spe-4171-pa predicting thermal conductivities of formations from other known properties

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Predicting Thermal onductivities of FormationsFrom Other Known Properties

J. ANAND*W, H. SOME RTON

E, GOMAA**

MEMBERS AlME ABSTRACT

,ifeasuying ~})e ~hertnal properties o rocks and

rock-fluid systems is di//icult ard time consuming,

and the resuIts /rorn such measurements ure 0/

limited value unless conlpIete dt scrip tions of the

rock and fluids are given. A need exists /or a

method of predicting thermal behavior from othermore eosily measurable properties. Presented here

a?e crn-relajions developed /or predicting the thermoI

conductivity o/ consolidated sandstones /rem a

knouledgc o/ density, porosity, permeability, and

~ormatio v resist ivity factor. Values for alI these

properties u..e auailahIe from well logs or core

urmiysis data. Also obtained l~ere correlations for

estimating the thermal conductivity O/ liquid-

saturrzted sandstones from a knowledge of the

conduct itities o/ dry srmdstones and tbe saturating

liquid mrd properties of tile sandstone, The thermal

conduct il’ity 0/ most rocks decreases with

increasing temperature and a method of estimating

thi.q elfcct is presented. Tbe e//ect of pressure on

conductivities is generally small, bat may be

Pre ficteri /rem a knouledge oi the buIk compressi-

bility of the rock.

INTRODUCTION

Although thermal recovery processes have been

applied in the petroleuc industry for many years,

there is still a lack of basic t>ermal data with

which to predict the performance of these processes.

Much of the thermal conductivity work reported in

the literature lacks a complete description of the

physical properties of the rocks used, and in

addition, most of the thermal conductivity

mettsurements have been made at room temperature

and at atmospheric pressure. The work reported in

this paper deals with the thermal conductivity of

typical porous rocks at simulated subsurface

conditions of temperature, pressure, and saturation.

Because thermal conductivity is difficult to

measure, emphasis has been placed here on

Paper SPE 4171) was presented at SPE-AIME 43rd Annual

California Regional Fall Meeting, held in Bakersfield, Nov. S-10,

1972. @copyright 1973 American Institute of Mining, Metallur-

gical, and Petroleum Engineers, Inc.

*Now at U. of Southern California, Los Angeles.

**Now wjth Standard Oil CO. of California, San Francisco.

preferences given at end of PaPer.

oCTOBER, 1973

U. OF CALIFORNIA AT BERKELEY

BERKELEY, CAL IF.

methods of predicting thermal conductivity from

other more easily measured properties as well as

on methods of predicting the effects of temperature,

pressure, and liquid saturation on thermal properties.

R EL ATI ONS HI P OF TH ER MAL C OND U CTI VI TY

TO OTH ER P HYSICAL P ROP ERTIE S

The thermal conductivities of dry rocks havebeep shown to be functions of density, porosity,

grain size and shape, cementation, and mineral

compositional The first two properties are easy to

measure and precise values may be assigned for

correlation purposes. Grain size and shape and

cementation are difficult to quantify. There are,

however, other related properties that can be used

to characterize these properties for use in

correlations. Permeability and formation resistivity

factors are probably most closely related to these

properties and are readily measurable as unique

values. Precise mineral composition values are

generally not available, ald even if they were, it

would be difficult to introduce them into

correlations. The high thermal conductivity of

quartz seems to have a predominating influence,

and thus for most sandstones containing quartz in

moderate amounts, the effects of other minerals can

be ignored.

Many efforts have been made to relate thermal

conductivity to the physical properties of porous

rocks. These efforts have been reviewed in rather

complete detai 1 by Scorer 1 and Anand.2

Unfortunately, most of the correlations developed

require a knowledge of the thermal conductivity of

the rock matrix or the dry rock at some known

porosity. Although some simple correlations have

been obtained, these are for specific systems and

are not applicable generally. Probably the most

useful work in this area is that reported by Zierfuss

and Van der Vliet. 3 Basing their analysis on 36

sandstones having a wide range in measured

properties, they obtained a correlation between

effective porosity and the product of thermal

conductivity and formation resist ivity factor. A

fourth-order polynomial fit of thermal conductivity

and fractional porosity was obtained by regression

analysis. Their data also seemed to indicate that

thermal conductivity increases with permeability,

this being attributed to conduction in wider pores.

