Simulation Of Elastic Wave Propagation Research Articles

A 2.5D hybrid SBM-MFS methodology for elastic wave propagation problems

This paper proposes a novel hybrid methodology that combines the singular boundary method (SBM) and the method of fundamental solutions (MFS) for the computational simulation of elastic wave propagation. Particularly, the methodology aims to address radiation or scattering problems of longitudinally invariant systems in the wavenumber–frequency domain involving boundaries with intricate geometries. The approach uses the SBM to deal with the complex parts of these geometries and the MFS for the smooth ones. The method is studied in the framework of three case studies involving longitudinally infinite cavities in a homogeneous full-space with circular, square-shaped and five-cusped hypocycloid cross-sections. These three examples are selected to assess the accuracy and robustness of the hybrid SBM-MFS approach in comparison with alternative modelling strategies. The comparisons show that the proposed method inherits the accuracy of the MFS while keeping the robustness of the SBM when dealing with complex geometries. The method is found to be computationally more efficient than the SBM or the boundary element method (BEM). Moreover, the hybrid approach naturally mitigates the effect of fictitious eigenfrequencies, a feature that neither conventional versions of the SBM nor the BEM have.

Study Investigates Seismic Monitoring for Carbon Storage Leak Detection

_ This article, written by JPT Technology Editor Chris Carpenter, contains highlights of paper IPTC 22980, “Carbon Storage Leak Detection Through Seismic FWI and RTM: Different Survey Analyses,” by Sajjad Amani, Kyoto University. The paper has not been peer reviewed. Copyright 2023 International Petroleum Technology Conference. Reproduced by permission. _ In the complete paper, marine seismic data processing is investigated as a tool for monitoring possible leakages in geological carbon storage. Because of the great importance of storage permanence, a precise leakage-monitoring strategy is crucial. The proficiency of seismic monitoring solutions for leakage monitoring can be affected by shallower layers as a result of structure, seismic wave attenuation, and leak size. The authors explore two popular seismic monitoring methods used in this application in different scenarios: full waveform inversion (FWI) and reverse-time migration (RTM). Introduction Among the various carbon capture and storage (CCS) options, underground storage in saline aquifers is the best-understood solution. To assure storage consistency and permanence, finding the best strategy to precisely detect possible carbon leaks is essential. A perfect method must demonstrate the difference between stored CO2 and injected CO2 to detect potential fast- and slow-leakage areas. The seismic monitoring technique is the most efficient approach in this respect. Two popular tools for seismic monitoring are FWI and RTM. Several studies of their use in this application have been conducted. However, previous research did not analyze different survey arrays for carbon-leak detection using FWI and RTM. The current research aims to investigate quantitative aspects of CCS monitoring to conduct sensitivity analysis of the three different survey arrays [vertical seismic profile (VSP), crosswell, and surface] for different amounts of CO2-storage leakage in a saline aquifer reservoir. The capability of seismic-imaging methods for small amounts of leakage was tested. Comparison of these three arrays using monitoring methods such as RTM and FWI reveals their pros and cons in providing detailed information about the reservoir. In this research, a simple synthetic model was built that closely fits actual reservoirs characterized by suitable physical features such as velocity and density. Subsequently, elastic wave propagation simulation was implemented by use of a finite-difference scheme over a physical model of the reservoir. Then, FWI was applied to enhance the accuracy of the model parameters. With an efficient forward-modeling and inversion scheme, RTM was used as a powerful imaging tool that provides final high-resolution results for monitoring CO2 migration and possible leakage. This process examined the synthetic model for different amounts of CO2 leakage in saline aquifers to evaluate the performance of CO2 leakage monitoring using the seismic method. The results of the different receiver and source arrays also were compared to establish their effectiveness.

On the enriched mixed Galerkin method with θ scheme for the elastic wave equation

In this paper, we propose the application of an enriched Galerkin (EG) method to solve the elastic wave equation in the “displacement pseudo-pressure” mixed form, which is known for being locking-free. Unlike the continuous Galerkin (CG) method and the discontinuous Galerkin (DG) method, the EG method is based on the weak form of the DG method, and the enriched space combines a continuous piecewise linear vector-valued function space with some discontinuous piecewise linear functions. Specifically, we employ the enriched space to discretize the displacement, while the pseudo-pressure is discretized using the P0 element space. As a result, compared to the DG method, the EG method significantly reduces the number of degrees of freedom. Furthermore, we adopt a more general θ-scheme for time discretization, which behaves as an implicit scheme for 14≤θ≤1 and as an explicit scheme for 0≤θ<14. We establish error estimates for both the semi-discrete and fully-discrete schemes. The accuracy of the proposed method is validated through numerical examples, which not only confirm the theoretical analysis but also provide simulations of elastic wave propagation.

