shown in Fig. 2.

Fig. 2
figure 2

Casing collapse strength under different borehole curvature

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It can be seen from Fig. 2 that when the thickness of casing wall is the same, the collapse strength of casing decreases with the increase in borehole curvature, which shows that the resist deformation ability of casing decreases with casing bending degree increasing, resulting in the reduction in casing collapse strength. When the borehole curvature is same, the casing collapse strength increases with casing thickness wall, which shows that the thickness of casing wall is helpful to enhance the collapse strength of casing.

Simulation and verification

Model building

Model size and parameter setting

In this paper, the finite element three-dimensional model is established with the help of ABAQUS finite element analysis software. In order to make the casing tensile stress analysis more reliable and reasonable, it is necessary to simplify the finite element model and make the following basic assumptions.

  1. 1.

    casing, cement and formation are in close contact with each other without cracks.

  2. 2.

    casing, cement and borehole are all ideal circles.

  3. 3.

    casing, cement and stratum are all isotropic elastic materials.

  4. 4.

    The relative displacement and friction are not considered in the contact in the model.

Taking the P110 casing with thickness of 9.17 mm as the research object, the yield strength is 758 MPa, the formation size (length × width × height) is 830 m × 400 m × 1200 m, the outer diameter of cement is 215.9 mm, the outer diameter of casing is 139.7 mm, and the inner diameter of casing is 121.36 mm. According to the data of a shale oil well, the required mechanical parameters are obtained. The specific material parameters are shown in Table 2, to set the corresponding parameters in the three-dimensional finite element model (Yin and Zhang 2016).

Table 2 Related parameters of formation, cement and casing
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In this paper, the casing tensile failure in the process of fracturing is taken as the working condition. The fracturing parameters are set as follows: the number of horizontal fracturing sections is 12, the length of fracturing sections is 55 m, and the wellhead pressure is 69 MPa (Yu et al. 2014; Liu et al. 2022; Zhao et al. 2022).

Grid division

The contact relationship between formation, cement and casing is surface contact, that is, the outer surface of casing is in contact with the inner surface of cement, and the outer surface of cement is in contact with formation, and the friction coefficient is 0.2. All the grid models are calculated and analyzed by hexahedral structured grid, and the shape of grid element is hexahedral. Because the casing is the main goal of the research, the grid is divided according to the principle that the casing is the densest, the cement is sparse and the formation is sparse from inside to outside. Because the whole string is divided into three sections, the vertical section, the bending sections and the horizontal section. The method of direct global seed layout cannot be used to divide the grid, the solid sweep is used to sweep along the whole borehole trajectory, and by mirroring, mapping, cutting and other ways. The meshing result is shown in Fig. 3a.

Fig. 3
figure 3

Model building results. a Three-dimensional finite element model of whole string, meshing. b Three-dimensional finite element model of whole string, boundary condition. c Three-dimensional finite element model of whole string, 3D perspective view

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Boundary conditions and load setting

The bottom surface of the model is fully constrained, and the model surface in the direction of X and Y axis of the model is displaced and fixed.

Because the depth of the well is about 1200 m, the influence of gravity on the casing cannot be ignored. Two kinds of loads are created: 1. Gravity load: 9.8 N/kg; 2. The pressure on the inner surface of casing.

Create in situ stress, apply three-dimensional force to the model at the initial step, refer to the average value of in situ stress field measurement data of Xushen gas field, take vertical in situ stress σh = 55 MPa, maximum horizontal in situ stress σmax = 65 MPa, minimum horizontal in situ stress σmin = 60 MPa.

The result of boundary condition constraint is shown in Fig. 3b, and the three-dimensional perspective view of casing–cement–formation is shown in Fig. 3c.

