Inversion of ultrasonic wave velocity measurements to obtain the microcrack orientation distribution function in rocks

Inversion of ultrasonic wave velocity measurements to obtain the microcrack orientation distribution function in rocks C.M.

Sayers

Koninklijke/Shell Exploratie en Produktie 2288 GD Rijswijk ZH, The Netherlands Received

8 June

Laboratorium,

Volmerlaan

6,

1987

Ultrasonic wave velocities in rock are reduced significantly by the presence of microcracks. In general, these microcracks are not randomly orientated and the rock displays an elastic anisotropy determined by the shape and content of the cracks and by the crack orientation distribution function. This function gives the probability of a crack having a given orientation with respect to a set of axes fixed in the rock, and is used here to calculate the variation of elastic wave velocity with propagation direction. The coefficients, W,,,,,, of a series expansion of the crack orientation distribution function in generalized spherical harmonics can be obtained to order /= 4 from the angular variation of the ultrasonic wave velocity. This allows construction of microfracture pole figures, which may be compared with those obtained by petrofabric examination. The theory is applied to the measurements of Thill, Willard and Bur on Salisbury granite. Since the anisotropy in strength properties originates from the preferred orientation of microstructural defects, the prediction of the microfracture orientation distribution will have an important application in the field of rock fracture. Keywords:

crack characterization;

ultrasound

Laboratory measurements of ultrasonic wave velocities in rock samples as a function of confining pressure have demonstrated that the presence of microcracks significantly decreases the elastic wave velocities in the rockl,‘. As the pressure is increased the measured velocities increase markedly. This behaviour is attributed to the closure of cracks with increasing pressure until they have no effect on the elastic properties of the rock. In general, these microcracks are not randomly orientated and the rock displays an elastic anisotropy determined by the shape and content of the cracks and by the crack orientation distribution function (CODF). This function gives the probability of a crack having a given orientation with respect to a reference frame fixed within the rock. Anderson, Minster and Cole3 have calculated the elastic wave velocities in a matrix containing aligned ellipsoidal fluid-filled cracks by using the results of Eshelby4. When the cracks are ellipsoids of revolution, the material becomes axially symmetric with five elastic constants. It was found that for aligned oblate spheroids the largest reduction in longitudinal wave velocity occurs for propagation along the axis of symmetry. Thill et ~1.~ have measured longitudinal ultrasonic wave velocities in a large number of directions on spherical samples of Yule marble, Newberry Crater pumice and Salisbury granite. Of particular interest for the present work are the measurements on Salisbury granite for which the observed pattern of longitudinal velocity is associated with a preferred orientation of microfractures in the rock. In this Paper the problem of inverting the measured variation in ultrasonic velocity with direction of propa-

004-624X/88/020073-05 $03.00 0 1988 Butterworth & Co (Publishers)

Ltd

velocities;

wave

scattering

gation and polarization to obtain the microcrack orientation distribution function is addressed. The crack orientation distribution function is introduced and is used to derive expressions for the anisotropy of elastic wave velocities in the long wavelength limit. The coefficients, K$,,,,, of a series expansion of the CODF in generalized spherical harmonics can be obtained to order I = 4 from the angular variation of the ultrasonic velocities. The results from the measurements of Thill et ~1.~ are used to plot microfracture pole figures and these are compared with the pole figures obtained by Thill et ~1.~ using conventional petrofabric techniques. On the basis of these petrofabric measurements, Thill et ~1.~ show that the mineral alignment is too weak to explain the measured velocity anisotropy and that this in fact arises from the preferred orientation of microcracks in the rock. This information is required in the present analysis since the measurements of Thill et al. were made at atmospheric pressure. However, because microcracks close up at relatively low confining pressures’g2, it is possible to separate the crack induced anisotropy from the anisotropy due to mineral alignment by making ultrasonic measurements as a function of an applied compressive stress. A petrofabric analysis of the mineral orientation is, therefore, not required in general.