A surrogate model for finite element simulation of elastic wave propagation in composite materials

Deep learning surrogate models can be employed in solid mechanics to forecast the behavior of structures subjected to various loading conditions, substantially decreasing the computational costs associated with simulations. In this letter, we have utilized convolutional neural networks and Fourier transform to predict the elastic wave output from composite bars. The microstructures of the bar are utilized as inputs to the deep learning model, while the output is the elastic wave response. The convolutional neural network learns to identify crucial input composite features and utilizes this information to predict the output elastic waves. Finally, the mean squared error of the predicted output signals is compared to the actual output signals, which was used to evaluate the model. The outcomes of this study demonstrate that the deep learning model can precisely and swiftly predict the output elastic waves of the composites, thus serving as a surrogate model for time-consuming finite element simulations.

Simulation of Elastic Wave Propagation Based on Meshless Generalized Finite Difference Method with Uniform Random Nodes and Damping Boundary Condition

When the grid-based finite difference (FD) method is used for elastic wavefield forward modeling, it is inevitable that the grid divisions will be inconsistent with the actual velocity interface, resulting in problems related to the stepped grid diffraction and inaccurate travel time of reflected waves. The generalized finite difference method (GFDM), which is based on the Taylor series expansion and weighted least square fitting, solves these problems. The partial derivative of the unknown parameters in the differential equation is represented by the linear combination of the function values of adjacent nodes. In this study, the Poisson disk node generation algorithm and the centroid Voronoi node adjustment algorithm were combined to obtain an even and random node distribution. The generated nodes fit the internal boundary more accurately for model discretization, without the presence of diffracted waves caused by the stepped grid. To avoid the instability caused by the introduction of boundary conditions, a Cerjan damping boundary condition was proposed for boundary reflection processing. The test results generated by the different models showed that the generalized finite difference method can effectively solve the problems related to inaccurate travel time of reflection waves and stepped grid diffraction.

Imaging and seismic modelling inside volcanoes using machine learning

Despite advances in seismology and computing, the ability to image subsurface volcanic environments is poor, limiting our understanding of the overall workings of volcanic systems. This is related to substantive structural heterogeneities which strongly scatters seismic waves obscuring the ballistic arrivals normally used in seismology for wave velocity determination. Here we address this constraint by, using a deep learning approach, a Fourier neural operator (FNO), to model and invert seismic signals in volcanic settings. The FNO is trained using 40,000+ simulations of elastic wave propagation through complex volcano models, and includes the full scattered wavefield. Once trained, the forward network is used to predict elastic wave propagation and is shown to accurately reproduce the seismic wavefield. The FNO is also trained to predict heterogeneous velocity models given a limited set of input seismograms. It is shown to capture details of the complex velocity structure that lie far outside the ability of current methods available in volcano imagery.

An open-source implementation of GPU-accelerated Finite-Difference Time Domain (FDTD) simulations in the Unity game engine to visualize elastic wave propagation and scattering for pedagogical or training scenarios

Simulations of elastic wave propagation through solid materials can be realized through industry-standard software or custom implementations in a variety of programming languages. These codes are generally optimized for numerical accuracy and can be difficulty to interact with, slow to run, and often require external processing of results to create 3D visualizations. This is especially true for parallelized or Graphical Processing Unit (GPU)-based simulations which require additional libraries. These factors tend to limit the use of numerical simulations of wave propagation in pedagogical applications to simple 2D implementations of acoustic propagation through fluids. Here we present a fully-3D open-source Finite-Difference Time Domain (FDTD) simulation of elastic wave propagation that is implemented in the Unity game development engine. Due to the nature of the engine the simulation code runs entirely on the GPU without additional coding. Additionally the solutions are rendered as the simulation runs which removes the need for saving and post-processing large data files, as well as allowing the user to rotate the viewpoint in real-time to better visualize the wave propagation and scattering in complex geometries.

