Numerical computation of individual far‐field arrivals excited by an acoustic source in a borehole

Abstract

In this paper, I model the acoustic logging problem and numerically compute individual arrivals at far-field receivers. The ability to compute individual arrivals is useful for examining the sensitivities of each arrival to various factors of interest, as opposed to examining the full waveform as a whole. While the numerical computation of the mode arrivals (Peterson, 1974) and the numerical computation of the first head waves (Tsang and Rader, 1979) have been previously reported, the numerical computation of the entire set of head-wave arrivals is new and is the major contribution of this paper.Following Roever et al. (1974) and others, the full wave field is represented as a sum of contributions from both poles and branchcuts in the complex wavenumber plane. The pole contributions correspond to mode arrivals while the branch cuts are associated with body waves (i.e., head waves). Both the pole and branch cut contributions are computed numerically and results are presented for the cases of a slow and a fast formation. The shear event in the slow formation is found to be relatively small, consistent with observations in measured data. Contrary to existing knowledge, the shear event in the fast formation is also relatively small. The apparent strong shear arrival in the full waveforms is due primarily to the trapped mode pole in the vicinity of cutoff.Acoustic well-logging waveforms recorded in a mud-filled borehole can be understood, in part, by studying the propagation of acoustic waves in a fluid-filled circular cylinder surrounded by a homogeneous solid. In such a model, the full acoustic field is formulated exactly in the frequencywavenumber domain and numerical methods are employed to obtain space-time domain waveforms (White and Zechman, 1968;Tsang and Rader, 1979). The study of full synthetic waveforms computed in this manner can develop intuition regarding the influence of a formation on measured waveforms (Cheng and Toksoz, 1981).Individual arrivals which make up the full waveforms are each controlled by one or more quantities, and these quantities determine the usefulness of a given arrival. To take a familiar example, the velocity of the compressional head wave across the receiver array is controlled primarily by the compressional velocity of the formation in the vicinity of the array and (to a much lesser extent) by the borehole radius, or other factors. Measurement of the compressional head-wave velocity is useful because there is only one factor controlling wave speed and that factor is of interest. Generally, a given arrival is controlled by several quantities, and unless all but one are known, the usefulness of such an arrival may be limited.To determine the quantities which control a given arrival, either an analytic or numerical means of analyzing the individual arrivals in isolation is required; analyzing full waveforms is not sufficient. For example, suppose we want to determine those quantities which control the frequency content of the shear wave. Because the shear event is often overlapped ...

Sonic waveform attenuation in gas hydrate‐bearing sediments from the Mallik 2L‐38 research well, Mackenzie Delta, Canada

“…We use the modeling technique described by Tsang and Rader [1979], Cheng and Toksöz [1981] and Kurkjian and Chang [1986]. Assuming that the borehole is an infinite fluid-filled cylinder in an infinite elastic medium, the excitation of a receiver located at a distance z from the transmitter is the sum of a direct pulse through the borehole fluid p i and of a formation response pulse p r given by…”

“…where X(w) is the Fourier transform of the source pulse x(t), V f is the sonic velocity in the fluid, k z is the along-axis wave number, and A(k z ,w) a function describing the response of the formation [Tsang and Rader, 1979;Kurkjian, 1985;Kurkjian and Chang, 1986]. The attenuation is introduced by using complex velocities:…”

Semi-analytical response of acoustic logging measurements in frequency domain

“…We verify our technique against the classical results of Kurkjian [6] and the latest finite-element approach proposed in [1].…”

“…One technique is known as the real axis integration (RAI) method, introduced by Rosenbaum [4] and Tsang & Rader [5]. The other technique employs a deformed integration path in the complex plane and the most common reference to this method is the work of Kurkjian [6].…”

“…On the other hand, the method that uses Cauchy theory [6] is well adapted to recover the frequency-domain solution, but it can be difficult to handle because it involves several integrals in the complex plane. Basically, the Fourier inversion integral is computed as the sum of residues given by the singularities, plus integrals over both sides of the brunch cuts where the integrated function is non-analytic.…”