Fracture-Induced Anisotropic Attenuation

Transcription

1 Rock Mech Rock Eng DOI.7/s y ORIGIAL PAPER Fracture-Induced Anisotroic Attenuation José M. arcione Juan E. Santos Stefano Picotti Received: 29 July 2 / Acceted: 28 February 22 Ó Sringer-Verlag 22 Abstract The triaxial nature of the tectonic stress in the earth s crust favors the aearance of vertical fractures. The resulting rheology is usually effective anisotroy with orthorhombic and monoclinic symmetries. In addition, the resence of fluids leads to azimuthally varying attenuation of seismic waves. A dense set of fractures embedded in a background medium enhances anisotroy and rock comliance. Fractures are modeled as boundary discontinuities in the dislacement u and article velocity v as ½j u þ g vš; where the brackets denote discontinuities across the fracture surface, j is a fracture stiffness, and g is a viscosity related to the energy loss. We consider a transversely isotroic background medium (e.g., thin horizontal lane layers), with sets of long vertical fractures. Schoenberg and Muir s theory combines the background medium and sets of vertical fractures to rovide the 3 comlex stiffnesses of the long-wavelength equivalent monoclinic and viscoelastic medium. Long-wavelength J. M. arcione (&) S. Picotti Istituto azionale di Oceanografia e di Geofisica Serimentale (OGS), Borgo Grotta Gigante 42c, 34 Sgonico, Trieste, Italy jcarcione@inogs.it equivalent means that the dominant wavelength of the signal is much longer than the fracture sacing. The symmetry lane is the horizontal lane. The equations for orthorhombic and transversely isotroic media follow as articular cases. We comute the comlex velocities of the medium as a function of frequency and roagation direction, which rovide the hase velocities, energy velocities (wavefronts), and quality factors. The effective medium ranges from monoclinic symmetry to hexagonal (transversely isotroic) symmetry from the low- to the high-frequency limits in the case of a article velocity discontinuity (lossy case) and the attenuation shows tyical Zener relaxation eaks as a function of frequency. The attenuation of the couled waves may show imortant differences when comuted versus the ray or hase angles, with trilication aearing in the Q factor of the qs wave. We have erformed a full-wave simulation to comute the field corresonding to the couled qp qs waves in the symmetry lane of an effective monoclinic medium. The simulations agree with the redictions of the lane-wave analysis. Keywords Fractures Anisotroy Attenuation Schoenberg Muir theory Boundary conditions J. E. Santos OIET, Instituto del Gas y del Petróleo, Facultad de Ingeniería, Universidad de Buenos Aires, 27AAR Buenos Aires, Argentina santos@math.urdue.edu J. E. Santos Universidad acional de La Plata, La Plata, Argentina J. E. Santos Deartment of Mathematics, Purdue University, W. Lafayette, I , USA List of Symbols v u x Z = ðj þ ixgþ r e I, e ij, I =,, 6, i, j =,, 3 c IJ, I, J =,, 6 Particle-velocity vector Dislacement vector Angular frequency Fracture comliance matrix Stress tensor Strain comonents Elasticity comonents

2 J. M. arcione et al. s IJ, I, J =,, 6 omliance comonents b Angle between the fracture lane and the y-axis P Stiffness tensor q Mass density k, l, E = k? 2l Lamé constants Kelvin hristoffel matrix v Phase velocity Q Quality factor v e Energy velocity h Phase velocity angle w Energy-velocity angle Introduction Wave roagation through fractures, faults, and cracks is an imortant subject in seismology, exloration geohysics, and mining. Faults in the earth s crust constitute sources of earthquakes (Pyrak-olte et al. 99) and hydrocarbon and geothermal reservoirs are mainly comosed of fractured rocks (akagawa and Myer 29). Alications in geotechnical engineering, such as analysis of the dynamic stability of rock sloes and tunnels, involve the study of imerfect joints in rock masses (Perino et al. 2; Fan et al. 2). In geohysical rosecting, knowledge of reservoirs fracture orientations, densities, and sizes is essential, since these factors control hydrocarbon roduction (Hansen 22; Hall and Kendall 23; Grechka and Tsvankin 23; Barton 27). The analysis of the data exloits the fact that seismic velocity and attenuation anisotroy due to the resence of fractures are sensitive to key roerties of the reservoir, such as orosity, ermeability, and fluid tye. A few frequency-deendent models have been develoed to describe anisotroy and attenuation. arcione (992) generalized Backus averaging to the anelastic case, obtaining the first model for Q-anisotroy (see arcione 27). Analyses on sequences of sandstone limestone and shale limestone with different degrees of anisotroy indicate that the quality factors (Q) of the shear modes are more anisotroic than the corresonding hase velocities, cuss of the qsv mode are more ronounced for low frequencies and midrange roortions, and, in general, attenuation is higher in the direction erendicular to layering or close to it, rovided that the material with lower velocity is the more dissiative. This model has been further analyzed by Picotti et al. (2), who shown how to obtain the medium roerties with quasi-static numerical exeriments. Other alternative models of Q-anisotroy were roosed by arcione and avallini (994) and arcione et al. (998). A brief descrition of all these henomenological models can be found in arcione (27). Zhu and Tsvankin (27) analyze in detail the attenuation in orthorhombic media, assuming homogeneous viscoelastic waves. They simlify the interretation for rocessing uroses by introducing a set of attenuation anisotroy arameters. A Backus tye model to describe wave roagation in fractures has been introduced by arcione (996a), where lane layers are searated by thin continuous layers of viscous fluid. A similar model is considered by Liu et al. (2), where the fracture is a very thin soft viscous layer. On the other hand, a recently develoed model (haman 23; Maultzsch 25) exlicitly describes the effects of cracks and fractures on wave roagation, since the elastic constants are derived in terms of microstructural arameters and, therefore, the model is redictive. It describes attenuation and velocity disersion at seismic frequencies and redicts how these effects are related to the fluid tye and size of the fractures. An aroach to exlicitly model cracks and fractures is roosed by Zhang and Gao (29). Their scheme treats the fractures as nonwelded interfaces that satisfy the linear-sli dislacement discontinuity conditions instead of using equivalent medium theories. Hence, the algorithm can be used to characterize the seismic resonse of fractured media and to test equivalent medium theories. Other oroelasticity models describing anisotroic attenuation are given by Krzikalla and Müller (2) and arcione et al. (2), who obtain the five comlex and frequency-deendent stiffnesses of an equivalent medium corresonding to thin oroelastic layers. Modeling fractures requires a suitable interface model for describing the dynamic resonse of the joint. Theories that consider imerfect contact were mainly based on the dislacement discontinuity model at the interface. Pyrak- olte et al. (99) roosed a non-welded interface model based on the discontinuity of the dislacement and the article velocity across the interface. The stress comonents are roortional to the dislacement and velocity discontinuities through the secific stiffnesses and one secific viscosity, resectively. Dislacement discontinuities conserve energy and yield frequency-deendent reflection and transmission coefficients. On the other hand, velocity discontinuities generate energy loss at the interface. The secific viscosity accounts for the resence of a liquid under saturated conditions. The liquid introduces a viscous couling between the two surfaces of the fracture (Schoenberg 98) and enhances energy transmission, but, at the same time, this is reduced by viscous losses. The model may account for sli and dilatancy effects. hichinina et al. (29a, b) describe anisotroic attenuation in a transversely isotroic (TI) medium using Schoenberg s

