The azimuth of fractures has long been a subject of interest for geophysicists, and it holds paramount importance in the exploration and development of oil and gas resources. However, traditional fracture azimuth prediction methods heavily rely on seismic data quality and well-logging data, often encountering severe noise interference and 90° ambiguity. This makes fracture azimuth prediction challenging in areas with complex geological structures. A method for seismic statistical prediction of fracture azimuth based on the Fourier series has been proposed to address these issues. Firstly, the Ruger approximation is rewritten into Fourier series form, combining parameters with high linear correlation to mitigate the ill-conditioning of the coefficient matrix. Secondly, construct a complex representation of fracture azimuth and initially adjust the sign based on the characteristic that the azimuthal period of the fourth-order Fourier coefficient is π/2. Thirdly, considering that the fourth-order Fourier coefficients are susceptible to noise, a directional statistical method is introduced to enhance the stability of fracture azimuth prediction. Then, by analyzing the relationship between second-and fourth-order Fourier coefficients under saturated fluid and gas-filled conditions, the Welch t-test, suitable for data with non-homogeneous variance, is introduced to eliminate the influence of fluid type on fracture azimuth prediction. Numerical experiments and field data demonstrate that the proposed method overcomes the 90° ambiguity inherent in conventional fracture azimuth prediction, proving its stability and effectiveness in areas with severe structural variations.

Elastic parameters such as compressional-wave velocity and shear-wave velocity are essential for characterizing and predicting oil-gas reservoirs. However, the current commonly used shear-wave velocity prediction methods have problems such as weak generalization of empirical formulas and difficulty in obtaining some rock parameters in various rock physics models. We proposed a deep learning network based on the self-attention mechanism to predict shear-wave velocity. First, we need to extract the spatial and temporal features of well logging data using convolutional neural network (CNN) and gated recurrent unit (GRU), respectively. However, the spatial and temporal features exhibit different correlations in the depth direction due to the gradual variation of sedimentary layers. Thus, we fuse the self-attention mechanism with the deep learning network to enhance the network’s sensitivity to crucial spatiotemporal features. Finally, we take the tight sandstone reservoir of Tarim Basin as the research object to estimate shear-wave velocity using CNN, GRU network, and our optimized method. The results show that the CNN-GRU fusion network based on the self-attention mechanism network we proposed is better than the other two networks in the prediction accuracy and generalization degree.

Seismic amplitude variation with offset/angle and azimuth (AVOZ/AVAZ) inversion driven by rock physics is an important tool for the estimation of fractured reservoir properties. Generally, the azimuthal AVO reflection coefficients and fractured reservoir properties are nonlinearly coupled, resulting in strong ill-posedness in azimuthal AVO inversion. We propose an azimuthal AVO probabilistic nonlinear inversion methodology based on a generalized Gaussian mixture model and an improved adaptive-particle-swarm Metropolis-Hastings (APS-MH) algorithm to estimate fractured reservoir properties and lithofacies for fractured reservoirs with a group of vertical fractures. We first derive a novel nonlinear azimuthal AVO reflection coefficient related to the rock-matrix shear modulus, porosity, fracture density and water saturation. The approximation accuracy and inversion feasibility of the derived formulation are verified by a series of theoretical numerical analyses of the stiffness coefficient matrix and azimuthal AVO reflection coefficients. With a generalized Gaussian mixture probability model (GGMM) and the hierarchical Bayesian inference, we establish a nonlinear azimuth-AVO objective function coupling the irregular shape a priori of fractured reservoir properties and the lithology a priori. Combining the global search capability of the adaptive-particle-swarm optimization and the local sampling characteristics of Metropolis-Hastings, a probabilistic azimuthal AVO nonlinear inversion algorithm is developed, which improves the convergence speed and inversion accuracy of reservoir properties. Besides, the methodology simulates the a posteriori probability distribution of rock matrix shear modulus, porosity, fracture density, saturation and lithofacies effectively, which also evaluates the inversion uncertainty of reservoir properties simultaneously. The stability and applicability of the methodology is illustrated by theoretical numerical tests and real seismic data application.

