Anisotropic Parameter Estimation Research Articles

A shortest‐path‐aided fast‐sweeping method to improve the accuracy of traveltime calculation in vertically transverse isotropic media

AbstractThe high accuracy and efficiency of traveltime calculation are critical in seismic tomography, migration, static corrections, source locations and anisotropic parameter estimation. The fast‐sweeping method is an efficient upwind finite‐difference approach for solving the eikonal equation. However, the fast‐sweeping method is accurate only along the axis directions. In two‐dimensional or higher dimensional cases, the accuracy is severely decreased in the diagonal directions due to the numerical errors in these directions. These similar numerical errors also arose in higher order fast‐sweeping method and anisotropic fast‐sweeping method. To improve the accuracy of traveltime calculation in two‐dimensional or higher dimensional space, a shortest‐path‐aided fast‐sweeping method is proposed. The shortest‐path‐aided solution is embedded into the sweeping process of the standard fast‐sweeping method to improve the traveltime accuracy in the diagonal directions. Shortest‐path‐aided fast‐sweeping method is very easy to implement nearly without additional computational cost and memory consumption. Furthermore, this method is easy to extend from two‐dimensional to higher dimensional, from low‐order to higher‐order and from isotropic to anisotropic cases.

Rock physics-based anisotropy parameters estimation of Marcellus Shale using log data acquired in a vertical well

The elastic properties of shale rock exhibit anisotropy due to the presence of highly anisotropic clay minerals and their alignment. This anisotropy has a significant impact on geophysical and geomechanical responses. However, the elastic anisotropy of shale is often overlooked due to the difficulty in measuring sufficient parameters in the field, resulting in a gap between theoretical understanding and practical implementation. To address this issue, a rock physics-based approach was tested on log data obtained from a vertical well in the Marcellus Shale to estimate anisotropy parameters. The approach utilized a rock physics model based on the Sayers-den Boer method with model parameters optimized using measured stiffness parameters C33, C55, and C66. Anisotropy parameters δ and ε were then calculated using these optimized model parameters. The optimized model parameters and estimated anisotropy parameters are consistent with existing studies, indicating the correctness of the model. Additionally, the rock physics model was used to investigate the impact of gas saturation on the vertical VP/VS ratio and ε/γ ratio. It was found that gas saturation has a significant impact on these ratios. This carries important implications for acoustic log data interpretation and seismic characterization of shales.

Seismic pre-stack inversion for physical and anisotropic parameters in fractured shale reservoirs

AbstractThe 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.

Azimuthal amplitude difference inversion constrained by azimuth velocity anisotropy

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.

Re-examination of Critical-angle Reflectometry Used in Ultrasonic Non-destructive Testing For Estimating Material Anisotropy

Seismic critical-angle reflectometry (SCR) can hardly be regarded as a reliable standalone method for anisotropic parameter estimation of strata in seismic exploration, but ultrasound critical-angle reflectometry (UCR) is useful in ultrasonic non-destructive testing (NDT) and medical ultrasound for the estimation of anisotropy in media. Therefore, there is a conflict between using critical-angle reflectometry (CR) in seismic exploration and applications of ultrasound. This study aims to find the reason as to why CR is useful in ultrasonic NDT and medical ultrasound but unreliable in seismic exploration for anisotropy estimation of media. The results of the study show that the critical angle can be accurately determined from the maximum amplitude of the reflections for a plane wave as used in ultrasonic NDT and medical ultrasound and thus the anisotropy of the medium can be successfully estimated by UCR. However, for the spherical waves used in seismic exploration, the maximum amplitude of the reflections is shifted towards post-critical offsets and the shift of the maximum amplitude relative to the critical point depends not only on the frequency and model parameters but also the azimuth. Therefore, SCR cannot currently be regarded as a reliable standalone method for the anisotropic parameter estimation of strata.

