Environmental Engineering Reference
In-Depth Information
a. Uniform Flow
b. Mobile-immobile water
c. Dual-porosity
d. Dual-permeability
Water
Water
Mobile
Water
Water
Water
Water
Water
Water
Imob.
Imob.
Mobile
Slow
Slow
Fast
Fast
Solute
Solute
Solute
Solute
Solute
Solute
Solute
Solute
Imob.
Imob.
Mobile
Mobile
Imob.
Imob.
Mobile
Mobile
Slow
Slow
Fast
Fast
=
=
θ im
+
+
θ m o
θ
=
=
θ im
+
+
θ m
θ
=
=
θ m
+
+
θ f
θ
θ
θ
θ mo
Fig. 18.14 Conceptual models of water flow and contaminant transport (
is the water content,
θ im in ( b )and( c ) are water contents of the mobile and immobile flow regions, respectively,
and
with
θ f in ( d ) are water contents of the matrix and macropore
(fracture) regions, respectively) (after Šimunek and Van Genuchten ( 2006 ))
θ mo /(
θ mo +
θ im )
= φ m ,and
θ m and
∂θ mo c mo
+
f
ρ
s mo
θ mo D mo
c mo
qc mo
=
φ mo s
t
t
z
z
z
(18.50)
∂θ im c im
+
(1
f )
ρ
s im
=− φ im + s
t
t
for the mobile (macropores, subscript mo ) and immobile (matrix, subscript im )
domains, respectively, where f is the dimensionless fraction of sorption sites in
contact with the mobile water [
φ im are reactions in the mobile and
immobile domains [ML 3 T 1 ], respectively, and
],
φ mo and
Γ s is the contaminant transfer rate
between the two regions [ML 3 T 1 ]. The same Eq. ( 18.50 ) can be used to describe
contaminant transport considering both the mobile-immobile and dual-porosity
models shown in Fig. 18.14b and c , respectively.
Dual-Permeability Model
One approach for implementing a dual-permeability formulation for contaminant
transport is to assume advection-dispersion type equations for transport in both the
fracture and matrix regions as follows (Gerke and Van Genuchten 1993 ):
∂θ f c f
+ ∂ρ
s f
θ f D f
c f
q f c f
φ f s
w
=
(18.51)
t
t
z
z
z
∂θ m c m
+ ∂ρ
s m
θ m D m
q m c m
s
c m
=
φ m
(18.52)
t
t
z
z
z
1
w
where the subscript f and m refer to the macroporous (fracture) and matrix pore
systems, respectively;
φ m represent sources or sinks in the macroporous and
matrix domains [ML 3 T 1 ], respectively; and w is the ratio of the volume of the
macropore (inter-aggregate) domain and that of the total soil system [-]. Equations
( 18.51 ) and ( 18.52 ) assume advective-dispersive type transport descriptions for both
φ f and
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