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

Authors: Hua Tong; Daqiang Guo; Xiaohua Zhu
Journal: Mathematical Problems in Engineering
 (2015)

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## Abstract

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 failure mechanism of well casing in deep complicated salt formation, but also acts as a new perspective of novel engineering applications of the multiphysics coupling FEM.


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## Body

## 1. 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 [5–8], 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.


## 2. 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 three-dimensional thermal conduction equation of wellbore under cylindrical coordinate system is shown as follows [12]: (1)∂2T∂r2+1r∂T∂r+1r2∂2T∂θ2+∂2T∂z2=ρCλ∂T∂t,where T is casing temperature, K; ρ is casing density, kg/m3; C is specific heat capacity, J/(kg·K); λ is casing heat transfer coefficient, W/(m·K); and t 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,(2)Tt=0=TΓ=TBHST,where TBHST 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:(3)-λ∂T∂nΓ=hT-TBHCTΓ,where TBHCT is bottom hole circulating temperature, K; h is convective heat transfer coefficient, W/(m2·K); n 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:(4)dεT=αdT+T-T0dα,where T is transient temperature; T0 is initial temperature; α is thermal expansion coefficient.


### 2.2. Salt Rock Creep Model

As a viscous, slowly flowing material, due to its sealing and low permeability, salt rock may easily develop to become the covering layer of the deep reservoir. Salt rock behaves in a visco-elastic manner is called 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 (ab), steady creep (bc) and accelerated creep (cd). 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:(5)dεc=Φσ¯,ε¯c,T,tdt,where σ¯ is equivalent stress; ε¯c is equivalent creep strain; T is temperature; t is time.Figure 1
Typical creep curve of salt rock.


### 2.3. 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:(6)dεe=dD-1dTσdT+D-1dσ(7)dεp=dε¯p∂σ¯∂σ(8)σ=σxσyσzτxyτyzτzxT(9)σ-=12σ1-σ22+σ2-σ32+σ3-σ12(10)ε¯p=23ε1p-ε2p2+ε2p-ε3p2+ε3p-ε1p2(11)D=E(1-μ)(1+μ)(1-2μ)×1μ1-μμ1-μ000μ1-μ1μ1-μ000μ1-μμ1-μ10000001-2μ2(1-μ)0000001-2μ2(1-μ)0000001-2μ2(1-μ)where T is temperature; σ1 is the maximum principal stress; σ2 is the intermediate principal stress; σ3 is the minimum principal stress; ε1p is the maximum principal plastic strain; ε2p is the intermediate principal plastic strain; ε3p is the minimum principal plastic strain; E is Young’s modulus; μ is Poisson’s ratio.


### 2.4. 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 [14–17], total strain increment is calculated as follows:(12)dε=dεe+dεp+dεT+dεc.Putting (4), (5), (6) and (7) into (12), we can calculate the stress increment as:(13)dσ=Dα+T-T0dαdT+dD-1dTσdTdε-dε¯p∂σ¯∂σ-Φ∂σ¯∂σdtHHHH-α+T-T0dαdT+dD-1dTσdT.As the incremental equation of yield surface is:(14)σ¯=H(∫dε¯p,T)dσ¯=∂H∂ε¯pdε¯p+∂H∂TdT,where H 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:(15)dσ=Depdε+dσ0-dε0-dεc,where Dep is elastoplastic matrix; dσ0 is initial stress increment; dε0 is initial strain increment, detailed expression of them can be written as follows:(16)Dep=D-D∂σ¯/∂σ∂σ¯/∂σTD∂H/∂ε¯p+∂σ¯/∂σTD∂σ¯/∂σdσ0=D∂H/∂TdT∂σ¯/∂σ∂H/∂ε¯p+∂σ¯/∂σTD∂σ¯/∂σdε0=α+(T-T0)dαdT+dD-1dTσdT.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 substituteΔT for dT, substitute Δσ for dσ, substitute Δε for dε, 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:(17)σΓ=σs-σ¯,where σs is the yield strength; σ¯ 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.