3 Fracture-Induced Anisotroy linear-sli model with comlex-valued normal and tangential fracture stiffnesses. The theory and laboratory exeriments show that, in the vicinity of the symmetry axis, P-wave attenuation is comarable to S-wave attenuation when the fracture is filled with a fluid. On the other hand, in the resence of dry fractures, P-wave attenuation is much greater than S-wave attenuation. In this work, we generalize the orthorhombic model given by Schoenberg and Helbig (997) to the anelastic monoclinic case, by introducing a article-velocity discontinuity in the fracture surface, allowing us to describe Q-anisotroy. The medium consists of sets of vertical fractures embedded in a TI background medium (generally, horizontal fine layering) to form a long-wavelength equivalent monoclinic medium. Using the theory of Schoenberg and Muir (989), we obtain the 3 comlex and frequency-deendent stiffnesses of this medium. We then obtain the quality factors and wave velocities as a function of frequency and roagation angle. 2 Interface Model Let us consider a lanar fracture. The non-ideal characteristics of the interface are modeled by imosing suitable boundary conditions. The model roosed here is based on the discontinuity of the dislacement and article-velocity fields across the interface. Then, the boundary conditions at the interface are j ½uŠþg ½vŠ ¼r n; ðþ (Pyrak-olte et al. 99; arcione 996b), where u and v are the dislacement and article-velocity comonents, resectively, r is the stress tensor, n is the unit normal to the fracture, j is the secific stiffness matrix, and g is the secific viscosity matrix (both of dimensions 3 9 3). They have dimensions of stiffness and viscosity er unit length, resectively. Moreover, the symbol indicates scalar roduct and the brackets denote discontinuities across the interface, such that for a field variable /, it is ½/Š ¼/ 2 / ; where and 2 indicate the two sides of the fracture. The article velocity is given by v ¼ _u; ð2þ where a dot above a variable indicates time differentiation. In the Fourier domain, v ¼ ixu; ð3þ where x is the angular frequency and i ¼ ffiffiffiffiffiffi : Equation () then becomes ½uŠ ¼Z ðrnþ; ð4þ where Z ¼ðjþixgÞ ð5þ is a fracture comliance matrix, whose dimension is length/ stress. This aroach is equivalent to the linear-sli model introduced by Schoenberg (98). In fact, Eq. (5), with g ¼ ; is given in oates and Schoenberg (995) and Schoenberg and Helbig (997). Three models have been studied by Liu et al. (2) to obtain the exression of Z for different fracture models. Two of the models only rovide the real art of Z.The third model describes the fracture as a thin and soft viscoelastic layer embedded in an isotroic elastic background medium, where Z can be obtained in terms of the thickness, Lamé constants, and viscosity of the soft material infill. The resulting equivalent medium has TI symmetry. Similarly, arcione (996a) described the infill by a viscous fluid. The comliance matrix Z of the set of fractures is the diagonal non-negative matrix Z Z Z 2 A Z 3 ðj þ ixg Þ ðj 2 þ ixg 2 Þ A; ðj 3 þ ixg 3 Þ ð6þ where Z is the normal comliance, Z 2 is the horizontal tangential comliance, and Z 3 is the vertical tangential comliance. The fact that Z 2 6¼ Z 3 means that the texture of the fracture surface has different roughnesses vertically and horizontally, while the fact that there are no off-diagonal comonents means that, across the fractures, the normal motion is uncouled from the tangential motion. A common situation in the earth s crust is to have a finely layered medium and vertical fractures. Figure shows such a case, where q = 2 sets of long vertical fractures are embedded in a TI medium with a vertical symmetry. This background medium is the long-wavelength equivalent of the finely layered medium, according to Backus (962). The mechanical (viscoelastic) reresentation of the fracture boundary condition by a Kelvin Voigt model is illustrated in the figure, according to the stress-dislacement relation r n ¼ðjþixgÞ½uŠ; ð7þ where the model simulates the fracture by a zero-width layer of distributed srings and dashots. The quantity j þ ixg is the comlex modulus er unit length of the Kelvin Voigt element (e.g., arcione 27). A dislacement discontinuity yields comliance, while a discontinuity in the article velocity imlies an energy loss at

