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Tight, low-porosity reservoirs can produce significantamounts of hydrocarbons if fracture networks providethe necessary permeability. Characterization of naturallyfractured reservoirs using surface seismic data is animportant exploration problem that has attracted muchattention in the literature. TLE, for example, had an entireissue (August 1996) on this subject. While the data andmodeling in existing publications demonstrate that frac-turing causes significant azimuthal variation in recordedseismic signatures, the problem is how to infer fractureparameters from these data.

The goal of this paper is to complement recent exper-imental results with an analytic and modeling study of theamplitude-variation-with-offset (AVO) response of frac-tured reservoirs. The question we address is: What doesP-wave reflectivity really tell us about the fractures? Tofind the answer, we introduce a convenient notation forazimuthally anisotropic models and present conciseapproximations for reflection coefficients in terms of theanisotropy parameters. Our analysis shows that P-waveamplitudes may be sensitive even to relatively weakanisotropy of the rock mass. We also discuss how theazimuthally dependent AVO signature can be interpretedand combined with normal-moveout and shear-wave-splitting analyses to constrain the crack density and othermedium parameters. The simple relationships betweenreflection amplitudes and anisotropic coefficients can beregarded as rules of thumb to quickly evaluate the impactof anisotropy in a particular play, plan acquisition, andguide more advanced numerical inversion.

Reflection coefficients and AVO. It is relatively straight-forward, albeit tedious, to compute reflection coefficients forinterfaces between anisotropic media. Inverting the reflec-tion response for the physical properties of the medium,

however, is much more involved due to the algebraic com-plexity of the reflection coefficients and the number of rel-evant medium parameters. To gain insight into anisotrop-ic reflectivity, it is helpful to derive approximate scatteringcoefficients by assuming that both the contrast in the elas-tic properties across the interface and the anisotropic para-meters are small. Shuey used the weak-contrast assumptionfor isotropic media to provide simple approximations illus-trating the dependence of the AVO gradient (the initialslope of the reflection-coefficient curves) on the jumps in theP- and S-wave velocities and density (see “A simplificationof the Zoeppritz equations,” GEOPHYSICS, 1985). Shuey’srelationship is widely used for lithologic interpretation inmodern (isotropic) AVO analysis.

To understand the reflectivity of fractured reservoirs ina similar way, we follow the approach of Stuart Crampin,Leon Thomsen and others and initially consider the first-order model of azimuthal anisotropy conventionally usedin shear-wave birefringence experiments. Assuming thatparallel vertical cracks are embedded in a homogeneousisotropic matrix, we obtain the transversely isotropicmodel with a horizontal axis of symmetry (HTI media),sketched in Figure 1. As indicated by the arrows,azimuthal anisotropy has a first-order influence on shearwaves that split into two components traveling with dif-ferent velocities. The fractional difference between thevelocities of split shear waves at vertical incidence (Thom-sen’s coefficient γ) is close to the crack density, an impor-tant parameter responsible for the permeability of thefracture network. Shear-wave methods, developed exten-sively during the last decade, are designed to obtain γ fromthe difference in shear-wave traveltimes and normal-inci-dence reflection amplitudes. This technology, however,has drawbacks (e.g., the cost and the difficulty in acquir-ing high-quality shear data). Hence, it is extremely impor-tant to obtain more information about fractured reser-voirs from 3-D P-wave data. Amplitude variation withoffset (AVO) analysis is particularly well-suited for frac-

Using AVO for fracture detection: Analyticbasis and practical solutions

ANDREAS RÜGER and ILYA TSVANKIN, Colorado School of Mines, Golden, Colorado

Figure 1. Sketch of an HTI model. Different P-wavereflection amplitudes in the two vertical symmetryplanes, here called the symmetry-axis plane and theisotropy plane, lead to azimuthally-dependent AVOresponse. As indicated by the arrows, shear wavespolarized parallel and normal to the isotropy planehave different vertical velocities.

Figure 2. The reflection response from fractured reser-voirs depends on two angles: (1) the polar (incidence)angle i between the vertical and the slowness vector ofthe incident wave and (2) the azimuthal angle φdefined with respect to the symmetry axis.

ture characterization because it provides local informationabout the anisotropy at the target horizon.

