Application of Multiphysics Coupling FEM on Open Wellbore Shrinkage and Casing Remaining Strength in an Incomplete Borehole in Deep Salt Formation

Drilling and completing wells in deep salt stratum are technically challenging and costing, as when serving in an incomplete borehole in deep salt formation, well casing runs a high risk of collapse. To quantitatively calculate casing remaining strength under this harsh condition, a three-dimensional mechanical model is developed; then a computational model coupled with interbed salt rock-defective cement-casing and HPHT (high pressure and high temperature) is established and analyzed using multiphysics coupling FEM (finite element method); furthermore, open wellbore shrinkage and casing remaining strength under varying differential conditions in deep salt formation are discussed. The result demonstrates that the most serious shrinkage occurs at the middle of salt rock, and the combination action of salt rock creep, cement defect, and HPHT substantially lessens casing remaining strength; meanwhile, cement defect level should be taken into consideration when designing casing strength in deep salt formation, and apart from the consideration of temperature on casing the effect of temperature on cement properties also cannot be ignored. This study not only provides a theoretical basis for revealing the failuremechanismofwell casing in deep complicated salt formation, but also acts as a new perspective of novel engineering applications of the multiphysics coupling FEM.


Introduction
With the deep complicated regions becoming the major battlefield of oil and gas exploitation on land, increasing exploration and production around the world require drilling through and completing wells in deep salt formation, which are technically challenging and costing.The introduction of a borehole in deep salt formation changes the existing stress field and displacement field, resulting in open wellbore shrinkage and time-dependent loading on the casing, which leads to severe well casing damage [1].The statistical data shows that the well having casing damage in the Zhongyuan oil field is 1123, while the number of casing damage incidents in the salt strata contributed by 68.03% of the total number [2].At greater depths, both stress and temperature increase; well cementing also faces great technical challenges.Statistical data collected in an oil region in western China shows that five out of seven wells are poorly cementing at the depths of 2990∼3460 m [3].Partial cementing and deboning at the casing/cement or cement/rock interfaces are potential causes of cement sheath failure [4].Lacking the intact support of cement sheath, when well casings serve in deep salt formation, their security will drop significantly.
In recent years, severe casing damage in deep salt strata has been carefully followed and comprehensively studied by many scholars.Salt rock creep leads to open wellbore shrinkage and casing collapse [5][6][7][8], the well condition and casing performance are getting worse under high pressure and high temperature [1,9], and another cause of casing damage is cement sheath failures [4,10,11].However, to simplify the calculation, prior work on casing damage in deep salt formation only considered the impact of a single factor, ignoring interaction between salt rock creep, defective cement, and HPHT, which has great deviations from actual situation.To quantitatively assess the service applicability of well casing in deep salt formation, we propose to use multiphysics coupling FEM to solve this problem.

Basic Theory of Multiphysics Coupling FEM on Casing Remaining Strength Calculation
2.1.Temperature Field Model.Considering the particularity of the wellbore, to simplify the calculation, the following assumptions are made in this paper: (1) the formation temperatures at initial moment and at infinity are uniform, which are kept as the in situ temperature, and there is no internal heat source inside the formation; (2) wellbore thermal conductivity is constant, and the thermal stress is considered to be in a single-valued function relation with the elastic modulus and temperature increment; (3) the influence of the drilling mud radial temperature gradient and axial thermal conduction on wellbore temperature distribution is ignored.The threedimensional thermal conduction equation of wellbore under cylindrical coordinate system is shown as follows [12]: where  is casing temperature, K;  is casing density, kg/m 3 ;  is specific heat capacity, J/(kg⋅K);  is casing heat transfer coefficient, W/(m⋅K); and  is heat transfer time, s.
In the process of heat conduction between salt rock and cement, the interior wall of borehole is considered as the first boundary condition; that is, where  BHST is bottom hole standard temperature, K, and Γ is boundary.
For heat convection between drilling mud and casing, the interior wall of casing is considered as the third boundary condition: where  BHCT is bottom hole circulating temperature, K; ℎ is convective heat transfer coefficient, W/(m 2 ⋅K);  is outside the normal direction on the boundary; and Γ is boundary.According to incremental theory, temperature strain increment of well casing can be written as follows: where  is transient temperature;  0 is initial temperature; {} is thermal expansion coefficient.creep, which changes the existing stress field and displacement field, resulting open wellbore shrinkage and time-dependent loading on well casing [13].Creep behavior is used to describe as a creep curve, Figure 1 shows an example of typical creep behavior, as it sequentially undergoes the initial creep (), steady creep () and accelerated creep ().However, salt rocks surround the wellbore mainly experience the steady creep stage during the service life of the oil and gas wells.Temperature, time and stress field are closely related to salt creep rate, creep strain increment is described as follows:

Salt
where  is equivalent stress;   is equivalent creep strain;  is temperature;  is time.

