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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]

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]

Seismic wave scattering dispersion and attenuation can be significant in cracked reservoirs. Many scattering models have been proposed, and the fractal (self-similar) features of the medium need to be further incorporated and analysed. We solve the P -wave scattering caused by fluid-saturated aligned cracks of finite thickness embedded in fractal media. The model is based on crack displacement discontinuities by using the Foldy approximation and representation theorem. The frequency dependence of velocity and attenuation are analysed as a function of the incidence angle and the crack and fluid properties. The results show that the crack density, thickness and radius can have a significant influence on the wave properties, as well as the fluid bulk modulus and saturation. The model requires three parameters to describe self-similar cracked media, and can be relevant in seismology, oil exploration and non-destructive testing of materials. [ABSTRACT FROM AUTHOR]

Laboratory experiments on partially saturated rocks show that seismic attenuation can be significant. The main mechanism, wave‐induced local fluid flow (WILFF), is affected by the spatial fluid distribution, especially in conditions of patchy saturation at different spatial scales. We propose a theory to obtain the seismic properties of partially saturated rocks based on fractal (self‐similar) patches, leading to an effective frequency‐dependent fluid modulus. The model combines the differential effective medium and Biot‐Rayleigh theories, where the patches are inclusions incrementally added, such that the effective fluid calculated in the current addition serves as host fluid in the next one. The analysis shows that adding identical inclusions in one or several steps produces nearly the same results, but the seismic properties depend on the scale range (radius) of the inclusions, fractal dimension Df of the self‐similar distribution, parameter θ $\theta $ of the exponential distribution, mean radius r0 and variance σr2 ${\sigma }_{r}^{2}$ of the Gaussian distribution. Forced‐oscillation experiments were performed on a limestone sample under partial water‐saturation conditions at seismic frequencies (2–500 Hz), to obtain the velocity dispersion and extensional attenuation. The proposed theory provided a reasonable description of these experimental data as well as other published measurements on tight carbonate and Berea sandstone. Plain Language Summary: Seismic wave velocity dispersion and attenuation in partially saturated rock are affected by the size of the fluid patches, and their fractal dimension. To study this phenomenon, we have developed a wave propagation theory, in which the final partially saturated material is incrementally constructed by adding inclusions in size order into a homogenous frame saturated by a host fluid. We analyze the effects of wave‐induced local fluid flow due to the multi‐scale fluid heterogeneities on wave attributes. The broadband wave anelasticity predicted by the model is strongly affected by the scale range of the heterogeneities and their fractal dimension. The theory was compared with experimental data measured on different rock specimens and the fluid distribution characteristics at different saturations were estimated. Key Points: We present a wave propagation theory for partially saturated rocks with a fractal (self‐similar) distribution of fluid patchesWe analyzed the effects of wave‐induced local fluid flow on seismic attributes due to multi‐scale fluid heterogeneitiesWe estimated variations of the fluid distribution characteristics in partially‐saturated rock samples by tying the data to the low frequency measurements [ABSTRACT FROM AUTHOR]

We analyse the reflection coefficient of an inhomogeneous plane wave incident on the thermally insulated surface of a thermo-poroelastic medium. The theory, which includes the classic Lord-Shulman (LS) and Green-Lindsay (GL) theories as well as a generalization of the LS model, predicts three inhomogeneous longitudinal waves and one transverse wave, described by potential functions specified by the propagation direction and inhomogeneity angle. The GL model can give a stronger P1-wave thermal attenuation and consequently a stronger velocity dispersion than the LS model. We investigate the influence of inhomogeneity angle, type of incident wave, frequency and surface boundary conditions. The generalized LS model exhibits increased P1-wave thermal attenuation with increasing Maxwell–Vernotte–Cattaneo relaxation time and consequently predicts more interference energy, irrespective if the surface is open or sealed. The inhomogeneity angle affects the energy partitions particularly near the grazing incidence, with a significant interference energy, which must be taken into account to satisfy the energy conservation. The thermal dispersion occurs at frequencies around the thermal relaxation peak, which moves to low frequencies when the conductivity increases. [ABSTRACT FROM AUTHOR]

The Lord–Shulman thermoelasticity theory combined with Biot equations of poroelasticity, describes wave dissipation due to fluid and heat flow. This theory avoids an unphysical behaviour of the thermoelastic waves present in the classical theory based on a parabolic heat equation, that is infinite velocity. A plane-wave analysis predicts four propagation modes: the classical P and S waves and two slow waves, namely, the Biot and thermal modes. We obtain the frequency-domain Green's function in homogeneous media as the displacements-temperature solution of the thermo-poroelasticity equations. The numerical examples validate the presence of the wave modes predicted by the plane-wave analysis. The S wave is not affected by heat diffusion, whereas the P wave shows an anelastic behaviour, and the slow modes present a diffusive behaviour depending on the viscosity, frequency and thermoelasticity properties. In heterogeneous media, the P wave undergoes mesoscopic attenuation through energy conversion to the slow modes. The Green's function is useful to study the physics in thermoelastic media and test numerical algorithms. [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]

