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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]

SUMMARY: The pressure sensitivity of stiffness in fractured rocks is closely related to fracture-surface geometries. The resulting stress-dependence of stiffness can be represented by the third-order elastic constants (3oECs). Fracture surfaces are generally rough at various scales, and can significantly affect the 3oECs of pre-stressed fractures as well as the wave-induced fluid flow (WIFF) induced by the Biot slow P -wave between fractures and the background medium. The WIFF usually depends on the fracture width relative to the slow P -wavelength and the fracture-surface roughness. We generate various fracture-surface geometries at different scales of random roughnesses parametrized by the surface standard deviation (SSD) of fracture-surface heights. With theoretical analyses and numerical simulations, we investigate the effect of fracture-surface geometries on the stress- and frequency-dependent stiffness through the 3oECs for pre-stressed rocks with aligned fractures. For the elastic wave in the low-frequency regime of Biot theory with the fracture scale much less than the wavelength, the induced WIFF significantly enhances the effect of fracture-surface geometries on the 3oECs and P - and S -wave moduli. The stiffness of fractured rocks increases with increasing SSDs, yielding a high sensitivity to pre-stresses. Toward the high-frequency limit, however, the fluid diffusion between fractures and the porous background decreases, which reduces the influence of fracture-surface roughnesses with the 3oECs much less than that in the low-frequency regime. The resulting P -wave modulus of aligned fluid-saturated fractures approximates to the background value. [ABSTRACT FROM AUTHOR]

Cracks widely exist in rocks and are important in various geophysical applications. However, although electrical methods are conventionally employed for the detection and characterization of cracks, the fundamental question whether cracks improve the conductive ability of porous rocks (a quantification of their electrical conductivity if their porosity and saturant are the same) remains largely unaddressed. We address this knowledge gap through theoretical models with confirmed validity. We show that the conductive ability of a rock containing non‐interacting penny‐shaped cracks with random orientation will be improved only in the case when the aspect ratio of the cracks is below a certain value, which is referred to as the critical crack aspect ratio. We also show that the critical crack aspect ratio is uniquely determined by the porosity and electrical conductivity (or cementation exponent) of the porous rock where the cracks reside. We further demonstrate that the critical crack aspect ratio is some representation of the pore structure, and using two times the critical crack aspect ratio as the pore geometry can give rise to a reasonable agreement between modelled and measured P‐ and S‐wave velocities. The critical crack aspect ratio offers a consistent microstructure for the joint elastic–electrical modelling for the improved characterization of cracks through integrated seismic and electrical surveys. [ABSTRACT FROM AUTHOR]

Understanding the electrical properties of reservoir rocks has important applications in oil and gas exploration. Small‐scale fractures that are widely existing in reservoir rocks are usually aligned along a certain preferential direction due to reasons such as tectonic movement, making the electrical conductivity of the rocks anisotropic. Since all rocks are experiencing geological pressure, it is theoretically and practically important to study the influence of pressure on the fractures and its control on the anisotropic conductivity of reservoir rocks. We first prepared artificial porous sandstone samples with and without penny‐shaped fractures and experimentally measured the anisotropic conductivity of the samples under differential pressure (the difference between confining pressure and pore fluid pressure). Then, we inverted from the experimental data for the fracture porosity and fracture aspect ratio at each differential pressure. The variation of the fracture parameters caused by the differential pressure and their influence on the anisotropic electrical conductivity of the rocks were further analysed theoretically. The results show that as the differential pressure increases, the measured anisotropic conductivity of the artificial samples with and without fractures all decreases exponentially. We also show that both the inverted fracture porosity and fracture aspect ratio decrease exponentially with the increase of differential pressure, and the fracture porosity and fracture aspect ratio are linearly correlated as an implicit function of differential pressure. The modelling results show that the differential pressure‐induced decrease in the fracture porosity reduces the electrical conductivity in all directions, and the conductivity reduction in the rock parallel to the direction of the fractures is most significant. Comparing with the influence of the fracture porosity resulting from the applied differential pressure, the pressure‐caused variation in the fracture aspect ratio was shown to play a secondary role on affecting the pressure‐dependent anisotropic electrical properties. [ABSTRACT FROM AUTHOR]

