The paper presents a model for predicting and analysing the behaviour of sucker rod pumping installations in the inclined wells. This model incorporates the dynamics of the curved sucker rod string, and the friction forces resulting from the contact of the rod with the tubing through a system of two partial differential equations. This system of equations is solved by the finite difference technique. The model predicts the polished rod and pump dynamometer cards and incorporates the effects pump dynamometer cards and incorporates the effects of fluid inertia and viscosity. The model is capable of simulating a wide variety of pump conditions, and geometry of the rod string. The examples and comparisons with the results obtained using a program for vertical wells to analyse the deviated well are presented. The information predicted by the model is useful in the design and predicted by the model is useful in the design and operation of sucker rod pumping installations.
Rod pumping is still the most widely used means for artificial lift in oil wells. At present many wells are designed as deviated wells with the setting angle reaching 60 deg. The dynamic behaviour of the sucker rod string in a deviated well is different from that in a vertical well due to several reasons. One of them is the friction between the rod and the tubing. The other is the curvature of the rod string. The curvature causes the lateral displacements of the rods. It also couples longitudinal vibrations with transverse vibrations. Due to the axial forces in the rods (particularly compressive forces at the bottom of the well which can produce the buckling of the rod) the problem is nonlinear and thus difficult to solve.
The application of powerful personal computers makes possible an effective numerical representation of the rod pumping system even for more complex problems such as the behaviour of the sucker rod problems such as the behaviour of the sucker rod string in the deviated well. An accurate prediction of the performance of the sucker rod string requires careful modelling of the dynamic behaviour of the rod which has to satisfy the dynamic boundary conditions at the polished rod and at the pump. polished rod and at the pump. Let us consider a deviated rod in the curved tubing in a vertical plane (x,z) (Fig. 1). If we assume that the rod is supported by the tubing only at the points where couplings are installed, then the reaction forces between the tubing and the rod are the concentrated forces. If the rod lies on the tubing, the reaction forces are distributed along the length of the rod. The radius of the curvature of the tubing is a function of the depth, and is defined as R. Considering the equations of equilibrium of the forces acting on the element of the rod, ds, in the axial direction and the perpendicular direction to the rod, we obtain the following perpendicular direction to the rod, we obtain the following system of differential equations.