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GEOPHYSICS, VOL. 71, NO. 3 �MAY-JUNE 2006�; P. C25–C36, 16 FIGS., 2 TABLES.10.1190/1.2197487

VO attribute inversion for finely layered reservoirs

lexey Stovas1, Martin Landrø1, and Per Avseth2

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ABSTRACT

Assuming that a turbidite reservoir can be approximatedby a stack of thin shale-sand layers, we use standard ampli-tude variaiton with offset �AVO� attributes to estimate net-to-gross �N/G� and oil saturation. Necessary input is Gass-mann rock-physics properties for sand and shale, as well asthe fluid properties for hydrocarbons. Required seismic in-put is AVO intercept and gradient. The method is basedupon thin-layer reflectivity modeling. It is shown that ran-dom variability in thickness and seismic properties of thethin sand and shale layers does not change significantly theAVO attributes at the top and base of the turbidite-reservoirsequence. The method is tested on seismic data from off-shore Brazil. The results show reasonable agreement be-tween estimated and observed N/G and oil saturation. Themethodology can be developed further for estimatingchanges in pay thickness from time-lapse seismic data.

INTRODUCTION

Our goal is to investigate the PP-wave and PS-wave reflectionesponse from a stack of thin shale-sand layers sandwiched be-ween two half-spaces. This medium might serve as a first-orderD approximation to a turbidite system. From the reflection re-ponse, we estimate the net-to-gross �N/G� ratio of this stack. Aey issue in this work is to include the effects of a finely layerededium and derive new seismic attributes that relate N/G anduid-saturation effects to these attributes.There are several papers related to the N/G-ratio estimation from

eismic amplitudes. Vernik et al. �2002� suggest using both P-wavend S-wave impedance to estimate N/G �Gulf of Mexico�. Theyse both intrinsic anisotropy �for shale� and anisotropic effective-edium modeling �Backus, 1962�. The linear equation was ob-

ained to estimate oil-sand fraction with the parameters: p and smpedances, and average slope and intercept of the shale and oil

Manuscript received by the Editor October 7, 2004; revised manuscript1Norwegian University of Science & Technology, Department of Petrol

lexey@ipt.ntnu.no; mlan@ipt.ntnu.no.2Norsk Hydro Research Center, Geophysics Department, P.O. Box 71902006 Society of Exploration Geophysicists. All rights reserved.

C25

and. The total N/G is estimated by integration over the grosseservoir thickness. For impedance inversion, they use prestackime-migrated 3D seismic data that is flattened using nonhyper-olic moveout. To highlight amplitude variations with angle, thisata is stacked in four angle ranges: near �0–22°�, medium �20–5°�, far �30–55°�, and ultra far �40–65°�. Impedances p and s wereomputed from the simultaneous inversion of angle stacks, usingeismic wavelets obtained from impedance logs �Connolly, 1999�.

An inversion of near- and far-offset stacks into N/G ratio wasxploited in MacLeod et al. �1999� and Dubucq et al. �2001�. Themplitude variation with offset �AVO� inversion for a thin-layerodel has been exploited in Mahob et al. �1999�. This technique is

ased on the optimization of the elastic parameters, so it needsery good a priori information about the model.

In all the papers referred to above �with exception of the lastne�, no thin-layer effects were accounted for. All relationshipsere obtained in a purely empirical way. Inversion of near- and

ar-offset stacks into impedances requires low-frequency data. Onhe other hand, all methods exploiting seismic attributes sufferrom weak physical justification of the interpretation of such at-ributes.

In our approach, we start from a thin-layer seismic model of aedium that we assume is representative for a turbidite system.hen we select realistic rock-physics properties for the turbiditehale and sand bodies. Finally, we derive a direct physical relation-hip between the N/G and water saturation and between the seis-ic intercept and gradient attributes. We also test the proposedethod against the perturbations of the binary-medium parameters

o confirm that even for very large changes in the parameters, theeflection amplitudes at the top of the turbidite system are veryimilar to the ones from the unperturbed medium. The method isested on a real 2D seismic data set from offshore Brazil.

BINARY MEDIUM

We assume that a binary medium is representative of a lowest-rder approximation of a turbidite reservoir. The binary medium ishe medium with the cyclic change of elastic parameters �one set

August 3, 2005; published online May 22, 2006.gineering and Applied Geophysics, Trondheim, N-7491, Norway. E-mail:

n, N-5020, Norway. E-mail: per.avseth@hydro.com.

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or each of two layer types� with depth. Much more complicatedhan this simplified model, real-life turbidites can be representedy well-resolved and unresolved stacks of sequences. We are ad-ressing our method to unresolved stacks that consist of two typesf thin layers.

We assume in this case that the binary medium turns into an ef-ective medium for seismic frequencies, and the N/G only controlshe properties of the medium. The sand and shale properties canary both laterally and along the depth within the unit, but the sta-istics �in terms of Backus averaging� are assumed to be the same.f we have any a priori information about the change of this modelver the field, we can use it in the inversion. One more advantagef using a binary model is that it gives an analytic solution for botheflected and transmitted waves that is very convenient for analy-is.

There are several papers related to the transmission effect in anely layered medium: Schoenberger and Levin �1974�, Marionnd Coudin �1992�, Marion et al. �1994�, Hovem �1995�, and Sha-iro and Treitel �1997�. The critical wavelength/layer thickness ra-io �/d �when the medium starts to behave as an effective medium�aries in these papers from three to eleven. Stovas and Arntsen2003� show that critical �/d depends on the reflectivity contrastetween shale and sand layers, and in the weak-contrast limit criti-al �/d = 4. The reflectivity contrast between shale and sand issually small. This fact allows us to use Backus averaging

able 1. Model parameters.

Parameters

hale VP = 2.25 km/s, VS = 0.9 km/s, =2.2 g/cm3

Sand Kfr = 4.0 GPa, Kma = 36.8 GPa, � = 3.5 GPa,�ma = 2.65 g/cm3 � = 0.3

Fluid Kw = 2.75 GPa, �w = 1.02 g/cm3,Ko = 1.57 GPa, �o = 0.832 g/cm3

able 2. Number of layers and thickness (m) of shale andand layers for asymmetric model.

M1 M2 M3 M4 M5 M6 M7

Shale 1 � 50 2 � 25 4 � 12.5 8 � 6.25 16 � 3.1 32 � 1.5 64 � 0.7

Sand 2 � 25 3 � 16.7 5 � 10 9 � 5.56 17 � 2.9 33 � 1.4 65 � 0.7

igure 1. Series of binary models.

Backus, 1962� to define effective medium properties �Stovas andrntsen, 2003�.Taking this into consideration, it is very important to investigate

ow the properties of an effective medium, composed of the binaryequence of isotropic elastic sand and shale layers, depend on theontrast in elastic parameters. The next step is to estimate the AVOttributes on the interface between the shale and this effective me-ium.

First we introduce the properties of shale and oil-saturated sandayers �Table 1�. To test the reflection response from the stack ofand-shale layers, we consider a set of various binary modelsTable 2�. These models consist of shale and sand layers sand-iched within two semi-finite shale layers �Figure 1�. Each model

Mj+1 is constructed by splitting each layer in the model Mj into twoayers, keeping the total thickness the same �Figure 1�. Because thehale-sand sequence for the reservoir is bounded, the total numberf layers within the reservior is always odd �Figure 1�.

The zero-offset reflection responses from the various models arehown in Figure 2a. We use the first derivative of a Gaussian as a

igure 2. Normal-incidence reflection responses from the series ofodels �N/G = 0.5�. Shown are �a� the full reflected wavefield, �b�

he convolution model �primaries only�, and �c� the difference be-ween these two models.

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AVO attribute inversion C27

avelet with a central frequency of 40 Hz. The first three modelsf the series M1–M3 can be interpreted as a time-average mediumi.e., the layered medium with clear reflections from each interfaceor a given frequency range�, the fourth model as a resonance �tun-ng� medium, and models M5–M7 as an effective medium. The re-ection response from the models M1–M3 can be interpreted easilyecause all internal reflections are clearly distinguished. It is veryifficult to interpret specific reflections for the resonance model M4

nd impossible to interpret the internal reflections for the transitionnd effective medium models.

To demonstrate the effect of multiples, we also compute therimaries-only model �Figure 2b� and the difference between thesewo models �Figure 2c�. The effect of multiples is negligible forime-average models, highly pronounced for resonance models,nd builds up the bottom reflection only for effective mediumodels. These numerical simulations confirm that for thin-layerodeling/inversion, the simple convolution procedure can produce

ncorrect results.We assume that the fine-layered medium �effective medium� can

epresent a real turbidite system, as illustrated by the outcrop photon Figure 3. The question becomes whether we can estimate the/G ratio �i.e., the total thickness of the sand layers divided by the

otal thickness of the whole stack� from the reflection response.ote that the transition into effective medium depends mainly on

he contrast between sand and shale layers. However, when thetack of the layers has reached an effective medium limit, the prop-rties of this effective medium, such as seismic velocities, averageensity, and anisotropic parameters, are controlled by N/G. We be-in by showing that the N/G ratio influences the AVO gradient andto some extent� the AVO intercept.

Because the normal-incidence reflection coefficient for a shale-and interface is normally weak �in our case r � 0.08� and the dif-erence between P-wave velocities for shale and sand is also small,t is hard to see any difference in theffective-medium velocity with changesf N/G. The frequency content of the re-ection response is also very similar.The zero-offset reflection response forodels M4–M7 is shown in Figure 4 for/G ratios ranging between 0.9 to 0.1.he reflections for models M5–M7 areymmetric. The amplitudes on both theop and bottom of the stack are very simi-ar and decrease as the N/G ratio de-reases. By varying N/G ratios, we findhat the amplitude distributions of the re-ections result in high N/G values when

he effective medium has properties simi-ar to sand, and low N/G values when theffective medium has properties similaro shale.

The contrast in the stiffness coeffi-ients and density between shale and sandayers within the turbidite reservoir cane expressed as

�c33 = 2c33,2 − c33,1

c33,2 + c33,1Figure 4. Reflectionright�, and M7 �top l

�c44 = 2c44,2 − c44,1

c44,2 + c44,1,

�� = 2�2 − �1

�2 + �1, �1�

here the stiffness coefficients c33,1, c44,1, and density �1 are relatedo shale layers, and the stiffness coefficients c33,2, c44,2, and density2 are related to sand layers. The similar contrasts in P- and S-waveertical velocities �v� can be computed from equation 1 in theeak-contrast approximation with

�VP =1

2��c33 − ���

�VS =1

2��c44 − ��� . �2�

The effective medium parameters in the weak-contrast approxi-ation can be computed from Backus averaging �Stovas and Arnt-

igure 3. Turbidite system from Ainsa Basin.

se from the models M4 �bottom right�, M5 �bottom left�, M6 �topth the change of N/G ratio from 0.1 at the top to 0.9 at the bottom.

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en, 2003�. By weak contrast, we mean that the P-wave velocitiesor sand and shale are very similar; as a result, the phase changettributed to propagation in sand and shale is approximatelyhe same. The expressions for effective medium parameters ver-us N/G computed in Backus �1962� averaging are given in Appen-ix A.

The effective medium defined in Appendix A is a transverselysotropic medium with a vertically symmetric axis �results similaro those obtained by Folstad and Schoenberg, 1992, and Brittan etl., 1995�. We can compute the effective-anisotropy parameters inhomsen �1986� notations

� =c11 − c33

2c33

� =�c13 + c44�2 − �c33 − c44�2

2c33�c33 − c44�. �3�

ote, that for a layered isotropic medium, � − � 0 �Berryman,999�.

The anisotropy parameters are given by substituting equation-4 into equation 3

� = 212�N/G��1 − �N/G���c44�1

� � �N/G�12�1 − 21

2�N/G���c44�3, �4�

here the contrast constants are

�1 =�c33�1 − �c44/2� − 1

2�c44�1 − �c33/2��1 − �c44/2�2�1 + �c33/2�

�2 =�c44�1 + �c33/2� − 1

2�c33�1 + �c44/2��1 + �c44/2��1 + �c33/2�

�3 =��c33 − �c44�

�1 − �c442 /2��1 + �c33/2�

, �5�

nd where 12 = c44,1/c33,1 is vs/vp ratio squared for shale. Note that

he anisotropic parameter � is symmetric with respect to N/G0.5, with minimum value � = 0.51

2�c44�1. In Figure 5, we showhe effective P- and S-wave velocities, density, and anisotropy pa-ameters as a function of N/G. The vertical P- and S-wave veloci-ies are nonlinear functions of N/G, while the density is a linearunction of N/G. The anisotropic parameters are the result of finelyayered isotropic layers, all of which are negative and can reachignificant values: ��N/G = 0.5� = −0.06 and ��N/G = 0.6�

−0.123.To compute the AVO attributes at the interface between the shale

nd anisotropic-effective medium �Stovas and Ursin, 2003�, we as-ume that the contrast is weak at this interface in all elastic param-ters. Since the elastic properties of effective medium are functionsf N/G �Appendix A�, the contrasts at this interface are also func-ions of N/G:

�33 =�N/G��c33

1 +�1 − �N/G��

2�c33

�44 =

1 +�1 − �N/G��

2�c44

�� =�N/G���

1 −�1 − �N/G��

2��

. �6�

herefore, the AVO attribute defined at the interface i �P-wave im-edance� is defined by

�0� =1

2

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1

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=�N/G�

4 � �c33

1 +�1 − �N/G��

2�c33

+��

1 −�1 − �N/G��

2��� , �7�

igure 5. Effective-medium parameters versus N/G ratio. Vertical- and S-wave velocities and density �top� and anisotropy param-ters �bottom�.

