Feasibility of seismic characterization of multiple fracture sets

GEOPHYSICS, VOL. 68, NO. 4 JULY-AUGUST 2003); P. 1399 1407, 4 FIGS., 3 TABLES. 10.1190/1.1598133 Feasiility o seismic characterization o multiple racture sets Vladimir Grechka and Ilya Tsvankin ABSTRACT Estimation o parameters o multiple racture sets is oten required or successul exploration and development o naturally ractured reservoirs. The goal o this paper is to determine the maximum numer o racture sets o a certain rheological type which, in principle, can e resolved rom seismic data. The main underlying assumption is that an estimate o the complete eective stiness tensor has een otained, or example, rom multiazimuth, multicomponent surace seismic and vertical seismic proiling VSP) data. Although typically only a suset o the stiness elements or some o their cominations) may e availale, this study helps to estalish the limits o seismic racture-detection algorithms. The numer o uniquely resolvale racture systems depends on the anisotropy o the host rock and the rheology and orientation o the ractures. Somewhat surprisingly, it is possile to characterize ewer vertical racture sets than dipping ones, even though in the latter case the racture dip has to e ound rom the data. For the simplest, rotationally invariant ractures emedded in either isotropic or transversely isotropic with a vertical symmetry axis VTI) host rock, the stiness tensor can e inverted or up to two vertical or our dipping racture sets. In contrast, only one racture set o the most general microcorrugated) type, regardless o its orientation, is constrained y the eective stinesses. These results can e used to guide the development o seismic racture-characterization algorithms that should address important practical issues o data acquisition, processing, and inversion or particular racture models. INTRODUCTION Seismic racture characterization is critically important in exploration and development o naturally ractured reservoirs which oten contain multiple, dierently oriented racture net- works Grimm et al., 1999; Lynn et al., 1999; Pérez et al., 1999). As an example, Figure 1 shows three open racture sets detected y a orehole imager at Weyurn ield in Canada. Since all three sets may play a signiicant role in the luid low through the reservoir, it is essential to e ale to resolve them using seismic methods. According to the linear-slip theory originally suggested y Schoenerg 1980, 1983), ractures can e treated as displacement discontinuities with the jump in displacement proportional to the traction and excess racture compliances. The compliance matrix o the eective anisotropic medium containing one or more systems o aligned ractures can e otained y adding the compliances o each racture set to the ackground compliances. Despite several limitations o the linear-slip theory such as neglecting the interaction etween racture sets), its simplicity and generality make it particularly attractive or seismic inversion. Since ractured reservoirs are azimuthally anisotropic, their comprehensive characterization requires acquisition o wideazimuth, multicomponent seismic data. Processing o relected waves produces such signatures as the NMO ellipses o dierent modes, amplitude variation with oset and azimuth, and shear-wave polarizations and splitting coeicients. Additionally, walkaway vertical seismic proiling VSP) data can e used to estimate slowness suraces and polarization vectors at receiver locations in oreholes. Availale cominations o those signatures can e inverted or the eective stiness or compliance) coeicients o ractured rock. Then the linear-slip theory can e employed to iner the racture compliances and orientations or a certain racture model rom the otained stiness compliance) matrix. This approach is discussed in detail y Bakulin et al. 2000a,,c, 2002), who develop practical seismic racture-characterization algorithms or typical anisotropic models with one or two vertical racture sets. Estimation o racture parameters rom seismic data, however, cannot always e accomplished in a unique ashion. As noted y Bakulin et al. 2000a), it is clear that the numer o racture parameters or multiple racture sets and low symmetries o the ackground rock can e much larger than 21, Manuscript received y the Editor June 20, 2002; revised manuscript received Feruary 18, 2003. Formerly Colorado School o Mines, Center or Wave Phenomena, Golden, Colorado 80401; presently Shell International Exploration and Production Inc., Bellaire Technology Center, 3737 Bellaire Blvd., Houston, Texas 77001-0481. Colorado School o Mines, Center or Wave Phenomena, Department o Geophysics, Golden, Colorado 80401-1887. E-mail: ilya@dix.mines.edu. c 2003 Society o Exploration Geophysicists. All rights reserved. 1399

