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The tight oil/gas reservoirs in China have showed the great exploration prospects and high production potential, with the characteristics of low porosity, low permeability, and significant heterogeneity in formation rocks. It remains a challenge to sort out the relations between reservoir wave responses and rock physical properties, and the further studies on the wave response patterns of tight reservoirs are in an urgent demand. The shear modulus and S- (shear) wave attenuation of rocks are affected by the properties of pore fluid and confining pressure. The ultrasonic wave experiments are performed on eight partially-saturated tight sandstone samples at different confining pressures, and we estimate S-wave attenuation with the spectral-ratio method. Results show that S-wave attenuation decreases with increasing confining pressure, and the water saturation case shows more loss compared to the oil saturation case, while the gas saturation case gives the lowest attenuation. We observe the S-wave relaxation peak at an intermediate water saturation for the gas-water partial-saturation case in general. S-wave attenuation increases with increasing porosity or permeability. Based on the measured rock physical properties, and combined with the Voigt–Reuss–Hill (VRH) average, differential effective medium (DEM) model and squirt-flow model, a tight rock attenuation model is proposed for analyzing the attenuation characteristics of fluid-saturated rocks at different confining pressures. The model reasonably describes the S-wave attenuation characteristics. The model predictions of S-wave attenuation show apparent pressure- and fluid-sensitivity at full saturation and partial saturation conditions. For sample TS1-19 at full saturation with different confining pressures, the S-wave peak attenuation predicted by the model ranges from 11.6 to 69.5, and decreases with confining pressure, while the relaxation frequency shifts to high frequency end. For the partial saturation condition of the sample, the predicted S-wave peak attenuation ranges from 15.5 to 39.8 at a confining pressure of 30 MPa and increases with water saturation, while the relaxation frequency shifts to low frequency end. For all the samples at 30MPa confining pressure, the predicted S-wave attenuation ranges from 5.6 to 38.6. At the full-saturation case, the predicted S-wave attenuation increases with porosity and decreases with confining pressure. For the partial saturation case, the S-wave attenuation predicted with the model and the Voigt and Reuss bounds generally increases with water saturation, whereas the experimentally-measured attenuation exhibits the peak attenuation at an intermediate saturation. [ABSTRACT FROM AUTHOR]

Tight gas sandstone reservoirs play an important role in the field of exploration and development of unconventional oil/gas resources. However, these reservoirs typically exhibit low-porosity and poor-permeability, with pore structures of strong heterogeneity comprising connected and isolated pores. The traditional rock physics models (such as the Gassmann model) are established under the assumption of fully connected pores, which cannot reasonably describe the complexity in tight reservoirs. In this regard, this work proposes a rock physics modeling approach aimed at tight sandstone reservoirs based on a reformulated Xu–White model. Specifically, the differential effective medium model and Xu–White model are combined to analyze the influences of isolated pores on the dry rock skeleton. To characterize the general effects due to the different pore types on rock elastic moduli, the volume content of connected pores is defined as a pore-connectivity parameter. The simulated annealing algorithm is applied to invert the parameter, which is treated as a weighting coefficient for correcting the wet rock elastic moduli, thereby improving the precision of rock physics modeling. The proposed approach is tested and verified by using the log data of the work area of Sichuan Basin, West China. The result shows that, compared with the conventional method, the P- and S-wave velocities predicted by the proposed method are consistent with the log data. In addition to the inversion results of the conventional petrophysical parameters, the inverted pore-connectivity parameter can be a good auxiliary attribute that assists locating potential targets for tight gas sandstone reservoirs. [ABSTRACT FROM AUTHOR]

The finite-difference frequency domain (FDFD) method is widely applied for simulating seismic wavefields, and a key to achieving successful FDFD simulation is to construct FDFD coefficients that can effectively suppress numerical dispersion. Among the existing FDFD coefficients for seismic wavefield simulation, adaptive FDFD coefficients that vary with the number of wavelengths per grid can suppress numerical dispersion to the maximum extent. The current methods for calculating adaptive FDFD coefficients involve numerical integration, conjugate gradient (CG) optimization, sequential initial value selection, and smooth regularization, which are difficult to implement and inefficient in calculations. To simplify the calculation of adaptive FDFD coefficients and improve the corresponding computational efficiency, this paper proposes a new method for calculating adaptive FDFD coefficients. First, plane-wave solutions with different discrete propagation angles are substituted in the FDFD scheme, and the corresponding least-squares problem is constructed. As this problem is ill-conditioned and obtaining smooth adaptive FDFD coefficients by the conventional solving method based on normal equations is difficult, this paper proposes solving the least-squares problem by solving the corresponding overdetermined linear system of equations through QR matrix decomposition. Compared with the existing methods for calculating adaptive FDFD coefficients based on numerical integration, CG optimization, and sequential initial value selection, the proposed method allows for a simplified computational process and considerably higher computational efficiency. Numerical wavefield simulation results show that the adaptive-coefficient FDFD method based on QR matrix decomposition can achieve the same accuracy as those based on numerical integration, CG optimization, and sequential initial value selection while requiring less computation time. [ABSTRACT FROM AUTHOR]

