A three-dimensional model for the numerical simulation of casing-cement behavior is used to investigate residual strength in the perforated casing of ultra deep wells. The influence of the hole diameter, hole density and phase angle on the residual strength of the casing under non-uniform stress and fracturing conditions is revealed through the consideration of different perforation parameters. It is shown that the residual strength of the casing increases with the hole diameter and periodically changes with the hole density; the phase angle is the main factor that affects the residual strength of the perforated casing, and the perforation should be avoided in the direction of the minimum principal stress to reduce stress concentration at the perforation hole. Moreover, as shown by a companion orthogonal experiment, the descending order of influence of the different influential parameters is: phase angle, hole diameter, hole density and the thickness of casing.

A high and nonuniform

Since casing-cement-formation under perforation conditions is not axisymmetric or centrally symmetrical, only a three-dimensional model can be selected to analyze the overall stresses under a nonuniform stress.

This model takes the 5-1/2” oil layer casing of Well G101 as the research object, which is at a depth of 6600 m, in a certain area of Xinjiang. Its basic perforation parameters are a diameter of 13 mm, 16 holes/m, phase angle of 90°, and hole depth of 500 mm. The formation is simplified from inside to outside, and there are four parts: perforated casing, perforated cement, perforated formation, and outer formation.

Certain assumptions are introduced into the model to lower the difficulty of modeling and calculation, considering that the specific condition of a real perforated casing is more complicated in the formation. The specific assumptions are as follows: there is no eccentricity of the perforation hole, and the central axis of the hole is perpendicular to and intersects with the axis of the casing; the projection on the vertical plane of the axis of the perforation is a circle and the burrs and cracks on the edge of the hole are not considered; the ellipticity and wall of the casing are ignored. The thickness unevenness, casing, cement, and formation are homogeneous elastomeric materials.

A three-dimensional model of casing-cement-formation is established based on the above assumptions. The length, width, and height are 5 m, 5 m, and 0.7 m, respectively, as shown in

To prevent rigid body displacement of the model from causing nonconvergence in the calculation, the author used heterogeneous ground stress data and fracturing construction data in the southern margin area to set the boundary conditions, and the x and y directions in the outer boundary of the model are set perpendicular to a single surface of the x- and y-axes. The displacement is fixed, minimum and maximum horizontal ground stresses are applied to opposite surfaces, and the z-axis direction displacement of the upper and lower surfaces is fixed. At the same time, pressure is applied to the inner wall of the casing and perforation hole; the direct interface of the formation, casing, and cement is connected by binding.

Part | Inner diameter/mm | Outer diameter/mm | Thickness/mm |
---|---|---|---|

Casing | 111.16 | 139.70 | 14.27 |

Cement | 139.70 | 215.90 | 38.10 |

Perforated formation | 215.90 | 1215.9 | 500 |

According to field data, the maximum horizontal ground stress is 167.88 MPa, the minimum horizontal ground stress is 144.51 MPa, and the internal pressure during fracturing is approximately 120 MPa. According to the finite element calculation steps, four geometric model components—perforation casing, perforated cement, perforated formation and external formation—are established in sequence; the actual data are selected to set the material parameters of each part of the model (see

Part | Elastic Modulus/GPa | Poisson’s ratio |
---|---|---|

Casing | 206.0 | 0.30 |

Cement | 3.5 | 0.26 |

Formation | 14.0 | 0.26 |

To verify the accuracy of the model, the collapse strength of the N80 casing under different perforation conditions was tested due to the limitation of the experimental conditions. The casing size parameters and perforation parameters are shown in

Number | Length/m | Thickness/mm | Hole diameter/mm | Hole density/(holes/m) | Phase angle/° |
---|---|---|---|---|---|

1 | 0.2 | 8.05 | 12 | 16 | 90 |

2 | 0.2 | 9.19 | — | — | — |

3 | 0.2 | 9.19 | 12 | 10 | 90 |

4 | 0.2 | 9.19 | 12 | 16 | 90 |

5 | 0.2 | 9.19 | 12 | 20 | 60 |

The results show that the simulated failure loads are higher than the experimental data, but the trend of the load line is basically the same (