1400 Grechka and Tsvankin the maximum possile numer o independent stiness coeicients descriing wave propagation in the eective ractured medium. Hence, there exists a inite suset o ractured models with a relatively small numer o racture sets) which can e unamiguously characterized ased on seismic data. In act, the class o such resolvale ractured models is limited not just y the numer o unknown racture parameters. Elaorate analysis o the stiness tensor shows that certain compliances o dierently oriented racture sets may contriute to the same eective stiness coeicient e.g., Bakulin et al., 2002). As a result, it is possile to resolve only particular cominations o those compliances rather than their individual values. The goal o this paper is to identiy racture sets whose compliances and orientations can e determined in principle) rom seismic data. We examine three dierent rheological types o ractures rotationally invariant, diagonal [called orthotropic y Schoenerg and Helig 1997)], and completely general Grechka et al., 2003; a description o all three types is given elow). The ractures are emedded in either isotropic or transversely isotropic with a vertical symmetry axis VTI) host rock, with each racture set aritrarily oriented in 3D space. Throughout the paper, we assume an ideal scenario when all 21 stiness coeicients have een estimated, or example, rom multiazimuth walkaway P- and S-wave VSP data or rom a comination o surace and VSP data e.g., Bakulin et al., 2000d; Horne and Leaney, 2000; Grechka et al., 2001; Dewangan and Grechka, 2002). By computing the Frechèt derivatives o the eective stiness coeicients with respect to the racture and ackground parameters, we determine how many dierent racture sets o a certain type along with the unknown ackground parameters) are constrained y the stiness matrix. This analysis reveals the dependence o the maximum numer o resolvale racture sets on the rheology and orientation o the ractures, as well as on the presence o anisotropy in the ackground. EFFECTIVE STIFFNESS OF FRACTURED ROCK General relationships According to the linear-slip theory Schoenerg, 1980, 1983; Nichols et al., 1989; Schoenerg and Muir, 1989; Schoenerg and Sayers, 1995), the eective compliance matrix s o a medium containing N sets o aligned ractures is given y s = s s i), 1) where s and s i) are the 6 6 symmetric compliance matrices o the unractured ackground and the ith racture set, respectively. By deinition, the eective stiness matrix c is the inverse o the compliance matrix: c = s 1 = s s i) ) 1 = c 1 ) 1 s i) ; 2) c is the stiness matrix o the host ackground) rock. Note that all inverse matrices used here exist ecause the ackground matrices c and s are positive deinite and the compliance matrices s i) are nonnegative deinite. Here, we examine two types o the host rock: isotropic denoted as ISO) and VTI. Using the Lamé parameters λ and µ, the isotropic stiness matrix can e written as λ 2µ λ λ 0 0 0 λ λ2µ λ 0 0 0 c ISO = λ λ λ2µ 0 0 0. 3) 0 0 0 µ 0 0 0 0 0 0 µ 0 0 0 0 0 0 µ For VTI media, the stiness matrix has the orm c 11 c 12 c 13 0 0 0 c 12 c 11 c 13 0 0 0 c VTI c 13 c 13 c 33 0 0 0 =, 4) 0 0 0 c 44 0 0 0 0 0 0 c 44 0 0 0 0 0 0 c 66 where c 12 = c 11 2c 66. The matrices c ISO and c VTI are written in a certain gloal Cartesian coordinate rame {x 1, x 2, x 3 } used to descrie oth the ackground and the emedded ractures. Whereas the orientation o this rame can e aritrary or isotropic host rock, in the case o the VTI ackground the x 3 -axis is taken parallel to the symmetry axis o the medium. FIG. 1. Rose diagram o open-racture azimuths otained y a orehole imager at Weyurn ield Canada). The plot is courtesy o the Reservoir Characterization Project at Colorado School o Mines. Excess racture compliance The excess compliance o a racture set o the most general rheological type with the normal in the x 1 -direction is descried

Inversion or Multiple Fracture Sets 1401 y the ollowing 6 6 symmetric matrix Schoenerg, 1980; Bakulin et al., 2000c; Grechka et al., 2001): K N 0 0 0 K NV K NH s GN,x 1 =. 5) K NV 0 0 0 K V K VH K NH 0 0 0 K VH K H The diagonal elements o the matrix s GN,x1 are responsile or the normal K N ) and tagential K V and K H ) compliances o the racture. The nonzero o-diagonal elements K NV,K NH, and K VH ) imply the existence o coupling etween the normal to the racture plane) traction and the tangential jump in displacement displacement discontinuity is oten called slip). Likewise, tangential traction components or general ractures are coupled to normal slips. This coupling may e caused y microcorrugation o racture suraces Schoenerg and Douma, 1988; Grechka et al., 2001) or misalignment o the racture strike and the principal axes o stress Nakagawa et al., 2000). It can e shown Berg et al., 1991) that any rotation o the racture plane around the x 1 -axis does not change the orm o the matrix s GN,x1. However, i the racture normal deviates rom the x 1 -axis, the excess racture compliance matrix s GN,n may no longer contain any vanishing elements. We descrie the racture normal n y its azimuth α and dip or tilt) β Figure 2): n ={cos α cos β,sin α cos β, sin β}. 