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AVO attribute inversion C29

nd the AVO gradient g is defined by

G =1

4��33 − ��� − 22�44 +

1

2�

=�N/G�

4 � �c33

1 +�1 − �N/G��

2�c33

−��

1 −�1 − �N/G��

2��

−

4�c4412�1 +

�1 − 2�N/G��2

�c33��1 +

�1 − 2�N/G��2

�c44��1 +�1 − �N/G��

2�c33�

+ 212�1 − 21

2�N/G���c44�3� , �8�

ith the average vs/vp ratio squared

2 = 12�1 +

�1 − �N/G��2

�c44��1 +�1 − 2�N/G��

2�c33�

�1 +�1 − 2�N/G��

2�c44��1 +

�1 − �N/G��2

�c33�� 1

2�1 −�N/G�

2��c33 − �c44� . �9�

Now we consider three cases depending on the contrast betweenhale and sand in the stack of the layers. If the contrast is weak, weeep only first-order terms; if contrast is medium, we keep up tohe second-order terms.

ase 1: Weak contrast in all elastic constantsetween shale and sand

For this case, we keep only first-order terms in contrast. Effec-ive medium becomes isotropic � = � = 0 �Bakulin and Grechka,003�, with

R�0� ��N/G�

4��c33 + ���

G ��N/G�

4��c33 − �� − 41

2�c44� . �10�

n terms of the contrast in velocities �equation 2�, equations for in-ercept and gradient take the form

R�0� ��N/G�

2��VP + ���

G ��N/G�

2��VP − 21

2��� + 2�VS�� . �11�

n this case, both intercept and gradient are linear functions of N/G.

ase 2: Weak contrast in c33 and �nd medium contrast in c44

Equation 5 is simplified to

�1 � − 12�c44

�2 � �c44

�3 � − �c44. �12�

nisotropy parameters now depend on the contrast in c44 and areiven by

� = − 214�N/G��1 − �N/G����c44�2

� = − 12�N/G��1 − 21

2�N/G����c44�2. �13�

ote that both anisotropy parameters are negative, but � − ��N/G�1

2�1 − 212���c44�2 � 0. Finally,

R�0� ��N/G�

4��c33 + ���

G =�N/G�

4�c33 − ��− 41

2�c44 + 2�1 − 12���c44�2

��1 − 2�1 + 12��N/G��� . �14�

he intercept is the same as in case 1, but the gradient is now theecond order with respect to N/G. With contrast in velocities

R�0� =�N/G�

2��VP + ���

G =�N/G�

2�VP − 21

2��� + 2�VS�+ �1 − 12�

���� + 2�VS�2�1 − 2�1 + 12��N/G��� . �15�

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ase 3: Medium contrast in all elastic constants

For this case, we keep all second-order terms in contrast. Thenisotropy parameters are given by

� = 212�N/G��1 − �N/G���c44��c33 − 1

2�c44� ,

� = �N/G�12�1 − 21

2�N/G���c44��c33 − �c44� , �16�

nd AVO attributes are given by

R�0� =�N/G�

4��c33�1 −

1 − �N/G�2

�c33�+ ���1 +

1 − �N/G�2

���G =

�N/G�4

��c33�1 −1 − �N/G�

2�c33�

− ���1 +1 − �N/G�

2��� − 41

2�c44

��1 −1 − 2�N/G�

2�c44 −

�N/G�2

�c33�+ 21

2�1 − 212�N/G���c44��c33 − �c44� . �17�

The graphs of the AVO attributes versus N/G ratio are shown inigure 6. Note that these parameters do not depend on the numberf layers. We also computed the PS-AVO parameter following theeak-contrast approximation given in Stovas and Ursin �2002�.nalysis of PP- and PS-AVO parameters shows that the PP-AVOradient g is most sensitive to changes in N/G. The least sensitivettribute is the PP-AVO intercept.

igure 6. PP- and PS-AVO parameters at the interface betweenhale and effective medium.

APPLICABILITY OF A BINARY-EFFECTIVEMEDIUM WITH RANDOMLY VARYING LAYER

THICKNESSES AND SEISMIC PROPERTIESITHIN A THIN-LAYER SAND-SHALE SEQUENCE

In our approach above, an effective medium is used to describehe properties of a turbidite system �sand-shale sequence boundedith two shale layers�. Furthermore, AVO attributes based on seis-ic reflectivity from the top interface of such a binary system were

esigned and directly related to N/G. A possible limitation of theescribed method is that neither the layer thicknesses nor the shalend sand properties vary in such a binary, systematic way. Binary-ffective medium does not produce internal reflections because pri-aries and multiples cancel out each other.Here, we demonstrate the robustness of the 1D effective-me-

ium approach based on a synthetic-reflectivity modeling example.e show that the random perturbations in thickness and elastic

roperties both for sand and shale result in internal reflections. Forbinary-effective medium, we get seismic reflectivity only at the

op and base of the stack of sand-shale layers. When the binaryroperties of medium are changed, either by changing the thick-esses of each individual thin layer or by changing the seismicroperties of each layer, we find seismic reflectivity also betweenhe top and base events. However, the amplitude changes for theop and base reflections are small. Thus, we suggest that the simpleinary-effective medium might still be very useful as a practicaleismic attribute for N/G and fluid-saturation estimation. In thisection, we will consider only zero-offset reflectivity modeling.

Figure 7a shows the velocity profile �as a function of N/G� thate used to define a simplified turbidite system from a sand-shale

equence �binary medium�, denoted in the previous section as M7.e modify this model by first perturbing the layer thicknesses.ext, we perturb the seismic properties of each layer �keeping the

ayer thickness fixed�, Finally, we perturb both layer thicknessesnd layer properties. Figure 7b shows the velocity profile as a func-ion of N/G value for a perturbed binary medium M7.

The range of thickness perturbations is 50% of the initial thick-ess, i.e., �zi = 0.5�z*RAND�−1, + 1�, where RAND denotes aandomly generated number. The range of velocity perturbations is.1 km/s; i.e., vi = v + 0.1*RAND�−1, + 1�. The range of densityerturbations is 0.1 g/cm3; i.e., �i = � + 0.1*RAND�−1, + 1�.To keep the total thickness of the sand-shale thin-layer stack and

he total traveltime �in the time-average sense� constant, the dcomponent was removed separately from each random series. Theesulting series can be denoted quasi-random because the smallcale medium parameters vary randomly, while the overall varia-ions remain zero.

The comparison indicates that there is no correlation evident be-ween these perturbation models. Moreover, because the initialensities for sand and shale are very close to each other, their per-urbed values are overlapping for some layers.

The synthetic traces versus N/G for the two models �initial anderturbed� are shown in Figures 8a and 8b, respectively. The mostvident difference between the seismograms is the presence of in-ernal reflections �between the top and the base reflection�. For theop and base reflections, we observe a significant increase in ampli-ude with increasing N/G, and a smaller amplitude change betweenonperturbed and perturbed models.

Figure 9a shows a comparison between top reservoir amplitudeevels for the initial and perturbed models versus N/G. Note that

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AVO attribute inversion C31

hese media result in different effective-medium properties despiteur efforts to keep the total thickness and traveltime constant. Theveraging in elastic properties for the effective medium is given asvolume average: � = ��i and �v2 = ��i

−1vi−2 −1. Hence, these two

ffective media are somewhat different.Figure 9b shows the effective velocity and effective density for

nitial and perturbed models M7 versus N/G. The resulting effec-ive parameters computed from this perturbed medium for the casef N/G = 0.5 are VP = 2.107 km/s, VS = 1.075 km/s, � = 2.151/cm3, � = −0.062, and � = −0.119, which can be compared with

he effective parameters computed from the unperturbed mediumP = 2.111 km/s, VS = 1.076 km/s, � = 2.15 g/cm3, � = −0.063,

nd � = −0.119. Note that the AVO attributes computed from thesewo media will be very similar.

If we use the amplitudes from the perturbed model and compute/G using the unperturbed-model dependence, we typically find

hat the error in our N/G prediction is relatively small comparedith uncertainties from other factors �e.g., overburden changes,ifferent noise types, etc.�.

igure 7. Velocity profiles versus N/G for initial �a� and perturbedb� model M .

