Two efficient modeling schemes for fractional Laplacian viscoacoustic wave equation

Geophysics ◽  
2016 ◽  
Vol 81 (5) ◽  
pp. T233-T249 ◽  
Author(s):  
Hanming Chen ◽  
Hui Zhou ◽  
Qingqing Li ◽  
Yufeng Wang
Keyword(s):  
Wave Propagation ◽  
Wave Equation ◽  
Computational Efficiency ◽  
Fractional Laplacian ◽  
Heterogeneous Media ◽  
Spatial Variable ◽  
Variable Order ◽  
Wide Range ◽  
Seismic Quality Factor ◽  
Efficient Modeling

Recently, a decoupled fractional Laplacian viscoacoustic wave equation has been developed based on the constant-[Formula: see text] model to describe wave propagation in heterogeneous media. We have developed two efficient modeling schemes to solve the decoupled fractional Laplacian viscoacoustic wave equation. Both schemes can cope with spatial variable-order fractional Laplacians conveniently, and thus are applicable for modeling viscoacoustic wave propagation in heterogeneous media. Both schemes are based on fast Fourier transform, and have a spectral accuracy in space. The first scheme solves a modified wave equation with constant-order fractional Laplacians instead of spatial variable-order fractional Laplacians. Due to separate discretization of space and time, the first scheme has only first-order accuracy in time. Differently, the second scheme is based on an analytical wave propagator, and has a higher accuracy in time. To increase computational efficiency of the second modeling scheme, we have adopted the low-rank decomposition in heterogeneous media. We also evaluated the feasibility of applying an empirical approximation to approximate the fractional Laplacian that controls amplitude loss during wave propagation. When the empirical approximation is applied, our two modeling schemes become more efficient. With the help of numerical examples, we have verified the accuracy of our two modeling schemes with and without applying the empirical approximation, for a wide range of seismic quality factor ([Formula: see text]). We also compared computational efficiency of our two modeling schemes using numerical tests.

Modeling viscoacoustic wave propagation using a new spatial variable-order fractional Laplacian wave equation

Geophysics ◽  
2021 ◽  
pp. 1-74
Author(s):  
Xinru Mu ◽  
Jianping Huang ◽  
Lei Wen ◽  
Subin Zhuang
Keyword(s):  
Wave Propagation ◽  
Wave Equation ◽  
Fractional Order ◽  
Fractional Laplacian ◽  
Spatial Variable ◽  
Variable Order ◽  
New Wave ◽  
Dissipation Term ◽  
Laplacian Operators ◽  
Dispersion Term

We propose a new time-domain viscoacoustic wave equation for simulating wave propagation in anelastic media. The new wave equation is derived by inserting the complex-valued phase velocity described by the Kjartansson attenuation model into the frequency-wavenumber domain acoustic wave equation. Our wave equation includes one second-order temporal derivative and two spatial variable-order fractional Laplacian operators. The two fractional Laplacian operators describe the phase dispersion and amplitude attenuation effects, respectively. To facilitate the numerical solution for the proposed wave equation, we use the arbitrary-order Taylor series expansion (TSE) to approximate the mixed domain fractional Laplacians and achieve the decoupling of the wavenumber and the fractional order. Then the proposed viscoacoustic wave equation can be directly solved using the pseudospectral method (PSM). We adopt a hybrid pseudospectral/finite-difference method (HPSFDM) to stably simulate wave propagation in arbitrarily complex media. We validate the high accuracy of the proposed approximate dispersion term and approximate dissipation term in comparison with the accurate dispersion term and accurate dissipation term. The accuracy of numerical solutions is evaluated by comparison with the analytical solutions in homogeneous media. Theory analysis and simulation results show that our viscoacoustic wave equation has higher precision than the traditional fractional viscoacoustic wave equation in describing constant- Q attenuation. For a model with Q < 10, the calculation cost for solving the new wave equation with TSE HPSFDM is lower than that for solving the traditional fractional-order wave equation with TSE HPSFDM under the high numerical simulation precision. Furthermore, we demonstrate the accuracy of HPSFDM in heterogeneous media by several numerical examples.

