Estimation of fracture parameters from reflection seismic data Part II: Fractured models with orthorhombic symmetry

Transcription

1 GEOPHYSICS, VOL. 65, NO. 6 (NOVEMBER-DECEMBER 2000); P , 5 FIGS., 1 TABLE. Estimation of fracture parameters from reflection seismic data Part II: Fractured models with orthorhomic symmetry Andrey Bakulin, Vladimir Grechka, Ilya Tsvankin ABSTRACT Existing geophysical geological data indicate that orthorhomic media with a horizontal symmetry plane should e rather common for naturally fractured reservoirs. Here, we consider two orthorhomic models: one that contains parallel vertical fractures emedded in a transversely isotropic ackground with a vertical symmetry axis (VTI medium) the other formed y two orthogonal sets of rotationally invariant vertical fractures in a purely isotropic host rock. Using the linear-slip theory, we otain simple analytic expressions for the anisotropic coefficients of effective orthorhomic media. Under the assumptions of weak anisotropy of the ackground medium (for the first model) small compliances of the fractures, all effective anisotropic parameters reduce to the sum of the ackground values the parameters associated with each fracture set. For the model with a single fracture system, this result allows us to eliminate the influence of the VTI ackground y evaluating the differences etween the anisotropic parameters defined in the vertical symmetry planes. Susequently, the fracture weaknesses, which carry information aout the density content of the fracture network, can e estimated in the same way as for fracture-induced transverse isotropy with a horizontal symmetry axis (HTI media) examined in our previous paper (part I). The parameter estimation procedure can e ased on the azimuthally dependent reflection traveltimes prestack amplitudes of P-waves alone if an estimate of the ratio of the P- S-wave vertical velocities is availale. It is eneficial, however, to comine P-wave data with the vertical traveltimes, NMO velocities, or AVO gradients of mode-converted (PS) waves. In each vertical symmetry plane of the model with two orthogonal fracture sets, the anisotropic parameters are largely governed y the weaknesses of the fractures orthogonal to this plane. For weak anisotropy, the fracture sets are essentially decoupled, their parameters can e estimated y means of two independently performed HTI inversions. The input data for this model must include the vertical velocities (or reflector depth) to resolve the anisotropic coefficients in each vertical symmetry plane rather than just their differences. We also discuss several criteria that can e used to distinguish etween the models with one two fracture sets. For example, the semimajor axis of the P-wave NMO ellipse the polarization direction of the vertically traveling fast shear wave are always parallel to each other for a single system of fractures, ut they may ecome orthogonal in the medium with two fracture sets. INTRODUCTION This is the second part of our series of three papers on characterization of naturally fractured reservoirs using surface seismic data. The main goal of the series is to provide a connection etween fracture detection methods rock physics models of fractured media, such as those developed y Hudson (1980, 1981, 1988), Schoenerg (1980, 1983), Thomsen (1995). In particular, we are interested in elucidating the dependence of reflection seismic signatures on the physical properties of the fractures in developing inversion algorithms for estimating fracture parameters using P-wave converted (PS) reflection data. In the first paper of the series (Bakulin et al., 2000; hereafter referred to as part I), we address these prolems for transverse isotropy with a horizontal axis of symmetry (HTI media), which descries a single set of vertical, parallel, rotationally invariant fractures in a purely isotropic ackground medium. HTI is the Manuscript received y the Editor April 20, 1999; revised manuscript received March 1, Formerly St. Petersurg State University, Department of Geophysics, St. Petersurg, Russia. Presently Schlumerger Camridge Research, High Cross, Madingley Road, Camridge, CB3 0EL, Engl. akulin@camridge.scr.sl.com. Colorado School of Mines, Center for Wave Phenomena, Department of Geophysics, Golden, Colorado vgrechka@dix.mines. edu; ilya@dix.mines.edu. c 2000 Society of Exploration Geophysicists. All rights reserved. 1803

2 1804 Bakulin et al. simplest azimuthally anisotropic model that provides valuale insights into the ehavior of seismic signatures over fractured formations. However, since the rock matrix usually exhiits some form of anisotropy, it is difficult to expect the HTI model to e adequate for typical fractured reservoirs. Here, we extend the approach of part I to more complicated, ut likely more realistic, orthorhomic models. The description of seismic signatures in orthorhomic media can e simplified significantly y applying Tsvankin s (1997) notation ased on the analogy etween the symmetry planes of orthorhomic TI media. For instance, P-wave velocities traveltimes are fully controlled y the P-wave vertical velocity five dimensionless anisotropic coefficients, rather than y nine stiffnesses in the conventional notation. Also, this parameterization yields concise exact expressions for the NMO velocities of pure modes reflected from a horizontal interface (Grechka Tsvankin, 1998). As shown y Grechka et al. (1999), eight out of the nine parameters of orthorhomic media with a horizontal symmetry plane can e otained from the NMO velocities of horizontal P- split PS-events, provided the vertical velocities or reflector depth are known. Reflection moveout of P-waves alone is sufficient to find the orientation of the symmetry planes, the NMO velocities within them, three anellipticity coefficients η. The parameters η, however, can e determined only if dipping events or nonhyperolic moveout is availale. While orthorhomic symmetry may e caused y a variety of physical reasons, we restrict ourselves to two orthorhomic models elieved to e most common for fractured reservoirs: (1) a single set of vertical cracks in a transversely isotropic ackground with a vertical symmetry axis (VTI) (2) two systems of vertical fractures orthogonal to each other in an isotropic ackground medium. If the intrinsic anisotropy of the host rock the anisotropy induced y the fractures are weak, the anisotropic coefficients of the ackground fractures can e algeraically added in descriing the effective medium. This result provides important insights into the influence of fractures on the effective anisotropy helps to generalize the parameter estimation methodology of part I for orthorhomic media. For oth orthorhomic models we relate Tsvankin s (1997) anisotropic coefficients to the fracture compliances introduce practical fracture characterization techniques operating with P- PS-waves. MODEL 1: ONE SET OF VERTICAL FRACTURES IN A VTI BACKGROUND Effective orthorhomic medium A single system of vertical fractures emedded in a VTI matrix yields an effective orthorhomic medium in which one of the symmetry planes coincides with the fracture plane. According to the linear-slip theory (Schoenerg, 1980, 1983), the effective compliance tensor of such a medium can e otained y simply adding the excess fracture compliances to the compliance of the ackground. [Part I compares the linear-slip theory with the effective medium theories of Hudson (1980, 1981, 1988) Thomsen (1995).] The inversion of the resulting compliance tensor yields the stiffness tensor ( the corresponding two-index stiffness matrix) of the fracture-induced orthorhomic model. If the fracture faces are perpendicular to the x 1 -axis, the effective stiffness matrix c has the following form (Schoenerg Helig, 1997; Appendix A): c 11 c 12 c c 12 c 22 c ) c 13 c 23 c ( c1 0 c = =, (1) c c c c 66 where 0 is the 3 3 zero matrix c 1 c 2 are given y c 1 = c 11 (1 N ) c 12 (1 N ) c 13 (1 N ) c 2 ( 12 c 12 (1 N ) c 11 c 12 N c 13 1 N c 11 c 11 ( ) c 13 (1 N ) c 13 1 N c 12 c 11 c 33 N c 2 13 c 11 ) (2) c c 2 = 0 c 44 (1 V ) 0. (3) 0 0 c 66 (1 H ) Here c ij are the stiffness coefficients of the VTI ackground (constrained y c 12 = c 11 2c 66 ); N, V, H are the dimensionless weaknesses of the fractures (Schoenerg Helig, 1997; Bakulin Molotkov, 1998), which change from zero (no fractures) to unity (extreme fracturing). The tangential weaknesses V H provide a measure of crack density, whereas the normal weakness N contains information aout the fluid content of the fractures possile fluid flow etween the fractures pore space (Schoenerg Douma, 1988; part I). Matrix (1) descries a special type of orthorhomic media with the stiffnesses satisfying the relation (Schoenerg Helig, 1997) c 13 (c 22 + c 12 ) = c 23 (c 11 + c 12 ). (4) The existence of the additional constraint (4) stems from the fact that while general orthorhomic media are characterized y nine independent values of c ij, the fracture-induced model considered here is defined y only eight quantities (for a fixed fracture orientation): five stiffness coefficients (c 11, c 13, c 33, c 44, c 66 ) of the VTI ackground three fracture weaknesses ( N, V, H ). The inversion of surface data for the physical parameters of the fractures requires relating seismic signatures to the fracture weaknesses. The results of Grechka Tsvankin (1999) Grechka et al. (1999) show that, in orthorhomic media, such commonly used signatures as NMO velocities AVO gradients are most concisely expressed through the dimensionless anisotropic coefficients introduced y Tsvankin (1997). The definitions of Tsvankin s parameters ɛ (1,2), δ (1,2,3), γ (1,2) in

