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Temperature dependence of thermal transport properties of crystalline rocks — a general law
TECTONOPHYSICS ELSEVIER
Tectonophysics291 (1998) 161-171
Temperature dependence of thermal transport properties of crystalline rocks a general law U, S e i p o l d * GeoForschungsZentrum Potsdam, Telegrafenberg A17, 14473 Potsdam, Germany
Received 31 December 1996; accepted 26 June 1997
Abstract The temperature dependence of the thermal conductivity K was fitted to the function K = 1/(A + B x T), T is absolute temperature, for 121 rock samples. Using our own measuring results for 64 samples it was found that there exists a linear relationship between the constants A and B: A = -(532 4- 45) x B + (0.448 4- 0.014). The checked number of literature data reflect almost the same result. An analogous equation was obtained for the temperature dependence of the thermal diffusivity. Using the relationship between the constants A and B the elimination of one of them results in a generalized temperature law: K = 1/(B x (T - 532) + 0.448). For many rock samples a temperature function K = T / ( E + F x T) takes us to a much better fit of the experimental results for the temperature dependence of the thermal conductivity. In this case as well the constants are correlated. The result is K = T / ( F x (T - 314) + 122). © 1998 Elsevier Science B.V. All rights reserved. Keywords: thermal conductivity; thermal diffusivity; temperature; crystalline rocks
1. Introduction Temperature plays the most important role among the various influences on the thermal conductivity in the earth's crust. The temperature effect on the thermal transport properties can be caused directly by the temperature dependence of the heat transfer mechanisms or otherwise indirectly by temperature-induced physical and chemical changes in the investigated rock samples. The influences of the latter depend on the specific mineralogical composition of the investigated rock type, and their rules can not be applied to all rocks. Therefore, only the direct temperature dependence is considered in this paper. *Fax: +49 331 288 1450; E-mail: [email protected]
The investigation of the temperature dependence of the thermal conductivity gives information on the heat transfer mechanisms for the studied rocks because the different mechanisms in general will cause different functions of the temperature dependence. These functions are known from theoretical studies of the heat transport process (Klemens, 1969, 1983). Thus, one can try to fit the experimental results to these functions to find the acting heat transfer mechanisms. In the past, the limited accuracy of the measuring results and the insufficient density of the measuring points often prevented that preference was given to one of the fit functions. But now the progress by computer supported measuring methods and data processing improved the possibilities. The knowledge of the acting heat transfer mechanisms is
0040-1951/98/$19.00 © 1998 Elsevier Science B.V. All rights reserved. PII S0040-1951(98)00037-7
162
U. Seipold/Tectonophysics 291 (1998) 161-171 4.0
\
35
E .E >,
3.0
.>_ "5 X3
~ O
I 25
[ E b-
20
/, 1.5
- -
'
200
Pyriclasile
L 400
~-
I
,--
600
! 800
r 1 O0
Temperature in K
Fig. 1. Temperature dependence of the thermal conductivity for rock samples with different room-temperature conductivities.
not only a theoretically interesting problem. It has also an important practical consequence in so far as it enables us to extrapolate the results outside the direct experimentally studied temperature range. In our experimental investigations of the temperature influence on the thermal transport properties we came to the conclusion that there must exist a general rule. Rock samples with a relatively high thermal conductivity showed a strong decrease of the thermal conductivity with rising temperature, and inverse rock samples with a small thermal conductivity at room temperature only showed a small variation with increasing temperature (Fig. l). From this observation we tried to find a general law which quantifies the qualitative impression. 2. Data sources
Surely it was necessary to test a possible relationship with a quite extended data collection. Therefore, we processed all experimental results obtained by our own measurements. This data set has the advantage that it was obtained under exactly the same conditions (measuring method, sample preparation, measuring regime) for all 64 rock samples. The investigated rock collection consists of gneisses and amphibolites from the KTB-bore-
hole Windisch-Eschenbach, granites from the Saxonian Erzgebirge and metamorphic rocks (granulites, pyroxenites, pyriclasites, amphibolites and serpentinites) from the Saxonian Granulitgebirge. Some of the results have already been published (Seipold, 1990, 1995). In the experimental studies a transient measuring method was applied allowing us to determine simultaneously thermal conductivity and thermal diffusivity. Cylindrical rock samples of 27 mm diameter and 43 mm length were prepared from bore cores or rock blocks. A heat pulse was created by a line source in the axis of the cylindrical rock sample. This pulse of three seconds length propagates in radial directions. At about half the distance between heater and boundary of the sample a thermocouple measured the arriving signal. The thermal diffusivity was calculated from the travelling time of the pulse and the distance between heater and thermocouple. For the calculation of the thermal conductivity the peak height of the temperature pulse and the energy of the heating pulse were also needed. For details of the method see Seipold (1988). In most cases the measurements were carried out up to about 800°C. But often only the results of the lower temperature range were used to avoid disturbances by temperature-induced changes of the samples. For the quartz-containing rocks the results were taken up to the temperature of the ot-~ phase transition (573°C). In amphibolites and serpentinites the upper temperature limitation was determined by the inset of dehydration processes. For comparison a second data set (57 rock samples) was taken from a number of authors (Birch and Clark, 1940; Kawada, 1964, 1966; Schatz and Simmons, 1972; Tkatsch and Jurtschak, 1972; Sakwarelidse, 1973; Beck et al., 1977; Sibbitt et al., 1979; Petrunin and Popov, 1981; Petrunin and Popov, 1983; Petrunin and Popov, 1988). The data were carefully selected taking into account the following conditions. (1) Original measuring data without smoothing or any other data processing were preferred. (2) Only data for igneous and metamorphic rocks without alteration products and amorphous or glassy components were used. Increasing thermal conductivity with rising temperature was considered as an indication for glassy components. (3) Data from a sufficiently wide temperature
163
u. SeipoM/Tectonophysics 291 (1998) 161-171
range and with four or more measuring points were selected. By these conditions the small number of applicable published data in literature is further reduced. One has to keep in mind that there is a great venture to comparing results from various sources obtained by quite different measuring methods and often of unknown accuracy. Frequently, only four or five data points were published for a special rock sample, whereas a number of 35 up to 60 data points is commonly measured in our studies. Further constraints come from the small temperature ranges investigated in earlier studies. In many papers the temperature variation is limited in the interval from room temperature up to 400°C. In all calculations SI-units were used: thermal conductivity was measured in W m -l K -1, thermal diffusivity in 10-7 m2/s, temperature in K. 3. Discussion on the effect of thermally induced cracks Investigations of the thermal transport properties at high temperature without the application of high pressure or water saturation provoke the discussion on erroneous results because, due to differences of the value and the anisotropy of thermal expansivity of the mineral grains, cracks are created at high temperatures. From this argumentation it is surprising that most high-temperature studies of thermal conductivity were carried out at ambient pressure (which also excludes water saturation). In the frequently cited paper of Birch and Clark (1940) the authors point out that their results up to 400°C "may be considered to represent reversible temperature effects". All values were omitted if "subsequent measurements at lower temperature failed to return to their previous values". This reversibility criterium for the confidence of results is a very hard condition (sufficient but not necessary). We have used a less hard condition. We consider the results obtained in a first temperature run as correct if after a temperature release the values of the first and second runs coincide at the highest temperature of the first run. This condition does not exclude the possibility that in a second run smaller values can be obtained at temperatures lower than the maximum temperature of the first run. This criterium is based on the following
assumption: if the temperature is raised, the mineral grains try to expand. Partly this expansion is possible by the volume increase of the whole rock sample and partly the expansion is blocked by the adjacent grains resulting in the formation of local stresses. Also intragranular closed cracks can occur by the effect of shear stresses. But it is unlikely that a high number of open cracks is created. Only such airfilled cracks would essentially diminish the thermal conductivity. The above imagination is supported by our studies of thin sections from rock samples which were heated up to temperatures of 500°C. A direct check of a possible influence of temperature-induced cracks is obtained by studies of the temperature dependence at various pressures. The application of sufficiently high confining pressure should suppress the thermal cracking. Appropriate results under hydrostatic conditions are very rare. A sufficient data set was taken from a study of quartzcontaining igneous rocks by Durham et al. (1987) in the range from room temperature up to 400°C at a pressure up to 200 MPa. The processing of these data showed us that, opposite to the expectations by Pribnow et al. (1996), measurements of dry rock samples do not provide a higher temperature dependence of the thermal transport parameters at ambient pressure compared with investigations at high pressure (see Fig. 2). On the contrary, at higher pressures the temperature dependence seems to be somewhat stronger. 1,8
A
o
•
~1,41
o
•
0.1 M P a
•
5 MPa 20 MPa
o
100 M P a
[~ 1,2-
~
•
~-N--1,0•
_ 0,8I"0,6
0
1~0
2~o
3~0
,~o
Temperature in o C
Fig. 2. Temperaturedependenceof thermal diffusivityat various pressures. Results taken from Durham et al. (1987) for Climax Stock quartz monzonite.
