Bayesian Markov chain Monte Carlo inversion for anisotropy of PP-and PS-wave in weakly anisotropic and heterogeneous media

A single set of vertically aligned cracks embedded in a purely isotropic background may be con- sidered as a long-wavelength effective transversely iso- tropy （HTI） medium with a horizontal symmetry axis. The crack-induced HTI anisotropy can be characterized by the weakly anisotropic parameters introduced by Thomsen. The seismic scattering theory can be utilized for the inversion for the anisotropic parameters in weakly aniso- tropic and heterogeneous HTI media. Based on the seismic scattering theory, we first derived the linearized PP- and PS-wave reflection coefficients in terms of P- and S-wave impedances, density as well as three anisotropic parameters in HTI media. Then, we proposed a novel Bayesian Mar- kov chain Monte Carlo inversion method of PP- and PS- wave for six elastic and anisotropic parameters directly. Tests on synthetic azimuthal seismic data contaminated by random errors demonstrated that this method appears more accurate, anti-noise and stable owing to the usage of the constrained PS-wave compared with the standards inver- sion scheme taking only the PP-wave into account.

Stability condition of finite difference solution for viscoelastic wave equations

The stability problem is a very important aspect in seismic wave numerical modeling. Based on the theory of seismic waves and constitutive equations of viscoelastic models, the stability problems of finite difference scheme for Kelvin- Voigt and Maxwell models with rectangular grids are analyzed. Expressions of stability conditions with arbitrary spatial accuracies for two viscoelastic models are derived. With approximation of quality factor Q≥5, simplified expressions are developed and some numerical models are given to verify the validity of the corresponding theoretical results. Then this paper summarizes the influences of seismic wave velocity, frequency, size of grid and difference coefficients, as well as quality factor on stability condition. Finally the prerequisite conditions of the simplified stability equations are given with error analysis.

Simulation and analysis of point-source surface wave fields

The complexity of near surface intensifies the diversity of seismic wave fields, which makes study on near surface wavefields important in many aspects. The strong absorption of low velocity layer can affect the resolution of seismic data, and free boundary can cause surface wave. Considering the above problems, we focus on the Rayleigh wavefields simulation using finite-difference wave equation of higher-order staggered grids and PML boundary conditions. Free boundary, buried source and overlying low velocity layer are taken into consideration and point explosion source is adopted. Through some numerical simulation with different parameters, we quantitatively analyze relationship between wave intensity and source depth, as well as the energy variation with propagation and obtain some practical knowledge and conclusions.

Finite-difference modeling with variable grid-size and adaptive time-step in porous media

Forward modeling of elastic wave propagation in porous media has great importance for understanding and interpreting the influences of rock properties on characteristics of seismic wavefield. However,the finite-difference forward-modeling method is usually implemented with global spatial grid-size and time-step; it consumes large amounts of computational cost when small-scaled oil/gas-bearing structures or large velocity-contrast exist underground. To overcome this handicap,combined with variable grid-size and time-step,this paper developed a staggered-grid finite-difference scheme for elastic wave modeling in porous media. Variable finite-difference coefficients and wavefield interpolation were used to realize the transition of wave propagation between regions of different grid-size. The accuracy and efficiency of the algorithm were shown by numerical examples. The proposed method is advanced with low computational cost in elastic wave simulation for heterogeneous oil/gas reservoirs.

Stress dependence of elastic wave dispersion and attenuation in fluid-saturated porous layered media

The fluid-saturated porous layered(FSPL)media widely exist in the Earth's subsurface and their overall mechanical properties,microscopic pore structure and wave propagation characteristics are highly relevant to the in-situ stress.However,the effect of in-situ stress on wave propagation in FSPL media cannot be well explained with the existing theories.To fill this gap,we propose the dynamic equations for FSPL media under the effect of in-situ stress based on the theories of poroacoustoelasticity and anisotropic elasticity.Biot loss mechanism is considered to account for the stress-dependent wave dispersion and attenuation induced by global wave-induced fluid flow.Thomsen's elastic anisotropy parameters are used to represent the anisotropy of the skeleton.A plane-wave analysis is implemented on dynamic equations yields the analytic solutions for fast and slow P waves and two S waves.Modelling results show that the elastic anisotropy parameters significantly determine the stress dependence of wave velocities.Vertical tortuosity and permeability have remarkable effects on fast and slow P-wave velocity curves and the corresponding attenuation peaks but have little effect on S-wave velocity.The difference in velocities of two S waves occurs when the FSPL medium is subjected to horizontal uniaxial stress,and the S wave along the stress direction has a larger velocity,which implies that the additional anisotropy other than that induced by the beddings appears due to horizontal stress.Besides,the predicted velocity results have the reasonable agreement with laboratory measurements.Our equations and results are relevant to a better understanding of wave propagation in deep strata,which provide some new theoretical insights in the rock physics,hydrocarbon exploration and stress detection in deep-strata shale reservoirs.