Theory Introduction

For an ellipsoidal crack with principal axes 2a, 2b, 2c (a 3 b > c) it is convenient to introduce a set of axes

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1988 Vol 26 March

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Inversion

of ultrasonic

wave

velocity

measurements:

C. M. Sayers

0X,X,X, with origin at the centre of the ellipsoid and OX,, OX, and OX, along the a, b, and c axes, respectively. If the axes 0X,X,X, for all cracks are aligned, the material will exhibit orthorhombic symmetry with three orthogonal planes of mirror symmetry having plane normals in the OX,, OX, and OX, directions. It is convenient to write the fourth-order elastic stiffness tensor of the cracked material, Cijkl, in the form Cijkl

=

Ctk,

+

Yijkl

where yijkl is the difference between stiffness tensor, CE,,, of the uncracked given by

and the elastic material. CE,, is

Cijkl

C~k, =

rb4,

are Tss

+

~L(bikGj,

+

axi

axj

axk

axl

ax,ax,ax,ax,

(4)

then, taking into account the orientation distribution of cracks, the first-order correction yijkl in Equation (1) is given by =

(5)

TjkhnJmnpq

fiilfijk) (2)

rl

1,

~22,

b

r12

=

rzl,

r23

=

and FG6 in the Voigt (two-index)

Crack orientation

distribution

r32,

r31

=

notation.

function

2n 277 1

Wt,ICI>4)dtWd4=

1

where 2n 2a

I

b b -1

(6)

r13,

In general, it cannot be assumed that the cracks will be perfectly aligned and a quantitative description of the elastic constants and elastic wave velocities in the material requires a knowledge of the orientation distribution of cracks. The orientation of an ellipsoidal crack with principal axes 0X,X,X, with respect to a set of axes 0x,x,x, fixed in the material may be specified by three Euler angles, $, fJ and 4, shown in Figure 1. The orientation distribution of cracks may then be specified by the crack orientation distribution function W( r, $, 4) where 5 = cos 0. W( r, II/, 4) d{ d$ d4 gives the fraction of cracks between 5 and 5 + d5, 4 and rj + d4 and II/ and $ + d$. Clearly

(3)

sss

0 0

_ Tjklmnpq

?ijkl R~ijskl

where dij is the Kronecker delta and I and p are the second-order elastic constants of the untracked medium. For aligned ellipsoidal cracks the material has orthorhombic symmetry, the yijkl being denoted in this case by Fijkl. In the crack reference frame 0X,X,X, the non-zero rijkl

It will be assumed that in the presence of cracks the material has orthorhombic symmetry. Expressions for the elastic constants Cijkl of the cracked material can be derived from the Fijkl and the crack orientation distribution as follows. If Tjklmnpq is given by the transformation rule for tensors of rank four, i.e.,

-1

The averaging process expressed by Equations (5) and (6) is the same as that used to determine the elastic stiffness tensor in the Voigt approximation for an orthorhombic aggregate of crystals with orthorhombic crystallographic symmetry6,7, with the crystallite orientation distribution function in these references replaced by the crack orientation distribution defined above. The expressions for the yijkl involve the Fijkr and the coefficients, yt;,,, of an expansion of the CODF as a linear combination of the generalized Legendre functions introduced by Roes-lo. For orthorhombic material symmetry and ellipsoidal cracks, the non-zero &,,,, are all real and are restricted to even values of I, m and n. For randomly orientated cracks the assumption of an elliptic rather than a circular shape for the cracks did not significantly change the results” and, in this Paper, the case of ellipsoids of revolution with a = b >>c are considered. Here wt;,,, = 0 unless n = 0 and the averaging process in Equations (5) and (6) is reduced to finding the elastic stiffness tensor of an orthorhombic aggregate of crystals with hexagonal crystallographic symmetry12. fourth-order Backus’ 3 has shown that an arbitrary tensor satisfying the 60 symmetry relations Yijkl