6) Exact expressions or the elements o the matrix s s GN,n in terms o the angles α and β are given in Appendix A [equations A-4) A-24)]. Those equations are essential or the prolem at hand ecause they represent the most general excess racture compliance matrix allowed y the linear-slip theory. The compliance matrices or racture sets with any par- ticular rheology or orientation can e otained as special cases o the matrix s. For instance, equations 2) o Schoenerg et al. 1999) or vertical ractures with a diagonal compliance matrix ollow rom equations A-4) A-24) y simply setting K NV =K NH =K VH =0 and β = 0. For ractures with a less complicated rheology, normal tractions are decoupled rom tangential slips, and tangential tractions are decoupled rom normal slips. Then, in a certain coordinate rame the excess racture compliance matrix 5) ecomes diagonal. Thereore, we call ractures o this type diagonal and denote them with the superscript DI. It should e noted, however, that the compliance matrix s DI,x1 o diagonal ractures acquires a nonzero o-diagonal element s56 DI, x1 ater an aritrary rotation around the racture normal i.e., the x 1 -axis). Thus, the compliance matrix o diagonal ractures is given y K N 0 0 0 0 0 s DI,x 1 =. 7) 0 0 0 0 K V K VH 0 0 0 0 K VH K H DI, x1 To make the matrix s diagonal, the coordinate rame has to e rotated y the angle ν = 1/2) tan 1 [2 K VH /K V K H )] around the x 1 -axis. Equations A-4) A-24) o Appendix A yield the compliance matrix o diagonal ractures orthogonal to the vector n [equation 6)] i one sustitutes K NV = K NH =0. 8) The simplest type o ractures, called rotationally invariant RI in our notation) y Schoenerg and Sayers 1995), corresponds to diagonal ractures with equal tangential compliances K V = K H then K VH =0): K N 0 0 0 0 0 s RI,x 1 =. 9) 0 0 0 0 K V 0 0 0 0 0 0 K V Any rotation around the racture normal i.e., around the x 1 - axis) does not change the orm o the matrix 9). Hence, the compliance s RI,n o a rotationally invariant racture set with the normal n can e ound rom equations A-4) A-24) ater the ollowing sustitutions: K NV = K NH = K VH =0 and K H = K V. 10) FIG. 2. Orientation o a racture set is deined y the azimuth α and dip β o the unit vector n orthogonal to the racture plane. Tale 1 lists the numer o independent physical model parameters or the two types o host rock ackground) and three types o racture systems considered aove. The dimension o the composite parameter space or the eective model is equal to the sum o the dimensions o the individual parameter spaces. For example, the medium ormed y two rotationally

1402 Grechka and Tsvankin invariant racture sets emedded in a purely isotropic ackground is descried y a total o 2 4) 2 = 10 independent parameters. Since racture characterization can e unique only i there are no more than 21 physical parameters the numer o stiness coeicients or the most general, triclinic symmetry), Tale 1 helps to identiy ractured models that cannot even in principle) e constrained y seismic data. For instance, it is impossile to estimate the parameters o three aritrarily oriented diagonal racture sets in a VTI host rock ecause the total numer o parameters in this case is 3 6) 5 = 23 which is greater than 21. FRECHÈT DERIVATIVES OF THE EFFECTIVE STIFFNESS MATRIX Although Tale 1 provides useul insight into the racturecharacterization prolem, it only shows that or a certain class o models the numer o constraints is smaller than the numer o unknowns, so the racture and ackground parameters cannot e resolved. However, even i the numer o eective stiness coeicients is larger than the numer o unknown physical parameters, the inversion or a particular parameter can still e amiguous. To study the uniqueness o this nonlinear inverse prolem, we apply the singular value decomposition SVD) to the matrix o Frechèt derivatives o the eective stinesses c with respect to the model parameters. The model parameter vector m can e represented as m 1)... N). 