4 J. M. arcione et al. β y n ρ c c 33 c3 c 55 c 66 Rock κ η Fracture Rock where 2c 66 ¼ c c 2 ; r I denotes the stress comonent and e I denotes the strain comonent in the Voigt notation (e.g., arcione 27), such that z the interface (arcione et al. 996b, 998, 27); j ¼ gives the article-velocity discontinuity model and g ¼ gives the dislacement discontinuity model. On the other hand, if j!or g!; the model gives a welded interface. 2. The Equivalent Monoclinic Medium Let us consider a background TI medium. The stress strain relation is r r c c 2 c 3 r 2 r 22 c 2 c c 3 r 3 r 33 c r ¼ 3 c 3 c 33 4 r ¼ 23 c 55 B B r 5 r 3 c 55 A r 6 r 2 c 66 e e e 2 e 2 e 3 e 3 e 4 e ; ð8þ 4 B A e 5 y x Fig. Two sets of vertical fractures embedded in a transversely isotroic (TI) medium. In this case, transverse isotroy is due to fine layering. If the two sets are orthogonal, the equivalent medium is orthorhombic; otherwise, the symmetry is monoclinic with a horizontal single lane of symmetry. The boundary condition at the fracture describes an imerfect bonding in terms of the secific stiffness j and secific viscosity g: Angle b is measured from the y-axis towards the strike direction e 5 ½e ; e 2 ; e 3 ; e 4 ; e 5 ; e 6 Š > ¼½ ; 22 ; 33 ; 2 23 ; 2 3 ; 2 2 Š > ; ð9þ where ij are the strain comonents, c IJ are the elasticity constants, and is the elasticity matrix. The background medium can easily be generalized to a viscoelastic medium by using one of the three models of anisotroic anelasticity roosed in hater 4 of arcione (27). The generalization imlies that the elasticity constants c IJ become comlex and frequency deendent. Here, the urose is to analyze the attenuation due to the fracture solely; however, in the last examle, we consider an effective HTI (TI with a horizontal symmetry axis) medium to test the commonly used equation by which the total dissiation factor is equal to the dissiation factor of the background medium lus the dissiation factor due to the fractures. Schoenberg and Muir (989) showed how to combine arbitrary sets of elastic thin layers and find their longwavelength equivalent medium roerties. The method is alicable also when a set of layers is infinitely thin and comliant, as it is the fracture set shown in Fig.. ichols et al. (989) and Hood (99) simlified the rocedure to include fractures by using comliance matrices instead of stiffness matrices. As shown below, the comliance matrix of the equivalent medium is found by the addition of exanded versions of the fracture comliance matrices to the comliance matrix of the background medium. Let us consider many sets of vertical fractures, so that the fracture normal of a given set makes an angle b with the x-axis (see Fig. ). We avoid the fracture index (q) for simlicity. The exanded fracture comliance matrix of each set is s s 2 s 6 s 2 s 22 s 26 S f ¼ s 44 s 45 ; ðþ s 45 s 55 A s 6 s 26 s 66 where e 6 e 6

5 Fracture-Induced Anisotroy s ¼ 3Z þ Z H þ Z 8 2 cos 2b þ Z Z H cos 4b; 8 s 2 ¼ Z Z H ð cos 4bÞ; 8 s 6 ¼ Z sin 2b þ Z Z H sin 4b; 2 4 s 22 ¼ 3Z þ Z H 8 Z 2 cos 2b þ Z Z H cos 4b; 8 s 26 ¼ Z sin 2b Z Z H sin 4b; 2 4 s 44 ¼ Z Vð cos 2bÞ ; 2 s 45 ¼ Z V sin 2b ; 2 s 55 ¼ Z Vð þ cos 2bÞ ; 2 s 66 ¼ Z þ Z H 2 Z Z H 2 cos 4b ðþ (Schoenberg et al. 999). The comutation of the comliance matrix in Eq. () can be derived by carrying out the matrix multilication given by ichols et al. (989) or directly using 4th-rank tensor notation. We have introduced Z ¼ Z L ; Z H ¼ Z 2 L ; Z V ¼ Z 3 L ; ð2þ where L is a characteristic length, such that these quantities have dimensions of comliance, and j ¼ Lj ; j H ¼ Lj 2 ; j V ¼ Lj 3 ; ð3þ g ¼ Lg ; g H ¼ Lg 2 ; g V ¼ Lg 3 ; such that these quantities have dimensions of stiffness and viscosity, resectively. The quantity L, called h in Grechka et al. (23) (see their Eq. 3), is the average fracture sacing, which has to be much smaller than the dominant wavelength of the ulse. It is Z ¼ j þ ixg ; Z H ¼ j H þ ixg H ; Z V ¼ j V þ ixg V : ð4þ Then, the comlex and frequency-deendent stiffness matrix of the equivalent medium is given by " PðxÞ ¼ þ X # S ðqþ f ðxþ ; ð5þ q where the sum is over the q sets of fractures. The equivalent homogeneous anisotroic medium (in the long wavelength limit) is a monoclinic medium with a horizontal mirror lane of symmetry. Its comlex and frequency-deendent stiffness matrix has the form P ¼ : ð6þ A At zero frequency, we obtain the lossless case, where Z ¼ =j ; Z H ¼ =j H ; and Z V ¼ =j V ; and the medium remains monoclinic. At infinite frequency, the fracture stiffnesses vanish ðs ðqþ f! Þ and we obtain the TI and lossless background medium. By infinite frequency, we mean frequencies such that the wavelength is much larger than the distance between single fractures, i.e., the longwavelength aroximation. It can be shown that the wave velocities at zero frequency are smaller than the wave velocities at infinite frequency, i.e., the medium is more comliant. 2.2 Orthorhombic Equivalent Media Assume two fracture sets with b = (fracture strike along the y-direction) and b = /2 (fracture strike oints along the x-direction). The fracture stiffness matrices are given by S ðþ f ¼ Z ðþ Z ðþ V Z ðþ H A ð7þ and Z ð2þ S ð2þ f ¼ Z ð2þ V ; ð8þ A Z ð2þ H resectively. The evaluation of Eq. (5) requires simle oerations with matrices. We obtain

