Amer[can Mineralogist, Volume 6], pages 560-568, 1976
Estimation of thermal diffusivity from field observations of temperature as a function of time
and depthl
W. M. Aoerras,
GnoncsWerrs nNnGEoRcEMesoN
Departmentof GeologyandGeophysics,
Uniuersityof Hawaii
Honolulu.Hawaii 96822
Abstract
Methods have been previously reported for estimating the thermal conductivity from field
observations;application of severalsuch methods gives resultsthat differ significantlyfrom
one another. The causeof the discrepancyappearsto be that each method only utilizessome
of the information contained in the data. Therefore,a method has beendevelopedthat usesall
of the information, giving a "best" answer,in this sense.The method has beenapplied to data
taken from beneath Lake Waiau, a tarn on Mauna Kea, Hawaii. Observationstaken over
more than two years at depths in the sedimentsto about ten meters are used to estimatethe
thermal diffusivity of the sediment.The thermal conductivity of some sedimentsampleshas
been measuredin laboratory apparatus for comparison with the field results.
Introduction
The thermal diffusivity is related ro the thermal
c o n d u c t i v i t yt h r o u g h t h e r e l a t i o ng i v e n i n K a p p e l m e y e r a n d H a e n e l ( 1 9 7 4 ,p . l 0 ) a s
K : Dpc
D:
(l)
where K is the thermal conductivity (heat flow per
unit time per unit distance per degree of temperature), D is the thermal diffusivity (area per unit
time), p is the density (massper unit volume) and c is
the specificheat (quantity of heat per unit weight per
degreeof temperature).
The therrnal diffusivity can be estimated from
measurementsof amplitude decrementand/or phase
differenceof the temperaturewaves betweenvarious
depths in the ground. The decay with depth of the
amplitude of the temperature wave is given, theoretically by
Tr:
fr9-lzz-z'lViTDP
Q\
where Ir is the amplitude of the temperaturewave in
the ground at depth zr, Tris the attenuatedamplitude
at depth z, in the ground, P is the period of the
temperaturewave, and D is the thermal diffusivity of
the ground. The phase shift, 4, of the attenuated
temperaturewave between depths z, and z, is
6 : Q, - z')GFDP.
From equations (2) and (3) the thermal diffusivity
can be expressedas a function of the temperature
a m p l i t u d e sb y
(3)
1Contribution No. C-751of the Hawaii
Instituteof Geoohvsics.
University of Hawaii
(2" - zr)'rf P
lln (Tr) - ln (Iz)1"
(4)
and as a function of phase shift by
D = (2, - zy)2r/Pg'.
(5)
This method is discussedin more detail by Kirkham
and Powers (1972) and by Kappelmeyer and Haenel
( 1 9 7 4 ,p . 8 7 ) .
The temperatureprofile curve (temperatureversus
d e p t h a t a g i v e n t i m e ) c a n b e o b t a i n e db y m e a s u r i n g
the temperatureat severaldepths within the ground.
If this temperatureprofile curve is acquireda number
of times during the penetration of the temperature
wave into the ground, the measurementof the depth
at the crossoverpoint of any two of the temperatureprofile curveswithin the same periodic cycle provides
input data for a method of calculatingthermal diffus i v i t y . T h e r e l a t i o n s h i pp r o v i d e d b y L o v e r i n g a n d
G o o d e ( 1 9 6 3 ,p . 2 7 ) i s ( t h e m i n u s - p l u sw a s e r r o n e ously givenas plus-minus.)
D:
42,'tr/ P
l(t, I
t")2r/P T (2n I
Dol'
(6)
where 11and t, are the times the measurementswere
taken from the beginningof the driving function, z" is
the depth at the crossoverof thesetwo curves,and r
560
THERMAL DIFFUSIVITY FROM FIELD OBSER'/ATIONS
561
i s 0 , 1 , 2 , e t c . , r e p r e s e n t i n gt h e c o r r e s p o n d i n gf i r s t ,
second, third, etc. crossoverof the two curves. The
a n n u a l t e m p e r a t u r ew a v e i s a s s u m e dt o b e a p e r i o d i c
driving function. The thermal diffusivity, D, is the
averagethermal-diffusivity value of the test material
between the crossing points and the level providing
the driving function. The thermal conductivity can be
indirectly obtained from thermal diffusivity measurements (deterrninedfrom periodic methods) by either
measuring or estimating the density and the specific
heat of the test material, thus introducing more measurement error into the thermal-conductivitydetermination.
