Linear Slip Model Research Articles

Amplitude variation with azimuth direct inversion for effective pressure-sensitivity parameters in horizontal transversely isotropic media

Effective pressure prediction is of great significance for fracturability evaluation in unconventional reservoirs. Quantitative characterization of effective pressure remains a great challenge in the inversion prediction of subsurface media. Our goal is to develop a novel Bayesian amplitude varying with azimuth (AVAZ) inversion method to estimate fracture weaknesses and effective pressure for a horizontal transversely isotropic (HTI) medium. Using Schoenberg’s linear slip model and pore space compressibility theory, an anisotropic saturated stiffness matrix directly characterized by effective pressure is derived based on a weak-contrast interface separating two weakly anisotropic half-spaces of HTI medium. Based on the perturbation stiffness matrix and scattering theory, we derive a PP-wave reflection coefficient approximation directly characterized by fracture weaknesses and effective pressure-sensitive parameters, which establishes a quantitative relationship between seismic reflection coefficient and effective pressure. The analysis of accuracy and sensitivity is performed to verify the reliability and applicability of the constructed reflection coefficient. And then, the azimuth-amplitude-difference forward solver is constructed based on seismic reflection coefficients to eliminate the influence of isotropic background perturbations on seismic reflection coefficients and reduce the condition number of the forward solver. Finally, we present a novel Bayesian azimuth-amplitude-difference AVAZ inversion method for transversely isotropic media constrained by Cauchy and Gaussian regularization to directly estimate the fracture weaknesses and effective pressure-sensitive parameters. The application of synthetic data and field data verifies that the inversion results are in good agreement with the actual data, indicating that the method can effectively realize the stable prediction of anisotropic parameters and effective pressure-sensitive parameters of complex media.

The in situ stress prediction of a fractured shale reservoir based on amplitude variation with angle and azimuth inversion: A case study from Southwest China

In situ stress estimation of a shale reservoir is critical for the design of shale hydraulic fracturing and wellbore stability analysis. Fractured shales have orthogonal anisotropy (OA) because of the presence of vertical fractures and lamination. However, the existing in situ stress estimation method mainly focuses on transversely isotropic media. To accommodate orthogonally anisotropic media, an in situ stress estimation method is developed by using amplitude variation with angle and azimuth inversion and linear-slip theory. First, an orthogonal anisotropic medium is introduced to characterize the fractured shale by using a linear-slip model, and then we formulate a reflection coefficient approximation for OA media to perform the seismic inversion. Next, a fracture weakness inversion strategy that incorporates the vertically transverse isotropic background is developed to obtain the stiffness matrix of the media. Finally, the in situ stresses of the OA medium are formulated by using a stress-strain relationship. Inversion tests and a real data application indicate that our method is effective.

Seismic wave modeling of fluid-saturated fractured porous rock: including fluid pressure diffusion effects of discretely distributed large-scale fractures

Abstract. The scattered seismic waves of fractured porous rock are strongly affected by the wave-induced fluid pressure diffusion effects between the compliant fractures and the stiffer embedding background. To include these poroelastic effects in seismic modeling, we develop a numerical scheme for discretely distributed large-scale fractures embedded in fluid-saturated porous rock. Using Coates and Schoenberg's local-effective-medium theory and Barbosa's dynamic linear slip model characterized by complex-valued and frequency-dependent fracture compliances, we derive the effective viscoelastic compliances in each spatial discretized cell by superimposing the compliances of the background and the fractures. The effective governing equations for fractured porous rocks are viscoelastic anisotropic and numerically solved by the mixed-grid-stencil frequency-domain finite-difference method. The main advantage of our proposed modeling scheme over poroelastic modeling schemes is that the fractured domain can be modeled using a viscoelastic solid, while the rest of the domain can be modeled using an elastic solid. We have tested the modeling scheme in a single fracture model, a fractured model, and a modified Marmousi model. The good consistency between the scattered waves off a single horizontal fracture calculated using our proposed scheme and the poroelastic modeling validates that our modeling scheme can properly capture the fluid pressure diffusion (FPD) effects. In the case of a set of aligned fractures, the scattered waves from the top and bottom of the fractured reservoir are strongly influenced by the FPD effects, and the reflected waves from the underlying formation can retain the relevant attenuation and dispersion information. The proposed numerical modeling scheme can also be used to improve migration quality and the estimation of fracture mechanical characteristics in inversion.