3 Fractured Orthorhomic Media 1805 terms of the stiffness elements c ij are given in Appendix B. Sustituting equations (1) (3) for the stiffnesses of the fracture-induced orthorhomic model into equations (B-1) (B-9) yields the anisotropic coefficients ɛ, δ, γ as functions of N, V, H. Weak-anisotropy approximation The exact expressions for Tsvankin s anisotropic parameters in terms of the weaknesses, however, are too complicated to give insight into the influence of the fractures on the effective anisotropic model. Therefore, we assume that Thomsen s (1986) coefficients ɛ, δ, γ of the VTI ackground, along with the weaknesses N, V, H, are small quantities of the same order. Linearization of the effective stiffness matrix in these small quantitites [equation (A-17)] shows that for weak anisotropy the effective anisotropic coefficients ɛ (1,2), δ (1,2,3), γ (1,2) should represent the sums of the VTI ackground parameters ɛ, δ, γ the corresponding anisotropic coefficients of the HTI medium resulting from the fractures in an imaginary isotropic host rock sufficiently close to the VTI ackground model. This isotropic medium can e characterized, for instance, y the P- S-wave velocities V P V S equal to the vertical velocities V P0 V S0 in the VTI ackground. Since this result provides a theoretical asis for most of the susequent discussion, we prove it for one of the anisotropic coefficients the parameter ɛ (2) defined y equation (B-3). Sustituting the stiffnesses from equations (1) (2) into equation (B-3) yields ( ) c 11 c 33 N c 11 c2 13 c 11 ɛ (2) = ( c 2 ). (5) 13 2c 33 1 N c 11 c 33 Using the definition of the coefficient ɛ in the VTI ackground medium [ɛ (c 11 c 33 )/(2c 33 ); see Thomsen (1986)], equation (5) can e rewritten as ɛ (2) = ɛ N c 2 11 c c 11 c 33 1 N c 2 13 c 11 c 33. (6) Linearizing this equation with respect to ɛ N, we drop the anisotropic term in the denominator to otain ɛ (2) ɛ N c 2 11 c c 11 c 33. (7) Since (c 2 11 c 2 13 )/(c 11 c 33 ) is multiplied y the already small weakness N, it can e replaced in the weak anisotropy approximation y the corresponding isotropic quantity 4g(1 g), where g V 2 S0 /V 2 P0 is the ratio of the squared vertical S- P-wave velocities in the ackground. Thus, equation (7) can e represented as ɛ (2) ɛ 2g(1 g) N. (8) The comination 2g(1 g) N can e recognized as the linearization of the anisotropic coefficient ɛ (V) (Rüger, 1997; Tsvankin, 1997a) in an HTI medium due to a single set of vertical fractures perpendicular to the x 1 -axis that are emedded in a purely isotropic rock with the squared S-to-P velocity ratio g (part I). Hence, equation (8) implies that ɛ (2) ɛ + ɛ (V). (9) We conclude that in the weak anisotropy limit the anisotropic coefficient ɛ (2) reduces to the sum of the Thomsen ackground parameter ɛ the coefficient ɛ (V) of the HTI model due to the fractures emedded in an isotropic medium close enough to the VTI ackground medium. Similar linearizations for the other anisotropic parameters of the effective orthorhomic model are given elow. Symmetry plane [x 2, x 3 ] parallel to the fractures. The linearized anisotropic coefficients in the symmetry plane [x 2, x 3 ], derived from equations (B-6) (B-8), have the following form: ɛ (1) = ɛ, (10) δ (1) = δ, (11) γ (1) = γ + V H. (12) 2 Therefore, for weak anisotropy coefficients ɛ (1) δ (1) coincide with those of the VTI ackground medium. This is an expected result; if rotationally invariant fractures are introduced into an isotropic matrix, the plane [x 2, x 3 ] represents the so-called isotropy plane of the effective HTI medium. In the isotropy plane the velocities of all waves are not influenced y the fractures remain constant for all propagation directions (e.g., Tsvankin, 1997a). The coefficient γ (1) does not coincide with γ only ecause, in contrast to the more conventional model from part I, the fractures considered here are not rotationally invariant (i.e., V H ). The difference etween V H, however, has no earing on parameters ɛ (1) δ (1). In addition to the asic set of anisotropic parameters introduced y Tsvankin (1997), it is useful to consider the anellipticity coefficients η (1,2,3) responsile for time processing of P-wave data (Grechka Tsvankin, 1999). For weak anisotropy, the η coefficient in the [x 2, x 3 ]-plane given y equation (B-10) is also equal to the ackground value: η (1) = η. (13) Symmetry plane [x 1, x 3 ] perpendicular to fractures. The weak-anisotropy approximations of the anisotropic coefficients in the plane [x 1, x 3 ] are given y ɛ (2) = ɛ 2g(1 g) N, (14) δ (2) = δ 2g[(1 2g) N + V ], (15) γ (2) = γ H 2, (16) η (2) = η + 2g[ V g N ]. (17) Each of the expressions (14) (17) contains two terms with a distinctly different physical meaning. The first term is the corresponding anisotropic coefficient of the VTI ackground,

