Seismic characterization of multiple fracture sets from ...€¦ · seismic data. Key words: multicomponent wide-azimuth seismic data, fracture characterization, multiple fracture

$Page 1: Seismic characterization of multiple fracture sets from ...€¦ · seismic data. Key words: multicomponent wide-azimuth seismic data, fracture characterization, multiple fracture$
CWP-533

Seismic characterization of multiple fracture sets

from multicomponent, multiazimuth, 3D data:

Rulison Field, CO

Ivan Vasconcelos†

and Vladimir Grechka?

†Department of Geophysics, Colorado School of Mines, Golden, CO 80401, USA? Shell International Exploration and Production, Inc., 3737 Bellaire Blvd., P.O.Box 481, Houston, TX, 77001-0481, USA

ABSTRACT

Conventional fracture-characterization methods assume the presence ofa single set of aligned, vertical cracks in the subsurface. We relax thisassumption and demonstrate the feasibility of seismic characterizationof multiple fracture sets. Our technique relies on recent theoretical find-ings indicating that multiple, differently oriented, possibly intersectingplanar cracks embedded in an otherwise isotropic host rock result in anapproximately orthotropic effective media. Here, the governing parame-ters of crack-induced orthotropy are estimated from 3D, wide-azimuth,multicomponent (P-, fast shear-, and slow shear-wave) seismic reflectiondata acquired over the tight-gas Rulison Field in Colorado.We use strong azimuthal variations of the normal-moveout velocities toinvert for the interval crack densities, fracture orientations, type of fluidinfill, and velocities of P- and S-waves in the unfractured rock. Our in-version procedure identifies a set of cracks approximately aligned in theWNW-ESE direction in the western part of the study area and multi-ple, likely intersecting fractures in its eastern part. We validate both ourunderlying theoretical model and the obtained estimates by a numberof independent measurements: (i) the estimated fluid-infill parameterindicates dry cracks as expected for the gas-producing sandstones atRulison; (ii) our model predicts almost elliptical anisotropy and, in-deed, the long-spread P-wave data exhibit no nonhyperbolic moveout;(iii) the obtained crack orientations are supported by well observations.As a by-product of fracture characterization, we build an anisotropicvelocity model of the Rulison reservoir which, we believe, is the firstexample of an orthotropic velocity model constructed from field surfaceseismic data.

Key words: multicomponent wide-azimuth seismic data, fracturecharacterization, multiple fracture sets, orthorhombic anisotropy

1 INTRODUCTION

Naturally fractured rocks are widely recognized aspotential hydrocarbon-bearing formations. Perhapsthis is why the 2004 Summer Research Workshop ofthe Society of Exploration Geophysics (SEG) andthe European Association of Geoscientists and En-

gineers (EAGE) was held to determine the stateof the art of seismic characterization of fracturedreservoirs. The workshop demonstrated that cur-rent industry capabilities are limited to a single setof parallel, vertical cracks embedded in an other-wise isotropic host rock. For circular cracks, thisarrangement leads to the effective horizontal trans-

116 I. Vasconcelos & V. Grechka

verse isotropy (HTI) whose parameters can be esti-mated from seismic data. The papers presented atthe workshop exploited HTI model and discussedvarious approaches to inferring the crack density,fracture orientation, and the type of fluid infill fromsuch seismic signatures as normal-moveout (NMO)velocities (Berthet et al., 2004), amplitude-versus-offset (AVO) variations (del Monte et al., 2004;Gray and Todorovic-Marinic, 2004; Minsley et al.,2004; Todorovic-Marinic et al., 2004), and shear-wave splitting coefficients (Gaiser and Van Dok,2004).

As all the above mentioned techniques arebased on measuring the magnitude of azimuthalanisotropy for a selected seismic signature andtranslating it into the parameters of a single setof cracks, they are bound to produce misleadingresults in the presence of differently oriented frac-tures. Such arrangements of multiple fracture setswith different orientations are often identified in theborehole televiewer data, and have been observed atnumerous outcrops throughout the globe. Given theubiquity of multiple fracturing, the inadequacy ofexisting fracture-characterization technology mightbe explained by an inherent nonuniqueness in in-verting the effective anisotropy (estimated fromseismic data) for the parameters associated withmultiple crack systems. Indeed, the number offracture-related parameters can be arbitrarily largewhereas the maximum number of independent ef-fective stiffness coefficients for any homogeneousrock volume is only 21. While the number 21 obvi-ously imposes an ultimate limit on the number Nof fracture sets that can be uniquely resolved fromlong-wavelength seismic data, this N is typicallygreater than one. Grechka and Tsvankin (2003)used the linear slip theory of Schoenberg (1980) toderive limits for N . They found, in agreement withthe earlier results of Bakulin et al. (2000), that seis-mic data can uniquely constrain the parameters ofup to two vertical, rotationally-invariant fracturesets embedded in isotropic background rock.

To make further progress in fracture charac-terization, one has to rely on certain microstruc-tural information related to the cracks. Such in-formation was utilized in the effective media the-ory proposed by Kachanov (1980). Being similarto Schoenberg’s (1980) linear slip theory for drycracks, Kachanov’s theory adds the following im-portant insight: the collective contribution of mul-tiple sets of dry fractures to the effective elastic-ity is largely controlled by a symmetric, second-rank crack-density tensor. [Schoenberg and Sayers(1995) call such fractures “scalar.”] The ability ofrepresenting multiple crack systems by a second-

rank tensor leads to orthotropic effective symmetryfor any orientation distribution of fractures and im-plies that arbitrarily oriented, vertical cracks can

be replaced by two mutually orthogonal fracturesets (called the principal ones) as far as propaga-tion of low-frequency seismic waves is concerned.Moreover, the effective orthotropy induced by drycracks residing in a purely isotropic host rock turnsout to be remarkably simple: it is fully describedby only four independent quantities instead of nineneeded for general orthotropy. These quantities arethe two principal crack densities, e1 and e2, and theLame constants of the isotropic background, λb andµb.

