Influences of Fracturing Fluid Injection on Mechanical Integrity of Cement Sheath under Four Failure Modes

The significant decreased wellbore temperature and increased casing pressure during fracturing fluid injection present a big challenge for the mechanical integrity of cement sheath in fracturing wells. Based on the theories of elastic mechanics, thermodynamics, and a multi-layer composed thick-wall cylinder, this paper proposed a new mechanical model of cement sheath for fracturing wells, coupling pressure, and thermal loads, which consider the failure modes of de-bonding, radial cracking, disking, and shear failure. The radial nonuniform temperature change and the continuous radial stress and radial displacement at two interfaces have been considered. With the proposed model, the radial distributions of failure stress and the corresponding safety factor for cement sheath during fracturing fluid injection have been analyzed and compared under four failure modes. Results show that the decreased wellbore temperature will produce significant tri-axial tensile stress and induce cement failure of de-bonding, radial cracking, and disking. The increased casing pressure will significantly lower the risk of de-bonding but also aggravate radial cracking and shear failure. For integrity protection of cement sheath, increasing the injected fluid temperature, maintaining higher circulation pumping pressures, and adopting cement sheath with a low elasticity modulus have been suggested for fracturing wells.


Introduction
For maintaining gas well's long-term and safe production, mechanical integrity of cement sheath has been given more emphasis in recent years.Field experiences and laboratory studies have both proven that cement sheath is very likely to fail after downhole operations, such as pressure testing, hydraulic fracturing, acidizing, steam injection, and more [1][2][3].The cement sheath fails mainly because of the loads from variations of wellbore temperature and pressure.For fracturing wells, because of the high displacement and pump pressure fracturing fluid injection into wellbore, the wellbore temperature decrease (by −70 • C) and casing pressure increase (by 70 MPa) are both very serious.In addition, this leads to cement sheath failure and continuous annular casing pressure.Presently, Portland cement is still frequently-used for wellbore purposes.However, because of its high elasticity and lower tensile strength, it is more likely to fail.Consequently, an analysis of the Energies 2018, 11, 3534 2 of 18 failure stress and the corresponding failure mode in the cement sheath during the fracturing process is significant for the wellbore safety of fracturing wells.
Currently, some theoretical mechanical models of cement sheath coupling pressure and temperature loads have been established for high pressure and high temperature (HPHT) gas wells and steam injecting wells.Thiercelin et al. [4] first proposed the mechanical model for the cement sheath in which the influence of thermo-elastic properties of materials on cement failure was verified.Li et al. [5] proposed a thermal stress model for a coupled casing-cement-formation system in a thermal well and investigated the casing failure behaviors.After that Li et al. [6] further established a mechanical model of cement sheath coupling the effects of temperature and pressure loads in a non-uniform in-situ stress field.Bois et al. [7] also proposed a cement sheath model.The failure modes were analyzed by investigating the casing deformation induced by wellbore temperature variation.Haider et al. [8] developed a composite axisymmetric casing-cement-formation model for a CO 2 storage well.The results indicated that the decrease in wellbore temperature could induce radial tensile stress in the cement sheath.Bui et al. [9] also presented a mechanical model for cement failure prediction, where the non-uniform stress field and the thermal stress were involved.Xi et al. [10] established a transient thermo-mechanical model of casing-cement sheath-formation assembly by using the analytical method and the numerical method comprehensively.Our previous study [11] proposed a mechanical model of casing-cement-formation system and investigated the influence of wellhead casing pressure on cement sheath integrity for HPHT gas wells.After that, we further proposed a model of thermal stress for cement sheath during hydraulic fracturing [12], where radial non-uniform temperature change was involved, and the main influencing factors were also investigated.All the models mentioned above could be used to simulate the stress and failure of cement sheath in some ways.However, because of the complexity of wellbore geometry and the diversity of cement sheath failure modes, the present models cannot provide a relatively accurate and comprehensive failure analysis for cement sheath.
In this paper, a new mechanical model of cement sheath, coupling pressure, and thermal loads and considering the failure modes of de-bonding, radial cracking, disking, and shear failure, has been proposed for fracturing wells, based on the elastic mechanics, thermodynamics theory, and the theory of multi-layer composed thick-wall cylinder.The radial non-uniform temperature change in the combined system has been considered and both the radial stress and radial displacement at the casing-cement sheath interface and the cement sheath-formation interface are supposed to be continuous.Based on the proposed model, the radial distributions of failure stress and corresponding safety factor for cement sheath during fracturing fluid injection have been analyzed and compared under four failure modes.In addition, some measures have been proposed for cement sheath protection, which can be of great significance for decreasing the risk of sustained casing pressure after fracturing operations.

