Crack field characterization by ultrasonic attenuation—preliminary study on rocks

Int. J. Rock Mech. Min. Sci. & Geomech. Abstr. Vo|. 19, pp. 267 to 278, 1982

0148-9062/82/060267-12503.00/0 Copyright © 1982 Pergamon Press Ltd

Printed in Great Britain. All rights reserved

Crack Field Characterization by Ultrasonic Attenuation Preliminary Study on Rocks J.-P. MOLINA* B. WACK*

The attenuation of an ultrasonic beam is used to characterize the crack field of a rock sample. The experimental results, supported by theoretical models, were compared with those obtained by the image analysis procedure. Due to experimental difficulties, only a qualitative comparison could be made at the final stage. This study shows clearly the good sensitivity of the ultrasonic attenuation to crack density.

INTRODUCTION It is now well known that the mechanical behaviour of rock material, in the brittle domain, is characterized, from the physical point of view, by a pre-existing crack network and by its evolution with applied load [1-3]. For this reason the experimental study of such a material's constitutive equation supposes that we are able to follow the crack field development along a loading path. The classical method of characterizing the crack field is to examine sections cut in the material (thin plates or polished faces). The quantification of the crack field parameters was done initially by manual methods [4]; now this is done more systematically since image analysis procedures have become available [5]. But this direct analysis is destructive and therefore greatly limits the possibilities of the experiments. To avoid this difficulty a non-destructive method seems a priori very interesting. The transmission of ultrasonic waves is considered. The transmission speed of longitudinal and transverse waves has already been used, but does not give sufficient information on the crack field [6, 7]. It has been shown [8] that the energy loss of ultrasonic waves propagating through the material is related to the ratio of the wavelength to the mean distance separating the discontinuities present in the material. Therefore, a study of the ultrasonic wave attenuation with the wave frequency should provide some precise information on the crack network existing in the rock material [9]. The scope of this study is to show the possibilities and the limits of this method, in comparison with the classical destructive method.

During the transmission of an ultrasonic wave through such a real rock material an important part of the wave energy is lost: we distinguish loss by absorption (for example loss by friction of the two crack faces), loss by diffusion on the discontinuity surfaces, beam diffraction and matrix absorption. Since we are only interested in the variation of energy loss between an initial state and a final state, we consider further only the loss by absorption and diffusion due to the additional cracks introduced between these two states. We characterize the energy loss by the attenuation coefficient ~ defined by = ~10 log10

(1)

where, Lis the theoretical path length of the ultrasonic wave in the sample, and E~ and E2 are the beam energy before and after the sample crossing respectively. A theoretical model was built based on the following hypothesis: --the rock material is constituted of a matrix with a few discontinuities; --the matrix is homogeneous, isotropic and has an elastic behaviour; --the discontinuities are plane and their concentration is low enough so that it is possible to neglect multiple diffractions; --the Rayleigh conditions are satisfied for the incident plane wave:

ka << 1

(2)

POSITION OF THE PROBLEM

where k is the wave number and a the crack characteristic dimension; --the friction between the crack faces is negligible.

Consider the grain scale cracks existing in a rock sample and their evolution with a compression load.

The theoretical results obtained can be written in the form: ~j --- Kjn ( a 6 ) f 4

* Institut de M+canique de Grenoble, Universit6 de Grenoble I, L.A. 6 C.N.R.S., Grenoble Cbdex 38041, France.

267

(3)

where Kj is a proportion coefficient depending for a

268

J.-P. Molina and B. Wack

given material on the crack geometry-type j; n is the number of cracks per unit of volume; (a ~) is the mean value of the sixth power of the crack radius a and f is the wave frequency. The value of j is related to the crack geometry type; the following cases were calculated by M. Piau* [10-19]: j = 1 parallel circular cracks; j = 2 axisymmetrically distributed circular cracks; j = 3 randomly distributed circular cracks.