6 J. M. arcione et al. ¼ 2 ¼ 3 ¼ 22 ¼ 23 ¼ þ Z ðþ þ Z ðþ þ Z ðþ þ Z ðþ þ Z ðþ c þ Z ð2þ ðc2 c2 2 Þ ðc2 c2 2 Þþc ðz ðþ Zð2Þ Zð2Þ c 2 ðc2 c2 2 Þþc ðz ðþ c 3 ½ þ Z ð2þ ðc c 2 ÞŠ ðc2 c2 2 Þþc ðz ðþ Zð2Þ c þ Z ðþ ðc2 c2 2 Þ ðc2 c2 2 Þþc ðz ðþ Zð2Þ c 3 ½ þ Z ðþ ðc c 2 ÞŠ ðc2 c2 2 Þþc ðz ðþ Zð2Þ 33 ¼ c 33 þðz ðþ c 55 þ Zð2Þ þ Zð2Þ Þ ; þ Zð2Þ Þ ; þ Zð2Þ Þ ; þ Zð2Þ Þ ; þ Zð2Þ Þ ; Þðc c 33 c 2 3 ÞþZðÞ þ Z ðþ 44 ¼ ; þ c 55 Z ð2þ V c ¼ ; þ c 55 Z ðþ V c ¼ þ c 66 ðz ðþ H þ Zð2Þ H Þ : Zð2Þ Zð2Þ ðc2 c2 2 Þþc ðz ðþ ðc c 2 Þ½c 33 ðc þ c 2 Þ 2c 2 3 Š þ Zð2Þ Þ ; ð9þ Since the two sets are orthogonal, the equivalent medium has orthorhombic symmetry. In the case of a single fracture set, e.g., b =, Eq. (9) gives the stiffness matrix of Schoenberg and Helbig (997): 2.3 HTI Equivalent Media c ð d Þ c 2 ð d Þ c 3 ð d Þ c 2 ð d Þ c ð d c 2 2 =c2 Þ c 3ð d c 2 =c Þ c P ¼ 3 ð d Þ c 3 ð d c 2 =c Þ c 33 ½ d c 2 3 =ðc c 33 ÞŠ c 55 ; ð2þ c 55 ð d V Þ A c 66 ð d H Þ where d ¼½þ=ðZ c ÞŠ ; d H ¼½þ=ðZ H c 66 ÞŠ ; d V ¼½þ=ðZ V c 55 ÞŠ : ð2þ If the background medium is isotroic ðc ¼ c 2 þ 2c 55 ; c 2 ¼ c 3 ; c 55 ¼ c 66 Þ and the fracture set is rotationally invariant, we have Z H ¼ Z V Z T ; and the equivalent medium is TI with a horizontal symmetry axis (HTI), whose stiffness matrix is c c c 2 c c 2 c c 2 c c c 2 2 Z c c 2 c 2 2 Z c c P ¼ 2 c c 2 c 2 2 Z c c c 2 2 Z c c 55 ; ð22þ c 55 c T A c 55 c T

7 Fracture-Induced Anisotroy where c ¼ð þ c Z Þ and c T ¼ð þ c 55 Z T Þ : ð23þ The stiffness matrix in oates and Schoenberg (995) is equivalent to Eq. (22) with a rotation of /2 around the y-axis and considering the lossless case ðg ¼ Þ: If g 6¼ ; Eq. (22) is equivalent to the medium studied by hichinina et al. (29b). The fractured medium defined by Eq. (22) can be obtained from Backus averaging (Backus 962) of a eriodic medium comosed of two isotroic constituents with roortions i ; and P-wave and S-wave moduli, resectively, given by E i and l i ; i ¼ ; 2: Backus effective stiffness constants are given by (arcione 27) ¼½E E 2 þ 4 2 ðl l 2 Þðk þ l k 2 l 2 ÞŠD; 2 ¼½k k 2 þ 2ðk þ k 2 2 Þðl 2 þ l 2 ÞŠD; 3 ¼ðk E 2 þ k 2 2 E ÞD 33 ¼ E E 2 D; 55 ¼ l l 2 ð l 2 þ 2 l Þ ; 66 ¼ l þ 2 l 2 ; D ¼ð E 2 þ 2 E Þ : ð24þ Equation (22) is then obtained by taking the limits! and 2! (or 2 Þ and setting E ¼ c and l ¼ c 55 (background medium) and E 2 ¼ 2 =Z and l 2 ¼ 2 =Z H (fracture). This is shown by Schoenberg (983) in the lossless case. 3 Proerties of the Effective Medium A general lane-wave solution for the dislacement field u ¼ðu x ; u y ; u z Þ > ¼ðu ; u 2 ; u 3 Þ > is u ¼ U ex ½ixðt s x s 2 y s 3 zþš; ð25þ where s i are the comonents of the slowness vector s ¼ ðs ; s 2 ; s 3 Þ; U is a comlex vector, and t is the time variable. We consider homogeneous viscoelastic waves, such that s ¼ sðl ; l 2 ; l 3 Þ; where s ¼ ffi s 2 þ s2 2 þ s2 3; and l i are the direction cosines defining the roagation (and attenuation) directions. The comlex velocity is v ¼ s : 3. Symmetry Plane of a Monoclinic Medium ð26þ In the symmetry lane of a monoclinic medium, there is a ure shear wave and two couled waves. The resective disersion relations in the (x, y)-lane are 33 qv 2 ¼ ; ð qv 2 Þð 22 qv 2 Þ 2 2 ¼ ; where q is the density of the background medium, ¼ l 2 þ 66l 2 2 þ 2 6l l 2 ; 22 ¼ 66 l 2 þ 22l 2 2 þ 2 26l l 2 ; 33 ¼ 55 l 2 þ 44l 2 2 þ 2 45l l 2 ; 2 ¼ 6 l 2 þ 26l 2 2 þð 2 þ 66 Þl l 2 ; ð27þ ð28þ (arcione 27), where v is the comlex velocity and l ¼ sin h and l 2 ¼ cos h; with h being the hase roagation angle. If we label the ure mode (the SH wave) and 2 and 3 the qs and qp waves, the corresonding comlex velocities are (e.g., arcione 27) qffi v ¼ q ð 55 l 2 þ 44l 2 2 þ 2 45l l 2 Þ; qffiffiffiffi v 2 ¼ð2qÞ =2 l 2 þ 22l 2 2 þ 66 þ 2l l 2 ð 6 þ 26 Þ A; qffiffiffiffi v 3 ¼ð2qÞ =2 l 2 þ 22l 2 2 þ 66 þ 2l l 2 ð 6 þ 26 ÞþA; qffiffiffiffiffiffiffi A ¼ ð 22 Þ 2 þ 4 2 2: ð29þ The hase velocity is given by v ¼ Re ð3þ v and the quality factor is simly Q ¼ Reðv2 Þ Imðv 2 ð3þ Þ (e.g., arcione 27). The values of the qp quality factor along orthogonal directions are Q P ðh ¼ =2Þ ¼ Reð Þ Imð Þ and Q Pðh ¼ Þ ¼ Reð 22Þ Imð 22 Þ ; ð32þ resectively, while those of the shear waves are Q SV ðh ¼ =2Þ ¼Q SV ðh ¼ Þ ¼Q SH ðh ¼ Þ ¼ Reð 44Þ Imð 44 Þ ; and Q SH ðh ¼ =2Þ ¼ Reð 55Þ Imð 55 Þ : ð33þ ext, we obtain the energy velocity at zero frequency (the lossless elastic limit) times one unit of time. The SH wave energy velocity is v e ¼ ½ðc 55 qv l þ c 45 l 2Þ^e þðc 44 l 2 þ c 45 l Þ^e 2 Š: ð34þ On the other hand, the qp and qs energy-velocity comonents v e and v e2 are