7

ESTIMATION OF FLUID SATURATION —TESTING ON A SEISMIC DATA SET

FROM OFFSHORE BRAZIL

For simultaneous estimation of N/G and fluid saturation, we canse the PP-AVO parameters �Appendix B�. To model the effect ofater saturation, we use the Gassmann model �Gassmann, 1951�.nother way of doing this is to apply the poroelastic Backus aver-

ging, based on the Biot model �Gelinsky and Shapiro, 1997�.Both N/G and water saturation can be estimated from the cross-

lot of AVO parameters. This method is applied on the seismic dataet from offshore Brazil. The poststack seismic section is shown inigure 10. The top-reservoir interface is interpreted as the negativeeflection at about 3.05–3.1 s. To build the AVO crossplot for thenterface between the overlaying shale and the turbidite channel,e used the rock-physics data estimated from well logs and given

n Table 1.The AVO crossplot contains the contour lines for intercept and

radient plotted versus N/G and water saturation �Figure 11�. Theiscrimination between the AVO attributes depends on the dis-

igure 8. Normal-incidence reflection responses versus N/G fornitial �a� and perturbed �b� model M .

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C32 Stovas et al.

rimination angle �i.e., the angle between the contour lines, pertovas and Landrø, 2004�. One can see that the best discrimination

s observed for high values of N/G and water saturation, while theoorest discrimination is for low N/G and water saturation �wherehe contour lines are almost parallel�. The inverted crossplots arehown in Figure 12.

Note that the inversion is performed in the diagonal band ofVO attributes. Zones outside this band relate to the values that areutside the chosen sand-shale model. We believe that the top-eservoir reflection should give relatively high values for N/G, re-ardless of water-saturation values. The arbitrary reflection shouldive either low values for N/G with large uncertainties in wateraturation �Figure 11�, or both N/G and saturation values outsidehe range for the chosen model �Figure 12�. Real AVO data arehown by stars. Data outside the diagonal band are considered asoise.

The section for AVO attributes is shown in Figure 13. For cali-ration purposes, we use well-log data from the well located atDP 7559. The P-wave velocity, density, and gamma-ray logs are

hown in Figure 14. Unfortunately, the shear-wave log is not avail-ble for this well. We compute S-wave velocity from the mudrockquation, making the model fairly weak for shear-wave analysis.

The top of the reservoir is located at a depth of 2720 m, and theariations in the sand properties are higher than those we tested inhe randomization model. Nevertheless, the range of the variations

igure 9. Reflection amplitudes picked from the top of initial anderturbed model M7 �top� and effective-medium parameters �bot-om�.

eflects more on the applicability of the Backus averaging �weakontrast approximation� than the value for the Backus statistics.

The AVO attributes were picked from the AVO sections �inter-ept and gradient�, calibrated to the well logs, and then placed onhe crossplot. The inverted results along the top reservoir arehown in Figure 15 �N/G and �S denote estimated values for N/Gnd water saturation, respectively�. The estimated oil content ocas computed along the top reservoir using the equation oc = �1�S� N/G �smoothed� and is shown together with the AVO sec-

ions and the values of estimated AVO attributes extracted �at topeservoir� in Figure 15.

To assess the robustness of this method, we applied the samerocedure to an arbitrary interface above the interpreted top-eservoir interface. The model used for this interface is differentrom what we used for the top reservoir because of different cali-ration. The results of this exercise are summarized in Figure 16.he N/G values are significantly lower. The saturation values are

epresented by scattered spikes only. The resulting oil-content val-es are negligible.

The method used here appears to be fairly robust because theredictions are significantly different for the two examples. How-

igure 10. Seismic section from the offshore Brazil.

igure 11. AVO crossplot for the interface between shale and tur-idite channel plotted versus N/G and water saturation. The inter-ept is the red line and gradient the blue line.

Fazv

Fe

F

Fs

AVO attribute inversion C33

igure 12. Inverse-AVO crossplot for the interface between shalend turbidite channel plotted versus intercept and gradient. Noteones beyond the chosen model. The real AVO data from top reser-oir shown by stars.

Fs

igure 13. AVO-attributes sections: intercept at the top and gradi-nt at the bottom.

igure 14. Well-log data from well located at CDP 7559.

igure 15. Results of inversion from AVO attributes to N/G andaturation for the top of reservoir.

igure 16. Results of inversion from AVO attributes to N/G andaturation for the arbitrary reflection above the reservoir.

etsrptts

rtbwrteltrtfir

dilppormdrad

foNdtstp

stsa

mfl

m

iNs

a

a

w

we

C34 Stovas et al.

ver, it should be noted that the AVO attributes are different for thewo cases. Thus, one could argue that the AVO attributes them-elves can be used as a hydrocarbon indicator — a method cur-ently utilized by industry. However, the attractiveness of the pro-osed method is that in a fully deterministic way, we convert thewo AVO attributes directly into N/G and saturation attributes. Fur-hermore, the results are quantitative, given the limitations andimplifications in the model used.

