Thermal expansion of rocks: some measurements at high pressure

Tectonophysics, 57 (1979) 95-117 @ Elsevier Scientific Publishing Company, Amsterdam - Printed in The Netherlands

THERMAL EXPANSION PRESSURE

TENG-FONG

OF ROCKS: SOME MEASUREMENTS

95

AT HIGH

WONG and W.F. BRACE

Department of Earth and Planetary Sciences, Massachusetts Institute of Technology, Cam bridge, Mass. 02139 (U.S. A.) (Submitted

April 27, 1978; revised version accepted July 5, 1978)

ABSTRACT Wong, T.-F. and Brace, W.F., 1979. Thermal expansion of rocks: some measurements high pressure. Tectonophysics, 57: 95-117.

at

The thermal expansion of rocks has customarily been measured at room pressure; it is typically irreversible after heating above room temperature and the coefficient of thermal expansion of a rock is usually much larger than the average coefficients for the minerals in the rock. We measured coefficients of thermal expansion of rocks under sufficient pressure that the strains were reversible; the coefficients obtained fell close to the theoretical bounds for polycrystalline aggregates calculated by Walsh. Thus, within the limit of assumption in the theory, Walsh’s theoretical bounds gave a good estimate of the intrinsic thermal expansion of rocks. The thermal expansions were measured by strain gauges attached to copper-jacketed samples of granite, diabase, marble, limestone, dunite and quartzite. The temperature range was 2’--38’ and confining pressure ranged up to 600 MPa. Confining pressure had a small effect as long as pressure was greater than some critical value which apparently was the minimum needed to prevent thermal cracking and other non-elastic effects. The critical pressure depended on rock type and on thermal history of a sample. We propose theoretical models to explain the effects of temperature on the opening of cracks at high confining pressure, and the propagation of open cracks at low pressure induced by thermal gradient or internal stresses.

INTRODUCTION

The thermal expansion of rocks is important in many geophysical, mining and geothermal applications. Estimates of the thermal expansion coefficients of rocks at pressure up to hundreds of GPa’s (several megabars) are needed for various geophysical calculations (Anderson, 1967; Birch; 1968). Among other factors, long-term structural stability of underground openings depends on the thermal stresses produced by temperature changes during geothermal energy generation (Thirumalai and Demou, 1973). Thermal expansion has been customarily measured at room pressure. Thermal cycles for rocks are usually observed to be irreversible, i.e., residual

96

strains remain after a heating and cooling cycle (Rosenholtz and Smith, 1949). Furthermore, the strain-temperature cycles are usually not reproducible (Thirumalai and Demou, 1973). It is difficult to characterize such behaviour with a unique set of thermal expansion coefficients; e.g. Skinner (1966) reported coefficient values of 8 f 4 - 10-6/“C for limestones and 8 f 3 . ~o-~/“c for granites at room temperature and pressure. Sample (Austin et al., 1940) and technique (Campbell, 1962) differences can result in divergent thermal expansion data. However, some other intrinsic factor apparently is involved when many spurious values of thermal expansion are observed for a single specimen tested with the same apparatus throughout the experiment (Richter and Simmons, 1974). Walsh (1973) derived expressions for theoretical bounds on thermal expansion of polycrystalline materials, and observed that measurements for rocks at room pressure generally are above the theoretical upper bounds. He suggested that irreversible effects discussed above and the high measured values were both consequences of microcrack formation induced by internal stresses. Sprunt and Brace (1974), in their scanning electron microscope studies of rocks, actually observed that thermal cycling caused new cracks to form and existing cracks to extend and intersect. Most microcracks are thought to be closed at a high enough confining pressure (Brace, 1965). If one can perform an experiment at such high pressure, the intrinsic aggregate expansion is what one measures, and the thermal expansion should be reversible and reproducible. Previously, Bridgman (1935) had measured the thermal expansion of three alkali metals at pressure up to 2 GPa. Yagi (1977) reported measurements for four alkali halides to 9 GPa and 800°C using the X-ray diffractometer. As for rocks, however, no previous attempt has been made to measure their thermal expansion at high pressure. In the present study, measurements were made with strain gauges put on copper-jacketed rock samples. Highest confining pressure was 600 MPa. The measured thermal expansion was compared with theoretical aggregate averages derived by Walsh (1973). Strains were measured at temperatures between 2°C and 38°C. Under such conditions, intrinsic thermal expansions of the rock and its constituent phases should be linear. Both non-linearity and residual strain in a thermal cycle are good indications of the onset of thermal cracking. We also tried to study the critical pressure needed to inhibit thermal cracking. Two series of measurements, one with successively decreasing confining pressure and a second at increasing pressure up to 600 MPa, were performed on some of the samples. Implications of these preliminary data on thermal cracking mechanisms were also investigated using several theoretical models. EXPERIMENTAL