P-waves in HTI media. The influence of azimuthalanisotropy on P-wave propagation is not as dramatic asfor shear waves, but it can be substantial. Let us discussP-wave reflectivity for the HTI model introduced above.Waves confined to the plane normal to the symmetry axis(the “isotropy plane”) do not experience any angularvelocity variation. For all other vertical planes, the veloc-ity does change with incidence angle; it also varies withazimuth (Figure 2), significantly complicating interpreta-tion of reflection data.

Figure 3 illustrates P-wave propagation in the verticalplane containing the symmetry axis (the symmetry-axisplane). The continuous white line marks the points ofequal P-wave traveltime on the seismic rays (wavefront),while the dashed white circle represents the referenceisotropic wavefront. Conventionally, anisotropic media aredescribed by the elements of the elastic stiffness tensor (amaximum of 21). While helpful in numerical modeling, thistensor notation is not optimal for analyzing and invertingseismic signatures. For example, it would benefit surfaceseismic applications to define a coefficient responsible forP-wave near-vertical velocity variations. We found thatAVO and normal moveout in HTI are best described byadapting Thomsen’s notation for transverse isotropy witha vertical symmetry axis (VTI media). Instead of using thegeneric Thomsen’s coefficients specified with respect to thehorizontal symmetry axis, however, we introduce theseparameters with respect to the vertical by taking advantageof the equivalence between the symmetry-axis plane ofHTI media and VTI media. Thus, five elastic coefficientscan be replaced with the vertical velocities of the P-wave(α) and the fast S-wave (β), the shear-wave splitting para-meter γ, and two more Thomsen-style anisotropic coeffi-cients, ε(v) and δ(V). The superscript “V” emphasizes that thecoefficients are computed with respect to the vertical andcorrespond to the “equivalent” VTI model that describeswave propagation in the symmetry-axis plane.

For the particular example in Figure 3, the continuouswhite wavefront falls behind the corresponding isotrop-

ic (dashed) wavefront as the ray deviates from vertical,indicating that the P-wave velocity near vertical decreas-es with angle (i.e., δ(V)

(2)

reveals that a maximum of six coefficients can be extract-ed from the dependence of Rp on the azimuthal and inci-dence angles. Since it is often difficult to extract reliableestimates of the high-angle reflectivity term sin2i tan2i, weconcentrate on the AVO gradient that determines the low-angle reflection response.

One of the most important messages of our approxi-mation is that the contrasts in anisotropy parameters δ(V)and γ have a non-negligible, first-order influence on theAVO gradient (see the term Bani). For example, for azimuthφ= 0 and a typical -β/-α ≈ 1/2, the contrast in the shear-wavesplitting parameter (∆γ=γ2−γ1) at the boundary has aweighting factor twice as large as that of the P-wave veloc-ity contrast ∆α-α . Hence, Bani introduces a measurableazimuthal variation in the AVO response.

The AVO gradient term B = Biso + Bani cos2 φ in equa-tions (1) and (2) is composed of the azimuthally invariantpart Biso and the anisotropic contribution Bani dependent onthe azimuthal angle φ with the symmetry axis. If the sym-metry-axis orientation is unknown, φ can be expressed asthe difference between the azimuthal direction φj of the jthobserved azimuth and the direction of the symmetry-axisplane φsym. The AVO gradient measured at azimuth φj canthen be written as

(3)

This equation is nonlinear with three unknowns (Biso,Bani, and φsym), so a minimum of three azimuthal measure-ments of the AVO gradient is needed to find the orienta-tion of the symmetry planes and reconstruct the low-anglereflection coefficient at all azimuths. If the direction of thesymmetry axis is known (for example from S-wave split-ting analysis), the equation becomes linear and two inde-pendent measurements are sufficient to recover Biso andBani. Plotting B as a function of φj, we find that the extremacorrespond to the symmetry-plane azimuths, even thoughmore information is needed to distinguish between thesymmetry-axis and the isotropy plane. The differencebetween the symmetry-plane AVO gradients yields thevalue Bani that contains information about anisotropy in theform of a combination of the coefficients δ(v) and γ. In gen-eral, δ(v) and γ are independent; however, for a tight for-mation with no equant porosity and thin fluid-filledcracks, δ(v) ≈ -γ. In this case, the azimuthal variation in theP-wave AVO gradient can be inverted directly for γ andthe crack density.

