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The rheological models of Lomnitz and Jeffreys have been widely used in earthquake seismology (to simulate a nearly constant Q medium) and to describe the creep and relaxation behavior of rocks as a function of time. Other similar models, such as those of Becker, Scott Blair and Kolsky, show similar properties, particularly the Scott Blair model describes a perfectly constant Q as a function of frequency. We first give a historical overview of the main scientists and the development and versions of the various models and priorities of discovery. Then, we clarify the relationship between the different versions of these models in terms of mathematical expressions of the complex modulus and calculate the phase velocity and quality factor Q as a function of frequency, illustrating the various special cases. In addition, we give useful hints for the numerical calculation of these moduli, which include special cases of the hypergeometric function. [ABSTRACT FROM AUTHOR]

The diffusive viscous (DV) model is a useful tool for interpreting low-frequency seismic attenuation and the influence of fluid saturation on frequency-dependent reflections. Among present methods for the numerical solution of the corresponding DV wave equation, the finite-difference frequency-domain (FDFD) method with complex-valued adaptive coefficients (CVAC) has the advantage of efficiently suppressing both numerical dispersion and numerical attenuation. In this research, the FDFD method with CVAC is first generalized to a 3D DV equation. In addition, the current calculation of CVAC involves the numerical integration of propagation angles, conjugate gradient (CG) iterative optimization, and the sequential selection of initial values, which is difficult and inefficient for implementation. An improved method is developed for calculating CVAC, in which a complex-valued least-squares problem is constructed by substituting the 3D complex-valued plane-wave solutions into the FDFD scheme. The QR-decomposition method is used to efficiently solve the least-squares problem. Numerical dispersion and attenuation analyses reveal that the FDFD method with CVAC requires ∼2.5 spatial points in a wavelength within a dispersion deviation of 1% and an attenuation deviation of 10% for the 3D DV equation. An analytic solution for 3D DV wave equation in homogeneous media is proposed to verify the effectiveness of the proposed method. Numerical examples also demonstrate that the FDFD method with CVAC can obtain accurate wavefield modelling results for 3D DV models with a limited number of spatial points in a wavelength, and the FDFD method with QR-based CVAC requires less computational time than the FDFD method with CG-based CVAC. [ABSTRACT FROM AUTHOR]

Temperature and pressure variations during the geologic diagenesis process can lead to complex pore structures in tight rocks. The effective-medium theory, based on the stress–strain relationship in combination with pore structure parameters, can be used to describe the elastic-wave responses of rocks. In this work, the differential effective medium (DEM) and self-consistent approximation (SCA) models are combined to invert the pore-crack spectrum. The Voigt–Reuss–Hill average is used to estimate the elastic moduli of the minerals. Then, based on SCA, the pore structures are incorporated into the rock matrix to create a new host phase. Subsequently, the DEM theory is used to add cracks with different volume fractions and aspect ratios to the host phase. To predict the structure of pores and cracks (crack density and aspect ratio), an objective function is defined as the sum of variances between experimentally measured and predicted wave velocities. The results show that the modeling predictions of P- and S-wave velocities at different temperatures and pressure ratios agree well with the experimental measurements. Variations in pore structure are determined at a zero effective pressure and different temperatures. We analyze the characteristics of how cracks change with variations in temperature and confining pressure, providing a theoretical basis for characterizing the structure of tight rocks. [ABSTRACT FROM AUTHOR]

The time-fractional Cattaneo (TFC) equation is a practical tool for simulating anomalous dynamics in physical diffusive processes. The existing numerical solutions to the TFC equation generally deal with the Dirichlet boundary conditions. In this paper, we incorporate the absorbing boundary condition as a complex-frequency-shifted (CFS) perfectly matched layer (PML) into the TFC equation. Then, we develop an adaptive-coefficient (AC) finite-difference frequency-domain (FDFD) method for solving the TFC with CFS PML. The corresponding analytical solution for homogeneous TFC equation with a point source is proposed for validation. The effectiveness of the developed AC FDFD method is verified by the numerical examples of four typical TFC models, including the different orders of time-fractional derivatives for both the homogeneous model and the layered model. The numerical examples show that the developed AC FDFD method is more accurate than the traditional second-order FDFD method for solving the TFC equation with the CFS PML absorbing boundary condition, while requiring similar computational costs. [ABSTRACT FROM AUTHOR]

