Finite element solution of a new formulation for gas flow in a pipe with source terms

A new formulation for isothermal gas flow in a pipeline with gravity and friction source terms is introduced and solved by finite element methods using FEniCS open-source computing platform. The proposed formulation can be easily adapted to simulate different types of real boundary conditions and involves only one scalar unknown instead of two ones (typically mass flux and pressure or density). The main motivation for presenting the work is to find a compact formulation which is able to be solved with a high order accuracy scheme just by changing the type of elements. Also, it can be extended to gas network easily. In order to check the performance of the numerical method, it has been applied to several test problems. At first, with the purpose of finding the order of accuracy, two manufactured tests for different types of boundary conditions are solved. The results show that, for nodal finite elements of degree one, the order of accuracy in space is more than two and for time is between two and three. Then, two test cases with step boundary conditions are solved. The corresponding numerical results show that, even with large time-step and coarse mesh, they are in good agreement with those obtained with a fine mesh. Finally, a real pipe from the Spanish gas network is solved and the results compared with on-line measurements.