Thomsen Anisotropy Research Articles

Comparison of elastic anisotropy in the Middle and Upper Wolfcamp Shale, Midland Basin

AbstractOrganic‐rich shales contain large amounts of oil and gas and are anisotropic because of fine‐scale layering and the partial alignment of organic matter and anisotropic clay minerals with the bedding. An example is the Wolfcamp Shale in the Permian Basin. Elastic anisotropy needs to be accounted for in the characterization of such formations using seismic data and plays a role in hydraulic fracturing and in the evaluation of stress changes and geomechanical effects resulting from production. Using extensive well log data acquired in the Midland Basin, the eastern sub‐basin of the Permian Basin, we estimate and compare the elastic anisotropy in the Middle and Upper Wolfcamp Shale by combining data from a vertical pilot well with two lateral wells, one (6SM) drilled in the Middle Wolfcamp and one (6SU) drilled in the Upper Wolfcamp. The data used were acquired at the Hydraulic Fracture Test Site 1, located in the eastern part of the Midland Basin. Thomsen's anisotropy parameter calculated from the fast and slow shear sonic is higher on average for the 6SM lateral than for 6SU, consistent with there being less carbonate content in 6SM than in 6SU. However, the anisotropy parameter in some regions with higher carbonate content in well 6SU is higher than in well 6SM. This may indicate the influence of natural fractures. The primary set of steeply dipping fractures observed in the lateral wells at Hydraulic Fracture Test Site 1 acts to increase if the ratio of the normal‐to‐shear fracture compliance is less than about 0.5. Sub‐horizontal fractures may also increase and could affect the vertical extent of hydraulic fractures. Relations between elastic moduli C33 and C55 in the Upper and Lower Wolfcamp in a vertical pilot well allow C33 to be predicted in a lateral well using measurements of C55 in that well. Comparison of Thomsen's anisotropy parameters and , with calculated from the measured values of C55 and C66 and calculated from the measured values of C11 and predicted values of C33, show that is mostly greater than .

Third‐order elasticity of transversely isotropic field shales

AbstractThe formations above a producing reservoir can exhibit large mechanical changes, creating a risk of significant subsidence and loss of rock integrity. These changes can be monitored by time‐lapse seismic acquisition, which measures the corresponding velocity changes via time‐shifts. Third‐order elastic theory can be used to connect subsurface strains and stress changes to these seismic attribute changes. Existing models assume isotropic strain dependence of the dynamic stiffness in shales. It is important to re‐evaluate this isotropic assumption considering the inherent anisotropy of shales and their abundance in the overburden. Thus, we instead propose a third‐order elastic model with a transversely isotropic strain dependence of the dynamic stiffness. When calibrated, this new model satisfactorily predicted P‐wave velocity changes determined in undrained laboratory experiments conducted on overburden field shales, covering a wide range of propagation directions and stress variations. The shales exhibit anisotropic dynamic strain sensitivity, resulting in a significantly higher strain sensitivity predicted for Thomsen's anisotropy parameters epsilon and delta subjected to a uniaxial strain parallel to the horizontal bedding plane compared to the vertical direction. Geomechanical modelling, considering a depleting disk‐shaped reservoir surrounded by shales, was employed to predict the dynamic stiffness changes of the overburden using the laboratory‐calibrated third‐order elastic model. The overburden time‐shifts increased with offset angle, peaking at about 45°, suggesting a strong influence of shear strains on the time‐shifts. In contrast, a corresponding model with an isotropic third‐order elastic tensor, calibrated to the same data, exhibited a significantly lower sensitivity to the shear strains. These results underscore the importance of considering the anisotropic strain dependence of the dynamic stiffness when studying shales. Interpreting offset‐dependent trends in pre‐stack time‐lapse seismic data, along with geomechanical modelling and an appropriate strain‐dependent rock physics model, can assist in quantifying subsurface strains and stress changes.

