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In a large body of rock-physics research, seismic wave velocity dispersion and attenuation in fluid-saturated porous rock are studied by constructing analytical or numerical models for time- or frequency-dependent dynamic (effective, or viscoelastic) moduli. A key and broadly used model of such kind is the Zener's, or the standard linear solid (SLS). This model is qualitatively successful in explaining many field and laboratory observations and serves as the key element of many generalizations such as the Burgers model for plastic deformations or the generalized SLS explaining band-limited or near-constant seismic attenuation. However, as a physical model of fluid-saturated porous rock, the SLS has several major limitations: disregard of inertial effects, absence of secondary wave modes and lack of key physical parameters such as porosity and Skempton coefficients. Grainy and porous rock is an unconsolidated material in which the effective density is frequency-dependent, and its effects on wave velocities may exceed those of the dynamic modulus. To overcome these limitations of the empirical SLS, we propose a rigorous rheologic model based on classical continuum mechanics and called the extended SLS, or eSLS. This rheology explains the available attenuation and dispersion observations equally well, but it is also close to Biot's model, honours all poroelastic relations, includes inertial effects, and reveals several new physical properties of the material. Detailed comparison of the eSLS and Biot's models gives a physical-mechanism-based classification of wave-induced fluid flow (WIFF) phenomena. In this classification, the so-called 'global-scale' flows occur in Biot's type structures within the material, whereas the 'local-scale' WIFF occurs in eSLS-type structures. Combining Biot's and eSLS models gives a broad class of rheologies for linear anelastic phenomena within rock with a single type of porosity. The model can be readily generalized to multiple porosities and different types of internal variables, such as describing squirt flows, wetting or thermoelastic effects. Modelling is conducted with relatively little effort, using a single matrix equation similar to a mechanical form of the standard SLS. By combining the eSLS and Biot's models, observations of dynamic-modulus dispersion and attenuation can be inverted for macroscopic mechanical properties of porous materials. [ABSTRACT FROM AUTHOR]

In forced-oscillation mechanical testing of rock samples, low-frequency attenuation is traditionally measured by the tangent of the strain-stress phase lag, which is interpreted as the frequency-dependent inverse quality factor. However, such phenomenological parameter only refers to harmonic waves in a homogeneous medium, lacking physical meaning in heterogeneous media or for finite bodies. It depends on specific boundary conditions and becomes insufficient for characterizing fluid-saturated porous rock. It is also sensitive to geometrical spreading, which is poorly known but can be significant in forcedoscillation experiments. To overcome these limitations and uncertainties of quality factor, one can use the temporal attenuation coefficient, a more fundamental quantity directly representing the relative mechanical-energy dissipation rate within the medium. Here, frequency-dependent attenuation coefficient is formulated from calibration experiments with Plexiglas and several published forced-oscillation measurements with fluid-containing porous rocks at variable temperatures. The resulting attenuation coefficient, unlike the quality factor, reveals important attenuation attributes: Effective geometrical attenuation, effective attenuation, relaxation time, and effective viscosity. The effective attenuation is related to the presence of pore fluids or melts, increases with temperature, and decreases with static pressure and pore-fluid viscosity. The effective geometrical attenuation is small in experiments with sandstone but becomes significant in high-temperature, torsionaldeformation experiments with olivine aggregates. Unlike the inverse quality factor, the peak in the residual attenuation coefficient yields additional quantitative parameters to characterize the elasticity and internal friction within the rock. This work provides a new way for studying seismic attenuation, which shall be helpful to oil and gas exploration. [ABSTRACT FROM AUTHOR]