The findings of Kachanov (1980; also review1993) are extremely valuable for fracture character-ization because they reduce any number of verticalfracture sets to just two and, thus, provide a theo-retical basis for estimating parameters of multiplesystems of fractures. Prior to using these theoret-ical predictions for such a parameter estimation,however, it is important to establish their accuracyand the range of applicability. This work has beendone by Grechka and Kachanov (2006a; 2006b) andGrechka et al. (2006), who performed finite elementsimulations of the effective elasticity for a wide vari-ety of crack arrays that contained planar, open andpartially closed, intersecting and non-intersecting,circular and irregular dry fractures. These papersdemonstrate that the deviations from effective or-thotropy never exceed 2% in the entire range ofthe crack densities expected in naturally fracturedformations. Moreover, the results of Grechka andKachanov (2006a) indicate that the effective or-thotropy holds for liquid-filled cracks even with abetter accuracy due to the stiffening of fracturesby fluids and a subsequent reduction of crack con-tribution to the overall elasticity. Following numer-ical verification of the effective crack-induced or-thotropy, Grechka and Kachanov (2006a) proposeda technique for characterization of multiple verti-cal fracture sets. Their method utilizes 3D, mul-tiazimuth, multicomponent seismic reflection dataand yields estimates of the orientations and crackdensities of two principal fracture sets, the density-normalized Lame constants of the background, andthe so-called fluid factor that ranges from 0 for drycracks to 1 for fractures filled with a stiff liquid.Bakulin et al. (2000) also proposed a method forcharacterizing two orthogonal crack sets in isotropicbackground. Because their approach is based onthe linear-slip theory, seismic signatures are ex-pressed in terms of the fracture weaknesses insteadof microstructural parameters, such as crack densi-ties and fluid infill. Since in this we follow the mi-crostructural approach from Kachanov (1980), wewill mostly refer to studies based on this approach.The work by Bakulin et al. (2000) provides com-prehensive results in the context of the linear-slipapproach. Grechka and Kachanov (2006a, 2006b)

Seismic characterization of multiple fracture sets at Rulison Field 117

also provide a thorough numerical comparison be-tween these approaches.

Here we apply this fracture-characterizationtechnique to a 3D, 9C, wide-azimuth data set ac-quired by the Reservoir Characterization Project(Colorado School of Mines) over the tight-gas Ruli-son Field located in Colorado. We invert the inter-val NMO ellipses of P, S1 (fast shear), and S2 (slowshear) waves for the two principal crack densities,fracture azimuths, and fluid factors. The fractureorientations obtained from seismics are supportedby direct borehole measurements, while the fluidfactors turn out to be less than 0.01 as expected forthe gas-bearing sandstones at Rulison. The theoryalso predicts almost elliptical anisotropy, and, in-deed, we do not observe nonhyperbolic moveout onthe P-wave long-spread data. Overall, the model ofcrack-induced orthotropy seems to adequately ex-plain our data and allow us to characterize multiplefracture sets in the context of effective orthotropyfor planar cracks (Kachanov, 1980, 1993; Grechkaand Kachanov, 2006a, 2006b; Grechka et al., 2006).

2 THEORETICAL BACKGROUND

2.1 Effective anisotropy induced by

vertical cracks

The model we assume here is based on a generalrepresentation of the effective compliance tensor,se, of a rock containing cracks as a sum,

se = sb + ∆s , (1)

of the compliance, sb, of the isotropic unfracturedbackground and the cumulative contribution, ∆s,of multiple fractures. In the non-interaction ap-proximation (Kachanov, 1980; 1993; Grechka andKachanov, 2006a), the nonzero elements of tensor∆s (in Voigt notation) are

∆s11 =16 e1 (1 − ν2

b )

3 Eb

(1 − ς) ,

∆s22 =16 e2 (1 − ν2

b )

3 Eb

(1 − ς) ,

∆s44 =32 e2 (1 − ν2

b )

3 Eb (2 − νb),

∆s55 =32 e1 (1 − ν2

b )

3 Eb (2 − νb),

∆s66 = ∆s44 + ∆s55 . (2)

Here e1 and e2 are the densities of the two principalvertical fracture sets, 0 ≤ ς ≤ 1 is the fluid factor(ς ≈ 0 for dry cracks, ς ≈ 1 for liquid-filled ones),and Eb and νb are the Young’s modulus and Pois-son’s ratio of the background, respectively. They

are given by standard expressions,

Eb = µb

3 λb + 2 µb

λb + µb

and νb =λb

2 (λb + µb), (3)

in terms of the Lame parameters λb and µb of thehost rock. The compliance components (2) are writ-ten in the coordinate frame whose x1 and x2 axesare normal to the two principal fracture sets, andthe x3 axis is vertical.

The predictions of the non-interaction effec-tive media theory [equations (1) and (2)] appearto be remarkably simple. Yet, their high accuracyhas been confirmed numerically for fracture arraysthat grossly violate the assumptions of the non-interaction approximation (Grechka and Kachanov,2006a; 2006b; Grechka et al., 2006). Specifically,formulations (1) and (2) remain sufficiently accu-rate for multiple sets of differently oriented, irreg-ularly shaped, partially closed cracks that mightintersect each other and form interconnected frac-ture networks. This means that long (compared tothe crack sizes) seismic waves propagating throughthose fractures “see” them as if they were isolatedand orthogonal to each other. As a result, the ef-fective symmetry sensed by such long seismic wavesis nearly orthorhombic even though no local sym-metry exists on the scale of a few fractures. Hence,the effective crack densities and their orientationsfor a medium with many fracture sets reflect re-spectively, the compliance change induced by thepresence of cracks and the corresponding principalcrack directions. As a result, best-fit crack densi-ties are approximately equivalent to projecting theactual crack densities corresponding to many dif-ferent fracture sets onto the principal directions ofthe orthotropic effective elasticity tensor. Anotherconclusion that follows from equations (1) and (2)is that the effective anisotropy caused by dry frac-tures (ς ≈ 0) is almost elliptical (Kachanov, 1993;Grechka and Kachanov, 2006a). Below we showthat seismic data acquired at Rulison support thistheoretical result.

