# Wellbore stability model based on iterative coupling method in water alternating gas injection

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## Abstract

Ensuring wellbore integrity is the most important factor in injection well design. The water alternating gas (WAG) injection is increasingly applied globally as the effective enhanced oil recovery (EOR) method in oil wells. High injection pressure or low injection temperature could lead to compressive wellbore failure. The rock stress around the wellbore is a function of the wellbore fluid flow and it should be precisely determined to avoid the wellbore failure. The purpose of this study is to propose a method to ensure the stability of the wellbore for the WAG process using iterative coupling method. The parameters of pressures, temperature, saturations and stresses are obtained for the multiphase flow condition using mathematical modeling. In this study, finite difference method is used to solve pressure, temperature and saturations; and finite volume method is acquired to solve the rock stresses. Iterative coupling method is employed to improve the accuracy of the results. This study introduces improved iterative coupling method between flow and stress models to reduce the processing time of obtaining corrected stress and failure results. Wellbore stability model is developed to determine the maximum pressure values, which lead to wellbore failure in WAG injection process for some different boundary conditions.

## Keywords

Geomechanics Wellbore Iterative coupling Failure index Stress## List of symbols

*G*Acceleration of gravity (vector) (m/s

^{2})*I*Identity tensor (dimensionless)

*K*Intrinsic permeability tensor (m

^{2})- Krψ
Relative permeability for flow in phase ψ − l, g (dimensionless)

- qh
Flux density for energy over all phases (J/(m

^{2}s))- Sψ
Saturation of phase ψ − g, l (dimensionless)

*t*Time (s)

*T*Absolute temperature (K)

*u*Velocity (m/s)

*z*Elevation (m)

*α*_{T}Linear thermal expansion

*α*Biot’s constant for a porous media (dimensionless)

*β*Turbulency factor

*ε*Total strain tensor (dimensionless)

- φ, φ
_{f} Porosity in general and porosity (dimensionless)

- μψ
Dynamic fluid viscosity of fluid phase ψ − l, g (Pa s)

- ρ
_{l}, ρ_{g} Liquid and gas density (kg/m

^{3})*σ*Macroscopic total stress tensor (tension positive) (MPa)

- ∇A
Gradient of a vector

*r*Radius (m)

*P*Pressure (Psi, MPa)

*Z*Real gas deviation factor

*θ*Angle

*i*Position indicator

*n*Time indicator

- W,
*M*_{w} Molecular weight

*P*_{c}Capillary pressure

*k*_{mul}Constant

*λ*Module of elasticity

*G*Module of rigidity

*u*,*v*,*w*Deformation in the direction of

*x*,*y*and*z*, respectively*F*,*G*,*H*Strain in the direction of

*x*,*y*and*z,*respectively (same as ε*x*, ε*y*and ε*z*)- ∆
*sx*,*y*,*z* Projection of the triangle in

*x*,*y*and*z*planes*k*Bulk modulus

*K*_{s}Bulk modulus of the solid phase

*K*_{f}Bulk modulus of the fluid phase

- ζ
Fluid strain

- FVM
Finite volume method

- FDM
Finite difference method

## Introduction

Rocks are a combination of different materials. However, rocks exhibit poroelastic response. The amount of the stress indexed by pore pressure depends on pore content. The study of stress in a two-phase medium and in void space is essential for well integrity in oil production. The study of temperature is also important in defining the stress. The theory of thermo-poroelasticity or porothermoelasticity is developed by combining the influence of thermal stress and the difference between solid and fluid expansion (Espinoza 1983; Fredrich et al. 2000; Zare 2012).

Enhanced oil recovery refers to several processes conducted to increase oil production from a reservoir after primary and secondary recoveries, which are typically performed by injecting water or gas. The injected fluid may push the oil or interact with the reservoir rock oil system to prepare suitable conditions for recovery. Injecting water alternating gas (WAG) shows better sweep efficiency than injecting water or gas alone (Irawan and Bataee 2014; Chalaturnyk and Li 2004; Chin et al. 2002).