Borehole curvature

At present, most oilfields use drilling parameters with borehole curvature radius of 400 m, 300 m, 200 m and 100 m, the borehole curvature radius of 400 m, 300 m, 200 m and 100 m is analyzed and studied. The radius of borehole curvature can be converted into borehole curvature by conversion. For this reason, in the established three-dimensional finite element analysis model of the whole string, the borehole curvature is 4.3°/30 m, 5.73°/30 m, 8.59°/ 30 m and 17.18°/30 m, respectively, and the variation law of casing tensile stress under different borehole curvature is calculated. Based on the on-site casing tensile failure data and simulation results, it can be found that the damaged parts of the whole casing string mainly appear in two parts, the casing tensile stress cloud images with borehole curvature of 4.3°/30 m and 8.59°/30 m are selected to highlight the main casing tensile failure parts. The casing tensile stress cloud map has a certain deformation bending to express the risk position in Fig. 4. The variation of casing tensile stress with borehole curvature is shown in Fig. 5.

Fig. 4
figure 4

Cloud map under different borehole curvature. a borehole curvature of 4.3°/30 m. b borehole curvature of 8.59°/30 m

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Fig. 5
figure 5

Casing tensile stress under different borehole curvature

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It can be seen from Fig. 4 that the tensile stress of the whole casing string is mainly concentrated in the bending sections, in which the casing deformation is most likely to occur at the tilting point and the target point.

It can be seen from Fig. 5 that the casing tensile stress increases with the increase in borehole curvature, and the casing tensile stress increases rapidly when the borehole curvature increases from 8.59°/30 m to 17.18°/30 m. When the borehole curvature is 17.18°/30 m, the casing tensile stress is 755.3 MPa, and the casing tensile stress approaches the yield stress value.

The variation trend of casing tensile stress with borehole curvature and the trend of casing collapse strength with borehole curvature verify each other. The greater the casing tensile stress is, the lower the casing collapse strength is. It is concluded that the greater the bending degree of borehole, the more obvious the tensile stress and deformation of casing.

Borehole curvature coupled in situ stress field

With the change of borehole curvature, there is a in situ stress field in the actual formation. The influence of ground stress in three directions on casing cannot be ignored. Therefore, it is necessary to study the casing tensile stress under the change of in situ stress field in order to find out the dangerous borehole curvature.

When σmin is 55 MPa, σmax and σh are the same as 65 MPa, 70 MPa and 80 MPa. The influence of the combination of the three principal stresses on the casing is studied, and the variation of casing tensile stress with borehole curvature under different in situ stress is shown in Fig. 6.

Fig. 6
figure 6

Coupling curve of casing tensile stress with ground stress field and wellbore curvature

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It can be seen from Fig. 6 that when the borehole curvature is higher than 5.73°/30 m, in situ stress difference has a significant effect on casing tensile stress, the larger in situ stress difference is, the faster the casing tensile stress increases; when the borehole curvature is less than 5.73°/30 m, in situ stress difference has little effect on casing tensile stress. When the borehole curvature is 42.9°/30 m and in situ stress difference is 20 MPa, the tensile stress value is 760 MPa, the casing tensile stress exceeds the yield limit, and the casing has plastic deformation.

To sum up, it is finally determined that the casing tensile failure may occur when the borehole curvature increases to 42.9°/30 m under the action of in situ stress and wellhead pressure.

Factors affecting casing tensile stress of the bending sections

The formation characteristics of deep shale oil are complex, the vertical and transverse heterogeneity of the formation is strong, and the formation pressure system is complex, which makes the borehole trajectory formed by some oil wells in actual drilling cannot be drilled completely according to the design trajectory, and the bending degree is much higher than expected. When the borehole curvature is close to 42.9°/30 m, the casing tensile failure occurs frequently. Because of this, when the borehole curvature is 42.9°/30 m, the study of casing tensile stress distribution is in line with the actual working conditions, and the research results can reduce the probability of casing tensile failure to a certain extent.

