IRP=12 Regular hysteresis

IRP = 12 Regular hysteresis

The hysteretic form of the van Genuchten model (Parker and Lenhard, 1987; Lenhard and Parker, 1987) has been implemented. Details of the implementation are described in Doughty (2013). The regular hysteresis model is invoked by setting both IRP and ICP to 12.

${k_{rl}} = \sqrt {{{\bar S}_l}} {\left[ {1 - \left( {1 - \frac{{{{\bar S}_{gt}}}}{{1 - \bar S_l^\Delta }}} \right){{(1 - {{({{\bar S}_l} + {{\bar S}_{gt}})}^{1/m}})}^m} - \left( {\frac{{{{\bar S}_{gt}}}}{{1 - \bar S_l^\Delta }}} \right){{(1 - {{(\bar S_l^\Delta )}^{1/m}})}^m}} \right]^2}$

${k_{rg}} = {k_{rg\max }}{(1 - ({\bar S_l} + {\bar S_{gt}}))^\gamma }{(1 - {({\bar S_l} + {\bar S_{gt}})^{1/m}})^{2m}}$

where

${\bar S_l} = \frac{{{S_l} - {S_{lr}}}}{{1 - {S_{lr}}}}$, $\bar S_l^\Delta = \frac{{S_l^\Delta - {S_{lr}}}}{{1 - {S_{lr}}}}$, ${\bar S_{gt}} = \frac{{S_{gr}^\Delta ({S_l} - S_l^\Delta )}}{{(1 - {S_{lr}})(1 - S_l^\Delta - S_{gr}^\Delta )}}$

$S_{_{gr}}^\Delta = \frac{1}{{1/(1 - S_l^\Delta ) + 1/{S_{gr\max }} - 1/(1 - {S_{lr}})}}$

$S_l^\Delta$ is the turning-point saturation, and $S_{gr}^\Delta$ is the residual gas saturation.

RP(1) = m; van Genuchten m for liquid relative permeability (need not equal CP(1) or CP(6)); $k_{rl}$ uses the same m for drainage and imbibition.

RP(2) = $S_{lr}$ : $k_{lr}(S_{lr})$ = 0, $k_{rg}(S_{lr}) = k_{rgmax}$. Must have $S_{lr}$ > $S_{lmin}$ in capillary pressure (CP(2)). $S_{lr}$ is minimum saturation for transition to imbibition branch. For $S_{l}$ < $S_{lr}$ , curve stays on primary drainage branch even if $S_{l}$ increases.

RP(3) = $S_{grmax}$; maximum possible value of $S_{gr}^\Delta$ . Note that the present version of the code requires that $S_{lr}$ + $S_{grmax}$ < 1, otherwise there will be saturations for which neither fluid phase is mobile, which the code cannot handle. Setting $S_{grmax}$ = 0 effectively turns off hysteresis. As a special option, a constant, non-zero value of Sgr may be employed by setting CP(10)>1 and making RP(3) negative. The code will set $S_{gr}^\Delta$ = -RP(3) for all grid blocks at all times.

RP(4) = $\gamma$; typical values 0.33 – 0.50.

RP(5) = $k_{rgmax}$

RP(6) = fitting parameter for krg extension for $S_{l}$ < $S_{lr}$ (only used when < 1); determines type of function for extension and slope of $k_{rg}$ at $S_{l}$ = 0.

≤ 0 use cubic spline for 0 < $S_{l}$ < $S_{lr}$ , with slope at $S_{l}$ = 0 of RP(6)

> 0 use linear segment for 0 < $S_{l}$ < RP(8)Slr and cubic spline for RP(8) $S_{lr}$ < $S_{l}$ < $S_{lr}$ , with slope at $S_{l}$ = 0 of –RP(6).

RP(7) = numerical factor used for krl extension to $S_{l}$ > $S_{l}^*$ ,. RP(7) is the fraction of Sl* at which krl curve departs from the original van Genuchten function. Recommended range of values: 0.95–0.97. For RP(7)=0, krl =1 for $S_{l}$ > $S_{l}^*$ (not recommended).

RP(8) = numerical factor used for linear krg extension to $S_{l}$ < $S_{lr}$ (only used when $k_{rgmax}$< 1). RP(8) is the fraction of $S_{lr}$ at which the linear and cubic parts of the extensions are joined.

RP(9) = flag to turn off hysteresis for $k_{rl}$ (no effect on Pc and $k_{rg}$ ; to turn off hysteresis entirely, set $S_{grmax}$ = 0 in RP(3)).

=0 hysteresis is on for krl

=1 hysteresis is off for krl (force $k_{rl}$ to stay on primary drainage branch ( $k_rl^{d}$) at all times)

RP(10) = $m_{gas}$ ; van Genuchten m for gas relative permeability (need not equal CP(1) or CP(6)); $k_{rg}$ uses same mgas for drainage and imbibition. If zero or blank, use RP(1) so that $m_{gas}$ = m.