IRP=7 van Genuchten-Mualem Model

IRP = 7 van Genuchten-Mualem Model (Mualem, 1976; van Genuchten, 1980)

$k_{rl}=\sqrt{{{S}^{*}}}\ {{\left\{ 1\ -\ {{\left( 1\ -\ {{[{{S}^{*}}]}^{{1}/{\lambda }\;}} \right)}^{\lambda }} \right\}}^{2}}\quad \text{ if}\ \text{ }{{S}_{l}}\ <\ {{S}_{ls}}$

$k_{rl}=1\quad \quad \quad \quad \quad \quad \quad \quad \quad \text{ }\quad \text{if}\ \text{ }{{S}_{l}}\ \ge \ {{S}_{ls}}$

Gas relative permeability can be chosen as one of the following two forms, the second of which is due to Corey (1954)

${{k}_{rg}}=1\ -\ {{k}_{rl}}\quad \ \quad \quad \quad \text{ if}\ {{S}_{gr}}\,=\,0$

${{k}_{rg}}={{\left( 1\ -\ \hat{S} \right)}^{2}}\ \left( 1\ -\ {{{\hat{S}}}^{2}} \right)\quad \text{ if}\ {{S}_{gr}}\,>\,0$

subject to the restriction $0\quad \le \quad {{k}_{rl}},\ {{k}_{rg}}\quad \le \quad 1$

Here ${{S}^{*}}\quad =\quad {\left( {{S}_{l}}\ -\ {{S}_{lr}} \right)}/{\left( {{S}_{ls}}\ -\ {{S}_{lr}} \right)}\;$and $\hat{S}\quad =\quad {\left( {{S}_{l}}\ -\ {{S}_{lr}} \right)}/{\left( 1\ -\ {{S}_{lr}}\ -\ {{S}_{gr}} \right)}\;$

Parameters: RP(1) = $\lambda$

RP(2) = Slr

RP(3) = Sls

RP(4) = Sgr

Notation: Parameter $\lambda$ is m in van Genuchten’s notation, with m = 1 - 1/n; parameter n is often written as $\beta$.