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We develop a numerical algorithm for simulation of wave propagation in anisotropic thermoelastic media, established with a generalized Fourier law of heat conduction. The wavefield is computed by using a grid method based on the Fourier differential operator and a first-order explicit Crank-Nicolson algorithm to compute the spatial derivatives and discretize the time variable (time stepping), respectively. The model predicts four propagation modes, namely, a fast compressional or (elastic) P wave, a slow thermal P diffusion/wave (the T wave), having similar characteristics to the fast and slow P waves of poroelasticity, respectively, and two shear waves, one of them coupled to the P wave and therefore affected by the heat flow. The thermal mode is diffusive for low values of the thermal conductivity and wave-like (it behaves as a wave) for high values of this property. As in the isotropic case, three velocities define the wavefront of the fast P wave, i.e, the isothermal velocity in the uncoupled case, the adiabatic velocity at low frequencies, and a higher velocity at high frequencies. The heat (thermal) wave shows an anisotropic behavior if the thermal conductivity is anisotropic, but an elastic source does not induce anisotropy in this wave if the thermal properties are isotropic. [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]

Geometric heterogeneities in tight reservoir rocks saturated with a fluid mixture may exhibit different scale distribution characteristics. Conventional models of rock physics based on poroelasticity, which usually consider single‐scale pore structure and fluid patches, are inadequate for describing elastic wave responses. A major challenge is to establish the relationship between the wave response at different spatial scales and frequencies. To address this problem, three sets of observational data over a wide frequency range were obtained from a tight oil reservoir in the Ordos Basin, China. Ultrasonic measurements were made on eight sandstone samples at partial oil‐water saturation at 0.55 MHz. Data from six borehole measurements and seismic profiles were acquired and analyzed at about 10 kHz and 30 Hz, respectively. Analysis of the cast thin sections shows that dissolution pores and microcracks generally develop, with fractal dimensions of the pores ranging from 2.45 to 2.67 for the samples with porosities between 5.1% and 10.2%. Compressional wave velocity and attenuation were estimated from the observed data. The results show that the velocity dispersion from seismic to ultrasonic frequencies is 10.02%, mostly occurring between sonic and ultrasonic frequencies. The attenuation is stronger at higher oil saturation. The relationships between velocity, attenuation, and wavelength were established and can be used for further forward modeling and seismic interpretation studies. A partial saturation model has been derived based on effective differential medium theory and a double double‐porosity model, assuming that the medium contains fractal cracks and fluid patches. The effects of scale and saturation on wave responses are prevalent. Modeling results consistent with observed data show that the radii of cracks and fluid patches range from 0.1 μm to 2.8 mm, affecting ultrasonic, acoustic, and seismic attenuation. The multiscale data and proposed model quantify the relationship between fracture and fluid distributions and attenuation and could be useful for upscaling to the reservoir scale. The study helps improve the understanding of seismic wave propagation in partially saturated rocks, which has potential applications in seismic exploration, hydrocarbon production in reservoirs, and CO2 sequestration in aquifers. Plain Language Summary: The physical properties of the rock and fluid can be inferred from the measured elastic wave responses and energy dissipation characteristics. However, the effects of heterogeneities of different sizes and at different frequencies can hinder studies to quantify wave responses in a partially saturated porous medium, which are usually based on laboratory measurements. A major problem is the difference between observed frequencies and scales: megahertz in the laboratory, 10 of kilohertz in the borehole scale, and hertz in the seismic exploration scale. In this work, the frequency‐ and saturation‐dependent compressional velocity and attenuation are investigated using three geophysical data sets from the same tight reservoirs. A strong velocity dispersion over the measured frequency range is observed. The stronger attenuation at partial saturation may be caused by the multiscale heterogeneities of the pore structure and fluid patch distribution. A fractal poroelasticity model is developed by gradually inserting inclusions of different sizes with compliant pores and liquid patches into a homogeneous host skeleton. The wave responses are significantly affected by scale distribution and saturation. The proposed model, verified by the measured data, can be useful in interpreting the anelasticity of tight heterogeneous reservoirs in a broadband range. Key Points: We investigate how fluid saturation affects wave attenuation and dispersion over a wide frequency range in tight sandstone layersA partial saturation model describes the fractal properties of the layersThe size of the cracks and fluid patches are determined by applying the fractal model to the multiscale data [ABSTRACT FROM AUTHOR]

