This paper presents the modelign of a sucker rod pumping system for both vertical and deviated wells. It focuses on the derivation and application of a new set of comprehensive equtions that incorporate both the dynamic rod motion as well as the fluid flow through the annulus between the rod and tubing. The mathematical model for a deviated well is developed using the principles of virtual work. partial differential equations, and their associate boundary conditions, have been derived to describe the dynamic performance of the system in great generality. The model incorporates all system components and most downhole pump inflow conditions. The effect of lateral rod deformation between the couplings is evaluated by introducing effective longitudinal stiffness and by the Garlakin method. The model for deviated wells incorporates not only viscous friction, but also Coulomb friction. The model' results are validated against actual measured data. Numerous examples are presented to demonstrate the application of the new technique for the design, analysis and optimization of sucker- rod pumped wells, specially deviated wells.
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