A Study of Sonic Logging in a Cased Borehole
Authors: Chang S.K. , Everhart A.H. 
Published: December 2007

Abstract
A study was undertaken to investigate the feasibility of sonic logging in a cased borehole. Results were obtained from a scale-model laboratory experiment and from computer simulations. The waveforms from the computer model indicate that sonic logging can be successful in bonded and unbonded cased holes. A slowness/time- semblance signal-processing technique is used to obtain wave velocities from waveforms.

Introduction
Through-casing sonic logs can provide essential data when additional information, such as lithology, is needed for an existing well. Meaningful measurement is difficult when the cement bonding of the casing is poor. The high-speed casing arrivals are usually very strong in a poorly bonded cased hole. These persistent arrivals tend to mask head waves and can render sonic logs indecipherable. This study is part of a research effort to develop techniques and tools to allow sonic logging of cased boreholes. In this effort, research on logging through casing has been carried out through laboratory modeling and computer simulations. Laboratory experiments provided the first waveform data available from cased holes under controlled conditions. These experiments also provided important data for developing a slowness/time-semblance signal-processing technique, This technique has been used extensively to obtain wave velocities from field data and computer-generated sonic waveforms. Computer simulation utilizing real-axis integration is the core of the study presented here. The mathematical modeling allowed synthetic waveforms to be studied under a wide parametric range not practically possible during the laboratory experiment. possible during the laboratory experiment. Simulation by Real-Axis Integration
For the computer simulation the cased hole was modeled as shown in Fig. 1. It is a four-medium model consisting of borehole mud, steel casing, a cement layer, and the formation. The transmitter and the receiver array are placed on the axis of the cylindrical borehole so that the problem is axially symmetric. The pressure response of receivers at a distance in the z direction from the transmitter and a time, t, is expressed in the following frequency/wave-number Fourier transform.
where is the amplitude of the incident pressure pulse at from the point pressure source, is the sound velocity in the borehole mud, x(t) and X( ) are the transmitter pulse shape and the pulse spectrum, is the angular wave number in the z direction, and is angular frequency. Angular wave number is defined as angular frequency multiplied by slowness. The frequency integral is along a Laplace contour parallel to and at a distance (where &gt;) above the real axis in the complex frequency domain. Tsang and Rader have shown the wave number integration can be done on the real axis without encountering singularities or branch cuts. The waveform simulation method is thus properly termed a real-axis integration (RAI) method, properly termed a real-axis integration (RAI) method, The function ( ) in the integrand is the solution of the acoustic waves in the borehole in the frequency/wave-number domain. It is expressed explicitly by Tsang and Rader for openhole sonic logging.
JPT
p. 1745