So(f) \

SIR(f)

,Aa-rm

We recall that similar results were obtained for wave attenuation by grain joints in a polycrystalline material without cracks and pores [8, 13, 14]: 0

l

f

Fig. 1. Definition of the attenuation coefficient variation.

= A f + BD3/"4

for

2 > 2rrD

= CDf 2

for

)~ < 27tD

= E f + G f z + HD-~ for

(4) length L of the unstressed sample and then

2 <

where A to H are constants depending on the material and D is the mean grain size. Based on the preceding model (3), the variation of the attenuation coefficient Acti between a reference state (R) and a final state (i) is of the following form: Ao~i = K 3 [ K i / k 3 ni ( a ~ ) -

nR (a6)]f

4

(5)

if we suppose a random crack distribution for the initial state. Ultrasonic wave attenuation in the real rock material is very important and thus it is only possible to use the measure of the first transmitted signal in a transmission type measurement. The ultrasonic signal is decomposed by a frequency analysis procedure which gives the spectrum of the pulse (Fig. I). Let us consider the frequency response So(f) given by the initial pulse; the response Se(f) of the reference pulse measured after transmision through the reference state is defined by the reference compression stress aR and the response Si(f) of the pulse after transmission through the state compressed by the stress (Fig. 1). If Le and Li are the sample length stressed by a compression aR and ai respectively, the variation of the attenuation coefficient Ac(i between the reference state and the final state is given by Aai(f) _= at(f) - ~e(f) - Soft) - Sift) Li

SOU) - SR(f) LR (6)

Since the rock deformation in the brittle domain is small, it is possible to replace Le and Li by the initial

* We are indebted to Dr M o n i q u e Piau who accepted, at the second author's request, to m a k e these calculations•

Aai(f) -

s.ff) - sift)

L

(7)

and the result does not depend on the So(f) value. The last aspect of the problem concerns the physical characterization of the crack field. The mean quantities used for that purpose can be the number, the dimensions and the orientations of cracks and the distances between cracks; all these values are theoretically defined in three dimensions as quantities per unit volume. But in practice it is only possible to make physical measurements on the traces of the crack field in different plane cuttings which give values per unit surface. Thus the image analyser can give the following quantities per unit surface, concerning the crack traces in a cutting: --the total surface occupied by the cracks, thus giving the mean porosity of cracks; if the traces can be considered of the same thickness this measure also gives the total length of the traces. - - t h e total perimeter of the traces; by neglecting the small sides of the perimeter, this measure gives the total length of the cracks. - - t h e histogram of some crack trace quantities, e.g. individual surfaces and lengths. - - t h e angular repartition of the crack directions. Since the use of the image analyser begins to be classical, we will not enter into the details of this technique; a valuable survey is given by Underwood [16] and examples of it's use with brittle materials are given by Stroeven [17, 18] and Karcz & Dickman [19]. THE EXPERIMENTAL M E T H O D

The propagation of the cracks is obtained by a compression test. During the test, ultrasonic measurements are made in three orthogonal directions; one is parallel-

Ultrasonic Attenuation in Cracked Rocks TABLE 1

Young's modulus (GPa) Poisson's ratio Compression strength (MPa) Mean grain diameter (ram)

Villette

Carrara

45-60 0.2-0.3 100-150 0.15-O.3

60-93 0.3 110 0.1-0.3

to the compression direction and the two other are perpendicular to the compression direction. Each sample is loaded to a compression stress value of kRi(O < k < 1), Ri being the ultimate compressive strength. After this test the sample is cut in pieces to permit the examination of the crack traces. This procedure necessitates the construction of a special cell test: it permits ultrasonic measurements during a compression test on a cubic specimen.

The rock type Two rock types are considered, a Villette and a Carrara limestone. The sample has a cubic form with a 4 cm side length. At this scale we consider the rock to be isotropic. The principal characteristic values of these well known rocks are given in Table l [l, 15]. All the samples were cut with regard to the normal direction of the stratification plane, which is the Z direction; the X and Y directions are always kept the same for the different samples.