In the work discussed here, multiple regression

267



analyses using all the common physical properties

of rocks were employed to obtain a useful

relationship for predicting thermal conductivity.

TP ERMAL CONDU CTIVI TIE S OF

L IQU ID -S ATU RATE D R OC KS

The thermal conductivity of fluid-saturated rocks

is dependent upon the conductivities of the dryrock and the saturating flu’id and physical properties

of the rock. Assad,4 Sugawara and Yoshizawa~ and

Van derVliet3 have dealt with this problem; however,

testing their relationships with data from the

literature and with the data we obtained did not

yield satisfactory results. The difficulty seems to

lie in the fact that, although liquid-saturated rocks

have higher conductivities than dry rocks, the

amount of increase is a complex iunction of the

amount, character, and distribution of pore space,

and the conductivity of the saturating fluid.

Earlier efforts2 to develop correlations using

regression analysis on common physical properties

of :ire rock fluid system were only partially

successful. In the present work, nonlinear multiple

regression analysis on dimensionless groupings of

properties of the system gave satisfactory

relationships.

EFFECT OF TEMPERATURE ON

THERh fAL CONDUCTIVITY

The thermal conductivity of most materials that

have crystall e structures decreases with increased

temperature. Theory indicates that the:mal

conductivity should vary with the reciprocal of

temperature. In mixed crystals and highly disordered

crystals, conductivity varies more slowly than T-l

and, in fact, may show a slight increase with

temperature. The thermal conductivity of glasses

and vitreous materials increases with temperature.

To predict the effect of temperature on the thermal

conductivity of rocks, Tikhomirov7 developed a

correlation equation based on experimental data. A

plot of this equation shows that moderate negative

gradients of thermal conductivity with temperature

will be predicted for high-conductivity rocks,

whereas small positive gradients will be predicted

for low-conductivity rocks. This agrees with theory

and with the experimental results of our

investigation.

For the work discussed here we developed a

modified expression of the same general form as

Tikhomirov’s equation. This expression predicts

the thermal conductivity-temperature behavior of

liquid-saturated rocks as well as of dry rocks.

EFFECT OF P RE .SURE ON

THERMAL CONDUCTIVITY

Several investigators have shown that thermal

conductivity increases with an increase in effective

stress on the rock.8-10 This should be expected

since increasing the stress improves the thermal

contact between grains and increases the over-all

density of the rock and, consequently, the thermal

conductivity.

In measuring thermal conductivity in the laboratory

26S

a large part of the apparent increase in conductivity

with pressure may actually be due to reduction in

thermal contact resistances between the several

contacting surfaces: heat source, heat sink, the

temperature measuring devices, the standards, and

the test specimen. When good thermal contact is

established, the change in thermal conductivity

with added stress is generally small.

A relation between the bulk compressibility of

porous rocks and the change in thermal conductivity

should be expected. Edmondsong showed that the

thermal conductivities of Bereaj Bandera, and

Boise sandstones increase by 7.8, 9.5, and 12.3

percent/1 ,000 psi, respectively, in the pressure

range of 900 to 3,600 psi. The compressibilities of

Berea, Bandera, and Boise sandstones are re orted by Lobreell to be 4.54 x 10-7/psi, 6.46 x 10- /psi,

and 9.0 x 1~7 /psi, respectively. If these data are

plotted as shown in Fig. 1, z linear relation is

obtained. Edmondson’s values for the increase in

conductivity with pressure are high compared with

results obtained by Woodside and Messmer1° and

with those obtained in our work. An increase of 11. spercent/1 ,000 psi in the pressure range of zero to

1,000 psi, and an increase of 2.5 percent/1 ,000 psi

in the pressure range of 2,000 to 4,000 psi for

Berea sandstone, were reported by Woodside and

Messmer.

The effect of pore pressure is to reduce the

effective stress on the rock. More realistically, a

reduction in fluid pore pressure results in increased

effective stress on the rock and thus an increase in

thermal conductivity. Pore pressure may also be

associated with the phase behavior of contained

fluids. Reduction in pore pressure may result in the

vaporization of some of the liquid components and

this may cause a large reduction in thermal

conductivity. This is a fluid saturation effect and

should not be attributed to pore pressure per se.

EXPERIMENTAL WORK

AP PARATU S AND P ROC ED URE

Correlation equations were developed mainly from

data taken from the literature.3 To test these

correlations, a new set of data was obtained and

the correlations were modified slightly in some

cases to fit these new data.