164
U. Seipold/Tectonophysics 291 (1998) 161-171
4. Theoretical background and data processing Diffusive heat transfer in crystalline rocks is determined by the interactions of phonons. In the quasi-particle picture of the heat transfer process the observed value of the thermal conductivity is determined by the resistivities against the propagation of the heat-carrying phonons. Phonons are scattered in phonon-phonon interactions caused by nonlinearities of the lattice potential and by interactions with the various types of imperfections of the crystal lattice. As the measure for the number of scattering processes the mean free path length of phonons was introduced. Its value decreases in most silicate minerals from the order of about 10 ~ at room temperature to a few hngstr6m at 1000 K (Petrunin and Popov, 1994). This means that the number of scattering processes within the mineral grains is by orders higher than the scattering number by cracks and grain boundaries. Therefore, it is expected that they play an essential role only if an additional macroscopic contact resistance appears at the grain boundaries or if open air-filled cracks occur. The temperature dependence of the thermal conductivity depends on the contributing heat transfer mechanisms. For simple ionic compounds the thermal conductivity is proportional to T -l, where T is the absolute temperature. The temperature dependence according to this so-called Eucken law (Eucken, 1911) is characteristic for a heat transfer dominated by 3-phonon Umklapp processes. But for rock samples in most cases the temperature dependence is essentially lower. Therefore, in the Eucken law a constant term was added: K ~ 1 / ( T + const). This additional constant is interpreted as being caused by the scattering of phonons by various types of lattice imperfections (Madarasz and Klemens, 1987; Ratsifaritana and Klemens, 1987). Another modification of the Eucken law often used was to change the exponent: K c< T -n with 0 < n < 1. For temperatures higher than about 600°C rapidly increasing contributions to the heat transfer by photons in the infrared range of the electromagnetic spectrum are expected from theoretical considerations and some experimental results (Cabannes, 1976; Clauser, 1988). In this case a third term proportional to T 3 has to be added. Corresponding to the above considerations the experimental results for the thermal conductivity K
were fitted to the following temperature functions, WLF1 to WLF3: WLF1 : K = 1 / ( A + B x T)
(1)
WLF2 : K = 1 / ( A ' + B' × T) + C' x T 3
(2)
WLF3 : K = D ' / T n with 0 < n < 1
(3)
The function WLF1 was preferred because it enables the above-mentioned physical interpretation. The fits were performed using the Levenberg-Marquardt algorithm. Before applying the fit procedure the data were smoothed by FFT-filtering. In the comparison of the various temperature functions the goodness of a fit was measured by the value of X2, defined by: X2 = Z ( y i
- f/)2/(n - p)
(4)
where Yi are measuring values, j5 are values of the fit function, n is number of measuring values, p is number of parameters.
5. The relationship between the constants A andB From the observed correlation between the value of the room-temperature thermal conductivity and the temperature-induced decrease of thermal conductivity we supposed that there exists a relationship between the parameters A and B of the temperature function WLF1. To test this assumption the values of the constants for all investigated rock samples were depicted in the same diagram. The result in Fig. 3 shows that the expected dependence exists. Some different 2-parametric functions (potential, logarithmic, exponential) were tested to fit the data set. The best fit was obtained by a linear function. Obviously, a better fit is possible by using a function with a higher number of parameters. But in this case the uncertainty in the determination of the coefficients was much higher. The following relationship between the constants A and B was obtained by linear regression: A = - ( 5 3 2 -4- 45) × B + (0.448 -4- 0.014)
(5)
with a correlation coefficient R = -0.84. After the expected relationship between the constants A and B was tested using our own results the same procedure was performed with the data set from the literature. As seen in Fig. 4 the same be-
165
U. Seipold/Tectonophysics 291 (1998) 161-171 0.60
the same equation:
Fit: A = -(532+.45) * B + 0.448+-0.014, R =- 0.84
. +
A ---- - ( 5 3 6 -t- 33) x B -t- (0.443 + 0.012)
4"0 4" 0.40
' 4,I 4, '
with a correlation coefficient R = - 0 . 8 5 . Further, the constants A and B were averaged for all samples of the same rock types (see Table 1). The mean values with the standard deviations are represented in Fig. 5. Again the earlier result was confirmed. The error bars show the ranges of the standard deviations for the constants A and B. The regression line is described by the equation:
4,0 0
~
2 "~
o
o
0
0.20
_~
Gnels o
•Ira
Amphibolite
0
OV~X~ O~ " ~
Granite n"
o.oo
O
A = - (564 -t- 116) x B -t- (0.447 -4- 0.031)
Serpenlinile Granutlle Pyroxenite
~"
'
I
I
'
2
'
I
4
'
6
Constant 8 * E 4
Fig. 3. Temperaturedependence of thermal conductivity. Relationship between the constants A and B in WLF1. Own results. haviour was observed and also the equation for the regression line agrees, with only small differences: A = - ( 5 6 0 + 54) x B -t- (0.447 -t- 0.022)
(6)
with a correlation coefficient R = - 0 . 8 5 . Thus, it is clear that also the resulting set from own data and those from the literature obeys about 0.60
'
•
I
'
I
'
I
0.50
'
0.40
AzX
0.30
~
pentinites
5Jasa~ts ~
~
AmpMbolites
Pyroxenltes
lites
c m o 0.20
....... =~
•
(8)
with a correlation coefficient R = - 0 . 9 1 . Some general rules can be derived from the distribution of the various rocks within the A - B plane concerning their temperature dependence. The granites are concentrated to the right of the figure. This position indicates high values of the thermal conductivity with a high temperature dependence. On the contrary, the serpentinites occupy the left end of the distribution. This means that generally they show small thermal conductivity which is only slowly influenced by temperature. The serpentinites are followed by the amphibolites and basalts also concentrated in the left half of the figure and thus with similar properties. The gneisses are distributed over a very broad
•
0.40 ~
(7)
I
~ ~
0.20
J ~Gneisses
I
0 • +"~
$ibbitt
¢
41~ e02~, ~ A 0.10
I
T O Bvine Rlocks
Pettunin/Popov
o.0o
C)
Others
•
OwnResufls ' I
O*
2
'
{
'
4
A = - (564 +- 116 ) * B + 0.447 +- 0.031, R =- 0.91
I
'
6
C o n s t a n t B * E4
Fig. 4. Temperature dependence of thermal conductivity. Relationship between constants A and B in WLF1. Own and literatureresults.
0.00
'
J
~
2
1
'
4
Constant B " E4
Fig. 5. Temperature dependence of the thermal conductivity. Relationship between the constants A and B in WLF1. Averages of all results for different rock types.
U. Seipold/Tectonophysics 291 (1998) 161-171
166
Table 1 Mean values of constants A and B for various rock types (literature and our own results)
Amphibolites Basalts Granites Granulites Gneisses Pyroxenites Serpentinites Olivine rocks
Number
A
16 4 15 8 26 6 7 13
0.357 0.359 0.203 0.271 0.241 0.311 0.427 0.110
+ 0.063 :k 0.036 ± 0.069 + 0.053 ± 0.079 ~k 0.081 ± 0.064 ± 0.061
range (see also Fig. 3) indicating a strong variation of their thermal properties with composition, structure and texture. But the centre corresponding to the high content of quartz is positioned in the vicinity of the granites and very near the granulites. The olivine-containing rocks occupy an extreme position in the A - B diagram. Looking at Table 1 we see that they have by far the smallest value of the ratio A / B . This means that they show the highest temperature dependence (see below). The relatively very small value of constant A is an indication for a high crystal perfection of the investigated samples and is the reason also for the high room-temperature thermal conductivity. In summary, the olivine-containing rocks obviously do not belong to the series of the investigated crystalline rocks. Therefore, they were omitted from the linear regression. On a first look the constants A and B should be independent because they are measures of the influence of completely different scattering processes. But an internal connection exists in the sense that phonons scattered by imperfections will not be involved at the same time in 3-phonon processes and vice versa. Thus, the increase of one of the constants will lower the value of the other one.
