5. Rock Fracture and Frictional Sliding

5. ROCK FRACTURE AND FRICTIONAL SLIDING

Hartmut Spetzler Department of Geological Sciences and Cooperative Institute for Research in Environmental Sciences University of Colorado Boulder, Colorado 80309

Symbols in Order of Appearance in the Chapter Ki Uij

C

Kic V

P d dn

W W,

r

CY

I A B I0

G t Y

dy/dt U to

W W,

b h M I E h L 1

Stress intensity factor for mode i Stress component Crack length Fracture toughness Poisson ratio Load Sample thickness; see Fig. 3 See Fig. 3 Sample width; see Figs. 3 and 7 see Fig. 3 Stress intensity correction factor for thick specimens 2d/ W Compliance of double torsion specimen Deflection of double torsion specimen Slope of compliance vs. crack length curve Compliance at zero crack length Shear modulus Time Displacement Displacement rate Crack tip velocity Specimen thickness; see Fig. 7 Specimen width; see Fig. 7 Specimen width between grooves; see Fig. 7 Specimen thickness; see Fig. 12 Specimen width; see Fig. 9 Applied moment; see Fig. 10 Moment of inertia of one arm; see Fig. 10 Young’s modulus Specimen half-width; see Fig. 11 See Fig. 12 See Fig. 12 131

METHODS OF EXPERIMENTAL PHYSICS Vol. 24, Part A

Copyright 0 1987 by Academic Press, Inc. All rights of reproduction in any form reserved.

132 d 4i kij

fl

dP/dx,

HARTMUT SPETZLER ( c d d ) ; see Fig. 12 and Eqs. (12) and (13)

Volume flow per unit time; see Eq. (14) Permeability tensor Viscosity of fluid Pressure gradient

1. Introduction In this chapter various aspects of experimental rock fracture are discussed. No attempt has been made to be complete nor has the subject been treated historically. New and novel ideas have received special attention, not because they are necessarily more important than the more conventional approaches, but because they are not as readily available t o the traditional rock-mechanician and because they make the arduous task of writing more palatable. Much duplication with the existing literature is thus avoided. References to rock mechanics books, containing nearly up-to-date references, are given throughout the chapter. An understanding of the propagation of single cracks is paramount to eventually being able to predict the large-scale failure of rocks. The study of single cracks is therefore treated first, followed by measurements that can be made within a pressure vessel. Optical holography and new strain gauge technology are introduced there. The section on acoustic emissions includes up-to-date developments in transducers. Potential and often ignored problems that plague the interpretation of frictional sliding are pointed out in the section on frictional sliding. The application of radioactive tracers in permeability and gas conduit studies is introduced in the last section.

2. Single-Crack Propagation Brittle failure in rocks occurs by the interaction of individual microcracks, which grow in response to the stress field, the temperature, and the chemical environment at their tips. It is paramount to understand the mechanisms that control single-crack propagation if we ever expect to reliably predict failure of rocks under engineering or geological conditions. The techniques for the study of single-crack propagation were developed initially for ductile materials like metals.' They were modified for brittle amorphous materials like glasses2 and more recently for ceramics and Before describing the experimental techniques that are involved in single crack propagation we must define the fracture parameters that can be measured. There are three modes of fracture. They are illustrated in Fig. 1. Mode I is by far the most important mode of propagation in brittle materials. In isotropic materials cracks have a tendency to propagate in a direction

5.

133

ROCK FRACTURE AND FRICTIONAL SLIDING

II

I

111

FIG.1 . The three modes of fracture: I, opening mode; 11, sliding mode; 111, tearing mode. (With permission from Cambridge Univ. Press; see reference 7.)

normal to the maximum tensile stress. Mode I1 and mode I11 propagation occur in general only in highly ductile materials such as polymers and highly plastic metals. Shear fractures also occur in geologic situations where the large confining pressures suppress large-scale tensile ruptures. On a small scale, however, microcrack growth may indeed occur under mode I Figure 2’ together with Eq. (1) gives the definition for the Y

X

FIG.2. Uniform loading configurations for plane cracks with straight fronts: internal crack in infinite plate (all modes operating). (With permission from Cambridge Univ. Press; see reference 7.)

134

HARTMUT SPETZLER

stress-intensity factors for all three modes for homogeneous loading in the infinite plate approximation.

Note that the stress-intensity factors depend linearly on the remotely applied stress (ai,)~ and on the crack length c to the 1/2 power. The form is the same for all three modes. The fracture toughness KICis a material property applicable under mode I loading. Crack growth occurs subcritically for values of K I < KIC and unstably for values of KI 1 KIC. Several techniques have been developed to measure crack growth parameters and fracture toughness.

3. Double Torsion Technique A double torsion (DT) specimen and associated loading arrangement are shown in Fig. 3. Thin plates, similar in geometry to microscope slides, are subjected to torsion which results in crack growth under mode I loading.

LOADING

I '

)

P/2

I

I 7

'/

I

/ / / / / / / /0 ///////r

P/2 FRONT

SIDE

FIG. 3. Double torsion specimen with loading arrangement. P, Applied load. Geometric parameters are defined in the figure, others in the text through Eqs. (2) and (3). (With permission from ASTM ; see reference 9.)

5.

ROCK FRACTURE A N D FRICTIONAL SLIDING

135

In general, a side groove guides the crack while it is propagating. Fuller* and Pletka et al. give evaluations and detailed descriptions of the double torsion technique. The specimen geometry is such that the stress-intensity factor is independent of crack length over most of the sample length; i.e., a DT specimen is a "constant K" specimen. From the change in torsional compliance with crack extension, the relationship between K I and the specimen dimensions is

K I = PWm[3/Wd3dn(l- v ) Q " ~

(2) where P is the load, v is Poisson's ratio, and the other terms are defined in Fig. 3 . The is a geometric correction factor for specimen thickness. It is given t o within 0.1% by

< = 1 - 0 . 6 3 0 2 ~+~1.2Oae-"'"

(3) where a is the thickness ratio, CY = 2d/ W . The independence of K I on crack length makes it possible to use opaque specimens. It is not necessary to observe the crack tip, which greatly facilitates the adaption of this technique to hostile environments such as high temperature and corrosive chemicals. Williams and Evans" and Evans" have shown that the compliance I of the DT specimens depends linearly on the crack length c, i.e.,

I

= A / P = (Bc

+ Io)

(4)

where A is the deflection of the plate, B the slope of the compliance versus crack length curve, and A0 the intercept, i.e., the compliance at zero crack length. The slope B may also be determined analytically as B

=

3 W;/ Wd3G

(5)

where G is the shear modulus. In practice, there are three ways of obtaining the crack velocity without visual observations. In all cases the specimen is precracked; that is, a crack is initiated either by increasing the load incrementally or applying a constant displacement rate until crack initiation is noted by a sudden decrease in the load. In constant load experiments the deflection of the sample is measured as the crack propagates. The crack tip velocity v can be obtained directly from the deflection rate through the compliance equation, Eq. (4). The constant load technique is especially useful at very low crack propagation rates and at very high temperatures, where extraneous relaxations of the apparatus can otherwise become a problem. Evans and Wiederhorn12 showed that a constant displacement rate dy/dt could be achieved at a load plateau such that v=-

dy/dt PB

136

HARTMUT SPETZLER

Thus when equilibrium crack propagation has been achieved the constant displacement rate and constant load experiments are equivalent. They are only different in the manner in which equilibrium is reached. Servo-controlled equipment is almost necessary if both of these techniques are to be used with the same apparatus. Constant load and constant displacement rate testing devices correspond to soft and stiff machines, respectively. The load relaxation method is generally the preferred method for obtaining KI-u data. Williams and Evans" also showed that for a fixed grip (constant displacement) experiment, the crack tip velocity depends on the instantaneous load and on the corresponding load relaxation rate ( d P / d t )

where Pi is the initial load, Pf the final load, ci the initial crack length, and cf the final crack length. The load decreases as the crack propagates. The crack tip velocity can then be calculated from a load versus time record (Fig. 4) and initial or final crack length according to Eq. (7). Data with a range of several orders of magnitude in velocity can thus be obtained in a single experiment. In general, the crack tip velocity is not a single-valued function of the stress-intensity factors. It also depends on the chemical environment of the crack tip. Corrosive agents such as water molecules enhance the breaking of

-

z : z

40

SINGLE CRACK RELAXATION T E S T

--

v)

0 W

Y

38 --

36 ..

p,--V II:

0

LL

34--

t L

I I

0

t

1

2

TIME

3

4

5

6

7

(SECONDS)

FIG.4. Data from a single-crack relaxation test performed on a double torsion specimen. The crack tip velocity is directly proportional to the rate at which the load decreases.