Fracture orientation and in particular the fracture strike play an important role in guiding the accurate estimation of the orientation of in situ stress fields. Fracture strike can be extracted based on the azimuthal seismic reflection responses. Thus, the characteristics of seismic reflection responses varying with fracture strike are critical for fracture strike estimation. In this work, a set of vertical fractures with different strikes is introduced into an isotropic host matrix based on the Schoenberg linear-slip theory, and the azimuthal seismic reflection response is modeled based on the reflection/transmission analysis at a plane interface. The effect of fracture parameters, i.e., fracture density, fracture aspect ratio, and fluid infill on the seismic reflection response, is analyzed. The influences of fracture strike on the stiffness matrix are simulated, and we further discuss the impacts on the P- and S-wave velocities. The influences of fracture strike on the reflection coefficient are depicted using the rose diagram method. The proposed workflow is applied to real seismic reflection data. The method is used to estimate the fracture strike with the seismic data in the Sichuan Basin work area by assuming that all fractures are vertically oriented, and the rose diagram is obtained close to the target layer. The results show that the fracture strike predicted by the method is in good agreement with the estimation based on the well-bore imaging data. [ABSTRACT FROM AUTHOR]

The production of shale gas plays an important role in the global oil/gas industries, which is considered to approach about 1/3 of the total natural gas production at 2040. In recent years, Wufeng–Longmaxi Formation shale in Sichuan Basin has become a key target of gas development in China. The detection of high-quality gas-producing areas based on the relation between pore structure, mineralogy and elastic properties is the main research direction. However, the effects of depositional environment and diagenetic processes are rarely considered in the studies of these petrophysical properties. We select core samples from the two boreholes to perform X-ray diffraction experiments together with another set of cores from the target formation to obtain scanning electron microscope images and ultrasonic velocities. The mineralogy, pore structure and elastic properties are analyzed. The formation is divided into two sections based on the reservoir characteristics and sedimentary environments. Then, we study the different diagenetic processes of the upper and lower sections. Three-dimensional rock-physics templates of the shale in the two sections are built by using equivalent medium theories and elastic attributes. The templates are calibrated with the samples and well-log data. A 2D survey line and a 3D seismic block of the upper and lower sections are selected, respectively, to map the mineral component, porosity and microcrack density. The good match between the predictions and log interpretation and actual gas content reports suggests that the rock-physics template based on differential diagenesis can be effectively used for further interpretation of seismic inversion results. [ABSTRACT FROM AUTHOR]

In recent years, the Yanchang shale-oil formations of the Ordos Basin are rich in reserves with complex lithology and structure characteristics, low porosity and low permeability, and weak anomalies for oil and water discriminations, have been the key targets of unconventional oil/gas resource exploration and development in the relevant areas. The joint acoustic-electrical (AE) properties can be used to interpret reservoir lithology, mineralogy, pore structure, and fluid saturation. To conduct tests of thin section analysis, X-ray diffraction, and ultrasonic and electrical experiments at different pressures and saturation degrees, cores from the shale-oil formations in the Q area of the basin are collected. The variations in AE properties with respect to clay content, porosity, pressure (microfracture), and saturation are analyzed. The experimental results indicate that the rock physics behaviors of sandstones with different clay contents vary significantly. The AE properties of clean sandstones are basically dependent on the microfractures (pressure), while for muddy sandstones, the clay content is an important factor affecting the responses. The target reservoir consists of interbedded sandstone and shale layers. The AE equivalent medium equations and the Gurevich theory are applied to establish the joint models for the different lithologies and simulate the variations in AE properties with respect to fluid type, pore structure, and mineral components. The three-dimensional joint templates of clean and muddy sandstones, as well as shale, are developed based on the elastic and electrical attributes and then calibrated using the experimental and well-log data. The reservoir properties are estimated with the templates and validated by the log data. The results indicate that the joint templates based on lithology characteristics can effectively characterize the properties of interbedded sandstone and shale layers. Furthermore, the combined application of AE data provides more beneficial information for the assessment of rock properties, leading to precise estimates that conform with the actual formation conditions. [ABSTRACT FROM AUTHOR]