WAVE PROPAGATION MODELING IN HIGHLY HETEROGENEOUS MEDIA BY A POLY-GRID CHEBYSHEV SPECTRAL ELEMENT METHOD

2012 ◽  
Vol 20 (02) ◽  
pp. 1240004 ◽  
Author(s):  
GÉZA SERIANI ◽  
CHANG SU
Keyword(s):  
Wave Propagation ◽  
Computational Efficiency ◽  
Spectral Element Method ◽  
Heterogeneous Media ◽  
Spectral Element ◽  
Small Scale ◽  
Computational Domain ◽  
Problem Size ◽  
Wide Range ◽  
Element Method

A wide range of applications requires the modeling of wave propagation phenomena in media with variable physical properties in the domain of interest, while highly accurate algorithms are needed to avoid unphysical effects. Spectral element methods (SEM), based on either a Chebyshev or a Legendre polynomial basis, have excellent properties of accuracy and flexibility in describing complex models, outperforming other techniques. In the standard SEM approach the computational domain is discretized by using very coarse meshes and constant-property elements, but in some cases the accuracy and the computational efficiency may be seriously reduced. For instance, a finely heterogeneous medium requires grid resolution down to the finest scales, leading to an extremely large problem dimension. In such problems the wavelength scale of interest is much larger but cannot be exploited in order to reduce the problem size. A poly-grid Chebyshev spectral element method (PG-CSEM) can overcome this limitation. In order to accurately deal with continuous variation in the properties, or even with small scale fluctuations, temporary auxiliary grids are introduced which avoid the need of using any finer global grid, and at the macroscopic level the wave field propagation is solved maintaining the SEM accuracy and computational efficiency.

On the construction and efficiency of staggered numerical differentiators for the wave equation

Geophysics ◽  
1990 ◽  
Vol 55 (1) ◽  
pp. 107-110 ◽  
Author(s):  
M. Kindelan ◽  
A. Kamel ◽  
P. Sguazzero
Keyword(s):  
Numerical Modeling ◽  
Wave Propagation ◽  
Wave Equation ◽  
Finite Difference ◽  
Computational Efficiency ◽  
Differential Operators ◽  
High Order

Finite‐difference (FD) techniques have established themselves as viable tools for the numerical modeling of wave propagation. The accuracy and the computational efficiency of numerical modeling can be enhanced by using high‐order spatial differential operators (Dablain,1986).

scholarly journals Simulation of elastic wave propagation across fractures using a nodal discontinuous Galerkin method—theory, implementation and validation

2019 ◽  
Vol 219 (3) ◽  
pp. 1900-1914 ◽  
Author(s):  
T Möller ◽  
W Friederich
Keyword(s):  
Wave Propagation ◽  
Wave Equation ◽  
Elastic Wave ◽  
Discontinuous Galerkin ◽  
Heterogeneous Media ◽  
Rheological Behaviour ◽  
Displacement Discontinuity ◽  
Signal Frequency ◽  
Elastic Wave Equation ◽  
Numerical Flux

SUMMARY An existing nodal discontinuous Galerkin (NDG) method for the simulation of seismic waves in heterogeneous media is extended to media containing fractures with various rheological behaviour. Fractures are treated as 2-D surfaces where Schoenberg’s linear slip or displacement discontinuity condition is applied as an additional boundary condition to the elastic wave equation which is in turn implemented as an additional numerical flux within the NDG formulation. Explicit expressions for the new numerical flux are derived by considering the Riemann problem for the elastic wave equation at fractures with varying rheologies. In all cases, we obtain further first order differential equations that fully describe the temporal evolution of the particle velocity jump at the fracture. Our flux formulation allows to separate the effect of a fracture from flux contributions due to simple welded interfaces enabling us to easily declare element faces as parts of a fracture. We make use of this fact by first generating the numerical mesh and then building up fractures by selecting appropriate element faces instead of adjusting the mesh to pre-defined fracture surfaces. The implementation of the new numerical fluxes into NDG is verified in 1-D by comparison to an analytical solution and in 2-D by comparing the results of a simulation valid in 1-D and 2-D. Further numerical examples address the effect of fracture systems on seismic wave propagation in 1-D and 2-D featuring effective anisotropy and coda generation. Finally, a study of the reflective and transmissive behaviour of fractures indicates that reflection and transmission coefficients are controlled by the ratio of signal frequency and relaxation frequency of the fracture.

