1 INTRODUCTION

Seismic frequency-dependent amplitude variations versus offset (AVO) is a technique proposed following the development of rock physics studies. The key of the method is to interpret the frequency-dependent abnormal seismic reflections in terms of rock physics. The basic assumption is that frequency-dependent elastic modulus leads to frequency-dependent reflection coefficients within the seismic b and width. The development of the frequency-dependent AVO relies on the improvement of rock physics methods. Currently,the rock physics models used in the method include the multi-scale equant porosity model proposed by Chapman (2003) and Chapman et al. (2003) ,the patch-saturated model proposed by White (1975) , and those models developed by Johnson (2001) ,Carcione et al. (2003) ,Carcione and Picotti (2006) and Quintal et al. (2007) based on the White model.

According to these rock physics models,Chapman and Liu (2005) ,Chapman et al. (2005,2006) and Liu et al. (2006) researched the applications of the frequency-dependent AVO on reservoir fluid identification. Quintal et al. (2007) ,Ren et al. (2009a,2009b) and Teng et al. (2013) classified frequency-dependent AVO based on forward modeling and the White model. Liu et al. (2011) investigated the frequency-dependent AVO according to the Biot's theory and Zhao et al. (2014) researched the issue from the theory of attenuation and viscoelasticity. Chen (2009) summarized the development of the frequency-dependent AVO method and prospected further applications of this technique. Xuan et al. (2010) ,Liu (2011) ,Cheng et al. (2012) and Lu (2013) discussed the applications of the method on the identification of reservoir fluids.

In spite of these progresses of the frequency-dependent AVO technique in fluids identification,deep mechanism of the frequency-dependent seismic reflections deserves further investigation. Current studies usually use conventional Zoeppritz equations that are based on the single interface model while have not considered the effects of layered structures. Liu et al. (2006) found that seismic reflections from oil- and gas-bearing reservoirs do not show the abnormal high amplitude at low-frequency. This may be due to tuning and interferences of the b and width-limited seismic reflections that lead to the frequency-dependency of the complicated waveforms (Gao et al., 2003,2006; Li et al., 2012,2013) . Thus,both the effects of attenuation and dispersion and the effect of layered structure need to be considered in the study of the frequency-dependent AVO.

As shown in Fig. 1,we compute frequency-dependent AVO responses based on the generalized propagator matrix method (Carcione,2001; Guo,2008; Guo et al. 2014) . Compared with traditional methods,the advantages of the proposed method include: First,based on the propagator matrix method,both the effects of dispersion and attenuation associated with lithology and the effects of thin bed and interbed associated with layered structures,as well as anisotropy can be considered in the calculation of frequency-dependent AVO responses.

Second,the method realizes the seamless link between the rock physics modeling and the calculation for frequency-dependent reflection coefficients. The propagator matrix method implements the calculation in the frequency domain and deals with complex modulus,thus the output frequency-dependent complex modulus from the rock physics model can be input straightforwardly to the propagator matrix method, and therefore dispersion and attenuation related to lithology can be accurately considered. By contrast,traditional Zoeppritz method is based on elastic assumption, and can only deal with the real part of the complex modulus while ignoring imaginary parts of the complex modulus.Third,the proposed method is able to consider dispersion and attenuation resulting from various mechanisms in the same model. For example,in an interbedded structure,s and stone may represent velocity dispersion and attenuation described by the Chapman's fluid-squirt model,while shale may show inelasticity and anisotropy described by the Carcione's theory. This considers the common feature of rock physics models for inelastic medium where the strain-stress relations are constructed in the frequency domain and the outputs of the models are complex frequency-dependent modulus. In contrast,it is difficult for the conventional methods for the calculation of reflection coefficients or the theories for seismic modeling to consider different mechanisms of dispersion and attenuation in the same model.

Fourth,the spectra of reflected waves can be calculated by multiplying the frequency-dependent reflection coefficients with the wavelet spectrum, and the synthetics can be obtained by implementing inverse Fourier transform on the reflection spectra.

2 AVO THEORY OF THE INTERBED

For the models of layered and interbedded structures in Fig. 2,reflection coefficients are not only associated with incidence angle,elastic parameters,but also with the frequency of incident waves,the thickness of reservoir layer and interbedded structures according to the theories of Kennett (1983) ,Rokhlin and Wang (1992) ,Ursin and Stovas (2002) and Liu and Schmitt (2003) . Meanwhile,inelasticity of the layer also contributes to dispersion and attenuation of seismic reflections. Generalized propagator matrix method considers these effects in the calculation of reflection coefficients (Carcione,2001) . For incident P wave,the vector of reflection and transmission coefficients r = [R_PP,R_PS,T_PP,T_PS]^T can be computed by

where the matrices A₁ and A₂ are the propagation matrices associated with elastic properties of upper and lower layers, and the matrix B_α (α = 1,· · ·,N) is the propagator matrix of the interbedded layer consisting of N fine layers. i_P represents the P wave incidence vector that is associated with elastic parameters of the incidence medium. Meanwhile,all these matrices and vectors are the functions of incidence frequency and slowness (or incidence angle) . The method considers the layered model,copes with frequency-dependent complex modulus, and computes reflection coefficients at each frequency.