6. A generalized temperature law The derived relationship between the constants A and B has important practical consequences. Eqs. 5 8 can be used to eliminate one of the constants in Eq. 1. Using Eq. 5 one gets an expression for the temperature dependence of the thermal conductivity containing only one parameter (B): K = 1 / ( B x ( T - 532 + 45) + 0.448 4- 0.014)
(9)
B x 104
A/(B x 104)
1.89 ± 0.85 1.43 ± 1.00 4.07 ± 1.00 3.66 ± 0.66 3.48 -k 1.20 1.44 ± 1.16 1.10 ± 0.40 3.18 ± 0.80
0.189 0.251 0.050 0.074 0.069 0.216 0.388 0.034
The last parameter can be determined from the knowledge of the thermal conductivity at any temperature, for instance at room temperature. Thus, we can calculate the whole temperature dependence of the thermal conductivity of a rock sample if we know only the room-temperature conductivity. This means also that all rock samples with the same room-temperature conductivity show the same temperature dependence of their thermal conductivity independent from their mineralogical composition. This conclusion shows that Eq. 9 is a very far-reaching generalization! But one should remember that it is exactly valid only for samples with (A, B) values directly on the regression line in Fig. 3 and that in fact there exists a wide scattering around the regression line measured by the standard deviations in Eq. 5. Analogous results were obtained for the temperature dependence of the thermal diffusivity a. If one defines corresponding constants C and D by an equation: a = 1/(C + D x T)
(10)
then one obtains C ---- -(490-4-47) × D + (0.166 + 0.011)
(11)
with a correlation coefficient R = -0.80. The corresponding data set is depicted in Fig. 6. The thermal diffusivity is related to the thermal conductivity according to the equation K = c p a (c is specific heat capacity, 9 is density). According to Debye's theory, the specific heat capacity increases with rising temperature. Therefore, the decrease of the thermal diffusivity is essentially stronger than that of the thermal conductivity. This behaviour is clearly reflected in the values of the constants. Comparing Eqs. 1 and 10, the corresponding constants B and D
U. SeipoM/Tectonophysics 291 (1998) 161-171
167
20
g
Fit: C = - (490.2+-47)* R =- 0 , 8 0
• •
E
D + 0.1665+-0.011,
.c_
._> 10
--
:t:+
0
÷ ~ t + ~ + ' ÷ l ' ÷~ l , 4• • 0 O 0
=
Z
o
o o
0 O
.E
+
5
0
F-
0
41. . . . . . s..p~u.,,
Measured Points ............Fit with WLF1 -Fit with WLF4
2
an Granite
x
~
~
o~ ~ ~o.O , ~ ~z~
~-
0 0
o
200
0
Fit: WLF4
Fit: WLF1
ChP,2 = 0,00728 E:-45,0+-1,72 F:0,443+-0,0037
ChP,2 = 0,01465 A: 0.2465-=-0,0070 B:1,97E-4+-l,28E-5
i
i
i
i
400
600
800
1000
1200
Temperature in K ~/~
Fig. 7. Temperature dependence of the thermal conductivity for a KTB-gneiss sample. Comparison of the fits with WLF1 and WLF4.
Pytoxenite 1
I 1
'
I 2 Constant
'
I 3
'
D * E4
Fig. 6. Temperature dependence of the thermal diffusivity. Relationship between the constants C and D in Eq. 10.
have very similar values. But the constant C from the thermal diffusivity is about one order smaller than the corresponding constant A from the thermal conductivity. If the temperature function WLF3 is used for the description of the measuring results, the exponents n are essentially higher for the thermal diffusivity results than for the thermal conductivity.
ferent curvatures of the functions WLF1 and WLF4 in Fig. 7 it can be concluded that extrapolations outside the fit range using WLF1 would provide much higher errors of the function values than the newly proposed function. Looking for a physical interpretation of the temperature function WLF4 a series development was carded out: K = T / ( E + F x T) = I / F -
E / ( F 2 x T)
Jr E2/(F 3 x T 2) - E3/(F 4 x T 3) J r . . .
(13)
7. An improved temperature function In the course of the data processing to determine the constants A and B we often had the impression that the temperature function WLF1 did not bring us to the optimal fit of the experimental data. Therefore, we tried to find a better function. It had to have a higher curvature in the range of lower temperatures and a smaller decrease at higher temperatures than WLFI. The search was restricted to two-parametric functions. The required results were obtained by a temperature function WLF4 defined by K = T/(E + F x T)
(3
~o o ~
100
O0
u?, 0
•
0
~,/ +
9
~" +-~
o
0~
+ -
+
,.,.,bo,,,o
O-
Serpentinffe
/
•
31¢
GnmuIlte Pyroxenlte
(12)
Fig. 7 shows a typical example for a better fit of the experimental data by WLF4 as compared with WLF1. A quantitative comparison was carried out by the calculation of X2 for both fit functions processing all our own data. In most cases the better approximation was reached by WLF4, From the dif-
Fit: E = - ( 3 1 4 +- 3 5 ) * R =-0,75
' 0.20
I 0.40
F + 1 2 2 +- 2 0
'
I 0.60 Constant
'
I 0.80
' 1,00
F
Fig. 8. Temperature dependence of the thermal conductivity. Relationship between the constants E and F in WLF4.
U. Seipold/Tectonophysics 291 (1998) 161-171
168
The convergence properties of the series were estimated for a sample with common values of E and F. In the worst case (lowest temperature) the first four terms contribute 67, 22, 7.4 and 2.4%, respectively, to the result. Therefore, it is sufficient to consider only the first three terms. The constant 1 / F and the term proportional to T -I were interpreted as in the case of WLF1. Additionally now appears a term proportional to T -2. Pomeranchuk (1941) found out that such temperature dependence is typical for 4-phonon interactions. For the new temperature function WLF4 also we checked whether there exists a correlation between the constants E and F. The result is shown in Fig. 8. The constants are related by the equation: E = - ( 3 1 4 -4- 35) × F + (122 ± 20)
x ( T - 314 4- 35) + 122 -4- 20)
8. Relations between the temperature functions WLF1 and WLF3 Using the values of X2 as a measure tor the goodness of a fit we looked for a decision between WLF1 and WLF3. But on the average for the investigated rock collection the values were in the same order for both temperature functions. Thus, WLF1 and WLF3 represent different equivalent approximations. In this sense, both functions were developed in Taylor series a t T = To: K(T) = D'/T n = K(To) - n x K(To) × To I
× (T - To)2 -
+..-
~'
/
To e (16)
~
5 o~ (D
0.20 0.40 Constant A / B * E4
0.60
Fig. 9. Relationship between 1/n and A/B. K(T) = I/(A + B x T) = K ( T o ) - B x K2(To) x ( T - To)
+ 2 x B 2 x K3(T0) x (T - To)2 -
+ ... (17)
(15)
We then also tried to fit the experimental data for the temperature dependence of the thermal diffusivity to the new function WLF4. Looking for the smallest X2-values it was found that clearly the temperature function WLF1 delivers a better approximation for the thermal diffusivity than WLF4. This is in contrast to the behaviour of the thermal conductivity.
×(T-To)+nx(n+l)×K(To)×
l f n = 1+23.64*AtB* E4
(14)
with a correlation coefficient R = -0.75. This means that the correlation coefficient is a little smaller than for the function WLFI. Of course, also Eq. 14 can be used to derive a generalized temperature law by the elimination of the constant E in Eq. 12: K = T/(F
16
Equating the terms linear in (T - To) results in: l / n = 1 + 1~To x A / B
(18)
In Fig. 9 the value of 1 / n was depicted versus A / B for the data set from 24 amphibolites and gneisses. From the fit of the data to the function 18 we obtained: l / n = 1 + (23.64 4-0.73) × 10-4 × A / B
(19)
T0 = (423 5: 13)K Eq. 19 indicates that the value of 1 / n , respectively, n can be used as a measure for the temperature dependence of the thermal conductivity like the ratio B/A.
9. Fits by the temperature function W L F 2 The clearly best fits of the experimental data were obtained with the temperature function WLF2, especially for wide temperature intervals. This is not surprising because this function has an additional parameter. However, there is the disadvantage that
U. Seipold/Tectonophysics 291 (1998) 161-171
the standard deviations of the parameters are, relatively seen, essentially higher than in the case of the two-parametric functions. Therefore, the chance is much smaller to find a general relationship between the parameters. The additional term C' x T 3 in the function WLF2 was introduced to take into account contributions to the heat transfer from the electromagnetic radiation. But we obtained in some cases a negative value for the parameter C'. Obviously this result contradicts the above interpretation of the T3-term.
