Table 2.1
Two-fluid capillary pressure and permeability models (Chen et al. 1999)
Models
Parameters Capillary pressure function Eqs.
Permeability function
Eqs.
VGM and VGB (van Genuchten–Mualem/Burdine)
[1], [2], and [3]
θ ws
θ rw
.
α
.
S
ew
¼
1
1þ
αhc
ð Þ
n
½
m
(2.7)
(a) VGM: (.
¼ 1À1/.)
(2.8)
k
rw
¼ S
0:5
ew 1
À 1
À S
1
m
ew
m
h
i 2
(2.9)
k
rn
¼ 1
À S
ew
ð
Þ
0:5
1
À S
1
m
ew
h
i
2m
(2.10)
(b) VGB: (.
¼ 1À2/.)
(2.11)
k
rw
¼ S
2
ew 1
À 1
À S
1
m
ew
m
h
i
(2.12)
k
rn
¼ 1
À S
ew
ð
Þ
2
1
À S
1
m
ew
h
i
m
(2.13)
BCM and BCB (Brook and Corey–Mualem/Burdine)
[3] and [4]
θ ws
θ rw
.
.
e
λ
η
S
ew
¼
he
hc
λ
(2.14) (c) BCM:
k
rw
¼ S
ηþ2þ2=λ
ew
(2.15)
k
rn
¼ 1
À S
ew
ð
Þ
η
1
À S
1þ1=λ
ew
Â
à 2
(2.16)
(4) BCB: (.
¼ 1À1/.)
(2.17)
k
rw
¼ S
3þ2=λ
ew
(2.18)
k
rn
¼ 1
À S
ew
ð
Þ
2
1
À S
1þ2=λ
ew
Â
Ã
(2.19)
LNM (Lognormal Distribution–Mualem)
[1] and [5]
θ ws
θ rw
.
.
m
σ b
η
S
ew
¼ F
ln h
hc
ð Þ
σb
!
(2.20)
k
rw
¼ S
η
ew F
n F
À1
n S
ew
ð Þþσ b
Â
Ã
È
É
2
(2.21)
k
rn
¼ 1
À S
ew
ð
Þ
η
1
À F
n F
À1
n S
ew
ð Þþσ b
Â
Ã
È
É
2
(2.22)
F
n x
ð Þ ¼
1
ffiffiffiffi
2π
p
R x
À1
exp
À
x 2
2
dx
(2.23)
¼
1
2 erfc
x
ffiffi
2
p
(2.24)
BRB (Brutsaert-Burdine)
[2] and [6]
θ ws ,
θ rw
.
β, γ
S
ew
¼
β
b
β
b þhc
γ
b
(2.25)
k
rw
¼ S
2
ew 1
À 1
À S
ew
ð
Þ
1À2=γ
b
h
i
(2.26)
k
rn
¼ 1
À S
ew
ð
Þ
3À2=γ
b
(2.27)
GDM (Gardner–Mualem)
[1] and [7]
θ ws ,
θ rw
.
α g
S
ew
¼ 1
þ
1
2 α g h
c
À
Á
e
À 1
2 αghc
(2.28)
k
rw
¼ e
Àαghc
(2.29)
k
rn
¼ 1
À e
À 1
2 αghc
2
(2.30)
[1]: Mualem (1976), van Genuchten (1980); [2]: Burdine (1953); [3]: Luckner et al. (1989); [4]: Brooks and Corey (1964); [5]: Kosugi (1994, 1996); [6]: Brutsaert (1967);
[7]: Gardner (1958)
74
S. Colombano et al.
Two-fluid capillary pressure and permeability models (Chen et al. 1999)
Models
Parameters Capillary pressure function Eqs.
Permeability function
Eqs.
VGM and VGB (van Genuchten–Mualem/Burdine)
[1], [2], and [3]
θ ws
θ rw
.
α
.
S
ew
¼
1
1þ
αhc
ð Þ
n
½
m
(2.7)
(a) VGM: (.
¼ 1À1/.)
(2.8)
k
rw
¼ S
0:5
ew 1
À 1
À S
1
m
ew
m
h
i 2
(2.9)
k
rn
¼ 1
À S
ew
ð
Þ
0:5
1
À S
1
m
ew
h
i
2m
(2.10)
(b) VGB: (.
¼ 1À2/.)
(2.11)
k
rw
¼ S
2
ew 1
À 1
À S
1
m
ew
m
h
i
(2.12)
k
rn
¼ 1
À S
ew
ð
Þ
2
1
À S
1
m
ew
h
i
m
(2.13)
BCM and BCB (Brook and Corey–Mualem/Burdine)
[3] and [4]
θ ws
θ rw
.
.
e
λ
η
S
ew
¼
he
hc
λ
(2.14) (c) BCM:
k
rw
¼ S
ηþ2þ2=λ
ew
(2.15)
k
rn
¼ 1
À S
ew
ð
Þ
η
1
À S
1þ1=λ
ew
Â
à 2
(2.16)
(4) BCB: (.
¼ 1À1/.)
(2.17)
k
rw
¼ S
3þ2=λ
ew
(2.18)
k
rn
¼ 1
À S
ew
ð
Þ
2
1
À S
1þ2=λ
ew
Â
Ã
(2.19)
LNM (Lognormal Distribution–Mualem)
[1] and [5]
θ ws
θ rw
.
.
m
σ b
η
S
ew
¼ F
ln h
hc
ð Þ
σb
!
(2.20)
k
rw
¼ S
η
ew F
n F
À1
n S
ew
ð Þþσ b
Â
Ã
È
É
2
(2.21)
k
rn
¼ 1
À S
ew
ð
Þ
η
1
À F
n F
À1
n S
ew
ð Þþσ b
Â
Ã
È
É
2
(2.22)
F
n x
ð Þ ¼
1
ffiffiffiffi
2π
p
R x
À1
exp
À
x 2
2
dx
(2.23)
¼
1
2 erfc
x
ffiffi
2
p
(2.24)
BRB (Brutsaert-Burdine)
[2] and [6]
θ ws ,
θ rw
.
β, γ
S
ew
¼
β
b
β
b þhc
γ
b
(2.25)
k
rw
¼ S
2
ew 1
À 1
À S
ew
ð
Þ
1À2=γ
b
h
i
(2.26)
k
rn
¼ 1
À S
ew
ð
Þ
3À2=γ
b
(2.27)
GDM (Gardner–Mualem)
[1] and [7]
θ ws ,
θ rw
.
α g
S
ew
¼ 1
þ
1
2 α g h
c
À
Á
e
À 1
2 αghc
(2.28)
k
rw
¼ e
Àαghc
(2.29)
k
rn
¼ 1
À e
À 1
2 αghc
2
(2.30)
[1]: Mualem (1976), van Genuchten (1980); [2]: Burdine (1953); [3]: Luckner et al. (1989); [4]: Brooks and Corey (1964); [5]: Kosugi (1994, 1996); [6]: Brutsaert (1967);
[7]: Gardner (1958)
74
S. Colombano et al.