GEOPHYSICS, VOL. 64, NO.1 (JANUARY-FEBRUARY 1999); P. 93-98,9 FIGS.

Stress-dependent seismic anisotropy of shales

Colin M. Sayers* as illustrated in Figure 1 (Swan et al., 1989; Hornby et al., 1994; Schoenberg et al., 1996). The normal n to the contacts between clay platelets varies from domain to domain, with a preferred orientation resulting from the depositional and stress history of the rock. Thus, the clay particles vary in orientation but are aligned locally (Swan et al., 1989; Hornby et al., 1994; Schoenberg et al., 1996). A simple model of the stress-dependent elastic stiffness of shales having a domain structure is presented in which the elastic stiffness tensor is expressed in terms of the elastic stiffness of the shale at high confining stress and second-rank and fourthrank tensors that depend on the orientation distribution of contacts between clay platelets. The theory allows the normal and shear stiffness of the contact regions between clay platelets to be obtained as a function of stress from measurements of the ultrasonic compressional- and shear-wave velocities for shales.

ABSTRACT

A simple theory for the stress-dependent seismic anisotropy of shales can be obtained in terms of a secondrank tensor and a fourth-rank tensor that depend on the orientation distribution of contacts between clay platelets. The theory allows the normal and shear stiffness of the contact regions between clay platelets to be obtained as a function of stress from measurements of seismic P- and S-wave velocities for shales. The ratio of the normal-to-shear compliance, B N /B T , of the contact regions between clay particles is found to be sensitive to the saturation state of the shale. Inversion of velocity measurements for fully saturated shales shows a low value of BNIBT when compared with measurements on air-dry shales, consistent with the expected reduction in normal compliance in a fluid-saturated, lowpermeability rock. For all shales considered, B N /B T is found to be less than unity. The contacts between clay particles are therefore more compliant in shear than in compression.

THEORETICAL MODEL

It is assumed the stress dependence of the elastic properties of a shale is because of deformation of the contact regions between clay platelets. At high confining stress, these contacts are assumed to close so that the shale may be treated as a homogeneous anisotropic elastic medium with elastic stiffness tensor C° I and elastic compliance tensor S° ki . At intermediate values of the stress, it is assumed the contact regions between clay particles will be partially open (see Figure 1). In Appendix A, the elastic compliance of the shale is

INTRODUCTION

Shales form a major component of sedimentary basins (Jones and Wang, 1981) and play an important role in fluid flow and seismic wave propagation because of their low permeability and anisotropic microstructure. For example, Banik (1984) studied 21 data sets from the North Sea and found an excellent correlation between the occurrence of depth errors obtained from surface seismic data and the presence of shales in the subsurface. Failure to account for anisotropy in seismic processing may also lead to errors in normal moveout (NMO) correction, dip moveout (DMO) correction, migration, and amplitude versus offset (AVO) analysis. The seismic anisotropy of shales results from a partial alignment of plate-like clay minerals (Kaarsberg, 1959; Tosaya, 1982; Sayers, 1994). Slow sedimentation of clay minerals from dilute suspension favors faceto-face aggregation to form domains of parallel clay platelets,

= S° kl + ASijki, (1) where the excess compliance, ASijkt , because of the contact Sijki

regions between clay platelets can be written as

S

ASijkl ='( 8 ika'jl + Sila'jk + jkail + SJIU'ik) + iijkl• ( 2

)

Here, a il is a second-rank tensor and , ijkl is a fourth-rank tensor defined by aij

=V

E BT n, ri

rl n jr) A lr) (3)

r

Manuscript received by the Editor July 7, 1997; revised manuscript received May 21, 1998. *Schlumberger Cambridge Res. Ltd., High Cross, Madingley Road, Cambridge CB3 OEL, England; E-mail: [email protected]. © 1999 Society of Exploration Geophysicists. All rights reserved. 93

Downloaded 25 Dec 2010 to 199.6.131.16. Redistribution subject to SEG license or copyright; see Terms of Use at http://segdl.org/