8 J. M. arcione et al. qv ð þ 22 2qv 2 Þv e ¼ð 22 qv 2 Þðc l þ c 6 l 2Þ þð qv 2 Þðc 66 l þ c 26 l 2Þ 2 ½2c 6 l þðc 2 þ c 66 Þl 2Š and qv ð þ 22 2qv 2 Þv e2 ¼ð 22 qv 2 Þðc 66 l 2 þ c 6 l Þ þð qv 2 Þðc 22 l 2 þ c 26 l Þ 2 ½2c 26 l 2 þðc 2 þ c 66 Þl Š; ð35þ ð36þ where c IJ are the zero-frequency limits of the IJ (e.g., arcione 27). 3.2 Orthorhombic Media The disersion relation has the same form as in the lossless case, but relacing the (real) elasticity constants by the comlex stiffnesses IJ ; i.e., the comonents of matrix P. Following Schoenberg and Helbig (997), we have v 6 a 2 v 4 þ a v 2 a ¼ ; ð37þ where a ¼ þ þ ; a ¼ þ 33 þ ; a 2 ¼ þ 22 þ 33 ; ð38þ with ¼ l 2 þ 66l 2 2 þ 55l 2 3 ; 22 ¼ 66 l 2 þ 22l 2 2 þ 44l 2 3 ; 33 ¼ 55 l 2 þ 44l 2 2 þ 33l 2 3 ; 2 ¼ð 2 þ 66 Þl l 2 ; 3 ¼ð 3 þ 55 Þl 3 l ; 23 ¼ð 44 þ 23 Þl 2 l 3 ð39þ being the comonents of the Kelvin hristoffel matrix (arcione 27). In the (three) symmetry lanes of an orthorhombic medium, there is a ure shear wave (labeled below) and two couled waves. The corresonding comlex velocities are given by a generalization of the lossless case (e.g., arcione 27) to the lossy case: (x, y)-lane: qffiffiffiffiffi v ¼ ðqþ ð 55 l 2 þ 44l 2 2 Þ ; qffiffiffiffiffiffiffiffiffiffi v 2 ¼ð2qÞ =2 l 2 þ 22l 2 2 þ 66 A; qffiffiffiffiffiffiffiffiffiffi v 3 ¼ð2qÞ =2 l 2 þ 22l 2 2 þ 66 þ A; qffiffiffiffiffiffiffiffiffi A ¼ ½ð Þl 2 2 ð 66 Þl 2 Š2 þ 4½ð 2 þ 66 Þl l 2 Š 2 ; ð4þ (x, z)-lane: qffiffiffiffiffi v ¼ ðqþ ð 66 l 2 þ 44l 2 3 Þ ; qffiffiffiffiffiffiffiffiffiffi v 2 ¼ð2qÞ =2 l 2 þ 33l 2 3 þ 55 A; qffiffiffiffiffiffiffiffiffiffi v 3 ¼ð2qÞ =2 l 2 þ 33l 2 3 þ 55 þ A; qffiffiffiffiffiffiffiffiffi A ¼ ½ð Þl 2 3 ð 55 Þl 2 Š2 þ 4½ð 3 þ 55 Þl l 3 Š 2 ; ð4þ (y, z)-lane: qffiffiffiffiffiffi v ¼ ðqþ ð 66 l 2 2 þ 55l 2 3 qffi Þ v 2 ¼ð2qÞ =2 22 l 2 2 þ 33l 2 3 þ 44 A; qffi v 3 ¼ð2qÞ =2 22 l 2 2 þ 33l 2 3 þ 44 þ A; qffiffi A ¼ ½ð Þl 2 3 ð Þl 2 2 Š2 þ 4½ð 23 þ 44 Þl 2 l 3 Š 2 : ð42þ In terms of angles, l ¼ sin h and l 2 ¼ cos h in the (x, y)- lane, l ¼ sin h and l 3 ¼ cos h in the (x, z)-lane, and l 2 ¼ sin h and l 3 ¼ cos h in the (y, z)-lane. The comlex velocities along the rincial axes are: (x, y)-lane: v ð Þ¼v S ð Þ¼ 44 =q v ð9 Þ¼v S ð9 Þ¼ 55 =q v 2 ð Þ¼v qs ð Þ¼ 66 =q v 2 ð9 Þ¼v qs ð9 Þ¼ ð43þ 66 =q v 3 ð Þ¼v qp ð Þ¼ 22 =q v 3 ð9 Þ¼v qp ð9 Þ¼ =q; (x, z)-lane: v ð Þ¼v S ð Þ¼ 44 =q v ð9 Þ¼v S ð9 Þ¼ 66 =q v 2 ð Þ¼v qs ð Þ¼ 55 =q v 2 ð9 Þ¼v qs ð9 Þ¼ 55 =q v 3 ð Þ¼v qp ð Þ¼ 33 =q v 3 ð9 Þ¼v qp ð9 Þ¼ =q; (y, z)-lane: ð44þ

