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In this study, we present a semi-analytical method to simulate the propagation of seismic waves in horizontally stratified double-porosity media. We solve the governing equations in the frequency–wavenumber domain and then compute the time–space domain solutions by Hankel and fast Fourier transforms. We conduct numerical simulations to study the properties of the seismic waves in the double-porosity media. The results show the existence of three P waves, one fast P wave and two slow P waves. The two slow P waves are highly attenuated and can be observed by assuming a low fluid viscosity. By comparing the fast P wave in a single-porosity medium and that in a double-porosity medium, we find that the double-porosity model predicts much higher attenuation of the fast P wave due to the local fluid flow between the background medium and the inclusions. We also study the parameters affecting the attenuation, such as the radius of the spherical inclusions, the viscosity of the pore fluid, the permeability of the background medium and the porosity of the inclusions. Finally, we present a cross-well survey model to investigate the seismic responses in the double-porosity media and compare them with the responses in the single-porosity media. We find that both the amplitude and velocity of the fast P wave decrease with the increase of the volume ratio of the inclusions. [ABSTRACT FROM AUTHOR]

An exact analytical solution is presented for the effective dynamic shear modulus in a heterogeneous fluid-filled poroelastic medium containing spherical inclusions. The complex and frequency-dependent properties of the derived shear modulus are solely caused by the physical mechanism of mesoscopic-scale wave-induced fluid flow whose scale is assumed to be smaller than wavelength but larger than the size of pores. Our model consists of three phases: a spherical inclusion, a shell of matrix material with different mechanical and/or hydraulic properties and an outer region of effective homogeneous medium of infinite extent. This three-phase model represents a self-consistent model or an approximate model of composite having periodically distributed inclusions. The behaviors of both the inclusion and the matrix are described by Biot's equations (1941) with standard conditions of Deresiewicz and Skalak (1963) at the inclusion-matrix interface. The effective medium is regarded as an equivalent elastic or viscoelastic material with complex and frequency-dependent moduli to be determined. The derived effective shear modulus is used to quantify the shear-wave attenuation and velocity dispersion. For the problem of fluid patchy saturation (inclusions with the same solid frame as the matrix but with a different pore fluid from that in the matrix), the gas pocket does not affect the shear attenuation and dispersion characteristic of the water-filled matrix medium at all. For the problem of double porosity structure (inclusions having a different solid frame than the matrix but the same pore fluid as the matrix), with the increase of frequency the heterogeneous medium transitions from a low-frequency state having drained inclusions and drained matrix with no pore pressure difference to a higher-frequency state having undrained inclusions and undrained matrix with no fluid communication at the inclusion's surface. The relaxation frequency at which the maximum value of inverse quality factor occurs moves to frequencies by two orders of magnitude lower if the size of a unit cell increases by one order of magnitude. Stiff inclusions imbedded in a relatively soft matrix can cause significant and observable attenuation at seismic frequency bands, but softer inclusions imbedded in a relatively stiff matrix cause very weak attenuation. The mixed heterogeneity in both the solid frame and pore fluid also has important influences on the frequency-dependent shear wave attenuation. [ABSTRACT FROM AUTHOR]

• The exact solutions of wave scattering by a rectangular permeable crack in a fluid-saturated porous medium are derived. • The analytic solutions are used to develop an effective medium model for a porous medium with aligned cracks. • The effects of wave-induced fluid diffusion as well as crack geometry on seismic attenuation and dispersion are shown numerically. • New results are useful for extracting crack information from seismic frequency-dependent signatures with applications in exploration geophysics. It is well known that mesoscopic-scale wave-induced fluid flow (WIFF) between cracks and surrounding porous matrix is one of significant mechanisms in causing seismic dispersion and attenuation. Previous theoretical models that were used to interpret the interaction between cracks and passing waves assumed the cracks to be circular or slit-like. However, in many actual cases, the cracks can have a shape more close to a rectangle. In this paper, we develop an effective medium model to estimate the P-wave attenuation and velocity dispersion in a saturated porous rock containing a random distribution of permeable, aligned, rectangular cracks of infinitesimal thickness. This is done by combining the far-field displacement representation of the solution of a normally incident P wave scattered by a single crack and Foldy's scattering theorem. The effective low-frequency P-wave velocity predicted by the present model is asymptotically consistent with the Gassmann's theory. It is shown that thorough knowledge of the crack shape is important for understanding seismic signatures. Specifically, the WIFF relaxation frequency is mainly determined by the crack intermediate dimension, while it is insensitive to the crack largest dimension. Moreover, P-wave velocity of the rectangular crack model can be quite different from those of the circular crack model and the slit crack model, although the overall frequency-dependent trends of the velocities for different models are very similar. Analogously to the circular crack model and the slit crack model, at the normal incidence of a fast-P wave, the WIFF is found to dominate the wave attenuation and velocity dispersion. These results are helpful to estimate formation parameters from different frequency components of observed seismic data. [ABSTRACT FROM AUTHOR]

