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<jats:p> Separation filtering is incomplete even under the ideal synthetic condition of known power spectra of the regional and residual fields. I have designed some Wiener filters, which minimize the inevitable separation error, from previous statistical source models of Naidu, and Spector and Grant. This formulation includes the classic separation filters of Strakhov and of Elkins as Wiener filters. A proposed generalization of Wiener filters, denoted as uniformly suboptimum filters, quantitatively supports the statement that a wide span of separation problems may be solved adequately using some convenient, small standard filter family. A uniform random‐source model without assumed vertical correlations invokes upward continuation filters. In addition to this role as a Wiener filter, the upward continuation operator is given by elementary functions in both space and wavenumber domains, is numerically stable, and is also physically comprehensible when applied to real, nonrandom anomalies. In view of these distinguishing features, I propose to use the upward continuation operator to build a convenient, standard family of separation filters. </jats:p>

<jats:p>To decouple qP and qSV sheets of the slowness surface of a transversely isotropic (TI) medium, a sequence of rational approximations to the solution of the dispersion relation of a TI medium is introduced. Originally conceived to allow isotropic P-wave processing schemes to be generalized to encompass the case of qP-waves in transverse isotropy, the sequence of approximations was found to be applicable to qSV-wave processing as well, although a higher order of approximation is necessary for qSV-waves than for qP-waves to yield the same accuracy. The zeroth‐order approximation, about which all other approximations are taken, is that of elliptical TI, which contains the correct values of slowness and its derivative along and perpendicular to the medium’s axis of symmetry. Successive orders of approximation yield the correct values of successive orders of derivatives in these directions, thereby forcing the approximation into increasingly better fit at the intervening oblique angles. Practically, the first‐order approximation for qP-wave propagation and the second‐order approximation for qSV-wave propagation yield sufficiently accurate results for the typical transverse isotropy found in geological settings. After only slight modification to existing programs, the rational approximation allows for ray tracing, (f-k) domain migration, and split‐step Fourier migration in TI media—with little more difficulty than that encountered presently with such algorithms in isotropic media.</jats:p>

<jats:p> Seismic reflections depend on the contrasts of the elastic properties of the subsurface and their 3D geometry. As a result, interpreting seismic data for petrophysical rock properties requires a theoretical rock-physics model that links the seismic response to a rock’s velocity and density. Such a model is based on controlled experiments in which the petrophysical and elastic rock properties are measured on the same samples, such as in the wellbore. Using data from three wells drilled through a clastic offshore gas reservoir, we establish a theoretical rock-physics model that quantitatively explains these data. The modeling is based on the assumption that only three minerals are present: quartz, clay, and feldspar. To have a single rock-physics transform to quantify the well data in the entire intervals under examination in all three wells, we introduced field-specific elastic moduli for the clay. We then used the model to correct the measured shear-wave velocity because it appeared to be unreasonably low. The resulting model-derived Poisson’s ratio is much smaller than the measured ratio, especially in the reservoir. The associated synthetic amplitude variation with offset response appears to be consistent with the recorded seismic angle stacks. We have shown how rock-physics modeling not only helps us to correct the well data, but also allows us to go beyond the settings represented in the wells and quantify the seismic signatures of rock properties and conditions varying in a wider range using forward seismic modeling. </jats:p>

<jats:p> In unconventional reservoirs with significant organic content, the Greenberg-Castagna (GC) S-wave velocity prediction method does not yield accurate S-wave velocity predictions, with observed mean errors varying from 6% to 16% in a variety of unconventional reservoirs rich in organic content. This is because kerogen content is not explicitly taken into account in the GC S-wave velocity prediction method. Two alternative approaches for bedding-normal S-wave velocity prediction from P-wave velocity and other well logs in relatively straight holes drilled in unconventional reservoirs are investigated. The first method is purely empirical, requiring minimal information such as P-wave velocity and total organic carbon content. This approach implicitly accounts for compositional and stress effects on mudrock elasticity. The second method can be classified as a hybrid technique, comprising the following three steps: (1) computing a nonkerogen phase P-wave velocity using the Vernik-Kachanov (VK) model, (2) determination of the nonkerogen S-wave velocity from the GC approach, and (3) using a simplified VK model to mix the nonkerogen matrix with nanoporous kerogen to predict the bedding-normal S-wave velocity of the organic mudrock. The second method explicitly takes into account all the variables that control elastic properties of organic mudrock reservoirs. Tests in nine wells from seven different oil and gas shale reservoirs indicate that both methods have prediction accuracy better than 3% error when input data are accurate. </jats:p>

