Wave-induced Fluid Flow Research Articles

Dynamics anisotropy in a porous solid with aligned slit fractures

Crustal rocks are commonly permeated by aligned fractures which may control the wave anisotropy and permeability pattern. In the presence of pore fluid the mechanical and hydraulic features of such rocks become more complex. Understanding the dynamic anisotropy in fluid-saturated fractured rocks is important for detecting and characterizing fractured reservoirs and fault zones with applications in geomechanics, hydrogeology, exploration geophysics and reservoir engineering. For waves propagating normal to the fractures, the effects of wave-induced fluid flow (WIFF) due to the presence of permeable fractures on seismic dispersion and attenuation are significant and have been quantified in earlier studies. But previous literatures are restricted to low frequency range within which the fracture size is much smaller than the incident wavelength. In this paper, we extend low-frequency normal incidence results to full-frequency oblique incidence. We first derive exact solutions of the scattering problem of obliquely incident plane waves by a single slit fracture in a poroelastic solid. Based on previous analysis, for ideal fractures with infinitesimal thickness, the fracture fluid can be modelled as an incompressible one. Then, based on the solutions and Foldy's scattering theorem we develop a dynamic-effective-medium model to estimate frequency-dependent anisotropy of wave propagation in a fluid-saturated poroelastic rock with a sparse set of aligned fractures. We find that for the oblique incidence problem apart from WIFF there exist another two important attenuation mechanisms, i.e., the elastic scattering (scattering into fast P and S waves via mode conversion at the fracture faces) and Biot's global flow, in causing velocity dispersion and attenuation. The mixed-boundary problem reveals that the WIFF is controlled by the normal displacement discontinuity that is determined by effective normal stress applied on the fracture faces, while the scattering effects by the tangential displacement discontinuity that is determined by effective shear stress. Because the effective normal and shear stresses depends on incident angles and frequency, the dispersion and attenuation of both P and S waves are anisotropic and frequency-dependent. In contrast, Biot's global flow is an intrinsic energy loss mechanism that can play a role in causing velocity dispersion and attenuation at higher frequency range but it is independent of the presence of fractures or incident angle.

Local and global fluid-flow effects on dipole flexural waves

Wave-induced fluid flow (WIFF) can significantly alter the effective formation velocities and cause increasing waveform dispersion and attenuation. We have used modified frame moduli from the theory of Chapman together with the classic Biot theory to improve the understanding of local- and global-flow effects on dipole flexural wave modes in boreholes. We investigate slow and fast formations with and without compliant pores, which induce local flow. The discrete wavenumber summation method generates the waveforms, which are then processed with the weighted spectral semblance method to compare with the solution of the period equation. We find compliant pores to decrease the resulting effective formation P- and S-wave velocities, that in turn decrease the low-frequency velocity limit of the flexural wave. Furthermore, depending on the frequency at which the local-flow dispersion occurs, different S-wave velocity predictions from the flexural wave become possible. This issue is investigated through changing the local-flow critical frequency. Sensitivity analyses of the flexural-wave phase velocity to small changes in WIFF parameters indicate the modeling to be mostly sensitive to compliant pores in slow and fast formations.

Semi-analytical solution to the problem of frequency dependent anisotropy of porous media with an aligned set of slit cracks

Fractures control fluid flow in geological formations because their presence can greatly affect effective permeability. Existence of cracks in a porous rock can induce seismic dispersion attenuation and frequency-dependent anisotropy, due to wave-induced fluid flow (WIFF) between the fractures and pores. Thus, dispersion and attenuation can be valuable seismic signatures for fracture characterization. In previous studies these effects were modeled by considering cracks as low aspect-ratio inclusions embedded in a porous background medium modeled by Biot's poroelasticity equations. However, previous works are limited to normal wave incidence. In this research, we study the P-wave dispersion and attenuation for oblique incidence in a saturated porous medium with aligned slit (two-dimensional) fractures, through solving a mixed boundary value problem for Biot's dynamic poroelasticity equations. The semi-analytical solution of this problem gives dispersion and attenuation as functions of the incidence angle. The strongest dispersion and attenuation occur when the P-wave propagates along the fracture normal and decrease with increasing incident angle (as measured from the fracture normal). For the frequency-dependent anisotropy at low frequencies, the sample behaves as nearly elliptical, and at high frequencies the medium exhibits a strong anellipticity. The attenuation anisotropy parameters reach their maximum when the fluid diffusion length is comparable to the crack radius. Comparison of the heuristic solution, which is s based on an interpolation between low- and high frequencies using a relaxation function, to the semi-analytical solution of the rigorous equations shows excellent agreement, demonstrating the validity of the heuristic approach.

