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In a fractured porous hydrocarbon reservoir, wave velocities and reflections depend on frequency and incident angle. A proper description of the frequency dependence of amplitude variations with offset (AVO) signatures should allow effects of fracture infills and attenuation and dispersion of fractured media. The novelty of this study lies in the introduction of an improved approach for the investigation of incident-angle and frequency variations-associated reflection responses. The improved AVO modeling method, using a frequency-domain propagator matrix method, is feasible to accurately consider velocity dispersion predicted from frequency-dependent elasticities from a rock physics modeling. And hence, the method is suitable for use in the case of an anisotropic medium with aligned fractures. Additionally, the proposed modeling approach allows the com- bined contributions of layer thickness, interbedded structure, impedance contrast and interferences to frequency-dependent reflection coefficients and, hence, yielding seismograms of a layered model with a dispersive and attenuative reservoir. Our numerical results show bulk modulus of fracture fluid significantly affects anisotropic attenuation, hence causing frequency- dependent reflection abnormalities. These implications indicate the study of amplitude versus angle and frequency (AVAF) variations provides insights for better interpretation of reflection anomalies and hydrocarbon identification in a layered reservoir with vertical transverse isotropy (VTI) dispersive media. Keywords Seismic anisotropy · Fractured media · Attenuation and dispersion · AVO responses · Frequency dependence 1 Introduction considered accurately during the traditional AVO response analysis. Numerous researchers have employed the Biot’s Quantitative analysis of seismic reflections suggests that poroelastic theory (Biot 1962a, b) to investigate the behav- hydrocarbon-bearing regions often show anomalously high iors of reflected and transmitted waves from an interface levels of wave attenuation and the associated strong disper- separating two poroelastic layers filled with different fluids. sion of phase velocity (e.g., Brajanovski et al. 2010; Cao The classic Biot’s theory, unfortunately, underestimates sub- et al. 2016; Carcione and Picotti 2006; Deng et al. 2020; stantially the wave attenuation; it yields negligible attenu- Gurevich et al. 2010; Li et al. 2020; Rubino and Holliger ation in the seismic-exploration band. Nevertheless, larger 2013; Wu et al. 2014; Zhang et al. 2017). The resulted effect than usual levels of attenuation can be predicted when con- on seismic reflection signatures, however, this has not been sidering heterogeneous media, such as partial saturation and fractured porous medium. Meanwhile, the fractured reser- voir systems are important for the storage of underground water and monitoring of the injected carbon dioxide. And Edited by Jie Hao and Chun-Yan Tang. thus, an appropriate characterization of natural fractures * Shang-Xu Wang using surface seismic observations may potentially provide firstname.lastname@example.org significant implications for an improved calculation of pore- fluid flow patterns and media permeability within a fractured State Key Laboratory of Petroleum Resources and Prospecting, CNPC Key Laboratory of Geophysical poroelastic reservoir (e.g., Maultzsch et al. 2003; Chapman Exploration, China University of Petroleum (Beijing), et al. 2006; Guo et al. 2018; Zhang et al. 2018; Cao et al. Beijing 102249, China 2019; Guo et al. 2019). Daqing Oilfield Exploration and Development Institute, Daqing 163000, Heilongjiang Province, China Vol:.(1234567890) 1 3 Petroleum Science (2021) 18:758–772 759 The philosophy that we hope to develop in this study sandstone having weak to moderate velocity anisotropy. Via holds that strong wave attenuation and the associated veloc- using a generalized Zoeppritz equation-based method, Jin ity dispersion anomaly is the measured ground truth and et al. (2018) explored the impacts of anisotropic attenuation we should, therefore, focus our attention on those physical and velocity dispersion on P-wave reflection signatures and mechanisms. Using the more specific mechanism, the exist- observed that fracture-related anisotropic velocity dispersion ence of excess attenuation and dispersion of phase velocity can substantially affect both the magnitude and phase vari- related to the hydrocarbon-bearing regions can be inter- ations of P-wave reflections as expected. preted with enhanced confidence. In what follows we study first the effects of fracture infills It is broadly accepted that periodically stratified rocks on angle- and frequency-dependent elastic properties of a show important VTI anisotropic property and are widely dis- porous medium permeated by sets of aligned fractures. And tributed within the crust. Previous authors have made plenty then we investigate numerically the influences of abnormally of attempts to determine the frequency-dependent anisotropy high levels of wave attenuation and velocity dispersion on considering wave attenuation and dispersion of phase veloc- seismic reflection signatures. In this study, the