Anisotropic elastic parameter estimation from multicomponent ground-motion observations: a theoretical study

SUMMARY We investigate the potential of multicomponent, single-point ground-motion observations (displacement, rotation and strain) to allow the estimation of near-receiver anisotropic elastic parameters. Based on full-space, plane-wave propagation analysis, we demonstrate that in (locally homogeneous) anisotropic media, the wave propagation direction and the velocities of quasi-P and quasi-S waves can—in principle—be determined from three components of displacements and three components of rotations. Mimicking the situation of a borehole setting, we formulate an inverse problem, estimating the full elastic tensor from multidirectional observations. We show that in the presence of noise it is beneficial to observe additionally a longitudinal strain component (e.g. along the borehole), further constraining the predominantly quasi-P related elastic tensor components.

An Improved Scheme of Azimuthally Anisotropic Seismic Inversion for Fracture Prediction in Volcanic Gas Reservoirs

Seismic prediction of natural fractures is essential for the characterization of unconventional reservoirs because fractures provide seepage paths for fluid migration and storage space for hydrocarbon accumulation. However, it remains challenging to robustly estimate anisotropic parameters for fracture characterization based on seismic inversion methods. We propose an improved inversion scheme for the robust and accurate estimation of anisotropic parameters using PP-wave azimuthal amplitude differences incorporated with a hybrid optimization algorithm. Modeling analysis indicates that by removing the effect of isotropic terms, azimuthal amplitude differences are more sensitive to anisotropic parameters than the traditional azimuthal reflection coefficient; this can avoid possible instabilities in anisotropic parameter estimates that may occur in conventional methods owing to unbalanced weighting coefficients between isotropic and anisotropic terms. Meanwhile, the proposed hybrid algorithm takes advantage of the global optimization ability of the simulated annealing algorithm and the fast convergence characteristics of the particle swarm optimization algorithm. Synthetic data tests indicate that the hybrid algorithm achieves reliable and stable estimates of anisotropic parameters with higher computational accuracy and efficiency than traditional methods. The improved inversion method is applied to characterize fractures in volcanic gas reservoirs. Based on azimuthal amplitude differences obtained after estimating fracture orientation using the Fourier series method, anisotropic parameters are computed and converted to weakness properties, which is more meaningful in terms of the fracture properties. Our results indicate that the obtained fracture tangential weakness exhibits an apparent correspondence with well-log permeability, justifying the applicability of the proposed method for fracture prediction.

Orthotropic anisotropy analysis and parameter estimation from 3-D vertical seismic profile data

SUMMARY Orthotropic velocity models are commonly considered as to have the lowest complexity (or highest symmetry) that can realistically describe the combined effect of Earth’s velocity heterogeneity in the vertical and in the horizontal (radial) directions. Velocity heterogeneity in the vertical direction is commonly caused by horizontal stratification of subsurface rock layers with different elastic properties or caused intrinsically by microlayered rocks such as shales. In the horizontal direction, azimuthal variation of the Earth’s stress field can initially lead to preferential deformation of pore space in the rocks and eventual development of aligned (vertical) fractures that are commonly regarded as the main cause for seismic azimuthal anisotropy. Building an appropriate orthotropic velocity model that can explain seismic velocity variations in vertical planes and along any azimuthal direction can therefore enhance the quality of seismic data processing and imaging. This, however, requires knowledge of several anisotropy parameters defined in the three symmetry planes of the orthotropic model and also the orientation of these planes. A combination of P- and S-wave slowness and polarization data derived from vertical seismic profile (VSP) measurements has been routinely used to estimate parameters of less complex anisotropic media such as transverse isotropy with vertical axes of symmetry (VTI). The accuracy of anisotropy parameter estimation by these methods depends primarily on the availability of data that are used in the inversion and the size of anisotropy parameters. In this study, we develop a benchmark for the accuracy of orthotropic parameters that are estimated from 3-D VSP measurements using P-wave slowness only method. Through numerical analysis, we model a wide range of fracture-induced orthotropic symmetries by adding vertical fractures to a background VTI media and show that the accuracy of our orthotropic parameter estimation method is dependent on anisotropy in the host media, intensity of fracturing, noise level in slowness data and slowness data coverage in both the dip and azimuthal planes. We then expand our accuracy analysis work to multilayer media by examining the accuracy of P-wave slowness method for orthotropic anisotropy parameter estimation by inverting finite-difference data generated in a pseudo 3-D VSP experiment. We show that, for a typical 3-D VSP experiment, reasonable estimates of orthotropic parameters in the model can only be obtained with slowness data taken from Xmax/Z ≥ 1 and line azimuth interval ϕ ≤ 10° where Xmax is the maximum source to receiver offset and Z is the receiver depth. We finally describe how field VSP measurements can be inverted for orthotropic anisotropy parameters.

