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With the development of 5D (3D + offset + azimuth) seismic technology, the stable acquisition of anisotropy information from wide-azimuth seismic data has become a key scientific problem in the seismic inversion of fractured reservoirs. The amplitude variation with incidence and azimuth (AVAZ) inversion method using wide-azimuth seismic data is an effective way to predict the anisotropic information of the subsurface medium. However, the conventional AVAZ inversion method suffers from too many parameters to be estimated, large variation in contribution, and inversion instability. Therefore, an AVAZ inversion method with coherence-attribute constraints is developed to solve the problem of unstable inversion of anisotropic parameters. First, we use seismic coherence attributes to build a fracture-probability-distribution model containing anisotropic information of the subsurface medium, which can be used to simulate large-scale subsurface fractures and faults. Then, it is added to the objective function as an anisotropic information constraint to improve the reliability and stability of the anisotropic inversion. Furthermore, an AVAZ inversion method in a Bayesian framework is implemented by using wide-azimuth seismic data. Gaussian distribution and a smoothing background model are added to the objective function to improve the reasonableness and stability of the inversion. In addition, we develop a stepwise optimization inversion method for isotropic and anisotropic parameters, prioritizing the inversion of parameters that contribute significantly to the reflection coefficient, and then using the results of the previous inversion as the initial values for the next inversion step to achieve multiparameter inversion. This method can reduce the number of the estimated parameters and thus improve the stability of the inversion of the anisotropic parameters. Field data examples indicate that this method produces suitable inversion results even at moderate levels of noise. Therefore, we can conclude that the proposed method has good applicability and stability in predicting the anisotropy parameters of fractured shale reservoirs.

The reservoir brittleness index can characterize the relative brittleness of shale oil and gas reservoirs, which provides guidance for hydraulic fracturing in the later stage of oil and gas exploration and development. It is a significant index to evaluate the sweet spot of oil and gas reservoirs. Most research on the brittleness index is about laboratory measurements and petrophysical quantitative analysis, but few studies directly predict the reservoir brittleness index from prestack seismic data. The Markov chain Monte Carlo (MCMC) probabilistic algorithm incorporating the delayed rejection adaptive Metropolis (DRAM) strategy has been developed to invert the reservoir brittleness index directly. The different brittleness index is analyzed and compared by the constructed rock-physics model of the shale gas exploration area to select the most sensitive one. Based on the chosen brittleness index, the approximate equation of the PP-wave reflection coefficient for the brittleness index is deduced, and the correctness and achievability of the modeling operator are indicated by theoretical analysis. To analyze the uncertainty of the brittleness index and enhance computational efficiency, an improved MCMC probabilistic inversion algorithm is introduced, which aims to optimize the sampling process of the general MCMC algorithm by delayed rejection and adaptive Metropolis strategies. Our study finds that the DRAM strategy improves the efficiency of the algorithm by nearly 30%. Synthetic examples and field seismic data verify the applicability and effectiveness of our inversion method.

Undercompaction and hydrocarbon generation are the main factors affecting pore pressure. The current seismic pore-pressure prediction (PPP) method is to obtain the overpressure trend by estimating the normal compaction trend (NCT) to predict the physical parameters during normal compaction and comparing the measured parameters. However, selecting a single parameter to indicate overpressure may cause insufficient consideration of factors, such as hydrocarbon generation. Because hydrocarbon generation requires specific temperature and other conditions, pore pressure is divided into two parts: undercompaction in the early stage and hydrocarbon generation after reaching the hydrocarbon generation threshold. A rock-physical model is established for estimating the NCT before hydrocarbon generation to modify the bulk modulus of the model and calculate the pressure generated by undercompaction; the pressure generated by hydrocarbon generation is added to obtain the final pore pressure. In the shale gas work area in the Sichuan Basin, the prediction results are more in line with the actual situation, and the petrophysical analysis indicates that the ratio of free hydrocarbon content and kerogen to water is the influencing factor indicating pore pressure. The practicality of the PPP formula considering hydrocarbon generation in oil and gas sweet spots is illustrated through an example in the research area.

