Earth Resources Laboratory :: Abstracts

Borehole Wave Propagation in Isotropic and Anisotropic Media:
Three-Dimensional Finite Difference Approach

Ningya Cheng

Submitted to the Department of Earth, Atmospheric, and Planetary Sciences on February, 1994 in partial fulfillment of the requirements for the degree of Doctor of Philosophy

Abstract

In this thesis we develop a three-dimensional method to simulate wave propagation in an isotropic as well as an anisotropic medium. The wave equation is formulated into the first-order hyperbolic equations by using velocity and stress. They are discretized on a staggered grid. The three-dimensional finite difference time domain scheme is second-order accurate in time and fourth-order accurate in space. The grid dispersion and anisotropy are analyzed and the stable condition of the scheme is obtained. Higdon's absorbing boundary condition is discussed and generalized to the anisotropic medium. The scheme provides realistic 3-D wave propagation simulation by the use of a parallel computer.

The finite difference method is first tested in homogeneous media. The finite difference results agree excellently with the analytic solutions of a point explosion source in the acoustic medium and a point force source in elastic and transversely isotropic media. The finite difference method accurately models not only the far field P and S waves, but also the near field term. The method is then tested in a fluid-filled borehole surrounded by a homogeneous elastic formation. The finite difference results are in good agreement with the discrete wavenumber solutions for both monopole and dipole sources in the hard as well as the soft formations. These tests also show the good performance of Higdon's absorbing boundary in isotropic and anisotropic media. It not only works for the body waves but also for the guided waves.

The 3-D finite difference time domain method is applied to fluid-filled borehole wave propagation problems in isotropic and anisotropic formations. The effects of the off-centered sources, the elliptic borehole, and the tilted layer on acoustic logs are investigated for the isotropic formation. The finite difference synthetics are compared with ultrasonic laboratory measurements in a scaled borehole in an orthorhombic phenolite solid. Both monopole and dipole logs agree well. In the anisotropic formation the different borehole orientations are considered for monopole and dipole logs. Due to shear wave anisotropy, there are shear-pseudo-Rayleigh wave arrivals on the monopole log between the P and Stoneley waves in the phenolite formation. Anisotropy can also cause shear wave splitting on the dipole log.

Field data sets collected by an array monopole acoustic logging tool and a shear wave logging tool were processed and interpreted. The P- and S-wave velocities of the formation are determined by threshold detection with cross-correlation correction from the full waveform and the shear wave log, respectively. The extended Prony's method is used to estimate the borehole Stoneley wave phase velocity and attenuation as a function of frequency. The formation between the depths of 2950 and 3150 ft can be described as an isotropic elastic medium. The inverted Vs from the Stoneley wave phase velocity is in excellent agreement with the shear wave log results. The formation between the depths of 3715 and 3780 ft is a porous, permeable and anisotropic medium. The shear wave velocity anisotropy is about 10% to 20%, and the symmetry axis is perpendicular to the borehole axis. The disagreement between estimated permeabilities from low frequency Stoneley wave velocity and attenuation data are in good agreement with the core measurements. Also, it is shown that the formation permeability is not the primary cause of the discrepancy between the shear wave velocity inverted from the Stoneley wave and measured by the shear wave logs. The 3-D finite difference synthetics in the anisotropic formation confirm that the discrepancy can be explained as shear wave anisotropy.