effect of ity due to the squirt flow mechanism (Chapman 2003; Tang velocity dispersion and attenuation was determined using an 2011), as well as the mesoscopic flow effect (e.g., Krzikalla equivalent effective medium theory based on the mesoscopic and Muller 2011; Carcione et al. 2013; Kudarova et al. 2013; wave-induced fluid flow concept of Norris (1993) (please Galvin and Gurevich 2015). Researchers have also addressed also read Brajanovski et al. 2005; Kong et al. 2017) for the sensitivity of reservoir acoustic properties to the fracture fractured porous rocks, in which the pores and fractures are size, fracture weakness and facture infills (e.g., Maultzsch filled with two different fluids. Using a frequency-domain et al. 2003; Chichinina et al. 2006; Rubino et al. 2016; Shi propagator matrix algorithm that allows anisotropic elas- et al. 2018). However, how the anisotropic poroelasticities tic tensors, in this presented work we perform our calcula- of fractured porous layers affect the frequency-dependent tions of seismic AVO responses from a hydrocarbon reser- seismic AVO response signatures, to the authors’ knowledge, voir with complex lithology (Guo et al. 2015a, b; He et al. has not been fully investigated. In addition, current analysis 2018). The numerically modeling results demonstrate that on seismic AVO signatures normally uses single interface the frequency-dependent responses are closely related to reflection model or normal-incidence reflection approach. seismic AVO signatures in an interbedded reservoir system This forward modeling has not accurately considered the and hence a potential effective approach to help to identify influences of both the stratified structure and the attenuation fracture-filled fluid variations inside an interbedded porous and dispersion of complex stiffness tensors from rock phys- reservoir. ics simulation in the computations of angle- and frequency- dependent reflection coefficients (e.g., Rüger et al. 1997; Ren et al. 2009; Liu et al. 2011; Dupuy and Stovas 2014a, 2 Methodology b; Zhao et al. 2015; Kumari et al. 2017; Kumar et al. 2018; He et al. 2019). 2.1 Anisotropic wave attenuation and velocity Recent studies have demonstrated that wave attenuation dispersion in a fluid‑saturated fractured and velocity dispersion may produce important influences poroelastic medium on seismic AVO analysis in fractured porous reservoirs (e.g., Guo et al 2015a, b; Guo et al 2016, 2017a, b). Neverthe- Although the geometries of rock fractures are very complex, less, few publications deal with the means to quantitatively they can be represented by highly porous permeable layers explore the seismic AVAF response anomalies of an ani- which are also noted as planer fractures, in case the fracture sotropic medium, particularly for the anisotropy variations radii are much smaller than the fracture spacing and seismic as a function of frequency. To the best of our knowledge, wavelengths. Thus, the porous rock having a set of aligned Chapman et al. (2006) have first introduced a methodology planer fractures can be treated as a layered poroelastic that was utilized to evaluate the influences of abnormally medium with infinite extents in lateral. Plenty of rock phys- high values of velocity dispersion and attenuation from the ics theories that have been derived specifically to explore the fluid-filled reservoir on seismic measurements. In their pre- frequency dependence in the elastic stiffness tensors press sented work, seismic reflections at an interface between a the main challenges in carrying out seismic AVO modeling nondispersive overburden and an equivalent dispersive layer analysis, principally porosity–anisotropic velocity transform, were obtained using a reflectivity technique. Additionally, velocity–pressure relation, fluid saturation and velocity pre- based on Brajanovski’s porous and fractured model (Bra- diction. In this study, an extended poroelastic theory based janovski et al. 2005), Yang et al. (2017) examined the char- on Carcione et al. (2013) and Kong et al. (2017) was utilized acteristics of seismic responses from an interface separat- for the determination of frequency-dependent full elastic ing a purely elastic isotropic overburden and a dispersive stiffness tensors. Additionally, the introduced theory allows 1 3 760 Petroleum Science (2021) 18:758–772 us to account for fluid-sensitive attenuation and velocity dis- Equations (3)-(6) are exact for an anisotropic VTI persion attributes in the frame of anisotropic viscoelasticity. medium. In these equations, the bulk density is defined as On the basis of the assumption that waves propagating ρ = (1–φ )ρ + φ ρ , where φ is porosity of the background b s b f b is always normal to the layering plane, all interlayer flow porous medium and ρ and ρ are grain and pore-fluid densi- s f models can be approximated using an unrelaxed (high- ties, respectively. Subsequently, we obtain the phase velocity frequency) and a relaxed (low-frequency) P-wave stiffness and attenuation of the three wave modes as (Mavko et al. value related by a frequency-dependent function (i.e., Krzi- 2009). kalla and Müller 2011; Carcione et al. 2013). For wave prop- −1 agation in a fractured poroelastic medium that is