Identification and prediction of high-quality source rocks is the key to obtaining new resources in the exploration area of Cenozoic basins in offshore China. We research the seismic response and distribution of hydrocarbon source rocks using seismic data, well curves, lithologic interpretation, and geochemical analysis. The target is the source rock development zone of the W Formation in a survey of the South China Sea. The results show that the seismic response of thick layer source rocks differ from surrounding rocks in the seismic profile (strong reflections with opposite polarity at the top and bottom and messy or chaotic reflections inside). Seismic reflections of interlayer source rocks have the characteristics of low frequency and continuous strong amplitude. The dominant frequency and maximum amplitude decrease as the number of mudstone layers increases. Through seismic petrophysical analysis, we have obtained three sensitive parameters of source rock in this survey: clay content, P-wave impedance, and elastic impedance. Different classification methods can realize the classification and prediction of hydrocarbon source rocks, among which the Kernel Fisher discriminant analysis method is the best. The prediction results are consistent with the geological background, geochemical information, and well curves.

The estimation of physical and anisotropic parameters is of great importance for the characterization of fractured reservoirs. Vertical fractures developing in a laterally isotropic (VTI) setting are equivalent to orthotropic anisotropic (OA) media common in stratified fractured shale reservoirs. To obtain independent anisotropic and physical information, a novel reflection coefficient approximation containing physical and anisotropic parameters is derived to improve the stability of the inversion for orthotropic media. To simplify the equation for the reflection coefficient, an approximate rock physics model is constructed using the approximate theory of rock modulus. The estimated parameters are reduced from nine to six. The accuracy analysis reveals that the new reflection coefficient is appropriate and suitable for inversion. In addition, a stepwise Bayesian amplitude versus angle and azimuth (AVAZ) inversion method with smooth background constraints is developed to estimate the anisotropic and physical parameters from the azimuthal seismic data. The smooth background constraint improves the robustness of the inversion. And the stepwise inversion strategy solves the problem that the contribution of the fracture parameter to the reflection coefficient is smaller than the other parameters. Synthetic cases show that the proposed stepwise Bayesian AVAZ inversion method is feasible in estimating the anisotropic parameters for OA media even when the signal-to-noise ratio is 2. The field cases show that the proposed inversion method has good stability and robustness in predicting shale reservoirs with vertical or near-vertical fractures.

Lithology discrimination plays an important role in characterizing deep-hydrocarbon reservoirs, particularly for tight sandstones with special petrophysical properties. Stable prediction of lithological-sensitive parameters for deep tight sandstones is a significant challenge. In this paper, a direct inversion method of elastic impedance is developed to estimate lithological-sensitive parameters from pre-stack seismic data to improve the stability of the inversion. Elastic and physical parameter models extracted from actual wells are used to analyze the influence of petrophysical parameters on amplitude variation with offset characteristics. Cross-plots and sensitivity analysis of elastic parameters illustrate that the elastic-sensitive parameter Poisson impedance (PI) can distinguish gas-bearing sandstone and abnormal limestone in deep tight-sandstone reservoirs. In addition, a pragmatic elastic impedance direct inversion under the framework of Bayesian theory is implemented for the lithology indicator PI from pre-stack seismic data. And the Cauchy regularization and low-frequency regularization constraints are used to construct the objective function for improving the robustness of inversion. Field data examples show that the inversion results are in good agreement with the well logging interpretation results, and validate the feasibility and stability of the proposed method in the estimation of inverted parameters. Finally, we can conclude that this method has great application potential in the lithology discrimination of deep tight-sandstone reservoirs.