Wave propagation in anelastic rocks is a relevant scientific topic in basic research with applications in exploration geophysics. The classical Biot theory laid the foundation for wave propagation in porous media composed of a solid frame and a saturating fluid, whose constitutive relations are linear. However, reservoir rocks may have a high-viscosity fluid, which exhibits a non-Newtonian (nN) behaviour. We develop a poroelasticity theory, where the fluid stress-strain relation is described with a Kelvin–Voigt mechanical model, thus incorporating viscoelasticity. First, we obtain the differential equations from first principles by defining the strain and kinetic energies and the dissipation function. We perform a plane-wave analysis to obtain the wave velocity and attenuation. The validity of the theory is demonstrated with three examples, namely, considering a porous solid saturated with a nN pore fluid, a nN fluid containing solid inclusions and a pure nN fluid. The analysis shows that the fluid may cause a negative velocity dispersion of the fast P (S)-wave velocities, that is velocity decreases with frequency. In acoustics, velocity increases with frequency (anomalous dispersion in optics). Furthermore, the fluid viscoelasticity has not a relevant effect on the wave responses observed in conventional field and laboratory tests. A comparison with previous theories supports the validity of the theory, which is useful to analyse wave propagation in a porous medium saturated with a fluid of high viscosity. [ABSTRACT FROM AUTHOR]

Seismic wave propagation is affected by wave‐induced local fluid flow between stiff pores and multiscale fractures. To investigate this phenomenon, forced vibration (1–100 Hz) and ultrasonic (106 Hz) measurements are performed on two tight sandstones with complex pore geometry in dry and water‐saturated scenarios. Porosity, permeability, and ultrasonic velocities were also measured at different differential pressures. The results indicate that the nonlinear behavior of these properties is strongly influenced by the presence of cracks, and the correlations between the permeability/ultrasonic velocities and porosity are different. A wave propagation model is then developed in which penny‐shaped cracks are inclusions introduced stepwise into a porous medium to describe the wave anelasticity in a wide frequency band at different pressures. As a result, the model provides good agreement with the measured P‐wave velocity dispersion, and the pore aspect ratio spectrum and crack radii are determined. We then compare the estimated crack radii and pore size distributions from nuclear magnetic resonance spectroscopy. Published data of a tight sandstone and a low porosity sandstone in the frequency range (2–200, 106) and (1–3,000) Hz at different differential pressures are also analyzed to validate the model. The aspect ratios, volume fractions, and radii of pores/cracks are used to describe the measured permeability. The present work can provide new insights into the geophysical properties of reservoir rocks with complex pore geometry. Plain Language Summary: Multiscale cracks exert an important influence on the dispersion and attenuation of broadband acoustic waves. To better understand the underlying mechanism, we measured frequency‐dependent elastic moduli and pressure‐dependent porosity, permeability, and ultrasonic wave velocities of two tight sandstones. These data show that velocity dispersion is observable in water‐saturated rock and that permeability and ultrasonic velocities are differentially correlated with porosity. A proposed wave propagation model, in which penny‐shaped cracks are inclusions introduced stepwise into a porous medium, successfully describes the experimental data and other published data on tight sandstone and low‐porosity sandstone, and aspect ratios, volume fractions, and radii of pores/cracks are determined. Key Points: We propose a model to investigate the effect of multiscale cracks on seismic waves in tight sandstonesThe pore aspect ratio and crack‐radius spectra are estimated at different differential pressuresThe permeability is obtained according to the pore geometry [ABSTRACT FROM AUTHOR]

We simulate variations in the P- and S-wave velocities due to CO2 injection in basalt, caused by water replacement and mineralization in the form of carbonate cements. Carbonization can be monitored with seismic data on the basis of changes in velocity. We consider basalts composed of (Fe, Ca and Mg) minerals mainly, and the pore space is assumed to be initially saturated with water. The mineralization is based on a model that estimates the volume of precipitated cement and resulting porosity loss, and the velocities as a function of injection time are calculated with a patchy cement model. In the example, there are two stages: 1 week of CO2 injection and 29 weeks of frame cementation. After 1 week, water has been replaced by CO2, where density and P-wave velocity decrease, and cementation implies an increase in bulk density, P-wave velocity, whereas porosity and surface area are reduced. To our knowledge there are no models relating the effects of carbon mineralization on seismic properties. The present model can be applied to other rocks, where CO2 injection induces mineralization and storage. [ABSTRACT FROM AUTHOR]

