35. AN ESTIMATE OF THE HEAT FLOW IN THE WESTERN NORTH ATLANTIC AT DEEP SEADRILLING PROJECT SITE 5341

Jeremy Henderson,2 Department of Earth and Planetary Sciences, Massachusetts Institute of Technology,Cambridge, Massachusetts

andEarl Davis, Pacific Geoscience Centre, P.O. Box 6000, Sidney, B.C., Canada V8L 4B2

ABSTRACT

A method employing both a simple relationship between total rock conductivity and the concentrations and conduc-tivities of constituents, and an estimate of the thermal gradient based on a downhole temperature log, yielded anestimated value between 1.01 (42 mW m~2) and 1.37 HFU (57 mW m~2) for the heat flow at Site 534. These measure-ments are in good agreement with other heat-flow measurements made in the same area, indicating that this methodmay have a wide application in estimating heat flow at well sites.

INTRODUCTION

Hole 534 was drilled in the Blake-Bahama Basin,northwest Atlantic (Fig. 1), where the crustal age (ca.150 m.y.) is great enough and the sediment cover thickand extensive enough that the heat flow should be unper-turbed by hydrothermal circulation. This hypothesis hasbeen verified by detailed heat-flow measurements atfour sites nearby (Fig. 1), where the heat flow has beenfound to be extremely uniform over distances up to sev-eral hundred kilometers (E. Davis et al., unpublisheddata). Thus we can reasonably assume that in the absenceof deep transients, the heat flow at Site 534 should bethe same as that measured at the survey sites nearby.

In this study we have estimated a thermal gradientfrom a temperature log of the hole. In the absence of de-tailed thermal-conductivity measurements, we have esti-mated the conductivities of the sediments at Site 534 us-ing two simple relationships between the bulk conduc-tivity and the conductivities of the constituent phases.The resulting heat-flow estimates are then compared toheat-flow measurements obtained nearby to check thevalidity of the conductivity estimates.

POROSITY AND CONDUCTIVITY

Gravimetric porosity determinations were made byshipboard scientists on samples recovered from Site 534cores at about 5-m intervals. Figure 2 shows porosityvalues taken from this data set at intervals of approxi-mately 10 m, plotted against sub-bottom depth. Alsoplotted in Figure 2 are the least-squares best fit to thecomplete data set; the line found by Le Douaran andParsons (in press) to fit data from a number of holes inthe North Atlantic; and the exponential curve quoted bySclater and Christie (1980) as being a good fit to datafrom the North Sea. As may be readily seen, the scatter

Sheridan, R. E., Gradstein, F. M., et al., Init. Repts. DSDP, 76: Washington (U.S.Govt. Printing Office).

2 Present address: Exxon Company, U.S.A., P.O. Box 2180, Houston, Texas.

of data from Site 534 around all these lines is large, andany smooth curve through the data would be only poor-ly defined. For our purpose of using porosity for estimat-ing conductivity, a mean porosity was calculated foreach of the lithologies identified by shipboard scientistsfrom the recovered cores for which porosity data wereavailable (Subunits 2a through 7a). Then, by inspectionof the smear-slide summaries, a representative mineralcomposition was estimated. For simplicity, the sedimentswere assumed to consist of sea water, quartz, clay, andcarbonate.

Two methods of estimating bulk conductivity wereused here. Budiansky (1970) presents a theoretical rela-tionship among the conductivity of a composite materi-al, K, and the volume fractions,/, and thermal conduc-tivities of the constituent phases, Kh as follows:

ft (2/3 + l/3[Ki/K\)-1 = 1

This relationship, derived from the solution to the anal-ogous electrostatic problem, assumes that the compositeconsists of a random mixture of n isotropic compo-nents, at least n - 1 of which are distributed in partic-ulate fashion. Traditionally the geometric mean, K =K{lK&,..., Kf

nn

t has been used with some success (Wood-side and Messmer, 1961), and although there is some-what less theoretical justification for its being used tomodel a multicomponent system, we have employed ithere also. In practice, the conductivity of sediments andsedimentary rocks is not likely to be described accurate-ly by either theoretical relationship over a very wide rangeof porosities. At high porosities, grain to grain contactswill be few, and the bulk conductivity will be controlledby the interstitial water. The correct mean value wouldperhaps best be described by a series (weighted har-monic) mean of the conductivities of the constituents.At low porosities, grain to grain contacts will be com-mon, and a Budiansky-type mean may be reasonable,