PROCEDURE

Popular methods include dilatometer,

of thermal expansion measurement at room pressure interferometric methods, optical comparator, and X-ray

97

diffractometer (Kirby, 1969; Yates, 1972). These are not easily adaptable to a high pressure environment. Strain gauges were used by Rosenfield and Averbach (1956), who measured thermal expansion of steel under tensile stress, and they reported the gauge performance to be satisfactory. In our study, the rock samples were prepared and jacketed in copper foil as described by Brace (1965). Electric resistance strain gauges were then epoxied to the copper jacket (Brace, 1964). Signals from the gauge were read by a strain indicator with resolution of 10 * 10b6. Both the temperature and pressure effects on strain gauges can be significant. The temperature effect is typically non-linear (Scott, 1962). Corresponding to a change of temperature AT, resistance change AR is given by: 1AR --= GR

where (Yand a0 are thermal expansion of the substrate and the gauge material respectively, and G is the gauge factor. q, which is the temperature coefficient of resistance of the gauge material, is generally a function of temperature and quite sensitive to the composition of the gauge alloy, its impurities and the degree of cold-working (Dally and Riley, 1965). Stress analysts usually want to minimize the temperature effect by choosing the appropriate special-melt gauge for which a0 - q/G = (Y. For our purpose, however, we want the signal from thermal expansion (Yto be maximized. This can be achieved by choosing gauges for which 0~~- q/G = 0, originally designed for stress analysis of materials with almost zero thermal expansion. Identical gauges from the same batch were put on the rock sample and on a standard of known thermal expansion. The difference in signal when the temperature changes by AT should be given by: = (a - a’) AT

where primes refer to the standard. As for the pressure effect, Brace (1964) demonstrated that it was approximately the same for a wide range of material. If subjected to the same pressure, the rock and the standard have identical pressure effect. By taking the differential signal, both the temperature and pressure effects are cancelled out if pressure is kept constant. This procedure has the additional advantage of reducing error associated with drift and noise. The standard used was either a sample of Armco iron or an aluminum alloy; the choice of standard for a particular rock was made such that the differential thermal expansion was greater than 5 + 10-6/“C. Linear thermal expansion of Armco iron is well documented: Touloukian (1975) screened through more than 80 sets of data and recommended the value 11.8 - 10m6/ “C at 20°C and room pressure. Possible error was stated to be 3%. Q!of the aluminum standard was obtained by measuring the differential thermal expansion against Armco iron.

98

Both the standard and the copper-jacketed rock sample were put into a pressure vessel. The pressure medium was kerosene. A mang~i~ coil was used for monito~ng pressure to about 0.3% absolute accuracy. Temperature was measured by a coppe~const~~n thermocouple soldered on to either the standard or the copper jacket. Resolution was about O.l”C. An exterior circulating system was used to vary the temperature. Init~ly at room temperature, pressure was increased to the desired constant value. The first reading was taken when the specimen had cooled and acquired thermal eq~lib~um at about 2°C. For both the heating and cooling runs, readings were taken at temperature inurement of about 10°C up to 38”C, and then back to 2°C. Pressure ~uctuations coupled with such temperature increment were up to 10 MPa. The pressure was adjusted back to the fixed value at each temperature increment, and sufficient care was taken to ensure that thermal equilibrium (indicated by a steady thermal strain) was established before each reading was taken. Heating rate was about lO”C/hr. A typical run lasted about eight hours. Total experiments error in the measured cy’sis estimated to be about 3.5%. In total, six rocks were studied (Table I). Four of these - Oak Hall limestone, Danby marble, Cheshire quartzite, and Twin Sisters dunite - are almost monominer~i~; Westerly granite and Frederick diabase contain two or more phases in almost equal abundance. Conf~ing pressure was initially fixed at a high value (300 or 400 MPa). All except the dunite were then subjected to two sequences of runs in the following manner: thermal cycles were performed at a pressure which was decreased in steps of 50 MPa until reversible behavior could be observed; one or two more runs were made at room pressure to thermally crack the rock further, and the sequence was then reversed with the pressure increased by increments of 50 MPa until reversible behavior was again observed.

