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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]

Deep carbonate reservoirs are subject to in-situ conditions of high temperature and high pressure. We consider six water-saturated dolomite specimens from these reservoirs and perform ultrasonic experiments to obtain the P- and S-wave waveforms and velocities under different pressure and temperature conditions. The P-wave attenuation is estimated using the spectral-ratio method. The results show that with the increase of temperature, the velocities slightly decrease and attenuation increases. At effective pressures <40 MPa, the P-wave velocities increase sharply with pressure and then approximately linearly at higher pressures, while attenuation decreases gradually with pressure. The crack porosity as a function of pressure is obtained from the experimental data. The P-wave velocities decrease with this porosity while attenuation shows an opposite behavior. Then, a multiscale poroelasticity model considering micro-, meso- and macro-scale fluid-flow mechanisms is proposed to analyze the effects of the fluid properties, temperature and crack content on the wave responses. The model results agree well with the experimental data at different pressures, which provides a theoretical basis for the analysis of broadband wave velocity dispersion and attenuation phenomena of carbonate reservoirs and underground porous media in general. [ABSTRACT FROM AUTHOR]

There is a mathematical analogy between the poroelastic and thermoelastic behavior of the compressional P wave, whose dissipation is due to the coupling between the elastic deformation and the phenomenon of pressure and heat diffusion, represented by Darcy's law and heat conduction, respectively. This attenuation effect is more pronounced in heterogeneous media, where conversion of the fast P wave to the slow (Biot) and thermal diffusive modes occurs at material interfaces. Specifically, the problem is to obtain the P-wave properties of a porous medium due to temperature gradients between the solid and fluid phases. Then, we consider a simplified one-dimensional porous medium and obtain the quality factor, Q, and phase velocity as a a function of frequency, based on an isostress condition and using Gassmann equation and the Kramers Kronig relations.The model allows for the incorporation of a relaxation time to simulate a proper wave-like behavior at high frequencies, avoiding in this way infinite velocities. Moreover, we perform a complete analysis varying the different parameters, namely, the heat conduction, the specific heat, the thermal expansion, the types of solid and fluid and the pore size. [ABSTRACT FROM AUTHOR]

We obtain the amplitude and energy reflection coefficients of seismic waves in porous media with penny-shaped inclusions, based on the generalized Biot-Rayleigh model that takes into account the attenuation due to mesoscopic local fluid flow (LFF). We consider two cases, including a contact between two porous media having either different fluids (gas–water contact) or crack density and aspect ratio, as well as a water half-space overlying a porous medium, and study the frequency-dependent reflection-transmission (scattering) coefficients for open- and sealed-pore boundary conditions. Our examples show that the LFF mechanism mainly reduces the reflection coefficients (amplitude and energy) at the gas–water contact and at a water–porous medium interface for frequencies less than 10 kHz, due to the fact that the velocity in the lower medium decreases. For the latter case, if the fluid is gas, the LFF effect becomes only important at frequencies between 0.0001 and 10 Hz for the open-pore case. This is due to the fact that the acoustic impedance contrast between water and gas is high. At frequencies less than 0.0001 Hz, the interface is equivalent to a water–elastic medium one, and hence the results are the same as those of the sealed-pore case. Moreover, the crack density and aspect ratio affect the mesoscopic attenuation and relaxation frequency, and therefore the reflection coefficients. [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]

Generally, local stress induced by individual crack hardly disturbs their neighbours for small crack densities, which, however, could not be neglected as the crack density increases. The disturbance becomes rather complex in saturated porous rocks due to the wave‐induced diffusion of fluid pressures. The problem is addressed in this study by the comparison of two solutions: the analytical solution without stress interactions and the numerical method with stress interactions. The resultant difference of effective properties can be used to estimate the effect of stress interactions quantitatively. Numerical experiments demonstrate that the spatial distribution pattern of cracks strongly affects stress interactions. For regularly distributed cracks, the resulting stress interaction (shielding or amplification) shows strong anisotropy, depending on the arrangement and density of cracks. It has an important role in the estimation of effective anisotropic parameters as well as the incident‐angle‐dependency of P‐ and SV‐wave velocities. Contrarily, randomly distributed cracks with a relative small crack density generally lead to a strong cancellation of stress interactions across cracks, where both the numerical and analytical solutions show a good agreement for the estimation of effective parameters. However, for a higher crack density, the incomplete cancellation of stress interactions is expected, exhibiting an incidence‐angle dependency, slightly affecting effective parameters, and differentiating the numerical and analytical solutions. [ABSTRACT FROM AUTHOR]

Abstract The variations of the static and dynamic moduli of porous rocks as a function of differential pressure have been described in terms of pore microstructure, i.e., the effects of cracks (soft pores) and intergranular (stiff) pores. Specifically, the first play a major role on the elastic properties of rocks. By using the Mori-Tanaka theory, this work relates rock elastic properties to the pore microstructure. Then, we estimate the distribution of pore aspect ratios by using the static bulk modulus instead of the dynamic one. The approach is applied to data from the Navajo and Weber sandstones. The results show that the cumulative crack porosity and density obtained from the static bulk modulus are higher than those estimated from the dynamic moduli, while the dominant crack aspect ratio is lower. The total porosity estimated from the static modulus agrees better with the experimental data. Furthermore, the cumulative crack (soft or compliant) porosity are estimated by the method, which also agrees with results from stress-strain relation and from bulk compressibility curves. [ABSTRACT FROM AUTHOR]