Poro-acoustoelasticity finite-difference simulation of elastic wave propagation in prestressed porous media

Insights into wave propagation in prestressed porous media are important in geophysical applications, such as monitoring changes in geo-pressure. This can be addressed by poro-acoustoelasticity theory, which extends the classical acoustoelasticity of solids to porous media. The relevant poro-acoustoelasticity equations can be derived from anisotropic poroelasticity equations by replacing the poroelastic stiffness matrix with an acoustoelastic stiffness matrix consisting of second-order and third-order elastic constants. The theory considers the poroelasticity equations to be nonlinear due to the cubic strain-energy function with linear strains under finite-magnitude prestresses. A rotated staggered-grid finite-difference method with an unsplit convolutional perfectly matched layer absorbing boundary is used to solve a first-order velocity-stress formulation of poro-acoustoelasticity equations for elastic wave propagation in prestressed porous media. Numerical solutions are partially verified by computing the velocities of fast P wave, slow P wave, and S wave as a function of hydrostatic prestress and are compared with the exact values. Numerical simulations of wave propagation are carried out for the model of poro-acoustoelastic homogeneous space under three states—prestress confining (hydrostatic), uniaxial, and pure shear—and for the model of two poro-acoustoelastic homogeneous half-spaces in the planar contact under confining (hydrostatic) prestress. The resulting wavefield snapshots show fast P-wave, slow P-wave, and S-wave propagations in poro-acoustoelastic media under loading prestresses, which illustrate that the stress-induced velocity anisotropy is of orthotropy strongly related to the orientation of prestresses. These examples demonstrate the significant impact of prestressing conditions on seismic responses in velocity and anisotropy.

Acoustoelastic FD Simulation of Elastic Wave Propagation in Prestressed Media

Insights into wave propagation in prestressed media are important in geophysical applications such as monitoring changes in geo-pressure and tectonic stress. This can be addressed by acoustoelasticity theory, which accounts for nonlinear strain responses due to stresses of finite magnitude. In this study, a rotated staggered grid finite-difference (RSG-FD) method with an unsplit convolutional perfectly matched layer absorbing boundary is used to solve the relevant acoustoelastic equations with third-order elastic constants for elastic wave propagation in prestressed media. We partially verify our numerical simulations by the plane-wave theoretical solution. Comparisons of theoretical and calculated wave velocities are conducted for both P-wave and S-wave as a function of hydrostatic prestresses. We discuss several aspects of the numerical implementation. Numerical acoustoelasticity simulations for wave propagation in single- and double-layer models are carried out under four states of prestresses, confining, uniaxial, pure-shear, and simple-shear. The results display the effective anisotropy of elastic wave propagation in acoustoelastic media, illustrating that the prestress-induced velocity anisotropy is of orthotropic features strongly related to the orientation of prestresses. These examples demonstrate the significant impact of prestressed conditions on seismic responses in both phase and amplitude.

Full-waveform centroid moment tensor inversion of passive seismic data acquired at the reservoir scale

SUMMARY Passive seismic inversion at the reservoir scale offers the advantages of low cost, negligible environmental impact and the ability to probe a target area with low-frequency energy not afforded by even the most modern active-source seismic technology. In order to build starting models suitable for full-waveform wave speed tomography, characterization of earthquake sources is an indispensable first step. We present a workflow for the centroid moment tensor (CMT) inversion of seismic events identified in a passive seismic data set acquired by a large and dense array of three-component broad-band seismic sensors in a mountainous setting in the Himalayan foothills. The data set comprised 256 instruments operating for 2×4 months over an area of 8000 km2. An initial 3-D wave speed model was determined for the region via the analysis of first-arriving traveltime picks. Of the 2607 identified seismic events that were well recorded at frequencies between 0.2–50 Hz, 86 with magnitudes 1.3 ≤ M ≤ 3.0 initially had their CMT focal mechanisms determined by a waveform fitting procedure built on a Green’s function approach in a 1-D layered average wave speed model, for stations within an offset of 10 km, in the frequency range 0.2–1.4 Hz. Here, we obtain updated CMT mechanisms for the 86 events in that catalogue via multicomponent full-waveform inversion in the 3-D wave speed model. Our workflow includes automated data- and model-driven data selection using a combination of different metrics derived from signal-to-noise considerations and waveform-fitting criteria, and relies upon spectral-element simulations of elastic wave propagation in the 3-D wave speed model, honouring topography. Starting from the initial CMT solutions, we seek improvement to the data fit within the frequency band 0.5–2.5 Hz by minimizing the waveform difference between observed and synthetic data, while accommodating wave speed-model errors by allowing for small time-shifts. We balance uneven data coverage and tune their contributions via data-space weighting functions. We quantify the improvements to the data fit in terms of different metrics. We summarize the changes to the CMT solutions, and present and analyse the resulting catalogue for the region, including their breakdown into double-couple and non-double couple components, and their relation to mapped faults.

