Commonly used methods at present for designing rod pumping unit presume vertical wells and are often applied to the design of deviated wells in field practice. This may be possible to cause substantial errors in the analysis of deviated wells because performance and load-deflection pattern of the rod in deviated wells are different from those in vertical wells. Due to borehole geometry, curved turbing makes the rod result in transverse displacements which is coupled by longitudinal displacements. In this case, load-deflection calculations become more complicated, and, in general, six components of stress resultant or internal forces may appear at any cross section along the rod. Besides, Coulomb friction between the rod and tubing and contact of the rod with tubing are also serious and difficult problems.
A new mathematical model based on 3D quasi-static analysis is presented and used to evaluate the sucker rod string in deviated wells. The model is illustrated as Fig. 1, which shows axial forces, Coulomb forces, side forces, shear forces, bending moments, touques, rod weight, and inertia forces applying to a rod element supported by two successive centralizers. Two types of deflection of the rod are considered. First, the rod is in contact with tubing along its whole length and deforms due to curved tubing. The solution for this case is much easier by using elastic deformation theory of beam. The cubic spine technique is also used to simulate borehole geometry through measurements in the deviation survey. Second, the rod is supported by tubing at certain points where centralizers are placed but it is also transversely loaded by its own weight. Therefore, lateral displacements of the rod are possible but are constrained by the wall of tubing. The problems for this case are non-linear and can be solved only by the iterative numerical techniques. The asympotic method for the solution of statically indeterminate continuous beams is used with combining iteraive method and constrained deflection condition, and finally the bending moments on the node of each element and rod/tubing side forces which may apply on the contact zones are determined approximately. By means of equilibrium equations of a rod element, all components of stress resultants at any depth are calculated recursively starting at their boundary value.
An efficient computer program is developed to design the sucker rod string of deviated pumping wells by trial-and-error method and to predict performance until a satisfactory configuration is obtained. Polished rod loads and stresses at any cross section by well depth increments are calculated together with evaluation of rod-tubing friction along the rod. In addition, it also suggests rod sizes and types as well as recommendation for the centralization of the rod. For example, Fig. 2 shows axial forces along the rod on the upstroke ans downstroke for an examined well. Corresponding friction forces between the rod and tubing are presented in Fig. 3. Table 1 presents comparisons of polished rod loads for 6 examined wells with field measurements to demonstrate the validity of the design method.