4 1806 Bakulin et al. while the second term asors the influence of the fractures. As discussed aove for ɛ (2), the fracture-related terms in equations (14) (17) are approximately equal (assuming H = V ) to the coefficients ɛ (V ), δ (V ), γ (V ), η (V ) (respectively) derived in part I for a fracture set emedded in an isotropic ackground. The constraint on the effective stiffnesses [equation (4)] leads to an additional relationship etween the anisotropic coefficients. In the weak anisotropy limit, equation (4) can e rewritten as γ (2) γ (1) = 1 4g [ δ (2) δ (1) ( ɛ (2) ɛ (1)) ] 1 2g. (18) 1 g It is interesting that the ackground parameters the tangential compliance H have no influence on constraint (18), which depends only on the differences etween the anisotropic coefficients in the vertical symmetry planes, i.e., on ɛ (V ), δ (V ), γ (V ). As a result, equation (18) is analogous to the constraint given y Tsvankin (1997a) part I for HTI media due to rotationally invariant fractures (i.e., for H = V ). Horizontal symmetry plane [x 1, x 2 ]. The only Tsvankin s (1997) anisotropic coefficient defined in the horizontal plane is δ (3) [equation (B-9)]. After linearization, it ecomes δ (3) = 2g[ N H ]. (19) The parameter δ (3) does not contain any ackground anisotropic coefficients ecause [x 1, x 2 ] is the isotropy plane of the VTI medium. Since δ (3) is defined with respect to the x 1 -axis, which is normal to the fractures, equation (19) coincides with the expression for the generic Thomsen coefficient δ otained for VTI media due to horizontal fractures y Schoenerg Douma (1988). The linearized anellipticity coefficient η (3) [equation (B-12)] has the form η (3) = 2g[ H g N ]. (20) Estimation of the anisotropic parameters from reflection data Inversion of reflection data for the effective parameters of orthorhomic media is discussed y Grechka Tsvankin (1998, 1999), Rüger (1998), Grechka et al. (1999). We now give a rief overview of their parameter-estimation methods operating with either wide-azimuth P-wave data or a comination of azimuthally dependent signatures of P- PS-waves. P-wave signatures. NMO velocity of P-waves in a horizontal orthorhomic layer is descried y an ellipse with the axes in the vertical symmetry planes (Grechka Tsvankin, 1998). Since for oth orthorhomic models considered here the orientation of the symmetry planes is determined y the strike of the fractures, the axes of the P-wave NMO ellipse yield the fracture azimuth(s). In the orthorhomic model due to a single fracture set, δ (1) >δ (2) [see equation (32)] the semimajor axis of the P-wave NMO ellipse points in the direction of the fracture plane. This result holds for horizontal transverse isotropy as well (part I). If δ (1) = δ (2) the P-wave NMO velocity from a horizontal reflector is azimuthally independent (i.e., the ellipse degenerates into a circle), the azimuths of the symmetry planes can e otained from the NMO ellipse of a dipping event (Grechka Tsvankin, 1999). P-wave reflection traveltimes from dipping interfaces or the azimuthal variation of nonhyperolic moveout (Al-Dajani et al., 1998) can also e inverted for the anellipticity coefficients η (1), η (2), η (3) [equations (13), (17), (20)]. Nonnegligile values of oth η (1) η (2) help to detect the presence of anisotropy in the ackground discriminate etween HTI orthorhomic models. For HTI media, all anisotropic coefficients (including η) in one of the symmetry planes should go to zero. The semiaxes V (1) (2) P,nmo V P,nmo of the P-wave NMO ellipse from a horizontal reflector are given y (Grechka Tsvankin, 1998) V (i) P,nmo = V P δ (i), (i = 1, 2). (21) Hence, V (i) P,nmo can e comined with the vertical velocity V P0 to determine δ (1) δ (2).IfV P0 is unknown, the P-wave NMO ellipse constrains the difference χ etween the two δ coefficients: ( (2) ) 2 ( (1) ) 2 V P,nmo V P,nmo χ ( (2) ) 2 ( (1) ) 2 = δ(2) δ (1) V P,nmo + V 1 + δ (2) + δ (1) δ(2) δ (1). P,nmo (22) Additional information for parameter estimation is provided y prestack amplitudes of P-waves. In the weak-anisotropy approximation, the variation of the P-wave amplitude variation with offset (AVO) gradient etween the vertical symmetry planes is governed y the expression δ (2) δ (1) 8g(γ (2) γ (1) ) (Rüger, 1998). Therefore, if the squared velocity ratio g is known δ (2) δ (1) has een found from the P-wave NMO ellipse [equation (22)], the AVO gradient yields an estimate of γ (2) γ (1). P- PS-wave signatures. Although P-wave data alone may e used to estimate a suset of the effective parameters of orthorhomic media, it is highly eneficial to comine P-wave traveltimes or amplitudes with the signatures of shear or converted waves. Since S-waves are not generated in most exploration surveys, we emphasize the joint inversion of P-waves P- to S-mode conversions. The vertical traveltimes of P- PS-waves give a direct estimate of the vertical-velocity ratio needed to compute the fracture weaknesses. The shearwave splitting parameter at vertical incidence, conventionally evaluated from the time delays etween the fast slow shear or converted waves, is close to γ (2) γ (1). Also, the difference etween the AVO gradients of the PS-wave in the vertical symmetry planes can e comined with the azimuthal variation of the P-wave AVO gradient or the P-wave NMO ellipse to otain another estimate of γ (2) γ (1). A more detailed discussion of the azimuthal AVO inversion of PS-waves is given in part I. Grechka et al. (1999) outline the following methodology of the joint moveout inversion of P- PS-waves. Similar to pure modes, the azimuthally varying NMO velocity of either split PS-wave in a horizontal orthorhomic layer is descried y an ellipse aligned with the vertical symmetry planes. The semiaxes of the NMO ellipses of the P- two split PS-waves (i.e., the symmetry-plane NMO velocities) can e used to reconstruct the NMO ellipses of the pure shear waves S 1 S 2. Assuming that wave S 1 represents an SV (in-plane polarized) mode in the [x 2, x 3 ]-plane [the superscript (1)], the semiaxes of

5 Fractured Orthorhomic Media 1807 the shear-wave ellipses are expressed as V (2) S1, nmo = V S γ (2) = V (1) S2, nmo, (23) V (1) S1, nmo = V S σ (1), (24) V (2) S2, nmo = V S σ (2), (25) V (1) S2, nmo = V S γ (1) = V (2) S1, nmo. (26) Here σ (1) σ (2) are the anisotropic coefficients ( ) 2 σ (1) VP0 ( ɛ (1) δ (1)) (27) V S1 σ (2) ( VP0 V S2 ) 2 ( ɛ (2) δ (2)), (28) V S1 V S2 are the vertical velocities of the split shear waves S 1 S 2 : 1 + 2γ V S1 = V (1) S2 (29) 1 + 2γ (2) V S2 = V S0. (30) If one of the vertical velocities or the reflector depth is known the anisotropic parameters δ (1,2) have een otained from P-wave moveout, the shear-wave NMO ellipses make it possile to find four additional coefficients ɛ (1,2) γ (1,2).The only anisotropic parameter not constrained y conventionalspread normal moveout in a horizontal orthorhomic layer is δ (3). Estimation of fracture parameters Inversion for the weaknesses. Since we are mostly interested in evaluating the properties of the fracture set, it is convenient to remove the influence of the ackground medium at the outset of the inversion procedure. Inspection of equations (10) (17) shows that the contriution of the VTI ackground parameters can e eliminated y computing the difference etween the anisotropic coefficients in the planes parallel orthogonal to the fractures: ɛ (2) ɛ (1) = 2g(1 g) N, (31) δ (2) δ (1) = 2g[(1 2g) N + V ] χ, (32) γ (2) γ (1) = V 2, (33) η (2) η (1) = 2g[ V g N ]. (34) Equations (31) (34) are identical to the expressions for ɛ (V ), δ (V ), γ (V ), η (V ) (respectively) in HTI media due to a single set of vertical rotationally invariant fractures with the weaknesses N V (part I). Therefore, the weaknesses can e determined from equations (31) (34) using the HTI expressions descried in detail in part I. We emphasize that otaining N V requires knowledge of any two of the differences ɛ (2) ɛ (1), δ (2) δ (1), γ (2) γ (1), η (2) η (1). As discussed, χ δ (2) δ (1) can e estimated from the elongation of the P-wave NMO ellipse for horizontal events [equation (22)]. Also, η (2) η (1) can e otained using the NMO velocities of dipping P-events or nonhyperolic moveout. Thus, P-wave moveout data provide sufficient information to determine the weaknesses N V from equations (32) (34): ( δ (2) δ (1)) + ( η (2) η (1)) N =, (35) 2g(1 g) V = 1 2(1 g) [ 1 2g ( η (2) η (1)) ( δ (2) δ (1))]. g (36) In principle, the inversion ased on equations (35) (36) requires knowledge of the squared vertical-velocity ratio g, which cannot e found without shear or converted-wave data. However, numerical tests ( the results of part I) show that even a relatively rough estimate of g is sufficient for recovering the weaknesses with acceptale accuracy. P-wave moveout data may provide not just the difference etween η (2) η (1) ut also the individual values of these coefficients. Therefore, in addition to sustituting η (2) η (1) into the equations for the weaknesses, we can use the approximate relation (13), η (1) = η, to determine the η coefficient in the ackground VTI model. If the P-wave moveout information is limited to the NMO ellipse from a horizontal reflector, it is possile to otain the difference γ (2) γ (1) y including the P-wave AVO gradients in the directions parallel perpendicular to the fractures. The algorithm ased on the NMO ellipses of horizontal events AVO gradients is identical to the one descried in part I for HTI media. The presence of anisotropy in the ackground has no influence either on estimating δ (2) δ (1) γ (2) γ (1) or on inverting these differences [equations (32) (33)] for the weaknesses N V. In multicomponent surveys, the vertical traveltimes of PSwaves (in comination with P-wave data) provide estimates of g the splitting coefficient γ (2) γ (1). Thus, another possile set of input parameters includes the P-wave NMO ellipse from a horizontal reflector (yielding δ (2) δ (1) ) the vertical traveltimes of converted modes. Note that the weakness H does not enter any of the differences (31) (34), despite the fact that it appears in equations (12) (16) for γ (1) γ (2). However, even the individual values of γ (1) γ (2) constrain the comination γ H /2 ut not H separately. The only source of information aout H is η (3) or δ (3) [equations (19) (20)], which can e estimated using the NMO velocities of P-wave reflections from dipping interfaces (Grechka Tsvankin, 1999): H = η(3) 2g + g N. (37) If dipping events are not availale, the data cannot e inverted for the coefficient η (3), therefore, for the weakness H. In this case, a reasonale simplifying assumption is that the fractures are rotationally invariant V = H. Then the numer of independent medium parameters reduces to seven, δ (3) η (3) can e expressed as δ (3) = δ (2) δ (1) 2 ( ɛ (2) ɛ (1)), (38) η (3) = η (2) η (1). (39)