## 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 three-dimensional thermal conduction equation of wellbore under cylindrical coordinate system is shown as follows [12]: (1)∂2T∂r2+1r∂T∂r+1r2∂2T∂θ2+∂2T∂z2=ρCλ∂T∂t,where T is casing temperature, K; ρ is casing density, kg/m3; C is specific heat capacity, J/(kg·K); λ is casing heat transfer coefficient, W/(m·K); and t 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,(2)Tt=0=TΓ=TBHST,where TBHST 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:(3)-λ∂T∂nΓ=hT-TBHCTΓ,where TBHCT is bottom hole circulating temperature, K; h is convective heat transfer coefficient, W/(m2·K); n 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:(4)dεT=αdT+T-T0dα,where T is transient temperature; T0 is initial temperature; α is thermal expansion coefficient.


## 2.2. Salt Rock Creep Model

As a viscous, slowly flowing material, due to its sealing and low permeability, salt rock may easily develop to become the covering layer of the deep reservoir. Salt rock behaves in a visco-elastic manner is called 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 (ab), steady creep (bc) and accelerated creep (cd). 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:(5)dεc=Φσ¯,ε¯c,T,tdt,where σ¯ is equivalent stress; ε¯c is equivalent creep strain; T is temperature; t is time.Figure 1
Typical creep curve of salt rock.


## 2.3. 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:(6)dεe=dD-1dTσdT+D-1dσ(7)dεp=dε¯p∂σ¯∂σ(8)σ=σxσyσzτxyτyzτzxT(9)σ-=12σ1-σ22+σ2-σ32+σ3-σ12(10)ε¯p=23ε1p-ε2p2+ε2p-ε3p2+ε3p-ε1p2(11)D=E(1-μ)(1+μ)(1-2μ)×1μ1-μμ1-μ000μ1-μ1μ1-μ000μ1-μμ1-μ10000001-2μ2(1-μ)0000001-2μ2(1-μ)0000001-2μ2(1-μ)where T is temperature; σ1 is the maximum principal stress; σ2 is the intermediate principal stress; σ3 is the minimum principal stress; ε1p is the maximum principal plastic strain; ε2p is the intermediate principal plastic strain; ε3p is the minimum principal plastic strain; E is Young’s modulus; μ is Poisson’s ratio.


## 2.4. 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 [14–17], total strain increment is calculated as follows:(12)dε=dεe+dεp+dεT+dεc.Putting (4), (5), (6) and (7) into (12), we can calculate the stress increment as:(13)dσ=Dα+T-T0dαdT+dD-1dTσdTdε-dε¯p∂σ¯∂σ-Φ∂σ¯∂σdtHHHH-α+T-T0dαdT+dD-1dTσdT.As the incremental equation of yield surface is:(14)σ¯=H(∫dε¯p,T)dσ¯=∂H∂ε¯pdε¯p+∂H∂TdT,where H 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:(15)dσ=Depdε+dσ0-dε0-dεc,where Dep is elastoplastic matrix; dσ0 is initial stress increment; dε0 is initial strain increment, detailed expression of them can be written as follows:(16)Dep=D-D∂σ¯/∂σ∂σ¯/∂σTD∂H/∂ε¯p+∂σ¯/∂σTD∂σ¯/∂σdσ0=D∂H/∂TdT∂σ¯/∂σ∂H/∂ε¯p+∂σ¯/∂σTD∂σ¯/∂σdε0=α+(T-T0)dαdT+dD-1dTσdT.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 substituteΔT for dT, substitute Δσ for dσ, substitute Δε for dε, 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:(17)σΓ=σs-σ¯,where σs is the yield strength; σ¯ 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.


## 3. 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 Figure2, where σv is the vertical stress in situ; σH is the max horizontal stress in situ; σh is the max horizontal stress in situ; F is casing axial tensile force; TBHST is bottom hole standard temperature; TBHST is bottom hole circulating temperature; and P is drilling mud pressure.Figure 2
The three-dimensional mechanical model.