Wave‐induced pore‐fluid flows are recognized as an important cause of seismic‐wave attenuation at frequencies below 1 kHz within heterogeneous rocks. By using Lagrangian continuum mechanics, wave‐induced pore‐fluid flow mechanisms can be classified strictly on the basis of macroscopic non‐Biot's material properties. In this classification, type I and II wave‐induced pore‐fluid flows are identified by local internal deformations being elastically coupled with either the whole Biot's rock or only with its pore fluid, respectively. Type III wave‐induced pore‐fluid flow is defined as a mixture of types I and II. For all types of wave‐induced pore‐fluid flows, the model predicts all rock properties observed in rock‐deformation experiments, such as the frequency‐dependent poroelastic moduli. The observed attenuation peaks and effective moduli can be used to invert for new, non‐Biot's material properties of porous rock. In data examples, we focus on the pore‐fluid coupled (type II) wave‐induced pore‐fluid flow mechanism and compare it to a previous analysis of type I. Non‐Biot's elastic and viscous rock properties are inverted for by fitting the effective drained bulk moduli measured in two previously published experiments: (1) with real sandstone including two saturating fluids and multiple confining pressures, and (2) a numerical experiment with heterogeneous sandstone containing mesoscopic‐ and microscopic‐scale wave‐induced pore‐fluid flows. Compared with a previous study of type I wave‐induced pore‐fluid flow, the data‐fitting method is improved by focusing on attenuation peaks and additional points in the observed spectra. For both of these experiments, both type I and II interpretations yield accurate fitting of the observed attenuation and dispersion spectra. Combinations of type I and II models (type III) yield a broad variety of acceptable mechanical models. This ambiguity of inversion shows that the different types of wave‐induced pore‐fluid flows cannot be differentiated in conventional attenuation/dispersion experiments. However, these WIFF types are physically meaningful and lead to rigorous equations of rock deformation that can be used in many applications. [ABSTRACT FROM AUTHOR]

In a companion article, Safarshahi and Morozov (2020) argued that construction of distance- and frequency-dependent models for seismic-wave amplitudes should include four general elements: (1) a sufficiently detailed (parametric or nonparametric) model of frequency-independent spreading, capturing all essential features of observations; (2) model parameters with well-defined and nonoverlapping physical meanings; (3) joint inversion for multiple parameters, including the geometrical spreading, Q, and source and receiver couplings; and (4) the use of additional dataset-specific criteria of model quality, while fitting the logarithms of seismic amplitudes. Some of these elements are present in existing models, but, taken together, they are poorly understood and require an integrated approach. Such an approach was illustrated by detailed analysis of an S-wave amplitude dataset from southern Iran. The resulting model is based on a frequency-independent Q, and matches the data closer than conventional models and across the entire epicentral-distance range. Here, we complete the analysis of this model by evaluating the uncertainties and trade-offs of its parameters. Two types of trade-offs are differentiated: one caused by a (possibly) limited model parameterization and the second due to statistical data errors. Data bootstrapping shows that with adequate parameterization, attenuation properties Q, and geometrical spreading parameters are resolved well and show moderate trade-offs due to measurement errors. Using the principal component analysis of these trade-offs, an optimal (trade-off free) parameterization of seismic amplitudes is obtained. By contrast, when assuming theoretical values for certain model parameters and using multistep inversion procedures (as commonly done), parameter trade-offs increase dramatically and become difficult to assess. In particular, the frequency-dependent Q correlates with the distribution of the source and receiver-site factors, and also with biases in the resulting median data residuals. In the new model, these trade-offs are removed using an improved parameterization of geometrical spreading, constant Q, and model quality constraints. [ABSTRACT FROM AUTHOR]

Empirical models of geometrical-, Q-, t-star, and kappa-type attenuation of seismic waves and ground-motion prediction equations (GMPEs) are viewed as cases of a common empirical standard model describing variation of wave amplitudes with time and frequency. Compared with existing parametric and nonparametric approaches, several new features are included in this model: (1) flexible empirical parameterization with possible nonmonotonous time or distance dependencies; (2) joint inversion for time or distance and frequency dependencies, source spectra, site responses, kappas, and Q; (3) additional constraints removing spurious correlations of model parameters and data residuals with source-receiver distances and frequencies; (4) possible kappa terms for sources as well as for receivers; (5) orientation-independent horizontal- and three-component amplitudes; and (6) adaptive filtering to reduce noise effects. The approach is applied to local and regional S-wave amplitudes in southeastern Iran. Comparisons with previous studies show that conventional attenuation models often contain method-specific biases caused by limited parameterizations of frequency-independent amplitude decays and assumptions about the models, such as smoothness of amplitude variations. Without such assumptions, the frequency-independent spreading of S waves is much faster than inferred by conventional modeling. For example, transverse-component amplitudes decrease with travel time t as about t-1.8 at distances closer than 90 km and as t-2.5 beyond 115 km. The rapid amplitude decay at larger distances could be caused by scattering within the near surface. From about 90 to 115 km distances, the amplitude increases by a factor of about 3, which could be due to reflections from the Moho and within the crust. With more accurate geometrical-spreading and kappa models, the Q factor for the study area is frequency independent and exceeds 2000. The frequency-independent and Q-type attenuation for vertical-component and multicomponent amplitudes is somewhat weaker than for the horizontal components. These observations appear to be general and likely apply to other areas. [ABSTRACT FROM AUTHOR]

