Introduction:

Gas transport in a porous medium occurs in several important applications including Oil and Gas exploration, carbon sequestration and some industrial processes involving heat transfer. Understanding the fundamental mechanisms and processes of gas and vapor transport in porous media allows models to be used to evaluate and optimize the performance and design of these systems.

Gas Transport Mechanisms:

Gas-phase momentum tranport in porous media consists of advective and diffusive components. In this article, only one-phase situations will be discussed. Two-phases, or unsaturated, will be discussed in another article coming soon.

First, the individual advective and diffusive components will be presented separately, then followed by an explanation of the combined mechanism.

Gas-phase advection: Is generally analyzed using Darcy's law (Darcy, 1856), which simply states that gas velocity ug, is directly proportional to the gas-phase pressure ∇Pg, by a factor called permeability 'kg', Darcy's law can be written as: 

where μg is the gas-phase viscosity and g is the gravitational constant.

Neglecting the gravity we obtain :

Note that the Darcy velocity, ug, is not a physical velocity. Rather, it is a superficial velocity based on the entire cross section of the flow, not just the fluid flow cross-section. The Darcy velocity is related to the pore velocity, Vg, through the porosity, φ.

Darcy’s law is applicable to low velocity flow, which is generally the case in porous media flow, and to regions without boundary shear flow, such as away from walls. When wall shear is important, the Brinkman extension can be used as discussed below. For turbulent flow conditions, the Forchheimer equation is appropriate. In some situations,the Brinkman and Forchheimer equations are both employed for a more complete momentum equation.

The Brinkman extension to Darcy’s law equation includes the effect of wall or boundary shear on the flow velocity

As you can see, here I chose a version where the gravity has been neglected, and this for clarity reasons. But it's not always the case.

So, the first term of the equation is recognizable as the Darcy's law, and the second term is a shear stress term such as it would be required by a no-slip condition in front of the walls. The coefficient μ that multipilies the Laplacian of the velocity is the effective viscosity at the walls, which in general is fearly different from gas viscosity μg. This extension is not widely used because the effect of shear stress is not significant. The effect can be clearly observed in a region close to the boundary whose thickness is of order of the square root of the gas permeability.

The Forchheimer extension is used for high velocities cases. When flow becomes turbulent, flow resistance becomes non-linear, and the Forchheimer equation is more appropriate.

In this case, gravity has been ignored again. cF is a constant. the first term is obviously the Darcy's law, and the second term is a non-linear flow resistance term. This equation is based on the work of Dupuit (1863) and Forchheimer (1901) and has been modified by Ward (1964) by his discover of the nature of the coefficient cF that is a function of the composition and roughness of the porous medium.

Gas-phase Diffusion: In a porous medium it consists of 2 ways; Continuum (or ordinary) diffusion and free-molecule diffusion.

Continuum Diffusion refers to the different motion of different gas species. Free-molecule diffusion, or Knudsen diffusion occurs when mean-free path of the gas molecule size is in the same order as the pore diameter. As the pore size decreases further, configurational diffusion is encountered where the gas molecule size is comparable to the pore diameter.

(the mean free path is the average distance traveled by a moving particle (such as an atom, a molecule, a photon) between successive impacts (collisions), which modify its direction or energy or other particle properties. Wikipedia)

Ordinary Diffusion is quantified for clear fluids by Fick's law, But we can apply it to porous media by the introduction of a porous media factor. So, Fick' law is actually 2 laws. The first law is the relationship of the diffusive flux of a gas component as a function of the concentration gradient under steady-state conditions. And the second law relates the unsteady diffusive flux to the concentration gradient. Both laws were originally derived for clear fluids (no porous media).

Free-Molecule Diffusion occurs when the gas molecular mean free path becomes of the same order as the tube dimensions, free-molecule, or Knudsen diffusion becomes important. Due to the influence of walls, Knudsen diffusion and configurational diffusion implicitly include the effect of the porous medium. Unlike ordinary (continuum) diffusion, there are no approaches for the free-molecule diffusion regime that use clear fluid approaches modified to include porous media effects. The molecular flux of gas i due to Knudsen diffusion is given by the equation :

where ni is the molecular density and DiK is the Knudsen diffusion coefficient.

As the dimensions of pores approach those of a single molecule, the flow mechanisms change, and configurational diffusion becomes important. Cunningham and Williams (1980) suggest that configurational diffusion may be encountered when the pore sizes are less than about 10 Å.

Combined Mechanisms: The interaction between advection and diffusion in porous media can be significant.cConsider two separate volumes connected by a tube containing light gas and a heavy gas. Diffusion of the light gas is faster than the heavy gas because of the higher molecule velocity. The net flow of molecules is toward the heavy gas volume, so the pressure rises in the heavy gas volume and decreases in the light gas volume. In turn, this pressure difference causes advection from the heavy gas volume to the light gas volume. Thus, diffusion directly leads to advection. Only in the case of equimolar gases will diffusion not result in advection.

For more informations about the combined mechanisms, I recommend you to read about the Dusty Gas Model in porous media (DGM).

References:

Clifford K. Ho, StephenW. Webb, 2006, ''Gas Transport in Porous Media'', Springer.

Dr. R.W. Zimmerman , '' Fluid Flow in Porous Media'', Imperial College, London.