Reflection and transmission coefficient approximation at weak-contrast interfaces for strong VTI media

Reflection and transmission(R/T)responses characterize the propagation and energy distribution of incident and reflected waves on both sides of an interface which is crucial for imaging,amplitude variation with offset(AVO),and seismic inversion techniques.Subsurface media are typically characterized by anisotropy which can have a significant impact on the R/T response,even at small incident angles.Currently,anisotropic media problems including reflection,transmission,and inversion are generally discussed under a weak anisotropy assumption.However,this assumption is no longer valid in cases of large angles where anisotropy enhancement exacerbates the error of the conventional R/T coefficient approximation.An R/T coefficient approximation method for strong VTI media was proposed based on the assumption of weak-contrast of the media.In contrast to the conventional approach,which simplifies the phase velocity and polarization in an anisotropic background,the phase velocity and polarization at the weak-contrast interface of the elasticity and anisotropy parameters were approximated using a combination of the anisotropic background and perturbation terms.Specifically,a first-order approximation of the R/T coefficients for the VTI media characterized by elastic and anisotropic parameters was derived using Cramer's law to invert the anisotropic background matrix,avoiding the assumption of weak anisotropy.Subsequently,the exact solution of the Zoeppritz equations was used to correct the isotropic part,improving the accuracy of the R/T coefficients at interfaces with high-velocity contrast.Modeling tests on four classes of typical interfaces showed that the derived equations can be degraded to the Aki approximation in isotropic media,while exhibiting high accuracy in strong VTI media.Uncertainty analyses showed that a linear approximation that facilitates seismic inversion can be obtained by taking the S-to P-velocity ratio and anisotropy parameters in the coefficient terms a priori.

Seismic rock physical modelling for gas hydrate-bearing sediments

There are ambiguities and uncertainties in the recognition of gas hydrate seismic reflections and in quantitative predictions of physical information of natural gas hydrate reservoirs from seismic data. Rock physical modelling is a bridge that transforms the seismic information of geophysical observations into physical information, but traditional rock physics models lack descriptions of reservoir micro-structures and pore-filling materials. Considering the mineral compositions and pore microstructures of gas hydrates, we built rock physical models for load-bearing and pore-filling gas hydrate-bearing sediments,describe the mineral compositions, pore connectivity and pore shape using effective media theory, calculated the shear properties of pore-filling gas hydrates using Patchy saturation theory and Generalized Gassmann theory, and then revealed the quantitative relation between the elastic parameters and physical parameters for gas hydrate-bearing sediments. The numerical modelling results have shown that the ratios of P-wave and S-wave velocities decrease with hydrate saturation, the P-wave and S-wave velocities of load-bearing gas hydrate-bearing sediments are more sensitive to hydrate saturation, sensitivity is higher with narrower pores, and the ratios of the P-wave and S-wave velocities of pore-filling gas hydrate-bearing sediments are more sensitive to shear properties of hydrates at higher hydrate saturations. Theoretical analysis and practical application results showed that the rock physical models in this paper can be used to calculate the quantitative relation between macro elastic properties and micro physical properties of gas hydrate-bearing sediments, offer shear velocity information lacking in well logging, determine elastic parameters that have more effective indicating abilities, obtain physical parameters such as hydrate saturation and pore aspect ratios, and provide a theoretical basis and practical guidance for gas hydrate quantitative predictions.

Facies-constrained prestack seismic probabilistic inversion driven by rock physics

Seismic Rock physics plays a bridge role between the rock moduli and physical properties of the hydrocarbon reservoirs.Prestack seismic inversion is an important method for the quantitative characterization of elasticity,physical properties,lithology and fluid properties of subsurface reservoirs.In this paper,a high order approximation of rock physics model for clastic rocks is established and one seismic AVO reflection equation characterized by the high order approximation(Jacobian and Hessian matrix)of rock moduli is derived.Besides,the contribution of porosity,shale content and fluid saturation to AVO reflectivity is analyzed.The feasibility of the proposed AVO equation is discussed in the direct estimation of rock physical properties.On the basis of this,one probabilistic AVO inversion based on differential evolution-Markov chain Monte Carlo stochastic model is proposed on the premise that the model parameters obey Gaussian mixture probability prior model.The stochastic model has both the global optimization characteristics of the differential evolution algorithm and the uncertainty analysis ability of Markov chain Monte Carlo model.Through the cross parallel of multiple Markov chains,multiple stochastic solutions of the model parameters can be obtained simultaneously,and the posterior probability density distribution of the model parameters can be simulated effectively.The posterior mean is treated as the optimal solution of the model to be inverted.Besides,the variance and confidence interval are utilized to evaluate the uncertainties of the estimated results,so as to realize the simultaneous estimation of reservoir elasticity,physical properties,discrete lithofacies and dry rock skeleton.The validity of the proposed approach is verified by theoretical tests and one real application case in eastern China.

Azimuthally pre-stack seismic inversion for orthorhombic anisotropy driven by rock physics

Based on the long-wavelength approximation,a set of parallel vertical fractures embedded in periodic thin interbeds can be regarded as an equivalent orthorhombic medium. Rock physics is the basis for constructing the relationship between fracture parameters and seismic response. Seismic scattering is an effective way to inverse anisotropic parameters. In this study,we propose a reliable method for predicting the Thomsen's weak anisotropic parameters and fracture weaknesses in an orthorhombic fractured reservoir using azimuthal pre-stack seismic data. First, considering the influence of fluid substitution in mineral matrix, porosity, fractures and anisotropic rocks, we estimate the orthorhombic anisotropic stiffness coefficients by constructing an equivalent rock physics model for fractured rocks. Further, we predict the logging elastic parameters, Thomsen's weak parameters, and fracture weaknesses to provide the initial model constraints for the seismic inversion. Then, we derive the P-wave reflection coefficient equation for the inversion of Thomsen's weak anisotropic parameters and fracture weaknesses.Cauchy-sparse and smoothing-model constraint regularization taken into account in a Bayesian framework, we finally develop a method of amplitude variation with angles of incidence and azimuth(AVAZ) inversion for Thomsen's weak anisotropic parameters and fracture weaknesses, and the model parameters are estimated by using the nonlinear iteratively reweighted least squares(IRLS) strategy. Both synthetic and real examples show that the method can directly estimate the orthorhombic characteristic parameters from the azimuthally pre-stack seismic data, which provides a reliable seismic inversion method for predicting Thomsen's weak anisotropic parameters and fracture weaknesses.