Thermo-poroelasticity describes the effect of the change in temperature and fluid flow on the stress in the borehole and reservoir. Injecting water results in changes in temperature, pore pressure, and stress in the reservoirs and affects the reservoir permeability and porosity. Most reservoir simulators undergo stress changes and rock deformations during production because of the considerable physical effect of the geomechanical aspects of reservoir behavior (Lewis et al. 1986; Geertsma 1973; Hansen et al. 1995).

Freeman et al. (2009) studied the geomechanics of bitumen formations. They had used two different simulators and compared the results (Freeman et al. 2009). Du and Wong (2005) had developed the coupled geomechanical thermal flow simulator. They had used the finite element method to express the reservoir model (whereas in most of the studies the flow is modeled by the finite difference method). The finite element formulation is explained well, but the result was not comprehensively expressed (Du and Wong 2005).

Yin et al. had developed a finite element model of stress coupled with finite difference model for the flow in the reservoir and done some examples related to the model. Although some of the examples are not applicable for the reservoir and one or two formulae were obsolete, the results are a good representative of what is actually and accurately happening in the reservoir (Yin et al. 2009).

The simulation study had used the iterative coupling method and studied the result of stability with the different initial values of parameters (Joseph et al. 2011).

Safari and Ghassemi (2011) analyzed the geomechanical aspect of huff and puff process. They had done the study for a fractured geothermal reservoir. Their model had shown a good agreement with the field measurements. They had analyzed different geomechanical and flow behavior of the fractures after some years (Safari and Ghassemi 2011).

Chiaramonte and Zoback had published some books on the subject of reservoir geomechanics and CO_{2} sequestration simulation. They had done another CO_{2}-EOR simulation project in a fractured reservoir (2012). They had investigated the mobility of CO_{2} in a fractured field (Chiaramonte 2012).

Tran et al. (2004) developed new iterative coupling method and had applied it in CMG reservoir simulator. They had also corrected the porosity formula for the method. They call it pseudo-coupling and their study had shown that the result of this model is like the fully coupling method, but with higher speed. One year later (2005), they compare the different methods of coupling and their results (Tran et al. 2004).

Mendes et al. (2012) had done a study of coupling with heterogeneity. They get their special boundary conditions and solve the two-phase flow problem using Monte Carlo algorithm. They reach to the result of locally conservative numerical solution and impress that there is an obvious change in production resulted by heterogeneity (Mendes et al. 2012).

Some research tried to model the sand production around the wellbore. Bianco and Halleck analyzed the mechanisms of arch instability and sand production in two-phase saturated poorly consolidated sandstones (Bianco and Halleck 2001). Wan and Wang starts to model the sand production within a continuum mechanics framework (Wan and Wang 2000). He continued his work on sand production and published a paper 4 years later on the topic of “Analysis of Sand Production in Unconsolidated Oil Sand, Using a Coupled Erosional-Stress-Deformation Model” (Wan and Wang 2004). Some years later he analyzed the borehole failure modes and the pore pressure effects on it (Papamichos 2010). Papamichos developed the relation between sand production rate under multiphase flow and water breakthrough (Papamichos 2010).

A flow simulation is based on time. This process determines initial conditions and goes through a time in a defined time step (Irawan and Bataee 2014a, b). Geomechanical calculations are not based on time, except for such phenomena as creep that are usually ignored (Lewis et al. 1989; Rutqvist et al. 2010). However, the deformation and pore volume changes feed back to the time-based flow results. The degree of frequency of this updating procedure (implicitness) significantly affects running speed and result accuracy (Pao et al. 2001; Rutqvist 2011; Edalatkhah 2010). Such frequency can be categorized as follows:

Full coupling: Flow and geomechanics variables (pressure, temperature, stresses, and strains) are solved simultaneously. Full coupling provides accurate solutions. However, this approach requires the solution of a large matrix and processing time is long.

Iterative coupling: Flow and geomechanics variables are solved separately and in sequence. This method has accuracy close to that of full implicit coupling but with higher speed.

