Leakage is an undesired abnormality that causes economical losses and impacts the environment. Leak detection tests in pipe networks are usually interpreted using water-hammer equations (WHE). These equations are nonlinear hyperbolic partial differential equations (PDEs) used to describe transient flows in pipes. The associated uncertainties in initial and boundary conditions, parameters, and leak strength and location increases the stochastic behavior of these equations. The method of distributions is used to derive a deterministic PDE for probability density function (PDF) of pressure head and flow velocity under uncertain initial conditions. The derivation requires a closure approximation that ensures its consistency with the mean and the variance of the state variables. A series of numerical experiments confirms the computational gain of this method over Monte Carlo simulations.

The PDF of pressure head obtained using the method of distributions serves as a prior PDF for data assimilation. Bayesian framework is used to update this distribution with a statistical model for observations obtained from the data collected by a pressure sensor. The result is posterior PDFs for leak location and leak strength. Series of numerical experiments are conducted for a single pipe and pipe networks under uncertain initial velocity and measurement noise to identify leak location and leak strength. The results are compared with the best fitness function that is used in inverse transient analysis.