719

J. HENDERSON, E. DAVIS

30c

25'

75 70° 65°

Figure 1. Map showing location of Site 534 (open circle) in relation to other heat-flow stations (solid tri-angles) and identified magnetic anomalies. (Dashed lines are contours of water depth at 1000-m inter-vals. Boxed numbers are heat-flow values.)

but the thermal contact resistance between the grainswill lower the effective conductivities of the constitu-ents. The effect of layering in sediments will be to re-duce the vertical thermal conductivity of a sedimentaryunit. Detritally preferred orientation of any isotropicmineral grains will influence the effective conductivityof that constituent. In summary, then, it is probablyunreasonable to approach in a purely theoretical waythe problem of predicting bulk conductivity accurately.A reasonable estimate may be obtained, however, usinga theoretical relationship with empirically established"effective" constituent conductivities.

We have used the following values as estimates for"effective" grain conductivities:

Wm-1K"1)(Beck, 1965)7 × I0-3 cal-cm-1-s-1-°C-1 (2.9W m~1 K"1) (Robertson, 1979)

W m~1 K~1) (value quoted belowreduced by a factor equivalent tothat by which KquaTtz and Calcitehave been reduced)1.5 × 10-3cal-cm-1-s-1-°C-1(1.6Wm-iK" 1 ) (Powell, 1958)

The values for quartz and calcite have been derivedfrom polycrystalline aggregates. It is interesting to com-

pare these values with typical single crystal values ofconductivity to emphasize the importance of grainboundary contact resistance:

K,quartz

K,calcite

= 17 × 10-3(W m-1 K1940)

= 8.6 × I0-3

1) (Birch and Clark,

1971)Kr

k quartz

C alcite

K,clay, feldspar, etc. ~~

^ seawater

clay, feldspar, etc. = 5.7 × I 0 " 3 Cal-Cm" 1-S" C " * (2.4W m-1 K~1) (Sass, 1965)

As a "calibration" we have plotted the estimatedconductivity of DSDP Sites 438 and 439 sediments inFigure 3a, using the average mineralogy of the fairlyuniform sediments at the sites (Scientific Party, 1980)

/quartz = 0.05/calcite = 0.02/clay, feldspar, etc. = " « " J

and the values of constituent conductivities given earli-er. These curves are compared to direct measurementsof porosity and conductivity (Carson and Bruns, 1980).The agreement is surprisingly good, in spite of the obvi-ous uncertainties in choosing appropriate constituentconductivities; the geometric mean provides as good afit to the data as the Budiansky curve over the full rangeof porosities. The latter relationship predicts conductiv-ities that are about 5% higher at high porosities (cf.,

720

HEAT FLOW IN THE WESTERN NORTH ATLANTIC

200

400

600

800

1000

1200

1400

1600

Porosity (%)

10 20 30 40 50 60\ i i i i f ,i '

/ ' 'I ' *If *r '

/ /7 '

// /// /

ft v Λ

• Ax ' $

v y "f 1 11 I f X

i / / ¥ -/ / / & . .

i l l .Xy */ JL• , X y "

Y ' / »X y

" * x i ' / ' ×xY X / / / X X

x * / ' ΛX

×× jjt

x / / xr x

y m vXx / x * -

x / Yx/» yXv 1 * Λ

- × \ X

Lithologicunit

1

2ab

c

d34a

b

cd

5a

b

c

d

6a

b

7a

b and c

Thickness(m)

545.8

20.828.471

2946.523

122.5

2736

148.5

94.7

66

74

86

67.6

54.2

85.5

Porosity(%)

(47)

48.3 "42.5336.93

24.2833.143.4

42.14

41.5834.18

23.73

22.84

22.08

18.29

18.21

15.9

17.42

(24)

