Environmental Engineering Reference
In-Depth Information
Papadopulos (
1965
) are then compared with
the data and Β is selected from the curve that
best matches the data. The value of
v
z
is then
determined from Equation (
8.5
). The drainage
rate is equal to the product of
v
z
and volumet-
ric moisture content. As an alternative to the
type-curve matching, a parameter estimation
routine, such as PEST (Doherty,
2004
), can be
applied to Equations (
8.3
) and (
8.4
) to determine
the Β value that produces the best match with
measured temperatures.
Constantz
et al
. (
2003
) determined tem-
perature profiles for drainage rates of 1, 10,
and 50 mm/yr (
Figure 8.4a
). Results for 1 and
10 mm/yr are nearly linear with depth and
appear to provide a reasonable match to meas-
ured temperatures. Results for 50 mm/yr are
more nonlinear with depth; the higher flux rate
results in a concave profile. The authors could
conclude only that the magnitude of the verti-
cal flux appeared to be less than 10 mm/yr.
Stallman (
1967
) proposed a slight modifica-
tion to the Bredehoeft and Papadopulos (
1965
)
approach that allows for finer resolution of low
vertical velocities. Instead of plotting dimen-
sionless temperature on the x axis, Stallman
(
1967
) plotted dimensionless depth minus
dimensionless temperature [
z/L
- (
T
(
z
) -
T
0
)/
(
T
L
-
T
0
)]. The data from PW-1 are replotted in
this format (
Figure 8.4b
), and although the data
points do not match theoretical curves exactly,
the curves for Β = 0.6 and Β = 1.2 completely
bracket the data. Velocities calculated from
Equation (
8.5
) are 18 and 36 mm/yr downward
for Β equal to 0.6 and 1.2, respectively, assum-
ing
K
T
= 0.8 W/m°C (Constantz
et al
.,
2003
). So
the modified approach produces a more refined
range of drainage estimates, between 1.8 and
3.6 mm/yr, assuming an average moisture con-
tent of 0.10 (Reynolds Electrical and Engineering
Company,
1994
).
Analysis of temperature data can provide a
quick and easy estimate of diffuse drainage rates
within the geothermal zone, but the method
is limited by uncertainty in values of thermal
conductivity and by assumptions of uniform
thermal conductivity and moisture content.
Stallman (
1967
) stated that sensitivity for the
method is about 0.1 mm/d when applied over
5 m thick semiconfining beds with tempera-
tures measured to an accuracy of 0.001°C.
As demonstrated by this example, however,
greater sensitivity can be obtained when meas-
urements are made over much wider depth
intervals.
8.3.2 Diffuse drainage in the surficial
zone
The majority of applications of heat to trace
diffuse drainage rates have centered on the
geothermal zone, where drainage rates are
assumed constant in time. A few studies
have examined the use of heat to trace dif-
fuse drainage rates within the surficial zone.
Cartwright (
1974
) identified groundwater dis-
charge and recharge zones on the basis of dif-
ferences in soil temperatures at a depth of 1 m
between winter and summer. Groundwater
discharge tends to moderate temperature
fluctuations; areas with small temperature
differences were assumed to be dominated by
discharge. Recharge zones were those areas
that displayed large differences. This qualita-
tive approach was extended to provide quan-
titative estimates of infiltration based on
differences (over periods of weeks to years) in
soil temperatures at single or multiple depths
down to 1 m (Tabbagh
et al
.,
1999
; Bendjoudi
et
al
.,
2005
; and Cheviron
et al
.,
2005
). The abun-
dance of available soil-temperature data and
the ease of application make these methods
attractive, but there are complications. Water
percolating downward within the surficial
zone may never reach the water table; it may
be withdrawn by plants and returned to the
atmosphere by evapotranspiration. Shallow
water-table depths may impact the soil tem-
perature profile and affect the calculations.
Cheviron
et al
. (
2005
) examined six 3-year peri-
ods for a site in central France. The predicted
infiltration rates were used with a water-
budget equation to estimate recharge. For
four of the periods, estimated recharge rates
were deemed acceptable; for the other two
periods, estimates were unacceptable because
total recharge exceeded precipitation. With
further research, these methods may prove to
be useful in recharge studies.
