Predicting multi-tapered sucker-rod pumping systems with the analytical solution

Abstract

The sucker-rod string is an important part of the sucker-rod pumping system. Many methods have been employed to solve the one-dimensional wave equation that describes the longitudinal vibration of the sucker-rod string. The analytical solution with the separation of variables method is an important method. However, the analytical solution is mainly applied for diagnostic analysis and is difficultly applied for predictive analysis. As a result, the characteristics of predictive analysis for multi-tapered sucker-rod string with the analytical solution are misunderstood.In this paper, the analytical solution easily applied for predicting the behaviors of multi-tapered sucker-rod pumping systems is proposed after the general analytical solution of the separation variable method is expressed with a matrix. Based on the same parameters and boundary condition, the prediction results between the analytical solution and the finite differential solution, which is extensively applied to solve the wave equation, are compared; the parameter sensitivity in the predictive analysis of the sampling point number, Fourier coefficient number and viscous damping factor value are explored; and the prediction characteristics of different tapered rod string are simulated.The simulation results show that the polished rod card predicted by the analytical solution has a smooth effect and that the sharp corners of the pump card can be precisely described compared with the finite difference method in the predicate analyses. With an increase in the stage number from 1 to 4 of the sucker-rod string, the program execution time ratio between the finite differential solution and the analytical solution is exponentially increasing from 4.7 to 10.5. The optimal Fourier coefficients number is half of the sample point number; otherwise, there will exist the Gibbs phenomenon in the pump load approximated by the truncated Fourier series. For the same viscous damping factor, the prediction results of the two methods differ, and the analytical solution is insensitive to the viscous damping factor. With an increase in the stage number of the sucker-rod string for the same total length, the polished rod load decreases while the pump stroke may decrease.Eventually, the predicted polished rod card calculated according to the field parameters shows a satisfactory correspondence with the measured card, indicating that the model presented in this paper is feasible.