The elastic stress–strain relation of deep source rocks must account for the presence of fractures and lamination due to thermally mature kerogen layers. Clay lamination describes the variation in transverse isotropy with a vertical symmetry (VTI) characteristic of shale background, and vertical fractures induce the transverse isotropy with a horizontal symmetry (HTI) perturbations. Thus, the fractured shale rock can be described as a medium with an orthorhombic symmetry in terms of effective anisotropy. We present a new workflow for calculating the elastic constants of a pressure-dependent fractured shale rock and the effects of clay lamination and microfractures. We investigate the shale fabric that can result in the orthorhombic elastic symmetry and the effect of pressure on elastic parameters, and propose an inversion methodology. Results based on well-log data show that the proposed methodology can be effectively applied to field data. [ABSTRACT FROM AUTHOR]

The acoustic-electrical (AE) response of subsurface hydrocarbon reservoirs is highly affected by rock heterogeneity. In particular, the characterization of the microstructure of tight (low-permeability) rocks can be aided by a joint interpretation of AE data. To this purpose, we evaluate cores from a tight-oil reservoir to obtain the rock mineralogy and pore structure by X-ray diffraction and casting thin sections. Then, ultrasonic and resistivity experiments are performed under different confining pressures to analyze the effects of pores, microcracks and mineralogy on the AE properties. We have developed acoustic and electrical models based on effective-medium theories, and the Cole–Cole and triple-porosity equations, to simulate the response to total and soft (crack) porosities and clay content. The results show that these properties play a significant role. Then, a 3D rock-physical template is built and calibrated by using the core samples and well-log data. The template is applied to tight-oil reservoirs to estimate the rock properties, which are validated with log data. The good match between the predictions and these data indicates that the model can effectively explain the effects of the heterogeneous microstructure on the AE data. Article Highlights: Tight rock microstructure is analyzed with X-ray diffraction, thin sections and ultrasonic and electrical resistivity tests Rock acoustic-electrical properties are obtained by the effective-medium and triple-porosity theories Practical application is given based on an acoustic-electrical rock physics template [ABSTRACT FROM AUTHOR]

Petrophysical seismic inversion, aided by rock physics, aims at estimating reservoir properties based on reflection events, but it is generally based on the Gassmann equation, which precludes its applicability to complex reservoirs. To overcome this problem, we present a methodology based on the double-porosity Biot–Rayleigh (BR) model, which takes into account the rock heterogeneities. The volume ratio of inclusions in the BR model is treated as a spatially varying parameter, facilitating a better description of the pore microstructure. The method includes the Zoeppritz equations to extract reservoir properties from prestack data. To handle the ill-posedness of the inversion and achieve a stable solution, the algorithm is formulated as a multi-objective optimization based on the Bayes theorem, where the reservoir-property estimation is jointly conditioned to seismic and elastic data with multiple prior terms. The method is validated with field data of a tight gas sandstone reservoir, illustrating its effectiveness compared to the Gassmann-based estimation, reducing uncertainties and improving the accuracy of identifying gas zones. Article Highlights: The petrophysical seismic inversion is based on the double-porosity Biot–Rayleigh model Spatially varying inclusion volumes are used to describe complex pore structures A multi-objective optimization with joint data misfit enables stable results [ABSTRACT FROM AUTHOR]

The complex seismic responses of heterogeneous reservoirs can be related to the fabric structure, pore/microcrack shape, mineral composition and fluid distribution of the rock in situ. The pore structure refers to the geometric shape, size, spatial distribution and interconnectedness of pores, microcracks and throats. It is closely related to the storage space of reservoirs and the spatial distribution of oil/gas. Understanding the pore structure is crucial for the development of processes to increase oil/gas production capacity. Six dolomite samples from the Gaoshiti-Moxi Longwangmiao Formation are sorted out for measurements, and the ultrasonic and seismic attenuation are determined by using the spectral ratio method and the enhanced frequency shift method, respectively. When predicting the pore structure, we assume that the aspect ratio and volume fraction of pores and microcracks correspond to a normal distribution. On this basis, a model with the Voigt–Reuss–Hill average (VRH), differential effective medium (DEM) theory and infinituple-porosity media (IPM) theory is proposed. The acoustic wave responses in terms of reservoir porosity and standard deviation of normal distribution are analyzed, and multiscale 3D rock physics templates (RPT) are created. The calibrations of the templates are performed with the ultrasonic and seismic data, and then the templates are applied to the field data. The results show that the estimated porosity and pore structure (corresponding to the mean aspect ratio and standard deviation of a normal distribution, respectively) are in substantial agreement with the log data and the actual gas production results. [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]

