Computing Gearbox Torque and Motor Loading For Beam Pumping Units With Consideration of Inertia Effects
Authors: Gibbs S.G. 
Published: December 2007

A method that considers inertia effects is presented for computing gearbox and motor loading on beam units. Previous methods neglected the inertia torques, which can be important when ultrahigh-slip prime movers are used. The method has practical application in determining the type of prime mover/beam unit combination to be used.

Introduction
During the past few years, ultrahigh-slip electric motors have been introduced for powering beam pumping units. These prime movers employ inertia effects to diminish equipment loading, particularly on the gearbox and the motor itself.
The torque factor method accepted in the API standards does not consider inertia effects and so should not be expected to give correct answers when ultrahigh-slip prime movers are used. To provide a suitable computation prime movers are used. To provide a suitable computation method, two additional terms have been added to the API technique - rotary and articulating inertia torques. To account for inertias, it is necessary to measure instantaneous motor speed, an item previously not required in conventional dynamometer analysis. Once motor speed variations are known, it is possible to account for torsional work and kinetic energy interchanges as the rotary components accelerate and decelerate. Also, the torsional effects related to accelerations of the articulating components, primarily the beam assembly, can be computed.
Two methods have been used to evaluate the various moments of inertia required in the analysis. The method most frequently used has been the direct calculation method, based on the engineering drawings of the unit components. The other method infers the moments of inertia by the manner in which the unit coasts to a stop after the motor switch is turned off.

Review of Torque Computation Methods
Two frequently used methods for inferring gearbox torque from a surface dynamometer card are the API torque factor method and an unnamed technique that is based on surface load range and unit stroke.
The API torque factor method is based on the following equation.
T = F (Q − Q) − M sin(0j +).........(1)
The net torque is the difference between the rod load torque and the counterbalance torque. Rod load torque, Fj(Q RL -Q SU), is the product of the torque factor and the polished rod load corrected for structural unbalance of the polished rod load corrected for structural unbalance of the unit. The counterbalance torque, M sin (0 j + beta), opposes the rod load torque with a sinusoidally varying effect with amplitude M.
Pumping-unit manufacturers usually publish a set of torque factors vs crank angle. To assist in relating crank angle to stroke position on the measured dynamometer card, corresponding stroke positions in nondimensional form are also published. Position zero refers to the bottom of the unit's stroke and position unity (1.0) denotes the top of the stroke. A sample set of torque factors and positions for a 120-in. unit equipped with a 456,000-in.-lb gearbox is shown in Table 1.
A dynamometer card for an example well is shown in Fig. 1. To illustrate the torque factor method, the net torque at a crank angle of 60 degrees is determined as follows. According to Table 1, the crank angle of 60 degrees produces a nondimensional, polished rod position of 0.304 on the upstroke and a torque factor of 61.08 in. The polished rod load at this position is 15,036 lb.
JPT
P. 1153