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.
Water‐hammer equations are used to describe transient flow in pipe networks. Uncertainty in model parameters, initial and boundary conditions, and location and strength of a possible leak renders deterministic predictions of this system untenable. When deployed in conjunction with pressure measurements, probabilistic solutions of the water‐hammer equations serve as a tool for detecting leaks in pipes. We use the method of distributions to obtain a probability density function (PDF) for pressure head, whose dynamics are described by the stochastic water‐hammer equations. This PDF provides a prior distribution for subsequent Bayesian data assimilation, in which data collected by pressure sensors are combined with this prior to obtain a posterior PDF of the leak location and size. We conduct a series of numerical experiments with uncertain initial velocity and measurement noise to ascertain the robustness and accuracy of the proposed approach. The results show the method's ability to identify the leak location and strength in a water transmission main.
We thank the Comment's authors for pointing out the need for further developments of our leak detection method. We identify strategies for dealing with uncertainty in wave speed. We thank the Comment's authors both for their kind words about our analysis and for pointing out the need for further developments of our leak detection method. We agree with the authors that the performance of our method, like that of any data assimilation technique, is affected by, and must handle, ubiquitous measurement errors. That is why the generation of observational data in Alawadhi and Tartakovsky (2020) includes white noise (t), which "accounts for measurement errors and ambient noise. This gives a Gaussian observation model with mean h obs (t) and variance 1."
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