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# Pipeline Hydraulics: Resolving the Paradox of Time Reversibility in Damped Transient

In the context of pipeline hydraulics, time-reversal has been successfully used historically for “valve stroking” (using the desired response to determine the necessary cause) and it has recently been shown by our research group that time-reversal can enable the development of efficient, robust, high resolution and noise-tolerant schemes for the detection of leaks, blockages, and bursts in real pipe systems where damping (e.g., friction and visco-elasticity) is present.

Herein lies a crucial and unresolved paradox; although time-reversal breaks down due to the inevitable damping of real waves, it still produces superior results than other defect detection techniques. This same paradox is described in the classic waterhammer book by Wylie et al (1993), page 223: “the [waterhammer] equations are remarkably robust in calculating backward in time.... [However] the idea of calculating backwards in time defies logic since it cannot be done physically. It cannot because of the irreversibility of losses in real systems.”

This proposed research will address this paradox using theoretical and experimental means. Theoretically, this research will seek a mathematical transformation that transforms the damped wave equation into an undamped one; that is, an approach to transform a non-time reversible problem (damped wave equation) into a time-reversible problem (undamped wave equation). The damping effects, that will be considered, include steady and unsteady friction and visco-elasticity. The resulting transformation will be solved to obtain the undamped response from the measured one which will be used in conjunction with the time-reversal schemes for defect detection. Physically, the required transformation must be an amplification operation that adds back the energy that is lost to damping. Unfortunately, this means that measurement errors will also be amplified under the transformation making the problem ill-posed. Two regularization techniques will be tested in this research. To supplement this theoretical work, the experimental aim is to provide the first-ever proof that time-reversal provides efficient, robust, high resolution, and noise-tolerant techniques for the detection of defects in pipes and to test the range of validity of the proposed theoretical transformation.

Objectives:

• Seek a mathematical transformation that transforms a non-time reversible problem (damped wave equation) into a time-reversible problem (damped wave equation). The damping effects that will be considered include steady and unsteady friction and viscoelasticity.

• Devise stable inversion techniques for the transformation derived in (1) so as to obtain damping-free pressure signals from the measured pressure signals.

• Experimentally test the time-reversible techniques by conducting transient tests in the lab with single and multiple defects. The testing will involve measuring the pressure, applying to it the inversion technique developed in (2) to obtain the un-damped pressure, and using the resulting un-damped pressure in conjunction with time-reversal techniques for defect detection.

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