2.2 Inversion strategy

As follows from equations (1)–(3), the effective elas-tic properties of rocks with multiple vertical frac-ture sets are described by the parameter vector

M = {λb, µb, e1, e2, ς} (4)

that might vary spatially over the seismic surveyarea. Grechka and Kachanov (2006a) demonstratedthat all components of M can be uniquely es-timated from 3D, multiazimuth, multicomponentseismic reflection data. (Of course, seismic dataconstrain the density-normalized background Lameconstants rather than λb and µb themselves.) In


[b]

Figure 1. Map of gas-producing fields in PiceanceBasin, Colorado (after Ellison, 2004).

particular, it has been shown that M can be un-ambiguously inverted from the data vector

D(M ) =

VS1

VP0,

VS2

VP0, W

P , WS1, W

S2

ff

, (5)

where VP0, VS1, and VS2 are the velocities of verti-cally propagating P- and two split shear-waves (fastS1 and slow S2). The velocity ratios entering D

can be computed from the zero-offset times afterestablishing the P-to-S event correspondence. The2×2 matrices W in equation (5) are the normal-moveout (NMO) ellipses of pure modes reflectedfrom a horizontal interface (Grechka and Tsvankin,1998).

Vertical cracks result in effective orthorhom-bic media with two vertical symmetry planes, andsince the reflector is horizontal, all three NMO el-lipses W

P , WS1, and W

S2 are always co-oriented:their axes point along the normals to two principalfracture sets. Therefore, data vector (5) containseight quantities: two velocity ratios and three pairsof the ellipse semi-axes, which are the symmetry-direction NMO velocities of P-, S1-, and S2-waves.Only seven quantities of those eight are generallyindependent because two out of four semi-axes ofthe shear-wave NMO ellipses coincide (Grechka etal., 1999). Thus, the functional dependence D(M )consists of seven equations and five unknowns. Toaid understanding of this dependence, Appendix Agives the weak-anisotropy (or small-crack-density)approximations of both the elements of D and theanisotropic coefficients of orthorhombic media thatwere introduced by Tsvankin (1997).

In practice, we estimate data vectors (5)from wide-azimuth P, S1, and S2 common-

midpoint (CMP) gathers. The NMO ellipses areobtained from 3D semblance analysis (Grechka andTsvankin, 1999; Vasconcelos and Tsvankin, 2006)that maximizes the semblance by fitting an ellipseto the azimuthal variation of the NMO velocity.Another technique used in the industry is basedon measuring the traveltime shifts of a given reflec-tion event after azimuthally-independent NMO cor-rection (Jenner, 2001). Once the azimuthal NMOanalysis is completed, the Dix-type differentiation(Grechka and Tsvankin, 1999) is applied to cal-culate the interval NMO ellipses W

Pint, W

S1int, and

WS2int and the interval data vector Dint. These el-

lipses typically exhibit some misalignments due avariety of factors (e.g., noise, possible azimuthalanisotropy of the host rock, reflector dip, fracturetilt, interaction of the closely spaced cracks) whichare not captured in our model. The data indi-cate, however, that those misalignments are usuallysmall (see Figure 9 and the corresponding discus-sion), therefore, we ignore them and estimate theparameter vector M at each CMP location thoughnonlinear minimization,

F = min(M)

||Dint − D(M )|| , (6)

that operates with the elliptical semi-axes only.In our inversion procedure, we use the exact ex-pressions for D(M ) rather than their small-crack-density approximations given in Appendix A.

Finally, we examine whether the inferred or-thorhombic model correctly predicts seismic signa-tures that were not used in the inversion and com-pare our estimates of the crack densities and orien-tations with the available borehole measurements.

3 RULISON FIELD DATA

3.1 Brief overview of the field

Rulison Field, located in the South Piceance Basin,Garfield county, Colorado (Figure 1), produces gasfrom a 1,200 ft pay section of channel sand lensesstacked within the Cretaceous Mesaverde sandstoneformation. Porosities (from 1% to 10%) and perme-abilities (from 1 µD to 60 µD) of these sand bod-ies characterize the formation as a tight gas reser-voir (Rojas, 2005). Because of such a low matrixpermeability, it is believed (Johnson et al., 1987;Spencer, 1989; Lynn, 1999) that gas production iscontrolled by open natural fractures. Fracture ori-entations vary over the field area (Jansen, 2005),and their intersections are thought to be of im-portance for the reservoir production. The averagethickness of sand lenses is about 10 ft, which makesthem impossible to image with surface seismic data.Since neither sand bodies nor fracture zones can be


0.4

0.6

0.8

1.0

1.2

1.4

Tim

e (s

)

10 30 50 70 90 110 130 150Inline1000 ft

Top

Bottom

Figure 2. P-wave seismic section along Crossline 60. Arrows mark the reflection events used for azimuthal velocityanalysis, which correspond the top and bottom of the reservoir.

individually isolated on seismic images, it is nec-essary to characterize the fractures collectively asan effective medium. Reflector dips at Rulison aresmall and can be neglected for the purposes of seis-mic data processing (Figure 2).

3.2 Data acquisition

A multicomponent seismic data set covering 5.5km2 was acquired by the Reservoir Characteriza-tion Project (Colorado School of Mines) in an effortto characterize Rulison Field with information frommultiple sources (surface seismic, VSP, geomechan-ics, well logs, production data) and monitor thechanges in production and stresses with time-lapsemeasurements. For our study, we use the baselinewide-azimuth, 3D, multicomponent seismic surveyacquired with 3C vibrators and 3C geophones.