We now show our attempts to apply theseconventional techniques to a practical problem and how
the inconsistent results of the different methods
prompted us to develop an improved method. We
then describethat method and its application.
Estimation of thermal diffusivity from field
observations
Lake Waiau is situatedin the Waiau cone, near the
s u m m i t o f t h e i n a c t i v ev o l c a n o M a u n a K e a , o n t h e
island of Hawaii. A more specificlocation is shown in
F i g u r e 1 .T h e e x i s t e n c e
o f a n e g a t i v et h e r m a lg r a d i e n t
under Lake Waiau has been determined by Woodc o c k a n d G r o v e s ( 1 9 6 9 ) .T h e c a u s eo f t h e a n o m a l o u s
gradient remains uncertain. The answer is contingent
on knowledge of the differencesin the relative thermal properties of the lake water, the lake sediment,
a n d t h e c i n d e r sa n d l a v a s u r r o u n d i n gt h e l a k e ,a s w e l l
as the thermal regime establishedin the area by natural processes.The purpose of this study is to determ i n e t h e t h e r m a l c o n d u c t i v i t yo f t h e l a k e s e d i m e n t s
i n w h i c h t h e n e g a t i v et h e r m a l g r a d i e n to c c u r s .T h i s
w i l l a l s o a l l o w u s t o o b t a i n a n e s t i m a t eo f t h e h e a t
flux through the sediments.
Woodcock et al. (1966) have describedthe upper
two meters of the sediments.This section contains
two coarse layers of black ash and severallayers of
finer gray ash comprising about 5 and 10 percent,
respectively,of the section.The remaining 85 percent
of the sedimentsare colorful shadesof red and olivegreen. These colorful layers consistprimarily of very
fine particles,lessthan 0.05 mm in diameter, believed
to be windblown from local sources, and about 5
percent is combustible organic materials.
Table I lists the thermal-probe data obtained by
A. H. Woodcock that were used in the thermal-gradient determination.These data have also been used
in this study to make estimatesof the thermal diffusivity using the non-steady-stateperiodic methods
LAKE
WAlAU
0
20
40 Meters
F t c I M a p o f L a k e W a i a u s h o w i n g e s t t m a t e dd e p t h c o n t o u r s
i n m e t e r s .T h e i n n e r s q u a r e m a r k s t h e l i m i t s o f t h e a r e a i n w h i c h
t h e t e m p e r a t u r e m e a s u r e m e n t sw e r e m a d e , a f t e r W o o d c o c k a n d
Groves ( 1969).
previously described.The estimateswere first made
without any attempt to smooth the data, therefore
large variancesare to be expected.
Estimates by amplitude decay and phase lag
Figure 2 is a temperature-time graph of the thermal-probe measurementsfrom Lake Waiau at the 3meter and the 5-meter depths below the iake surface.
The amplitudes Z, and T, are taken as one half of the
differenceof the maximum and the minimum temperature valuesin each data set.The amplitude is 2.93'C
at the 3-meterlevel and 0.73'C at the 5-meterlevel.If
we use equation (4) for temperature variation, the
estimate of the thermal diffusivity in the sediment
layer between3 and 5 metersbelow the lake surfaceis
0.00205cm2lsec.