6 1808 Bakulin et al. Interpretation of the weaknesses in terms of the physical properties of the fractures is discussed in part I. For rotationally invariant penny-shaped cracks, the tangential weakness V = H gives an estimate of the crack density, while the normal weakness N represents a sensitive, aleit nonunique, indicator of fluid content. Numerical examples. Although the weak-anisotropy approximations provide useful insight into the parameter estimation prolem, it is preferale to use the exact equations for actual inversion. In the examples elow, we compute the fracture weaknesses N, V, H, along with the anisotropic coefficient η of the ackground, y inverting the values of χ [equation (22)], η (1), η (2), η (3) presumaly extracted from P-wave seismic data. The goal of this numerical study is to examine the sensitivity of the inverted parameters to errors in the input data in the parameters of the ackground medium. Our nonlinear inversion algorithm is ased on equations (1) (3) for the stiffness elements on the exact expressions for the anisotropic coefficients given in Appendix B. The weakanisotropy approximations (13) (35) (37) are used only to otain the initial guesses for the weaknesses the coefficient η. The parameters of the ackground medium are g = V 2 S0 /V 2 P0 = 0.25, δ = 0.2, γ = 0.1(ɛ can e expressed through δ η ). Although the only ackground parameter in equations (35) (37) is the squared velocity ratio g, the anisotropic coefficients δ γ of the VTI medium are contained in the exact equations for the stiffnesses effective anisotropic parameters must e specified for the inversion. As shown elow, however, the inversion results are insensitive to variations of g, δ, γ within the range important in practice. The fracture set is defined y the weaknesses N = 0.5 V = H = 0.2, which approximately correspond to values for penny-shaped gas-filled cracks (see part I). The inversion results are displayed in Figure 1. For the correct input parameters ( the correct ackground parameters g, δ, γ ), the inversion algorithm produces accurate values of the weaknesses η. Errors in the measured anisotropic parameters of up to ±0.1 result in similar (often smaller) errors in the inverted values, indicating that the parameter estimation procedure is reasonaly stale. As expected from the analytic results, different inverted parameters are most sensitive to errors in different measured quantities. For example, errors in χ produce comparale distortions in all three weaknesses, reaching sometimes exceeding ±0.1 (Figure 1a). The anellipticity coefficient η is quite sensitive to errors in η (1) (Figure 1) is almost independent of the other input parameters, in accordance with the weak-anisotropy approximation (13). Errors in η (1) η (2) of aout ±0.1 cause smaller errors (±0.05) in the tangential weaknesses V H than in the normal weakness N (±0.1; Figures 1,c). The weakness H is sensitive primarily to errors in η (3) (Figure 1d), as suggested y equation (37). In the weak-anisotropy limit, N V are independent of η (3) [equations (35) (36)], which is confirmed y Figure 1d. In the second test, we examine the influence of errors in the ackground parameters g, δ, γ (which were assumed known a priori in the previous example) on the inversion FIG. 1. Numerical inversion of χ, η (1), η (2), η (3) for the fracture weaknesses N (dotted line), V (dash-dotted), H (dashed), the coefficient η of the ackground medium (solid). Errors of ±0.1 were introduced in (a) χ, () η (1), (c) η (2), (d) η (3), with the other input parameters held at the correct values. In the asence of errors, the inversion yields the correct values of the weaknesses η (large dots).

7 Fractured Orthorhomic Media 1809 results. Sustantial errors in g of ±0.1, or up to 40%, lead to relatively small distortions of only aout ±0.03 in all inverted parameters except N (for N the errors reach ±0.1; Figure 2a). The anisotropic coefficients δ γ have an even smaller influence on the estimated quantities, especially on the tangential compliances (Figure 2,c). This result is especially encouraging ecause δ γ are difficult to estimate from surface seismic data without information aout the vertical velocities or reflector depth. Estimation of ackground parameters. As discussed aove, the individual values of the anisotropic parameters ɛ (1,2), δ (1,2), γ (1,2) can e found y comining P-wave moveout data with the reflection traveltimes of PS-waves (provided the reflector depth is known). Then it is possile to estimate not just the fracture weaknesses ut also the squared vertical-velocity ratio g the ackground anisotropic parameters ɛ, δ, γ [equations (10) (12); to find γ, we assume V = H ]. For the numerical example in Figure 3, the input data included the vertical NMO velocities of P- split S-waves, where the shear-wave signatures were supposedly determined from P PS data. The NMO velocity of each mode was computed in three azimuthal directions with a step of 45 approximated with an ellipse to find the NMO velocities in the symmetry planes. Then the vertical velocities symmetryplane NMO velocities were comined to determine ɛ (1,2), δ (1,2), γ (1,2) from equations (21) (23) (26). To check the influence of measurement errors, we added Gaussian noise with a stard deviation of 2% to the vertical NMO velocities performed the inversion for 200 sets of the distorted input parameters. The algorithm is ased on the exact expressions for the anisotropic coefficients derived from equations (1) (3), with the starting model computed using the weak-anisotropy approximation. Since the inversion errors in Figure 3 are limited y ±0.05, the estimation of oth weaknesses all ackground parameters is sufficiently stale. FIG. 3. Inversion of the vertical NMO velocities of the P- split S-waves for the parameters of the fractures VTI ackground. The dots mark the correct values of the parameters; it is assumed that V = H. The ars correspond to ± one stard deviation in the inverted quantities caused y Gaussian noise in the input data with a stard deviation of 2%. FIG. 2. Influence of the ackground parameters (a) g = V 2 S0 /V 2 P0, () δ, (c) γ on the inversion results from Figure 1 for the correct values of χ η (1,2,3). Dotted line N ; dash-dotted V ; dashed H ; solid η.