169
3.20
~cE2.40Mst ?__ ,IIII~SSS Jti r N~:~\\ :=
~
~ -- K(T:293K)=2,4
o 2.00
-
E
~ ~ ...--2.8
a= I-1.60
I0. Comparison with other temperature laws In an earlier data compilation Zoth and Haenel (1988) subdivided the data from literature corresponding to fundamental rock types and derived empirical equations for each type: K = 474/(T + 77) + 1.18 for basic rocks
(20)
K = 807/(T + 77) + 0.64 for acid rocks
(21)
K = 705/(T + 77) + 0.75 for metamorphic rocks (22) where T is temperature in °C. In Fig. 10 the curves for acid, metamorphic and basic rocks are compared with our curves for 2.8, 2.6 and 2.4 W m -1 K -1 room-temperature conductivity. A much better coincidence was obtained by a small variation of the constant term in Eq. 9: K = 1/(B
x (T -
532) + 0.5)
(23)
The result is shown in Fig. 11. A perfect coincidence cannot be expected because the rock samples were grouped by Zoth and Haenel using an other principle than we did. An other general law for the temperature dependence of the thermal conductivity was derived by Sass et al. (1992) based on the experimental results of Birch and Clark (1940) for a wide spectrum of very different rock types. A comparison of the Sass law with our generalization using the temperature function WLF1 is shown in Fig. 12. For a rock with 2.8 W m -1 K -1 room-temperature conductivity we have exactly the same temperature dependence of the thermal conductivity. For smaller initial values
1.20 -
'
I
200
'
400
[ 600
800
Temperature in K Fig. 10. Comparison o f the curves f r o m Zoth and Haenel (1988)
for different rock types with curves from the generalized temperature law (Eq. 9) (dashed lines).
our curves have a smaller slope whereas for higher initial values than 2.8 W m -l K -l our curves show a stronger temperature dependence than the curves calculated by using the Sass law. Another behaviour was observed when we used our generalized temperature law (Eq. 15) based on 3.00 cid Rocks ~etarnorphtc R
E
2,50
ks
•
Basic Rocks
.__.
\~\
•"ac 2.00 O0
K(T=293K}=2.4
1.50
K~X=29~=2.e
' 200
I 400
'
I 000
Temperature in K
'
I 800
Fig. 11. Comparison of the curves from Zoth and Haenel (1988) for different rock types with curves from the varied temperature law (Eq. 23) (dashed lines).
170
U. Seipold/Tectonophysics 291 (1998) 161-171 3.20
ences concerning the temperature dependence of the thermal conductivity. It is the typical behaviour that the decrease of the thermal conductivity with rising temperature is concentrated in the interval from room temperature up to about 600 K.
Acid Rocks 2.80 p ic roc ~ \
\ \
E Bas~c ~ ._=
"-.
2.40
:=,_ .>_
11.
== o o
2.00
E .c= 1.60
/ '
1,20
I
200
'
t
400
, I
600
~ 800
Temperature in K
Fig. 12. Comparison of the curves from Sass et al. (1992) with curves from the generalized temperature law (Eq. 9) (dashed lines). the temperature function W L F 4 (see Fig. 13). In this case we generally observed a stronger temperature dependence in the lower temperature range than Sass and a smaller temperature influence in the high temperature range. The general course of these curves corresponds better to all our experimental experi3,20 K(T=293K)=3.1 \
\ \
~E
2.80
= 2.8
\
~-\ \\
,,
--
\ \
-o c
\ \\
\
2.40
\\.. \ ~. "" "
---
Curves from Sass
-- -- -' 200
Curves from the A u t h o r I 300
'
I
With the increasing number of published experimental results for the temperature dependence of the thermal conductivity there is obviously a need for a concentration and generalization of the data. The presentation of the constants A and B of the temperature function WLF1 in an A - B diagram is an appropriate tool to display the behaviour of an individual rock sample and to compare the properties of different rock types. In our investigations of an extended collection of crystalline rocks a linear relationship between the constants A and B was obtained. The application of this result carried us to a general temperature law (Eq. 9). Equivalent results were obtained for the much higher temperature dependence of the thermal diffusivity. In many cases the newly proposed temperature function W L F 4 provided a better approximation to the experimental results of the temperature dependence of the thermal conductivity than the commonly used function WLF1. The observed relationship between the constants E and F of the function W L F 4 was applied to derive a second generalized temperature law (Eq. 15) for crystalline rocks. The temperature functions WLF1 and WLF3 represent equivalent approximations. A relationship (Eq. 18) between the corresponding constants A, B and n was derived and tested. References
I-.-
2.00
Summary
'
400
I 500
r
I 600
Temperature in K
Fig. 13. Comparison of the curves from Sass et al. (1992) with curves from the generalized temperature law (Eq. 15) (dashed lines).
Beck, A.E., Darbha, D.M., Schloessin, H.H., 1977. Lattice conductivities of single-crystal and polycrystalline materials at mantle pressures and temperatures. Phys. Earth Planet. Inter. 17, 35-53. Birch, E, Clark, H., 1940. Thermal conductivity of rocks and its dependence upon temperature and composition. Am. J. Sci. 238, 529-558; 613-635. Cabannes, E, 1976. Thermal conductivity and emissivity of oxide ceramics at high temperatures. High Temp. High Pressures 8, 155-166. Clauser, C., 1988. Opacity - - the concept of radiative thermal conductivity. In: Haenel, R., Rybach, L., Stegena, L. (Eds.),
U. Seipold / Tectonophysics 291 (1998) 161-171 Handbook of Terrestrial Heat-Flow Density Determination. Kluwer, Dordrecht, pp. 143-165. Durham, W.B., Mirkovich, V.V., Heard, H.C., 1987. Thermal diffusivity of igneous rocks at elevated pressure and temperature. J. Geophys. Res. 92 (Bll), 11615-11634. Eucken, A., 1911. Ueber die Temperaturabhaengigkeit der Waermeleitfaehigkeit fester Nichtmetalle. Ann. Phys. Leipzig 34, 186-221. Kawada, K., 1964. Studies of the thermal state of the earth. Variation of thermal conductivity of rocks, Part 1. Bull. Earthquake Res. Inst. 42, 631-647. Kawada, K., 1966. Studies of the thermal state of the earth. Variation of thermal conductivity of rocks, Part 2. Bull. Earthquake Res. Inst. 44, 1071-1091. Klemens, P.G., 1969. Theory of the thermal conductivity in solids. In: Tye, R.P. (Ed.), Thermal Conductivity. Academic Press, London, Vol. 1, pp. 2-65. Klemens, P.G., 1983. Theory of thermal conductivity of solids at high temperatures. High Temp. High Pressures 15, 249-254. Madarasz, EL., Klemens, P.G., 1987. Reduction of lattice thermal conductivity by point defects at intermediate temperatures. Int. J. Thermophys. 8, 257-262. Petrunin, G.I., Popov, V.G., 1981. Vlijanie serpentinizacii na teplovye svojstva giperbazitov (The influence of the serpentinization on the thermal properties of ultrabasites). Fiz. Zemli 4, 18-24. Petrunin, G.I., Popov, V.G., 1983. Teplovye svojstva kvarcoder2a~ich intruzivnych gornych porod (Thermal properties of quartz-containing intrusive rocks). Fiz. Zemli 3, 75-81. Petrunin, G.I., Popov, V.G., 1988. Teplovye svojstva porod iz Kol'skoj sverchglybokoj skva~iny (Thermal properties of rocks from the Kola superdeep borehole). Fiz. Zemli 1, 44-51. Petrunin, G.I., Popov, V.G., 1994. Osobennosti temperaturuogo povedenija re~eto~noj teploprovodnosti mineral'nogo ve~stva zemli (Peculiarities of the temperature dependence of the lattice thermal conductivity of mineralic materials). Fiz. Zemli 7/8, 35-41. Pomeranchuk, I., 1941. On the thermal conductivity of di-
171
electrics. Phys. Rev. 60, 820-821. Pribnow, D., Williams, C.F., Sass, J.H., Keating, R., 1996. Thermal conductivity of water-saturated rocks from the KTB pilot hole at temperatures of 25 to 300°C. Geophys. Res. Lett. 23, 391-394. Ratsifaritana, C.A., Klemens, P.G., 1987. Scattering of phonons by vacancies. Int. J. Thermophys. 8, 737-751. Sakwarelidse, E.A., 1973. Thermophysical properties of rocks in the temperature interval 20-500°C. In: Heat Flow from the Crust and Upper Mantle of the Earth. Nauka, Moscov, pp. 125-136. Sass, J.H., Lachenbruch, A.H., Moses, T.H., 1992. Heat flow from a scientific research well at Cajon Pass, California. J. Geophys. Res. 97 (B4), 5017-5030. Schatz, J.E, Simmons, G., 1972. Thermal conductivity of earth materials at high temperatures. J. Geophys. Res. 77, 69666983. Seipold, U., 1988. Simultaneous measurements of thermal diffusivity and thermal conductivity under high pressure using thermal pulses of finite length. High Temp. High Pressures 20, 609-613. Seipold, U., 1990. Pressure and temperature dependence of thermal transport properties of granites. High Temp. High Pressures 22, 541-548. Seipold, U., 1995. The variation of thermal transport properties in the earth's crust. J. Geodyn. 20, 145-154. Sibbitt, W.L., Dodson, J.G., Tester, J.W., 1979. Thermal conductivity of crystalline rocks associated with energy extraction from hot dry rock geothermal systems. J. Geophys. Res. 84, 1117-1124. Tkatsch, G.F., Jurtschak, R.P., 1972. Izmerenie teplovych parametrov gornych porod v ~irokom intervale temperatur (Measurements of the thermal parameters of rocks in a wide temperature range). Fiz. Zemli 5, 81-83. Zoth, G., Haenel, R., 1988. Thermal conductivity. In: Haenel, R., Rybach, L., Stegena, L. (Eds.), Handbook of Terrestrial Heat Flows Density Determination. Kluwer, Dordrecht, pp. 449-468.
