June 2011 SPE Journal 451 Asymptotic Model of the 3D Flow in a Progressing-Cavity Pump S.F.A. Andrade, Petrobras, Pontifícia Universidade Católica do Rio de Janeiro; and J.V. Valério and M.S. Carvalho, Pontifícia Universidade Católica do Rio de Janeiro Copyright © 2011 Society of Petroleum Engineers Original SPE manuscript received for review 23 December 2009. Revised manuscript received for review 16 August 2010. Paper (SPE 142294) peer approved 26 August 2010. Summary Fundamental understanding of the flow inside progressing-cavity pumps (PCPs) represents an important step in the optimization of the efficiency of these pumps, which are largely used in artificial- lift processes in the petroleum industry. The computation of the flow inside a PCP is extremely com- plex because of the transient character of the flow, the moving boundaries, and the difference in length scale of the channel height between the stator and rotor. This complexity makes the use of computational fluid dynamics (CFD) as an engineering tool almost impossible. This work presents an asymptotic model to describe the single-phase flow inside PCPs using lubrication theory. The model was developed for Newtonian fluid, and lubrica- tion theory was used to reduce the 3D Navier-Stokes equations in cylindrical coordinates to a 2D Poisson’s equation for the pressure field at each timestep, which is solved numerically by a second- order finite-difference method. The predictions are close to the experimental data and the results obtained by solving the complete 3D, transient Navier-Stokes equations with moving boundaries, available in the literature. Although the accuracy is similar to the complete 3D model, the computing time of the presented model is orders of magnitude smaller. The model was used to study the effect of geometry, fluid properties, and operating parameters in the pump-performance curves and can be used in the design of new pumping processes. Introduction Almost every fluid-like material can be pumped with PCPs. Since the pumping principle was conceived by Moineau (1930), a large number of different industrial applications make this positive- displacement pump one of the more used technologies for mov- ing fluids, from sandy crude oil to waste sludge. PCPs in their simplest form consist of a single-threaded screw (rotor) turning inside a double-threaded nut (stator), forming consecutive cavities separated by seal lines, as illustrated in Fig. 1. This simple configu- ration belongs to the single-lobe category of PCP. More-complex geometries with a larger number of lobes are also used, and any combination is possible as long as the stator has one more lead than the rotor. The first generation of PCPs had a metallic rotor and stator, forming rigid moving cavities in the space between the two surfaces. The following generations presented a rubber- covered stator, which is now common. The deformable stator cre- ates a compression fit with the rotor, in contrast with the metallic pump, where there is a small clearance leading to a larger leakage between consecutive cavities. Considering that deformable stators have operational limitations related to temperature and mechanical resistance of the elastomers, metallic PCPs have been recognized by the oil industry as an important technological alternative for heavy-oil production. This apparently simple mechanism produces an almost-pulsation-free positive-displacement flow without the need for valves that is based on the movement of the cavities from the suction end to the discharge end of the pump as the rotor turns inside the stator. The volumetric flow delivered by a PCP at a constant rotor speed and pressure difference depends on three design features: rotor diameter, rotor eccentricity, and stator pitch. The pump pressure rating depends on the number of stages. A peculiar characteristic of PCPs is the occurrence of backflow, also called slipflow, derived from nonperfectly sealed cavities. Because of their unique design and principle of operation, PCPs provide many benefits in oilfield applications, such as high solid-content tolerance, best efficiency with high-viscosity fluids, and simple installation and operation. Oil production with PCPs is generally designed over the knowledge of characteristic pump curves provided by the manufacturer, but several variables can affect and change the volumetric efficiency of both metallic and elastomeric-stator pumps. It is common knowledge that the char- acteristic curves change significantly with liquid viscosity and gas content; therefore, pump curves provided by manufacturers usually do not represent the real pump performance at downhole condi- tions. Moreover, in order to design PCPs that can be operated at extreme conditions, it is important to understand the effect of each geometric design parameter on the pump performance. These are the main reasons behind research efforts dedicated to study the flow inside PCPs. The performance of PCPs is a function of the volumetric pump displacement and the slipflow (i.e., the backward flow between consecutive cavities because of the adverse pressure gradient along the pump). The limitation of simple models in predicting pump performance is related to the difficulties in calculating the internal backflow. For any type of stator, rigid or deformable, slippage is a function of the fluid characteristics, the differential pressure, the dimensions of the different components, and the rotor’s kinemat- ics. In the case of elastomeric stators, the problem becomes even more complex because the geometry of the flow channel becomes a function of the pressure field and the elastomer properties. The volumetric displacement associated with the rotor move- ment can be calculated easily from the pump components’ geom- etry, but calculating the backflow is not a trivial problem. Available Models for Flow Inside PCPs The first and simplest numerical model to describe the flow inside a PCP was presented by Moineau (1930), and it is based on calculat- ing the backflow across the pump, considering a Hagen-Poiseuille flow through the seal lines, which is subtracted from the volume displaced by the rotating rotor, giving the volumetric flow rate. As the differential pressure across the pump rises, so does the slippage, and the relation between differential pressure and net volumetric flow pumped can be calculated. Because the slippage gap area is not clearly defined, the model is able to describe only the qualita- tive behavior of the flow. Quantitative predictions can be made only by determining experimentally the proportionality constant that relates the slippage flow and the pressure difference. In order to improve Moineau’s model, Gamboa et al. (2003) have modeled the slip as the superposition of two different mechanisms: one from the rotor’s movement and the other from the differential pressure between two cavities. The backward flow was evaluated on the basis of a friction factor inside the pump. However, limitations of the model were recognized, and the authors discuss the need to improve the evaluation of the friction factor and the gap between the stator and rotor in order to obtain a definitive model for flow inside a PCP. In order to estimate the backflow, Gamboa et al. (2002) and Olivet et al. (2002) performed an experimental study and obtained