Summary
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.
Introduction
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.
© 2009. Society of Petroleum Engineers
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History
- Original manuscript received:
2 August 2007
- Meeting paper published:
11 November 2007
- Revised manuscript received:
11 June 2008
- Manuscript approved:
3 July 2008
- Published online:
16 March 2009
- Version of record:
1 March 2009
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