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Logistic fit

In addition to the previously introduced constitutive models for the relative permeability, we included a logistic fit function to capture the saturation dependence of the hydraulic conductivity. Note that this approach is pure curve fitting and has no physical interpretation. The relative permeability function of the pore water and pore air read β={w,a}:

kβ=1kminβ1+eχβ(Seζ)+kminβ   with:  χw=χ  and  χa=χ.

Therein χ and ζ are curve fitting parameter. kminβ is the minimum relative permeability of pore water (β=w) and pore air (β=a), respectively.

The input line for the Logistic fit model takes the following form:

*Relative permeability = Logistic-fit
k^w_min, k^a_min, chi, zeta

where kminw and kminw are the minimum relative permeability for the pore water and pore air phase, respectively. χ and ζ are fitting parameter.

Theory

The contributions of to the Jacobian read:

kwSe=(1kminw)χeχ(Seζ)(1+eχ(Seζ))2,kaSe=(1kmina)χeχ(Seζ)(1+eχ(Seζ))2.

The development of relative permeability kβ with effective degree of saturation Se is presented here: