GEOPHYSICS, VOL. 70, NO. 2 (MARCH-APRIL 2005); P. C1–C5, 7 FIGS., 3 TABLES.10.1190/1.1884825

AVO for one- and two-fracture set models

He Chen1, Raymon L. Brown2, and John P. Castagna1

ABSTRACT

A theoretical comparison is made of PP and PSangle–dependent reflection coefficients at the top oftwo fractured-reservoir models using exact, general,anisotropic reflection coefficients. The two vertical-fracture models are taken to have the same total crackdensity. The primary issue investigated is determina-tion of the fracture orientation using azimuthal AVOanalysis. The first model represents a single-fracture setand the second model has an additional fracture setoblique to the first set at an angle of 60◦. As expected,the PP-wave anisotropy is reduced when multiple frac-ture sets are present, making the determination of ori-entation more difficult than for the case of a single-fracture set. Long offsets are required for identificationof dominant fracture orientations using PP-wave AVO.PS-wave AVO, however, is quite sensitive to fractureorientations, even at short offsets. For multiple-fracturesets, PS signals can potentially be used to determine ori-entations of the individual sets.

INTRODUCTION

The use of reflected P-wave and P-SV-wave amplitude-versus-offset analysis (AVO) can potentially aid in the charac-terization of fractured reservoirs (Thomsen, 1988). Cores andwell logs offer methods of describing fractures at a very finescale. The seismic method potentially offers a method of char-acterizing the reservoir at a larger scale that can be used forpredicting reservoir properties. Typically, the estimation offracture properties, especially those for multiple-fracture sets,requires multiple types of data (e.g., Shen et al., 2002; Grechkaet al., 2000). A recent study by DeVault et al. (2002) indicatesthat S-waves have great potential application for fracturedreservoirs. The utility of PS-waves is examined here for theparticular problem of fracture-set orientation when multiple-fracture sets are present. In this paper, the offset-dependent

Manuscript received by the Editor August 25, 2002; revised manuscript received August 8, 2004; published online March 22, 2005.1University of Oklahoma, Institute of Exploration and Development Geosciences, Norman, Oklahoma 73069. E-mail: che@gcn.ou.edu;

jcastagna@ou.edu.2Oklahoma Geological Survey, 100 E. Boyd Street, Norman, Oklahoma 73069. E-mail: raybrown@ou.edu.

c© 2005 Society of Exploration Geophysicists. All rights reserved.

reflectivity for the top of a fractured reservoir is studied witha view toward identifying those approaches best suited for de-termining fracture-set orientations within the reservoir.

We examine the azimuthal AVO variations of two fracturedmodels using exact anisotropic reflection coefficients (Rokhlinet al., 1986). Model 1 is designed containing a single-fractureset, and model 2 consists of two-fracture sets. The single-fracture-set model exhibits a stronger azimuthal AVO varia-tion. The results of the modeling study illustrate the potentialproblems and approaches that can be used in the interpreta-tion of multiple-fracture sets.

SYNTHETIC FRACTURED-RESERVOIR MODELS

Two models of fractured reservoirs are set up for syntheticazimuthal AVO studies in this paper. The models studied aresimple and limited to a constant fracture density so that the ef-fects of fracture orientation could be studied. See the paper byLiu et al. (2000) for a discussion of multiple-fracture sets. Thematrix material of the reservoirs is assumed to be limestonewith 10% porosity, and the overlying seal layer is assumedto be shale. Both the incident shale and background (matrix)for the fractured reservoir layers are assumed to be isotropic.Only the presence of fractures contributes to the anisotropyof the models examined. Table 1 lists the petrophysical pa-rameters for the isotropic portions of the models. All of theseparameters are set within the range of the laboratory andempirical relationships between Vp, Vs, and ρ described byCastagna et al. (1993) for the rocks considered.

Model 1 is assumed to have one set of vertical fractureswhose planes have an azimuth measured from north of 0◦. TheX2-axis points north, and the X1-axis points east. A positiveazimuth angle is measured toward the east from the X2-axis.This means that the normals to the first set of fractures arepointed parallel to the X1-axis (Figure 1b). Model 2 has twosets of fractures whose planes are at azimuths of 0◦ and 60◦

(Figure 1c). The crack density of the fracture set in model 1 isset equal to 0.1. For the purpose of comparison between mod-els, the total crack density in model 2 is also set as 0.1, with the

C1

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C2 Chen et al.

crack density of each fracture set equal to 0.05. In this way,the total crack density is the same for models 1 and 2, whiletheir anisotropy differs. Table 2 lists the fracture parametersdescribed above.

The stiffness matrices (Table 3) are calculated assumingthat the excess compliance caused by each fracture set can

Table 1. Model matrix information.

Reservoir matrix Overlying shale

Vp (m/s) 5877 3700Vs (m/s) 3039 1982Density (g/cm3) 2.44 2.41Porosity 10%

Figure 1. (a) Azimuthal direction of incident plane, describedby angle φ. (b) X1X2-plane sketch of fracture orientation inmodel 1. (c) X1X2-plane sketch of fracture orientations inmodel 2.

simply be added to find the change in compliance caused bythe presence of both fracture sets. This assumption neglectsany interaction between the fractures. As a first-order approx-imation, we use the “scalar” cracks described by Schoenbergand Sayers (1995) to estimate the compliance for each frac-ture set. Using scalar cracks simplifies the stiffness tensor forthe models, but it is not expected to change the major conclu-sions for the paper.