94



Colin M. Sayers

and q

= V E

1

ijkl N

( B (r) N

B (r)) n (r) n (r } n (r) n(r)A(rl T

i 7

(4)

k l

r

where and B (r) are the normal and shear compliances of the rth contact (see Figure 1 and Appendix A), n the ith component of the normal to the contact, and A is the area of the contact plane. BN' characterizes the displacement discontinuity normal to the contact produced by a normal traction, while BT' S characterizes the shear displacement discontinuity produced by a shear traction applied at the contact. BTr) is assumed to be independent of the direction of the shear traction within the contact plane. Note that a, and Pijki are symmetric with respect to all rearrangements of the indices so that, for example, $1122 = $1212, $1133 = $1313, etc. If BN = BT for all contacts, OS;jkl is completely determined by the second-rank tensor i3 . (

r

)

)

a

INVERSION OF ULTRASONIC VELOCITY MEASUREMENTS

The components a and $i jkl of the second- and fourthrank tensors introduced above may be obtained from equation (2) by inverting elastic stiffnesses obtained from measured P- and S-wave velocities. The shales considered are transversely isotropic (Hornby, 1994; Johnston and Christensen, 1993; Vernik, 1993). If the symmetry axis is chosen to lie along x3i the nonvanishing AS ;Jki obtained from equation (2) are given by equations (A-14)-(A-19) of Appendix A. For the analysis of the experimental data, it is more convenient to use the conventional (two-subscript) condensed 6 x 6 matrix notation in which 11-31, 22-* 2, 33-* 3, 23-* 4, 13 -> 5, and 12-> 6 while factors 2 and 4 are introduced as follows (Nye, 1985): ;;

Sijkl > S pq when both p, q are 1, 2, or 3; SPq when one of p, q are 4, 5, or 6;

2Sijkl

4Sijkl _> Spq when both p, q are 4, 5, or 6.

Factors 2 and 4 are absent in the condensation of the stiffnesses tensor components so that Cijkl ± C pq (i, j, k, t = 1, 2, 3;

S(r)

n

p, q = 1, ... , 6)

(r)

FIG. 1. A region within the shale, showing a local alignment of

clay platelets.

Hornby (1994) measured ultrasonic wave velocities for two fully saturated shales of Jurassic age. Figure 2 shows the elastic stiffnesses calculated by Hornby (1994) from the measured ultrasonic wave velocities using a reference set of axes 0x1x 2 x 3 with Ox 3 perpendicular to the bedding plane. These values were inverted to obtain the elastic compliance components S ij plotted in Figure 3. Figure 4 shows the components of the second- and fourthrank tensors a 11 and $ ijkl obtained from S ij plotted in Figure 3 by using equation (2) and assuming the variation in the elastic stiffnesses of the clay particles with stress can be neglected. The elastic compliances S, corresponding to the case when the contacts are closed, were taken to be the values corresponding to the highest stress used in the experiment. These values are S I = 29.3, S33 = 45.1, S° _ -13.1, S55 112.4, and S66 = 70.4 TPa -1 for Figure 4a and S 1 = 23.1, S 3 = 39.9, S13 = - 11.0, S°5 97.1, and S 66 = 53.2 TPa -1 for Figure 4b. The components of the second- and fourth-rank tensors shown therefore correspond to the change in grain contacts between the lowest stress used (5 MPa) and the highest (80 MPa). Figure 4 shows that all components of a 1j and i; jkl except a33 and ,83333 are small. It follows from equations (3) and (4) that most contacts are aligned normal to the symmetry axis Ox3 The opposite sign of a 33 and X3333 implies that B N /B T <1 [see equations (3) and (4)]. The contacts between clay particles are therefore more compliant in shear than in compression. If the contacts between clay particles are assumed to be perpendicular to the symmetry axis, a comparison of equations (3) and (4) with the values of a 33 and X3333 plotted in Figure 4 allows the ratio B N /B T of normal-to-shear compliance of the contact regions to be estimated for each value of the confining stress. Between 5 and 60 MPa confining stress, B N /B T takes values in the range 0.26 < B N /B T < 0.30 for Figure 4a, with an average value of 0.29. For Figure 4b the values lie in the range 0.33 < B N /B T < 0.41, with an average value of 0.37. It is interesting to compare these results with those obtained for shales under air-dry conditions. Johnston and Christensen (1993) calculate the elastic stiffnesses for airdry shales from the Millboro and Braillier members of the Devonian-Mississippian Chattanooga Formation from their ultrasonic velocity measurements (see Figure 5). These values were inverted to obtain the elastic compliance components S ij shown in Figure 6. The components of the second- and fourthrank tensors a;j and 6iikt were then obtained by using equation (2). The results are shown in Figure 7. B N /B T was estimated for these shales from the values of a 33 and X3333 plotted in Figure 7 by assuming the grain contacts to be parallel to the bedding plane. Between 20 and 100 MPa confining stress, B N /B T takes values in the range 0.47 < B N /B T < 0.58 for the Millboro sample, with an average value of 0.52. For the Brailler sample the values lie in the range 0.54 < B N /B T < 0.63, with an average value of 0.58. The contacts between clay particles are again more compliant in shear than in compression, although the ratio is increased compared to that for fluid-saturated shales. This difference is consistent with the expected reduction in normal compliance in a fluidsaturated, low-permeability rock. Vernik (1993) measured ultrasonic velocities for an airdry mature, kerogen-rich shale; the elastic stiffnesseses and compliances obtained from these measurements are shown in