We can also plot the data for the S-meterlevel on
an exaggeratedvertical scaleso the curve is about the
same size as the curve for the 3-meter plot. Then by
placing one curve on the other and sliding it along the
time axis until the curvesmatch (or by cross-correla-
ADAMS, WATTSAND MASON
o nooofdt icmo ec ;k a n d G r o v e s ( 1 9 6 9 )
T 4 e l e L T e m p e r a t u r e ( ' C ) a t s t a n d a r d d e p t h s i n L a k e W a i a u s e d i m e n t s a s a f u n c t iW
Depth*
( ' o)
JuLy
7 ,1965
3.0
8.9
B.l+
l+.0
4.>
5.0
6.0
7.0
8.8
B.r
7.5
7.0
6.7
9.3
B.o
7')+
' l. o
6.7
o. )
o. J)
6.)+5
6.2,
July
)7
Nov.
l
Dec.
l-
b.4
0.J
7.7
7. t
7.0
o,I
7)7)
b.4
7.0
7.r
6.2
b. o>
o, o)
o.l
6.3
6.8
a6
6.8
5.8
6. l+
June
f
July
)
3.0
9.0
8.0
9.0
7.r
5.8
*Depth
o. {
6.0
,.9
o.u
o.l
be70w
6.2
water
ac
2a
Depth *
(n)
\.0
4.>
5.0
6.0
7.o
9.6
8.3
Aug.
11
8.2
o. I
Nov,
Aug .
)'t
o)
7.0
Sept.
I ).t
July
surfacei
for
depth
l.+
6.8
6.6
belour
tion), we can estimatethe phaselag by measuringthe
time difference between the zero-time axes. The estimated phase shift between these two curves is 60
days. And with equation (5) based on the phase lag,
we obtain an estimate of 0.00374 cm2/sec for the
thermal diffusivity of the same2-meter layer.There is
a differenceof 45 percentbetweenthesetwo methods
() -^
o)
=
o
o
cl
E
6,0
r,6
o
200
400
Time,in doys
600
800
Frc. 2 Temperature, from thermal probes in Lake Waiau
s e d i m e n t ,p l o t t e d a g a i n s tt i m e f o r m e a s u r e m e n t tsa k e n a t t h r e e a n d
five meters below the water surface
9
Jan.
6,1966
May
1
Feb.
Mar.
15
19
5, 3
l+.8
)+9
.
5.0
5.\
6.0,
6.2
6,2
,.9
5.B
5.9
5.0
6.2
6.3
7A
\t
a?
o.a
7.0
6.8
o. o>
6.t4
o.l
3.)+
)+.0
5.r
5.6
6.0
6.2
6.rj
Jan.
Mar .
APr.
May
2 6 , :-967
3
B
L
5.3
)+.8
)+.9
5.\
5.8
5.r
5.9
,.8
5.8
6.0
6.5
5.9
, .9
6.r
7.9
7.3
6.95
5.6
^
<
6.3'
6.)+
aL
Dec.
zB
) +. 2
7^
"R
E1
o.J
4.
6.8
6.9
6.9
>.o
I
o.f
6.)1
o.1
sediment
surface
subtract
-
2.85
a^Aa
?
w ,
I
"c
n.
for this layer of sediments,even though the resultsare
basedon the very same set of data.
The data in Table I have been evaluated for each
level of measurementscombined with every other
level. The results are shown in Table 2. The differencesbetweenthe amplitude estimatesand the phase
estimates have a mean value of 50 percent. If we
assume that these sediments are isotropic and
homogeneousin the layer between the 3-meter and
the 7-meter depths, then the thermal diffusivity
should be the same for each of the above estimates.
The mean thermal-diffusivity value from the amplitude computations is 0.00298 cmz/sec with a standard deviation of 0.00212,and the mean of the phase
computations is 0.00410cm2/secwith a standard deviation of 0.0103.The differencebetweenthesephase
and amplitude estimatesis 38 percentlThis difference
was attributed to the methodology being inadequate
to resolve the unsmoothed data, rather than the assumptions;an alternatemethodology was usedto try
to obtain a more reliable value for the thermal diffusivity.