The test cell The test cell is placed between the two platens of a classical hydraulic compression machine. The transducers of the compression axis are placed in an intermediate piece, called the anvil, whose section is equal to

that of the sample (Fig. 2). The four other transducers are in direct contact with the sample. The ultrasonic energy loss through the 30 mm thickness of the anvil bottom, along the compression line, is supposed constant when the compression stress varies. The test cell is essentially constituted by a heavy steel cylinder (Fig~ 3), the external and internal diameters being equal to 250 and 190mm respectively; the cylinder supports the four horizontal transducer attachments and the bottom transducer. The setting of a transducer in an anvil is shown in Fig. 4. The anvil design was chosen so that the contact surface deflection is less than 1 pm for a mean compression stress of 400 kN. To obtain a good plane definition for the contact between the transducer and the anvil, this latter piece was fabricated in two parts and glued together after the machine-finishing of the surface. The contacts between transducers, anvils and sample are kept lubricated when it is possible. This is the case on the two horizontal lines, as the pressure contact between the transducers and the sample is low; this is also possible between the transducers and the anvils along the compression line. But the high stress on the compression line does not allow the presence of grease on the contact between the sample and the anvil, to avoid injection of the grease into the sample. Furthermore, to best maintain the normality of the compression stress we use a sheet of cadmium as an insert between the anvil and the sample; the contact between the cadmium sheet and the anvil is lubricated, but not the contact between the sample and the cadmium sheet. The disadvantage of this choice is the necessity of having a compression stress higher than a given value in order to make ultrasonic measurement, as we will see later.

J

Fig. 2. Scheme of the experimental procedure. R.M.M.S. 19/6

t~

269

270

J.-P. Molina and B. Wack

Fig. 3. View inside the pressurc cell.

The ultrasonic measurement A diagram of the ultrasonic measurement device is shown in Fig. 5. The transducers have a quartz element of 12.7 mm dia and a large frequency rate. For the first experiments a resonance frequency of 10MHz was chosen and was used for the measurements with the Villette limestone. As maximum energy transmitted was for lower frequencies, we used for the Carrara limestone, on the compression axis, two transducers of 2.25 MHz resonance frequency. The ultrasonic pulser-receiver emits electrical impulses of the following features: the frequency range is 100 Hz-30 MHz, the repetition frequency is adjustable between 0.2 and 5.0 kHz; an energy attenuator in the range 0-68 dB gives the possibility of limiting the amplitude of the received electrical signal to an acceptable level. The stepless gate gives the possibility of separating the desired part of the received electrical signal before its frequency analysis. The tuning of the ultrasonic pulser-receiver and the stepless gate is done by the use 'of an oscilloscope. The isolated electrical

signal is analysed by a frequency analyser which gives the energy repartition of that signal with respect to the frequency. The physical observation of the cracks After the partial compression test, each sample is carefully cut into 27 small cubes. To reduce the amount of figure analyses only the three cuttings of the central cube were examined. The preparation of these faces was conducted in such a way as to reduce the artifacts produced by the different operations [20]. The small cube is injected with a liquid monomer (methyl methacrylate) with the addition of a fluorescent pigment; the polymerization is done at 65'C during 48 hr. To reduce the disturbance of the crack field produced by the injection, the entire small cube is subjected to injection pressure of 1 MPa, giving a sufficient depth of injection. The polishing takes place after polymerization so that all the cracks reaching the surface are filled with the polymer during this operation. Thus the artifacts produced by careful polishing can be considered as negli-

Fig. 4. The anvil and its transducer.