Ultra-deep carbonate reservoirs have low porosity, a complex pore space with microcracks of varying aspect ratio and dissolved pores, that affect the seismic and transport properties. We propose a rock-physics model based on penny-shaped inclusions in the framework of the double-porosity theory to estimate rock features, such as the crack porosity and aspect ratio, and stiff porosity. Based on this model, a 3-D rock-physics template is built, calibrated at the ultrasonic and seismic frequency bands, from attenuation, P-wave impedance and VP/VS ratio to quantitatively evaluate the effect of those features. Attenuation is estimated by using the spectral-ratio and improved frequency-shift methods. The template is applied to ultra-deep carbonates of the S work area of the Tarim Basin (China). The predictions agree with the well-log data and field production reports. In general, the higher the crack aspect ratio, the higher the storage and transport capacity of the reservoir. Therefore, these crack features can be used as indicators of these reservoir properties. [ABSTRACT FROM AUTHOR]

Pre-stack seismic inversion of the P- and S-wave velocities and bulk density is important in seismic exploration for evaluating lithological units and fluid properties. Generally, this inversion is based on ray-tracing modeling, which introduces errors and requires substantial pre-processing for stratified models due to its oversimplified single-interface assumption. To overcome those problems, we propose a pre-stack inversion method, using wave-equation-based modeling as a forward engine. Most wave-equation-based pre-stack inversions are based on the reflectivity method and adopt nonlinear optimization algorithms, although accurate, but computationally expensive. Hence, we use a fast propagator matrix (PM) method valid for layered media. To improve the stability and accuracy, the PP data inversion is extended to joint PP and PS PM-based inversion (JPMI). A linear inversion scheme is adopted to reduce the computational cost, and the Fréchet derivatives are computed accordingly. Moreover, to obtain an optimal model solution, the L-BFGS (Limited-memory Broyden–Fletcher–Goldfarb–Shanno) optimization algorithm and L-curve criterion, an adaptive regularization parameter acquisition method, are implemented. A posterior probability analysis shows that the method has a higher parameter sensitivity than the joint exact Zoeppritz-based inversion and gives better estimations than the single-data inversion. We discuss the effects of dataset weight, internal multi-reflections, time window setting, noise level and initial model by using model tests. Synthetic and real-data examples demonstrate that the algorithm is better than the single PP inversion in terms of stability and accuracy, especially for S-wave velocity and density estimations. [ABSTRACT FROM AUTHOR]

Finite-difference (FD) methods are popularly utilized to achieve precise and efficient simulations of acoustic wavefields in large-scale 3D seismic inversion and imaging. General cuboid grid (i.e., different grid spacings in different spatial directions)-based space-domain FD is only second-order accurate. The standard time–space-domain FD is based on cubic grids (i.e., the same spacing for all directions), and only achieves high-order accuracy along 48 propagation directions. We developed two novel cuboid grid-based high-order time–space FD for 3D acoustic modeling. Two new stencils containing grid points both on and off the axis are developed to achieve fourth- and sixth-order discretization, respectively, for solving second-order temporal derivatives on general cuboid grids. The joint time–space dispersion relations for wave extrapolation using the proposed temporal FD and spatial central FD are derived by plane-wave analysis, and the analytical expressions for the coefficients used are derived by Taylor expansion. The two schemes yield arbitrary even-order accuracy in space and fourth- or sixth-order accuracy in time. Dispersion analysis, stability analysis, and numerical examples on cuboid grids confirm that the two schemes have better accuracy and stability than both the space-domain FD and the Lax–Wendroff FD. Because they allow for larger time steps, the two schemes are more efficient. They show superiority over the standard time–space-domain FD as well when cubic grids are used. Our new methods thus provide more general and effective tools for studying 3D acoustic wave propagation. [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]

