AVO attribute inversion for finely layered reservoirs

GEOPHYSICS, VOL. 71, NO. 3 MAY-JUNE 006 ; P. C5 C36, 16 FIGS., TABLES. 10.1190/1.197487 Downloaded 09/1/14 to 84.15.159.8. Redistribution subject to SEG license or copyright; see Terms of Use at http://library.seg.org/ AVO attribute inversion for finely layered reservoirs Alexey Stovas 1, Martin Landrø 1, and Per Avseth ABSTRACT Assuming that a turbidite reservoir can be approximated by a stack of thin shale-sand layers, we use standard amplitude variaiton with offset AVO attributes to estimate netto-gross N/G and oil saturation. Necessary input is Gassmann rock-physics properties for sand and shale, as well as the fluid properties for hydrocarbons. Required seismic input is AVO intercept and gradient. The method is based upon thin-layer reflectivity modeling. It is shown that random variability in thickness and seismic properties of the thin sand and shale layers does not change significantly the AVO attributes at the top and base of the turbidite-reservoir sequence. The method is tested on seismic data from offshore Brazil. The results show reasonable agreement between estimated and observed N/G and oil saturation. The methodology can be developed further for estimating changes in pay thickness from time-lapse seismic data. INTRODUCTION Our goal is to investigate the PP-wave and PS-wave reflection response from a stack of thin shale-sand layers sandwiched between two half-spaces. This medium might serve as a first-order 1D approximation to a turbidite system. From the reflection response, we estimate the net-to-gross N/G ratio of this stack. A key issue in this work is to include the effects of a finely layered medium and derive new seismic attributes that relate N/G and fluid-saturation effects to these attributes. There are several papers related to the N/G-ratio estimation from seismic amplitudes. Vernik et al. 00 suggest using both P-wave and S-wave impedance to estimate N/G Gulf of Mexico. They use both intrinsic anisotropy for shale and anisotropic effectivemedium modeling Backus, 196. The linear equation was obtained to estimate oil-sand fraction with the parameters: p and s impedances, and average slope and intercept of the shale and oil sand. The total N/G is estimated by integration over the gross reservoir thickness. For impedance inversion, they use prestack time-migrated 3D seismic data that is flattened using nonhyperbolic moveout. To highlight amplitude variations with angle, this data is stacked in four angle ranges: near 0, medium 0 45, far 30 55, and ultra far 40 65. Impedances p and s were computed from the simultaneous inversion of angle stacks, using seismic wavelets obtained from impedance logs Connolly, 1999. An inversion of near- and far-offset stacks into N/G ratio was exploited in MacLeod et al. 1999 and Dubucq et al. 001. The amplitude variation with offset AVO inversion for a thin-layer model has been exploited in Mahob et al. 1999. This technique is based on the optimization of the elastic parameters, so it needs very good a priori information about the model. In all the papers referred to above with exception of the last one, no thin-layer effects were accounted for. All relationships were obtained in a purely empirical way. Inversion of near- and far-offset stacks into impedances requires low-frequency data. On the other hand, all methods exploiting seismic attributes suffer from weak physical justification of the interpretation of such attributes. In our approach, we start from a thin-layer seismic model of a medium that we assume is representative for a turbidite system. Then we select realistic rock-physics properties for the turbidite shale and sand bodies. Finally, we derive a direct physical relationship between the N/G and water saturation and between the seismic intercept and gradient attributes. We also test the proposed method against the perturbations of the binary-medium parameters to confirm that even for very large changes in the parameters, the reflection amplitudes at the top of the turbidite system are very similar to the ones from the unperturbed medium. The method is tested on a real D seismic data set from offshore Brazil. BINARY MEDIUM We assume that a binary medium is representative of a lowestorder approximation of a turbidite reservoir. The binary medium is the medium with the cyclic change of elastic parameters one set Manuscript received by the Editor October 7, 004; revised manuscript received August 3, 005; published online May, 006. 1 Norwegian University of Science & Technology, Department of Petroleum Engineering and Applied Geophysics, Trondheim, N-7491, Norway. E-mail: alexey@ipt.ntnu.no; mlan@ipt.ntnu.no. Norsk Hydro Research Center, Geophysics Department, P.O. Box 7190, Bergen, N-500, Norway. E-mail: per.avseth@hydro.com. 006 Society of Exploration Geophysicists. All rights reserved. C5

C6 Stovas et al. Downloaded 09/1/14 to 84.15.159.8. Redistribution subject to SEG license or copyright; see Terms of Use at http://library.seg.org/ for each of two layer types with depth. Much more complicated than this simplified model, real-life turbidites can be represented by well-resolved and unresolved stacks of sequences. We are addressing our method to unresolved stacks that consist of two types of thin layers. We assume in this case that the binary medium turns into an effective medium for seismic frequencies, and the N/G only controls the properties of the medium. The sand and shale properties can vary both laterally and along the depth within the unit, but the statistics in terms of Backus averaging are assumed to be the same. If we have any a priori information about the change of this model over the field, we can use it in the inversion. One more advantage of using a binary model is that it gives an analytic solution for both reflected and transmitted waves that is very convenient for analysis. There are several papers related to the transmission effect in a finely layered medium: Schoenberger and Levin 1974, Marion and Coudin 199, Marion et al. 1994, Hovem 1995, and Shapiro and Treitel 1997. The critical wavelength/layer thickness ratio /d when the medium starts to behave as an effective medium varies in these papers from three to eleven. Stovas and Arntsen 003 show that critical /d depends on the reflectivity contrast between shale and sand layers, and in the weak-contrast limit critical /d = 4. The reflectivity contrast between shale and sand is usually small. This fact allows us to use Backus averaging Table 1. Model parameters. Parameters Shale V P =.5 km/s, V S = 0.9 km/s, =. g/cm 3 Sand K fr = 4.0 GPa, K ma = 36.8 GPa, = 3.5 GPa, ma =.65 g/cm 3 = 0.3 Fluid K w =.75 GPa, w = 1.0 g/cm 3, K o = 1.57 GPa, o = 0.83 g/cm 3 Table. Number of layers and thickness (m) of shale and sand layers for asymmetric model. M 1 M M 3 M 4 M 5 M 6 M 7 Shale 1 50 5 4 1.5 8 6.5 16 3.1 3 1.5 64 0.7 Sand 5 3 16.7 5 10 9 5.56 17.9 33 1.4 65 0.7 Backus, 196 to define effective medium properties Stovas and Arntsen, 003. Taking this into consideration, it is very important to investigate how the properties of an effective medium, composed of the binary sequence of isotropic elastic sand and shale layers, depend on the contrast in elastic parameters. The next step is to estimate the AVO attributes on the interface between the shale and this effective medium. First we introduce the properties of shale and oil-saturated sand layers Table 1. To test the reflection response from the stack of sand-shale layers, we consider a set of various binary models Table. These models consist of shale and sand layers sandwiched within two semi-finite shale layers Figure 1. Each model M j+1 is constructed by splitting each layer in the model M j into two layers, keeping the total thickness the same Figure 1. Because the shale-sand sequence for the reservoir is bounded, the total number of layers within the reservior is always odd Figure 1. The zero-offset reflection responses from the various models are shown in Figure a. We use the first derivative of a Gaussian as a Figure 1. Series of binary models. Figure. Normal-incidence reflection responses from the series of models N/G = 0.5. Shown are a the full reflected wavefield, b the convolution model primaries only, and c the difference between these two models.

AVO attribute inversion C7 Downloaded 09/1/14 to 84.15.159.8. Redistribution subject to SEG license or copyright; see Terms of Use at http://library.seg.org/ wavelet with a central frequency of 40 Hz. The first three models of the series M 1 M 3 can be interpreted as a time-average medium i.e., the layered medium with clear reflections from each interface for a given frequency range, the fourth model as a resonance tuning medium, and models M 5 M 7 as an effective medium. The reflection response from the models M 1 M 3 can be interpreted easily because all internal reflections are clearly distinguished. It is very difficult to interpret specific reflections for the resonance model M 4 and impossible to interpret the internal reflections for the transition and effective medium models. To demonstrate the effect of multiples, we also compute the primaries-only model Figure b and the difference between these two models Figure c. The effect of multiples is negligible for time-average models, highly pronounced for resonance models, and builds up the bottom reflection only for effective medium models. These numerical simulations confirm that for thin-layer modeling/inversion, the simple convolution procedure can produce incorrect results. We assume that the fine-layered medium effective medium can represent a real turbidite system, as illustrated by the outcrop photo in Figure 3. The question becomes whether we can estimate the N/G ratio i.e., the total thickness of the sand layers divided by the total thickness of the whole stack from the reflection response. Note that the transition into effective medium depends mainly on the contrast between sand and shale layers. However, when the stack of the layers has reached an effective medium limit, the properties of this effective medium, such as seismic velocities, average density, and anisotropic parameters, are controlled by N/G. We begin by showing that the N/G ratio influences the AVO gradient and to some extent the AVO intercept. Because the normal-incidence reflection coefficient for a shalesand interface is normally weak in our case r 0.08 and the difference between P-wave velocities for shale and sand is also small, it is hard to see any difference in the effective-medium velocity with changes of N/G. The frequency content of the reflection response is also very similar. The zero-offset reflection response for models M 4 M 7 is shown in Figure 4 for N/G ratios ranging between 0.9 to 0.1. The reflections for models M 5 M 7 are symmetric. The amplitudes on both the top and bottom of the stack are very similar and decrease as the N/G ratio decreases. By varying N/G ratios, we find that the amplitude distributions of the reflections result in high N/G values when the effective medium has properties similar to sand, and low N/G values when the effective medium has properties similar to shale. The contrast in the stiffness coefficients and density between shale and sand layers within the turbidite reservoir can be expressed as c 44 = c 44, c 44,1 c 44, +c 44,1, = 1 + 1, where the stiffness coefficients c 33,1,c 44,1, and density 1 are related to shale layers, and the stiffness coefficients c 33,,c 44,, and density are related to sand layers. The similar contrasts in P- and S-wave vertical velocities v can be computed from equation 1 in the weak-contrast approximation with V P = 1 c 33 V S = 1 c 44. The effective medium parameters in the weak-contrast approximation can be computed from Backus averaging Stovas and Arnt- Figure 3. Turbidite system from Ainsa Basin. 1 c 33 = c 33, c 33,1 c 33, +c 33,1 Figure 4. Reflection response from the models M 4 bottom right, M 5 bottom left, M 6 top right, and M 7 top left with the change of N/G ratio from 0.1 at the top to 0.9 at the bottom.

C8 Stovas et al. Downloaded 09/1/14 to 84.15.159.8. Redistribution subject to SEG license or copyright; see Terms of Use at http://library.seg.org/ sen, 003. By weak contrast, we mean that the P-wave velocities for sand and shale are very similar; as a result, the phase change attributed to propagation in sand and shale is approximately the same. The expressions for effective medium parameters versus N/G computed in Backus 196 averaging are given in Appendix A. The effective medium defined in Appendix A is a transversely isotropic medium with a vertically symmetric axis results similar to those obtained by Folstad and Schoenberg, 199, and Brittan et al., 1995. We can compute the effective-anisotropy parameters in Thomsen 1986 notations = c 11 c 33 c 33 = c 13 +c 44 c 33 c 44. 3 c 33 c 33 c 44 Note, that for a layered isotropic medium, 0 Berryman, 1999. The anisotropy parameters are given by substituting equation A-4 into equation 3 = 1 N/G 1 N/G c 44 1 N/G 1 1 1 N/G c 44 3, where the contrast constants are 1 = c 33 1 c 44 / 1 c 44 1 c 33 / 1 c 44 / 1+ c 33 / = c 44 1 + c 33 / 1 c 33 1+ c 44 / 1 + c 44 / 1 + c 33 / 4 c 33 c 44 3 = 1 c 44 / 1 + c 33 /, 5 and where 1 =c 44,1 /c 33,1 is v s /v p ratio squared for shale. Note that the anisotropic parameter is symmetric with respect to N/G = 0.5, with minimum value = 0.5 1 c 44 1. In Figure 5, we show the effective P- and S-wave velocities, density, and anisotropy parameters as a function of N/G. The vertical P- and S-wave velocities are nonlinear functions of N/G, while the density is a linear function of N/G. The anisotropic parameters are the result of finely layered isotropic layers, all of which are negative and can reach significant values: N/G = 0.5 = 0.06 and N/G = 0.6 = 0.13. To compute the AVO attributes at the interface between the shale and anisotropic-effective medium Stovas and Ursin, 003,weassume that the contrast is weak at this interface in all elastic parameters. Since the elastic properties of effective medium are functions of N/G Appendix A, the contrasts at this interface are also functions of N/G: 44 = = N/G c 44 1+ 1 N/G c 44 N/G 1 1 N/G. 6 Therefore, the AVO attribute defined at the interface i P-wave impedance is defined by R 0 = 1 I 1 Ī 4 33 + = N/G c 33 4 1+ 1 N/G c 33 + 1 1 N/G, 7 33 = N/G c 33 1+ 1 N/G c 33 Figure 5. Effective-medium parameters versus N/G ratio. Vertical P- and S-wave velocities and density top and anisotropy parameters bottom.

AVO attribute inversion C9 Downloaded 09/1/14 to 84.15.159.8. Redistribution subject to SEG license or copyright; see Terms of Use at http://library.seg.org/ and the AVO gradient g is defined by G= 1 4 33 44 + 1 = N/G c 33 4 1+ 1 N/G c 33 1 1 N/G 4 c 44 1 1 + 1 N/G c 33 1 + 1 N/G + 1 1 1 N/G c 44 3, with the average v s /v p ratio squared 1 N/G + = 1 1 1 + 1 N/G 1 1 N/G c 33 c 44. c 44 1 + 1 N/G c 33 c 44 1 + 1 N/G c 44 1 + 1 N/G 8 c 33 c 33 Now we consider three cases depending on the contrast between shale and sand in the stack of the layers. If the contrast is weak, we keep only first-order terms; if contrast is medium, we keep up to the second-order terms. Case 1: Weak contrast in all elastic constants between shale and sand For this case, we keep only first-order terms in contrast. Effective medium becomes isotropic = =0 Bakulin and Grechka, 003, with R 0 N/G c 33 + 4 G N/G c 33 4 4 1 c 44. 9 10 In terms of the contrast in velocities equation, equations for intercept and gradient take the form R 0 N/G V P + G N/G V P 1 + V S. 11 In this case, both intercept and gradient are linear functions of N/G. Case : Weak contrast in c 33 and and medium contrast in c 44 Equation 5 is simplified to 1 1 c 44 c 44 3 c 44. 1 Anisotropy parameters now depend on the contrast in c 44 and are given by = 1 4 N/G 1 N/G c 44 = 1 N/G 1 1 N/G c 44. 13 Note that both anisotropy parameters are negative, but = N/G 1 1 1 c 44 0. Finally, R 0 N/G c 33 + 4 G= N/G c 33 4 4 1 c 44 + 1 1 c 44 1 1 + 1 N/G. 14 The intercept is the same as in case 1, but the gradient is now the second order with respect to N/G. With contrast in velocities R 0 = N/G V P + G= N/G V P 1 + V S + 1 1 + V S 1 1 + 1 N/G. 15

C30 Stovas et al. Downloaded 09/1/14 to 84.15.159.8. Redistribution subject to SEG license or copyright; see Terms of Use at http://library.seg.org/ Case 3: Medium contrast in all elastic constants For this case, we keep all second-order terms in contrast. The anisotropy parameters are given by = 1 N/G 1 N/G c 44 c 33 1 c 44, = N/G 1 1 1 N/G c 44 c 33 c 44, 16 and AVO attributes are given by R 0 = N/G c 33 1 1 N/G 4 + 1 + 1 N/G c 33 c 33 G= N/G c 33 1 1 N/G 4 1 + 1 N/G 4 1 c 44 1 1 N/G c 44 N/G c 33 + 1 1 1 N/G c 44 c 33 c 44. 17 The graphs of the AVO attributes versus N/G ratio are shown in Figure 6. Note that these parameters do not depend on the number of layers. We also computed the PS-AVO parameter following the weak-contrast approximation given in Stovas and Ursin 00. Analysis of PP- and PS-AVO parameters shows that the PP-AVO gradient g is most sensitive to changes in N/G. The least sensitive attribute is the PP-AVO intercept. Figure 6. PP- and PS-AVO parameters at the interface between shale and effective medium. APPLICABILITY OF A BINARY-EFFECTIVE MEDIUM WITH RANDOMLY VARYING LAYER THICKNESSES AND SEISMIC PROPERTIES WITHIN A THIN-LAYER SAND-SHALE SEQUENCE In our approach above, an effective medium is used to describe the properties of a turbidite system sand-shale sequence bounded with two shale layers. Furthermore, AVO attributes based on seismic reflectivity from the top interface of such a binary system were designed and directly related to N/G. A possible limitation of the described method is that neither the layer thicknesses nor the shale and sand properties vary in such a binary, systematic way. Binaryeffective medium does not produce internal reflections because primaries and multiples cancel out each other. Here, we demonstrate the robustness of the 1D effective-medium approach based on a synthetic-reflectivity modeling example. We show that the random perturbations in thickness and elastic properties both for sand and shale result in internal reflections. For a binary-effective medium, we get seismic reflectivity only at the top and base of the stack of sand-shale layers. When the binary properties of medium are changed, either by changing the thicknesses of each individual thin layer or by changing the seismic properties of each layer, we find seismic reflectivity also between the top and base events. However, the amplitude changes for the top and base reflections are small. Thus, we suggest that the simple binary-effective medium might still be very useful as a practical seismic attribute for N/G and fluid-saturation estimation. In this section, we will consider only zero-offset reflectivity modeling. Figure 7a shows the velocity profile as a function of N/G that we used to define a simplified turbidite system from a sand-shale sequence binary medium, denoted in the previous section as M 7. We modify this model by first perturbing the layer thicknesses. Next, we perturb the seismic properties of each layer keeping the layer thickness fixed, Finally, we perturb both layer thicknesses and layer properties. Figure 7b shows the velocity profile as a function of N/G value for a perturbed binary medium M 7. The range of thickness perturbations is 50% of the initial thickness, i.e., z i = 0.5 z * RAND 1, + 1, where RAND denotes a randomly generated number. The range of velocity perturbations is 0.1 km/s; i.e., v i =v+0.1 * RAND 1, + 1. The range of density perturbations is 0.1 g/cm 3 ; i.e., i = + 0.1 * RAND 1, + 1. To keep the total thickness of the sand-shale thin-layer stack and the total traveltime in the time-average sense constant, the dc component was removed separately from each random series. The resulting series can be denoted quasi-random because the small scale medium parameters vary randomly, while the overall variations remain zero. The comparison indicates that there is no correlation evident between these perturbation models. Moreover, because the initial densities for sand and shale are very close to each other, their perturbed values are overlapping for some layers. The synthetic traces versus N/G for the two models initial and perturbed are shown in Figures 8a and 8b, respectively. The most evident difference between the seismograms is the presence of internal reflections between the top and the base reflection. For the top and base reflections, we observe a significant increase in amplitude with increasing N/G, and a smaller amplitude change between nonperturbed and perturbed models. Figure 9a shows a comparison between top reservoir amplitude levels for the initial and perturbed models versus N/G. Note that