=

Yjikl

=

Yijlk

=

Yklij

may be uniquely represented as a linear combination of 21 canonical harmonic tensors of degree 0, 2 and 4 with components denoted by y,,lrn4. Here, 1 is the degree of the spherical harmonic and may take the values 0, 2 and 4, m is the order of the spherical harmonic with value 0 < m < 1and 4 is either ‘c’ for cosine or ‘s’ for sine. The subscript 0 may be either ‘S’ for symmetric or ‘A’ for antisymmetric. Expressions for the yLrn+ are given in Reference 12 in terms of the w,,,, and three anisotropy factors a,, a2 and a3. In terms of the rijkl defined above, the a, are given by

a,=r,,+r,,-2r,,-4r,, a,=r,,- 3r,,+2r,,-2r,,

(7)

a3=4rl, - 3r3,-r13-2r,, Figure 1 Orientation of co-ordinate system 0X,X,X, with respect to the material co-ordinate system 0x,x2x3 specified by the Euler angles I), 0 and C#J.0X,X,X, is reached from 0~1~2x3 by a rotation of I) about 0x3, a rotation of 19about Ox>, the new x2 axis, and a rotation of 4 about 0Xx. ---, Intermediate positions of the axes

74

Ultrasonics

1988

Vol 26 March

Calculation

of the elastic

wave

velocity

In this Paper only longitudinal waves will be considered since this is the mode of propagation most commonly

Inversion

of ultrasonic

wave velocity measurements:

Determination

of microcrack

C. M. Sayers

pole figures

Consider the normal, 5, to a microfracture plane with the orientation shown in Figure 2. In a conventional petrofabric analysis a microscope is used to determine the orientation of normals for a set of microfractures which are then plotted and contoured by projecting onto a convenient plane, giving a microfracture pole figure. Let n( [, 9) d[ dq be the number of cracks with normal between [ and [ + d[ and ?J and q + dr]. The microcrack normal orientation distribution, q(1, q), is defined by (13)

This may be expanded

as a series of spherical

harmonics (14)

Figure 2 co-ordinate

Orientation of a vector system 0x1~2,~

,” with

respect

to the material

used in both ultrasonic and seismic studies. The velocity for propagation in the direction rz, shown in Figure 2, may be evaluated to first order in the number density of cracks using the variational methodi4-i7. In Figure 2, x and q are the polar and azimuthal angles of the vector 2 in the reference frame 0x,x,x, fixed in the rock. It is convenient to introduce a function r([,q) defined by r(i,r)

velocity in the direction

n 1

4710, =

(9)

n,(i> 9) di drl ss 0

-1

and [ = cos x. r([, q) may be expanded as a linear combination of spherical harmonics with expansion coefficients R,,

r(i,r) = i

i

hA”(i)exp(

l=Om= -1

--iv)

where P;“(c) are the normalized functions’. It follows that 2n

%(i,rl)=1+4~CQ20PB(I)+2Q22P~(l)cos2rl + Q40P%I) + 2QuP:(i)

cos 29

(15)

+ 2QuP$(i)COS4q1 For circular cracks the Q[,,, are given in terms of the I&,,, by Q~m=2~Wt;mo.

(8)

= u,(i, r)/471G

where u,([, r~) is the longitudinal 2rr

where the P;“(i) are the normalized associated Legendre functions. Only the coefficients up to I = 4 can be obtained from the angular variation of the elastic wave velocitiesr8. Expanding Equation (14) to order I = 4 gives

associated

Application granite

of the

theory

to Salisbury

Thill et al.’ have compared longitudinal wave velocity with structural subfabrics for Yule marble, Newberry Crater pumice and Salisbury granite. Figure 3 shows a contoured equal area projection of the longitudinal velocities measured by Thill et al. for Salisbury granite.

(10) Legendre

1

R,, = (277-l

r(i,~)P~(i)exp(im?)didq

(11)

ss 0 -1

R,, values are real and vanish for odd values of I since u,(<, x) = ol( -c,x + rc). In terms of the II$,,, defined earlier the non-zero R,, are found to be given by the following equations R,, = -47ta,W2,,/105pv~i$ R,, = 8na, W,,,/3

lSpv$,

(12)

The Km;,, may, therefore, be obtained from measurements of v,([,q) using Equation (11) provided that the a, given by Equation (7) are known and the crack-free velocity VP can be estimated. Calculations of the IV,,,,, for Salisbury granite are presented in the section on application of the theory.

t Xl Figure 3 Contoured equal velocities (km s-’ ) measured

area projection of the longitudinal by Thill et a/.5 on Salisbury granite

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1988

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Inversion

of ultrasonic

wave velocity measurements:

C. M. Sayers

The velocity distribution shows an orthorhombic symmetry, the co-ordinate planes being approximately parallel to the three mirror planes. The orientation distribution of microcrack normals was measured by Thill et aL5 using a petrofabric microscope and correlates closely with the longitudinal velocity symmetry pattern. Two sets of microfractures were found to be present. The principal set consisted of well developed microcracks frequently extending through many of the grains. For this set the crack normals showed a marked alignment along the direction Ox, (see Figure 3). The second set had crack normals aligned in the direction Ox, (see Figure 3) and consisted of relatively few, discontinuous, weakly defined microcracks. Thill et al5 concluded that the effect of this second set on the measured velocity is negligible, the principal set controlling the velocity anisotropy of the rock. The coefficients R,,, R,,, R,,, R,, and R,,in Equation ( 11) and the value of V, defined by Equation (9) were obtained by integration over the measured velocity distribution to obtain a microfracture pole figure based on the ultrasonic measurements. The results are given in Table 1. It is seen that the fourth-order terms R,,, R,, and R,, are substantially smaller than the second-order terms R,, and R,,. This is expected from the expressions for the anisotropy factors, ai, occurring in Equation (12). Hudson” has calculated the Fij in Equation (7) by considering the interaction of an elastic wave with a random spatial distribution of cracks in the longwavelength limit. The cracks are assumed to be circular and may be empty or filled with a solid or fluid material. The results are obtained using the solution for an ellipsoidal inclusion under static stress and are correct to first-order in n0a3, where n, is the number density of cracks and a is their radius. It is found that

rzl = -n,a3~~2U33/p r13= r31= rz3= r3*= -n,a31,(A + 2p)u,,/p r33= -n,a3(l + 2~)~U,,/p ra4= rss= -n,a3pLU,, re6=o rII

= rzz = r12 =

(16)

The quantities U, 1 and U,, depend primarily on the conditions imposed on the surface of the crack. For the dry cracks appropriate to the measurements of Thill et aL5 the stress-free condition gives U,,=16(1-v)/3(2-v)

20

15

i

00

where v = A/2(,? + p) is the Poisson’s ratio of the uncracked rock. Figure 4 shows plots of a,, a2 and a3 for this case. It can be seen that a, is much smaller than a3 in agreement with Equation (12) and the values of R,, given in Table 1.

02

0.3

04

05

Y

Figure 4 Variation of the anisotropy Equations (7) with Poisson’s ratio, Y

a; defined by

factors

Because of the small value of a,, the coefficients W,,, make only a small contribution to the measured longitudinal wave velocity and can not, therefore, be obtained accurately from the data of Thill et al.’ for which an error in the measured velocities of 1.4% is quoted. Therefore, only W,,, and W,,, were evaluated and the microfracture pole figure, plotted in Figure 5, was calculated from Equation (15) using the values W,,, = -0.0105 and W,,, = 0.00646 determined from Equation (12) and the values of R,, and R,, given in Table 1. For this calculation the quantity noa occurring in Equation (16) is required. This was obtained from the velocity fiI defined in Equation (9) and the longitudinal and shear wave velocities up and u,” in the absence of cracks. For these the values no = 6.4 km s 1 and u” = 3.7 km s- 1 measured by Birch’ &rd Simmons2’ at 16 kbar for Barre granite were used.

Discussion

U,, = 8( 1 - v)/3

01

and conclusions

The ultrasonic pole figure shown in Figure 5 shows a preferred orientation of crack normals along Ox, in agreement with the results of Thill et al.‘. The ultrasonic pole figure is an approximation to that obtained by petrofabric analysis since only the first few coefficients, w5,,, in an expansion of the crack orientation distribution

Table 1 Coefficients R20, R22, R40, RQ. and R+I in Equation (11 ) and the value of P, in Equation (9) determined from the longitudinal wave velocity measurements of Thill et a/.5 (km s-‘)

Rzo

R22

R40

R42

R44

V,

0.002581

-0.001588

- 0.000239

- 0.000162

0.000074

5.591

76

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Inversion of ultrasonic wave velocity measurements:

C. M. Sayers

detail could be obtained by including the terms WbOO, WA and WA,. However, this would require either more accurate measurements of the longitudinal velocity or, preferably, a measurement of the shear wave velocities.