9 Fracture-Induced Anisotroy v ð Þ¼v S ð Þ¼ 66 =q v ð9 Þ¼v S ð9 Þ¼ 55 =q v 2 ð Þ¼v qs ð Þ¼ 44 =q v 2 ð9 Þ¼v qs ð9 Þ¼ 44 =q v 3 ð Þ¼v qp ð Þ¼ 33 =q v 3 ð9 Þ¼v qp ð9 Þ¼ 22 =q: ð45þ The energy velocity can be comuted for each frequency comonent, with the wavefront corresonding to the energy velocity at infinite frequency. Let us consider the (x, z)-lane of symmetry. The energy-velocity vector of the qp and qs waves is given by v e ¼ðl þ l 3 cot wþ ^e þðl tan w þ l 3 Þ ^e 3 ð46þ v (arcione 27; eq. 6.58), where tan w ¼ Reðc X þ n WÞ Reðc W þ n ð47þ ZÞ defines the angle between the energy-velocity vector and the z-axis, c ¼ ffi A B; ffi n ¼v A B; ð48þ B ¼ l 2 33l 2 3 þ 55 cos 2h; where the uer and lower signs corresond to the qp and qs waves, resectively. Moreover, W ¼ 55 ðnl þ cl 3 Þ; X ¼ c l þ n 3 l 3 ; Z ¼ c 3 l þ n 33 l 3 ð49þ (arcione 27; Eqs ), where v denotes the rincial value, which has to be chosen according to established criteria. On the other hand, the energy velocity of the SH wave is v e ¼ and v h qreðvþ l Re 66 v ^e þ l 3 Re 44 v tan w ¼ Reð 66=vÞ Reð 44 =vþ tan h (arcione 27; eq. 4.5). In general, we have the roerty v ¼ v e cosðw hþ; i ^e 3 ð5þ ð5þ ð52þ where v e ¼jv e j: The quality factor exressions for each symmetry lane are similar to the equations obtained for the symmetry lane of the monoclinic medium. Along ure mode directions, we have Q II ¼ Reð IIÞ ; I ¼ ;...; 6: Imð II Þ ð53þ The attenuation has a maximum for a given value of the secific viscosity. Let us consider, for instance, the P-wave and Eq. (9). A calculation yields Q ¼ c xg j ðj þ c Þþx 2 g 2 : ð54þ This function has a maximum at xg ¼ ffiffiffiffi j ðj þ c Þ j if j c : On the other hand, ð55þ Q JJ ¼ ðc 2 J =c JJÞxg ðj þ c Þðj þ c c 2 J =c JJÞþx 2 g 2 ; J ¼ 2; 3; ð56þ with a maximum at qffiffiffiffiffiffi xg ¼ ðj þ c Þðj þ c c 2 J =c JJÞ: ð57þ The attenuation of the shear waves (the 44, 55, and 66 comonents) have similar exressions. Two different values of the viscosity may give the same value of the quality factor, although the hase velocities differ. 3.3 Transversely Isotroic Media The analysis for the HTI medium follows as a articular case of one of the symmetry lanes of the orthorhombic medium. One can use the exact velocity exressions or the following aroximations: v 2 ¼ðc 55=qÞð d H sin 2 hþ; v 2 ¼ðc 55=qÞð d H cos 2 2h ðc 55 =c Þd sin 2 2hÞ; v 2 3 ¼ðc =qþ ½ ðc 55 =c Þd H sin 2 2h ð 2ðc 55 =c Þd cos 2 hþ 2 Š; ð58þ where h is the angle between the wavenumber vector and the vertical direction. Similar aroximations have been obtained by Schoenberg and Douma (988) (their Eq. 26) for VTI media (horizontal fractures). Schoenberg and Douma (988) have written those exressions in terms of the coefficients d =ð d Þ and d H =ð d H Þ; instead of d and d H ; resectively, assuming d and d H : Without imosing these conditions, more accurate exressions are obtained by using d and d H ; as in hichinina et al. (29a, b).

10 J. M. arcione et al. 4 Examles (a) We consider the TI background medium studied by Schoenberg and Helbig (997), reresenting a tyical shale, 4 2:5 4 2:5 2:5 2:5 6 ¼ q 2 ð59þ 2 A 3 (in MPa), where q = 2,3 kg/m 3, e.g., c = 23 GPa. We assume a frequency f ¼ x=ð2þ ¼5 Hz and the low- and high-frequency limits for comarison. First, we consider two orthogonal sets of fractures with b ¼ and b 2 ¼ 9 : The fracture stiffnesses of the first set are given by j ðþ ¼ 9c ; j ðþ H ¼ 8 3 c 66; and k ðþ V ¼ 4c 55: In the ¼ :; dðþ ¼ 3=; and dðþ ¼ lossless case, we obtain d ðþ H V =5; i.e., the values used by Schoenberg and Helbig (997). Moreover, we set j ð2þ ¼ bjðþ ; jð2þ H ¼ bjðþ H and kð2þ V ¼ ; with b =.5. The fracture viscosities are assumed to be bk ðþ V g ðqþ ¼ ajðqþ ; gðqþ H ¼ ajðqþ H ; and gðqþ V ¼ akðqþ V ; where a =-3 s, for both fracture sets. The stiffness-matrix comonents, given by Eq. (9), are ¼ð2:34; :7Þ; 3 ¼ð4:87; :22Þ; 23 ¼ð4:6; :29Þ; 44 ¼ð3:3; :3Þ; 66 ¼ð3:32; :53Þ; 2 ¼ð6:93; :56Þ; 22 ¼ð8:83; :5Þ; 33 ¼ð3:44; :9Þ; 55 ¼ð3:73; :22Þ; ð6þ in GPa. Figure 2 shows the dissiation factors as a function of frequency along ure mode directions. The attenuation behaves as relaxation eaks, similar to the Zener model. Figure 3 shows the energy velocity at the (x, z)-lane for Hz (a), 5 Hz (b), and infinite frequency (c). The high-frequency limit corresonds to the unfractured TI case. The dissiation factors versus hase and ray angle are reresented in Figure 4, where the ure mode (the SH wave) shows more attenuation. The attenuation of the couled waves have a different behavior versus the ray angle, corresonding to the roagation of wave ackets, comared to the reresentation versus the hase (roagation) angle, corresonding to the roagation of lane waves. In articular, the trilication aears also in the Q factor of the qs wave. Exerimental setus should consider these facts. ext, we consider two sets of fractures with b =2 and b 2 =65 and erform the analysis in the symmetry axis of the effective monoclinic medium. The fracture arameters (b) Fig. 2 Dissiation factors (53) as a function of frequency, where (a) corresonds to the P waves and (b) corresonds to the S waves corresonding to the first examle are considered. The stiffness-matrix comonents, given by Eq. (5), are ¼ð8:5; :Þ; 2 ¼ð8:98; :29Þ; 3 ¼ð4:83; :23Þ; 6 ¼ð :7; :3Þ; 22 ¼ð7:27; :26Þ; 23 ¼ð4:69; :26Þ; 26 ¼ð :5; :5Þ; 33 ¼ð3:44; :9Þ; ð6þ 36 ¼ð :22; :3Þ; 44 ¼ð3:37; :25Þ; 45 ¼ð :67; :Þ; 55 ¼ð3:7; :9Þ; 66 ¼ð4:53; :46Þ; in GPa. Figure 5 shows the low-frequency limit energy velocities (a) and dissiation factors (b) in the (x, y) symmetry lane as a function of the hase (roagation) angle. The frequency in Fig. 5b isf = 5 Hz. In this case, the qs wave shows the maximum attenuation. In order to verify the shae of the wavefronts shown in Fig. 5, we erform a 2D full-wave numerical simulation of qp qs roagation in the symmetry lane, where the effective medium is defined by the low-frequency elasticity