Effective properties of a porous medium with aligned cracks containing compressible fluid

SUMMARYUnderstanding the wave propagation in fluid-saturated cracked rocks is important for detecting and characterizing cracked reservoirs and fault zones with applications in geomechanics, hydrogeology, exploration geophysics and reservoir engineering. In sedimentary rocks, microscopic-scale pores are usually filled with fluid. One logical means of modelling the essential features of such rocks is to use poroelasticity theory. But previous models of wave propagation in cracked porous medium are either restricted to low frequencies at which effects of the elastic scattering (scattering into fast-P and S waves via mode conversion at the crack faces) are negligible or to the case that the crack-filling fluid is assumed to be incompressible. To overcome these restrictions, we consider the effects of crack fluid compressibility by extending spring condition into poroelasticity and derive exact solutions of the scattering problem of an incident P wave by a circular crack containing compressible fluid in a porous medium. Based on the solutions, we develop two different effective medium models to estimate frequency-dependent effective velocity and attenuation in a fluid-saturated porous rock with a set of aligned cracks. The mixed-boundary value problem reveals that both the wave-induced fluid flow (WIFF) and elastic wave scattering can cause important velocity dispersion and attenuation. The diffusion-type WIFF dominates the velocity change and attenuation for the low frequency range, while the elastic scattering dominates them for the relatively higher frequency range. The dependences of the P-wave velocity on the crack fluid compressibility are different at different frequencies. For the WIFF-dominated frequency range and Rayleigh-scattering frequency range, the P-wave velocity decreases with the crack fluid compressibility. In contrast, for the Mie scattering frequency range, the opposite occurs (the P-wave velocity increases with the crack fluid compressibility).

Wave propagation characteristics at the welded interface of double-porosity solid and double-porosity dual-permeability materials

ABSTRACT In the present study, propagation characteristics (reflection/refraction phenomena) at the welded interface of two dissimilar double-porosity media are investigated. Both the double-porosity materials behave dissipative to wave propagation due to the presence of viscosity in pore fluid. All the waves (i.e. incident, reflected and refracted) are considered as inhomogeneous (i.e. different directions of propagation and attenuation). Reflection and refraction coefficients are calculated theoretically in the form of eight inhomogeneous linear equations. Reflected and refracted waves energy shares are represented with two different energy matrix. For the incidence of and SV waves, the partition of incident energy among various reflected and refracted waves is calculated numerically for particular numerical examples. The impacts of various physical properties (like, pore-fluid viscosity, double-porosity structure, wave-induced fluid flow and permeability) of double-porosity dual-permeability materials on the partition of incident energy are depicted graphically at each angle of incidence. Numerically, the conservation of incident energy during the reflection/refraction process is obtained at each angle of incidence.

Seismic attenuation and dispersion in a cracked porous medium: An effective medium model based on poroelastic linear slip conditions

Mechanical and hydraulic properties of a crack can significantly affect seismic wave propagation. To explore these effects, we developed an effective medium model that describes the P-wave dispersion and attenuation in inhomogeneous porous media containing a distribution of aligned cracks. Although there are numerous theoretical models for quantifying the seismic dispersion and attenuation, some of them are restricted to low frequencies at which only effects of the wave-induced fluid flow (WIFF) are considered but the influences of elastic scattering are ignored. Others only consider crack mechanical properties without incorporating crack permeability. The others describe the crack by a thin layer with infinitely lateral extension, neglecting the crack length information. To improve the applicability of previous theoretical models and overcome some of these restrictions, we consider a crack of finite size as a porous medium having different poroelastic properties from the matrix material and use poroelastic linear slip conditions to describe the jumps in the stress and displacement across the crack. We first study the scattering of a normally incident fast-P wave by a single circular crack. Then, by combining the theoretical solution and Foldy's scattering method, we develop an effective medium model that can relate seismic characteristics to mechanical compliance and hydraulic permeability of the cracks. Finally, we perform comprehensive parametrical analysis to study the role played by different characteristics of crack on the seismic signatures. We show that the P-wave phase velocity and attenuation are sensitive to the crack mechanical properties, fluid mobility inside the crack and crack size. The findings provide deep understandings on seismic characteristics in cracked rocks and may allow for extracting these properties from seismic data.