Explicit coupling: Pressures, saturations, and temperatures data are called from the flow simulator to the stress calculations. However, the change in porosity, permeability and hence, the corrected pressures is not calculated. This method is called one-way or explicit coupling. Explicit coupling is fast, and lots of the geomechanics simulators use this method. However, the accuracy of this method is low because the flow characteristics depend on geomechanics and it is ignored in this method.

Pseudo-coupling: Some correlations between porosity and stress are used in flow calculations to identify compaction and dilation. However, this method does not process geomechanics (e.g., stress field), and simple formulas are used in a reservoir simulator to calculate subsidence during the process. The running speed of this method is high (Bataee and Irawan 2014; Bataee and Kamyab 2010; Settari et al. 2001).

## Methodology

In most coupling studies, the parameters of pressure, temperature, saturation, stress, and strain are integrated. The full coupling can be performed by several methods, such as finite difference method (FDM), finite element method, and finite volume method (FVM). Thus, a large matrix can be solved. In this study, stress and strain solved using FVM, which is a suitable method for large meshes and able to deal with mesh concentration in high-stress sensitive parts. The other parameters (i.e., pressure, temperature, and saturation) calculated by FDM.

The relation between the change in porosity and strain change was considered. The new values of porosity and permeability are integrated into the flow equation to obtain the corrected values of pressure, temperature, and saturation. The stress and strain are updated. This procedure continues until the convergence condition is obtained under a certain level of accuracy.

Step 1—Flow model: In this step, pressures and saturations around the wellbore are calculated. Temperature values are calculated after the results of pressures and saturations.

Step 2—Stress model: Stress and strain parameters are obtained in this part, based on the mechanical properties of wellbore rock. The stress equation is the main equation that should be discretized. The change in pressure and temperature is ignored in this part and its effect processed in next step to find the corrected stress values.

Step 3—Iterative coupling between two models: The fluid flow in porous media alters the stresses, porosity, and permeability. This effect results in the change in pressure distribution, which requires using two-way coupling method. The coupling study conducted after obtaining the required parameters. The nodes in the FDM and FVM are in different positions. Thus, the values are obtained using bi-linear interpolation of the nearby nodes. The relation between the change in porosity and permeability change is considered.

Step 4—Wellbore Stability model: Based on the results of stress values, which are obtained

Stability model is based on the result of corrected stress values (from step 3). Proper failure criterion should be used to calculate the failure index around the wellbore.

### Step 1: flow and energy modeling

FDM employed to solve the parameters, namely, pressure, temperature, and saturation. The continuity equation for the water flow expanded with following conditions: cylindrical model, considering gravity, considering turbulence effect attributed to high injection rate, incompressible water, and slightly compressible oil and compressible gas with the use of implicit pressure–explicit saturation (IMPES) method while ignoring tangential flow. The turbulence effect is only considered in the wellbore with the Muscat equation. The condition for the pressure and temperature is chosen for immiscible WAG (iWAG). The water is injected, and the gas is injected subsequently under an immiscible condition. The constants for the equation related to rock properties were chosen for the sandstone reservoir. The energy balance equation for the three phases can be expanded after the saturation (from the flow part) is obtained to calculate the temperature values. The conditions for solving the temperature are as follows: cylindrical, implicit method and convection and conduction with the tangential flow ignored.

If we want to deal with the wellbore flow, the equations should be defined first and then discretized for the FDM. Thereafter, a program that can be used to determine the pressure and saturation for each node should be developed. The energy balance equation that uses the saturation obtained in the previous part should be discretized. After obtaining the temperature, the heterogeneity study can be conducted with the random matrix for porosity and permeability distribution through special random functions. The boundary condition may be changed to find the values in different cases.

#### Pressures and saturations

The steps in this part include water injection, gas injection, water–oil system, and three-phase flow. The continuity equation for the water flow expanded for the finite difference method with the following conditions: cylindrical model, considering gravity, considering turbulence effect attributed to high injection rate, incompressible water, and slightly compressible oil and compressible gas with the use of implicit pressure–explicit saturation (IMPES) method while ignoring tangential flow.