simulated V , = Re ( ) (7) as a periodically stratified system of two different layers as described above, we compute the complex and frequency- dependent VTI stiffnesses C according to the poroelastic 2 ij Im v −1 Backus averaging based on the equation Q (, ) =− (8) Re v high high low C = C − R() ⋅ C − C (1) ij ij ij ij where i indicates either the P-, the SV- or the SH-wave mode. Precise forms of the above relationships are also pre- where ω is the angular frequency (= 2πf), and the subscript sented in Kong et al. (2017) and He et al. (2020). pair (ij) indicates (11), (13), (33), (44) and (66), respectively. To examine the influences of fracture-filled fluid vari- The scalar complex relaxation function R(ω) can be obtained ations on the phase velocity and seismic attenuation (or through normalizing the normal-incidence P-wave stiffness −1 inverse quality Q ) behaviors, we consider a poroelastic C with its relaxed and unrelaxed limits as sandstone with water saturation. The assumed fractured high sandstone has 20% rock porosity and consists of quartz as C − C R() = the grain material (bulk modulus K = 37 GPa and density (2) high low C − C 33 33 ρ = 2650 kg/m ), which is adapted from Kong et al. (2017). The pore-fluid properties of water are density ρ = 1000 kg/ With Eq. (1), we derive all effective stiffness components, 3 –3 m , viscosity η = 10 Pa·s and bulk modulus K = 2.25 GPa. f f and the anisotropic properties are determined fully by the According to the proposed approach in Krief et al. (1990), high low high-frequency and low-frequency limits C and C . The 33 ij the bulk and shear moduli of the dry background media frequency dependence of the elastic stiffnesses C is con- ij can be estimated using the rock porosity with the following trolled solely by the scalar complex relaxation function equation. R(ω). In such a medium, attenuation and phase velocity of 3∕(1− ) the plane and homogeneous waves are both frequency- and b b = = 1 − (9) angle-dependent, as well as the anisotropy parameters K s s (Thomsen 1986). From the elastic stiffnesses, complex P- where K and μ are bulk and shear modulus of the grain, s s and S-wave velocities of an equivalent VTI medium will be respectively. We set the fracture normal weakness δ = 0.2. determined as a function of incident angle θ using the fol- The dependence of rock permeability κ of the dry back- lowing formulation (e.g., Mavko et al. 2009). ground medium on porosity is assumed to follow the empiri- � � � � 2 cal Kozeny–Carman relation. C + C sin + C + C cos + D 11 44 33 44 v = , 15 − 0.035 100 (3) = (10) 1.035 − √ b � � � � C + C sin + C + C cos − D 11 44 33 44 v = , In Fig. 1, the variations of phase velocity and inverse SV quality factor as functions of incident angle and frequency (4) are demonstrated for the P- and SV-waves propagating in a fractured porous equivalent medium with pores and fractures C sin + C cos 66 44 (5) v = , SH filled by two different fluids. We observe apparent veloc- ity dispersion and attenuation over the seismic exploration 2 2 2 2 (6) D= C − C sin − C − C cos + 4 C + C sin cos . 11 44 33 44 13 44 1 3 Frequency, Hz Frequency, Hz Frequency, Hz Frequency, Hz Petroleum Science (2021) 18:758–772 761 2.7 0.12 (a) (b) 0.10 2.8 0.12 2.6 0.08 2.6 0.08 2.5 0.06 2.4 0.04 2.4 0.04 2.2 0 200 200 2.3 0.02 150 150 90 90 100 100 60 60 30 50 30 2.2 0 0 0 0 0 -4 ×10 (c) (d) 1.30 -4 ×10 1.35 1.28 1.30 1.26 1.25 1.24 1.20 1.22 1.15 1.20 0 200 200 150 150 90 1.18 90 100 100 60 60 50 50 30 30 1.16 0 0 0 0 0 −1 Fig. 1 Variations of the a, c phase velocities and the b, d inverse quality factors (Q ) with frequency and incident angle in a fractured porous medium, where pores and fractures are filled with two different fluids. In addition, graphs a and b correspond to the P-wave and, c and d corre- spond to the SV-wave, respectively band. Please note attenuation of the SV-wave is quite small (i.e., F > F*), we observe from Fig. 2(a) that the levels of compared to that of the P-wave in the low frequencies. P-wave velocity dispersion (differences between velocities In Fig. 2, the estimated phase velocity and attenuation at the high-limit and low-limit frequencies) at an incident of P- and SV-waves are displayed, respectively, for P- and angle of 30° are weaker than that at an incident angle of 0°. SV-waves as functions of frequency and incident angle. In Nevertheless, when the fracture fluid compressibility is very Fig. 2, velocity variations, as well as the maximum attenu- large (i.e., F < F*), we find slight variations between the ation peaks, are very large for both liquid-filled (F = 100 magnitudes of velocity dispersion from incident angles of corresponding to fracture fluid bulk modulus K = 16 GPa) 0° and 30°. Meanwhile, we see the frequency-dependent SV- fc and nearly dry (F = 0.01 corresponding to highly compress- wave velocity and attenuation exhibits an apparent increase ible gases) fractures. The variations, nevertheless, become with increasing F values at 30° incident angle. Conversely, fairly smaller at intermediate values of the fracture fluid acoustic properties with a 0° incident angle show frequency compressibility, particularly approaching zero when F is independence for varying F values. Based on the numerical