Macroscopic mechanical properties of porous rock with one saturating fluid

Effective frequency-dependent moduli and [Formula: see text]-factors are broadly used for characterizing the behavior of earth media in laboratory and field seismic observations. However, such properties are wavemode- and experiment-dependent and are often incomplete and/or inaccurate for modeling realistic situations. For example, viscoelastic moduli for porous fluid-saturated rock are usually derived for primary waves, but they may not apply to cases in which secondary waves are important, such as reflections in finely layered poroelastic media or quasistatic pore-fluid flows in laboratory experiments. To obtain a model applicable to all cases, equations of mechanics should be used, and mechanical properties of the material must be identified. To reveal and measure such properties for fluid-saturated porous rock, we have developed a Biot-consistent model based on Lagrangian continuum mechanics. The model is “minimal,” purely macroscopic, and independent of the macrostructure or patterns of pore-fluid flows; thus, it could represent many existing wave-induced fluid-flow (WIFF) as well as non-WIFF models. Due to its mechanical definition, the model should be applicable to all rock-physics experiments (linear creep, pore flow, low-frequency, resonant, or ultrasonic), any waves in the field (primary, secondary, standing, surface, etc.) under arbitrary boundary conditions, and also finite-difference and finite-element numerical modeling. When based on this model, numerical simulations require no integral equations, fractional derivatives, memory variables, or additional kinetic equations. We further use the model to invert for detailed elastic and viscous properties of fluid-saturated Berea and Fontainebleau sandstones from recently published low-frequency laboratory experiments. All rock properties inferred in the model are time- and frequency-independent, comparable to other physical observations, and the model closely predicts the data. The model can be approximately pore-fluid independent, which allows performing rigorous fluid substitution with viscous pore fluids. As an illustration, P-wave velocity dispersion and attenuation in water-, oil-, and gas-saturated sandstone are simulated.

Self-consistent and squirt flow modelling of velocity dispersion and attenuation for effective-stress dependent experimental data

ABSTRACTWe evaluate acoustic scattering and Wave Induced Fluid Flow (WIFF) mechanisms as a source of velocity dispersion in experimental compressional P- and shear S- waves. The latter was carried out by implementing the Self Consistent (SC) approach and the Chapmańs squirt flow model over a data set consisting of frequency- and pressure dependent velocities obtained by Sothcott, McCann, and O'Hara (2000, The influence of two different pore fluids on the acoustic properties of reservoir sandstones at sonic and ultrasonic frequencies. 70th SEG Meeting, Expanded s, 1883–1886) on brine saturated Clashach sandstones. Modelled velocities are further used to estimate attenuation values for both P- and S- waves. Error estimation between experimental data and theoretical modelling suggest that WIFF is the main mechanism in defining dispersion and attenuation on the studied samples.

Wave-velocity dispersion and rock microstructure

Porosity, fluid type and rock texture significantly affect acoustic wave propagation, since velocity dispersion and attenuation in fluid saturated rocks are mainly caused by wave-induced local fluid flow between microcracks and intergranular pores. We analyze P-wave velocity dispersion as a function of porosity to obtain information about the rock microstructure. The P-wave velocities in water-saturated rocks are predicted from measurements in gas-saturated rocks, using the Gassmann fluid substitution equation (the relaxed state). The dispersion is estimated from the difference between this predicted velocity and the measured one, where the latter corresponds to the unrelaxed state. We evaluate the wave dispersion as a function of porosity for 112 carbonates, 128 sandstones and 56 volcanic rocks, including our measurements for 86 tight rocks, showing that dispersion increases with porosity in the low porosity range, but decreases in the high porosity range. The dispersion peak occurs at a porosity of approximately 15%. Double-porosity poroelasticity modeling based on the local fluid-flow mechanism confirms this behavior. The microcrack radius has a peak in the porosity range 15–19% for all the lithologies from our collection, while the behavior of microcrack porosity is less evident. The dispersion peak may reveal the characteristics of lithological units, in particular porosity, fluid type and rock microstructure.

