ICP=12 Regular hysteresis

ICP = 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 hysteretic model is invoked by setting both IRP and ICP to 12.

${P_c} = - P_0^p{\left[ {{{\left( {\frac{{{S_l} - {S_{l\min }}}}{{1 - S_{gr}^\Delta - {S_{l\min }}}}} \right)}^{ - \left( {\frac{1}{{{m^p}}}} \right)}} - 1} \right]^{(1 - {m^p})}}$

where

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

CP(1) = $m^d$; van Genuchten m for drainage branch $P_{c}^d(S_l)$.

CP(2) = $S_{lmin}$; saturation at which original van Genuchten Pc goes to infinity.

Must have $S_{lmin}$< $S_{lr}$ , where $S_{lr}$ is the relative permeability parameter RP(2).

CP(3) = $P_{0}^d$; capillary strength parameter for drainage branch $P_{c}^d(S_1)$ [Pa].

CP(4) = $P_{c,max}$; maximum capillary pressure [Pa] obtained using original van Genuchten $P_c$. Inverting the original van Genuchten function for $P_{c,max}$ determines $S_m$, the transition point between the original van Genuchten function and an extension that stays finite as $S_l$ goes to zero. For functional form of extension, see description of CP(13) below.

CP(5) = scale factor for pressures for unit conversion (1 for pressure in Pa).

CP(6) = $m^w$; van Genuchten m for imbibition branch $P_{c}^w(S_l)$. Default value is CP(1) (recommended unless compelling reason otherwise).

CP(7) = $P_w^0$; capillary strength parameter for imbibition branch $P_{c}^w(S_l)$ [Pa]. Default value is CP(3) (recommended unless compelling reason otherwise).

CP(8) = parameter indicating whether to invoke non-zero Pc extension for $S_l$ > $S_l^*$

= 1 – $S_{gr}^\Delta$

=0 no extension; $P_c$ = 0 for $S_l$ > $S_l^*$

>0 power-law extension for $S_l^*$ <$S_l$ <1, with $P_c$ = 0 when $S_l$ = 1. A non-zero CP(8) is the fraction of $S_l^*$ at which the Pc curve departs from the original van Genuchten function. Recommended range of values: 0.97–0.99.

CP(9) = flag indicating how to treat negative radicand, which can arise for $S_l$> $S_l^{\Delta23}$ for second-order scanning drainage curves (ICURV = 3), where $S_l^{\Delta23}$ is the turning-point saturation between first-order scanning imbibition (ICURV = 2) and second-order scanning drainage. None of the options below have proved to be robust under all circumstances. If difficulties arise because $S_l$> $S_l^{\Delta23}$ for ICURV = 3, also consider using IEHYS(3) > 0 or CP(10) < 0, which should minimize the occurrence of $S_l$> $S_l^{\Delta23}$ for ICURV = 3.

=0 radicand = max(0,radicand) regardless of $S_l$ value

=1 if $S_l$> $S_l^{\Delta23}$ , radicand takes value of radicand at $S_l$= $S_l^{\Delta23}$

=2 if $S_l$> $S_l^{\Delta23}$ , use first-order scanning imbibition curve (ICURV = 2)

CP(10) = threshold value of $\vert \Delta S \vert$ (absolute value of saturation change since previous time step) for enabling a branch switch (default is 1E-6; set to any negative number to do a branch switch no matter how small $\vert \Delta S \vert$ is; set to a value greater than 1 to never do a branch switch). See also IEHYS(3).

CP(11) = threshold value of $S_{gr}^\Delta$. If value of $S_{gr}^\Delta$ calculated from $S_{l}^\Delta$(Equation (2)) is less than CP(11), use $S_{gr}^\Delta$ = 0. Recommended value 0.01–0.03; default is 0.02.

CP(12) = flag to turn off hysteresis for $P_c$ (no effect on $k _{rl}$ and $k _{rg}$; to turn off hysteresis entirely, set $S_{grmax}$ = 0 in RP(3)).

=0 hysteresis is on for $P_c$

=1 hysteresis is off for $P_c$ (switch branches of $P_c$ as usual, but set $S_{gr}$ = 0 in $P_c$ calculation. Make sure other parameters of $P_c^d$ and $P_c^w$ are the same: CP(1) = CP(6) and CP(3) = CP(7))

CP(13) = parameter to determine functional form of Pc extension for $S_l$>< $S_{lmin}$ (i.e., $P_c$ > $P_{cmax}$ )

=0 exponential extension

>0 power-law extension with zero slope at $S_l$ = 0 and $P_{c}(0)$ =CP(13). Recommended value: 2 to 5 times CP(4)=$P_{cmax}$ . Should not be less than or equal to CP(4).