Ultrasonic Attenuation in Cracked Rocks

271

tuning f~1

t~

signal

Pulser Receiver

IAI

Stepless gate

Frequence analyser

E

Tfull sig. eot. si,. f'c D

L

Oscilloscope

Fig. 5. Diagram of the ultrasonic measurement device.

gible, in spite of the strength difference between the rock material and the polymer. The choice of the fluorescent pigment gives a better contrast between the matrix and the crack traces, but this contrast reproduced by photography is not sufficient for a direct analysis by the image analyser. To give correct information to the analyser it was necessary to draw the figure of the crack traces (black ink on white paper). This was done under a microscope with an ultraviolet light. The magnification between the small cube faces and the drawing is 200. Thus it was not possible to use all the information included in a small cube face and a sampling procedure was necessary: 25 drawings corresponding to regularly spaced small squares were done for one face of a cube, corresponding to a sampling fraction of approximatively 1/10. We checked that the difference between results was negligible for sampling factors of 1/5 and 1/10. The image analyser obtains its information by means of the horizontal sweeping of a digital figure of 800 x 625 points; the apparatus does not give back individual indications but only mean values calculated by its associated computer. The ratio of black points to white points gives the total surface occupied by the cracks, or the crack porosity. By working with the transitions between black and white points the apparatus gives the total perimeter of the traces; if we neglect the small sides of the perimeter this result gives the total length of all cracks in the image. By comparing the black points of two adjacent sweeping lines the apparatus gives the number of black regions, i.e. the number of cracks in our case. Thus the mean length of individual cracks can be calculated. By examining the image in different orientations, defined by the angle 0 between a physical axis and a screen axis, we obtain the number of transitions between black

and white zones D(O) for each direction 0. As shown by Maire & Gilles [21] we have the following relation: l

02D

p(O) = D(O) + 00---T

(8)

where p(O) is the cumulated length of cracks contained between the angles 0 and 0 + dO; k is a characteristic coefficient of the apparatus. To obtain the second derivative of D, we adjust locally the D(O) curve with a parabola defined by 2n + 1 points, measured with an angle variation dO = ft. The results shown later were obtained for n = 5 and fl = 2, or 4 °. RESULTS The tests on the Villette limestone are preliminary tests and concern only the acoustic measurements. Complete results are obtained with the Carrara limestone and concern acoustic measurement as well as field trace measurement.

Preliminary acoustic measurement with the Villette limestone [22] Figure 6 shows typical results obtained with the Villette limestone and measured along the compression line. The reference stress of 69 MPa corresponds to the maximum ultrasonic transmitted energy. Keeping in mind the theoretical results, we see first that the interpretation is limited to the frequency range of 1-3 MHz. Secondly we see that in the stress range 69-94 MPa some phenomena other than a crack increase occur: it is probable that the contact continues to improve with the beginning of crack propagation. In the stress range 94-119 MPa, the variation of A0t with frequency follows a variation of power four; for a compressive stress of 125 MPa, the variation is of power three. This confirms

272

J.-P. Molina and B. Wack

A AO'z 6~/

6

Ref

5

] AUll 2

, 68.75

MPo

4

2

5

f ,

4

alIM Pa)

;,~

1

56 60 64 68 72 76

0.72 0,77 0,82 0.~7 0,92 0.97

2 3 4 5 6

E

i

Number

5

6

MHz

Fig. 6. Variation of the attenuation coefficient ('gillette limestone).

that, in the range of 0.7-0.9 of the ultimate compressive strength, the circular crack model seems to be adequate, whereas for a narrow range near the ultimate stress the Griffith model is adequate. For this rock, the two horizontal ultrasonic measuring lines were identical to the compressive line and did not give significant results since the contact pressure was not sufficient. This made us decide to change the device and put these four transducers in direct contact with the sample. The range of frequency sensitivity induced us to replace on the compression line the 10 M H z transducers by 2.25 MHz transducers. Acoustic measurement with the Carrara limestone