Tectonophysics291 (1998) 161-171
Temperature dependence of thermal transport properties of crystalline rocks a general law U, S e i p o l d * GeoForschungsZentrum Potsdam, Telegrafenberg A17, 14473 Potsdam, Germany
Received 31 December 1996; accepted 26 June 1997
Abstract The temperature dependence of the thermal conductivity K was fitted to the function K = 1/(A + B x T), T is absolute temperature, for 121 rock samples. Using our own measuring results for 64 samples it was found that there exists a linear relationship between the constants A and B: A = -(532 4- 45) x B + (0.448 4- 0.014). The checked number of literature data reflect almost the same result. An analogous equation was obtained for the temperature dependence of the thermal diffusivity. Using the relationship between the constants A and B the elimination of one of them results in a generalized temperature law: K = 1/(B x (T - 532) + 0.448). For many rock samples a temperature function K = T / ( E + F x T) takes us to a much better fit of the experimental results for the temperature dependence of the thermal conductivity. In this case as well the constants are correlated. The result is K = T / ( F x (T - 314) + 122). © 1998 Elsevier Science B.V. All rights reserved. Keywords: thermal conductivity; thermal diffusivity; temperature; crystalline rocks
1. Introduction Temperature plays the most important role among the various influences on the thermal conductivity in the earth's crust. The temperature effect on the thermal transport properties can be caused directly by the temperature dependence of the heat transfer mechanisms or otherwise indirectly by temperature-induced physical and chemical changes in the investigated rock samples. The influences of the latter depend on the specific mineralogical composition of the investigated rock type, and their rules can not be applied to all rocks. Therefore, only the direct temperature dependence is considered in this paper. *Fax: +49 331 288 1450; E-mail: [email protected]
The investigation of the temperature dependence of the thermal conductivity gives information on the heat transfer mechanisms for the studied rocks because the different mechanisms in general will cause different functions of the temperature dependence. These functions are known from theoretical studies of the heat transport process (Klemens, 1969, 1983). Thus, one can try to fit the experimental results to these functions to find the acting heat transfer mechanisms. In the past, the limited accuracy of the measuring results and the insufficient density of the measuring points often prevented that preference was given to one of the fit functions. But now the progress by computer supported measuring methods and data processing improved the possibilities. The knowledge of the acting heat transfer mechanisms is
0040-1951/98/$19.00 © 1998 Elsevier Science B.V. All rights reserved. PII S0040-1951(98)00037-7
162
U. Seipold/Tectonophysics 291 (1998) 161-171 4.0
\
35
E .E >,
3.0
.>_ "5 X3
~ O
I 25
[ E b-
20
/, 1.5
- -
'
200
Pyriclasile
L 400
~-
I
,--
600
! 800
r 1 O0
Temperature in K
Fig. 1. Temperature dependence of the thermal conductivity for rock samples with different room-temperature conductivities.
not only a theoretically interesting problem. It has also an important practical consequence in so far as it enables us to extrapolate the results outside the direct experimentally studied temperature range. In our experimental investigations of the temperature influence on the thermal transport properties we came to the conclusion that there must exist a general rule. Rock samples with a relatively high thermal conductivity showed a strong decrease of the thermal conductivity with rising temperature, and inverse rock samples with a small thermal conductivity at room temperature only showed a small variation with increasing temperature (Fig. l). From this observation we tried to find a general law which quantifies the qualitative impression. 2. Data sources
Surely it was necessary to test a possible relationship with a quite extended data collection. Therefore, we processed all experimental results obtained by our own measurements. This data set has the advantage that it was obtained under exactly the same conditions (measuring method, sample preparation, measuring regime) for all 64 rock samples. The investigated rock collection consists of gneisses and amphibolites from the KTB-bore-
hole Windisch-Eschenbach, granites from the Saxonian Erzgebirge and metamorphic rocks (granulites, pyroxenites, pyriclasites, amphibolites and serpentinites) from the Saxonian Granulitgebirge. Some of the results have already been published (Seipold, 1990, 1995). In the experimental studies a transient measuring method was applied allowing us to determine simultaneously thermal conductivity and thermal diffusivity. Cylindrical rock samples of 27 mm diameter and 43 mm length were prepared from bore cores or rock blocks. A heat pulse was created by a line source in the axis of the cylindrical rock sample. This pulse of three seconds length propagates in radial directions. At about half the distance between heater and boundary of the sample a thermocouple measured the arriving signal. The thermal diffusivity was calculated from the travelling time of the pulse and the distance between heater and thermocouple. For the calculation of the thermal conductivity the peak height of the temperature pulse and the energy of the heating pulse were also needed. For details of the method see Seipold (1988). In most cases the measurements were carried out up to about 800°C. But often only the results of the lower temperature range were used to avoid disturbances by temperature-induced changes of the samples. For the quartz-containing rocks the results were taken up to the temperature of the ot-~ phase transition (573°C). In amphibolites and serpentinites the upper temperature limitation was determined by the inset of dehydration processes. For comparison a second data set (57 rock samples) was taken from a number of authors (Birch and Clark, 1940; Kawada, 1964, 1966; Schatz and Simmons, 1972; Tkatsch and Jurtschak, 1972; Sakwarelidse, 1973; Beck et al., 1977; Sibbitt et al., 1979; Petrunin and Popov, 1981; Petrunin and Popov, 1983; Petrunin and Popov, 1988). The data were carefully selected taking into account the following conditions. (1) Original measuring data without smoothing or any other data processing were preferred. (2) Only data for igneous and metamorphic rocks without alteration products and amorphous or glassy components were used. Increasing thermal conductivity with rising temperature was considered as an indication for glassy components. (3) Data from a sufficiently wide temperature
163
u. SeipoM/Tectonophysics 291 (1998) 161-171
range and with four or more measuring points were selected. By these conditions the small number of applicable published data in literature is further reduced. One has to keep in mind that there is a great venture to comparing results from various sources obtained by quite different measuring methods and often of unknown accuracy. Frequently, only four or five data points were published for a special rock sample, whereas a number of 35 up to 60 data points is commonly measured in our studies. Further constraints come from the small temperature ranges investigated in earlier studies. In many papers the temperature variation is limited in the interval from room temperature up to 400°C. In all calculations SI-units were used: thermal conductivity was measured in W m -l K -1, thermal diffusivity in 10-7 m2/s, temperature in K. 3. Discussion on the effect of thermally induced cracks Investigations of the thermal transport properties at high temperature without the application of high pressure or water saturation provoke the discussion on erroneous results because, due to differences of the value and the anisotropy of thermal expansivity of the mineral grains, cracks are created at high temperatures. From this argumentation it is surprising that most high-temperature studies of thermal conductivity were carried out at ambient pressure (which also excludes water saturation). In the frequently cited paper of Birch and Clark (1940) the authors point out that their results up to 400°C "may be considered to represent reversible temperature effects". All values were omitted if "subsequent measurements at lower temperature failed to return to their previous values". This reversibility criterium for the confidence of results is a very hard condition (sufficient but not necessary). We have used a less hard condition. We consider the results obtained in a first temperature run as correct if after a temperature release the values of the first and second runs coincide at the highest temperature of the first run. This condition does not exclude the possibility that in a second run smaller values can be obtained at temperatures lower than the maximum temperature of the first run. This criterium is based on the following
assumption: if the temperature is raised, the mineral grains try to expand. Partly this expansion is possible by the volume increase of the whole rock sample and partly the expansion is blocked by the adjacent grains resulting in the formation of local stresses. Also intragranular closed cracks can occur by the effect of shear stresses. But it is unlikely that a high number of open cracks is created. Only such airfilled cracks would essentially diminish the thermal conductivity. The above imagination is supported by our studies of thin sections from rock samples which were heated up to temperatures of 500°C. A direct check of a possible influence of temperature-induced cracks is obtained by studies of the temperature dependence at various pressures. The application of sufficiently high confining pressure should suppress the thermal cracking. Appropriate results under hydrostatic conditions are very rare. A sufficient data set was taken from a study of quartzcontaining igneous rocks by Durham et al. (1987) in the range from room temperature up to 400°C at a pressure up to 200 MPa. The processing of these data showed us that, opposite to the expectations by Pribnow et al. (1996), measurements of dry rock samples do not provide a higher temperature dependence of the thermal transport parameters at ambient pressure compared with investigations at high pressure (see Fig. 2). On the contrary, at higher pressures the temperature dependence seems to be somewhat stronger. 1,8
A
o
•
~1,41
o
•
0.1 M P a
•
5 MPa 20 MPa
o
100 M P a
[~ 1,2-
~
•
~-N--1,0•
_ 0,8I"0,6
0
1~0
2~o
3~0
,~o
Temperature in o C
Fig. 2. Temperaturedependenceof thermal diffusivityat various pressures. Results taken from Durham et al. (1987) for Climax Stock quartz monzonite.