Wave-induced fluid flows (WIFF) can be viewed as cases of broader local-deformation (LD) phenomena and represent the principal causes of seismic-wave attenuation in fluid-saturated porous rock. Most existing WIFF models refer to greatly simplified microstructures and specific flow patterns, such as planar divergent flows within thin cracks (squirt flows, SF) or flows within patchy-saturation zones. However, such microstructures represent only idealized mathematical models that may be impossible to consistently identify within a given rock. At the same time, most details of such microstructures are insignificant for seismic waves, which are only sensitive to averaged properties of the medium. To perform microstructure-independent modelling of LD effects, we develop a simple yet general approach based entirely on a macroscopic local-deformation variable. This variable is broadly analogous to Biot's fluid content and is illustrated for two specific microstructural models. The macroscopic model is Biot-consistent and uses only time- and frequency-independent material properties. Both local and global (Biot's) pore flows and all types of waves and deformations are explained in a rigorous and consistent manner. The model allows constraining a minimal set of material properties responsible for all observed elastic and anelastic effects in porous rock. Because of making no assumptions about the microstructures and their spatial scales, this approach should comprise at least some of the existing WIFF models. In particular, this model accurately reproduces all attenuation and velocity dispersion spectra predicted by a broadly used SF model. The model also contains effects not considered previously, such as bulk viscosity of pore fluid and viscous coupling between the rock frame and fluid-filled pores. The model offers straightforward extensions to multiple porosities and cases of viscous fluids in primary pores. Based on the resulting differential equations, physically consistent schemes for numerical modelling of seismic wavefields can be developed for porous rock with arbitrary LD effects. [ABSTRACT FROM AUTHOR]

Linear and non-linear viscoelastic (VE) models such as the standard linear solid (SLS) and the generalized SLS (GSLS) are broadly used to represent the anelasticity of materials and Earth's media. However, although the VE approach is often satisfactory for any given observation, the inferred physical causes of anelasticity may be significantly misrepresented by this paradigm, and its predictions may be wrong or inaccurate in other cases. This problem is particularly important in heterogeneous media, including most cases of interest for seismology. For example, in homogenous media, VE and mechanics-based models predict identical quality-factor Q (f) and phase velocity c (f) spectra, but in heterogenous media, these models yield different time-stepping equations and interactions with material–property boundaries. The commonly used VE algorithms for modelling seismic waves rely on postulated convolutional integrals in time, whereas physically, models of rock rheologies should still be based on spatial interactions. To understand how VE models relate to mechanics, it is instructive to consider which physical properties of the medium are constrained reliably and which of them remain unconstrained by a pair of Q (f) and c (f) spectra, that is by VE properties. Despite its popular association with 'attenuation,' the peak value of Q −1(f) is actually a purely elastic property representing the existence of two (for SLS) or multiple (for GSLS) elastic moduli. These moduli are analogous to the drained and undrained moduli in poroelasticity or isothermal and adiabatic moduli in thermodynamics. By virtue of the Kramers–Krönig relations, the peak Q −1 is related to the total velocity dispersion, which is also caused by the difference between elastic moduli. By contrast, true anelasticity-related physical properties like viscosity are represented not by Q −1 values but by the frequencies of Q −1(f) peaks in the data. However, these frequencies also depend on multiple material properties that are not recognized or arbitrarily selected in the SLS and GSLS models. Inertial, body-force friction and the corresponding boundary effects are also ignored in VE models, which may again be improper for layered media. Thus, for physically accurate interpretation of laboratory experiments and numerical modelling of seismic waves, first-principle equations of mechanics should be used instead of VE models. [ABSTRACT FROM AUTHOR]