SUMMARYThe stress-dependent wave reflection and transmission (R/T) coefficients for the layered transversely isotropic media with a vertical symmetry axis (VTI) are seldom investigated in the published literature. To fill this gap, we propose the exact formulas for the plane wave R/T coefficients on the welded and non-welded boundaries between two distinct VTI half-spaces under the effect of initial stress. The theory of acoustoelasticity is used to describe the influence of initial applied stress on the overall elastic properties of VTI media and to represent two different boundary conditions. The normal and tangential compliances are used to characterize the discontinuity of non-welded boundary based on the linear-slip model, and their stress dependences are ideally considered according to the effect of stress on fracture aperture. Then the plane-wave displacement equations are substituted into the boundary conditions to yield the analytic formulas for frequency-independent R/T coefficients for the welded interface and frequency-dependent R/T coefficients for the non-welded interface. The stress-dependent wave slowness vector and polarization vector embedded in R/T coefficients can be directly computed with the Christoffel equation given by the acoustoelastic equation. Modelling results graphically show the effects of initial stress on the angle-dependent wave velocities, Thomsen elastic anisotropy parameters, fracture compliances, the R/T coefficients and seismic reflection responses for welded and non-welded interfaces in detail. The R/T coefficients are more sensitive to initial stress at relatively large incidence angles for the designed two-layer model with welded or non-welded boundary. The proposed R/T coefficient formulas and modelling results are relevant to in situ stress detection, fracture characterization, and exploration for oil and gas in shale reservoirs in high-stress fields.

Joint inversion of seismic amplitude and frequency information for fluid discrimination works as a popular approach for reservoir fluid identification. However, ignorance of the factor caused by fluid flow and the viscosity of oil and gas in current frequency-dependent inversion approaches leads to the inaccuracy of oil and gas prediction. Therefore, based on squirt flow and viscoelastic theory, a new frequency-dependent viscoelastic squirt-flow fluid factor is established, which is compared with other conventional fluid factors to verify its advantages in fluid detection. In addition, a new linearized reflectivity equation related to frequency-dependent viscoelastic squirt-flow fluid factor is derived. Finally, to verify the feasibility of the new fluid factor in predicting oil/gas, a frequency-dependent elastic impedance inversion method is introduced. Synthetic data and field data examples find that frequency-dependent viscoelastic squirt-flow fluid factor can eliminate the interference of “false bright spots” in conventional fluid factors inversion and has a higher vertical resolution, which demonstrated the effectiveness and stability of our method in fluid prediction.

SUMMARYAmplitude variation with incident angle and azimuth (AVAZ) inversion is usually used for estimating subsurface medium properties from azimuthal stacked seismic data for fracture prediction. And weak anisotropy parameters play an important role in the fracture prediction of shale reservoirs. However, the low contribution of the anisotropic parameters to the reflection coefficient and too many condition numbers of forward solver lead to insufficient inversion stability of the anisotropic parameters. Therefore, a new azimuthal-amplitude-difference inversion method with the azimuth-velocity-anisotropic constraints is proposed to improve the stability of inversion and obtain a reasonable estimation of anisotropic parameters. The azimuthal amplitude difference AVAZ forward solver in the anisotropic medium is initially established from a rewritten approximate reflectivity equation and variable substitution. Sensitivity analysis of estimated parameters of the rewritten approximate reflectivity indicates that our approach is sufficiently suitable for seismic inversion of anisotropic parameters. Furthermore, an azimuthal amplitude difference Bayesian AVAZ inversion approach with azimuth-velocity-anisotropic constraint is developed to improve the stability of the inversion. We add Gaussian noise to the synthetic seismic records and assume the Cauchy distributions as a prior constraint on the model parameters. The anisotropic information obtained from the azimuth-velocity-anisotropic inversion is utilized as a constraint for improving the stability of inversion. Synthetic data tests reveal that the anisotropic parameters can be estimated stably even with moderate noise. Field data tests illustrate the feasibility and reliability of the presented inversion method for estimating anisotropic parameters in a shale reservoir with vertical or near-vertical fractures.

