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An accurate analysis of casing stress distribution and its variation regularities present several challenges during hydraulic fracturing of shale gas wells. In this paper, a new analytical mechanical-thermal coupling method was provided to evaluate casing stress. For this new method, the casing, cement sheath, and formation (CCF) system was divided into three parts such as initial stress field, wellbore disturbance field, and thermal stress field to simulate the processes of drilling, casing, cementing, and fracturing. The analytical results reached a good agreement with a numerical approach and were in-line with the actual boundary condition of shale gas wells. Based on this new model, the parametric sensitivity analyses of casing stress such as mechanical and geometry properties, operation parameters, and geostress were conducted during multifracturing. Conclusions were drawn from the comparison between new and existing models. The results indicated that the existing model underestimated casing stress under the conditions of the geostress heterogeneity index at the range of 0.5–2.25, the fracturing pressure larger than 25 MPa, and a formation with large elastic modulus or small Poisson’s ratio. The casing stress increased dramatically with the increase of in situ stress nonuniformity degree. The stress decreased first and then increased with the increase of fracturing pressure. Thicker casing, higher fluid temperature, and cement sheath with small modulus, large Poisson’s ratio, and thinner wall were effective to decrease the casing stress. This new method was able to accurately predict casing stress, which can become an alternative approach of casing integrity evaluation for shale gas wells.

During the multistage fracturing process, fracturing fluids are pressed into a borehole with a high pump rate and pressure. The complex downhole environments—high pressure and large temperature variation—increase the risk of casing deformation. The volume fracturing technique effectively reconstructs shale reservoirs; however, frequent and serious casing deformation failures occur [

Analytical solutions have been developed under different conditions for casing-cement sheath-formation system. In-plane and out-of-plane analyses for the stress field around an internally pressurized, cased, cemented, and remotely loaded circular hole were developed [

A finite element method (FEM) had been proposed to achieve a better understanding of the ultimate collapse strength of casing [

However, the above analytical or FEM solutions are obtained by setting the casing and cement together instantly at the beginning of the analysis. The strains induced by the initial stress are included in the model, which is not in-line with the actual situation. In fact, initial stress has already existed in the formation before wellbore excavation. The wellbore stress redistributed after removing the rocks that originally occupied the borehole volume and just affected the stress and displacement near wellbore zones. In view of this, more sophisticated solutions have been developed with additional parameters and appropriate assumptions regarding the drilling, completion, and fracturing phases.

Pattillo and Kristiansen [

In this work, to evaluate the thermal and mechanical stresses of casing, an analytical model of casing-cement sheath-formation system was established considering wellbore construction. The boundary stresses in the wellbore coordinate system were obtained through three-dimensional rotations from principal in situ stresses. The casing stress was obtained by dividing the model into three parts such as initial stress field, disturbance stress field, and thermal stress field. The continuous homogeneous isotropy and linear elasticity were taken into consideration. Solutions were validated by a finite element method. Sensitivity analyses were conducted to estimate the influence of different factors on casing stress such as property of cement sheath and casing, fracturing pressure, fluid temperature, and initial geostress. Useful countermeasures were put forward to decrease casing stress during the fracturing operation.

For existing method, the casing, cement sheath, and formation were set together at the beginning of analysis. Then, temperature boundary and loads were added to the system to calculate wellbore stress distribution as shown in Figure

Loading process of the existing method. Outer boundary temperature, _{f}; inner boundary temperature, _{n}; normal stress, _{x} and _{y}; shear stress _{xy}.

To exactly predict the wellbore stress distribution, a casing-cement sheath-formation model using a new analysis method was provided. The analysis process was divided into four steps (Figure _{m} was applied to the internal face of the wellbore. During casing and cementing, casing and cement sheath elements were added simultaneously to the model and the cement hardening procedure was not considered. Initial stress and strain in casing and cement sheath were ignored. The cement slurry pressure _{c} was applied to the internal casing wall. During fracturing, a low temperature _{n} and the fracturing pressure _{f} were assigned in the internal casing wall. The wellbore stress field was obtained under the condition of thermal-mechanical coupling.

Loading process of the new method. Outer boundary temperature, _{f}; inner boundary temperature, _{n}; normal stress, _{x} and _{y}; shear stress, _{xy}; drilling mud pressure, _{m}; cement slurry pressure, _{c}; fracturing pressure, _{f}.