SUMMARY: We consider surface-wave propagations at an interface separating a fluid layer and a double-porosity medium embedded with cracks. The theory is based on a generalization of the Biot-Rayleigh model from spherical cavities to penny-shaped cracks randomly embedded into a host medium, where mesoscopic local fluid flow (LFF) plays an important role. We derive closed-form dispersion equations of surface waves, based on potentials and suitable boundary conditions (BCs), to obtain the phase velocity and attenuation by using numerical iterations. Two special cases are considered by letting the thickness of the fluid (water) layer to be zero and infinity. We obtain pseudo-Rayleigh and pseudo-Stoneley waves for zero and infinite thickness and high-order surface modes for finite nonzero thickness. Numerical examples confirm that the LFF affects the propagation at low frequencies, causing strong attenuation, whereas the impact of BCs is mainly observed at high frequencies, due to the propagation of slow wave modes. The crack density mainly affects the level of attenuation, whereas the aspect ratio the location of the relaxation peak. The fundamental mode undergoes a significant velocity dispersion, whose location moves to low frequencies as the thickness increases. In all cases, there also exist two slower surface modes that resemble the two slow body waves, only present for sealed BCs. [ABSTRACT FROM AUTHOR]

The wave-induced local fluid flow mechanism is relevant to the complex heterogeneity of pore structures in rocks. The analysis of the local fluid flow mechanism is useful for accurately describing the wave propagation characteristics in reservoir rocks. In the exploration and production of hydrocarbon reservoirs, the real stratum may be partially saturated with a multi-phase fluid mixture in general. Therefore, it is of great significance to investigate the wave velocity dispersion and attenuation features in relation to pore structures and fluids. In this work, the characteristics of fabric microstructures are obtained on the basis of pressure dependency of dry rock moduli using the effective medium theory. A novel anelasticity theoretical model for the wave propagation in a partially-saturated medium is presented by combining the extended Gurevich squirt-flow model and White patchy-saturation theory. Numerical simulations are used to analyze wave propagation characteristics that depend on water saturation, external patchy diameter, and viscosity. We consider a tight sandstone from the Qingyang area of the Ordos Basin in west China and perform ultrasonic measurements under partial saturation states and different confining pressures, where the basic properties of the rock are obtained at the full gas saturation. The comparison of experimental data and theoretical modeling results shows a fairly good agreement, indicating that the new theory is effective. [ABSTRACT FROM AUTHOR]

Underground carbonate deposits are widespread worldwide and have considerable hydrocarbon potential. They are generally characterized by a complex microscopic structure that affects the properties of the macroscopic fluid flow and the relevant petrophysical behavior. In recent years, advances in digital technology have helped reveal the microstructures (i.e., pore connections, cracks, pore size and radius, etc.) of rocks in the subsurface. In this work, drill cores (cylinder) are taken from a deep carbonate deposit in the Sichuan Basin in western China to perform computed tomography (CT) scans, thin sections and mineral analysis. The characteristics of lithology and pore structure are investigated. Ultrasonic experiments with different fluid types and pressures are conducted to determine rock samples’ wave velocities, attenuation and crack porosity. The experimental data show that the rocks have low porosity/permeability and a complex pore/crack system, leading to significant pressure, crack and fluid type effects on the velocities, dispersion and attenuation. We develop a model of multiple pore-crack structures for carbonates by considering the complex structure and fluid properties. Digital cores are reconstructed based on CT scans, image processing and threshold segmentation. The aspect ratios of pores and cracks are extracted with their volume fractions to simulate the rock skeleton with the differential effective medium theory. The Biot–Rayleigh wave propagation equations are applied to simulate the effects of different pore and fluid types on the velocity and attenuation of P-waves. The agreement between the modeling results and the ultrasonic and log data confirms that the model can validly reproduce the wave responses. [ABSTRACT FROM AUTHOR]

