Environmental Engineering Reference
In-Depth Information
and
p
* <<
p
atm
. Under these conditions, the viscosity of
a vapor is also nearly constant, and Equations (5.94) and
(5.95) combine to give
can be applied to give the airflow,
Q
, extracted by the
well as
k
[(
p
/
p
R r
)
2
−
1
]
a
atm
w
Q L
=
π
µ
p
(5.104)
*
w
θµ
a
∂
∂
p
t
ln(
/
)
= ∇
2
p
*
(5.96)
w
k p
a
atm
where
L
(
= b
) is the length of the well intake (screen)
and the pressures are absolute pressures. The pressure,
p
w
, at the well, is generally determined by the blower
characteristics, and based on
p
w
, Equation (5.104) gives
the corresponding airflow. Typically,
p
w
is 5-10 kPa
(0.7-1.5 psi) below atmospheric pressure, and the
radius of influence,
R
, is usually estimated in the field
from a plot of
p
versus
r
. Typical values of
R
are in
the range 10-30 m (30-90 ft) (Charbeneau, 2000),
depending on soil conditions, being smaller for sandy
soils and larger for silty and clayey formations (Prakash,
2004). Fortunately, Equation (5.104) is not very sensi-
tive to
R
, and, if no data are available, a value of 12 m
(40 ft) can be used without a significant loss of accuracy
(Johnson et al., 1990). Typically, if the intrinsic air per-
meability,
k
a
, is less than 1 darcy (10
−12
m
2
), flow rates
may be too low to achieve successful remediation in a
reasonable time frame (Cohen and mercer, 1993). Air
permeability tests are utilized in predesign studies. In
air permeability tests, air is removed from an extraction
well, measurements are made of the subsurface pres-
sure distribution, and the measured distribution com-
pared with Equation (5.101) to determine the air
permeability,
k
a
. This approach is almost identical to
the procedure used with the Theis equation to deter-
mine the transmissivity and storage coefficient in the
saturated zone (Chin, 2013).
Equation (5.104) assumes that the airflow is steady
and horizontal. However, the source of air is usually
directly from the atmosphere at the ground surface, so
vertical flow components might be significant. Since the
ideal conditions associated with analytical solutions
seldom exist in reality, analytical relations, such as Equa-
tion (5.104), are most useful for screening purposes and
for exploring the relationships between variables, and
their practical applicability is limited to simple prob-
lems. For more complex problems, numerical models
are generally required (e.g., Poulsen et al., 1996;
Rathfelder et al., 1991). Removal rates,
M
, for contami-
nants with concentration
c
in the vapor phase can be
approximated by
where the intrinsic permeability,
k
a
, for airflow has been
assumed constant. In the common case where air is
extracted from a well, the pressure distribution sur-
rounding the well is radially symmetric, and Equa-
tion (5.96) can conveniently be written in radial
coordinates as
θµ
a
∂
∂
p
t
*
1
∂
∂
∂
∂
p
r
*
(5.97)
=
r
k p
r
r
a
atm
This equation is to be solved subject to the following
initial and boundary conditions:
p r
*( , 0
=
(5.98)
p
*( , )
∞
t
= 0
(5.99)
*
∂
∂
p
r
Q
k
lim
r
= −
(5.100)
2π
(
a
/
µ
)
b
r
→
0
where
b
is the thickness of the airflow zone and
Q
is air
pumpage rate. The solution of Equation (5.97) subject
to Equations (5.98-5.100) is (Johnson et al., 1989, 1990)
Q
b k
p
*
=
W u
( )
(5.101)
4π
(
a
/
µ
)
where
W
(
u
) is the well function, and
r
k p t
2
θµ
a
(5.102)
u
=
4
a
atm
Typically, for sandy soils the pressure distribution
approximates a steady state in 1-7 days (Bedient et al.,
1999), and the steady-state pressure distribution is given
by (Johnson et al., 1989)
ln(
r r
R r
/
/
)
(
)
w
p r
( )
2
−
p
2
=
p
2
−
p
2
(5.103)
w
atm
w
M fQc
ln(
)
=
(5.105)
w
where
p
w
is the pressure at the well with radius
r
w
, and
p
=
p
atm
at the radius of influence
R
. Using the pressure
distribution given by Equation (5.103), Darcy's law
where
f
is the fraction of pumped air that flows through
contaminated soil. The contaminant concentration,
c
, in
the pumped air can be estimated using a combination
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