Geofluid identification from seismic data are crucial for understanding reservoir characteristics. However, fluid indicators based on elastic parameter combinations show strong ambiguity in terms of geofluid identification. Although the effective pore-fluid bulk modulus proves to be the superior fluid indicator in geofluid discrimination, it is limited to empirical models such as the critical porosity model. The consolidation parameter model can evaluate the consolidation and compaction of sediments and is widely used due to its better applicability. Therefore, a novel fluid indicator considering consolidation parameter is proposed and a boundary-constrained inversion strategy for geofluid identification is developed. First, the novel fluid indicator is defined based on the poroelasticity theory. Then, the linearized AVA (amplitude variation with angle) approximation equation related to the new fluid indicator is derived to directly link seismic data and properties of pore-filling materials. The model analysis verifies the accuracy of the derived approximation equation at moderate incident angles, which can be used for parameter prediction by pre-stack seismic inversion. Furthermore, the contribution of each attribute of the novel equation to the reflectivity is analyzed to validate the feasibility of new fluid indicator inversion. Finally, a boundary-constrained AVA pre-stack inversion method is presented to enhance the robustness of the inversion results of the model parameter. The reliability of the improved method is proved by accurate inversion results of synthetic seismic records. After determining the consolidation parameter using well-log data combined with BGT (Biot–Gassmann Theory), the field data further demonstrated that the proposed approach is accurate and effective.

The PP-wave reflection coefficient (i.e., the ratio of the amplitude of reflected P wave to the amplitude of incident P wave) has crucial practical applications in many field scenarios, such as seismic inversion and in-situ stress monitoring. However, the effect of biaxial stress, frequently encountered in the earth’s interior, on the PP-wave reflection coefficient is seldom studied. To fill this gap, we first formulate the P- and S-wave moduli and density in a biaxial-stress-induced anisotropic medium based on the acoustoelasticity theory. Then, the stress-dependent anisotropy parameters and normal/tangential rock weaknesses are proposed in the context of the stress-induced anisotropy model to obtain the corresponding effective elastic stiffness tensor and its spatial perturbation. Two stress-induced anisotropy factors (SIAF) are established to quantify the magnitude of anisotropy in a stress-induced anisotropic medium. The media with large SIAF tend to demonstrate strong anisotropy when stressed and tend to be more easily fractured from the perspective of engineering application. Furthermore, an approximate formula for the PP-wave reflection coefficient for biaxial-stress-induced anisotropic media is developed with the scattering theory. Numerical results illustrate how the reflection coefficients are affected by horizontal biaxial stress and demonstrate the feasibility and accuracy of our formula. Modeled and field inversion examples illustrate that SIAF can be reasonably inverted with our amplitude variation with incident angle and azimuth inversion method from azimuthal seismic data. Our results help to better understand the effect of biaxial stress on anisotropy and wave propagation in stress-induced anisotropic media.

SUMMARY It has been widely recognized that seismic attenuation and dispersion can be induced by various mechanisms. Wave induced fluid flow (WIFF) is one of the popular mechanisms to interpret seismic wave attenuation and dispersion, especially at microscopic scale. In this study, we focus on the local flow induced by different compliances of adjacent pores and develop a unified model called BISSQ (Biot-spherical squirt) model based upon the Biot and spherical squirt flow theories. The proposed model is an extension of BISQ (Biot-squirt) model. The P-wave velocity and attenuation of BISSQ model are influenced by two mechanisms: the Biot and spherical squirt-flow mechanisms. To evaluate the impact of these two mechanisms on seismic wave propagation, we use different parameters (e.g. squirt flow length, fluid viscosity, permeability, saturation and so on) to simulate wave attenuation and dispersion. We then apply the proposed model to Fox Hills Sandstone, which shows that the model can realistically predict P-wave velocity dispersion. To sum up, our proposed model offers an alternative squirt flow attenuation mechanism, which can describe fluid state of squirt flow reasonably and help us to assess the effect of different parameters on seismic wave attenuation.

