stylistic:style_transfer, linguistic:translation
Porous media

For diffusion in porous media the basic equations are (if Φ is constant):

where D is the diffusion coefficient, Φ is porosity, n is the concentration, m > 0 (usually m > 1, the case m = 1 corresponds to Fick's law).

Care must be taken to properly account for the porosity (Φ) of the porous medium in both the flux terms and the accumulation terms. For example, as the porosity goes to zero, the molar flux in the porous medium goes to zero for a given concentration gradient. Upon applying the divergence of the flux, the porosity terms cancel out and the second equation above is formed.

For diffusion of gases in porous media this equation is the formalization of Darcy's law: the volumetric flux of a gas in the porous media is

where k is the permeability of the medium, μ is the viscosity and p is the pressure.

The advective molar flux is given as

J = nq

and for p ∼ n γ {\displaystyle p\sim n^{\gamma }} Darcy's law gives the equation of diffusion in porous media with m = γ + 1.

In porous media, the average linear velocity (ν), is related to the volumetric flux as:

υ = q / ϕ {\displaystyle \upsilon =q/\phi }

Combining the advective molar flux with the diffusive flux gives the advection dispersion equation

∂ n ∂ t = D Δ n m − ν ⋅ ∇ n m, {\displaystyle {\frac {\partial n}{\partial t}}=D\,\Delta n^{m}\ -\nu \cdot \nabla n^{m},}

For underground water infiltration, the Boussinesq approximation gives the same equation with m = 2.

For plasma with the high level of radiation, the Zeldovich –Raizer equation gives m > 4 for the heat transfer.