Heat flow at standard depth

Journal o f Volcanology and Geothermal Research, 9(1981)77--85

77

Elsevier Scientific Publishing Company, Amsterdam -- Printed in Belgium

HEAT FLOW AT STANDARD DEPTH

J.P.CULL

Bureau o f Mineral Resources, P.O.Box 378, Canberra City, A.C.T. 2601 (Australia) (Accepted June 10, 1980)

ABSTRACT Cull, J.P., 1981. Heat flow at standard depth. J. Volcanol. Geotherm. Res., 9: 77--85. Secular and long-term periodic changes in surface temperature cause perturbations to the geothermal gradient which may be significant to depths o f at least 1000 m, and major corrections are required to determine absolute values of heat flow from the Earth's interior. However, detailed climatic models remain contentious and estimates of error in geothermal gradients differ widely. Consequently, regions o f anomalous heat flow which could contain geothermal resources may be more easily resolved by measuring relative values at a standard depth (e.g. 100 m) so that all data are subject to similar corrections. Regional heat flow data obtained in existing deep holes show reasonable correlation with values determined at shallow depth. Hence geothermal resources o f low enthalpy can be characterised by extrapolating temperatures from relative heat flow data readily obtained from shallow boreholes. Regional control can be provided by casing deep boreholes drilled for other purposes. For routine geothermal exploration, borehole temperatures can be measured using gradient probes with f'Lxed sensor separation (e.g. 5 m), allowing very accurate determinations of the geothermal gradient at a single depth. Values of relative heat flow can then be obtained after determining the thermal resistivity of the corresponding core interval. Sampling errors can be minimised by multiple determinations o f thermal conductivity over the complete interval.

INTRODUCTION

Heat is generated within the Earth principally b y the decay of trace element radioactivity ( R o y et al., 1968). In addition it is probable that some heat remains from whole Earth accretion and core segregation (MacDonald, 1963). However, these sources o f heat m a y n o t be uniformly distributed, and anomalies in the geothermal field can be caused b y mantle convection cells associated with global tectonism, geochemical inhomogeneities in the lithosphere, episodic crustal intrusions, and refraction o f heat in crustal structures of contrasting thermal conductivity. A geothermal energy prospect resulting from perturbations o f this t y p e can be detected and assessed b y observations of surface heat flow. At depths less than 1000 m, a direct relation defining values of heat flow should normally exist between the geothermal gradient and the thermal 0377-0273/81/0000--0000/$02.50 © 1981 Elsevier Scientific Publishing Company

78 conductivity of the rocks in which the gradient is established. However, in regions of recent glaciation, systematic trends in heat flow have been observed with values depending on the depth of determination (Crain, 1968). These trends may result from transient perturbations in the geothermal gradient caused by surface warming subsequent to the retreat of glacier sheet ice. Other more recent secular and periodic changes in surface temperature have been detected using thermal anomalies at shallow depths (Cull, 1980). Numerous values of heat flow are required to define a geothermal energy prospect; these data must be obtained at sufficient depths to avoid or to normalise the effects of any transient anomalies caused b y the suggested perturbations in climate. Climatic histories and thermal propagation characteristics must therefore be considered in specifying depths at which adequate estimates o f anomalous heat flow can be determined. SHORT-PERIOD SOLAR HEATING Temperatures measured near the surface of the Earth depend on diurnal, seasonal and secular perturbations in solar heating. Carslaw and Jaeger (1959, p. 65) considered the problem of periodic surface heating and they derived the solution: T = To exp ( - - e z ) sin ( c o t -

(1)

ez)

where co= 2 ~ / P , P is the period, e = (co/2K )1~, and K is the thermal diffusivity. The time-dependent term results in an effective wavelength (4~xP) 1~ approximately equal to 1 m for the diurnal wave and 20 m for the seasonal wave {assuming a representative K = 10 -4 m 2 Is). The maximum amplitude of the disturbance is governed by the exponential term in equation (1). At a depth of one wavelength the surface amplitude is reduced b y a factor exp (--27r) = 0.0019; consequently if geothermal fields are mapped using auger holes 1 m deep, the diurnal wave can be neglected and corrections calculated for the seasonal perturbation alone. The magnitude of the seasonal periodicity at any depth can be calculated by differentiating equation (1) to give the expression: 5T = To exp(--ez) (--e) [sin ( c o t - - e z ) 5z

+ cos ( c o t -

ez)]

(2)

A minimum (zero) disturbance occurs at any depth when: sin ( w t - - e z ) = - - s i n (~/2 - - cot + e z )

