Velocity-stress Finite-difference Research Articles

The velocity-stress finite-difference method with a rotated staggered grid applied to seismic wave propagation in a fractured medium

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.

Wave propagation in pyramidal tip-like structures with cubic material properties

The three-dimensional wave propagation in cubic materials like silicon AFM tips is investigated numerically. The used VS-FDTD (Velocity Stress Finite-Difference Time-Domain) code is explained starting with the fundamental equations for the wave propagation in cubic materials. The discretisation, the boundary conditions, and the stability conditions are considered. As a validation of the numerical code, first the simulated wave velocities are compared with analytically calculated results for different polarizations and propagating directions. Second the code is validated by monitoring the internal energy of the simulated structure. Third, the results of the numerical code for long wavelengths agree well with a simplified analytical model for wave propagation in horn geometries. With the presented code pyramidal focusing tips are investigated regarding their focusing and detection properties and regarding the influence of the crystallographic orientation. It turns out that the focusing effect for the primary waves is strong in tips with lateral dimensions at the sharp end which are smaller than the dominant wave length. In contrast, the detection properties of a tip are better if the lateral dimensions at the sharp end of the tip are larger than the dominant wave length.

Detection of defects in cylindrical structures using a time reverse method and a finite-difference approach

The detection and characterization of defects in structures is an important issue in non-destructive testing. To avoid the scanning of large samples, guided elastic waves, which propagate along the structure, are excited. These waves interact with a defect, which results in a scattered wave field. In an experiment, the displacements of these scattered waves are recorded over time for a fixed axial coordinate at a number of circumferential positions of a circular cylindrical tube. Since in complex structures it is difficult to determine the axial and particularly circumferential position of the defect directly from the time signals, a time reversed numerical simulation is performed. There the measured displacement histories are reversed in time and used as displacement excitations in a simulation of the tested structure. A three-dimensional code in cylindrical coordinates, based on a velocity-stress finite-difference method, is used to simulate the wave propagation. As long as the geometric and material parameters are chosen equivalent to the performed experiment, the scattered waves travel back through the simulated structure and interfere, even if no defect is present in the numerical model. The result is an increase of the amplitudes of the stress and displacement components at the location where the defect was in the tested sample.

Accuracy of the Explicit Planar Free-Surface Boundary Condition Implemented in a Fourth-Order Staggered-Grid Velocity-Stress Finite-Difference Scheme

We compute the accuracy of two implementations of the explicit planar free-surface boundary condition for 3D fourth-order velocity-stress staggered-grid finite differences, 1/2 grid apart vertically, in a uniform half-space. Due to the stag- gered grid, the closest distance between the free surface and some wave-field com- ponents for both implementations is 1/2-grid spacing. Overall, the differences in accuracy of the two implementations are small. When compared to a reflectivity solution computed at the staggered positions closest to the surface, the total misfit for all three components of the wave field is generally found to be larger for the free surface colocated with the normal stresses, compared to that for the free surface colocated with the xz and yz stresses. However, this trend is reversed when compared to the reflectivity solution exactly at the free surface (the misfit encountered in staggered-grid modeling). When the wave field is averaged across the free surface, thereby centering thestaggered wave field exactly on the freesurface,thefree-surface condition colocated with the xz and yz stresses generates the smallest total misfit for increasing epicentral distance. For an epicentral distance/hypocentral depth of 10, the total misfit of this condition is about 15% smaller than that for the condition colocated with the normal stresses, mainly controlled by the misfit on the Rayleigh wave.

Stability and Grid Dispersion of the P-SV 4 th -Order Staggered-Grid Finite-Difference Schemes

Stability and grid dispersion in the P-SV 4th-order in space, 2nd-order in time, displacement-stress staggered-grid finite-difference scheme is investigated in the case of a homogeneous unbounded medium. All results, however, also apply to the velocity-stress and displacement- velocity-stress finite-difference schemes. Independent stability conditions for the P and S waves are obtained by exact separation of equations for the two types of waves. Since the S-wave group velocity can differ from the actual velocity as much as 5% for the sampling ratio 1/5, commonly used in numerical modelling, the sampling of the minimum S wavelength by 6 grid spacings (with the velocity difference not larger than 2.5%) is recommended. Grid dispersion is strongest for a wave propagating in a direction of a coordinate axis and weakest for a wave propagating along a plane diagonal. Grid dispersion in the 4 th -order scheme for the sampling ratios s = 1/5 and s = 1/6 is smaller than grid dispersion in the 2 nd -order scheme for s = 1/10 and s = 1/12, respectively.

Quadrangle-grid velocity-stress finite-difference method for elastic-wave-propagation simulation

SUMMARY I present a 2-D numerical-modelling algorithm based on a first-order velocity-stress hyperbolic system and a non-rectangular-grid finite-difference operator. In this method the velocity and stress are defined at different nodes for a staggered grid. The scheme uses non-orthogonal grids, thereby surface topography and curved interfaces can be easily modelled in the seismic-wave-propagation stimulation. The free-surface conditions of complex geometry are achieved by using integral equilibrium equations on the surface, and the stability of the free-surface conditions is improved by introducing local filter modification. The method incorporates desirable qualities of the finite-element method and the staggered-grid finite-difference scheme, which is of high accuracy and low computational cost.