Estimation of parameters and thickness of a horizontal layer of arbitrary anisotropy from P-wave moveout

We present an inversion scheme to estimate anisotropy parameters and thickness of a homogeneous layer of arbitrary anisotropy from traveltimes of a reflected P wave. The scheme is based on the weak-anisotropy approximation of P-wave moveout formula. For the description of anisotropy, we use the so-called A-parameters, which represent an alternative to standard elastic moduli specifying anisotropic media, and allow specification of anisotropy of any symmetry, orientation and strength. The proposed scheme allows estimation of, at most, nine of fifteen A-parameters, which describe P-wave propagation in the weak-anisotropy approximation. Therefore, generally, the reconstruction of the P-wave phase-velocity surface from the estimated parameters is impossible. If we deal, however, with higher-symmetry anisotropy (transversely isotropic, orthorhombic) and if information about the orientation of its symmetry elements is available, reconstruction of the phase-velocity surface is possible. Synthetic tests indicate that the inversion scheme works well even in situations, in which the use of other commonly used moveout formulae leads to unsatisfactory results.

Rock Physics Model of Shale: Predictive Aspect

AbstractShales often show strong elastic anisotropy that originate from the alignment and platy nature of its constituent minerals. Despite its impact on amplitude variation with offset response and seismic time‐shifts, elastic anisotropy of shales is, however, often ignored since it is difficult to measure enough parameters in the field. Being able to correctly estimate anisotropy parameters of shales can therefore significantly improve seismic reservoir characterization. A predictive model is developed by combining existing theories. Properties of locally aligned clay platelets, called domains, are calculated using a rock physics model based on the anisotropic Hashin‐Shtrikman estimates. The effect of domain orientation is then accounted for by the orientation distribution function of domains. The applicability of the model was investigated using existing core measurements. Interesting findings include: (a) most of modeled anisotropy parameters are consistent with the measured values even though only limited information was used for the model parameters optimization, and (b) most of optimized interplatelet medium properties are consistent with the saturated fluid and small interplatelet medium shear modulus suggested by existing studies. These findings imply that the model can be used to predict anisotropy parameters from limited information.

Effect of Sr - Transition Metal Substitution on Electronic and Mechanical Properties of Mg2Si: A DFT Study

Abstract: Recently, magnesium alloys have attracted scientific interest due to their technological importance in thermoelectric, piezoelectric, photo-voltaic and infrared photonics applications. The electronic and elastic properties of MgXSi (X = Mg, Sr) compounds were investigated in this work, using the density functional theory (DFT) with pseudo-potential plane-waves (PPW) approach as implemented in Quantum Espresso code. The results of the elastic constants of Mg2Si are in agreement with the previous theoretical results and favourably compared with experimental data. The electronic band structures of these semiconductors were calculated to give narrow indirect and direct band gaps of Mg2Si and MgSrSi, respectively. Our results show that the two compounds are mechanically stable. The Pugh’s ratio, B/G, indicated that Mg2Si and MgSrSi are brittle and ductile in nature. The estimated anisotropy parameter, A, shows that Mg2Si has a higher degree of elastic isotropy in comparison to MgSrSi. Three-dimensional (3D) projection of Young’s modulus and area modulus of the compounds was presented. Keywords: Electronic structure, Elastic constants, Mechanical properties, Mg2Si, MgSrSi. PACS: 31.15.A-, 62.20.F-, 71.55.-i, 71.20.Be.

Rock-physics constrained seismic anisotropy parameter estimation

Compared with isotropic media, at least two extra parameters are involved in common P-wave seismic data processing and interpretation for transversely isotropic media. Previous synthetic model testing has shown that it is challenging to estimate anisotropy parameters even using extremely low noise level seismic data from a simple geologic setting. Although theoretically independent, anisotropy parameters are not free variables for organic-rich mudrocks whose elastic properties are often approximated by transverse isotropy. One potential approach to improve the accuracy in the estimated anisotropy parameters is to consider the physical relationships between them during the inversion process. To test this proposition, we first modify a commonly used nonhyperbolic reflection moveout equation as a function of the interval anisotropy velocities so that rock-physics constraints could be effectively applied to each layer. The rock-physics constraints are established from data analysis of selected laboratory anisotropy measurement data. The laboratory data are then used to parameterize hundreds of 15-layer transverse isotropy models using a Monte Carlo simulation. The synthetic model testing indicates that the accuracy of the estimated anisotropy parameters can be improved if the relationships between the anisotropy parameters are considered during the inversion process.