5. ROCK

FRACTURE AND FRICTIONAL SLIDING

137

I

-4

MOISTURE INDEPENDENT CRACKING

REACTION RATE LIMITED VELOCITY

I -

I

I I I

FLUID TRANSPORT LIMITED VELOCITY

INCREASING PHzO

STRESS INTENSITY FACTOR

KIC

FIG. 5 . Single-crack propagation is a sensitive function of the chemical environment and stress. Three regions are identified. In region I the rate of crack propagations is directly proportional to the moisture content and exponentially dependent on the stress intensity. In region I1 the crack growth is independent of stress but depends on the rate at which moisture can be transported to the crack tip. In region I11 the rate of crack growth is again exponentially dependent on stress, but independent of moisture. K K ,fracture toughness, a material property.

atomic bonds at the crack tip. The higher the concentration of moisture at the crack tip, the greater is the crack tip velocity at constant stress-intensity factor. Figure 5 illustrates the form of KI-v data found for glass. In region I crack propagation is controlled by the moisture at the crack tip. In region I1 the crack tip is outrunning the moisture and the velocity is limited by the rate at which moisture can be transported to the crack tip. Propagation in region I11 is very rapid and mostly independent of moisture. At K I C ,the crack velocity is unstable and propagates close to elastic wave speed. The double torsion technique has yielded consistent results for glasses. Data from the work of Wiederh~rn'~ spanning all three regions are shown in Fig. 6. Reliable data for rocks are sparse.

138

HARTMUT SPETZLER

fat:

g’

I

I

4.0

5.0

a

0

I I

I

6.0

7.0

8.0

STRESS INTENSITV FACTOR (K,,N/rn3’2 x 10’) FIG.6. Crack velocity vs. stress intensity data for soda lime glass. Data are shown for all three regions illustrated in Fig. 5. (With permission from Plenum Publishing Corp. ; see reference 3.)

4. Double Cantilever Beam Technique The double cantilever beam (DCB) technique has been used in fracture A DCB specimen is shown in Fig. 7 and a typical mechanics experimental setup in Fig. 8. The stress-intensity factor for this arrangement is given by

5.

ROCK FRACTURE AND FRICTIONAL SLIDING

139

FIG.7. Double cantilever beam specimen. (With permission from ASTM; see reference 14.)

FIG.8. Experimental setup for a compact double cantilever beam specimen. (With permission from ASTM; see reference 15.)

140

HARTMUT SPETZLER

KI = [ 1~ F ( c )W/ Wmto]'"'P

(8)

where the dimensional terms are given in Fig. 7. P is the load, and F(c) = c2 + 1.32ct0 + 0.542ti. The specimens are usually grooved on both sides to provide guidance for the crack. A precrack of minimum length 2td6 is established before KICtesting commences. It is often convenient to arrange the test so that the crack propagation is horizontal. In such a case the torque on the specimen may not be negligible and it may be necessary to provide a support on the end opposite the loading pins to ensure that the initial load P is zero. Variations of the DCB technique are the tapered double cantilever beam (TDCB) technique" and the constant moment18 (CM) configuration. Specimens for these techniques are shown in Figs. 9 and 10, respectively. Equations (9) and (10)give the KIvs. geometry and loading relationships for the TDCB and CM techniques, respectively.

where M is the applied moment and Z the moment of inertia of one arm. The constant moment and TDCB specimens are constant K specimens (like the DT specimens) provided the latter is shaped so that rn is constant. Another variation on the DCB method is the loading of the specimen with a wedge." This arrangement is shown in Fig. 11. It is inherently a stiff

I

m=-t3h

c2 h3

FIG.9. Tapered double cantilever beam specimen. Constant K conditions are obtained by shaping the specimen to give constant m. (With permission from Plenum Publishing Corp. ; see reference 25.)

5.

ROCK FRACTURE A N D FRICTIONAL SLIDING

i_,-

141

T I

M

KI=-

rn

FIG.10. Constant moment specimen. M , applied moment; I , moment of inertia of one arm. (With permission from Plenum Publishing Corp. ; see reference 25.) rn

-7h

t FIG.11. Wedge-loaded double cantilever beam specimen. (With permission from Plenum Publishing Corp. ; see reference 25.)

loading system and can rather easily lead to crack arrest. With this loading arrangement

fi

Eyh3I2 2 c (1 + 0.64h/c)’

KI=-

where E i s the Young’s modulus and the other parameters are defined in Fig. 11. The analysis that leads to Eq. (1 1) is based on the work of Kanninen. 2o The effect of axial loading has so far been ignored and is not included in Eq. (1 1). Its importance has recently been recognized by Swanson” and Peck et dz2 In a given experimental arrangement, as the wedge angle and therefore the axial force are increased, the propagation of the crack in its own plane becomes more likely.

142

HARTMUT SPETZLER

5. Notched Bending Beam Technique Variations of the bending beam configuration are the four- or three-point bend (FPB)23and the single-edged notched beam (SENB)24p25 arrangements. A specimen of the SENB method is shown in Fig. 12. The stress-intensity factor in terms of geometry and loading is

P/ 2

PI2

FIG. 12. Schematic of test specimen geometry for single-edge notched beam tests. (With permission from ASTM ; see reference 24.)

TRANSLATING STAGE

-I

MICROFOCUS X - RAY SOURCE RECORDING PLATE

0

b

FIG. 13. (a) Schematic representation of the principle of X-ray microradiography and (b) loading fixture with built-in load cell (LC) and constant moment DCB test arrangement. (With permission from ASTM; see reference 28.)

5.

ROCK FRACTURE AND FRICTIONAL SLIDING

143

where Y ( d ) = [3.86 - 6.15~6+ 21.7(~$)~]”~ and

c6 = co/d

(13)

The SENB technique is very convenient for KIC determinations, but because of the K dependence on crack length it is difficult to obtain reliable K vs. v data. Fracture toughness determinations based on controlled surface flaws have been adapted for the characterization of ceramicsz6’27 but have not been used extensively for rocks. Because of the inherent anisotropy and variation of grain size in rocks, it is unlikely that these techniques will soon be adapted for rocks. Wu et aLZ8used X-rays in conjunction with the DCB arrangement to image the region of the crack tip while the specimen was under stress. Their experimental arrangement is shown in Fig. 13. A collimated X-ray beam passes through the specimen and produces an image on film. Cracks, being filled with air only, attenuate the X-rays less than the intact rock and are therefore imaged as highly exposed areas. Figure 14, from Wu et al., shows X-ray microradiographs of AlzO3. Wu et aLZ8also introduced a technique whereby cracks can be viewed in situ in a scanning electron microscope (SEM) while the specimen is under

FIG.14. (a) Composite micrograph of a grooved specimen before propagation of a rock. A and B are in the groove. The specimen has large grains (50-2000bm) and preexisting microcracks. (b) Micrograph showing the area left (note C and D) of the groove after a crack propagated from A to B and then left the groove near C . Note that there is little or no effect of the spherical pores (dark circular features) on crack propagation in contrast to the microcracks (dark elongated features). With permission from ASTM ; see reference 28.)