With the help of the established full string three-dimensional finite element combination model of formation, cement and casing, the high-risk the casing tensile failure area (the bending sections) in the whole string model with borehole curvature of 42.9°/30 m is further studied. The finite element three-dimensional model of the borehole section is shown in Fig. 7. The effects of cement loss, the casing wall pressure, temperature change and Young’s modulus of elasticity of rock coupling cement on casing tensile stress distribution are analyzed and studied.

Fig. 7
figure 7

Three-dimensional finite element model of the bending sections. a Stratum grid. b Cement grid. c Casing grid. d Two-dimensional plan of casing–cement–formation. e Three-dimensional combination model of casing–cement–formation in the bending sections

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Cement missing

In the actual construction process, due to the poor quality of cement, there is often a shortage of cement. Once the cement is missing, it will lead to non-uniform load distribution of casing, and excessive local load will cause the casing tensile failure. On the other hand, the deformation part of casing in the bending sections is often concentrated in the tilting point (In a directional well, the position where directional deflections start) and the target position. For this purpose, the tilting area (In a directional well, the area where the well is deflected) of the bending sections, the middle area of the bending sections, and the target area of the bending sections are missing cement in Fig. 8. At the same time, considering the occurrence of in situ stress difference, σmin is set to 55 MPa, and σmax and σh are the same and increased from 60 to 100 MPa, in situ stress difference (σ = σmax − σmin = σh − σmin) is increased from 5 to 45 MPa.

Fig. 8
figure 8

Cement missing area. a Tilting area missing. b Central area missing. c Target area missing

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When in situ stress difference is 10 MPa, the cloud map of casing tensile stress distribution under the different missing parts of cement is shown in Fig. 9. Under the different in situ stress difference, the variation law of casing tensile stress with the missing position of cement is shown in Fig. 10.

Fig. 9
figure 9

The cloud map of casing tensile stress distribution under different missing parts of cement. a Absence of tilting area. b Absence of central area. c Absence of target area d Complete cement

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Fig. 10
figure 10

Change curve of casing tensile stress with missing cement at different parts

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It can be seen from Fig. 9 that the cement of tilting area is missing, and the casing tensile stress concentration occurs in the target area. When the cement of middle area is missing, or the cement of target area is missing, or the cement is complete, the casing tensile stress concentration appears in the inclined area, indicating that the stress concentration area of casing tensile varies with the different cement missing parts.

It can be seen from Fig. 10 that under the condition of constant in situ stress difference, the casing tensile stress with complete cement is far less than missing cement, which is due to the uneven the casing tensile stress caused by the absence of cement. Tensile stress concentration also occurs. When the cement in the tilting area is missing, the casing tensile stress is greater than the cement in the target area, because the tilting area is affected by the minimum horizontal combined tensile stress, while the target area is mainly affected by the horizontal and vertical stress.

When the cement of tilting area is missing, the casing tensile stress decreases with the increase in in situ stress difference, which is because the horizontal in situ stress increasing in the target area is offset by the casing pressure to a certain extent. When the cement is missing in the middle region or the target area, the casing tensile stress tends to increase with in situ stress difference increasing, and the tensile stress concentration is located in the tilting region and the value is close. This is because the combined force of the minimum horizontal in situ stress and the horizontal in situ stress is increasing in the tilting area, which makes the external load of casing become larger and larger. When in situ stress difference reaches 25 MPa, the stress value of casing reaches the yield strength 758 MPa, and the casing changes from elastic deformation to plastic deformation. The damage probability of casing is greatly increased due to the stress damage caused by the wellhead pressure and in situ stress of casing.

Casing wall pressure

The fracturing construction of shale oil usually uses a large wellhead pressure to make the fracturing fluid break the rock to achieve the purpose of fracture; however, it also produces great pressure on the casing in the fracturing section. In order to study the influence of wellhead pressure on casing tensile stress distribution, the internal pressure range of casing is set to be increased from 0 to 120 MPa, in which the sudden drop of the pressure on the casing is considered after the pump is stopped during the fracturing operation, and in extreme cases the internal pressure of casing is 0 MPa. At the same time, considering that the cement with different Poisson’s ratio affects the degree of looseness and fragmentation, the Poisson’s ratio of cement is 0.12, 0.27 and 0.37, respectively.

