Mathematical Problems in Engineering

Volume 2018, Article ID 9205251, 7 pages

https://doi.org/10.1155/2018/9205251

## 3D-Dynamic Modelling and Performance Analysis of Service Behavior for Beam Pumping Unit

^{1}Machine Science and Engineering, Northeast Petroleum University, Daqing, China^{2}Research Institute of Oil Production Engineering of Daqing Oilfield Company Limited, China

Correspondence should be addressed to Feng Zi-Ming; moc.361@mzfnauyeux

Received 6 October 2017; Revised 10 April 2018; Accepted 16 May 2018; Published 12 July 2018

Academic Editor: Jian G. Zhou

Copyright © 2018 Feng Zi-Ming et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

In consideration of the rod, the tube, and the liquid column, a 3D dynamic model was established, which could be expressed as a set of partial differential equations. A measured torque curve and a calculated torque curve of Nan1-2-22 oil-well in Daqing oilfield were contrasted with each other which improved the rationality of this model. At last, we researched the influence of the stroke and the frequency of stroke on the displacement of rod, suspension velocity, suspension acceleration, polish rod load, and net torque of gearbox. This 3D dynamic model has a higher calculated accuracy and veracity and could be used to design and optimize the structure of rod pumping and the working process of pumping wells.

#### 1. Introduction

A fast and accurate calculation of polished rod load was the prerequisite for optimization design of type selection, the reasonable matching design, fault diagnosis, and intelligent control of the beam pumping well system.

The earliest rod pumping system design was semiempirical design methods, such as “API Specification for Pumping Units”. This method relied on the plunger diameter, the static fluid load, the rod size(s), the dynamic force, and the friction loads, which were all calculated by the semiempirical formulation. The semiempirical formulation had a lot of limitation for refined calculation. The first semiempirical modelling of rod pumping system was done by Kemelor [1]. Their efforts were then concentrated on the construction of an electrical analogue. Halderson [2] also established an analogue model. Many analogue models were built, but none of them completely and accurately solved the sucker-rod pump problem.

Norton [3] developed an analytical solution by an approximation method. This method was partially successful but was never widely applied. The next major development was by Snyder [4], who used a numerical technique to calculate the down hole force and displacement. This method was primarily for diagnosis of operating problem in existing wells. In 1963, Gibbs [5] presented a method that suitably generalized for a wider range of application. Many companies, such as Lufkin Foundry and Machine Company, made this method to be available as a commercial serviced by Sucker Rod Pump Analysis. It is a numerical technique programmed for solution on a digital computer.

Chacin and Purcupile [6] dispersed the rod string to be the spring mass system with damping, and they established the ordinary differential equations to describe the lumped mass element vibration law, and they established the system simulation model by use of the numerical integral method. In 1983, Doty and Schmidt [7] established a two-dimensional model of the coupled vibration of rod string and liquid column.

Lelia and Evans [8] had conducted further research about dynamic forecasting model of rod string and liquid column with coupling motion. They had carried on two hypotheses to the liquid column: one is that the liquid column was a single phase incompressible liquid, two is that the liquid column was a single-phase microcompressible, and it could be used difference method to calculate the numerical solution. In 1988, Yu [9] considered the fact of general unanchor of oil tubing in China; he comprehensively considered the sucker rod, liquid column, and the tubing string and established a 3D vibration model for prediction of polished rod load, but its limit condition for calculation was limiting its scope of application.

In 2010, Shardakov et al. [10] developed a new mathematical model of longitudinal vibrations of the sucker rod string and its numerical implementation. Its model can take into account strongly nonlinear forces resulting from pump valves and Coulomb’s friction. The numerical results indicated that it shows rather good agreement between the numerical and experimental results. In 2013, Jiang et al. [11] developed a coupling mathematical optimized model of the prime motor and the walking beam pumping units to analyze the dynamics performance. In 2015, Ferrigno et al. [12] proposed a calculation algorithm using a combination of the motor performance analysis and the calculations based on API 11E, allowing us to determine the real torque of units in real time from telemetry, eliminating the need to make on-site surveys of the counterweighting of the unit.

In 2016, Lao and Zhou [13] developed a new direct calculation method that directly calculates the fluid force acting on the curved cylindrical surface of the rod element. They used the pump dynamometer cards as boundary conditions and solved the wave equations for sucker-rod pumping with more accuracy than before. In 2016, considering the limitation of simulation model of the deep well and ultradeep well, Xing [14] developed an improved simulation model of sucker rod pumping system. Xing concluded that, comparing with the steel rod, carbon fiber, and fiberglass rod, the resonant phenomenon of single rod is more likely to happen for the wire rope sucker rod. However, when sucker rod is double stage rod, the resonant phenomenon of fiberglass rod is more likely to happen than other rods.

In this paper, considering the static and dynamic and friction forces of the rod, the tube, and the liquid column, we established a mathematical calculation model named 3D dynamic wave model which was made up of a set of partial differential equations. By programming this dynamic model, we researched the impact of stroke and frequency of stroke on suspension velocity, suspension acceleration, polished rod load, and net torque of gearbox of CYJY10-3-37HB rod pumping. At the same time, the calculated data by this dynamic model had good meeting with the measured data. It provided an optional theoretical support for the type-selection and optimization design, the reasonable matching design, the fault diagnosis, and the intelligent control of the oil pumping well system.

#### 2. Dynamics Model of Pumping Unit System

The polished rod load of pumping unit was the main basis factor for the optimum design and the swabbing technology design. The polished rod load was made up of the weight of the sucker rod and the liquid column, the inertia load, the load of vibration load, and the friction load. In order to calculate these loads accurately, the mechanical model of rod string was established, and it analyzed the influence factors, respectively. The suspended displacement, the suspended velocity, and the suspended acceleration were depicted in the literatures [15].

##### 2.1. Rod-Tube-Fluid 3D Coupling Vibration Model

In order to simplify the dynamic model of rod pump system, the following were assumed:(i)The beam pumping units are our research object, where rotate speed of gearbox crank is a constant variable.(ii)There are no gassy bubbles in the liquid column and the liquid density is a constant.(iii)The resistance of valve is ignored.(iv)The dynamic viscosity of liquid column , the back pressure of wellhead , and the intake pressure of pump are all constants.

###### 2.1.1. Dynamic Model of the Oil Rod

The rod element force diagram was as shown in Figure 1(a); we can see from the dynamic force equilibrium: By the Hooke’s law, the inner force relation of oil rod can be achieved: The following equation can be achieved by a partial derivative with time to the above equation: where is rod velocity, m/s; is inner force of rod, N; is rigidity of rod, N/m^{2}; is sectional area of rod, m^{2}; is density of rod, kg/m^{3}; is liquid density, kg/m^{3}; is resistance met with rod, N/m; is resistance acted on the liquid, N/m.