(3)

train = (3~/4 + ez)/co In practice, although periodicity is well known, the thermal diffusivity of near-

79

surface layers is ill determined and consequently tmh~ cannot be calculated with any great precision. Furthermore, gradients are measured in terms of AT/Az and consequently tra= is relevant only for one of the two observation points. For these reasons measurements are best made at depths where the m a x i m u m possible amplitude of the annual perturbation is negligible. In equation (2) maxima occur at times: tmax = ( v / 4 +

ez)l~

(4)

and the amplitudes of the maxima are expressed as:

(5)

= To e x p ( - - e z ) ( ~ e ) (~f2) = A~ 8z

max

8O

60

4o

%

,q

I

20

I

o 0

10

I

I 20

OEPTH (m)

Fig. 1. M a x i m u m a m p l i t u d e s o f seasonal p e r t u r b at i on ( ~ ) gradient (~ ~ 3 0 ° C / k i n ) .

30

I

I 40 G4 6 0 - - 8 9 A

a d d e d to n o m i n a l g e o t h e r m a l

80

These maxima are plotted as a function o f depth in Fig. 1. Observations must be made at depths greater than 30 m if gradients are required with an accuracy better than 1% independent o f the time of measurement. Gradients obtained at depths less than 20 m m a y be subject to errors greater than 25%. S E C U L A R C H A N G E S IN C L I M A T E

Secular changes in climate are not well determined. Large changes in surface temperature can be expected to result from the retreat of sheet ice associated with periods o f glaciation (Beck, 1977), b u t variations in insolation may also be caused b y recent cultural changes to the land surface (Hyndman and Everett, 1968). Changes of this type can be modelled as a series of discrete events each with a separate step function in surface temperature for which Carslaw and Jaeger (1959, p. 59) have given the solution:

T=

To e l f

(6)

(z/2~Kt)

Where surface temperatures have increased b y 10nc after glaciation at 10,000 years B.P., errors of 20% may result in geothermal gradients measured at depths to 1000 m (Crain, 1968). Few data have been obtained at depths greater than 1000 m and consequently it may be concluded that absolute determinations of heat flow are rare: existing data have been obtained at various depths and are therefore subject to corrections o f different magnitudes. Furthermore, the amplitude of any secular change may vary according to location as well as depth; maximum amplitudes can be expected in regions of recent glaciation, decreasing near more temperate zones (Cull, 1979). Any such correction will be even more complicated where several periods of glaciation can be detected. 14"8Calculated temperatures c o r r e s p o n d i n g to i n s e t c I i m a t i c m ° d e I --~L..~......,.-.="lrer

---~ 14-6 -~_k 14"4-

•

•

Borehole d a t a - - - t

• •

O~O•

•

14"2 -* ~

TIME ('I0 3BP)

,4"0-

0

C~ 1 3 ' 8 ~

3

6

9

'Z 15 18

n

13"61 0

I 40

I

I 60

I

~ 4 I I I 120 160 D E P T H (m)

I

I 200

I

I 240

I

I 280 6460 - 88A

Fig. 2. B o r e h o l e t e m p e r a t u r e s n e a r C a n b e r r a indicating the effects o f glacier r e t r e a t in the S n o w y M o u n t a i n s , s o u t h e a s t e r n Australia.

81

Complex climatic models based on geothermal data have been proposed for the northern hemisphere, but the magnitude of each perturbation remains contentious. Beck (1977) has reviewed the principal facts of climatic history in the northern hemisphere; he concludes that climatic perturbations in the geothermal gradient should remain significant to depths of 2000 m, causing errors of 20% at depths less than 400 m. However, no systematic trends were detected in heat flow data obtained at depths approaching 3000 m in Canada (Sass et al., 1971). It is probable therefore that successive climatic events in the Pleistocene have resulted in compensating perturbations, causing only small residual changes. Alternatively, because of continual ice cover in regions of high latitude, the present surface temperature may not be significantly warmer than during the Pleistocene glacial periods (Gates, 1976); 19 '8

19 '72

-

19 "7

93o-

\

~.