Investigation on interfacial shear performance of concrete strengthened with bonded steel plate

This study investigated the interfacial shear performance of concrete strengthened with bonded steel plate through tests, fracture energy based theoretical analysis and numerical simulation. Firstly, the interfacial shear bearing capacity of concrete strengthened with bonded steel plate having various lengths was experimentally investigated, and the failure mode and bearing capacity were discussed. Secondly, based on fracture theory, fictitious crack hypothesis, concrete failure criteria and bilinear linear softening shear slip model, the interfacial shear capacity was theoretically derived, including the effective interfacial shear stress transfer length, fracture energy and peak interfacial shear bearing capacity. Moreover, the interfacial shear failure behavior was also numerically simulated, and the interfacial shear stress and normal stress during the whole loading and failure process were discussed. The results confirmed that the numerical simulation using the theoretically derived interfacial fracture energy and peak shear strength could give simulation results in good agreement with experimental results. Also, the calculated interfacial bearing capacity using the theoretical approach showed excellent agreement with the experimental results. Furthermore, the influence of bonding length was also theoretically derived using the numerical parametric simulation results, and a revised interfacial shear bearing capacity considering bonding length was further proposed, which was also in good agreement with both experimental and numerical simulation results.

Reflection and transmission of plane waves in stressed media with an imperfectly bonded interface

SUMMARY The reflection and transmission of elastic waves are of great significance for predicting reservoir physical properties, interpreting seismic data and detecting crustal structures. Most studies only consider the initial vertical stress state when studying the reflection and transmission of elastic waves at imperfectly bonded interfaces, but few studies consider the influence of initial stress on boundary conditions. Moreover, the effect of initial stress on the energy distribution of elastic waves at imperfectly bonded interfaces has rarely been investigated. We propose a unified method to calculate the energy reflection and transmission coefficients for different incident waves at welded or imperfectly bonded interfaces in stressed media. The effects of initial stress on the equation of motion, the elastic properties of the medium and the boundary conditions at the interface are considered. The elastic properties of rocks under initial stress are described by acoustoelasticity theory. In addition, we define a new stress tensor to modify the linear-slip model for describing boundary conditions at the imperfectly bonded interface in the presence of initial stress. Numerical results show that the energy reflection and transmission coefficients at the non-welded interface in stressed media depend on the elastic properties of the incident and transmitted media, the initial stress, the type and magnitude of the interfacial compliance and the frequency and propagation direction of the incident wave. The initial vertical and horizontal stresses dominate the reflection and transmission coefficients at small and large angles, respectively. The discontinuity in displacement across the imperfectly bonded boundary results in the frequency dependence of the reflection and transmission coefficients. Imperfect bonding enhances P-wave and SV-wave energy reflection and weakens P-wave energy transmission. However, imperfect bonding can enhance the energy transmission of the SV wave for the imperfectly bonded interface with high contrast between tangential and normal compliance and a resonance peak appears at a specific frequency. We also notice that imperfectly bonded interfaces with interfacial compliance less than $1.0 \times {10}^{ - 11}$ m Pa−1 can be regarded as welded interfaces in the seismic frequency band (lower than 100 Hz). In addition, the initial stress greatly influences the reflection coefficients at high frequencies and the transmission coefficients at low frequencies. The initial vertical stress can reduce the energy transmission of SV waves at imperfectly bonded interfaces. In contrast, the initial horizontal stress can significantly increase the energy transmission of low-frequency SV waves and may lead to the disappearance of the resonant peak in the transmission coefficients.