Figure 11 is the cloud map of casing tensile stress distribution under different internal pressure when the cement Poisson ratio is 0.27; under the condition of different cement Poisson ratio, the casing tensile stress varies with casing inner wall pressure as shown in Fig. 12.

Fig. 11
figure 11

Casing tensile stress cloud map of casing under different internal pressure. a No internal pressure. b Casing pressure, 60 MPa. c Casing pressure, 120 MPa

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Fig. 12
figure 12

Variation curve of casing tensile stress with casing inner wall pressure

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As can be seen from Fig. 11, with casing internal pressure increasing, the casing tensile stress concentration moves from the tilting area to the target area. When the internal pressure of casing is 0 MPa or 60 MPa, the tensile stress concentration occurs on the inner wall and outer wall of casing. When the internal pressure of casing is 120 MPa, the casing tensile stress is mainly concentrated in the inner wall of casing.

It can be seen from Fig. 12 that the casing tensile stress decreases first and then, increases with casing internal pressure increasing. When the casing internal pressure increases to 120 MPa and the Poisson’s ratio of cement is 0.37, the casing tensile stress is 738.9 MPa, and the casing tensile stress approaches the yield strength. The fracturing fluid produces pressure on the casing inner wall, which to some extent offsets some additional loads imposed by in situ stress on the casing wall. When the casing internal pressure is 20 MPa, the additional loads are offset the most, making the casing tensile stress the least.

When the casing internal pressure is lower than 20 MPa, the casing tensile stress decreases with Poisson’s ratio of cement increasing. Because the greater the Poisson’s ratio of cement, the looser cement is, allowing cement to remove some remote in situ stress and reduce the additional load of casing. When the internal pressure of casing is 20 MPa, the influence of Poisson’s ratio of cement on casing tensile stress can be ignored. When the internal pressure of casing is higher than 20 MPa, the casing tensile stress increases with Poisson’s ratio of cement increasing. This is because the greater the Poisson’s ratio of cement, the greater cement unloads part of far-field in situ stress, resulting in a greater resultant force on the casing wall, that is, an increase in the additional load.

Temperature change

In the process of fracturing, continuous injection of a large amount of fracturing fluid through the casing will cause a great change in the bottom hole temperature. The additional stress caused by temperature change will reduce the strength of casing, which has an important impact on borehole integrity. At present, the bottom hole temperature of most wells is in the range of 45–145 ℃, and the temperature of fracturing fluid injected on the surface is usually not lower than 45 ℃, the limit temperature difference of bottom hole is not more than 100 ℃, and the situation of exceeding 100 ℃ does not accord with the actual working conditions. Because of this, the case of more than 100 ℃ is not considered in this paper.

For this reason, when the model bottom hole temperature is increased from 45 to 145 ℃, the bottom hole temperature difference is increased from 0 to 100 ℃, in which the initial bottom hole temperature of 145 ℃ is equal to the formation temperature. At the same time, considering the difference of soft and hard surrounding rock added by the casing, Young’s modulus of elasticity of rock is set to 10 GPa (soft stratum) and 35 GPa (hard stratum), respectively. Figure 13 shows the temperature distribution when the casing temperature drops to 116 ℃. The casing temperature stress varies with the temperature difference as shown in Fig. 14.

Fig. 13
figure 13

Temperature distribution cloud map (Young’s modulus of elasticity of rock is 35 GPa)

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Fig. 14
figure 14

Variation curve of casing temperature stress with temperature difference

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As can be seen from Fig. 13, due to the injection of a large amount of fracturing fluid into the borehole, the bottom hole temperature decreases sharply and transfers heat to the surrounding rock near the borehole. The highest temperature appears in the central region.