## 4. Numerical Calculations of Well Casing in an Incomplete Borehole in Salt Formation

### 4.1. Behavior of 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.(i)
The steady creep stage accounts for a large proportion of the entire process of salt rock creep, and the creep strain is unrecoverable.(ii)
The steady creep fitted curve can be obtained by uniaxial or triaxial creep test of salt rock specimens in the laboratory, which has certain applicability in engineering.(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:(18)ε˙cr=A(eBσ-e-Bσ)2e-Q/RT,where ε˙cr is the steady-state creep rate, s−1; Q is the activation energy, J/mol; R is the universal gas constant, J/mol; σ is the deviatoric stress, MPa; T is the thermodynamic temperature, K; A,B   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.Table 1
The steady-state creep parameters obtained through laboratory test.
Rock
Depth (m)
Horizon
Mineral constituent
Steady-state creep parameters
NaCl
CaSO4
A
B
Q (105 J/mol)
Salt
3922.30~3922.44
130/56
93.0%
6.9%
0.0684
0.0928
1.3294


### 4.2. 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:(19)EN80=2.132080-0.0025902T×105,T≥293.15KσN80=0.98016-0.00034878Tσs,T≥293.15K,where EN80 is Young’s modulus of N80 casing, σs is yield strength of N80 casing, and T is temperature. Consider(20)σbc=-0.053T2+10.82T-520.93,363.15K≤T≤383.15K0.0009T2-0.3465T+54.081,383.15K≤T≤423.15KEc=0.0194σbc1.4787,where Ec is Young’s modulus of class G cement, MPa; σbc is compressive strength of class G cement, MPa; and T is temperature, K.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.Table 2
Comparisons between the formula calculation and experimental data.
Number
Temperature(°C)
Item
Experiment(MPa)
Calculation(MPa)
Deviation(%)
1
100.0
σ
bc
30.5
31.1
1.97
2
135.0
σ
bc
24.1
23.7
1.66
3
225.0
σ
N80
582.0
581.6
0.07
4
325.0
σ
N80
551.0
559.1
1.47


### 4.3. 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 (Figure3), in which γ is defined as the defective circumferential angle of the cement sheath, C is defined as the defective depth of cement sheath, and L is defined as the defective length of cement sheath. The other parameters adopted in the computational model are presented in Table 3.Table 3
The parameters for computational model.
Casing property
Density (kg/m3)
7800
Poisson’s ratio
0.28
Expansion coefficient (×10−6/K)
12.0
Thermal conductivity (W/(m·K))
45.0
Specific heat capacity (J/(kg·K))
460.0
Specification (mm)
Φ
146.1
×
12.65
Cement property
Density (kg/m3)
1380
Poisson’s ratio
0.25
Expansion coefficient (×10−6/K)
10.5
Thermal conductivity (W/(m·K))
0.9
Specific heat capacity (J/(kg·K))
865.0
Thickness (mm)
9.5
Sandstone property
Density (kg/m3)
2570
Poisson’s ratio
0.16
Expansion coefficient (×10−6/K)
10.1
Thermal conductivity (W/(m·K))
2.5
Specific heat capacity (J/(kg·K))
762.0
Thickness (mm)
800
×
2
Salt rock property
Density (kg/m3)
2280
Poisson’s ratio
0.35
Expansion coefficient (×10−6/K)
10.3
Thermal conductivity (W/(m·K))
4.8
Specific heat capacity (J/(kg·K))
869.0
Thickness (mm)
400
Drilling mud property
Density (kg/m3)
1608
Thermal conductivity (W/(m·K))
1.6
Specific heat capacity (J/(kg·K))
1674.8
Model depth (m)
3950–3952
Wellbore diameter (mm)
Φ165.1
Sandstone stress in situ (MPa)
σv
121.25
σH
50.94
σh
45.88
Salt rock stress in situ (MPa)
σv
121.25
σH
58.54
σh
53.45Figure 3
Schematic diagram and FEM mesh of salt formation-defective cement-casing.
(a)
Schematic diagram of computational model
(b)
FEM mesh of salt formation-defective cement-casing