Estimate by crossouer of temperature profiles
Figure 3 is a graph of the temperature-depthprofiles for the measurementsmade on 27 July 1966and
28 December 1966in the sedimentsof Lake Waiau.
THERMAL
DIFFUSIVITY
FROM FIELD OBSERVATIONS
TneLr2. Thermal-diffusivitycomputationsfromamplitudedecayandphaselagofthethermal-probe
data taken in Lake waiau. 1965-1967
!r!
r uDr
v r
uJ
!rr
r
urf
v a uJ
llnnc-
T,owcr
i- l!n9n" e r
Danf.h
lenth
Annl
(cm)
(crn)
(oc)
jluflq
(cmz/ see)
(cmzlsec )
0. 00281+
0 . 0 0 1 75
0.0020]+
0.00205
0. 0026].l
0.0030?
0.00840
0.00350
0.00391
0.0037]+
O .O O 2 \ 1
0. 00282
300
300
300
300
300
300
350
Loo
)+50
500
5oo
700
2.93
2 .93
r.a2
0. T3
o. )+6
0. 30
O .O O f f B
0.00fT6
O .O O ] - 8 6
0. 00260
0.003t1
o.0o[29
0.00309
0.00197
o ,00202
0.00233
350
350
350
350
l+00
2.IB
2.IB
2.TB
2.LB
2.LB
t.38
L,02
0.73
o. l+6
0. 30
o. o02Bg
0.002)+3
o.00336
o. oo387
o .40259
0.0011+O
0.00205
0. 0026)+
l , ^
r
1
)+oo
0. 00208
0.0035t
o. o0)+r2
0.00r46
0.00197
0.00278
4>U
0.oo\95
0.005f2
0.00328
0.00rT\
o.00129
o.o277B
3ro
^
4ro
\ro
500
6oo
700
)+50
500
6oo
700
2 .r B
2.93
"A
r3B
fL
33
ol
ro2
133
1B
)19
B1
107
i)
24
500
500
6oo
700
0.73
5oo
700
O .) + 6
1
OU
LT2
0.73
O.l+5
0.30
1.38
r
"8
L.02
1.02
The crossoverdepth of thesecurvesoccursat 447 cm.
II we assumethe temperaturewave detectedat the 3meter depth to be a periodic driving function with a
period of 365 days, then the thermal diffusivity of the
sedimentsbetween300 and 417 cm can be estimated
b y e q u a t i o n( 6 ) t o b e 0 . 0 1 3 8c m 2 / s e c .
Table 3 is a listing of 43 pairs of curvesevaluatedin
a similar manner. The mean thermal diffusivity is
estimatedto be 0.00765cm2lsecwith a standard deviation of 0.0135, and the distribution was very
strongly skewed toward the lower values.The large
valuesare due to poor determinationof the crossover
depth, which occurs when the curvescross at a small
angle instead of at nearly right angles-the ideal case
depicted in Figure 3. A better statistic for this situation is the mode, rather than the mean. The mode of
t h i s d i s t r i b u t i o n i s 0 . 0 0 1 3 0c m 2 l s e cw i t h a s t a n d a r d
deviation of 0.00030. The computations are made
under the assumption that the sediment layer between 300 and 682 m below the lake surface is isotropic and homogeneous.
Such large differencesamong the various ways of
estimating the thermal diffusivity make evident that
the answer was more dependentupon the method of
analysisthan on the data! If each method were theo-
l0
31
l+)+
I.38
500
6oo
700
)r q^
Phase
Lag
(days)
Lover
Anplitude
(oC)
o.lr6
0.30
0)
o1
B7
o,l+6
0. 30
BB
0. 30
l0
retically correct and usedthe samedata base,then the
resultsshould be identical. Each method did seemto
have a correct theoretical basis but used different
portions of the data, e.g., the temperature amplitudes, the phase differencebetweendepths,and only
the crossoverpoints. The large differencebetweenthe
results thus indicated the need to use all of the data
available.
("C)
Temperoture
4.O
5.O
6.0
7.O
8.0
o
o
o
E
.E
o
(,
o
t
I
o
o
'
o
q)
ll
E
cl
q)
o
Frc. 3. Temperatureprofile of two thermal-probe
measurementsmadein the sediments
of LakeWaiau.
ADAMS, WATTSAND MASON
564
Estimate by an improued method: the least-squares
technique
Figure 4 is a map of the temperature-probemeasurementson the time-depth plane. Two distinct features are expressedin the characterof the isotherms.