Elastic and electrical properties can be jointly interpreted for better characterizations of rocks with cracks that are common in geological rocks. However, the cross-property relationship between the elastic and electrical properties of cracked rocks, which forms the key to the successful joint elastic–electrical interpretation, remains poorly understood. We investigate the pressure effects on the joint elastic–electrical properties in brine-saturated artificial sandstones with aligned non-interacting penny-shaped cracks that are far from percolation. We measured and compared the anisotropic electrical conductivity and ultrasonic velocity in the artificial sandstones with and without aligned cracks under applied effective stress from 5 to 50 MPa. We showed that the existence of aligned cracks significantly enhanced the elastic and electrical anisotropies of the rocks, and the difference in the elastic and electrical anisotropies between the cracked and intact samples reduced as the effective stress increased. We also showed that the pressure-dependent electrical conductivity and ultrasonic velocity exhibited strong linear correlations in both the intact and cracked samples cored in different directions, and a difference existed in the slopes between the samples with and without aligned cracks. The distinct contributions of the pressure-induced deformation of cracks to the anisotropic elastic and electrical properties, as well as their different sensitivity to the cracks, were found to plausibly explain the observed experimental results. Theoretical modelling for quantitative interpretation of the experimental data is presented in a companion paper. [ABSTRACT FROM AUTHOR]

Cracks are widely existing in natural rocks, and the effects of cracks on the elastic and electrical properties of rocks make it possible to establish appropriate joint interpretation to detect the crack characteristics more accurately. However, the premise is the knowledge of the joint elastic–electrical properties of cracked rocks. We aim to study theoretically the pressure-dependent joint elastic–electrical properties of porous rocks with aligned penny-shaped cracks. To do so, we developed a model describing the deformation of cracks in a transversely isotropic (TI) background with stress, and combined with existing models for the anisotropic elastic and electrical properties of TI rocks with aligned cracks. The impacts of crack deformation on the anisotropic elastic and electrical properties of cracked rocks, and the effects of crack porosity and crack aspect ratio on the pressure-dependent joint elastic–electrical properties were then studied. We showed that the modelling results compared satisfactorily with experimental data. The modelling results demonstrated that the crack compaction with stress reduced the elastic and electrical anisotropy. We also found that the anisotropic elastic wave velocities and electrical conductivities of cracked rocks exhibited approximately negative linear correlations when linked by the effective stress, and the slopes and intercepts of the linear relationships were shown to be systematically dependent on the porosity and aspect ratio of the aligned cracks. The above knowledge of the pressure-dependent joint elastic–electrical properties of TI rocks with aligned cracks can be employed for quantitative interpretation of the joint elastic and electrical exploration data. [ABSTRACT FROM AUTHOR]

Seismic waves propagating in a fractured porous rock with fluids can be greatly attenuated due to the wave‐induced fluid flow, including the macroscopic Biot flow, the mesoscopic interlayer fluid flow and the microscopic squirt flow. However, the comprehensive effects of the above wave‐induced fluid flows on the frequency‐dependent seismic properties in layered and fractured rocks with partial saturation are still poorly understood. To obtain such knowledge, we present a superposition approach accounting for the dispersion and attenuation induced by the multi‐scale wave‐induced fluid flows in the above‐mentioned target rocks, and study the influences of the water saturation, layer thickness ratio and fracture properties (i.e. fracture density and fracture aspect ratio) on the frequency‐dependent vertical P‐wave velocity (VP0). The results demonstrate that the increasing water saturation enhances the dispersion and attenuation of VP0 in the full frequency band, and the varying layer thickness ratio not only affects the dispersion and attenuation induced by the interlayer fluid flow, but also influences those caused by the Biot flow and squirt flow. The fracture properties are also found to significantly impact the seismic properties of the target rock. In addition to its effects on the squirt flow, the increasing fracture density is shown to weaken the dispersion and attenuation induced by the Biot flow, and enhance those caused by the interlayer fluid flow. The obtained results provide new insights into the frequency‐dependent seismic properties in the partially saturated layered rocks with fractures, and the knowledge gained from these results has important practical applications for the successful interpretation of fluid and fracture properties applying the seismic exploration data acquired in fractured reservoirs. [ABSTRACT FROM AUTHOR]

Natural fractures are usually multiscale in size, orientations and distribution, resulting in complex anisotropic characteristics. Analytical methods for quantifying the associated effective elastic properties are based on some assumptions, such as dilute fracture concentration and regular-shaped fractures, which do not occur in actual reservoirs. Here, we conduct anisotropic finite-element modeling of effective elastic properties of complex fractured rocks using the least-square fitting method. The algorithm is developed for a 3D case and validated by classical effective medium theories for models with different fracture densities. The results of the 3D numerical method agree well with the theoretical predictions at low fracture density. The model also considers the interactions among fractures to calculate equivalent stiffness tensors at high fracture density. Three 2D fracture models are simulated to demonstrate the basic behavior of stress interactions and their effect on the overall elasticity under different fracture densities. We applied the developed model to 3D natural fractures built from a real outcrop, and we found that the fracturing pattern significantly affects the effective anisotropy properties. The resultant P-wave phase velocities as functions of the incidence angle and frequency are anisotropic. This study provides a great potential to calculate equivalent stiffness tensors and anisotropic properties of 3D multiscale natural fractures. [ABSTRACT FROM AUTHOR]