3.3 Data processing

Our input data consist of three CMP-sorted vol-umes (P, S1 and S2) corrected for the source-receiver statics and with source signature decon-volution applied. These pre-processing steps havebeen done by Veritas prior to our work. In addition,ground-roll suppression turned out to be necessary.Strong surface waves, present in all three data vol-umes, are particularly harmful for the shear reflec-

tions (Figure 3). As the S-wave events come later intime with the moveout slower than that of P-waves,a greater offset interval is covered by the ground-rollas opposed to P-data (compare Figures 3a and 3c).To suppress it, we resort our irregularly sampled(in offsets and azimuths) data into shot and re-ceiver lines, where the sampling is regular, and ap-ply an f -kx-ky filter that preserves the slopes up of150 ms/kft. This approach to ground-roll suppres-sion does not destroy azimuthal variations of trav-eltimes and results in a significant improvement ofthe reflected events (Figure 3).

Another pertinent issue concerns the twoshear-wave volumes (S1 and S2) that we have atour disposal. The shear data have been rotated tothe azimuth of the principal regional stress N45E.Treating this azimuth as the orientation of one ofthe principal fracture sets implies a constant frac-ture azimuth throughout the field, in contradictionwith the laterally variable orientations of the P-wave NMO ellipses that we observe from the bot-tom of the producing interval. To mitigate thisdiscrepancy, we rotated the shear volumes to theprincipal directions of these P-wave NMO ellipsesat each CMP location prior to performing the S-wave velocity analysis. This is done because themodel assumes shear-waves polarizations and NMOellipses to be aligned. Because P-wave azimuthalanisotropy above the reservoir is small (the eccen-tricities of the P-wave NMO ellipses are consistently


smaller than 3%) we assume that the ellipse orien-tations are predominantly dictated by anisotropyat the reservoir level, so we use P-wave NMO el-lipse orientations estimated from the bottom hori-zon events. Rotating shear-wave data in such man-ner will introduce errors in the estimated parame-ters wherever the model is inadequate.

After the ground-roll suppression and shear-wave rotation, we resort the data back to CMPgeometry, create 9×9 (440×440 ft) superbins andextract the NMO ellipses over the survey area. Ateach superbin location, the NMO-ellipse estimationis performed for a window centered around the zero-offset times that correspond to the reservoir top andbottom horizons (Figure 2) interpreted from the P,S1, and S2 image volumes. The NMO ellipses arecomputed in 5 ms time increments over a windowcentered around the horizon times. The averaginglengths of the semblance windows are 35 ms for P-and 105 ms for shear-wave data. Our use of the in-terpreted horizons for velocity analysis ensures re-sults that are consistent with the geologically-basedtime structure.

We obtain reliable (see Section 4.1) estimatesof the P-waves NMO ellipses over the entire surveyarea (the corresponding semblance values range be-tween 0.55 and 0.65). As the shear data are noisier(compare Figure 3b with 3d), the S-wave NMO el-lipses have been estimated for only about one thirdof the survey area. We discarded the S-wave NMOellipses whenever the semblances happened to besmaller than 0.25. Indeed, the reflection events can-not be visually distinguished on the gathers withsuch low semblance values. On the other hand, inthe higher-quality data areas, the azimuthal travel-time variations are clearly seen on the gathers cor-rected with an azimuthally-constant NMO velocity(Figure 4).

After estimating the P, S1, and S2 NMO el-lipses for the top and bottom horizons (located ap-proximately at 4,000 ft and 7,500 ft depths) over theentire survey area, we proceed with the Dix-typedifferentiation (Grechka and Tsvankin, 1999) thatyields the interval ellipses W

Pint, W

S1int, and W

S2int.

Finally, minimizing the objective function (6), weinvert W

Pint, W

S1int, and W

S2int and the interval zero-

offset time ratios for the fracture parameters.

3.4 Fracture characterisation

Figure 5 displays the inverted background velocityfields VP, b and VS, b [approximations (A9), (A10)]and the principal crack densities e1 and e2 (by def-inition, e1 ≥ e2). The crack densities in Figure 5care considerably greater than those in Figure 5dsuggesting that the fracturing is dominated by thecracks trending in the WNW-ESE direction. Our

results indicate that the western part of the areais controlled by a single fracture set that has den-sity e1, while the eastern part has a non-negligiblecontribution of other, differently oriented fracturesthat exhibit themselves as the set with crack den-sity e2 (Figures 5c and 5d). We also see that theareas of the highest crack densities do not neces-sarily coincide with the largest P-wave NMO ellipseeccentricities. We will revisit this issue below.

The background velocities (Figures 5a and 5b)show a sizeable contrast between the eastern andwestern portions of the study area, with both P-and S-wave velocities being slower in the west. Thefluid factors ς have been estimated also but we donot display them because all the obtained values ofς are smaller than 0.01. These estimates are consis-tent with the fact that Rulison Field produces onlygas.

3.5 Orthotropic model of the reservoir

The estimated background velocities and crack den-sities (along with the fluid factors ς ≈ 0) are suf-ficient for building an orthorhombic depth modelof the reservoir. Figures 6 and 7 show the verti-cal velocities VP0 and VS0 and Tsvankin’s (1997)anisotropic coefficients (the corresponding small-crack-density approximations can be found in Ap-pendix A). We also display the anellipticity coeffi-cients η(1, 2, 3) (right column in Figure 7) to demon-strate that the predicted anisotropy is nearly ellip-tical.