AVO attribute inversion C31 Downloaded 09/1/14 to 84.15.159.8. Redistribution subject to SEG license or copyright; see Terms of Use at http://library.seg.org/ these media result in different effective-medium properties despite our efforts to keep the total thickness and traveltime constant. The averaging in elastic properties for the effective medium is given as a volume average: = i and v = i 1 v i 1. Hence, these two effective media are somewhat different. Figure 9b shows the effective velocity and effective density for initial and perturbed models M 7 versus N/G. The resulting effective parameters computed from this perturbed medium for the case of N/G = 0.5 are V P =.107 km/s, V S = 1.075 km/s, =.151 g/cm 3, = 0.06, and = 0.119, which can be compared with the effective parameters computed from the unperturbed medium V P =.111 km/s, V S = 1.076 km/s, =.15 g/cm 3, = 0.063, and = 0.119. Note that the AVO attributes computed from these two media will be very similar. If we use the amplitudes from the perturbed model and compute N/G using the unperturbed-model dependence, we typically find that the error in our N/G prediction is relatively small compared with uncertainties from other factors e.g., overburden changes, different noise types, etc.. ESTIMATION OF FLUID SATURATION TESTING ON A SEISMIC DATA SET FROM OFFSHORE BRAZIL For simultaneous estimation of N/G and fluid saturation, we can use the PP-AVO parameters Appendix B. To model the effect of water saturation, we use the Gassmann model Gassmann, 1951. Another way of doing this is to apply the poroelastic Backus averaging, based on the Biot model Gelinsky and Shapiro, 1997. Both N/G and water saturation can be estimated from the crossplot of AVO parameters. This method is applied on the seismic data set from offshore Brazil. The poststack seismic section is shown in Figure 10. The top-reservoir interface is interpreted as the negative reflection at about 3.05 3.1 s. To build the AVO crossplot for the interface between the overlaying shale and the turbidite channel, we used the rock-physics data estimated from well logs and given in Table 1. The AVO crossplot contains the contour lines for intercept and gradient plotted versus N/G and water saturation Figure 11. The discrimination between the AVO attributes depends on the dis- Figure 7. Velocity profiles versus N/G for initial a and perturbed b model M 7. Figure 8. Normal-incidence reflection responses versus N/G for initial a and perturbed b model M 7.

C3 Stovas et al. Downloaded 09/1/14 to 84.15.159.8. Redistribution subject to SEG license or copyright; see Terms of Use at http://library.seg.org/ crimination angle i.e., the angle between the contour lines, per Stovas and Landrø, 004. One can see that the best discrimination is observed for high values of N/G and water saturation, while the poorest discrimination is for low N/G and water saturation where the contour lines are almost parallel. The inverted crossplots are shown in Figure 1. Note that the inversion is performed in the diagonal band of AVO attributes. Zones outside this band relate to the values that are outside the chosen sand-shale model. We believe that the topreservoir reflection should give relatively high values for N/G, regardless of water-saturation values. The arbitrary reflection should give either low values for N/G with large uncertainties in water saturation Figure 11, or both N/G and saturation values outside the range for the chosen model Figure 1. Real AVO data are shown by stars. Data outside the diagonal band are considered as noise. The section for AVO attributes is shown in Figure 13. For calibration purposes, we use well-log data from the well located at CDP 7559. The P-wave velocity, density, and gamma-ray logs are shown in Figure 14. Unfortunately, the shear-wave log is not available for this well. We compute S-wave velocity from the mudrock equation, making the model fairly weak for shear-wave analysis. The top of the reservoir is located at a depth of 70 m, and the variations in the sand properties are higher than those we tested in the randomization model. Nevertheless, the range of the variations reflects more on the applicability of the Backus averaging weak contrast approximation than the value for the Backus statistics. The AVO attributes were picked from the AVO sections intercept and gradient, calibrated to the well logs, and then placed on the crossplot. The inverted results along the top reservoir are shown in Figure 15 N/G and S denote estimated values for N/G and water saturation, respectively. The estimated oil content oc was computed along the top reservoir using the equation oc = 1 S N/G smoothed and is shown together with the AVO sections and the values of estimated AVO attributes extracted at top reservoir in Figure 15. To assess the robustness of this method, we applied the same procedure to an arbitrary interface above the interpreted topreservoir interface. The model used for this interface is different from what we used for the top reservoir because of different calibration. The results of this exercise are summarized in Figure 16. The N/G values are significantly lower. The saturation values are represented by scattered spikes only. The resulting oil-content values are negligible. The method used here appears to be fairly robust because the predictions are significantly different for the two examples. How- Figure 10. Seismic section from the offshore Brazil. Figure 9. Reflection amplitudes picked from the top of initial and perturbed model M 7 top and effective-medium parameters bottom. Figure 11. AVO crossplot for the interface between shale and turbidite channel plotted versus N/G and water saturation. The intercept is the red line and gradient the blue line.