CONCLUSIONS

We find that for a finely layered shale-sand medium, there is aelationship between the N/G ratio of the medium and the reflec-ion characteristic composed from the reflection amplitudes —oth from the top and the base of the stack — depending on withhat the medium is sandwiched. We also show that the AVO pa-

ameters for PP �intercept and gradient� and PS �gradient� reflec-ion depend on the N/G ratio. In addition, we demonstrate that theffective-medium parameters, computed from the stack of finelyayered sand-shale sequence, are strongly dependent on the con-rast in elastic parameters between shale and sand layers. We de-ive the exact and approximate equations to compute these effec-ive parameters, including the anisotropy parameters caused byne layering. We show that the anisotropy parameter � is symmet-ic with respect to N/G while the � parameter is nonsymmetric.

The applicability of a binary-effective medium has been ad-ressed by changing the layer thicknesses and medium parametersn a sand-shale stack of horizontal layers. For an average change inayer thickness of 25% and a corresponding change in the elasticarameters of 2.5%, we find that random variations result in an ap-roximately 5% error in N/G prediction �checked only for zero-ffset reflectivity�. The N/G prediction is performed by using top-eservoir seismic amplitudes. Compared to thin-layer reflectivityodeling of a binary-effective medium, a random-effective me-

ium gives multiple reflections between the top and the base of theeservoir section. If in addition to the random variations a system-tic trend in the medium parameters is introduced, more significanteviations are expected.

We also define the AVO attributes for the effective medium as aunction of N/G and water saturation in the sand layers. We dem-nstrate an inversion procedure to convert the AVO attributes into/G and water saturation. Since the water saturation values may beifferent for each sand layer, we estimate an effective water satura-ion that can be estimated approximately by weighted averaging ofaturation in the individual layers. The only limitation related tohe model data is the applicability of the chosen sand and shaleroperties.

When our method was tested on a real seismic data set from off-hore Brazil, the results were encouraging. For the top reservoir in-erface, we find high N/G values, as well as high probability of oilaturation. For another interface above the reservoir, the N/G rationd oil content are found to be negligible.

Future work will include the effect of uncertainties in both theodel and AVO attributes on discrimination between N/G anduid saturation, 3D tests, and time-lapse seismic application.

ACKNOWLEDGMENTS

We thank Norsk Hydro for their financial support and for per-ission to use and present the seismic data. Aart-Jan Wijngaarden

s acknowledged for valuable discussions and suggestions. Theorwegian Research Council �NFR� is acknowledged for financial

upport.

APPENDIX A

BACKUS AVERAGING FOR A BINARY MEDIUM(EFFECTIVE MEDIUM PROPERTIES VERSUS

CONTRAST IN ELASTIC PARAMETERS)

For the stack of isotropic layers, the propagation constraints are given by Backus �1962� as

A1 = 4�c44,1�1 −c44,1

c33,1��1 − �N/G��

+ c44,2�1 −c44,2

c33,2��N/G�

A2 =�1 − �N/G��

c33,1+

�N/G�c33,2

,

A3 = �1 − 2c44,1

c33,1��1 − �N/G�� + �1 − 2

c44,2

c33,2��N/G� ,

nd

A4 =�1 − �N/G��

c44,1+

�N/G�c44,2

, �A-1�

here N/G is the volume ratio �N/G ratio for our case�.Substituting equation 1 into equation A-1 results in

A1 = 4c44,1�1 − 12 + �N/G�� �c44

1 − �c44/2

+ 12�c33 − 2�c44 + �c33��c44�2/4

�1 − �c44/2�2�1 + �c33/2� �A2 =

1

c33,1�1 − �N/G�

�c33

1 + �c33/2 ,

A3 = 1 − 212 + 2�N/G�1

2 �c33 − �c44

�1 − �c44/2��1 + �c33/2�

A4 =1

c44,1�1 − �N/G�

�c44

1 + �c44/2 , �A-2�

here 12 = c44,1/c33,1 is vs/vp ratio squared for shale. The effective

lastic parameters are given by

c11 = A1 +A3

2

A2

c13 =A3

A

2

S

c

T

r

w

wsiKswa

e

wSo

a

e

T

AVO attribute inversion C35

c33 =1

A2

c44 =1

A4. �A-3�

ubstituting equation A-2 into equation A-3, we obtain

c11 = c33,1

1 + 412�N/G��1 − �N/G��

�c44�c33�1 − �c44/2� − 12�c44�1 − �c33/2��

�1 − �c44/2�2�1 + �c33/2�

1 − �N/G��c33

�1 + �c33/2�

13 = c33,1

1 − 212 + 21

2�N/G��c33 − �c44

�1 − �c44/2��1 + �c33/2�

1 − �N/G��c33

�1 + �c33/2�

c33 = c33,11

1 − �N/G��c33

�1 + �c33/2�

c44 = c44,11

1 − �N/G��c44

�1 + �c44/2�

. �A-4�

he effective density is given by

� = �1�1 + �N/G���

1 − ��/2� . �A-5�

APPENDIX B

THE FLUID EFFECT

Using the Gassmann equation �Gassmann, 1951�, the elastic pa-ameters for an isotropic sand can be written as

