Modeling a Finite-Length Sucker Rod Using the Semi-Infinite Wave Equation and a Proof to Gibbs' Conjecture
- Jeffrey J. DaCunha (Lufkin Automation) | Sam Gibbs
- Document ID
- Society of Petroleum Engineers
- SPE Journal
- Publication Date
- March 2009
- Document Type
- Journal Paper
- 112 - 119
- 2009. Society of Petroleum Engineers
- 5.4.6 Thermal Methods, 5.3.2 Multiphase Flow, 2.2.2 Perforating, 4.1.5 Processing Equipment, 3.1 Artificial Lift Systems, 5.6.4 Drillstem/Well Testing, 4.1.2 Separation and Treating, 3.1.1 Beam and related pumping techniques
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In this paper, we study semi-infinite-spatial-domain wave equations modeling the real-world problem of longitudinal waves propagating along a long, slender, homogeneous elastic rod of finite length. A practical conclusion from the paper is that precision of the downhole card can be increased by improving the accuracy of the friction law in the wave equation. To conclude, we provide a rigorous proof of Gibbs' theorem and illustrate its validity with an existing well.
In the paper and patent by S.G. Gibbs (Gibbs 1963 and Gibbs 1967, respectively), new methods are presented for diagnosing and predicting the behavior of sucker-rod pumping systems. In Gibbs (1963), the method employs two boundary conditions. Because it is a prediction method, the two boundary conditions are specified at both the polished rod and at the pump. The first boundary condition is the surface position of the polished rod, while the second is a Robin boundary condition (i.e., it involves position and load at the pump), which simulates the downhole pump condition.
However, it is the solution in Gibbs (1967) that is the subject of this paper. The result uses a separation-of-variables technique to solve the viscous-damped-wave equation on a semi-infinite domain and uses the measured surface position and load as the two necessary boundary conditions to model a rod string of finite length. Therefore, there is no explicit information about the position or load at the pump. In light of this fact, how are the reflections from the end of the rod string accounted for and measured in this model of a semi-infinite rod string? It was realized that embedded in these measured surface-position and load boundary conditions are the reflected tension and compression waves that travel back and forth along the rod string between the pump and the polished rod. Using this discovery and employing the semi-infinite model, with the boundary conditions of the pump embedded in the boundary conditions of the polished rod, the solution to the semi-infinite-viscous-damped-wave equation is found to be exactly the solution to the model of the viscous-damped-wave equation of a finite-length rod string. Thus, using the semi-infinite model, we can correctly model the wave propagation and reflection along a finite-length rod string.
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