AVO attribute inversion C33 Downloaded 09/1/14 to 84.15.159.8. Redistribution subject to SEG license or copyright; see Terms of Use at http://library.seg.org/ Figure 1. Inverse-AVO crossplot for the interface between shale and turbidite channel plotted versus intercept and gradient. Note zones beyond the chosen model. The real AVO data from top reservoir shown by stars. Figure 14. Well-log data from well located at CDP 7559. Figure 15. Results of inversion from AVO attributes to N/G and saturation for the top of reservoir. Figure 13. AVO-attributes sections: intercept at the top and gradient at the bottom. Figure 16. Results of inversion from AVO attributes to N/G and saturation for the arbitrary reflection above the reservoir.

C34 Stovas et al. Downloaded 09/1/14 to 84.15.159.8. Redistribution subject to SEG license or copyright; see Terms of Use at http://library.seg.org/ ever, it should be noted that the AVO attributes are different for the two cases. Thus, one could argue that the AVO attributes themselves can be used as a hydrocarbon indicator a method currently utilized by industry. However, the attractiveness of the proposed method is that in a fully deterministic way, we convert the two AVO attributes directly into N/G and saturation attributes. Furthermore, the results are quantitative, given the limitations and simplifications in the model used. CONCLUSIONS We find that for a finely layered shale-sand medium, there is a relationship between the N/G ratio of the medium and the reflection characteristic composed from the reflection amplitudes both from the top and the base of the stack depending on with what the medium is sandwiched. We also show that the AVO parameters for PP intercept and gradient and PS gradient reflection depend on the N/G ratio. In addition, we demonstrate that the effective-medium parameters, computed from the stack of finely layered sand-shale sequence, are strongly dependent on the contrast in elastic parameters between shale and sand layers. We derive the exact and approximate equations to compute these effective parameters, including the anisotropy parameters caused by fine layering. We show that the anisotropy parameter is symmetric with respect to N/G while the parameter is nonsymmetric. The applicability of a binary-effective medium has been addressed by changing the layer thicknesses and medium parameters in a sand-shale stack of horizontal layers. For an average change in layer thickness of 5% and a corresponding change in the elastic parameters of.5%, we find that random variations result in an approximately 5% error in N/G prediction checked only for zerooffset reflectivity. The N/G prediction is performed by using topreservoir seismic amplitudes. Compared to thin-layer reflectivity modeling of a binary-effective medium, a random-effective medium gives multiple reflections between the top and the base of the reservoir section. If in addition to the random variations a systematic trend in the medium parameters is introduced, more significant deviations are expected. We also define the AVO attributes for the effective medium as a function of N/G and water saturation in the sand layers. We demonstrate an inversion procedure to convert the AVO attributes into N/G and water saturation. Since the water saturation values may be different for each sand layer, we estimate an effective water saturation that can be estimated approximately by weighted averaging of saturation in the individual layers. The only limitation related to the model data is the applicability of the chosen sand and shale properties. When our method was tested on a real seismic data set from offshore Brazil, the results were encouraging. For the top reservoir interface, we find high N/G values, as well as high probability of oil saturation. For another interface above the reservoir, the N/G ratio and oil content are found to be negligible. Future work will include the effect of uncertainties in both the model and AVO attributes on discrimination between N/G and fluid saturation, 3D tests, and time-lapse seismic application. ACKNOWLEDGMENTS We thank Norsk Hydro for their financial support and for permission to use and present the seismic data. Aart-Jan Wijngaarden is acknowledged for valuable discussions and suggestions. The Norwegian Research Council NFR is acknowledged for financial support. 