Environmental Engineering Reference
In-Depth Information
This equation requires that the integral of the quan-
tity in parentheses must be equal to zero for any arbi-
trary control volume, and this can be true only if the
integrand itself is equal to zero. Following this logic,
Equation (5.11) requires that
discrete solutions to the advection-diffusion equation
in space and time and are most useful in cases of complex
geology and irregular boundary conditions. Analytic
models provide continuous solutions to the advection-
diffusion equation in space and time and are most useful
in cases of simple geology and simple boundary condi-
tions. most groundwater contamination problems can
be analyzed assuming steady-state flow conditions,
which implies that the flow velocity and diffusion char-
acteristics remain constant with time. Several useful
analytic diffusion models are described in the following
sections.
∂
∂
cn
t
0
(5.12)
+ ∇⋅ −
q
S n
=
m
This equation can be combined with the expression
for the mass flux given by Equation (5.7) and written in
the expanded form
∂
∂
cn
t
5.3.1
Instantaneous Point Source
(5.13)
+ ∇⋅
(
n c
V D
− ∇
nc
)
=
S n
m
In the case where a mass,
M
(m), of conservative con-
taminant is injected instantaneously over a depth,
H
(L),
of a uniform aquifer with mean seepage velocity,
V
(LT
−1
), the resulting concentration distribution,
c
(
x
,
y
,
t
),
is given by the fundamental solution to the two-
dimensional advection-diffusion equation, which can be
written in the form
Assuming that the porosity,
n
, is invariant in space
and time, Equation (5.13) simplifies to
∂
∂
c
t
+
V
⋅∇ + ∇⋅
c
c
(
V D
)
= ∇ +
2
c S
m
(5.14)
2
2
M
tHn D D
(
x Vt
D t
−
)
y
D t
c x y t
( ,
, )
=
exp
−
−
In the case of incompressible fluids, conservation of
fluid mass requires that
4
4
4
π
L
T
L
T
(5.18)
∇⋅
V
0
(5.15)
where
t
is the time since the injection of the contami-
nant (T),
n
is the porosity (dimensionless),
x
is the coor-
dinate measured in the direction of the seepage velocity
(L),
y
is the transverse (horizontal) coordinate (L), the
contaminant source is located at the origin of the coor-
dinate system, and
D
L
and
D
T
are the longitudinal and
transverse dispersion coefficients (LT
−2
). Equation
(5.18) is more commonly applied in cases where a con-
taminant is initially mixed over a depth,
H
, of the aquifer,
not the entire depth of the aquifer, and vertical disper-
sion is negligible compared with longitudinal and
horizontal-transverse dispersion. Contaminants are
seldom released instantaneously into the groundwater,
however, if the duration of release is short compared
with the time of interest, and if the volume released is
small enough not to influence the groundwater flow
pattern significantly near the release point, the instan-
taneous release assumption is justified. A contaminant
mass cannot be added realistically at a point over a
depth
H
. If the contaminant mass is added over an area
A
0
and the initial concentration is
c
0
, the following sub-
stitution into Equation (5.18) is appropriate,
and combining Equations (5.14) and (5.15) yields the
following advection-diffusion equation:
∂
∂
c
t
+
V
⋅∇ = ∇ +
c
D
2
c S
m
(5.16)
The dispersion coefficient,
D
, in porous media is
generally anisotropic, and denoting the principal com-
ponents of the dispersion coefficient as
D
i
, the advection-
diffusion equation can be expressed in the form
3
3
2
∂
∂
c
t
∂
∂
c
x
∂
∂
c
∑
∑
+
V
=
D
+
S
m
(5.17)
i
i
x
2
i
i
i
=
1
i
=
1
where
x
i
are the principal directions of the diffusion
coefficient tensor. This form of the advection-diffusion
equation, which is appropriate for flow through porous
media, is identical to the form of the advection-diffusion
equation used in surface waters.
Solutions to the advection-diffusion equation for
specified initial and boundary conditions can be either
analytic or numerical. numerical models provide
M
Hn
=
c A
0
(5.19)
0
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