164
U. Seipold/Tectonophysics 291 (1998) 161-171
4. Theoretical background and data processing Diffusive heat transfer in crystalline rocks is determined by the interactions of phonons. In the quasi-particle picture of the heat transfer process the observed value of the thermal conductivity is determined by the resistivities against the propagation of the heat-carrying phonons. Phonons are scattered in phonon-phonon interactions caused by nonlinearities of the lattice potential and by interactions with the various types of imperfections of the crystal lattice. As the measure for the number of scattering processes the mean free path length of phonons was introduced. Its value decreases in most silicate minerals from the order of about 10 ~ at room temperature to a few hngstr6m at 1000 K (Petrunin and Popov, 1994). This means that the number of scattering processes within the mineral grains is by orders higher than the scattering number by cracks and grain boundaries. Therefore, it is expected that they play an essential role only if an additional macroscopic contact resistance appears at the grain boundaries or if open air-filled cracks occur. The temperature dependence of the thermal conductivity depends on the contributing heat transfer mechanisms. For simple ionic compounds the thermal conductivity is proportional to T -l, where T is the absolute temperature. The temperature dependence according to this so-called Eucken law (Eucken, 1911) is characteristic for a heat transfer dominated by 3-phonon Umklapp processes. But for rock samples in most cases the temperature dependence is essentially lower. Therefore, in the Eucken law a constant term was added: K ~ 1 / ( T + const). This additional constant is interpreted as being caused by the scattering of phonons by various types of lattice imperfections (Madarasz and Klemens, 1987; Ratsifaritana and Klemens, 1987). Another modification of the Eucken law often used was to change the exponent: K c< T -n with 0 < n < 1. For temperatures higher than about 600°C rapidly increasing contributions to the heat transfer by photons in the infrared range of the electromagnetic spectrum are expected from theoretical considerations and some experimental results (Cabannes, 1976; Clauser, 1988). In this case a third term proportional to T 3 has to be added. Corresponding to the above considerations the experimental results for the thermal conductivity K
were fitted to the following temperature functions, WLF1 to WLF3: WLF1 : K = 1 / ( A + B x T)
(1)
WLF2 : K = 1 / ( A ' + B' × T) + C' x T 3
(2)
WLF3 : K = D ' / T n with 0 < n < 1
(3)
The function WLF1 was preferred because it enables the above-mentioned physical interpretation. The fits were performed using the Levenberg-Marquardt algorithm. Before applying the fit procedure the data were smoothed by FFT-filtering. In the comparison of the various temperature functions the goodness of a fit was measured by the value of X2, defined by: X2 = Z ( y i
- f/)2/(n - p)
(4)
where Yi are measuring values, j5 are values of the fit function, n is number of measuring values, p is number of parameters.
5. The relationship between the constants A andB From the observed correlation between the value of the room-temperature thermal conductivity and the temperature-induced decrease of thermal conductivity we supposed that there exists a relationship between the parameters A and B of the temperature function WLF1. To test this assumption the values of the constants for all investigated rock samples were depicted in the same diagram. The result in Fig. 3 shows that the expected dependence exists. Some different 2-parametric functions (potential, logarithmic, exponential) were tested to fit the data set. The best fit was obtained by a linear function. Obviously, a better fit is possible by using a function with a higher number of parameters. But in this case the uncertainty in the determination of the coefficients was much higher. The following relationship between the constants A and B was obtained by linear regression: A = - ( 5 3 2 -4- 45) × B + (0.448 -4- 0.014)
(5)
with a correlation coefficient R = -0.84. After the expected relationship between the constants A and B was tested using our own results the same procedure was performed with the data set from the literature. As seen in Fig. 4 the same be-
165
U. Seipold/Tectonophysics 291 (1998) 161-171 0.60
the same equation:
Fit: A = -(532+.45) * B + 0.448+-0.014, R =- 0.84
. +
A ---- - ( 5 3 6 -t- 33) x B -t- (0.443 + 0.012)
4"0 4" 0.40
' 4,I 4, '
with a correlation coefficient R = - 0 . 8 5 . Further, the constants A and B were averaged for all samples of the same rock types (see Table 1). The mean values with the standard deviations are represented in Fig. 5. Again the earlier result was confirmed. The error bars show the ranges of the standard deviations for the constants A and B. The regression line is described by the equation:
4,0 0
~
2 "~
o
o
0
0.20
_~
Gnels o
•Ira
Amphibolite
0
OV~X~ O~ " ~
Granite n"
o.oo
O
A = - (564 -t- 116) x B -t- (0.447 -4- 0.031)
Serpenlinile Granutlle Pyroxenite
~"
'
I
I
'
2
'
I
4
'
6
Constant 8 * E 4
Fig. 3. Temperaturedependence of thermal conductivity. Relationship between the constants A and B in WLF1. Own results. haviour was observed and also the equation for the regression line agrees, with only small differences: A = - ( 5 6 0 + 54) x B -t- (0.447 -t- 0.022)
(6)
with a correlation coefficient R = - 0 . 8 5 . Thus, it is clear that also the resulting set from own data and those from the literature obeys about 0.60
'
•
I
'
I
'
I
0.50
'
0.40
AzX
0.30
~
pentinites
5Jasa~ts ~
~
AmpMbolites
Pyroxenltes
lites
c m o 0.20
....... =~
•
(8)
with a correlation coefficient R = - 0 . 9 1 . Some general rules can be derived from the distribution of the various rocks within the A - B plane concerning their temperature dependence. The granites are concentrated to the right of the figure. This position indicates high values of the thermal conductivity with a high temperature dependence. On the contrary, the serpentinites occupy the left end of the distribution. This means that generally they show small thermal conductivity which is only slowly influenced by temperature. The serpentinites are followed by the amphibolites and basalts also concentrated in the left half of the figure and thus with similar properties. The gneisses are distributed over a very broad
•
0.40 ~
(7)
I
~ ~
0.20
J ~Gneisses
I
0 • +"~
$ibbitt
¢
41~ e02~, ~ A 0.10
I
T O Bvine Rlocks
Pettunin/Popov
o.0o
C)
Others
•
OwnResufls ' I
O*
2
'
{
'
4
A = - (564 +- 116 ) * B + 0.447 +- 0.031, R =- 0.91
I
'
6
C o n s t a n t B * E4
Fig. 4. Temperature dependence of thermal conductivity. Relationship between constants A and B in WLF1. Own and literatureresults.
0.00
'
J
~
2
1
'
4
Constant B " E4
Fig. 5. Temperature dependence of the thermal conductivity. Relationship between the constants A and B in WLF1. Averages of all results for different rock types.