TESTING OF AN OPTIMIZATION ALGORITHM FOR AVAZ INVERSION ON SYNTHETIC DATASET

In the paper, we study an optimization algorithm for a nonlinear AVAZ inversion of PP reflections from an anisotropic media. The algorithm is based on the exact formulas for PP wave reflection coefficient for an anisotropic HTI medium and could be applied in the case of strong-contrast boundary and various anisotropy degree. Algorithm testing on synthetic dataset for radial survey system shows that estimation of anisotropy parameters γ , δ and HTI medium symmetry axis is robust in the case of signal to noise ratio ≥ 5. For estimation of parameter ε far offset data is needed.

Model parameterization and amplitude variation with angle and azimuthal inversion in orthotropic media

Horizontal layered formations with a suite of vertical or near-vertical fractures are usually assumed to be an approximate orthotropic medium and are more suitable for estimating fracture properties with wide-azimuth prestack seismic data in shale reservoirs. However, the small contribution of anisotropic parameters to the reflection coefficients highly reduces the stability of anisotropic parameter estimation by using seismic inversion approaches. Therefore, a novel model parameterization approach for the reflectivity and a pragmatic inversion method are proposed to enhance the stability of the inversion for orthotropic media. Previous attempts to characterize orthotropic media properties required using four or five independent parameters. However, we have derived a novel formulation that reduces the number of parameters to three. The inversion process is better conditioned with fewer degrees of freedom. An accuracy comparison of our formula with the previous ones indicates that our approach is sufficiently precise for reasonable parameter estimation. Furthermore, a Bayesian inversion method is developed that uses the amplitude variation with angle and azimuth (AVAZ) of the seismic data. Smooth background constraints reduce the similarity between the inversion result and the initial model, thereby reducing the sensitivity of the initial model to the inversion result. Cauchy and Gaussian probability distributions are used as prior constraints on the model parameters and the likelihood function, respectively. These ensure that the results are within the range of plausibility. Synthetic examples demonstrate that the adopted orthotropic AVAZ inversion method is feasible for estimating the anisotropic parameters even with moderate noise. The field data example illustrates the inversion robustness and stability of the adopted method in a fractured reservoir with a single well control.

Effect of stress interactions on anisotropic P‐SV‐wave dispersion and attenuation for closely spaced cracks in saturated porous media

ABSTRACTGenerally, local stress induced by individual crack hardly disturbs their neighbours for small crack densities, which, however, could not be neglected as the crack density increases. The disturbance becomes rather complex in saturated porous rocks due to the wave‐induced diffusion of fluid pressures. The problem is addressed in this study by the comparison of two solutions: the analytical solution without stress interactions and the numerical method with stress interactions. The resultant difference of effective properties can be used to estimate the effect of stress interactions quantitatively. Numerical experiments demonstrate that the spatial distribution pattern of cracks strongly affects stress interactions. For regularly distributed cracks, the resulting stress interaction (shielding or amplification) shows strong anisotropy, depending on the arrangement and density of cracks. It has an important role in the estimation of effective anisotropic parameters as well as the incident‐angle‐dependency of P‐ and SV‐wave velocities. Contrarily, randomly distributed cracks with a relative small crack density generally lead to a strong cancellation of stress interactions across cracks, where both the numerical and analytical solutions show a good agreement for the estimation of effective parameters. However, for a higher crack density, the incomplete cancellation of stress interactions is expected, exhibiting an incidence‐angle dependency, slightly affecting effective parameters, and differentiating the numerical and analytical solutions.