Young’s modulus and Poisson ratio are essential parameters for rock brittleness evaluation. At present, prestack inversions of these parameters mostly adopt linear approximation, whereas the nonlinear inversion methods that are based on the exact Zoeppritz equation are seldomly conducted. Nevertheless, in the case of large incident angles and high-impedance contrast, the nonlinear equation has higher accuracy than the linear approximation. It is more in line with the nature of geophysical inversion problems, and it describes the variation of formations’ elastic parameters better than the linear approximation does. Therefore, based on the exact solution of the Zoeppritz equation, we have parameterized and derived a new nonlinear reflection coefficient equation using nonlinear Young’s modulus, Poisson ratio, and density (YPD) as variables. Subsequently, by combining the inverse operator algorithm with the YPD equation, we develop a new nonlinear amplitude-variation-with-offset inversion scheme. This approach can robustly extract Young’s modulus and Poisson ratio from seismic data and identify shale-gas sweet spot locations within the reservoir, as well as predict reservoir fluid properties. We have tested the accuracy and applicability of this approach with synthetic and real data examples.

The estimation of P- to S-wave velocity ratio (PSR) in the subsurface has many applications in gas-bearing reservoir prospecting, lithology discrimination, and anomalous pore-pressure prediction. Conventionally, it is estimated with the P- and S-wave velocities/moduli/impedances that are directly obtained from prestack seismic data using the existing reflection coefficient equation (e.g., the Aki-Richards approximation). However, this indirect inversion method creates cumulative errors for the estimated PSR results. To eliminate cumulative errors, we first develop a novel generalized elastic impedance that has more explicit physical meaning compared to conventional elastic impedance. Then, we derive a linear P-wave reflection coefficient equation in terms of the PSR, P-wave velocity, and density under the assumption of weak contrast. Furthermore, a robust amplitude-variation-with-offset inversion method is constructed with the proposed reflection coefficient equation in the Bayesian framework. Cauchy and Gaussian probability distributions are used as the prior probability distribution of the model parameter and likelihood function, respectively, to predict the maximum posterior probability solution for PSR, P velocity, and density. Synthetic and field examples illustrate that the proposed direct inversion method performs with higher accuracy compared with the indirect method that inverts for the P- and S-wave velocities and also illustrate the feasibility of our method for inverting the three parameters even with strong noise. Our inverted results conform to the drilling results, which validate the robustness of the proposed direct inversion method.

SUMMARY The elastic properties of the subsurface, such as density, P-velocity and S-velocity, can be estimated in pre-stack seismic inversion. In recent research, the deep neural network is widely used in pre-stack seismic inversion for its strong non-linear fitting and feature extraction ability. However, the label data is generally inadequate due to high drilling costs and strong data-sharing barriers in the field of exploration geophysics. In order to reduce the dependence of network performance on label data and ensure the accuracy of inversion mostly, semi-supervised learning is adopted. Here, we develop a cooperative multinetworks semi-supervised pre-stack seismic inversion method. In the cooperative multinetworks inversion framework, the inversion network, mapping network, and modification network are adopted to complete the inversion task cooperatively. A forward network is constructed to automatically generate seismic data from density, P-velocity and S-velocity, which can assist the above networks to complete semi-supervised learning. Compared with some published deep learning pre-stack inversion methods, the spatio-temporal correlation of data can be fully mined, the prior geological structure and low-frequency information can be utilized effectively, and reflectivity is adopted as an intermediate output parameter to improve the robustness of the method. The experiments on the Marmousi2 model demonstrate that cooperative multinetworks semi-supervised inversion strategy is superior to conventional semi-supervised inversion methods in both inversion accuracy and antinoise performance. In addition, the susceptibility experiments of the initial model indicate that the proposed method can maintain a high inversion accuracy with little effective information in the initial model. Finally, the proposed method is successfully applied to the field data and obtains a high-resolution inversion result.