The spatial structure of subsurface media usually exhibits anisotropic characteristics due to geological factors such as depositional environment, tectonic movement and overlying pressure, and the spatial structure characteristics are different at different spatial points. The traditional Kriging interpolation method based on global spatial range is difficult to accurately describe the anisotropic spatial structure characteristics of each spatial point in the subsurface medium, and the obtained Kriging interpolation results have larger errors. In this paper, we estimate the statistical characteristic parameters of random media from seismic data, which correspond to the anisotropic spatial structure characteristics of each spatial point of the subsurface medium, so as to accurately describe the anisotropic spatial structure characteristics of each spatial point of the subsurface medium. Then the spatial correlation length of each spatial point of the underground medium in different directions is calculated from the statistical characteristic parameters of the random media, and then the anisotropic spatial covariance function is calculated, furthermore the anisotropic Kriging weight coefficients are obtained, and finally the final anisotropic spatial structure Kriging interpolation results are obtained. Compared with the traditional Kriging interpolation method, the Kriging interpolation results obtained by the method in this paper are more reliable, better match the actual anisotropic spatial structure characteristics of the subsurface medium, with smaller errors and less uncertainty. Model tests and field data application demonstrates the effect of this method and verify its feasibility and practicality.

Rocks with a set of vertical fractures embedded in a vertical transverse isotropic (VTI) background can be regarded as an effective long-wavelength orthorhombic (ORT) medium, common in fractured reservoirs. Seismic inversion using wide-azimuth data is an effective tool for fracture detection. However, azimuthal seismic inversion in ORT medium is inherently ill-posed and uncertain, largely due to the considerable number of parameters involved in the reflection coefficients and the relatively minor contribution of fracture parameters to the reflection coefficients. Therefore, in this article, a modified PP-wave reflection coefficient equation was derived through parameter combination, comprising only five model parameters and exhibiting near-equivalent accuracy to the existing equation. Furthermore, each parameter in the equation has a clear physical meaning. These include attribute A (P-wave impedance), attribute B (anisotropic shear modulus), attribute C (horizontal P-wave phase velocity), normal fracture weakness, and tangential fracture weakness. Subsequently, an azimuthal seismic inversion method was developed to estimate the above five model parameters based on Bayesian inference, combined with Cauchy prior information and low-frequency regularization constraints. The method was validated through synthetic tests, which demonstrated its efficacy and feasibility even with moderate noise. Field data also illustrated the stability and effectiveness of the method. Finally, based on field data, the interpretation capability of attribute parameters was discussed. The modified reflection coefficient equation and the inversion method proposed in this article serve to enhance the inversion robustness of the ORT medium, which in turn guides the prediction of the orthogonally fractured reservoir using azimuth seismic data.

Accurate description and characterization of faults are essential in the exploration and development of oil and gas reservoirs. The multi-azimuth seismic coherences are effective edge-detection attributes, which can quantitatively characterize fault structures at different observation azimuths by quantifying the multi-azimuth similarities of neighborhood seismic traces. The multi-azimuth discontinuous features can be integrated to interpret the fault structures more comprehensively using the multi-azimuth seismic coherences fusion algorithm. However, the accuracy degradation problem generally exists in current coherences fusion algorithms because of the original information loss in fusion and the introduced calculation errors in decomposition and reconstruction. Here, we propose a multi-azimuth seismic coherence fusion method based on azimuth focus detection, which can search the focus regions in multi-azimuth coherences and apply it as a decision map to guide the fusion of multi-azimuth coherences. Compared with previous works, the original information loss problem in the fusion process can be alleviated, and the resolution and discontinuity characteristics of fault structures in the fused coherence can be further enhanced by fusing the information at advantageous observation azimuth. Finally, the field data application indicates that the proposed method is superior to the weighted-average-based and principal-component-analysis (PCA)-based fusion methods in terms of the resolution, the retention of the discontinuity characteristics, and consistency with the well-logging interpretation.