3 ROCK PHYSICS MODEL OF INELASTIC MEDIUM

Figure 2 illustrates that the interbed may consist of s and stone and shale that represent dispersion and attenuation due to different mechanisms. In this study,we describe the fluid-saturated s and stone using multi-scale fracture model (Chapman,2003 and Chapman et al., 2003) and model the shale using viscoelastic anisotropic theory (Carcione,1997) .

3.1 Multi-scale Fracture Model for S and stone

In the multi-scale fracture rock physics model,fluids-squirt between pores and fractures leads to dispersion and attenuation of seismic waves. The complex and frequency-dependent modulus is represented as

where λ and μ are Lame parameters,ω is frequency,τ is the time scale parameter,φ is porosity,ε is fracture density, and κ_f is the bulk modulus of fluids.

Elastic tensor C_ijkl is obtained by introducing disturbance C_ijkl¹ to isotropic elastic background C_ijkl^iso:

If we define $\bar{\lambda }$ and ${\bar{\mu }}$ the Lame parameters for a certain porosity φ, where ρ is density,V_P and V_S are P and S wave velocities at reference frequency, where κ_f⁰,ε₀ and τ₀ are the bulk modulus of fluids,fracture density, and time scale parameter at the reference frequency ω₀. Then,elastic tensor can be further defined as where Λ and M are frequency-independent Lame parameters.

Then,for an arbitrary frequency ω,time scale τ,porosity φ,fracture density ",fluid bulk modulus κ_f,the anisotropic tensor for dispersion and attenuation can be represented by

Adequate descriptions of the theory were given by Chapman (2003) and Chapman et al. (2003) .

Figure 3 shows the calculated results based on the rock physics model where P wave velocities and attenuation factors vary with frequency for gas and water-saturated situations. Within the seismic b and width,dispersion of the P wave velocity results in the frequency-dependence of the reflection coefficients, and the frequency-dependent attenuation further complicates seismic reflections. In Fig. 3,the quality factor Q of P wave is given by

where Re and Im indicate real and imaginary part of a complex number,respectively.

According to the rock physics model,S wave shows no velocity dispersion and ignorable attenuation.

The properties used in the model include that P- and S wave velocities are 2790 m·s^-1 and 1463 m·s^-1,densities for gas- and water-saturated s and stone are 2080 kg·m^-3 and 2060 kg·m^-3. Other parameters associated with multi-scale factures include that crack density for micro-cracks is ε_c = 0,porosity φ = 0.1,fracture density ε_f = 0.05, and the velocity and density are 1710 m·s^-1 and 1100 kg·m^-3 for water, and 620 m·s^-1 and 65 kg·m^-3 for gas. Time scale parameter is τ = 4 × 10-6 s.

3.2 Viscoelastic Anisotropic Model for Shale

On the basis of the St and ard linear solid model and Boltzmann theory, Carcione (1997) calculated VTI (Transverse isotropy with a vertical axis) anisotropic complex modulus p₁₁,p₃₃, p₅₅, and p₁₃,

where c₁₁,c₃₃,c₅₅ and c₁₃ are real modulus at the high frequency elastic limits. For Q → ∞,we have p_IJ → c_IJ .

From Eq. (11) to (14) ,the complex Zener modulus is

where v = 1,2 correspond to the relaxation mechanism for P- and S wave, and associated relaxation time for stress and strain are where τ₀ is the relaxation time at the central frequency and Q_v (v = 1,2) are the quality factors for P- and S wave,respectively.

Figure 4 is the computed P- and S wave and corresponding attenuation factors that vary with the frequency. We can observe dispersion and attenuation within the seismic b and width which lead to the frequency-dependent reflection coefficients. The properties of shale used in the model include that vertical velocities are V_P = 2743 m·s^-1 and VS=1394 m·s^-1,density is ρ = 2060 kg·m^-3,anisotropy parameters are ε = 0.2,γ = 0.3,δ = -0.1, and P- and S wave quality factors are Q_P = 20 and Q_S = 10.

4 NUMERICAL STUDY AND ANALYSIS

Then,we calculate the frequency-dependent AVO responses from the inelastic thin bed and interbed in Fig. 2, and analyze the frequency-dependent features associated with layered structure,attenuation and dispersion.

4.1 Frequency-Dependent AVO Responses from Thin S and Layer

First,we consider the thin s and model shown in Fig. 2a,where the dispersion and attenuation are displayed in Fig. 3. We compute and compare seismic AVO responses from elastic and inelastic s and stones, and from the s and stones with varied layer thickness and layered structure. According to the improved method in Fig. 1,we obtain the calculation results shown in Fig. 5.