around the critical value F , at which there is no fluid low at modeling results, we can draw a simple conclusion that wave all. Here, the dimensionless parameter F, which is associ- attenuation (or inverse quality factor) and velocity dispersion ated with the fluid compressibility, is defined as the ratio always coexist and vary with incident angles, indicating an of the bulk modulus of the fracture fluid ( K ) and the dry effect referred as frequency-dependent anisotropy. fc P-wave modulus of fracture medium (L ). We also find that Behaviors of the inverse quality factor and phase veloc- all curves in Fig. 2 exhibit similar features of trends and ity variations with the incident angle and frequency can be shapes for both cases of incident angle θ = 0° and θ = 30°. characterized ulteriorly through examining the frequency- When the fluid compressibility of fracture is relatively small dependent anisotropy attributes, whose concise and apparent 1 3 Incident angle, degree Incident angle, degree Incident angle, degree Incident angle, degree Phase velocity, km/s Phase velocity, km/s 1/Q 1/Q 762 Petroleum Science (2021) 18:758–772 2.9 (a) (b) 12 F = 0.001 F = 0.01 F = 0.1 2.7 F = 1 F = 10 F = 100 2.5 2.3 F = 0.001 F = 1 F = 0.01 F = 10 F = 0.1 F = 100 2.1 0 -1 0 2 3 3 4 5 -1 0 2 3 3 4 5 10 10 10 10 10 10 10 10 10 10 10 10 10 10 Frequency, Hz Frequency, Hz 1.35 1.5 (c) (d) F = 0.001 F = 0.01 F = 0.1 F = 1 1.30 1.0 F = 10 F = 100 F = 0.001 1.25 F = 0.01 0.5 F = 0.1 F = 1 F = 10 F = 100 1.20 0 -1 0 2 3 3 4 5 -1 0 2 3 3 4 5 10 10 10 10 10 10 10 10 10 10 10 10 10 10 Frequency, Hz Frequency, Hz Fig. 2 Illustrations of the frequency dependence of a, c phase velocity and b, d scaled inverse quality factor (100/Q) of P-wave (top) and SV- wave (base) for different values of parameter F that indicates the fracture fluid compressibility. The dashed lines indicate the calculated acoustic properties with the incident angle 30° and solid lines with the incident angle 0° expressions can be obtained by computing the frequency C − C 11 33 = . (11) dependence of Thomsen’s anisotropy parameters (Thomsen 2C 1986). As the elastic stiffness coefficients are complex val- Here, we refer the real and imaginary compartments of ued, the P-wave anisotropic parameters are also complex. parameter ε as the respective phase velocity and attenuation Accordingly, the Thomsen’s anisotropy parameter ε can be anisotropy parameters. For the different values of F , Fig. 3 computed using (e.g., Thomsen 1986; Kong et al. 2017). displays the frequency dependence characteristics of the real 0.15 (a) (b) 0.01 F =0.001 F =1 F =0.01 F =10 F =0.1 F =100 0.10 F =0.001 F =1 F =0.01 F =10 F =0.1 F =100 0.05 -0.01 -1 0 1 2 3 4 5 -1 0 1 2 3 4 5 10 10 10 10 10 10 10 10 10 10 10 10 10 10 Frequency, Hz Frequency, Hz Fig. 3 Illustrations of frequency dependence of the a real compartment of the complex anisotropy parameter ε and the b alternative attenuation anisotropy parameter ε for varying F values. In addition, note that different values of parameter F correspond to varied fracture fluid compress- ibility 1 3 Phase velocity, km/s Phase velocity, km/s Real Attenuation anisotropy 100/Q 100/Q Petroleum Science (2021) 18:758–772 763 and imaginary components for the complex parameter ε. In represents the P-wave incidence vector related to the elastic addition, we observe from Fig. 3 that parameters ε and ε properties of the incident medium, respectively, which can are separated into two distinct areas, corresponding to dif- be expressed by ferent F values. In Fig. 3a, we find that the pore pressure is , , −Z , −W i = i , (13) equilibrated between pores and fractures, when fractures are P P P P P 1 1 1 1 filled with a very compressible fluid (i.e., gas) and F > F . Moreover, an alternative way to evaluate the magnitudes of ⎛ ⎞ P S 1 1 wave attenuation is to use a dimensionless parameter ε = 1/ Q ⎜ ⎟ − − 00 P S Q –1/Q (Galvin and Gurevich 2015). We can observe ⎜ 1 1 ⎟ A = i , 33 11 (14) −Z −Z 00 ⎜ ⎟ P S 1 1 from Fig. 3b that parameter ε tends to zero in the low- and ⎜ ⎟ W W 00 P S high-frequency limits, and produces a peak magnitude in the ⎝ 1 1 ⎠ intermediate frequencies for varied F values. � � � � ⎛ ⎞ 00 exp −is h exp −is h P Z S S 2 P 2 2 2.2 Frequency‑dependent poroelastic seismic ⎜ � � � � ⎟ ⎜ ⎟ reflection coefficient 00 exp −is h exp −is h P Z S Z 2 P 2 S 2 2 ⎜ � � � � ⎟ A = i , ⎜ ⎟ 00 −Z exp −is h −Z exp −is h P Z S Z 2 P 2 S As illustrated in Figs. 1, 2 and 3, effective elastic stiff- ⎜ 2 2 ⎟ � � � � ⎜ ⎟ 00 −W exp −is h −W exp −is h ness tensors of the fluid-filled fractured porous sandstone P Z S Z ⎝ 2 P 2 S ⎠ 2 2 reservoirs show significant incident angle and frequency (15) dependence and, hence, will yield frequency- and angle- −1 B = T(0)T h , (16) j j dependent characteristics of seismic AVO responses. In a poroelastic medium with stratified structures, seismic reflec- where parameters W, Z, s , β and γ with the scripts of P and tions embody the complex combined impacts of dynamic S correspond to the quasi-longitude and shear waves, respec- information regarding the impedance contrast across the tively; the subscripts 1 and 2 represent the upper and lower interfaces, incident angles and interferences, as well as the media; β and