Wave reflection and transmission coefficients for layered transversely isotropic media with vertical symmetry axis under initial stress

SUMMARYThe stress-dependent wave reflection and transmission (R/T) coefficients for the layered transversely isotropic media with a vertical symmetry axis (VTI) are seldom investigated in the published literature. To fill this gap, we propose the exact formulas for the plane wave R/T coefficients on the welded and non-welded boundaries between two distinct VTI half-spaces under the effect of initial stress. The theory of acoustoelasticity is used to describe the influence of initial applied stress on the overall elastic properties of VTI media and to represent two different boundary conditions. The normal and tangential compliances are used to characterize the discontinuity of non-welded boundary based on the linear-slip model, and their stress dependences are ideally considered according to the effect of stress on fracture aperture. Then the plane-wave displacement equations are substituted into the boundary conditions to yield the analytic formulas for frequency-independent R/T coefficients for the welded interface and frequency-dependent R/T coefficients for the non-welded interface. The stress-dependent wave slowness vector and polarization vector embedded in R/T coefficients can be directly computed with the Christoffel equation given by the acoustoelastic equation. Modelling results graphically show the effects of initial stress on the angle-dependent wave velocities, Thomsen elastic anisotropy parameters, fracture compliances, the R/T coefficients and seismic reflection responses for welded and non-welded interfaces in detail. The R/T coefficients are more sensitive to initial stress at relatively large incidence angles for the designed two-layer model with welded or non-welded boundary. The proposed R/T coefficient formulas and modelling results are relevant to in situ stress detection, fracture characterization, and exploration for oil and gas in shale reservoirs in high-stress fields.

Modeling Method to Characterize the Pore Structure of Fractured Tight Reservoirs

The study of unconventional reservoirs has gained increasing attention with the deepening of exploration and development especially in deep-buried tight sandstone reservoirs. We could not obtain the accurate elastic parameters of reservoirs using the conventional rock physics model, since tight sandstone reservoirs have the characteristics of strong heterogeneity, complex lithology and storage space. In order to better describe tight sandstone reservoirs, we improved the traditional tight sandstone rock physics model by combining the dual-connected pore model and the linear slip model. Since the combined modeling process subtly considers four characteristics including the diversity of tight sandstone matrix minerals, the irregularities of pores structure, the connectivity between pores, and the anisotropy caused by fractures, multiple reservoir characteristic parameters can be derived from the limited logging information by the improved model. These reservoir characteristic parameters could account for the difference of diagenesis, which are useful to distinguish different pore types and eliminate ineffective reservoirs. The practical application shows that the improved model can extract microscopic reservoir information hidden in the logging data more effectively than the traditional model. It provides a reliable tool for identifying effective reservoirs in tight sandstone.

Comparison of wave-propagation simulations in fractured domains using discrete fractures and equivalent media

SUMMARY Fractures largely control reservoir permeability and, therefore, it is of immediate importance to know the geometrical parameters of fracture sets and their effects on seismic data. To understand the interaction between the fractures and seismic wavefield, we performed numerical simulations of elastic wave propagation in fractured digital rocks (FDRs) using the discontinuous Galerkin method, the linear-slip model and sets of randomly distributed fractures. We compared the results with those obtained using Hudson’s equivalent media theory (HEM) and observed that, when the fracture density is 0.08 or less, there is no statistical difference between the FDR and HEM results; however, when the fracture density is higher than 0.08, the results of HEM diverge from those of FDR. Furthermore, HEM accuracy depends not only on the fracture density but also on the P- to S-wave velocity ratio. The P-wave anisotropy induced by the fractures is observed as a delay, which can be due to fracture density, length or a mixture of both. The Pwave is delayed in all directions, but mostly in the direction perpendicular to the fractures’ orientation.