It is instructive to point out that the modelin Figures 6 and 7 was obtained from reflectiondata only without using any borehole informationto constrain the vertical depth scale. Although well-log or checkshot data are usually necessary forbuilding orthorhombic subsurface models in depthdomain (Grechka et al., 2005), we did not needsuch data in our study. The reason is that we ex-plicitly targeted the crack-induced rather than gen-eral orthotropy. The former is significantly simplerbecause it is governed by fewer (five rather thannine) independent parameters. This reduction inthe number of unknowns allowed us to rely solelyon surface seismic data for building the uniqueorthorhombic model of Rulison reservoir. As Fig-ures 6 and 7 indicate, the reservoir is noticeablyanisotropic: the magnitudes of anisotropic coeffi-cients ε(2) and δ(2) reach 0.2 at the highest totalcrack density e = e1+e2 = 0.17 (Figures 5c and 5d).


db

Tim

e (s

)

Tim

e (s

)T

ime

(s)

Tim

e (s

)

caTrace numberTrace number

Trace numberTrace number

1.0

1.5

500 1000 1500 20001.5

2.0

2.5

1000 2000 3000

1.5

2.0

2.5

1000 2000 3000

1.0

1.5

500 1000 1500 2000

Figure 3. Representative input [(a) and (c)] and f-kx-ky-filtered [(b) and (d)] CMP gathers of P- (left column) andS1-waves (right column). Offsets increase with increasing trace number. Both offsets and azimuths are irregularlysampled in the CMP domain.

3.6 Convention approaches to fracture

characterization

Having estimated the reservoir parameters fromboth P- and S-wave data, we can predict whatwould happen if we relied S- or P-waves alone forfracture characterization. In a typical pure shear-wave survey, one would measure the shear-wavesplitting coefficient, γ(S) ≈ γ(1) − γ(2), and inter-pret it as the crack density of a single fracture set.The result of this interpretation (Figure 8a) sug-gests that the western part of the study area ismore fractured than the eastern one, quite oppositeto the conclusion drawn from Figures 5c and 5d.

If only P-wave data were used (e.g., Grechkaand Tsvankin, 1999), one would estimate the ec-centricity of the P-wave NMO ellipses, which isquantified by the difference of two δ coefficients,δ(1) − δ(2). This difference, shown in Figure 8b,yields a similar result that the western part of the

area is apparently more fractured than the easternpart.

The origin of these mutually contradictory con-clusions can be understood from the approxima-tions given by equations (A25) and (A26). Clearly,both quantities γ(S) and δ(1)−δ(2) are proportionalto the difference e1−e2 between the crack densitiesof two principal fracture sets. As a consequence, theshear-wave splitting coefficients and the eccentrici-ties of the P-wave NMO ellipses are useful for frac-ture characterization only if one fracture set dom-inates, that is, when e1 � e2. If multiple sets ofcracks resulting in comparable e1 and e2 are presentin the subsurface, both differences γ(1) − γ(2) andδ(1) − δ(2) become ambiguous. In particular, if thetwo principal crack densities coincide, e1 = e2 6= 0,both differences vanish, γ(1)−γ(2) = δ(1)−δ(2) = 0,and one would arrive to an obviously incorrect con-clusion that the fractures are absent.


Figure 4. P- (a) and S1-wave (b) gathers after azimuthally-invariant NMO correction. The traces are sorted byazimuth. The apparent cosine-type dependence of the residual moveout is indicative of azimuthal anisotropy. Thereflection events at approximately at 1.27 s in (a) and at 2.35 s in (b) correspond to the reservoir bottom. Theoffset-to-depth ratios are between 0.8 and 1.0.

4 VERIFICATION OF RESULTS

An important part of any inversion procedure is as-sessing the errors in the estimated quantities. Thissection describes out efforts in this direction.

4.1 Variances in data and estimated

parameters

We estimated the data variances associated withuncertainties in traveltime picking. A plausiblepicking interval was defined in accordance with theautocorrelation function of traces in a CMP gather.Specifically, we take this interval to be equal to thetime lag where the autocorrelation drops by a fac-tor of 10 from its zero-lag value. Such a definitionyields the picking uncertainties equal to 8 ms forP- and 14 ms for S-waves. Next, a standard lin-ear error-propagation technique is applied to trans-late these traveltime errors into uncertainties in theNMO ellipses. Finally, using the Frechet-derivativematrix ∂Dj/∂Mi, where Mi and Dj are the compo-nents of vectors M and D [equations (4) and (5)],we propagate the errors into the estimated quanti-

ties shown in Figure 5. This yields uncertainties ofabout 7% in the velocities VP, b and VS, b and 0.01in the principal crack densities e1 and e2 within the90% confidence interval.

Estimation of the model variances is not theonly resource for validating our results. As we havealready mentioned, the fluid factors ς turned outto be nearly zero at all CMP locations which isexpected because of the absence of production ofliquids (oil or water) at Rulison.

Another way of assessing the quality of ourmodel assumptions is to analyze the misalignmentsin the orientations of the interval NMO ellipses fordifferent wave modes. The histograms in Figure 9exhibit pronounced peaks at 0◦ indicating that theellipses are perfectly aligned at many CMP loca-tions. Even though the presence of outliers (prob-ably caused by noise in the data) makes the av-erage differences in the ellipse orientations of P-and S1-waves and P- and S2-waves be equal to 12◦

and 17◦, respectively, their significant portion canbe attributed to fracture interaction or, in seismicterminology, to multiple scattering of elastic energyon the closely spaced cracks. For instance, Grechka


0 1000 ft

c d

ba

Inline Inline

Inline

Crossline

InlineCrossline

Crossline

Crossline

N

Figure 5. Output of fracture characterization: the background velocities VP, b and VS, b (in kft/s) of P- (a) andS-waves (b), and the principal crack densities e1 (c) and e2 (d). The directions of the principal fracture sets areshown with ticks. Their lengths are proportional to the eccentricities of the interval P-wave NMO ellipses (shown in

Figure 8b).

and Kachanov (2006a) found this mechanism to beresponsible for the NMO ellipse misalignments upto 12◦ at the total crack density e = 0.14. Thus,we conclude that orientations of the NMO ellipsesare consistent with our fracture model over most ofthe survey area. In our parameter estimation pro-cedure we do not take into account the distortionsthat might be introduced by the manner in whichS-wave data was rotated.