### 4.4. Open Wellbore Shrinkage

To investigate open wellbore shrinkage in deep salt formation, a computational model without casing and cement is performed, as shown in Figure4, 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.Figure 4
The open wellbore shrinkage with time.
(a)
1 month
(b)
10 months
(c)
20 months
(d)
40 months


### 4.5. Effects of Defective Cement on Casing Remaining Strength

Figure5 shows the effects of cement defective depth and defective circumferential angle on casing remaining strength; in this computational model, L 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 C and decreases rapidly when γ varies from 20° to 100° compared to other conditions. Specifically, when γ 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 5
Effects ofC and γ on casing remaining strength.
(a)
Defect at first cemented interface
(b)
Defect at second cemented interfaceFigure6 shows the effects of cement defective length on casing remaining strength; with L increase, casing remaining strength declines gradually; however, when L>0.2 m, 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.Figure 6
Effects ofL on casing remaining strength.


### 4.6. Effects ofTBHCT on Casing Remaining Strength

From Figure7 we can clearly see the effects of TBHCT on casing remaining strength; with TBHCT increase, casing remaining strength drops gradually, and C exacerbated the downward trend. It is noteworthy that casing remaining strength drop slows down after TBHCT greater than 393.15 K, 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, Ec 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 TBHCT greater than 393.15 K.Figure 7
Effects ofTBHCT on casing remaining strength.


### 4.7. Effects of Salt Rock Creep on Well Casing

As shown in Figure8, 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.Figure 8
Effects of salt rock creep on additional load and displacement of casing.


## 4.1. Behavior of 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.(i)
The steady creep stage accounts for a large proportion of the entire process of salt rock creep, and the creep strain is unrecoverable.(ii)
The steady creep fitted curve can be obtained by uniaxial or triaxial creep test of salt rock specimens in the laboratory, which has certain applicability in engineering.(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:(18)ε˙cr=A(eBσ-e-Bσ)2e-Q/RT,where ε˙cr is the steady-state creep rate, s−1; Q is the activation energy, J/mol; R is the universal gas constant, J/mol; σ is the deviatoric stress, MPa; T is the thermodynamic temperature, K; A,B   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.Table 1
The steady-state creep parameters obtained through laboratory test.
Rock
Depth (m)
Horizon
Mineral constituent
Steady-state creep parameters
NaCl
CaSO4
A
B
Q (105 J/mol)
Salt
3922.30~3922.44
130/56
93.0%
6.9%
0.0684
0.0928
1.3294


## 4.2. 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:(19)EN80=2.132080-0.0025902T×105,T≥293.15KσN80=0.98016-0.00034878Tσs,T≥293.15K,where EN80 is Young’s modulus of N80 casing, σs is yield strength of N80 casing, and T is temperature. Consider(20)σbc=-0.053T2+10.82T-520.93,363.15K≤T≤383.15K0.0009T2-0.3465T+54.081,383.15K≤T≤423.15KEc=0.0194σbc1.4787,where Ec is Young’s modulus of class G cement, MPa; σbc is compressive strength of class G cement, MPa; and T is temperature, K.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.Table 2
Comparisons between the formula calculation and experimental data.
Number
Temperature(°C)
Item
Experiment(MPa)
Calculation(MPa)
Deviation(%)
1
100.0
σ
bc
30.5
31.1
1.97
2
135.0
σ
bc
24.1
23.7
1.66
3
225.0
σ
N80
582.0
581.6
0.07
4
325.0
σ
N80
551.0
559.1
1.47