Anisotropic Poroelasticity and AVAZ Inversion for in Situ Stress Estimate in Fractured Shale-Gas Reservoirs

Knowledge of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">in situ</i> stress is of great significance for hydraulic fracture stimulating of unconventional reservoirs. Quantitative estimation of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">in situ</i> stress, especially in the far field, is a major challenge. This research mainly focuses on developing a novel Bayesian AVAZ inversion approach to estimate <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">in situ</i> stress of fractured shale-gas reservoirs with horizontal transversely isotropic (HTI) symmetry. Using the generalized Hooke’s law and Schoenberg’s linear slip model, we first deduce the anisotropic horizontal stress equation in an HTI medium formed by a single set of vertically aligned fractures embedded in an isotropic background rock. Based on the critical porosity model, we then obtain the saturated stiffness matrix with vertical effective stress-sensitive parameters and dry fracture weaknesses using the relationship between vertical effective stress and porosity. Combining the scattering function and the perturbed stiffness matrix, we deduce a linearized PP-wave reflection coefficient as a function of fluid bulk modulus, vertical effective stress-sensitive parameter, dry-rock P- and S-wave moduli, density, and fracture density. Finally, we propose a novel Bayesian amplitude variation with azimuth (AVAZ) inversion constrained by the Cauchy regularization and low-frequency regularization to estimate these model parameters, which are further used to calculate <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">in situ</i> stress. The analysis of synthetic data demonstrates that <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">in situ</i> stress can be reasonably estimated even with moderate noise. A field data set test reveals that the inversion results agree well with the well log interpretation, and the proposed approach can generate meaningful results that are useful for seismic identification of potential fracturing barriers during fracturing stimulation of shale-gas reservoirs.

Inversion Method of Elastic and Fracture Parameters of Shale Reservoir With a Set of Inclined Fractures

The inversion of reservoir elastic parameters and fracture parameters is of great significance to oil and gas production. In shale reservoirs with inclined fractures, using the reflection coefficient equation of vertical fractures or horizontal fractures under the vertical transverse isotropy (VTI) background has certain limitations. For this reason, based on the linear-slip model, this article establishes the approximate stiffness matrix of the monoclinic medium with a set of inclined fractures under the background of VTI and combines the Born scattering theory to further derive the PP-wave linear reflection coefficient equation of the monoclinic medium. The equation includes parameters, such as Young’s modulus, Poisson’s ratio, density, fracture weaknesses, and bedding weaknesses. Then, a rock physics model that comprehensively considers horizontal bedding and inclined fractures is established, and the influence of the inclination of the fractures on the reflection coefficient is analyzed. Finally, taking the advantage of Bayesian theory, the fracture weakness inversion method based on the azimuth amplitude difference is established, and the inversion accuracy is improved by adding Cauchy constraints and smoothing model regularization terms. Then, take the fracture weaknesses as inputs to invert the parameters, such as Young’s modulus, Poisson’s ratio, density, and bedding weaknesses. The inversion results of theoretical data show that the method can invert the fracture information well, and it can also be effectively applied to low-to-medium signal-to-noise ratio (SNR) data. The inversion results of theoretical and actual data prove that the inversion equation derived in this article can be used to obtain more accurate elastic and fracture parameters in fractured shale reservoirs.

Seismic Characterization of Decoupled Orthorhombic Fractures Based on Observed Surface Azimuthal Amplitude Data

Commonly-used seismic fracture characterization method assumes one single fracture set, and this article aims to relax the limitation and demonstrate the feasibility of seismic characterization for two sets of orthorhombic fractures. Elastic properties of orthorhombic fractures can be studied using a model of horizontal and vertical fractures as linear-slip interfaces embedded in an isotropic background. High fracture density in hydrocarbon reservoirs is the “sweet spot” of effective permeability for fluid flow, and seismic characterization of orthorhombic fractures using the variations in reflection amplitude versus offset and azimuth (AVOAz) helps to optimize the production of naturally fractured reservoirs. Based on an effective orthorhombic linear-slip model, we first construct the effective elastic stiffness matrix in terms of elastic moduli and decoupled horizontal and vertical fracture densities under the assumption of weak anisotropy (WA) or small fracture densities. Using the scattering function and first-order perturbation in effective elastic stiffness components, we derive a WA and linearized PP-wave reflection coefficient containing decoupled orthorhombic fracture densities to describe the characteristics of horizontal and vertical fracture sets. We then propose an efficient AVOAz inversion method to perform the seismic characterization of decoupled orthorhombic fracture densities based on observed surface azimuthal amplitude data in a Bayesian framework. We finally apply the inversion approach to synthetic and field datasets to demonstrate its feasibility and reasonability in orthorhombic fracture characterization.

