A Graphical Method To Size Submersible Pumps for Variable Speed Application
Authors: Collins L.C. , Autrey J.W. 
Published: December 2007

Abstract
A graphical method is presented to size submersible pumps for variable speed motor controller application. This will optimize the selection of a pump of broad operating range for a given well inflow performance. The method was used to successfully design a number of submersible pump installations in the Gulf Coast Region. Two field examples are presented.

Introduction
Variable speed drive motor controllers have increased the flexibility of the submersible pump to operate over a broader range of conditions than was possible with the standard 60 Hz configuration.
Previous to the introduction of the variable Previous to the introduction of the variable speed drive, the only method to determine a given well's ability to produce high volumes of fluid was to size a pump of sufficient lift capacity to produce at the desired rate with the most produce at the desired rate with the most pessimistic inflow performance capabilities. The pessimistic inflow performance capabilities. The production rate was kept within the operating production rate was kept within the operating range by applying wellhead tubing pressure using a fluid choke. This results in changing the operating point on the pump head vs. flowrate curve. Figure 1 shows a typical 60 Hz pump head vs. flowrate curve for a multi-stage centrifugal pump. The well performance curve, which includes pump. The well performance curve, which includes well productivity, friction loss in the tubing and wellhead tubing pressure expressed in terms of liquid head (Hi) is overlayed. The intersection of the two curves determines the operating point of the system. Note that the operating point of the system. Note that the operating point is moved from 1 to 2 by choking the well. point is moved from 1 to 2 by choking the well. The vertical distance between 2 and 2a represents the increase in wellhead tubing pressure in terms of head. In contrast, the variable speed motor controller permits the use of a family of pump head output curves, each being unique to a given motor speed. The well's production rate can be increased or decreased by changing the nature of the pump's head curve rather than by changing the operating point on the 60 Hz curve.
Variable Speed Drive Basics
Reviewing centrifugal pump fundamentals, the pump flowrate (Q), head output (H), and horsepower requirements (hp), at speeds (N), other than 3600 rpm (60Hz) can be related to the 60 Hz performance parameters as follows: performance parameters as follows: These relationships accurately describe pump performance for all practical purposes over a performance for all practical purposes over a fairly wide range of motor speeds. Using these relationships, a family of pump curves can be generated as shown in figure 2.

Design Techniques
As with the single speed pump curve, the well performance may be overlayed. The dashed lines performance may be overlayed. The dashed lines represent the recommended operating range of the pump, which changes with the change in speed. pump, which changes with the change in speed.