The sloping of the troughs and ridgesfronl the left at
the top toward the right at the bottom is an indication of the phase lag throughout the layer. And
the isotherm gradient decreasesmarkedly from the
top to the bottom, indicating the decayof the temper-
T n s l - r 3 T h e r m a l - d i f f u s i v i t yc o m p u t a t i o n s f r o m t e m p e r a t u r e - p r o f i l ec r o s s i n g so f t h e r m a l p r o b e s
in Lake Waiau
Therrnal
Cros sover
IJrltUSr-v].ry
(cm2lsec;
n^-+1
2nd
T ime
( aays )
!eprn
(cm)
0.001f0
0.001f5
0. 00099
0 .00f22
317
337
339
3\2
21+7
r-Bl
o.oooBS
o.o]+982
3\3
\2'
r81
2IT
3?l
rB9
,9
59
r89
0.a6r73
375
3BI
383
3B)+
189
238
2L7
105
o.a2\93
o. o29r9
o.0or-rI
0.001r5
387
387
390
390
o.o1o67
O .O O r O I +
o.ot699
o. orB3o
0 . 02 311+
0.001f6
?qn
2LI
238
Summer
Cyele
( d a . w s)
287
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THERMAL DIFFUSIVITY FROM FIELD OBSERVATIONS
56s
o
q)
O
J4
LrJ
lrE
=
a
rd5
Y
=
o
Ll
co6
I
F
(L
UJ
o
T l ME ( d o y s )
F t c 4 . M a p o f t h e t e m p e r a t u r e - p r o b em e a s u r e m e n t so n t h e t i m e - d e p t h p l a n e . I s o t h e r m a l c o n t o u r s a r e s h o w n
ature fluctuation with depth in the layer. Thus, the
isotherm map is a more continuous and total representation of the propagation of the annual temperature wavs through the sedimentlayer over the time
span of the measurementthan any two selecteddepth
o r t i m e s e c t i o n st h r o u g h t h i s m a p .
A well-definedtrough on the right side of the map
in Figure 4 shows a phaseshift of 94 days in the fourmeter layer. The thermal diffusivity is estimatedto be
0.00609 cm2/sec using the phase-lag computation.
However, this particular phaselag is not representative of the entire map, and there is no other welldefined ridge or trough that would be a more representativephaselag. If we look at the amplitude decay
of the temperature range with depth, the computations give an approximatethermal-diffusivityvalue of
0.0023 cm2,/secthroughout the sediment layer. Another isotherm map, similar to Figure 4, was constructed for the idealizedcase of a constant thermal
diffusivity of 0.0023 cmz/sec.
The temperature, 7, at any point in the sediment
layer can be representedby the following relation
f r o m L o v e r i n ga n d G o o d e ( 1 9 6 3 )
D
z
I
@
-thermal diffusivityof sedimentlayer,
-depth of temperature
measurement,
-time of temperature
measurement,
and
-phase displacement
in time.
This equation is then used as a model to obtain a
least-squaresfit of the observed data in Table I, as
representedby the isothermal map in Figure 4, to
various idealizeddata sets.
The temperature, I, the depth, z, and the time, I,
are taken to be the known parametersin this model.
The period, P, is set to 365 days, and the mean
temperature, T*, is set to 6.3oC from valuesobtained
in the previous calculations.This leavesthree parameters, the temperature range, 7", the phase displacement, S, and the thermal diffusivity, D, to be
fitted in the estimation.
The program for calculating least squares was
checked and debugged using an artificial data set.
Values at the points T(t, z) of the observeddata in
Figure 4 were taken from the idealizedmap with D :
0.0023 cm"/sec by superimposing the two maps.
When the least-squares
program was run on the artificial data set, it convergedon the expectedvaluesof
T: T^ t r,s-zvc7DP sin [(r + 6)2tr/p - zy'i7DP1 1t1 the input parameters,including D : 0.0023cm"/sec.