γ denote the horizontal and vertical complex anisotropy, velocity dispersion and attenuation of the vis- polarization, respectively; h denotes the total layer thickness coelastic reservoir. Hence, it is necessary to introduce an of the interbedded model ( h ), with each of them having j=1 improved modeling scheme for calculating such complex a thickness h . In addition, note that the above parameters are seismic reflections using methods based on the conventional complex functions of incident wave frequency and slowness. single interface reflection model methods (e.g., Chapman In this case, the reflection and transmission coefficients are et al. 2006; Liu et al. 2011; Zhao et al. 2015). related to the elastic stiffness and incident angle, as well as We introduce a seamless procedure in this work that the incident wave frequency, layer thickness and stratified directly links rock physics modeling and seismic reflection structures of a reservoir. computations on the basis of an extended propagator matrix According to Eqs. 12 to 16, angle- and frequency-depend- method in Carcione (2001). The forward modeling algo- ent reflection coefficients are estimated for the fractured rithm can accurately allow in this improved workflow the porous reservoir model having an interbedded structure. frequency-dependent complex stiffnesses (anisotropic veloc- Sequentially, synthetic seismic waveforms can be yielded ity and attenuation) predicted from rock physics modeling via multiplying the complex reflection coefficients R and PP and yield angle- and frequency-dependent reflection coef- R with the spectrum of an incident wavelet and then imple- PS ficients and produce synthetic seismograms of a stratified menting inverse Fourier transform, model in the frequency and slowness domain. For a propa- gating P-wave with an incident angle θ, the corresponding it reflection and transmission coefficients vector R = [R , R , S = W()R (;)e d PP PS (17) PQ PG 2π T , T ] is written as (e.g., Carcione 2001). PP PS −∞ −1 where G denotes P- or SV-waves, and W(ω) represents the R =− A − B A i , (12) wavelet in frequency domain. 1 j 2 j=1 Here, on the basis of the proposed methodology of rock physics simulation and the propagator matrix approach, we where A and A represent the propagation matrices associ- 1 2 simulate numerically and assess the effects of fracture infills ated with the rock elastic properties in the upper and lower (in liquid and gas phases) on frequency-dependent seismic media, B (j = 1,..., N) represents the propagator matrix AVO signatures of PP- and PS-waves in a fractured porous for the stratified media consisting of N thin beds, and i 1 3 764 Petroleum Science (2021) 18:758–772 model having significant wave attenuation, velocity disper - Although the simulated results apparently demonstrate sion and anisotropic properties, correspondingly. how the seismic AVO behaviors can be affected by veloc - ity dispersion and attenuation of the reservoir layer, evalua- tion of the reservoir anisotropy influences on seismic AVO 3 Numerical modeling and analysis: seismic responses is also important via considering the fractured res- amplitude‑versus‑angle and frequency ervoir as an equivalent VTI medium. In Fig. 5, the PP- and responses PS-wave reflection coefficients of models 1 and 2 are illus- trated for the isotropic and anisotropic cases, respectively. 3.1 Seismic response sensitivity Due to medium anisotropy, anomalies of AVO response to frequency‑dependent anisotropy tend to be more obvious than their equivalent isotropic case. These implications indicate that the presence of rock anisot- We calculate the seismic responses for the models, in which ropy can produce substantial impact on seismic AVO behav- an attenuative and dispersive sandstone reservoir is either iors. In addition, the presence of layered structure results in overlaid by a shale half-space (model 1) or interbedded more significant AVO response anomalies. And thus, it is between two shale half-spaces (model 2), with only small necessary to consider rock anisotropy effects during seismic differences in rock elasticities across the interface. In this AVO analysis of fractured hydrocarbon-bearing reservoirs. study, we concentrate on assessing the impact of attenua- tive and dispersive elasticity anomalies of an effective VTI 3.2 Layer thickness effect on frequency‑dependent anisotropic medium on seismic reflection signatures, and AVO signatures thus the overburden shale is assumed to represent purely elastic and isotropic properties, whose P- and S-wave veloci- We then explore the effect of varied reservoir layer thickness ties are set to 3650 and 1830 m/s, respectively. Due to wave on the frequency-dependent seismic AVO response behav- attenuation and velocity dispersion of the P- and S-waves, iors. Figures 6 and 7 illustrate the frequency dependence of seismic reflection coefficients at the interface separating the computed reflection coefficients for the PP- and PS-waves, nondispersive shale and the dispersive reservoir layer will and the associated amplitude spectra and seismic waves for be a function of the frequency and incident angle. two cases of the interbedded reservoir with a layer thickness In Fig. 4, the magnitudes of