Equations are derived from first principles for predicting the behavior of sucker-rod pumping systems including the effects of rod and fluid dynamics, and kinematics of the surface pumping unit. Equations are also developed for both incompressible and slightly compressible fluid flow scenarios. The resulting composite rod and fluid dynamic model is solved using the MacCormack Explicit Numerical Scheme. Example problems used to validate this model are presented in a companion paper.
This paper presents the derivations of a new set of composite models for rod and fluid dynamics in sucker-rod pumped wells. The major components of a sucker-rod pumping well system are fully integrated into the model presented in this work. These components are: (1) the prime mover, (2) the speed (gear) reducer, (3) the surface pumping unit, (4) the sucker-rod string, (5) the well fluid column, (6) the subsurface sucker-rod-driven pump, and (7) the necessary strings of tubing and/or casing within which the sucker rods and the downhole pump operate and through which the fluid is pumped.
Several methods have been presented in the literature and are used to design a sucker-rod pumping well systems. Some of these methods include the semi-empirical and trial and error methods of the years prior to the 1940's. The semi-empirical methods were replaced by the analogue models of the 1950's, which following the availability of high speed digital computers, have been replaced by mathematical models that are solved by numerical methods. The development of predictive methods for sucker-rod pumping well systems has evolved considerably, but at the present time the difficulty of coupling all the integral parts (enumerated above) into one predictive model necessitated the neglect of some vital parts. The fluid motion which depends on the motion of the sucker-rod string and which is very important in the determination of the viscous damping effects on polished rod loads was not explicitly treated in the Gibbs and the Chacin models. Only in the last decade has a predictive model considered some aspects of the fluid dynamics in sucker-rod pumped wells. The models of References 8 - 10 accounted for the kinematics of the surface pumping unit through the incorporation of the work of Gray. Gray used the four-bar linkage theory to derive an equation for polished rod displacement as a function of crank angle. From Gray's equation one can derive expressions for polished rod velocity and acceleration as a function of crank cycle time or angular position.