The stress state and coordinate transformation system are shown in Figure _{v}-axis and rotating clockwise _{v}-axis; second, rotating anticlockwise

The coordinate rotation processes: (a) stress state of a horizontal well and (b) coordinate transformation system. Principal in situ stress coordinate system,

In the principal stress coordinate system, the principal horizontal stress matrix is σ^{0}, whose components are maximum principal stress, _{H}; the minimum principal stress, _{h}; and the overburden pressure, _{v}, shown in equation (_{x}, _{y}, and _{z} are the rotation angles in a counterclockwise direction when looking towards the origin coordinate.

A thermo-pressure coupling model of casing-cement sheath-formation (CCF) system was established (Figure _{x}, _{y}, and _{xy} were obtained by using equation (

CCF composite assembly. Formation boundary temperature, _{f}; internal casing temperature, _{n}; internal casing pressure, _{i}; radius, _{i},

For simplicity, some assumptions have been made [

Geometry: the casing, cement, and borehole were concentric circles, which were assumed to be perfectly bonded to each other at each interface. The perfect bonding mathematically indicated that the continuity of radial stress and displacement was satisfied at each interface.

Thermal effect: the stress induced by wellbore temperature variation was assumed to be steady state and the time effect was ignored.

Material: to simplify the complex property of strong anisotropy and well-developed bedding planes of shale formation [_{4} ⟶ ∞). The cement sheath was also a complex material. The 3D images revealed the evolution of a large connected pore network with characteristic widths on the micrometer scale as hydration proceeded [

The boundary compression normal stresses (−_{x}, −_{y}) were decomposed into uniform stress (_{0}) and deviator stress (_{xy}):

According to the basic hypotheses in Section

The stress distribution of the thermal-pressure coupling model around a wellbore was decomposed into four parts as shown in Figure

Stress decompositions. Inner casing pressure, _{i}; thermal stress, _{T}.

It was convenient to convert the Cartesian coordinate system into the polar coordinate system to calculate wellbore stress. The normal boundary stresses in the polar coordinate system were expressed as follows under the conditions of the infinite outer boundary radius of _{4}:

Since temperature and stress were coupled, the stress distribution around a cased wellbore induced by temperature variation was hard to solve in the closed form. However, the steady-state condition made the temperature and stress decouple and the problem analytically solvable [

Under the condition of the uniform internal pressure and external stress, the stress and displacement in a thin wall cylinder were obtained by using the following equations shown in Figure _{i} is the material elastic modulus; _{i}, _{i+1} were the interfacial pressure, positive in the radial increase direction, _{i} (

Stress induced by the uniform stress.

Before drilling the borehole, the initial geostress field already existed in the formation. When the rock was removed from the borehole, the wellbore stress field redistributed to produce a disturbance field, which only affected the near-wellbore zones [

Wellbore stress components under the condition of uniform stress.

In the polar coordinate system, the initial conditions of _{4} approached to infinity,

The pressures at casing-cement sheath interface and cement sheath-formation interface were

Interface pressures induced by uniform stress.

Substituting these initial and boundary conditions in equations (

According to the hypotheses that cement sheath-formation interface and casing-cement sheath interface were perfectly bonded to each other, the interfacial displacement continuity conditions were expressed in the following equation:

Substituting equation (

The interfacial pressures

The deviator stress boundary conditions are shown in Figure

Stress induced by deviator stress. Outer stress,

The stress and strain under the condition of nonuniform stress are

From the geometric equations,

The radial displacement

Similar to that of uniform stress, the actual stress field,

Formation stress components under the nonuniform stress condition.

In the polar coordinate system, initial stresses were

For casing and cement sheath in the polar coordinate system, initial stresses were

The interfacial displacement and stress continuity and boundary conditions were expressed in the following equation:

Substituting equations (

The stress induced by shear stress was _{x} and

Stress distribution induced by shear stress.

It could be found that the stress distribution induced by shear stress was similar with that by deviator stress when counterclockwise rotating the angle of

The thermal field was obtained by using the steady temperature distribution model to calculate the thermal stress. When fracturing fluids were pumped into a wellbore with a high pump rate, they were always in the turbulent state. The heat transfer coefficient between casing and fluid was calculated using the Marshall model [^{−2}·°C^{−1}), ^{−3}), ^{n}), ^{3}·min^{−1}), _{m} is the coefficient of heat conductivity (W·m^{−1}·°C^{−1}), and _{m} is the fluid specific heat capacity (J·kg^{−1}·°C^{−1}).

The temperature distribution among casing, cement sheath, and formation is shown in Figure

The distribution of interface temperatures.