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]

Ultrasonic P-wave attenuation was measured in tight oil siltstones, carbonates and a tight sandstone with two independent estimation methods. The dependence on saturation in gas-water partially-saturated siltstones at in-situ conditions shows a different behavior compared to the other rocks. The siltstones in our experiments exhibit a behavior characterized by a gradual decrease of attenuation with increasing water saturation in the presence of gas, and full-gas saturation shows more attenuation than full-oil and full-water saturations. However, previous theoretical and experimental studies show that gas-water saturated carbonates and sandstones have the highest attenuation at high water saturations, and generally, a liquid-saturated rock shows more attenuation than a gas-saturated one. Poroelasticity theory shows that the two dominant loss mechanisms (due to fabric heterogeneity and patchy saturation) have peaks at different frequencies for siltstones, resulting in a gradual decrease of attenuation with water saturation, while these mechanisms overlap at ultrasonic frequencies for carbonates and sandstones, leading to an attenuation peak at high water saturations. The predicted attenuation dependence on fluid type agrees with the measurement for most samples. Regarding the tight oil siltstones, although the model fails to explain the experimental results for oil-water saturation, it can be concluded that for gas-water saturation the squirt flow caused by fabric heterogeneity dominates the attenuation, which differs from carbonates and sandstones. Experimental studies show that the attenuation dependence on saturation in tight oil reservoirs can be associated with fabric texture. The theory describes these behaviors, which can potentially improve the practices of detecting and monitoring multi-phase fluids in the reservoirs. • Ultrasonic P-wave attenuation decreases with water saturation in in-situ water-gas partially-saturated siltstones. • Attenuation behavior with fluid type and saturation in tight oil siltstones differ from those of carbonates and sandstone. • Poroelasticity modeling explains the observed phenomena in water-gas saturated siltstones. [ABSTRACT FROM AUTHOR]

Shale reservoir formations are porous rocks of low permeability composed of fluid-saturated illite–smectite and kerogen layers, which behave as viscoelastic transversely isotropic (VTI) media at long wavelengths, that is, much larger than the average layer thickness. Seismic waves travelling across these heterogeneous materials induce wave-induced fluid flow (WIFF) and Biot slow waves generating energy loss (mesoscopic loss) and velocity dispersion. When these formations are saturated by two-phase fluids, the presence of capillary forces—interfacial tension—and interaction between the two fluids as they move within the pore space need to be taken into account. This can be achieved using a Biot model of a poroelastic solid saturated by a two-phase fluid that includes capillary pressure and relative permeability functions and supports the existence of two slow waves. An upscaling finite-element method is used to analyse the WIFF, which determines an effective VTI medium predicting higher attenuation and (Q) anisotropy than the classical single-phase (single fluid) models. [ABSTRACT FROM AUTHOR]

Backus and Wyllie equations are used to obtain average seismic velocities at zero and infinite frequencies, respectively. Here, these equations are generalized to obtain averages of the seismic quality factor (inversely proportional to attenuation). The results indicate that the Wyllie velocity is higher than the corresponding Backus quantity, as expected, since the ray velocity is a high-frequency limit. On the other hand, the Wyllie quality factor is higher than the Backus one, following the velocity trend, i.e., the higher the velocity (the stiffer the medium), the higher the attenuation. Since the quality factor can be related to properties such as porosity, permeability, and fluid viscosity, these averages can be useful for evaluating reservoir properties. [ABSTRACT FROM AUTHOR]

The wave-induced interlayer flow between mesoscopic-scale heterogeneities (i.e., larger than pore scale but smaller than the predominant wavelengths) is the most important cause of attenuation at frequencies below 1 kHz. According to the White's layered model ( White et al., 1975 ), the wave-induced flow at an interface is symmetrical in periodically layered porous media due to its symmetrical structure. However, in one-dimensional (1-D) randomly layered porous rocks, layers of various thicknesses lead to different types of the interlayer flow. The coexistence of various wave-induced interlayer flow patterns at different interfaces results in interactions. The numerical creep test and a volume average of White's analytical solution were used to analyse the interactions in the two kinds of models (i.e., representative elementary volumes and randomly layered models). One of the parameters used to build a randomly layered model is the standard deviation of the thickness of the water-saturated layer, which could control the interaction strength of the interlayer flow. The detailed characteristics of the interlayer flow interaction were demonstrated by analysing the relative fluid velocity, while the statistical characteristics of the interlayer flow interaction were demonstrated by analysing the average solid velocity. We investigated the effect of the interaction on the amplitude versus the deviation at the interface between a non-dispersive medium and a patchy-saturated dispersive medium in three types of reservoirs. These results are beneficial for describing the distribution of oil/gas patches based on the statistical seismic properties in a layered formation with random disorder. [ABSTRACT FROM AUTHOR]