11) Here, the vector contains the independent ackground stiness coeicients c ; or instance, = [ 1, 2 ] = [λ, µ] when the host rock is isotropic. Similarly, the vectors 1),..., N) are composed o the independent excess racture compliances s [equations 5), 7), or 9)] and the orientation angles o each racture set. Note that even though the elements o c and s are conventionally written in the orm o 6 6 matrices, we include them in the single vector m to calculate the Frechèt derivatives. As discussed aove, the necessary prerequisite or a unique inversion is dim m = dim dim i) 21, 12) where dim v denotes the dimension o each vector v. The matrix o Frechèt derivatives can e otained y dierentiating Tale 1. Numer o independent parameters needed to descrie isotropic ISO) and VTI ackground media, as well as rotationally invariant RI), diagonal DI), and general GN) ractures. The numers or each racture set include the independent compliances and orientation angles α and β. Host rock Fracture set ISO VTI RI DI GN 2 5 4 6 8 equation 2): F c m = c 1 m ) 1 = c 1 c 1 s i) s i) s i) c 1 ) 1 c m c 1 s i) ) m ) 1. 13) Applying equation 2) again yields F = c c 1 c s i) ) m c 1 c m c = c s m s s i) ) c. 14) m Note that all derivatives in equation 14) are easy to ind in explicit orm ecause the stiness matrix c is dierentiated only with respect to the stiness coeicients o the host rock [i.e., to the portion o the vector m; see equation 11)], while the compliance matrices s i) are dierentiated only with respect to the racture compliances and orientations i.e., to the portions i) o m). Indeed, c is independent o s i),so the corresponding derivative c / i) = 0 or any i; likewise, s i) / = 0 and, i i j, s i) / j) = 0. Clearly, the Frechèt matrix F, which has the dimension 21 dim m, is sparse ecause so many derivatives in equation 14) vanish. ANALYSIS OF THE FRECHÈT MATRIX To answer the question whether it is possile to estimate the parameter vector m rom the measured stiness coeicients c, we perorm SVD o the matrix F. According to the standard SVD criterion, i the condition numer κ deined as the ratio o the greatest singular value to the smallest one) o F is inite, κ cond F <, 15) the inversion or m is theoretically unique or noise-ree data). Estimation o m ecomes amiguous nonunique) i κ =. 16) To identiy the maximum possile numer o racture sets which can e resolved using the eective stiness coeicients, we use the ollowing simple approach. Ater choosing the symmetry o the host rock isotropic or VTI), we keep adding racture sets o a certain type to the model until the condition numer ecomes ininite. In the tests elow, κ is considered ininite i it exceeds the numerical limit set in the Matla sotware 2 210 1.8 10 308 ). Aritrarily oriented dipping) ractures The analysis o the condition numer or racture sets with aritrary dip and azimuth is summarized in Tale 2. The results show an intuitively ovious trend: the more complicated the rheology o the racture systems, the ewer such systems can

Inversion or Multiple Fracture Sets 1403 e uniquely estimated rom the eective stinesses. It is interesting that or rotationally invariant and diagonal ractures, it should e possile to invert or as many racture sets as allowed y the dimensionality constraint 12). In contrast, the stiness matrix can e inverted or the parameters o only one general racture set, even or the simplest isotropic ackground rock. Oviously, a inite value o the condition numer indicates the possiility o inverting a complete, 21-element stiness matrix or the medium parameters. Estimating the stiness matrix itsel is a highly challenging prolem that typically requires acquisition and processing o multicomponent, multiazimuth data. Despite a signiicant progress recently achieved in the inversion or transversely isotropic e.g., Bakulin et al., 2000a; Horne and Leaney, 2000; Tsvankin and Grechka, 2000a,; Tsvankin, 2001; Grechka et al., 2002), orthorhomic Grechka et al., 1999; Bakulin et al., 2000), monoclinic Bakulin et al., 2000c; Grechka et al., 2000) and even triclinic Grechka et al., 2003; Dewangan and Grechka, 2002) media, parameter estimation or lower anisotropic symmetries requires urther development. In particular, seismic signatures needed or characterization o dipping ractures have seldom een discussed in the literature. An interesting possiility explored in the work o Angerer et al. 2002) is to use relection traveltimes o mode-converted PS-waves in the detection o racture dip. The presence o dipping ractures in an otherwise laterally homogeneous, isotropic ackground medium makes PS traveltime asymmetric with respect to the source and receiver locations i.e., the traveltime does not remain the same i one interchanges the source and receiver). Angerer et al. 2002) introduce a measure o this traveltime asymmetry that can e inverted or the racture dip and apply their methodology to 3D multicomponent data rom the Emilio ield in the Adriatic Sea. A relatively simple model with a single set o dipping rotationally