Elastic waves in complex radially symmetric media

2008

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.

2002

The boundary element method (BEM) is used to fully simulate the propagation of waves between two fluid-filled boreholes. The sources are placed in one of the boreholes while the receivers are placed in the other. This model is frequently used in cross-hole seismic prospecting techniques to assess the characteristics of the elastic medium between the two boreholes. This work studies the dependence of the wave propagation patterns on the distance between the source and the receiver, their location and orientation relative to the axis of a circular borehole and type of elastic formation (fast and slow formations). In addition, this BEM model is used to compute the influence of the deformed boreholes whose cross-section is not circular. Both the spectra responses and the time-domain responses are computed to elucidate the main physical features of the problem solved.

2003

SEG Technical Program Expanded Abstracts 2007, 2007

At low frequencies tube or Stoneley waves represent a dominant arrival propagating along boreholes. They can be excited by the source in a well or by external source due to conversion from other wave types. Tube wave experiences reflection at the bed boundaries, borehole diameter changes and fractures or permeable zones. It was proven in previous studies that 1D effective wavenumber approach provides simple and accurate low-frequency description of tubewave propagation in open boreholes surrounded by radially homogeneous formation. Tube waves become even more dominant in cased boreholes, but casing further modifies wave propagation and reflection/transmission phenomena. In this study we apply 1D effective wavenumber approach to radially inhomogeneous media and demonstrate that it still provides excellent description of low-frequency tubewave propagation. In particular, we focus on three models representative of cased boreholes: reflection from geological interfaces behind casing, reflection from corroded casing section and reflection from idealized diskshaped perforation in cased hole. In all three cases frequency-dependent reflection coefficient obtained by 1D effective method and by finite-difference computations show excellent agreement.

Geophysical Prospecting, 2009

Many tasks in geophysics and acoustics require estimation of mode velocities in cylindrically layered media. For example, acoustic logging or monitoring in open and cased boreholes need to account for radial inhomogeneity caused by layers inside the borehole (sand screen, gravel pack, casing) as well as layers outside (cement, altered and unaltered formation layers). For these purposes it is convenient to study a general model of cylindrically layered media with inner fluid layer and free surface on the outside. Unbounded surrounding media can be described as a limiting case of this general model when thickness of the outer layer is infinite. At low frequencies such composite media support two symmetric modes called Stoneley (tube) and plate (extensional) wave. Simple expressions are obtained for these two mode velocities valid at zero frequency. They are written in a general form using elements of a propagator matrix describing axisymmetric waves in the entire layered composite. This allows one to apply the same formalism and compute velocities for n-layered composites as well as anisotropic pipes. It is demonstrated that the model of periodical cylindrical layers is equivalent to a homogeneous radially transversely isotropic media when the number of periods increases to infinity, whereas their thickness goes to zero. Numerical examples confirm good validity of obtained expressions and suggest that even small number of periods may already be well described by equivalent homogeneous anisotropic media.