the PP-wave reflection coef- of 10 and 35 m, correspondingly. Here, to simulate the real ficient as a function of frequency and incident angle are petrophysical properties of the fractured porous reservoirs, demonstrated, along with the AVO curves extracted for the we assume the sandstone medium exhibits significant attenu- low- and high-frequency limits. PP reflection coefficients ation and velocity dispersion versus each incident angle. from the model 1 are shown in Fig. 4a for a dispersive sand- In Figs. 6a, b, 7a, b, we find that the layer thickness has stone layer, and in Fig. 4c for the sandstone with anisotropic an important influence in causing the frequency-dependent velocity and attenuation predicted in Fig. 1. Within seismic seismic AVO response anomalies. For the case of the inter- exploration bands, we observe that seismic reflection coef- bedded sandstone with a layer thickness of 35 m, seismic ficients of two cases show similar dependence characteris- reflection coefficients exhibit obvious oscillation with vary - tics on the frequency. Meanwhile, the class III AVO curves ing frequencies for each incident angle, and the period of separate for the low- and high-frequency limits, while curves the oscillation increases with the increasing incident angle. of the anisotropic model coincide at lager incident angle as Meanwhile, we also obtain the reflection amplitude spec- displayed in Fig. 4b, d. The numerically modeling results tra, through multiplying frequency-dependent reflection also imply that the normal incidence reflection coefficients coefficients with the spectrum of a source Ricker wavelet in the low-frequency bands are larger than that of the high- with the peak frequency of 30 Hz. Additionally, we observe frequency band. Moreover, we observe in Fig. 4e, g that from Figs. 6 and 7 that the reflection amplitudes present considering the existence of stratified structure drastically marked behaviors of variations with incident angle for both alters frequency-dependent AVO signatures for the isotropic the interbedded reservoirs with the layer thicknesses of 10 m and anisotropic cases. We also note that the absolute value of and 35 m, respectively. It is also interesting to find that the the reflection coefficient tends to increase and then decrease energy of reflected waves from the reservoir of 35 m layered with increasing frequencies. This phenomenon indicates thickness is larger than that from the reservoir with a 10 m that the existence of the stratified structure may provide thickness, and reflections for the two models show two dif- potential interpretations for reservoir hydrocarbon identifi- ferent types of AVO behaviors. cations. Accordingly, seismic AVO responses calculated at Through applying the inverse Fourier transform for reflec- the low- and high-frequency limits show more complicated tion amplitude spectra, we obtain seismic waveforms of PP- variations. and PS-waves, respectively. Here, we can observe that the synthetic waveforms become more complex at large angle 1 3 Petroleum Science (2021) 18:758–772 765 (a) (b) -0.1 -0.10 -0.15 -0.2 -0.20 -0.25 15 Hz 40 Hz -0.3 01020304050 01020304050 Incident angle, degree Incident angle, degree (c) (d) -0.1 -0.15 -0.20 -0.2 -0.25 15 Hz 40 Hz -0.3 01020304050 01020304050 Incident angle, degree Incident angle, degree 0.4 (e) (f) 0.3 0.2 0.2 0.1 -0.1 15 Hz 40 Hz -0.2 01020304050 01020304050 Incident angle, degree Incident angle, degree 100 0.4 (g) (h) 0.3 0.2 0.2 0.1 -0.1 15 Hz 40 Hz -0.2 01020304050 01020304050 Incident angle, degree Incident angle, degree Fig. 4 Estimated (left column) PP-wave reflection coefficients and the (right column) corresponding AVO curves extracted for the low- and high-frequency limits are illustrated. a represents the dispersive case of a sandstone half-space, c represents anisotropic and dispersive sandstone half-space, e represents layered and dispersive sandstone with a thickness of 15 m, and g represents the corresponding model to e but consider- ing medium anisotropy effects. Colors: the blue lines correspond to the low-frequency limit of 15 Hz, while red lines correspond to the high- frequency limit of 40 Hz 1 3 Frequency, Hz Frequency, Hz Frequency, Hz Frequency, Hz Reflection coefficients Reflection coefficients Reflection coefficients Reflection coefficients 766 Petroleum Science (2021) 18:758–772 0.16 0.16 (a) (b) 0.12 0.12 0.08 0.08 0.04 0.04 0 0 01020304050 01020304050 Incident angle, degree Incident angle, degree Fig. 5 Variations of the reflection coefficient as a function of incident angle, corresponding to the (solid lines) isotropic and (hollow circles) ani- sotropic cases for the a anisotropic and dispersive sandstone half-space, and the b layered and anisotropic sandstone reservoir with a layer thick- ness of 15 m. In addition, the blue curves represent PP-wave and the red represent PS-wave of incident waves. It is interesting to observe from Fig. 7c sandwiched between two elastic isotropic half-spaces, we that reflections from the top interface of the thick reservoir concentrate on seismic response anomalies associated with layer show the typical class III AVO behaviors. Reflected frequency-dependent anisotropic dispersion and attenuation wavetrains at the