° =

°

Downloaded 25 Dec 2010 to 199.6.131.16. Redistribution subject to SEG license or copyright; see Terms of Use at http://segdl.org/

°

=

.

Anisotropy of Shales

Figure 8. This sample contained bedding-parallel microcracks caused by the process of hydrocarbon generation. Figure 9 shows the components of the second- and fourth-rank tensors a ;j and 8ijkl obtained from S;j plotted in Figure 8 by using equation (2). The ratio B N /B T was estimated from the values of a33 and ,83333 plotted by assuming the grain contacts and microcracks to be parallel to the bedding plane. BN /B T takes values between 5 and 30 MPa confining stress in the range 0.6 < B N /B T < 0.8, with an average value of 0.68. CONCLUSION A simple theory has been presented in which the stressdependent seismic anisotropy of shales is obtained in terms of components a, j and ,& jkl of a second-rank and a fourth-rank tensor that depend on the orientation distribution of contacts

95

between clay platelets. The theory allows the normal and shear stiffnesses of the contact regions between clay platelets to be obtained as a function of stress from measurements of seismic P- and S-wave velocities for shales. For the shales considered, all components of a ;, and ,8 i jkt except a 33 and ,83333 are small. It follows, from equations (3) and (4) that most contacts are aligned normal to the symmetry axis. The opposite sign of a33 and f33 33 implies that RN/ B r <1 [see equations (3) and (4)]. The contacts between clay particles are therefore more compliant in shear than in compression. The ratio BN /B T of the normal-to-shear compliance of the contact regions between clay particles is sensitive to the saturation state of the shale. Inversion of measurements of Hornby (1994) for fully saturated samples shows a reduced value of B N /B T compared to air-dry shales, consistent with the expected reduction in normal compliance in a fluid-saturated, low-permeability rock.

60

50

250

(b) • . . • (a) • (a) 50 40

•

• '^

•



n

30 a. 020

0

40

0

0 0 o 0

❑ 10

°o°

0 °

rCa n



n n

0 50 i• 0

20

n n • n

10

50 0 ❑❑ ❑ ❑

0

0

0



I

I

0 1

-50 0 20 40 60 80 100

0 20 40 60 80 100

FIG. 2. Elastic stiffnesses C11(.), C33(0), C 32 (x), C13(+), C5 5(n), and C66( ❑) for the shales studied by Hornby (1994).

0 0 0

25

• • • • •

0

xx x x

-25 0 20 40 60 80 100

(MPa) Stress (MPa)

Stress Stress (MPa)

Stress (MPa)

FIG. 3. Elastic compliances S11(•), 533(0), S 12 (x ), S13(+), S55(fi), and S66 (❑) for the shales studied by Hornby (1994).

