The coefficient of coherence between two stationary time series was introduced by Wiener in 1930. It is related to the signal‐to‐noise ratio, to the minimum prediction error, and has important invariance properties. As an estimate of this parameter, most geophysicists have used the so‐called “sample coherence.” An approximate distribution of the sample coherence for Gaussian data has been derived by N. R. Goodman. We have tested this distribution by means of Monte Carlo experiments for validity and robustness (insensitivity to the Gaussian assumption). It has passed the tests. The Goodman distribution provides a means of constructing estimates of the true coherence which are better than the widely used sample coherence. It can also be used to calculate confidence intervals. Finally, it forms a basis for choosing the lag window and data window necessary for best estimation of the true coherence. For good estimates of the true coherence, two precautions must be observed: 1. The cross‐spectrum and power spectra of the two time series must be smoothly varying over the width of the spectral window. 2. The ratio of the length of the data window to the lag window must be large. For most seismic work the second requirement severely limits the spectral resolution. Examples show that large errors can result if this resolution is not sufficient to satisfy the first requirement. In many geophysical studies the parameter of interest is the signal‐to‐noise ratio. Because of its relation to the coherence, the Goodman distribution provides a basis for its estimation as well.
Time‐lapse crosswell seismic data acquired with a cemented receiver cable have been processed into P‐ and S‐wave tomograms which image heavy oil sand lithofacies and changes as a result of steam injection. Twenty‐seven crosswell surveys were acquired between two wells over a 3.5 month period before, during, and after a 34‐day, 30 MBBL [Formula: see text] steam injection cycle. Interpretation was based on correlations with reservoir data and models, observation well data, and engineering documentation of the production history and steam cycle. Baseline S‐ and P‐wave tomograms image reservoir sand flow units and areas affected by past cyclic steam injection. S‐wave tomograms define lithology and porosity contrasts between the excellent reservoir quality, “high flow” turbidite channel facies and the interbedded “low to moderate flow” bioturbated levee facies. The reservoir dip of approximately 20° is defined by the velocity contrast between lithofacies. P‐wave baseline tomograms image lithology, porosity, structure, and several low velocity zones caused by past steam injection. Previous steam‐heat injection caused the formation of gas which reduced velocities as much as several thousand ft/s (600 m/s), an amount which obscures the velocity contrast between lithofacies and smaller velocity reductions as a result of temperature alone. Time‐lapse and difference P‐wave tomograms document several areas with small decreases in velocity during steam injection and larger decreases after cyclic steam injection. Velocity reductions range from 300 to 900 ft/s (90 to 270 m/s) adjacent to and above injectors located 20 to 50 feet (6 to 15 m) from the tomogram cross‐section. Poisson’s ratio tomograms show a significant decrease (.10) in the same area, and include low values indicative of gas saturation. Continuous injectors located 50 to 350 feet (15 to 100 m) from the survey area also caused a progressive decrease in velocity of the “high flow” channel sands during the time‐lapse survey. Interdisciplinary interpretation indicates that tomograms not only complement other borehole‐derived reservoir characterization and temperature monitoring data but can be used to quantitatively characterize interwell reservoir properties and monitor changes as a result of the thermal recovery process. Monitoring results over 3.5 months confirms that stratification has controlled the flow of steam, in contrast to gravity override. This suggests that tomographic images of reservoir flow‐units and gas‐bearing high temperature zones should be useful for positioning wells and optimizing injection intervals, steam volumes, and producing well completions.
S‐wave, P‐wave, and Poisson’s ratio tomograms have been used to interpret the 3-D distribution of rock and fluid properties during an early phase of a California heavy oil sand steamflood. Four lines of good quality crosswell seismic data, with source to receiver offsets ranging from 287 to 551 ft (87 to 168 m), were acquired in a radial pattern around a high temperature cemented receiver cable in four days. Processing, first‐arrival picking, and good quality tomographic reconstructions were completed despite offset‐related variations in data quality between the long and short lines. Interpretation was based on correlations with reservoir models, log, core, temperature, and steam injection data. S‐wave tomograms define the 3-D distribution of the “high flow” turbidite channel facies, the “moderate‐low flow” levee facies, porosity, and structural dip. The S‐wave tomograms also define an area with anomalously low S‐wave velocity, which correlates with low shear log velocities and suggests that pressure‐related dilation and compaction may be imageable. P‐wave tomograms define the same reservoir lithology and structure as the S‐wave tomograms and the 3-D distribution of low compressional velocity zones formed by previous steam‐heat injection and the formation of gas. The low P‐wave velocity zones, which are laterally continuous in the “high flow” channel facies near the top of most zones, indicate that the steam‐heat‐gas distribution is controlled by stratification. The stratigraphic control of gas‐bearing zones inferred from P‐wave tomograms is confirmed by Poisson’s ratio tomograms which display low Poisson’s ratios indicative of gas (<0.35) in the same zones as the low P‐wave velocities. The interpretation results indicate that radial survey tomograms can be tied at a central well and used to develop an integrated 3-D geoscience‐engineering reservoir model despite offset‐related variations in data quality. The laterally continuous, stratification‐controlled, low P‐wave velocity zones, which extend up‐dip, suggest that significant amounts of steam‐heat are not heating the surrounding reservoir volume but are flowing updip along “high flow” channels.
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