Effective diffusivity in porous media

Laws for calculating effective diffusion coefficients. More...

Description

When averaging over a given volume of a porous medium, diffusion appears effectively restricted since not all volume is accessible to particles and diffusion is hindered by the solid matrix acting as obstacles. Effective diffusivity laws provide constitutive relations for the effective diffusion coefficients based on the solid matrix material parameters and the fluid configuration in the pore space.

The effective diffusion coefficient of component \( \kappa \) in phase \( \alpha \) can be modeled as

\begin{equation} D^\kappa_{\text{eff},\alpha} = \phi S_\alpha \tau D^\kappa_\alpha, \end{equation}

where \( \phi \) is the porosity (volume fraction of the pore space), \( S_\alpha \) is the saturation of phase \( \alpha \) (the volume fraction of phase \( \alpha \) being \( n_\alpha = \phi S_\alpha \)), \( D^\kappa_\alpha \) denotes the binary diffusion coefficient of component \( \kappa \) in phase \( \alpha \), and \( \tau \) is the tortuosity coefficient.

Bear [13] reports values of \(\tau\) in the range of 0.4 to 0.8. Note that in some literature the tortuosity \( \lambda \) is used instead of the tortuosity coefficient \(\tau\). The two quantities are related by \( \lambda = \sqrt{1/\tau} \).

The following laws are implemented:

Constant Tortuosity

For DiffusivityConstantTortuosity, \( \tau = \text{const.} \), with default value 0.5, empirically obtained by Carman [19]. The value can be changed at runtime by setting parameter "SpatialParams.Tortuosity". This will change the value of \( \tau \), and therefore the effective diffusion coefficient wherever the function effectiveDiffusionCoefficient is used.

Millington Quirk

For DiffusivityMillingtonQuirk, the tortuosity coefficient is estimated after Millington and Quirk (1961) [55] by

\[ \tau = \frac{1}{\phi^2} \left(\phi S_\alpha\right)^{7/3}. \]

See also Helmig (1997) [39] [page 129].

Classes
class	Dumux::DiffusivityConstantTortuosity< Scalar >
	Relation for the effective diffusion coefficient with constant tortuosity. More...
class	Dumux::DiffusivityMillingtonQuirk< Scalar >
	Relation for the effective diffusion coefficient after Millington and Quirk. More...