After the general region of the least-squaresminwhere
i m u m w a s d e t e r m i n e d ,i t w a s p o s s i b l et o d e t e r m i n e
T, -temperaturerangeat drivinglevel,
the extreme value for the sum of the squared differT--mean temperature
of sediments,
ences between the observed and the idealized data
566
ADAMS, WATTSAND MASON
o
(l)-
(|)
E
-4
trl
o
tL
E
l
a
bJ5
Y
=
J
LrJ
co6
T.
c
UJ
o
T lM E (doys)
that best-fitthe data in Fig' 4
Frc. 5 ldealizedmap on the time-depthplane,usingthe valuesof parameters
setsby varying the valuesof the input parameters,f'
g and D, by small amounts. The minimum value for
the sum of the squareswas found to be 30.3462("C)
for 147data points. The correspondingparametersat
t h i s m i n i m u m a r e T , : 2 . 6 5 5 o C , 0 : 7 0 . 6 7d a y s a n d
D : 0.OO2l2cm'/sec. This calculation then representsthe fit of the best regressionsurfaceof temperature to the observed temperature surface on the
sametime-depth plane. An idealizedmap of the temperature over the same time interval and layer thicknesswas constructed using theseparameters,including D : 0.00212cm'/sec, shown in Figure 5. For the
computer program used in this estimate see Watts
( 1 9 7 5 )A p p e n d i x .
We now describe laboratory measurementsmade
to obtain values of thermal conductivity for comparison with the values for thermal diffusivity from
the field measurements.
Thermal conductivity of sedimentsunder a Hawaiian
alpine lake
A 2-meter core sample from the sediment layer
between3 and 5 metersbelow Lake Waiau's surface
was obtained at the temperature-probemeasurement
s i t e b y A . H . W o o d c o c k o n 5 M a y 1 9 6 7 .A m o r e
specificlocation of the selectionsite is shown in Figure I (see the square). Two specimenswere sliced
from the ends of this core. one from the 3-meterand
the other from the 5-meter level. Each sample was
placed inside a cast acrylic annulus in order to, be able
to use the steady-stateapparatus (Watts, 1975).The
t o t a l h e a t l o s s i n t h e m e a s u r e m e not f t h e s e d i m e n t si s
a p p r o x i m a t e l y8 . 5 p e r c e n t( s e eW a t t s , 1 9 7 5 ,C h a p t e r
3 ) . T h e h e a t - f l u xv a l u e s c o m p u t e d f o r t h e t h e r m a l conductivity measurementshave been adjusted to
compensatefor this loss. The results of the thermal
conductivity measurements in the laboratory are
listed in Tables 4 and 5. The measurementsof the 5meter sample are consideredunreliable lor the later
t i m e s . T h e m e a n o f t h e f i r s t f o u r m e a s u r e m e n t si s
oC and is consideredthe thermal0.0029 cal/sec cm
c o n d u c t i v i t y m e a s u r e m e n to f t h e 5 - m e t e r s a m p l e '
The mean value of the thermal-conductivitymeasurements for the 3-metersampleis 0.0024cal,/seccm "C.
The difference between the steady-statemeasurements of the 3-meter and the 5-meter levels is 20
percent. The mean particle density was found to be
2.40 gm/cm3 and the moisture content of the sample
from the 3-meter sample was 76.7 percent, following
p r o c e d u r e so f L a m b e ( 1 9 5 1 ) .T h e b u l k d e n s i t yf o r t h e
3-meter sample is computed to be I .36 gm/cm'.
