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.
p c = − 1 α [ ( S e c ) ( γ − 1 ) / m − 1 ] 1 / n {p_c} = - \frac{1}{\alpha }{\left[ {{{\left( {{S_{ec}}} \right)}^{(\gamma - 1)/m}} - 1} \right]^{1/n}} p c = − α 1 [ ( S ec ) ( γ − 1 ) / m − 1 ] 1/ n S l ≥ ( S l r c + ε ) {S_l} \ge ({S_{lrc}} + \varepsilon ) S l ≥ ( S l rc + ε )
p c = − 1 α [ S e c ∗ ( γ − 1 ) / m − 1 ] 1 / n − β ⋅ ( S l − S l r c − 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) p c = − α 1 [ S ec ∗ ( γ − 1 ) / m − 1 ] 1/ n − β ⋅ ( S l − S l rc − ε ) S l < ( S l r c + ε ) {S_l} < ({S_{lrc}} + \varepsilon ) S l < ( S l rc + ε )
with
linear extension: β = − ( 1 − γ ) α n m ⋅ 1 ( 1 − S l r c ) ⋅ ( S e c ∗ ( γ − 1 ) / m − 1 ) 1 n − 1 S e c ∗ ⋅ ( γ − 1 − m m ) \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)} β = − α nm ( 1 − γ ) ⋅ ( 1 − S l rc ) 1 ⋅ ( S ec ∗ ( γ − 1 ) / m − 1 ) n 1 − 1 S ec ∗ ⋅ ( m γ − 1 − m )
p c = − 1 α [ S e c ∗ ( γ − 1 ) / m − 1 ] 1 / n ⋅ 1 0 β ⋅ ( S l − S l r c − 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)}} p c = − α 1 [ S ec ∗ ( γ − 1 ) / m − 1 ] 1/ n ⋅ 1 0 β ⋅ ( S l − S l rc − ε ) S l < ( S l r c + ε ) {S_l} < ({S_{lrc}} + \varepsilon ) S l < ( S l rc + ε )
with
log-linear extension: β = − l o g 10 ( e ) ⋅ ( 1 − m m ⋅ γ − 1 ε ⋅ 1 S e c ∗ ( 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) β = − l o g 10 ( e ) ⋅ ( m 1 − m ⋅ ε γ − 1 ⋅ S ec ∗ ( 1 − γ ) / m − 1 1 )
p c ≥ − p c , max {p_c} \ge - {p_{c,\max }} p c ≥ − p c , m a x
where
S e c = S l − S l r c 1 − S l r c {S_{ec}} = \frac{{{S_l} - {S_{lrc}}}}{{1 - {S_{lrc}}}} S ec = 1 − S l rc S l − S l rc S e c ∗ = ε 1 − S l r c {S_{ec*}} = \frac{\varepsilon }{{1 - {S_{lrc}}}} S ec ∗ = 1 − S l rc ε
Parameters:
CP(1) : n (parameter related to pore size distribution index,see also CP(4) )
CP(2) : 1 / 1 α α {1 \mathord{\left/ {\vphantom {1 \alpha }} \right. } \alpha } 1 / 1 α α
USERX(4,N) >0: 1 / α i 1/{\alpha _i} 1/ α i USERX(4,N)
USERX(4,N) <0: 1 / α i 1/{\alpha _i} 1/ α i USERX(4,N)·CP(2)
if CP(2) is negative, apply Leverett scaling rule:
1 / α i = 1 / α r e f ⋅ k i / k r e f 1/{\alpha _i} = 1/{\alpha _{ref}} \cdot \sqrt {{k_i}/{k_{ref}}} 1/ α i = 1/ α re f ⋅ k i / k re f
where:
1 / α r e f 1/\alpha _{ref} 1/ α re f |CP(2)|
k r e f k _{ref} k re f PER(NMAT)
USERX(1,N) >0: k i k _{i} k i USERX(1,N)
USERX(1,N) <0: k i k _{i} k i USERX(1,N) ·PER(NMAT)
CP(3) : ε or P c , m a x P_{c,max} P c , ma x
if CP(3) = 0, P c , m a x P_{c,max} P c , ma x 1 0 50 10^{50} 1 0 50 ε = − 1 \varepsilon = - 1 ε = − 1
if 0 < CP(3) < 1, =CP(3) and use linear extension
if CP(3) => 1, P c , m a x P_{c,max} P c , ma x CP(3) , ε = − 1 \varepsilon = - 1 ε = − 1
if –1< CP(3) <0, ε \varepsilon ε 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) : T r e f T_{ref} T re f
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) : S l r c S_{lrc} S l rc
if zero, then S l r c = R P ( 1 ) = S l r k {S_{lrc}} = {\rm{ RP(1) }} = {S_{lrk}} S l rc = RP ( 1 ) = S l r k