U. Seipold/Tectonophysics 291 (1998) 161-171
166
Table 1 Mean values of constants A and B for various rock types (literature and our own results)
Amphibolites Basalts Granites Granulites Gneisses Pyroxenites Serpentinites Olivine rocks
Number
A
16 4 15 8 26 6 7 13
0.357 0.359 0.203 0.271 0.241 0.311 0.427 0.110
+ 0.063 :k 0.036 ± 0.069 + 0.053 ± 0.079 ~k 0.081 ± 0.064 ± 0.061
range (see also Fig. 3) indicating a strong variation of their thermal properties with composition, structure and texture. But the centre corresponding to the high content of quartz is positioned in the vicinity of the granites and very near the granulites. The olivine-containing rocks occupy an extreme position in the A - B diagram. Looking at Table 1 we see that they have by far the smallest value of the ratio A / B . This means that they show the highest temperature dependence (see below). The relatively very small value of constant A is an indication for a high crystal perfection of the investigated samples and is the reason also for the high room-temperature thermal conductivity. In summary, the olivine-containing rocks obviously do not belong to the series of the investigated crystalline rocks. Therefore, they were omitted from the linear regression. On a first look the constants A and B should be independent because they are measures of the influence of completely different scattering processes. But an internal connection exists in the sense that phonons scattered by imperfections will not be involved at the same time in 3-phonon processes and vice versa. Thus, the increase of one of the constants will lower the value of the other one.
6. A generalized temperature law The derived relationship between the constants A and B has important practical consequences. Eqs. 5 8 can be used to eliminate one of the constants in Eq. 1. Using Eq. 5 one gets an expression for the temperature dependence of the thermal conductivity containing only one parameter (B): K = 1 / ( B x ( T - 532 + 45) + 0.448 4- 0.014)
(9)
B x 104
A/(B x 104)
1.89 ± 0.85 1.43 ± 1.00 4.07 ± 1.00 3.66 ± 0.66 3.48 -k 1.20 1.44 ± 1.16 1.10 ± 0.40 3.18 ± 0.80
0.189 0.251 0.050 0.074 0.069 0.216 0.388 0.034
The last parameter can be determined from the knowledge of the thermal conductivity at any temperature, for instance at room temperature. Thus, we can calculate the whole temperature dependence of the thermal conductivity of a rock sample if we know only the room-temperature conductivity. This means also that all rock samples with the same room-temperature conductivity show the same temperature dependence of their thermal conductivity independent from their mineralogical composition. This conclusion shows that Eq. 9 is a very far-reaching generalization! But one should remember that it is exactly valid only for samples with (A, B) values directly on the regression line in Fig. 3 and that in fact there exists a wide scattering around the regression line measured by the standard deviations in Eq. 5. Analogous results were obtained for the temperature dependence of the thermal diffusivity a. If one defines corresponding constants C and D by an equation: a = 1/(C + D x T)
(10)
then one obtains C ---- -(490-4-47) × D + (0.166 + 0.011)
(11)
with a correlation coefficient R = -0.80. The corresponding data set is depicted in Fig. 6. The thermal diffusivity is related to the thermal conductivity according to the equation K = c p a (c is specific heat capacity, 9 is density). According to Debye's theory, the specific heat capacity increases with rising temperature. Therefore, the decrease of the thermal diffusivity is essentially stronger than that of the thermal conductivity. This behaviour is clearly reflected in the values of the constants. Comparing Eqs. 1 and 10, the corresponding constants B and D
U. SeipoM/Tectonophysics 291 (1998) 161-171
167
20
g
Fit: C = - (490.2+-47)* R =- 0 , 8 0
• •
E
D + 0.1665+-0.011,
.c_
._> 10
--
:t:+
0
÷ ~ t + ~ + ' ÷ l ' ÷~ l , 4• • 0 O 0
=
Z
o
o o
0 O
.E
+
5
0
F-
0
41. . . . . . s..p~u.,,
Measured Points ............Fit with WLF1 -Fit with WLF4
2
an Granite
x
~
~
o~ ~ ~o.O , ~ ~z~
~-
0 0
o
200
0
Fit: WLF4
Fit: WLF1
ChP,2 = 0,00728 E:-45,0+-1,72 F:0,443+-0,0037
ChP,2 = 0,01465 A: 0.2465-=-0,0070 B:1,97E-4+-l,28E-5
i
i
i
i
400
600
800
1000
1200
Temperature in K ~/~
Fig. 7. Temperature dependence of the thermal conductivity for a KTB-gneiss sample. Comparison of the fits with WLF1 and WLF4.
Pytoxenite 1
I 1
'
I 2 Constant
'
I 3
'
D * E4
Fig. 6. Temperature dependence of the thermal diffusivity. Relationship between the constants C and D in Eq. 10.
have very similar values. But the constant C from the thermal diffusivity is about one order smaller than the corresponding constant A from the thermal conductivity. If the temperature function WLF3 is used for the description of the measuring results, the exponents n are essentially higher for the thermal diffusivity results than for the thermal conductivity.
ferent curvatures of the functions WLF1 and WLF4 in Fig. 7 it can be concluded that extrapolations outside the fit range using WLF1 would provide much higher errors of the function values than the newly proposed function. Looking for a physical interpretation of the temperature function WLF4 a series development was carded out: K = T / ( E + F x T) = I / F -
E / ( F 2 x T)
Jr E2/(F 3 x T 2) - E3/(F 4 x T 3) J r . . .
(13)
7. An improved temperature function In the course of the data processing to determine the constants A and B we often had the impression that the temperature function WLF1 did not bring us to the optimal fit of the experimental data. Therefore, we tried to find a better function. It had to have a higher curvature in the range of lower temperatures and a smaller decrease at higher temperatures than WLFI. The search was restricted to two-parametric functions. The required results were obtained by a temperature function WLF4 defined by K = T/(E + F x T)
(3
~o o ~
100
O0
u?, 0
•
0
~,/ +
9
~" +-~
o
0~
+ -
+
,.,.,bo,,,o
O-
Serpentinffe
/
•
31¢
GnmuIlte Pyroxenlte
(12)
Fig. 7 shows a typical example for a better fit of the experimental data by WLF4 as compared with WLF1. A quantitative comparison was carried out by the calculation of X2 for both fit functions processing all our own data. In most cases the better approximation was reached by WLF4, From the dif-
Fit: E = - ( 3 1 4 +- 3 5 ) * R =-0,75
' 0.20
I 0.40
F + 1 2 2 +- 2 0
'
I 0.60 Constant
'
I 0.80
' 1,00
F
Fig. 8. Temperature dependence of the thermal conductivity. Relationship between the constants E and F in WLF4.
U. Seipold/Tectonophysics 291 (1998) 161-171
168
The convergence properties of the series were estimated for a sample with common values of E and F. In the worst case (lowest temperature) the first four terms contribute 67, 22, 7.4 and 2.4%, respectively, to the result. Therefore, it is sufficient to consider only the first three terms. The constant 1 / F and the term proportional to T -I were interpreted as in the case of WLF1. Additionally now appears a term proportional to T -2. Pomeranchuk (1941) found out that such temperature dependence is typical for 4-phonon interactions. For the new temperature function WLF4 also we checked whether there exists a correlation between the constants E and F. The result is shown in Fig. 8. The constants are related by the equation: E = - ( 3 1 4 -4- 35) × F + (122 ± 20)
x ( T - 314 4- 35) + 122 -4- 20)
8. Relations between the temperature functions WLF1 and WLF3 Using the values of X2 as a measure tor the goodness of a fit we looked for a decision between WLF1 and WLF3. But on the average for the investigated rock collection the values were in the same order for both temperature functions. Thus, WLF1 and WLF3 represent different equivalent approximations. In this sense, both functions were developed in Taylor series a t T = To: K(T) = D'/T n = K(To) - n x K(To) × To I
× (T - To)2 -
+..-
~'
/
To e (16)
~
5 o~ (D
0.20 0.40 Constant A / B * E4
0.60
Fig. 9. Relationship between 1/n and A/B. K(T) = I/(A + B x T) = K ( T o ) - B x K2(To) x ( T - To)
+ 2 x B 2 x K3(T0) x (T - To)2 -
+ ... (17)
(15)
We then also tried to fit the experimental data for the temperature dependence of the thermal diffusivity to the new function WLF4. Looking for the smallest X2-values it was found that clearly the temperature function WLF1 delivers a better approximation for the thermal diffusivity than WLF4. This is in contrast to the behaviour of the thermal conductivity.
