Environmental Engineering Reference
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solution to Equation ( 8.1 ) for the special case of
steady vertical water flow through a semicon-
fining bed of uniform hydraulic and thermal
properties. The approach requires measure-
ment of the temperature-depth profile across
the semiconfining bed and an estimate of
average thermal conductivity. The measured
profile is plotted in dimensionless form and
matched to a type curve. Once the appropriate
type curve is selected, vertical velocity can be
calculated. Flux is then determined as veloc-
ity times porosity. Numerous studies have
used the Bredehoeft and Papadopulos ( 1965 )
approach for estimating vertical rates of water
flow (Sorey, 1971 ; Silliman and Booth, 1993 ;
Constantz et al ., 2003 ).
The Bredehoeft and Papadopulos ( 1965 )
solution was developed for estimating flow
through a semiconfining bed under saturated
conditions. However, the approach is applic-
able over any interval of the saturated zone that
is dominated by steady, vertical water move-
ment. It is also applicable in certain intervals
of thick unsaturated zones where water perco-
lates downward at a steady rate (see the discus-
sion on the unit-hydraulic gradient method in
Chapter 5 ; many of the assumptions for that
method are also applicable here) and hydraulic
and thermal properties (including moisture
content) are uniform (Constantz et al ., 2003 ).
Shan and Bodvarsson ( 2004 ) developed an ana-
lytical solution for the case of steady, vertical
flow through multilayered geologic media; this
solution is applicable for both the saturated and
unsaturated zones. Using temperature logs that
spanned a 400 m thick segment of unsatur-
ated zone at borehole SD-12 at Yucca Mountain,
Nevada, Shan and Bodvarsson ( 2004 ) estimated
that water was percolating through the unsat-
urated zone at the rate of about 15 mm/yr.
Numerical models of water and heat transport
also have been used to determine the drainage
rates that are required to reproduce tempera-
ture profiles in the unsaturated zone (Kwicklis,
1999 ; Flint et al ., 2002 ; Constantz et al ., 2003 ).
Numerical models may require more effort to
apply, but they are not limited by assumptions
inherent in analytical solutions to the heat-flow
equation (as described in Section 8.2 ).
20
Upward
40
60
80
Downward
100
18
18.5
19
19.5
20
Temperature (°C)
Figure 8.3 Hypothetical vertical temperature gradients
within interval of geothermal zone of uniform thermal
properties for cases of no vertical water movement (solid
line), steady downward water movement (dotted line), and
steady upward water movement (dashed line). Depth is
measured from top of interval.
movement and if thermal conductivity is uni-
form ( Figure 8.3 ). In reality, thermal conductiv-
ity is rarely uniform, but under steady heat-flow
conditions, within each layer of uniform prop-
erties, a constant vertical temperature gradient
exists. The occurrence of upward or downward
water movement adds an advective component
to heat flow, thus altering the temperature
gradient. Groundwater moving upward car-
ries with it heat from deeper in the profile,
enhancing heat flow toward land surface and
producing a convex pattern of higher tempera-
tures relative to the conduction-only condition
( Figure 8.3 ). Water at land surface has a cooler
temperature, on average, than that in the sub-
surface, so water movement downward from
land surface tends to inhibit the upward flow of
energy, resulting in a concavity in the tempera-
ture profile ( Figure 8.3 ).
Analysis of temperature-depth profiles to
estimate rates of vertical groundwater move-
ment was first proposed by Suzuki ( 1960 ) and
Stallman ( 1965 ). Perturbations from the profile
expected under pure conductance are assumed
to be related to water percolation, and models
are used to determine the flux rate that pro-
duces the best agreement between simulated
and measured temperatures. Bredehoeft and
Papadopulos ( 1965 ) developed an analytical
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