Multiscale modeling of acoustic wave propagation in 2D media

Geophysics ◽  
2014 ◽  
Vol 79 (2) ◽  
pp. T61-T75 ◽  
Author(s):  
Richard L. Gibson ◽  
Kai Gao ◽  
Eric Chung ◽  
Yalchin Efendiev
Keyword(s):  
Wave Propagation ◽  
Wave Equation ◽  
Acoustic Wave ◽  
Heterogeneous Media ◽  
Coarse Grid ◽  
Basis Functions ◽  
Seismic Survey ◽  
Fine Scale ◽  
Acoustic Wave Propagation ◽  
Fine Grid

Conventional finite-difference methods produce accurate solutions to the acoustic and elastic wave equation for many applications, but they face significant challenges when material properties vary significantly over distances less than the grid size. This challenge is likely to occur in reservoir characterization studies, because important reservoir heterogeneity can be present on scales of several meters to ten meters. Here, we describe a new multiscale finite-element method for simulating acoustic wave propagation in heterogeneous media that addresses this problem by coupling fine- and coarse-scale grids. The wave equation is solved on a coarse grid, but it uses basis functions that are generated from the fine grid and allow the representation of the fine-scale variation of the wavefield on the coarser grid. Time stepping also takes place on the coarse grid, providing further speed gains. Another important property of the method is that the basis functions are only computed once, and time savings are even greater when simulations are repeated for many source locations. We first present validation results for simple test models to demonstrate and quantify potential sources of error. These tests show that the fine-scale solution can be accurately approximated when the coarse grid applies a discretization up to four times larger than the original fine model. We then apply the multiscale algorithm to simulate a complete 2D seismic survey for a model with strong, fine-scale scatterers and apply standard migration algorithms to the resulting synthetic seismograms. The results again show small errors. Comparisons to a model that is upscaled by averaging densities on the fine grid show that the multiscale results are more accurate.

Interface conditions for acoustic and elastic wave propagation

Geophysics ◽  
1991 ◽  
Vol 56 (2) ◽  
pp. 168-181 ◽  
Author(s):  
J. S. Sochacki ◽  
J. H. George ◽  
R. E. Ewing ◽  
S. B. Smithson
Keyword(s):  
Wave Propagation ◽  
Wave Equation ◽  
Elastic Wave ◽  
Heterogeneous Media ◽  
Elastic Wave Propagation ◽  
Integration Scheme ◽  
Acoustic Wave Equation ◽  
Sh Wave ◽  
Numerical Schemes ◽  
Tangential Stresses

The divergence theorem is used to handle the physics required at interfaces for acoustic and elastic wave propagation in heterogeneous media. The physics required at regular and irregular interfaces is incorporated into numerical schemes by integrating across the interface. The technique, which can be used with many numerical schemes, is applied to finite differences. A derivation of the acoustic wave equation, which is readily handled by this integration scheme, is outlined. Since this form of the equation is equivalent to the scalar SH wave equation, the scheme can be applied to this equation also. Each component of the elastic P‐SV equation is presented in divergence form to apply this integration scheme, naturally incorporating the continuity of the normal and tangential stresses required at regular and irregular interfaces.

