Applicability of the Linearized Governing Equation of Gas Flow in Porous Media

Abstract

In order to make the non-linear gas flow Equation tractable, the linearization treatment has been commonly applied in many subsurface gas flow problems such as natural gas production, soil vapor extraction, barometric, and pneumatic pumping. In this study, the accuracies of two representative linearization methods denoted as the conventional and the Wu solutions (Wu et al. Transp. Porous Media 32(1):117–137, 1998), are investigated quantitatively based on a numerical solution. The conventional solution uses a linearized constant gas diffusivity, while the Wu solution employs a spatially averaged but time-dependent gas diffusivity. The numerical solution is obtained by implementing the stiff solver ODE15s in MATLAB to deal with the time derivative and using the finite-difference method to approximate the spatial derivative in the non-linear gas flow equation. Two scenarios, the one-dimensional gas flow with constant pressure difference between two boundaries and the one-dimensional radial gas flow with constant mass injection rate at the origin of the coordinate system, are considered. The percentage error, defined as the ratio of difference between the numerical solution and the linearization solution to the ambient pressure, is calculated. It is founded that the Wu solution generally provides more accurate pressure evaluation than the conventional solution. The conventional solution always underestimates the pressure, while the Wu solution generally underestimates the pressure near the higher pressure boundary and overestimates the pressure near the lower pressure boundary. The maximal percentage error of the conventional solution is insensitive to time. This observation can be explained through the property of the complementary error function involved in the convention solution. For the one-dimensional flow example, the maximal percentage error of the conventional solution is 1.7, 25.5, and 90% when the pressure at one boundary suddenly rises above the ambient pressure by 50, 200, and 400%, respectively. While for the same example, the maximal percentage error of the Wu solution is 1.1, 14, and 44%, respectively.