Accounting for pressure-dependent ultrasonic beam skew in transversely isotropic rocks: combining modelling and measurement of anisotropic wave speeds

SUMMARY The intrinsic anisotropy of rock influences the paths of propagating seismic waves and indicates mineralogical texture and strains; and as such it is important that laboratory measurements of such properties be fully understood. Usually, when studying anisotropy, ultrasonic wave speeds are measured in a variety of strategic directions and, subsequently transformed to the dynamic elastic moduli using symmetry-appropriate formula. For transversely isotropic rocks the moduli are ideally found by measuring wave speeds in directions vertical, parallel and oblique to the foliation or bedding using finite-width ultrasonic transducers. An important, but ignored, complication is that at oblique angles the ultrasonic beam unavoidably deviates, or skews, away from the transmitter's normal axis making proper wave speed determinations difficult. The pressure dependence of the wave speeds further confounds finding a solution as skew angles, too, vary with confining pressure. We develop a new technique that incorporates dual ultrasonic receivers to account for and mitigate the effects of the pressure-dependent beam skew problem. Anisotropy measurements to 200 MPa hydrostatic confining pressure combined with recent beam modeling algorithms illustrate the errors obtained in the determined wave speeds that are subsequently magnified in calculating the full set of elastic stiffnesses. In materials with P-wave anisotropies near 30 per cent the error introduced by ignoring beam skew exceeds the transit time picking errors by more than a factor of three, these propagate to much larger errors in the stiffnesses particularly for C13 and the dynamic elastic moduli referred to C13. Meanwhile, shortening the sample or enlarging the transmitter size is not suggested to counter the beam skew issue because it reduces the beam skew effect but increases the diffraction effect.

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The transversely isotropic poroelastic wave equation including the Biot and the squirt mechanisms: Theory and application

Geophysics ◽

10.1190/1.1444132 ◽

1997 ◽

Vol 62 (1) ◽

pp. 309-318 ◽

Cited By ~ 47

Author(s):

Jorge O. Parra

Keyword(s):

Wave Equation ◽

Fluid Flow ◽

Analytical Solution ◽

Seismic Waves ◽

Transversely Isotropic ◽

P Wave ◽

Fluid Properties ◽

Permeability Anisotropy ◽

Attenuation And Dispersion ◽

Maximum Permeability

The transversely isotropic poroelastic wave equation can be formulated to include the Biot and the squirt‐flow mechanisms to yield a new analytical solution in terms of the elements of the squirt‐flow tensor. The new model gives estimates of the vertical and the horizontal permeabilities, as well as other measurable rock and fluid properties. In particular, the model estimates phase velocity and attenuation of waves traveling at different angles of incidence with respect to the principal axis of anisotropy. The attenuation and dispersion of the fast quasi P‐wave and the quasi SV‐wave are related to the vertical and the horizontal permeabilities. Modeling suggests that the attenuation of both the quasi P‐wave and quasi SV‐wave depend on the direction of permeability. For frequencies from 500 to 4500 Hz, the quasi P‐wave attenuation will be of maximum permeability. To test the theory, interwell seismic waveforms, well logs, and hydraulic conductivity measurements (recorded in the fluvial Gypsy sandstone reservoir, Oklahoma) provide the material and fluid property parameters. For example, the analysis of petrophysical data suggests that the vertical permeability (1 md) is affected by the presence of mudstone and siltstone bodies, which are barriers to vertical fluid movement, and the horizontal permeability (1640 md) is controlled by cross‐bedded and planar‐laminated sandstones. The theoretical dispersion curves based on measurable rock and fluid properties, and the phase velocity curve obtained from seismic signatures, give the ingredients to evaluate the model. Theoretical predictions show the influence of the permeability anisotropy on the dispersion of seismic waves. These dispersion values derived from interwell seismic signatures are consistent with the theoretical model and with the direction of propagation of the seismic waves that travel parallel to the maximum permeability. This analysis with the new analytical solution is the first step toward a quantitative evaluation of the preferential directions of fluid flow in reservoir formation containing hydrocarbons. The results of the present work may lead to the development of algorithms to extract the permeability anisotropy from attenuation and dispersion data (derived from sonic logs and crosswell seismics) to map the fluid flow distribution in a reservoir.