Estimation of anisotropy parameters VTI (vertical transverse isotropy) using velocity variation with offset (VVO) method, case study: BS oil field

Implementation of anisotropy value in seismic data processing greatly affect seismic cross section image. In addition to enhanced seismic imagery, anisotropy can be used for identification of hydrocarbons. In this study, the vertical transverse isotropy (VTI) anisotropy parameter estimation by the VVO method has been done. The magnitude of the velocity value of the offset/angle is due to the ever-greater velocity with a large incidence angle being the basis of the VVO method to demonstrate the anisotropy effect. The residual moveout result of the time transfer correction caused by the velocity increase will be used as input data. Validity of the VVO method is tested using a synthetic forward modelling to obtain anisotropy parameters that describe the subsurface conditions of the target area which are then applied to the actual data. The results found that residual values continued to increase due to the anisotropy effect and there after the angle of 30 degrees with the value of ɛ = 0.14 and δ = 0.049.

Estimation of in situ anisotropy parameters from two perpendicular walkaway VSP lines in South Pars field

Abstract Vertical phase slowness and polarization angle are the in situ parameters of P-wave propagation that can be derived from walkaway vertical seismic profiling (VSP) data. To use these data for estimating anisotropy parameters, we obtain an explicit equation of vertical slowness as a function of polarization angle for P-wave propagation in transversely isotropic with vertical symmetry axis (VTI) media. We use this equation to estimate anisotropy parameters of a target layer in the South Pars field, Iran. This field is one of the world's largest gas fields. We show that the orthorhombic symmetry is a reasonable assumption for this layer, providing some geological and petrophysical information. Two walkaway VSP lines along the symmetry axes of the presumed orthorhombic layer are used to estimate its parameters. Seven is the maximum number of parameters that can be estimated using P-wave data in this acquisition pattern. Of those estimated parameters are six of the Tsvankin style parameters for orthorhombic media, plus an approximate combination of two others that define vertical S-wave splitting. We show that a previous method, which is based on weak anisotropy approximation, leads to considerable errors, even in models where the magnitude of anisotropy parameters do not exceed 0.1. We design a numerical experiment to study the reliability of the estimated parameters by the exact approach and show the importance of acquisition pattern in this regard. To show applicability, these parameters are used to estimate the in situ fracture properties of the studied layer.

A new approach in physical modeling for velocity anisotropy study

Special automated equipment is constructed for ultrasonic velocity measurements at various incidence angles of waves that is necessary for estimation of anisotropy parameters of rocks. The equipment includes hydraulic press for simulation of vertical pressure in the Earth, at a range up to 40 MPa and computer controlled positioning system that provides precise independent positioning of ultrasonic source and receiver. This setup enables us to perform reliable velocity measurements at the angle interval from 15° to 90° and additional single measurement at 0°. The minimum step of a transducer position in 15°-90° range is 8.1'. As an example on anisotropy study, we perform physical modeling of wave propagation in artificial sample of transversely isotropic (TI) symmetry, which represents a model of fractured rock. Using experimental setup, we acquire P-wave velocity data for plate-stack model with vertical axis of symmetry formed by acrylic 1 mm thick sheet plates. Using measured velocities, we estimate Thomsen’s anisotropy parameters e and δ, and get dependence between the velocity of P-waves and symmetry axis of the model. . In our oncoming research, the upgraded Ultrasonic Measuring System will be supplemented by multicomponent transducers similar to those described in Chichinina et al. (2009), which allow simultaneous recording of P, SV and SH waves at each data point.

Quantitative inversion of azimuthal anisotropy parameters from isotropic techniques

Exploration and development of unconventional reservoirs, where fractures and in situ stress play a key role, calls for improved characterization workflows. In this work, we present a method for quantitative estimation of anisotropic parameters related to stress and fracture detection that makes use of standard isotropic modeling and inversion techniques in anisotropic media. Based on the Rüger reflectivity equations for horizontal transverse isotropic media, we build a set of transforms that map the elastic parameters used in prestack inversion into effective anisotropic elastic parameters. When used in isotropic forward modeling and inversion, these effective parameters accurately mimic the anisotropic reflectivity behavior of the seismic data, thus closing the loop between well-log data and seismic inversion results in the anisotropic case.

Estimation of anisotropy parameters for shales based on an improved rock physics model, part 2: case study

Estimation of anisotropy parameters for shales based on an improved rock physics model, part 2: case study