Numerical test of the Schoenberg-Muir theory

Carcione, José M.; Picotti, Stefano; Cavallini, F.; Santos, Juan E.

doi:10.1190/geo2011-0228.1

Cited by 26 publications

(11 citation statements)

References 14 publications

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“…This approach has been further developed into an algorithm that calculates the rock compliance instead of the rock stiffness by Hood (1991) and Nichols, Muir and Schoenberg (1989). Using the Backus (1962) theory, a numerical simulation was conducted to show that the Schoenberg-Muir theory is valid from the kinematic viewpoint (involving travel times) and the dynamic viewpoint (involving amplitudes) for very long flat parallel fractures and thin layered media (Carcione et al 2012). In the theory, a fracture is treated as an infinitely extended weakness plane and is used as an element for assembling a fractured medium.…”

Section: T H E O R E T I C a L M E T H O D Smentioning

confidence: 99%

Seismic modelling for geological fractures

Cui

Lines

Krebes

2017

Geophysical Prospecting

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A B S T R A C TBased on knowledge of a commutative group calculation of the rock stiffness and on some geophysical assumptions, the simplest fractured medium may be regarded as a fracture embedded in an isotropic background medium, and the fracture interface can be simulated as a linear slip interface that satisfies non-welded contact boundary conditions: the kinematic displacements are discontinuous across the interface, whereas the dynamic stresses are continuous across the interface. The finite-difference method with boundary conditions explicitly imposed is advantageous for modelling wave propagation in fractured discontinuous media that are described by the elastic equation of motion and non-welded contact boundary conditions. In this paper, finite-difference schemes for horizontally, vertically, and orthogonally fractured media are derived when the fracture interfaces are aligned with the boundaries of the finite-difference grid. The new finite-difference schemes explicitly have an additional part that is different from the conventional second-order finite-difference scheme and that directly describes the contributions of the fracture to the wave equation of motion in the fractured medium. The numerical seismograms presented, to first order, show that the new finite-difference scheme is accurate and stable and agrees well with the results of previously published finite-difference schemes (the Coates and Schoenberg method). The results of the new finite-difference schemes show how the amplitude of the reflection produced by the fracture varies with the fracture compliances. Later, comparisons with the reflection coefficients indicate that the reflection coefficients of the fracture are frequency dependent, whereas the reflection coefficients of the impedance contrast interface are frequency independent. In addition, the numerical seismograms show that the reflections of the fractured medium are equal to the reflections of the background medium plus the reflections of the fracture in the elastic fractured medium.

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Section: T H E O R E T I C a L M E T H O D Smentioning

confidence: 99%

Seismic modelling for geological fractures

Cui

Lines

Krebes

2017

Geophysical Prospecting

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“…Full-scale simulation can be used to study the peculiarities of wave propagation in complex models, such as anisotropic [1,2], viscoelastic [1], and poroelastic models [3,4], and models with irregular topography [5][6][7], etc. Moreover, numerical simulation is an essential element in seismic imaging procedures such as Reverse Time Migration and Full Waveform Inversion [8].…”

Section: Introductionmentioning

confidence: 99%

Local time–space mesh refinement for simulation of elastic wave propagation in multi-scale media

Kostin¹,

Lisitsa

Reshetova

et al. 2015

Journal of Computational Physics

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“…For simplicity, we take the two sets of test parameters presented by Carcione et al (2012). In their tests, they use the SchoenbergMuir theory to derive the effective medium parameters and verify the accuracy by comparing the spatial wavefields calculated by the Fourier pseudospectral method for the original layered medium and the effective medium.…”

Section: Horizontally Layered Mediummentioning

confidence: 99%

“…The effective elasticity constants from Backus averaging or Schoenberg-Muir theory are given as (Carcione et al, 2012) C Backus ¼ C Because Carcione et al (2012) verify the results of the Schoenberg-Muir theory by means of wave equation modeling, we take the results from Schoenberg-Muir theory to be the "true'' solution. We compare the results from Backus averaging, Schoenberg-Muir theory, and our method, and we see that for such a layer composition, Backus averaging and Schoenberg-Muir theory give exactly the same result, and due to the numerical solution nature of our method, we obtain slightly different values.…”

Section: Horizontally Layered Mediummentioning

confidence: 99%

“…Later, by applying matrix and group theory, Schoenberg and Muir (1989) develop a more general effective medium theory for horizontally aligned elastic layers, with general anisotropy in which all 21 independent elasticity constants might be nonzero in the elasticity matrix (we will refer to this method as the Schoenberg-Muir theory). Carcione et al (2012) verify the Schoenberg-Muir theory by comparing the analytical solutions and the solutions calculated with a finite-difference method for the elastic wave equation, and they find that the Schoenberg-Muir theory can accurately determine effective elasticity parameters for finely layered elastic VTI, HTI, and TTI media as well.…”

Section: Introductionmentioning

confidence: 99%

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A numerical homogenization method for heterogeneous, anisotropic elastic media based on multiscale theory

et al. 2015

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The development of reliable methods for upscaling finescale models of elastic media has long been an important topic for rock physics and applied seismology. Several effective medium theories have been developed to provide elastic parameters for materials such as finely layered media or randomly oriented or aligned fractures. In such cases, the analytic solutions for upscaled properties can be used for accurate prediction of wave propagation. However, such theories cannot be applied directly to homogenize elastic media with more complex, arbitrary spatial heterogeneity. Therefore, we have proposed a numerical homogenization algorithm based on multiscale finite-element methods for simulating elastic wave propagation in heterogeneous, anisotropic elastic media. Specifically, our method used multiscale basis functions obtained from a local linear elasticity problem with appropriately defined boundary conditions. Homogenized, effective medium parameters were then computed using these basis functions, and the approach applied a numerical discretization that was similar to the rotated staggered-grid finite-difference scheme. Comparisons of the results from our method and from conventional, analytical approaches for finely layered media showed that the homogenization reliably estimated elastic parameters for this simple geometry. Additional tests examined anisotropic models with arbitrary spatial heterogeneity in which the average size of the heterogeneities ranged from several centimeters to several meters, and the ratio between the dominant wavelength and the average size of the arbitrary heterogeneities ranged from 10 to 100. Comparisons to finite-difference simulations proved that the numerical homogenization was equally accurate for these complex cases.

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Numerical test of the Schoenberg-Muir theory

Cited by 26 publications

References 14 publications

Seismic modelling for geological fractures

Seismic modelling for geological fractures

Local time–space mesh refinement for simulation of elastic wave propagation in multi-scale media

A numerical homogenization method for heterogeneous, anisotropic elastic media based on multiscale theory

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