A specific model is elaborated for stick-slip and bit-bounce vibrations, which are dangerous dynamic phenomena typically encountered in the context of rotary drilling applications. Such a model takes into account two coupled degrees of freedom of drill-string vibrations. Moreover, it assumes a state-dependent time delay and a viscous damping for both the axial and torsional vibrations and relies on a sawtooth function to account for the cutting force fluctuation. In the frame of this theoretical approach, the influence of rock brittleness on the stability of the drill string is calculated via direct integration of the model equations. The results show that the rock brittleness has a great influence on the rotational speed and bit depth.

During drilling, the vibration of drill string in borehole mainly includes axial vibration, torsional vibration, and lateral vibration. The axial vibration is the bit-bounce, torsional vibration shows stick-slip vibration, lateral vibration is the collision between the bit and wellbore [

The contact force between bit and rock is consist of cutting force and friction force. One of the factors affecting stick-slip vibration of drill string is the velocity nonlinearity of contact force [

Since rock brittleness has significant influence on the fluctuation of torque and the stability of drill string, saw teeth wave function is introduced into cutting force function to simulate rock brittleness. The influence of rock brittleness on the stability of drill string system is studied numerically.

_{t}. The rotation speed of the top of drill string is assumed to be uniform rotation speed Ω_{0}, and the rotation speed of the bit is represented by Ω. Torque T is generated by the tangential component of the bit-rock contact force acting on the moment of inertia I. C_{a,} C_{t} are the axial and torsional damping of the drill string, respectively.

The model ignores the axial yield of drill string and simplifies the BHA and drill string mass into lumped mass _{a}; the displacement at the top of the drill string is assumed to be uniform velocity _{0}, the displacement of bit is

where:

where,

where

Therefore, the instantaneous cutting depth of a single blade (as shown in

where

The delay time

The total depth of n blades per revolution is:

When the bit rotates at a uniform speed, the speed is a constant value

Therefore, the dynamic differential equation of the drill string is shown as follows:

where:

_{0} represents the effective WOB and is the negative of the bit-rock contact force, which is a drilling control parameter, and bit-rock contact force is negative of each other.

It should be noted that _{0} and _{0} are independent parameters. _{0} can be expressed as follows:

where H(.) is Heaviside function,

sgn(.) is sgn function,

Since _{c} in torsional dynamics model _{c} in axial dynamics model

Richard et al.’s research [

For the convenience of numerical calculation, the following assumptions will be made in this paper:

①Ignore the clearance between adjacent cutters on the same blade.

②The arrangement of teeth of different blades is exactly the same.

③PDC teeth with the same distance from the axis of the bit belong to the same row. The PDC teeth closest to the axis of the bit are the first row, and the second row and the number as increases the distance.

④The adjacent cutting teeth of the same blade do not affect each other in the cutting process.

⑤The fluctuation of cutting force is sawtooth.

Based on the above assumptions, the cutting torque _{ck} and cutting WOB _{ck} are generated by the

In the process of PDC tooth cutting rock, the bulk cracking caused by rock brittleness is the main factor of its cutting force fluctuation. To simulate the fluctuation of cutting force [_{ck} and cutting WOB _{ck} are multiplied by a sawtooth function _{ck}. The torque _{ck} and _{ck} generated by the raw n tooth are expressed as follows:

According to references [

where,

The wave function of cutting force of the row

where:

_{1} is the bit rotation angle when the fluctuation of cutting force begins, and

There are many ways to solve the vibration system. In this paper, the direct integration method is used to solve the equation from the physical equation, without solving the inherent characteristics of the equation, and without decoupling the equation. In general, the direct integration method firstly discretizes the time with equal interval method, and only satisfies the motion differential equation at time points after discretization, and there is no requirement between time points. Secondly, the first discrete point is integrated according to the discrete incremental equilibrium equation in the time interval, and then the following points are integrated step by step. In the process of integration, different calculation methods can be obtained according to different assumptions. These methods have different convergence and stability. The commonly used methods mainly include linear acceleration method, Wilson-θ method, etc. In this paper, linear acceleration method is adopted.

Parameters required for calculation are shown in

Parameter | Symbol | Value | Unit |
---|---|---|---|

Axial stiffness of drill string | K_{a} |
7.0 * 10^{5} |
N/m |

Mass of drill string | M_{dp} |
2.8 * 10^{4} |
Kg |

Torsional stiffness of drill string | K_{t} |
940 | Nm/rad |

Moment of inertia of drill string | I_{dp} |
97 | Kg.m^{2} |

Mass of the BHA | M_{bha} |
2.5 * 10^{4} |
Kg |

Moment of inertia of BHA | I_{bha} |
83 | Kg.m^{2} |

Radius of the bit | a | 0.108 | m |

Length of wear surface | l | 0.0012 | m |

Rock strength | ε | 60 | MPa |

Contact pressure | σ | 60 | MPa |

The coefficient of friction | μ | 0.6 | |

Tool skew factor | ξ | 0.6 | |

Axial damping coefficient | ξ | 0.1 | |

Torsional damping coefficient | κ | 0.1 | |

Axial torsional frequency ratio | β | 1.58 | |

Blade number | n | 4 |

When

There is a process of entering and exiting stick-slip vibration in the process of numerical calculation. Assume the current time step is _{i}, the axial velocity of bit _{i}.

Enter torsional stick:

If _{i} = _{i-1}, the bit enters torsional stick. Then _{i} = 0, _{i} = 0.

Exit torsional stick:

The model will automatically exit torsional stick.

The comparison of calculation results between the original model and the improved model is shown in

Reference [ |
Current paper | |||
---|---|---|---|---|

Parameter | ν | ω_{0} |
_{V0} |
Ω_{0} |

Unit | – | – | m/s | rad/s |

Parameter 1 |
0.6416 | 7.336 | 0.00063 | 20.95 |

1.401 | 9.17 | 0.00137 | 26.18 | |

1.099 | 2.934 | 0.001075 | 8.377 |

Set number | Rotation speed (rad/min) | Cutting depth (mm) | ||
---|---|---|---|---|

Reference [ |
Current paper | Reference [ |
Current paper | |

Parameter 1 | 0.50 | 0.78 | 0.0036 | 0.012 |

Parameter 2 | 58 | 78 | 7 | 49 |

Parameter 3 | 33 | 19 | 75 | 4.1 |

In this paper, a lot of numerical simulation is carried out based on the improved model, and the parameter boundary of stationary drilling in _{0}

The method in this paper is verified by the experimental equipment in reference [_{0} = 6 and _{0}_{0} = 0.98 mm/s, Ω_{0} = 17.13 rad/s. The experimental results are shown in

(1) The analysis of the calculated results show that brittleness leads to the increase of the amplitude of rotation velocity and cutting depth of parameter 1 and parameter 2; Brittleness results in reduced fluctuation of rotation speed and cutting depth of parameter 3.

(2) The brittleness of rock affects not only the fluctuation amplitude of rotation speed and cutting depth, but even the motion state of the bit. The parameter 2 is stick-slip only in the original model, while stick-slip and bit bouncing are shown in the improved model.

(3) The improved model’s stationary drilling parameters are zonal, and it is much more different from the original model and more like the field results.

Rock brittleness is an important factor affecting the motion state of drill bit. It can improve the accuracy of model calculation and provide a reference for the selection of drilling parameters.