Seismic reflection coefficient equation for the fluid-saturated porous reservoirs under the effect of in situ stress is of great importance to broad fields such as geofluid discrimination, in situ stress prediction, and safe production. However, the stress effect on seismic reflection coefficient in porous reservoirs is poorly understood. To fill this knowledge gap, an approximate seismic reflection coefficient equation for fluid-saturated porous reservoirs under horizontal stress was proposed to model the natural effect of horizontal stress on seismic reflection response. We first revisited the acoustoelasticity (AE) theory and used it to characterize the impact of horizontal stress on skeleton anisotropy. Then, the effective elastic stiffness tensor and the corresponding spatial perturbation in the stressed fluid-saturated porous reservoirs were established under the assumption of fluid incompressibility, which were further employed to derive the approximate seismic reflection coefficient equation based on the elastic inverse scattering theory. By comparing our equation to the exact one, we confirmed its validity within the moderate incidence angles and stresses. The effects of horizontal stress on the P-wave amplitude variation with angle and azimuth (AVAZ) characteristics and seismic response were thoroughly investigated. It was shown that the horizontal stress significantly influenced the amplitude magnitude and seismic phases. Furthermore, the derived reflection coefficient equation was inserted into the Bayesian inversion scheme to estimate the geofluid indicator and other elastic parameters. Synthetic test and filed application showed a reasonable agreement between the inverted result and drilling data, which illustrated the feasibility and stability of our inversion method.

The vertically transverse isotropic (VTI) reservoirs subjected to horizontal in situ stress are frequently encountered in the subsurface, which can be approximately treated as an orthorhombic medium in the framework of acoustoelasticity. However, the seismic estimation for anisotropy parameters in such stress-induced reservoirs is still poorly studied. To address this issue, we derive a linearized reflection coefficient equation in stress-induced orthorhombic media by means of the theories of acoustoelasticity and elastic inverse scattering. The acoustoelasticity theory is utilized to characterize the effective elastic stiffness tensor in a stress-induced orthorhombic medium. The introduction of two dimensionless stress-induced anisotropy (SIA) parameters eliminates the need for third-order elastic constants (3oECs). Then, the stiffness perturbation is presented under the weak-anisotropy hypothesis and is substituted into the scattering function to derive the linearized reflection coefficient equation in stress-induced orthorhombic media. The feasibility of our reflection coefficient equation within the range of moderate stress (or moderate SIA) is confirmed by comparing it to the exact solution. Incorporating the wavelet effect, our reflection coefficient equation as a forward operator is utilized to establish a step-wise Bayesian inversion approach to estimate the anisotropy parameters. Specifically, the SIA parameters are inverted from the amplitude differences in seismic data at different azimuths in the first step. Next, the obtained parameters as the prior dataset are input into the second-step procedure to predict the VTI parameters with partial angle-stacked seismic data. Synthetic and field tests illustrate the robustness and effectiveness of our approach.

Accurate and stable identification of oil and gas reservoirs based on seismic data can effectively improve exploration success rates, enhance production efficiency, and reduce exploration and development costs. Limited by uncertainties in seismic data and inadequate label samples, problems of overfitting and instability generally exist in current deep-learning reservoir discrimination studies. A semi-supervised physics-informed workflow for reservoir discrimination is herein proposed. The approach synthesizes rock physics theory, elastic forward modeling, prior geological information, and deep learning algorithms. Furthermore, to establish a connection between seismic data and reservoir types, a geofluid parameter is employed, selected for its sensitivity to oil and gas reservoirs and its reliable extraction from seismic data. Accordingly, the reservoir classification network, geofluid inversion network, and elastic forward network are designed to complete the reservoir prediction cooperatively with a task-decomposed strategy. Finally, the established networks are optimized based on the constructed “seismic-geofluid-reservoir” training dataset with the proposed multi-step cooperative semi-supervised training strategy, which can improve the learning ability of the model by capturing explicit physics knowledge from labeled data, mining implicit knowledge from massive unlabeled data, and incorporating geophysics domain knowledge simultaneously. The proposed reservoir discrimination workflow is successfully applied to a field survey. The precision, recall, and f1-score of the predicted gas reservoirs can reach about 55%, 84%, and 67%.