Combination of AVO and moveout data. A complicationin the AVO inversion is the anisotropic parameter δ(v) inthe gradient term Bani that prevents (except for the specialcase discussed above) obtaining the shear-wave splittingparameter and estimating the crack density using the P-wave AVO response. However, δ(v) can be found fromazimuthally dependent P-wave moveout data. The P-wave normal moveout (short-spread) velocity in a hori-zontal HTI layer is given by

(4)

where φ is the azimuth of the CMP line with respect to thesymmetry axis. Equation (4) describes an ellipse in thehorizontal plane and is valid for HTI models with anystrength of the anisotropy. Since the NMO velocity con-

tains just three unknowns, three measurements of Vnmo atdifferent azimuthal angles can be inverted for the verticalvelocity, the axis orientation, and the parameter δ(v) (pro-vided the fractured interval is not too thin). Hence, P-wave moveout data can identify the crack orientation andrecover the parameter δ(v) that is required for the AVOinversion. Then two azimuthal measurements of the AVOgradient are sufficient to obtain the AVO gradient in thesymmetry planes and estimate the shear-splitting para-meter γ.

Fluid-filled vs. dry cracks: a modeling example. Figure4 shows three models with a different target layer and thesame isotropic overburden. The first target layer is isotrop-

1432 THE LEADING EDGE OCTOBER1997

Figure 4. Three models used to study the differencesin the AVO response for fluid-filled and dry cracks.Table 1 lists the model parameters.

B B Bjiso ani

j symcos .φ φ φ2( ) = + −( )

Vnmo2 = αα δ

δ φ2

2

1 2

1 2

++

( )( )

V

V sin,

Table 1. Parameters of the reflecting layers used toinvestigate the azimuthally varying AVO response.Model 1 is isotropic; Models 2 (wet cracks) and 3 (drycracks) have the same host rock, but also containvertical cracks with different contents.

isotropic wet cracks dry cracks

α 4.500 4.498 4.388β 2.530 2.530 2.530ρ 2.800 2.800 2.800

γ 0.000 0.085 0.085ε(v) 0.000 -0.003 -0.150δ(v) 0.000 -0.088 -0.155

Table 2. Percentage changes in the elastic parametersfor reflection from the top of the target layer. Theparameters of the isotropic overburden are α = 3.67km/s, β = 2.0 km/s, ρ = 2.41 g/cm3.

isotropic wet cracks dry cracks∆α-α 0.203 0.203 0.180∆Ζ-Ζ 0.350 0.350 0.328

∆GG 0.601 0.601 0.601

R i A B B iPiso ani, cos sinφ φ2 2( ) = + +{ } +

C C Ciso ani anicos sinφ φ4 21 2+ +{ icos sin tφ2 2} itan ,2

ic with the elastic parameters given in the first column ofTable 1; the other two layers have the same host rock, butalso contain vertical aligned cracks (causing HTI symme-try) with an aspect ratio of 0.001 and a crack density of 7%.The second column is computed for water-filled cracks,the third for dry cracks.

Before evaluating the reflection response for the threemodels, let us discuss the difference between the modelparameters and use equation (1) to predict the behavior ofthe three AVO responses. All three models have the samedensity and the same fast vertical shear-wave velocity(the shear wave polarized within the isotropy plane is notinfluenced by the fracturing). Also, the fractures are toothin and their density is too small to influence the bulkdensity of the host rock. As a result, the vertical P-wavevelocity is almost the same for the unfractured rock andthe medium with water-filled cracks, and is only slightlysmaller (by about 2.5 %) for the medium with dry cracks.The anisotropy parameters, however, are substantiallyinfluenced by the presence and contents of the cracks. Forwater-filled cracks, the coefficient ε(v) is negligibly small,while the shear-wave-splitting parameter γ and coefficientδ(v) have almost the same magnitude but opposite sign. Ifthe cracks are dry, both ε(v) and δ(v) are negative and ε(v) ≈δ(v); the absolute value of ε(v) and δ(v) is approximatelytwice as large as the shear-wave splitting parameter γ.