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 a are given by Backus 196 as and A 1 =4 c 44,1 1 c 44,1 N/G c 33,1 1 +c 44, 1 c 44, c 33, N/G A = 1 N/G c 33,1 + N/G c 33,, A 3 = 1 c 44,1 c 33,1 1 N/G + 1 c 44, c 33, N/G, A 4 = 1 N/G c 44,1 + N/G c 44,, A-1 where N/G is the volume ratio N/G ratio for our case. Substituting equation 1 into equation A-1 results in A 1 =4c 44,1 1 c 1 + N/G 44 1 c 44 / + c 33 c 44 + c 33 c 44 /4 1 1 c 44 / 1+ c 33 / A = 1 c 33 N/G c 33,1 1 1+ c 33 /, A 3 =1 c 33 c 44 1 + N/G 1 1 c 44 / 1 + c 33 / A 4 = 1 c 44 N/G A- c 44,1 1 1+ c 44 /, where 1 =c 44,1 /c 33,1 is v s /v p ratio squared for shale. The effective elastic parameters are given by c 11 =A 1 + A 3 c 13 = A 3 A A

c 33 = 1 A c 44 = 1 A 4. AVO attribute inversion A-3 W= K ma 1 K w 1 K o 1 K fr + K ma K ma K o, C35 Downloaded 09/1/14 to 84.15.159.8. Redistribution subject to SEG license or copyright; see Terms of Use at http://library.seg.org/ Substituting equation A- into equation A-3, we obtain 1+4 1 N/G 1 N/G c 44 c 33 1 c 44 / 1 c 44 1 c 33 / 1 c 44 / 1+ c 33 / c 11 =c 33,1 c 33 1 N/G 1 + c 33 / 1 1 + c 33 c 44 1 N/G 1 c 44 / 1 + c 33 / c 13 =c 33,1 c 33 1 N/G 1 + c 33 / 1 c 33 =c 33,1 c 33 1 N/G 1 + c 33 / 1 c 44 =c 44,1. A-4 c 44 1 N/G 1 + c 44 / The effective density is given by = 1 1 + N/G 1 /. APPENDIX B THE FLUID EFFECT A-5 Using the Gassmann equation Gassmann, 1951, the elastic parameters for an isotropic sand can be written as where U= dry c 33, =c 1+U S 33 1+W S c 44, =c dry 44, = dry 1+F S, c dry 33 =K fr + 4 3 + K fr K ma K ma 1 K fr + K ma K ma K o K fr + ma 4 3 K 1 + K ma K ma K o + K fr K ma K ma K fr + 4 3 1 K fr K w 1 K o B-1 dry = o + 1 ma F= w o o + 1 ma, B- where S is water saturation, is porosity, K fr is bulk modulus of solid framework, is shear modulus of solid framework, and K ma is intrinsic modulus of solid, i.e., bulk modulus for zero porosity. K o and K w denote the pure fluid bulk moduli for oil and water, respectively. ma is the matrix density and o and w denote oil and water densities, respectively. Note that both U and W are negative and U W. From equation B-1, we compute the contrast in elastic parameters as a function of water saturation with c 33 = c 33 dry +D 0 S 1+D 1 S c 44 = c dry 44, = dry +R 0 S 1+R 1 S, B-3 dry dry where c 33, c 44, and dry are the contrasts in P-wave and S-wave modulus and density between the dry sand and shale. The other parameters are dry D 0 =U W+ c U+W 33 D 1 = U 1 dry +W+ c U W 33, R 0 = 1 + dry F R 1 = 1 1 + dry F B-4 and the contrast in c 44 is saturation independent. For simplicity, we use the weak contrast case only. Therefore, equation 10 takes the form R 0 N/G c 33 dry +D 0 S + dry +R 0 S 4 1+D 1 S 1+R 1 S G N/G c 33 dry +D 0 S 4 1+D 1 S 4 1 c 44 dry. dry +R 0 S 1+R 1 S To solve equation B-5, first we get quadratic equation B-5

C36 Stovas et al. Downloaded 09/1/14 to 84.15.159.8. Redistribution subject to SEG license or copyright; see Terms of Use at http://library.seg.org/ a R 0 +g G S + a 1 R 0 +g 1 G S +a 0 R 0 +g 0 G=0 B-6 with a 0 = c dry 33 +4 1 c dry 44 + dry a 1 = c dry 33 R 1 +4 1 c dry 44 D 1 +R 1 + dry D 1 +R 0 D 0 and a =D 1 R 0 D 0 R 1 +4 1 c dry 44 D 1 R 1 g 0 = c dry 33 + dry g 1 =D 0 +R 0 + c dry 33 R 1 + dry D 1 g =D 0 R 1 +D 1 R 0 being the model parameters, while R 0 angle = 0 and G are the real data. The solution of B-6 gives two values for water saturation S 1, = a 1R 0 +g 1 G ± a1 R 