bottom reservoir interface, nevertheless, of the reservoir layer, as well as the stratified structure. represent a combining contribution of the elastic imped- In Fig. 8a, b, the predicted amplitude spectra of PP ree fl c - ances, velocity dispersion and attenuation of the anisotropic tions, where apparent derivations existing in terms of band- reservoir, layer thickness and incident angle. These numeri- width, energy and amplitude variation with incident angel cal results reveal that it is inappropriate to treat the bottom are frequency, are illustrated for the models with fractures interface reflection as a typical AVO response. Moreover, filled by liquid and gas, respectively. Correspondingly, the for the thinner reservoir in Fig. 6c, the seismic reflections synthetic waveforms in Fig. 8c, d show differences in the embody the combined effects regarding velocity dispersion waveform and AVO behaviors. Nevertheless, the PS-wave and attenuation within the fractured reservoir, as well as reflected waveforms for the models with fractures filled by the tuning and interference reflections. In this case, thus, liquid and gas exhibit insignificant deviations, as shown in it would be more appropriate to study seismic responses Fig. 9. These modeling results imply that it can be challeng- of the fractured porous reservoir in terms of the integrated ing to accurately discriminate seismic reflection signature reflection waveforms rather than from the viewpoint of the alternations induced by fracture infill variations using the reflected waves at the top and bottom reservoir interfaces, PS-wave reflections solely. respectively. 3.4 Impacts of fracture weakness 3.3 Eec ff ts of fracture infill on frequency‑dependent seismic reflection on frequency‑dependent seismic reflection signature variations signatures According to previous studies that the magnitude of wave An effective fractured VTI medium that is saturated with attenuation and velocity dispersion relies significantly on two different fluids can cause varied frequency and angle the degree of fracturing of a porous medium (e.g., Chapman dependence values of the phase velocity and quality factor, et al. 2003; Brajanovski et al. 2005), and we consider in this hence resulting in distinct seismic AVO response anomalies. work the ee ff ct of fracture weakness of the porous sandstone To demonstrate the impact of fracture infills on frequency- on frequency-dependent seismic AVO signatures. dependent seismic reflection signatures, we calculate seis- In Fig. 10, the seismic amplitude spectra and synthetic mic reflection coefficients and the corresponding waveform waveforms of PS-wave reflections from the single-layer responses for a fractured porous sandstone with varied interbedded model, corresponding to different fracture fracture fluids (i.e., different F values here). Via assum- weaknesses, are demonstrated via assuming the fractured ing a stratified model in which the viscoelastic reservoir is porous sandstone reservoir with a 30 m thickness. We see 1 3 Reflection amplitude Reflection amplitude Petroleum Science (2021) 18:758–772 767 (a) (b) 0.4 0.3 0.2 0.3 0.1 0.2 0.1 -0.1 01020304050 01020304050 Incident angle, degree Incident angle, degree 150 0.5 (c) (d) 0.4 0.4 100 0.3 0.3 0.2 0.2 0.1 0.1 0 0 01020304050 01020304050 Incident angle, degree Incident angle, degree (e) Incident angle, degree (f) Incident angle, degree 01020304050 01020304050 0 0 50 50 100 100 Fig. 6 Illustrations of a, b the frequency-dependent reflection coefficients, c, d the amplitude spectra and e, f the associated seismograms of (left column) PP- and (right column) PS-waves, respectively. The interbedded poroelastic sandstone layer having a thickness of 10 m is saturated by two different fluids with corresponding medium anisotropy and velocity dispersion that apparent variations exist in energy, time and waveforms the PP-wave reflections in Fig. 11 exhibit obviously influ- between the two fracturing cases. Meanwhile, seismic ampli- ences of wave attenuation and velocity dispersion of the tudes of the spectra from the fractured layer with fracture fractured porous reservoir, particularly for the base reservoir weakness δ = 0.1 are larger than those from the sandstone reflections that are further complicated by waves traveling with δ = 0.01. Correspondingly, synthetic seismograms of through the dispersive and attenuative sandstone layer. In addition, the fracture tangential weakness δ levels varying 1 3 Travel-time, ms Frequency, Hz Frequency, Hz Travel-time, ms Frequency, Hz Frequency, Hz 768 Petroleum Science (2021) 18:758–772 150 150 (a) (b) 0.4 0.3 0.2 0.3 100 100 0.1 0.2 50 50 0.1 -0.1 01020304050 01020304050 Incident angle, degree Incident angle, degree 150 150 0.5 (c) (d) 0.4 0.4 100 0.3 100 0.3 0.2 0.2 50 50 0.1 0.1 0 0 010203040 50 01020304050 Incident angle, degree Incident angle, degree (e) Incident angle, degree (f) Incident angle, degree 01020304050 01020304050 0 0 50 50 100 100 Fig. 7 Illustrations of a, b the frequency-dependent reflection coefficients, c, d the amplitude spectra and e, f the associated seismograms for (left column) PP- and (right column) PS-waves for the model in Fig. 6, for which the porous sandstone reservoir has a layer thickness of 