The mechanical model of transverse vibration of sucker rod string (SRS) in directional well is simplified to the transverse vibration model of longitudinal and transverse curved beam with initial bending under borehole constraints. In this paper, besides considering the excitation of alternating axial load on the transverse vibration of SRS, it is proposed for the first time that curved borehole is also the main excitation for the transverse vibration when the SRS moves reciprocating axially in the borehole. Based on the elastic body vibration theory, the transverse vibration mathematical model of SRS with initial bending under borehole constraints is established. In this model, the curved borehole excitation caused by the axial motion and the alternating axial load excitation is considered. Besides, the elastic collision theory is applied to describe the constraint of tube on the SRS transverse vibration in this model. Then the fourth-order Runge–Kutta method is used to calculate the transverse vibration of SRS in directional wells. The simulation results show the following: (1) The simulation results of the three simulation models in this paper are different. The results indicate that the curved borehole excitation caused by the axial motion and the alternating axial load excitation is the main excitation for the SRS transverse vibration. (2) In directional wells, the rod and tube contact along the well depth, and the dangerous sections locate at the deviation section of the borehole and the compression section of the rod. On the whole, the contact force between rod and tube in deviation section of borehole is larger. The transverse vibration of the compression section of the rod is the most violent.

Pumping unit lifting is the main way of artificial lifting [

The bending deformation of SRS in tube is similar to that of drill string in borehole. It can be simplified as a mechanical problem of bending deformation of slender rod string in borehole. At present, there are many researches on the dynamic behavior of drill string [

Both the drill string and the SRS have axial motion relative to the borehole. Especially in pumping wells, there is axial reciprocating motion of SRS with large displacement and velocity. As the axial position of the borehole bump acting on the SRS changes with time, the curved borehole trajectory is a transverse vibration excitation for the axially moving SRS. This problem is similar to the dynamic problem of up-down vibration of vehicles under the excitation of rough road surface. At present, the research on transverse vibration of SRS excited by borehole trajectory has not been reported in literature. In this paper, considering the initial bending of SRS caused by the bending of borehole trajectory and considering the excitation of borehole trajectory to the transverse vibration of SRS, a transverse vibration simulation model of SRS in tube is established based on the theory of elastic body vibration.

This paper is organized as follows: In Section

Figure

Models for describing borehole trajectory and SRS spatial shape.

The borehole trajectory is a spatial curve, which can be described in rectangular coordinate system. The

Any point A on the borehole trajectory can be expressed as

According to the test data of oil well, the inclination angle _{A}, principal normal vector _{A}, binormal vector _{A}, curvature _{A}, and torsion _{A} at point A on borehole trajectory axis can be calculated.

The formulas for calculating tangent vector, principal normal vector, and binormal vector are

and the formulas for calculating borehole trajectory axis curvature and torsion are

Based on the description of borehole trajectory, the new coordinate system is established at the origin of coordinate A point on borehole trajectory, where the direction of _{C}(

According to _{C}(_{C}(_{C}(_{C}(_{C}(

The formulas for calculating tangent vector, principal normal vector, and binormal vector of point C on rod string axis can be written as_{s} is the first-order partial derivative of _{ss} is the second-order partial derivative of

The formulas for calculating SRS axis curvature and torsion can be expressed as follows:_{sss} is the third-order partial derivative of _{A})_{s} is the first-order partial derivative of _{A} to s.

Figure

In order to facilitate the research and highlight the focus of this paper, the following simplifications and assumptions are made: (1) The rod string is an elastic body, and the rod string moves at the same speed as the suspension point along the borehole trajectory. (2) The rod string is a homogeneous single-stage rod. (3) The influence of longitudinal and torsional vibration of SRS on transverse vibration is not considered. The transverse vibration of SRS is studied only. (4) Do not consider the warping of the cross section of the SRS. The cross section perpendicular to the axis of the SRS before deformation is still perpendicular to the axis of the SRS after deformation. (5) The shear stress of each cross section of the SRS is always in the normal plane of the SRS axis. (6) The centralized axial load acting on the coupling is simplified as the axially distributed load. (7) The position of any point on the axis of SRS is represented by the curvilinear coordinates along the borehole axis.

Under the above assumptions, the mechanical model of the SRS transverse vibration shown in Figure _{ub}(_{ub}(_{S}(

Mechanical model of transverse vibration of SRS. (a) Mechanical model. (b) Element model force. (c) Schematic diagram of borehole trajectory excitation.