For the six rocks studied, reversible thermal cycles were observed at or above a pressure of about 200 MPa. A typical set of data for Cheshire quartzite at three different pressures is presented in Fig. 1. The thermal straintemperature curves at the higher pressures are perfectly linear and reversible. In general, standard deviation of the slope calculated with a linear regression scheme was about 3%. The measured thermal expansion coefficients were typically reproducible to within +2 * lo-‘1°C. Some of these data are tabulated in Table II. Below a critical pressure fP*), the thermal strain-temperature curve was no longer linear, and appreciable residual strain could be observed after a thermal cycle. P* of the rocks for the first (with pressure decreasing) and second (with pressure ~creasing) sequence of meas~ement are listed in Table III. Crack-closing pressure estimated from ~ompressib~ity and resistivity me~~ernen~ (Brace, 1965; Brace et al., 1965) is also included for comparison.

27.5 quartz 35.4 microcline 31.4 Ani7 4.9 mica 48 An67 49 augite-hypersthene 1 mica

0.75

0.175

0.011

0.001

2646

3020

Granite, Westerley, R.I.

Diabase, Frederick, Md.

Brace (1965)

Brace (1965)

Birch (1960) Brace and Change (1968) 92 Fais 7 pyroxene 0.5 serpentine

0.001

3320

Dunite, Twin Sisters Mtn., Washington

10.0

91 quartz 7 orthoclase 2 microcline

0.30

0.006

2643

Quartz&e, (Cheshire) Rutland, Vt.

Brace (1965) Brace (1965)

99 calcite

0.20

0.003

2712

Marble, Danby, Vt.

Brace (1965)

99 calcite

0

2712

Limestone, Oak Hall, Pa.

0.08

Grain size (mm)

Reference

Porosity

Density tkg/m3)

Specimen Modal analysis

Description of materials

TABLE I

‘7.0 + 0.24

0.03 0.08 0.35 0.17 0.24

3.1 6.7 8.6 10.6

-

2 k * + -

0‘13 0.06 0.28 0.17

* + + + +

0.09 0.07 0.30 0.17 0.24

7.0 + 0.24

3.4 5.5 8.9 10.6 7.7

average

5.13

4.61

5.44 5.23

3.15 3.52 8.65 10.66

lower

3.79 4.14 8.72 10.69

upper

Calculated values (.106 “C)

The o]l and ~~1’sare computed by linear regression. The standard deviation of the experimental data with respect to the least-squares slope is also included. When more than one set of data under the same condition are measured, only the average value is listed here. Average (Yis calculated by oAv = (l/3) (d/l f 2011).

Frederick diabase

4.0 3.1 9.4 10.6 7.7

+ + f + +

300 300 300 300 100 Room 100 Room

Oak Hall limestone Danby marble Twin Sisters dunite Cheshire quartzite Westerly granite

0L.i.

Measured values (*lo6 “C)

Pressure (MPa)

Specimen all

thermal expansion coefficients

TABLE II

Comparison of measured and theoretied

101

T, ‘C Fig. 1. Thermal expansion of Cheshire quartzite at three different confining pressures.

TABLE III Critical confining pressure for irreversibility P* and crack-closing pressure PC Specimen

Oak Hall limestone Danby marble Cheshire quartzite Westerly granite Frederick diabase

p2*

p1*

(first run, pressure decreasing) (MPa)

(second run, pressure increasing) (MPa)

Rm-50 150-200 200-250 Rm-50 50-100

100-150 no data 250-300 100-l 50 100-150

2

Crack-closing pressure PC l (MPa) 50 100 200 200 100

PI*, P2*: The pressure is given to within 50 MPa. When PI * for the limestone is listed as Rm-50 MPa, it means that reversible thermal expansion was observed at 50 MPa, and when pressure was reduced to room pressure, residual strain was observed after a thermal cycle. r The crack-closing pressure PC was estimated from compressibility and resistivity measurements (Brace, 1965; Brace et al., 1966). 2 The Danby marble sample had problems of jacket leak during this first sequence. It could have undergone thermal cracking at low effective pressure.

102 DISCUSSION

Comparison of measurement and theoretical bounds The measured expansions were compared with theoretical values calculated from the mineral thermoelastic moduli. Most averaging schemes in the literature (Turner, 1946; Kerner, 1956; Budiansky, 1970; Lakkad et al., 1973) placed restrictive assumptions on the internal stress field or phase geometry. Rosen and Hashin (1970), and Walsh (1973) proposed a variational approach making none of the above assumptions. By minimizing the “thermoelastic potential energy”, expressions for thermal expansion of an isotropic aggregate consisting of anisotropic phases were derived, assuming all the phases are randomly oriented. The derivation is rigorous so long as linear thermoelasticity applies. Following Walsh’s notation, the theoretical upper and lower bounds (CQ,and CQ,)are, respectively, given by: 01~= a,

f Ae

ffL=(Y,-A(Y where: 3a,

-pu _-ppL -3(c(r) = C(u)+ pv (c) - (Q) i (

3Aa

=

(0 - Pd”‘&J -Lo”’ pu_pL

[(pu-p,)(cca~~-~)-(~-~~~)‘]“’

