Finite‐difference modeling of faults and fractures

Geophysics ◽  
1995 ◽  
Vol 60 (5) ◽  
pp. 1514-1526 ◽  
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.

Semi-inverse method for predicting stress–strain relationship from cone indentations

2003 ◽  
Vol 18 (9) ◽  
pp. 2068-2078 ◽  
Author(s):  
A. DiCarlo ◽  
H. T. Y. Yang ◽  
S. Chandrasekar
Keyword(s):  
A Priori ◽  
Model Material ◽  
Material Behavior ◽  
The Finite Element Method ◽  
Stress Strain ◽  
Indentation Tests ◽  
Strain Relationship ◽  
Initial Force ◽  
Relationship Of ◽  
Force Displacement

A method for determining the stress–strain relationship of a material from hardness values H obtained from cone indentation tests with various apical angles is presented. The materials studied were assumed to exhibit power-law hardening. As a result, the properties of importance are the Young's modulus E, yield strength Y, and the work-hardening exponent n. Previous work [W.C. Oliver and G.M. Pharr, J. Mater. Res. 7, 1564 (1992)] showed that E can be determined from initial force–displacement data collected while unloading the indenter from the material. Consequently, the properties that need to be determined are Y and n. Dimensional analysis was used to generalize H/E so that it was a function of Y/E and n [Y-T. Cheng and C-M. Cheng, J. Appl. Phys. 84, 1284 (1999); Philos. Mag. Lett. 77, 39 (1998)]. A parametric study of Y/E and n was conducted using the finite element method to model material behavior. Regression analysis was used to correlate the H/E findings from the simulations to Y/E and n. With the a priori knowledge of E, this correlation was used to estimate Y and n.

Simulation of a seismic survey with combined surface and in-mine data acquisition for imaging steeply dipping ore veins

2021 ◽  
Author(s):  
Olaf Hellwig ◽  
Stefan Buske
Keyword(s):  
Finite Difference ◽  
Message Passing Interface ◽  
Seismic Survey ◽  
Difference Operators ◽  
Mining District ◽  
Prestack Depth Migration ◽  
Data Set ◽  
Depth Migration ◽  
Difference Grid ◽  
Triangulated Surfaces

<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° and 85° 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>

Mimetic seismic wave modeling including topography on deformed staggered grids

Geophysics ◽  
2014 ◽  
Vol 79 (3) ◽  
pp. T125-T141 ◽  
Author(s):  
Josep de la Puente ◽  
Miguel Ferrer ◽  
Mauricio Hanzich ◽  
José E. Castillo ◽  
José M. Cela
Keyword(s):  
Finite Difference ◽  
Seismic Wave ◽  
Seismic Waves ◽  
Numerical Solutions ◽  
Surface Condition ◽  
Finite Difference Methods ◽  
Staggered Grid ◽  
Grid Deformation ◽  
Large Dispersion ◽  
Difference Methods

Finite-difference methods for modeling seismic waves are known to be inaccurate when including a realistic topography, due to the large dispersion errors that appear in the modelled surface waves and the scattering introduced by the staircase approximation to the topography. As a consequence, alternatives to finite-difference methods have been proposed to circumvent these issues. We present a new numerical scheme for 3D elastic wave propagation in the presence of strong topography. This finite-difference scheme is based upon a staggered grid of the Lebedev type, or fully staggered grid (FSG). It uses a grid deformation strategy to make a regular Cartesian grid conform to a topographic surface. In addition, the scheme uses a mimetic approach to accurately solve the free-surface condition and hence allows for a less restrictive grid spacing criterion in the computations. The scheme can use high-order operators for the spatial derivatives and obtain low-dispersion results with as few as six points per minimum wavelength. A series of tests in 2D and 3D scenarios, in which our results are compared to analytical and numerical solutions obtained with other numerical approaches, validate the accuracy of our scheme. The resulting FSG mimetic scheme allows for accurate and efficient seismic wave modelling in the presence of very rough topographies with the advantage of using a structured staggered grid.

scholarly journals Unsplit complex frequency-shifted PML implementation using auxiliary differential equations for seismic wave modeling

Geophysics ◽  
2010 ◽  
Vol 75 (4) ◽  
pp. T141-T154 ◽  
Author(s):  
Wei Zhang ◽  
Yang Shen
Keyword(s):  
Free Surface ◽  
Finite Difference ◽  
Differential Equations ◽  
Seismic Wave ◽  
Numerical Scheme ◽  
Complex Frequency ◽  
Order Accuracy ◽  
First Order ◽  
Time Marching ◽  
Interior Domain

The complex-frequency-shifted perfectly matched layer (CFS-PML) technique can efficiently absorb near-grazing incident waves. In seismic wave modeling, CFS-PML has been implemented by the first-order-accuracy convolutional PML technique or second-order-accuracy recursive convolution PML technique. Both use different algorithms than the numerical scheme for the interior domain to update auxiliary memory variables in the PML and thus cannot be used directly with higher-order time-marching schemes. We work with an unsplit-field CFS-PML implementation using auxiliary differential equations (ADEs) to update the auxiliary memory variables. This ADE CFS-PML results in complete first-order differential equations. Thus, the numerical scheme for the interior domain can be used to solve ADE CFS-PML equations. We have implemented ADE CFS-PML in the finite-difference time-domain method and in anonstaggered-grid finite-difference method with the fourth-order Runge-Kutta scheme, demonstrating its straightforward implementation in different numerical time-marching schemes. We have also theoretically analyzed the role of the scalingfactor of CFS-PML; it transforms the PML to a transversely isotropic material, reducing the effective wave speed normal to the PML layer and bending the wavefront toward the normal direction of the PML layer. Our numerical tests indicate that the optimal value reduces the points per dominant wavelength at the outermost boundary to three, about half the value required by the numerical scheme. We also have found that the PML equations should be derived taking the free-surface boundary condition into account in finite-difference methods. Otherwise, the free surface in the PML layer causes instability or ineffective absorption of surface waves. Tests show that we can use a narrow-slice mesh with ADE CFS-PML to simulate full wave propagation efficiently in models with complex structure.

A Truly Generalized Plane Strain Meshless Model for Combined Normal and Shear Loading of Fiber Reinforced Materials

2010 ◽  
Author(s):  
Eisa Ahmadi ◽  
M. M. Aghdam
Keyword(s):  
Direct Method ◽  
Micromechanical Model ◽  
Computation Time ◽  
Integral Form ◽  
Shear Loading ◽  
Interface Condition ◽  
Fiber Reinforced ◽  
Equilibrium Equations ◽  
Normal Loading ◽  
Fiber Reinforced Materials

A truly meshless method based on the integral form of equilibrium equations is formulated. A micromechanical model is developed to study micro-stresses in normal and shear loading of unidirectional fiber reinforced composites. A small repeating area of composite including a fiber surrounded by matrix called representative volume element (RVE) is considered as solution domain. A direct method is proposed for enforcement of the appropriate periodic boundary conditions for shear and normal loading. Especially transverse shear loading is considered in this analysis. Fully bonded interface condition is investigated and the continuity of displacements and traction is imposed to the fiber-matrix interface. Comparison of the predicted results shows excellent agreement with results in available literature. Results of this study also revealed that the presented model can provide highly accurate predictions with respectively small number of nodes and small computation time without the complexity of mesh generation.

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