Thermal conductivities were measured in a

steady-state comparator apparatus (described in

detail in Ref. 2). The apparatus, shown schemati-

cally in Fig. 2, consists essentially of a stack

containing a holder for the disc-shaped sample of

unknown conductivity. The sample is sandwiched

between two holders containing standards of thesame geometry as the unknown sample, but for

which the conductivities are accurately known.

Thermocouples are mounted in the centers of sample

and standard itoIder plates so that the temperatures

across the sample and the standards may be

measured. The heat source at the top of the stack

is a tank in which heated silicone oil is circulated

from a constant-temperature bath. The heat sink at

SOCIETY OF PET ROLEIIM ENGINEERS JOURNAL



the bottom of the stack is a plate through which

silicone oil is circulated from a second constant-

temperature bath. Thus heat flows vertically

downward to minimize convective heat transfer. The

total temperature drop across the stack is small —

about 40 to 50% — and radiative heat transfer is

considered to be negligible.

The sample and standard holders are made ofBakelite, which has a thermal conductivity about

an order of magnitude less than that of most of the

samples measured. To minimize radial heat losses,

however, a Bakelite ring on which heating tapes

are mounted surrounds the stack and is maintained

at the midpoint temperature of the stack. The space

between the guard heater and the stack is filled

with a ceramic fiber insulation. The entire apparatus

c44

at

/- CALCULATEDEc24 I j

I rEXpERIMENTAL OATA ,,II

I2 –

---- ----

--—-—--— -

0 1 I 1 I , I

2 4 6 8 10

i3ULK COMPRESSIBILITY, PSI ’. IO’ ILOBREE DATA [:1]

FIG . 1 — EFFECT OF PRESSURE ON THERMAL

C OND UC TI VI TY VS R OC K C OMP RE SS IB IL ITY.

N

t . ..1

I II ~ LOADINGEAR

HEATER TAPE ‘ ~ HEAT SINK

OETAIL S OF STACK

FIG , 2 — SCHEMATIC DIAGRAM OF THERMAL

CONDUCTIVITY APPARATUS .

is mounted in a loading frame so that controlled

axial stress may be applied. On the sample holder

is a pore pressure fitting and a line that connects

to a pressure recording and control system.

Pyroceram glass ceramic code 9609 was used for

the standards. Its conductivity is quite close to

that of sandstones; it is capable of withstanding

the loads to which it must be subjected; and it is

quite stable at temperatures used in the experiments.

A multipoint strip-chart recorder was used to

monitor the temperatures of the six thermocouples.

When the apparatus reached steady-state conditions

(in 3 to 4 hours), differential temperatures across

the two standards and the sample were recorded.

Thermal conductivities were calculated on the

basis of these data and of the known conductivities

of the standards.

The test specimens and the standards were discs

2 in. in diameter and 0.625 in. thick. All samples

were run dry and saturated with brine (0.1 N

potassium chloride). Several samples were also run

saturated with silicone oil (a dimethyl polysiloane),

Stoddard solvent, and ?Z.hexane. A high-temperature

version of this apparatus was used to extend the

temperature correlation.

Other physical properties of the sandstone

samples were measured with stanr;ard laboratory

equipmen” These properties ioclude density,

porosity, permeability, and electrical resist ivity

factor.

E XP ER IME NTAL RE SU LTS

Typical results of the thermal conductivity

measurements are shown in Figs. 3 through 5. The

maximum error expected in these tests is f5 percent

as calculated by error analysis,2 In general, thermal

conductivity decreases linearly with increase in

temperature, the decrease being much more

pronounced for liquid-saturated samples than fordry samples.

Liquid saturation increases thermal conductivity

substantially. The amount of increase is related to

the thermal conductivity of the saturating fluid and

to the properties of the rock, including porosity and

thermal conductivity in the dry state. It should be

pointed out that all liquid-saturated samples were

L --+llll I I50 200 250 300

TEMPERATURE . ‘F

‘FIG . 3 — THERMAL CONDUCTIVITY OF BEREA

SANDSTONE — EFFECTS OF TEMPERATURE AND

FLUID SATURATION.