8 1810 Bakulin et al. MODEL 2: TWO ORTHOGONAL FRACTURE SETS IN ISOTROPIC ROCK Another practically important fractured model with orthorhomic symmetry is composed of two orthogonal vertical fracture sets emedded in an isotropic or VTI ackground. To simplify the parameter estimation procedure, we assume that the fractures are rotationally invariant the ackground matrix is purely isotropic. Choosing the x 1 -axis to e perpendicular to the first set of fractures, we otain the compliance matrices for oth fracture systems in equations (A-3) (A-14) where, ecause of the rotational invariance, the compliances satisfy K V 1 = K H1 K T 1 K V 2 = K H2 K T 2. Building the effective matrix of the excess compliance [equation (A-12)] adding the compliance of the isotropic ackground yields the effective stiffness matrix [equation (A-1)] c 11 c 12 c c 12 c 22 c c 13 c 23 c c = = c, c c c c 66 (40) where 0 is the 3 3 zero matrix the matrices c 1 c 2 are given y c 1 = (λ + 2µ)l 1 m 3 λl 1 m 1 λl 1 m 2 1 λl 1 m 1 (λ + 2µ)l 3 m 1 λl 2 m 1, d λl 1 m 2 λl 2 m 1 (λ + 2µ)(l 3 m 3 l 4 ) (41) c 2 = µ(1 T 2 ) µ(1 T 1 ) µ (1 T 1)(1 T 2 ). (1 T 1 T 2 ) (42) Here, λ µ are the Lamé parameters of the ackground medium l 1 = 1 N1, l 2 = 1 r N1, l 3 = 1 r 2 N1, l 4 = 4r 2 g 2 N1 N2, m 1 = 1 N2, m 2 = 1 r N2, m 3 = 1 r 2 N2, g = µ/(λ + 2µ) = V 2 / S V 2 P, r = 1 2g, (43) d = 1 r 2 N1 N2. The values Ni Ti (i=1,2) are the normal tangential fracture weaknesses (see part I) related to the fracture compliances y the equations Ni = K Ni(λ+2µ) (44) 1+ K Ni (λ+2µ) Ti = K Tiµ 1+ K Ti µ, (45) which represent a special case (valid for an isotropic ackground) of equations (A-4). Since oth fracture planes the horizontal plane constitute three orthogonal planes of mirror symmetry, the effective medium must e orthorhomic with the nine stiffness elements [equation (40)]. Our model, however, represents a special case of general orthorhomic media with only six independent parameters: λ, µ, N1, T 1, N2, T 2. As follows from equations (41) (42), the three additional relationships (constraints) etween the stiffnesses are c 12 (c 33 + c 23 ) = c 13 (c 22 + c 23 ), (46) 2 (c 11 + c 13 ) = ( c 11 c 33 c13 2 ) ( c 44 + c 55 1 ), (47) c 44 c 55 c 66 2(c 22 + c 23 ) = ( c 22 c 33 c 2 23 ) ( c 44 + c 55 1 ). (48) c 44 c 55 c 66 Models with two identical orthogonal fracture sets, characterized y fewer independent parameters, are discussed in Appendix C. Anisotropic coefficients in the weak-anisotropy limit Equations (40) (43), comined with the definitions from Appendix B, can e used to express Tsvankin s (1997) anisotropic coefficients in terms of the fracture weaknesses Ti Ni. Here we restrict ourselves to linearized expressions otained in the limit of small fracture compliances T 1,2 1 N1,2 1. The result of this linearization can e predicted from equation (A-13): since in the linear approximation fracture systems are not influenced y each other, the effective orthorhomic medium is composed of two HTI media produced y each fracture set individually. Symmetry plane [x 2, x 3 ]. The anisotropic parameters with the superscript (1) are defined in the symmetry plane [x 2, x 3 ], which is parallel to the first set of fractures orthogonal to the second one. In the asence of the second set, the [x 2, x 3 ]-plane would coincide with the isotropy plane of the HTI medium associated with the first fracture system. Therefore, we can expect these parameters to e largely influenced y the second set of fractures orthogonal to the x 2 -axis. Indeed, the linearized ɛ, δ, γ, η coefficients in the [x 2, x 3 ] plane depend only on the weaknesses N2 T 2 : ɛ (1) = 2g(1 g) N2, (49) δ (1) = 2g[(1 2g) N2 + T 2 ], (50) γ (1) = T2 2, (51) η (1) = 2g[ T 2 g N2 ], (52) where g VS 2/V P 2 is the ratio of the squared S- P-wave velocities in the ackground. Equations (49) (52) coincide with the expressions for the anisotropic coefficients ɛ (V), δ (V), γ (V), η (V) of the HTI model associated with the second fracture set.

9 Fractured Orthorhomic Media 1811 Symmetry plane [x 1, x 3 ]. Likewise, the linearized anisotropic coefficients in the symmetry plane [x 1, x 3 ] are governed y the weaknesses of the first fracture set. As follows from the symmetry of the model, the expressions for ɛ (2), δ (2), γ (2), η (2) are fully analogous to equations (49) (52): ɛ (2) = 2g(1 g) N1, (53) δ (2) = 2g[(1 2g) N1 + T 1 ], (54) γ (2) = T1 2, (55) η (2) = 2g[ T 1 g N1 ]. (56) Symmetry plane [x 1, x 2 ]. The remaining anisotropic coefficient δ (3) defined in the horizontal symmetry plane [x 1, x 2 ]is given y δ (3) = 2g[ N1 T 1 ] 2g[(1 2g) N2 + T 2 ]. (57) The linearized parameter δ (3) is equal to the sum of the generic Thomsen coefficient δ caused y horizontal fractures in VTI media (the term 2g[ N1 T 1 ]; see Schoenerg Douma, 1988) the HTI coefficient δ (V ) is due to vertical fractures (part I). To explain this result, recall that δ (3) is defined with respect to the x 1 -axis, which is orthogonal to the first fracture set (so this set ecomes horizontal if the x 1 -axis is made vertical in VTI media) lies in the planes of the second set of fractures (making this set vertical). The anellipticity coefficient η (3) [equation (B-12] in the horizontal plane is η (3) = η (1) + η (2). (58) Relations etween anisotropic coefficients. The nine stiffnesses of the effective orthorhomic model contain only six independent quantities, which leads to the three constraints (46) (48). Rewriting equations (46) (47) through Tsvankin s (1997) parameters yields γ (i) = 1 4g [ δ (i) ɛ (i) 1 2g 1 g ], (i = 1, 2), (59) where quadratic higher-order terms in the anisotropic coefficients were dropped. The same result can e otained from the linearized expressions for the anisotropic coefficients given aove. Equation (59) is identical to the relationship etween ɛ (V ), δ (V ), γ (V ) of HTI media (Tsvankin, 1997a; part I). The remaining constraint (48) takes the form δ (3) = δ (1) + δ (2) 2ɛ (2). (60) Equations (59) (60) show that in Tsvankin s notation our orthorhomic model can e fully descried y the two vertical velocities four anisotropic coefficients δ (1,2) ɛ (1,2). In the special case of penny-shaped gas-filled fractures, the normal tangential compliances are equal to each other (K N1 = K T 1 K N2 = K T 2 ), as noticed y Schoenerg Sayers (1995), the anellipticity parameters in the symmetry planes go to zero: Estimation of fracture parameters η (1) = η (2) = η (3) = 0. (61) The structure of equations (49) (56) for the anisotropic coefficients suggests that estimation of the weaknesses of the orthogonal fracture sets can e decomposed into two HTItype inversions in the vertical symmetry planes discussed in part I. Here, we verify the accuracy of the weak-anisotropy approximations outline two possile strategies for otaining the weaknesses from surface reflection data. One is ased on azimuthally dependent P-wave reflection moveout alone; the other uses oth P- converted-wave data. In contrast to the model with a single fracture set, we must know one of the vertical velocities (or reflector depth) ecause the inversion algorithm requires the individual values of Tsvankin s anisotropic coefficients rather than their differences. P-wave inversion. As for the model with a single fracture set, the fracture orientation can e determined directly from the P-wave NMO ellipse (or the NMO ellipses of converted waves), unless the ellipse degenerates into a circle. This some other special cases are discussed in Appendix D. Suppose the parameters δ (1) δ (2) have een found using the semiaxes of the P-wave NMO ellipse from a horizontal reflector the vertical velocity [equations (21)]. Comining the δ coefficients with the anellipticity parameters η (1) η (2) (also determined from P-wave moveout) allows us to solve equations (50), (52), (54), (56) for the weaknesses N1 = δ(2) + η (2) 2g(1 g), (62) [ 1 1 2g T 1 = η (2) δ ], (2) 2(1 g) g (63) N2 = δ(1) + η (1) 2g(1 g), (64) T 2 = [ 1 1 2g η (1) δ ]. (1) (65) 2(1 g) g The equations for each pair of weaknesses [(62) (63) (64) (65)] are identical to those otained in part I for HTI media due to a single set of rotationally invariant fractures. To test the accuracy of the weak anisotropy approximations (62) (65), we computed the exact anisotropic coefficients δ (1,2) η (1,2) for three fractured models in Tale 1 [using equations (40) (42) the definitions from Appendix B] inverted them for the weaknesses in the limit of weak anisotropy. Tale 1 shows that approximations (62) (65) give reasonaly good estimates of the fracture weaknesses. The only significant error, in the tangential weakness of the second fracture set, is unrelated to the first set ecause it remains the same for the corresponding HTI model (third row in Tale 1). Inversion of P- PS-wave data. If dipping events are not availale CMP spreads are not sufficiently long for using nonhyperolic moveout, it may e possile to find the anisotropic coefficients ɛ (1,2), δ (1,2), γ (1,2) y comining P- PS-wave data [equations (23) (26)]. Since these anisotropic parameters depend on only four fracture weaknesses, P- PS-wave data (including the vertical velocities or reflector depth) provide useful redundancy in the inversion procedure. In the weak-anisotropy approximation, the

