Fluid–structure–soil interaction during water hammer in leaky pipes: A theoretical investigation

For transient modeling of water supply pipes that consider pipe-wall vibration impacts, a four-equation model, which describes the fluid–structure interaction mechanisms, should be used. However, little attention is paid to the role of the surrounding soil on the transient response of the piping system in previous studies. To fill this gap, this study considers the soil's mechanical resistance toward the pipe movement using a linear elastic Winkler description and thus modifies the pipe's axial movement equation. To investigate the transient response, including the surrounding soil resistance, the transfer matrix method is applied to the modified four-equation model. The proposed transient response formulation in terms of pressure, flow rate, longitudinal stress, and velocity of the pipe is scrutinized and validated by removing the soil-related term (to compare with those of the literature), and then including it to study various soil types, especially extreme soil stiffness. Following this validation, we further analyze the soil's various longitudinal patterns and stiffness in intact/leaky pipes, demonstrating that the soil's restraint impact on the pipe's axial movement gets stronger with the increase in its contact length and stiffness, i.e., approaching the asymptotic conditions of a fully restrained pipe. It is argued that the inclusion of the soil properties in the transient modeling enhances the accuracy of the transient-based leakage detection, recognizing that soil changes both the amplitude and phase shift of the system.