Elastic wave propagation using cylindrical coordinates

Kessler, David; Kosloff, Dan

doi:10.1190/1.1443020

Cited by 36 publications

(16 citation statements)

References 9 publications

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“…Two adjacent meshes are combined by decomposing the wave field into incoming and outgoing wave modes at the interface between the media and modifying these modes on the basis of the fluid/solid boundary conditions [6]. The inward propagating waves depend on the solution exterior to the subdomains and, therefore, are computed from the boundary conditions (continuity of stress and particlevelocity components), while the behavior of the outward propagating waves is determined by the solution inside the subdomain [3,5,15]. The approach involves the following equations for updating the field variables at the grid points defining the interface between the fluid and the solid, such that the upper sign corresponds to fluid (1) /solid (2) and the lower sign to solid (1) / fluid (2):…”

Section: Domain Decomposition and Boundary Conditionsmentioning

confidence: 99%

“…The problem of obtaining a realistic VSP survey by using pseudospectral differential operators was considered by Kessler and Kosloff in two papers [14,15]. They solve for 2-D acoustic and elastic wave propagation in a (horizontal) plane perpendicular to the axis of symmetry of the hole by using Chebychev and Fourier differential operators in the radial and The ranges of the four meshes are r a − for the inner mud ( is the minimum radius of the inner mesh), r b − r a for the drill-string, r c − r b for the outer mud, and r d − r c for the casingcement-formation system angular directions, respectively.…”

Section: Introductionmentioning

confidence: 99%

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Simulation of axis-symmetric seismic waves in fluid-filled boreholes in the presence of a drill string

2008

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A numerical algorithm for simulation of 2-D (axis-symmetric) wave propagation using a multidomain approach is proposed. The method uses a cylindrical coordinate system, Chebyshev and Fourier differential operators to calculate the spatial derivatives along the radial and vertical direction, respectively, and a Runge-Kutta time-integration scheme. The numerical technique is based on the solution of the equations of momentum conservation combined with the stress-strain relations of the fluid (drilling mud) and isotropic elastic media (drill string and formation). Wave modes and radiated waves are simulated in the borehole-formation system. The algorithm satisfies the reciprocity condition and the results agree with an analytical solution and low-frequency simulation of wave-propagation modes reported in the literature. Examples illustrating the propagation of waves are presented for hard and soft formations. Moreover, the presence of casing, cement, and formation heterogeneity have been considered. Since the algorithm is based on a direct (grid) method, the geometry and the properties defining the media at each grid point, can be general, i.e., there are no limitations such as planar interfaces or uniform (homogeneous) properties for each medium.

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Section: Domain Decomposition and Boundary Conditionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Simulation of axis-symmetric seismic waves in fluid-filled boreholes in the presence of a drill string

2008

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“…As the grid nodes are equally spaced in the azimuthal direction, we can use a discrete Fourier transform to perform this interpolation. The same kind of boundary treatment in Cartesian and polar coordinates was used by Carcione (1991), Kessler & Kosloff (1991) and Tessmer et al (1992) for elastic media and by Sidler et al ( , 2013 for poroelastic media. The derivation of the characteristic vector for wave propagation in radial direction for a poroelastic medium in cylindrical coordinates is shown in Appendix B.…”

Section: Boundary Conditionsmentioning

confidence: 99%

“…Pseudospectral methods are efficient and highly accurate techniques for the modelling of complex wave propagation phenomena (e.g., Kessler & Kosloff 1991;Fornberg 1996;Carcione 2007;Liu et al 2011). They can be viewed as the limit of finite differences with infinite order of accuracy, as the spatial derivatives are calculated in the wavenumber domain using a forward and backward discrete Fourier transform (Fornberg 1988;Boyd 2001).…”

Section: N U M E R I C a L S O L U T I O Nmentioning

confidence: 99%

A pseudospectral method for the simulation of 3-D ultrasonic and seismic waves in heterogeneous poroelastic borehole environments