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Empirical Formula for Dynamic Biot Coefficient of Sandstone Samples from South-West of Poland

Energies ◽

10.3390/en14175514 ◽

2021 ◽

Vol 14 (17) ◽

pp. 5514

Author(s):

Dariusz Knez ◽

Mohammad Ahmad Mahmoudi Zamani

Keyword(s):

Elastic Moduli ◽

Confining Pressure ◽

Open Pit ◽

High Porosity ◽

Empirical Correlations ◽

Biot Coefficient ◽

South West ◽

Dynamic Elastic Moduli ◽

Skempton Coefficient ◽

Low Porosity

In this research, two empirical correlations have been introduced to calculate the dynamic Biot coefficients of low-porosity and high-porosity sandstone samples from two open pit mines located in South-West Poland. The experiments were conducted using an acoustic velocity measurement apparatus. Under the undrained condition, firstly, the confining pressure was increased in increments of 200 psi, and the values of pore pressure and dynamic elastic modulus were recorded. This experiment was continued until the Skempton coefficient remained in the range of 0.98–1. Secondly, an experiment on the same sample was conducted under drained conditions, and the corresponding dynamic elastic moduli were calculated. Then, using the calculated dynamic elastic moduli, the dynamic Biot coefficient was determined for each sample under different confining pressure. Finally, two empirical correlations were formulated for each sandstone category. The results demonstrate that, as the confining pressure increases, the Biot coefficient decreases from 0.79 to 0.50 and from 0.84 to 0.45 for low-porosity and high-porosity samples, respectively. Furthermore, as the porosity increases, the sandstone behavior increasingly approaches that of soil. The empirical correlations can be used for sandstone formations with the same porosity in projects where there is not a measurement method for the Biot coefficient.

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Relationships between Dynamic Elastic Moduli in Shale Reservoirs

Energies ◽

10.3390/en13226001 ◽

2020 ◽

Vol 13 (22) ◽

pp. 6001

Author(s):

Sheyore John Omovie ◽

John P. Castagna

Keyword(s):

Organic Carbon ◽

Shear Modulus ◽

Elastic Moduli ◽

Volume Fraction ◽

P Wave ◽

Wave Velocities ◽

Shear Wave Velocities ◽

Average Formation ◽

Dynamic Elastic Moduli ◽

Shale Reservoirs

Sonic log compressional and shear-wave velocities combined with logged bulk density can be used to calculate dynamic elastic moduli in organic shale reservoirs. We use linear multivariate regression to investigate modulus prediction when shear-wave velocities are not available in seven unconventional shale reservoirs. Using only P-wave modulus derived from logged compressional-wave velocity and density as a predictor of dynamic shear modulus in a single bivariate regression equation for all seven shale reservoirs results in prediction standard error of less than 1 GPa. By incorporating compositional variables in addition to P-wave modulus in the regression, the prediction standard error is reduced to less than 0.8 GPa with a single equation for all formations. Relationships between formation bulk and shear moduli are less well defined. Regressing against formation composition only, we find the two most important variables in predicting average formation moduli to be fractional volume of organic matter and volume of clay in that order. While average formation bulk modulus is found to be linearly related to volume fraction of total organic carbon, shear modulus is better predicted using the square of the volume fraction of total organic carbon. Both Young’s modulus and Poisson’s ratio decrease with increasing TOC while increasing clay volume decreases Young’s modulus and increases Poisson’s ratio.

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Upscaling in orthorhombic media: Behavior of elastic parameters in heterogeneous fractured earth

Geophysics ◽

10.1190/geo2015-0392.1 ◽

2016 ◽

Vol 81 (3) ◽

pp. C113-C126 ◽

Cited By ~ 12

Author(s):

Yuriy Ivanov ◽

Alexey Stovas

Keyword(s):

Seismic Waves ◽

Homogeneous Medium ◽

Transversely Isotropic ◽

Processing Parameters ◽

P Wave ◽

Orthorhombic Symmetry ◽

Weak Anisotropy ◽

Fluid Saturation ◽

Fractured Medium ◽

Anisotropic Layers

A stack of horizontal homogeneous elastic arbitrary anisotropic layers in welded contact in the long-wavelength limit is equivalent to an elastic anisotropic homogeneous medium. Such a medium is characterized by an effective average description adhering to previously derived closed-form formalism. We have used this formalism to study three different inhomogeneous orthorhombic (ORT) models that could represent real geologic scenarios. We have determined that a stack of thin orthorhombic layers with arbitrary azimuths of vertical symmetry planes can be approximated by an effective orthorhombic medium. The most suitable approach for this is to minimize the misfit between the effective anisotropic medium, monoclinic in that case, and the desirable orthorhombic medium. The second model is an interbedding of VTI (transversely isotropic with a vertical symmetry axis) layers with the same layers containing vertical fractures (shales are intrinsically anisotropic and often fractured). We have derived a weak-anisotropy approximation for important P-wave processing parameters as a function of the relative amount of the fractured lithology. To accurately characterize fractures, inversion for the fracture parameters should use a priori information on the relative amount of a fractured medium. However, we have determined that the cracks’ fluid saturation can be estimated without prior knowledge of the relative amount of the fractured layer. We have used field well-log data to demonstrate how fractures can be included in the interval of interest during upscaling. Finally, the third model that we have considered is a useful representation of tilted orthorhombic medium in the case of two-way propagation of seismic waves through it. We have derived a weak anisotropy approximation for traveltime parameters of the reflected P-wave that propagates through a stack of thin beds of tilted orthorhombic symmetry. The tilt of symmetry planes in an orthorhombic medium significantly affects the kinematics of the reflected P-wave and should be properly accounted for to avoid mispositioning of geologic structures in seismic imaging.

