MATHEMATICAL MODEL AND ANALYTICAL SOLUTIONS FOR UNSTEADY FLOW IN NATURAL GAS RESERVOIRS

The effects of pressure on the gas viscosity and compressibility factor lead to a nonlinear partial differential equation for the flow in a gas reservoir even if the flow process follows Darcy's law at isothermal conditions. For further study on gas flow performances in gas reservoirs, a mathematical model of gas flow is developed in this article. Exact analytical solutions of one-dimensional unsteady gas flow at low and high pressures in gas reservoirs are obtained by transferring the nonlinear partial differential equation into a nonlinear ordinary differential equation. The numerical solutions obtained by finite difference for two cases of low- and high-pressure condition are given to validate the analytical solutions presented in this work. The key parameters, such as viscosity index, permeability, and porosity, to determine the characteristic of pressure distribution in porous media are analyzed in this work. The solutions at high pressures imply that it leads to obvious errors for prediction pressure distribution when ignoring pressure's effects on gas viscosity and compressibility factor for gas flow at high pressures in deep gas reservoirs. Both the increase in viscosity index and the decrease in permeability lead to an increase in pressure gradients along the distance.