Figure

Figure

The calculation model of the suspension displacement _{S}(_{ub}(_{ub}(_{S}(_{ub}(_{ub}(

Figure _{eτ}(_{en}(_{eb}(_{eτ}(_{en}(_{eb}(_{eτ})_{s} is the first-order partial derivative of _{eτ} to _{en})_{s} is the first-order partial derivative of _{en} to _{eb})_{s} is the first-order partial derivative of _{eb} to

As for the restriction of tube, the transverse vibration of rod string has little effect on the longitudinal vibration, but the longitudinal motion of the rod string will significantly affect its transverse vibration. Therefore, in this paper, the longitudinal vibration of rod string is solved separately through the model established in [

According to the force diagram of the element, the moment balance equation is established. Then the relationship between forces can be obtained:

It is simplified to

Introducing formulae (_{eb} and _{en} can be obtained:

Formula (_{ssss} is the fourth-order partial derivative of _{A})_{s} is the first-order partial derivative of _{A} to _{A})_{ss} is the second-order partial derivative of _{A} to

Distributed external forces acting on rod string can be expressed by the following formula:_{l} is the liquid density in tubing string;

The relationship between the static position curvilinear coordinates and the dynamic position curvilinear coordinates of axially moving SRS is as follows:

In order to analyze the effect of curved borehole trajectory excitation on the transverse vibration of axially moving SRS, the comparative simulation model 2 is obtained by ignoring the axial movement of SRS:

Formulae (

SRS is constrained by transverse displacement and angular displacement at the wellhead, so the suspension to the wellhead can be simplified as a sliding fixed constraint. The transverse displacement and angular displacement of the bottom of the rod string are constrained by the pump barrel, so the bottom of the rod string can be simplified as sliding fixed constraints too. The boundary conditions of SRS can be expressed as

Assuming that the SRS axis coincides with the borehole trajectory axis in the initial state, the suspension point is located at the top dead center. The initial conditions can be expressed as

The rod-tube contact force consists of the impact force _{s} caused by the rod-tube collision and the transverse force _{n} caused by the bending and axial tension of the rod string.

The rod-rube collision condition is as follows:_{r} is the radius of sucker rod; _{c} is the radius of coupling; when

The position of the corresponding node after collision is

The velocity of the corresponding nodes after collision is_{s} is the collision recovery coefficient, and its value depends on the material of the collision body [

As it is difficult to determine the instantaneous value of the collision force, this paper describes the collision force by the change of the impulse. Assuming that the rod-tube collision is completed in _{s} [

The formula for calculating the transverse force of rod string is

The rod-tube contact force can be expressed as

The simulation model of the transverse vibration of SRS is a fourth-order partial differential equation with variable coefficients, so the analytical solution of the equation cannot be obtained. Therefore, the fourth-order Runge–Kutta method is used to solve the problem. The solution forms are as follows:

The initial condition of the formula is

Combining with the borehole constraints in Section

Computational flow chart.

Basic simulation parameters are as follows: rod column diameter is 22 mm; coupling diameter is 46 mm; rod string length is 1305 m; rod string density is 7800 kg/m^{3}; elastic modulus of rod string is 210 GPa; inner diameter of tube is 62 mm; stroke length of pumping unit is 6 m; stroke frequency of pumping unit is 6 min^{−1}. The borehole trajectory, suspension point motion law, and axial force of SRS are shown in Figure

Curves of well trajectory, suspension displacement, and axial stress. (a) Borehole trajectory. (b) Suspension displacement. (c) Axial force of node.

Based on the above three simulation models, the transverse vibration simulation system of SRS in directional well is developed. The transverse vibration law of SRS and the contact force between rod and tube are obtained. Figure

Simulation results of model 1. (a) Distance between the node position and the borehole axis. (b) Contact force between rod and tube.

Statistical table of rod-tube contact force characteristic values in one cycle.

Model | Average contact force (N/m) | Position of max contact force (m) | Time of max contact force (s) | The max contact force (N/m) | Standard deviation of difference with model 1 (N/m) |
---|---|---|---|---|---|

Model 1 | 14.91 | 405 | 0.807s | 1361.1 | |

Model 2 | 14.99 | 396 | 0.798s | 1370.2 | 17.72 |

Model 3 | 14.76 | 386 | 0.762s | 536.1 | 56.10 |

Figure

The coupling node simulation results at 3 s. (a) Distance between node position and borehole axis. (b) Contact force between rod and tube.