Wave propagation in heterogeneous media: Effects of fine-scale heterogeneity

Geophysics ◽  
2008 ◽  
Vol 73 (3) ◽  
pp. T37-T49 ◽  
Author(s):  
Florence Delprat-Jannaud ◽  
Patrick Lailly
Keyword(s):  
Wave Propagation ◽  
Wave Equation ◽  
Multiple Scattering ◽  
Stability Condition ◽  
Heterogeneous Media ◽  
Fine Scale ◽  
Fine Grid ◽  
Scale Heterogeneity ◽  
Scattering Effects ◽  
Multiple Scattering Effects

We study the multiple scattering effects caused by fine-scale heterogeneity. For this purpose, it is unreasonable to rely on a linearization of the dependency of the wavefield in the parameters that describe the medium. Therefore, the only tools that correctly model wave propagation are based on the numerical solution of the (two-way) wave equation by finite differences or finite elements. A fine grid provides a straightforward approach to account for fine-scale heterogeneity. In this situation, there is no need for high-order schemes. Variational methods allow us to exhibit a numerical scheme that accounts for heterogeneity and that is, by construction, stable, provided a stability condition is fulfilled. This condition is a sufficient-stability condition contrary to the classical necessary-stability conditions. In addition, general mathematical results prove the finite-difference solution is close to the solution of the wave equation when the grid is fine enough. The multiple scattering effects caused by fine-scale heterogeneity are very important. In particular, we observe that imaging the so-computed synthetic data by standard migration techniques (that assume a linearization of the above-mentioned dependency) shows a strongly noise-corrupted image. This illustrates the importance of preprocessing data to remove the effect of multiple scattering. We try to improve the signal-to-noise ratio by removing multiples related to the free surface. Although significant noise reduction is achieved, even more sophisticated preprocessing is required to obtain a clear image of the subsurface.

Efficient modeling of wave propagation in a vertical transversely isotropic attenuative medium based on fractional Laplacian

Geophysics ◽  
2019 ◽  
Vol 84 (3) ◽  
pp. T121-T131 ◽  
Author(s):  
Tieyuan Zhu ◽  
Tong Bai
Keyword(s):  
Wave Propagation ◽  
Wave Equation ◽  
Fractional Laplacian ◽  
Numerical Solutions ◽  
Pseudospectral Method ◽  
Transversely Isotropic ◽  
Seismic Attenuation ◽  
Fourier Pseudospectral Method ◽  
Frequency Independent ◽  
Theoretical Predictions

To efficiently simulate wave propagation in a vertical transversely isotropic (VTI) attenuative medium, we have developed a viscoelastic VTI wave equation based on fractional Laplacian operators under the assumption of weak attenuation ([Formula: see text]), where the frequency-independent [Formula: see text] model is used to mathematically represent seismic attenuation. These operators that are nonlocal in space can be efficiently computed using the Fourier pseudospectral method. We evaluated the accuracy of numerical solutions in a homogeneous transversely isotropic medium by comparing with theoretical predictions and numerical solutions by an existing viscoelastic-anisotropic wave equation based on fractional time derivatives. To accurately handle heterogeneous [Formula: see text], we select several [Formula: see text] values to compute their corresponding fractional Laplacians in the wavenumber domain and interpolate other fractional Laplacians in space. We determined its feasibility by modeling wave propagation in a 2D heterogeneous attenuative VTI medium. We concluded that the new wave equation is able to improve the efficiency of wave simulation in viscoelastic-VTI media by several orders and still maintain accuracy.

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

2020 ◽  
Vol 223 (3) ◽  
pp. 1794-1804
Author(s):  
R Ponomarenko ◽  
D Sabitov ◽  
M Charara
Keyword(s):  
Wave Propagation ◽  
Heterogeneous Media ◽  
Co2 Storage ◽  
Carbon Capture And Storage ◽  
Spectral Element ◽  
Slip Model ◽  
Theoretical Formulation ◽  
Small Aperture ◽  
Wide Range ◽  
Linear Slip Model

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.

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