Composition

quartz

.02"^

.02

.03

.03

.04

.04

.15

.06

.02

.02

.02

.03

.01

0

.02

.03

fclay

.21

.29

.28

.36

.36.526

.43

.49

.40

.35

.40

.46

.51

.59

.53

.53

calcite

.29

.26

.32

.36

.260

0

.03.24

.39

.36

.29

.30

.23

.29

.27

Estimated conductivity(T C I' I

Budiansky

(3.15)

3.343.513.89

4.503.983.11

3.66

3.303.78

4.52

4.48

4.41

4.42

4.28

4.62

4.56

(4.25)

Geometric

(2.9)

3.093.213.57

4.143.622.92

3.37

3.073.53

4.2

4.25

4.16

4.27

4.09

4.39

4.36

(4)

Figure 2. Graph of porosity values (X), taken from the Site 534 data set at intervals of about 10 m, plotted against sub-bottom depth. (The heavy lineis the least-squares best fitting straight line to the whole data set, the dashed line is that found by Le Douaran and Parsons [in press] to be a gooddescription of porosity-depth data for a number of deep drill holes in the North Atlantic, and the dotted-dashed line is Sclater and Christie's[1980] exponential fit to data from the North Sea. The tabular material gives, for each lithologic unit, the thickness, average porosity, estimatedcomposition [as a volume fraction of the wet rock], and two estimates of conductivity. Porosities estimated for Unit 1 and Subunits 7b and c areshown in parentheses.)

preceding discussion). We have thus gained confidenceto apply the geometric mean with the appropriate con-stituent conductivities to Site 534 sediments. A similarplot, with estimated formation conductivities (listed inFig. 2) and smooth curves of conductivity calculatedover the full range of porosities for a sediment of aver-age mineral composition, using both Budiansky's rela-tionship and the geometric mean, is shown in Figure 3Bfor Site 534.

With the thermal conductivities thus estimated, an es-timate of the total average thermal conductivity can bederived using the series, or weighted harmonic, mean:

where Sj and Kj are the thickness and conductivity of theyth layer. Using the geometric mean formation conduc-tivities, a conductivity of 3.35 × 10~3 cal-cm^-s~1-°C- l (1.4 W m- * K-l) is obtained for the sediment col-umn down to a depth of 1325 m (see the next section). Ifwe were to use the conductivities calculated on the basisof Budiansky's relationship, the total average conduc-tivity would be marginally higher.

721

J. HENDERSON, E. DAVIS

I 3

40 50 60Porosity (%)

90 100 10 20 30 40 50 60Porosity (%)

70 80 90 100

Figure 3. A. Graph of measured conductivity values plotted against measured porosities at Sites 438 and 439. (Curves are estimates of conductivityfor a rock of average mineral composition [see text]. The solid line indicates conductivities estimated using the relationship described by Budian-sky [1970], and the broken line indicates conductivities estimated as the geometric mean of the mineral conductivities.) B. Graph of conductivityof Site 534 Subunits 2a to 7a plotted against porosity. (Open circles show conductivities estimated via geometric mean, and crosses those estimat-ed on the basis of the relationship delineated by Budiansky [1970]. Curves are estimates of conductivity using a rock of average mineral composi-tion.)

TEMPERATURE LOG ANALYSIS

Approximately four days after drilling operationswere terminated, a section of pipe was washed into thehole to a sub-bottom depth of 944 m in order to enablethe logging tool to pass a known constriction at theBlake-Bahama Formation contact. Logging of the holewas then accomplished with a Gearhart-Owen thermallog down to a depth of 1414 m, where a bridge in theCat Gap Formation stopped the tool. The hole had beendrilled to a total depth of 1666 m sub-bottom. The result-ing temperature-depth plot is shown in Figure 4.

