Civil Engineering Reference
In-Depth Information
Most deterministic methods are based on general flow and transport balance
(conservation) equations for the modeling domain with corresponding boundary
conditions determining a boundary or initial-boundary (in the transient case)
problem, using a system of partial differential (or integral) equations. These
boundary problems mathematically describe the principal physical processes deter-
mining contaminant transport in a water-bearing system, the most important of
which are
advection
(transport with groundwater flow velocity),
dispersion
of the
contaminant front (caused by different deviations of contaminant particles from
their “advection” paths and positions), and
sorption
of the contaminant by water-
bearing rock.
The partial differential equation describing the contaminant transport in
groundwater in saturated conditions can be written in the form [
Ciang and
Kinzelbach
, 2001]
∂ ∂ ∂ ∂
∂∂ ∂ ∂
C
C
q
N
( )
∑
1
=
D
vC
−
s
C
+
R
,
+
(1.14)
ij
i
s
k
t
x
x
x
n
k
=
i
j
i
where
C
is the concentration of a dissolved contaminant in groundwater (in units
of mass or activity per unit volume,
M
/
L
3
),
t
is time (
T
),
x
i
are linear distances
along the corresponding axes of the Cartesian coordinate system (
L
),
D
ij
is the
hydrodynamic dispersion tensor (
L
2
T
−1
),
v
i
is the real flow velocity (
LT
−1
),
q
s
is the
volume water flow rate per unit volume of water-bearing medium representing
sources of water recharge and discharge (
T
−1
),
C
s
is the contaminant concentration
in the recharge and discharge sources (
ML
−3
),
n
is the porosity (dimensionless),
and
k
∑
1
is a chemical reaction term, or the contaminant mass recharge or
discharge sources (
ML
−3
T
−1
).
When only the equilibrium sorption and irreversible reactions of first-order
chemical reactions are considered, the chemical reaction term in equation (1.14)
can be represented in the form [
Grove and Stollenwerk
, 1984]
N
R
k
N
ρ
∂
C
ρ
∑
1
(1.15)
R
−
=
b
−
λ
+
C
b
C
,
k
nt
∂
n
k
=
where
ρ
b
is the specific weight of rock (
mL
-3
), C
̅
the concentration of contaminant
sorbed by rock per unit rock mass (
Mm
-1
), and
λ
the first-order chemical reaction
rate constant (
T
-1
). The contaminant transport equation (1.14) is coupled with
the groundwater flow equation by the relation
K
∂
∂
h
v
= −
ii
,
(1.16)
i
nx
i
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