Viscous compressible flow from a semi-sealed porous system: Mechanisms and wave-mediated diffusion

Abstract

It is widely known that a low Mach number gas flow can be approximated as incompressible at the leading order in a small Mach number expansion of the full solution. However, it has been shown recently that such an incompressible approximation becomes invalid for a semi-sealed system with no inlets and no boundary movements. Studies based on linearized compressible Navier–Stokes equations for such a system made of small capillary tubes have revealed some counter-intuitive flow characteristics such as no-slip flow with a slip-like mass flow rate. Here, we extend these works to gas drainage from a semi-sealed porous system. It is shown that gas drainage is solely driven by its volumetric expansion, and it cannot be approximated as an incompressible flow modulated by a small compressible effect. At the pore-scale, gas motion is governed by a damped rarefaction wave, and multiple scattering of the wave leads to an effective diffusion of the gas at the macroscopic scale. A large number of pore-scale simulations are performed for various pore structures, and the results are used to extract the macroscopic wave-mediated effective diffusion coefficient. Experiments are carried out to validate the wave-mediated effective diffusion model. The flow rates predicted by the effective diffusion model agree well with the experiments while results based on Darcy's law severely underestimate the flow rate. It is established that Darcy's law cannot be applied to a semi-sealed system, and no-slip flow with a slip-like flow rate is inherently embedded in the compressible Navier–Stokes equations.