144

HARTMUT SPETZLER

Fro. 15. (a) In situ SEM specimen stage and (b) detail of specimen loading arrangement. Note the tapered specimen. (With permission from ASTM; see reference 28.)

5.

ROCK FRACTURE AND FRICTIONAL SLIDING

145

FIO.16. Crack branching seen in Lucalox; (b) is a higher-magnificationmicrograph. (With permission from ASTM ; see reference 28.)

146

HARTMUT SPETZLER

stress (see Fig. 15). The tapered DCB arrangement was used. A mechanical feed-through normally used for positioning is modified to force a tungsten carbide wedge into the machined notch. (See Fig. 9 for the TDCB method.) While only surface phenomena can be seen, they can be resolved to a fraction of a micrometer. The big advantage of viewing the cracks in situ rather than in a recovered sample is that it is possible to observe the progress of crack growth and to see the cracks while they are open. The linking of cracks and the role that preexisting cracks play become readily apparent. An example of crack branching in A1203 is shown in Fig. 16.

6. In Situ Measurements In this section we will examine the in situ measurements that can be made during rock deformation experiments. These range from externally imposed variables such as temperature, confining pressure, and differential stress to physical properties of the rock such as its elastic and anelastic moduli, shape, electrical and thermal conductivity, acoustic emissions, and permeability. In general, the measurement and control of the externally imposed variables can be accomplished with commercially available equipment. When using thermocouples under confining pressure, care must be taken to correct for the pressure effect on their ~ a l i b r a t i o n . ~ ~ Sample deformation can be measured in a number of different ways. In confined experiments, where the sample is in a pressure vessel, the axial shortening of the sample can be measured by measuring the advance of the piston. A simple mechanical dial gauge or, if an electrical output is desired, a linear variable differential transformer can be used. A careful calibration is necessary, taking the distortion of the entire apparatus into consideration. This is readily accomplished if a stiff (i.e., much stiffer than the rock sample) dummy sample of known elastic moduli is compressed over the entire anticipated force range. The deformation perpendicular to the axial direction, the tangential strain, is usually measured within the pressure chamber. Notable exceptions are the indirect measurements of the volume change in the pressure ~ h a m b e r . ~ ' - ~ ~ In the first technique one measures the volume of the confining medium, which must be added or subtracted from the chamber in order to keep the pressure constant; a calibrated piston is advanced or retracted. In the second,32the volume change is measured by measuring the level change of the liquid with an ultrasonic interferometer. Commercially available solid-state or resistance strain gauges are most commonly used. Such gauges come with known gauge factors and temperature corrections. When used under confining pressure, they must be

5.

ROCK FRACTURE A N D FRICTIONAL SLIDING

147

calibrated against materials with known properties. Solid-state gauges have the highest sensitivity, i.e., the largest gauge factors, but are limited by breakage to a small range of strain (typically 0.2%). Metal foil strain gauges can be used up to about 2% strain and are by far the most common. The bonding of strain gauges is slowly emerging from the realm of magic into that of art. Especially if the gauges are to be used at high temperatures, the bonding problems become severe and the useful range of the gauges is limited to about 1.5010. Commercially available gauges can be used to 300°C. Bonner and Heard33have developed a technique that has been successful to 800°C. A strain gauge without the conventional plastic backing material was attached to the metal jacket by spraying porous ceramic over the gauge. The strain measurements were very reproducible and, in spite of the very small useful range (< 1Yo), quite adequate for the anelasticity measurements for which they were designed. The preparation of the gauges was quite elaborate and consequently expensive. A different, as yet untried technique that should be useful to very high temperatures is suggested in Fig. 17. Tubes or rods of A1203 are glued on the jacket as shown. Two loops of strain gauge wire are threaded and tied through holes in the ceramic pieces. This will further ensure that the ceramic pieces do not twist during an experiment. Approximately 30 windings of 0.12-mm-diameter strain gauge wire such as Constantan" or KarmaTM on a 20-mm-diameter sample will produce a standard 120Q gauge. The wire should be wound under constant tension and the ends secured by threading and typing them to the ceramic rods through holes in the rods. Transverse changes in the specimen dimensions will be translated into strains in the wire. With a wire diameter about one-half of the jacket thickness, there will be no appreciable added constraint on the sample in the transverse direction. As the temperature is raised the glue that was originally used to attach the ceramic pieces will fail and the tension on the wire will hold the gauge on the rock. The strain gauge can be calibrated using materials with well-known thermal expansion coefficients and elastic moduli. Other lateral SAMPLE CERAMIC SPACERS

STRAIN GAUGE WIRE

FIG. 17. Tangential strain gauge for use at high temperatures.

148

HARTMUT SPETZLER

deformation gauges that do not require bonding of transducers to the sample were developed by S c h ~ l e and r ~ ~Holcomb and M ~ N a m e e . ~ ~ In recent years optical holographic interferometry has been adapted to Since this technique is not in measure deformation of rock wide use, but offers some unique advantages over other methods, it will be described in some detail. When viewing a hologram one sees a three-dimensional image. It is equivalent to looking through a window at an object. The image is generally recorded on a photographic emulsion on a flat glass plate and becomes visible when the developed emulsion is illuminated with a laser beam. The image appears behind the glass plate and the glass plate becomes the window through which the three-dimensional viewing can be done. It is not possible to distinguish visually between a good holographic image and the original object, when it is illuminated by laser light. The optical record of the image is an interference pattern formed by combining a coherent reference beam with the light that is scattered from the object, as shown in Fig. 18. Two or more images can be recorded on one emulsion. Let us assume that two images of the same object are recorded on one plate and the object has been strained between the two exposures. After developing the hologram and viewing it, the two very similar images can be seen. Since the images were formed with coherent laser light, the light from them will interfere and the observer sees the object with superimposed interference fringes, which are a measure of the distortion that occurred between the two exposures. It is also possible to make one exposure, develop the plate, and superimpose the image over the real object. Any subsequent distortion of the object will result in fringes that are visible in real time. This method is referred to as stored beam holography. The developing and repositioning BEAMSPLITTER (pass I a L y MICROSCOPE SLIDE r OBJECT BEAM

0

t

LASER

OBJECT

+

DIVERGING LENS-

/"OLO-PLATC REFERENCE BEAM

MIRROR

FIG. 18. Simple arrangement for making holograms, including provisions for an object and a reference beam. (With permission from Pure and Applied Geophysics; see reference 36.)

5.

ROCK FRACTURE AND FRICTIONAL SLIDING ,CORNER

149

REFLECTOR

ILLUMINATING EL VIEWING DIRECTION IHe-Ne OR

I

HOLO- PLATE’

FIG.19. Holography system as used in the author’s laboratory. Illuminating the object with a parallel beam and ensuring that the illuminating and viewing directions are coincident make the interpretation of fringes very simple. (With permission from Pure and Applied Geophysics; see reference 36.)

of the photographic plate must be done with utmost care to minimize the number of fringes due to shrinking emulsion and positioning errors. With stored beam holography the sample distortion can be recorded continuously on film or videotape. A new hologram must be made when the fringe density becomes too great to be resolvable. To make holographic interferometry quantitative, Heflinger et developed a technique whereby the illumination and viewing directions are coincident and the sample is illuminated with a parallel beam. In our laboratory we have adapted this experimental technique, using the arrangement shown in Fig. 19. Small rigid body translations do not result in fringes, and rotation results in straight fringes regardless of the shape of the body (for and Meyer and S p e t ~ l e rThese . ~ ~ straight fringes details see Heflinger et are parallel to the axis of rotation. Their spacing is a measure of the angle of rotation. The interpretation of the fringe patterns obtained by using the arrangement of Fig. 19is very simple. The pattern corresponds to a topographic map with a contour interval equal to one-half the wavelength of the illuminating laser light. The value of the wavelength must be taken in the medium in which the change in optical path length occurred. In the case of measuring the distortion of a sample while it is under confining pressure it is necessary to know the index of refraction of the pressure medium. We have measured the index of our pressure medium (Dow Corning 200 fluid) as a function of pressure.42 The data are shown in Fig. 20. For a He-Ne laser with a wavelength of 638 nm (in vacuum) the contour intervals for holograms taken of

150

HARTMUT SPETZLER

1 X

w 1.43-

.