Basic Assumptions
To maintain the wellbore mechanical and hydraulic integrity during fracturing and long-term production, the open hole is usually cemented with a steel casing.When the high pump pressure fluid with continuous large displacement is injected into the wellbore, the coupled loads from increased casing pressure p i and decreased wellbore temperature T(r) will simultaneously exert on the casing-cement sheath-formation combined system, which induces cement sheath stresses.The schematic diagram of the casing-cement-formation combined system during fracturing fluid injection is shown in Figure 1.Before developing the mechanical model of cement sheath, to make the present model tractable, the following assumptions are made [11,12].
(2) The casing-cement sheath-formation system is completely cemented and axisymmetric.(3) Both casing-cement sheath and cement sheath-formation interface satisfy the continuity conditions for radial stress and radial displacement.(4) The radial non-uniform temperature varies along the combined casing-cement sheath -formation system except for casing as consideration of its thin wall and well heat conduction performance.( 5) The temperature at the inner wall of casing is equal to the wellbore fracturing fluid temperature and that at the outer wall of formation maintains an original formation temperature during fracturing.
Energies 2018, 11, x 3 of 20 (3) Both casing-cement sheath and cement sheath-formation interface satisfy the continuity conditions for radial stress and radial displacement.(4) The radial non-uniform temperature varies along the combined casing-cement sheath -formation system except for casing as consideration of its thin wall and well heat conduction performance.( 5) The temperature at the inner wall of casing is equal to the wellbore fracturing fluid temperature and that at the outer wall of formation maintains an original formation temperature during fracturing.

Cement Sheath Stress Induced by the Decrease of Wellbore Temperature
The continuous fracturing fluid injection will significantly decrease wellbore temperature and constrict the combined system, which will lead to thermal stresses in the cement sheath.Our previous study investigated thermal stresses for cement sheath during hydraulic fracturing based on the elastic mechanics and thermodynamics theory [13], and the general solutions of radial thermal displacement and thermal stresses for thick wall cylinder were given as follows [12,13].
In Equation (1), T(r) stands for the radial temperature change at radius of r in the cylinder after and before fracturing fluid injection.Considering wellbore temperature varies radially constant in casing and logarithmically in cement sheath and formation, if the temperature at the inner boundary of casing is Ti before fracturing and it decreased to Tt after fracturing fluid injection, while the temperature at the outer boundary of formation is constantly Te, the radial wellbore temperature change T(r) can be determined by the following [12].

Cement Sheath Stress Induced by the Decrease of Wellbore Temperature
The continuous fracturing fluid injection will significantly decrease wellbore temperature and constrict the combined system, which will lead to thermal stresses in the cement sheath.Our previous study investigated thermal stresses for cement sheath during hydraulic fracturing based on the elastic mechanics and thermodynamics theory [13], and the general solutions of radial thermal displacement and thermal stresses for thick wall cylinder were given as follows [12,13].
In Equation (1), T(r) stands for the radial temperature change at radius of r in the cylinder after and before fracturing fluid injection.Considering wellbore temperature varies radially constant in casing and logarithmically in cement sheath and formation, if the temperature at the inner boundary of casing is T i before fracturing and it decreased to T t after fracturing fluid injection, while the temperature at the outer boundary of formation is constantly T e , the radial wellbore temperature change T(r) can be determined by the following [12].
Energies 2018, 11, 3534 4 of 18 For the integral expression r a T(r)rdr in Equation ( 1), it can be deduced as: Submitting Equations ( 2) and (3) into Equation (1), the thermal radial displacement and thermal stresses for combined system can be determined, and the detailed derivative results are shown in Appendix A.

Cement Sheath Stresses Induced by the Increase of Casing Pressure
Taking wellbore temperature as no change during fracturing fluid injection and, since only the increased casing pressure acts on the casing-cement sheath-formation combined system, casing expansion and induced stresses in the cement sheath will occur.Assuming the plane strain of the combined system (ε z ≈ 0), the following can be obtained [13].
Furthermore, the radial displacement of a cylinder induced by the increased inner pressure can be expressed below.
If the contact pressure induced by increased casing pressure is p c1 at the casing-cement sheath interface and p c2 at the cement sheath-formation interface respectively, the induced stress in the cement sheath can be calculated with the Lame Formula [13].Now, we will solve p c1 and p c2 with the interfacial continuity conditions.
For the casing, because it is subjected to the increased casing pressure p i at its inner wall and the contact pressure p c1 at its outer wall, its radial and tangential stresses can be obtained with the Lame Formula as follows: By submitting Equation (6) into Equation ( 5), the radial displacement of the casing at its outer wall (r = r 1 ) can be obtained.
For the cement sheath, because it is subjected to the contact pressure p c1 at its inner wall and the contact pressure p c2 at its outer wall, its radial and tangential stresses can also be obtained with the Lame Formula below.
By submitting Equation (8) into Equation ( 5), the radial displacement of the cement sheath at its inner wall (r = r 1 ) can be obtained.
Energies 2018, 11, 3534 5 of 18 According to the continuity condition at the casing-cement sheath interface, there exists u P s | r=r 1 = u P c | r=r 1 .Thus, we can get the following.
Similarly, the radial displacement of the cement sheath at its outer wall (r = r 2 ) can also be obtained as follows.
For the formation, because it is subjected to the contact pressure p c2 at its inner wall and the changed formation pressure p f at its outer wall, its radial and tangential stresses can also be obtained with the Lame Formula below.
Then, by submitting Equation (12) into Equation ( 5), the radial displacement of the formation at its inner wall (r = r 2 ) can be obtained.
According to the continuity condition at the cement sheath-formation interface, there exists u P c | r=r 2 = u P f | r=r 2 .Thus, we can get the following.
Solving Equations ( 10) and ( 14) simultaneously, the contact pressure p c1 and p c2 can be obtained.Lastly, from Equation ( 8) we can easily get the radial stress σ P rc and tangential stress σ P θc respectively.Furthermore, the axial stress σ P zc in the cement sheath caused by the increased casing pressure can be calculated by Equation (4) as:

Combined Stresses of Cement Sheath during Fracturing Fluid Injection
In fact, the cement sheath will simultaneously suffer from induced stresses caused by decreased wellbore temperature and increased casing pressure during the fracturing fluid injection.Therefore, the combined radial, tangential, and axial stress of cement sheath during fracturing fluid injection can be expressed as follows.