0 +g 1 G 4 a R 0 +g G a 0 R 0 +g 0 G, a R 0 +g G B-7 with 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 c dry 33 +D 0 S + dry +R 0 S 1+D 1 S 1+R 1 S. B-8 From equation B-7 and B-8, we can see that if both intercept and gradient go to zero, N/G also goes to zero, but saturation is not defined. The denominator in equation B-7 can be very small because of uncertainties in AVO attributes R 0 and G and model parameters a and g, which leads to instability in computation of water saturation. It is always the case when N/G is very low. The AVOattribute contour lines Figure 11 are almost parallel to the water saturation axis the discrimination between N/G and water saturation is very low. With increase of N/G values, discrimination also increases. REFERENCES Backus, G. E., 196, Long-wave elastic anisotropy produced by horizontal layering: Journal of Geophysical Research, 67, 447 4440. Bakulin, A., and V. Grechka, 003, Effective anisotropy of layered media: Geophysics, 68, 1817 181. Berryman, J. G., 1999, Transversely isotropic elasticity and poroelasticity arising from thin isotropic layers, in Y.-C. Teng, E.-C. Shang, Y.-H. Pao, M. H. Schultz, and D. Pierce, eds., Theoretical and Computational Acoustics 97: Proceedings of the Third International Conference on Theoretical and Computational Acoustics, 457 474. Brittan, J., M. Warner, and G. Pratt, 1995, Anisotropic parameters of layered media in terms of composite elastic properties: Geophysics, 60, 143 148. Connolly, P., 1999, Elastic impedance: The Leading Edge, 18, 438 45. Dubucq, D., S. Busman, and P. V. Riel, 001, Turbidite reservoir characterization: Multi-offset stack inversion for reservoir delineation and porosity estimation; a Gulf of Guinea example: 71st Annual International Meeting, SEG, Expanded Abstracts, 609 61. Folstad, P. G., and M. Schoenberg, 199, Low frequency propagation through fine layering: 6nd Annual International Meeting, SEG, Expanded Abstracts, 178 181. Gassmann, F., 1951, Über die Elastizität poroser Medien, Vierteljahrsschrift der Naturforschenden Gesellschaft in Zürich, 96, 1 3. Gelinsky, S., and S. A. Shapiro, 1997, Poroelastic Backus averaging for anisotropic layered fluid- and gas-saturated sediments: Geophysics, 6, 1867 1878. Hovem, J. M., 1995, Acoustic waves in finely layered media, Geophysics, 60, 117 11. MacLeod, M. K., R. A. Hanson, C. R. Bell, and S. McHugo, 1999, The Alba Field ocean bottom cable seismic survey: Impact on development, The Leading Edge, 18, 1306 131. Mahob, P. N., J. O. Castagna, and R. A. Young, 1999, AVO inversion of a Gulf of Mexico bright spot A case study: Geophysics, 64, 1480 1491. Marion, D., and P. Coudin, 199, From ray to effective medium theories in stratified media: An experimental study: 6nd Annual International Meeting, SEG, Expanded Abstracts, 1341 1343. Marion, D., T. Mukerji, and G. Mavko, 1994, Scale effects on velocity dispersion: From ray to effective theories in stratified media: Geophysics, 59, 1613 1619. Schoenberger, M., and F. K. Levin, 1974, Apparent attenuation due to intrabed multiples: Geophysics, 39, 78 91. Shapiro, S. A., and S. Treitel, 1997, Multiple scattering of seismic waves in multilayered structures: Physics of the Earth and Planetary Interiors, 104, 147 159. Stovas, A., and B. Arntsen, 003, Low frequency waves in finely layered media: EAGE Annual Meeting, Extended Abstracts, 16. Stovas, A., and M. Landrø, 004, Optimal use of PP and PS time-lapse stacks for fluid-pressure discrimination: Geophysical Prospecting, 5, 301 31. Stovas, A., and B. Ursin, 003, Reflection and transmission responses of a layered transversely isotropic viscoelastic media: Geophysical Prospecting, 51, 1 31. Thomsen, L., 1986, Weak elastic anisotropy: Geophysics, 51, 1954 1966. Vernik, L., D. Fisher, and S. Bahret, 00, Estimation of net-to-gross from P and S impedance in deepwater turbidite: The Leading Edge, 1, 380 387.