Thomsen Anisotropy Parameters 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 .

Linearized waveform inversion for vertical transversely isotropic elastic media: Methodology and multi-parameter crosstalk analysis

Seismic migration and inversion are closely related techniques to portray subsurface images and identify hydrocarbon reservoirs. Seismic migration aims at obtaining structural images of subsurface geologic discontinuities. More specifically, seismic migration estimates the reflectivity function (stacked average reflectivity or pre-stack angle-dependent reflectivity) from seismic reflection data. On the other hand, seismic inversion quantitatively estimates the intrinsic rock properties of subsurface formulations. Such seismic inversion methods are applicable to detect hydrocarbon reservoirs that may exhibit lateral variations in the inverted parameters. Although there exist many differences, pre-stack seismic migration is similar with the first iteration of the general linearized seismic inversion.Usually, seismic migration and inversion techniques assume an acoustic or isotropic elastic medium. Unconventional reservoirs such as shale and tight sand formation have notable anisotropic property. We present a linearized waveform inversion (LWI) scheme for weakly anisotropic elastic media with vertical transversely isotropic (VTI) symmetry. It is based on two-way anisotropic elastic wave equation and simultaneously inverts for the localized perturbations (ΔVp0/Vp0,ΔVs0/Vs0,Δϵ,Δδ) from the long-wavelength reference model. Our proposed VTI-elastic LWI is an iterative method that requires a forward and an adjoint operator acting on vectors in each iteration. We derive the forward Born approximation operator by perturbation theory and adjoint operator via adjoint-state method. The inversion has improved the quality of the images and reduces the multi-parameter crosstalk comparing with the adjoint-based images. We have observed that the multi-parameter crosstalk problem is more prominent in the inversion images for Thomsen anisotropy parameters. Especially, the Thomsen parameter δ is the most difficult to resolve. We also analyze the multi-parameter crosstalk using scattering radiation patterns.The linearized waveform inversion for VTI-elastic media presented in this article provides quantitative information of the rock properties that has the potential to help identify hydrocarbon reservoirs.

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.

Two‐dimensional anisotropic acoustic wave modelling using the support operator method

AbstractWe developed an algorithm to simulate two‐dimensional frequency domain acoustic‐wave response in a transversely isotropic medium with a tilted symmetry axis. The algorithm employs a support operator finite‐difference method for modelling. This method constructs a nine‐point stencil finite‐difference scheme for second‐order elliptic equations for generalized anisotropic physical properties. The medium's properties are described as P‐wave velocity on the symmetric axis, density, Thomsen's anisotropic parameters (epsilon and delta) and the tilt angle. The benchmarking analysis of the modelled amplitude is illustrated using an isotropic whole‐space model. Several synthetic experiments are conducted to evaluate the accuracy of the scheme for anisotropic models. The results suggest that the developed algorithm simulates the P‐wave solution and the fictitious S‐wave mode as reported in the literature. Simulation for a heterogeneous model with a spatially varying tilt angle of the medium symmetry axis is performed to ascertain the algorithm's robustness. The outcomes of the numerical experiments demonstrate that the developed algorithm can accurately simulate the frequency domain response of acoustic waves in the tilted anisotropic media.

Spherical-wave elastic inversion in transversely isotropic media with a vertical symmetry axis

SUMMARY Although subsurface media are usually assumed to be isotropic, anisotropy is ubiquitous in crustal rocks and leads to the variation of seismic response with direction. Transversely isotropic media with a vertical symmetry axis (VTI media) are widely found in the real world, such as in textured shale reservoirs. Plane-wave reflection coefficients (PRCs) in VTI media have been widely exploited in amplitude variation with offset (AVO) inversion to estimate the elastic and anisotropy parameters of subsurface media. However, the PRCs in VTI media meet some fundamental problems, especially at near-critical or post-critical incidence angles where the spherical-wave effect is significant. To consider the wave front curvature, a complex spherical-wave reflection coefficient (SRC) in VTI media is derived. To better understand the spherical-wave seismic response in VTI media, we investigate the dependence of the complex SRC on frequency, reflector depth and Thomsen anisotropy parameters ($\varepsilon $ and $\delta $). Based on a complex convolution model, a spherical-wave AVO inversion approach in VTI media is proposed to estimate the vertical (symmetry-axis) compressional and shear wave velocities (P and S waves), density and Thomsen anisotropy parameters from observed seismic data with different incidence angle and frequency components. Synthetic data with Gaussian random noise are used to verify the robustness of the spherical-wave AVO inversion approach in VTI media. Field data examples show that the proposed approach can produce reasonable inversion results that match well with the well-logging data.

