1. Field of the Invention
This invention relates generally to the logging of earth formations. More particularly, this invention relates to techniques for quantifying errors associated with the determination of values for formation parameters via sonic logging. The invention has particular application to sonic well logging techniques where the slowness of a formation is determined via processing of semblances of sonic wave information.
2. State of the Art
Sonic well logs are typically derived from tools suspended in a mud-filled borehole by a cable. The tools typically include a sonic source (transmitter) and a plurality (M) of receivers which are spaced apart by several inches or feet. Typically, a sonic signal is transmitted from one longitudinal end of the tool and received at the other, and measurements are made every few inches as the tool is slowly drawn up the borehole. The sonic signal from the transmitter or source enters the formation adjacent the borehole, and the arrival times and other characteristics of the received responses are used to find formation parameters.
One sonic log of the art which has proved to be useful is the slowness-time coherence (STC) log. Details of the techniques utilized in producing an STC log are described in U.S. Pat. No. 4,594,691 to Kimball et al., as well as in Kimball, et al., "Semblance Processing of Borehole Acoustic Array Data"; Geophysics, Vol. 49, No. 3, (March 1984) pp. 274-281 which are hereby incorporated by reference in their entireties herein. Briefly, the slowness-time coherence log utilizes the compressional, shear, and Stoneley waves detected by the receivers. A set of time windows is applied to the received waveforms with the window positions determined by two parameters: the assumed arrival time at the first receiver, and an assumed slowness. For a range of values of arrival time and slowness, a scalar semblance is computed for the windowed waveform segments by backpropagating and stacking the waveforms and taking the normalized ratio of the stacked energies to the unstacked energies. The semblance may be plotted as a contour plot with slowness and arrival times as axes, with maximum semblance values indicating the determined formation slowness value. In addition, local maxima of the semblance function are identified by a peak-finding algorithm, and the corresponding slowness values may be plotted as gray-scale marks on a graph whose axes are slowness and borehole depth, or processed further to yield continuous, labelled results. The intensity of the gray-scale marks is proportional to the height of the semblance peak.
As indicated in the aforementioned article and U.S. Pat. No. 4,594,691 to Kimball et al., the same backpropagation and stacking techniques are used regardless of whether the wave being analyzed is a P-wave, S-wave, or a Stoneley wave; i.e., regardless of whether the wave is non-dispersive or dispersive. However, while such backpropagation and stacking techniques may be optimal for non-dispersive waves, they are not optimal for dispersive waves such as flexural and Stoneley waves. In response to this problem, at least two different approaches have been utilized.
A first approach known as "bias-corrected STC" which is currently commercial, is disclosed in A. R. Harrison, et al., "Acquisition and Analysis of Sonic Waveforms from a Borehole Monopole and Dipole Source . . . ", SPE 20557, pp. 267-282 (Society of Petroleum Engineers, Inc. 1990) and U.S. Pat. No. 5,229,939 which are both hereby incorporated by reference herein in their entireties, is to process the flexural mode waveforms as before, but to correct the non-dispersive processing results by a factor relating to the measured slowness; i.e., to post-process the STC results. In particular, correction values are obtained by processing model waveforms with the STC techniques and comparing the measured slowness with the formation shear slowness of the model. The model waveforms assume a particular source and are bandlimited to a prescribed band (typically 1 to 3 KHz) before STC processing. Tables of slowness corrections are designated by a particular source and processing bandwidth, and contain corrections as percentage-of-measured-value factors functions of measured value and hole diameter. The percentage correction required increases with hole diameter and is a function of formation slowness, and ranges from less than one percent to as much as fifteen percent. This approach has the drawback that the waveform spectra often disagree with those of the model which causes the pre-calculated correction to be in error, leading to an error in the measured slowness. A second drawback of the bias-corrected STC approach is that the analysis band may exclude the majority of the flexural mode energy even though it reduces sensitivity to environmental parameters.