A preliminary compression test gave the ultimate compressive strength for this rock under the particular test conditions (cubic form of the sample, presence of the cadmium sheet). A value of Ri = 78 M P a was found, representing 71% of the value corresponding to a classical cylindrical form. A series of six tests was done, each test being defined by its maximum compressive stress value ~ri = k i Ri (Table 2). For most of the tests the ultrasonic measurements were made along the loading and the unloading branches. For test no. 6, only measurements on the loading branch could be made, since it was necessary to unload rapidly in order to avoid the destruction of the sample. Due to the time necessary to make all the ultrasonic measurements for one stress value (about a quarter of an hour), the loading and unloading branches present dwell-time of about 15 min. For samples nos 5 and 6, Fig. 7 gives the evolution of the transmitted energy with the compression stress for a loading-unloading cycle; this maximum energy is defined independently of the frequency. In a first stage

* For all the tests the same orientation of the X and Y axes were maintained with respect to the physical axes of the rock mass.

we see an almost linear variation which probably corresponds to an improvement of the contact quality. In a second stage the energy decreases non-linearly until the maximum stress: this stage corresponds to the propagation of cracks. During the unloading, the energy decreases quasi-linearly, with the same slope as in the first stage; for one case this stage is preceded by a short range of non-linear decrease, showing that the crack propagation may continue at the beginning of the unloading branch. The variations on the X and Y axes are larger than on the Z compression axis. This result corresponds to the fact that more cracks are parallel than perpendicular to the Z axis. The energy spectrums for test 5, for example, are given in Fig. 8 for the stress range of crack propagation. We notice that the resonance frequency corresponding to the maximum transmitted energy decreases as well as the energy level. The values of the increase of the scattering coefficient Ac~ are then calculated with reference to the value measured for a compressive stress of 56 M P a (Fig. 9). A first interpretation of the influence of frequency and the crack field on the attenuation coefficient, can be represented by the variation of A~, for a given frequency, with the maximum compressive stress; these results are given in Fig. 10 for the frequency l, 1.5 and 2 MHz. The attenuation coefficient increases regularly with the compressive stress as expected. The variation is important for the X and Yaxes and small for the Z axis. This indicates that most of the cracks, which propagate after a compressive stress of 56 MPa, are orientated parallel to the Z axis. The results also show an unexpected general difference between the X and Y values; this can be explained by a small anisotropy between the two axes*. A more general interpretation may be made by the use of the theoretical results giving a relation between the attenuation coefficient and the frequency. The best correlation results are given by a function of two terms: A~ = a f +

bf 4

(9)

The linear term accounts for energy loss by friction between the crack faces; the term of the fourth power of the frequency accounts for energy loss due to the scattering of the cracks. The values of the coefficients a and b were computed by the statistical method for each sample and the results are given in Table 3; the range of frequency indicated on Table 3 corresponds to the left-hand part of the curve (A~,f); the correlation is calculated including the origin, taken as an ordinar3,

Ultrasonic Attenuation in Cracked Rocks

273

Z 45

50

55

I 60

I 65

I

I

t

55

60

65

I 70

I 75

5 Y

Y 0 lid

I

I

I

45

5O

=1 ~'

I ~

I

0

tin

45

50

~

I

7 0 ~

'O

-5

-5 ¢

-I0

_il

x

5I

45

X

45

50

F-"--J

I

55

I

60

l

o"

(MPa)

o"

(a)

(MPa)

(b)

Fig. 7. Variation of the transmitted ultrasonic energy with the compressive stress (samples 5 and 6).

measured point. The evolution of coefficients a and b with the compressive stress is represented in Fig. 11. As expected, this shows an increase of a and b with a; the difference between the axes X and Y is made obvious; the measurement along the X axis for sample 3 has to be considered as erroneous.

The linear term in equation (9) is of the same order as the fourth power term. This indicates that the loss of energy by the ultrasonic beam is due to the scattering phenomenon as well as to the friction phenomenon. This fact is not surprising and was expected, for example by Johnston et al. [23] who noted that for a

I

I

6 0 MPa

~ 65 MPa 7 0 MPa 72 MPa

I

X

I

2

3

4

Y

I

2

3

f (MHz)

Fig. 8. Typical energy spectrums (sample 5).