An approximate method for calculating anisotropy parameters and reflectivity of shales with horizontal fractures

Abstract Shale is a typical medium of transverse isotropy with a vertical axis of symmetry (VTI), and its strong anisotropy is mainly due to the combined effect of intrinsic anisotropy and that induced by horizontal fractures. To calculate the anisotropy parameters of shale, a physical rock model is built based on Hudson's thin-coin fracture model and Schoenberg's linear-sliding model, and an approximate theoretical calculation method for Thomsen's anisotropy parameters of VTI media with horizontal fractures is proposed. These calculation results using the proposed method confirm that this anisotropy contributed to by horizontal fractures cannot be ignored to the overall anisotropy of shale. To simplify Rüger's formula that is an approximate theoretical formula for calculating the anisotropic reflection coefficients of VTI media, a new four-term approximate formula is derived in a standard reflectivity form based on Rüger's and Aki–Richards’ formulas. The simulation results of a VTI theoretical model and logging data of shale reservoirs show that there is only a small difference between the newly derived four-term formula and Rüger's formula for incidence angles <40°, and the new four-term formula can correctly reveal the seismic amplitude-versus-offset (AVO) characteristics of VTI media and fully retain the corresponding anisotropic seismic responses. Compared to Rüger's formula, the proposed new formula only has four terms of unknown parameters and can directly decouple Thomsen's anisotropy parameter ε from them, which helps to alleviate the ill-posed problems of simultaneous inversion of multiple parameters and enhance its application potential in seismic inversion of VTI media as shale.

Least‐squares reverse time migration based on an efficient pure qP‐wave equation

AbstractSeismic waves propagating in anisotropic earth media suffer from waveform distortion, which leads to degraded resolution in seismic data imaging if this adverse effect is not corrected. Additionally, the wavefields simulated by the traditional coupled pseudo‐acoustic wave equations contain undesired shear wave artefacts and are unstable when Thomsen's anisotropy parameter ε is less than δ. To address these issues, many pure qP‐wave equations are proposed to describe wave propagation in anisotropic media. However, these equations are either low precision or high accuracy but computationally expensive. Considering the trade‐off between the computation efficiency and simulation accuracy, we derive a pure qP‐wave equation from the exact dispersion formula in tilted transverse isotropic media. The newly derived equation has only two higher order partial derivatives, which can better balance the accuracy and efficiency than the previous pure qP‐wave equations. The least‐squares reverse time migration method can balance amplitudes, suppress migration artefacts and improve imaging resolution. Then, based on the newly derived pure qP‐wave equation, we derive the Born modelling operator and adjoint migration operator and develop an anisotropic least‐squares reverse time migration method. To achieve the accelerated convergence, we introduce a generalized minimum residual algorithm to implement the anisotropic least‐squares reverse time migration method. Numerical simulation results show that the proposed pure qP‐wave equation outperforms the previous equations in balancing the trade‐off between accuracy and efficiency. In addition, synthetic examples demonstrate the effectiveness and robustness of our proposed anisotropic least‐squares reverse time migration method in correcting the deviations due to anisotropic effects.

Regularized seismic amplitude inversion via variational inference

AbstractOver the years, seismic amplitude variation with offset has been successfully applied for predicting the elastic properties of the subsurface. Nevertheless, the solution of the amplitude inversion is not stable due to insufficient information in the recorded seismic data. To enhance the stability, various amplitude inversions under a Bayesian framework have been introduced and most of which are based on single‐component a priori probability densities, which can be solved deterministically. For multi‐component a priori densities, Monte Carlo sampling methods are often used to obtain the posterior distribution of the inverted parameters. The variational methods, on the other hand, provide the gradients of the parameters with respect to the data. Here, we present an inversion framework that uses the gradients of the variational inference as regularization. Due to the ill‐posedness of the amplitude inversion, solving the likelihood of the data leads to solutions that may exist within the low‐probability regions of the a priori density. The rule of the regularization (variational inference gradient) is, therefore, to force solutions within the high‐probability regions that enhance the stability of the inversion process. The performance of the regularization is tested using synthetic and field data examples to jointly estimate the elastic and Thomsen's anisotropy parameters. The proposed constrained optimization successfully predicts the P‐wave velocity, S‐wave velocity and P‐wave anisotropy and preferably constrains the density and near‐vertical anisotropy parameters.