4

EXPERIMENTAL DATA

4

0 W

2 1.42I-

LORENTZ- LORENZ CURVE

PRESSURE, IN BARS

FIG.20. Refractive index n vs. pressure for the pressure fluid (Dow Corning 200 fluid). The manufacturer’s compression data ( p = density) were used to calculate the index in accordance with the Lorentz-Lorenz law K p = ( n Z - l)/(nz + 2), where K is a constant. Maximum error in holographic strain measurements introduced by use of the Lorentz-Lorenz extrapolation is less than 0.5%. (With permission from Plenum Publishing Corp. ; see reference 42.)

a sample in our pressure vessel under 100 MPa confining pressure is approximately 210 nm. The possibilities for extending the usefulness of holography seem endless. Any diffusely reflecting surface, i.e., a nonmirror finish, can serve as a reference surface without the need for special preparation. With Q-switched lasers the exposure time can be a fraction of a nanosecond, which makes it possible to capture phenomena of very short duration. With mirrors the sample may be viewed from most directions. In many cases, motions that do not in themselves involve any differential motion in the viewing direction may be used to generate displacements with a component in that direction. An example of how the axial strain of a sample was measured is illustrated in Fig. 21. Note that the inherent sensitivity due to the laser light wavelength can be increased or decreased by choosing an appropriate gauge geometry. The major benefit that is derived from using holography is that the entire surface deformation in the viewing direction is obtained and not just a spot measurement. The photo insert in Fig. 21 is an example in which the tilt of the sample is obtained in addition to the axial strain. In certain cases it is convenient to use optical holography to check on and modify design criteria of equipment. An example of this is shown in Fig. 22a-c, where finite-element calculations and deformation measurements by optical holography were used in the design of an optical window43for a highpressure vessel. The window allows us to perform holographic deformation

~

6-i 5.

151

ROCK FRACTURE AND FRICTIONAL. SLIDING

7. 0

5.0 fi

0

a cn \

a

v

b tr i ong 1 e 2.0

w

L3 3

<

(3

t

laO 0.0

-

m

viewing direct ion

-

-

-

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . d rn t ul D c N d

d

d

d

GAUGE LENGTHISAMPLE LENGTH

m

m

d

d

m

N‘

(L/S)

FIG.21. Gauge sensitivity vs. gauge length to sample length ratio. Photo insert shows the response of two sinusoidal gauges to a nominal sample shortening of 3 pm. Sample length is 44mm and gauge length/sample length ratios for a and b are 1.13 and 1.86, respectively. Nonparallelism of fringes is the result of a tilt or less than 0.2 km across the sample. (With permission from Plenum Publishing Corp. ; see reference 42.)

measurements under confining pressure. In this case the entire inside (highpressure side) surface of the window was painted white with diffusely reflecting paint and a white cross was painted on the outside (atmospheric pressure) surface. With the window installed, the pressure was increased and the window deformation measured by interpreting the fringes on doubly exposed holograms (Fig. 22b). A bulge in rocks6i38* 39i44 that precedes failure as shown in Fig. 23 is another example of a subtle deformation that would be difficult to detect and measure with more conventional techniques. Liu and Livanos4’ used slit diffraction to detect a precursory bulge on an unconfined sample. Their experimental setup is shown in Fig. 24. A diffraction pattern is formed between a sharp edge and the sample. The pattern is recorded on film, and as the sample deforms the width of the slit changes nonuniformly and shortly before failure reveals a precursory bulge.

152

HARTMUT SPETZLER

FIG.22. (a) Design of optical windows (A) and (B)installed in high-pressure chambers. (b) Double exposed hologram showing fringes produced by a load of 6.9 MPa applied to inner surface of window (A). (c) Displacement of the surface of window (A), calculated by finiteelement analysis with a load difference of 6.9 MPa. Open circles represent the holographic results. (With permission from Review of Scientific Instuments; see reference 43.)

5.

153

ROCK FRACTURE AND FRICTIONAL. SLIDING

0

I cm

I I I WINDOW SCALE

(C)

FIG.22-continued

7. Acoustic Emissions By acoustic emissions (AE) we mean the elastic waves generated when rapid localized failure occurs in a stressed material. The stresses may be externally applied or they may be internal residual stresses which are the result of the thermal and stress history of the sample. The recording and evaluation of acoustic emissions are useful in nondestructive testing of engineering materials and in understanding the failure mechanisms of engineering as well as geological materials. There are a multitude of techniques for recording AE. These range from simply recording their approximate number above some threshold to obtaining many displacement vs. time records, similar to seismograms, for an individual AE event. The former gives an indication of the total AE activity and can be very useful in a statistical sense for failure prediction. It is

154

HARTMUT SPETZLER

FIO. 23. Bulge on a Westerly granite sample ( - 18 x 18 x 40 mm) outlining the intense deformation zone which precedes failure and delineates the eventual fracture plane. During low stain rate experiments (c < IO-’/s) the intense deformation zone first becomes apparent at about 60% of the axial strain at failure. This hologram was made through an optical window while the sample was under a confining pressure of 50 MPa. The axial strain was 96% of that at failure. (With permission from Academic Press; see reference 44.)

5.

ROCK FRACTURE AND FRICTIONAL SLIDING

,~

LOAD ~

~

/TOOL

155

STEEL ANVIL

10.16cm Dia. x 2.54cm

SPHERICAL SEAT 5.00 crn Dia. Base

=,I---& - -;

STRAIGHTEDGE

WESTERLY GRANITE CYL.INDER 2.453011Dia. x 5.047 crn

.HARDENED BERYLLIUM COPPER ENDPIECE 2.449 crn Dla. x 4.605 cm

FIG.24. Uniaxial compression test arrangement using slit diffraction method to map the rock sample surface deformation. (With permission from American Geophysical Union ; see reference 45 .)

furthermore useful even at very high AE rates. The recording of entire AE events is appealing because of the vast amount of information that is contained in such records. In ideal cases the seismograms yield the location of the AE event, its fault plane solution, its radiated energy, and the power spectrum of the radiated energy. The device that detects the elastic wave energy and sends a corresponding electrical signal is referred to as a transducer. The most common transducers used in AE work are made of piezoelectric materials. In response to a change in shape these materials become electrically charged and vice versa. Thin disks are cut from the piezoelectric material to make transducers, the thickness being determined by the desired resonance frequency. The disk’s diameter is typically 5 to 10 times its thickness. The book “Fundamentals of ultrasonic^"^^ gives many practical hints for the selection, bonding, and damping of piezoelectric transducers. Figure 25 shows a simple schematic for recording AE. In this case three transducers are used to allow for discrimination against electrical noise and false AE signals that originate outside the sample. A typical AE in rock is shown in Fig. 26. When counting such signals one has to decide where to set an amplitude threshold as well as a time duration (dead time) before a new event will be counted. In the case of large events the amplitude may still be above the threshold after the dead time and the event will be recorded as more than one event. The AE that occur outside the sample can be recognized by their delay between transducers A and C (see Fig. 25). Electrical noise will arrive at the transducers virtually simultaneously and can thus be recognized. All three transducers must be checked, however, since a simultaneous arrival at any two of them could result from an event appropriately located within

156

HARTMUT SPETZLER

i

STRIP

i1

,

CHART I -I L--RECORDER,C!