These differences in the anisotropy parameters lead toa different azimuthal variation in the reflection responsefor each of the models. Taking into account the percent-age changes in the elastic parameters (see Table 2), equa-tion (1) predicts that the initial slope of the reflectioncoefficient (AVO gradient) for the isotropic Model 1 isidentical to the AVO gradient in the isotropy plane ofModel 2, and is slightly less negative than the isotropy-plane AVO gradient in Model 3. For water-filled cracks,equation (1) predicts a substantial azimuthal difference inthe AVO gradients, with a more negative value in theisotropy plane [recall that the difference in the symmetryplane gradients is approximated by (δ(v) + 2γ)]. In con-trast, for dry cracks the combination δ(v) + 2γ ≈ 0, and wecannot expect a measurable azimuthal variation in the

AVO gradient. In this case, reflection coefficients com-puted at different azimuths start to diverge only at largeangles of incidence.

Figures 5-7 show the exact and approximate reflectioncoefficients for the three models (Tables 1 and 2). To facil-itate the comparison, the vertical scale is the same on allthree plots. Approximation (1) is sufficiently close to theexact reflection coefficients for all three models; specifi-cally, it predicts the different azimuthal variation in thesmall- and large-angle reflection response for the dry andwater-filled cracks.

Hence, this study confirms the physical insight pro-vided by equation (1) and shows that P-wave AVO has thepotential of discriminating between gas- and fluid-filledcrack systems. Also, if the AVO response at differentazimuths diverges substantially only at large angles ofincidence, it is a strong indication of a large value of theparameter ε(v). In this case, the less negative AVO gradi-ent corresponds to the isotropy-plane reflection because∆ε(v) is negative for the interface between an isotropic layerand a fractured medium with the HTI symmetry.

Another important message is that the azimuthal dif-ference in AVO gradients cannot be related directly to thecrack density (it may even be close to zero for dry cracks),and that knowledge of the anisotropy parameter δ(v) is cru-cial to any meaningful azimuthal AVO analysis. As men-tioned above, δ(v) can be obtained from P-wave NMOvelocity; alternatively, if γ is available from S-wave data,this study suggests combining the shear-wave-splittingand azimuthal-AVO analysis to obtain δ(v) for study of thecontents of the cracks.

Discussion. This analysis was limited to interpretationand inversion of the reflection coefficients. In practice, werecord the amplitude-versus-offset signature at the surfacerather than the reflection coefficients at the target horizon.Various methods to correct for the propagation phenom-ena have been discussed in the literature, but most areapproximate even for isotropic media. The situationbecomes more complex if the reflected wave propagatesthrough an anisotropic overburden, which may happen, for

OCTOBER1997 THE LEADING EDGE 1433

Figure 5. The exact reflectioncoefficient (solid line) and theapproximation (dashed) for theunfractured target layer. Modelparameters are given in the leftcolumn of Tables 1 and 2.

Figure 6. Same as Figure 5, but thereflecting layer contains water-filledcracks (the central column in Tables 1and 2). The reflection coefficients areshown for azimuths of 0˚, 30˚, 60˚, and90˚ measured from the symmetry axis,with decreasing line thickness (i.e., thethickest line corresponds to 0˚).

Figure 7. Same as Figures 5 and 6,but the reflecting layer containsdry cracks (the right column inTables 1 and 2) .

instance, if the cracks are not confined to the reservoirlayer. The presence of anisotropy above the reflector leadsto the so-called wavefront focusing phenomena, or dis-tortions in the amplitude distribution along the wave-fronts of the incident and reflected waves. This can be seenin Figure 3, where focusing and defocusing of seismic raysmark areas of high and low energy, respectively.Azimuthal variation in the transmission coefficients inthe anisotropic overburden may also contribute to theoverall amplitude response at the surface. If not correct-ed, these distortions can propagate into the AVO signatureand distort the inversion results.

