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 \(\beta= \{w,a\}\):

\[\begin{equation*} \label{eq:rel_perm_logistic} k^{\beta} = \frac{1-k^\beta_{min}}{1+e^{\chi^\beta(S^e-\zeta)} } + k^\beta_{min} ~~~\mbox{with:}~~ \chi^w = - \chi ~~\mbox{and}~~ \chi^a = \chi. \end{equation*}\]

Therein \(\chi\) and \(\zeta\) are curve fitting parameter. \(k^\beta_{min}\) is the minimum relative permeability of pore water (\(\beta = w\)) and pore air (\(\beta = a\)), 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 \(k^{w}_{min}\) and \(k^{w}_{min}\) are the minimum relative permeability for the pore water and pore air phase, respectively. \(\chi\) and \(\zeta\) are fitting parameter.

Theory

The contributions of to the Jacobian read:

\[\begin{align*} \frac{\partial k^w}{\partial S^e} &= \left(1-k^w_{min}\right) \chi \frac{e^{-\chi(S^e-\zeta)}}{\left( 1+e^{-\chi(S^e-\zeta)} \right)^2}, \label{eq:rel_perm_deriv_logistic_1} \\ \frac{\partial k^a}{\partial S^e} &= -\left(1-k^a_{min}\right) \chi \frac{e^{\chi(S^e-\zeta)}}{\left( 1+e^{\chi(S^e-\zeta)} \right)^2}. \label{eq:rel_perm_deriv_logistic_2} \end{align*}\]

The development of relative permeability \(k^\beta\) with effective degree of saturation \(S^e\) is presented here: