# Eccentric Wear Mechanism and Centralizer Layout Design in 3D Curved Wellbores
## 1 Introduction  
Deep oil reservoirs, low permeability reservoirs, and high water cut reservoirs are deeply developed, resulting in a continuous increase in the number of deep wells, highly deviated wells, horizontal wells, and cluster wells. Concurrently, as the oil field enters the middle and later stages of production, the water cut increases year by year, and the injection medium becomes more complex. Under complex well conditions, the problems of eccentric wear of SRS and shorter pump inspection cycles are becoming more frequent. In 2018, China National Petroleum Corporation had 235,000 oil wells; the proportion of wells less than 300 days old is 25%, and the proportion of wells that exist with failure of rod, tube, and the pump is 83%. Different kinds of failure of rod, tube, and pump are shown in Figure 1. All kinds of failure of rod and tube belong to the dynamics problem of rod string in a wellbore, which is the key for optimal design and fault diagnosis of a sucker rod pumping system. As the power transmission and bearing mechanism of oil production machinery, SRS is the core equipment of the sucker rod pump. The eccentric wear andfracture of SRS impact the safety and economy of oil wells. Therefore, the study of sucker rod string mechanics has received wide attention. The semi-empirical formula method was used to predict the load of the polished rod before 1950, but its accuracy is not very high. In 1963, [PERSON] simplified the sucker rod as an elastomer and first established the longitudinal vibration equation to predict the axial load and displacement on any section of the sucker rod [1]. The mechanical model was very successful and has been widely used in oil wells. In 1990, [PERSON] used the concept of equivalent stiffness and equivalent density to improve the longitudinal vibration equation of rod string from single-stage sucker rod string to multi-stage sucker rod string [2]. But the error of the computational model was large. [PERSON] aimed for further improvement in the [PERSON] model [3]; the Taylor series method was used to deal with the continuous conditions at the interface between adjacent rod strings, which reduced the error of calculation. In 2021, [PERSON] used the separated variable method to solve the longitudinal vibration equation of the multistage rod string [4]. [PERSON] established a longitudinal vibration model for sucker rods based on a simplified thermo-solid coupled pattern and the finite element method for solving [5]. In 1983, [PERSON] established the sucker rod and liquid longitudinal vibration coupled equation [6]. In 1995, [PERSON], based on the coupled equation proposed by [PERSON] [7], supposed that the liquid column is a single-phase incompressible liquid and developed the MacCor-mack explicit numerical algorithm. The calculated results were in good agreement with the measured results [8]. Aiming at the engineering problem that the casing in the oil well is not anchored, [PERSON] established the sucker rod [9], tube, and liquid longitudinal vibration mechanics model in 1988. In 1997, [PERSON] found the results from the sucker rod [10], tube, and the liquid three-dimensional vibration mechanics model were close to the measured results, which proves that the model was suitable for SRS mechanics of vertical wells. In 2016, [PERSON] successfully applied the sucker rod [11], tube, and liquid vibration equation to directional wells by using the implicit difference method at the continuous position of the sucker rod and the variable step method at couplings. In 2020, [PERSON] successfully solved the coupled model between the high-slip motor and pumping unit according to the external characteristic function of the motor [12]. However, the above longitudinal vibration models are difficult to reveal the mechanical essence of sucker rod and tube eccentric wear, thread off of sucker rod, and other relevant problems because of having the limitations.