Stress Field Model.
At greater depths, stress and temperature increase, considering the influence of temperature, elastic strain increment and plastic strain increment are described as follows: Mathematical Problems in Engineering 3 where  is temperature;  1 is the maximum principal stress;  2 is the intermediate principal stress;  3 is the minimum principal stress;   1 is the maximum principal plastic strain;   2 is the intermediate principal plastic strain;   3 is the minimum principal plastic strain;  is Young's modulus;  is Poisson's ratio.

Multi-Physics Coupling Field Model.
In this computational model coupled with HTHP and creep, we propose to use multiphysics coupling FEM to calculate casing remaining strength [14][15][16][17], total strain increment is calculated as follows: Putting ( 4), ( 5), ( 6) and ( 7) into (12), we can calculate the stress increment as: As the incremental equation of yield surface is: where  is a strengthening function in the plastic flow theory, which closely related to stress, strain, plastic deformation history and the selection of hardening model.
Hence, we can calculate the stress increment as follows: where [] ep is elastoplastic matrix; { 0 } is initial stress increment; { 0 } is initial strain increment, detailed expression of them can be written as follows: Based on finite element incremental theory, to calculate casing remaining strength in this multiphysics coupling model, the thermal elastoplastic creep deformation can be linearized during calculate process.When loading, after substitute Δ for , substitute Δ{} for {}, substitute Δ{} for {}, we can calculate each parameter by the finite element incremental control equations.
Assuming failure of well casing obedience to Mises yield criterion, the remaining strength of the casing can be calculated as follows: where   is the yield strength;  is equivalent stress.Thus, comparing with the traditional one, multiphysics coupling FEM can take interaction between salt rock creep, defective cement and HPHT into consideration, which is in accordance with actual situation.Furthermore, the another advantage of the multiphysics coupling FEM on investigate the open wellbore shrinkage and casing remaining strength in an incomplete borehole is that we can quantitatively assess the service applicability of well casing in deep salt formation, which indicates that the multiphysics coupling FEM may act as a new perspective of novel engineering applications in oil and gas industry.

The Mechanical Model
Based on the multiphysics coupling FEM previously proposed, to reflect actual stress state of well casing as far as possible, in this paper, a three-dimensional mechanical model is developed and applied at the near wellbore scale, as shown in Figure 2, where  V is the vertical stress in situ;   is the max horizontal stress in situ;  ℎ is the max horizontal stress in situ;  is casing axial tensile force;  BHST is bottom hole standard temperature;  BHST is bottom hole circulating temperature; and  is drilling mud pressure.(iii) The steady creep rate is mainly determined by the partial stress and temperature; with the partial stress and temperature increase, the creep rate increases nonlinearly.
When the creep behavior is mainly caused by lattice dislocation gliding, the steady-state creep of salt rock described by Heard model is generally accepted, which can be written as follows: where ε cr is the steady-state creep rate, s −1 ;  is the activation energy, J/mol;  is the universal gas constant, J/mol;  is the deviatoric stress, MPa;  is the thermodynamic temperature, K; ,  are exponents dependent on creep mechanism, which can be determined through laboratory experiments.In this study, the steady-state creep parameters are obtained through the creep experiment performed on TAV-1000 [19] and are listed in Table 1.

Effect of Temperature on Cement and Casing Property.
At greater depths, stress and temperature increase; for alloy material and silicate composites material in deep formation, the impact of temperature on cement and casing mechanical property cannot be ignored.According to experimental study [12,20], there is a function relation between Young's modulus, yield strength, compressive strength, and temperature, which are described as follows: where  80 is Young's modulus of 80 casing,   is yield strength of 80 casing, and  is temperature.Consider   To verify the reliability of the above formulas, when compared with the experimental data [21,22], the maximum deviation of the result calculated by the above formulas is less than 2% (Table 2), which is acceptable in engineering.