The rod node simulation results at 3 s. (a) Distance between node position and borehole axis. (b) Contact force between rod and tube.

The coupling node simulation results at 9 s. (a) Distance between node position and borehole axis. (b) Contact force between the rod and tube.

The rod node simulation results at 9 s. (a) Distance between node position and borehole axis. (b) Contact force between the rod and tube.

Statistical table of the coupling nodes and sucker rods, which contact with tube string at 3 s and 9 s.

Time (s) | Type | Model | The coupling nodes/sucker rods contact with tube string |
---|---|---|---|

3 | Coupling node | Model 1 | 6, 8∼11, 14∼22, 24∼34, 36, 37, 41∼47, 49∼92, 94∼96, 99∼105, 107, 109∼126, 128, 133, 136, 139∼143 |

Model 2 | 12, 14, 15, 17∼22, 24, 27, 29∼31, 33∼71, 75∼77, 80∼101, 103, 104, 106∼123, 125∼128, 134∼136, 141 | ||

Model 3 | 2∼8, 10∼16, 18∼29, 32, 33, 35∼37, 39∼43, 46∼59, 61∼64, 68∼74, 76∼99, 101∼104, 106∼115, 117∼125, 127∼136, 138∼144 | ||

Sucker rod | Model 1 | 42∼47, 49, 51∼55, 70∼75, 113∼119 | |

Model 2 | 42∼48, 50∼55, 70∼74, 113∼119 | ||

Model 3 | 36∼129, 134∼137, 139∼145 | ||

9 | Coupling node | Model 1 | 11, 12, 17∼21, 23∼28, 31, 33, 35∼42, 44∼56, 58∼66, 68, 72∼74, 76∼78, 80, 81, 83∼106, 108∼111, 113∼115, 117∼124, 126∼131, 133, 134, 136, 137, 139∼144 |

Model 2 | 14, 17, 19, 20, 22, 36∼40, 42∼44, 47∼49, 52∼69, 71, 73∼118, 120∼128, 131∼134, 136∼138, 140∼144 | ||

Model 3 | 2∼5, 8, 12∼19, 21∼24, 26∼29, 34∼37, 39∼47, 49∼56, 58∼74, 76∼102, 104∼107, 109∼129, 131∼141, 143∼144 | ||

Sucker rod | Model 1 | 42∼48, 51∼55, 68∼75, 77, 111, 113, 115, 116, 118, 121∼145 | |

Model 2 | 42∼47, 50∼55, 57, 69∼74, 92, 106, 111, 114∼119, 121, 122, 124∼127, 129∼145 | ||

Model 3 | 37∼145 |

Statistical table of rod-tube contact force characteristic values at 3 s and 9 s.

Time (s) | Node type | Model | Average contact force (N/m) | Position of max contact force (m) | The max contact force (N/m) | Standard deviation of difference with model 1 (N/m) |
---|---|---|---|---|---|---|

3 | Coupling node | Model 1 | 319.13 | 378 | 1126.0 | |

Model 2 | 321.10 | 405 | 1129.4 | 74.31 | ||

Model 3 | 141.61 | 387 | 472.6 | 223.83 | ||

Rod node | Model 1 | 1.17 | 500.5 | 92.79 | ||

Model 2 | 1.16 | 391.5 | 94.25 | 2.25 | ||

Model 3 | 11.77 | 411.5 | 122.73 | 23.20 | ||

9 | Coupling node | Model 1 | 152.70 | 396 | 560.80 | |

Model 2 | 158.06 | 396 | 573.03 | 48.22 | ||

Model 3 | 95.80 | 414 | 293.04 | 98.83 | ||

Rod node | Model 1 | 3.02 | 1294.5 | 196.75 | ||

Model 2 | 3.06 | 1297.5 | 217.97 | 12.86 | ||

Model 3 | 5.49 | 375 | 69.61 | 16.67 |

Figure

Through the observation of Figure

Figure

Figure

From the above observations, the following can be found: (1) At the same time, the contact state and the distribution law of the contact force obtained by the three models are different. (2) The simulation results of the three models show that the transverse vibrations of SRS obtained by model 1 and model 2 are more obvious. (3) Most of the coupling nodes on the SRS contact with the tube in the whole well, and the contact force is much larger than that of the rod node.

