Environmental Engineering Reference
In-Depth Information
D
h
∂
c
J
=
qc
−
θ
(18.33)
∂
z
where
D
h
is the hydrodynamic dispersion coefficient [L
2
T
−
1
] that accounts for both
molecular diffusion and mechanical dispersion.
Mechanical dispersion in most subsurface transport problems dominates molec-
ular diffusion in the liquid phase, except when the fluid velocity becomes relatively
small or is negligible. Diffusion dominated transport occurs in low permeability
media, such as clays, rock matrices, and man-made structures such as concrete.
18.3.2 Advection-Dispersion Equations
18.3.2.1 Transport Equations
The equation governing transport of dissolved contaminants in the vadose zone
is obtained by combining the contaminant mass balance (Eq. (
18.23
)) with equa-
tions defining the total concentration of the contaminant (Eq. (
18.24
)) and the
contaminant flux density (Eq. (
18.33
)) to give
∂
(
ρ
b
s
+
θ
c
+
ag
)
∂
∂
D
h
∂
c
∂
∂
z
(
aD
g
∂
g
−
∂
(
qc
)
∂
=
z
(
θ
z
)
+
z
)
−
φ
(18.34)
∂
t
∂
∂
z
where
D
h
and
D
s
g
are the hydrodynamic dispersion coefficient in the liquid and
gaseous phases [L
2
T
−
1
], respectively.
Several alternative formulations of Eq. (
18.34
) can be found in the literature. For
example, for one-dimensional transport of non-volatile contaminants, Eq. (
18.34
)
simplifies to
∂
(
ρ
b
s
+
θ
c
)
=
∂
(
θ
Rc
)
∂
∂
D
h
∂
c
−
∂
(
qc
)
∂
=
θ
−
φ
(18.35)
∂
∂
∂
t
t
z
z
z
where
q
is the vertical water flux density [LT
−
1
] and
R
is the retardation factor [
−
]:
+
ρ
b
θ
ds
(
c
)
dc
R
=
1
(18.36)
For transport of inert, non-adsorbing contaminants during steady-state water flow
a further simplification is possible:
2
c
∂
c
D
h
∂
v
∂
c
t
=
−
(18.37)
∂
∂
z
2
∂
z
The above equations are usually referred to as advection-dispersion equations
(ADEs), or convection-dispersion equations (CDEs).
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