10 1812 Bakulin et al. weaknesses can e computed, for example, as ( (2) ) 2 V T 1 = 2γ (2) S1,nmo = 1, (66) V S1 ( ) (1) 2 V T 2 = 2γ (1) S2,nmo = 1, (67) V S2 [ N1 = 1 T1 + δ(2) 1 2g 2g ], (68) [ N2 = 1 T2 + δ(1) 1 2g 2g ]. (69) Figure 4 shows the results of numerical inversion ased on NMO equations (23) (26) the exact expressions for Tsvankin s anisotropic coefficients in terms of the fracture weaknesses. Similar to the numerical example in Figure 3, the vertical velocities azimuthally varying NMO velocities of P- S-waves (input data) were distorted y Gaussian noise with a stard deviation of 2%. The initial model was found from the weak-anisotropy approximations (66) (69). The stard deviations for the inverted weaknesses are relatively small, with errors in the tangential compliances T 1 T 2 eing somewhat lower than those in N1 N2. DISCUSSION AND CONCLUSIONS Applying the linear-slip theory developed y Schoenerg (1980, 1983), we studied two types of orthorhomic media elieved to e representative of naturally fractured reservoirs. The first model contains a single set of vertical fractures emedded in a VTI ackground (e.g., the ackground Tale 1. Comparison of the actual fracture parameters with those estimated y inverting the exact anisotropic coefficients [equations (40) (42) Appendix B] using the linearized equations (62) (65); the squared velocity ratio g = The first model is composed of two orthogonal fracture sets in an isotropic ackground; the second third models contain only one fracture set (i.e., they have the HTI symmetry). Model Fracture parameters N1 T 1 N2 T 2 1 Actual Estimated Actual Estimated Actual Estimated FIG. 4. Inversion of the P- S-wave vertical NMO velocities for the model with two orthogonal fracture sets. The input data were contaminated y Gaussian noise with a stard deviation of 2%, the inversion was repeated 200 times for different realizations of the input parameters. The ars correspond to ± one stard deviation in the inverted quantities. anisotropy may result from fine layering), while the second is produced y two orthogonal systems of rotationally invariant vertical fractures in an isotropic host rock. For oth effective models we otained Tsvankin s (1997) anisotropic parameters, which capture the cominations of the stiffness coefficients responsile for commonly measured seismic signatures. To gain a etter understing of the influence of fractures on the effective medium, we simplified the anisotropic parameters under the assumption of weak ackground fractureinduced anisotropy. For the model with a single fracture set, the anisotropic parameters of the HTI medium due to the fractures are added to the ackground coefficients to produce the linearized effective parameters of the orthorhomic medium. Therefore, y computing the difference etween the effective coefficients defined in the vertical symmetry planes, we eliminate the influence of the ackground reduce the inverse prolem to that for horizontal transverse isotropy (part I). The information necessary for this inversion procedure can e otained from azimuthally dependent P-wave reflection traveltimes alone if dipping events or nonhyperolic moveout are availale (also, it is necessary to have an estimate of the ratio of the P- S-wave vertical velocities). Alternatively, the inversion can e performed y comining the P-wave NMO ellipse from a horizontal reflector with other data, such as the azimuthally varying P-wave AVO gradient or the vertical traveltimes, possily, NMO velocities of the split converted (PS) modes. For weak anisotropy, the algorithm ased on the NMO ellipses AVO gradients of P- PS-waves reflected from horizontal interfaces is identical to that outlined in part I for HTI media (i.e., it is independent of the presence of anisotropy in the ackground). In the case of two orthogonal fracture sets, the linearized expressions for the effective anisotropic coefficients in each vertical symmetry plane contain only the contriution of the fractures orthogonal to this plane. As a result, the inversion for the fracture weaknesses splits into two separate inversion procedures in the symmetry planes, which can e carried out using the HTI algorithm of part I. In contrast to the model with a single fracture set, however, determination of the weaknesses of oth fracture sets requires knowledge of the vertical velocities in addition to the azimuthally varying surface seismic signatures. Although oth effective media considered here have orthorhomic symmetry, the results of our analysis indicate several possile ways to identify the correct underlying physical model. In oth cases, the stiffness tensor is descried y fewer independent parameters than for general orthorhomic media, the anisotropic coefficients for the models with one two fracture sets satisfy different constraints. Another useful criterion is the sign of the anisotropic parameters. The coefficients ɛ (1,2), δ (1,2), γ (1,2) for the model with two orthogonal systems of fractures in an isotropic ackground are negative [equations (49) (51) (53) (55)]. In contrast, ɛ γ ( often δ as well) defined in the fracture plane of the model with a single fracture set in a VTI ackground are usually positive. Also, we can distinguish etween the two models y comparing the polarization direction of the vertically traveling fast shear wave with the orientation of the semimajor axis of the P- wave NMO ellipse. For one set of fractures (in either isotropic or VTI ackground), they always coincide with each other; for two systems of fractures with different fluid content, they may ecome orthogonal. This happens, for example, if the two

11 Fractured Orthorhomic Media 1813 inequalities T 1 > T2 (1 2g) N1 + T 1 < (1 2g) N2 + T 2 (or, equivalently, δ (2) >δ (1) ) are satisfied simultaneously. ACKNOWLEDGMENTS This research was carried out during a visit of Andrey Bakulin to the Center for Wave Phenomena (CWP), Colorado School of Mines, We are grateful to memers of the A(nisotropy)-team of CWP for helpful discussions to Andreas Rüger (Lmark) for his review of the manuscript. The support for this work was provided y the memers of the Consortium Project on Seismic Inverse Methods for Complex Structures at CWP y the U.S. Department of Energy (award #DE-FG03-98ER14908). REFERENCES Al-Dajani, A., Tsvankin, I., Toksöz, N., 1998, Nonhyperolic reflection moveout in azimuthally anisotropic media: 68th Ann. Internat. Mtg., Soc. Expl. Geophys., Exped Astracts, Alkhalifah, T., Tsvankin, I., 1995, Velocity analysis in transversely isotropic media: Geophysics, 60, Bakulin, A. V., Molotkov, L. A., 1998, Effective models of fractured porous media: St. Petersurg Univ. Press (in Russian). Bakulin, A., Tsvankin, I., Grechka, V., 2000, Estimation of fracture parameters from reflection seismic data Part I: HTI model due to a single fracture set: Geophysics, 65, , this issue. Grechka, V., Tsvankin, I., 1998, 3-D description of normal moveout in anisotropic inhomogeneous media: Geophysics, 63, , 3-D moveout velocity analysis parameter estimation for orthorhomic media: Geophysics, 64, Grechka, V., Theophanis, S., Tsvankin, I., 1999, Joint inversion of P- PS-waves in orthorhomic media: Theory a physicalmodeling study: Geophysics, 64, Hudson, J. A., 1980, Overall properties of a cracked solid: Math. Proc. Cam. Phil. Soc., 88, , Wave speeds attenuation of elastic waves in material containing cracks: Geophys. J. Roy. Astr. Soc., 64, , Seismic wave propagation through material containing partially saturated cracks: Geophys. J., 92, Molotkov, L. A., Bakulin, A. V., 1997, An effective model of a fractured medium with fractures modeled y the surfaces of discontinuity of displacements: J. Math. Sci., 86, Nichols, D., Muir, F., Schoenerg, M., 1989, Elastic properties of rocks with multiple sets of fractures: 59th Ann. Internat. Mtg., Soc. Expl. Geophys., Exped Astracts, Rüger, A., 1997, P-wave reflection coefficients for transversely isotropic models with vertical horizontal axis of symmetry: Geophysics, 62, , Variation of P-wave reflectivity with offset azimuth in anisotropic media: Geophysics, 63, Schoenerg, M., 1980, Elastic wave ehavior across linear slip interfaces: J. Acoust. Soc. Am., 68, , Reflection of elastic waves from periodically stratified media with interfacial slip: Geophys. Prosp., 31, Schoenerg, M., Douma, J., 1988, Elastic wave propagation in media with parallel fractures aligned cracks: Geophys. Prosp., 36, Schoenerg, M., Helig, K., 1997, Orthorhomic media: Modeling elastic wave ehavior in a vertically fractured earth: Geophysics, 62, Schoenerg, M., Muir, F., 1989, A calculus for finely layered anisotropic media: Geophysics, 54, Schoenerg, M., Sayers, C., 1995, Seismic anisotropy of fractured rock: Geophysics, 60, Thomsen, L., 1986, Weak elastic anisotropy: Geophysics, 51, , Elastic anisotropy due to aligned cracks in porous rock: Geophys. Prosp., 43, Tsvankin, I., 1997a, Reflection moveout parameter estimation for horizontal transverse isotropy: Geophysics, 62, , Anisotropic parameters P-wave velocity for orthorhomic media: Geophysics, 62, Winterstein, D. F., 1990, Velocity anisotropy terminology for geophysicists: Geophysics, 55, APPENDIX A COMPLIANCE FORMALISM FOR FRACTURED MEDIA Here we review oth exact approximate methods of otaining effective parameters of fractured media using the results of Schoenerg (1980, 1983), Schoenerg Douma (1988), Schoenerg Muir (1989), Nichols et al. (1989), Molotkov Bakulin (1997). A discussion of different approaches to effective medium theory for fractured models can e found in part I. To simplify the derivation of the effective elastic parameters of fractured media, it is convenient to use the compliances s instead of the stiffnesses c. The effective compliance matrix of a fractured medium can e written as the sum of the ackground compliance s the so-called matrix of excess fracture compliance s f : c 1 s = s + s f. (A-1) If the ackground is VTI, then the stiffness matrix is given y c 11 c 12 c c 12 c 11 c c s 1 c 13 c 13 c =, c c c 66 (A-2) where c 12 = c 11 2c 66. The matrix s f of the excess compliance of a fracture set with the normal in the x 1 -direction can e written as K N s f =, (A-3) K V K H where K N is the normal fracture compliance K V K H are the two shear compliances in the vertical horizontal directions. The matrix s f is no longer diagonal if the fractures are corrugated (part I; for more details, see part III of this series). Since K V K H, fractures descried y matrix (A-3) are sometimes called orthorhomic (Schoenerg Douma, 1988). It is convenient to replace the excess fracture compliances K N, K V, K H y the following dimensionless quantities introduced y Schoenerg Helig (1997): N = K N c K N c 11, V = K Vc K V c 44, H = K H c K H c 66. (A-4)