The transversely isotropic media with a vertical symmetry (VTI) has been widely used in shale reservoirs, and the reflection response and prestack seismic inversion have been of interest to geophysicists. Petrophysical experiments have shown that shale reservoirs may exhibit moderate or even stronger anisotropy, causing limitations in the weak anisotropy-based reflectivity approximation. Existing weak-contrast approximations are usually highly nonlinear equations characterized by the stiffness parameters and slowness, which is not conducive to AVO analysis and prestack seismic inversion of anisotropic parameters. We proposed a novel approach for weak-contrast PP-reflectivity based on the P- and S-elastic impedance matrices and applied it to prestack seismic inversion. First, by simplifying the phase velocities and polarization vectors, the elastic impedance matrices are linearly characterized in the weak-contrast half-spaces. Then, the perturbation theory is used to derive the weak-contrast PP-reflectivity, which consists of two terms, one directly resulting from the difference between the parameters of upper and lower media, and the other resulting from the first-order perturbation of the phase velocity and polarization. It has a clear physical meaning and can degenerate to the Aki-approximation in isotropic media. In particular, the isotropic second-order terms are introduced to significantly improve its accuracy at the interfaces with high-speed contrast. Furthermore, we designed a coefficient matrix to linearize the high-order reflectivity terms and developed an improved algorithm based on the alternating direction method of multipliers (ADMMs). Finally, the applications on logging curves and field seismic data for the shale prove the feasibility and effectiveness of the proposed method.

Shale is a critical hydrocarbon reservoir generally considered a typical transverse isotropy with a vertical axis of symmetry vertical transverse isotropic (VTI) media. Petrophysical experiments indicated that shales may exhibit moderate or strong anisotropy, exacerbating uncertainties in seismic reflection analysis and pre-stack seismic inversion. In addition, the significant difference in contribution between the elastic and anisotropic parameters to the reflection coefficients limits the stability of anisotropy predictions. A quadratic reflectivity-based joint PP and PS inversion for VTI media was proposed, which indirectly predicts the anisotropy using the stiffness matrix. First, based on the elastic impedance matrix, we derived the weak-contrast PP and PS reflectivities characterized by the stiffness matrix. The contribution analysis indicates that the PP wave is sensitive to the stiffness matrix C11, and the PS wave is sensitive to C13, which implies that the joint PP and PS reflection was more effective in analyzing the effects of anisotropic parameters. Then, to improve the accuracy of the reflectivity at the interface with high-contrast velocity, quadratic PP and PS reflectivities for isotropic media were derived. Tests on different typical models showed that the reflectivity with quadratic terms performed well in accuracy, even for strong anisotropy. Finally, a joint inversion constrained by the nonlinear relationship between the stiffness matrix and the anisotropic parameters was developed, which can overcome the ill-posedness of indirectly calculating the anisotropic parameters using the stiffness matrix. The model and field seismic data applications prove the feasibility and effectiveness of the proposed method.

Reservoir lithology identification is critical to reservoir characterization, reserves calculation, and geological modeling. The deep learning lithology identification method is a data-driven algorithm for establishing the relationship between lithology-sensitive properties and litho-types from a large amount of observed data. The lithology label is inadequate for high drilling and core recovery costs. Consequently, we propose a reservoir lithology identification method based on improved adversarial learning to relieve the overfitting problem and the multi-solution problem caused by inadequate labeled data and massive learnable parameters in training. Firstly, a probabilistic lithology classification neural network (PLCNN) is constructed to predict lithology from density, P-velocity, and S-velocity. In addition, we design an improved adversarial learning (IAL) lithology identification workflow to train the PLCNN with limited labeled data and large-scale unlabeled data. In the workflow, a lightweight discrimination network is established to ensure that the prediction result of the PLCNN is consistent with the data distribution characteristics of real underground lithology. Finally, the proposed method is successfully applied to the Book cliffs model. Compared with the conventional supervised learning workflow, the misclassification of sand and sandy shale can be relieved efficiently with the IAL workflow, and the classification accuracy can be improved to 92.71%.

Matching pursuit (MP) is an algorithm which can reconstruct signal accurately, and is widely used in signal processing. However, MP algorithm has efficiency problem in processing large amount of data like five-dimensional (5D) seismic data. Generalized orthogonal matching pursuit with singular value decomposition (SVD_GOMP) is an algorithm which can improve the calculation efficiency a lot, and keeps the advantage of high accuracy. In this study, a redundant atom dictionary includes incident angles and azimuth is built. Then the five-dimensional seismic data is reconstructed efficiently and accurately by the generalize orthogonal matching pursuit with singular value decomposition algorithm. Compared with traditional matching pursuit method, the proposed method decomposes the five-dimensional seismic data at the same time and recovers the angle information efficiently. The reconstructed results of synthetic and field data examples are utilized to demonstrate the feasibility, computational efficiency and precision of the proposed method.