A 2.5-D time-domain elastodynamic equation for a general anisotropic medium

SUMMARY In order to provide a quantitative analysis of real seismic records from complex regions we need to be able to calculate the wavefields in three dimensions. However, full 3-D modelling of seismic-wave propagation is still computationally intensive. An economical approach to the modelling of seismic-wave propagation which includes many important aspects of the propagation process is to examine the 3-D response of a model where the material parameters vary in two dimensions. Such a configuration, in which a 3-D wavefield is calculated for a 2-D medium, is called the ‘2.5-D problem’. Recently, Takenaka & Kennett ( 1996) proposed a 2.5-D time-domain elastodynamic equation for seismic wavefields in models with a 2-D variation in structure but obliquely incident plane waves in the absence of source. This approach is useful even for non-plane waves. In the presence of source a new 2.5-D elastodynamic equation for general anisotropic media can be derived in the time domain based on the Radon transform over slowness in the direction with constant medium properties. The approach can also be formulated in terms of velocity-stress, a representation which is well suited to the use of numerical techniques for 2-D time-domain problems such as velocity-stress finite-difference or velocity-stress pseudospectral techniques.

Modeling of a piezoelectric transducer and its application to full waveform acoustic logging

A velocity-stress finite-difference (VS-FD) model is presented for a cylindrical piezoelectric transducer in a borehole. The borehole may be irregular and the surrounding formation inhomogeneous. The model is two dimensional in that azimuthal symmetry is assumed. The description of the tranducer is a full elasto-electromagnetic one, including transverse isotropy in the elastic, dielectric, and piezoelectric parameters, and dissipation in the piezoelectric material. The borehole propagation portion of the model is verified by comparison with a standard transform technique. Predictions of the model for a piezoelectric cylinder radiating into a fluid medium are compared to experimental results with excellent agreement. The radiation patterns of a bare transducer near resonance frequencies are found to be anisotropic. Acoustic waveforms in a borehole excited by a finite sized cylindrical transducer are displayed and are quite different from those excited by an ideal point pressure source. The effect of borehole loading upon the impedance of the transducer is shown to be small.

Multipole sources in boreholes penetrating anisotropic formations: Numerical and experimental results

A 2.5-D velocity-stress finite-difference code is described that models acoustic propagation in a borehole penetrating a generally anisotropic formation. The excitation may be a dilatation (monopole), or a point force (dipole) in an arbitrary direction. The anisotropic formation is homogeneous along the axis of the borehole but may be inhomogeneous in the transverse plane. The borehole cross section and location of the source in the borehole are arbitrary. Synthetic time-domain waveforms are displayed for arrays of monopole and dipole receivers deployed along the borehole axis in both fast and slow anisotropic formations. The specific anisotropy model employed for the numerical results is a transversely isotropic (TI) formation with its axis of symmetry inclined with respect to the borehole axis by an arbitrary angle. Flexural/shear and Stoneley wave slowness and attenuation estimates are extracted from the synthetic waveforms using a variant of Prony’s method for a range of borehole inclinations relative to the formation axis of symmetry. In a fast formation, with borehole and formation symmetry axes perpendicular, flexural mode dispersion curves for quasi-SV and SH polarizations separate only at low frequency where moderate attenuation is also observed. In a slow formation, distinct dispersion curves are obtained for quasi-SV and SH polarizations over the entire frequency range. Moderate attenuation is again observed at low frequency. Scaled laboratory experiments confirm the numerical procedure. Experimental and numerical waveforms for monopole and several polarizations of dipole excitation in a transversely isotropic model formation overlay with excellent agreement.

Multipole acoustic waveforms in nonaxisymmetric boreholes and formations

Using a 2.5-dimensional velocity-stress finite-difference (vs-fd) technique, waveforms are calculated for multipole excitation of azimuthally nonsymmetric boreholes and formations. Axial uniformity of the borehole cross section and formation is assumed. The vs-fd waveforms agree with previous transform methods for circular boreholes, and the modes found for elliptical boreholes agree with those derived by a boundary integral technique. Borehole ellipticity substantially reduces monopole compressional headwave amplitudes in fast formations. Low-frequency dipole waveforms in elliptical boreholes show marked sensitivity to source dipole orientation in fast formations, but small sensitivity in slow formations. Waveforms from dipole receivers perpendicular to the source dipole may be comparable in amplitude to those recorded by parallel dipole receivers. Dipole waveforms from an exaggerated ‘‘keyhole’’-shaped borehole are quite similar to those from an elliptical borehole of similar aspect ratio in a slow formation. Finally, the behavior of early compressional headwaves arising from a nearby faster bed parallel to the borehole axis is qualitatively similar to that for a point source above a simple planar interface.