Seismic numerical analysis for partially gas-saturated porous rocks by an extended local Rytov Fourier approximation based on the diffusion-viscous theory

The frequency-dependent attenuation and velocity dispersion of seismic responses are closely related to hydrocarbon reservoirs. To further investigate the characteristics of seismic responses caused by pore fluid-bearing reservoirs, the role of gas saturation is analyzed in seismic responses of sand reservoirs characterized by the patchy saturation model. To this end, a novel wave extrapolation method is developed based on the diffusive-viscous wave equation (DVWE) as well as a scheme for an extended local Rytov Fourier (ELRF) approximation within the extrapolation depth interval. Our proposed method considers the presence of fluid mixtures in the porous media, resulting in seismic attenuation and dispersion by the mechanism generally known as wave-induced fluid flow (WIFF). This method enables an accommodation for the lateral variations in slowness, diffusion coefficient and viscosity. Subsequently, the extrapolation is adopted to model the synthetic seismic data of a distributary channel model. During this modeling, a gas-water saturated sand reservoir embedded into one of the channels was used to comparatively analyze the distinct features on its seismic synthetic data. We exhibited the numerical simulation results using the proposed wave extrapolation method here and the traditional acoustic wave equation (AWE) method. A comparison of the simulation results, demonstrates that our proposed numerical method can depict the seismic dispersion and frequency-dependent attenuation as well as the phase delay effects associated with gas-water-saturated sand reservoirs. Furthermore, we compare the seismic responses by changing the gas saturations of the sand reservoir. The gas saturation of the reservoir has significant effects on the seismic characteristics of the numerical modeling data. The numerical modeling method improves our understanding of the mechanisms of seismic frequency-dependent characteristics associated with gas saturations and potentially contributes to better insights into gas reservoir indicators derived from seismic field data.

Reflection dispersion signatures due to wave-induced pressure diffusion in heterogeneous poroelastic media

ABSTRACTWe have studied the reflection dispersion signatures of a heterogeneously patchy saturated reservoir exhibiting attenuation and velocity dispersion in a seismic range of frequency. A modified patchy saturation model is employed to predict velocity dispersion and intrinsic attenuation behaviour associated with wave-induced pressure diffusion due to the presence of both mesoscopic and microscopic heterogeneities. Understanding the combined effects of microscopic squirt flow and wave-induced fluid flow related to mesoscopic heterogeneities on wave attenuation and associated dispersion would be important for determining the relative contribution of both of the interdependent energy loss mechanisms. The acoustic properties from the modified patchy saturation model were then employed in combination with an approximate expression for the frequency-dependent normal-incidence reflection coefficient of a dispersive reservoir rock, to give insights into the impacts of attenuation (and its associated velocity dispersion) and acoustic impedance contrast on the reflection dispersion characteristics. Analysis of the results indicates that the dispersion attribute of reflection coefficients and the associated cross-plot are more sensitive to fluid saturation and porosity than the magnitude of the reflection coefficient, and hence provide a promising approach to characterise fluid saturation and porosity in heterogeneous hydrocarbon reservoirs.

A unified poroviscoelastic model with mesoscopic and microscopic heterogeneities

The wave-induced fluid flow (WIFF) is considered to be the main cause of dispersion and attenuation of seismic waves in fluid-saturated porous media. Among numerous theories, the mesoscopic and microscopic heterogeneities are considered to be the primary mechanisms causing the WIFF. Furthermore, in most rocks, the mesoscopic and microscopic heterogeneities exist simultaneously and can cause obvious transitions of the fast P-wave velocity, which means it is necessary to consider the influence of the two mechanisms on the dispersion and attenuation simultaneously. Numerous results have shown that the dispersions and attenuations caused by these two mechanisms can be approximated in terms of the Zener model. To combine the two mechanisms into a unified model, we introduce a new generalized Zener model into the Biot poroelasticity theory to obtain a new poroviscoelastic model. Comparisons between the numerical results and two groups of experimental data further confirm the validity of our new model.

Laboratory study of oil saturation and oil/water substitution effects on a sandstone’s modulus dispersion and attenuation

Using the stress–strain oscillation method, we conducted experiments on a sandstone over a frequency range of 1–100 Hz at a differential pressure of 5 MPa to investigate the effects of oil saturation and oil/water substitution on modulus dispersion and attenuation. We first saturated the sandstone with a low viscosity oil (2#) at different saturation degrees, then replaced the 2# oil with a more viscous oil (68#) and finally injected distilled water in the oil-saturated sample. Young’s modulus, Poisson’s ratio and extensional attenuation were measured during these saturation processes. The measured moduli of this sandstone, when saturated with the 2# oil at different saturation degrees (0% to 100%) or when fully saturated with the 68# oil, manifest little dispersion over the frequency range of 1–100 Hz, and low attenuation was observed. The measured bulk modulus for the oil-saturated sandstone agrees well with the prediction of Gassmann’s fluid-substitution theory, suggesting that wave-induced fluid flow (WIFF) plays an insignificant role in control of the bulk modulus when the sandstone is filled with oil. However, as oil saturation and viscosity increase, the shear modulus keeps increasing, violating the Gassmann’s fluid-substitution theory, which may be caused by the viscous coupling mechanism. When water was injected into sandstone fully saturated with the 68# oil, distinct modulus dispersion and attenuation were observed. The characteristics of the modulus dispersion and attenuation are consistent with mesoscopic WIFF theory, indicating that mesoscopic WIFF is the main cause of modulus dispersion and attenuation for this sandstone saturated with oil–water mixture at seismic frequencies.