Rock brittleness directly affects reservoir fracturing and its evaluation is essential for establishing fracturing conditions prior to reservoir reforming. Dynamic and static brittleness data were collected from siltstones of the Qingshankou Formation in Songliao Basin. The brittle-plastic transition was investigated based on the stress-strain relation. The results suggest that the brittleness indices calculated by static elastic parameters are negatively correlated with the stress drop coefficient and the brittleness index B2, defined as the average of the normalized Young’s modulus and Poisson’s ratio, is strongly correlated with the stress drop. The brittleness index B2, Young’s modulus, and Poisson’s ratio correlate with the brittle minerals content; that is, quartz, carbonates, and pyrite. We also investigated the correlation between pore fluid and porosity and dynamic brittle characteristic based on index B2. Pore fluid increases the plasticity of rock and reduces brittleness; moreover, with increasing porosity, rock brittleness decreases. The gas-saturated siltstone brittleness index is higher than that in oil- or water-saturated siltstone; the difference in the brittleness indices of oil- and water-saturated siltstone is very small. By comparing the rock mechanics and ultrasonic experiments, we find that the brittleness index obtained from the rock mechanics experiments is smaller than that obtained from the ultrasonic experiments; nevertheless, both decrease with increasing porosity as well as their differences. Ultrasonic waves propagate through the rock specimens without affecting them, whereas rock mechanics experiments are destructive and induce microcracking and porosity increases; consequently, the brittleness of low-porosity rocks is affected by the formation of internal microcrack systems. [ABSTRACT FROM AUTHOR]

Seismic wavefield modeling is important for improving seismic data processing and interpretation. Calculations of wavefield propagation are sometimes not stable when forward modeling of seismic wave uses large time steps for long times. Based on the Hamiltonian expression of the acoustic wave equation, we propose a structure-preserving method for seismic wavefield modeling by applying the symplectic finite-difference method on time grids and the Fourier finite-difference method on space grids to solve the acoustic wave equation. The proposed method is called the symplectic Fourier finite-difference (symplectic FFD) method, and offers high computational accuracy and improves the computational stability. Using acoustic approximation, we extend the method to anisotropic media. We discuss the calculations in the symplectic FFD method for seismic wavefield modeling of isotropic and anisotropic media, and use the BP salt model and BP TTI model to test the proposed method. The numerical examples suggest that the proposed method can be used in seismic modeling of strongly variable velocities, offering high computational accuracy and low numerical dispersion. The symplectic FFD method overcomes the residual qSV wave of seismic modeling in anisotropic media and maintains the stability of the wavefield propagation for large time steps. [ABSTRACT FROM AUTHOR]

In heterogeneous natural gas reservoirs, gas is generally present as small patchlike pockets embedded in the water-saturated host matrix. This type of heterogeneity, also called 'patchy saturation', causes significant seismic velocity dispersion and attenuation. To establish the relation between seismic response and type of fluids, we designed a rock physics model for carbonates. First, we performed CT scanning and analysis of the fluid distribution in the partially saturated rocks. Then, we predicted the quantitative relation between the wave response at different frequency ranges and the basic lithological properties and pore fluids. A rock physics template was constructed based on thin section analysis of pore structures and seismic inversion. This approach was applied to the limestone gas reservoirs of the right bank block of the Amu Darya River. Based on poststack wave impedance and prestack elastic parameter inversions, the seismic data were used to estimate rock porosity and gas saturation. The model results were in good agreement with the production regime of the wells. [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]

The rheological models of Lomnitz and Jeffreys have been widely used in earthquake seismology (to simulate a nearly constant Q medium) and to describe the creep and relaxation behavior of rocks as a function of time. Other similar models, such as those of Becker, Scott Blair and Kolsky, show similar properties, particularly the Scott Blair model describes a perfectly constant Q as a function of frequency. We first give a historical overview of the main scientists and the development and versions of the various models and priorities of discovery. Then, we clarify the relationship between the different versions of these models in terms of mathematical expressions of the complex modulus and calculate the phase velocity and quality factor Q as a function of frequency, illustrating the various special cases. In addition, we give useful hints for the numerical calculation of these moduli, which include special cases of the hypergeometric function. [ABSTRACT FROM AUTHOR]

