Unsteady gas dynamics in relief-influenced pipelines: analytical modeling and pressure inversion

This study presents a novel analytical framework for modeling unsteady gas flow in long-distance pipelines laid across nonuniform terrain. Unlike existing approaches that typically neglect gravitational and geometric complexities, the proposed model incorporates slope-dependent gravitational effects through exponential modulation coefficients and accounts for transient leakage using a time-dependent source term located at an arbitrary point along the pipeline. Using Charny's linearization method, the nonlinear governing equations are reduced to a solvable diffusion-type partial differential equation. A closed-form solution for pressure distribution P(x,t) is derived by applying a strategic substitution that separates terrain and friction effects, enabling Laplace-based inversion to yield physically interpretable results. The derived pressure expression reveals how leakage intensity (Gut), terrain slope (a1), and leak location (l) interact to produce non-intuitive pressure patterns-such as pressure inversion, where downstream pressure loss becomes smaller than that at the inlet. The model is validated through extensive simulations, and key pressure dynamics are analyzed across multiple operating scenarios. Results indicate that increasing slope angle magnifies the gravitational effect, leading to early pressure crossover phenomena and stronger downstream stability, even for high leakage rates. This finding has substantial implications for sensor deployment, leak localization, and the development of SCADA-integrated decision-making tools. This work constitutes the first known analytical solution to the unsteady gas dynamics problem in sloped pipeline systems with leakage and provides a critical foundation for next-generation monitoring and control strategies in energy transport infrastructure.