Failure Criterion and Safety Factor for Cement Sheath
Four kinds of failure modes may lead to the loss of zonal isolation for cement sheath including inner debonding, outer debonding, radial cracking, shear damage, and disking.Bois et al. [14] illustrated that in Figure 2. Debonding, radial cracking, and disking can be determined with the maximum tensile stress criterion, while shear failure can be determined with the Mohr-Coulomb Criterion or the Mogi-Coulomb Criterion.Compared to the Mogi-Coulomb Criterion, because the Mohr-Coulomb Criterion does not consider the effect of intermediate principal stress on the strength of the material, it cannot reflect the real failure under a true tri-axial stress state in the rock [15,16].The Mogi-Coulomb Criterion was selected in the paper.
the combined radial, tangential, and axial stress of cement sheath during fracturing fluid injection can be expressed as follows.

Failure Criterion and Safety Factor for Cement Sheath
Four kinds of failure modes may lead to the loss of zonal isolation for cement sheath including inner debonding, outer debonding, radial cracking, shear damage, and disking.Bois et al. [14] illustrated that in Figure 2. Debonding, radial cracking, and disking can be determined with the maximum tensile stress criterion, while shear failure can be determined with the Mohr-Coulomb Criterion or the Mogi-Coulomb Criterion.Compared to the Mogi-Coulomb Criterion, because the Mohr-Coulomb Criterion does not consider the effect of intermediate principal stress on the strength of the material, it cannot reflect the real failure under a true tri-axial stress state in the rock [15,16].The Mogi-Coulomb Criterion was selected in the paper.If the radial tensile stress at the casing-cement sheath interface or the cement sheath-formation interface exceeds the tensile strength of cement sheath, the de-bonding failure can be described as follows [17].
rc r r or r tc (17) If the tangential tensile stress in the cement sheath surpasses its tensile strength, cracking will begin to form and propagate along the radial direction.When multiple radial cracking are linked, the leakage path can be generated.The radial cracking failure can be expressed by Equation (18) below.
Similarly, if the axial tensile stress in the cement sheath exceeds its tensile strength, disking will begin to form and run across the cement wall.The disking failure can be described below.
Shear failure is a critical failure mode, which typically causes an entire failure for cement sheath.The Mogi-Coulomb Criterion assumes that shear failure will occur when the octahedral shear stress τoct in the cement sheath exceeds the maximum allowable shear stress τcmax, which can be expressed as the following equation [15,16].If the radial tensile stress at the casing-cement sheath interface or the cement sheath-formation interface exceeds the tensile strength of cement sheath, the de-bonding failure can be described as follows [17].
If the tangential tensile stress in the cement sheath surpasses its tensile strength, cracking will begin to form and propagate along the radial direction.When multiple radial cracking are linked, the leakage path can be generated.The radial cracking failure can be expressed by Equation ( 18) below.
Similarly, if the axial tensile stress in the cement sheath exceeds its tensile strength, disking will begin to form and run across the cement wall.The disking failure can be described below.
Shear failure is a critical failure mode, which typically causes an entire failure for cement sheath.The Mogi-Coulomb Criterion assumes that shear failure will occur when the octahedral shear stress τ oct in the cement sheath exceeds the maximum allowable shear stress τ cmax , which can be expressed as the following equation [15,16].
If a linear form is considered for the Mogi-Coulomb criterion and, noting that the compressive stress is negative in this paper, the maximum allowable shear stress τ cmax in Equation ( 20) can be determined by the equation below [18].
Energies 2018, 11, 3534 7 of 18 Considering the following relationship between the uniaxial compressive strength of cement sheath σ cc and its cohesion c c , the internal friction angle φ c is shown below.
If σ cc and φ c are both known, by combining Equations ( 20)-( 22), we can express the Mogi-Coulomb Criterion below.
According to the failure criteria above, the safety factors for four failure modes can be calculated respectively, as shown in Table 1.
Table 1.Safety factor for cement sheath.

Failure Mode
Failure Criterion Safety factor (SF)

Basic Calculation Parameters
A shale gas fracturing well in the Sichuan Basin in China was selected for the analysis of mechanical integrity of cement sheath during fracturing fluid injection.The 215.9 mm open hole was cemented with 139.7 mm × 9.17 mm P110 casing.The basic calculation parameters, including wellbore geometry parameters, material property parameters, and operation parameters, are listed in Table 2.