Geometry effects on flow characteristics of micro-scale planar nozzles

A numerical investigation of geometry effects on the flow characteristics associated with converging-diverging 2D planar micronozzles is reported. The classical N–S with linear slip model, DSMC method, and a hybrid N–S/DSMC based on the continuum breakdown concept are employed for the simulations. The various methods adopted are validated with available experimental data. The study addresses the issue of heterogeneous findings on optimized half divergence angle recommended by previous studies. The analysis is conducted for various throat dimensions (2–200 µm thoat height), divergence angles (5∘–30∘), and divergence length. The performance and flow characteristics are compared at two distinct operating conditions, i.e. when the flow is mostly subsonic and supersonic in the divergent section. Results show an optimum divergence angle and length, maximizing the performance for a specific operating condition and nozzle size. Literature data suggest various divergence angles for optimum performance. This is due to the differences in operating conditions and the size of the micronozzles used. The current study shows that differences in flow behavior are mainly due to the pressure difference across the nozzle. When the micronozzle is operating under low-pressure differences, the flow is primarily subsonic in the divergent section. However, it accelerates due to the effect created by the growing thick boundary layer. The flow at higher pressure differences is characterized by clear flow separation and the occupation of the separated flow to a considerable portion of the divergent section. The separated phenomenon becomes less pronounced as the pressure difference and nozzle size decreases. As the nozzle size decreases to the nanoscale, Re is very low, and the subsonic layer fully occupies the divergent nozzle section even at very high-pressure differences. The results show that the performance is significantly influenced by the wall thermal conditions. The study highlights the differences in flow behaviour between micro and nano-scale nozzles at various operating conditions. Based on numerical data obtained in this present work, a new correlation is proposed to predict thrust per unit width for micronozzles.

Simulation of borehole acoustic wavefields in fractured media by combining the spectral-element method and linear-slip model

We have used the spectral-element method (SEM) to simulate 3D acoustic wavefields in the fluid-filled borehole embedded in fractured media. The fractures are characterized by the linear-slip model (LSM), which is incorporated into the surface integral of the SEM weak form, avoiding meshing individual fractures, thus reducing the degrees of freedom of the fractures compared with meshing each fracture directly. For the fracture-free case, we validate SEM through the comparison with the real-axis integration method for monopole and dipole sources. For the case with a fracture, we compare the SEM-LSM solutions with the reference numerical solutions of a thin-layer model using the finite-difference method. Good agreement is achieved between the results from our method and the reference finite-difference solutions. We find that the acoustic wavefields excited by a dipole source are more sensitive to the fractures than those by a monopole source. To show the ability of the approach to handle complex problems, we simulate the cases with a tilted fracture and multiple fractures. Based on the simulated results, we investigate the influence of the fracture parameters (e.g., stiffness, tilt angle, azimuth, thickness, number, and spatial intervals of fractures) on the scattered wavefields. We find that the tilt angle has an obvious influence on the scattered waveforms and amplitudes. The results also demonstrate that the wavefields are quite sensitive to the number of fractures. The magnitudes of the horizontal components’ transmitted wavefields decrease linearly with the number of fractures. Through analyzing the synthetic data in the time and frequency domains, we discuss how to evaluate the properties of fractures intersected by a borehole.