The reflection-coefficient calculations are made using themethod described by Chen (2000) and Rokhlin et al. (1986).First, the horizontal slowness of the incident wave is deter-mined. Next, this horizontal slowness is used to set up twosixth-order polynomials that can be solved to find the reflectedand refracted slownesses for the two media across the reflect-ing boundary. The eigenvectors for each of the reflected andrefracted waves are determined next. Finally, the boundaryconditions are set up, and the reflection and refraction coeffi-cients are determined.

According to Hudson (1980, 1981), one vertical set of frac-tures can introduce transverse isotropy with a horizontal sym-metry axis (HTI), while multiple sets of fractures can causearbitrary anisotropy. An example of what happens with mul-tiple fracture sets can be examined in two simple models dis-cussed below. The reader is referred to the inversion work byBakulin et al. (2002) for a study of two orthogonal-fracturesets. An incident P-wave is assumed for each model. The ver-tical planes of incidence for the P-waves will be described bythe azimuths of the planes. These planes will all contain theX3-axis, and the azimuth is measured positively in a clockwisemovement from north (Figure 1a) (the same direction as theX2-axis). The three expected reflections are a quasi-P (PP)-wave, a quasi-in-plane S (PSI)-wave and a quasi-out-of-planeS-wave. The incident phase and/or ray (group) angle (bothare the same for incident medium, which is isotropic) rangefor the study varies from 0◦ to 45◦. This angle is measuredfrom the X3-axis in the plane of incidence. The variation ofreflection amplitudes with angle of incidence is observed atfour different azimuth directions: 0◦, −30◦, −60◦, and −90◦,respectively. In other words, the plane of incidence is varied,although the fracture geometry is the same; i.e., the stiffnesstensor remains the same while the azimuthal angle changes.

AZIMUTHAL INTERPRETATION OFAVO FROM TOP OF RESERVOIR

The first wave mode to be considered is a downgoingP-wave reflected from the reservoir as a P-wave (PP).Figures 2 and 3 show the exact azimuthal PP reflection ampli-tudes from models 1 (single-fracture set) and 2 (two-fracture

Table 2. Model fracture information.

Crackdensity

Azimuthalangle

Dipangle

Model 1 Fracture set no. 1 0.10 0◦ 90◦Model 2 Fracture set no. 1 0.05 0◦ 90◦

Fracture set no. 2 0.05 60◦ 90◦

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Fractured AVO C3

sets), respectively. First, there are only slight differences onthe normal-incidence PP reflection coefficients, showing thatthe total crack density may be the primary factor related tofractures that influence the zero-offset PP reflections.

Secondly, the azimuthal variations of the PP reflection canonly be clearly observed within each curve family when theincidence angles exceed 25◦. This is why long offsets are ex-pected to be required to detect vertical-fracture orientationswith P-waves. Note that model 1 (Figure 2), with a single frac-ture set, shows more azimuthal AVO than model 2 (Figure 3).

Table 3. Stiffness matrix information.

Cij (GPa) 1 2 3 4 5 6

Isotropic background:1 84.24 39.08 39.08 0 0 02 39.08 84.24 39.08 0 0 03 39.08 39.08 84.24 0 0 04 0 0 0 22.57 0 05 0 0 0 0 22.57 06 0 0 0 0 0 22.57

Single-fracture set(model 1):

1 26.99 12.52 12.52 0 3.3 × 10−7 02 12.52 71.91 26.75 0 3.8 × 10−7 03 12.52 26.75 71.91 0 8.7 × 10−7 04 0 0 0 22.57 0 2.4 × 10−7

5 3.3 × 10−7 3.8 × 10−7 8.7 × 10−7 0 13.47 06 0 0 0 2.4 × 10−7 0 13.47

Two-fracture sets(model 2):

1 49.73 18.44 21.60 7.0 × 10−8 3.0 × 10−7 3.252 18.44 57.28 23.99 2.0 × 10−7 1.6 × 10−7 3.293 21.61 23.99 73.92 2.7 × 10−7 4.6 × 10−7 2.074 7.0 × 10−8 2.0 × 10−7 2.7 × 10−7 20.18 1.19 7.0 × 10−8

5 3.0 × 10−7 1.6 × 10−7 4.6 × 10−7 1.19 18.81 2.0 × 10−8

6 3.25 3.29 2.07 7.0 × 10−8 2.0 × 10−8 17.56

Figure 2. Azimuthal P-wave reflection coefficients for model 1.

Note also that, in Figure 3, the azimuth −90◦ curve is identicalto the azimuth −30◦ curve. For these two situations, the in-cidence plane happens to be perpendicular to one of the twofracture sets. As a result, the P-wave sees the same reflectionresponse at these azimuths, because the two fracture sets havethe same crack density.