The exploration and exploitation of shale oil is an important aspect in the oil industry. Seismic properties and well‐log data are essential to establish wave‐propagation models. Specifically, the description of wave dispersion and attenuation under complex geological conditions needs proper lithological and petrophysical information. This complex physical mechanism has to be considered if a traditional modeling approach is adopted. In this sense, machine learning (ML) techniques provide new possibilities for this purpose. We compare two deep‐neural‐network (DNN)‐based wave propagation models. In the first (pure data‐driven), a DNN is trained to connect seismic attributes, such as wave velocities, to multivariate functions of rock‐physics properties. By training DNNs with different initial parameters, the uncertainty of the proposed method can be quantified. The second method assumes the form of the wave equations. Then, the elastic constants of the constitutive relations are predicted by DNNs. The resulting dynamical equations describe the dispersion and attenuation and wavefield simulations can be performed to obtain more information. On the basis of a test, the two kinds of wave‐propagation models yield acceptable estimations of the seismic properties, with the second approach showing a broader application because the DNN is trained without S wave data. The methodologies illustrate that the new wave‐propagation model based on ML has high precision and can be general in terms of rheological description. Plain Language Summary: We develop a model to estimate the underground conditions, based on two machine‐learning techniques by using measurement data as input. The first uses pure data‐driven surrogate models, while the second is based on the classical wave propagation model in fluid‐saturated porous medium proposed by Biot, where deep neural networks predict the elastic coefficients so that the theoretical seismic properties match the actual measurements. The two models are tested on the basis of data from five wells located in a shale‐oil reservoir area. Compared with the pure data‐driven model, the wave‐equation method may be used to predict more reservoir properties. Key Points: Machine‐learning methods are proposed to drive wave propagation based on seismic dataOptimal prediction results of seismic attributes are obtained by feeding rock‐physics properties into the trained modelsThe proposed method has the potential to obtain additional physical quantities and shows a broad application prospect [ABSTRACT FROM AUTHOR]

The fractal texture (or fabric) of porous media, which supports fluid flow at different scales, is the main cause of wave anelasticity (dispersion and attenuation) on a wide range of frequencies. To model this phenomenon, we develop a theory of wave propagation in a fluid saturated infinituple‐porosity media containing inclusions at multiple scales, based on the differential effective medium (DEM) theory of solid composites and Biot‐Rayleigh theory for double‐porosity media. The dynamical equations are derived from first principles, that is, based on the strain (potential), kinetic, and dissipation energies, leading to generalized stiffness and density coefficients. The scale of the inclusions can be characterized by different distributions. The theory shows that the anelasticity depends on the size (radius) of the inclusions, parameter θ (exponential distribution), mean radius r0 and variance σr2 (Gaussian distribution) and the fractal dimension Df (self‐similar distribution). When Df = 2, θ = 1 and σr2 = 4, the three distributions give the same P‐wave velocities and attenuation, since each added inclusion phase has nearly the same volume fraction. For the modeling results, the range of anelasticity of Df = 2.99/θ = 1/σr2 = 4 is broader than that of Df < 2.99/θ < 1/σr2 < 4. To confirm the validity of the model, we compare the results with laboratory measurements on tight sandstone and carbonate samples in the range 1 Hz–1 kHz, Fox Hill sandstone (5 Hz–800 kHz) and field measurements of marine sediments (50 Hz–400 kHz). This comparison shows that the model successfully describes the observed anelasticity. Key Points: We develop a theory of wave propagation in infinituple‐porosity media based on multiple‐scale inclusionsWe consider different distributions of the size of the inclusions aimed to model the anelasticityThe model provides a reasonable explanation of broadband laboratory and field data [ABSTRACT FROM AUTHOR]

In thermoelastic wave attenuation, such as that caused by heterogeneities much smaller than the wavelength, e.g., Savage theory of spherical pores, the shape of the relaxation peak differs from that of the Zener (or standard linear solid) mechanical model. In these effective homogeneous media, the anelastic behavior is better represented by a stress-strain relation based on fractional derivatives; particularly, P- and S-wave dispersion and attenuation is well described by a Cole–Cole equation. We propose a time-domain algorithm for wave propagation based on the Grünwald–Letnikov numerical derivative and the Fourier pseudospectral method to compute the spatial derivatives. As an example, we consider Savage theory and verify the algorithm by comparison with the analytical solution in homogeneous media based on the frequency-domain Green function. Moreover, we illustrate the modeling performance with wave propagation in a two half-space medium where one section is lossless and the other is a Cole–Cole medium. This apparently simple example, which does not have an analytical solution, shows the complexity of the wavefield that characterizes a single flat interface. [ABSTRACT FROM AUTHOR]

