Transient two-phase flow in low velocity hilly terrain pipelines

For a hilly terrain pipeline operating at low flow rates of liquid and gas, the liquid tends to accumulate in the valleys while the gas tends to accumulate in the peaks. Under such conditions frictional pressure losses can be neglected and the system is totally gravity controlled. Liquid and gas is supplied at the entrance and is collected at the exit. The system can exhibit a complex transient behavior of the gas and the liquid in the pipe, even though the input flow rates of the liquid and gas are constant. The transient behavior eventually results in either a stable steady-state flow, with two-phase bubble or slug flow in the upward sections and stratified flow in the downwards sections, or complex indefinite fluctuations in pressure and fluids distributions when the liquid in the upward sections is unstable. In this work we propose a model and present the equations for predicting the behavior of such a system. Example solutions are provided for both the stable and the unstable cases.