invariant ractures in a VTI host rock marked y a diamond in Tale 2) is analyzed y Grechka and Tsvankin 2003). The eective medium in this case has monoclinic symmetry with a vertical symmetry plane that coincides with the dip plane o the ractures. Grechka and Tsvankin 2003) show that all racture and ackground parameters can e otained rom the vertical velocities and NMO ellipses o PP-waves and two split SS-waves SS traveltimes can e otained rom PP and PS data; see Grechka and Tsvankin, 2002) relected rom horizontal interaces. The analysis o the Frechèt derivatives indicates that it is possile to resolve a total o our systems o rotationally invariant ractures in a VTI ackground. Such a model has the maximum possile numer o independent physical parameters 21), all o which can e estimated rom the stiness matrix o the eective triclinic medium. Figure 3 shows a typical example o the relationship etween the condition numer κ and the dip β o the racture normal i.e., β is the racture tilt away rom the vertical; see Figure 2) or a model that includes three rotationally invariant racture sets. For a wide range o dips, the curve is almost lat with inite values o κ, which indicates that the inverse prolem is well posed according to our criterion). Although the condition numer or the lat part o the curve is relatively large 10 2 <κ<10 3 ), our previous results Bakulin et al., 2000a,,c) indicate that parameter estimation or some models with κ o such magnitude is not only theoretically possile ut also suiciently stale in the presence o errors in the data. Only or values o β extremely close to 0 near-vertical ractures) and 90 near-horizontal ractures) does the condition numer rapidly go to ininity. The parameter-estimation prolem or vertical ractures is discussed in detail in the next section. Horizontal ractures are not typical or naturally ractured reservoirs and will not e analyzed urther. Vertical ractures Most existing papers on racture characterization treat vertical racture networks, which are elieved to e common or naturally ractured reservoirs. To estalish the maximum numer o vertical racture sets that can e characterized y seismic data, we set the racture tilt β Figure 2) to zero and assume that it is known a priori. Although it seems that making the ractures vertical should help in racture detection ecause each set is descried y one ewer parameter, Tale 3 proves this expectation to e wrong. For instance, or rotationally invariant ractures in oth isotropic and VTI ackground, the maximum numer o resolvale vertical sets is just two compared to our dipping sets in Tale 2. Tale 2. Maximum numer o dipping racture sets that can e uniquely resolved rom the error-ree stiness matrix. The diamond marks the model studied y Grechka and Tsvankin 2003). RI DI GN ISO 4 3 1 VTI 4 2 1 FIG. 3. The inluence o the dip β o the racture normal on the condition numer κ o the Frechèt matrix. The model includes three rotationally invariant racture sets with the same β emedded in isotropic rock with V P /V S = 2β=0 corresponds to vertical ractures). For the irst set, K 1) N = 0.11 all compliances are density normalized and given in s 2 /km 2 ), K 1) V = 0.18, and α 1) = 0 α is the azimuth o the racture normal); or the second set, K 2) 2) N = 0.15, K V = 0.13, and α2) = 60 ; or the third set, K 3) 3) N = 0.16, K V = 0.19, and α 3) = 120.

1404 Grechka and Tsvankin To explain this puzzling result, note that making the ractures vertical β = 0) removes one degree o reedom in the description o each racture set. As a consequence, certain excess compliances o dierent racture systems can map into the same element o the eective stiness matrix, which makes estimation o these individual compliances impossile. An example o this type o amiguity is discussed y Bakulin et al. 2002) or two orthogonal vertical racture sets in a VTI host rock see elow). The simplest ractured model is that o vertical rotationally invariant ractures in an otherwise isotropic host rock one star in Tale 3). For a single racture set, the eective medium is transversely isotropic with a horizontal symmetry axis e.g., Schoenerg and Sayers, 1995; Bakulin et al., 2000a). Fracturecharacterization algorithms or this model see Bakulin et al., 2000a) can e ased entirely on surace relection data. Two vertical racture systems making an aritrary angle with each other lower the eective medium symmetry to monoclinic the only symmetry plane o this model is horizontal). In the special cases o orthogonal or identical racture sets, the eective model is orthorhomic. For oth orthorhomic and monoclinic models with two vertical racture sets, the racture and ackground parameters can e estimated using the vertical velocities and NMO ellipses o PP- and