12 1814 Bakulin et al. Then equation (A-3) ecomes s f = N c 11 (1 N ) V c 44 (1 V ) H c 66 (1 H ). (A-5) We call N, V, H the normal, vertical, horizontal weaknesses introduced y the fractures (Bakulin Molotkov, 1998). The weaknesses vary from 0 to 1, with the zero value corresponding to unfractured media unity descriing heavily fractured media in which the P- (for N = 1) or S-wave (for V = 1or H =1) velocity vanishes for propagation across the fractures (part I). Sustituting equations (A-2) (A-5) into equation (A-1) yields the effective stiffness matrix for a single fracture set emedded in a VTI ackground (Schoenerg Helig, 1997): ) ( c1 0 c =, (A-6) 0 c 2 where c 1 = c 11 (1 N ) c 12 (1 N ) c 13 (1 N ) c 2 ( 12 c 12 (1 N ) c 11 c 12 N c 13 1 N c 11 c 11 ( ) c 13 (1 N ) c 13 1 N c 12 c 11 c 33 N c 2 13 c 11 c c 2 = 0 c 44 (1 V ) 0, 0 0 c 66 (1 H ) ), (A-7) (A-8) 0 is the 3 3 zero matrix. Alternatively, the same result can e otained using the series c [s + s f ] 1 = [( 1 I + s f s ) ] 1 s = c [I + s f c ] 1 = c ( s f c ) k, (A-9) k=0 where I is the 6 6 identity matrix. Series (A-9) converges to equation (A-6) if all eigenvalues of the matrix s f c have asolute values less than 1. This is always the case if the weaknesses N, V, H, which define nonzero eigenvalues of the matrix s f [see equation (A-5)], are sufficiently small. Equation (A-9) can serve as the asis for developing useful approximations for the effective stiffness matrix c. If the crack density (or fracture intensity) is small the weaknesses { N, V, H } 1, we may truncate series (A-9) y keeping only linear terms with respect to N, V, H : c c c s f c. (A-10) It is interesting to note that equation (A-10) ecomes exact if we replace the matrix s f [equation (A-5)] y its linearized version: N /c s lin f = V /c H /c 66 (A-11) Approximation (A-10) can e used to otain two important results. First, it can e extended in a straightforward way to multiple fracture sets (Nichols et al., 1989). If the effective compliance represents the sum of the excess compliances of N fracture sets, N s f = s fi, (A-12) i=1 then for the effective stiffness c we have from equation (A-10) N c c c s fi c. (A-13) i=1 The compliance matrix s fi cannot e descried y equation (A-3) if the normal n i to the ith fracture set does not coincide with the x 1 -axis. The matrices s fi for aritrary orientation of n i may e otained from equation (A-3) using the so-called Bond rotation (Winterstein, 1990). For example, the compliance matrix for a fracture set with the normal n i = [0, 1, 0] has the form 0 K N s fi =. (A-14) K V K H Second, equation (A-10) is well suited for deriving the weakanisotropy approximation for the stiffnesses of effective media formed y fractures emedded in an anisotropic ackground. We assume that the ackground medium is weakly anisotropic, so that c = c iso + εc ani, (A-15) where c iso is the stiffness matrix of a purely isotropic solid that approximates c in some sense, c ani is the anisotropic portion

13 Fractured Orthorhomic Media 1815 of c, ε 1. We also assume that the elements of the fracture compliance matrix are of the same order as ε can e denoted as εs f. If we represent the effective stiffness matrix c in a form similar to equation (A-15), equation (A-10) can e rewritten as c iso + εc ani c iso + εc ani ( c iso Collecting the terms linear in ε yields + εc ani ) ( εs f c iso + εc ani ). (A-16) c ani = c ani c iso s f c iso. (A-17) This equation, valid in the weak anisotropy limit, can e loosely interpreted in the following way: the anisotropy c ani of an effective medium containing fractures in an anisotropic ackground descried y c is equal to the sum of the ackground anisotropy c ani the anisotropy caused y the fractures emedded into any isotropic medium with c iso sufficiently close to c [see equation (A-13)]. The matrix c iso must satisfy the inequality ( c / c iso ) 1 <ε. APPENDIX B ANISOTROPIC PARAMETERS FOR ORTHORHOMBIC MEDIA The asic set of the anisotropic parameters for orthorhomic media has een introduced y Tsvankin (1997). His notation contains the vertical velocities of the P-wave one of the S-waves seven dimensionless Thomsen-type (1986) anisotropic coefficients. The definitions of those parameters in terms of the stiffnesses c ij density ρ are given elow. V P0 the P-wave vertical velocity: c33 V P0 (ρ is density). (B-1) ρ V S0 the velocity of the vertically traveling S-wave polarized in the x 1 -direction: c55 V S0 ρ. (B-2) ɛ (2) the VTI parameter ɛ in the [x 1, x 3 ] symmetry plane normal to the x 2 -axis [this explains the superscript (2)]: ɛ (2) c 11 c 33 2c 33. (B-3) δ (2) the VTI parameter δ in the [x 1, x 3 ] plane: δ (2) (c 13 + c 55 ) 2 (c 33 c 55 ) 2. (B-4) 2c 33 (c 33 c 55 ) γ (2) the VTI parameter γ in the [x 1, x 3 ] plane: γ (2) c 66 c 44 2c 44. (B-5) ɛ (1) the VTI parameter ɛ in the [x 2, x 3 ] symmetry plane: δ (3) the VTI parameter δ in the [x 1, x 2 ] plane (x 1 plays the role of the symmetry axis): δ (3) (c 12 + c 66 ) 2 (c 11 c 66 ) 2. (B-9) 2c 11 (c 11 c 66 ) These nine parameters fully descrie wave propagation in general orthorhomic media. In particular applications, however, it is convenient to operate with specific cominations of Tsvankin s parameters. For example, P-wave NMO velocity from dipping reflectors depends on three coefficients η, which determine the anellipticity of the P-wave slowness in the symmetry planes (Grechka Tsvankin, 1999). The definitions of η (1,2,3) are analogous to that of the Alkhalifah Tsvankin (1995) coefficient η in VTI media: η (1) the VTI parameter η in the [x 2, x 3 ] plane: η (1) ɛ(1) δ (1). (B-10) 1 + 2δ (1) η (2) the VTI parameter η in the [x 1, x 3 ] plane: η (2) ɛ(2) δ (2). (B-11) 1 + 2δ (2) η (3) the VTI parameter η in the [x 1, x 2 ] plane: η (3) ɛ(1) ɛ (2) δ (3)( 1 + 2ɛ (2)) ( )( 1 + 2ɛ (2) 1 + 2δ (3)). (B-12) Vertical transverse isotropy may e considered as a special case of orthorhomic media, where ɛ (1) c 22 c 33 2c 33. (B-6) δ (1) the VTI parameter δ in the [x 2, x 3 ] plane: δ (1) (c 23 + c 44 ) 2 (c 33 c 44 ) 2. (B-7) 2c 33 (c 33 c 44 ) γ (1) the VTI parameter γ in the [x 2, x 3 ] plane: γ (1) c 66 c 55 2c 55. (B-8) ɛ (1) = ɛ (2) = ɛ, δ (1) = δ (2) = δ, γ (1) = γ (2) = γ, δ (3) = 0,, as a consequence, η (1) = η (2) = η, η (3) = 0. (B-13) (B-14) (B-15) (B-16) (B-17) (B-18)