Understanding the stress-dependent elastic moduli of fractured rocks is essential for monitoring geopressure and tectonic stress. The elastic nonlinearity with finite strains can be described by the theory of acoustoelasticity through the third-order elastic constants (3oECs) that are strictly valid for an isotropic homogeneous medium. The extension to fractured rocks becomes particularly interesting because of the sensitivity of fractures to prestress, but remains largely unaddressed. We address this issue through theoretical modeling with confirmed validity by considering a group of homogeneously distributed fractures (ellipses) embedded into a homogeneous background medium subject to a uniform confining pressure. We incorporate both the David-Zimmerman (DZ) and Mori-Tanaka (MT) models into the theory of acoustoelasticity. The DZ model accounts for the closure influence of cracks with different aspect ratios, whereas the MT model is used to estimate the combining effect of the background acoustoelasticity and the elastic nonlinearity due to the closure of cracks. We validate the acoustoelastic DZ-MT model of fractured rocks by experiment data with Fontainebleau and Vosges sandstones. We show that the effective elastic moduli of fractured rocks mainly depend on the background acoustoelasticity and nonlinear elastic deformations induced by closing cracks. The inclusions inside fractures have too low 3oECs that the associated acoustoelastic effect could be neglected for both interconnected and isolated cracks. The effective elastic moduli of rocks change significantly during the closing period of cracks with increasing pressure under low pressure. The influence of cracks can be reduced only when cracks are closed completely, where, unlike the theoretical prediction by the DZ model, the effective elastic moduli of rocks still increase along with increasing pressure because of the acoustoelastic effect of the background. [ABSTRACT FROM AUTHOR]

Wave‐induced fluid flow plays an important role in affecting the seismic dispersion and attenuation of fractured porous rocks. While numerous theoretical models have been proposed for the seismic dispersion and attenuation in fractured porous rocks, most of them neglect the wave‐induced fluid flow resulting from the background anisotropy (e.g. the interlayer fluid flow between different layers) that can be normal in real reservoirs. Here, according to the theories of poroelasticity, we present an approach to study the frequency‐dependent seismic properties of more realistic and complicated rocks, i.e. horizontally and periodically layered porous rock with horizontal and randomly orienting fractures, respectively, distributed in one of the two periodical layers. The approach accounts for the dual effects of the wave‐induced fluid flow between the fractures and the background pores and between different layers (the interlayer fluid flow). Because C33 (i.e., the modulus of the normally incident P‐wave) is directly related to the P‐wave velocity widely measured in the seismic exploration, and its comprehensive dispersion and attenuation are found to be most significant, we study mainly the effects of fracture properties and the stiffness contrast between the different layers on the seismic dispersion and attenuation of C33. The results show that the increasing stiffness contrast enhances the interlayer fluid flow of the layered porous rocks with both horizontal and randomly orienting fractures and weakens the wave‐induced fluid flow between the fractures and the background pores, especially for the layered porous rock with horizontal fractures. The modelling results also demonstrate that for the considered rock construction, the increasing fracture density reduces the interlayer fluid flow while improves the dispersion and attenuation in the fracture‐relevant frequency band. Increasing fracture aspect ratio is found to reduce the dispersion and attenuation in the fracture‐relevant frequency band only, especially for the layered porous rock with horizontal fractures. [ABSTRACT FROM AUTHOR]

Poro-acoustoelastic theory has made a great progress in both theoretical and experimental aspects, but with no publications on the joint research from theoretical analyses, experimental measurements, and numerical validations. Several key issues challenge the joint research with comparisons of experimental and numerical results, such as digital imaging of heterogeneous poroelastic properties, estimation of acoustoelastic constants, numerical dispersion at high frequencies and strong heterogeneities, elastic nonlinearity due to compliant pores, and contamination by boundary reflections. Conventional poro-acoustoelastic theory, valid for the linear elastic deformation of rock grains and stiff pores, is modified by incorporating a dual-porosity model to account for elastic nonlinearity due to compliant pores subject to high-magnitude loading stresses. A modified finite-element method is employed to simulate the subtle effect of microstructures on wave propagation in prestressed digital cores. We measure the heterogeneity of samples by extracting the autocorrelation length of digital cores for a rough estimation of scattering intensity. We conductexperimental measurements with a fluid-saturated sandstone sample under a constant confining pressure of 65 MPa and increasing pore pressures from 5 to 60 MPa. Numerical simulations for ultrasound propagation in the prestressed fluid-saturated digital core of the sample are followed based on the proposed poro-acoustoelastic model with compliant pores. The results demonstrate a general agreement between experimental and numerical waveforms for different stresses, validating the performance of the presented modeling scheme. The excellent agreement between experimental and numerical coda quality factors demonstrates the applicability for the numerical investigation of the stress-associated scattering attenuation in prestressed porous rocks. [ABSTRACT FROM AUTHOR]

