An optimal method for a staggered-grid finite-difference solution of elastic wave equations including rotational deformation

To enable a mathematical description, geologic fractures are considered as infinitely thin planes embedded in a homogeneous medium. These fracture structures satisfy linear slip boundary conditions, namely, a discontinuous displacement and continuous stress. The general finite-difference (FD) method described by the elastic wave equations has challenges when attempting to simulate the propagation of waves at the fracture interface. The FD method expressed by velocity-stress variables with the explicit application of boundary conditions at the fracture interface facilitates the simulation of wave propagation in fractured discontinuous media that are described by elastic wave equations and linear slip interface conditions. We have developed a new FD scheme for horizontal and vertical fracture media. In this scheme, a fictitious grid is introduced to describe the discontinuous velocity at the fracture interface and a rotated staggered grid is used to accurately indicate the location of the fracture. The new FD scheme satisfies nonwelded contact boundary conditions, unlike traditional approaches. Numerical simulations in different fracture media indicate that our scheme is accurate. The results demonstrate that the reflection coefficient of the fractured interface varies with the incident angle, wavelet frequency, and normal and tangential fracture compliances. Our scheme and conclusions from this study will be useful in assessing the properties of fractures, enabling the proper delineation of fractured reservoirs.

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Staggered-Grid High-Order Differencemethod for the First-Order Elastic Wave Equations

Chinese Journal of Geophysics ◽

10.1002/cjg2.54 ◽

2000 ◽

Vol 43 (3) ◽

pp. 441-449 ◽

Cited By ~ 20

Author(s):

Liang-Guo DONG ◽

Zai-Tian MA ◽

Jing-Zhong CAO

Keyword(s):

Elastic Wave ◽

Wave Equations ◽

High Order ◽

Staggered Grid ◽

First Order ◽

Elastic Wave Equations

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Numerical modelling for elastic wave equations including rotational deformation and strain gradient

IOP Conference Series Earth and Environmental Science ◽

10.1088/1755-1315/660/1/012144 ◽

2021 ◽

Vol 660 (1) ◽

pp. 012144

Author(s):

Zhiyang Wang ◽

Chaopu Chen ◽

Youming Li ◽

Hong Liu ◽

Wenlei Bai

Keyword(s):

Numerical Modelling ◽

Elastic Wave ◽

Strain Gradient ◽

Wave Equations ◽

Elastic Wave Equations ◽

Rotational Deformation

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Three‐dimensional induction logging problems, Part 2: A finite‐difference solution

Geophysics ◽

10.1190/1.1468608 ◽

2002 ◽

Vol 67 (2) ◽

pp. 484-491 ◽

Cited By ~ 85

Author(s):

Gregory A. Newman ◽

David L. Alumbaugh

Keyword(s):

Finite Difference ◽

Electric Field ◽

Krylov Subspace ◽

Inverse Operator ◽

Three Dimensional ◽

Staggered Grid ◽

Linear System Of Equations ◽

Order Of Magnitude ◽

Finite Difference Solution ◽

Difference Solution

A 3‐D finite‐difference solution is implemented for simulating induction log responses in the quasi‐static limit that include the wellbore and bedding that exhibits transverse anisotropy. The finite‐difference code uses a staggered grid to approximate a vector equation for the electric field. The resulting linear system of equations is solved to a predetermined error level using iterative Krylov subspace methods. To accelerate the solution at low induction numbers (LINs), a new preconditioner is developed. This new preconditioner splits the electric field into curl‐free and divergence‐free projections, which allows for the construction of an approximate inverse operator. Test examples show up to an order of magnitude increase in speed compared to a simple Jacobi preconditioner. Comparisons with analytical and mode matching solutions demonstrate the accuracy of the algorithm.

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Pseudo-analytical finite-difference elastic-wave extrapolation based on the k-space method

Geophysics ◽

10.1190/geo2017-0088.1 ◽

2018 ◽

Vol 83 (1) ◽

pp. T1-T14 ◽

Cited By ~ 2

Author(s):

Xufei Gong ◽

Qizhen Du ◽

Qiang Zhao ◽

Chengfeng Guo ◽

Pengyuan Sun ◽

...

