While the size of the four-bar linkage is the basis of kinematic performance analysis in a beam pumping unit, there is still a lack of effective and direct measurement of it. Since the motor input power and the polished rod position are commonly used production data, a size identification algorithm of the four-bar linkage based on the motor input power and the polished rod position is proposed in this paper. Firstly, the kinematic model of a beam pumping unit, the speed model of a motor, the initial value, and the state space model are established. Secondly, a Cubature Kalman filter with nonlinear constraints is designed and the size identification algorithm is worked out. Lastly, the model and the size identification algorithm are validated based on the simulated and measured data, and the characteristics of identification with different measurement covariance are studied. The results demonstrate that both the model and the algorithm are feasible. The maximum relative error between the identified and the real size increases from 0.23% to 9.37% when the covariance increases from 10⁻⁶ to 10⁻¹. With the measured covariance of the polished rod velocity, the maximum relative error is 7.09%. A comparison of several current identification methods demonstrates that the proposed algorithm is more accurate.

Fault dynamometer cards are the basis of the diagnosis technique for sucker rod pumping systems. Predicting fault cards with a pumping condition model is an economical and effective method. The usual model is described by a mixed function of the pump displacement and pump load, and it is difficult to use in the prediction method based on the analytical solution of the sucker rod string wave equation. In this paper, a normal pumping condition model described by a function of polished rod velocity is proposed. For the analytical solution of the sucker rod wave equation, an iterative prediction algorithm with pumping condition models is proposed, its convergence is analyzed, and then it is validated by classical finite difference method simulated cards and measured surface dynamometer cards. The results show that the proposed algorithm is accurate. The algorithm has a maximum relative error of 0.10% for the classical method simulated card area and 1.45% for the measured card area. The research of this paper provides an effective scheme for the design, prediction, and fault diagnosis of a sucker rod pumping system with an analytical solution.

Simulating surface conditions by solving the wave equation of a sucker-rod string is the theoretical basis of a sucker-rod pumping system. To overcome the shortcomings of the conventional finite difference method and analytical solution, this work describes a novel hybrid method that combines the analytical solution with the finite difference method. In this method, an analytical solution of the tapered rod wave equation with a recursive matrix form based on the Fourier series is proposed, a unified pumping condition model is established, a modified finite difference method is given, a hybrid strategy is established, and a convergence calculation method is proposed. Based on two different types of oil wells, the analytical solutions are verified by comparing different methods. The hybrid method is verified by using the finite difference method simulated data and measured oil data. The pumping speed sensitivity and convergence of the hybrid method are studied. The results show that the proposed analytical solution has high accuracy, with a maximum relative error relative to that of the classical finite difference method of 0.062%. The proposed hybrid method has a high simulation accuracy, with a maximum relative area error relative to that of the finite difference method of 0.09% and a maximum relative area error relative to measured data of 1.89%. Even at higher pumping speeds, the hybrid method still has accuracy. The hybrid method in this paper is convergent. The introduction of the finite difference method allows the hybrid method to more easily converge. The novelty of this work is that it combines the advantages of the finite difference method and the analytical solution, and it provides a convergence calculation method to provide guidance for its application. The hybrid method presented in this paper provides an alternative scheme for predicting the behavior of sucker-rod pumping systems and a new approach for solving wave equations with complex boundary conditions.