Another complication is the difference between group(ray) and phase angles in anisotropic media. Analyzing theextracted reflection coefficient as a function of the source-receiver offset (rather than as a function of the incidentphase angle) can lead to a misleading interpretation of theazimuthal AVO variation. In HTI media, the group andphase angles coincide for waves propagating in theisotropy plane, but may differ significantly in the sym-metry-axis plane. In other words, the rays connectingsources and receivers at any fixed offset have anazimuthally dependent incidence phase angle at the reflec-tor. To correctly interpret the AVO signatures in anisotrop-ic media using the analytic reflection coefficients givenhere, it is essential to represent them as a function ofphase angle. Unfortunately, in practice the anisotropicparameters of the overburden may not be known with suf-ficient accuracy to account for various propagation phe-nomena.

Conclusions. Linearized approximations can provide

valuable insight into the otherwise incomprehensiblycomplex reflection coefficients in anisotropic media. Themain assumptions used in this study — small jumps inthe elastic parameters across the reflecting interface, sub-critical incidence, and weak anisotropy — are geologicallyand geophysically reasonable and have proved useful inmany exploration contexts. We showed that the azimuthalvariation in the P-wave AVO gradient is controlled by twoanisotropic parameters — the shear-wave splitting coef-ficient γ and the “moveout” parameter δ(v). Both analyticand modeling results indicate that P-wave reflectivitycan detect fracture orientation, estimate crack density,and discriminate between fluid-filled and dry cracks.However, unambiguous interpretation of P-wave AVOresults in terms of the medium parameters requires com-bination with moveout data or shear-wave splittinganalysis. A big challenge in the implementation ofanisotropic AVO is reliable recovery of reflection coeffi-cients from surface data by amplitude-preservinganisotropic processing.

Certainly, the simple HTI model discussed here maybe inadequate for realistic models of fractured reservoirs,and it is important to study whether the reflectionresponse in media of lower symmetry (e.g., orthorhombic)differs significantly from the HTI case. However, it hasalready been shown that although wave propagation inorthorhombic media is significantly more complex, thereflection coefficients in the symmetry planes have essen-tially the same form as in HTI models and vary smoothlywith azimuth.

Suggestions for further reading. Papers of StuartCrampin (“Evidence for aligned cracks in the Earth’scrust”, First Break, 1985, and many others) and LeonThomsen (“Reflection seismology over azimuthallyanisotropic media”, GEOPHYSICS, 1988) give a good expo-sition of the implications of azimuthal anisotropy in seis-mology. Thomsen’s work (“Weak elastic anisotropy,”GEOPHYSICS, 1986; “Weak anisotropic reflections” in Off-set Dependent Reflectivity, SEG, 1993) provided a basis fordefining anisotropy parameters and deriving linearizedscattering coefficients. The models shown in this paperare computed using Hudson’s fracture theory (“A high-er order approximation to the wave propagation con-stants for a cracked solid,” Geophysical Journal of the RoyalAstronomical Society, 1986). Derivations of the AVO equa-tions and extensions to P- and S-wave AVO in orth-orhombic media are given in Rüger ’s articles “Variationof P-wave reflectivity with offset and azimuth inanisotropic media” (GEOPHYSICS, accepted for publica-tion) and “Analytic insight into shear-wave AVO for frac-tured reservoirs” (to appear in Proceedings of 71WSA,special SEG volume on seismic anisotropy, Miami, 1996).Reflection from the bottom of a fractured layer was stud-ied by Sayers and Rickett in “Azimuthal variation inAVO response for fractured gas sands” (GeophysicalProspecting, 1997). The anisotropic energy-focusing phe-nomena and normal-moveout velocity in HTI mediawere described by Tsvankin in “Body-wave radiationpatterns and AVO in transversely isotropic media” (GEO-PHYSICS, 1995) and “Reflection moveout and parameterestimation for horizontal transverse isotropy” (GEO-PHYSICS, 1997). LE

Corresponding author: Andreas Rüger, at Landmark Graphics,arueger@lgc.com

1434 THE LEADING EDGE OCTOBER1997