4.2 Nonhyperbolic moveout

Our next step is to analyze the character of long-offset data in order to verify the ellipticity conditionof the model of approximate effective orthotropyin isotropic background. Small values of the anel-lipticity coefficients η in the reservoir (right col-umn of Figure 7) and nearly isotropic overburdenimply the absence of nonhyperbolic moveout. Fig-ure 10, which displays a typical long-spread P-wavegather, clearly supports this model conclusion. In-deed, no ’hockey-sticks’ indicative of the nonhyper-bolic moveout are observed in Rulison Field.

4.3 Borehole measurements

The survey area contains well B. RWF 542-20 (lo-cated at Inline 92, Crossline 76), where an FMI logwas run and fractures in the entire reservoir werecounted. Figure 11 compares the borehole fracturecount with our estimate of the crack orientation. Asa dominant set of cracks has been identified at In-line 92, Crossline 76 (Figures 5c and 5d), we do notdisplay the second set in Figure 11 because it hasnearly zero crack density. The fractures observedin FMI form two sets oriented approximately atN70W and N73E (blue in Figure 11). Clearly, thesesets are not orthogonal to each other. And yet,our results suggest that their influence on propa-gation of long (compared to the fracture sizes) seis-mic waves is equivalent to that of two orthogonalsets. The dashed red line in Figure 11 shows theazimuth of the dominant equivalent set estimatedfrom seismics.

Figure 11 can be regarded as an illustration ofthe resolution achievable from seismic data: whilelong seismic waves cannot resolve each individualfracture set, they are sensitive to the influence ofall fractures simultaneously. This cumulative influ-


b

dc

e f

a

Crossline

InlineInline CrosslineCrossline

Inline Inline

Inline Inline

CrosslineCrossline

Crossline

Figure 6. Vertical velocities VP0 and VS0 (in kft/s) and anisotropic coefficients ε(1, 2) and γ(1, 2) at Rulison reservoir.

ence of all cracks on elastic wave propagation isdescribed by the crack-density tensor (Kachanov1980; Grechka and Kachanov, 2006a, their equa-tion (2)) that can be represented in terms of contri-butions of mutually orthogonal (or principal) frac-ture sets. It is those principal sets that control theseismic signatures and, therefore, can be estimatedfrom them. Thus, returning to Figure 11, we canstate that our fracture-characterization results areconsistent with the FMI log in well B. RWF 542-20.

4.4 Production data

There is a number of gas-producing wells drilled inRulison Field, and, naturally, we attempted to es-tablish a correlation between their Estimated Ulti-

mate Recovery (EUR) and the obtained crack den-sities. As we observe no correlation, we would liketo offer our thoughts on this issue because we feelthat they might be relevant to other tight-gas fields.

First, the wells in Rulison Field have beendrilled at different times in the past and, conse-quently, completed differently. Specifically, noveltechniques of hydraulic fracturing and modernproppants significantly improved the gas recovery.Seismic data, however, are sensitive to both nat-ural and man-made fractures and, obviously, donot allow us to separate a portion of the estimatedcrack densities primarily responsible for the pro-duction. Second, the EURs, being largely influ-enced by the completion techniques, vary rapidlythroughout the field, whereas the crack densities


b

dc

e f

a

Crossline

InlineInline CrosslineCrossline

Inline Inline

Inline Inline

CrosslineCrossline

Crossline

Figure 7. Anisotropic coefficients δ(1, 2, 3) and η(1, 2, 3) at Rulison reservoir.

a b

Crossline Crossline InlineInline

Figure 8. The shear-wave splitting coefficient γ(S) (a) and the eccentricity of the P-wave NMO ellipses δ(1)− δ(2)

(b) at Rulison reservoir.


−40 −20 0 20 400

500

1000

1500

2000

2500

3000

Misalignment of NMO ellipses (degrees)

a

−40 −20 0 20 400

200

400

600

800

1000

1200

1400

1600

1800

Misalignment of NMO ellipses (degrees)

b

Figure 9. Histograms of azimuthal misalignments of the interval NMO ellipses of P- and S1-waves (a) and P- andS2-waves (b).

0.5

1.0

1.5

1000 3000Trace number

Tim

e (s

)

Figure 10. Typical long-spread P-wave gather af-ter azimuthally-varying hyperbolic moveout correction.The offsets are irregularly sampled. Maximum offset-to-depth ratio is 2.0.

estimated from seismic data are relatively smooth(Figures 5c and 5d) given the lateral resolution ofseismic data. This difference in the rate of changealso hinders the correlation between the two.

The third and perhaps main reason for the ab-sence of correlation can be sought in the physics ofeffective elasticity and permeability in low-porosityrocks. The quantity governing their overall elastic-ity is the crack density. Being the average of thecubed half-lengths of fractures, it carries no infor-mation about the crack widths or their openings.In contrast, the effective, crack-related permeabil-ity is essentially controlled by the fracture open-ings. Consequently, a short, wide-open crack (likethe man-made ones) significantly contributes to thepermeability and little to the elasticity. Thus, the

Figure 11. Fracture count (blue) in wellB. RWF 542-20 located at Inline 92, Crossline 76and the 90% confidence interval (dashed red) corre-sponding to the azimuth of the fracture set with densitye1 estimated from seismic data.

effective permeability and elasticity are governedby different quantities that, in general, cannot beinferred from each other. However, with increasingpore space, an effective fractured porous mediumis likely to show a direct cross-property relation-ship between its effective elasticity and permeabil-ity tensors. This cross-property connection wouldarise essentially from the microstructural interac-tion between cracks and pores.