##### Water flow

*z*over

*z*is equal to 1. The equation should then be prepared for the finite difference study:

##### Gas flow

##### Two-phase immiscible water–oil

To solve the two-phase problems, four equations require obtaining the four unknowns, namely, *P* _{w}, *P* _{o}, *S* _{w}, and *S* _{o}. The four equations are water flow, oil flow, capillary, and total saturation equations.

*S*

_{w}* is obtained from the following equation:

##### Three-phase immiscible water–oil–gas

Six equations require to obtain the six unknowns, namely, *P* _{w}, *P* _{o}, *P* _{g}, *S* _{w}, *S* _{o}, and *S* _{g}, to solve the three-phase problems. These six equations are water flow, oil flow, gas flow, wo-capillary, OG-capillary, and total saturation equations.

#### Temperature

### Step 2: stress modeling

FVM employed to determine the stress and strain parameters. The stress equation is the main equation that should be discretized. The change in pressure and temperature are ignored in this part.

FVM is employed to develop the stress model in the wellbore. The procedure consists of transforming the equations to weak form, meshing the defined shape, and programming to obtain the values for each node.

#### Weak form of equations

*f*” is the body force and is assumed zero in this case.

*ɛ*

_{vol}is as follows:

*x-, y-,*and

*z*-directions are expressed as in Eqs. (34–36).

*u*,

*v*, and

*w*are the deformations toward the

*x-, y-,*and

*z*-directions.

*x*-direction) are written. The procedure for the others is the same. The factor is taken from the equations, and the derivation of

*u*over

*x*is transformed to

*F*(

*F*= δ

*u*/δ

*x*). The procedure used to obtain

*N*and

*H*is the same.

After rearranging these formulas, Eqs. (45–47) obtained. Therefore, they should be solved for every single node with the summation equation for all the nearby elements for *F*, *N*, and *H* (Eqs. 48–50).

##### Tetrahedral meshed shapes

The body shape meshed to find the values by FVM. The program in this study developed for only tetrahedral shapes. The advantage of this method is that any shape can be applied. The requirements are only meshed positions, connectivity, and boundary condition. Therefore, the mesh node positions can be imported from any software to the program.

*x*,

*y*, and

*z*planes. The values for ∆

*sx*, ∆

*sy*, and ∆

*sz*should be known for the main Eqs. (45–47).

The program reads the data of the node positions, connectivity, and initial and boundary conditions. Subsequently, the program calculates all the element volumes and surface projections. The program then uses the iteration method to solve the matrix of Eqs. (45–50) for all nodes.

Rock and fluid properties and wellbore data used in this study

Model parameter | Values |
---|---|

Poisson’s ratio | 0.45 |

Permeability (md) | 35 |

Porosity (%) | 20 |

Bulk modulus (MPa) | 1100 |

Solid bulk modulus (MPa) | 32,600 |

Fluid bulk modulus (MPa) | 3290 |

Gas bulk modulus (MPa) | 330 |

Thermal expansion coefficient of fluid (1/k) | 3.0×10 |

Thermal expansion coefficient of solid (1/k) | 1.8×10 |

Thermal expansion coefficient of gas (1/k) | 0.001 |

Wellbore initial temperature (°C) | 50 |

Reservoir temperature (°C) | 70 |

Friction angle (°) | 40 |

Initial wellbore pressure (Psi) | 3600 |

Injection pressure (Psi) | 5000 |

Connate water saturation (%) | 0.22 |

Residual oil saturation (%) | 0.4 |

Water density (g/cm | 1 |

Oil density (g/cm | 0.8 |

Gas density (g/cm | 0.00184 |

Wellbore radius (cm) | 10.795 |

The simple explanation about the case in this study is that there is a wellbore shape as in Fig. 6 (meshed as in Fig. 7 for FVM study). The input parameters are as in Table 1 and the boundary conditions are as in Eqs. 51–55. For the process of the flow around the wellbore, firstly, water injected into fully saturated oil medium. Then, gas injected to this OW two-phase medium. The stresses applied are horizontal maximum, minimum, and vertical stresses. Horizontal maximum in situ stress is in the direction of north–south.