The Three-Dimensional Finite Element Model. In order
to analyze open wellbore shrinkage and casing remaining strength in an incomplete borehole in deep salt formation, based on the mechanical model developed foregoing, the three-dimensional finite element model is established by ANSYS (Figure 3), in which  is defined as the defective circumferential angle of the cement sheath,  is defined as the defective depth of cement sheath, and  is defined as the defective length of cement sheath.The other parameters adopted in the computational model are presented in Table 3.

Open Wellbore Shrinkage.
To investigate open wellbore shrinkage in deep salt formation, a computational model without casing and cement is performed, as shown in Figure 4, with the creep of salt rock and fixation of sandstone at both ends the most serious shrinkage occurs at the middle of salt rock; furthermore, the open wellbore shrinkage worsens with time.Therefore, when designing casing strength in deep salt formation, according to the designed life of well casing, the effects of open wellbore shrinkage need to be considered.

Effects of Defective Cement on Casing Remaining Strength.
Figure 5 shows the effects of cement defective depth and defective circumferential angle on casing remaining strength; in this computational model,  is set as 0.1 m; besides, cement defect at the first cemented interface (casing/cement interface) and second cemented interface (cement/rock interface) are calculated.As shown in Figure 5, the effects of defective cement at the first and second cemented interfaces on the casing strength are basically the same; that is, casing remaining strength declines gradually with increasing  and decreases rapidly when  varies from 20 ∘ to 100 ∘ compared to other conditions.Specifically, when  values at the first and second cemented interfaces are 45 ∘ and 63 ∘ , respectively, casing remaining strength falls to its lowest point; that is because the casing needs to withstand the most serious nonuniform loading under this condition.Based on the contrast calculation above, for the convenience of modeling, the cement defect is set at the first cemented interface in the following studies.
Figure 6 shows the effects of cement defective length on casing remaining strength; with  increase, casing remaining strength declines gradually; however, when  > 0.2m, it tends to be basically stable.The result indicates that cement defect level should be taken into consideration when we design the casing strength in deep salt formation.8, in the first year of the service life, the additional load casing withstands growing rapidly, but one year later it increases uniformly.With increasing creep time, the additional displacement of the well casings grows nonlinearly, and after serving for eight years, it tends to be basically stable.Moreover, the time-dependent displacement on well casing is insensitive to the defect size of the cement sheath.Therefore, the behavior of salt creep exacerbates the deformation and damage on well casing, which runs a higher risk of casing collapse combined cement defect.

Conclusions
In this paper a three-dimensional computational model coupled with interbed salt rock-defective cement-casing and HPHT is established and analyzed using multiphysics coupling FEM; moreover, open wellbore shrinkage and casing remaining strength under varying differential conditions in deep salt formation are discussed, which leads to the following conclusions.
(1) The most serious shrinkage occurs at the middle of salt rock, and the open wellbore shrinkage worsens with time.Therefore, when designing casing strength in deep salt formation, according to the designed life  (2) Defective cement changes stress distribution of wellbore, which resulting in stress concentration on well casing; salt rock creep leading to additional load and displacement on casing with time; high temperature in deep formation reduced the yield strength of casing and generating temperature load on casing; the combination action of these factors substantially lessens casing remaining strength.

Figure 3 :
Figure 3: Schematic diagram and FEM mesh of salt formation-defective cement-casing.

Figure 5 :Figure 6 :
Figure 5: Effects of  and  on casing remaining strength.

Figure 7 :
Figure 7: Effects of  BHCT on casing remaining strength.

Table 1 :
The steady-state creep parameters obtained through laboratory test.

Numerical Calculations of Well Casing in an Incomplete Borehole in Salt Formation
[1,6,13,18,19]f Salt Rock.The wellbore mainly experiences the steady creep stage of salt rock during the service life of the oil and gas wells.Many laboratory works have been devoted to the creep of salt rock[1,6,13,18,19]; despite the complexity of the behavior of salt creep, many scholars agree on several main features of the steady-state creep of salt rock.

Table 2 :
Comparisons between the formula calculation and experimental data.
trend.It is noteworthy that casing remaining strength drop slows down after  BHCT greater than 393.15K, which is different from previous research conclusions; this is mainly because previous studies only consider the effect of temperature on the performance of the casing while ignoring the effect of temperature on the performance of cement.According to the foregoing study,   decreases after 393.15 K, and low stiffness cement has a protective effect on casing; that is why casing remaining strength drop slows down after  BHCT greater than 393.15K.