In this section, the simulation results of the two coupling nodes at 531 m and 1080 m are compared; and the simulation results of the two rod nodes at 500 m and 1283 m are compared. Combining with Figure

Simulation results of the coupling node at 531 m. (a) Node position. (b) Contact force between the rod and tube.

Simulation results of the coupling node at 1080 m. (a) Node position. (b) Contact force between the rod and tube.

Simulation results of the rod node at 500 m. (a) Node position. (b) Contact force between the rod and tube.

Simulation results of the rod node at 1283 m. (a) Node position. (b) Contact force between the rod and tube.

Statistical table of rod-tube contact force characteristic values of specific nodes in one cycle.

Node type | Position (m) | Model | Average contact force (N/m) | Time of max contact force (s) | The max contact force (N/m) | Standard deviation of difference with model 1 (N/m) |
---|---|---|---|---|---|---|

Coupling node | 531 | Model 1 | 459.97 | 0.879 | 826.0 | |

Model 2 | 455.65 | 0.750 | 813.2 | 51.80 | ||

Model 3 | 195.99 | 1.022 | 339.0 | 131.50 | ||

1080 | Model 1 | 221.40 | 1.290 | 575.8 | ||

Model 2 | 199.49 | 1.283 | 560.1 | 103.55 | ||

Model 3 | 119.69 | 1.152 | 202.8 | 105.75 | ||

Rod node | 500 | Model 1 | 0.13 | 9.625 | 38.7 | |

Model 2 | 0.51 | 0.136 | 37.4 | 3.54 | ||

Model 3 | 34.50 | 0.757 | 54.0 | 13.09 | ||

1283 | Model 1 | 15.48 | 8.236 | 138.0 | ||

Model 2 | 12.38 | 7.250 | 205.4 | 18.96 | ||

Model 3 | 7.25 | 0.170 | 27.9 | 21.84 |

According to Figures

By observing Figures

From the above observations, the following can be found: (1) In the same period, the instantaneous contact states and the contact forces obtained by the three models are different. (2) Compared with the tension section, the transverse vibration is more obvious in the compression section, and the contact force is larger. (3) Compared with the rod node, the rod-tube contact force of coupling node is larger.

Through the analysis of Sections

According to the above researches, the following conclusions can be obtained:

The transverse vibration of SRS is one of the main factors leading to eccentric wear between SRS and tube string, so the research on SRS transverse vibration has always been the focus in oil field. In this paper, based on the previous studies (model 3), the transverse vibration simulation model 2 of SRS is established considering the geometric stiffness excitation caused by the alternating axial force. On the basis of model 2, it is proposed that the curved borehole trajectory is a main excitation of the SRS transverse vibration, and a more comprehensive transverse vibration simulation model 1 of SRS is established. Then the fourth-order Runge–Kutta method is used to simulate the transverse vibration of SRS in directional wells. The simulation results show that the dangerous section of eccentric wear between SRS and tube string occurs in the deviation section of oil well and the compression section of SRS. The rod-tube contact force is relatively larger in the deviation section of oil well, and the transverse vibration of SRS is relatively obvious in the compressed section of SRS.

Through the above quantitative and qualitative analysis, it is found that although the average contact forces between SRS and tube string obtained by the three models are similar in the whole cycle, the change rules of the instantaneous contact state and instantaneous contact force between SRS and tube string are different. It shows that the main excitation of SRS transverse vibration includes the geometric stiffness excitation caused by the alternating axial force and the curved borehole trajectory excitation caused by the SRS axial movement. So the transverse vibration simulation model 1 is more comprehensive.

The simulation of the instantaneous contact position and contact force between SRS and tube string is the basis of the prediction of the eccentric wear position and the wear life of SRS. According to the research results of this paper, it is found that the geometric stiffness excitation caused by the alternating axial force and the curved borehole trajectory excitation caused by the SRS axial movement will significantly affect the simulation results of the instantaneous contact position and contact force between SRS and tube string. Therefore, the transverse vibration simulation model 1 of SRS established in this paper is helpful to predict the eccentric wear position and the wear life of SRS more accurately, and it is of great significance and economic value to prevent eccentric wear of SRS.

The data used to support the findings of this study are included within the article.

The authors declare that there are no conflicts of interest regarding the publication of this paper.

This work was supported by the National Natural Science Foundation of China (51974276).