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.

Combination of Grid-Characteristic Method on Regular Computational Meshes with Discontinuous Galerkin Method for Simulation of Elastic Wave Propagation

Combination of Grid-Characteristic Method on Regular Computational Meshes with Discontinuous Galerkin Method for Simulation of Elastic Wave Propagation

Formulation and Stability Analysis of a Multi-Scale Modeling Approach for Simulation of Elastic Wave Propagation

In this work, we report on the formulation and detailed stability analysis of a dynamic multi-scale scheme involving two different local computational strategies for modeling of elastic wave propagation. The coupled model involves the Local Interaction Simulation Approach and Cellular Automata for Elastodynamics, however the presented analysis approach is general and applies to other numerical techniques. This scheme is capable of coupling two numerical models with possibly dissimilar spatial discretization lengths and material properties, hence it is appealing for a multi-scale and/or multi-resolution analysis. The method developed in this paper employs an interface force–displacement coupling to yield the multi-scale model equations. It is shown that the governing equations contain a self-coupling term that affects the stability of the system, as it contributes to additional stiffness at the interface. Stability analysis is presented in terms of rotations of two vectors in [Formula: see text] space, where each vector represents individual model’s stability. Three model configurations of practical interest were investigated, analytical formulae derived and used to analyze stability. These analytical formulae were compared against results from numerical simulations and perfect agreement was observed.

Review and Recent Developments on the Perfectly Matched Layer (PML) Method for the Numerical Modeling and Simulation of Elastic Wave Propagation in Unbounded Domains

This review article revisits and outlines the perfectly matched layer (PML) method and its various formulations developed over the past 25 years for the numerical modeling and simulation of wave propagation in unbounded media. Based on the concept of complex coordinate stretching, an efficient mixed displacement-strain unsplit-field PML formulation for second-order (displacementbased) linear elastodynamic equations is then proposed for simulating the propagation and absorption of elastic waves in unbounded (infinite or semi-infinite) domains. Both time-harmonic (frequency-domain) and time-dependent (timedomain) PML formulations are derived for two-and three-dimensional linear elastodynamic problems. Through the introduction of only a few additional variables governed by low-order auxiliary differential equations, the resulting mixed timedomain PML formulation is second-order in time, thereby allowing the use of standard time integration schemes commonly employed in computational structural dynamics and thus facilitating the incorporation into existing displacementbased finite element codes. For computational efficiency, the proposed timedomain PML formulation is implemented using a hybrid approach that couples a mixed (displacement-strain) formulation for the PML region with a classical (displacement-based) formulation for the physical domain of interest, using a standard Galerkin finite element method (FEM) for spatial discretization and a Newmark time scheme coupled with a finite difference (Crank-Nicolson) time scheme for time sampling. Numerical experiments show the performances of the PML method in terms of accuracy, efficiency and stability for two-dimensional linear elastodynamic problems in single-and multi-layer isotropic homogeneous elastic media.

Discontinuous finite element method for efficient three-dimensional elastic wave simulation

AbstractThe existing discontinuous Galerkin (DG) finite element method (FEM) for the numerical simulation of elastic wave propagation is primarily implemented in two dimensions. Here, a discontinuous FEM (DFEM) for efficient three-dimensional (3D) elastic wave simulation is presented. First, the velocity–stress equations of 3D elastic waves in isotropic media are transformed into first-order coefficient-changed partial differential equations. A DG discretisation method for wave field values on a unit boundary is then defined using the local Lax–Friedrichs flux format. The equations are first transformed into equivalent integral equations, and subsequently into a spatial semi-discrete ordinary differential equation system using a hierarchical orthogonal basis function. The DFEM is extended to an arbitrary high-order accuracy in the time domain using the exponential integrator technique and the explicit optimal strong-stability-preserving Runge–Kutta method. Finally, an efficient method for selecting the calculation area of the geometry of the current shot record is realised. For the computation, a multi-node parallelism with improved resource utilisation and parallelisation efficiency is implemented. The numerical results show that the proposed method can improve both the accuracy of the simulation and the efficiency of the calculation compared with existing methods.

Maximizing the accuracy of finite element simulation of elastic wave propagation in polycrystals.