Thermoelastic attenuation is similar to wave-induced fluid-flow attenuation (mesoscopic loss) due to conversion of the fast P wave to the slow (Biot) P mode. In the thermoelastic case, the P - and S -wave energies are lost because of thermal diffusion. The thermal mode is diffusive at low frequencies and wave-like at high frequencies, in the same manner as the Biot slow mode. Therefore, at low frequencies, that is, neglecting the inertial terms, a mathematical analogy can be established between the diffusion equations in poroelasticity and thermoelasticity. We study thermoelastic dissipation for spherical and cylindrical cavities (or pores) in 2-D and 3-D, respectively, and a finely layered system, where, in the latter case, only the Grüneisen ratio is allowed to vary. The results show typical quality-factor relaxation curves similar to Zener peaks. There is no dissipation when the radius of the pores tends to zero and the layers have the same properties. Although idealized, these canonical solutions are useful to study the physics of thermoelasticity and test numerical algorithm codes that simulate thermoelastic dissipation. [ABSTRACT FROM AUTHOR]

The classical Zener model of thermoelasticity can be represented by a mechanical (or viscoelastic) model based on two springs and a dashpot, commonly called standard-linear solid, whose parameters depend on the thermal properties and a relaxation time, and yield the isothermal and adiabatic velocities at the low- and high-frequency limits. This model differs from the more general Lord-Shulman theory of thermoelasticity, whose low-frequency velocity is the adiabatic one. These theories are the basis of thermoelastic attenuation in inhomogeneous media, with heterogeneities much smaller than the wavelength, such as Savage's theory of thermoelastic dissipation in a medium with spherical pores. In this case, the shape of the relaxation peak differs from that of the Zener and Lord-Shulman models. In these effective homogeneous media, the anelastic behavior of real materials can better be described by using a stress-strain relation based on fractional derivatives. In particular, wave propagation (dispersion and attenuation) is well described by a Cole-Cole stress-strain equation, as illustrated by the agreement with Savage's theory. We propose a time-domain algorithm based on the Grünwald-Letnikov numerical approximation of the fractional time derivative involved in the time-domain representation of the Cole-Cole model. The spatial derivatives are computed with the Fourier pseudospectral method. We verify the results by comparison with the analytical solution, based on the Green function. The numerical example illustrates wave propagation at an interface separating a porous medium and a purely solid phase. [ABSTRACT FROM AUTHOR]

Thermoelasticity extends the classical elastic theory by coupling the fields of particle displacement and temperature. The classical theory of thermoelasticity, based on a parabolic-type heat-conduction equation, is characteristic of an unphysical behaviour of thermoelastic waves with discontinuities and infinite velocities as a function of frequency. A better physical system of equations incorporates a relaxation term into the heat equation; the equations predict three propagation modes, namely, a fast P wave (E wave), a slow thermal P wave (T wave), and a shear wave (S wave). We formulate a second-order tensor Green's function based on the Fourier transform of the thermodynamic equations. It is the displacement–temperature solution to a point (elastic or heat) source. The snapshots, obtained with the derived second-order tensor Green's function, show that the elastic and thermal P modes are dispersive and lossy, which is confirmed by a plane-wave analysis. These modes have similar characteristics of the fast and slow P waves of poroelasticity. Particularly, the thermal mode is diffusive at low thermal conductivities and becomes wave-like for high thermal conductivities. [ABSTRACT FROM AUTHOR]

We develop a numerical algorithm for simulation of wave propagation in linear nonisothermal poroelastic media, based on Biot theory and a generalized Fourier law of heat transport in analogy with Maxwell model of viscoelasticity. A plane wave analysis indicates the presence of the classical P and S waves and two slow waves, namely, the Biot and the thermal slow modes of propagation, which present diffusive behavior under certain conditions, depending on viscosity, frequency, and the thermoelastic constants. The wavefield is computed with a direct meshing method using the Fourier differential operator to calculate the spatial derivatives. We propose two alternative time‐stepping algorithms, namely, a first‐order explicit Crank‐Nicolson method and a second‐order splitting method. The Fourier differential operator provides spectral accuracy in the calculation of the spatial derivatives. Modeling the thermal diffusive mode is relevant for high‐temperature high‐pressure fields and since it leads to mesoscopic attenuation by mode conversion of the fast waves to the thermal waves. Key Points: The differential equations of dynamic thermoporoelasticity are solved with a direct grid methodThe solver is based on the Fourier pseudospectral method to compute the spatial derivativesThe simulation reveals the behavior of four wave modes, namely, the fast P, slow P, S, and the thermal wave-diffusion field [ABSTRACT FROM AUTHOR]

