Agriculture Reference
In-Depth Information
transient water flow in a variably saturated rigid porous medium are (Simunek and
a
v
D
g
∂θ
c
1
+
∂ρ
s
1
+
∂
a
v
g
1
∂
∂
∂
D
ij
,1
∂
c
1
∂
∂
∂
ij
,1
∂
g
1
−
∂
q
i
c
1
∂
=
θ
+
−
Sc
r
,1
−
∂
t
∂
t
t
x
i
x
j
x
i
∂
x
j
x
i
μ
w
,1
+
μ
w
,1
)
μ
s
,1
+
μ
s
,1
)
μ
g
1
+
μ
g
,1
)
a
v
g
1
+
γ
w
,1
θ
+
γ
s
,1
ρ
+
γ
g
,1
a
v
(7.7)
(
θ
c
1
−
(
ρ
s
1
−
(
a
v
D
ij
,
k
∂
∂θ
c
k
+
∂ρ
s
k
+
∂
a
v
g
k
∂
∂
∂
D
ij
,1
∂
c
k
∂
∂
∂
g
k
∂
−
∂
q
i
c
k
∂
=
θ
+
−
Sc
r
,
k
−
∂
t
∂
t
t
x
i
x
j
x
i
x
j
x
i
μ
w
,
k
+
μ
w
,
k
)
μ
s
,
k
+
μ
s
,
k
)
μ
g
,
k
+
μ
g
,
k
)
a
v
g
k
+
μ
w
,
k
−
1
θ
(
θ
c
k
−
(
ρ
s
k
−
(
c
k
−
1
+
μ
s
,
k
−
1
ρ
s
k
−
1
+
μ
g
,
k
−
1
a
v
g
k
−
1
+
γ
w
,
k
θ
+
γ
s
,
k
ρ
+
γ
g
,
k
a
v
k
ε
(2,
n
s
)
(7.8)
where c, s, and g are solute concentrations in the liquid [ML
-3
], solid [MM
-3
],
and gaseous [ML
-3
] phases, respectively;
q
i
is the
i
th component of the volumetric
flux density [LT
-1
];
μ
g
are first-order rate constants for solutes in the
liquid, solid, and gas phases [T
-1
], respectively;
μ
w
,
μ
s
, and
μ
g
are similar first-
order rate constants providing connections between individual chain species;
μ
w
,
μ
s
, and
γ
w
,
γ
g
are zero-order rate constants for liquid [ML
-3
T
-1
], solid [T
-1
], and gas
phases[ML
-3
T
-1
], respectively;
γ
s
, and
is the soil bulk density [ML
-3
],
a
v
is the air con-
tent [L
-3
L
-3
],
S
is the sink term in the water flow,
c
r
is the concentration of the sink
term [ML
-3
],
D
y
w
is the dispersion coefficient tensor [L
2
T
-1
] for the liquid phase,
and
D
y
g
is the diffusion coefficient tensor [L
2
T
-1
] for the gas phase. The subscript
w, s, and g correspond with the liquid, solid, and gas phases, respectively; while the
subscript
k
represents the
k
th chain number, and
n
s
is the number of solutes involved
in the chain reaction.
ρ
7.4.4.2 One-Dimensional Transport
Solute transport equation for a homogeneous, isotropic porous medium during
steady-state unidirectional groundwater flow reduces to the following:
2
c
2
c
D
T
∂
D
L
∂
v
∂
c
R
∂
c
x
2
+
−
z
−
λ
Rc
=
(7.9)
∂
∂
z
2
∂
∂
t
where
is a first-order degradation constant,
D
L
and
D
T
are the longitudinal and
transverse dispersion coefficients, respectively;
λ
ν
is the average pore water velocity
(
q
z
/
) in the flow direction,
R
is the solute retardation factor, and
z
and
x
are the
spatial coordinates parallel and perpendicular to the direction of flow. The initial
solute-free medium is subjected to a solute source,
c
0
, of unit concentration. The
θ
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