Effective elasticity of a medium with many parallel fractures

Summary We consider an alternative way of obtaining the effective elastic properties of a cracked medium. Similarly, to the popular linear-slip model, we assume flat, parallel fractures, and long wavelengths. However, we do not treat fractures as weakness planes of displacement discontinuity. In contrast to the classical models, we represent fractures by a thin layer embedded in the background medium. In other words, we follow the Schoenberg-Douma matrix formalism for Backus averaging, but we relax the assumptions of infinite weakness and marginal thickness of a layer so that it does not correspond to the linear-slip plane. To represent the properties of a fracture, we need a fourth order elasticity tensor and a thickness parameter. The effective tensor becomes more complicated, but it may describe a higher concentration of parallel cracks more accurately. Apart from the derivations of the effective elasticity tensors, we perform numerical experiments in which we compare the performance of our approach with a linear-slip model in the context of highly fractured media. Our model becomes pertinent if filled-in or empty cracks occupy more than one per cent of the effective medium.

Estimates of Individual Fracture Compliances Along Boreholes From Full‐Waveform Sonic Log Data

AbstractAn important seismic attenuation mechanism in fractured environments is related to energy conversion into reflected and transmitted waves at fracture interfaces. Using full‐waveform sonic (FWS) log data, we show that it is possible to quantify transmission losses across a set of fractures from time delays and amplitude differences of the critically refracted P‐wave as compared to intact sections along the borehole. In the presence of fractures, the transmission coefficient associated with a given fracture is obtained by combining information on transmission losses from multiple receivers and source positions into a linear system of equations for all fractures intersecting the borehole. Fracture compliance is computed from the inferred transmission coefficient based on a linear slip model. For validation, we use synthetic FWS log data obtained from numerical simulations of wave propagation in a water‐filled borehole surrounded by low‐permeability rocks with discrete fractures. The methodology is then applied to field data acquired along boreholes penetrating multiple fractures in a granodioritic host rock. We show that our estimations of mechanical compliance are consistent with previously reported values, which were estimated for individual fractures intersecting one of the boreholes with a related method valid only for isolated single fractures. Comparison between our estimates of fracture compliance and transmissivity profiles from previous hydraulic characterizations of the fractures suggests that the proposed method may also allow to locate the most permeable fractures along a borehole, which, in turn, opens the perspective of enhancing the design and effectiveness of subsequent hydraulic testing and fracturing experiments.

Azimuthal amplitude variation with offset parameterization and inversion for fracture weaknesses in tilted transversely isotropic media

The characterization of fracture-induced tilted transverse isotropy (TTI) seems to be more suitable to actual scenarios of geophysical exploration for fractured reservoirs. Fracture weaknesses enable us to describe fracture-induced anisotropy. With the incident and reflected PP-wave in TTI media, we have adopted a robust method of azimuthal amplitude variation with offset (AVO) parameterization and inversion for fracture weaknesses in a fracture-induced reservoir with TTI symmetry. Combining the linear-slip model with the Bond transformation, we have derived the stiffness matrix of a dipping-fracture-induced TTI medium characterized by normal and tangential fracture weaknesses and a tilt angle. Integrating the first-order perturbations in the stiffness matrix of a TTI medium and scattering theory, we adopt a method of azimuthal AVO parameterization for PP-wave reflection coefficient for the case of a weak-contrast interface separating two homogeneous weakly anisotropic TTI layers. We then adopt an iterative inversion method by using the partially incidence-angle-stacked seismic data with different azimuths to estimate the fracture weaknesses of a TTI medium when the tilt angle is estimated based on the image well logs prior to the seismic inversion. Synthetic examples confirm that the fracture weaknesses of a TTI medium are stably estimated from the azimuthal seismic reflected amplitudes for the case of moderate noise. A field data example demonstrates that geologically reasonable results of fracture weaknesses can be determined when the tilt angle is treated as the prior information. We determine that the azimuthal AVO inversion approach provides an available tool for fracture prediction in a dipping-fracture-induced TTI reservoir.