According to field perforation plan data, the parameters of different hole diameters, hole densities, phase angles, etc., were selected within a certain range, and the basic perforation parameters of the oil layer casing of the G101 well were compared to analyze each perforation under nonuniform

To visually characterize the change law of the perforating casing strength, the normal form equivalent stress of the fourth strength theory (von Mises stress) is used as an evaluation index. The normal form equivalent stress uses stress contours to represent the stress distribution inside the model. The most dangerous area in the model can be quickly determined, and the specific expression is shown in

In the formula,

The difference between the yield stress of the casing material and the maximum equivalent stress under simulated conditions is regarded as the residual strength of the casing [

The researchers select diameters of 7, 10, 13, and 16 mm to establish the casing-cement-formation 3D models, simulate and calculate the equivalent stress cloud diagram of the perforated casing (

Casing-cement-formation 3D models under different hole densities of 8, 12, 16, 20 24 holes/m are established. The maximum equivalent stress of the casing under different hole densities is obtained according to the equivalent stress cloud diagram, and the curves of the influence of the hole density on the maximum equivalent stress and residual strength of the perforated casing are drawn (

The perforation phase angles are 45°, 60°, 90°, 120°, and 180°, and the model building method and curve drawing method are the same as those for the hole diameter and hole density. The influence of the phase angle on the strength of the perforated casing is shown in

Under the condition of a certain hole diameter or hole density, 40 sets of perforation schemes are designed to establish a three-dimensional model and calculate the maximum equivalent stress. According to the equivalent stress cloud diagram under different phase angle conditions, the maximum equivalent stress drawing curve is obtained (see

It can be seen from the calculation results that the influence of the phase angle on the strength of the casing will not change when the hole diameter or hole density is constant, the maximum equivalent stress of the casing is the smallest at a phase angle of 180°, the remaining strength of the casing is the highest, and the stress in each area of the casing under each scheme does not reach its material yield stress of 1040 MPa. Taking the perforation parameters of a diameter of 13 mm, 16 holes/m, and phase angle of 90° as an example, the stress cloud diagram shows that the stress concentration on the inner wall of the casing is the most serious (see

Since the minimum principal stress direction of this model is the x-axis direction, there will be holes in the direction of the minimum principal stress when the phase angle is 45° and 90°, resulting in a larger equivalent stress. There is no perforation with a small principal stress direction in the perforation direction when the phase angle is 60°, 120° and 180°, and perforations are perpendicular to the phase angle when the phase angle is 180°. Therefore, in principle, no matter what phase angle is used, the perforation direction toward the minimum principal stress direction should be avoided, which can greatly improve the stress concentration phenomenon at the perforation.

To further quantify the influence of various factors on the strength of the perforated casing and to judge the influence of each factor on the residual strength of the perforated casing, this paper chooses the method of combining orthogonal experiments and range analysis. An orthogonal test is an efficient and concise test method to explore optimal parameter combinations for multifactor tests. For most complicated tests with more than 3 parameters that need to be optimized, if a comprehensive test is carried out, the test scale is large, the implementation is difficult, and the analysis processes are very cumbersome [

Based on the aforementioned research and considering the influence of wall thickness on the strength of the casing, the factors of the orthogonal experiment are determined as the casing wall thickness, hole diameter, hole density, and phase angle. Four factors are selected for each factor. For the four-level experiment, the specific factor level table is shown in

