An Investigation of Analytical and Numerical Sucker Rod Pumping Mathematical Models

Abstract

SPE Members Abstract This paper details a study made of the analytical (diagnostic) and numerical (predictive) models commonly used today in the analysis and design of sucker rod pumping systems. The results should help the user of this technology in selecting appropriate values for the several parameters used in applying the models. Introduction The walking beam sucker rod pumping system is one of the oldest mechanical systems known to man, having been used by the Chinese at least 3000 years ago. The standard analysis and design procedure for these systems used by the industry for the past 20 to 25 years was completed by the Midwest Research Institute in 1962. The Institute, under contract to Sucker Rod Pumping Research, Inc., developed a method for computing forces and displacements using an analog computer simulation. This method, called the API RP11L method, has been a powerful engineering tool but has several limitations. These limitations include a simplified pumping unit geometry and polished rod motion, low slip prime movers and full pump fillage. Pioneering work by Gibbs in the 1960s resulted in the development of two mathematical models which form the basis for most of the technology today. These two models, which include solutions to the damped wave equation. are a diagnostic analytical model and a predictive finite difference model. The diagnostic analytical model is a classical Fourier Series solution of the damped wave equation. Stated simply, the input to this model is the surface dynamometer card, and the output is the pump dynamometer card. The primary use of this model is to diagnose downhole pump problems. The predictive finite difference model is a solution to the same equation but with a different statement of the boundary conditions. The input to this model is the surface polished rod motion and load at the pump, and the output is load at the surface and motion at the pump. The primary use of this model, when used together with the kinematic description of the surface unit and characteristics of the prime mover, is to predict loads at any point in the system so that the suitability of a particular design may be verified. The purpose of this paper is to present a discussion of the effect of the various parameters on the solution of the problem by these two methods. This should aid the user in selecting appropriate values of these parameters for use in both models. THE ROD PUMPING SYSTEM The sucker rod pumping system, represented mathematically as a boundary value problem, can be divided into three sections for analysis: the surface equipment, the rod string and the pump. Complete analysis involves a study of the interaction of these three sections. The motion described by the surface unit has been analyzed by Gray and more recently by Svinos. In this study, the method described by Svinos was used to solve the four-bar linkage problem. Gibbs and others have considered the effect to prime mover speed variations on the problem; we only considered constant prime mover speed in this study. P. 405^