60

60

nn • •

75

X100 o0 0 ❑ ❑ ❑

0 1I I I I 0 20 40 60 80 100

IOU

100-

x n

n

150

30 oo 0

* X

(b)

125

200

• •

150

(a)

VV

50



. • 50

40 0 100 r(6 I-

a.

( 30

0

U

10

V C co 0 L3

m

0

20

50

40

40

30

0

.30

0

0

0 20 -0^ ❑ ❑

20

0 -10

10

-20

0



10

-50 I I I 1 I 0 50 100 150 200 250

an (MPa) 0 1020304050607080

-100 Stress 0 1020304050607080 -

Stress (MPa)

Stress (MPa)

FIG. 4. Components a33(•), a11(0), 133333(•), $1133( ❑) , and P1111(+) of the second- and fourth-rank tensors a,, and ,8i jkl for the shales studied by Hornby (1994).

0

1

0 50 100 150 200 250

Stress (MPa)

FIG. 5. Elastic stiffnesses C11(.), C 33 (o), CIO), C1 3 (+), C55 (•), and C66 (❑) of a shale from (a) the Millboro member and (b) the Braillier member of the Devonian—Mississippian Chattanooga Formation studied by Johnston and Christensen (1993).

Downloaded 25 Dec 2010 to 199.6.131.16. Redistribution subject to SEG license or copyright; see Terms of Use at http://segdl.org/

Colin M. Sayers

96 140

140

I

3U

I (b)

(a)

25

120

120

4U

30 20

100

100

r.. 80

..80

a 20

~ 15 e2 10

CO

10

V

60

I-

C 5 40

U 40

20

20

0

0

0

0 -5

-10

-10 _i c

-20

20 0 50 100 150 200 250

0 50 100 150 200 250

Stress (MPa)

Stress (MPa)

FIG. 6. Elastic compliances S11(.), S33(0), S12 (x), S13(+), S55( •), and S66( ❑) of a shale from (a) the Millboro member and (b) the Braillier member of the Devonian—Mississippian Chattanooga Formation studied by Johnston and Christensen (1993).

60

40 80 120 160

•

•

•

40 80 120 160

0

Stress (MPa)

Stress (MPa)

FIG. 7. Components a33(S), a'll(a), 33333(1), 01133( 13 ). and

11111(+) of the second- and fourth-rank tensors cr,, and Aikl for (a) the Millboro member and (b) the Braillier member of the Devonian—Mississippian Chattanooga Formation studied by Johnston and Christensen (1993).

100

(a)

CO

-20 0

40

• n

•

(b) nn

50

80

40

60 0

30

•

CO d 0



0

a-

CO

0 0

930

U

0

20 ❑ ❑ ❑ ❑ ❑

F" 20 •

❑8 ❑ ❑ ❑ ❑

r

40 07 20

0 0

• .s,

10 o

0

•

•

•

CO

0

o a 13

10

0

0 1020304050607080

-20 1

-^ 0 1020304050607080

Stress (MPa)

I

I

I

1

0 10 20 30 40 50 60 70

Stress (MPa)

FIG. 8. (a) Elastic stiffnesses C 1 1(!), C33(0), C 12 (x), C13(+), C55(n), and C 66 (❑) and (b) elastic compliances SIl(S), S33(0), S12 ( x), S13(+), S55 (•), and S66 (D) for a sample of mature kerogen-rich shale with bedding parallel cracks studied by Vernik

n

n

-10

0 xxx+

13

Stress (MPa)

FIG. 9. Components c33(•), a11(0) , f33333 (M), and fi1133(❑), and thin (+) of the second- and fourth-rank tensors a, and i8ijkl for a sample of mature, kerogen-rich shale with bedding parallel cracks studied by Vernik (1993).

(1993).