We performed ultrasonic experiments in specimens from a tight oil reservoir. The P-wave attenuation of fluid-saturated specimens was estimated by the spectral ratio method. The results suggest that at ultrasonic frequencies, most specimens have stronger attenuation under gas-saturated conditions than at water- or oil-saturated conditions. The P-wave attenuation positively correlates with permeability. Scanning electron microscopy observations and the triple-porosity structure model were used to simulate the wave propagation. The P-wave velocity dispersion and attenuation are discussed on the basis of the Biot, Biot-Rayleigh double-porosity medium, and the triple-porosity structure models. The results suggest that the Biot and Biot-Rayleigh models cannot explain the attenuation, whereas the triple-porosity structure model is in agreement with the experimental data. Furthermore, we infer that microcracks are common in a porosity of 5%–10%, and the size of microcracks increases in samples with higher porosity. However, the volume ratios of microcracks and clay inclusions remain constant regardless of porosity variations. The size of microcracks is significantly larger than the clay inclusions, and the bulk modulus of microcracks is lower than the bulk modulus of clays. [ABSTRACT FROM AUTHOR]

The correct simulation of wave propagation with direct grid methods at fluid–solid interfaces requires a proper implementation of the boundary condition, which involves continuity of the normal component of the particle velocity and traction. The horizontal component of the particle velocities can differ (slip) and the tangential traction vanishes. The challenge is to model the interface waves, which are relevant when the source and receiver are located close to the interface, in particular the Scholte and leaky (or pseudo, or generalized) Rayleigh waves. The explicit modelling is based on a domain-decomposition technique (one mesh for the fluid and another mesh for the solid) and the Fourier and Chebyshev differential operators along the horizontal and vertical directions, respectively. On the other hand, two implicit modelling algorithms are based on a single mesh and the fluid is described by setting to zero the shear wave velocity above the interface. The modelling is verified by comparisons with the analytical solution for a fluid–solid interface in lossless media, with source and receiver away from and at the interface. To our knowledge, the latter comparison, mainly involving the interface waves (Scholte and leaky-Rayleigh), has not been performed with other grid methods so far. It is shown that the implicit simulation yields erroneous results. [ABSTRACT FROM AUTHOR]

Taking into account three important porous media mechanisms during wave propagation (the Biot-flow, squirt-flow, and solid-skeleton viscoelastic mechanisms), we introduce water saturation into the dynamic governing equations of wave propagation by analyzing the effective medium theory and then providing a viscoelastic Biot/squirt (BISQ) model which can analyze the wave propagation problems in a partially viscous pore fluid saturated porous media. In this model, the effects of pore fluid distribution patterns on the effective bulk modulus at different frequencies are considered. Then we derive the wave dynamic equations in the time-space domain. The phase velocity and the attenuation coefficient equations of the viscoelatic BISQ model in the frequency-wavenumber domain are deduced through a set of plane harmonic solution assumptions. Finally, by means of numerical simulations, we investigate the effects of water saturation, permeability, and frequency on compressional wave velocity and attenuation. Based on tight sandstone and carbonate experimental observed data, the compressional wave velocities of partially saturated reservoir rocks are calculated. The compressional wave velocity in carbonate reservoirs is more sensitive to gas saturation than in sandstone reservoirs. [ABSTRACT FROM AUTHOR]

We study the wave anelasticity (attenuation and velocity dispersion) of a periodic set of three flat porous layers saturated by two immiscible fluids. The fluids are very dissimilar in properties, namely gas, oil, and water, and, at most, three layers are required to study the problem from a general point of view. The sequence behaves as viscoelastic and transversely isotropic (VTI) at wavelengths much longer than the spatial period. Wave propagation causes fluid flow and slow P modes, inducing anelasticity. The fluids are characterized by capillary forces and relative permeabilities, which allow for the existence of two slow modes and the presence of dissipation, respectively. The methodology to study the physics is based on a finite-element uspcaling approach to compute the complex and frequency-dependent stiffnesses of the effective VTI medium. The results of the experiments indicate that there is higher dissipation and anisotropy compared to the widely used model based on an effective fluid that ignores the effects of surface tension (capillarity) and viscous flow interference between the two fluid phases. [ABSTRACT FROM AUTHOR]