ПРОЯВЛЕНИЕ ИЗМЕНЕНИЙ ТРАНСПОРТНЫХ СВОЙСТВ ТРЕЩИНОВАТО-ПОРИСТЫХ СРЕД В СЕЙСМИЧЕСКИХ ВОЛНОВЫХ ПОЛЯХ

In this paper we research the response of carbonates dissolution when interacting with carbon dioxide in the seismic wave fields in a fractured-porous reservoir. We numerically estimate the change of limestone physical properties due to CO2 sequestration based on the analysis of samples CT-scans. Obtained estimations is then used to model a poroelastic material, which we use as fracture-filling material in statistically generated fractured porous fluid-saturated media models. A numerical modeling of wave propagation is performed to estimate a velocity dispersion and attenuation caused by a wave-induced fluid flow.

Rock-physics templates based on seismic Q

We build rock-physics templates (RPTs) for reservoir rocks based on seismic quality factors. In these templates, the effects of partial saturation, porosity, and permeability on the seismic properties are described by generalizing the Johnson mesoscopic-loss model to a distribution of gas-patch sizes in brine- and oil-saturated rocks. This model addresses the wave-induced fluid flow attenuation mechanism, by which part of the energy of the fast P-wave is converted into the slow P (Biot) diffusive mode. We consider patch sizes, whose probability density function is defined by a normal (Gaussian) distribution. The complex bulk modulus of the composite medium is obtained with the Voigt-Reuss-Hill average, and we show that the results are close to those obtained with the Hashin-Shtrikman average. The templates represent the seismic dissipation factor (reciprocal of seismic quality factor) as a function of the P-wave velocity, acoustic impedance, and [Formula: see text] (P to S velocity ratio), for isolines of saturation, porosity, and permeability. They differentiate between oil and brine on the basis of the quality factor, with the gas-brine case showing more dissipation than the gas-oil case. We obtain sensitivity maps of the seismic properties to gas saturation and porosity for brine and oil. Unlike the gas-brine case, which shows higher sensitivity of attenuation to gas saturation, the gas-oil case shows higher sensitivity to porosity, and higher acoustic impedance and [Formula: see text] sensitivity values versus saturation. The RPTs can be used for a robust sensitivity analysis, which provides insights on seismic attributes for hydrocarbon detection and reservoir delineation. The templates are also relevant for studies related to [Formula: see text]-storage monitoring.

Seismic Signatures of Fractured Reservoirs: Theory Versus Numerical Simulations

Seismic attenuation and dispersion usually occur in the fractured reservoirs. The wave-induced fluid flow (WIFF) is recognized as an important mechanism for these phenomena. In this work, we study the seismic attenuation and dispersion due to WIFF in saturated rocks containing two orthogonal sets of intersecting fractures. Based on the existing unified model for the WIFF, we proposed the theoretical model for three types of fractures: periodic planar fractures, randomly-spaced planar fractures, and penny-shaped cracks. The 2D synthetic rock sample with intersecting fractures is then studied by both numerical simulations and the proposed theoretical model. The numerical simulations are carried out using an upscaling method based on Biot’s quasi-static equations of poroelasticity. We find a good agreement between the theoretical predictions and the numerical simulations. For the WIFF between fractures and the background, the seismic dispersion and attenuation predicted by the theoretical model for penny-shaped cracks are in best agreement with the numerical simulation results. On the other hand, for the WIFF between connected fractures, it turns out that the theoretical model for periodic planar fractures is best. The proposed theoretical approach can be applied to both 2D and 3D fracture systems, which can thus constitute a useful tool for the characterization of reservoirs with intersecting fractures.