The effects of fluid circulation during the wash-inprior to logging is clearly evident, and temperaturesdown to the maximum depth of the pipe are totally un-reliable. Below that, temperatures are still likely to beaffected by earlier drilling fluid circulation, but therethe effects are we hope minor, because several days hadelapsed since the most recent circulation. The tempera-ture-depth profile in the bare part of the hole is some-what erratic, particularly below about 1350 m; whichmay simply reflect differences in the rate of return tothermal equilibrium down the hole caused by variationsin the hole size. Under ideal conditions, the hole diam-eter should be little more than 25 cm, the diameter ofthe bit. Where caving takes place, the hole "diameter"may be several times this size, and the thermal effects ofthe circulating fluid will be correspondingly greater.

Temperature Conductivity<°C> (T.C.U.)

0 10 20 0 2 4

Conductivity(T.C.U.)

0 2 4 6

Figure 4. Sub-bottom temperatures recorded by the Gearhart-Owenlogging tool, and estimated conductivities. (Note the jump in tem-perature at 944 m as the tool enters the bare, and less thermally dis-turbed, hole.)

722

HEAT FLOW IN THE WESTERN NORTH ATLANTIC

Many techniques have been used for estimating themagnitude of drilling fluid disturbances. Hyndman etal. (1977) review several; in this study we have adoptedthe one chosen for their data analysis and presented byJaeger (1961). He considers the problem to be one inwhich the hole of some diameter remains at some dis-turbed temperature for the duration of fluid circulation.The thermal effects of the disturbance during the circu-lation and during the subsequent decay to equilibriumare calculated for a depth of 1325 m, assuming that thethermal diffusivity of the material filling the hole (fluid,chips, and drill pipe) is the same as that of the rock. Inusing this correction, we have assumed that the fluid issupplied at the temperature of the bottom water, 2.8°G(justified for high rates of circulation by Jaeger, 1961),and that the disturbance time may be represented by theactual drilling time, when pumping takes place. Table 1summarizes the timing of the disturbances we have con-sidered, and shows their effects at the time of logging.

For the purpose of calculating a gradient, we haveused the maximum observed temperature of 42.5°C at adepth of 1325 m. Applying the total correction of 27%implied by the results shown in Table 1, a gradient of43.5 m°C m~1 results. With no correction, the gradientwould be 30 m°C m~1. This lower value is undoubtablytoo low, but it does provide a lower limit for the truegradient value over the 1325-m interval. The true valueis probably closer to the "fully corrected" figure, al-though the latter may be too high, because the tempera-ture of the circulating fluid was bound to be a certainamount warmer than that assumed (the bottom watertemperature), and because circulation times are prob-ably overestimated. If subsequent logs were available, acorrection could be more confidently applied.

DISCUSSION

The heat flow, calculated as the product of the esti-mated thermal conductivity of the sediment column,3.35 × I0-3 cal/cm-s-°C (1.4 W m~1 K~1), and theminimum estimated temperature gradient, 30 m°C/m,is 1.01 HFU (42 mW m~2). Using the maximum esti-mated temperature gradient of 41 m°C/m, we find thatthe estimated heat flow is 1.37 HFU (57 mW m~2).

These limiting values are compared to the other near-by heat-flow determinations mentioned earlier (Davis etal., unpublished data; Galson and Von Herzen, 1981) inTable 2. It is encouraging that these values determined

Table 1. Summary of postdrilling disturbances of temperature in Hole534A at a sub-bottom depth of 1325 m caused by water circula-tion.

Table 2. Comparison of estimated heatflow at Site 534 with other heat-flowmeasurements in the same area (E. Da-vis et al., unpublished data; Galsonand Von Herzen, 1981).

Duration ofwater circulation

(hr.)

- 2 052

-242

Time elapsedsince disturbance

(days)

25184.53

Xto/3L2

2.46.02.80.24

Log n

1.480.920.651.56

Fraction ofdisturbanceremaining

(Jaeger, 1961)

0.030.070.140.03

Site 534 andheat-flow stations

Site 534A2 97KN77 4, 6, 11KN77 17KN77 14, 15

Age(m.y.)

153115141148151

HFU

1.01-1.371.13 ±0.051.16 ±0.141.16 ±0.121.15 ±0.06

Note: a 12.5 cm; × = 0.005 cm2-s ' ; t0 = disturbance duration (s); n= time elapsed/disturbance duration.