Geofísica Internacional, 2011

Se presenta una formulación analítica orientada al entendimiento de la difracción, dispersión y atenuación de los modos de propagación en pozo. El principal objetivo de este artículo es el de comunicar y promover en la comunidad científica los fundamentos sobre la respuesta sísmica en la vecindad de pozos. Se considera una fuente puntual interna (monopolar y dipolar) y se presentan resultados novedosos de simulaciones comparando con datos reales. Una técnica importante que no ha sido ampliamente explotada en la investigación cuidadosa de la propagación de ondas elásticas en pozos petroleros es el registro de formas de onda sónicas. El tratamiento apropiado y el procesamiento adecuado de los sendos grupos de micro-sismogramas permiten la extracción de información útil en la caracterización y entendimiento de las formaciones de roca, y es crucial en la toma de decisiones dentro de la cadena de producción de hidrocarburos. En este trabajo se estudia la propagación de ondas en el pozo por medio de simulaciones numéricas con el "denominado" método del Número-de-Onda Discreto aplicado a varios casos de pozos representativos en yacimientos mexicanos. Las contribuciones de esta investigación son: (1) evidenciar en los sismogramas el fuerte efecto de difracción y dispersión de ondas elásticas, aún trabajando en el caso homogéneo e isótropo, (2) describir en los dominios del tiempo y la frecuencia, la propagación de ondas generadas por una fuente puntual en un pozo cilíndrico lleno de fluido y (3) comparar resultados de simulaciones controladas con datos reales de registros geofísicos. Para validar nuestros cálculos se comparan historias de tiempo y curvas de dispersión contra formas de onda procesadas en profundidad para varios tipos de litologías. El grupo de resultados que aquí se reportan pueden ser útiles en el entendimiento y predicción de los efectos producidos por la presencia de fracturas y heterogeneidades sobre la propagación de ondas. Se espera que en un futuro cercano se establezca al modelado matemático de registros sónicos de onda completa como una técnica digna de confianza en la simulación e interpretación de datos de campo.

1996

Borehole related seismic measurements, such as acoustic logging, vertical seismic profiling, crosswell profiling, and single borehole profiling, provide high resolution data which are useful for reservoir characterization and locating new well sites. Study of elastic waves in boreholes is important for understanding and interpreting these borehole seismic data. Real boreholes are usually embedded in a layered formation, and may also have near borehole alterations, e.g., casing, cement, and invaded zones. In this thesis, two new semi-analytical approaches using the generalized reflection/transmission matrices method are developed to simulate these complicated boreholes. The first method simulates boreholes with radial layers; the second method deals with more realistic boreholes which have radial multi-layers with vertical variations in each layer. Because of their efficiency, stability, and accuracy, these new approaches have many applications to borehole seismic problems. In this t...

The Journal of the Acoustical Society of America

This work studies the wave motion in a fluid-filled borehole in the presence of drill string and geological formation. The synthetic waveforms are obtained by a three-dimensional axis-symmetric full-wave numerical simulation in a two-dimensional multi-domain where the medium is uniform with respect to the azimuth. The discretization is performed in cylindrical coordinates. In order to simulate the waves at the origin (axis of the polar radius), a very small radius is used to avoid the singularity. The free-surface and rigid boundary conditions are tested and it is shown that the rigid one constitutes the best approximation. The simulations provide the amplitude distribution and motion diagrams in the borehole vertical cross-sections and at the outer boundary, away from the borehole. Propagation in the presence of hard and soft formations is analysed. The dispersion, the amplitude, and the orbital polarization of the modes excited by a point source acting in the fluid inside a drillstring are considered and examples of comparison with literature results obtained using multi-modal analysis are shown. The proposed approach is more general than the multi-modal analysis, since it allows for arbitrary variations of the properties in the plane of symmetry.

Wave Motion, 2002

This paper studies wave propagation in the vicinity of a cylindrical solid formation submerged in an acoustic medium generated by point blast loads placed outside the inclusion. The full 3D solution is obtained first in the frequency domain as a discrete summation of responses for 2D problems defined by a spatial Fourier transform. Each 2D solution is computed using the Boundary Element Method, which makes use of two-and-a-half-dimensional Green's functions. This model is implemented to obtain Fourier spectra responses which make it possible to identify the behavior of both the axisymmetric and non-axisymmetric guided wave modes, when the cross-section of the elastic inclusion changes from circular to smooth oval.

Journal of Computational Acoustics, 2003

Borehole drillstring waves play an important role in geotechnical and exploration applications. Coupled waves are simulated inside and outside a real drillstring in borehole. The drillstring is considered a system with variable geometry. We simulated coupled waves in three domains: inner mud, drillstring, and annular mud. Each mode has a characteristic wave propagation velocity. A consequence of this coupling effect is that waves from the downhole tool can be partially reflected and converted to other modes at a discontinuity in the formation stiffness or in the annular cross section that can cause a significant reduction of the transmitted wave energy. Examples of application are reported.