# Study on Wellbore Stability of Multilateral Wells under Seepage-Stress Coupling Condition Based on Finite Element Simulation

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Mathematical Model

#### 2.1. Seepage Model

^{3}·s

^{−1}; A is the cross-sectional area of flow, m

^{2}; gradH is the hydraulic gradient; z is the height above a reference surface, m; ${\rho}_{w}$ is the fluid density, kg·m

^{−3}; H is the piezometric head, m; k is the permeability, m·s

^{−1}; and $\overline{v}$ is the average seepage velocity, m·s

^{−1}.

#### 2.2. Effective Stress Model

#### 2.3. Equilibrium Equation

#### 2.4. Continuity Equation

#### 2.5. Yield Criterion

#### 2.6. Boundary Conditions

## 3. Numerical Model

## 4. Wellbore Stability of Multilateral Wells

^{3}. When the drilling fluid density is lower than the value at the intersection, the maximum equivalent plastic strain under the non-seepage condition is larger than that under the seepage condition. If the drilling fluid density exceeds the value at the intersection, the maximum equivalent plastic strain under the non-seepage condition is lower than that under the seepage condition. Because the formation pressure is 12.6 MPa, the two curves under seepage and non-seepage conditions are intersected when the fluid column pressure of drilling fluids is 12.6 MPa. In the FEA model, the boundary conditions of pore pressure under the two conditions are the same and the fluid column pressure of drilling fluids in wellbores is identical to the formation pressure, so the calculated maximum equivalent plastic strain is also same.

^{3}under seepage and non-seepage conditions. In the drilling process, the maximum and minimum principal stresses under the seepage condition are lower than those under the non-seepage condition. So, the Mohr stress circle shifts leftwards to approach the failure envelope in the process and the wellbores are more prone to failure.

## 5. Influencing Factors of Wellbore Stability of Multilateral Wells

#### 5.1. Influences of Wellbore Diameters of Multilateral Wells

#### 5.2. Influences of the Angle between Main Wellbore and Branches

#### 5.3. Influences of Azimuth of Multilateral Wells

## 6. Conclusions

- (1)
- Stress concentration is most serious at multilateral junctions of multilateral wells, where wellbore instability is most likely to occur.
- (2)
- The maximum plastic strain at multilateral junctions increases slightly with the enlargement of wellbore diameter of multilateral wells, and the wellbore diameter exerts slight influences of the wellbore stability.
- (3)
- The larger the angle between main wellbore and branches, the more stable the multilateral wells. When the azimuth of multilateral wells is parallel to the direction of the minimum horizontal principal stress, the equivalent plastic strain is lowest and wellbores are most stable.
- (4)
- Appropriately increasing the drilling fluid density can effectively reduce the risk of wellbore instability at multilateral junctions.
- (5)
- When the angle between main wellbore and branches is larger than or equal to 45°, the regions at the risk of wellbore instability transfer from multilateral junctions to the inner areas of multilateral wellbores.

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 3.**Nephogram for equivalent plastic strain under different drilling fluid densities during seepage. (

**a**) 0.9 g/cm

^{3}, (

**b**) 1.0 g/cm

^{3}, (

**c**) 1.05 g/cm

^{3}, (

**d**) 1.1 g/cm

^{3}, (

**e**) 1.2 g/cm

^{3}, (

**f**) 1.3 g/cm

^{3}, (

**g**) 1.33 g/cm

^{3}.

**Figure 4.**Change trends of the equivalent plastic strain with different drilling fluid densities under seepage and non-seepage conditions.

**Figure 6.**Change trends of the principle stress under different drilling fluid densities. (

**a**) minimum principle stress, (

**b**) maximum principle stress.

**Figure 7.**Nephogram for the equivalent plastic strain under different wellbore diameters of multilateral wells. (

**a**) Wellbore diameter of 0.20 m, (

**b**) Wellbore diameter of 0.22 m, (

**c**) Wellbore diameter of 0.24 m, (

**d**) Wellbore diameter of 0.26 m, (

**e**) Wellbore diameter of 0.28 m.

**Figure 8.**Change trend of the maximum equivalent plastic strain under different wellbore diameters of multilateral wells.

**Figure 9.**Nephogram for Mises stress distribution, Pa. (

**a**) The angle between main wellbore and branches is 30°, (

**b**) The angle is 45°.

**Figure 10.**Nephograms for distribution of the equivalent plastic strain under different angles between main wellbore and branches. (

**a**) 15°, (

**b**) 30°, (

**c**) 45°, (

**d**) 60°, (

**e**) 75°.

**Figure 11.**Change trend of the maximum equivalent plastic strain with different angles between main wellbore and branches.

**Figure 12.**Nephogram for Mises stress distribution, Pa. (

**a**)The relative azimuth is 0°, (

**b**) The relative azimuth is 90°.

**Figure 14.**Nephograms for equivalent plastic strain under different azimuths of multilateral wells. (

**a**) Azimuth of multilateral wells of 0°, (

**b**) Azimuth of multilateral wells of 30°, (

**c**) Azimuth of multilateral wells of 45°, (

**d**) Azimuth of multilateral wells of 60°, (

**e**) Azimuth of multilateral wells of 90°.

**Figure 15.**Change trend of the maximum equivalent plastic strain under different azimuths of multilateral wells.

Parameters | Values | Parameters | Values |
---|---|---|---|

Rock density | 2300 kg/m^{3} | Well depth | 1300 m |

Elastic modulus | 6000 MPa | Overburden pressure | 30 Mpa |

Poisson’s ratio | 0.25 | Maximum horizontal principal stress | 27 Mpa |

Internal frictional force | 32° | Minimum horizontal principal stress | 23 Mpa |

Cohesion | 5 Mpa | Formation pressure | 12.6 Mpa |

Drilling fluid density | 1.1 g/cm^{3} | Porosity ratio | 0.5 |

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**MDPI and ACS Style**

Xu, H.; Cao, J.; Dong, L.; Yan, C.
Study on Wellbore Stability of Multilateral Wells under Seepage-Stress Coupling Condition Based on Finite Element Simulation. *Processes* **2023**, *11*, 1651.
https://doi.org/10.3390/pr11061651

**AMA Style**

Xu H, Cao J, Dong L, Yan C.
Study on Wellbore Stability of Multilateral Wells under Seepage-Stress Coupling Condition Based on Finite Element Simulation. *Processes*. 2023; 11(6):1651.
https://doi.org/10.3390/pr11061651

**Chicago/Turabian Style**

Xu, Hao, Jifei Cao, Leifeng Dong, and Chuanliang Yan.
2023. "Study on Wellbore Stability of Multilateral Wells under Seepage-Stress Coupling Condition Based on Finite Element Simulation" *Processes* 11, no. 6: 1651.
https://doi.org/10.3390/pr11061651