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

Geophysics ◽  
1997 ◽  
Vol 62 (1) ◽  
pp. 309-318 ◽  
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.

Migration of P‐wave reflection data in transversely isotropic media

Geophysics ◽  
1994 ◽  
Vol 59 (4) ◽  
pp. 591-596 ◽  
Author(s):  
Suhas Phadke ◽  
S. Kapotas ◽  
N. Dai ◽  
Ernest R. Kanasewich
Keyword(s):  
Wave Equation ◽  
Dispersion Relation ◽  
Synthetic Data ◽  
Transversely Isotropic ◽  
Anisotropic Media ◽  
Limited Range ◽  
Horizontal Velocity ◽  
P Wave ◽  
Transversely Isotropic Media ◽  
Isotropic Media

Wave propagation in transversely isotropic media is governed by the horizontal and vertical wave velocities. The quasi‐P(qP) wavefront is not an ellipse; therefore, the propagation cannot be described by the wave equation appropriate for elliptically anisotropic media. However, for a limited range of angles from the vertical, the dispersion relation for qP‐waves can be approximated by an ellipse. The horizontal velocity necessary for this approximation is different from the true horizontal velocity and depends upon the physical properties of the media. In the method described here, seismic data is migrated using a 45-degree wave equation for elliptically anisotropic media with the horizontal velocity determined by comparing the 45-degree elliptical dispersion relation and the quasi‐P‐dispersion relation. The method is demonstrated for some synthetic data sets.

scholarly journals Seismic wave attenuation and dispersion resulting from wave-induced flow in porous rocks — A review

Geophysics ◽  
2010 ◽  
Vol 75 (5) ◽  
pp. 75A147-75A164 ◽  
Author(s):  
Tobias M. Müller ◽  
Boris Gurevich ◽  
Maxim Lebedev
Keyword(s):  
Fluid Flow ◽  
Velocity Dispersion ◽  
Wave Attenuation ◽  
Field Measurements ◽  
Theoretical Models ◽  
Seismic Attenuation ◽  
P Wave ◽  
Induced Flow ◽  
Wave Induced ◽  
Attenuation And Dispersion

One major cause of elastic wave attenuation in heterogeneous porous media is wave-induced flow of the pore fluid between heterogeneities of various scales. It is believed that for frequencies below [Formula: see text], the most important cause is the wave-induced flow between mesoscopic inhomogeneities, which are large compared with the typical individual pore size but small compared to the wavelength. Various laboratory experiments in some natural porous materials provide evidence for the presence of centimeter-scale mesoscopic heterogeneities. Laboratory and field measurements of seismic attenuation in fluid-saturated rocks provide indications of the role of the wave-induced flow. Signatures of wave-induced flow include the frequency and saturation dependence of P-wave attenuation and its associated velocity dispersion, frequency-dependent shear-wave splitting, and attenuation anisotropy. During the last four decades, numerous models for attenuation and velocity dispersion from wave-induced flow have been developed with varying degrees of rigor and complexity. These models can be categorized roughly into three groups ac-cording to their underlying theoretical framework. The first group of models is based on Biot’s theory of poroelasticity. The second group is based on elastodynamic theory where local fluid flow is incorporated through an additional hydrodynamic equation. Another group of models is derived using the theory of viscoelasticity. Though all models predict attenuation and velocity dispersion typical for a relaxation process, there exist differences that can be related to the type of disorder (periodic, random, space dimension) and to the way the local flow is incorporated. The differences manifest themselves in different asymptotic scaling laws for attenuation and in different expressions for characteristic frequencies. In recent years, some theoretical models of wave-induced fluid flow have been validated numerically, using finite-difference, finite-element, and reflectivity algorithms applied to Biot’s equations of poroelasticity. Application of theoretical models to real seismic data requires further studies using broadband laboratory and field measurements of attenuation and dispersion for different rocks as well as development of more robust methods for estimating dissipation attributes from field data.

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

Geophysics ◽  
2016 ◽  
Vol 81 (3) ◽  
pp. C113-C126 ◽  
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.

