Computer Diagnosis of Down-Hole Conditions In Sucker Rod Pumping Wells
Authors: Gibbs S.G. , Neely A.B. 
Published: December 2007

Abstract
Sucker rod equipment is the most widely used means for artificially lifting oil wells. Owing to this wide application, a large potential exists for increasing income through better methods for analyzing and trouble-shooting sucker rod installations. This paper describes a computer-oriented method whereby subsurface dynamometer cards can be determined from polished rod dynamometer data. The subsurface cards are useful in calculation of intermediate rod stress, estimation of pump intake pressures, detection of unanchored tubing and detection of leaking production strings and down-hole pumps. To date, this method has been used as a production engineering tool by Shell Oil Co. in over 500 wells in the United States, Canada and Venezuela.
Introduction
Through the years the polished rod dynamometer has been the principal tool for analyzing the operation of rod- pumped wells. The dynamometer is an instrument which records a curve of polished rod load vs displacement. The shape of this curve is affected by the down-hole operating conditions. Ideally, these conditions would be apparent from the dynamometer card by visual interpretation. Owing to the complex behavior of sucker rod systems and the diversity of card shapes, however, visual diagnosis of down-hole conditions is not always possible. Though much information can be gained from visual interpretation Of surface data, this information is qualitative in nature. Furthermore, the success in visual interpretation is directly linked with the skill and experience of the dynamometer analyst, and even the most experienced analysts are frequently misled into an incorrect diagnosis. The analytical technique described in this paper was developed to bridge the gap which arises when visual interpretation is inconclusive or when quantitative downhole data are needed. The method is based on a mathematical model of the pumping system which is virtually free from simplifying assumptions. The polished rod data are analyzed on a digital computer with a rigorous solution to the mathematical model. As a result, the technique provides a rational, quantitative method for calculating downhole conditions which is independent of the skill and experience of the analyst. It is no longer necessary to guess at down-hole pump conditions on the basis of recordings taken several thousands of feet above at the polished rod. By use of the analytical method, direct calculations of subsurface conditions can be made. The analytical method has been extensively used in field operations. Discussed in this paper are calculation of intermediate rod stresses, estimation of pump intake pressures, detection of unanchored tubing, detection of excessive rod friction, detection of leaking pumps and production strings, detection of gas separation problems and computation of gearbox torque.
DEVELOPMENT OF EQUATIONS
The theory underlying the analytical method is not particularly simple but the basic idea behind it is. It is helpful to think of the. sucker rod string as a transmission line. At the lower end of the transmission line, is a transmitter (the down-hole pump) and at the upper end is a receiver (an instrument installed at the polished rod). Information about down-hole pump conditions is transmitted along the sucker rod in the form of strain waves. These waves travel at the acoustical velocity in the rod material (about 16,000 ft/sec in steel). In its role as a transmission line, the sucker rod continually transmits information about downhole operations to the surface. But the information received at the surface is in code. Information received at the polished rod must be decoded so that quantitative deductions regarding down-hole operating conditions can be made. The technique of interpreting or decoding the information received at the polished rod is developed by solving a boundary value problem based on the wave equation.* The pertinent boundary conditions are the signals which are received at the polished rod, namely, time histories of polished rod load and displacement. To develop the basic equations, let z(x, t) be a complex variable in the wave equation:
(1)
Assume product solutions z(x, t)=X(x)T(t) and obtain the ordinary differential equations:
(2)
(3)
JPT
P. 91ˆ