two split SS-waves or converted PSwaves) rom horizontal interaces Bakulin et al., 2000,c). Two vertical, rotationally invariant racture sets in a VTI ackground two stars in Tale 3) also yield an eective monoclinic medium with a horizontal symmetry plane. Bakulin et al. 2002) show that i the racture sets are orthogonal then the eective medium is orthorhomic), the racture and ackground parameters cannot e estimated in a unique ashion. Although the numer o the physical parameters in this model is equal to the numer o nonzero eective stinesses nine), there is an additional relation constraint) etween the stinesses or Tsvankin s 1997) anisotropic coeicients that causes the amiguity. While this conclusion is correct, our study reveals that the orthogonal orientation o the racture systems is the only special case or this model when the Frechèt matrix F degenerates. As illustrated y Figure 4, the condition numer κ rapidly increases only when the angle α etween the ractures is in a narrow vicinity o 0 parallel sets) or 90 orthogonal sets). Even i α deviates y just a ew degrees rom 90, the inversion should e easile, and the stinesses o the eective monoclinic medium this model is similar to the one descried y Bakulin et al., 2000c) can e used to constrain the racture and ackground parameters. A single system o vertical diagonal ractures [equation 7)] with K VH = 0 in a VTI host rock one dagger in Tale 3) creates an eective orthorhomic medium Schoenerg and Helig, 1997). Bakulin et al. 2000) show that the racture compliances and orientations or this model can e estimated Tale 3. Maximum numer o vertical racture sets that can e uniquely resolved rom the error-ree stiness matrix. The stars and daggers mark models studied in the literature see the main text). RI DI GN ISO 2 2 1 VTI 2 2 1 rom multiazimuth PP and PS or SS) relection data, although one o the tangential compliances K H ) remains unconstrained i dipping events are not availale. The inversion or the VTI ackground parameters also requires knowledge o the vertical velocities or relector depth. Moreover, Tale 3 indicates that the parameter estimation is still possile or two sets o diagonal ractures in oth isotropic and VTI ackground. The compliances o two racture sets, however, cannot e determined without using multiazimuth VSP data. Characterization o vertical ractures o the most general type emedded in a purely isotropic host rock dagger in Tale 3) has een studied y Grechka et al. 2003). Despite the lowest possile triclinic) symmetry o the eective model, Grechka et al. 2003) prove that the racture and ackground parameters can e determined rom a comination o relection and orehole data that includes the vertical velocities and NMO ellipses o PP- and SS-waves supplemented with P-wave multiazimuth walkaway VSP data. According to Tale 3, the addition o another general racture system makes the inversion amiguous, although the numer o independent physical parameters or either isotropic or VTI) ackground still does not exceed 21. DISCUSSION AND CONCLUSIONS Seismic estimation o the parameters o multiple racture sets emedded in otherwise isotropic or VTI ackground host) rock is an important practical issue in characterization and development o naturally ractured reservoirs. Although our approach can e applied to other types o host rock i.e., orthorhomic or monoclinic), such low ackground symmetries can e caused only y the intrinsic anisotropy o the rock-orming crystals. Since crystal anisotropy is known to e atypical or relatively shallow strata important in seismic exploration, we decided to exclude those more complicated ackground models rom consideration. FIG. 4. The condition numer κ or the model o two vertical, rotationally invariant racture sets in VTI host rock. α denotes the dierence etween the racture azimuths. The compliances are K 1) N = 0.15, K 1) V = 0.14, K 2) N = 0.13, and = 0.12. The density-normalized ackground stinesses in km 2 /s 2 ) are c 11 = 3.90, c 33 = 4.00, c 44 = 1.00, c 66 = 1.19, and c 13 = 1.71. K 2) V