Modeling the Effect of Microscopic and Mesoscopic Heterogeneities on Frequency-Dependent Attenuation and Seismic Signatures

At seismic and sonic frequencies, the major cause of wave attenuation and dispersion in fluid-saturated rocks might be the wave-induced fluid flow on microscopic and mesoscopic scales. However, it is challenging to assess these effects as the attenuation mechanisms related to both heterogeneities that cannot be expected to be independent. This is due to the fact that, fluid flow taking placing at the mesoscopic scale may be impacted by squirt flow mechanism in the presence of microscopic heterogeneities via modifying the dry rock to be frequency-dependent complex moduli. Understanding the integrated effects, related to microscopic squirt flow and wave-induced fluid flow of mesoscopic heterogeneities, would be important for quantifying the relative contribution of the interdependent energy loss mechanisms. We introduce a procedure in this letter to estimate the frequency-dependent seismic attenuation and dispersion by considering the combined presence of microscopic and mesoscopic heterogeneities. The corresponding seismic reflections of a finely stratified model with a dispersive reservoir are calculated using a propagator matrix method in the frequency domain to study the sensitivity of seismic signatures to pore-fluid mobility and rock heterogeneities.

Effects of ellipsoidal heterogeneities on wave propagation in partially saturated double-porosity rocks

Reservoir rocks are heterogeneous porous media saturated with multiphase fluids, in which strong wave dissipation and velocity dispersion are closely associated with fabric heterogeneities and patchy saturation at different scales. The irregular solid inclusions and fluid patches are ubiquitous in nature, whereas the impact of geometry on wave dissipation is still not well-understood. We have investigated the dependence of wave attenuation and velocity on patch geometry. The governing equations for wave propagation in a porous medium, containing fluid/solid heterogeneities of ellipsoidal triple-layer patches, are derived from the Lagrange equations on the basis of the potential and kinetic energies. Harmonic functions describe the wave-induced local fluid flow of an ellipsoidal patch. The effects of the aspect ratio on wave velocity are illustrated with numerical examples and comparisons with laboratory measurements. The results indicate that the P-wave velocity dispersion and attenuation depend on the aspect ratio of the ellipsoidal heterogeneities, especially in the intermediate frequency range. In the case of Fort Union sandstone, the P-wave velocity increases toward an upper bound as the aspect ratio decreases. The example of a North Sea sandstone clearly indicates that introducing ellipsoidal heterogeneities gives a better description of laboratory data than that based on spherical patches. The unexpected high-velocity values previously reported and ascribed to sample heterogeneities are explained by varying the aspect ratio of the inclusions (or patches).

Analysis of Dynamic Fracture Compliance Based on Poroelastic Theory. Part II: Results of Numerical and Experimental Tests

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.

3D seismic modeling in geothermal reservoirs with a distribution of steam patch sizes, permeabilities and saturations, including ductility of the rock frame

Seismic propagation in the upper part of the crust, where geothermal reservoirs are located, shows generally strong velocity dispersion and attenuation due to varying permeability and saturation conditions and is affected by the brittleness and/or ductility of the rocks, including zones of partial melting. From the elastic-plastic aspect, the seismic properties (seismic velocity, quality factor and density) depend on effective pressure and temperature. We describe the related effects with a Burgers mechanical element for the shear modulus of the dry-rock frame. The Arrhenius equation combined to the octahedral stress criterion define the Burgers viscosity responsible of the brittle-ductile behaviour.The effects of permeability, partial saturation, varying porosity and mineral composition on the seismic properties is described by a generalization of the White mesoscopic-loss model to the case of a distribution of heterogeneities of those properties. White model involves the wave-induced fluid flow attenuation mechanism, by which seismic waves propagating through small-scale heterogeneities, induce pressure gradients between regions of dissimilar properties, where part of the energy of the fast P-wave is converted to slow P (Biot)-wave. We consider a range of variations of the radius and size of the patches and thin layers whose probability density function is defined by different distributions. The White models used here are that of spherical patches (for partial saturation) and thin layers (for permeability heterogeneities). The complex bulk modulus of the composite medium is obtained with the Voigt-Reuss-Hill average. Effective pressure effects are taken into account by using exponential functions.We then solve the 3D equation of motion in the space-time domain, by approximating the White complex bulk modulus with that of a set of Zener elements connected in series. The Burgers and generalized Zener models allows us to solve the equations with a direct grid method by the introduction of memory variables. The algorithm uses the Fourier pseudospectral method to compute the spatial derivatives. It is tested against an analytical solution obtained with the correspondence principle. We consider two main cases, namely the same rock frame (uniform porosity and permeability) saturated with water and a distribution of steam patches, and water-saturated background medium with thin layers of dissimilar permeability. Our model indicates how seismic properties change with the geothermal reservoir temperature and pressure, showing that both seismic velocity and attenuation can be used as a diagnostic tool to estimate the in situ conditions.