### Step 3: iterative coupling

In this study, stress and strain are solved using FVM, and the other parameters (i.e., pressure, temperature, and saturation) calculated using FDM. The coupling study is conducted after obtaining the required parameters. The nodes in the FDM and FVM are in different positions. Thus, the values are obtained using bi-linear interpolation of the nearby nodes. The relation between the change in porosity and strain change and that between porosity and permeability was considered. The new values of porosity and permeability are integrated into the flow equation to obtain the corrected values of pressure, temperature, and saturation. The stress and strain are updated. This procedure continues until convergency obtained under a certain level of accuracy (0.01 Psi).

The new values of pressure, saturation, and temperature are calculated with the updated porosity and permeability. This iterative coupling procedure continues until it converges. The new values of pressure, saturation, and temperature are calculated with the updated porosity and permeability. This iterative coupling procedure continued until it converged.

### Step 4: Wellbore stability model

The proper failure criterion for this study should have the capability of calculation the stress for three-dimensional stress direction (capable of considering intermediate stress), and it should be suitable for wellbore rock model. Drucker–Prager is the most suitable rock failure criterion among all failure criteria regarding the wellbore modeling. Therefore, failure index values will be calculated using calculated rock stresses around the wellbore.

## Results of the program

In this study, some different in situ stresses are applied to check the result of the failure index values in the wellbore. All the cases are investigated for the boundary condition equations of 51–55 and input parameters of Table 1. To find the results of the stress values, pressures and temperatures should be calculated. Temperature values are related to the saturations of each phase, therefore, saturations also should be defined for each node during the time of WAG injection. Calculation of these parameters is crucial in the first step of injection as they have a sharp change in the wellbore.

The stress values and the failure are studied in three different cases. The three cases show different failure conditions, based on in situ stress values and injection pressures. The first case shows the normal failure (as in fracturing). The second case expresses the compressive failure and the last case describes the shear failure (as in sand production phenomenon).

### Case 1

In situ stress values and injection pressure for case 1

Case parameter | Value |
---|---|

Horizontal maximum stress (MPa) | 35 |

Horizontal minimum stress (MPa) | 30 |

Vertical stress (MPa) | 36 |

Injection pressure (MPa) | 34.47 |

To sum it up, fracturing will happen if the injection pressure becomes very high. In this case, the wellbore wall cannot withstand the induced normal stress. The direction of the fracturing is the same as the direction of the maximum horizontal in situ stress.

### Case 2

In situ stress values and injection pressure for case 2

Case parameter | Value |
---|---|

Horizontal maximum stress (MPa) | 39 |

Horizontal minimum stress (MPa) | 30 |

Vertical stress (MPa) | 36 |

Injection pressure (MPa) | 28 |

### Case 3

In situ stress values and injection pressure for case 3

Case parameter | Value |
---|---|

Horizontal maximum stress (MPa) | 35 |

Horizontal minimum stress (MPa) | 30 |

Vertical stress (MPa) | 36 |

Injection pressure (MPa) | 15 |

### Sensitivity analysis

The sensitivity of wellbore fracturing to different parameters is investigated in this part. The parameters of interest in this part are injection pressure, temperature and in situ stresses. The results show the effect of each parameter on wellbore stability.