Results and Discussion
To make clear the failure laws of cement shear during fracturing fluid injection, the proposed cement sheath stress model has been validated by the FEA method and then the corresponding failure stresses and safety factors of cement sheath under a different failure mode have been calculated according to the basic parameters in Table 2. Simultaneously, when seeking the measures of protecting cement sheath, the safety factors have also been investigated under a different elasticity modulus of the cement sheath E c .Up to now, as the complexity of the casing-cement sheath-formation combined system and the difficulty of applying coupled loads, fewer experimental studies were conducted about the in situ cement sheath integrity during fracturing.To validate the cement sheath stress model proposed in this paper, a 2D axisymmetric casing-cement sheath-formation model has been established with Ansys Workbench 14.5.Figure 3 shows the finite element analysis model of casing-cement sheath-formation combined system and the element type of Plane 183 was selected for this analysis.For simulating cement sheath stresses induced by increased casing pressure and decreased wellbore temperature during fracturing, a Steady-State Thermal analysis followed by a Static Structural analysis has been conducted.The axial displacement on the upper and lower bounds of the whole model and the radial displacement on the outer boundary of formation are all set to be zero.Furthermore, 70 MPa net increased casing pressure is exerted on the inner boundary of casing.The initial temperature is uniform at 100 • C on the whole model and it decreased to 30 • C on the casing inner wall while still maintaining 100 • C on the formation outer wall during fracturing.

Results and Discussion
To make clear the failure laws of cement shear during fracturing fluid injection, the proposed cement sheath stress model has been validated by the FEA method and then the corresponding failure stresses and safety factors of cement sheath under a different failure mode have been calculated according to the basic parameters in Table 2. Simultaneously, when seeking the measures of protecting cement sheath, the safety factors have also been investigated under a different elasticity modulus of the cement sheath Ec.

Cement Sheath Stresses Validated by the FEA Method
Up to now, as the complexity of the casing-cement sheath-formation combined system and the difficulty of applying coupled loads, fewer experimental studies were conducted about the in situ cement sheath integrity during fracturing.To validate the cement sheath stress model proposed in this paper, a 2D axisymmetric casing-cement sheath-formation model has been established with Ansys Workbench 14.5.Figure 3 shows the finite element analysis model of casing-cement sheath-formation combined system and the element type of Plane 183 was selected for this analysis.For simulating cement sheath stresses induced by increased casing pressure and decreased wellbore temperature during fracturing, a Steady-State Thermal analysis followed by a Static Structural analysis has been conducted.The axial displacement on the upper and lower bounds of the whole model and the radial displacement on the outer boundary of formation are all set to be zero.Furthermore, 70 MPa net increased casing pressure is exerted on the inner boundary of casing.The initial temperature is uniform at 100 °C on the whole model and it decreased to 30 °C on the casing inner wall while still maintaining 100 °C on the formation outer wall during fracturing.Figure 4 shows the comparison of cement sheath stresses between analytical model results and finite element analysis ones.It can be seen that the radial stress, tangential stress, and axial stress in the cement sheath calculated with two methods are all very close.The average relative errors of three kinds of stress are 7%, 5%, and 2% respectively, which indicated that the proposed model can be applicable for cement sheath stress calculation during fracturing to some extent.Figure 4 shows the comparison of cement sheath stresses between analytical model results and finite element analysis ones.It can be seen that the radial stress, tangential stress, and axial stress in the cement sheath calculated with two methods are all very close.The average relative errors of three kinds of stress are 7%, 5%, and 2% respectively, which indicated that the proposed model can be applicable for cement sheath stress calculation during fracturing to some extent.

De-Bonding
De-bonding occurs when radial tensile stress at the cement interfaces exceeds its own tensile strength, which may mainly result from decreased wellbore temperature during fracturing fluid injection [18].Figure 5 shows the radial distribution of the radial stress in the cement sheath considering increased casing pressure pi of 0-70 MPa and decreased temperature at the casing inner wall of 70 °C.It can be seen from Figure 5 that the decreased wellbore temperature will induce significant radial tensile stress, which is very harmful to cement sheath and may cause de-bonding.However, the increased casing pressure will produce radial compressed stress in the cement sheath at the same time and offset the radial tensile stress.In addition, with the increase of casing pressure pi, the radial stress in the cement sheath gradually changes from tensile to compressed stress, which indicates that the risk of de-bonding decreases.When the increased casing pressure is 50 MPa, the radial stress is nearly compressive.Consequently, maintaining higher circulation pumping

De-Bonding
De-bonding occurs when radial tensile stress at the cement interfaces exceeds its own tensile strength, which may mainly result from decreased wellbore temperature during fracturing fluid injection [19].Figure 5 shows the radial distribution of the radial stress in the cement sheath considering increased casing pressure p i of 0-70 MPa and decreased temperature at the casing inner wall of 70 • C. It can be seen from Figure 5 that the decreased wellbore temperature will induce significant radial Energies 2018, 11, 3534 9 of 18 tensile stress, which is very harmful to cement sheath and may cause de-bonding.However, the increased casing pressure will produce radial compressed stress in the cement sheath at the same time and offset the radial tensile stress.In addition, with the increase of casing pressure p i , the radial stress in the cement sheath gradually changes from tensile to compressed stress, which indicates that the risk of de-bonding decreases.When the increased casing pressure is 50 MPa, the radial stress is nearly compressive.Consequently, maintaining higher circulation pumping pressures, avoiding pumping off during injection, and lowering the pump pressure slowly during the operation completion will all be beneficial for cement sheath protection.