The hydrocarbon-bearing shale reservoirs can be approximately modeled as the transversely isotropic media with the vertical symmetry axis (VTI), usually exhibiting strong anisotropy and significant contrast among the reservoirs. Therefore, the effects of strong anisotropy should be considered when conducting the seismic interpretation for this type of reservoir through the AVO (amplitude variation with offset) technique. However, the existing approximately linear PP-wave reflection coefficient equations derived based on the assumption of weak anisotropy and weak contrast in elastic parameters will create considerable errors in the case of strong anisotropy and contrast. Quadratic approximation has also been proposed in the case of strong contrast in elastic parameters. To overcome the problem of strong anisotropy and contrast in AVO inversion, we propose a novel PP-wave reflection coefficient equation (expressed by Lamé constants) in terms of five reflectivity parameters (i.e., the ratio of the difference to the mean value between the parameters of two half-spaces) based on the quasi-Zoeppritz equation for the VTI media. Combining the novel reflection coefficient equation and Bayesian inversion framework, we constructed the nonlinear inversion objective function. We chose the multivariate Gaussian distribution as the prior function to reduce the inversion multiplicity. The MCMC sampling algorithm was adopted to estimate the five reflectivity parameters for the nonlinear inversion problem.

Periodic horizontal thin interlayers, or formations developing horizontally layered fractures in a homogeneous background, are commonly studied as a transverse isotropy with a vertical axis of symmetry (VTI) media. Anisotropy needs to be considered in predicting such special reservoirs, and can be inferred from the amplitude variation with offset (AVO) by the exact or linear reflectivity equations. However, the linear approximation is usually derived based on the assumptions of weak anisotropy and weak contrast, the accuracy suffers when the anisotropy or the contrast of two adjacent media increases. In addition, due to the high nonlinearity of the Graebner equation, the stability and efficiency of the exact inversion are limited. To address these issues, we propose a robust pre-stack seismic inversion for VTI media using quadratic PP-reflectivity approximation. Firstly, by deriving quadratic approximations for phase velocity and polarization, we analyze the advantage of the second-order terms at a higher-contrast interface. Then, based on perturbation theory, we derive a quadratic PP-reflectivity approximation from the exact equation. It consists of first- and second-order terms with isotropic and anisotropic components, and the accuracy is significantly improved even with enhanced anisotropy. Finally, we introduce the Hadamard product operator to linearly characterize the objective function, and develop an improved Alternating Direction Method of Multipliers (ADMM) algorithm to handle the nonlinearity of the quadratic equations. The feasibility and stability of the proposed method are tested on two datasets, that is, the logging curves used as numerical experiments and a field dataset from a shale reservoir.

Matching pursuit (MP) is an algorithm that can represent signal sparsely, and this advantage makes MP popular in signal processing. However, MP algorithm is a greedy algorithm which means it cannot deal with a large family of signals like seismic data which becomes larger and larger with development of data acquisition technologies. Generalized orthogonal MP (GOMP) is an improved algorithm which helps to reduce the cost of the calculation greatly. Fast MP algorithm is a method that can build dynamic dictionary by making full use of the characteristics of the original signal. In this study, singular value decomposition (SVD) is involved into the GOMP algorithm with dynamic dictionary to improve its efficiency. Compared with conventional MP, the proposed method picks multiatoms at each iteration. It has advantage in calculation speed and can reconstruct the original signal more exactly. Synthetic and field data examples are utilized to demonstrate the feasibility, computational efficiency, and precision of the proposed method.