The next reflected phase to be considered is the incidentP-wave, which reflects from the reservoir as an S-wave (PS).The polarization of S-waves for anisotropic media can be ex-pressed in many ways. To simplify the description, we describe

Figure 3. Azimuthal P-wave reflection coefficients for model 2.

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C4 Chen et al.

the S-wave polarization as in-plane (near-SV) and out-of-plane (near-SH). In this way, the reader gets an intuitivefeeling for the polarization without actually plotting the po-larization or having to keep track of fracture alignments.Figures 4 and 5 show the out-of-plane shear reflection (ornear-SH reflection) amplitudes from models 1 and 2, respec-tively. This mode is unique to the anisotropic circumstancesand is caused by coupling between the in-plane (near-SV) andout-of-plane (near-SH) components of motion at the inter-face. It can be seen that:

1) The azimuthal variation of the converted PS reflectionsis much more sensitive to fractures than PP reflections(Figures 2 and 3). Even at small incidence angles, Figures4 and 5 show a distinct azimuthal AVO response for bothmodels 1 and 2.

Figure 4. Azimuthal P-SH reflection coefficients for model 1.

Figure 5. Azimuthal P-SH reflection coefficients for model 2.

2) The normal incidence reflections are always zero for allquasi-PS-waves, because all the fracture sets are vertical.There is no converted shear wave when the P-wave is nor-mally incident upon the reflection surface for the modelassumed.

3) In Figure 4, for model 1, the azimuth –90◦ and 0◦ AVOcurves are zero. With only one set of vertical fractures atazimuth –90◦, the reservoir model is an HTI medium witha single vertical-symmetry plane perpendicular to the frac-tures (Tsvankin, 1995, 1996). One of the symmetry planesis the isotropic plane at azimuth 0◦ (X2X3 plane) and theother is an effective VTI plane at azimuth –90◦ direction(X1X3 plane) (Tsvankin, 1995, 1996). These models showthat the converted shear-wave reflections can be used tofind the orientation of the fracture set if only one fractureset exists.

4) In Figure 5, for model 2, a zero-reflection coefficient curveis observed at an azimuth of –60◦, which equally splits thetwo fracture systems. The two reflection-coefficient curvesfor azimuths –90◦ and −30◦ show up as mirror images ofeach other reflected about the azimuth –60◦ curve. Theseobservations indicate that the azimuth –60◦ plane acts as asymmetry plane, but that it is not parallel or perpendicu-lar to any fracture sets. In this two-fracture-set model, the–60◦ azimuth plane can be misinterpreted as an “apparenteffective” orientation of one fracture set.

Figures 6 and 7 show the quasi-in-plane shear reflections (ornear-SV) for models 1 and 2, respectively. Once again, as inthe out-of-plane S-wave case, the in-plane S-wave reflectioncurves show strong azimuthal variations for fracture modelsunder small incidence-angle conditions. Even more impor-tantly, the azimuthal AVO variations are larger than those ofout-of-plane S-wave curves (compare to Figures 4 and 5). An-other observation is that the azimuthal AVO from the single-fracture-set model (model 1) is larger than those of the two-fracture-set model (model 2).

Figure 6. Azimuthal P-SV coefficients for model 1.

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Fractured AVO C5

Figure 7. Azimuthal P-SV coefficients for model 2.

CONCLUSIONS

Exact reflection-coefficient versus angle-of-incidence cur-ves for P- and PS-waves were calculated for models with oneor two sets of fractures. We find that:

1) The azimuthal variation of converted PS reflections ismore sensitive to fractures than PP reflections. Even atsmall angles of incidence, the PS reflection coefficientsshow distinct azimuthal AVO variations. The azimuthalvariations of the PP reflection caused by fractures canbe clearly observed only when the incidence angle islarge (greater than 25◦ for the models in this syntheticstudy).

2) For the same total crack density, a model containing asingle-fracture set exhibits the largest variation of az-imuthal AVO. Multiple-fracture sets tend to weaken theazimuthal anisotropic effects (especially for PP-waves).

3) With large-offset data, the azimuthal variation in P-wavereflections may indicate the existence of fracture zones.For larger relative-amplitude variations with azimuth,there is a greater possibility for a single-fracture set or adominant fracture orientation.

4) The out-of-plane/in-plane PS-wave azimuthal AVOs canbe used, when they can be observed, to identify symmetryplanes as well as the orientation of individual fracture sets.This offers a powerful tool for understanding the fracturemakeup of a reservoir.

If the ideal fracture sets contribute equally to fracture per-meability, then one of the symmetry planes identified by PSreflections is in the direction of maximum horizontal perme-ability, while the other represents a minimum direction of hor-izontal permeability. If there is no PS reflection at normal inci-dence, the existing fracture sets can be assumed to be normalto the reflection boundary.

ACKNOWLEDGMENTS

H. Chen received financial support from the OU IntegratedReservoir Characterization Consortium sponsors. Apprecia-tion is expressed to Bill Lamb for his discussions of the topic.H. Chen and R. Brown acknowledge the support of this workby the U.S. Department of Energy through project DE-AC26-99BC15212.

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