(A typographical error in the original reference has been corrected here.) In the equation above, /3 is the aggregate compressibility; if the bars denote volume averaging, then: (CYY) = (yii where aij is the single crystal

thermal

expansion

(c)

= Ciijj where Cijkl is the single crystal

stiffness

(s)

= siiji where Sijhl is the single crystal

compliance

Pu = (s) where flu is the Reuss bound; PL = 9/(c) where & is the Voigt bound; bd

=

kxY2)=

tensor;

tensor; tensor;

and

Cijkkaij; cijkl”ijakl*

To calculate the theoretical bounds, we need (Yij, and volume fraction, V, of each of the minerals. Also, bulk compressibility, /3, of the rock has to be known. To calculate the bounds at high pressure, the variation of all these quantities with pressure has to be considered too. The data used are given in Appendix I. Cijkl

103 117

I

I Cheshire quartzite

I

I

G1

lo-

9Twin sisters dunite

5 I

au <

7-

W

4 d

6-

Frederick diabase

5-

4-

3’

I

1

RM.

100 Pressure,

200 MPa

Fig. 2. Comparison of intrinsic thermal expansion with theoretical bounds. The solid bars represent the measured value with standard deviation; the arrows are the calculated theoretical bounds.

Table II and Fig. 2 compare some of our data with the theoretical bounds. Measurements in both the longitudinal (~~1) and transverse ((Ye) directions had been made for the limestone, marble, quartzite and dunite. The average values (l/3 (‘Y,,+ 2a,)) were compared with the theoretical bounds. It can be seen that for the relatively isotropic quartzite, dunite and limestone, measured and theoretical thermal expansions are close. Discrepancy observed for the very anisotropic Danby marble probably reflects that for this case the model’s assumption of an isotropic aggregate is not appropriate. Elastic moduli of the Twin Sisters dunite are remarkably anisotropic (Christensen and Ramananantoandro, 1971), whereas those of Oak Hall limestone have insignificant anisotropy (Brace, 1965). However, anisotropy in thermal expansion for the two rocks is comparable, probably owing to the fact that the thermal expansion of calcite is extremely anisotropic, and so a slight

preferred orientation of grains gives rise to a large anisotropy. It is very difficult to estimate theoretical thermal expansion of the granite and diabase. Thermoelastic moduli for some of their minerals such as feldspar and pyroxene are not well known. Data needed for calculation of the pressure compensation are almost non-existent. The best we can do is merely to calculate the theoretical bounds at room pressure. A significant discrepancy exists between our measurement and the theoretical values (Fig. 2). Error in the input data could have contributed to this. Another possibility is that for these two rocks or more complicate miner~ogy, the assumption -

‘2c I 5

I

I

e

libII

Cheshire quartzite

Oak hall

limestone

31 ’

RM.

I 100

1 x)0

I

I

300

400

500

60(

3

Pressure, M Pa Fig. 3. Effect of confining pressure on intrinsic thermal expansion. 1 = intrinsic thermal expansion (with standard deviation) measured in the first sequence when pressure was decreased from a high value; 2 = measurement in the second sequence when pressure W~.S increased. The irreversible data at pressure lower than PT and P; are not included. Arrows represent theoretical upper and lower bounds for Cheshire quartzite.

105

that all the minerals are randomly oriented does not hold. Detailed petrofabric analysis of the rocks would be necessary to check on this. Figure 3 illustrates the typical effects of pressure and thermal history on thermal expansion. Two sequences (with pressure increasing and decreasing, respectively) are shown for the quartzite and limestone. Measurements were made only in the longitudinal direction. For the perfectly isotropic Cheshire quartzite, we can take (~11to be the average thermal expansion, and compare them with the theoretical bounds. The agreement is good, considering possible errors associated with the input data and approximation in calculating the pressure compensation for the aggregate. The effect of pressure on the intrinsic thermal expansion is small (Fig. 3). For a pressure increase of 100 MPa, (Yusually decreases by about 5%. This gradual chang;e is, however, perturbed by irreversible effects at pressure lower than P*, e.g., 0111of Oak Hall limestone increased drastically from 4.7 * 10e6/“C at 50 MPa to 7.9 - 10-6/oC at room pressure. Figure 3 also illustrates our observation that generally thermal history does not seem to have any effect on the intrinsic thermal expansion. The second sequence of measurements were made after the rock had undergone thermal cracking at room pressure. The slight discrepancy between measured values of the first and second sequence is within the experimental error. Effects