Pre-stack seismic inversion based on sensitive elastic parameters is critical in reservoir lithology prediction and geofluid identification. The ability of a single elastic attribute to identify a reservoir depends on its sufficient sensitivity to distinguish the target reservoir from the surrounding sediments. In general, high-dimensional data space composed of multiple elastic attributes is more conducive to describing reservoir characteristics. Therefore, a multiple elastic attribute fusion method using affinity propagation clustering strategy for gas hydrate reservoir identification is proposed. Rock-physics modeling is the most effective tool to determine the influence of microscopic physical parameters on macroscopic elastic response and to quantitatively evaluate the sensitivity of elastic attributes. Consequently, a rock-physics model of hydrate sediments considering the non-negligible shear properties of hydrates is constructed to clarify reservoir-sensitive elastic parameters. Additionally, a clustering evaluation indicator is defined to determine the optimal data clustering dimension in terms of feasibility and economy of the proposed method, and to avoid bias in the results due to data redundancy. It is shown that the 3D elastic attribute space consisting of shear modulus, Young's modulus, and S-wave velocity has the best discrimination ability for hydrate reservoirs. The logging data are used to verify the feasibility and effectiveness of the proposed method. Finally, the hydrate reservoir development is accurately discriminated by using the multiple elastic attributes yield from the pre-stack seismic inversion and combined with the fusion strategy.

Compared with the plane-wave reflection coefficient, the spherical-wave reflection coefficient (SRC) can more accurately describe the reflected wavefield excited by a point source, especially in the case of low seismic frequency and short travel distance. However, unlike the widely used plane-wave amplitude-variation-with-offset/frequency (AVO/AVF) inversion, the practical application of spherical-wave AVO/AVF inversion in multilayer elastic media is still in the exploratory stage. One of the difficulties is how to fully use the amplitude and phase information of the complex-valued SRC and the spherical-wave response property of each frequency component to obtain the spherical-wave synthetic seismogram in multilayer elastic media. In view of this, we have developed a complex convolution model considering the amplitude and phase information of an SRC to obtain the complex synthetic seismogram of a certain frequency component. A simple harmonic superposition method is further developed. By superposing the complex synthetic seismograms of different frequency components, the synthetic seismogram of the full-frequency band can be obtained. In addition, a novel three-parameter SRC in terms of P- and S-wave moduli and density is derived. Based on the SRC and complex seismic traces with different offsets (or incidence angles) and frequency components, an inversion approach of complex spherical-wave amplitude and phase variation with offset and frequency is proposed. A noisy synthetic data example verifies the robustness of our complex spherical-wave inversion approach. Field data examples indicate that the P- and S-wave moduli estimated by the complex spherical-wave inversion approach can reasonably match the filtered well-logging data. Considering spherical waves rather than plane waves can improve the accuracy of seismic inversion results.

The delineation of shale oil sweet spots is a crucial step in the exploration of shale oil reservoirs. A single attribute such as total organic carbon (TOC) is conventionally used to evaluate the sweet spots of shale oil. This study proposes a probabilistic Fisher discriminant approach for estimating shale oil sweet spots, in which the probabilistic method and Gaussian mixture model are incorporated. Statistical features of shale oil facies are obtained based on the well log interpretation of the samples. Several key parameters of shale oil are projected to data sets with low dimensions in each shale oil facies. Furthermore, the posterior distribution of different shale oil facies is built based on the classification of each shale oil facies. Various key physical parameters of shale oil facies are inversed by the Bayesian method, and important elastic properties are extracted from the elastic impedance inversion (EVA-DSVD method). The method proposed in this paper has been successfully used to delineate the sweet spots of shale oil reservoirs with multiple attributes from the real pre-stack seismic data sets and is validated by the well log data.