5 DISCUSSION AND CONCLUSIONS

We have discussed seismic characterization of frac-tures at Rulison Field, Colorado. To the best ofour knowledge, Rulison is the first case studywhere multiple fracture sets have been quantita-tively characterized by not only their effective ori-entations but by their microstructural parameters:


two principal crack densities, type of infill, andbackground parameters. The key of our parameter-estimation methodology is the understanding (con-firmed with extensive numerical modeling) that dif-ferently oriented, multiple vertical sets of cracks ap-pear as two orthogonal sets for long seismic waves.Following this understanding, we have targeted theparameters of those two orthogonal (or principal)sets in our inversion. The obtained results pointto the presence of fractures approximately alignedin the WNW-ESE direction in the western part ofthe study area. In contrast, at least two principalsets of cracks exist in its eastern part. Their crackdensities reach e1 = 0.11 and e2 = 0.06 indicatingthat these fractures likely form an interconnectednetwork. This follows from a straight forward geo-metrical consideration: it would be difficult to placemany cracks corresponding to two sets of orthogo-nal fractures that have random locations and crackdensities of e1 = 0.11 and e2 = 0.06 into a rock vol-ume in a manner in which those fractures wouldnot intersect each other. Thus, our inversion hasidentified places in the reservoir where swarms ofintersecting cracks are likely to be present.

Under the assumption of approximate effec-tive orthotropy in isotropic background (Kachanov,1980; Grechka and Kachanov, 2006a, 2006b;Grechka et al., 2006), we have built an orthotropicdepth model of the Rulison reservoir. The recon-structed orthorhombic velocity model explicitly at-tributes the observed anisotropy to the presenceof fractures. This rock-physics constraint reducesthe number of independent quantities to be esti-mated and obviously improves stability of the in-version. In particular, it allows the vertical P- andS-wave velocities and all anisotropic coefficients tobe uniquely determined solely from 3D, multiaz-imuth, multicomponent seismic reflection data. Webelieve that Rulison is the first example of buildinga subsurface velocity model that has orthorhombicsymmetry.

Special attention was paid in our study to ver-ifying the obtained results. Here are the evidencesthat support our findings.

(1) We propagated uncertainties in the pickedzero-offset times and NMO ellipses into the quan-tities that characterize the fractures. This yieldederrors of about 7% in the P- and S-wave velocitiesin the host rock and around 0.01 in the principalcrack densities.

(2) Rulison Field produces dry gas. While know-ing this beforehand, we intentionally included thefluid factor into our unknowns. Its estimated valuescame out to be nearly zero in the entire study area,consistent with our information about of the typeof fracture infill.

(3) The NMO ellipses of P-, S1-, and S2-wavesrecorded from horizontal reflectors are supposed

to be co-oriented according to the adopted non-interaction theory of crack-induced anisotropy. Inspite of the fact that we cannot account for errorsinduced by the rotation of shear-wave data, the el-lipse misalignments observed in seismic data turnedout to be comparable to those predicted by numer-ical modeling that takes into account interaction ofclosely spaced fractures.

(4) Anisotropy caused by multiple sets of drycracks is almost elliptical. And, indeed, no nonhy-perbolic moveout has been observed in the long-spread P-wave data.

(5) The available FMI log shows the fracture di-rections that are consistent with the effective prin-cipal crack orientations estimated from seismics.

Despite the above assurances that the obtainedresults not only exhibit internal consistency butalso fit some independent external data, our esti-mates of the crack densities show little correlationwith the EUR of wells in the field. While this poorcorrelation can be partially attributed to a varietyof completion techniques used in the course of de-velopment of Rulison Field, we feel that its rootcause is the physics of fluid flow through fracturedmedia. For low-porosity (1% to 10%) sandstonesat Rulison, fracture networks appear to providethe main pathways for the gas flow. The effective,crack-related permeability of these networks is es-sentially controlled by the fracture widths, whereasthe seismic signatures are governed by the crackdensities or the fracture lengths. If a relationshipbetween the crack widths and lengths is absent, theeffective permeability and elasticity are uncorre-lated. Perhaps this is what seismic and productiondata at Rulison tell us. Still, even if this inferenceis correct, a more general question remains: underwhat geologic conditions does the crack density be-come a proxy for the permeability which would beuseful for the reservoir development?

6 ACKNOWLEDGEMENTS

We thank the Reservoir Characterization Project(Colorado School of Mines) for providing seismicdata and Shell E & P for permission to publishthis paper. We are grateful to Jon Sheiman (Shell)for helping with data processing and Jorge Lopez(Shell) for his comments on the manuscript. Thework described in our paper has been carried outduring summer internship of I. V. at Shell E & Pin 2005. We also thank Ilya Tsvankin and RodrigoFuck for their careful review of the manuscript andtheir enriching comments.


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APPENDIX A: SMALL-CRACK-DENSITY APPROXIMATIONS OF SEISMIC

SIGNATURES

The goal of this Appendix is to develop an intuitive understanding of kinematic seismic signatures in for-mations containing multiple sets of vertical fractures. To arrive at tractable expressions, we assume thatthe principal crack densities are small, {e1, e2} � 1, and linearize all pertinent quantities in e1 and e2. Asstated in the main text, arbitrarily oriented fractures embedded in an otherwise isotropic host rock causenearly orthorhombic effective symmetry, therefore, Tsvankin’s (1997) anisotropic coefficients introduced fororthorhombic media are of obvious importance for our study. We begin with deriving these anisotropic co-efficients in terms of the governing parameters M [equation (4)] and then turn our attention to the NMOellipses of pure modes reflected from horizontal interfaces.