\ "~ 19"6 -

Q : 73 mWm - 2

\

= 3.2 Wm-IK -I

\ .\ \ ~ 19'5

I

19.21-

-

I

14.20

I

16 I00

\

14000

YEARS BP

\

~ 19-4 -Ae • •

B o r e h o l e data

19"3 -,,I

~ 19.2 C u r v a t u r e c a u s e d by i n d i v i d u a l c l i m a t i c event 19.1

/ D [ Calculated temperatures \ c o r r e s p o n d i n g to i n s e t c l i m a t i c model. R e p r e s e n t s sum of i n d i v i d u a l c u r v a t u r e s

--

E

19'0

I 0

I

I 60

I

I

DEPTH(m)

I 120

I

L

I 180

XSe/B=O-=O-tA

Fig. 3. Perturbations in the geothermal gradient at Berrigan, southeast Australia. Borehole penetrates granite of uniform composition giving constant conductivity. Correction of 20% required to apparent heat flow at depth of 100 m.

82 the magnitude of any perturbation in surface temperature may therefore be negligible even in the upper 1000 m. Climatic models for Australia during the Quaternary have been reviewed by Bowler et al. (1976). Geological data indicate glacier retreat commencing a b o u t 15,000 years B.P. b u t subsequent warming rates are poorly known. The geothermal consequences have been considered b y Cull (1979). Data from t w o boreholes near Canberra are consistent with continuous warming in the period 1 5 , 0 0 0 - 9 0 0 0 years B.P. (Fig. 2). However, more detailed models could n o t be formulated because of extreme variations in thermal conductivity at shallow depths. In a subsequent survey geothermal data have been obtained in a horehole 300 km west of Canberra at Berrigan, N.S.W. (35.7°S, 145.8°E). This hole penetrates granite o f uniform composition to a depth of 200 m, and the data are consistent with the previous Canberra models, indicating an increase in surface temperatures of 5°C at 15,000 years B.P. However, the curvature in gradient caused b y the primary climatic event is insufficient to accommodate data in the interval 80--140 m (segment CD in Fig. 3). The greater departure from linearity in this shallow interval m a y indicate an abrupt cooling of 0.1°C at 100 years B.P. Increased curvature persisting to depths o f 80 m (segment BC) is readily attributed to a secular increase in surface temperature of 0.5°C at 16 years B.P. This event corresponds with the starting date of short-term quarry operations which resulted in rapid removal of 3 m o f surface rock. The geothermal record of this well d o c u m e n t e d cultural event imposes a valuable internal constraint on models of climatic change. LONG-PERIOD VARIATIONS -- THE MILANKOVITCH CYCLE The Berrigan geothermal data suggest relatively stable surface temperatures in New South Wales from 9000 years B.P., a result consistent with data from the Antarctic (Lorius et al., 1979). In contrast, variations o f 0.5°C have been suggested for New Zealand with periodicities from 50 to 100 years (Wilson et al., 1979). Propagation wavelengths near 150 m would be expected with such periodicity, together with a characteristic exponential decay. No such anomalies with magnitudes greater than 0.1°C have been detected in the Berrigan data, and it can be concluded that only those perturbations associated with glacial histories are significant in determinations of heat flow. Variation in solar insolation can be expected from the geometrical relations describing the orbit of the Earth a b o u t the sun. The principal features related to obliquity, precession, and eccentricity have been reviewed by Hays et al. (1976); they demonstrate that the original Milankovitch concepts are broadly consistent with the geological records of climatic change. In particular, spectral analysis can be used to demonstrate a dominant 100,000 year periodicity with other peaks at 23,000 and 42,000 years. The amplitude of each excursion in surface temperature m a y depend on non-linear atmospher. ic multipliers, b u t maxima should not exceed 5°C.

83 The perturbations to the geothermal gradient resulting from these periodicities can be predicted from equations (1) and (5). The propagation wavelength (6000 m) is much greater than the depth of most boreholes and it is clear from the results shown in Fig. 4 that observations must be made at depths greater than 1500 m if absolute values of heat flow are to be measured with errors less than 5%. For relative values, however, it may not be necessary to calculate the exact corrections required for data at shallower depths. For example, observations at depths of 200 m and 400 m may require corrections of 20% and 17%, respectively. The difference between these shallow hole corrections may not be very important for the purpose of geothermal exploration. Exact compensation for the Milankovitch cycle may be impossible to implement because of difficulties in establishing a phase origin for the periodicity. However, a trend to further glaciation has been predicted (Hays et

0 BP I0-1

~

fl

!.

I

I

I

xl ,

5 0 , 0 0 0 BP I I

I

/

I00,000 BP I

~,,=,-,~

I

N

I

I

I

I

PI

P.5

Po

5

3 I...

200 o

400 I

6 0 ~ i

800 -'~,..A

I000 I

1200

1400 1600 I I DEPTH (m,I

1800 i

k.