I

sAo

PISTON

I

PULSESHAPER OR DEMODULATOR

68

SAMPLE

-

COUNTER

I !

I I

&ON IF SIMULTANEOUS

( I)

IF DELAY LONGER THAN SAMPLE TRANSIT TIME, SUBTRACT ONE

FIG.25. A simple system for recording acoustic emissions incorporating both simultaneity and delay checks. The simultaneity check discriminates against electrical noise and the delay check against events that occur outside the sample. The dashed lines show optional analog recording. A, B, and C are piezoelectric transducers.

-o.200

' 10

20

30

I

I

I

50 60 /O MICROSECONDS 40

I

80

1

90

#

lo(

FIG. 26. Typical acoustic emission events from a large sample (79 x 70 x 70 cm) showing both the compressional and shear waves.

the sample. If the number of AE events is of secondary importance but a measure of the energy that was radiated is desired, the dashed circuit in Fig. 25 may be more desirable. It may, of course, be used in addition to the counting circuit. In the demodulator box the signal is amplified, demodulated, and integrated. This can be done simply by charging a capacitor and

5.

ROCK FRACTURE AND FRICTIONAL SLIDING

157

A Strip Chart Recorder

-

FIG.27. Demodulator circuit for analog recording

(see Fig. 25).

recording the output on a strip chart recorder. Figure 27 shows an appropriate circuit. The time constant RC must be chosen so that multiple events do not saturate the system. Signals from the delay and simultaneity checks can be recorded on a second pen to ensure recognition and discrimination of false events. This analog recording has the advantage over digital recording that it does not tie up expensive digital equipment and that the progress of the experiment is readily visible at any time. The logic described here can also be applied to purely digital recording. When more than AE counts and some sense of their energy is desired, the instrumentation and its calibration and the interpretation of the data become much more critical. In order to locate AE events the location of the transducers and the velocity within the test specimen must be known. A minimum of four transducers must be used to be able to determine the three spatial coordinates and the time of the event. In practice it is best to use as many transducers as feasible, since some redundancy in the location determinations is desirable and not all transducers will yield usable signals for all events. A state-of-the-art system for recording AE events is described by Sondergeld and Estey4’ and shown schematically in Fig. 28. This system incorporates an effective noise discriminator4* that accepts only acoustic signals which are in a predetermined frequency range and cross a minimum amplitude threshold for a specified number of cycles, typically 6 to 10. A typical sample and transducer arrangement is shown in Fig. 29.49 Unlike the case in seismology, where the size of the transducer (the seismometer) is a very small fraction of a wavelength, when piezoelectric transducers are used for AE recordings, the diameter of the transducer may actually be several wavelengths. This introduces considerable errors if the transducers are considered as single points located at the center of their contact with the sample. The transducers have an effective radius for sensing an incoming signal. The more oblique the incidence of the signal, the larger is the effect of the finite size of the transducer. This effect is shown in Fig. 30” for a transducer with an effective radius of 3 mm. The effective size of

158

HARTMUT SPETZLER

I

I I I I I

Cumulative Event Counter I N

I-,

Transducers

Rate Counter dnMt

G=50db

Roc

1024x 32blt

-

XL Controlled Clock

I I I

Nlcolet Olqltal OKlllorcoper

FIG. 28. Block diagram of the acoustic\emission monitoring system. Signals from piezoelectric transducers (1-8) are amplified by 50dB with the FET input amplifiers. Master transducers (1) is filtered digitally by the discriminator and used to trigger four Nicolet digital oscilloscopes. The four oscilloscopes all trigger within 10 ns of the main trigger and, after capturing the transient, can record ther waveforms on floppy disks built into the system. Waveforms are analyzed after the experiment. (From R~ecken:~p. 36.)

5.

ROCK FRACTURE A N D FRICTIONAL SLIDING

159

UI

FIG.29. Typical sample with transducers and strain gauge locations. The receiving transducers have numbers 1 to 8. The strain gauge locations are indicated. The source transducers or pulsers are indicated at locations P1, P2, P3, and P4. The origin of the coordinate system used for the locations of the hypocenters is in the center of the sample. (FromR~ecken,~’p. 31 .)

the transducer must depend on the physical size of the real transducer as well as on the wavelength of the incoming signal. In the results reported by Roecken et aLsothe effective transducer sizes were between 50 and 70% of their physical sizes. During deformation experiments the elastic wave velocity often becomes anisotropic. To obtain accurate locations it becomes necessary to incorporate a velocity model in the location determination. This velocity model must be continuously updated to reflect the velocity changes, which may be as large as a factor of 2. An example of the changes that are involved when Westerly granite is slowly stressed to failure is shown in Fig. 31.39 To find the location of an AE one may choose to follow the procedure outlived by Sondergeld and Estey4’ with some modifications. This was done by R ~ e c k e nFirst . ~ ~a trial hypocenter (location of an AE event) and an origin

160

HARTMUT SPETZLER

6.0

Effective transducer radius 3mm

5.6 h

u)

\

5 5.2

Y

> t

0 4.8

9 w >

4.4

4.0

-90 -75

-60

-45

-30

-15

0

15

30

45

60

75

90

ANGLE from NORMAL,OL (deg) FIG.30. Apparent velocity dependence on the angle of incidence on the receiver. The parameter that changes from curve to curve is the distance between the source and the center of the receiver. The greatest apparent velocity anomaly can be seen for a distance of 1Omm. The other apparent velocities are calculated for distances of 20-100 mm. (From R ~ e c k e n , ~ ~ p. 74.)

time are selected and errors are calculated for these values. By iterations the errors are reduced and a preliminary hypocenter is found. With the preliminary location, the directions and distances to the transducers can be calculated and the travel times corrected for the effective transducer sizes and the velocity anisotropy in the sample. Further iteration in the location of the AE event including these corrections will further reduce errors and increase the precision of the hypocenter locations. An example of the average location errors at various stages of deformation is shown in Fig. 32. The data are from Roecken et al. ;” the velocity model and transducer locations are given in Figs. 33 and 29, respectively. The accuracy with which hypocenters can be located also depends on their locations relative to the transducers. Highest accuracy is achieved when the hypocenter is located within a volume circumscribed by the transducers. Figure 3449 shows this effect. As described above, the technology now exists for obtaining accurate locations for AE events. But there is potentially much more information

5.

ROCK FRACTURE AND FRICTIONAL SLIDING

161

STRESS AXIS

0

FIG.3 1. Normalized velocity for a sample of Westerly granite as a function of angle from the stress axis at 0, 25, 50, 70, and 100% final tangential strain. At 25% the velocities have decreased in the transverse direction, but little change is seen in other directions. By 50% all observed velocities have decreased, some as much as 30%. The ratio of major to minor axes continually increases as tangential strain increases. At 25% it is 1.13 and by failure it is 1.75. The confining pressure was 50 MPa. (With permission from American Geophysical Union; see reference 39.)

-

available from recorded waveforms. Focal mechanisms may be determined if the direction of the first motion is clear. The polarity response of the transducers may be found by checking their response, for example, to the drop of a ball bearing or to the breaking of the synthetic lead of a mechanical pencil, the former sending a first arrival that is compressional and the latter a mostly dilatational one, With data obtained with the system shown in Fig. 28, Sondergeld and Estey’’ plotted first arrivals on focal spheres to obtain focal mechanisms for AE events in Westerly granite. While the response of transducers to first arrivals is easily determined for normal or near-normal

162

HARTMUT SPETZLER

Id0

0

I50

200

STRESS (MPa) FIG. 32. Average location errors for the source transducers with various corrections. The leftmost bars are based on a constant isotropic velocity. The center bars are based on the stressdependent anisotropic velocity model shown in Fig. 33. The righmost bars are based on the anisotropic velocity model with the inclusion of transducer size effects shown in Fig. 30.