De-Bonding
De-bonding occurs when radial tensile stress at the cement interfaces exceeds its own tensile strength, which may mainly result from decreased wellbore temperature during fracturing fluid injection [18].Figure 5 shows the radial distribution of the radial stress in the cement sheath considering increased casing pressure pi of 0-70 MPa and decreased temperature at the casing inner wall of 70 °C.It can be seen from Figure 5 that the decreased wellbore temperature will induce significant radial tensile stress, which is very harmful to cement sheath and may cause de-bonding.However, the increased casing pressure will produce radial compressed stress in the cement sheath at the same time and offset the radial tensile stress.In addition, with the increase of casing pressure pi, the radial stress in the cement sheath gradually changes from tensile to compressed stress, which indicates that the risk of de-bonding decreases.When the increased casing pressure is 50 MPa, the radial stress is nearly compressive.Consequently, maintaining higher circulation pumping pressures, avoiding pumping off during injection, and lowering the pump pressure slowly during the operation completion will all be beneficial for cement sheath protection.Figure 6 shows the safety factors of cement sheath for de-bonding at two interfaces under different increased casing pressures p i .It can be seen form Figure 6 that the safety factors at the casing-cement interface and the cement-formation interface both increase exponentially with the increase of casing pressure p i .When p i is less than 32 MPa, both interfaces will separate.When p i is between 32 MPa and 42 MPa, only the cement-formation interface will separate.When p i is larger than 42 MPa, both interfaces will not separate.Generally, the outer de-bonding is more relatively easy compared to inner de-bonding.This is because increased casing pressure p i has a larger effect on the casing-cement interface.For the integrity of cement sheath, the value of p i must be maintained above 42 MPa during the whole fracturing fluid injection stage.Figure 6 shows the safety factors of cement sheath for de-bonding at two interfaces under different increased casing pressures pi.It can be seen form Figure 6 that the safety factors at the casing-cement interface and the cement-formation interface both increase exponentially with the increase of casing pressure pi.When pi is less than 32 MPa, both interfaces will separate.When pi is between 32 MPa and 42 MPa, only the cement-formation interface will separate.When pi is larger than 42 MPa, both interfaces will not separate.Generally, the outer de-bonding is more relatively easy compared to inner de-bonding.This is because increased casing pressure pi has a larger effect on the casing-cement interface.For the integrity of cement sheath, the value of pi must be maintained above 42 MPa during the whole fracturing fluid injection stage.

Radial Cracking
Radial cracking appears when tangential tensile stress in the cement surpasses its own tensile strength, Figures 7 and 8 shows the radial distribution of the tangential stress and safety factors for radial cracking in the cement sheath considering increased casing pressure pi of 0-70 MPa and

Radial Cracking
Radial cracking appears when tangential tensile stress in the cement surpasses its own tensile strength, Figures 7 and 8 shows the radial distribution of the tangential stress and safety factors for radial cracking in the cement sheath considering increased casing pressure p i of 0-70 MPa and decreased temperature at the casing inner wall of 70 • C. It can be seen from Figure 7 that the decreased wellbore temperature and increased casing pressure will both induce significant tangential tensile stress, which leads to the risk of radial cracking.Figure 8 shows that the safety factors are all less than one under the present operation parameters, which means that radial cracking occurs completely.To some extent, this answers for the sustaining annular pressure after gas production.Meanwhile, we can also know easily that the maximum tangential tensile stress is nearly located at the casing-cement sheath interface and is caused by a decreased temperature of 70 • C, which is nearly 10 MPa less than that caused by increased casing pressure of 70 MPa.This indicates that radial cracking is mainly induced by increased casing pressure p i and it occurs from the inner wall to the outer wall.However, p i cannot be reduced during fracturing fluid injection because the need of fracture generation.Consequently, some other measures must be taken for protecting cement sheath, such as selecting cement with low elasticity and high tensile strength.

Disking
Disking occurs when axial tensile stress in the cement sheath exceeds its tensile strength when the gas leaking will run across the cement wall.Figures 9 and 10 show, respectively, the radial distribution of the axial stress and safety factors for disking in the cement sheath considering increased casing pressure p i of 0-70 MPa and decreased temperature at the casing inner wall of 70 • C. It can be seen from Figure 9 that the decreased wellbore temperature will induce significant axial tensile stress, which may cause disking and is very threatening for cement sheath.Yet, the increased casing pressure will produce axial compressed stress and offset a portion of the axial tensile stress but is very Figure 10 shows that the safety factors for disking are all less than one under the present operation parameters, which indicates that disking occurs through the cement sheath.This is also the reason for sustaining annular pressure.Taking Figures 9 and 10 into consideration simultaneously, it can be found that disking mainly results from the decreased wellbore temperature and it will generate from the inner wall to the outer wall.Consequently, increasing the temperature of injected fracturing fluid can reduce the risk of disking.11 that the decreased wellbore temperature and increased casing pressure induce significant octahedral shear stress.When the decreased temperature at the casing inner wall is 70 • C with the increase of p i , the octahedral shear stress τ oct decreases first and then increases.The maximum τ oct is located at the casing interface.From Figure 12, can conclude that the shear failure of cement sheath nearly does not occur if the increased casing pressure p i is less than 50 MPa while the shear failure will initiate from the inner wall to the outer wall.Considering the present operation parameters, some measures must be taken, such as selecting cement with low elasticity, for protecting cement sheath.
Energies 2018, 11, x 13 of 20 casing pressure pi is less than 50 MPa while the shear failure will initiate from the inner wall to the outer wall.Considering the present operation parameters, some measures must be taken, such as selecting cement with low elasticity, for protecting cement sheath.