A1 Anisotropic coefficients

As follows from equations (1)–(3), definition of the effective stiffness tensor, ce = s−1e , and definitions (16)–

(19), (23)–(26) of Tsvankin (1997), the vertical velocities VP0, VS0 and relevant anisotropic coefficients, fullylinearized in the crack densities, are:

VP0 = VP, b

»

1 +2 λ2

b (ς − 1) (e1 + e2)

3 µb (λb + µb)

–

, (A1)

VS0 = VS, b

„

1 −8

3e1

λb + 2 µb

3 λb + 4 µb

«

, (A2)

ε(2) =8

3e1 (ς − 1) , (A3)

ε(1) =8

3e2 (ς − 1) , (A4)

δ(2) =8

3e1

»

(ς − 1) λb

λb + µb

−4 µb

3 λb + 4 µb

–

, (A5)

δ(1) =8

3e2

»

(ς − 1) λb

λb + µb

−4 µb

3 λb + 4 µb

–

, (A6)

γ(2) = −8

3e1

λb + 2 µb

3 λb + 4 µb

, (A7)

γ(1) = −8

3e2

λb + 2 µb

3 λb + 4 µb

. (A8)

Here

VP, b =

s

λb + 2 µb

ρ(A9)

and

VS, b =

r

µb

ρ(A10)

are the velocities in the isotropic unfractured rock, ρ is the density, and the quantities λb, µb, e1, e2, andς are described in the main text. Note that the anisotropic coefficients ε(2), δ(2), and γ(2), defined in theplane [x1, x3] that contains the normal to the first fracture set with the crack density e1, are proportionalto e1 and independent of e2. Similarly, the coefficients ε(1), δ(1), and γ(1) are controlled by e2 only. The sameobservations were made earlier by Bakulin et al. (2000).


A2 NMO ellipses and ratios of vertical velocities

Now we present the small-crack-density approximations for components of the data vector D [equation (5)].The ratios of the vertical velocities are

VS1

VP0=

VS0

VP0

s

1 + 2 γ(1)

1 + 2 γ(2)=

VS, b

VP, b

(

1 −2

3 µb (λb + µb)

"

λ2b (ς − 1) e1 (A11)

+(3 λb + 2 µb) (λ2

b ς + 4 µ2b) − λ2

b (3 λb − 2 µb ς)

3 λb + 4 µb

e2

#)

and

VS2

VP0=

VS0

VP0=

VS, b

VP, b

(

1 −2

3 µb (λb + µb)

"

λ2b (ς − 1) e2 (A12)

+(3 λb + 2 µb) (λ2

b ς + 4 µ2b) − λ2

b (3 λb − 2 µb ς)

3 λb + 4 µb

e1

#)

.

The pure-mode NMO ellipses from a horizontal reflector beneath a homogeneous orthorhombic layerare given by (Grechka and Tsvankin, 1998)

1

V 2Q, nmo(φ)

= WQ, 11 cos2 φ + WQ, 22 sin2 φ =cos2 φ

h

V(1)

Q, nmo

i2 +sin2 φ

h

V(2)

Q, nmo

i2 . (A13)

Here Q = P, S1, or S2 is the wave type, φ is the azimuth measured from the normal to the first fracture set,WQ, 11 and WQ, 22 are the nonzero elements of 2×2 matrices W that describe general NMO ellipses, and

V(i)

Q, nmo are the symmetry-direction NMO velocities. These velocities are conveniently expressed in terms ofthe P- and S-wave vertical velocities and the anisotropic coefficients listed above (Grechka et al., 1999):

V(i)

P, nmo = VP0

p

1 + 2 δ(i) , (i = 1, 2) , (A14)

V(1)

S1, nmo = VS1

p

1 + 2 σ(1) , (A15)

V(2)

S1, nmo = V(1)

S2, nmo = VS1

p

1 + 2 γ(2) = VS2

p

1 + 2 γ(1) , (A16)

V(2)

S2, nmo = VS2

p

1 + 2 σ(2) , (A17)

where

σ(1) =

„

VP0

VS1

«2

(ε(1) − δ(1)) , (A18)

σ(2) =

„

VP0

VS2

«2

(ε(2) − δ(2)) . (A19)

Substitution of equations (A1)–(A10) into equations (A14)–(A19) and subsequent linearization yields

V(1)

P, nmo = VP, b

(

1 +2

3 µb (λb + µb)

"

λ2b (ς − 1) e1 (A20)

+1

3 λb + 4 µb

`

3 λ3b (ς − 1) + 16 µb

`

λ2b (ς − 1) + λb µb (ς − 2) − µ2

b

´´

e2

#)

,

V(2)

P, nmo = VP, b

(

1 +2

3 µb (λb + µb)

"

λ2b (ς − 1) e2 (A21)

+1

3 λb + 4 µb

`

3 λ3b (ς − 1) + 16 µb

`

λ2b (ς − 1) + λb µb (ς − 2) − µ2

b

´´

e1

#)

,


V(1)

S1, nmo = VS, b

1 +8 (λb + 2 µb)

3 (λb + µb)e2

»

ς −µb

3 λb + 4 µb

–ff

, (A22)

V(2)

S1, nmo = V(1)S2, nmo = VS, b

»

1 −8 (λb + 2 µb)

3 (3 λb + 4 µb)(e1 + e2)

–

, (A23)

V(2)

S2, nmo = VS, b

1 +8 (λb + 2 µb)

3 (λb + µb)e1

»

ς −µb

3 λb + 4 µb

–ff

. (A24)

It is a simple matter now to construct the Frechet-derivative matrix, F , of the quantities given byequations (A11), (A12), (A20)–(A24) with respect to the elements of parameter vector M [equation (4)]and verify that all singular values of F are nonzero.

A3 Shear-wave splitting coefficient and the eccentricity of the P-wave NMO ellipse

Finally, we use the already derived approximations to demonstrate that such well-known fracture indicatorsas the shear-wave splitting coefficient and the eccentricity of P-wave NMO ellipse are controlled by thedifference of two principal crack densities, e1 −e2, and, therefore, insufficient for characterization of multiplefracture sets.

The eccentricity (or normalized elongation) of the P-wave NMO ellipse is just the difference of two δcoefficients. As follows from equations (A5) and (A6), the eccentricity is

δ(1) − δ(2) =8

3(e1 − e2)

»

(1 − ς) λb

λb + µb

+4 µb

3 λb + 4 µb

–

. (A25)

Similarly, definition of the shear-wave splitting coefficient, γ(S) ≈ γ(1) − γ(2), and equations (A7), (A8) yield

γ(S) =8

3(e1 − e2)

λb + 2 µb

3 λb + 4 µb

. (A26)

Clearly, these signatures unambiguously constrain the crack density only for a single fracture set, that is,when one of the principal crack densities is zero.


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