and analysis of thermal cracking

Thermal history has significant effect on P*. Table III lists the critical pressure for both the first and second sequence of measurement (PT and Pt). Pl is generally greater than Pr by 50-100 MPa. This indicates that after a rock has undergone thermal cracking, a higher critical pressure is needed to eliminate the irreversible effects. It is seen that for some rocks, PT is close to the crack-closing pressure; for others, such as Westerly granite, it can be quite different. Reversible thermal expansion could be observed for Westerly granite at 50 MPa, although the compressibility data indicate many cracks were open at such a low pressure. We try to explain these observations by considering several theoretical models of cracking mechanisms. Mathematical details are given in Appendix II. At or above the crack-closing pressure, P,, there should only be almost equi-dimensional pores in the rock. For such a system, internal stresses are due to mismatch of thermoelastic moduli across the grains. If E is the Young’s modulus and Aa the thermal expansion contrast across the grains, an estimate of the internal stress u induced by a temperature change AT is given by: (J - EA(rAT Estimate of (7 appropriate for our experimental set-up was about 1O-fN MPa, and one would not expect significant crack formation to be induced by differential stress of such small magnitude. The temperature range has to be

106

extended to hundreds of degrees before any crack formation occurs. Below the crack-closing pressure, cracks of various con~g~ations act as sources of stress concentration. The theoretical models in Appendix II indicate that for a crack of length a, a temperature variation AT will change the stress intensity factor at the crack tip by: where G is the rigidity. Hence for a given rock and a fixed AT, only cracks longer than a critical length will have attained the critical stress intensity for further crack prop~a~on. In the transient stage preceding therms eq~lib~um, local thermal gradients can induce cracks to grow. Goodier and Florence (1964) showed that for a Griffith crack of length Q, and specific fracture-su~ace energy S, the critical temperature gradient is given by :

The thermal gradient is dependent on the heating rate, and geometry and thermal properties of the rock sample. If all three are kept fixed in the experiment, the model indicates that there is a critical length such that only cracks longer than it are favorable to crack growth. Whether those cracks favorable to further propagation are open or closed at a given pressure depends on the aspect ratio y (Walsh, 1965). Pressure required to close a crack is approximately given by Ey. At a pressure slightly lower than the crack-closing pressure PC, some cracks with aspect ratio equal to or larger than P,./E are open. But if the crack geometry is such that the open cracks are all with length less than the critical value, no thermal cracking would develop. Tbe pressure has to be reduced further before there is any crack propagation. Hadley’s measurement (reported in Brace, 1977) of crack dimensions of Westerley granite seems to indicate that longer cracks tend to have lower aspect ratio. If this is indeed true, then the critical cracks are open only if the pressure is relatively low, and it would not be surp~ing that no cracking was detected for some rocks until the pressure was lowered to a value si~~ficantly less than PC_ After the rock has undergone thermal cracking, the crack geometry is modified. Cooper and Simmons (1978) observed changes in the “crack spectra” when Westerly granite was subjected to thermal cycling at room pressure. If the width of some of the cracks is increased, the aspect ratio also increases and a higher pressure is required to close them. This is a possible explanation for the increase of Pz over PT. As pointed out in the Appendix, these models can only offer us qualitative understanding of thermal cracking mechanisms. Physical interpretation and exact magnitude of some of the parameters in the models are still not clearly understood. Further studies should be performed with extended

107

temperature ranges and different heating rates. Data on rock samples saturated with various fluids of different thermodynamic properties would also be useful. The dependence of thermal expansion on mineralogy (Kern, 1978; Cooper and Simmons, 1978), grain size (Harvey, 1967; Kuszyk and Bradt, 1973), and pore porosity (Coble and Kingery, 1956) have been discussed in the literature. We have focussed on the effect of pressure in our study. Further studies are necessary to resolve the question as to whether these other factors could have significant effects on the intrinsic thermal expansion or thermal cracking mechanism in rocks. APPENDIX BOUND

I:

INPUT

DATA

AND

ERROR

ANALYSIS

OF THE

THEORETICAL

CALCULATIONS

Thermal

expansion

If more than one set of cyij is available, we choose the one that has most data points near room temperature such that we can have more confidence in the value of the slope reported for 20°C. The values we used are listed below: Calcite (Clarke, 1928): a1 I =

-5.5

* lO+‘/“C; 0!33= 25.37 - 1O-6/“C

Quartz (Kirby et al., 1972): a,1 = 13.6 . 1O-6/“C; a,33 = 7.4 * 1O-6/“C Olivine Fog,,Falo (Singh and Simmons, all = 4.52 - lo-‘j/C; Microcline