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.

Nondestructive measurement of anisotropic elastic constants of selective laser melted 316L based on a tri-mode ultrasonic method

Selective laser melting (SLM) technology can rapidly form metal parts with complex geometry, but the internal microstructure and material properties are significantly different from those of traditionally manufactured parts, such as obvious anisotropy and inhomogeneity. This paper proposed a tri-mode ultrasonic system to nondestructively measure the anisotropic elastic constants of 316L stainless steel prepared by SLM. Firstly, the quantitative relationship between ultrasonic wave velocities and anisotropic elastic constants was derived for elliptical anisotropic materials. The anisotropic elastic constants and Thomsen anisotropic parameters can be determined by measuring four ultrasonic wave velocities , , , and . Therefore, a tri-mode ultrasonic transducer was designed and fabricated to build the elastic constant ultrasonic measurement system. Then, the elastic constants and Thomsen anisotropic parameters of SLM samples were measured, and the key process parameters’ influence was discussed. The standard static tensile test validated the result of the ultrasonic test, and the maximum relative error was 6.94%. This study will provide a novel nondestructive measurement method and system of anisotropic elastic constants to ensure internal quality and improve the additive manufacturing process.

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 shales: The roles of crack alignment and compliance ratio

Cracks, broadly defined as compliant discontinuities, are a major cause of elastic anisotropy. However, few models are available for quantifying crack properties relevant to anisotropy. We developed a rock-physics model to quantify crack angular distribution and normal-to-tangential compliance ratio from pressure-dependent acoustic velocities measured in the laboratory. Our model uses a rectangular function of variable width and amplitude to extract the maximum dip angle of cracks, which is a direct quantification of crack alignment relative to the bedding plane. We tested the model on an organic-rich shale data set and confirmed that crack alignment and compliance ratio strongly impact Thomsen anisotropy parameters, thus demonstrating the model as a useful tool for better understanding how cracks affect elastic anisotropy.

Bayesian Deterministic Inversion Based on the Exact Reflection Coefficients Equations of Transversely Isotropic Media With a Vertical Symmetry Axis

Unconventional reservoirs usually have strong anisotropy. Generally, they can be regarded as transversely isotropic media with a vertical symmetry axis [(VTI) media] in the absence of fractures. Therefore, it is of great significance to develop high-accuracy inversion method for VTI media. The three elastic parameters and two Thomsen anisotropy parameters of VTI media are usually obtained by indirect calculation or inversion methods based on the approximate formulas. However, the cumulative errors caused by indirect calculation and low calculation accuracy of the approximate formulas limit the estimation accuracy of these parameters. In this article, we propose a new method based on the exact reflection coefficients equations (ERCEs) of VTI media to improve the inversion accuracy of these elastic and anisotropy parameters. The new method adopts the Bayesian deterministic inversion (BDI) strategy to solve the inversion problem, which becomes highly nonlinear when using the ERCEs. We analyze the feasibility of the new method using the residual function maps (RFMs) and eigenvalue analysis of Hessian matrix. The analysis results show that the BDI strategy can well solve a series of problems brought by the ERCEs for nonlinear inversion, such as strong nonlinearity and more target parameters need to invert. Both synthetic and field data examples show that the proposed method can accurately estimate the elastic and anisotropy parameters, which verifies the feasibility and effectiveness of the method.