Abstract: The heterogeneous distribution of fluids in patchy-saturated rocks generates significant velocity dispersion and attenuation of seismic waves. The mesoscopic Biot–Rayleigh theory is used to investigate the relations between wave responses and reservoir fluids. Multiscale theoretical modeling of rock physics is performed for gas/water saturated carbonate reservoirs. Comparisons with laboratory measurements, log and seismic data validate the rock physics template. Using post-stack and pre-stack seismic inversion, direct estimates of rock porosity and gas saturation of reservoirs are obtained, which are in good agreement with oil production tests of the wells. [Copyright &y& Elsevier]

ABSTRACT We generalize the classical theory of acoustoelasticity to the porous case (one fluid and a solid frame) and finite deformations. A unified treatment of non-linear acoustoelasticity of finite strains in fluid-saturated porous rocks is developed on the basis of Biot's theory. A strain-energy function, formed with eleven terms, combined with Biot's kinetic and dissipation energies, yields Lagrange's equations and consequently the wave equation of the medium. The velocities and dissipation factors of the P- and S-waves are obtained as a function of the 2nd- and 3rd-order elastic constants for hydrostatic and uniaxial loading. The theory yields the limit to the classical theory if the fluid is replaced with a solid with the same properties of the frame. We consider sandstone and obtain results for open-pore jacketed and closed-pore jacketed 'gedanken' experiments. Finally, we compare the theoretical results with experimental data. [ABSTRACT FROM AUTHOR]

S-wave attenuation in porous media is affected by the presence and properties of saturating fluids. We measured the ultrasonic S-wave velocities of three types of tight rocks (siltstone, dolomite, and sandstone) under partial fluid saturation conditions using the pulse-transmission technique. The S-wave attenuation is estimated by the centroid frequency shift and spectral-ratio methods. The fluid-saturated test results show that the attenuation of most water-saturated dolomites is stronger than that of gas- or oil-saturated dolomites, whereas no apparent relationship exists between attenuation and fluid type in siltstone. The attenuation of siltstone and dolomite increases with increasing porosity and permeability, as well as with clay content for siltstone with clay contents less than 6%. The maximum attenuation of rocks partially saturated with gas and water occurs at high water saturation (55%–100%) in most siltstones. The attenuation of most dolomites shows an increasing trend with water saturation, but remains nearly unchanged in partially saturated sandstone. The attenuation of rocks partially saturated with oil and water is strongest at the partial saturation state in siltstone, but remains essentially constant with water saturation in dolomite. A squirt flow model is used to predict the S-wave attenuation of the fluid-saturated sample. Based on rock property and microstructure analyses, wave attenuation in the fluid-saturated rocks measured in this study can be explained by squirt flow mechanisms and the scattering effect owing to heterogeneities of water patches and grain clusters. [ABSTRACT FROM AUTHOR]

We study the anelastic properties (attenuation and velocity dispersion) of surface waves at an interface between a finite water layer and a porous medium described by Biot theory including the frequency-dependent effects due to mesoscopic flow. A closed-form dispersion equation is derived, based on potential functions and open and sealed boundary conditions (BC) at the interface. The analysis indicates the existence of high-order surface modes for both BCs and a slow true surface mode only for sealed BC. The formulation reduces to two particular cases in the absence of water and with infinite-thickness water layer, with the presence of pseudo-versions of Rayleigh and Stoneley waves. The mesoscopic flow affects the propagation of all the pseudo-surface waves, causing significant velocity dispersion and attenuation, whereas the effect of the BC is mainly evident at high frequencies, due to the presence of the slow Biot wave. The mesoscopic-flow peak moves to low frequencies as the thickness of the water layer increases. In all cases, the true surface wave resembles the slow P2 wave, and is hardly affected by the flow. [ABSTRACT FROM AUTHOR]

We compare the exact normal-incidence PP reflection coefficient [Geertsma–Smit expression] to approximations reported by several authors, based on open-pore boundary conditions at a plane interface between two porous media. The approximations correspond to low frequencies. Two of them are derived from the low-frequency Biot theory below the Biot characteristic frequency, but the results show significant differences much below the Biot frequency. Then, we extend the Geertsma–Smit equations by including the high-frequency viscodynamic operator (i.e., the full-frequency range Biot theory), showing that there are additional substantial differences at the high-frequency range. Use of this latter expression is required to honor the physics in the whole frequency range. We further generalize the Geertsma–Smit equations to the case of general boundary conditions other than the open-pore interface. At the seismic band, it is shown that the lossless (elastic) expression based on the Gassmann P-wave impedance is the reflection coefficient to use for practical applications. It is inferred that interpretations based on the frequency dependency of these approximations can be misleading, since this dependency does not provide a suitable description of the physics. [ABSTRACT FROM AUTHOR]