Environmental Engineering Reference
In-Depth Information
3
w, o, m, and a indicate water, organic, min-
eral, and air, respectively. Typical values for C s
fall in the range of 1.1 × 10 6 to 3.2 × 10 6 J/m 3 °C
(Stonestrom and Blasch, 2003 ).
Estimates of specific flux in the vertical
direction (i.e. drainage, D ) can be obtained by
simultaneously solving Equation ( 8.1 ) and the
Richards equation ( Section 3.3.2 ) for water
movement. Ranges in values of thermal prop-
erties are much less on a relative basis than the
range in magnitudes of hydraulic conductivity,
a parameter in the Richards equation. Hence,
analytical and numerical solutions to Equation
( 8.1 ) and the Richards equation are typically
much more sensitive to uncertainty in values
of hydraulic conductivity than to uncertainty
in values of thermal properties (Constantz et al .,
2003 ), a feature that underscores the usefulness
of subsurface temperature data for estimating
water fluxes.
Analytical solutions to the one-dimensional
vertical form of Equation ( 8.1 ) for vertical flow
are available. Stallman ( 1965 ) developed a solu-
tion to predict temperatures at any depth and
time under the assumption of a sinusoidal vari-
ation in surface temperature. Silliman et al .
( 1995 ) adapted that solution to predict steady
exchange between a stream and an aquifer.
Keery et al . ( 2007 ) and Schmidt et al . ( 2007 ) pre-
sented alternate solutions for the same problem.
Bredehoeft and Papadopulos ( 1965 ) developed
an analytical solution for the case of steady ver-
tical flow through a semiconfining layer.
Numerical models offer more flexibility
than analytical models. They typically are not
bound by assumptions of one-dimensional
steady flow, constant boundary conditions,
or uniform thermal and hydraulic properties.
Lapham ( 1989 ) developed a numerical model to
solve the one-dimensional heat-flow equation
for the case of a stream connected to an aquifer.
A multitude of numerical models are available
that simultaneously solve Equation ( 8.1 ) and the
groundwater flow equation ( Section 3.5 ) or the
Richards equation for variably saturated flow;
these include VS2DH (Healy and Ronan, 1996 ),
HYDRUS (Simunek et al ., 1999 ), FEHM (Zyvoloski
et al ., 1997 ), HST3D (Kipp, 1997 ), SUTRA (Voss
and Provost, 2002 ), and TOUGH2 (Pruess et al .,
2.5
2
Sand
1.5
1
Clay
Peat
0.5
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Water content (m 3 /m 3 )
Figure 8.2 Thermal conductivity of soils as a function
of volumetric water content; points were experimentally
determined (de Vries, 1966 ), and lines are empirical fits to
the data for the three soil types (Stonestrom and Blasch,
2003 ).
component allows determination of specific
flux.
Solution of Equation ( 8.1 ) requires infor-
mation on thermal conductivity, heat capaci-
ties of water and rock matrix, water flux, and
boundary conditions, in addition to measured
temperatures. Thermal conductivity, the abil-
ity of a material to conduct heat, varies with
soil texture and moisture content ( Figure 8.2 )
and ranges from approximately 0.2 W/m°C
for a dry soil to 2.5 W/m°C for a saturated soil
(Stonestrom and Blasch, 2003 ). A number of
methods are available for measuring K T in the
laboratory or the field (Bristow, 2002 ); values
for K T also can be estimated on the basis of
moisture content and other physical prop-
erties such as texture and mineral content
(Campbell, 1985 ).
Heat capacity of water, C w , is the product of
water density, ρ w , and specific heat, c w . A typi-
cal value for C w is 4.2 × 10 6 J/m 3 °C (Carslaw and
Jaeger, 1959 ); this value varies little over the
temperature range from 0 to 100°C. C s is the
product of solid density and specific heat. On
the basis of known heat capacities of constitu-
ent materials, de Vries ( 1966 ) presented a simple
equation to calculate C s :
xC
(8.2)
C
=
x C
+ +
xC
x C
+
s
wwo
o
mma
a
where x is volume fraction, C is heat capacity
of the specific constituents, and subscripts
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