Petrophysics and fluid-flow simulations are used to build a realistic pre- and post- CO 2 injection geological model for the Utsira formation at the Sleipner field, and the Fourier pseudospectral method is employed to compute synthetic seismograms. The methodology can be used to perform a seismic sensitivity analysis for the detection of carbon dioxide. We built the model solely based on the porosity and clay content of the formations with the aid of fluid-flow and seismic simulations. The pressure map before the injection is assumed to be hydrostatic for which a reference porosity map is defined. The injection induces pore pressure variations and partial saturation, which affect the poroelastic properties and hence the associated seismic response. A proper porosity–permeability–clay content relation is one of the key factors since permeability determines the preferential flow directions and the distribution of the CO 2 plume. The petrophysical model is based on a shaly sandstone (or sandy shale) to represent the caprock, Utsira Sand and embedded mudstone layers. The composite permeability (anisotropic) is analogous to the inverse electrical resistance model. Gas viscosity depends on pressure and temperature. The P- and S-wave velocities are obtained from Gassmann equation (pre-injection) and White's mesoscopic model (post-injection), which also yields the P-wave quality factor in the case of partial (patchy) saturation. To model a realistic situation, we implement a fractal variation of the porosity and clay content, based on the von Kármán correlation function. We then compare the real and synthetic seismograms (pre-injection and post-injection) and show the effect of attenuation on the seismic data. Simulations and real data show a remarkable match. [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 obtain the reflection and transmission coefficients for inhomogeneous plane waves incident on a flat interface separating two double-porosity media described by the Biot–Rayleigh model, which takes into account the effect of local fluid flow (LFF). Three longitudinal and one transverse waves are reflected and transmitted, represented by potential functions specified by the propagation and attenuation directions. The continuity of the energy at the interface for sealed and open-boundary conditions yields a system of equations for the coefficients, and the expressions of the energy ratios for the reflected and refracted waves are derived in closed form. Numerical examples showing the magnitude, phase and energy ratio as a function of frequency and incidence angle are carried out to investigate the influence of the inhomogeneity angle, boundary condition, type of incidence wave and LFF effect. The results confirm that the LFF affects the reflection and transmission behaviors for the incident P1 and SV waves, irrespective of whether the interface is open or sealed. The effect causes interference fluxes between different waves, a consequence of energy conservation at the interface. We also perform full-waveform simulations to validate the results. [ABSTRACT FROM AUTHOR]

We provide a new derivation of the Kramers-Kronig relations on the basis of the Sokhotski-Plemelj equation with detailed mathematical justifications. The relations hold for a causal function, whose Fourier transform is regular (holomorphic) and square-integrable. This implies analyticity in the lower complex plane and a Fourier transform that vanishes at the high-frequency limit. In viscoelasticity, we show that the complex and frequency-dependent modulus describing the stiffness does not satisfy the relation but the modulus minus its high-frequency value does it. This is due to the fact that despite its causality, the modulus is not square-integrable due to a non-null instantaneous response. The relations are obtained in addition for the wave velocity and attenuation factor. The Zener, Maxwell, and Kelvin-Voigt viscoelastic models illustrate these properties. We verify the Kramers-Kronig relations on experimental data of sound attenuation in seabottoms sediments. [ABSTRACT FROM AUTHOR]

In Part I, a dynamic fracture compliance model (DFCM) was derived based on the poroelastic theory. The normal compliance of fractures is frequency-dependent and closely associated with the connectivity of porous media. In this paper, we first compare the DFCM with previous fractured media theories in the literature in a full frequency range. Furthermore, experimental tests are performed on synthetic rock specimens, and the DFCM is compared with the experimental data in the ultrasonic frequency band. Synthetic rock specimens saturated with water have more realistic mineral compositions and pore structures relative to previous works in comparison with natural reservoir rocks. The fracture/pore geometrical and physical parameters can be controlled to replicate approximately those of natural rocks. P- and S-wave anisotropy characteristics with different fracture and pore properties are calculated and numerical results are compared with experimental data. Although the measurement frequency is relatively high, the results of DFCM are appropriate for explaining the experimental data. The characteristic frequency of fluid pressure equilibration calculated based on the specimen parameters is not substantially less than the measurement frequency. In the dynamic fracture model, the wave-induced fluid flow behavior is an important factor for the fracture-wave interaction process, which differs from the models at the high-frequency limits, for instance, Hudson’s un-relaxed model. [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]

It becomes increasingly clear that non-uniform distribution of immiscible fluids in porous rock is particularly relevant to seismic wave dispersion. White proposed a patchy saturation model in 1975, in which spherical gas pockets were located at the center of a liquid saturated cube. For an extremely light and compressible inner gas, the physical properties can be approximated by a vacuum with White's model. The model successfully analyzes the dispersion phenomena of a P-wave velocity in gas-watersaturated rocks. In the case of liquid pocket saturation, e.g., an oil-pocket surrounded by a water saturated host matrix, the light fluid-pocket assumption is doubtful, and few works have been reported in White's framework. In this work, Poisson's ratio, the bulk modulus, and the effective density of a dual-liquid saturated medium are formulated for the heterogeneous porous rocks containing liquid-pockets. The analysis of the difference between the newly derived bulk modulus and that of White's model shows that the effects of liquid-pocket saturation do not disappear unless the porosity approaches zero. The inner pocket fluid can no longer be ignored. The improvements of the P-wave velocity predictions are illustrated with two examples taken from experiments, i.e., the P-wave velocity in the sandstone saturated by oil and brine and the P-wave velocity for heavy oils and stones at different temperatures. [ABSTRACT FROM AUTHOR]