Temperature field distribution solutions were obtained according to integral and boundary conditions ^{−1}·°C^{−1}); _{i} is the radius (m); and

The heat flow density continuity conditions were expressed as

The temperatures at interfaces of casing-cement sheath and cement sheath-formation system were defined as _{2} and _{3} and were calculated by using the following equation:

Interfacial temperature of

The actual thermal stress field,

Thermal stress field. (a) Formation stress components. (b) Interface pressures. _{i} is the interface pressure,

The initial stresses were _{i} is the interface pressure (Pa), and _{i} is the material thermal expansion coefficient,

The temperatures were known, and the boundary was free at internal casing and external formation. So, radial stress at inner and outer boundaries equals to zero, and radial displacement at the outer boundary equals to zero as well. The boundary and interfacial displacement continuity conditions were expressed as

Substituting equations (

The constants of

The total stresses were obtained using the following equation:

It is generally accepted that the yield of isotropic material such as casing has nothing to do with hydrostatic pressure, while hydrostatic pressure is not considered in von Mises yield criterion. So, this criterion was adopted to determine the casing failure:

For uniaxial tension,

From 2009 to 2017, PetroChina has drilled 141 fracturing wells (including 112 horizontal wells) in the Changning-Weiyuan National Shale Gas Demonstration Area. The geometrical dimensions of the CCF model were a wellbore diameter of 8.5 in, casing diameter of 5.5 in, and casing thickness of 9.17 mm. According to the Saint-Venant principle, a formation boundary dimension should be five to six times larger than that of the wellbore geometry to avoid the influence of boundary effect on wellbore stress. In view of this, the model geometry was 2,000 × 2,000 mm, while the corresponding wellbore diameter was 215.9 mm. The direction of horizontal in situ stress was N120°E. The well deviation angle was 90°, and the wellbore azimuth was N30°E, indicating that the horizontal trajectory was along the minimum in situ stress direction. The internal casing pressure was calculated from the pump pressure plus the downhole hydrostatic fluid pressure. The external boundary stress was obtained from the geostress data of the shale reservoir. The thermal and mechanical properties of different materials are presented in Table

Thermal and mechanical Parameters of fluid-casing-cement sheath-formation system.

Property | Casing | Cement sheath | Formation | Fluid |
---|---|---|---|---|

Elastic modulus, _{i} (GPa) |
210 | 5 | 35 | — |

Poisson’s ratio, _{i} |
0.3 | 0.15 | 0.25 | — |

Coefficient of thermal expansion, _{i} (10^{−5}·°C^{−1}) |
1.5 | 1.0 | 1.0 | — |

Thermal conductivity, _{i} (W·m^{−1}·°C^{−1}) |
58.2 | 1.0 | 1.0 | 1.73 |

Specific heat, _{pi} (J·kg^{−1}·°C^{−1}) |
460 | 1830 | 1043 | 3935 |

Density, _{i} (kg·m^{−3}) |
7850 | 1800 | 2500 | 1080 |

Note: properties in parenthesis were used in the parametric study.

The applied maximum horizontal stress _{H} was 82 MPa, the minimum horizontal stress _{h} was 55 MPa, the vertical stress _{v} was 57 MPa, the inner casing pressure _{i} was 75 MPa, the boundary temperature _{4} was 100°C, the fluid temperature _{a} was 20°C, and the convective heat transfer coefficient was obtained by using equation (^{−2}·°C^{−1}) at the pump rate of 20 m^{3}/min.

The finite element analysis method was adopted to validate the results of the analytical models. A steady-state thermal analysis followed by a static structural analysis was conducted to calculate the stress considering thermal-pressure coupling. The solutions of radial stress, circumferential stress, and Mises stress are compared in Figure

Comparison of numerical and analytical solutions. (a) Radial stress along the radial directions of 0° and 90°. (b) Radial stress at the internal casing wall. (c) Circumferential stress along the radial directions of 0° and 90°. (d) Circumferential stress at the internal casing wall. (e) Mises stress at inner and outer casing walls.

The analytical solutions of radial stress, circumferential stress, and Mises stress were in good agreement with the results obtained by a finite element method, which indicates the validity of the analytical method. The maximum deviation between analytical and finite element results was 1.4–13.9%, indicating that the analytical model could provide an accurate calculation of stress distribution for the CCF system.

From Figures

The radial displacements along the 0° direction calculated by the new model and existing model under the same conditions were shown in Figure

Radial displacements of the wellbore assembly along the 0° direction.

