Method of Distributions for Water Hammer Equations With Uncertain Parameters

Abstract

Water hammer equations (WHE) are routinely used to interpret leak detection tests in pipe networks. Assimilation of pressure data into model predictions is typically done within the probabilistic framework, which treats uncertain model parameters (e.g., initial and boundary conditions, location, and intensity of a leak) as random variables so that solutions of the WHE are given in terms of probability density functions (PDFs) of fluid pressure and velocity. These are usually estimated with computationally expensive Monte Carlo simulations. We use the method of distributions to derive a deterministic equation for the (joint) PDFs of the pressure and flow rate governed by the WHE. This PDF equation employs a closure approximation that ensures the self‐consistency in terms of the mean and variance of the state variables. Our numerical experiments demonstrate the agreement between solutions of the PDF equation and Monte Carlo simulations, and the computational efficiency of the former relative to the latter.

An efficient solver for cumulative density function-based solutions of uncertain kinematic wave models

“…To solve the stochastic kinematic wave equation (1) and obtain a statistical description of k at any space-time point, one can employ the CDF method [33,34,3]. In the deterministic setting, we introduce the concept of (one-point, one-time, Eulerian) fine-grained cumulative density function (CDF) of k:…”

“…The initial condition of the Burgers' equation may lead to shocks at later time. To address such issue, we follow earlier works [34,1] and employ the Rankine-Hugoniot condition [26] to determine the shock location x s at each realization. Here the fine-grained CDF solution would be divided by two parts: behind x s , the Π satisfies the governing equation whereas it remains its initial condition ahead of x s :…”

Comments on “Bayesian Update and Method of Distributions: Application to Leak Detection in Transmission Mains” by Alawadhi and Tartakovsky

“…In Alawadhi and Tartakovsky (2020), the Bayesian leak detection method is realized by solving the PDF of the pressure head h given the PDF of u 0 , based on the method of distributions (Alawadhi et al, 2018). However, it would be far more powerful and practical if the proposed Bayesian approach were to be extended to account for the uncertainty in wave speed by solving the PDF given the PDF f a of the wave speed a .…”

“…We further demonstrate the influence of a and u 0 via the leak localization result using the matched-field processing (MFP) method (Wang & Ghidaoui, 2018;Wang et al, 2019). Here, the MFP leak localization error in the case of imprecise u 0 and a is plotted in Figure 2b In Alawadhi and Tartakovsky (2020), the Bayesian leak detection method is realized by solving the PDF h|x leak of the pressure head h given the PDF u 0 of u 0 , based on the method of distributions (Alawadhi et al, 2018). However, it would be far more powerful and practical if the proposed Bayesian approach were to be extended to account for the uncertainty in wave speed by solving the PDF h|x leak given the PDF f a of the wave speed a.…”

An Integration Method Using Kernel Principal Component Analysis and Cascade Support Vector Data Description for Pipeline Leak Detection with Multiple Operating Modes

“…From the perspective of measurement signal acquisition, pipeline leak detection and positioning systems can be broadly classified into external detection methods and internal detection methods [2]. Externally-based methods monitor external pipeline parameters, such as acoustic signals [9][10][11] and fiber-optic cables [12][13][14], while internally-based methods typically collect pressure, flow, and temperature signals, such as real time transient modeling [15][16][17][18], negative pressure wave method [19,20], pressure point analysis, and the flow balance method. Recently, some scholars have also pointed out that integrating multi-source signals, including internal sensors and external sensors, is also an effective way to improve the performance of pipeline leak detection and localization [21].…”