Linearized Frequency-Dependent Reflection Coefficient and Attenuated Anisotropic Characteristics of Q-VTI Model

Seismic wave exhibits the characteristics of anisotropy and attenuation while propagating through the fluid-bearing fractured or layered reservoirs, such as fractured carbonate and shale bearing oil or gas. We derive a linearized reflection coefficient that simultaneously considers the effects of anisotropy and attenuation caused by fractures and fluids. Focusing on the low attenuated transversely isotropic medium with a vertical symmetry axis (Q-VTI) medium, we first express the complex stiffness tensors based on the perturbation theory and the linear constant Q model at an arbitrary reference frequency, and then we derive the linearized approximate reflection coefficient of P to P wave. It decouples the P- and S-wave inverse quality factors, and Thomsen-style attenuation-anisotropic parameters from complex P- and S-wave velocity and complex Thomsen anisotropic parameters. By evaluating the reflection coefficients around the solution point of the interface of two models, we analyze the characteristics of reflection coefficient vary with the incident angle and frequency and the effects of different Thomsen anisotropic parameters and attenuation factors. Moreover, we realize the simultaneous inversion of all parameters in the equation using an actual well log as a model. We conclude that the derived reflection coefficient may provide a theoretical tool for the seismic wave forward modeling, and again it can be implemented to predict the reservoir properties of fractures and fluids based on diverse inversion methods of seismic data.

Prediction of sweet spots in tight sandstone reservoirs based on anisotropic frequency-dependent AVO inversion

Abstract The frequency-dependent amplitude-versus-offset (FAVO) method has become a practical method for fluid detection in sand reservoirs. At present, most FAVO inversions are based on the assumption that reservoirs are isotropy, but the application effect is not satisfactory for fractured reservoirs. Hence, we analyse the frequency variation characteristics of anisotropy parameters in tight sandstone reservoirs based on a new petrophysical model, and propose a stepwise anisotropic FAVO inversion method to extract frequency-dependent attributes from prestack seismic field data. First, we combine the improved Brie's law with the fine-fracture model to analyse frequency-dependent characteristics of velocities and Thomsen anisotropy parameters at different gas saturations and fracture densities. Then, we derive an anisotropic FAVO inversion algorithm based on Rüger's approximation formula and propose a stepwise anisotropic FAVO inversion method to obtain the dispersions of anisotropy parameters. Finally, we propose a method that combines the inversion spectral decomposition with the stepwise anisotropy FAVO inversion and apply it to tight sand reservoirs in the Xinchang area. We use P-wave velocity dispersion and anisotropy parameter ε dispersion to optimise favourable areas. Numerical analysis results show that velocity dispersion of the P-wave is sensitive to fracture density, which can be used for fracture prediction in fractured reservoirs. In contrast, anisotropic parameter dispersion is sensitive to gas saturation and can be used for fluid detection. The seismic data inversion results show that velocity dispersion of the P-wave and anisotropic parameter dispersion are sensitive to fractured reservoirs in the second member of Xujiahe Group, which is consistent with logging interpretation results.

Estimation of transversely isotropic formation parameters using flexural and quadrupole dispersion data in the fast formation

Borehole acoustic logging plays an important role in inverting the five Thomsen parameters of many formations characterized as a transversely isotropic medium with a vertical axis of symmetry (VTI). In general, these parameters are obtained under certain assumptions, and the formation type defined by the relation of Thomsen parameters is not taken into consideration. We have developed a method to determine the Thomsen parameters of all kinds of fast VTI formations from the first-order flexural and first-order quadrupole dispersion data by dividing the Thomsen parameters inversion process into two parts: low-frequency asymptotic slowness inversion of the flexural or quadrupole dispersion and Thomsen anisotropy parameter inversion by a constrained evolutionary optimization algorithm. Compared with traditional approaches, the new method is not only independent of any assumed correlation among the Thomsen anisotropy parameters but it also provides a more accurate result than the inversion of either of them alone for both kinds of fast VTI formations. The inversion results obtained from either the flexural or quadrupole dispersion data alone are incorrect if the formation type is incorrectly identified, such as misidentifying an isotropic formation as a special VTI formation. The new method can not only identify the type of fast VTI formation, but it also obtains the shear slowness in a special VTI formation, which cannot be obtained by existing methods. Accordingly, the application to synthetic examples validates the significance and necessity of the proposed joint method in inversion of the Thomsen parameters.

