Most current diagnostic techniques that stablish downhole conditions from polished rod measurements, are based on simplified analytical solutions and thus have inherent limitations. In the present investigation these limitations are fully explored and a new diagnostic approach, based on the method of characteristics, is introduced in order to overcome said limitations. This new approach is more flexible (since it can be easily extended to include deviated and directional wells), allows the estimation of computational error, yields a more accurate downhole load estimation and compares very favorably with state-of-the-art codes.
A specially designed Downhole Instrumentation Unit (DIU) was used to obtain downhole measurements (at the pump) in a test performed in a 5625 feet deep rod pumped well. The DIU is a performed in a 5625 feet deep rod pumped well. The DIU is a microprocessor based 5 feet long "intelligent sucker rod" that can be screwed in anywhere along the rod string. The results of this test show that the diagnostic technique using the method of characteristics provides a very good estimation of actual loads along the rod string and at the subsurface pump.
The standard diagnostic tool of the production engineer is the polished rod dynamometer card. It is a plot of the measured force polished rod dynamometer card. It is a plot of the measured force vs position at the polished rod through the pumping cycle. It has evolved as the preferred means of presenting polished rod data, since experienced engineers can recognize specific downhole operating conditions from particular dynamometer card shapes.
Interpretation of polished rod dynamometer cards, however, requires quite a bit of experience on the part of the operator and it is sometimes difficult to determine what is occurring at the downhole pump by looking at tophole measurements. This fact prompted Shell researchers in the early 1960's to develop what is prompted Shell researchers in the early 1960's to develop what is commonly known as the "Diagnostic Technique (DT)". In simple terms, the DT involves the use of measured polished rod force and position time histories to obtain direct numerical estimates of position time histories to obtain direct numerical estimates of force and position at any other point of the rod string, in particular at the downhole pump. particular at the downhole pump. The DT was first implemented by Gibbs and later by Adamache. Both investigators used different analytical means to obtain approximate solutions for which no error estimates can be found. Furthermore, their approaches can only be applied to rod string models where the dissipative behavior is of the viscous type, which is good for straight wells, but which is probably not realistic in deviated and directional wells.
The main objective of this work is to develop a numerical approach to the DT that allows the estimation of computational error and provides the flexibility to diagnose wells where dissipative mechanisms, other than viscous friction, are responsible for wave attenuation along the string. A further objective is to verify the Diagnostic Technique, using the proposed numerical approach, with actual downhole proposed numerical approach, with actual downhole measurements recorded in a pumping well.
Gibbs modelled the rod string with a one dimensional damped wave (or Telegrapher's) equation of the form
Assuming periodic solutions of the form
(which is reasonable, since rod pumped wells in stable conditions show cyclic very repeatable dynamometer cards) and approximating the measured polished rod motion, u(0,t), and dynamic strain., with Fourier series. i.e.