Elastic anisotropy of the Marcellus Shale

ABSTRACTThe Marcellus Shale is one of the largest shale gas resources in the world and is anisotropic due to fine layering and the partial alignment of anisotropic clay minerals and organic matter with the bedding. This anisotropy can be approximated as Vertically Transversely Isotropic (transversely isotropic with a vertical axis of rotational symmetry) with five independent density‐normalized elastic stiffnesses A11, A13, A33, A55 and A66 in the two‐index notation and with axis of rotational symmetry along x3. Compressional and dipole shear‐wave data acquired in a horizontal well allows estimation of A11, A55 and A66, while the same data in a vertical pilot well allows estimation of A33 and A55. The ratio of vertically propagating P‐ to S‐velocity is and has a quadratic dependence on in the Upper Marcellus with minimum occurring for the largest volume fraction of kerogen. This relation allows estimation of A33 along a lateral well using measured values of A55 in the well. Comparison of Thomsen's anisotropy parameter ε is found to be mostly greater than anisotropy parameter γ. Estimating A13 as the average of A33 – 2A55 and A11 – 2A66 as proposed recently by Yan and Vernik allows Thomsen's anisotropy parameter δ and a parameter K0 that relates the horizontal and vertical effective stresses to be estimated. The results are expected to help in estimating horizontal stress needed for design of hydraulic fractures, and in interpreting seismic amplitude variation with offset required for reservoir characterization.

Influence of bedding structure on stress-induced elastic wave anisotropy in tight sandstones

To understand the evolution of stress-induced elastic wave anisotropy, three triaxial experiments were performed on sandstone specimens with bedding orientations parallel, perpendicular, and oblique to the maximum principal stress. P-wave velocities along 64 different directions on each specimen were monitored frequently to understand the anisotropy change at various stress levels by fitting Thomsen's anisotropy equation. The results show that the elastic wave anisotropy is very sensitive to mechanical loading. Under hydrostatic loading, the magnitude of anisotropy is reduced in all three specimens. However, under deviatoric stress loading, the evolution of anisotropic characteristics (magnitude and orientation of the symmetry axis) is bedding orientation dependent. Anisotropy reversal occurs in specimens with bedding normal/oblique to the maximum principal stress. P-wave anisotropy ε ′ is linearly related to volumetric strain S v and dilatancy, indicating that stress-induced redistribution of microcracks has a significant effect on P-wave velocity anisotropy. The closure of initial cracks and pores aligned in the bedding direction contributes to the decrease of the anisotropy. However, opening of new cracks, aligned in the maximum principal direction, accounts for the increase of the anisotropy. The experimental results provide some insights into the microstructural behavior under loading and provide an experimental basis for seismic data interpretation and parameter selection in engineering applications.

Creation of synthetic samples for physical modelling of natural shale

ABSTRACTNatural shale samples, particularly well‐preserved, drilled core samples, are extremely difficult to obtain for laboratory research. Multiple tests must be carried out on one sample, and some samples are disposed after destructive tests. Therefore, rarity and non‐reusability of samples strongly restrict shale studies. In this study, based on statistical data from the world's major shale block, a new type of synthetic shale was physically constructed via a process of interfusion, stuffing, and compaction using quartz, clay, carbonate, and kerogen as the primary materials, according to statistical data from the world's major shale blocks. Further evaluation of the synthetic shale involved the use of scanning electron microscopy imagery and analysis of its anisotropic characteristics in comparison with natural shale. The synthetic shale had a laminated microstructure similar to natural shale, and its velocity anisotropy corresponded to Thomsen's anisotropy of a transversely isotropic medium. The results of tests for homogeneity and repeatability indicated that the construction process was stable and that several identical synthetic samples, which were satisfactorily similar to natural shale, could be produced for both iterative and destructive tests. The composition of each mineral, as well as the density, porosity, permeability, and anisotropy of the samples, were all variable. Therefore, a series of synthetic samples could be obtained with properties set to meet the requirements of petrophysical experimentation. Moreover, gas or oil saturation was also considered in the construction of the synthetic shale, meaning that the characteristics of gas or oil saturation (or the complete range of data from dry to saturated samples) could be tested using the synthetic shale.

Measurements of seismic anisotropy and fracture compliances in synthetic fractured media