×(T-To)+nx(n+l)×K(To)×
l f n = 1+23.64*AtB* E4
(14)
with a correlation coefficient R = -0.75. This means that the correlation coefficient is a little smaller than for the function WLFI. Of course, also Eq. 14 can be used to derive a generalized temperature law by the elimination of the constant E in Eq. 12: K = T/(F
16
Equating the terms linear in (T - To) results in: l / n = 1 + 1~To x A / B
(18)
In Fig. 9 the value of 1 / n was depicted versus A / B for the data set from 24 amphibolites and gneisses. From the fit of the data to the function 18 we obtained: l / n = 1 + (23.64 4-0.73) × 10-4 × A / B
(19)
T0 = (423 5: 13)K Eq. 19 indicates that the value of 1 / n , respectively, n can be used as a measure for the temperature dependence of the thermal conductivity like the ratio B/A.
9. Fits by the temperature function W L F 2 The clearly best fits of the experimental data were obtained with the temperature function WLF2, especially for wide temperature intervals. This is not surprising because this function has an additional parameter. However, there is the disadvantage that
U. Seipold/Tectonophysics 291 (1998) 161-171
the standard deviations of the parameters are, relatively seen, essentially higher than in the case of the two-parametric functions. Therefore, the chance is much smaller to find a general relationship between the parameters. The additional term C' x T 3 in the function WLF2 was introduced to take into account contributions to the heat transfer from the electromagnetic radiation. But we obtained in some cases a negative value for the parameter C'. Obviously this result contradicts the above interpretation of the T3-term.
169
3.20
~cE2.40Mst ?__ ,IIII~SSS Jti r N~:~\\ :=
~
~ -- K(T:293K)=2,4
o 2.00
-
E
~ ~ ...--2.8
a= I-1.60
I0. Comparison with other temperature laws In an earlier data compilation Zoth and Haenel (1988) subdivided the data from literature corresponding to fundamental rock types and derived empirical equations for each type: K = 474/(T + 77) + 1.18 for basic rocks
(20)
K = 807/(T + 77) + 0.64 for acid rocks
(21)
K = 705/(T + 77) + 0.75 for metamorphic rocks (22) where T is temperature in °C. In Fig. 10 the curves for acid, metamorphic and basic rocks are compared with our curves for 2.8, 2.6 and 2.4 W m -1 K -1 room-temperature conductivity. A much better coincidence was obtained by a small variation of the constant term in Eq. 9: K = 1/(B
x (T -
532) + 0.5)
(23)
The result is shown in Fig. 11. A perfect coincidence cannot be expected because the rock samples were grouped by Zoth and Haenel using an other principle than we did. An other general law for the temperature dependence of the thermal conductivity was derived by Sass et al. (1992) based on the experimental results of Birch and Clark (1940) for a wide spectrum of very different rock types. A comparison of the Sass law with our generalization using the temperature function WLF1 is shown in Fig. 12. For a rock with 2.8 W m -1 K -1 room-temperature conductivity we have exactly the same temperature dependence of the thermal conductivity. For smaller initial values
1.20 -
'
I
200
'
400
[ 600
800
Temperature in K Fig. 10. Comparison o f the curves f r o m Zoth and Haenel (1988)
for different rock types with curves from the generalized temperature law (Eq. 9) (dashed lines).
our curves have a smaller slope whereas for higher initial values than 2.8 W m -l K -l our curves show a stronger temperature dependence than the curves calculated by using the Sass law. Another behaviour was observed when we used our generalized temperature law (Eq. 15) based on 3.00 cid Rocks ~etarnorphtc R
E
2,50
ks
•
Basic Rocks
.__.
\~\
•"ac 2.00 O0
K(T=293K}=2.4
1.50
K~X=29~=2.e
' 200
I 400
'
I 000
Temperature in K
'
I 800
Fig. 11. Comparison of the curves from Zoth and Haenel (1988) for different rock types with curves from the varied temperature law (Eq. 23) (dashed lines).
170
U. Seipold/Tectonophysics 291 (1998) 161-171 3.20
ences concerning the temperature dependence of the thermal conductivity. It is the typical behaviour that the decrease of the thermal conductivity with rising temperature is concentrated in the interval from room temperature up to about 600 K.
Acid Rocks 2.80 p ic roc ~ \
\ \
E Bas~c ~ ._=
"-.
2.40
:=,_ .>_
11.
== o o
2.00
E .c= 1.60
/ '
1,20
I
200
'
t
400
, I
600
~ 800
Temperature in K
Fig. 12. Comparison of the curves from Sass et al. (1992) with curves from the generalized temperature law (Eq. 9) (dashed lines). the temperature function W L F 4 (see Fig. 13). In this case we generally observed a stronger temperature dependence in the lower temperature range than Sass and a smaller temperature influence in the high temperature range. The general course of these curves corresponds better to all our experimental experi3,20 K(T=293K)=3.1 \
\ \
~E
2.80
= 2.8
\
~-\ \\
,,
--
\ \
-o c
\ \\
\
2.40
\\.. \ ~. "" "
---
Curves from Sass
-- -- -' 200
Curves from the A u t h o r I 300
'
I
With the increasing number of published experimental results for the temperature dependence of the thermal conductivity there is obviously a need for a concentration and generalization of the data. The presentation of the constants A and B of the temperature function WLF1 in an A - B diagram is an appropriate tool to display the behaviour of an individual rock sample and to compare the properties of different rock types. In our investigations of an extended collection of crystalline rocks a linear relationship between the constants A and B was obtained. The application of this result carried us to a general temperature law (Eq. 9). Equivalent results were obtained for the much higher temperature dependence of the thermal diffusivity. In many cases the newly proposed temperature function W L F 4 provided a better approximation to the experimental results of the temperature dependence of the thermal conductivity than the commonly used function WLF1. The observed relationship between the constants E and F of the function W L F 4 was applied to derive a second generalized temperature law (Eq. 15) for crystalline rocks. The temperature functions WLF1 and WLF3 represent equivalent approximations. A relationship (Eq. 18) between the corresponding constants A, B and n was derived and tested. References
I-.-
2.00
Summary
'
400
I 500
r
I 600
Temperature in K
Fig. 13. Comparison of the curves from Sass et al. (1992) with curves from the generalized temperature law (Eq. 15) (dashed lines).
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electrics. Phys. Rev. 60, 820-821. Pribnow, D., Williams, C.F., Sass, J.H., Keating, R., 1996. Thermal conductivity of water-saturated rocks from the KTB pilot hole at temperatures of 25 to 300°C. Geophys. Res. Lett. 23, 391-394. Ratsifaritana, C.A., Klemens, P.G., 1987. Scattering of phonons by vacancies. Int. J. Thermophys. 8, 737-751. Sakwarelidse, E.A., 1973. Thermophysical properties of rocks in the temperature interval 20-500°C. In: Heat Flow from the Crust and Upper Mantle of the Earth. Nauka, Moscov, pp. 125-136. Sass, J.H., Lachenbruch, A.H., Moses, T.H., 1992. Heat flow from a scientific research well at Cajon Pass, California. J. Geophys. Res. 97 (B4), 5017-5030. Schatz, J.E, Simmons, G., 1972. Thermal conductivity of earth materials at high temperatures. J. Geophys. Res. 77, 69666983. Seipold, U., 1988. Simultaneous measurements of thermal diffusivity and thermal conductivity under high pressure using thermal pulses of finite length. High Temp. High Pressures 20, 609-613. Seipold, U., 1990. Pressure and temperature dependence of thermal transport properties of granites. High Temp. High Pressures 22, 541-548. Seipold, U., 1995. The variation of thermal transport properties in the earth's crust. J. Geodyn. 20, 145-154. Sibbitt, W.L., Dodson, J.G., Tester, J.W., 1979. Thermal conductivity of crystalline rocks associated with energy extraction from hot dry rock geothermal systems. J. Geophys. Res. 84, 1117-1124. Tkatsch, G.F., Jurtschak, R.P., 1972. Izmerenie teplovych parametrov gornych porod v ~irokom intervale temperatur (Measurements of the thermal parameters of rocks in a wide temperature range). Fiz. Zemli 5, 81-83. Zoth, G., Haenel, R., 1988. Thermal conductivity. In: Haenel, R., Rybach, L., Stegena, L. (Eds.), Handbook of Terrestrial Heat Flows Density Determination. Kluwer, Dordrecht, pp. 449-468.