# 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)* = $$\lambda$$

&#x20;           *RP(2)* = Slr

&#x20;           *RP(3)* = Sls

&#x20;           *RP(4)* = Sgr

&#x20;

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