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ICP=11 Modified van Genuchten Model

The van Genuchten model (Luckner et al., 1989) has been modified to prevent the capillary pressure from decreasing towards negative infinity as the effective saturation approaches zero. The approach is identical to that in ICP=10, except that two extensions (linear and log-linear) are available. The modified van Genuchten model is invoked by setting both IRP and ICP to 11.

pc=−1α[(Sec)(γ−1)/m−1]1/n{p_c} = - \frac{1}{\alpha }{\left[ {{{\left( {{S_{ec}}} \right)}^{(\gamma - 1)/m}} - 1} \right]^{1/n}} for Sl≥(Slrc+ε){S_l} \ge ({S_{lrc}} + \varepsilon )

pc=−1α[Sec∗(γ−1)/m−1]1/n−β⋅(Sl−Slrc−ε){p_c} = - \frac{1}{\alpha }{\left[ {{S_{ec*}}^{(\gamma - 1)/m} - 1} \right]^{1/n}} - \beta \cdot \left( {{S_l} - {S_{lrc}} - \varepsilon } \right) for Sl<(Slrc+ε){S_l} < ({S_{lrc}} + \varepsilon )

with

linear extension: β=−(1−γ)αnm⋅1(1−Slrc)⋅(Sec∗(γ−1)/m−1)1n−1Sec∗⋅(γ−1−mm)\beta = - \frac{{(1 - \gamma )}}{{\alpha nm}} \cdot \frac{1}{{(1 - {S_{lrc}})}} \cdot {\left( {S_{ec*}^{(\gamma - 1)/m} - 1} \right)^{\frac{1}{n} - 1}}{S_{ec*}}^{ \cdot \left( {\frac{{\gamma - 1 - m}}{m}} \right)}

pc=−1α[Sec∗(γ−1)/m−1]1/n⋅10β⋅(Sl−Slrc−ε){p_c} = - \frac{1}{\alpha }{\left[ {{S_{ec*}}^{(\gamma - 1)/m} - 1} \right]^{1/n}} \cdot {10^{\beta \cdot \left( {{S_l} - {S_{lrc}} - \varepsilon } \right)}} for Sl<(Slrc+ε){S_l} < ({S_{lrc}} + \varepsilon )

with

log-linear extension: β=−log10(e)⋅(1−mm⋅γ−1ε⋅1Sec∗(1−γ)/m−1)\beta = - lo{g_{10}}(e) \cdot \left( {\frac{{1 - m}}{m} \cdot \frac{{\gamma - 1}}{\varepsilon } \cdot \frac{1}{{S_{ec*}^{(1 - \gamma )/m} - 1}}} \right)

pc≥−pc,max⁡{p_c} \ge - {p_{c,\max }}

where

Sec=Sl−Slrc1−Slrc{S_{ec}} = \frac{{{S_l} - {S_{lrc}}}}{{1 - {S_{lrc}}}}, Sec∗=ε1−Slrc{S_{ec*}} = \frac{\varepsilon }{{1 - {S_{lrc}}}}

Parameters:

CP(1): n (parameter related to pore size distribution index,see also CP(4))

CP(2): 1/1αα{1 \mathord{\left/ {\vphantom {1 \alpha }} \right. } \alpha } (parameter related to gas entry pressure [Pa])

USERX(4,N)>0: 1/αi1/{\alpha _i}= USERX(4,N)

USERX(4,N)<0: 1/αi1/{\alpha _i}= USERX(4,N)·CP(2)

if CP(2) is negative, apply Leverett scaling rule:

1/αi=1/αref⋅ki/kref1/{\alpha _i} = 1/{\alpha _{ref}} \cdot \sqrt {{k_i}/{k_{ref}}}

where:

1/αref1/\alpha _{ref}=|CP(2)|

krefk _{ref}= PER(NMAT)

USERX(1,N)>0: kik _{i}= USERX(1,N)

USERX(1,N)<0: kik _{i}= USERX(1,N)·PER(NMAT)

CP(3): ε or Pc,maxP_{c,max}

if CP(3) = 0, Pc,maxP_{c,max}= 105010^{50}, ε=−1\varepsilon = - 1

if 0 < CP(3) < 1, =CP(3) and use linear extension

if CP(3) => 1, Pc,maxP_{c,max}=CP(3), ε=−1\varepsilon = - 1

if –1< CP(3) <0, ε\varepsilon =|CP(3)| and use log-linear extension

CP(4): m

if zero then m=1-1/CP(1), else m=CP(4) and n=1/(1-m)

CP(5): TrefT_{ref}

if negative, |CP(5)| is reference temperature to account for temperature dependence of capillary pressure due to changes in surface tension

CP(6): γ parameter of Active Fracture Model (see Appendix C)

CP(7): SlrcS_{lrc}

if zero, then Slrc=RP(1)=Slrk{S_{lrc}} = {\rm{ RP(1) }} = {S_{lrk}}

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