As can be seen from Fig. 14, the additional temperature stress acting on the casing increases with bottom hole temperature difference increasing, and the influence of formation softness and hardness on temperature stress is also increasing. Compared with that without considering temperature difference, when the temperature difference is less than 5 ℃, the soft and hard degree of the surrounding rock near the bottom hole has little influence on the casing additional temperature stress. When the temperature difference is more than 5 ℃, the influence of the soft and hard formation on the casing additional temperature stress begins to increase obviously. The temperature difference of 5 ℃ is the critical point. When the temperature difference is constant, the casing additional temperature stress increases with the increase in Young’s modulus of elasticity of formation, and the formation begins to contract due to the decrease in temperature, the casing begins to be extruded. Compared with the soft formation, the casing is squeezed more in the hard formation, which increases the load on the outer wall of casing.

Young’s modulus of elasticity of rock coupling cement

Young’s modulus of elasticity of rock reflects the ability of rock to resist deformation by external force. Once the large-scale fracturing of the formation begins, the internal quasi-equilibrium of the rock is broken, so that Young’s modulus of elasticity of rock in this area decreases sharply and the rock plasticity increases.

In order to study the influence of different rock mechanical properties on the casing tensile stress, the Young’s modulus of elasticity of rock is set to be increased from 5 to 65 GPa. At the same time, considering that cement with different softness and hardness will also affect the casing tensile stress distribution, Young’s modulus of elasticity of cement is set to be increased from 5 to 65 GPa. Figure 15 is the cloud map of casing tensile stress distribution under different Young’s modulus of elasticity of rock when Young’s modulus of elasticity of cement is 15 GPa; Figure 16 shows the two-dimensional curve of casing tensile stress changing with Young’s modulus of elasticity of cement under different Young’s modulus of elasticity of rock; the three-dimensional variation curve of casing tensile stress is shown in Fig. 17, where A surface is the casing plastic deformation surface that begins to occur the casing tensile failure surface.

Fig. 15
figure 15

Stress cloud map of casing under different Young’s modulus of elasticity of rock. a Young’s modulus of elasticity of rock, 5 GPa. b Young’s modulus of elasticity of rock, 45 GPa. c Young’s modulus of elasticity of rock, 85 GPa

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Fig. 16
figure 16

Variation curve of casing tensile stress with Young’s modulus of elasticity of cement versus different Young’s modulus of elasticity of rock

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Fig. 17
figure 17

Three-dimensional variation curve of casing tensile stress versus Young’s modulus of elasticity of rock-coupled cement

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As can be seen from Fig. 15, with Young’s modulus of elasticity of rock increasing, the casing tensile stress concentration moves from the tilting area to the target area; from the point of view of the inner and outer wall of casing, the casing tensile stress is mainly concentrated in the inner wall of casing and is not affected by the change of Young’s modulus of elasticity of rock.

It can be seen from Fig. 16 that when the Young’s modulus of elasticity of rock is greater than 55 GPa, the casing tensile stress decreases with Young’s modulus of elasticity of cement increasing. When the Young’s modulus of elasticity of rock is 5 GPa, the change trend of casing decreases with Young’s modulus of elasticity of cement increasing. However, the decrease is greater. When the Young’s modulus of elasticity of rock is increased from 15 to 55 GPa, because the softer cement can be used as a buffer zone and the harder cement can play a supporting role, the casing tensile stress increases at first and then, decreases with Young’s modulus of elasticity of cement increasing, in which the Young’s modulus of elasticity of cement is 15 GPa, and the casing tensile stress is the highest.