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The Effective Elastic Moduli of Porous Granular Rocks

Journal of Applied Mechanics ◽

10.1115/1.3157738 ◽

1981 ◽

Vol 48 (4) ◽

pp. 803-808 ◽

Cited By ~ 296

Author(s):

P. J. Digby

Keyword(s):

Elastic Wave ◽

Elastic Moduli ◽

Confining Pressure ◽

Spherical Particles ◽

Effective Elastic Moduli ◽

Wave Speeds ◽

Small Areas

A porous granular rock is modeled by an aggregate of identical, randomly stacked, spherical particles. Contacting particles are initially bonded together across small areas. A theory is developed for the deformation of two such spherical particles under equal and opposite forces acting through the line joining their centers. The theory is used to calculate the effective elastic moduli of the medium. The dependence of the derived elastic wave speeds on the confining pressure and adhesion radius of the contacting particles is then predicted.

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The effect of the extent of freezing on seismic velocities in unconsolidated permafrost

Geophysics ◽

10.1190/1.1442181 ◽

1986 ◽

Vol 51 (6) ◽

pp. 1285-1290 ◽

Cited By ~ 82

Author(s):

Robert W. Zimmerman ◽

Michael S. King

Keyword(s):

Elastic Moduli ◽

Seismic Waves ◽

Compressional Wave ◽

Seismic Velocities ◽

Wave Speeds ◽

Direct Measurements ◽

Two Parameters ◽

Decreasing Functions ◽

Increasing Function ◽

Quartz Grains

We develop a model to relate velocities of seismic waves in unconsolidated permafrost is idealized as an assemblage of spherical quartz grains imbedded in a matrix composed of spherical water inclusions in ice. The theory of Kuster and Toksöz, based on wave‐scattering considerations, is used to determine the effective elastic moduli, and hence the wave speeds. The Hasin-Shtrikman theoretical bounds on the elastic moduli of heterogeneous materials and considerations establish the plausibility of the model. The model predicts [Formula: see text] and [Formula: see text] to be decreasing functions of both the porosity and the water‐to‐ice ratio, and the ratio [Formula: see text] to be an increasing function of these two parameters. The theory is then applied to laboratory measurements of shear‐ and compressional‐wave velocities in 23 permafrost samples from different sites in the Beaufort Sea, Mackenzie River Valley, and Canadian Arctic islands. Although no direct measurements were made of the extents of freezing in these samples, the data are consistent with the predictions of the model. We show the theory can be used to predict the extent of freezing of the water in the pore spaces, based on knowledge of the porosity and either of the two wave speeds.

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PERTURBATION OF PHASE VELOCITIES IN ELASTIC ORTHORHOMBIC MEDIA

Geophysics ◽

10.1190/geo2020-0820.1 ◽

2021 ◽

pp. 1-109

Author(s):

Alexey Stovas ◽

Yuriy Roganov ◽

Vyacheslav Roganov

Keyword(s):

Seismic Waves ◽

Transversely Isotropic ◽

Second Order ◽

P Wave ◽

Order Perturbation ◽

Perturbation Approach ◽

S Wave ◽

Anisotropic Models ◽

S Waves ◽

Phase Velocities

The parameterization of anisotropic models is very important when focusing on specific signatures of seismic waves and reducing the parameters crosstalk involved in inverting seismic data. The parameterization is strongly dependent on the problem at hand. We propose a new parameterization for an elastic orthorhombic model with on-axes P- and S-wave velocities and new symmetric anelliptic parameters. The perturbation approach is well defined for P waves in acoustic orthorhombic media. In the elastic orthorhombic media, the P-wave perturbation coefficients are very similar to their acoustic counterparts. However, the S-waves perturbation coefficients are still unknown. The perturbation coefficients can be interpreted as sensitivity coefficients, and they are important in many applications. We apply the second-order perturbation in anelliptic parameters for P, S1 and S2 wave phase velocities in elastic orthorhombic model. We show that using the conventional method some perturbation coefficients for S waves are not defined in the vicinity of the singularity point in an elliptical background model. Thus, we propose an alternative perturbation approach that overcomes this problem. We compute the first- and second-order perturbation coefficients for P and S waves. The perturbation-based approximations are very accurate for P and S waves compared with exact solutions, based on a numerical example. The reductions to transversely isotropic and acoustic orthorhombic models are also considered for analysis. We also show how perturbations in anelliptic parameters affect S-wave triplications in an elastic orthorhombic model.

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