OCTOBER, 197S 269



TABLE 1 —PHYsICAL PROPERTIES OF SANDSTONES

Density Resist ivity Permeability Thermal Conductivitiesot 68° F (Btu/hr. ft.° F)

Sompl e (gin/cc) Porosity Foctor (red) ~ ~ Silicone Oil Stodd.rd Solv~~— — —

Berea 2.15 0.162 13,0 190 1.35 3.00 2.55 1,98

Bandero 2.10 0.208 13.0 38 0.98 2.06 1.90 1.57

Boise 1.84 0.292 7*9 2,513 0.85 1.78 1.22 1.19

SS No. 1 2.22 0.160 18,5 152 1.47 2,96 1.65

:S Nc,, 2 2.00 0,250 10.9 557 1 1 3,26 2.24

S S NO. 3 2.26 0.149 12,0 34 0.92 1.42

Venango 2.34 0,122 35.9 437 2.39 - —

Gatchel I 2,04 0.227 13.3 858 1.19 -

fully saturated and that pore pressure was

maintained high enough to avoid vapor formation at

the temperature of the test.

.Most thermal conductivity tests were run at an

axial stress of 565 psi, this stress level being high

enough to minimize the effect of contact resistance.

Fig. 6 shows thermJ conduct ivities of Berea,

Boise, and SS 2 for different axial stress levels.

The apparent thermal conductivity increases sharply

from a stress of zero to approximately 400 to 500

psi and then rises at 2 low and constant rate. For

Berea and Boise sandstones, thermal conductivities

increase by 1.25 percent/1,000 psi and 2.0 percent

/1 ,000 psi, respectively. These values are much

lower than those reported earlier by Edmondson.g

Other physical properties of the sandstones,

including density, porosity, permeability, and

formation resistivity factor, are given in Table 1.

Thermal conductivity was measured on one shale

11

FIG . 4 — THERMAL CONDUCTIVITY OF BOISE

SANDSTONE — EFFECTS OF TEMPERATURE AND

FLU ID SATt J RATION .

~/- —-. D 1 l -a

. ----

a —- -*

z 41R DRIED (PO, 220 1 \.

:0 I I I

50 i00 I 50 Zbo 250 300

TEMPERATuRE , “F

FIG , 5 — THERMAL CONDUCTIVITY OF SHALE —

EFFECT OF TEMPERATURE AND FLU ID SATURA-

TION.

sample that was first air dried and then brine

saturated (see Fig. 5). The low conductivity values

and the slightly positive gradients with temperature

are typical for fine-grained rocks. The air-dried’

and brine-saturated densities are given in Fig. 5

for comparison purposes .

CORRELATIONS

It should be apparent from the foregoing that the

measurement of thermal con?~ctivity of rocks is

difficult, time consuming, and of limited practical

value unless some generalized relationships ro

other physical prc~erties and to changes in

environmental conditions can be developed. In the

following section, the results of our investigation

are used to test such relationships developed from

literature data. Also presented are methods of

predicting the thermal behavior of rocks from other

more easily measured properties.

TH ERMAL C OND UC TI VI TI ES OF D RY S AND STONE S

Multiple regression analyses run on thermal

conductivity and physics property data from the

literature and from our work yielded the fo lowing

equation.

A = 0.340p - 3.20~ + 0.53Gk0’10 + 0.0130/’

- 0.031, . . . . . . . . . . . . .(1)

o.eo~I 50 J

AXIAL STRESS , IISI

FIG . 6 — EFFECT OF AXIAL STRESS ON THERMAL

CONDUCTIVITY.

270 SOCIETY OF PET ROLE~M ES GISEERS JO CRXAL



where A =

p.

4’

k=

F=

thermal conductivity, Btu/hr-ft-° F

bulk density, gin/cc

fractional porosity

permeability, md

formation resistivity factor

in our correlation because of the usual lack of

knowledge of mineral composition and the difficulty

of assigning conductivity values.

THE RMAL COND UCTIVITI ES OF

L IQU I D-SATUR ATE D SAN DSTON ES

For 38 data points, the standard deviation was

0.139 for a conductivity range of 0.4 to 2.2 Btu/hr-

ft-°F. The agreement between measured thermal

conductivity values and calculated values is shown

in Fig. 7. The solid line represents perfect

agreement and the broken lines show the limits of

one standard deviation. With one exception, data

from our work were well within these limits. A

deviation of less than IO percent was obtained for

74 percent of the data points and less than 15

percent for 87 percent of the data points.

Further analysis of Eq. 1 indicates that porosity

is the most important variable; density and

permeability have about equal effect; and formation

resistivity factor is th{) least important variable.