<jats:p> The mechanical basis of the popular Cole-Cole rheological model in viscoelasticity is investigated by using Lagrangian mechanics with nonlinear energy dissipation. The Cole-Cole model is usually viewed as a convenient way to fit the observed frequency-dependent attenuation and velocity-dispersion spectra, but its time-domain and numerical formulations are complex and contradict standard physical principles. For example, time-domain modeling of Cole-Cole media requires special mathematical tools such as fractional derivatives, convolutional integrals, and/or memory variables. Nevertheless, we find that Cole-Cole spectra naturally arise from conventional mechanics with nonlinear internal friction (non-Newtonian viscosity). The Lagrangian mechanical formulation is applied to a finite body (a rock sample in a laboratory experiment) and a wave-propagating medium, in both cases providing rigorous differential equations of motion and revealing the time- and frequency-independent material properties. The model also leads to a generalized Cole-Cole (GCC) model with multiple internal variables (relaxation mechanisms), similar to the generalized standard linear solid (GSLS). As a practical application, the GSLS and GCC models are compared on interpretations of recent P-wave attenuation and dispersion measurements on bitumen-sand samples in the laboratory. The GSLS and GCC models can be used to predict the observed strain/stress ratios with adequate accuracy. However, each of these models offers certain advantages, which are the linearity (for GSLS) and potentially smaller number of dynamic variables and broader peaks in attenuation spectra (for GCC). Therefore, additional experiments focusing on linearity of internal friction are required to establish which of these models may be preferable for rock. The Lagrangian approach provides a simple and physically meaningful way for comparing all types of observations, formulating numerical modeling schemes, and predicting the propagation of waves and behavior of other deformations of earth media. </jats:p>

<jats:p> We have revisited the debate about whether flexural waves from dipole sonic tools and standard processing algorithms measure group or phase velocities in anisotropic formations. We observe that much of the confusion arises from a failure to understand the different meanings of group and phase velocities. Using a transversely isotropic medium with a vertical axis of symmetry that exhibits a triplication in its S-wave group slowness surface, we generate synthetic flexural sonic waveforms corresponding to boreholes at angles of 0°–90° with respect to the anisotropy symmetry axis in 1° increments. We processed these synthetic data using standard time- and frequency-domain semblance methods. The results conclusively demonstrate that dipole sonic logs measure the group slowness for the group angle corresponding to the angle between the borehole and the anisotropic symmetry axis. In addition, data that we have evaluated suggest that current tool geometries and semblance processing may not always be sensitive enough to resolve all branches of the group slowness triplication surface. </jats:p>

<jats:p> To establish a consistent framework for seismic wave propagation that accommodates the effects of stress changes, it is critical to take into account the different definitions of stress and their corresponding effects on seismic quantities (e.g., wave speeds) as dictated by continuum mechanics. Revisiting this fundamental theoretical foundation, we first emphasize the role of stress within various forms of the wave equation resulting from different choices of stress definitions. Subsequently, using this basis, we investigate connections among existing theories that describe the variation of elastic moduli as a function of changes in stress. We find that there is a direct connection between predicting stress-induced elastic changes with the well-known third-order elasticity tensor and the recently proposed adiabatic pressure derivatives of elastic moduli. However, each of these approaches has different merits and drawbacks in terms of experimental validation as well as in their use. In addition, we investigate the connection with another general approach that relies on micromechanical structures (e.g., cracks and pores). Although this can be done algebraically, it remains unclear as to which definition of stress and which corresponding constitutive relationship should be considered in practical scenarios. We support our analysis with validations using previously published benchmark experimental data. </jats:p>

