OGS
[case] RelativePermeabilityNonWettingPhaseVanGenuchtenMualem

van Genuchten-Mualem relative permeability function for non-wetting phase in terms of effective wetting-phase saturation [23] :

$k_{rel}^n= (1 - S_e)^{1/2} (1 - S_e^{1/m})^{2m}$

with

$S_e=\frac{S^L-S^L_r}{S^L_{\mbox{max}}-S^L_r}$

where

\begin{eqnarray*} &S^L_r& \mbox{residual saturation of wetting phase,}\\ &S^L_{\mbox{max}}& \mbox{maximum saturation of wetting phase,}\\ &m\, \in (0, 1) & \mbox{ exponent.}\\ \end{eqnarray*}

The derivative of the relative permeability with respect to saturation is computed as

$\frac{\mathrm{d} k_{rel}^n}{\mathrm{d}S^L}= -(\dfrac{[1-S_e^{1/m}]^{2m}}{2\sqrt{1-S_e}}+2\sqrt{1-S_e} {(1-S_e^{1/m})}^{2*m-1} S_e^{1/m-1})/(S^L_\mbox{max}-S^L_r)$

As $$S^L \to S^L_\mbox{max}$$, or $$S_e \to 1$$, $$\dfrac{[1-S_e^{1/m}]^{2m}}{2\sqrt{1-S_e}}$$ has a limit of zero.