35 m from 0 and 1 have also an important impact on the predicted 4 Conclusion complex stiffness tensors, hence resulting in abnormal reflection coefficient variations. Sensitivity of fracture tan- We have introduced an improved seismic AVO modeling gential weakness to the acoustic properties will be examined method in this work to calculate frequency- and incident- in a further study. angle-dependent reflection coefficients. The numerically modeling method allows linking the propagator matrix algo- rithm in the frequency domain and the rock physics simula- tion that can predict frequency-dependent anisotropic elastic stiffness tensors. Based on the modified methodology, we obtain the frequency-dependent PP- and PS-wave reflection 1 3 Travel-time, ms Frequency, Hz Frequency, Hz Travel-time, ms Frequency, Hz Frequency, Hz Petroleum Science (2021) 18:758–772 769 150 0.4 150 0.4 (a) (b) 0.3 0.3 100 100 0.2 0.2 50 50 0.1 0.1 0 10 20 30 40 50 01020304050 Incident angle, degree Incident angle, degree (c) Incident angle, degree (d) Incident angle, degree 01020304050 01020304050 0 0 50 50 100 100 Fig. 8 Illustrations of the a, b PP-wave reflection amplitude spectra, and the c, d associated seismograms for the interbedded sandstone of 30 m layer thickness in Fig. 4g with fractures filled by liquid a, c and gas b, d, respectively coefficients and the corresponding synthetic seismograms from an interbedded model with a porous sandstone layer containing sets of aligned fractures. In particular, we explore seismic attenuation and velocity dispersion variations of the fractured media. We have also illustrated the sensitivity of the seismic AVO anomalies to rock elastic properties, ani- sotropy, velocity dispersion and attenuation, and the layered structure, which cannot be studied using either normal-inci- dent reflection technique or the conventional Zoeppritz equa- tion-based interface reflection model algorithm. Numerical results of this study also indicate the feasibility of the modi- Incident angle, degree fied frequency-dependent seismic AVO analysis approach for the detection of fracture infills, as well as the degree Fig. 9 Obtained seismograms of PS-wave reflections, corresponding of fracturing, in a stratified reservoir system. The extended to the layered sandstone model in Fig. 8. Here, the black trace corre- research may allow various kinds of effective media which sponds to fractures filled with liquid and the red trace corresponds to can be determined using varied rock physics theories in the gas-filled fractures, respectively 1 3 Travel-time, ms Travel-time, ms Frequency, Hz Travel-time, ms Frequency, Hz 770 Petroleum Science (2021) 18:758–772 150 150 0.5 0.5 (a) (b) 0.4 0.4 100 100 0.3 0.3 0.2 0.2 50 50 0.1 0.1 0 0 01020304050 01020304050 Incident angle, degree Incident angle, degree (c) Incident angle, degree (d) Incident angle, degree 01020304050 01020304050 0 0 50 50 100 100 Fig. 10 Illustrations of the a, b PS-wave reflection amplitude spectra and the c, d associated seismic waveforms obtained using a 30 Hz Ricker wavelet, for the interbedded sandstone reservoir model of 30 m layer thickness in Fig. 1g. The fractured layer is saturated with two different flu- ids and has dry fracture normal weakness of a, c δ = 0.01 and b, d δ = 0.1, respectively N N same model, such as to explore frequency-dependent seismic AVO signatures of a stratified system with more complex lithology and yield more accurate predictions of fracture fluids in a heterogeneously fractured reservoir. Acknowledgements This work was financially supported by the 50 Science Foundation of China University of Petroleum (Beijing) (2462020YXZZ008), the National Natural Science Foundation of China (41804104, 41930425, U19B6003-04-03, 41774143), the National Key R&D Program of China (2018YFA0702504), the Pet- roChina Innovation Foundation (2018D-5007-0303) and the Science Foundation of SINOPEC Key Laboratory of Geophysics (33550006- 20-ZC0699-0001). The authors are grateful to the four anonymous reviewers, for their constructive comments and suggestions. Incident angle, degree Open Access This article is licensed under a Creative Commons Attri- bution 4.0 International License, which permits use, sharing, adapta- tion, distribution and reproduction in any medium or format, as long Fig. 11 Illustration of synthetic seismic waveforms of PP-wave reflec- as you give appropriate credit to the original author(s) and the source, tions, corresponding to the interbedded sandstone reservoir model of provide a link to the Creative Commons licence, and indicate if changes 30 m layer thickness in Fig. 10. In addition, the fractured layer is sat- were made. The images or other third party material in this article are urated with two different fluids and has dry fracture normal weakness included in the article’s Creative Commons licence, unless indicated of (green) δ = 0.01 and (red) δ = 0.1, respectively N N otherwise in a credit line to the material. 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Petroleum Science – Springer Journals
Published: May 19, 2021
Keywords: Seismic anisotropy; Fractured media; Attenuation and dispersion; AVO responses; Frequency dependence
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