Three-dimensional finite element (FE) modelling, with representation of materials at grain scale in realistic sample volumes, is capable of accurately describing elastic wave propagation and scattering within polycrystals. A broader and better future use of this FE method requires several important topics to be fully understood, and this work presents studies addressing this aim. The first topic concerns the determination of effective media parameters, namely, scattering induced attenuation and phase velocity, from measured coherent waves. This work evaluates two determination approaches, through-transmission and fitting, and it is found that these approaches are practically equivalent and can thus be used interchangeably. For the second topic of estimating modelling errors and uncertainties, this work performs thorough analytical and numerical studies to estimate those caused by both FE approximations and statistical considerations. It is demonstrated that the errors and uncertainties can be well suppressed by using a proper combination of modelling parameters. For the last topic of incorporating FE model information into theoretical models, this work presents elaborated investigations and shows that to improve agreement between the FE and theoretical models, the symmetry boundary conditions used in FE models need to be considered in the two-point correlation function, which is required by theoretical models.

Dispersion curves and identification of elastic wave modes for fiber metal laminates

The current work is devoted to the problem of the validation of dispersion curves and numerical simulations of fundamental Lamb wave and shear horizontal elastic wave modes for fiber metal laminates (FML). In the present study, two types of FM laminates are considered. In the first case, the studied composite material consists of 7 layers: one layer is made of aluminum alloy AW-6060 while the rest of the layers are made of carbon/epoxy resin. In the second case, the FML is constructed of one layer of aluminum alloy and the rest six layers are made of glass/epoxy. In order to extract the dispersion curves for the fundamental S0, SH0, and A0 modes, as well as higher modes from the range 5 up to 1000 [kHz], the stiffness matrix method, is employed. In the first case the computations are performed for the following cross-ply stacking sequences: [Al, ±5°6], [Al, ±15°6], [Al, ±25°6], … , [Al, ±85°6]. Next, the FEM simulations of elastic wave propagation are performed in the case of plates, which are made of analyzed FML composites. In the first studied FML material, the fundamental symmetric mode S0 is generated while in the second case the fundamental antisymmetric mode A0 is excited. It occurred that in the case of aluminum alloy/carbon fibers the generation of the pure fundamental symmetric mode S0 is very difficult. In addition, the fundamental symmetric mode S0, the shear horizontal SH0, and the antisymmetric A0 wave modes are also present. In the case of aluminum alloy and carbon/epoxy, the pure antisymmetric mode A0 can be relatively easily generated. However, a significant dispersion phenomenon is visible. For the aluminum alloy and glass/epoxy, the obtained results are experimentally verified.

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.

An adaptive cellular automata approach with the use of radial basis functions for the simulation of elastic wave propagation

A physics-based adaptive approach constructed on cellular automata is proposed for the analysis of two-dimensional (2D) elastodynamic initial boundary value problems. Although any 2D cell may be used in this approach, because of easy construction of arbitrary geometry, triangular plane cells are used as primary reference. The analysis of the problem includes two principal parts. In the first part, after discretization of the problem domain, a physics-based configuration considering the conservation of mass and the balance of momentum is developed as reference for cellular automata used in the next step. In the second part, radial basis functions (RBFs) are employed for an adaptive solution. For this purpose, after analysing the problem based on an arbitrary coarse mesh in the first step as reference, the problem domain may be discretized by iterative schemes, where RBFs indicate the lack and redundancy of computational points of the domain, from which some computational points may be added or eliminated for the next analysis. This process may be repeated to get the most appropriate mesh with the most accurate results. The principle of causality is taken into account in the proposed method, meaning that the unexcited parts of the domain are not in the solution domain. Similarly, those parts of the domain where the wave has already passed from would be deducted from the solution domain. The results of the proposed approach are compared with numerical and analytical results available in the literature. The comparison illustrates very good agreements between the results of the proposed approach and the results of those already reported in the literature.

Numerical simulation of elastic wave propagation in masonry compared with acoustic emission experimental results

It has been shown in acoustic emission literature that the distance between a possible cracking event and a receiver is affecting the signal parameters providing a wrong image of the real fracture characteristics. In this study, lab-scale experimental tests on masonry components are performed in order to determine the disturbance of the acoustic emission wave properties and verify the experimental observations via numerical wave propagation analysis. The investigation is extended to different geometries including “couplets,” “triplets” and masonry walls. The simulations allow to understand, verify and predict the acoustic emission signal properties alterations in many different types of masonry experiments as well as the correct characterization of the fracture mode.