Keyword(s):

Finite Difference ◽

Elastic Wave ◽

Wave Equations ◽

Computational Cost ◽

Superior Performance ◽

First Order ◽

Time Migration ◽

Elastic Wave Equations ◽

Temporal Dispersion ◽

Space Method

Cost-effective elastic-wave modeling is the key to practical elastic reverse time migration and full-waveform inversion implementations. We have developed an efficient elastic pseudo-analytical finite-difference (PAFD) scheme for elastic-wave extrapolation. The elastic PAFD scheme is based on a modified pseudo-spectral method, k-space method, in which a pseudo-analytical operator is used to ensure the high accuracy of elastic-wave extrapolation. However, the k-space method is motivated for a pure wave mode, and thus its application in coupled first-order elastic-wave equations may cause the elastic pseudo-analytical operators to suffer from crosstalk between the P- and S-wavefields. The approaches presented attempt to overcome these shortcomings by introducing two improvements to achieve the goal. This is done, first, by performing a predictor-corrector strategy in first-order elastic-wave equations to eliminate those errors during wave extrapolation. Considering the massive computational cost in the spectral domain, we have developed an efficient elastic PAFD implementation, in which an innovative model-adaptive finite-difference coefficient-predicted scheme is provided to reduce the computational cost of elastic pseudo-analytical operator differencing. Dispersion analysis demonstrates the flexibility with varying velocity and superior performance of our PAFD scheme for spatial and temporal dispersion suppression than the existing Taylor-expansion-based scheme. Under the same simulation parameters, several numerical examples prove that the elastic PAFD scheme can provide more accurate simulation results, whereas the conventional scheme suffers from spatial or temporal dispersion errors, even in complex heterogeneous media.

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Stability condition of finite difference solution for viscoelastic wave equations

Earthquake Science ◽

10.1007/s11589-009-0479-2 ◽

2009 ◽

Vol 22 (5) ◽

pp. 479-485 ◽

Cited By ~ 3

Author(s):

Chengyu Sun ◽

Yunfei Xiao ◽

Xingyao Yin ◽

Hongchao Peng

Keyword(s):

Finite Difference ◽

Stability Condition ◽

Wave Equations ◽

Viscoelastic Wave ◽

Finite Difference Solution ◽

Difference Solution

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Accuracy of finite‐difference and finite‐element modeling of the scalar and elastic wave equations

Geophysics ◽

10.1190/1.1441689 ◽

1984 ◽

Vol 49 (5) ◽

pp. 533-549 ◽

Cited By ~ 570

Author(s):

Kurt J. Marfurt

Keyword(s):

Finite Element ◽

Finite Difference ◽

Elastic Wave ◽

Frequency Domain ◽

Poisson’S Ratio ◽

Wave Equations ◽

Cost Effective ◽

Poisson's Ratio ◽

Solution Technique ◽

Elastic Wave Equations

Numerical solutions of the scalar and elastic wave equations have greatly aided geophysicists in both forward modeling and migration of seismic wave fields in complicated geologic media, and they promise to be invaluable in solving the full inverse problem. This paper quantitatively compares finite‐difference and finite‐element solutions of the scalar and elastic hyperbolic wave equations for the most popular implicit and explicit time‐domain and frequency‐domain techniques. In addition to versatility and ease of implementation, it is imperative that one choose the most cost effective solution technique for a fixed degree of accuracy. To be of value, a solution technique must be able to minimize (1) numerical attenuation or amplification, (2) polarization errors, (3) numerical anisotropy, (4) errors in phase and group velocities, (5) extraneous numerical (parasitic) modes, (6) numerical diffraction and scattering, and (7) errors and transmission coefficients. This paper shows that in homogeneous media the explicit finite‐element and finite‐difference schemes are comparable when solving the scalar wave equation and when solving the elastic wave equations with Poisson’s ratio less than 0.3. Finite‐elements are superior to finite‐differences when modeling elastic media with Poisson’s ratio between 0.3 and 0.45. For both the scalar and elastic equations, the more costly implicit time integration schemes such as the Newmark scheme are inferior to the explicit central‐differences scheme, since time steps surpassing the Courant condition yield stable but highly inaccurate results. Frequency‐domain finite‐element solutions employing a weighted average of consistent and lumped masses yield the most accurate results, and they promise to be the most cost‐effective method for CDP, well log, and interactive modeling.

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