Electrical surveys are among the most important geophysical applications to understand the subsurface Earth, where anisotropic rock matrix is frequently encountered. Knowledge of the electrical properties of such rocks and their relationships with information of geological features included in the anisotropic rocks is crucial for the interpretation of electrical survey data. For this purpose, we first derive rigorous theoretical models for the electrical anisotropy of transversely isotropic (TI) rocks with spheroidal inclusions. The models assume the anisotropic background behaves isotropically in the direction whey the electrical field is applied parallel to the TI axes, and account for both aligned and randomly distributed inclusions and reduce to classical electrical models in the case of isotropic background. We then study systematically, through the obtained new models, the combined effects of geometry, concentration and alignment of both conductive and insulating spheroidal inclusions on the electrical anisotropy of TI rocks. We show that for aligned inclusions, the impact of inclusion volume fraction and aspect ratio is most significant in the direction when the electrical conductivity is measured along and perpendicular to the inclusions for conductive and insulating inclusions, respectively. For randomly distributed inclusions, both horizontal and vertical conductivity increases with decreasing aspect ratio of conductive inclusions and decreases with reducing aspect ratio of insulating inclusions. Most importantly, it is shown that the effects of randomly distributed conductive and insulating inclusions on the electrical anisotropy are similar although the rocks with insulating inclusions exhibit greater electrical anisotropy. The results provide new insights into the roles of spheroidal inclusions on the electrical anisotropy of TI rocks and pave a way for the successful interpretation of electrical survey data acquired in anisotropic rocks. [ABSTRACT FROM AUTHOR]

Fractures are commonly existing in the subsurface rocks, and understanding the seismic anisotropy in fractured rocks is of great importance in a range of geophysical applications. However, although all the fractured rocks are suffering from geological stresses and most of them are experiencing partial water saturation, the combined effects of confining pressure and water saturation on the seismic anisotropy in fractured rocks remain poorly understood. To obtain such knowledge, we measured and compared the anisotropic velocities and corresponding seismic anisotropy in two artificial porous sandstone samples with and without aligned fractures, respectively, as functions of varying confining pressure and degree of water saturation. We showed that the existence of aligned fractures significantly reduced the five measured velocities but dramatically enhanced the seismic anisotropy in the fractured sample. We also showed that confining pressure and water saturation had an interdependent complex effect on the seismic anisotropy of the fractured sample. The seismic anisotropy of the dry sample showed the highest‐pressure sensitivity, and the saturation effect was most significant at low confining pressure. Integrated interpretation of the experimental data indicated that wave‐induced fluid flow due to pore‐scale and mesoscale fluid distribution in the rocks is responsible for most of the observed pressure and saturation‐dependent anisotropic behaviors in the rocks. The results have helped to gain new knowledge on the seismic anisotropy in partially saturated fractured rocks subject to varying pressures and suggest the requirement for advanced theoretical models to fully understand the acquired data. Key Points: Combined pressure and water saturation effects on the seismic anisotropy in artificial porous sandstone with aligned fractures are studiedExistence of aligned fractures reduces the measured velocities but enhances the seismic anisotropy in the fractured sampleSeismic anisotropy of dry sample shows the highest pressure sensitivity and saturation effect is most significant at low confining pressure [ABSTRACT FROM AUTHOR]

Cracks that are widely existing in geological formations play an important role in affecting the elastic and electrical rock properties, which are remotely measured for detecting and characterizing cracked rocks. Joint interpretation of the complementary elastic and electrical data is expected to give a better assessment of the cracks. However, the correlations between the elastic and electrical rock properties, which are the key to the successful joint interpretation, remain poorly understood in layered porous rocks with cracks of varying orientations. We reviewed and summarized the most recent advances in the theoretical models for the elastic and electrical properties of layered porous rocks with inclined and rotated cracks as well as with randomly orienting cracks. The limitations of the models were identified by comparing the modeling results with available experimental data, and we applied the models within their validity range to obtaining theoretically the correlations between the elastic and electrical properties in layered porous rocks with cracks of the above-mentioned orientations, in the horizontal and vertical directions, respectively. We showed that the elastic and electrical properties are systematically correlated with cracks of varying crack densities and aspect ratios, different cracks orientations affect the correlations in varying ways. The results not only illustrate the link between the elastic and electrical properties in cracked rocks but also have important practical applications when the seismic and electromagnetic exploration data are jointly interpreted to gain better knowledge about the subsurface Earth, although future work is required to extend the models to more realistic cracked formations and to experimentally test the validity of the models and the theoretically obtained joint correlations. [ABSTRACT FROM AUTHOR]