I

I

.I

I

I

2

,3

4

274

J.-P. M o l i n a a n d B. W a c k

<]

2

'~

2 f , MHZ

2

o

3

I

2

3 f,

MHz

T

go

I

<~

3 f ,

MHz

Fig. 9. Typical evolution of Act with the stress and the frequency (sample 5).

1 1.5

1.5 2

MHz

•

/ (e)

/ /

(-1

E line X

0

<~

(i)

072

0.77

Ql~

fine Y

/

/

./

rn

"o

/

MHI

2

/

I:l

<1

/

// / .///

./

/-

Y / /l

/ "/,

/.

/o

0,,

0.87

i'l

1.5

o77

~

MHt

2

Line Z

~2

/.

j

Q92

Qe7

Fig. 10. Values of Act determined at the maximum stress value.

~

;"L, " ' ° "

Ultrasonic Attenuation in Cracked Rocks TABLE 3

Axis

a (10ZdBm - 1 M H z - l )

h ( 1 0 Z d B m - 1 M H z -4)

X Y

Sample (1.135) 1.488 Sample 0.457 1.684

N o . 4, ~ =

X Y

Sample 0.682 2.362

No.

X Y

N o . 6, ~ =

X Y Z

Sample 2.208 2.904 0.075

No.

3, (7 =

5, ~ =

R a n g e of frequency (MHz)

64 MPa (0.995) 0.119

0; 0.8-1.2 0; 1.2-2.0

68 MPa 0.067 0.293

0; 1.5-2.4 0; 0.8-2.0

72 MPa 0.192 0.415

0; 1.0-2.1 0; 1.0-1.8

275

The results found for the variation of the coefficient a, which represents the energy dissipated by the phenomenon of friction is in agreement with the results given and the interpretation made by Lockner et al. [25]: the attenuation of the X and Y compressional wave increases at high compressional stress (k > 0.7) due to the energy loss by friction on two sets of vertical cracks oriented approximately at 45 ° to the direction of propagation. The energy loss by scattering, given by the coefficient b, increases also due to the wave scattering on the set of vertical cracks perpendicular to the direction of propagation.

Results of the image analysis 76 MPa 0.251 0.165 0.070

0; 0.9-1.8 0; 1.2-2.0 0; 1.0-2.0

NB For the samples 3, 4 and 5 the measurements made along the Z axis were not significant.

frequency higher than 1 MHz scattering effects can be important since this phenomenon varies very rapidly with the frequency. Thus the last hypothesis of the theoretical model (See Position of the Problem, p. 000) has to be avoided, since friction is still playing a nonnegligible role. The friction phenomenon has been taken into account in a few attenuation models, such as the one by Walsh [24]. But such a model requires knowledge of the friction law between the two faces of the cracks; in general one utilizes the well-known linear Coulomb equation, characterized by its friction coefficient. This hypothesis is difficult to verify: in the case of marble the friction coefficient may vary between 0.1-0.2 for smooth calcite single crystals and 0.7 for fractured surfaces of marble [24]. Thus the comparison between a theoretical model and experimental results can only give a value of a macroscopic friction coefficient by a general identification, and in general we do not have another set of experimental results (which are not a repetition of the first set) to control this value. Finally we can only claim that such a theoretical model is acceptable since the macroscopic value found for the friction coefficient is physically correct. The fact that friction and scattering both play a role in the attenuation of cracked media, in the frequency range considered here, is to be compared with results obtained in polycrystalline media where the same equation is found valid for the Rayleigh domain (2 > 2nD). Instead, with a propagation speed of 5900msec, the wavelength is 0.6 and 2.6 mm for frequencies of 10 and 2.25 MHz, respectively; these values are much greater than the crack dimension as we will see in the following paragraph.