We simulate the effects of diagenesis, cementation and compaction on the elastic properties of shales and sandstones with four different petro-elastic theories and a basin-evolution model, based on constant heating and sedimentation rates. We consider shales composed of clay minerals, mainly smectite and illite, depending on the burial depth, and the pore space is assumed to be saturated with water at hydrostatic conditions. Diagenesis in shale (smectite/illite transformation here) as a function of depth is described by a fifth-order kinetic equation, based on an Arrhenius reaction rate. On the other hand, quartz cementation in sandstones is based on a model that estimates the volume of precipitated quartz cement and the resulting porosity loss from the temperature history, using an equation relating the precipitation rate to temperature. Effective pressure effects (additional compaction) are accounted for using the Athy equation and the Hertz–Mindlin model. The petro-elastic models yield similar seismic velocities, despite the different levels of complexity and physics approaches, with increasing density and seismic velocities as a function of depth. The methodology provides a simple procedure to obtain the velocity of shales and sandstones versus temperature and pressure due to the diagenesis-cementation-compaction process. [ABSTRACT FROM AUTHOR]

We present a novel numerical upscaling technique for modeling the wave response of gas-hydrate bearing sediments composed of a rock frame, gas-hydrate and water, where the hydrate consists of ice-like lattice of water molecules with methane trapped inside. These sediments are highly heterogeneous at mesoscopic scales, much smaller than the wavelength but much larger than the pore size, inducing substantial seismic wave attenuation and dispersion due to mode conversions. The proposed numerical upscaling procedure simulates the wave-induced fluid-flow loss mechanism by computing an average effective viscoelastic medium having the same behavior of the original sediment. The method determines the complex stiffness coefficients associated with the viscoelastic medium by solving numerically boundary value problems formulated in the space-frequency domain, representing compressibility and shear experiments. The procedure is applied to composite media with regions of different amounts of hydrate with patchy or periodic-layer distributions, which define an anisotropic effective viscoelastic medium, respectively. The examples demonstrate that variations in hydrate content induce strong attenuation and dispersion effects on seismic waves due to the mesoscopic loss mechanism. [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]

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]

We first compare two apparently dissimilar expressions for the quality factor Q of inhomogeneous plane waves in isotropic viscoelastic media, where Q is defined as the result of an energy-balance relation. In this case, it is the ratio of twice the strain energy to the dissipated energy. Both expressions give the same Q for P-waves and Q for S-waves is independent of the inhomogeneity angle γ (isotropic media). Then, we consider the more general balance equation, which holds for anisotropic viscoelastic media, where anomalous behaviors are observed when γ exceeds some critical value (forbidden directions of propagation). This problem has already been solved analytically for SH-waves. Here, we consider the qP–qS case, which requires a numerical solution of the dispersion equation to obtain the wavenumber and attenuation factor, i.e., the real and imaginary parts of the wave vector, respectively. The forbidden directions appear when the phase velocity approaches a zero value. Generally, the phase velocity of homogeneous waves (γ = 0) exceeds that of inhomogeneous waves, while the latter show stronger attenuation (lower quality factor). In the vicinity of the forbidden directions, the opposite behavior may occur. [ABSTRACT FROM AUTHOR]

The acoustic properties of rocks depend on porosity, pressure, and pore fluid and also on pore geometry. Anelasticity (attenuation and velocity dispersion) is more affected by crack aspect ratio and fraction (soft pores) than by equant (stiff) pores. To study this fact, we have performed ultrasonic measurements on two dolomite samples under variable pressure and fluid content, and used the EIAS (equivalent inclusion-average stress) model to obtain the crack aspect ratio and fraction from the bulk and shear moduli of the rock. The theory has an excellent agreement with the experimental data, and the results show that the crack attributes decrease with increasing differential pressure and are higher for a stiffer fluid. In fact, the interpretation of the experiments with the model shows that crack fraction and aspect ratio increase with the bulk modulus of the fluid (water and oil). Then, by extending the theory to all frequencies, using the Zener mechanical model, we obtain the phase velocities and quality factors as a function of frequency. Our findings reveal the importance of considering differential pressure and fluid type to analyze pore geometry and rock anelasticity. [ABSTRACT FROM AUTHOR]