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STRESS (MPa) FIG.33. Axial and transverse velocity models for an experiment by Roecken.” These models were used to calculate the errors shown in Fig. 32.

5.

ROCK FRACTURE AND FRICTIONAL SLIDING

163

FIG.34. Location errors and their dependence on source location along six different profiles through the samples. The location errors are calculated from synthetic data for the locations where the centers of the error crosses are plotted. The size of the error crosses is drawn in scale to the rock sample. (From R ~ e c k e n ?p.~ 130.)

incidence, when the arrival becomes oblique the transducer response of the effective size becomes more complicated. Roecken, in his of the transducers, also noted that for angles of about 60" off the normal, the first arrival could not be detected anymore and a later arrival would show a polarity reversal leading to incorrect arrival times and erroneous focal mechanism determinations. The same problem is also reflected in the velocity determinations shown in Fig. 30. As a guideline, for hypocenter and focal mechanism determinations one should choose the location and the number of transducers so that signals that make an angle greater than 60" with the normal of any particular transducer can be ignored in the analysis. From the waveforms shown by Sondergeld and E ~ t e y ~ ' ,it' ~is clear not only that focal mechanism determinations are possible, but also that power

164

HARTMUT SPETZLER

spectra can potentially be wrung from the records. Here the transducer problem that plagued hypocenter and focal mechanism determinations becomes even more severe but may also be overcome. The convenience of piezoelectric transducers suggests their use again. They are usually categorized as compressional or shear transducers and as having a specific resonance frequency. These specifications refer to their response for plane waves with normal incidence. When these conditions do not hold the mode of the transducers is no longer pure shear or compressional and other resonances are excited. Uncalibrated, they can only be used for relative changes in spectra for events that occur at the same locations. Various attempts have been made to calibrate piezoelectric transducer^,^^ usually for normal incidence. Ideally, transducers would have a flat frequency response and be as sensitive as a piezoelectric transducer while being much smaller than a wavelengthgood luck. While the above dream has not been realized entirely, approximations to it do exist and can at least be used in the calibration of piezoelectric transducers. One such device is based on optical interferometry. It was developed by Palmer and Greens3 and used and evaluated by Kline.54 The device is

COLLIMATOR

-

BEAM SPLITTER SPECIMEN

REFERENCE BEAM

FIO. 35. Schematic of the optical interferometer AE transducer developed by Palmer and Green.’3 Vibration of the sample surface results in variation of the optical path length of the laser beam incident on the sample, while the path length of the reference beam remains constant. (With permission from R. A. K l i ~ ~ e . ’ ~ )

5. ROCK FRACTURE A N D FRICTIONAL SLIDING

/\

,,/---

A

165

PLATINUM WIRE CAPACITANCE PROBE

FIG. 36. Capacitance AE transducer probe positioned above the grounded, conducting sample surface. Incoming waves from an AE event change the gap and hence the capacitance between the probe and the sample surface.

shown schematically in Fig. 35. A He-Ne laser is used as the light source in a modified Michelson interferometer. The object beam returns from the sample and the reference beam from a piezoelectrically driven reference mirror which compensates for vibrations within the experimental setup. The advantages of an optical system are its inherent calibration, since the wavelength of the laser light is precisely known, its flat frequency response, and the fact that no contact with the sample is required. The components are somewhat cumbersome and not readily mountable on the sample, which requires a stability for positioning on the order of one wavelength between the sample and the rest of the interferometer. For more detail on the optical interferometer the reader is referred to the original Capacitance microphones, for e ~ a m p l e , ’and ~ electromagnetic sensorss6 offer other means of obtaining AE signals that can be interpreted in terms of their power spectra. The sensing areas have been on the same order as those of piezoelectric transducers and have thus yielded complicated signals for nonnormal incidence. A new device based on the capacitive pickup of the RCA Video-Disc5’. 58 has been developed.s9* The main advantage of this device is its small sensing surface, as shown in Fig. 36, and its portability. Figure 37 is a photograph of the present version of the capacitive transducer (CAP). The physical design is such that the signal from the AE event reaches the pickup capacitor approximately 20 ps before the signal that traveled through the structure. Thus 20ps of undistorted signal is available for

166

HARTMUT SPETZLER

FIG. 37. Photograph of the capacitance transducer on a semicylindrical sample used for piezoelectric transducer calibration.6' The micrometer head provides for coarse positioning of the capacitance probe. The transducer height including micrometer head is 14.4 cm. The diameter of the horizontal supports is 8.2 cm.

5.

ROCK FRACTURE AND FRICTIONAL SLIDING

167

915 MHz DRIVE OSCILLATOR

STRIPLINE RESONATOR

DETECTOR

T-

\SIGNAL

FRINGING CAPACITANCE RESONATOR OUTPUT

1 A

i

CAPACITANCE

FIG.38. Capacitance pickup circuitry designed by RCA57*58 using stripline (transmission line)

circuit techniques. The resonator center frequency (910 MHz) changes as a result of the changing signal capacitance. The 915-MHzdrive oscillator injects a 4-Vpeak-to-peak signal. The detector senses the changes of that signal induced in the resonator.

analysis. The schematic of the resonance circuit is shown in Fig. 38 and the principle of operation illustrated in Fig. 39. It is important that the average distance between the stylus and the sample be maintained such that the condition illustrated in Fig. 39 is maintained. This requires extremely close proximity, which is accomplished by using a small modified voice coil from a commercial headphone in a low-frequency (<20 Hz)feedback circuit. The output from the resonance circuit (Fig. 39) is kept at a steady value of about 2 V by controlling the current through the voice coil. The stylus is attached to the voice coil. Surface displacements of a fraction of an angstrom (tens of picometers) are measurable above the noise.57s6o A comparison between a piezoelectric transducer and the CAP was made with the arrangement shown in Fig. 40. The results for a ball bearing drop and the dynamic propagation of a single crack are given in Fig. 41. It is clear how the signal from the piezoelectric transducer is dominated by its resonances. Piezoelectric transducers are now being calibrated with the help of the CAP.61These calibrations will be used to obtain power spectra and moment tensor representations of AE events.62*63

168

HARTMUT SPETZLER FREQUENCY DEVIATION 115 KHt

915 MHz DRIVE 4-

a

P Y IW

n RESONANT FREQUENCY

FREQUENCY FIG. 39. Capacitance pickup resonator center frequency variation with changing signal capacitance. The magnitude of the output signal depends on the slope of the resonance curve at the 915-MHz drive frequency. For a bell-shaped resonance curve, the slope is a function of the peak voltage (Vp) and the bandwidth (BW). SIGNAL SOURCE

FIG. 40. Apparatus used for comparison of piezoelectric and capacitance transducer responses at an early stage of capacitance transducer development. Both sensors are positioned equidistant from the axis of the cylindrical medium. Signals from a source at the center of the upper cylinder surface are nominally identical on arrival at each receiver. The levers and height adjustment are for optimum positioning of the medium surface relative to the capacitance pickup styles.

5.