Wave propagation in rocks induces fluid pressure gradients and flow between stiff intergranular pores and compliant microfractures. This, in turn, leads to wave anelasticity, i.e., attenuation and velocity dispersion. Three known squirt-flow models attempt to explain this phenomenon. Based on these theories, we reformulate a new one, where we derive the dry-rock moduli on the assumption that the boundary fluid pressure is constant. The new model improves the predictions of the other models, e.g., a better description at high frequencies and the inclusion of permeability effects. The numerical example shows the response due to the characteristic squirt-flow length and fluid viscosity on wave propagation. The reformulated model is applied to experiments made on water-saturated tight sandstones, where the squirt-flow length is determined, showing that it increases with increasing permeability and porosity at full saturation. • A new squirt-flow model by deriving the dry-rock moduli on the assumption of the constant boundary fluid pressure. [ABSTRACT FROM AUTHOR]

The double-porosity theory of wave propagation in rocks considers a single scale based on soft and stiff pores, and wave attenuation occurs by local fluid flow between these pores. However, rocks exhibit more complicated fabric structures, with heterogeneities presented at multiple scales, having a self-similar fractal nature as reported. We develop a poroelasticity model to describe wave anelasticity in a fluid-saturated medium consisting of infinite components, i.e., an infinituple-porosity medium. Numerical modeling is achieved with infinite iterations, where at each iteration one component is added (a porous inclusion embedded in a porous host), analogous to the differential effective medium (DEM) theory of solid composites. By considering the self-similar structure, the properties of each component (inclusions) are scale-dependent. The example shows that the P-wave velocity dependence on frequency depends on the fractal dimension. Application of the model to sandy sea-bottom sediments shows a good agreement, with fractal dimension of 2.88. [ABSTRACT FROM AUTHOR]

We establish a generalization of the thermoelasticity wave equation to the porous case, including the Lord–Shulman (LS) and Green–Lindsay (GL) theories that involve a set of relaxation times ( τ i , i = 1 , ... , 4 ). The dynamical equations predict four propagation modes, namely, a fast P wave, a Biot slow wave, a thermal wave, and a shear wave. The plane-wave analysis shows that the GL theory predicts a higher attenuation of the fast P wave, and consequently a higher velocity dispersion than the LS theory if τ 1 = τ 2 > τ 3 , whereas both models predict the same anelasticity for τ 1 = τ 2 = τ 3 . We also propose a generalization of the LS theory by applying two different Maxwell–Vernotte–Cattaneo relaxation times related to the temperature increment ( τ 3 ) and solid/fluid strain components ( τ 4 ), respectively. The generalization predicts positive quality factors when τ 4 ≥ τ 3 , and increasing τ 4 further enhances the attenuation. The wavefields are computed with a direct meshing algorithm using the Fourier pseudospectral method to calculate the spatial derivatives and a first-order explicit Crank–Nicolson time-stepping method. The propagation illustrated with snapshots and waveforms at low and high frequencies is in agreement with the dispersion analysis. The study can be useful for a comprehensive understanding of wave propagation in high-temperature high-pressure fields. [ABSTRACT FROM AUTHOR]

In this paper second-order seismic wave equations are adopted to simulate the wave propagation in the presence of a free-surface. In the simulations, horizontal and vertical spatial derivatives are solved by Fourier and Chebyshev pseudo-spectral algorithms respectively. In addition, Fourier–Chebyshev pseudo-spectral method is also used to solve the boundary conditions of the free-surface, which are Neumann boundaries. That makes results at the free-surface keep same high precision with inner domain. The new scheme is tested against Lamb's problem in a uniform elastic half space. Agreement between numerical and analytical results is every good. Compared with the former simulation scheme based first-order velocity–stress equations, the new scheme based on second-order equations is more efficient because it has fewer controlling equations. [ABSTRACT FROM AUTHOR]