Poroelastic model to relate seismic wave attenuation and dispersion to permeability anisotropy

Geophysics ◽  
2000 ◽  
Vol 65 (1) ◽  
pp. 202-210 ◽  
Author(s):  
Jorge O. Parra
Keyword(s):  
Seismic Waves ◽  
Velocity Dispersion ◽  
Wave Attenuation ◽  
Test Site ◽  
Compressional Wave ◽  
P Wave ◽  
Low Velocity ◽  
Stiffness Constant ◽  
Principal Axes ◽  
Permeability Anisotropy

A transversely isotropic model with a horizontal axis of symmetry, based on the Biot and squirt‐flow mechanisms, predicts seismic waves in poroelastic media. The model estimates velocity dispersion and attenuation of waves propagating in the frequency range of crosswell and high‐resolution reverse vertical seismic profiling (VSP) (250–1250 Hz) for vertical permeability values much greater than horizontal permeability parameters. The model assumes the principal axes of the stiffness constant tensor are aligned with the axes of the permeability and squirt‐flow tensors. In addition, the unified Biot and squirt‐flow mechanism (BISQ) model is adapted to simulate cracks in permeable media. Under these conditions, the model simulations demonstrate that the preferential direction of fluid flow in a reservoir containing fluid‐filled cracks can be determined by analyzing the phase velocity and attenuation of seismic waves propagating at different azimuth and incident angles. As a result, the fast compressional wave can be related to permeability anisotropy in a reservoir. The model results demonstrate that for a fast quasi-P-wave propagating perpendicular to fluid‐filled cracks, the attenuation is greater than when the wave propagates parallel to the plane of the crack. Theoretical predictions and velocity dispersion of inter‐well seismic waves in the Kankakee Limestone Formation at the Buckhorn test site (Illinois) demonstrate that the permeable rock matrix surrounding a low‐velocity heterogeneity contains vertical cracks.

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

2020 ◽  
Vol 221 (1) ◽  
pp. 231-250 ◽  
Author(s):  
Wei Li ◽  
Douglas R Schmitt ◽  
Xiwei Chen
Keyword(s):  
Elastic Moduli ◽  
Seismic Waves ◽  
Transversely Isotropic ◽  
Confining Pressure ◽  
P Wave ◽  
Diffraction Effect ◽  
Finite Width ◽  
Ultrasonic Beam ◽  
Wave Speeds ◽  
Dynamic Elastic Moduli

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.

PERTURBATION OF PHASE VELOCITIES IN ELASTIC ORTHORHOMBIC MEDIA

Geophysics ◽  
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.

scholarly journals Frequency-dependent attenuation and dispersion caused by squirt flow: Three-dimensional numerical study

Geophysics ◽  
2020 ◽  
Vol 85 (3) ◽  
pp. MR129-MR145 ◽  
Author(s):  
Yury Alkhimenkov ◽  
Eva Caspari ◽  
Boris Gurevich ◽  
Nicolás D. Barbosa ◽  
Stanislav Glubokovskikh ◽  
...  
Keyword(s):  
Fluid Flow ◽  
Analytical Solution ◽  
Analytical Solutions ◽  
Numerical Study ◽  
Numerical Models ◽  
Pore Space ◽  
Fluid Pressure ◽  
Frequency Dependent ◽  
Attenuation And Dispersion ◽  
Dispersion And Attenuation

Seismic waves may exhibit significant dispersion and attenuation in reservoir rocks due to pore-scale fluid flow. Fluid flow at the microscopic scale is referred to as squirt flow and occurs in very compliant pores, such as grain contacts or microcracks, that are connected to other stiffer pores. We have performed 3D numerical simulations of squirt flow using a finite-element approach. Our 3D numerical models consist of a pore space embedded into a solid grain material. The pore space is represented by a flat cylinder (a compliant crack) whose edge is connected with a torus (a stiff pore). Grains are described as a linear isotropic elastic material, whereas the fluid phase is described by the quasistatic linearized compressible Navier-Stokes momentum equation. We obtain the frequency-dependent effective stiffness of a porous medium and calculate dispersion and attenuation due to fluid flow from a compliant crack to a stiff pore. We compare our numerical results against a published analytical solution for squirt flow and analyze the effects of its assumptions. Previous interpretation of the squirt flow phenomenon based mainly on analytical solutions is verified, and some new physical effects are identified. The numerical and analytical solutions agree only for the simplest model in which the edge of the crack is subjected to zero fluid pressure boundary condition while the stiff pore is absent. For the more realistic model that includes the stiff pore, significant discrepancies are observed. We identify two important aspects that need improvement in the analytical solution: the calculation of the frame stiffness moduli and the frequency dependence of attenuation and dispersion at intermediate frequencies.

