On the Existence of Steady-State Solutions to the Equations Governing Fluid Flow in Networks

The steady-state solution of fluid flow in pipeline infrastructure networks driven by junction/node potentials is a crucial ingredient in various decision support tools for system design and operation. While the non-linear system is known to have a unique solution (when one exists), the absence of a definite result on existence of solutions hobbles the development of computational algorithms, for it is not possible to distinguish between algorithm failure and non-existence of a solution. In this letter we show that a unique solution exists for such non-linear systems if the term \emph{solution} is interpreted in terms of \emph{potentials} and flows rather than \emph{pressures} and flows. The existence result for flow of natural gas in networks also applies to other fluid flow networks such as water distribution networks or networks that transport carbon dioxide in carbon capture and sequestration. Most importantly, by giving a complete answer to the question of existence of solutions, our result enables correct diagnosis of algorithmic failure, problem stiffness and non-convergence in computational algorithms.