The positive and negative numerical coefficients

for density and porosity, respectively, are as

expected. The positive coefficient for permeabilityis probably a reflection of the effect of grain size.

Other factors being equal, permeability and thermal

conductivity both increase with increased grain

size. A study2 of .Zierfuss and Van der Vliet data

confirms this observation. The positive coefficient

for formation resist ivity factor is a ppa rent ly

a ssocia t ed w ith it s rela t ion t o bulk densit y a nd

porosity.

Some quest ion ma y a r ise as to the need for both

density and porosity in the correlation equation

since they are interrelated. By including both terms

rather than using either terms alone, the correlation

was definitely improved This may be an expression

of the effect of mineral composition since t he less

dense feldspars and clays are known to have lower

thermal conductivities than quartz. AS was pointedout earlier, we did not include matrix conductivity

t- 1 ///

<~

LITERATURE VALUES [31 /

PRESENT WORK,k /

//

ONE STANDARD , , ‘

OEVIATION/

/

.B’ , ‘

/,/”

Y

/.

+x

/ x/

//

//

5’12 LJ0.4 08 1.2 1.6 20 24

PREI)ICTED THERMAL CONDUCTIVITY, Btu/hr-ft-”F

FIG, 7 — AG REEMENT BETWEEN MEASURED AND

C ALC UL ATE D TH ERMAL C OND UC TI VI TY VALU ES .

oCTOBER. 1973

Several efforts were made to obtain correlations

for predicting thermal conductivities oi liquid-

saturated sandstones. The most sat is factory

correlation was obtained with the following

dimensionless groupings or quantities.

where A = thermal conductivity

~ = fractional porosity

p = bulk density of rock

m = Archie’s cementation factor

and where subscripts are defined as follows:

d = dry rockI = saturation fluid

s = liquid-saturated rock.

When a nonlinear, multiple regression computer

program was applied to literature data,3 the

foJIowing equation gave the best fit.

A,

~= Looto.30~ -LOO]033+4.57[,+Z

[lo 4sm. +..~o

. —. . . . . . . .Au. .

(2)

For the 52 literature data points used in the

correlation, the standard deviation was O. 179 for

the range of As/Ad ratio values of 1.20 to 2.sO. The

agreement between literature values and calculated

values was within 10 percent for 56 percent of the

values and within 15 percent for 85 percent of the

values.

The agreement between literature values of the

thermal conductivity ratio and values calculated

from Eq. 2 is ‘wn in Fig. 8. The solid line

represents perf dgre’ement and the broken lines

show the limits of one standard deviation.

Experimental and calculated values from our work

are plotted as X’s in Fig. 8. Eleven of the 14

points are within one standard deviation, one point

was out of the range of the correlation, and two

points showed differences greater than one standard

deviation. The low experimental point may be due

to incomplete saturation of the test sample with the

viscous silicone oil.

Since the correlation equation is expressed in

terms of dimensionless ratios, m (the exponent in

Archie’s equation relating porosity and formation

resistivity factor) was used in the correlation rather

than resist ivity factor itself. The effect of m in Eq.

2 is similar to the effect of F in Eq. 1 in that

271



increasing both values increases the conductivity.

The conductivity of the saturating liquid has the

dominant effect on the conductivity of the liquid-

saturated rock. It is difficult to assess the relative

importance of the other parameters in Eq. 2 because

of their complicated interre lationships ,

In the case of saturation with two liquids or

liquids and a gas, the conductivity of the wettingphase has a dominant effect on the conductivity of

the rock -fluid system. Thus for water-wet

sandstones, the value of liquid conductivity to be

used in the correlation should be biased toward the

value of conductivity for the water. This matter is

the subject of current investigations.

It would, of course, be possible to combine Eqs.

I and 2 so that the thermal conductivity of liquid-

saturated samples could be estimated directly from

one equation. Regression analyses run on combined

data for dry and liquid-saturated samples gave poor

results. Part of the reason for this is that there is

a difference in the heat transfer mechanism between

solid and gas and solid and liquid. Assad4 observed

a substantial difference in the thermal conductivities

of the same sandstone saturated with a gas andwith a liquid of equal conductivity. For this reason,

Eq. 2 should not be used in estimating the thermal

conductivity of gas-saturated sandstones. The dry

(air-saturated) value would be more suitable for

this purpose.