Correction of second-order slip condition for higher Knudsen numbers by approximation of free-molecular diffusion

The computational predictions of channel and pipe flows with classical models and no-slip condition at the wall reach excellent results for lower Knudsen numbers (Kn) only. Linear slip models reach a very good approximation of measurement results over the region of 10−3 &amp;lt; Kn &amp;lt; 10−1. The numerical results of higher-order slip models match experimental data up to Kn ≈ 1. The present work derives an analytical model for the transition from the slip regime to the free-molecular flows by the superposition of diffuse molecular boundary reflection and the molecular diffusion inside the bulk flow. The methodology of the present publication models the mass flow resulting from the molecular diffusion for the approximation of the mass flow in microchannels and micropipes for the regime of molecular mass flows (1 &amp;lt; Kn &amp;lt; 100) in an excellent way. The present model shows good agreement with the former models, measurement data, and direct simulation Monte Carlo results for the complete region from the transitional regime up to free-molecular flow (10−2 &amp;lt; Kn &amp;lt; 102).

Study on S-wave propagation through parallel rock joints under in situ stress

In situ stress is an important feature of underground rock masses. This paper extends the time-domain recursive method to the study of S-wave propagation through in situ stressed rock masses containing parallel joints. A linear elastic model and hyperbolic nonlinear slip model (HNSM) are first used to establish equations for wave propagation across jointed rock masses under a combination of gravitational and tectonic stress. Then, a comparison is made of the waveforms generated using the HNSM and Mohr–Coulomb slip model. Their differences are investigated and the wave propagation equation verified. Finally, parametric studies are conducted to investigate the effect of joint angle, joint number, in situ stress, and lateral pressure coefficient. The results show that the HNSM can describe the dynamic changes in the stress and their effect on the deformation behavior of the joint during the propagation of the S-wave. The effect of in situ stress on wave propagation is related to the joint angle and lateral pressure coefficient, which determine the initial stress state and contact state of the joint.

Spectral element simulation of elastic wave propagation through fractures using linear slip model: microfracture detection for CO2 storage

SUMMARY Simulation of seismic wave propagation through fracture has a wide range of applications in environmental sciences. In this paper, we propose an efficient tool to compute accurate seismic response from a fracture within a reasonable time frame. Its theoretical formulation is based on the spectral element method (SEM) and extended to Schoenberg’s linear slip model (LSM). SEM is very effective in terms of accuracy and stability criteria. LSM is treated as a boundary condition and perfectly fits for modelling fractures with a small aperture. The method is implemented for 3-D heterogeneous media on GPU, which allows calculating the tasks with large and complex geometries. The validation of the numerical method shows good agreement with the theory. Finally, we applied the method to the task that illustrates the possibility of the proposed solution to handle real problems. We model sonic logging for a well with a microfracture in a cement sheath. Based on synthetic seismograms, strong connections between wave mode parameters and the fracture parameters were established. This task is of high importance for carbon capture and storage, as microfractures provide the path for long-term CO2 migration.

Generalized Multiscale Finite Element Method for Elastic Wave Propagation in the Frequency Domain

In this work, we consider elastic wave propagation in fractured media. The mathematical model is described by the Helmholtz problem related to wave propagation with specific interface conditions (Linear Slip Model, LSM) on the fracture in the frequency domain. For the numerical solution, we construct a fine grid that resolves all fracture interfaces on the grid level and construct approximation using a finite element method. We use a discontinuous Galerkin method for the approximation by space that helps to weakly impose interface conditions on fractures. Such approximation leads to a large system of equations and is computationally expensive. In this work, we construct a coarse grid approximation for an effective solution using the Generalized Multiscale Finite Element Method (GMsFEM). We construct and compare two types of the multiscale methods—Continuous Galerkin Generalized Multiscale Finite Element Method (CG-GMsFEM) and Discontinuous Galerkin Generalized Multiscale Finite Element Method (DG-GMsFEM). Multiscale basis functions are constructed by solving local spectral problems in each local domains to extract dominant modes of the local solution. In CG-GMsFEM, we construct continuous multiscale basis functions that are defined in the local domains associated with the coarse grid node and contain four coarse grid cells for the structured quadratic coarse grid. The multiscale basis functions in DG-GMsFEM are discontinuous and defined in each coarse grid cell. The results of the numerical solution for the two-dimensional Helmholtz equation are presented for CG-GMsFEM and DG-GMsFEM for different numbers of multiscale basis functions.