c33,2 = c33dry 1 + U�S

1 + W�S

c44,2 = c44dry,

�2 = �dry�1 + F�S� , �B-1�

here

c33dry = Kfr +

4

3� +

�Kfr − Kma�2

Kma�1 − � −Kfr

Kma+

Kma

Ko�

U =�Kfr +

4

3���Kma� 1

Kw−

1

Ko�

��Kfr +4

3���1 − � −

Kfr

K+

Kma

K� +

�Kfr − Kma�2

K

ma o ma

W =

�Kma� 1

Kw−

1

Ko�

�1 − � −Kfr

Kma+

Kma

Ko� ,

�dry = ��o + �1 − ���ma

F =���w − �o�

��o + �1 − ���ma, �B-2�

here �S is water saturation, � is porosity, Kfr is bulk modulus ofolid framework, � is shear modulus of solid framework, and Kma

s intrinsic modulus of solid, i.e., bulk modulus for zero porosity.o and Kw denote the pure fluid bulk moduli for oil and water, re-

pectively. �ma is the matrix density and �o and �w denote oil andater densities, respectively. Note that both U and W are negative

nd �U� � �W�.From equation B-1, we compute the contrast in elastic param-

ters as a function of water saturation with

�c33 =�c33

dry + D0�S

1 + D1�S

�c44 = �c44dry,

�� =��dry + R0�S

1 + R1�S, �B-3�

here �c33dry, �c44

dry, and ��dry are the contrasts in P-wave and-wave modulus and density between the dry sand and shale. Thether parameters are

D0 = U − W + �c33dryU + W

2

D1 =1

2�U + W + �c33

dryU − W

2� ,

R0 = �1 + ��dry�F

R1 =1

2�1 +

��dry

2�F �B-4�

nd the contrast in c44 is saturation independent.For simplicity, we use the weak contrast case only. Therefore,

quation 10 takes the form

R�0� ��N/G�

4��c33

dry + D0�S

1 + D1�S+

��dry + R0�S

1 + R1�S�

G ��N/G�

4��c33

dry + D0�S

1 + D1�S−

��dry + R0�S

1 + R1�S

− 412�c44

dry� . �B-5�

o solve equation B-5, first we get quadratic equation

w

a

br

w

Fgfioesasti

B

B

B

B

CD

F

G

G

H

M

M

M

M

S

S

S

S

S

TV

C36 Stovas et al.

�a2R�0� + g2G���S�2 + �a1R�0� + g1G��S + a0R�0�

+ g0G = 0 �B-6�

ith

a0 = − �c33dry + 41

2�c44dry + ��dry

a1 = − �c33dryR1 + 41

2�c44dry�D1 + R1� + ��dryD1 + R0 − D0

a2 = D1R0 − D0R1 + 412�c44

dryD1R1

nd

g0 = �c33dry + ��dry

g1 = D0 + R0 + �c33dryR1 + ��dryD1

g2 = D0R1 + D1R0

eing the model parameters, while R�0� �angle = 0� and G are theeal data. The solution of B-6 gives two values for water saturation

��S�1,2 =− �a1R�0� + g1G� ± ��a1R�0� + g1G�2 − 4�a2R�0� + g2G��a0R�0� + g0G�

2�a2R�0� + g2G�,

�B-7�

ith a sign that obtains positive value for saturation.Substituting B-7 into B-5 gives the estimated value for N/G

N/G =4R�0�

�c33dry + D0�S

1 + D1�S+

��dry + R0�S

1 + R1�S

. �B-8�

rom equation B-7 and B-8, we can see that if both intercept andradient go to zero, N/G also goes to zero, but saturation is not de-ned. The denominator in equation B-7 can be very small becausef uncertainties in AVO attributes R�0� and G and model param-ters a2 and g2, which leads to instability in computation of wateraturation. It is always the case when N/G is very low. The AVO-ttribute contour lines �Figure 11� are almost parallel to the wateraturation axis �the discrimination between N/G and water satura-ion is very low�. With increase of N/G values, discrimination also

ncreases.

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