ACKNOWLEDGMENTS

I thank Brian Hornby, Joel Johnston, and Lev Vernik for providing the velocity measurements used in this work. REFERENCES

Banik, N. C., 1984, Velocity anisotropy in shales and depth estimation in the North Sea basin: Geophysics, 49,1411-1419. Hill, R., 1963, Elastic properties of reinforced solids: Some theoretical principles: J. Mech. Phys. Solids, 11, 357-372. Hornby, B. E., 1994, The elastic properties of shales: Ph.D. thesis, Univ. of Cambridge. Hornby, B. E., Schwartz, L. M., and Hudson, J. A., 1994, Anisotropic effective medium modeling of the elastic properties of shales: Geo-

physics, 59,1570-1583. Johnston, J. E., and Christensen, N. I., 1993, Compressional to shear velocity ratios in sedimentary rocks: Int. J. Rocks Mech., 30, 751754.

Jones, L. E. A., and Wang, H. F., 1981, Ultrasonic velocities in Cretaceous shales from the Williston basin: Geophysics, 46, 288-297. Kaarsberg, E. A., 1959, Introductory studies of natural and artificial argillaceous aggregates by sound propagation and X-ray diffraction methods: J. Geol., 67,447-472. Kachanov, M., 1992, Effective elastic properties of cracked solids: Critical review of some basic concepts: Appl. Mech. Rev., 45, 304335.

Nye, J. F., Physical properties of crystals: Oxford Univ. Press, Inc. Sayers, C. M., 1994, The elastic anisotropy of shales: J. Geophys. Res. B, 99, 767-774.

Sayers, C. M., and Kachanov, M., 1991, A simple technique for finding effective elastic constants of cracked solids for arbitrary crack

Downloaded 25 Dec 2010 to 199.6.131.16. Redistribution subject to SEG license or copyright; see Terms of Use at http://segdl.org/

Anisotropy of Shales

orientation statistics: Internat. J. Solids Struct., 12, 81-97. 1995, Microcrack-induced elastic wave anisotropy of brittle rocks: J. Geophys. Res. B, 100, 4149-4156. Schoenberg, M., Muir, F., and Sayers, C. M., 1996, Introducing ANNIE: A simple three-parameter anisotropic velocity model for shales: J. Seis. Expl., 5, 35-49.



97

Swan, G., Cook, J., Bruce, S., and Meehan, R., 1989, Strain rate effects in Kimmeridge Bay shale: Int. J. Rock Mech., 26,135-149. Tosaya, C. A., 1982, Acoustical properties of clay-bearing rocks: Ph.D. thesis, Stanford Univ. Vernik, L., 1993, Microcrack-induced versus intrinsic elastic anisotropy in mature HC-source rocks: Geophysics, 58, 1703-1706.

APPENDIX A EXCESS COMPLIANCE FROM CONTACT REGIONS BETWEEN CLAY PLATELETS

It is assumed that the stress dependence of the elastic properties of a shale is because of deformation of the contact regions between clay platelets (see Figure 1). Consider a volume V containing N contact regions (with surfaces S ( r ) , r = 1, ... , N). The strain tensor c ij is defined in terms of the displacement vector u i by 1 au i au j (A-i) E{j = – ax i

2 ax j +

Using Gauss's theorem, the average strain tensor in the solid phase in volume V may be written (Hill, 1963) as r

S (r)

S

)

J

(A-2) where n, is the ith component of the outward normal to the solid, Vs is the volume of the solid phase, and E ij is defined by 1 /' = — (u i n j + u j n i ) dS (A-3)

J 2V S

(

(uin+ujni)dS,

E i1dV =Eij+^f —

Vs2Vs fv,

expressed in terms of the average displacement discontinuities [u i ] ( r ) _ (1/A ' ) f A( r ) [u i ] dA. The problem is reduced to finding [u ; ] (r ) in terms of the applied stresses. It is assumed that the stress interactions between contacts may be neglected so each contact region may be considered as subjected to the average stress field d ;3 . Introducing a symmetric second-rank compliance tensor B (Kachanov, 1992) that expresses the vector of average displacement discontinuity in terms of the uniform traction, with ith component t i , applied at the faces of the contact region between clay platelets gives, for the rth contact, [uiJ (r) = BiJ^t1 ,

(A - 6)

ti = Qjknk.