14 1816 Bakulin et al. APPENDIX C TWO IDENTICAL FRACTURE SETS If the model includes two identical orthogonal fracture sets in a purely isotropic ackground, only four effective stiffnesses out of nine remain independent ecause the stiffness matrix depends on just four quantities: λ, µ, N1 = N2 N, T 1 = T 2 T. From equations (40) (43) it follows that in this case c 11 = c 22, c 13 = c 23, c 44 = c 55. Constraints (47) (48) ecome identical, together with equation (46), may e reduced to c 12 (c 33 + c 13 ) = c 13 (c 11 + c 13 ) (C-1) 2(c 11 + c 13 )c 44 c 66 = (2c 66 c 44 ) ( c 11 c 33 c 2 13). (C-2) We can call this medium quasi-vti ecause its stiffness matrix is close to the one for vertical transverse isotropy. Unlike real VTI media, however, the stiffnesses of the quasi-vti model satisfy constraints (C-1) (C-2), which replace the VTI relationship c 11 = c c 66. Indeed, the comination of the stiffnesses that must vanish in VTI media can e written in the exact form as c 11 2c 66 c 12 = 4µ 2 (K T K N ) (1 + 2µK N )(1 + 2 µk T ). (C-3) Hence, ecause of the difference etween the normal shear compliances, the model with two identical fracture sets does not have the VTI symmetry. The anisotropic coefficients of quasi-vti media are descried y equations (B-3) (B-5), (B-13) (B-15), δ (3) = 4µ 2 (λ+µ)(k T K N ) λ 2µ (λ + µ)k [ N µ + 2µ(λ + µ)(k T K N ) + λ(1 + 4µK N ) 1 + 2µK N + 2µ 2 K T 1 + 2(λ + µ)k N 1 + 2µK N ] 1. (C-4) Interestingly, quasi-vti media have two additional vertical symmetry planes at 45 with respect to planes [x 1, x 3 ] [x 2, x 3 ]. Clearly, expressions for anisotropic coefficients ɛ (1,2), δ (1,2), γ (1,2), which could e defined within these symmetry planes, are different from those given y equations (49) (58). Two identical scalar fracture sets If we further assume that oth fracture sets are filled with gas [i.e., K N = K T ; Schoenerg Sayers (1995); part I], the effective medium ecomes VTI. The effective stiffnesses given elow depend only on the ackground parameters λ µ the weakness T = (K T µ)/(1 + K T µ) [equation (45)]: where c 11 = (λ + 2µ) (1 T )[1 + T (3 4g)], (C-5) D c 13 = λ 1 2 T D, (C-6) c 33 = (λ + 2µ) 1 T (8g 5), D (C-7) c 44 = µ(1 T ), (C-8) c 66 = µ 1 T, 1 + T (C-9) D (1 + T )[1 + T (2 3g)]. (C-10) The five stiffness elements of this three-parameter VTI medium are related y two constraints that can e otained y sustituting the VTI relation c 11 = c c 66 into equations (C-1) (C-2). In terms of Thomsen s (1986) anisotropic coefficients, these constraints take the form ɛ = δ = 4γ (1 + 2γ )(1 g). (C-11) The equality ɛ = δ indicates that the effective medium is elliptically anisotropic. In the weak-anisotropy limit, equation (C-11) yields ɛ = δ 4γ (1 g). For a typical value g = 0.25, ɛ = δ 3γ. (C-12) APPENDIX D FRACTURE CHARACTERIZATION FOR TWO ORTHOGONAL FRACTURE SETS: SPECIAL CASES Azimuthally independent P-wave NMO velocity In contrast to the case of a single fracture set in an isotropic or VTI ackground, the orientation of two orthogonal fracture systems cannot always e found from the P-wave NMO ellipse from a horizontal reflector. If δ (1) = δ (2) or, equivalently, (1 2g) N1 + T 1 = (1 2g) N2 + T 2 (D-1) [equations (50) (54)], the P-wave NMO ellipse degenerates into a circle. This equation does not necessarily imply that the two systems of fractures are identical ecause it can e satisfied for fracture sets with different crack densities (e.g., T 1 > T2 ) different fluid saturations ( N1 < N2 ). The azimuths of fractures in this case can e found from the shearwave polarization directions or y using P-wave reflections from dipping interfaces /or the azimuthal variation of P- wave nonhyperolic moveout. Equal tangential weaknesses If the tangential weaknesses are equal to each other ( T 1 = T 2 ), the anisotropic coefficients γ (1) γ (2) also ecome identical [see equations (51) (55)] there is no shear-wave splitting at vertical incidence. Hence, the fracture orientation cannot e determined from shear-wave splitting. However, if the fracture sets have different normal weaknesses ( N1 N2 ), the fracture azimuths can e found using P-wave NMO ellipses from a horizontal or a dipping reflector.

15 Fractured Orthorhomic Media 1817 Identical fracture sets (quasi-vti medium) In the case of two identical orthogonal fracture sets, neither the P-wave NMO ellipse from a horizontal reflector (it degenerates into a circle ecause δ (1) = δ (2) ) nor the shear-wave polarization directions (γ (1) = γ (2), see Appendix C) can e used to detect the fracture orientation. Still, since δ (3) (or η (3) ) differs from zero, the orientation can e found from normal moveout of dipping P events or from P-wave nonhyperolic moveout. The rest of the inversion procedure is similar to that for the general case of two different fracture systems. Despite the asence of shear-wave splitting in the vertical direction, S-waves can still e used for estimating fracture parameters. The wavefront of the slow shear wave in quasi-vti media always has cusps at 45 with respect to the symmetry planes [x 1, x 3 ] [x 2, x 3 ] (Figure D-1) if c 11 2c 66 c 12 > 0 [or, equivalently, K T > K N ; see equation (C-3) part I]. This wavefront structure y itself can e used to identify the symmetry directions (, therefore, the fracture strikes) if a sufficient numer of azimuthal measurements is availale. Once those directions are found, we can use equations (23) (29) for the NMO velocities of the S 1 - S 2 -waves in the vertical symmetry planes estimate the fracture parameters ased on equations (66) (68). It is interesting that in quasi-vti media the S-wave NMO velocities from a horizontal reflector are defined, strictly speaking, only within the vertical symmetry planes ecause the shear-wave singularity makes the S-wave offset-traveltime curve nondifferentiale in any other azimuth. Identical gas-filled fracture sets (VTI medium) Finally, if two identical fracture sets are gas filled, the effective medium ecomes VTI (Appendix C). Since the VTI model is azimuthally isotropic, it is impossile to find the fracture orientation from seismic data. The single parameter T (or K T ) needed to descrie the fractures can e found from any anisotropic coefficient given y equations (49) (56) if the ackground V S /V P ratio is known. Estimating several anisotropic coefficients provides a redundancy that can e used to verify the validity of this model [see equation (C-11)]. Indeed, this VTI medium is elliptical (ɛ = δ) has a specific relationship (C-12) etween ɛ γ. FIG. D-1. Group velocity surfaces of the (a) fast () slow shear waves for the model with two identical orthogonal fracture sets. The ackground velocities are V P = 2 km/s V S = 1 km/s. The fracture weaknesses N = 0 T = 0.15 approximately correspond to fluid-filled penny-shaped cracks with a crack density of 7%.