Wellbore injection pressure which fractures the formation

Case parameter | Value |
---|---|

Horizontal maximum stress (MPa) | 20 |

Horizontal minimum stress (MPa) | 17 |

Vertical stress (MPa) | 23 |

Formation pressure (psi) | 1000 |

Injection pressure (psi) | 3165.4 |

The effect of horizontal maximum stress change on wellbore failure

Horizontal maximum stress | 17.5 | 18 | 18.5 | 19 | 19.5 | 20 | 20.5 | 21 | 21.5 | 22 | 22.5 |

Injection pressure limit | 3440.602 | 3386.37 | 3332.139 | 3277.907 | 3223.676 | 3169.444 | 3115.213 | 3060.981 | 3006.75 | 2952.518 | 2898.287 |

The effect of horizontal minimum stress change on wellbore failure

Horizontal minimum stress | 14.5 | 15 | 15.5 | 16 | 16.5 | 17 | 17.5 | 18 | 18.5 | 19 | 19.5 |

Injection pressure limit | 2084.814 | 2247.508 | 2410.203 | 2572.897 | 2735.592 | 2898.286 | 3060.981 | 3223.675 | 3386.370 | 3549.064 | 3711.758 |

## Validation of failure results

The proper equipment that can serve the polyaxial test is rare and the test is very costly. Therefore, there are limited studies in this case. There are two experimental studies that can be used as the reference for this study, because of the injection condition and core characteristics. The result of this study is not completely matched the experimental results; some different facts cause this difference which is explained in each case.

### Case 1

Comparison between the results of polyaxial tests and the model

| | | Experimental fracking P (psi) | Model fracking P (psi) |
---|---|---|---|---|