scholarly journals Anisotropic viscoacoustic wave modelling in VTI media using frequency-dependent complex velocity

2020 ◽  
Author(s):  
Yabing Zhang ◽  
Yang Liu ◽  
Shigang Xu
Keyword(s):  
Wave Propagation ◽  
Wave Equation ◽  
Wave Equations ◽  
Transversely Isotropic ◽  
P Wave ◽  
Low Rank ◽  
Transversely Isotropic Media ◽  
Isotropic Media ◽  
Rank Decomposition ◽  
Low Rank Decomposition

Abstract Under the conditions of acoustic approximation and isotropic attenuation, we derive the pseudo- and pure-viscoacoustic wave equations from the complex constitutive equation and the decoupled P-wave dispersion relation, respectively. Based on the equations, we investigate the viscoacoustic wave propagation in vertical transversely isotropic media. The favourable advantage of these formulas is that the phase dispersion and the amplitude dissipation terms are inherently separated. As a result, we can conveniently perform the decoupled viscoacoustic wavefield simulations by choosing different coefficients. In the computational process, a generalised pseudo-spectral method and a low-rank decomposition scheme are adopted to calculate the wavenumber-domain and mixed-domain propagators, respectively. Because low-rank decomposition plays an important role in the simulated procedure, we evaluate the approximation accuracy for different operators using a linear velocity model. To demonstrate the effectiveness and the accuracy of our method, several numerical examples are carried out based on the new pseudo- and pure-viscoacoustic wave equations. Both equations can effectively describe the viscoacoustic wave propagation characteristics in vertical transversely isotropic media. Unlike the pseudo-viscoacoustic wave equation, the pure-viscoacoustic wave equation can produce stable viscoacoustic wavefields without any SV-wave artefacts.

scholarly journals An efficient Helmholtz solver for acoustic transversely isotropic media

Geophysics ◽  
2018 ◽  
Vol 83 (2) ◽  
pp. C75-C83 ◽  
Author(s):  
Zedong Wu ◽  
Tariq Alkhalifah
Keyword(s):  
Wave Equation ◽  
Transversely Isotropic ◽  
Anisotropic Media ◽  
P Wave ◽  
Seismic Exploration ◽  
S Wave ◽  
P Waves ◽  
Transversely Isotropic Media ◽  
Isotropic Media ◽  
Separable Approximation

The acoustic approximation, even for anisotropic media, is widely used in current industry imaging and inversion algorithms mainly because P-waves constitute most of the energy recorded in seismic exploration. The resulting acoustic formulas tend to be simpler, resulting in more efficient implementations, and they depend on fewer medium parameters. However, conventional solutions of the acoustic-wave equation with higher-order derivatives suffer from S-wave artifacts. Thus, we separate the quasi-P-wave propagation in anisotropic media into the elliptic anisotropic operator (free of the artifacts) and the nonelliptic anisotropic components, which form a pseudodifferential operator. We then develop a separable approximation of the dispersion relation of nonelliptic-anisotropic components, specifically for transversely isotropic media. Finally, we iteratively solve the simpler lower-order elliptical wave equation for a modified source function that includes the nonelliptical terms represented in the Fourier domain. A frequency-domain Helmholtz formulation of the approach renders the iterative implementation efficient because the cost is dominated by the lower-upper decomposition of the impedance matrix for the simpler elliptical anisotropic model. In addition, the resulting wavefield is free of S-wave artifacts and has a balanced amplitude. Numerical examples indicate that the method is reasonably accurate and efficient.

Sign in / Sign up

Export Citation Format

Share Document