As shown in Fig. 5a,for thick and elastic s and stone layer,reflection model can be represented as a single interface, and the reflection coefficients represent no frequency-dependence that means reflection coefficients at all frequencies show the same trends (middle) . Accordingly,the amplitudes stacked from 0 to 60° do not vary with the frequency (left) . At the same time,AVO curves at 10 Hz and 40 Hz are overlapped (right) .

In Fig. 5b,if the thick s and stone represents dispersion and attenuation shown in Fig. 3 due to the presence of fluids saturated within pores and fractures,corresponding reflection coefficients vary with frequency (middle) . The stacked amplitudes 0 to 60° have high amplitudes at low frequency (left) . Meanwhile,AVO curves at 10 Hz and 40 Hz separate (right) .

In Fig. 5c,for a s and stone layer that has a thickness of 15 m and represents dispersion and attenuation shown in Fig. 3,the reflection coefficients illustrate complicated frequency-dependence (middle) . The stacked amplitudes 0 to 60° show a trend of increase followed by decrease with frequency (left) . Thus,for the thin layer,oil and gas reservoir do not necessarily show the abnormity of high amplitudes at low frequency. The variations in reflection amplitudes are also associated with the layered structures. These results in Fig. 5c may offer an interpretation for the observations of Liu et al. (2006) .According to the workflow in Fig. 1,the spectra of reflected waves can be calculated by multiplying the frequency-dependent reflection coefficients with the spectrum of wavelet, and the reflected waveforms can be further obtained by implementing inverse Fourier transformation to the reflection spectra. Fig. 6 illustrates the reflection spectra computed from Fig. 5 and Fig. 7 shows corresponding waveforms. Fig. 8 shows the Ricker wavelet that has a dominant frequency of 20 Hz. Comparing Fig. 6a and Fig. 6b, and Fig. 7a and Fig. 7b,we can observe that attenuation and dispersion lead to the decrease in reflection amplitudes, and alter corresponding AVO features. Meanwhile,Fig. 6c and Fig. 7c show the impact of thin layer structure on reflection amplitudes and AVO features,which are quite different from the model of the thick layer and single interface.

4.2 Frequency-Dependent AVO of the Interbedded S and stone and Shale Layer

The interbedded model of s and stone and shale is shown in Fig. 2b. The overlying and underlying shale is elastic,while the interbedded layer is inelastic due to the saturation of oil and gas. The reservoir s and stone in the interbed is described by the multi-scale fracture model in Fig. 3, and the source shale represents viscoelasticity and anisotropy illustrated in Fig. 4. Total thickness of the interbed is 40 m where single s and stone layer has a thickness of 2 m and shale layer 3 m,corresponding to 40% s and content.

Figure 9a illustrates the frequency-dependent PP reflection coefficients of the inelastic interbed. Fig. 9b indicates the case for the high frequency elastic limit. For the Ricker wavelet of 40 Hz dominant frequency,the calculated reflection spectra are shown in Fig. 10a and 10b. By comparing Fig. 10a and 10b,we can observe that the inelasticity leads to the decrease in reflection amplitudes,especially at the high frequency. Corresponding to the spectra in Fig. 10,similar AVO features can also be observed in the reflected waveforms shown in Fig. 11.

Accordingly,Fig. 12 shows the reflection coefficients of the mode-converted PS waves,Fig. 13 illustrates corresponding reflection spectra for the Ricker wavelet with a dominant frequency of 40 Hz, and Fig. 14 shows corresponding waveforms. The effects of the inelasticity on reflected amplitudes and AVO features can be observed in these Figures.

5 CONCLUSIONS

We have investigated the frequency-dependent AVO responses,considering both the effect of frequency and attenuation associated with lithology, and interbed related to stratified structure.

The method realized a seamless link between the rock physics modeling and the calculation for reflection coefficients. We use the features of the propagator matrix method that implements calculations in frequency domain and copes with frequency variables,which makes it possible to treat the predicted frequency-dependent complex modulus by the rock physics model directly as the inputs for the calculation of reflection coefficients. While conventional methods usually only consider real parts of the complex modulus and neglect the imaginary parts that are also associated with dispersion and attenuation. Another advantage of the improved method is to consider distinct mechanism of dispersion and attenuation because anelasticity of rocks predicted by different models are all described by the frequency-dependent complex stiffnesses.

Results of numerical stimulations indicate that both inelasticity and layered structure contribute to the frequency-dependence of seismic AVO. In the presence of thin layer,fluid-saturated reservoir does not present low-frequency anomalies. The amplitudes of reflections may increase and then decrease with frequency. For the interbed model,dispersion and attenuation also lead to obvious variations in the frequency-dependent reflection coefficients,spectra, and AVO features for both PP and PS wave. Finally,we have improved the algorithm for the calculation of frequency-dependent AVO responses which may help for the modeling and analysis of seismic reflections in the presence of dispersion,attenuation and layered structure.

6 ACKNOWLEDGMENTS

This work was supported by the National Natural Science Foundation of China (41404090,41430322 and U1262208) .