1510 | 2510 | 3510 | 1550 | 1399.6 |

### Case 2

Comparison between hydraulic fracturing field data and the model results

| | | Field results (MPa) | Model results (MPa) |
---|---|---|---|---|

52.89 | 74.98 | 66.95 | 55.29 | 53.64897 |

30.8 | 33.78 | 49.67 | 34.48 | 38.75516 |

57.9 | 75 | 65.8 | 68.19 | 65.13034 |

34.6 | 53.7 | 51.64 | 39.53 | 31.94429 |

49.13 | 61.89 | 63.81 | 58.37 | 55.70289 |

45.76 | 58.84 | 59.43 | 50.35 | 51.40017 |

46.02 | 71.58 | 56.81 | 48.21 | 43.04055 |

42.8 | 60.23 | 49.78 | 49.96 | 45.87237 |

58.94 | 98.61 | 56.67 | 57.27 | 51.84377 |

60.2 | 85.58 | 59.02 | 70.91 | 63.89058 |

60.31 | 85.78 | 67.02 | 64.82 | 62.20345 |

46.5 | 47.16 | 65.5 | 62.64 | 60.44108 |

48.39 | 77.64 | 56.26 | 52.42 | 43.94844 |

64.48 | 108.8 | 59.16 | 61.38 | 56.09691 |

50.12 | 78.6 | 56.95 | 53.16 | 46.95784 |

62.84 | 86.5 | 67.57 | 72.39 | 67.21834 |

44 | 70.5 | 56.4 | 47.02 | 39.40782 |

50.13 | 89.43 | 56.96 | 45.16 | 38.87909 |

64.48 | 79.5 | 62.6 | 82.74 | 77.24095 |

59.37 | 90.77 | 57.09 | 64.12 | 58.57775 |

31.12 | 49.5 | 51.02 | 34.34 | 27.41614 |

25.77 | 31.77 | 35.34 | 29.65 | 32.16981 |

36.83 | 48.6 | 51.16 | 39.64 | 40.86822 |

31 | 34.46 | 49.23 | 34.82 | 38.79348 |

31.56 | 38.81 | 51.74 | 33.44 | 36.23694 |

40.1 | 51.07 | 54.92 | 45.5 | 45.51862 |

38.53 | 62.1 | 57.5 | 43.71 | 33.17238 |

### Case 3

Comparison between the results of polyaxial tests and the model of this study

| | | Experimental | Model |
---|---|---|---|---|

94 | 9 | 5 | 41 | 53.4 |

97 | 15 | 5 | 41 | 54.6 |

88 | 29 | 5 | 35 | 55.26667 |

109 | 44 | 5 | 43 | 60.06667 |

94 | 65 | 5 | 37 | 60.86667 |

109 | 12 | 8 | 47 | 56.2 |

129 | 27 | 8 | 53 | 60.86667 |

132 | 41 | 8 | 53 | 63.13333 |

135 | 50 | 8 | 53 | 64.73333 |

127 | 79 | 8 | 49 | 67.53333 |

147 | 15 | 15 | 62 | 62.6 |

157 | 29 | 15 | 64 | 65.8 |

165 | 62 | 15 | 63 | 71.26667 |

162 | 82 | 15 | 60 | 73.53333 |

159 | 88 | 15 | 59 | 73.93333 |

168 | 97 | 15 | 63 | 76.33333 |

178 | 20 | 20 | 74 | 68.06667 |

183 | 30 | 20 | 75 | 70.06667 |

173 | 41 | 20 | 68 | 70.2 |

185 | 50 | 20 | 72 | 73 |

177 | 57 | 20 | 67 | 72.86667 |

197 | 68 | 20 | 75 | 77 |

194 | 82 | 20 | 72 | 78.46667 |

193 | 97 | 20 | 71 | 80.33333 |

185 | 100 | 20 | 67 | 79.66667 |

197 | 30 | 30 | 79 | 73.26667 |

218 | 47 | 30 | 85 | 78.33333 |

224 | 69 | 30 | 84 | 82.06667 |

232 | 88 | 30 | 85 | 85.66667 |

229 | 109 | 30 | 82 | 88.06667 |

241 | 129 | 30 | 86 | 92.33333 |

227 | 150 | 30 | 81 | 93.26667 |

215 | 171 | 30 | 79 | 94.46667 |

224 | 40 | 40 | 87 | 79.53333 |

244 | 60 | 40 | 92 | 84.86667 |

252 | 70 | 40 | 93 | 87.26667 |

253 | 79 | 40 | 92 | 88.6 |

252 | 100 | 40 | 89 | 91.26667 |

274 | 99 | 40 | 99 | 94.06667 |

265 | 118 | 40 | 93 | 95.4 |

279 | 138 | 40 | 98 | 99.93333 |

274 | 159 | 40 | 95 | 102.0667 |

231 | 50 | 50 | 85 | 83.13333 |

## Conclusions

To ensure the wellbore stability, stresses values should be obtained. The stress values interact with pressures, temperatures and saturations regarding the change in porosity and permeability. Therefore, iterative coupling method implemented to calculate the corrected values of stress around the wellbore for WAG injection. The developed model speeds up the operation because only the parameters of stress and strain were solved by FVM, whereas the pressures, temperatures, and saturations solved by FDM.

To sum up the results of flow around the wellbore, it is recorded that the wellbore pressures and saturations changed very fast. It is due to the small wellbore area and high injection pressure. After the gas injection, OW bank pushed out of the wellbore, however, some amounts of oil and water remained in the pores and need chemical treatment to be removed. Temperature values affect the wellbore stress; in the case of injection, wellbore cooling will happen and might cause stability problems. Temperature values are related to the saturation distribution around the wellbore for each phase. Therefore, these values calculated after flow study had completed.

Stress redistribution will happen around the wellbore after the injection. The stress value is a function of in situ stresses, pressures, and temperatures. Maximum values of radial stress are in the direction of horizontal maximum in situ stress. The values are important in wellbore failure because fracturing will happen in this direction. Maximum values of tangential stress are in the direction of horizontal minimum in situ stress. Vertical stress around the wellbore is not related to horizontal maximum and minimum in situ stresses. It is a function of wellbore radius, pressures, and temperatures. The direction of the maximum shear stress is 45° with respect to maximum horizontal in situ stress.

Three different cases are investigated to show the three different failure types. In the first case, high injection pressure leads to normal failure as fracturing; it started in the direction of maximum in situ stress. In the second case, compaction failure occurred which is caused by the high difference between in situ maximum and minimum pressure and lack of well pressure support. It started in the direction of minimum in situ stress. The third case investigated the shear failure as in sand production. The low wellbore pressure caused the layers of the sand separated from the wall; this type of the failure is common in production wells.

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