Note: HFU = heat-flow unit.

by Davis et al. (unpublished data) and Galson and VonHerzen (1981) fall well within the range of values esti-mated at Site 534. This similarity lends great credence tothe technique of determining conductivity indirectlyfrom porosity and mineralogy data, although the ac-curacy of the technique could have been better assessedif the temperature log had been made more careful-ly (cf., Burch and Langseth, 1981). Nevertheless, thismethod appears to provide a good estimate of the aver-age conductivity, and it should be useful in obtainingheat-flow estimates at sites where no direct conductivitymeasurements are made, oil wells, for example.

There is little difference between the bulk conductiv-ity calculated as the geometric mean of the constituentconductivities and that using a more "rigorous" deriva-tion such as that given by Budiansky (1970). We favorthe geometric mean because of its simplicity.

ACKNOWLEDGMENTS

We are grateful to John Sclater and Dick Von Herzen, who re-viewed this chapter. The manuscript, typed by Dorothy Frank, isEarth Physics Branch contribution number 1008.

REFERENCES

Beck, A. E., 1965. Techniques of measuring heat flow on land. InLee, W. H. K. (Ed.), Terrestrial Heat Flow. Am. Geophys. UnionMonogr. Ser., 8:24-57.

Birch, F., and Clark, F., 1940. The thermal conductivity of rocks andits dependence upon temperature and composition. Am. J. Sci.,238:529-588 and 613-635.

Budiansky, B., 1970. Thermal and thermoelastic properties of isotrop-ic composites. J. Composite Materials, 4:286-295.

Burch, T. K., and Langseth, M. L., 1981. Heat flow determination inthree DSDP boreholes near the Japan trench. J. Geophys. Res.,86:9411-9419.

Carson, B., and Bruns, T., 1980. Physical Properties Appendix. InScientific Party, Init. Repts. DSDP, 56, 57, Pt. 1: Washington(U.S. Govt. Printing Office), 615-629.

Galson, D., and Von Herzen, R. P., 1981. A heat flow survey on anom-aly M-O south of the Bermuda Rise. Earth Planet Sci. Lett., 53:296-306.

Horai, K., 1971. Thermal conductivity of rock forming minerals. /.Geophys. Res., 76:1278-1308.

Hyndman, R. D., Von Herzen, R. P., Erickson, A. J., and Jolivet, J.,1977. Heat flow measurements, DSDP Leg 37. In Aumento, F.,Melson, W. G., Init. Repts. DSDP, 37: Washington (U.S. Govt.Printing Office), 347-362.

Jaeger, J. C , 1961. The effect of the drilling fluid on temperaturesmeasured in boreholes. J. Geophys. Res., 66:536-569.

Le Douaran, S., and Parsons, B., in press. A note on the correction ofocean floor bathymetry and heat flow with age. J. Geophys. Res.

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J. HENDERSON, E. DAVIS

Powell, P. W., 1958. Thermal conductivity and expansion coefficient Sclater, J. G., and Christie, P., 1980. Continental stretching: an exof water and ice. Adv. Phys., 7:276-297. planation of the Post Mid-Cretaceous subsidence of the Centra

Robertson, E., 1979. Thermal conductivities of rocks. U.S. Geol. Surv. North Sea basin. /. Geophys. Res., 85:3711-3739.Open File Rep., 79:356. Woodside, W., and Messmer, J. H., 1961. Thermal conductivity o

Sass, J. H., 1965. The thermal conductivity of fifteen feldspar speci- porous media. 1, Unconsolidated sands. 2, Consolidated rocks. Jmens. / . Geophys. Res., 70:4064-4065. Appl. Phys., 32:1688-1706.

Scientific Party, 1980. Sites 438 and 439: Japan Deep Sea Terrace, Leg57. In Scientific Party, Init. Repts. DSDP, 56, 57, Pt. 1: Washing-ton (U.S. Govt. Printing Office), 23-192. Date of Initial Receipt: April 7, 1982

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