It should be noted that . ew is calculated as follows (Eq. 2.31):
S ew ¼
S w À S rw
1 À S rw
¼
θ w À θ rw
θ ws À θ rw
ð2:31Þ
where,
θ w : porosity of water (–)
. ew : effective saturation (–)
. w : water saturation in the porous medium (–)
. rw : residual water saturation in the porous medium (–)
2.2.3 Numerical Expression: Multiphase Modeling
Numerical modeling is widely used in the NAPL remediation industry, in order to
optimize costs, and improve predictive capability in terms of remediation performance and residual risks. In the past, numerical simulation has been used to model
recovery processes in the oil industry. Prediction and simulation in reservoir engineering are often performed using the Buckley–Leverett analytical solution, which
has helped to solve problems in the secondary recovery stage, i.e., waterflooding in
oil recovery projects. Most NAPL soil remediation technologies and related technologies, such as chemical and thermal treatments, derived from the oil recovery
applications. Concerns regarding environmental issues only started to appear during
the second half of the twentieth century, and provided the impetus to develop new
technologies for environmental applications.
2.2.3.1 General Model of Multiphase/Multicomponent Flow
and Transport in Porous Media
The local-scale governing equations for multiphase and multicomponent flow in
porous media were derived by considering averaging over the pore scale. This means
that all physical and chemical procedures need to consider pore-scale phenomena.
The mass balance equation for component . in phase α is derived as in Eq. (2.32)
(Huyakorn and Pinder 1983; Miller et al. 1998; Crichlow 1977; Bear 1972):
∂
∂t
ε α ρ α ϖ
α
k
À
Á þ
∂
∂x
ε α ρ α ϖ
α
k v α,x
À
Á À
∂
∂x
J
α
k,x ¼ I
α
k þ E
α
k
ð2:32Þ
where,
ε α : volume fraction of each α phase (–)
ρ α :density of each α phase (kg m
À3 )
ω
/
k : mass fraction of component . in α phase (–)
. α,x : Darcy velocity of phase α in the . direction (m s
À1 )
2 Free Product Recovery of Non-aqueous Phase Liquids in Contaminated Sites:. . .
75
S ew ¼
S w À S rw
1 À S rw
¼
θ w À θ rw
θ ws À θ rw
ð2:31Þ
where,
θ w : porosity of water (–)
. ew : effective saturation (–)
. w : water saturation in the porous medium (–)
. rw : residual water saturation in the porous medium (–)
2.2.3 Numerical Expression: Multiphase Modeling
Numerical modeling is widely used in the NAPL remediation industry, in order to
optimize costs, and improve predictive capability in terms of remediation performance and residual risks. In the past, numerical simulation has been used to model
recovery processes in the oil industry. Prediction and simulation in reservoir engineering are often performed using the Buckley–Leverett analytical solution, which
has helped to solve problems in the secondary recovery stage, i.e., waterflooding in
oil recovery projects. Most NAPL soil remediation technologies and related technologies, such as chemical and thermal treatments, derived from the oil recovery
applications. Concerns regarding environmental issues only started to appear during
the second half of the twentieth century, and provided the impetus to develop new
technologies for environmental applications.
2.2.3.1 General Model of Multiphase/Multicomponent Flow
and Transport in Porous Media
The local-scale governing equations for multiphase and multicomponent flow in
porous media were derived by considering averaging over the pore scale. This means
that all physical and chemical procedures need to consider pore-scale phenomena.
The mass balance equation for component . in phase α is derived as in Eq. (2.32)
(Huyakorn and Pinder 1983; Miller et al. 1998; Crichlow 1977; Bear 1972):
∂
∂t
ε α ρ α ϖ
α
k
À
Á þ
∂
∂x
ε α ρ α ϖ
α
k v α,x
À
Á À
∂
∂x
J
α
k,x ¼ I
α
k þ E
α
k
ð2:32Þ
where,
ε α : volume fraction of each α phase (–)
ρ α :density of each α phase (kg m
À3 )
ω
/
k : mass fraction of component . in α phase (–)
. α,x : Darcy velocity of phase α in the . direction (m s
À1 )
2 Free Product Recovery of Non-aqueous Phase Liquids in Contaminated Sites:. . .
75