SUMMARY A detailed knowledge of wave reflectivity parametrization for non-welded interface plays an important role in fracture characterization and reservoir prediction. However, the effect of in situ stress on reflectivity formulae for non-welded contact interface remains unclear. Here based on the linear-slip theory and acoustoelasticity theory, we propose the accurate P-wave reflectivity and transmissivity formulae for planar P wave incident upon the non-welded interface between two elastic half-spaces under the in situ vertical stress. The discontinuous boundary conditions coupled with vertical stress is used to represent the non-welded contact interface for the elastic and viscous cases. Then accurate formulae are derived by solving the boundary conditions. Numerical results show that the proposed formulae are frequency-dependent and the non-welded interface performs as high- and low-pass filters for reflected and transmittedP wave at low stress, respectively. Moreover, initial vertical stress can change the maximum amplitudes of reflection and transmission coefficients and the corresponding critical angles. Particularly, vertical stress weakens the fracture-induced anomalous seismic reflections in zones lack of significant impedance contrasts at low incidence angles. The sensitivity analysis suggests a non-welded interface transforms into a welded interface at normal compliance ${\Delta _N} \le 5 \times {10^{ - 11}}$ m Pa−1. The phases and amplitudes of the four coefficients reach the maximum values at the specific viscosity $\eta \approx {10^6}$ Pa·s m−1 for the case of P wave obliquely incident upon the viscous interface at incidence angle 30°. Finally, we describe a method for simulating seismic reflection at the interface of two elastic media with experimental measurement.

The estimation of petrophysical and fluid-filling properties of subsurface reservoirs from seismic data is a crucial component of reservoir characterization. Seismic amplitude-variation-with-offset (AVO) inversion driven by rock physics is an effective approach to characterize reservoir properties. In general, PP-wave reflection coefficients, elastic moduli, and petrophysical parameters are nonlinearly coupled, especially in multiple-type pore-space reservoirs, which makes seismic AVO petrophysics inversion ill-posed. We have developed a new approach that combines Biot-Gassmann’s poroelasticity theory with Russell’s linear AVO approximation, to estimate the reservoir properties including elastic moduli and petrophysical parameters based on multitrace probabilistic AVO inversion algorithm. We first derive a novel PP-wave reflection coefficient formulation in terms of porosity, stiff-pore volume fraction, rock-matrix shear modulus, and fluid bulk modulus to incorporate the effect of pore structures on elastic moduli by considering the soft and stiff pores with different aspect ratios in sandstone reservoirs. Through the analysis of the four types of PP-wave reflection coefficients, the approximation accuracy and inversion feasibility of the derived formulation are verified. Our stochastic inversion method aims to predict the posterior probability density function in a Bayesian setting according to a prior Laplace distribution with vertical correlation and prior Gaussian distribution with lateral correlation of model parameters. A Metropolis-Hastings stochastic sampling algorithm with multiple Markov chains is developed to simulate the posterior models of porosity, stiff-pore volume fraction, rock-matrix shear modulus, and fluid bulk modulus from seismic AVO gathers. The applicability and validity of our inversion method is illustrated with synthetic examples and a real data application.

A Gaussian mixture Hamiltonian Monte Carlo (HMC) Bayesian method has been developed for the inversion of petrophysical parameters such as pyrolysis parameter S1, which is driven by a statistical shale rock-physics model. Pyrolysis parameter S1 can be used to indicate the content of free or adsorbed hydrocarbons in source rock, and it is an important indicator to evaluate the production of shale oil reservoirs. However, most studies on pyrolysis parameters are based on pyrolysis experiments and there is no relevant study to inverse pyrolysis parameter S1 from seismic data. In addition, compared to the total organic carbon content, pyrolysis S1 is more accurate for evaluating gas and oil in shale. In particular, high values of pyrolysis S1 can directly indicate the content of shale oil. We have developed a strategy for assessing shale oil sweet spots through estimating pyrolysis S1 and other petrophysical parameters. Based on the Gaussian mixture assumptions for the prior distribution of the model, we build a joint distribution to link the pyrolysis parameter S1 with elastic attributes, and then we derive a formulation to inverse S1 with the Bayesian model. Due to the components of the Gaussian mixture, the HMC method has been used to sample the posterior distribution. Our study finds that the HMC method for sampling can improve the efficiency and allow a more robust quantification of the uncertainty; also, application to real seismic data sets indicates that the delineation of sweet spots is more accurate combined with pyrolysis S1.