Sidler¹,

Carcione²,

Holliger³

2013

Geophysical Journal International

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S U M M A R YWe present a novel approach for the comprehensive, flexible and accurate simulation of poroelastic wave propagation in 3-D cylindrical coordinates. An important application of this method is the realistic modelling of complex seismic wave phenomena in fluid-filled boreholes, which represents a major, as of yet largely unresolved, problem in exploration geophysics. To this end, we consider a numerical mesh consisting of three concentric domains representing the borehole fluid in the centre followed by the mudcake and/or casing, and the surrounding porous formation. The spatial discretization is based on a Chebyshev expansion in the radial direction and Fourier expansions in the vertical and azimuthal directions as well as a Runge-Kutta integration scheme for the time evolution. Trigonometric interpolation and a domain decomposition method based on the method of characteristics are used to match the boundary conditions at the fluid/porous-solid and porous-solid/porous-solid interfaces as well as to reduce the number of gridpoints in the innermost domain for computational efficiency. We apply this novel modelling approach to the particularly challenging scenario of near-surface borehole environments. To this end, we compare 3-D heterogeneous and corresponding rotationally invariant simulations, assess the sensitivity of Stoneley waves to formation permeability in the presence of a casing and evaluate the effects of an excavation damage zone behind a casing on sonic log recordings. Our results indicate that only first arrival times of fast modes are reasonably well described by rotationally invariant approximations of 3-D heterogenous media. We also find that Stoneley waves are indeed remarkably sensitive to the average permeability behind a perforated PVC casing, and that the presence of an excavation damage zone behind a casing tends to dominate the overall signature of recorded seismograms.

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“…The application of the PSM to the solution of wave propagation problems in 2‐D cylindrical or polar coordinate systems has already been conducted for the forward modelling of geophysical exploration experiments for boreholes (Kessler & Kosloff 1990, 1991; Tessmer et al . 1992) and around circular objects such as a cavity.…”

Section: Introductionmentioning

confidence: 99%

Seismic wavefield calculation for laterally heterogeneous whole earth models using the pseudospectral method

Furumura¹,

Furumura²

1998

Geophysical Journal International

100

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S U M M A RYA method for simulating seismic wave propagation in a laterally heterogeneous whole earth model is presented by solving the elastodynamic equations in 2-D cylindrical coordinates using the pseudospectral method (PSM). The PSM is an attractive timedomain technique that uses the fast Fourier transform for an accurate di¡erentiation of ¢eld variables in the equations. Since no dispersion error arises in Fourier di¡er-entiation, even when using a large grid spacing, computer memory and time are reduced by several orders of magnitude compared to traditional ¢nite-di¡erence methods. In order to examine body-wave phases with current computing resources, a slice through the sphere is approximated with a 2-D cylindrical model. An irregular grid spacing is used in the vertical coordinate to improve the treatment of the various structural boundaries appearing in the earth model by matching the heterogeneity in the model. Synthetic seismograms obtained by the PSM calculation are compared with those calculated from an exact simulation method for a spherically homogeneous (1-D) earth model and achieve good agreement.The PSM method is illustrated by constructing the seismic P^SV wave¢eld for strongly heterogeneous earth models including a shield structure near the free surface and velocity perturbations just above the core^mantle boundary. The visualization of the evolution of seismic P and S waves in time and space is tracked using a sequence of snapshots and synthetic seismograms. These displays allow direct insight into the nature of the complex seismic wave behaviour in the Earth's interior.

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Elastic wave propagation using cylindrical coordinates

Cited by 36 publications

References 9 publications

Simulation of axis-symmetric seismic waves in fluid-filled boreholes in the presence of a drill string

Simulation of axis-symmetric seismic waves in fluid-filled boreholes in the presence of a drill string

A pseudospectral method for the simulation of 3-D ultrasonic and seismic waves in heterogeneous poroelastic borehole environments

Seismic wavefield calculation for laterally heterogeneous whole earth models using the pseudospectral method

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