11 Fracture-Induced Anisotroy (a) (a) (b) (b) (c) Fig. 3 Energy velocities in the (x, z) symmetry lane of the equivalent orthorhombic medium, where (a) f = Hz, (b) f = 5 Hz, and (c) f = Hz constants: c = 7.8 GPa, c 22 = 7 GPa, c 2 = 8.9 GPa, c 6 = -.8 GPa, c 26 = -.6 GPa, and c 66 = 4.44 GPa. The density is q = 2,3 kg/m 3. The algorithm solves the article-velocity/stress formulation based on the Fourier seudosectral method for comuting the satial Fig. 4 Quality factors in the (x, z) symmetry lane as a function of the hase (roagation) angle (a) and ray (energy) angle (b). The equivalent medium has orthorhombic symmetry and the frequency is f =5Hz derivatives and a 4th-order Runge Kutta technique for calculating the wavefield recursively over time (e.g., arcione 27). The source is a vertical force with a Ricker time history, located at the center of the mesh. The simulation uses a mesh with -m grid sacing and the central frequency of the source is 8 Hz. The algorithm has a time ste of. ms and a snashot of the verticalarticle velocity is comuted at 8 ms (see Fig. 6). It is verified that the results of the modeling algorithm and lane-wave analysis are in agreement. Finally, we test the aroximation Q ðhþ Q b ðhþþq f ðhþ; ð62þ where Q b and Q f are the quality factors of the background medium and fracture set in the same lossless background medium, resectively. Equation (62) is commonly used in the literature to obtain the total quality factor due to different attenuation mechanisms (e.g., hichinina et al. 29a, b). We assume for simlicity an isotroic

12 J. M. arcione et al. (a) (b) Fig. 6 Snashot in the (x, y) symmetry lane of the monoclinic effective medium. The elasticity constants are those of the lowfrequency limit Fig. 5 Low-frequency limit energy velocities (a) and dissiation factors (b) in the (x, y) symmetry lane as a function of the hase (roagation) angle. The equivalent medium has monoclinic symmetry and the frequency in b is f =5Hz background medium, and, therefore, Q b is indeendent of h. The effective medium has HTI symmetry and the stiffness matrix is given by Eq. (22), with c IJ comlex. Let us consider the simlest model, i.e., c! c ðq P þ iþ and c 55! c 55 ðq S þ iþ; where Q P and Q S are the P-wave and S-wave quality factors of the background medium. (ote that c 2 ¼ c 2c 55 :Þ Let us assume that Q P = 2 and Q S = 5. Figure 7 comares the exact dissiation factors to the aroximate dissiation factors, corresonding to the qp and qs waves, resectively. As can be areciated, Eq. (62) is a rough aroximation and should be used with caution. The aroximation imroves if Q increases. In order to verify that the equivalent-medium theory of fractures is correct, we consider an isotroic background medium defined by c 2 = GPa, c 55 = 3.9 GPa, and q = 2,3 kg/m 3 containing a horizontal and lane fracture set defined by j = 5,5 GPa/m and g = 36 GPa s/m (tangential stiffnesses er unit length), and j 3 = 34, Fig. 7 Exact (solid lines) and aroximate (dashed lines) dissiation factors of the qp and qs waves as a function of the roagation angle. The frequency is 5 Hz GPa/m and g 3 = 8 GPa s/m (normal stiffnesses er unit length). The fracture lanes are arallel to the (x, y)-lane and erendicular to the z-axis. The novel methodology consists of alying time-harmonic oscillatory tests at a finite number of frequencies. Each test is based on the wave equation of motion exressed in the sace frequency domain, imlementing exlicitly the fracture boundary conditions, and solved with a finite-element method. A similar algorithm considering multile thin layers is given by Santos et al. (2). Figure 8 shows the hase velocity v (a) and dissiation factor, Q - along the horizontal direction (comonent, circles) and vertical direction

