Study on Competitive Adsorption and Displacing Properties of CO₂ Enhanced Shale Gas Recovery: Advances and Challenges

Table 3

Characteristics of various flow regime under different Knudsen number [111].

Flow regime	range^a,b	Models	Comments
Continuum flow		Darcy’s equation for laminar flow and Forchheimer’s equation for turbulent flow	Assumes immobile fluid at the pore wall, and no permeability correction is generally required.
Slip flow		Darcy’s equation with Klinkenberg’s correction or Knudsen’s correction	Klinkenberg’s correction is generally implemented due to its easiness, and Knudsen’s correction is more accurate.
Transition flow		Darcy’s law with Knudsen’s correction can be applied. Alternative method is Burnett’s equation with slip boundary condition^c	Knudsen’s diffusion equation is a more reliable approach, especially when is closer to 10.
Free molecular flow		Knudsen’s diffusion equation^d, alternative methods are DSMC and lattice Boltzmann^c	Usually occurs in shale formations where the pore throat radius is very small.

^aThis Knudsen number classification is taken from Karniadakis et al. [112]. ^bSome reference suggests () as a cut off for continuum flow [109]. ^cFor more details, see Agarwal et al. [113]. ^dKnudsen’s diffusion can coexist with bulk and surface diffusion.