## 4.3. 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 (Figure3), in which γ is defined as the defective circumferential angle of the cement sheath, C is defined as the defective depth of cement sheath, and L is defined as the defective length of cement sheath. The other parameters adopted in the computational model are presented in Table 3.Table 3
The parameters for computational model.
Casing property
Density (kg/m3)
7800
Poisson’s ratio
0.28
Expansion coefficient (×10−6/K)
12.0
Thermal conductivity (W/(m·K))
45.0
Specific heat capacity (J/(kg·K))
460.0
Specification (mm)
Φ
146.1
×
12.65
Cement property
Density (kg/m3)
1380
Poisson’s ratio
0.25
Expansion coefficient (×10−6/K)
10.5
Thermal conductivity (W/(m·K))
0.9
Specific heat capacity (J/(kg·K))
865.0
Thickness (mm)
9.5
Sandstone property
Density (kg/m3)
2570
Poisson’s ratio
0.16
Expansion coefficient (×10−6/K)
10.1
Thermal conductivity (W/(m·K))
2.5
Specific heat capacity (J/(kg·K))
762.0
Thickness (mm)
800
×
2
Salt rock property
Density (kg/m3)
2280
Poisson’s ratio
0.35
Expansion coefficient (×10−6/K)
10.3
Thermal conductivity (W/(m·K))
4.8
Specific heat capacity (J/(kg·K))
869.0
Thickness (mm)
400
Drilling mud property
Density (kg/m3)
1608
Thermal conductivity (W/(m·K))
1.6
Specific heat capacity (J/(kg·K))
1674.8
Model depth (m)
3950–3952
Wellbore diameter (mm)
Φ165.1
Sandstone stress in situ (MPa)
σv
121.25
σH
50.94
σh
45.88
Salt rock stress in situ (MPa)
σv
121.25
σH
58.54
σh
53.45Figure 3
Schematic diagram and FEM mesh of salt formation-defective cement-casing.
(a)
Schematic diagram of computational model
(b)
FEM mesh of salt formation-defective cement-casing


## 4.4. Open Wellbore Shrinkage

To investigate open wellbore shrinkage in deep salt formation, a computational model without casing and cement is performed, as shown in Figure4, 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.Figure 4
The open wellbore shrinkage with time.
(a)
1 month
(b)
10 months
(c)
20 months
(d)
40 months


## 4.5. Effects of Defective Cement on Casing Remaining Strength

Figure5 shows the effects of cement defective depth and defective circumferential angle on casing remaining strength; in this computational model, L 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 C and decreases rapidly when γ varies from 20° to 100° compared to other conditions. Specifically, when γ 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 5
Effects ofC and γ on casing remaining strength.
(a)
Defect at first cemented interface
(b)
Defect at second cemented interfaceFigure6 shows the effects of cement defective length on casing remaining strength; with L increase, casing remaining strength declines gradually; however, when L>0.2 m, 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.Figure 6
Effects ofL on casing remaining strength.


## 4.6. Effects ofTBHCT on Casing Remaining Strength

From Figure7 we can clearly see the effects of TBHCT on casing remaining strength; with TBHCT increase, casing remaining strength drops gradually, and C exacerbated the downward trend. It is noteworthy that casing remaining strength drop slows down after TBHCT greater than 393.15 K, 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, Ec 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 TBHCT greater than 393.15 K.Figure 7
Effects ofTBHCT on casing remaining strength.


## 4.7. Effects of Salt Rock Creep on Well Casing

As shown in Figure8, 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.Figure 8
Effects of salt rock creep on additional load and displacement of casing.


## 5. 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 of well casing, the effects of open wellbore shrinkage need to be considered.(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.(3)
Cement defect plays an important role on casing remaining strength; to ensure the safety of well casing, cement defect level should be taken into consideration when designing casing strength in deep salt formation.(4)
In deep complicated formation, the impact of temperature on wellbore is crucial; apart from the consideration of temperature on casing, the effect of temperature on cement properties also cannot be ignored.(5)
The multiphysics coupling FEM may act as a new perspective of novel engineering applications in oil and gas industry.



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*Source: 575492-2015-05-19.xml*