(A-7)

where tj is given by

e

The change in compliance because of the presence of the

(Sayers and Kachanov, 1991,1995). Here, Se is the solid portion contacts is then obtained from equation (A-5) as of the exterior boundary. The macroscopic strains E, j are related to the macroscopic 1 ASijk16k1 2 V ([ui]nj + [uj]ni)dA stress components d ij [defined by dij = (1/ V) J, of j dV ] by= r) r the effective compliance tensor S,Jkl :

L

(A-4)

= Siik1 &kl •

= The strain Eij in the solid phase is given by E, j = S°klokl, where S° kl is the compliance tensor of the solid. For thin contact regions, the integral on the right side of equation (A-2) may be evaluated by taking the integration over the area A r of the contact and replacing displacements by displacement jumps across A(r) while putting Vs = V. This gives (

)

1N

j7r (B^p ) apg nqr l nor ) + B^ p1 Qpg nqr l n^ rl) Al'l. (A-8)

The shear compliance of the contact regions is assumed independent of direction in the plane of the contact so that Bij = BNninj +BT(Sij — ninj).

(A-9)

= Sgki^kl + Z V E fA(r) ([ui]n^^l + [uj]n r ^/ dA r=1

1

It follows that

_ (s g, + ASiik1)dkl

(A-5)

(r) (

(r^ (r) (r^

^ r ))

— 2 V BT °ignq nj ^ ajgnq ni (Sayers and Kachanov, 1995). Here [u ; ] denotes the ith comr ponent of the displacement discontinuity at the contact and n i is the ith component of the unit normal to the contact. The +2(BNr) — B Tr) )ang n^ r1 n^r ) npr inqr l^Al r l. first term in equation (A-5) is from the deformation of the anisotropic matrix (S° kl are the matrix compliances), and the (A-10) second term reflects the additional compliance because of the contact regions between clay platelets. To obtain the individual components OS^ Jk/ , consider a test It is assumed that the contacts are flat with unit normals stress n( r) that are constant along each A r . The value n;r) may then be taken out of the integral on the right side of equa(A-11) Qi; _ (Sik 8 il + Siksil). tion (A-5), and the extra compliance from contacts may be

OSijk1011

(

-

)

Downloaded 25 Dec 2010 to 199.6.131.16. Redistribution subject to SEG license or copyright; see Terms of Use at http://segdl.org/

98

Colin M. Sayers

This gives [

BTr (

)

(bikn^')n(P) +

ASijkl = 1

This equation quantifies the excess compliance from the contacts between clay particles in terms of the second- and fourthrank tensors a ll and ,8ijkl. For a general transversely isotropic orientation distribution of contacts between clay platelets, all = a22, ,Bull =,82222, fii212 = fii122 =thlli/3, and equation (A-13) reduces to

(r) (r)

P

+ S jkn,' ) ni` ) + 3 jln, ) nk

j

AS1111 = AS2222 = all + P1111,

(A-14)

AS3333 = a33 + p3333,

(A-15)

AS1212 = all/ 2 + p1111/ 3 ,

(A-16)

(

+ (BNP) — BTP) )n ' ) n^' ) n kP) n (' ) ]A ( P ) (A-12) upon substituting in equation (A-10) (Sayers and Kachanov, 1995).

Defining a second-rank tensor a i j and a fourth-rank tensor by equations (3) and (4) in the text, it then follows from equation (A-12) that

AS2323 = AS3131 = (U'11 + a33)/4 + 11133, (A-17)

18 1jk1

ASijkl =

1 (Sikajl + silajk + 6 jk 1 il + Sjlaik) + 8ijkl•

(A-18)

A 52233 = A 53311 = 11133 •

(A-19)

and

(A-13)

AS1122 = thin/3,



Downloaded 25 Dec 2010 to 199.6.131.16. Redistribution subject to SEG license or copyright; see Terms of Use at http://segdl.org/