The anelastic properties of porous rocks depend on the pore characteristics, specifically, the pore aspect ratio and the pore fraction (related to the soft porosity). At high frequencies, there is no fluid pressure communication throughout the pore space and the rock becomes stiffer than at low frequencies, where the pore pressure is fully equilibrated. The models considered here include explicit pore geometry information in determining the poroelastic parameters. They are extensions of the EIAS (equivalent inclusion-average stress) and CPEM (cracks and pores effective medium) models to the whole frequency range, based on the Zener model. Knowing the degree of stiffness dispersion between the low- and high-frequency limits, we fit experimental data in the whole frequency range and obtain the average crack aspect ratio and soft porosity as a function of effective pressure. Then, we compute the dispersion and quality factor of the bulk, shear and Young moduli, and the P- and S-wave seismic velocities and quality factors as a function of frequency. However, when measuring axial or volumetric motions along a cylindrical sample, there is fluid flow at the ends of the sample in the experiments considered here. This generates dispersion and attenuation due to axial flow of the pore fluid, which does not occur for a plane wave in unbounded media. This phenomenon is called "drained/undrained transition" Pimienta et al. (J Geophys Res Solid Earth; 10.1002/2017JB014645, 2017). Actually, it is an axial version of the Biot–Gardner (BG) effect, and implies an "artificial" (mesoscopic) attenuation peak (and dispersion) due to the generation of slow (diffusion) Biot modes at the cylinder boundary, inducing a global flow at the scale of the sample. The classical BG effect is due to fluid flow along the radial direction, on the basis of open-pore conditions at the sides of the sample. In this case, the sides are sealed. To use the EIAS and CPEM models, the BG effect has to be removed to obtain the intrinsic Q of the rock. The models are applied here for measurements on sandstone. The axial BG effect is more evident if the intrinsic attenuation is weak or absent. An example is Lavoux limestone, which has a bimodal porosity distribution, with an equal proportion of intragranular microporosity and intergranular macroporosity (round pores). In this case, the attenuation and dispersion are related to the BG effect, since no squirt flow is detected due to the absence of cracks. We verified that the bulk and Young moduli obtained from the axial and hydrostatic oscillations are consistent with each other, and that the theoretical description of the axial BG effect shows some discrepancies with the data. [ABSTRACT FROM AUTHOR]

An elastic two-phase composite, with no restriction on the shape of the two phases, has stiffness bounds given by the Reuss and Voigt equations, and a narrower range determined by the Hashin-Shtrikman bounds. Averages are given by the Voigt-Reuss-Hill, Hashin-Shtrikman, Gassmann, Backus and Wyllie equations. To obtain stiffness bounds and averages, we invoke the correspondence principle to compute the solution of the viscoelastic problem from the corresponding elastic solution. Then, seismic velocities and attenuation are established for the above — physical and heuristic — models which account for general geometrical shapes, unlike the Backus average. The approach is relevant to the seismic characterization of solid composites such as hydrocarbon source rocks. [ABSTRACT FROM AUTHOR]

The absolute permeability of rock depends on several factors, including porosity, ϕ , the geometry of the pore network (tortuosity), and the grain geometry, dimension and composition. The mineralogical composition plays an important role, mostly with respect to clay, which involves several components including illite, smectite, kaolinite and chlorite. The presence of quartz and feldspar increases permeability, while clay minerals and calcite tend to have the opposite effect. Essentially, permeability decreases with a smaller grain radius, increasing tortuosity of the pore space and decreasing porosity. As the specific surface area of the pores increases, permeability decreases. Here, we compare four expressions for permeability based on clay content, grain dimension, tortuosity and mineral composition. All the expressions somehow contain the Kozeny–Carman (KC) factor ϕ 3 / (1 - ϕ) 2 , which is obtained on physical grounds, and relies on fitting parameters related to the geometric characteristics of the rock and its composition. The Herron model is based on geochemical mineralogy composition. Despite the highly idealized parameters on which these models are based, the results support the predictive power of the Kozeny–Carman equation, provided that proper calibration is performed. [ABSTRACT FROM AUTHOR]

Tight gas sandstone reservoirs in Guang'an are characterized by wide distribution and low abundance. Sandstone samples from this area usually have low porosity and poor connectivity. We analyze the observed velocity data of tight sandstone samples with the Mori-Tanaka model, and give the sandstone framework physical model in this area based on theory and experiment analysis. The matrix modulus was obtained by an empirical relationship and then the experiment data were compared with the values predicted by the Mori-Tanaka model with different pore shapes. The results revealed that the experiment data were close to the model with low pore aspect ratio. Considering the matrix modulus and pore shape variation, we find that, under the condition of small mineral composition change, the effective pore aspect ratio of these samples increased with porosity evidently. [ABSTRACT FROM AUTHOR]