Environmental Engineering Reference
In-Depth Information
Short Exercise 11: Estimation of Seepage Flux Using Temperature Data
Diurnal oscillation of temperature in wetland-bed sediments can be used to estimate
groundwater seepage flux based on mathematical analysis of vertical heat transfer.
When the temperature at the sediment-water interface oscillates in a sinusoidal
manner with a fixed period (
), (here we will assume 1 day), and amplitude
A
0
(
C),
then the temperature
T
(
C) of the sediment at depth
z
(m) is given by:
τ
Tz
ðÞ¼
;
t
T
m
ð
z
Þþ
A
0
exp
ð
az
Þ
ð
π
t
τ
bz
Þ
sin 2
=
(3.59)
where
T
m
(
z
) is the time-averaged temperature profile representing the effects of a
long-term temperature gradient,
t
is time, and
a
(m
1
) and
b
(m
1
) are constants
defined by the thermal properties of the sediment and the magnitude and direction
of seepage flux (Stallman
1965
, equation 4; Keery et al.
2007
, equation 2).
Equation
3.59
indicates that the amplitude of oscillation decreases with depth,
and the phase delay of the sinusoidal signal increases with depth. Both amplitude
and phase delay are dependent on the thermal properties of the saturated sediment
and seepage flux. Suppose that the data recorded at two temperature sensors located
at depth
z
1
and
z
2
(
z
1
<
z
2
) have amplitudes of
A
1
and
A
2
, and a phase shift (i.e., time
difference of peak temperatures between two depths) of
t
(s). Seepage flux
q
(m s
1
) is positive for downward seepage in this example, which is the opposite
of its definition elsewhere in this chapter. Seepage is defined this way in this
exercise to be consistent with the construct used by Keery et al. (
2007
). Seepage
flux is related to temperature amplitude by (Keery et al.
2007
):
Δ
!
5
H
2
D
2
4
z
2
2
HD
3
z
2
π
2
c
2
ρ
2
D
4
z
2
H
3
D
4
z
2
q
3
2
q
2
þ
3
q
þ
2
¼
0 (3.60)
λ
e
2
4
ð
z
1
Þ
ð
z
1
Þ
ð
z
1
Þ
τ
ð
z
1
Þ
where
c
(J kg
1
K
1
) and
(kg m
3
) are the specific heat capacity and density,
respectively, of bulk sediment,
ρ
λ
e
is the effective thermal conductivity of bulk
sediment, and
c
w
(J kg
1
K
1
) and
ρ
w
(kg m
3
) are the specific heat capacity
and density, respectively, of water. In addition,
H
¼
c
w
ρ
w
=
λ
e
and
D
¼
ln
A
1
A
2
ð
=
Þ
(3.61)
It also follows that the magnitude of
q
is related to
Δ
t
by (Keery et al.
2007
):
s
c
2
2
λ
e
2
ρ
2
ð
z
2
z
1
Þ
16
π
2
Δ
t
2
jj¼
q
(3.62)
2
c
w
2
Δ
t
2
c
w
2
ρ
w
2
τ
2
ð
z
2
z
1
Þ
ρ
w
2
Therefore,
q
can be estimated from the analysis of temperature signals using
Eqs.
3.60
,
3.61
and
3.62
.