EFFECT OF TEMPERATURE ON

THERMAL CONDUCTIVITIES

The correlations discussed above have all been

based on thermal conductivity values measured at

or near room temperature. Our test data were

extrapolated to 68°F (20°C) to obtain values for use

in testing the correlations.

The effect of temperature on the thermal

conductivities of the sandstones used was tested

against Tikhomirov’s correlat ion. 7 The results werenot entirely satisfactory. Guided by Tikhomirov’s

correlation, we obtained a new famiiy of curves but

2.6 I I 1 1 I A/

LITERATURE VALUES [3] /

2.4 - X PRESENT wORK / /

/

//

/’

22 – //

7./

20 - x/

mm0

:018“7

\ In

516 -ONE STANDARD

9EvIA1 ON

I .4 -

/

I 2~

1.0 I

1.0 1,2 1.4 16 18 20 22 24

‘A Stxo)cALc

FIG , 8 — AGREEMENT BETWEEN MEASURED AND

C AL CU LATE D C OND UC TI VI TY RATIOS .

272

modified them according to the conductivity-

temperature trends we had obtained. The equation

of this family of curves is as follows.

AT = ~680- 0.71 X 10-3(T - 528) (~680- 0.80)

. [A68dT x 10-3)+-55*68” + 0.741, . . (3)

where AT = thermal conductivity at temperature T

~~so = thermal conductivity at temperature 68°F

T = temperature, “R = ‘F + 460°

A plot of Eq. 3 based on even values of thermal

conductivities at 68° F is shown as Fig. 9. The

resrdts of our conductivity-temperature measurements

are plotted on the same figure. Althou~,\ there is

some scatter in the data, the general ag.cement is

quite good. Additional data are needed in the higher

conductivity and higher temperature rangt,s. A few

results from tests using the high-temperature thermal

conductivity apparatus have been included in Fig.

9 as X’s joined by dashed lines.Tikhomirov’s correlation was developed for dry

rocks, but the present correlation seems to beequally valid for liquid-saturated sandstones.

Unusual thermal properties of some liquid saturants

could cause some deviation of behavior from that

predicted by Eq. 3, particularly for high-porosity

rocks. In addition, phase changes of saturants may

result in discontinuities, but this is a fluid

saturation effect rather than a temperature effect

per se.

The conductivity-temperature equation may need

to be modified if it is to be applied to a greater

range of rock- fluid systems. However, it can

certainly be used as a general guide for predicting

the thermal conductivity-temperature behavior of

most sandstones.

EFFECT OF P RESSURE ONTHERMAL CONDUCTIVITY

Most of the present thermal conductivity tests

were run at an axial stress high enough to minimize

the effect of contact resistance and at pore

pressures high enough to keep all fluid saturants in

liquid phase. Limited work at higher stresses

indicated that thermal conduct ivities increase by

L

~30

s

>25

m

> 20.~

~

Ulo r ~.— ... . . .. . .

: e====~

;05— —

3

Ot, I I I I I ~

68 m ?00 300 400 5C0 600T fMPERATu Rf , “F

FIG , 9 — EFFECT OF TEMPERATURE ON THERMAL

C OND U CTI VI TY OF S AND STON E.

SOCIETY OF PET ROLEt M E~Cr XEERS JOrlR~AL



only 1 to 2 percent for every 1,000 psi increase in

effective stress. These values are somewhat lower

than values reported in the Literature.s-lo In

estimating the magnitude of the effect of stress on

thermal conductivity, the known effects of stress

on other properties were considered. Eq. 1 was first

differentiated with respect to effective stress:

0.34 ~ -3.20 ~ + 0.053k-O”g0= ~

@

3F

‘0”013 ~””””””””””” 4)

The derivative terms for density, porosity,

permeability, and formation resistivity factor have

been evaluated by Dobrynin 12 in terms of

compressibilities. Assuming linear approximations,

the foIlowing substicutiorts have been made for the

derivative terms in Eq. 4.

where /1, /2, /3 and /4 are functions of compressi-

bility. Since numerical magnitudes of thecompressibility of rocks are generally not known,

numerical coefficients for formations of high,medium and low compressibility may be substituted

in the following general equation.

a

~=1()-5[A p~+ Br -CkO’10 +DF] , , .(5)

where dA/Jp = change in thermal conductivity, Btu

/hr-ft-°F-psi

p = bulk density, gin/cc

4 = fractional porosity

k = permeability, md

F = formation resistivity factor

Cb A— —

High 0,51

Medium 0.25

Low 0.13

Assuming Berea sandstone

compressibility and Boise

medium-high compressibility,

BCD—. —

5.75 0.37 0.12

3.51 0.18 0.071.44 0,09 0.034

to be of medium-low

sandstone to be of

0.8 and 1.3 percent

increases in thermal conductivity per 1,000 psi

increase in stress, respectively, are calculated

from Eq. 5. These compare with 1.25 and 2,0 percent

increases obtained experimentally, (See Fig. 1.)