The core sample was not stored in a sealedcontainer over the past eight years, therefore, the moisture-contentvalue cannot be representativeofthe "in
situ" situation. Woodcock and Groves (1969) report
values of moisture content at the 4-meter and the 6-
THERMAL
DIFFUSIVITY
FROM FIELD OBSERVATIONS
Tegt-r4. Thermal-conductivitymeasurementsofasedimentsamplefrom3metersbelowthesurfaceof
Lake Waiau
Tiroe
of
Day
Dat e
4.a,pr? 5
5Apr75
5Apr75
5 A p r T5
5Aprl)
6Apr75
6Apr75
6tpr75
6tpr75
6AprT5
Anbient
Air
Temper atur e
(oc)
SanpJ-e Mean
Tenperature
la*A"a*ir'il"
(oc)
qn
zo\,
oB\ 5
1300
I7o,
2I]-,
O5)+O
ogro
13OO
16oo
t9I5
Thermal
22.8
?t
0.0025
0 . 0 02 3
0.0023
o,oo22
o .0026
0.0023
o .oo25
0.0023
o .oo2,
0.002)+
,o.95
2r.o
2r.9
2 1. B
>z.yo
)z.y+
23.0
23.4
33.\T
34 .11+
?t on
60
" )1 ' 7 n ,
?
)cA
26.O
crn oC)
( cal/sec
Regtession equation:
k = 2.5 - 0 .OO7 T mcal,/sec cm oC;
= 0.72 ncaT/sec cm oC; Mean
Standard error
of estimate
A;<fhialz^^^reproducibiTitg
error = 3.6%; eennia
L.240 cm; Sample surface
area = 75.459 cmz.
meter levels to be 74 and 48 percent, respectively.
This indicates that the moisture content decreases
with increasingdepth. The varying amounts of water
with respectto the sediment particles will cause the
bulk density and the specificheat of the sedimentto
also vary with depth. These parametersare used in
the conversionof thermal diffusivity to thermal conductivity through the relation K = Dpc.
If we take the thermal diffusivity to be 0.00212
cm2/sec, as measured, the particle density as 2.40
gm/cmg, as measured,and assumethe specificheat of
the particles to be 0.22 cal/gm oC, then the corre-
sponding values of bulk density, specific heat, and
t h e r m a l c o n d u c t i v i t ya r e o b t a i n e d a n d a r e g i v e n i n
Table 6 for three values of moisture content (weight
percent). This table shows that for decreasingmoisture content, the bulk density increases,and the specific heat and thermal conductivity both decrease.
H o w e v e r ,w i t h i n t h e m o i s t u r ec o n t e n tl i m i t s o f 4 8 - 7 4
percent, the thermal conductivity varies about the
value 0.00205 cal,/seccm oC, differing by no more
than 4.3 percent.
The measurementsmade by the steady-statelaboratory apparatusdiffer from the above estimateby 18
T n s L r 5 T h e r m a l - c o n d u c t i v i t ym e a s u r e m e n t so f a s e d i m e n ts a m p l ef r o m 5 m e t e r s
b e l o w t h e s u r f a c eo f L a k e W a i a -
Date
Tine
of
Anbient
T a n n 6 F ' + , , r 6
(oc)
z9[ar75
2 9 M a r 75
29Mar75
29Mar7,
3OMar75
3OMar75
3oMar15
30Mar75
3oMar75
30Mar75
Regression
Standard
1620
I025
I55o
2I3A
0715
1030
r305
162,
2L20
2330
23 .O
Sarnl
c
Vcan
Temperature
(oc)
ThFrrPl
Conducti vity
(cal/sec
cn oC)
5 8. 2 9
57.85
0.0031
o.oo28
o.oa29
)J.OO
2 ) 4. 6
. l
^
25.8
)7
2a
1
?
of
0.0029
0 ,0021
0.0020
0,0020
0 ,0020
o.oo2l+
a.oo22
)4.O)
[3.)+2
\z.rz
40.80
\2.7o
50.82
40.93
k = - 0.33
+ 0.06
T
= 0.12
estimate
ncaf/sec
=
errot
3.9%;
SanpJe
disc
= f5.459
atea
cnz,
equation:
ertot
teproducibilitg
SampLe sutface
Air
oC;
cm
Mean
= .1.648
thickness
mcal/sec
cm aC;
cm;
ADAMS, WATTS AND MASON
568
T , q s L s6 . F o r f i x e d t h e r m a l d i f f u s i v i t y a n d c o n s t a n tp a r t i c l ed e n s i t y ,v a r i a t i o n o f
t h e r m a l c o n d u c t i v i t vv e r s u sm o i s t u r ec o n t e n t
Moisture
Co n t e n t
(%)
?l+*
oa
)+8*
*Moistute-cantent
Bul k
Density
(gnlcn3)
r .or
1Aa
r.95
vafues
Heat
(callgn
oC)
Thermal
Conductivity
(cal/sec cn oC)
o.522
o.516
0,00212
0.00205
0.00196
U. +J IJ
teoorted
and 4l percent.This differencecan be accountedfor,
in part, by the remolding that was necessaryto fit the
sample into the annuluses.Also, small slices from
within the core sample are probably not representative of the entire sedimentlayer.The most representative value for the "in situ" thermal conductivity of
the sedimentsof Lake Waiau found in this study is
0.00205cal,/seccm oC.