Ultrasonic velocities were measured on a stack of synthetic materials (Plexiglass plates), at low (90/120 kHz, long wavelength range: 12–23 mm) and high (431/480 kHz, short wavelength range: 1–6 mm) frequencies. The Plexiglass plates were pressed together with uniaxial normal stress. The ultrasonic measurements were repeated under different uniaxial stresses with and without circular rubber inclusions between the plates. The stress dependence of measured P-wave velocities is more pronounced (about 200 m s−1 difference) at low frequency compared to the high frequency measurements. S-wave velocity measurements in the vertical direction (perpendicular to plates) and horizontal direction with polarization perpendicular to the plates show that, without rubber inclusions, the model can be approximated as a vertical transverse isotropy (VTI) medium (difference of about 20 m s−1between the two S-wave velocities for stresses higher than 6 MPa), whereas, for the case with inclusions, there is a significant difference (about 350 m s−1 for stresses higher than 6 MPa). The difference in vertically propagating and horizontally propagating vertically polarized S-wave velocities indicates that the medium is inhomogeneous at these wavelengths. Ultrasonic velocity measurements were used to calculate fracture compliances and VTI anisotropy parameters. For both experiments (with and without rubber disc inclusion), fracture compliances and also absolute values of VTI anisotropy parameters, decrease with increasing stress and frequency. For our experiments, Thomsen's anisotropy parameter γ is highly correlated to ε and η (with average correlation coefficients of about 95 per cent). For all measurements, the medium showed positive anellipticity (i.e. ε–δ > 0). In all cases, the tangential compliance was greater than the normal compliance. For experiments without inclusions, the normal to tangential compliance ratios show an increasing trend with increasing normal stress. For experiments with inclusions, these ratios show a decreasing trend, especially at higher frequency ranges.

Thomsen's parameters from p‐wave measurements in a spherical sample‡

ABSTRACTThe aim of this paper is to understand the seismic anisotropy of the overburden shale in an oilfield in the North West Shelf of Western Australia. To this end, we first find the orientation of the symmetry axis of a spherical shale sample from measurements of ultrasonic P‐wave velocities in 132 directions at the reservoir pressure. After transforming the data to the symmetry axis coordinates, we find Thomsen's anisotropy parameters δ and ɛ using these measurements and measurements of the shear‐wave velocity along the symmetry axis from a well log. To find these anisotropy parameters, we use a very fast simulated re‐annealing algorithm with an objective function that contains only the measured ray velocities, their numerical derivatives and the unknown elasticity parameters. The results show strong elliptical anisotropy in the overburden shale. This approach produces smaller uncertainty of Thomsen parameter δ than more direct approaches.

Non-Linear Joint AVO Inversion of PP and PS Waves in a VTI Medium

This paper aims to estimate elastic parameters such as P-wave and S-wave velocity, Thomsen's anisotropy parameters (d, e), density, and layer thickness of a layered medium using pre-stack seismic data and some prior information. Three different algorithms have been developed in an object oriented environment: Travel time inversion, waveform inversion, and joint traveltime and reflectivity inversion. These algorithms can use compressional or joint compressional and converted shear waves data respectively. We have implemented exact parametric travel time equations in a VTI medium for a stack of either anisotropic or isotropic homogeneous layers. Reflection and transmission coefficients for the whole ray path have been calculated. A convolution model is used to calculate synthetic seismograms. To minimize an objective function a non-linear conjugate gradient (CG) algorithm has been implemented. All of the partial derivatives have been calculated analytically which results in a fast and robust differentiation of traveltimes and reflection and transmission coefficients. A numerical example of a stack of 10 layers has been tested for different elastic parameters and thicknesses. Compressional and shear waves velocities and thicknesses have been varied 25 percent from true values while d and e have been varied 100 percent from their true values. Joint traveltime inversion of P ® P & P ® Sv has successfully estimated the true values of the parameters. Waveform inversion is highly non-linear; hence only anisotropic parameters and density while varying up to 100 percent are estimated. Waveform inversion results in a full estimation of the true values of d and e and a fairly good estimate of density.

Stress induced elastic anisotropy of the Etnean basalt: Theoretical and laboratory examination

A theory, the stress‐sensitivity approach, has been developed, which relates the elastic moduli of anisotropic rocks to the stress tensor and pore pressure for an arbitrary symmetry of the applied load. The theory explains the stress‐induced changes of seismic velocities in terms of stress‐induced changes of the pore space geometry. The stress dependent anisotropy is described in terms of Thomsen's anisotropy parameters ε, γ and δ. To test the theory we analyze the laboratory (high frequency) results of deformation of an isotropically crack damaged dry lava flow basalt from Mt. Etna volcano. The theory states that, under an anisotropic (i.e. axisymmetric triaxial) load and in the case of an initially isotropic rock, (1) the anisotropy parameters are linear functions of the stress exponentials (i. e. exponential functions of principal stress components) and (2) the ratio of these anisotropy parameters as a function of the stress is constant. In order to verify these relationships, the stress exponentials and the anisotropy parameters based on the measured velocities are computed as well as the expected ratio of the Thomsen's parameters. Our experimental results are in very good agreement with the theoretically predicted relations.