A second approach known as "QSTC" (also now known as "dispersive STC" or "DSTC"), which is also currently commercial, is described in U.S. Pat. No. 5,278,805 which is hereby incorporated by reference herein in its entirety. In QSTC, detected signals resulting from flexural waves are Fourier transformed in a specified frequency band, and the Fourier transformed signals are backpropagated according to equations using different dispersion curves. The backpropagated signals are then stacked, and semblances are found in order to choose the dispersion curve of maximum semblance from which can be found the shear slowness of the formation. According to preferred aspects of the QSTC processing, prior to Fourier transforming, the signals are windowed according to a previous estimate of slowness, where the time position of the window is found from STC or by maximizing either energy or semblance as a function of time for a predefined slowness. The reduced set of data in the window are then extracted for Fourier transformation, and prior to backpropagation, are corrected for the window slowness estimate to prepare them for backpropagation and stacking.
While STC, bias-corrected STC, and QSTC logs all provide valuable information, they are subject to different types of error. For example, as described above, STC processing does not account for, and hence is not optimal for dispersive waves such as flexural waves. In fact, because processing may occur in a frequency band near the low frequency limit, the signal to noise (S/N) ratio may be low, as little energy propagates in that band. Bias-corrected STC, while attempting to account for dispersion, is subject to error because the waveform spectra often disagree with the model, and because the model waveforms are bandlimited to a particular prescribed band in which there may be limited flexural mode energy. Furthermore, bias-corrected STC and QSTC are both subject to errors in the processing of a dispersive wave which arise from errors in the waveforms (i.e., S/N error) as well as errors in formation and borehole parameters (e.g., hole diameter) required in the processing of the dispersive wave information.
In order to reduce the error in the STC-type logs, in a first embodiment of the parent application hereto, the flexural waves are processed in a frequency band having a center frequency which is a function of the shear slowness, and a product of the shear slowness and the borehole diameter. The bandwidth of the filter is normalized by scaling the bandwidth to a specified fraction of the center frequency, while the window length used in the STC-type processing is likewise normalized to scale with the center frequency. According to a second embodiment of the parent application, the processing band can be chosen by obtaining the error of a plurality of formation and/or borehole parameters as a function of frequency, obtaining the signal to noise ratio of the flexural wave signal as a function of frequency, finding total error of different frequency bands as a function of frequency, and choosing a frequency band of minimum total error. Again, once the frequency band is identified, the bandwidth of the filter is normalized by scaling the bandwidth to a specified fraction of the center frequency, and the window length used in the STC-type processing is likewise normalized to scale with the center frequency.
While the parent application hereto is effective in finding an optimal frequency band for the processing of sonic data in the STC-type logs in order to reduce error in the slowness measurements, there is presently no effective manner of determining the variance of the slowness measurements obtained. While the semblance determination is currently used as a quality indicator of the STC-type processing, semblance only indicates quantitatively how well the waveforms stack, but not the variance of the slowness measurement itself. To appreciate this, a case where processing having the same bandwidth and window duration is applied to two sets of waveform data can be considered, where the second set of data has its energy concentrated at twice the frequency of the first set of data. If the signal to noise ratios of both sets of data are the same, the mean semblance values on the two sets of data will be the same. However, the variance on the first set of data will be four times that of the second set of data because of the increased slowness resolution of the higher frequency data. Thus, semblance indicates the slowness variance only if the frequency content of the signals does not change; and also, as can be shown, only if the number of receivers, the length of the time window, and the bandwidth of the data remains the same.
Another prior art technique for measuring the variance of the slowness measurement has been to divide the received waveform into a first window where no signal is expected and a second window where the signal is expected. The signal energy in the first window is taken as noise, while the signal energy in the second window is taken as a function of the true signal and noise. Using the signal energies in both windows, a signal to noise ratio (S/N) is derived, which serves as a quality indicator. This prior art technique suffers from the difficulty of choosing the position of the windows; and if any signal is found in the noise window, the resulting calculations will be unreliable. In addition, because the first and second windows are artificially chosen, they will not correspond to the window in which the slowness is being calculated. Further, the measured signal to noise ratio S/N does not directly indicate slowness variance.