The negative thermal gradient determined by
W o o d c o c k a n d G r o v e s ( 1 9 6 9 )i s 0 . 0 5 2 o C , / m . I f w e
c o m b i n e t h i s v a l u e w i t h t h e t h e r m a l - c o n d u c t i v i t ye s timate above, we obtain a heat flux of 0.0107cal/sec
m2 downward through the sedimentsat the area of
the lake from which the measurementswere taken.
Discussion
The estimateof the thermal diffusivity of the Lake
W a i a u s e d i m e n t sc o u l d b e i m p r o v e d u p o n b y i n corporating another parameter into the new leastsquares method. The annual temperature wave was
a s s u m e dt o b e a s i n ew a v e w i t h a p e r i o d o f 3 6 5 d a y s ,
which is not the case in reality. A more realistic
d r i v i n g f u n c t i o n c o u l d b e m o d e l e di n t o t h e m e t h o d
from the continuous temperaturedata that are monitored at the Mauna Kea weather station.
Conclusions
T h e o b j e c t i v eo f t h i s s t u d y w a s t o o b t a i n a v a l u e
for the thermal conductivity of the sedimentsunder
bq
woodcock
and
Groves
(f969)
Lake Waiau. This was accomplishedby estimating
the thermal diffusivity to be 0.00212cm'/sec using a
least-squaresmethod applied to temperature data
collectedby A. H. Woodcock. The thermal conductivity of thesesedimentsis derived from this estioC. This result commate to be 0.00204cal/sec cm
bined with the results of Woodcock and Groves
(1969) indicates a heat flux of 1.07 pcal/sec cm2,
downward, flows through the 4-meter layer of sedim e n t s i n t h e c e n t e ro f t h e l a k e .
References
K e . p p er . n t v c n , O , q N o R . H , q s N s l -( t 9 7 4 ) G e o t h e r m i c sw i t h s p e cial referenceto application. Geoexploration Monographs Series
1 - N o 4 , G e b r u d e r B o r n t r a e g e r ,2 3 8 p .
Krnrn,rn, DoN ,tNo W. L Powens (1972) AduancedSoil Physics
W i l e y -I n t e r s c i e n c e .
L r M e r , W . T . ( t 9 5 1 ) S o i l T e s t i n gf o r E n g i n e e r s J o h n W i l e y a n d
S o n s .l n c
L o v r n r N c , T . S n N o H D . G o o o r ( 1 9 6 3 )M e a s u r i n gg e o t h e r m a l
g r a d i e n t si n d r i l l h o l e sl e s st h a n 6 0 f e e t d e e p ,E a s t T i n t i c D i s t r i c t ,
Utah U S Geological Suruey Bulletin No I172.
Wnrrs, Geonce P. (1975) Methods for measuring the thermal
conductiuity of earth materials with application lo anistropy in
basalt and heat.flur through lake sedimenls M S. Thesis University of Hawaii,Honolulu,Hawaii.
Wooococr, A H ,qNo GonnoN W GRovEs (1969) Negative
thermal gradient under an alpine lake in Hawaii. Deep-Sea
Research,Supplement to Vol 16, p 393-405
MEvrn RusrN nNo R. A Ducr (1966) Deep layer of
s e d i m e n t si n a n a l p i n e l a k e i n t h e t r o p i c a l m i d - P a c i f i c .S c i e n c e ,
t54,647-648