This difference is probably due to the difference in

stress levels in the two cases — about 1,000 psi

average stress in the experimental determinationand 2,000 psi average stress for calculated values.

SUMMARY AND CONCLUSIONS

The correlations given by Eqs. 1, 2, 3 and 5 are

based on the limited amount of reliable data

available and therefore must be considered as

tent~tive. However, the general form of the relations

OCTOBER. 1973

is believed to be valid and useful for estimation

purposes. And it is felt that even this step

represents significant progress, considering the

wide spread of thermal conductivity values

appearing in the literature. These correlations will

be improved and extended as more reliable data are

obtained for a greater variety of rock types and over

a broader range of environmental conditions.

Work is currently in progress at higher pressures

and temperatures, on unconsolidated sands, and for

multiple liquids and partial gas saturations.

ACKNOWLEDGMENTS

This work was performed as part of API Research

Project 117. We thank API for this support. Also,

for their support and encouragement, we express

our sincere apprecia t ion to the P roject G uida nce

Committee, R. L. B a iley , Cha irman, Chevron Oil

F ield Resea rch Co.; E . J . Couch, Mobil Research

and Development Corp.; and Vaughn Jones, Getty

Oil Co. Several other students have assisted in the

project — Doug G1..ndt, Ko Chen, and Jeff Keese —

and their assistance as well as that of David White,

Laboratory .Mechanician, is acknowledged.

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8.

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REFERENCES

S corer , J . D . T.: “The Rela tionship B et ween Ther ma l

Conduct ivit y a nd Ot her Rock proper t ies, ” MS thesis,

U. of California, Berkeley 1964).

Anand, J.: ~~Thermal Conductivity of Fluid Saturated

Rocks at Elevated Pressures and Temperatures,’~

MS thesis, U. of California, Berkeley SetJt. 1971).

Zierfuss, H. and Van der Vliet, G.: “Laboratory

Measurements of Heat Conductivity of Sedimentary

Rocks, “ Bull., AAPG 1956) Vol. 40, No. 10, 2475.

Assad, Y.: “A Study of the Thermal Conductivity of

Fluid Bearing Porous Rocks, ” PhD dissertation, U.

of Californi~, Berkeley 1955).

Sugawara and Yoshizaw?: “An Investigation of the

Thermal Conductivity of Porous Rocks, ” AustruliatzJ. o/ Physics 1961) Vol. 14, No. 4,469.

Powell, R. W., Ho, C. Y, and Liley, P. E.: Tbcrmul

Conductivity of Selected Materials, National Bureau

of S ta nda rds, Wa shingt on Nov. 1966 .

Tikhomirov, V. M.: t~conduc[ivity of Rocks and

Their Rela t ionship w ith Densit y, S a tura t ion and

Temperature, e‘ Ne/tianoe Kboziaisfro in Russian)

1968) Vol. 46, No. 4, 36.

Khan, A. M.: ~IA fiermoelectric Method for Measure-

ment of Steady State Thermal Conductivity of Rocks,”

MS thesis, U. of California, Berkeley 1961).

Edmondson, T. A.: {‘Thermal Diffusivity of

Sedimentary Rocks Subjected to Simulated Over-

burden Pressures, ” MS thesis, U. of California,

B erkeley 1961 ).

Woodside, W. and Messmer, J. H.: “’Thermal

Conductivity Porous Media , ” ]. pp[ ~bys

1961) Vol. 39, No. 9, 1688.

Lobree, D. T.: “Measurement of Compress ibi li ties

of Reservoir Type Rock at Elevated Temperatures, ”

MS thesis, U. of California, Berkeley 1968).

Dobrvnin, V. M.: “Effect of Overburden Pressure on

Some- Properties of Sandstones, ” Sot. Pet, Eng. ].

Dec. 1962) 360-366; Trans., AIME, VO1. 225.

27s

spe-4171-pa predicting thermal conductivities of formations from other known properties

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