* The use of such a value, having a poor theoretical definition, does not have important consequences, since qualitative comparison will only be made.

A first group of results includes the number of cracks and their mean dimension. For each sample and each face the values are obtained by the analysis of 25 small examination fields (See the physical observation of cracks, p. 000). A difficulty arises about the definition of the crack length mean value, as the theory considers the mean value of power six. The internal logic of the image analyser gives only as a result the linear mean value. In order to take partially into account the distribution of the crack length, we will use the mean value of power six of the 25 elementary results, each elementary result being a linear mean value* noted ( a ' 6 ) . The results measured on the faces X, Y and Z are given in Table 4 and represented on Figs 12 and 13; these results are values per unit surface (n', ( a ' 6 ) ) , as in the theory the parameters introduced are values per (a) 3--

/ A/

/

/'

I T

2

"/

E ¢n -o

I

/'

/

/

/

"/ , /

,4.

/

.~/

/ 56 0.72

I

? 60 0.77

(b)

64

68

72

0.82

0.87

092

76 cr , M PO 097 k

(if

i

T

• i]--'-E ¢,n 0 . 5

%

/ 56 0.72

60 0.77

64 0.82

../:" 68 0.87

72 0.92

76 o" t 0.97 k

MPo

Fig, 11. Variations of the coefficients a and b.

J.-P. Molina and B. Wack

276 I00

TABLt! 4

--

• A

line line

X I Y

•

line

Z

I

n' ~

Plane

/

/' 60

-

/

e~

/

i

E E c

60

/

-

•

(a'~' 5 '~ /,tmi

Z (103 # m 4)

X Y Z

Sample No. 1, a = 5 6 M P a 25 17 0.6 ] 48 2O 3.1 2.6 36 22 4.1

X Y Z

36 33 29

X Y Z

Sample No. 3, o = 6 4 M P a 54 25 13.2 ] 52 28 25.1 / 14.4 57 21 4.9

X Y Z

30 56 53

X Y Z

Sample No. 5, a = 7 2 M P a 22 26 6.8 ] 47 36 102.0 I 46.0 40 30 29.2

/ ' Y.o (11) 3 lira a)

0

Sample No. 2, a = 6 0 M P a

/

,/

/

:// •

(mm 2)

e/•

•

12 14 15

0.1 ] 0.2 0.3

0.2

-2.4

11.8

e

,o

Sample No. 4, a = 6 8 M P a ~A

ZO

I

56 0.72

60 077

1

I

64 082

68 087

~

I

72 0.92

76 cr , 097 k

MPo

22 32 35

3.4 ) 60.1 } 53.8 97.1J

51.2

43.4

Fig. 12. The number of cracks per unit surface. Sample No. 6, a = 7 6 M P a

unit of volume (n, ( a 6)). First we realize that there is a very important scattering concerning the number of cracks per unit surface of observation; this is probably due to the difference between the initial state of each sample. Nevertheless, the evolution tendency suggests a parabolic increase of n' with o. The evolution of the crack mean length (a '6) is less dispersed and clearly shows a parabolic increase of (a'6) 1/6 with or, the initial mean length being equal to about 20 pm. The second group of results concerns the orientation. For samples 1-5 inclusive, the cracks have an isotropic orientation in all planes (see for example sample 5, Fig. 14). For the sample 6, compressed to 76MPa, the cracks always have an isotropic orientation in the X and Y planes, but have a marked anisotropic orientation in the Z plane (Fig. 15). This result justifies the hypothesis of axisymmetry used in a theoretical model. To compare the image analysis results with the attenuation coefficient, the theory indicates for the scattering

60

line line

X ~ --.-Y

line

Z

I

•

-----

I:::

.//

:::L

:f'