Level | Experimental factors | ||||
---|---|---|---|---|---|

Hole diameter/mm | Hole density |
Phase/° | Casing wall thickness/mm | ||

1 | 7 | 12 | 60 | 13.72 | |

2 | 10 | 16 | 90 | 14.27 | |

3 | 13 | 20 | 120 | 15.11 | |

4 | 16 | 24 | 180 | 15.88 |

Serial number | A |
B |
C |
D |
E |
X |
---|---|---|---|---|---|---|

1 | 1 | 1 | 1 | 1 | 1 | 360.2 |

2 | 1 | 2 | 2 | 2 | 2 | 166.8 |

3 | 1 | 3 | 3 | 3 | 3 | 485.6 |

4 | 1 | 4 | 4 | 4 | 4 | 633.9 |

5 | 2 | 1 | 2 | 3 | 4 | 307.5 |

6 | 2 | 2 | 1 | 4 | 3 | 502.3 |

7 | 2 | 3 | 4 | 1 | 2 | 642.7 |

8 | 2 | 4 | 3 | 2 | 1 | 495.4 |

9 | 3 | 1 | 3 | 4 | 2 | 537.9 |

10 | 3 | 2 | 4 | 3 | 1 | 635.1 |

11 | 3 | 3 | 1 | 2 | 4 | 506.0 |

12 | 3 | 4 | 2 | 1 | 3 | 341.8 |

13 | 4 | 1 | 4 | 2 | 3 | 675.8 |

14 | 4 | 2 | 3 | 1 | 4 | 496.7 |

15 | 4 | 3 | 2 | 4 | 1 | 207.1 |

16 | 4 | 4 | 1 | 3 | 2 | 511.2 |

K_{1} |
1646.5 | 1881.4 | 1879.7 | 1841.4 | ||

K_{2} |
1947.9 | 1800.9 | 1023.2 | 1844.0 | ||

K_{3} |
2178.3 | 1841.4 | 2015.6 | 1939.4 | ||

K_{4} |
2311.1 | 1982.3 | 2587.5 | 1881.2 | ||

k_{1} |
411.6 | 470.4 | 469.9 | 460.4 | ||

k_{2} |
487.0 | 450.2 | 255.8 | 461.0 | ||

k_{3} |
544.6 | 460.4 | 503.9 | 484.9 | ||

k_{4} |
577.8 | 495.6 | 646.9 | 470.3 | ||

Range R | 166.2 | 45.4 | 391.1 | 24.5 | ||

Primary and secondary order | Phase angle > hole diameter > hole density > wall thickness | |||||

Optimal level | A_{4} |
B_{4} |
C_{4} |
D_{3} |
||

Optimal combination | A_{4}B_{4}C_{4}D_{3} |

This paper compares the magnitude of range R of different factors and judges the influence of each factor on the residual strength. From the calculation results, it can be seen that R_{C} > R_{A} > R_{B} > R_{D} > R_{E}, so the influence on residual strength in descending order is the phase angle, hole diameter, hole density, and wall thickness. At the same time, the higher the k value is, the better the level of a certain factor. The calculation results show that the initial hole diameter is 16 mm, the hole density is 24 holes/m, the phase angle is 180°, and the casing wall thickness is 15.11 mm under the experimental conditions. This is a combination of advantages. However, according to engineering experience, a phase angle of 180° is not widely used, which may cause problems such as insufficient formation opening and streamline concentration. It is recommended that the k value should be slightly lower than the phase angle of 180°, and a perforation phase angle of 120° is used more commonly. In the end, the preferred exit perforation scheme is a diameter of 16 mm, 24 holes/m, phase angle of 120°, and casing wall thickness of 15.11 mm. The final scheme calculates the residual strength to be 517.9 MPa, which can meet engineering requirements.

(1) To study the influence of perforation parameters on casing strength under the condition of a heterogeneous

(2) This paper analyzes the influence of perforation parameters on the casing strength, such as the hole diameter, hole density, and phase angle. The study found that under high and nonuniform ground stress and fracturing construction conditions, due to internal pressure, the increase in the large hole diameter and hole density will not greatly reduce the strength of the casing, and the perforation phase angle is the main factor affecting the strength of the casing. It is recommended that the direction of perforation toward the direction of the minimum principal stress should be avoided to prevent stress concentration at the perforation.

(3) This paper quantitatively analyzes the influence of each perforation parameter and casing wall thickness on the residual strength through orthogonal experiments and obtains the phase angle, hole diameter, hole density, and casing wall thickness in descending order of influence. Combined with range analysis, the final optimal plan is a hole diameter of 16 mm, hole density of 24 holes/m, phase angle of 120°, and casing wall thickness of 15.11 mm. The residual strength of the casing under this plan is calculated as 522.1 MPa, which can meet engineering requirements.

Thanks are due to the supported by Engineering Technology Research Institute of Petro China Xinjiang Oilfield Company and China University of Petroleum (East China).