Frequency-dependent anisotropy in partially saturated porous rock with multiple sets of mesoscale fractures

SUMMARY Accurate modelling of the frequency-dependence of seismic wave velocity related to fracture system and fluid content is crucial to the quantitative interpretation of seismic data in fractured reservoirs. Both mesoscale fractures and patchy saturation effects can cause significant velocity dispersion and attenuation in the seismic frequency band due to wave-induced fluid flow (WIFF) mechanism. Considering the coupled impact of ‘mesoscale fractures’ and ‘patchy saturation’, we derive expressions for the frequency-dependent anisotropy in partially saturated porous rock containing two fracture sets with different orientations, sizes and connectivities. Especially, we simplify the rock-physics model as an orthorhombic (ORT) media by assuming the mesoscale fractures to be orthogonal and give the explicit expressions for frequency-dependent elastic constants. Finally, we give the expressions for the frequency-dependent phase velocity in patchy saturated and fractured ORT media and investigate the effect of patchy saturation on P-wave velocity at different polar and azimuth angles. In this paper, we investigate the effects of fluid saturation and fluid pressure on frequency-dependent velocities and Thomsen anisotropy parameters. Also, the effect of the relative permeability is very noticeable. The relaxation frequency can be lower in partially saturated fractured rocks compared with the fully saturated case, which makes the rock have a larger stiffness. The non-monotonic relationships between frequency-dependent anisotropy and fluid saturation add complexity to seismic forward modelling and inversion in reservoirs with complex fracture patterns.

Inversion of Fracture Parameters in Anisotropic Coal Seam

Inversion of coal seam fracture parameters is of great significance to identify and predict the distribution state of coal seam fractures, and also provides certain theoretical support for the detection and identification of coalbed methane. According to the formula of PP wave reflection coefficient, this paper uses genetic algorithm to carry out inversion of coal seam fracture parameters and Thomsen anisotropy parameters, and analyzes the accuracy and stability of inversion results. The results show that the genetic algorithm has high inversion accuracy for density, shear wave velocity and fracture density of background medium, and has certain reference significance for fracture identification.

Modeling of Compaction Trends of Anisotropic Elastic Properties of Shales

AbstractModeling of anisotropic elastic moduli of shales is a long‐standing problem in rock physics. Every year, industry and academia conduct hundreds of measurements of elastic and geomechanical properties of shales complemented with detailed mineralogical analysis, electron microscopy, and microtomographic images. However, a predictive theoretical model, which would be able to predict elastic properties of shale based on its detailed characterization, is still lacking. This is due to the complex multicomponent nature of shales which leads to multiparametric character of the modeling. It is extremely challenging to separate and investigate effects of these multiple parameters individually on natural shale samples. In this work, we analyze a unique data set of elastic properties measured on two artificial shales compacted with carefully controlled mineralogy, microstructure, and porosity. We developed a new modeling approach that consists of four steps that take into account effects of (1) the orientation of clay particles, (2) porosity, (3) silt fraction, and (4) the stiffness of contacts between clay particles. Three of these four steps require only measurable parameters and, only at the final step, the effect of unknown parameters (namely, contact properties of clay platelets) are accounted for. Further, the obtained contact properties are shown to be in a good agreement with the contact properties estimated for natural shale samples. Finally, we demonstrate the effect of each of these identified parameters on elastic coefficients and Thomsen's anisotropic parameters.

Velocity anisotropy analysis for shale lithology of the complex geological section in Jaisalmer sub-basin, India

Measurement of velocity anisotropy is an essential parameter for capturing the heterogeneity of sub-surface geology to characterise the hydrocarbon-bearing reservoir. The incorporation of velocity anisotropy parameters during the preparation of the 3D velocity model represents a robust result in a challenging geological set-up during interpretation. Generally, we can observe that the shale formation is more sensitive to velocity anisotropy response in comparison with other formations such as sandstone, siltstone for clastic reservoir or limestone and dolomite for carbonate reservoir. This study was performed mostly in the high amount shale section mixed with limestone and claystone of the Jaisalmer sub-basin area which lies in the western part of India. The preparation of the velocity model for frequent changes of lithology in the clastic and carbonate reservoir is challenging due to several changes of velocity which show a limitation in the result of the gridded velocity model. The objective of this study is to capture the changes of compressional and shear wave velocity in mixed lithology of the significant shale formation. The idea was due to the inclusion of the anisotropy incorporated changed velocity during the preparation of the gridded velocity model for correctly identified lithology. The shale formation which is the zone of investigation of the current study is situated over a carbonate sequence, and an estimated velocity anisotropy factor of this shale formation will contribute significantly during the cumulative study of velocity modelling of all formation. The current study shows that shale formation shows the character of orthorhombic anisotropy; however, this study was performed based on significant changes of well log data and related effects of vertically transverse isotropic parameters of the shale formation. The fundamental Thomsen anisotropy parameters were estimated by capturing the deviation of five independent stiffness coefficients. Significant changes in evaluated shale velocity were observed after the incorporation of the estimated Thomsen parameter in velocity values.