Inversion or Multiple Fracture Sets 1405 The key assumption in our study was that all elements o the eective stiness matrix c ij can e recovered rom seismic data. By expressing the stinesses through the racture and ackground parameters and analyzing the condition numer o the corresponding Frechèt matrix, we determined the maximum numer o racture sets with a given rheology which can e uniquely resolved under those ideal conditions. I the racture sets are tilted away rom the vertical i.e., the ractures are dipping), it may e possile to characterize up to our rotationally invariant racture sets in either isotropic or VTI ackground. Surprisingly, parameter estimation ecomes more amiguous or vertical ractures which create simpler i.e., higher-symmetry) eective anisotropic models. Because o the trade-os etween dierent racture parameters which contriute to the eective stinesses only in certain cominations, no more than two vertical racture sets can e resolved rom the matrix c ij. Even a small racture dip, however, is suicient to sharply reduce the condition numer and make the inverse prolem much etter posed. In practice, estimation o racture parameters has to e ased on availale incomplete) data which usually cannot constrain all stiness coeicients individually. This is particularly true or lower anisotropic symmetries such as triclinic) descried y up to 21 independent stinesses. Inversion o ield data may yield only cominations o some stiness elements, while inormation aout other stinesses may e missing entirely. Such an incomplete stiness matrix may constrain ewer racture sets than indicated y our analysis i any), so it may ecome necessary to impose restrictions on the racture rheology e.g., assume rotationally invariant ractures) and orientation, or ignore the ackground anisotropy. In principle, a easiility study similar to the one descried here can e carried out or any given set o availale seismic data or eective stiness coeicients. Another reason or our results to e overly optimistic is that we deem the inversion to e nonunique only when the condition numer o the Frechèt matrix goes to ininity. Thereore, the entries in Tales 2 and 3 ormulated in a yes/no ashion should e regarded with caution. For the vast transitional set o models with relatively large condition numers, the easiility o inversion depends on the magnitude o errors in the data. As an example, according to the view taken in our paper, two racture sets making a small angle with each other may still e resolvale i the corresponding condition numer is inite see Figure 4). However, this is not necessarily the case in practice, when the eective stiness coeicients are estimated with an error. It is clear rom oth the physics and mathematics o the linear-slip theory that i the angle etween two racture sets is smaller than a certain value which depends on errors in c ij ), they appear as a single set with the total excess compliance equal to the sum o the corresponding individual compliances. This issue can e accounted or in our study y using a more conservative condition cond F > M, where M is a positive numer that depends on data uncertainty) to deine nonuniqueness. The main signiicance o this work, however, is in estalishing the limits o seismic racture-characterization algorithms and identiying the class o ractured models which merits urther investigation. Thereore, our results can provide a road map or uture applications o seismic methodologies to naturally ractured reservoirs. ACKNOWLEDGMENTS We are grateul to memers o the Anisotropy)-Team o the Center or Wave Phenomena CWP), Colorado School o Mines, or helpul discussions and to Roel Snieder and Reynaldo Cardona oth CSM) or their reviews o the manuscript. We also thank R. Cardona or providing Figure 1. This work was supported y the Consortium Project on Seismic Inverse Methods or Complex Structures at CWP and y the Chemical Sciences, Geosciences, and Biosciences Division, Oice o Basic Energy Sciences, U.S. Department o Energy. REFERENCES Angerer, E., Horne, S. A., Gaiser, J. E., Walters, R., Bagala, S., and Vetri, L., 2002, Characterization o dipping ractures using PS modeconverted data: 72nd Ann. Internat. Mtg., Soc. Expl. Geophys., Expanded Astracts, 1010 1013. 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1406 Grechka and Tsvankin Schoenerg, M., and Douma, J., 1988, Elastic wave propagation in media with parallel ractures and aligned cracks: Geophys. Prosp., 36, 571 590. Schoenerg, M., and Helig, K., 1997, Orthorhomic media: Modeling elastic wave ehavior in vertically ractured earth: Geophysics, 62, 1954 1974. Schoenerg, M., and Muir, F., 1989, A calculus or inely layered anisotropic media: Geophysics, 54, 581 589. Schoenerg, M., and Sayers, C., 1995, Seismic anisotropy o ractured rock: Geophysics, 60, 204 211. Schoenerg, M., Dean, S., and Sayers, C. M., 1999, Azimuth-dependent tuning o seismic waves relected rom ractured reservoirs: Geophysics, 64, 1160 1171. Tsvankin, I., 1997, Anisotropic parameters and P-wave velocity or orthorhomic media: Geophysics, 62, 1292 1309. 2001, Seismic signatures and analysis o relection data in anisotropic media: Elsevier Science Pul. Co., Inc. Tsvankin, I., and Grechka, V., 2000a, Dip moveout o converted waves and parameter estimation in transversely isotropic media: Geophys. Prosp., 48, 257 292. 