There exist two types of commonly studied stochastic elliptic models in literature: (I) � r � (a(x,x)r u(x,x)) = f(x) and (II) � r �ð aðx; xÞ}ruðx;xÞÞ ¼ f ðxÞ, where x indicates randomness, } the Wick product, and a(x,x) is a positive random process. Model (I) is widely used in engineering and physical applications while model (II) is usually studied from the mathematical point of view. The difference between the above two stochastic elliptic models has not been fully clarified. In this work, we discuss the difference between models (I) and (II) when a(x,x) is a log-normal random process. We show that the difference between models (I) and (II) is mainly characterized by a scaling factor, which is an exponential function of the degree of perturbation of a(x,x). We then construct a new stochastic elliptic model (III): � r �ð ða

/pdf/comput-methods-appl-mech-engrg-qn80ahqrxx.pdf

Comput. Methods Appl. Mech. Engrg.

A new 3D cylindrical perfectly matched layer (PML) formulation is developed for elastic wave propagation in a pressurized borehole surrounded by a triaxially stressed solid formation. The linear elastic formation is altered by overburden and tectonic stresses that cause significant changes in the wave propagation characteristics in a borehole. The 3D cylindrical problem with both radial and azimuthal heterogeneities is suitable for numerical solutions of the wave equations by finite-difference time-domain (FDTD) and pseudospectral time-domain (PSTD) methods. Compared to the previous 2.5D formulation with other absorbing boundary conditions, this 3D cylindrical PML formulation allows modeling of a borehole-conformal, full 3D description of borehole elastic waves in a stress-induced heterogeneous formation. We have developed an FDTD method using this PML as an absorbing boundary condition. In addition to the ability to solve full 3D problems, this method is found to be advantageous over the previously reported 2.5D finite-difference formulation because a borehole can now be adequately simulated with fewer grid points. Results from the new FDTD technique confirm the principle of superposition of the influence of various stress components on both the borehole monopole and dipole dispersions. In addition, we confirm that the increase in shear-wave velocity caused by a uniaxial stress applied in the propagation direction is the same as that applied parallel to the radial polarization direction.

A 3D cylindrical PML/FDTD method for elastic waves in fluid‐filled pressurized boreholes in triaxially stressed formations

This paper presents a numerical program for the simulation of elastic wave propagation and scattering in three-dimensional (3-D) cylindrical coordinates based on the first-order velocity-stress finite-difference scheme on staggered grids. Both Liao’s and Lindman’s absorbing boundary conditions are implemented for the exterior boundaries to efficiently truncate the computation domain for elongated 3-D well logging problems. Symmetric and anti-symmetric boundaries in azimuthal and axial directions are also implemented in the code to further reduce the size of the problem. Included for the first time with this code are very large and complex geometrical structures such as the whole slotted sleeve housing of a sonic well-logging tool which typically involves hundreds of millions of unknowns. The calculation for such a large problem only takes a couple of days on a four-processor SGI Power Challenge machine. Different types of slotted sleeve models are studied for sonic logging tools. Simulation results show that different slotted sleeves vary widely in delaying and attenuating the pipe waves which travel along the tool housing. A new slotted sleeve structure with three horizontal slot sections for every vertical slot period is proposed for better performance. A dipole source is found to produce much cleaner waveforms than a monopole source.

A three-dimensional finite difference code for the modeling of sonic logging tools

The development of dipole shear-wave imaging technology requires understanding of the effects of radiation, reflection, and reception of elastic waves from a borehole dipole source, for which we provide a comprehensive analysis. We first show that the radiation of the dipole source can be accurately computed using its far-field asymptotic solution when the radiation distance is greater than wavelength. We then demonstrate the reciprocity relationship between shear-wave radiation and reception of the dipole source. Consequently, the borehole radiation pattern can be used to compute the borehole reception directivity. The use of the reciprocity relationship and the asymptotic solution greatly facilitates the modeling of the wavefield for the borehole shear-wave reflection survey. The modeling results agree well with those from a 3D finite-difference elastic wave simulation. The modeling of SH- and SV-wave radiation/reception of the dipole source also demonstrates that the SH-wave component is far gr...

Shear-wave radiation, reception, and reciprocity of a borehole dipole source: With application to modeling of shear-wave reflection survey

SUMMARY Dipole acoustic fields in an arbitrarily deviated well penetrating a homogeneous as well as a stratified transversely isotropic formation are simulated using a 3-D finite-difference timedomain algorithm in cylindrical coordinates. The modelling results show that a dipole source can excite a fast- and a slow-flexural mode due to the shear wave anisotropy when the borehole is inclined with respect to the symmetry axis of transverse isotropy. Both flexural slownesses change with the wellbore deviation angle. The splitting of flexural modes is prominent in full wave arrays when the shear anisotropy is strong enough. It is revealed that the dipole orientation influences the relative amplitudes of the fast- and slow-flexural waves but it has no effect on their slownesses or phases. In a vertical well parallel to the symmetry axis, the two flexural waves degenerate and propagate at the same speed. The degenerated flexural wave travels approximately at the shear speed along the borehole wall except in a few formations. Our study shows, for example, that it is about 10 per cent slower than the shear wave in Mesaverde clayshale 5501. Even in that kind of formations, however, extraction of the fastand slow-shear velocities from the flexural modes is still possible if the borehole deviation is large enough. To examine the effect of layering, we modelled the full waves in a formation with a sandwich. When the well is perpendicular to the layer interfaces, reflection is obvious and can be recognized. It becomes weaker or even invisible as the deviation angle increases, so it is difficult to detect a thin layer embedded in a formation directly from reflected waves. The sandwich can, instead, be recognized from the irregularity in the spectra of the full waveforms displayed versus depth.

/pdf/fast-and-slow-flexural-waves-in-a-deviated-borehole-in-3hy2hl7x3j.pdf

Fast and slow flexural waves in a deviated borehole in homogeneous and layered anisotropic formations

In this paper the three‐dimensional finite difference method is used to simulate borehole wave propagations in an isotropic as well as an anisotropic formation. The finite difference results agree excellently with the analytic solutions of a point force source in the transversely isotropic medium. The finite difference synthetics are also in very good agreement with the discrete wave‐number solutions for fluid‐filled borehole wave propagation. The finite difference synthetics are compared with ultrasonic lab measurements in a scaled borehole model. The borehole is drilled along the X axis in an orthorhombic phenolite solid. Both monopole and dipole logs agree well. The observations of the shear wave splitting in the dipole logs are confirmed by the finite difference simulations. The 3‐D finite difference method is applied to the fluid‐filled borehole wave propagation in the tilted isotropic formation and in the orthorhombic phenolite formation. In a borehole drilled along the Z axis in a phenolite formati...

Borehole wave propagation in three dimensions

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Borehole wave propagation in three dimensions