• The mode-I stress intensity factor (SIF) of a periodic array of collinear cracks in a porous medium is investigated. • This paper studies the case of fluid-saturated permeable cracks. • The interacting cracks and crack surficial permeability can significantly affect the magnitude and the frequency-dependent trend of the SIF. The dynamic (frequency-dependent) mode-I stress intensity factor (SIF) is obtained for a porous medium containing an array of equally distributed collinear cracks. Unlike previous work in which the cracks are assumed to be dry and impermeable, this paper studies the case that the crack is fluid-saturated and permeable. The cracks are under the action of a normal incidence of a time-harmonic plane longitudinal (P) wave. The interaction between the collinear cracks and the mechanism of wave-induced fluid flow (scattered slow P wave) can significantly affect the magnitude and frequency-dependent trend of the SIF, in particular at low frequencies. At low frequencies, the magnitude of the SIF increases monotonously with the decreasing distance between the adjacent cracks and can be much larger than that of a single crack, implying that the material strength can be greatly lowered by the interaction of cracks. However, at higher frequencies, the impacts of interacting cracks are negligible. The comparison with the counterpart of dry impermeable cracks reveals that the magnitude of SIF is lower than that of dry impermeable cracks owing to effective normal stress effect. Moreover, unlike the SIF of dry impermeable cracks for which a peak value is observed around elastic wave resonance frequency, the corresponding SIF of fluid-saturated permeable cracks monotonously decreases with the increasing frequency. This frequency-dependent trend is attributed to the conversion of incident energy into the slow P wave. The obtained results reveal significant influences of the presence of pore fluid and interaction of collinear cracks upon the SIF, both magnitudes and frequency-dependent trends. Such information is useful in predicting the fracture strength of saturated porous materials subjected to oscillating loads. [ABSTRACT FROM AUTHOR]

• Analytic solutions of the stress intensity factors (SIFs) of a fluid-saturated permeable crack in a poroelastic material subjected to a time-harmonic elastic wave have been derived. • Insights of effects of incident wave types, material permeability, incident angle and crack surficial permeability on the frequency-dependent behaviors of the SIFs have been obtained through numerical examples. • Because of the wave-induced fluid flow, the crack fluid can significantly change the effective normal stress so that the magnitude of mode-I SIF can be lower than that of a dry impermeable crack. In many practices, cracks are filled with fluid and their background materials can be modelled within the framework of poroelasticity. Understanding the crack-tip mechanical behaviors of such a cracked medium subjected to dynamic loads is of importance for prediction of fracture failure. In this paper, based on Biot's theory of poroelastodynamics, solutions of mode-I and mode-II dynamic stress intensity factors (SIFs) of a Griffith crack are derived for the scattering problem of oblique incidence of in-plane time-harmonic P-SV waves. The crack surfaces are assumed to be permeable, and the crack is saturated by fluid. With the aid of Fourier integral transforms, the problem is reduced to a system of coupled Fredholm integral equations with special emphasis placed on finding the near-field solution. Impacts of incident wave type, material permeability and incident angle on the frequency-dependent behaviors of the SIFs have been shown graphically. It is found that the induced fluid pressure inside the crack can significantly change the effective normal stress so that the magnitude of mode-I SIF is much lower than that of a dry impermeable crack. Furthermore, the diffusion-type fluid flow (scattered slow P wave) can greatly lower the magnitude of mode-I SIF at low frequencies (much smaller than the resonance frequency). The characteristic frequency at which the curves of SIFs decline strongest occurs when the characteristic diffusion length is of the same order of the crack size. The fluid flow can also lower mode-II SIF at low frequencies and the effects are strong in the low permeability case. Specifically, in the case of scattering of SV waves, it is shown that the maximum drop of mode-II SIF in peak with respect to that of the corresponding dry crack in an elastic medium is about 10%. The obtained results reveal significant influences of the presence of pore fluid upon the SIFs, both magnitudes and frequency-dependent trends. Such information is useful in predicting the fracture strength of saturated porous materials subjected to oscillating loads. [ABSTRACT FROM AUTHOR]