<jats:p> We have developed a workflow for computing the seismic-wave moduli dispersion and attenuation due to squirt flow in a numerical model derived from a micro X-ray computed tomography image of cracked (through thermal treatment) Carrara marble sample. To generate the numerical model, the image is processed, segmented, and meshed. The finite-element method is adopted to solve the linearized, quasistatic Navier-Stokes equations describing laminar flow of a compressible viscous fluid inside the cracks coupled with the quasistatic Lamé-Navier equations for the solid phase. We compute the effective P- and S-wave moduli in the three Cartesian directions for a model in dry conditions (saturated with air) and for a smaller model fully saturated with glycerin and having either drained or undrained boundary conditions. For the model saturated with glycerin, the results indicate significant and frequency-dependent P- and S-wave attenuation and the corresponding dispersion caused by squirt flow. Squirt flow occurs in response to fluid pressure gradients induced in the cracks by the imposed deformations. Our digital rock-physics workflow can be used to interpret laboratory measurements of attenuation using images of the rock sample. </jats:p>

<jats:p> The squirt flow model developed by Mavko and Jizba has been widely applied to quantify elastic moduli/velocities of fluid-saturated rocks at ultrasonic frequencies and the related modulus/velocity dispersion between ultrasonic and seismic frequencies. In the model, the high-frequency saturated bulk modulus is obtained by taking the unrelaxed frame bulk modulus as the drained/dry one as the input to Biot’s or Gassmann’s formula. However, when using Gassmann’s formula, the “new” rock matrix contains rock mineral matrix and fluid-saturated soft pores, which is heterogeneous at the microscopic scale and thus breaks the fundamental assumption of microhomogeneity for Gassmann’s formula. Therefore, the high-frequency saturated bulk modulus computed by Gassmann’s formula is inaccurate, especially when the soft-pore fraction (SPF) (the ratio of soft porosity to total porosity) or crack density is large. To this end, we have derived the Brown and Korringa’s expression (Mavko-Jizba-Gurevich [MJG]-BK model) for the high-frequency saturated bulk modulus in Mavko and Jizba’s model based on Berryman and Milton’s generalized Gassmann’s equations for the composite porous media, which correctly characterizes the microheterogeneity of new rock matrix. The parameters in the MJG-BK model are totally determined by measured quantities in the laboratory. Numerical example indicates that the MJG model is consistent with the MJG-BK model at small SPFs or crack densities. When the SPF/crack density becomes large, the difference between them becomes large, the MJG model loses its accuracy, and the MJG-BK model is preferred. Furthermore, experiment data from the laboratory validate the effectiveness of the proposed MJG-BK model. In summary, the model develops Brown and Korringa’s expression for Mavko and Jizba’s squirt flow model and can be used to calculate the high-frequency saturated bulk modulus at different SPFs or crack densities more accurately. </jats:p>

<jats:p> Coal is a complex viscoelastic porous medium with fractal characteristics at different scales. To model the macroscale structure of coal, a fractal viscoelastic model is established, and the P-wave velocity dispersion and attenuation characteristics are discussed based on the complex modulus derived from this model. The numerical simulation results indicate that the fractional order [Formula: see text] and relaxation time [Formula: see text] greatly affect the P-wave velocity dispersion and attenuation. The fractal viscoelastic model indicates a full-band velocity dispersion between 1 Hz and 10<jats:sup>4</jats:sup> Hz. Meanwhile, the P-wave velocity has a weaker dispersion with the fractal viscoelastic model than with the Kelvin-Voigt model and Zener model between 1 Hz and 10<jats:sup>4</jats:sup> Hz for the same relaxation time and elastic modulus, but the velocity at 1 Hz based on the fractal viscoelastic model is higher with the Kelvin-Voigt model and Zener model. Simultaneously, the velocities of five coal samples are tested, and the attenuation factor is calculated using a low-frequency system. The experimental results indicate a strong dispersion in coal in the range of 10–250 Hz. The classic Kelvin-Voigt model and Zener model cannot describe the dispersion characteristics of coal, but the fractal viscoelastic model can describe them well by using the appropriate fractional order and relaxation time. </jats:p>