A

X Y Z

55 84 105

5t 46 68

96.8 79.6 1.04

104

phenomenon that 7 has to be compared to the function g defined by Z = n'

and A7 to the variation of Z from its reference value (the reference value is taken here for a -- 56 MPa). As shown by the crack orientation results, samples 1-5 have an isotropic crack orientation and thus the function z(a) can be considered in the interval 56-72 MPa (Fig. 16). We verify the general trend of increase of Z with regard to a, which can be considered as parabolic. Nevertheless it should be noticed, that a linear variation for o ~> MPa, Z being constant for a < 61 MPa, can also be considered. Because of the small number of experimental points it is not possible to make a definite choice, but the first interpretation is considered as more probable. The results obtained with sample 6 show that the Z value for the X and Ydirections continues to increase in the extension of the other samples with a mean value of about 88 x 10 3 ~ m 4, while the Z value indicates a very important jump to 1 x 10 7 # m 4. This result makes the non-isotropy of the crack field clearly evident.

,.I"

30 v

CONCLUDING REMARKS

1

o 56

60

O.72

0.77

1 64 0.82

1 68 0.87

I

I

72 0.92

1'6 0.97

o" ,

MPo

k

Fig. 13. The mean value of the crack length.

A notable dispersion of the results is obvious. This can be explained by the difference between the initial state of each sample. On the other hand, and for the image analysis results, the information was taken by an operator; thus, another source of error is introduced by

Ultrasonic Attenuation in Cracked Rocks

___~__.

/

S:4.

L~

.//"~~

597

277

.

/ //

--...

2QO~

',X

//

.

.

-

/ /" /,

. ~ .4-

,

,,

\ o

-5OO

'

~o ~

-z6o

-i~

(o)

'

o.

.

.,oo .

I,,

z~o

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(b)

Fig. 14. Orientation of the cracks for sample 5 in the plane Z(a) and in the plane X(b); the units are arbitrary.

,8,2"

2OOO,

1500~

i

"

-~

•

"

~

"'"

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=2 °

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6

'

~

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~

~o ~

(a)

-IOIX~

- 5O0

0

500

I000

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Fig. 15. Orientation of the cracks for sample 6 in the plane Z(a) and in the plane X(b); the units are arbitrary• the operator's subjectivity when he defines the crack traces. The results obtained show a good sensitivity of the ultrasonic beam with regard to the field of cracks, introduced by the compression stress. The sensitivity of the geometrical measurement seems less important. This can be partially explained by the measurement

"rE

Loo

%

o

o ,

I 50

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/

I

•

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64

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Fig. 16. E v o l u t i o n o f the Z function w i t h stress.

difficulties; the principal one is the cutting of the sample which may destroy its initial state and the second one is represented by the subjectivity of the operator who has to interpret what he is seeing, before giving the information to the image analyser. Nevertheless the goal of the study can be considered as partially reached, i.e. from the qualitative point of view. The attenuation of an ultrasonic beam is sensitive to the evolution of the crack field and is able to give some information on the geometry of the crack field. But a useful quantification of such a relation was not obtained. A greater number of results is necessary for that purpose in connection with improvements of the experimental procedure. Important progress will be obtained if it becomes possible to transfer directly to the image analyser the information lying in a cutting or a polished face; the last improvements about the image contrast seem quite hopeful. From the theoretical point of view concerning the ultrasonic attenuation it seems necessary to avoid the hypothesis of neglecting the friction between the faces of the cracks. Indeed a first interpretation of the attenuation shows that the energy loss by scattering and by friction are of the same order of magnitude.

278

J.-P. Molina and B. Wack

Acknowledgements--This study was carried out with the financial support of the Centre National de la Recherche Scientifique (Contract: A.T.P. No. 2478) and with experimental facilities offered by the Institut de Recherche lnterdisciplinaire de G6ologie et de M6canique. The image analysis procedure was used through the helpful advice of Dr Morleva and Mr Jolivet of the Centre d'Etudes Nucl6aires de Grenoble~

Receited 7 April 1982: revised 22 July 1982.

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