Comparison and Discussion
Figures 13 and 14 show the radial distribution of the failure stress and corresponding strength in the cement sheath considering increased casing pressure pi of 0 or 70 MPa and decreased temperature at the casing inner wall of 70 °C, respectively.It can be seen from Figure 13 that the decreased wellbore temperature has produced significant tri-axial tensile stress in the cement sheath

Comparison and Discussion
Figures 13 and 14 show the radial distribution of the failure stress and corresponding strength in the cement sheath considering increased casing pressure pi of 0 or 70 MPa and decreased temperature at the casing inner wall of 70 °C, respectively.It can be seen from Figure 13 that the decreased wellbore temperature has produced significant tri-axial tensile stress in the cement sheath

Comparison and Discussion
Figures 13 and 14 show the radial distribution of the failure stress and corresponding strength in the cement sheath considering increased casing pressure p i of 0 or 70 MPa and decreased temperature at the casing inner wall of 70 • C, respectively.It can be seen from Figure 13 that the decreased wellbore temperature has produced significant tri-axial tensile stress in the cement sheath and three of them exceed its own tensile strength, which will induce cement failure of de-bonding, radial cracking, and disking, especially de-bonding and disking.While the octahedral shear stress is less than the maximum allowable shear stress for cement sheath, it shows that shear failure will not appear.Consequently, increasing properly the injected fluid temperature can lower the failure risks of cement sheath for fracturing wells.Figure 14 shows that, when considering the increased casing pressure pi of 70 MPa simultaneously, the radial stress has turned into compressive but tangential and axial stresses, which are still tensile and still clearly larger than the tensile strength of cement sheath, which will lead to cement failure of radial cracking and disking.Meanwhile, the octahedral shear stress has also exceeded the maximum allowable shear stress near the casing interface, which indicates that shear failure occurs at this location.Combining the analysis above, we know that the increased casing pressure will significantly lower the de-bonding risk but aggravate radial cracking and shear failure.For protecting cement sheath integrity, higher circulation pumping pressures should be maintained in order to avoid de-bonding during whole fracturing operations.Figure 14 shows that, when considering the increased casing pressure pi of 70 MPa simultaneously, the radial stress has turned into compressive but tangential and axial stresses, which are still tensile and still clearly larger than the tensile strength of cement sheath, which will lead to cement failure of radial cracking and disking.Meanwhile, the octahedral shear stress has also exceeded the maximum allowable shear stress near the casing interface, which indicates that shear failure occurs at this location.Combining the analysis above, we know that the increased casing pressure will significantly lower the de-bonding risk but aggravate radial cracking and shear failure.For protecting cement sheath integrity, higher circulation pumping pressures should be maintained in order to avoid de-bonding during whole fracturing operations.Figure 14 shows that, when considering the increased casing pressure p i of 70 MPa simultaneously, the radial stress has turned into compressive but tangential and axial stresses, which are still tensile and still clearly larger than the tensile strength of cement sheath, which will lead to cement failure of radial cracking and disking.Meanwhile, the octahedral shear stress has also exceeded the maximum allowable shear stress near the casing interface, which indicates that shear failure occurs at this location.Combining the analysis above, we know that the increased casing pressure will significantly lower the de-bonding risk but aggravate radial cracking and shear failure.For protecting cement sheath integrity, higher circulation pumping pressures should be maintained in order to avoid de-bonding during whole fracturing operations.
In order to lower the risks of radial cracking, disking, and shear failure for cement sheath under a high pumping pressure from fracturing, the common method is cemented with low elasticity modulus cement sheath.Figures 15-17 show, respectively, the radial distribution of the cement sheath safety factors for three kinds of failure modes under a different elasticity modulus of cement sheath E c .The considered increased casing pressure p i is 60 MPa and the decreased temperature at the casing inner wall is 50 • C. It can be seen from Figures 15-17 that, with a decrease of E c , three kinds safety factors increase gradually and the increasing range is larger and larger.Therefore, lowering the elasticity modulus of cement sheath is very effective for cement protection.When E c is about lower than 4 GPa, cement sheath will not occur to radial cracking and disking.Meanwhile, when E c is about lower than 8 GPa, the shear failure may not occur.In conclusion, cement sheath with a low elasticity modulus must be adopted in fracturing wells.In order to lower the risks of radial cracking, disking, and shear failure for cement sheath under a high pumping pressure from fracturing, the common method is cemented with low elasticity modulus cement sheath.Figures 15-17 show, respectively, the radial distribution of the cement sheath safety factors for three kinds of failure modes under a different elasticity modulus of cement sheath Ec.The considered increased casing pressure pi is 60 MPa and the decreased temperature at the casing inner wall is 50 °C.It can be seen from Figures 15-17 that, with a decrease of Ec, three kinds of safety factors increase gradually and the increasing range is larger and larger.Therefore, lowering the elasticity modulus of cement sheath is very effective for cement protection.When Ec is about lower than 4 GPa, cement sheath will not occur to radial cracking and disking.Meanwhile, when Ec is about lower than 8 GPa, the shear failure may not occur.In conclusion, cement sheath with a low elasticity modulus must be adopted in fracturing wells.In order to lower the risks of radial cracking, disking, and shear failure for cement sheath under a high pumping pressure from fracturing, the common method is cemented with low elasticity modulus cement sheath.Figures 15-17 show, respectively, the radial distribution of the cement sheath safety factors for three kinds of failure modes under a different elasticity modulus of cement sheath Ec.The considered increased casing pressure pi is 60 MPa and the decreased temperature at the casing inner wall is 50 °C.It can be seen from Figures 15-17 that, with a decrease of Ec, three kinds of safety factors increase gradually and the increasing range is larger and larger.Therefore, lowering the elasticity modulus of cement sheath is very effective for cement protection.When Ec is about lower than 4 GPa, cement sheath will not occur to radial cracking and disking.Meanwhile, when Ec is about lower than 8 GPa, the shear failure may not occur.In conclusion, cement sheath with a low elasticity modulus must be adopted in fracturing wells.