Velocity anisotropy estimation for brine-saturated sandstone and shale

Velocity anisotropy is an important parameter in seismic analyses, especially for deviated wells in offshore fields when the velocity anisotropy effect must be corrected to obtain a realistic sonic log response equivalent to seismic data for small incident angles. However, there is no method for an anisotropy correction when only standard log data from a single well are available. We compiled experimental data of core velocity anisotropy from several recent publications to explore heuristic relationships between anisotropy parameters and general well log information and found a clear trend of velocity anisotropy in the crossplot domain between velocity and porosity. Subsequently, we developed a method to estimate Thomsen's anisotropy parameters e and γ by a regression in this domain. In this paper, we first show an application result of the method to demonstrate the significance of the correction. Then we summarize how we developed the method. We also discuss an alternative approach using neural networks....

Analysis of Thomsen parameters for finely layered VTI media

Since the work of Postma and Backus, much has been learned about elastic constants in vertical transversely isotropic (VTI) media when the anisotropy is due to fine layering of isotropic elastic materials. Nevertheless, there has continued to be some uncertainty about the possible range of Thomsen's anisotropy parameters ε and δ for such media. We use both Monte Carlo studies and detailed analysis of Backus' equations for both two‐ and three‐component layered media to establish the results presented. We show that ε lies in the range −3/8 ε ½[〈v2p〉〈v−2p〉−1], for finely layered media; smaller positive and all negative values of ε occur for media with large fluctuations in the Lamé parameter λ in the component layers. We show that δ can also be either positive or negative, and that for constant density media, sign (δ) = sign (〈v−2p〉 − 〈v−2s〉〈v2s/v2p〉). Monte Carlo simulations show that among all theoretically possible random media, positive and negative δ are equally likely in finely layered media. (Of course, the δs associated with real earth materials may span some smaller subset of those that are theoretically possible, but answering this important question is beyond our present scope.) Layered media having large fluctuations in λ are those most likely to have positive δ. This is somewhat surprising since ε is often negative or a small positive number for such media, and we have the general constraint that ε − δ > 0 for layered VTI media. Since Gassmann's results for fluid‐saturated porous media show that the mechanical effects of fluids influence only the Lamé parameter λ, not the shear modulus μ, these results suggest that small positive δ occurring together with small positive ε (but somewhat larger than δ) may be indicative of changing fluid content in a layered earth.

Anisotropic parameters and P‐wave velocity for orthorhombic media

Although orthorhombic (or orthotropic) symmetry is believed to be common for fractured reservoirs, the difficulties in dealing with nine independent elastic constants have precluded this model from being used in seismology. A notation introduced in this work is designed to help make seismic inversion and processing for orthorhombic media more practical by simplifying the description of a wide range of seismic signatures. Taking advantage of the fact that the Christoffel equation has the same form in the symmetry planes of orthorhombic and transversely isotropic (TI) media, we can replace the stiffness coefficients by two vertical (P and S) velocities and seven dimensionless parameters that represent an extension of Thomsen's anisotropy coefficients to orthorhombic models. By design, this notation provides a uniform description of anisotropic media with both orthorhombic and TI symmetry. The dimensionless anisotropic parameters introduced here preserve all attractive features of Thomsen notation in treating wave propagation and performing 2-D processing in the symmetry planes of orthorhombic media. The new notation has proved useful in describing seismic signatures outside the symmetry planes as well, especially for P‐waves. Linearization of P‐wave phase velocity in the anisotropic coefficients leads to a concise weak‐anisotropy approximation that provides good accuracy even for models with pronounced polar and azimuthal velocity variations. This approximation can be used efficiently to build analytic solutions for various seismic signatures. One of the most important advantages of the new notation is the reduction in the number of parameters responsible for P‐wave velocities and traveltimes. All kinematic signatures of P‐waves in orthorhombic media depend on just the vertical velocity [Formula: see text] and five anisotropic parameters, with [Formula: see text] serving as a scaling coefficient in homogeneous media. This conclusion, which holds even for orthorhombic models with strong velocity anisotropy, provides an analytic basis for application of P‐wave traveltime inversion and data processing algorithms in orthorhombic media.