It can be seen from Fig. 17 that under the coupling effect of Young’s modulus of elasticity of reservoir rock and Young’s modulus of elasticity of cement, with Young’s modulus of elasticity of rock increasing, the three-dimensional plane becomes smoother, which indicates that the influence of the change of the hardness of cement on casing tensile stress is decreasing(Hardness of cement means that the Young’s modulus of elasticity of cement reflects the hardness of cement; and the higher the Young’s modulus of elasticity of cement, the stronger the hardness of cement). When the Young’s modulus of elasticity of rock is 15 GPa and the Young’s modulus of elasticity of cement is 15 GPa, the casing tensile stress reaches 758 MPa under the action of in situ stress and the internal pressure of casing, the casing tensile stress breaks through the yield limit, the casing undergoes plastic deformation and local damage. The point with Young’s modulus of elasticity of rock 15 GPa, Young’s modulus of elasticity of cement 15 GPa and the casing tensile stress 758 MPa is on the casing tensile failure surface A of the three-dimensional coordinate system. The coordinate in the three-dimensional chart is higher than the casing tensile failure surface A, then the casing will have plastic deformation. In general, fracturing in the reservoir with Young’s modulus of elasticity of rock higher than 15 GPa can effectively reduce the casing tensile failure caused by the coupling of wellhead pressure and in situ stress.

Case demonstration

Figure 18 shows the actual well trajectory in Daqing Oilfield. Considering the main part of casing tensile failure deformation, that is, the casing part below the tilting point, a three-dimensional finite element model of formation-casing-cement failure from the tilting point to the horizontal section is established.

Fig. 18
figure 18

Borehole trajectory map

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The vertical stress of the reservoir is 68.2 MPa, the principal stress 70 MPa, the minimum principal stress 65 MPa, and the casing pressure 80 MPa. The cloud map of casing tensile stress distribution of actual reservoir is shown in Fig. 19.

Fig. 19
figure 19

Cloud map of casing tensile stress distribution under tilting area missing

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It can be seen from Fig. 19 that under the action of in situ stress and wellhead pressure, the casing tensile stress is the largest at the tilting area. This is due to the combined action of the fixed constraint of the vertical initial position, the cumulative effect of in situ stress change and the internal pressure of the casing. For the casing in the horizontal section, it can be seen that the force of casing (External load on casing) is smaller, which is due to the smaller axial load and friction resistance of casing.

According to the calculation results, the risk of casing tensile failure can be reduced by using high strength material the casing as far as possible and good cement quality.

Conclusions

On the basis of the influence of borehole curvature change on wellbore structure, the finite element three-dimensional model of the tilting point-entering target area is established. The effects of the absence of cement, the casing wall pressure, temperature stress and Young’s modulus of elasticity of rock coupling cement on casing tensile stress are studied. The conclusions are as follows:

  1. 1.

    When the cement of tilting area is missing, the casing tensile stress decreases with in situ stress difference increasing, while when there is a cement missing in the middle region or the target area, it tends to increase.

  2. 2.

    When the internal pressure of casing is lower than 20 MPa, the casing tensile stress decreases with casing internal pressure increasing. However, when the internal pressure of casing is higher than 20 MPa, the casing tensile stress increases with casing internal pressure increasing, and 20 MPa is the minimum point.

  3. 3.

    When the internal pressure of casing is lower than 20 MPa, the casing tensile stress decreases with Poisson’s ratio of cement increasing. When the internal pressure of casing is higher than 20 MPa, the casing tensile stress increases with Poisson’s ratio of cement increasing. When the internal pressure of casing is 20 MPa, the influence of Poisson’s ratio on casing tensile stress can be ignored.

  4. 4.

    When the temperature difference is more than 5 ℃, the influence of soft and hard strata on casing additional temperature stress begins to increase obviously. When the temperature difference is constant, the thermal stress on the casing increases with young’s modulus of elasticity of formation increasing.

  5. 5.

    When Young’s modulus of elasticity of rock is higher than 55 GPa or lower than 5 GPa, the casing tensile stress decreases with Young’s modulus of elasticity of cement increasing. When Young’s modulus of elasticity of rock is increased from 15 to 55 GPa, the casing tensile stress increases first and then decreases with Young’s modulus of elasticity of cement increasing.

  6. 6.

    Under the coupling effect of wellhead pressure and in situ stress, selecting the reservoir with Young’s modulus of elasticity higher than 15 GPa for fracturing can effectively reduce the casing tensile failure.