We perform seismic and ultrasonic measurements in carbonate and shaley sandstone samples as a function of differential pressure. The velocities show a strong frequency and pressure dependence. The dispersion disappears with increasing pressure and the squirt flow in turn inhibits the pressure dependence. To model these effects, we combine the Gurevich's squirt-flow model with the Mori–Tanaka scheme and the David Zimmerman model, and extend it with third-order elastic constants, to obtain a frequency-dependent acoustoelasticity model. Comparisons between measurements from this study and literature and modelling results show that the P -wave velocity increases non-linearly first and then nearly linearly, dominated by crack closure and acoustoelasticity, respectively. The pressure dependence of wave velocities is reduced by liquid substitution and further by the squirt-flow mechanism. The effects of fluid properties and crack closure on P -wave velocity decrease with differential pressure. The results will feed a new model and help better understanding the wave propagation in pre-stressed rocks at different scales. [ABSTRACT FROM AUTHOR]

Mesoscopic P-wave attenuation in layered, partially saturated thermo-poroelastic media is analyzed by combining the theories of Biot poroelasticity and Lord–Shulman thermoelasticity (BLS). The attenuation is quantified by estimating the quality factor Q. The mesoscopic attenuation effect, commonly referred to as wave-induced fluid flow (WIFF), is the process that converts fast compressional and shear waves into slow diffusive Biot waves at mesoscopic heterogeneities larger than the pore scale, but much smaller than the dominant wavelengths. This effect was first modeled in White's isothermal theory by quantifying the seismic response of a periodic sequence of planar porous layers that are alternately saturated with gas or water. This work presents a numerical extension of White's theory for the non-isothermal case in this type of sequence. For this purpose, an initial-boundary-value problem (IBVP) for the BLS wave propagation equations is solved using the finite element method, where the particle velocity field is recorded at uniformly distributed receivers. The quality factor is estimated using spectral-ratio and frequency-shift methods. The Q-estimates show that thermal effects influence the attenuation of the P-wave and the velocity dispersion compared to the isothermal case. [ABSTRACT FROM AUTHOR]

Knowledge of how temperature affects the oil–sand acoustic response is useful to exploit these reservoir rocks with seismic methods. We propose three models: double-porosity coherent potential approximation (CPA), lower-bound Hashin–Shtrikmann (HS-), and contact cement (CC), based on different spatial distributions of heavy oil and temperature and frequency-dependent empirical equations. The shear modulus and S-wave velocity are affected by temperature in all the cases. Moreover, the properties of oil sands with heavy oil as a continuous matrix and higher viscosity are more sensitive to temperature. The models can provide theoretical support for the establishment of rock physical models in heavy oil reservoirs so as to quantitatively characterize the seismic response changes caused by thermal mining. [ABSTRACT FROM AUTHOR]

The quality factor Q is a dimensionless measure of the energy loss per cycle of a wave field, and a proper understanding of this factor is important in a variety of fields, from seismology, geophysical prospecting to electrical science. Here, the focus is on viscoelasticity. When interpreting experimental values, several factors must be taken into account, in particular the shape of the medium (rods, bars or unbounded media) and the fact that the measurements are made on stationary or propagating modes. From a theoretical point of view, the expressions of Q may differ due to different definitions, the spatial dimension and the inhomogeneity of the wave, i.e. the fact that the vectors of propagation (or wavenumber) and attenuation do not point in the same direction. We show the difference between temporal and spatial Q, the relationships between compressional and shear Q, the dependence on frequency, the case of poro-viscoelasticity and anisotropy, the effect of inhomogeneous waves and various loss mechanisms, and consider the analogy between elastic and electromagnetic waves. We discuss physical theories describing relaxation peaks, bounds on Q and experiments showing the behaviour of Q as a function of frequency, saturation and pore pressure. Finally, we propose an application example where Q can be used to estimate porosity and saturation. [ABSTRACT FROM AUTHOR]