Figure 1 .
Figure 1.The schematic diagram of casing-cement sheath-formation combined system during fracturing fluid injection.

Figure 1 .
Figure 1.The schematic diagram of casing-cement sheath-formation combined system during fracturing fluid injection.

Figure 3 .
Figure 3.The 2D axisymmetric finite element analysis model of casing-cement sheath-formation system.

Figure 3 .
Figure 3.The 2D axisymmetric finite element analysis model of casing-cement sheath-formation system.

Figure 4 .
Figure 4. Comparison of stresses calculated with the analytical model and the finite element analysis model.

Figure 4 .
Figure 4. Comparison of stresses calculated with the analytical model and the finite element analysis model.

Figure 4 .
Figure 4. Comparison of stresses calculated with the analytical model and the finite element analysis model.

Figure 5 .
Figure 5. Radial distribution of the radial stress in the cement sheath under different increased casing pressures p i and a decreased temperature of 70 • C at the casing inner wall.

Energies 2018, 11 , x 10 of 20 Figure 5 .
Figure 5. Radial distribution of the radial stress in the cement sheath under different increased casing pressures pi and a decreased temperature of 70 °C at the casing inner wall.

Figure 6 .
Figure 6.Safety factor of cement sheath for de-bonding at two interfaces under different increased casing pressures pi and decreased temperature of 70 °C at the casing inner wall.

Figure 6 .
Figure 6.Safety factor of cement sheath for de-bonding at two interfaces under different increased casing pressures p i and decreased temperature of 70 • C at the casing inner wall.

20 Figure 7 .
Figure 7. Radial distribution of the tangential stress in the cement sheath under different increased casing pressure pi and decreased temperature of 70 °C at the casing inner wall.

Figure 8 .
Figure 8. Radial distribution of the cement sheath safety factors for radial cracking under different increased casing pressures pi and decreased temperature of 70 °C at the casing inner wall.3.2.4.Disking Disking occurs when axial tensile stress in the cement sheath exceeds its tensile strength when the gas leaking will run across the cement wall.Figures 9 and 10 show, respectively, the radial distribution of the axial stress and safety factors for disking in the cement sheath considering increased casing pressure pi of 0-70 MPa and decreased temperature at the casing inner wall of

Figure 7 . 20 Figure 7 .
Figure 7. Radial distribution of the tangential stress in the cement sheath under different increased casing pressure p i and decreased temperature of 70 • C at the casing inner wall.

Figure 8 .Figure 8 .
Figure 8. Radial distribution of the cement sheath safety factors for radial cracking under different increased casing pressures pi and decreased temperature of 70 °C at the casing inner wall.3.2.4.Disking Disking occurs when axial tensile stress in the cement sheath exceeds its tensile strength when the gas leaking will run across the cement wall.Figures 9 and 10 show, respectively, the radial distribution of the axial stress and safety factors for disking in the cement sheath considering increased casing pressure pi of 0-70 MPa and decreased temperature at the casing inner wall of

Energies 2018 ,
11, x 12 of 20wellbore temperature and it will generate from the inner wall to the outer wall.Consequently, increasing the temperature of injected fracturing fluid can reduce the risk of disking.

Figure 9 .
Figure 9. Radial distribution of the axial stress in the cement sheath under different increased casing pressures pi and a decreased temperature of 70 °C at the casing inner wall.

Figure 10 .
Figure 10.Radial distribution of the cement sheath safety factors for disking under different increased casing pressures pi and a decreased temperature of 70 °C at the casing inner wall.

Figure 9 .
Figure 9. Radial distribution of the axial stress in the cement sheath under different increased casing pressures p i and a decreased temperature of 70 • C at the casing inner wall.

Figure 9 .
Figure 9. Radial distribution of the axial stress in the cement sheath under different increased casing pressures pi and a decreased temperature of 70 °C at the casing inner wall.

Figure 10 .
Figure 10.Radial distribution of the cement sheath safety factors for disking under different increased casing pressures pi and a decreased temperature of 70 °C at the casing inner wall.

Figure 10 .
Figure 10.Radial distribution of the cement sheath safety factors for disking under different increased casing pressures p i and a decreased temperature of 70 • C at the casing inner wall.