1976):

eZ2 = 11.63 - 10-6/“C; (y33= 10.72 * 10-6/oC

0rR3Ab1? (Skinner,

1966):

(Y,, = 6.56 + 10-6/“C; (y33= 0.22 * 10-6/oC Plagioclase

Ab7,AnZ3 (Skinner,

(Y,, = 5.33 . lO+/‘C;

1966):

cZZ = 2.67 * 10-6/oC; cy33= 4 . 1O-6/“C

Abs6A&,

(Skinner,

1966):

(y11= 5.2 - 10-6/oC; a22 = 2.6 * 10-6/“C; Augite (Skinner,

cc33 =

5.2 * 10-6/oC

1966):

cyI1 = 3.6 . 1O-6/“C; a22 = 9.6 - 1O-6/oC; (y33= 4.8 + 10-6/oC Iron (Touloukian, CY = 11.8 * 1o-6/oc

1975):

108

Elastic moduli We list below references from which the elastic data were taken. Adiabatic moduli were converted to isothermal values using the appropriate formulae: calcite (Dandekar, 1968a); quartz (McSkimin et al., 1965); olivine Fog3Fa, (Kumazawa and Anderson, 1969); microcline, plagioclase (24% An), plagioclase (56% An), augite (Simmons and Wang, 1971). Bulk compressibility Compressibility data for non-porous aggregates at low pressure are not readily available. Except for quartzite (Chung and Simmons, 1969), the bulk compressibility for all the rocks at low pressure is approximated by the Voigt-Reuss-Hill average based on Cijkl (sources above). Values at high pressure (above crack-closing pressure P,) are from the references below: Oak Hall limestone (Brace, 1965); Danby marble (Brace, 1965); Cheshire quartzite (Brace, 1965); Twin Sisters dunite (Christensen and Ramananantoandro, 1971). Pressure derivatives of elastic moduli Data for calcite, quartz and olivine were taken from the same references as in the section on elastic moduli. Data for feldspars and pyroxenes are not available. The adiabatic data were not corrected to isothermal values since the difference would in any case be small. Pressure derivatives of thermal expansion Little work has been done on pressure effects on thermal expansion. However, we know from thermodynamics that the change of thermal expansion eij corresponding to stress Uij is given by:

i.e.: &j aoh, T = &S&Z,

l(J

where Szkl is the isothermal -_P6ij, then:

compliance

tensor.

(no summation where /3T denotes direction.

the isothermal

If stress is hydrostatic,

Uij =

on i)

linear compressibility

in the i-th principal

109

If we take an approximation similar to the Reuss averaging scheme, assuming the state of stress locally in all the grains is hydrostatic, the pressure derivative of thermal expansion will be equal to the temperature derivative of the isothermal compressibility. This first-order approximation is justified, considering the relatively small correction we have to make. In general, the pressure effect is to reduce the thermal expansion. For the pressure range we are in, it is justified to use the following linear relationship: &ii@) = QFi+

t

$0:1Pf no summation

on i)

where & is the room-pressure thermal expansion tensor. The references from which we got the temperature derivatives of elastic moduli, and the calculated pressure derivatives of thermal expansion are listed below: Calcite (Dandekar, 1968 b): aa,, ap=

-6.6 . 10-7/“C/GPa (-0.066

-

10-6/aC/kb)

aaX _ - --EL4 0 lO”/“C/GPa ap Quartz (Koga et al., 1958): &l --= ap

-21.5

-ae33 = -16.3 aP

* 10-7/?Z/GPa

*

lo-‘/“C!/GPa

Olivine Fog3Fa7 (Kumazawa and Anderson, 1969): 3%1 _.._= -2.2 - 10-7/“C/GPa ap aa

-5.0 1)10-7/oC/GPa

--= ap

aa -= an

-2.3 - 10-7/“C/GPa

Iron (Hughes and Maurette, 1956): $-= -3.8 * IO-‘J”C/GPa Volume fmcfion

Data for some of the constituent minerals are not available, and appropriate adjustment has to be made in the averaging scheme. We took the

110

marble, limestone, quart&e and dunite to be monomineralic. The diabase and granite were assumed to be without mica, and volume fractions of the other minerals were multiplied by a fixed factor so that their sum added to 100%. Possible errors in the calculation Experimental error for Lyij, and Cijkl or Sijkl is quite well-documented. That for the derivatives is, however, difficult to estimate. Furthermore, the theoretical bounds are complicated algebraic expressions. It is therefore almost impossible to estimate realistic error bounds on the calculated averages due to experimental error. An additional source of error is the approximation made in calculating the pressure derivatives of thermal expansion. The error may be small, but we have not been able to estimate the order of magnitude. APPENDIX