From the observed seismic data, the estimation of elastic parameters and lithofacies of subsurface layers with seismic amplitude variation with offset (AVO) inversion plays an important role in reservoir characterization. The sequential estimation of elastic parameters and lithofacies in AVO inversion usually affect the prediction accuracies of model parameters without considering the prior information related to lithofacies. We propose a novel probabilistic AVO inversion approach in hierarchical Bayesian framework that combines a hierarchical iterative strategy with Gibbs-IA2RMS sampling, to estimate elastic parameters and lithofacies simultaneously. From the exact PP-wave reflection coefficient, we first derive the posterior distribution of model parameters involving a Bayesian hyper-parameter of lithofacies through the hierarchical prior probability distributions and the likelihood distribution of observed data. Based on Monte Carlo Markov chain approximation, we then introduce an iterative Gibbs-IA2RMS sampling to achieve the high sampling stability of the high-dimensional posterior PDF in AVO inversion. The algorithm realizes the adaptive construction of proposal distribution, and has a higher acceptance rate than the generic Metropolis-Hastings algorithm. The numerical results show that the approach effectively simulates the posterior distributions and inversion uncertainties of model parameters, and improves the computational efficiency of probabilistic AVO inversion. The applicability and validity of the proposed approach are also demonstrated with a field data application.

The rock with a vertical group of aligned fractures in the tilted transverse isotropy (TTI) background medium is regarded as an effective monoclinic medium. Thomsen parameters and fracture weaknesses can reflect the development degree of TTI background anisotropy and vertical fractures, respectively. Thus, this paper first establishes the approximate stiffness matrix of the monocline medium through the Bond transformation from Thomsen weak anisotropy theory and effective medium theory. Moreover, considering the Born scattering theory, a linearized approximate formula of PP wave reflection coefficient is further deduced. This approximate formula contains the parameters of incident angle, azimuth, tilted angle, compressional and shear moduli, density, fracture weaknesses, and Thomsen parameters. Then, the contribution of each parameter on the reflection coefficient by numerical simulation is analyzed respectively. Finally, from the perspective of Bayesian inference, an azimuthal seismic amplitude difference inversion approach incorporating Cauchy prior information and low-frequency regularization of model parameters is proposed to realize the simultaneous estimation of fracture weaknesses and Thomsen parameters. The synthetic test demonstrates that the inversion approach proposed in this article can obtain information on vertical fractures and TTI background medium and can be applied to azimuthal seismic data with moderate noise well. Furthermore, the inversion results of the synthetic test and field data illustrate the stability and feasibility of the inversion approach and also confirm the fracture weaknesses and Thomsen parameters of the monoclinic medium can be reasonably estimated by the inversion equation derived in this paper.

Time-frequency analysis of the nonstationary signals is one of the hot researches in geophysics, environmental science, geology, and other fields. Matching pursuit (MP) has been widely used to reveal the characteristics in time-frequency domain, which helps to study the changes and differences of subtle geological structures. Compared with traditional time-frequency analysis methods, MP has higher time-frequency resolution. The synchrosqueezing transforms (SSTs) have been proven to be an effective methodology to improve the time-frequency resolution by squeezing and redistributing the time-frequency coefficients in the frequency direction. Considering the superiority of the SSTs, this letter proposes a novel time-frequency analysis method called synchrosqueezing MP to enhance the energy aggregation of the time-frequency plane. The proposed method can squeeze the time-frequency distribution into near the center time and frequency of the selected wavelet by a 2-D Gaussian function. We used synthetic and field seismic data to demonstrate the feasibility of our method. The results prove the superiority of our method in describing reservoir boundaries.

Seismic inversion is an important approach in parameters estimation in the fields of geosciences. The low-frequency component of the model parameter plays an important role in seismic inversion. The emergence of broadband seismic data acquisition and processing technologies is pushing the attention of the low frequency to a new level. With the review of the research status of the estimation of the low-frequency models in history and the low-frequency component contained in the complex frequency domain, a novel broadband seismic Bayesian inversion approach in the complex frequency domain is proposed to implement the estimation of the low-frequency component of the model parameter. The proposed approach makes full use of the advantage of broadband seismic data and the low-frequency component of the damped wave fields in the complex frequency domain. The kernel function of the proposed inversion approach is built with Bayesian inference. Synthetic examples demonstrate the feasibility and robustness of the proposed inversion approach in the estimation of the low-frequency component of the model parameter. A field data example verifies the feasibility and reasonability of the proposed inversion approach in application. Finally, the estimated low-frequency component of the model parameter is utilized as the initial model for the suggested seismic Bayesian inversion method in time domain. Model and field data examples further verify the effectiveness and superiority of the proposed inversion method in the final estimation of the model parameter by comparing with the conventional inversion approach.