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Title:3D rotated and standard staggered finite-difference solutions to Biot's poroelastic wave equations: Stability condition and dispersion analysis
Authors:O'Brien, G.S., 2010
Abstract: A fourth-order in space and second-order in time 3D staggered 􏰀SG􏰁 and rotated-staggered-grid 􏰀RSG􏰁 method for the solution of Biot’s equation are presented. The numerical dispersion and stability conditions are derived using a von Neumann analysis. The exact stability condition is calculated from the roots of a 12th-order polynomial and therefore no nontrivial expression ex- ists. To overcome this, a 1D stability condition is usually general- ized to three dimensions. It is shown that in certain cases, the 1D approximate stability condition is violated by a 3D SG method. The RSG method obeys the approximate 1D stability condition for the material properties and spatiotemporal scales in the exam- ples shown. Both methods have been verified against an analyti- cal solution for an infinite homogeneous porous medium with a misfit error of less than 0.5%. A free surface has been implemented to test the accuracy of this boundary condition. It also serves as a test of the methods to include high material contrasts. The methods have been compared with a quasi-analytical solution. For the specific material properties, spatial grid scaling, and propagation distance used in the test, a maximum error of 3.5% for the SG and 4.1% for the RSG was found. These errors depend on the propagation distance, temporal and spatial scales, and ac- curacy of the quasi-analytical solution. No discernable differ- ence was found between the two methods except for time steps comparable with the stability-criteria threshold time step, the SG was found to be unstable. However, the RSG remained stable for a homogeneous half-space. Time steps, comparable to the stabili- ty criteria, reduce the computational time at the cost of a reduc- tion in accuracy. The methods allow wave propagation to be modeled in a porous medium with a free surface.
ICHEC Project:Seismic source modelling and propagation in complex 3D Earth models
Publication:Geophysics, Vol 75, No. 4, T111-119,
URL: http://dx.doi.org/10.1190/1.3432759
Keywords: porosity, porous materials, seismic waves, seismology
Status: Published

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