~-4 -5

-6 -7

-~

G 460-90A

Fig. 4. Maximum amplitudes of perturbation caused by Milankovitch cycle (inset). Present perturbation estimated to range in the periods ( 0 . 1 - - 0 . 2 5 ) .

2o~o

2C

84 al., 1976), and it can be assumed that the present cycle is at least one quarter complete (i.e. t = 0.25P}. RECOMMENDATIONS Although absolute values of heat flow may not be accurately determined with conventional techniques even at depths of 1000 m, data useful for exploration can be obtained in shallower holes at a suitably chosen standard depth; constant corrections are then applicable but they need not be specified for relative heat flow. These values can then be used in modelling the local thermal structure which may then indicate a geothermal resource. A suitable standard depth can be specified from the propagation characteristics of the Milankovitch cycle. However, less contentious results may be obtained by assuming that Quaternary glacier retreat can be modelled as a purely secular change in climate. In either case the magnitude of the necessary correction varies only slowly with depth, and a constant correction can be assumed for small intervals. Stable temperatures used in determinations of heat flow are readily obtained in existing boreholes drilled for routine mineral exploration, stratigraphic sampling, or water table observations. In general these holes must be logged with tools of diameter less than 5 cm, but temperature probes housing a thermistor sensor can be constructed small enough to be suitable in most cases. Usually a single thermistor is lowered to successive depths, and temperatures are recorded at discrete intervals after a period of equilibration. Thermal gradients are then computed from linear segments using a leastsquares reduction method. Data are usually obtained at intervals greater than 10 m, and linear segments may comprise intervals exceeding 100 m; as a result thermal conductivity must be measured on numerous core samples to characterise the segment before heat flow can be determined. Additionally, individual climatic corrections must be specified for temperatures measured at each depth. For geothermal exploration it is preferable to measure heat flow at shallow depth {100 m) with a reduced data interval to ensure a constant climatic correction. If a single thermistor probe is used as described above, the depth between measurement points may contain errors resulting from wire stretch, counting loss, irregular drillholes, etc. This difficulty can be avoided by recording differential temperatures by means of a logging tool with two sensors at a fixed separation (e.g. 5 m). Linear regression between observations is not required with probes of this type and thermal gradients can be determined with great precision over small intervals. If drill core is extracted for the complete interval between the sensors, thermal conductivity (or resistivity) can be specified precisely for more reliable calculations of relative heat flow.

85 REFERENCES

Beck, A.E., 1977. Climatically perturbed temperature gradients and their effect on regional and continental heat flow means. Tectonophysics, 41 : 17--39. Bowler, J.M., Hope, G.S., Jennings, J.N., Singh, G. and Walker, D., 1976. Late Quaternary climates of Australia and New Guinea. Quaternary Res., 6 : 359--394. Carslaw, H.A. and Jaeger, J.C., 1959. Conduction of Heat in Solids. Oxford University Press, Oxford, 2nd ed. Crain, I.K., 1968. The glacial effect and the significance of continental terrestrial heat flow measurements. Earth Planet. Sci. Lett., 4: 69--72. Cull, J.P., 1979. Climatic corrections to Australian heat flow data. Bur. Miner. Resour., J. Aust. Geol. Geophys., 4: 303--307. Cull, J.P., 1980. Geothermal records of climatic change in New South Wales. Search, 11 (in press). Gates, W.L., 1976. Modelling the ice age climate. Science, 191: 1138--1144. Hays, J.D., Imbrie, J. and Shackleton, N.J., 1976. Variations in the Earth's orbit: pacemaker of the ice ages. Science, 194: 1121--1132. Hyndman, R.D. and Everett, J.E., 1968. Heat flow measurements in a low radioactivity area of the Western Australian Precambrian Shield. Geophys. J., 14: 479--486. Lorius, C., Merlivat, L., Jouzel, J. and Pourchet, M., 1979. A 30,000 yr isotope climatic record from Antarctic ice. Nature, 280 : 644--648. MacDonald, G.J.F., 1963. The deep structure of continents. Rev. Geophys., 1 : 587. Roy, R.F., Blackwell, D.D. and Birch, F., 1968. Heat generation of plutonic rocks and continental heat flow provinces. Earth Planet. Sci. Lett., 5: 1--12. Sass, J.H., Lachenbruch, A.H. and Jessop, A.M., 1971. Uniform heat flow in a deep hole in the Canadian Shield and its palaeoclimatic implications. J. Geophys. Res., 76:

8586--8596. Wilson, A.T., Hendy,C.H. and Reynolds, C.P., 1979. Short-term climate change and New Zealand temperatures during the last millennium. Nature, 279: 315--317.