2000, Two approaches to anisotropic velocity analysis o converted waves: 70th Ann. Internat. Mtg., Soc. Expl. Geophys., Expanded Astracts, 1193 1196. Winterstein, D. F., 1990, Velocity anisotropy terminology or geophysicists: Geophysics, 55, 1070 1088. APPENDIX A COMPLIANCE MATRIX OF A GENERAL, ARBITRARILY ORIENTED FRACTURE SET Here, we give exact expressions or the compliance matrix s o a set o general e.g., microcorrugated) ractures with aritrary orientation. The racture planes are orthogonal to the unit vector n deined y the azimuth α and dip β in the Cartesian coordinate rame {x 1, x 2, x 3 } Figure 2): n ={cos α cos β,sin α cos β, sin β}. A-1) To otain s, we apply an appropriate rotation to the compliance matrix s GN,x1 [equation 5)] o a general racture set orthogonal to the x 1 -axis n x1 ={1,0,0}). This operation includes two steps the rotation A β y the angle β around the x 2 -axis and another rotation A α y the angle α around the x 3 - axis and is descried y the matrix A = A α A β cos α sin α 0 cos β 0 sin β = sin α cos α 0 0 1 0 0 0 1 sin β 0 cos β cos α cos β sin α cos α sin β = sin α cos β cos α sin α sin β. A-2) sin β 0 cos β The corresponding transormation o the compliance matrix known as the Bond transormation) has the orm s GN,x1 s = Bs GN,x 1 B T. A-3) An explicit expression or the 6 6 matrix B in terms o the elements o the matrix A is given in Winterstein 1990); B T denotes the transposed matrix. Evaluating equation A-3) yields the ollowing 6 6 symmetric matrix s : s 11 = cos 2 α cos 2 β [ K H sin 2 α sin 2αK NH cos β K VH sin β) cos 2 α K N cos 2 β 2K NV sin 2β K V sin 2 β )], A-4) s 12 = cos α cos 2 β sin α [ cos 2αK NH cos β K VH sin β) cos α sin α K N cos 2 β 2K NV sin 2β K V sin 2 )] β K H, A-5) s 13 = cos α cos β sin β { sin αk VH cos β K NH sin β) cos α [K NV cos 2β K V K N ) sin β cos β] }, A-6) s 14 = cos α cos β { cos 2 α sin βk NH cos β K VH sin β) sin α cos α[k NV cos 3β sin βk H K N K V K N ) cos 2β)] sin 2 αk NH sin 2β K VH cos 2β) }, A-7) s 15 = cos α cos β { cos α cos β sin α3k NH sin β K VH cos β) cos 2 α[k NV cos 3β sin βk N K N K V ) cos 2β)] sin β K VH sin 2α sin β K H sin 2 α )}, A-8) s 16 = cos α cos 2 β [ K H sin 3 α cos α1 4 sin 2 α) K NH cos β K VH sin β) cos 2 α sin α 2K N cos 2 β 2K NV sin 2β 2K V sin 2 )] β K H, A-9) s 22 = cos 2 β sin 2 α K H cos 2 α K N cos 2 β sin 2 α K NH cos β sin 2α K VH sin 2α sin β K V sin 2 α sin 2 β K NV sin 2 α sin 2β ), A-10) s 23 = cos β sin α sin β { cos αk NH sin β K VH cos β) sin α[k NV cos 2β K V K N ) sin β cos β] }, A-11) cos β sin α { s 24 = 4K NV cos 3β sin 2 α 4 sin β 4 [ K H cos 2 α K N sin 2 α K N K V ) sin 2 α cos 2β ] sin 2α3K VH cos 2β 3K NH sin 2β K VH ) }, A-12) s 25 = cos β sin α { sin α sin β[k H cos α K NH cos β K VH sin β) sin α] 2 cos α cos β sin β [K NH cos α K N cos β K NV sin β) sin α] cos α cos 2β[K VH cos α K NV cos β K V sin β) sin α] }, A-13) s 26 = cos 2 β sin α { sin α4 cos 2 α 1)K NH cos β K VH sin β) cos α sin 2 α[k H K N K V K V K N ) cos 2β 2K NV sin 2β] K H cos 3 α }, A-14)

Inversion or Multiple Fracture Sets 1407 s 33 = sin 2 β K V cos 2 β K NV sin 2β K N sin 2 β ), A-15) s 34 = sin β { K V cos 3 β sin α 3K NV cos 2 β sin α sin β sin 2 βk NH cos α K NV sin α sin β) cos β sin β[k VH cos α K V 2K N ) sin α sin β] }, A-16) s 35 = sin β { sin α sin βk VH cos β K NH sin β) cos α 2 [K V K N ) cos β K V K N ) cos 3β 2K NV sin 3β] }, A-17) s 36 = sin β cos β{cos 2αK NH sin β K VH cos β) sin 2α[K NV cos 2β K V K N ) sin β cos β]}, A-18) s 44 = K V cos 4 β sin 2 α 4K NV cos 3 β sin 2 α sin β 4 cos β sin α sin 2 βk NH cos α K NV sin α sin β) 2 cos 2 β sin α sin β[k VH cos α K V 2K N ) sin α sin β] sin 2 β [ K H cos 2 α K VH sin 2α K V sin 2 α sin β ) sin β ], A-19) s 45 = cos 2α sin βk VH cos 2β K NH sin 2β) sin 2α [ KV K N K V K N ) cos 4β 4 2K H sin 2 β 2K NV sin 4β ], A-20) s 46 = cos β { cos 2α [ K VH cos 2 β sin α 2K NH cos β sin α sin β sin βk H cos α K VH sin α sin β) ] 2 cos α sin α [ 2K N K V ) cos 2 β sin α sin β K NV cos 3 β sin α cos β sin βk NH cos α 3K NV sin α sin β) sin 2 βk VH cos α K V sin α sin β) ]}, A-21) s 55 = sin 2α sin 2 β2k NH cos β K VH sin β) sin β K VH cos 2 β sin 2α K H sin 2 α sin β ) cos2 α [K V K N K V K N ) cos 4β 2 2K NV sin 4β], A-22) s 56 = cos β { K H cos 2α sin α sin β cos 2 α sin α [2K NV cos 3β 2K N K N K V ) cos 2β) sin β] cos α [ 2 sin 2 α sin βk NH cos β K VH sin β) cos 2αK VH cos 2β K NH sin 2β) ]}, A-23) s 66 = cos 2 β { K H cos 4 α sin 4 α) 2 sin 2αK NH cos β K VH sin β) sin2 2α [K H K N K V 2 K V K N ) cos 2β 2K NV sin 2β] }. A-24) Equations A-4) A-24) are derived or the most general racture rheology descried y six independent excess racture compliances. The matrices s or diagonal [equation 7)] or rotationally invariant [equation 9)] ractures can e otained y sustituting the corresponding constraints 8) and 10).