Acknowledgement This Paper is published by permission of Shell Research BV.

References

Figure 5 Contoured equal area projection of the orientation distribution of microcrack normals calculated from Equation (15) using the values W200 = -0.0105 and L&c = 0.00646 obtained from the ultrasonic measurements of Thill et aL5 on Salisbury granite

in the generalized spherical harmonics, Z,,,(l), by Roe9 are included. K,,,, is given by 2n2n

i&;,,=(

defined 9

1

4972)-l sss 0 0 -1

5

W(~,~,~)zrm,(5)exp(im~)

x exp(in4)dt[dll/dq5

Therefore, W&,,represents the value of a polynomial of trigonometrical functions of 8, $ and 4 averaged over all crack orientations. The evaluation of a limited number of KM therefore, corresponds to the specification of the distribution function by its first few moments9 and represents the maximum information that can be obtained using ultrasonic waves with wavelengths greater than the crack size. However, for many purposes this will be sufficient. For example, in the Voigt approximation, in which the angular dependence of a physical property described by a tensor of order p is given by an average over crack orientations, only those coefficients w,,,, for 1

10 11 12 13 14 15

16 17

18 19

20

Birch, F. The velocity of compressional waves in rocks to 10 kilobars. Part 1 J Geophys Res (1960) 65 1083-1102 Birch, F. The velocity of compressional waves in rocks to 10 kilobars. Part 2 J Geophys Res (1961) 66 2199-2224 Anderson, D.L., Minster, B. and Cole, D. The effect of orientated cracks on seismic velocities J Geophys Res (1974) 79 4011-4015 Eshelbv. J.D. The determination of the elastic field of an ellinsoidal inch&n and related problems Proc Roy Sot Ser A (19<7) 241 376-396 Tbill, R.E., Willard, R.J. and Bur, T.R. Correlation of longitudinal velocity variation with rock fabric J Geophys Res (1969) 74 4897-4909 Morris, P.R. Averaging fourth-rank tensors with weight functions J Appl Phys (1969) 40 447-448 Sayers, C.M. Seismic anisotropy in the upper mantle Geophys J Roy Astr Sot (1987) 88 417-424 Roe, R.J. and Krighaum, W.R. Description of crystallite orientation in polycrystalline materials having fibre texture J Chem Phys (1964) 40 2608-2615 Roe, R.J. Description of crystalline orientation in polycrystalline materials - III: General solution to pole figure inversion J Appl Phys (1965) 36 2024-2031 Roe, R.J. Inversion of pole figures for materials having cubic crystal symmetry J Appl Phys (1966) 37 2069-2072 O’Connell, R.J. and Budiansky, B. Seismic velocities in dry and saturated cracked solids J Geophys Res (1974) 79 5412-5426 Sayers, C.M. Angular dependent ultrasonic wave velocities in aggregates of hexagonal crystals UItrnsonics (1986) 24 289-291 Backus, G.E. A geometrical picture of anisotropic elastic tensors Rev Geophys Space Phys (1970) 8 633-671 Jeffreys, H. Small corrections in the theory of surface waves Geophys J Roy Astr Sot (1961) 6 115-117 Smith, M.L. and Dahlen, F.A. The azimuthal dependence of Love and Rayleigh wave propagation in a slightly anisotropic material J Geophys Res (1973) 78 3321-3333 Aki, K. and Richards, P.G. Quantitative seismoloyy. Theory and Methods Vol 1, W.H. Freeman and Company, UK (1980) Sayers, C.M. Angular dependence of the Rayleigh wave velocity in polycrystalline metals with small anisotropy Proc Roy Sot A (1985) 400 175-182 Sayers, C.M. Ultrasonic velocities in anisotropic polycrystalline aggregates J Phys D (1982) 15 2157-2167 Hudson, J.A. Wave speeds and attenuation of elastic waves in material containing cracks Geophys J Roy Astr Sot (1981) 64 133-150 Simmons, G. Velocity of shear waves in rocks to 10 kilobars J Geophys Res (1964) 66 1123-1131

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