Modeling of elastic waves in fractured media using the rotated staggered finite‐difference grid

<p>The polymetallic, hydrothermal deposit of the Freiberg mining district in the southeastern part of Germany is characterised by ore veins that are framed by Proterozoic orthogneiss. The ore veins consist mainly of quarz, sulfides, carbonates, barite and flourite, which are associated with silver, lead and tin. Today the Freiberg University of Mining and Technology is operating the shafts Reiche Zeche and Alte Elisabeth for research and teaching purposes with altogether 14 km of accessible underground galleries. The mine together with the most prominent geological structures of the central mining district are included in a 3D digital model, which is used in this study to study seismic acquisition geometries that can help to image the shallow as well as the deeper parts of the ore-bearing veins. These veins with dip angles between 40&#176; and 85&#176; are represented by triangulated surfaces in the digital geological model. In order to import these surfaces into our seismic finite-difference simulation code, they have to be converted into bodies with a certain thickness and specific elastic properties in a first step. In a second step, these bodies with their properties have to be discretized on a hexahedral finite-difference grid with dimensions of 1000 m by 1000 m in the horizontal direction and 500 m in the vertical direction. Sources and receiver lines are placed on the surface along roads near the mine. A Ricker wavelet with a central frequency of 50 Hz is used as the source signature at all excitation points. Beside the surface receivers, additional receivers are situated in accessible galleries of the mine at three different depth levels of 100 m, 150 m and 220 m below the surface. Since previous mining activities followed primarily the ore veins, there are only few pilot-headings that cut through longer gneiss sections. Only these positions surrounded by gneiss are suitable for imaging the ore veins. Based on this geometry, a synthetic seismic data set is generated with our explicit finite-difference time-stepping scheme, which solves the acoustic wave equation with second order accurate finite-difference operators in space and time. The scheme is parallelised using a decomposition of the spatial finite-difference grid into subdomains and Message Passing Interface for the exchange of the wavefields between neighbouring subdomains. The resulting synthetic seismic shot gathers are used as input for Kirchhoff prestack depth migration as well as Fresnel volume migration in order to image the ore veins. Only a top mute to remove the direct waves and a time-dependent gain to correct the amplitude decay due to the geometrical spreading are applied to the data before the migration. The combination of surface and in-mine acquisition helps to improve the image of the deeper parts of the dipping ore veins. Considering the limitations for placing receivers in the mine, Fresnel volume migration as a focusing version of Kirchhoff prestack depth migration helps to avoid migration artefacts caused by this sparse and limited acquisition geometry.</p>

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Staggered Grid with Collocated Grid Points of Velocities for Modeling Propagation of Elastic Waves in Anisotropic Solids by Finite-Difference Time Domain Method

Japanese Journal of Applied Physics ◽

10.1143/jjap.51.07gb04 ◽

2012 ◽

Vol 51 ◽

pp. 07GB04 ◽

Cited By ~ 7

Author(s):

Koji Hasegawa ◽

Takao Shimada

Keyword(s):

Finite Difference ◽

Time Domain ◽

Elastic Waves ◽

Finite Difference Time Domain ◽

Time Domain Method ◽

Staggered Grid ◽

Collocated Grid ◽

Grid Points ◽

Propagation Of Elastic Waves ◽

Difference Time

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Finite‐difference modeling of faults and fractures

Geophysics ◽

10.1190/1.1443884 ◽

1995 ◽

Vol 60 (5) ◽

pp. 1514-1526 ◽

Cited By ~ 171

Author(s):

Richard T. Coates ◽

Michael Schoenberg

Keyword(s):

Finite Difference ◽

Seismic Wave ◽

Computation Time ◽

Anisotropic Materials ◽

Model Material ◽

Material Behavior ◽

Interface Condition ◽

Medium Theory ◽

Difference Grid ◽

Receiver Array

For the purposes of seismic propagation, a slip fault may be regarded as a surface across which the displacement caused by a seismic wave is discontinuous while the stress traction remains continuous. The simplest assumption is that this slip and the stress traction are linearly related. Such a linear slip interface condition is easily modeled when the fault is parallel to the finite‐difference grid, but is more difficult to do for arbitrary nonplanar fault surfaces. To handle such situations we introduce equivalent medium theory to model material behavior in the cells of the finite‐ difference grid intersected by the fault. Virtually identical results were obtained from modeling the fault by (1) an explicit slip interface condition (fault parallel to the grid) and (2) using the equivalent medium theory when the finite‐difference grid was rotated relative to the fault and receiver array. No additional computation time is needed except for the preprocessing required to find the relevant cells and their associated moduli. The formulation is sufficiently general to include faults in and between arbitrary anisotropic materials with slip properties that vary as a function of position.

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