169

ROCK FRACTURE A N D FRICTIONAL SLIDING

BALL BEARING S l l E 2 . 3 8 mrn

1 O h sec

I

10

I

20

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30 time, p

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I

50

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FIG.41. (a) Results of dropping a 2.38-mm-diameter ball bearing from a height of 2 cm to generate a signal in the medium shown in Fig. 40. PZT, Piezoelectric transducer output; CAP, capacitance transducer output. The PZT signal is dominated by transducer resonances. (b) Seismic waveforms from a single thermal fracture in a thin glass plate, observed at a distance of 1SOmm and at an angle of 90" to the propagation direction of the crack. The PZT signal shows the first arrival at a. Later phases are not recognizable. The signal from the capacitance transducer shows the acceleration phase (a to b), the propagation phase (b to c). and the stopping phase starting at c. A dispersed wave plate arrives at d.

170

HARTMUT SPETZLER

8. Frictional Sliding Stable and unstable sliding between rock surfaces and between rock surfaces and gouge may be responsible for failures on a scale covering several orders of magnitude, from small man-made structures to large earthquakes. The variables that determine stability, stable sliding, or stick-slip behavior are many. M. S. Paterson in his recent book “Experimental Rock Deformation-The Brittle Field”64 devotes a chapter to friction and sliding phenomena. “Fundamentals of Rock Mechanics”65 by Jaegar and Cook also has a chapter on friction. The author recommends both books highly. References to many earlier works may be found there. In this section we will only discuss some apparatuses that are suitable for friction measurements. Friction is defined as the ratio of the shear stress to the normal stress at which sliding starts for the static case and which is necessary to maintain sliding without acceleration in the dynamic case. In a friction apparatus it must therefore be possible to measure the normal load, the shear load, and the displacement as a function of time. The principles of various friction measurements are shown in Fig. 42. Those in Fig. 42a and b are suitable for static friction measurements on saw cuts and previously induced shear fractures under triaxial conditions. Rummel el d3’and Stesky66among other, 62*65 made friction measurements on saw cuts and induced fractures, respectively. Both sets of experiments were performed in a gas high-pressure environment and over a range of temperatures. Stesky66also measured acoustic emissions at temperatures up to 700°C. This experimental configuration is shown in Fig. 43. Once there has been a slip on the fault, the distribution of forces becomes complicated. This is illustrated in Fig. 44. If sliding is to occur on the saw cut or fault and the two sample halves are to remain relatively undeformed, there must be motion on some interfaces that are perpendicular to the

t (a)

(b)

(C)

(d)

(el

FIG.42. Common types of friction tests: (a) sliding on saw cut in triaxial test ; (b) sliding on previously induced shear fracture in triaxial test; (c) conventional shear test; (d) double shear test ; (e) shear test on thin-walled hollow cylinders. (With permission from Springer-Verlag ; see reference 64.)

5.

ROCK FRACTURE AND FRICTIONAL SLIDING

171

TRANSDUCER

FIG.43. Experimental arrangement used by Stesky66for frictional sliding experiments at high temperature and confining pressure. Note the transducer (on the steel piston) for acoustic emission studies. (With permission from Pure and Applied Geophysics; see reference 66.)

compression axes. As soon as sliding has commenced, the cross section over which the force is applied and that over which sliding takes place become variables. Scholz et al. 67 have avoided some of these difficulties by applying the force to the sample through rollers. This arrangement is shown in Fig. 45. It was also used in the author's laboratory to study the uniformity or lack thereof of strain accumulation before stick-slip and as a result of stick-slip. Using holography, it was found that prior to stick-slip the saw cut did not appear anomalous; i.e., the fringes as well as their spacing and curvature were continuous across the saw cut. At this resolution, the surface deformation in the direction perpendicular to the sample did not reveal any nonuniformity. The change in surface deformation that accumulates during stick-slip events is very nonuniform, revealing evidence of nonuniform slip and locking of the fault. Repeated force applications show the same uniform deformation before stick-slip and the nonuniform deformation spanning the stick-slip events. The other experimental arrangements shown in Fig. 42c, d, and e do not share the difficulties of those of Fig. 42a and b that were illustrated in Fig. 44. They are, however, difficult to adapt to work under confining pressure. The arrangements in c and d can easily be adapted to large specimens and are amenable to static, dynamic, and cycling tests. During cycling

172

HARTMUT SPETZLER

APPLIED LOAD

JACKET

+ 4-

4-

W

a

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v)

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FIG.44. Complicated distribution of forces and geometry once sliding has commenced in simple triaxial friction tests (Fig. 42a and b). RAM

111111111

PLATEN FIG.45. Schematic diagram of loading configuration used by Scholz et al. :67 (1) specimen; (2) steel end caps; (3) roller bearings. (With permission from American Geophysical Union; see reference 67.)

5.

ROCK FRACTURE AND FRICTIONAL SLIDING

173

experiments, as gouge is generated from the large blocks, their surfaces become nonplanar and the geometry changes. Most of the problems mentioned above are avoided if the arrangement in Fig. 42e is used. The cylipders must be thin-walled so that the differences in linear speed as a function of radius between the inside and the outside of the cylinders becomes negligible. Continuous dynamic friction and many multiple cycle tests can easily be performed with or without fault gouge.

9. Permeability The permeability of a substance is a measure of its ability t o transmit a fluid. Its mathematical definition is analogous to those of electrical conductivity (Ohm’s law) and of thermal conductivity. Darcy’ law states

where qi is the volume flow per unit time, p the viscosity of the fluid, aP/ax the pressure gradient, and kij the tensor form of the permeability. In most cases the permeability is quoted according to Eq. (15), that is, as a scalar quantity. A convenient unit for the permeability is the darcy. One darcy (d) = 0.907 x 10-’2m2. A knowledge of permeability values of rocks is important in many engineering applications, such as in petroleum, hydrological, and mining engineering. Underground storage and retrieval of natural gas, safe storage or disposal of toxic and nuclear wastes, and underground gasification of coal and gas are examples of such applications. Before embarking on any rock permeability measurements, be they in the laboratory or in situ, the reader is encouraged to read a paper by Brace,68 where the available data are analyzed. The range of permeabilities at a single site may vary over six orders of magnitude while still not showing a recognizable trend with depth. Similarly, large ranges in values for laboratory measurements cast some doubt on the usefulness of absolute permeability values for any specific rocks. Relative changes in permeability values as functions of externally applied variables may indeed be quite useful. In the following we will examine some techniques for studying the permeability of rocks. For permeability values above about a microdarcy measurements are made under steady-state conditions; i.e., the fluid flow is measured under a constant pressure T o measure smaller values, Brace et al.”

174

HARTMUT SPETZLER

LHigh pressure vessel

FIG.46. Schematic diagram of the permeability measurement apparatus. V1 and VZ,volumes of upstream and downstream reservoirs, respectively; Pl and 9 , pressures in these reservoirs; P c , confining pressure. (With permission from Lawrence Livermore Laboratory Report; see reference 74.)

developed the transient method, which was refined by Lin72and Heard etal. 73 to allow the determination of values as low as d. A schematic for the low-permeability measurement technique is shown in Fig. 46 and a schematic for the appropriate apparatus in Fig. 47. These figures are from Lin,72who used water for the permeating fluid. Darcy’s law holds for fluids of varying v i s ~ o s i t y including ,~~ gases, as long as the viscosity of the fluid is taken into consideration. The sample is initially fully saturated by applying pore pressure to both the upstream and downstream reservoirs. A small increase of pressure in the upstream reservoir results in a pressure front that migrates into the sample, thus reducing the pressure difference between the two reservoirs. The decrease of the pressure is a function of the sample dimensions, the reservoir sizes, the viscosity of the fluid, and of course the permeability of the sample. Lin74developed a computer code for generating pressure vs. time curves for constant values of viscosity. Such curves can be generated for any sample, fluid, and reservoir arrangement. Figure 48 shows such a set of curves and some experimental data. For details of the instrumentation the reader is referred to the original p a p e r ~ . ~ l - ~ ~

5.