II: THERMAL

CRACKING

We describe here some theoretical models of thermal cracking. Mechanisms for new crack formation and extension of existing cracks are both considered. Formation

of new cracks

When a rock is subjected to a high enough pressure, most of the cracks are closed. For a homogeneous, isotropic material, thermal stress is absent no matter how high a temperature is applied (Boley and Weiner, 1960). Only if the material is anisotropic, and if there is mismatch in thermoelastic behavior, will there be internal thermal stresses. An estimate of the internal stresses can be made by considering a simple model of an inclusion in a matrix of different thermoelastic moduli. Myklestad (1942) had considered the case of an ellipsoidal inclusion with different thermal expansion but the same elastic moduli. Both inclusion and matrix are isotropic. A similar problem with an anisotropic inclusion can be solved using Eshelby’s solution (1957). Edwards (1951) solved the problem for an isotropic inclusion and matrix with different thermal and elastic moduli. These models invariably gave an estimate of the internal stress o resulting from a temperature change AT to be: e - EA(rAT where A(Y is the thermal expansion difference between the matrix and inclusion, and E is the Young’s modulus of the inclusion. Edwards (1951) showed (in his fig. 6) that only when the rigidity ratio (Ginciusion/Gmatrix) increased to about 10, and aspect ratio of the inclusion decreased to approaching zero, was the internal stress increased to about 10E ((Y,,trix - ainclusion) AT.

111

Taking E = 50 GPa, Aa = 5 * 10-6/oC and 30 - 10-6/0C (the worst situation in calcite rocks), and AT = 40°C an estimate of the internal stress is u N 10 MPa and GOMPa, respectively. If the pressure is higher than the crack-closing pressure P,, differential stress C’ required to induce dilatancy is generally of the order of hundreds of MPa’s (Brace et aL, 1966). Local internal stress can be higher because of mismatch of elastic moduli across the grains. For Westerly granite, C’ is about 600 MPa at a confining pressure of 200 MPa. Taking this to be a lower bound on the necessary internal stress, we calculate AT to be about 800°C. Recently, Kern (1978) reported the measurement of V, at high temperature and pressure. His data for a granite showed that at a pressure of 300 MPa or higher, decrease of V, with temperature was small and almost linear as temperature was increased from room value to a value of about 600°C. However, there was a sharp decrease in the slope beyond this value, and the V,-temperature curve became highly nonlinear, indicating onset of thermal crack formation. He also estimated the thermal expansion at different pressure by measuring the piston movement in his solid-medium apparatus. At high confining pressure, the change of thermal strain with temperature was gradual until at about 6OO”C, when a sharp change in the slope became evident. He did not check on the residual strain after a thermal cycle, and since part of the drastic change observed near 600°C could be attributed to the non-linear effect of temperature on thermal expansion, it is difficult to locate accurately the critical temperature from Kern’s data. The rough estimate is reasonably close to the prediction of the theoretical model. Propagation of existing cracks Our measurement indicates that at low confining pressure, a relatively small temperature change can result in appreciable thermal cracking. It is well-known that at pressure lower than PC, cracks of various configurations may be open. Here we consider two mechanisms by which these open cracks can propagate further. By thermal gradient Preceding thermal equilibrium, a local thermal gradient exists in the rock. Goodier and Florence (1964) considered the problem of a Griffith crack (in plain strain) subjected to a uniform thermal gradient 7, assuming the necessary surface energy for extension of an existing crack to be supplied by the heat flow. If the matrix is homogeneous and isotropic, a crack of length a will propagate if:

S, the specific

fracture-surface

energy,

has

been measured

for rocks (Fried-

112

man et al., 19’72) and some of their constituent minerals (Brace and Walsh, 1962). Measurement on rocks is usually one or two orders of magnitude higher than that for single crystals of quartz, calcite and feldspar. The high value is suspect because microcracks around the crack tip render it difficult to estimate the true surface area, and it is dubious whether linear elastic fracture mechanics still applies. Friedman et al, (1972) argued that the single crystal values might be a better estimate, and that for grain boundary cracks the value might actually be less than that for the individual mineral phases. S for minerals compiled by Brace and Walsh (1962) ranged from 230 ergs/ cm2 or 0.23 Jms2 (calcite) to 1.03 JmT2 (quartz). It is difficult to estimate the local thermal gradient. Walsh and Decker (1966) pointed out that since thermal conductivity of air is two orders of magnitude lower than that of the minerals, cracks in a dry rock form effective barriers to the flow of heat. The barrier effect is negligible when the rock is saturated or when the cracks are all closed. However, in our experiments, the rocks are dry and Fourier’s law would imply a jump in local thermal gradient of two orders of magnitude across an interface between grains and cracks. For comparison we consider a homogeneous circular cylinder of radius b, with thermal diffusi~ty tc = 1.2 - 1Om6m2/sec {average value for igneous rocks; Stacey, 1969). Russell (1936) showed that if temperature on the surface was increased at a uniform heating rate 8, then at time t, temperature distribution at radial distance r from the center was given by:

where 4, is the m-th root of the Bessel function J,(x). Russell’s computation showed that when to = b’/K, contribution from the third term (series of Bessel unctions) decayed to less, than l%, and the temperat~e gradient at time longer than to can be approximated by:

-aT ar

lci

=--_r

2~

with maximum thermal gradient near the surface of value 0.5 ib/~. The longest crack length of Westerly granite observed under scanning electron microscope was about 600 pm (Hadley, 1976). Assuming E = 50 GPa, (Y= 10-SfoC and S = 0.75 Jn.zm2, Goodier and Florence’s formulae would imply r > 313”C/mm. Taking typical sample dimension b = 10 mm, Russell’s solution would give a characteristic time of 80 seconds and a minimum heating rate of 75”C/sec. This is two to three orders of magnitude higher than was observed for rocks. Todd (1973) and Thirumalai and Demou (1973) reported that at a heating rate as low as 5”C/min, both acoustic emission and residual strain increased with heating rate, indicating significant cracking induced by the thermal gradient.

113

(b)

Fig. 4. Interaction

of an inhomogeneity

and a crack.

Unless we have better understanding of the temperature distribution near microcracks, Goodier and Florence’s model can only give us a qualitative picture of the thermal gradient cracking mechanism. Experiments with a range of heating rates and performed on rocks saturated with fluids of different thermal properties would be useful in resolving this problem.

By in ternal stresses In an isotropic and homogeneous matrix which is in thermal equilibrium, no stress concentration is set up in crack tips even if the temperature is increased significantly. To estimate the internal stresses near microcracks in rocks, we consider the simple model of an inclusion and a crack in an isotropic, homogeneous matrix. Khaund et al. (1977) have considered the interaction of a spherical inclusion and a semi-infinite crack (Fig. 4a). An approximate analytic expression for the change of stress intensity factor AK, at the crack tip induced by a temperature change AT is given by:

(In the above formulae we corrected an error in the original reference. Ki is here defined by the following equation for the near crack tip singular stress field in mode I:

114

Atkinson (1972) a circular inclusion biaxial tension. We mal stress case, and AK* = 0.3183

+

solved the two-dimensional problem of a finite crack and exactly (Fig. 4b). He gave the solution for an applied have been able to modify Atkinson’s solution to the therit can be shown that:

AK, [(1 - &)r - (1 - ZV,)] G&z - a,)AT 7#_z2-YZ~)“~(~T)“~

where r = G2/G,, and K* is the normalized stress intensity factor given in Atkinson. His numerical computation (Atkinson, fig. 5) showed that if the length of the crack was one-tenth of the inclusion radius, and if 7 = l/3, u1 = u2 = 0.25, maximum of AK* would be about 0.5 when the crack was very near the inclusion. Substituting into the above expression, we obtain: AK1 = 9.45GA~AT~u The stress intensity factor has to be above a critical value K,, before the crack can propagate. KI, for rocks has been measured. Peng and Johnson (1972) gave a value of 1.38 MNvz-~‘~ (500\/2a psi dlin) for Chelmsford granite. As discussed above for fracture-surface energy, such measurements are most probably not applicable to microcracks. We can convert the single crystal surface energy S used above to Kle, using the appropriate formulae relating the Griffith and Irwin criteria (Lawn and Wilshaw, 1975): 2s

=

(’

-

‘1

2G

K2

Ie

(plane strain)

If we again take the intermediate value 0.75 Jme2 of S for quartz as above, we obtain K,, = 0.35 MNm-3’2. If G = 30 GPa, Ae = 5 * 10e6/“C and AT = 4O”C, Atkinson’s data would indicate that crack-length a 2 37 ym, which is reasonable for Westerly granite (Hadley, 1976). Our model with one crack and one inhomogeneity infracting should be considered as just a first step to understand the more complicated situation in a rock of many microcracks interacting in a locally inhomogeneous and anisotropic matrix. The model does, however, give a qualitative picture of the mechanism, and brings out the parameters important to the problem. ACKNOWLEDGMENTS

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