13 Fracture-Induced Anisotroy (a) (b) (energy) angle are obtained for homogeneous viscoelastic lane waves (wavenumber and attenuation directions coincide). We consider the symmetry lane of a monoclinic medium and the three symmetry lanes of an orthorhombic medium. The examles show that the effective media have high anisotroy and show relaxation attenuation eaks, similar to Zener viscoelastic models. Moreover, we have tested the commonly used equation stating that the dissiation factor of the effective medium is equal to the sum of the dissiation factors of the background medium and fractured background (lossless) medium. The results indicated that this equation is not a good aroximation for realistic Q values of the background medium. Finally, we erformed a cross-check of the theory with the comutations of a finiteelement algorithm, showing that both hase velocity and dissiation factors agree with the numerical results. The novel model can be imortant in determining the orientation of fractures in the reservoir and the overlying ca rock. This lays an imortant role during roduction and other alications, such as O 2 injection and monitoring. References Fig. 8 umerical hase velocity (a) and dissiation factor (b) along the horizontal direction (circles) and vertical direction (triangles) corresonding to a fracture set embedded in a homogeneous and isotroic medium. The solid lines corresond to the theoretical curves (comonent 33, triangles). The match between the theory (solid lines) and numerical results is very good. 5 onclusions We have resented a theory to obtain the wave velocities and attenuation of many sets of vertical fractures embedded in a transversely isotroic medium. The anisotroic effective medium has monoclinic symmetry. Fractures are modeled as boundary discontinuities in the dislacement and article velocity fields. The theory generalizes an existing model describing the acoustic roerties of a single set of fractures embedded in an isotroic background medium. The exressions of the comlex and frequency-deendent stiffness constants corresonding to two orthogonal sets of fractures are obtained exlicitly, where the effective medium has orthorhombic symmetry. The hase, energy, and quality factors as a function of the roagation and ray Backus GE (962) Long-wave elastic anisotroy roduced by horizontal layering. J Geohys Res 67: Barton (27) Fracture-induced seismic anisotroy when shearing is involved in roduction from fractured reservoirs. J Seism Exlor 6:5 43 arcione JM (992) Anisotroic Q and velocity disersion of finely layered media. Geohys Prosect 4: arcione JM (996a) Plane-layered models for the analysis of wave roagation in reservoir environments. Geohys Prosect 44:3 26 arcione JM (996b) Elastodynamics of a non-ideal interface: alication to crack and fracture scattering. J Geohys Res : arcione JM (27) Wave fields in real media: wave roagation in anisotroic, anelastic, orous and electromagnetic media. In: Handbook of geohysical exloration (revised and extended), vol 38, 2nd edn. Elsevier, Amsterdam, arcione JM, avallini F (994) A rheological model for anelastic anisotroic media with alications to seismic wave roagation. Geohys J Int 9: arcione JM, avallini F, Helbig K (998) Anisotroic attenuation and material symmetry. Acustica 84: arcione JM, Santos JE, Picotti P (2) Anisotroic oroelasticity and wave-induced fluid flow: harmonic finite-element simulations. Geohys J Int 86: haman M (23) Frequency-deendent anisotroy due to mesoscale fractures in the resence of equant orosity. Geohys Prosect 5: hichinina TI, Obolentseva IR, Ronquillo-Jarillo G (29a) Anisotroy of seismic attenuation in fractured media: theory and ultrasonic exeriment. Trans Porous Media 79: 4 hichinina TI, Obolentseva IR, Gik L, Bobrov B, Ronquillo-Jarillo G (29b) Attenuation anisotroy in the linear-sli model: interretation of hysical modeling data. Geohysics 74:WB65 WB76

14 J. M. arcione et al. oates RT, Schoenberg M (995). Finite-difference modeling of faults and fractures. Geohysics 6: Fan LF, Ren F, Ma GW (2) An extended dislacement discontinuity method for analysis of stress wave roagation in viscoelastic rock mass. J Rock Mech Geotech Eng 3:73-8 Grechka VA, Bakulin A, Tsvankin I (23) Seismic characterization of vertical fractures described as general linear-sli interfaces. Geohys Prosect 5:7 3 Grechka V, Tsvankin I (23) Feasibility of seismic characterization of multile fracture sets. Geohysics 68: Hall SA, Kendall J-M (23) Fracture characterization at Valhall: alication of P-wave amlitude variation with offset and azimuth (AVOA) analysis to a 3D ocean-bottom data set. Geohysics 68:5 6 Hansen BRH (22) Evaluating the imact of fracture-induced anisotroy on reservoir rock roerty estimates made from seismic data. Reort no.: GPH 7/2, urtin University of Technology Hood JA (99) A simle method for decomosing fracture-induced anisotroy. Geohysics 56: Krzikalla F, Müller T (2) Anisotroic P-SV-wave disersion and attenuation due to inter-layer flow in thinly layered orous rocks. Geohysics 76:WA35 WA45. doi:.9/ Liu E, Hudson JA, Pointer T (2) Equivalent medium reresentation of fractured rock. J Geohys Res 5:298 3 Maultzsch S (25) Analysis of frequency-deendent anisotroy in VSP data. PhD thesis, University of Edinburgh akagawa S, Myer LR (29) Fracture ermeability and seismic wave scattering oroelastic linear-sli interface model for heterogeneous fractures. SEG Exanded Abstracts 28, 346. doi:.9/ ichols D, Muir F, Schoenberg M (989) Elastic roerties of rocks with multile sets of fractures. In: Proceedings of the 63rd Annual International Meeting of the Society of Exloration Geohysicists, Extended Abstracts, Perino A, Zhu, JB, Li J, Barla G, Zhao J (2) Theoretical methods for wave roagation across jointed rock masses. Rock Mech Rock Eng 43: Picotti S, arcione JM, Santos JE, Gei D (2) Q-anisotroy in finely layered media. Geohys Res Lett 37:L632. L doi:.29/29gl4246 Pyrak-olte LJ, Myer LR, ook GW (99) Transmission of seismic waves across single natural fractures. J Geohys Res 95: Santos JE, arcione JM, Picotti S (2) Viscoelastic-stiffness tensor of anisotroic media from oscillatory numerical exeriments. omut Methods Al Mech Eng 2: Schoenberg M (98) Elastic wave behavior across linear sli interfaces. J Acoust Soc Am 68:56 52 Schoenberg M (983) Reflection of elastic waves from eriodically stratified media with interfacial sli. Geohys Prosect 3: Schoenberg M, Douma J (988) Elastic wave roagation in media with arallel fractures and aligned cracks. Geohys Prosect 36:57 59 Schoenberg M, Helbig K (997) Orthorhombic media: modeling elastic wave behavior in a vertically fractured earth. Geohysics 62: Schoenberg M, Muir F (989) A calculus for finely layered anisotroic media. Geohysics 54: Schoenberg M, Dean S, Sayers (999) Azimuth-deendent tuning of seismic waves reflected from fractured reservoirs. Geohysics 64:6 7 Zhang J, Gao H (29) Elastic wave modelling in 3-D fractured media: an exlicit aroach. Geohys J Int 77:3 24 Zhu Y, Tsvankin I (27) Plane-wave attenuation anisotroy in orthorhombic media. Geohysics 72:D9 D9