Laplace-domain comparison of linear models of a reservoir-pipe-valve system with a leak

Abstract

The presence of a leak within a single line system alters the dynamic behavior of the system when subjected to transient pressure fluctuations. This change in dynamic behavior is seen in the time-domain as an increased rate of energy loss (damping rate) of the pressure response and in the frequency-domain as an attenuation in the magnitudes of the eigenfrequencies (harmonics) of the system. Two recent developments in the literature on leak detection, based on exploiting the time- and frequency-domain behavior are the transient damping method, and the frequency response method respectively. These methods are both utilize linearized approximations of the original nonlinear water hammer equations. The frequency-domain methods are essentially based on the transfer matrix method (TMM) of modeling the transient pipeline system, whereas the transient damping method utilizes a modal damping method (MDM) of modeling the system. The methods adopt the same approach to linearizing the original nonlinear water hammer equations, however their main difference lies in the manner in which they incorporate the internal boundary condition associated with the leak. These different approaches give rise to different linear systems. The focus of this paper is the comparative analysis of these different systems and the nature of how their dynamics change for different leak parameter values. As is convenient for linear systems, the TMM and the MDM are analyzed in the Laplace-domain, where this analysis takes the form of deriving the Laplace-domain pressure response functions for each method and studying the dependency of the pole locations of these functions on the leak parameters. The Laplace-domain is a useful way to consider and compare the methods as it is shown that the location of the poles of the pressure response functions are directly related to the magnitude of the eigenfrequencies (as used by the frequency-domain methods to estimate the leak properties) and are also directly related to the modal damping rates (as used by the transient damping method to estimate the leak properties). Both methods behave similarly for small leaks, however, for large leaks vastly different behaviors are observed. The TMM yields a solution that is consistent with the underlying equations for the asymptotically large leak case, but the MDM does not, therefore a difference in the behaviors is attributed to limitations on the MDM. Additionally, from the study of the TMM, insight is given into a contention that has existed in the literature on the impact that a leak has on the frequency response of a system. Through studying the structure of the poles for the TMM, it is shown that a leak only effects the eigenfrequencies of a system and that no new eigenfrequencies are introduced as a result of the leak. This paper was presented at the 8th Annual Water Distribution Systems Analysis Symposium which was held with the generous support of Awwa Research Foundation (AwwaRF)