The sensitivity analyses were carried out to study the influences of cement sheath properties, geostress, fracturing pressure, fluid temperature, casing thickness, and cement sheath thickness on casing stress. During analyzing, only one parameter was variable and others were constants. Unless otherwise mentioned, the parameters were set as mentioned in Section

Cement sheath properties is crucial for casing safety. To evaluate the effect of elastic modulus on casing stress, the cement sheath elastic modulus of _{2} was set at the range from 2 GPa to 50 GPa, and the formation elastic modulus of _{3} was set as 5 and 35 GPa to simulate a soft and hard formation. The Mises stresses at internal casing are shown in Figure

Casing Mises stress. (a, b) _{3} = 5 GPa and (c, d) _{3} = 35 GPa.

From Figures

To evaluate the effect of Poisson’s ratio on casing stress, cement sheath Poisson’s ratio, _{2}, with a range from 0.05 to 0.45, was adopted and the formation Poisson’s ratio, _{3}, was set as 0.05 and 0.45 to simulate a hard and soft formation. The casing Mises stresses are shown in Figure

Casing Mises stresses for different Poisson’s ratios. (a) _{3} = 0.05. (b) _{3} = 0.45.

From Figures

During the multifracturing operation for shale gas wells, the fracturing fluid was pressed into the formation and the in situ stress field changed abruptly to increase the nonuniformity of the stress around the wellbore. To evaluate the effect of in situ stress nonuniformity on casing stress, the nonuniformity index was defined as

Casing Mises stress for different stress nonuniformity indexes: (a) casing internal stress and (b) maximum casing stress.

As seen from Figure

A fracturing fluid with high pressure was used to fracture a shale formation. The high pressure depended on the formation regional tectonic stress; the larger the tectonic stress, the higher the pressure. Moreover, a high fracturing pressure posed a great potential challenge to casing failure. Different fracturing pressures with a range of 5–105 MPa were adopted to evaluate the effect of fracturing pressure on casing stress. The maximum casing Mises stresses are shown in Figure

Maximum casing Mises stress for different pressures.

It can be seen from Figure

During the cycle injection of fracturing fluid, the heat transfer coefficient ^{3}/min. The corresponding casing internal Mises stress was calculated under different fluid temperatures at a range of 10–100°C to evaluate the effect on casing stress. Figure

Casing Mises stress for different temperatures.

From Figure

The thickness of cement sheath and casing was curial for casing safety. To evaluate the effect of thickness on the casing stress, the cement thickness was set at a range of 2–50 mm and the casing thickness was set at a range of 5–15 mm. The comparisons of maximum casing Mises stress obtained by the two models are shown in Figure

Casing Mises stress for different thicknesses of casing and cement sheath. (a) Different cement thickness. (b) Different casing thickness.

As shown in Figure

A new analytical model considering drilling construction was established to assess the casing stress under different conditions considering thermal-pressure coupling. The solutions were obtained by dividing the model into three parts such as initial stress field, wellbore disturbance field, and thermal stress field. Sensitivity analyses of different factors were conducted to evaluate the influences on casing stress. Some conclusions were drawn from the comparisons between new model and existing model:

The results of radius stress, tangential stress, and casing Mises stress calculated by the analytical method were in good agreement with the solutions by a finite element solution. The minor deviations did not exceed 13.9%. Moreover, the analytical solutions were in-line with the actual boundary conditions of shale gas wells.

The casing stress calculated by the existing model was smaller than that by the new model for hard formation with larger modulus or low Poisson’s ratio, geostress heterogeneity index at a range of 0.5–2.25, and fracturing pressure larger than 25 MPa.

The casing stress increased with the increase of the in situ stress nonuniformity index. With the increase of fracturing pressure, casing stress decreased first and then increased.

Cement sheath with appropriate modulus and larger Poisson’s ratio, thinner cement sheath, thicker casing, and higher fluid temperature were effective to decrease the casing stress.

In conclusion, the new analytical model can accurately predict casing stress and become an alternative method of casing integrity evaluation for shale gas wells. It is a useful and efficient method for a preliminary design, being capable of simulating the actual situations in order to assess the casing stresses and integrity.

The data of each figure used to support the findings of this study are available from the corresponding author upon request.

The authors declare that they have no conflicts of interest.

This research was financially supported by the National Natural Science Funds of China (51674272), the Key Program of National Natural Science Foundation of China (U1762211), and China Petrochemical Corporation (HX20180001). The assistance of Dr. Wei Lian in contribution to modify the language of the manuscript and the pictures and editable figure files is gratefully acknowledged.