Factorized wave propagation model in tree-type pipe networks and its application to leak localization

Leak Localization in Looped Pipe Networks Based on a Factorized Transient Wave Model: Theoretical Framework

“…Therefore, if q and h at x 1 U and x 3 U are known, the q and h at x 2 D can be represented as (X. Wang et al, 2021):…”

“…In T Ω , there is unique wave propagation path between each V ex \ V s and the wave source V s . Therefore, the linear superposition‐based wave propagation model proposed for tree networks (X. Wang et al., 2021) is also valid in T Ω of the five‐loop network.…”

“…Therefore, the frequency domain mass and momentum equations across the internal Node ( v J ) are expressed as (X. Wang et al., 2021): $$$\left(\begin{array}{l}q\left({x}^{JD}\right)\hfill \\ h\left({x}^{JD}\right)\hfill \end{array}\right)={\mathbf{P}}_{22}\left(\begin{array}{l}q\left({x}_{1}^{JU}\right)\hfill \\ h\left({x}_{1}^{JU}\right)\hfill \end{array}\right)+\sum\limits _{i=1}^{{I}_{C}}{(-1)}^{{\alpha }_{i}}{\mathbf{P}}_{11}\left(\begin{array}{l}q\left({x}_{i}^{JU}\right)\hfill \\ h\left({x}_{i}^{JU}\right)\hfill \end{array}\right)$$$ where α i = 0 if the discharge in the i th child pipe is in the opposite direction with the mother pipe toward Node ( v J ), otherwise α i = 1; and Σ = summation operator. Equation indicates that, when across an interior node, the discharge of the mother pipe is the summation of all its child pipes, while the pressure head of the mother pipe only takes the value of the first child pipe.…”

“…A leak is a physical discontinuity of the pipe system and induces reflected waves; thus, the measured pressure wave echoes, which are modified by, and contain information on, properties of the leak (Duan et al., 2010; Lee et al., 2013; Meniconi et al., 2015). From the viewpoint of inverse problem, the transient wave‐based method can be classified into two main categories (Che et al., 2021; X. Wang et al., 2021), namely (i) local feature‐based methods, and (ii) full‐waveform inversion (FWI) methods. Category (i) can be further divided into three sub‐categories, namely (i‐a) reflection‐based method (Beck et al., 2005; Brunone, 1999; Ferrante & Brunone, 2003; Ferrante et al., 2007; Jönsson & Larson, 1992; Lee et al., 2007; Meniconi et al., 2021; Zeng et al., 2020), (i‐b) damping‐based method (Nixon et al., 2006), and (i‐c) frequency response function (FRF) peak pattern‐based method (Duan, 2017; Gong et al., 2013, 2014; Lee et al., 2006; Sattar & Chaudhry, 2008).…”

Uncertainty Quantification and Global Sensitivity Analysis for Transient Wave Propagation in Pressurized Pipes

“…An engineer should always assess these risks, where at least a rough forecast of the water hammer pressure rise is needed and can be computed via the Joukowsky equation (Keramat & Haghighi, 2014). Transient waves can also be employed to detect defects and anomalies in pipes, such as leakage, blockage, and deterioration (Colombo et al., 2009; Duan et al., 2013; Ferrante et al., 2007; Gong et al., 2013; Huang et al., 2020; Lee et al., 2005; Meniconi et al., 2015; Nguyen et al., 2018; Wang & Ghidaoui, 2018a, 2018b, 2019; Wang et al., 2021; Wang, 2021; Wang, Ghidaoui, & Lin, 2019). This is realized by matching the transient measurement with its physical model where defects are assumed to be free parameters to be optimized.…”