Figure 11 .
Figure 11.Radial distribution of the octahedral shear stress τoct in the cement sheath under different increased casing pressures pi and a decreased temperature of 70 °C at the casing inner wall.

Figure 12 .
Figure 12.Radial distribution of the cement sheath safety factors for shear failure under different increased casing pressures pi and a decreased temperature of 70 °C at the casing inner wall.

Figure 11 .
Figure 11.Radial distribution of the octahedral shear stress τ oct in the cement sheath under different increased casing pressures p i and a decreased temperature of 70 • C at the casing inner wall.

Figure 11 .
Figure 11.Radial distribution of the octahedral shear stress τoct in the cement sheath under different increased casing pressures pi and a decreased temperature of 70 °C at the casing inner wall.

Figure 12 .
Figure 12.Radial distribution of the cement sheath safety factors for shear failure under different increased casing pressures pi and a decreased temperature of 70 °C at the casing inner wall.

Figure 12 .
Figure 12.Radial distribution of the cement sheath safety factors for shear failure under different increased casing pressures p i and a decreased temperature of 70 • C at the casing inner wall.

Energies 2018 ,
11, x 14 of 20 appear.Consequently, increasing properly the injected fluid temperature can lower the failure risks of cement sheath for fracturing wells.

Figure 13 .
Figure 13.Radial distribution of the failure stress and corresponding strength of the cement sheath under only a decreased temperature of 70 °C at the casing inner wall.

Figure 14 .
Figure 14.Radial distribution of the failure stress and corresponding strength of the cement sheath under an increased casing pressure pi of 70 MPa and a decreased temperature of 70 °C at the casing inner wall.

Figure 13 .
Figure 13.Radial distribution of the failure stress and corresponding strength of the cement sheath under only a decreased temperature of 70 • C at the casing inner wall.

Figure 13 .
Figure 13.Radial distribution of the failure stress and corresponding strength of the cement sheath under only a decreased temperature of 70 °C at the casing inner wall.

Figure 14 .
Figure 14.Radial distribution of the failure stress and corresponding strength of the cement sheath under an increased casing pressure pi of 70 MPa and a decreased temperature of 70 °C at the casing inner wall.

Figure 14 .
Figure 14.Radial distribution of the failure stress and corresponding strength of the cement sheath under an increased casing pressure p i of 70 MPa and a decreased temperature of 70 • C at the casing inner wall.

Figure 15 .
Figure 15.Radial distribution of the cement sheath safety factors for radial cracking under a different elasticity modulus of cement sheath.

Figure 16 .
Figure 16.Radial distribution of the cement sheath safety factors for disking under a different elasticity modulus of cement sheath.

Figure 15 .
Figure 15.Radial distribution of the cement sheath safety factors for radial cracking under a different elasticity modulus of cement sheath.

Figure 15 .
Figure 15.Radial distribution of the cement sheath safety factors for radial cracking under a different elasticity modulus of cement sheath.

Figure 16 .
Figure 16.Radial distribution of the cement sheath safety factors for disking under a different elasticity modulus of cement sheath.

Figure 16 .
Figure 16.Radial distribution of the cement sheath safety factors for disking under a different elasticity modulus of cement sheath.
Radial, tangential, and axial thermal stress of cylinder, respectively, MPa Radial, tangential, and axial stress of cylinder caused by increased inner pressure, respectively, MPa Radial, tangential, and axial stress of formation caused by increased casing pressure, respectively, MPa σ rc , σ θc , σ zc Combined radial, tangential, and axial stress of cement sheath caused by decreased wellbore temperature and increased casing pressure during fracturing fluid injection, respectively, MPa σ 1 , σ 2 , σ 3 Maximum, intermediate, and minimum principal stresses in the cement sheath, respectively, MPa u T Radial thermal displacement in the cylinder, mm u P Radial displacement in the cylinder caused by increased inner pressure, mm u T s , u T c , u T f Radial thermal displacement for casing, cement sheath, and formation, respectively, mm Radial displacement caused by increased casing pressure for casing, cement sheath, and formation, respectively, mm E, E s , E c , E f Elasticity modulus of ordinary cylinder, casing, cement sheath, and formation, respectively, MPa µ, µ s , µ c , µ f Poisson's ratio of ordinary cylinder, casing, cement sheath, and formation, respectively, dimensionless α, α s , α c , α f Linear thermal expansion coefficient of ordinary cylinder, casing, cement sheath, and formation, respectively, 1/ • C a Internal radius of thick wall cylinder, mm r Radial distance from the axis of wellbore, mm r i Inside radius of casing, mm r 1 Outside radius of casing or inside radius of cement sheath, mm r 2 Outside radius of cement sheath or inside radius of formation, mm r o Outside radius of formation, mm T(r) Temperature change value at the radius of r after fracturing fluid injection, • C T i Wellbore temperature before fracturing fluid injection, • C T t Wellbore temperature after fracturing fluid injection, • C T e Formation temperature, • C C 1 , C 1s , C 1c , Contact pressure at the casing-cement sheath interface induced by increased casing pressure, MPa p c2 Contact pressure at the cement sheath-formation interface induced by increased casing pressure, MPa p f Changed formation pressure, MPa SF Safety factor of cement sheath, dimensionless