Azimuthally anisotropic elastic impedance parameterisation and inversion for anisotropy in weakly orthorhombic media

ABSTRACTA single set of vertically aligned fractures embedded in a horizontal fine layering may be considered a long-wavelength effective orthorhombic medium. Estimation of Thomsen anisotropic parameters and fracture weaknesses is of vital importance to characterise the orthorhombic anisotropy in orthorhombic media. Our goal is to demonstrate an azimuthally anisotropic elastic impedance parameterisation and Bayesian inversion approach to estimate orthorhombic anisotropy in a fractured reservoir. Previously derived reflection coefficient equations are limited to the assumption of a horizontally layered medium, which may be not suitable for orthorhombic media. Based on elastic inverse scattering theory, we first derive a linearised PP-wave reflection coefficient equation in terms of orthorhombic anisotropic parameters containing Thomsen anisotropic parameters and fracture weaknesses in a weakly orthorhombic medium. Using a new parameterisation for these orthorhombic anisotropic parameters, we derive an anisotropic elastic impedance equation variation with angles of incidence and azimuth. We perform anisotropic inversion in a Bayesian framework with the regularisation of Cauchy-sparse constraint and smoothing-model constraint. Using the nonlinear iteratively reweighted least squares algorithm, we iteratively estimate the orthorhombic anisotropy characterised by Thomsen anisotropic parameters and fracture weaknesses. Tests on both synthetic and real data validate the proposed approach.

Thomsen anisotropy parameters calculation in the Baltic Basin shale gas formations based on the measurement of well logging data, some attempts

Thomsen anisotropy parameters calculation in the Baltic Basin shale gas formations based on the measurement of well logging data, some attempts

Experimental study of fracture size effect on elastic-wave velocity dispersion and anisotropy

We have designed transversely isotropic models containing penny-shaped rubber inclusions, with the crack diameters ranging from 2.5 to 6.2 mm to study the influence of fracture size on seismic velocity under controlled conditions. Three pairs of transducers with different frequencies (0.5, 0.25, and 0.1 MHz) are used for P- and S-wave ultrasonic sounding, respectively. The P-wave measurements indicate that the scattering effect is dominant when the waves propagate perpendicular to the fractures. Our experimental results demonstrate that when the wavelength-to-crack-diameter ratio ([Formula: see text]) is larger than 14, the P-wave velocity can be described predominantly by the effective medium theory. Although the ratio is larger than four, the S-wave velocity is close to the equivalent medium results. When [Formula: see text] < 14 or [Formula: see text] is < 4, the elastic velocity is dominated by scattering. The magnitudes of the Thomsen anisotropic parameters [Formula: see text] and [Formula: see text] are scale and frequency dependent on the assumption that the transversely isotropic models are vertical transversely isotropic medium. Furthermore, we compare the experimental velocities with the Hudson theory. The results illustrate that there is a good agreement between the observed P-wave velocity and the Hudson theory when [Formula: see text] > 7 in the directions parallel and perpendicular to the fractures. For small fracture diameters, however, the P-wave velocity perpendicular to the fractures predicted from the Hudson theory is not accurate. When [Formula: see text] < 4, there is good agreement between the experimental fast S-wave velocity and the Hudson theory, whereas the experimental slow S-wave velocity diverges with the Hudson theory. When [Formula: see text] > 4, the deviation of fast and slow S-wave velocities with the Hudson prediction is stable.