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175

ROCK FRACTURE AND FRICTIONAL SLIDING

cmputei

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I I

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Pressure

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oil pump

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Pore pressure (water) system

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Confining pressure (oil) system

FIG. 47. Simplified diagram showing connections among the three main parts of the permeability apparatus : specimen assembly, confining pressure system, and pore pressure system. (With permission from Lawrence Livermore Laboratory Report ; see reference 74.)

In addition to the techniques mentioned above for measuring bulk rock properties, it is often of interest to know where and in what quantities the permeating medium resided within the rock and via what paths it flowed through the specimen. This is especially interesting when the rock is under stress and at elevated temperature. The use of radioactive tracers in conjunction with holography makes this possible. The experimental arrangement shown in Fig. 49, similar to that in Fig. 47, has provisions for a superimposed axial stress in addition to confining and pore pressures. A small reservoir containing a radioactive tracer gas is connected into the upstream part of the pore pressure system. When conventional permeability measurements are being made the trace gas reservoir is closed. When knowledge of the gas distribution is desired, the tracer gas is introduced into the sample and allowed to come to equilibrium. The sample is then removed from the pressure vessel and the gas distribution determined.

I I

176

HARTMUT SPETZLER 2.2

,

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Time ( s e c ) FIG. 48. Comparison of corrected observation (0)with calculated pressure decay curves (solid and dashed lines) for specimen 1. Permeability ( K ) is determined to be 1.52-2.0 x lo-” cm2. (Insert) Configuration of model study. b and K , volumes of the upstream and downstream reservoirs, respectively; fluid flows from 6 to K . (With permission from Lawrence Livermore Laboratory Report ; see reference 74.)

The ideal tracer gas is chemically inert and has a half-life that is very long in comparison to the duration of the experiment. It decays into a radioactive daughter that is a solid and has a half-life on the order of the duration of the experiment, hours to at most days. While the tracer gas is in the rock its decay product, the radioactive solid, is deposited in the cracks and pores that are occupied by gas. As the sample is removed from the high-pressure, hightemperature environment within the pressure vessel, the gas will at least partially escape. The rest may be driven off by heating the sample. The

5.

177

ROCK FRACTURE AND FRICTIONAL SLIDING

PRESSURE GAUGE WITH TRANSDUCERS

LINEAR VARIABLE DIFFERENTIAL TRANSFORMER

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PRESSURE GAUGE WITH TRANSDUCERS

POROUS HIGH STRENGTH A10 0 3

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FIG.49. Experimental arrangement for permeability and gas conduit measurements. A threestage bifilarly wound furnace ensures good temperature distribution and low electrical interference from the heater.

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178

HARTMUT SPETZLER

radioactive decay product, a solid, will remain in the sample in the same proportions in which the gas had occupied the rock. There are several means of measuring the radioactivity within the sample, each yielding unique information. If there is sufficient gamma ray activity, it is possible to obtain the bulk distribution of the spaces that were occupied by the gas. Gamma rays have a range of several centimeters in rock and can thus escape easily from the interior of typical laboratory samples. One possible arrangement for detecting the former gas distribution is shown in Fig. 50. The sample is inserted into a perforated lead shield, which in turn is wrapped in film with a radiation enhancement backing. The gamma radiation reaches the film from small solid angles, resulting in exposures of the film that are proportional to the activity in those solid angles. If alpha or beta radiation is present, a copper jacket of appropriate thickness may be placed around the sample to shield against that radiation. The exposure on the film will indicate the region or regions where the gas permeated the sample and can be correlated with the bulging of the sample as observed by holography; see Fig. 51. More detailed information can be obtained on the microscopic scale by sectioning the sample and using autoradiography to record the images. A FILM WITH RADIATION ENHANCEMENT

PATH OF )’ RADIATION

FILM IS CUT AWAY

7

FIG. 50. Arrangement for determining gas distribution. Radial holes in a cylindrical lead shield allow y-radiation to reach the film only from a small part of the sample. The exposed film gives an indication of the distribution of the radioactive tracer in the sample.

5.

ROCK FRACTURE AND FRICTIONAL SLIDING

179

possible procedure for this is indicated in Fig. 51. After the sample has been removed from the pressure vessel-or, if the experiment was performed in a vessel with an optical window, while the sample is still in place-the bulge is recorded by holography. A cutting plan is made for the sample and autoradiographs are produced by placing the appropriate film in contact with the cut faces of the sample. Film sensitive to beta or alpha radiation should be used. The resolution of the film is typically a fraction of a micrometer. The image on the film will show the location where the gas was and the image intensities will be proportional to the quantity of the gas that was in the pores and cracks. Comparisons of the autoradiographs with scanning electron photomicrographs can yield important data on crack volumes. Since the integrated intensity for a single crack on the autoradiographs is proportional to the volume of the crack as it was when the gas was in it, the crack aspect ratio (lengthlwidth) may be calculated from the SEM and autoradiography data. In the following we will examine several candidates for tracer gases. In searching through the nuclear chart, it seems that Ar42 has nearly ideal properties. It has a half-life of 33 years and decays into K42, which has a halflife of 12.4 hours. The Ar42decays directly to the ground state of K42 with a beta particle with a maximum energy of 0.6 MeV. Of the K42 nuclei, 82% go directly to the ground state of Ca42 with a beta of maximum energy 3.52-MeV and 18% decay to an excited state of Ca42,which then emits a prompt 1.5 MeV y ray. While the Ar42 is in a rock sample, it may be kept at high temperature and pressure for several half-lives (or a large fraction of one half-life) of K42 to allow K42 to come to secular equilibrium (or a reasonable fraction thereof). Since Ar42has no y-radiation, a whole-body count of the 1S7-MeV y from the K42 will be a measure of the total porosity that the rock sample had when it was at high temperature and pressure. Argon-42 may be produced by double neutron capture, Ar4'(n, y)Ar4l(n, y)Ar42,75or by triton capture, Ar4'(t, p)Ar42.76 Because of the small neutron cross sections of A40 and Ar41(both less than 1 barn) and because of the short halflife of A41( 1.3 hours), a large neutron flux is needed to produce Ar42 by double neutron capture. Another possibility where an inert gas has an intermediate radioactive solid daughter exists in the decay chain of radium. Radon-222 can be milked continuously from radium, such that its short half-life of 3.82 days is not a detriment. As Rn222decays, Pa'' grows in. It has a half-life of 138.4 days and is primarily an alpha emitter and thus very useful for autoradiography. Should neither Ar42nor Rn222prove to be practical because of difficulties in obtaining them, it may be necessary to use radioactive tracer gases that do not have a radioactive solid daughter. In that case the whole-sample

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HARTMUT SPETZLER

FIG. 51. (a) Doubly exposed hologram clearly showing the surface deformation associated with the development of the failure zone. (b) Cutting plan made for scanning electron microscopy and autoradiography studies.

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FIG. 51-continued

autoradiography (Fig. 49) can still be done quantitatively. It must be done within a pressure vessel while the sample is under stress and the cracks are open. The microautoradiography must then rely on the gas that was trapped within the sample. It thus becomes a qualitative rather than a quantitative method. Several gases, some commercially available, are attractive in this case. Xenon-133 is commercially used in hospitals for lung scans. It has a halflife of 5 days, is a /3 emitter ( - 0.4 MeV), and has an abundant 80 keV ~ ~ which it decays. The X-ray that results from an excited state of C S ’into range of 80-keV X-rays is somewhat less than 20 mm in rock, which makes Xe’33 still useful although somewhat less desirable because radiation from different ?arts of the sample would contribute with significantly different efficiencies. Krypton-85 has a half-life of about 11 years, which avoids the supply problem with Xe’33. It decays to the ground state of RbS5with a probability in excess of 99%, emitting a/3 particle with a maximum energy of 0.7 MeV.

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