Throughout the past century, the mechanical wave has shared the spotlight with various complex phenomena. In 1905, Albert Einstein shook the physics world when through his analysis of the photoelectric effect he showed that light must exhibit properties of both a wave and a particle, contradicting the previous 300 years of science. This wave-particle duality of light was ultimately explained through quantum mechanics, using tools such as the Schrodinger Equation, which is a parallel to Newton’s second law in classical mechanics.

Prior to the 1970s, the general consensus among dynamicists was that the simple harmonic

Figure 1. The red curve is the reflected P wave (RP), the green curve is the reflected PS wave (RPS) and both curves are plotted as AOI versus their respective reflection coefficient. (All figures courtesy of Griffin Phillips)
oscillator was exactly that, a predictable, perfectly harmonic mechanical system. However, throughout the 1970s it was shown that some very basic mechanical oscillators had a very complex underlying sensitivity to their initial conditions, ultimately leading to the discovery of the chaotic dynamical system. Although it is difficult to understand these chaotic sensitivities to the initial conditions, simply being aware of their outcomes can be sufficient. Chaos theory is currently being implemented to understand the apparent randomness in epileptic seizures.

In the course of the past year, the mechanical wave has received much attention in the form of multicomponent seismic data. The fundamental principle is that when a seismic wave crosses a boundary between two different media, a portion of the wave will undergo a reflection and, if the angle of incidence (AOI) is not too large, a portion of the wave will refract as well. However, each reflection or refraction can also split into two different modes of propagation, one being longitudinal and the other being transverse. This effect was described in a paper by Karl Zoeppritz, released 11 years after his death in 1919. Although fairly complicated, the Zoeppritz equations give an accurate account of the amplitudes and propagation modes of seismic waves once they are reflected or refracted. The advantage to multicomponent seismic is that even with a longitudinal source signal, the compressional wave (P) data is recorded along with the mode converted shear wave (PS) data, resulting in two seismic datasets. Once processed, each dataset contains its own special attributes resulting from the respective wave properties. For example, the velocity ratio (VP/VPS) has proven to be a valuable indicator for the shale content of a sand play, helping to identify the higher porosity and permeability regions of the formation.

Another recent development for the mechanical wave is spectral decomposition (SD). SD exists in many forms resulting from several different algorithms, but the results are often similar and equally puzzling. Among the first results from SD were the observations of low-frequency gas shadows below gas-bearing formations. For the Mississippian Limestone
Figure 2. The red line is the actual seismic ray path, and the blue dashed line is the linear approximation.
formation in northern Texas, the SD results observed for oil-producing reservoirs were the same for both solution gas-drive and water-drive mechanisms, suggesting that it was only the reservoir porosity that caused the frequency anomaly. These SD anomalies were always approximately half of the median frequency of the data; they existed about 40% deeper in the dataset than the actual reservoir, and they required a linear ray path AOI of at least 17°. While these observations may seem a bit peculiar, the physical explanation may be coupled to multicomponent seismic. More specifically, these SD low-frequency anomalies may actually be mode-converted shear waves.

To further explore this relationship, the Zoeppritz equations mentioned in the introduction can be used to predict what types of wave will be reflected. If the upper medium is a limestone with a density of 156 lb/cu ft (2,500 kg/cu m) and VP of 3,000 m/sec, and the lower medium is a fluid with a density of 62.4 lb/cu ft (1,000 kg/cu m) and a VP of 1,500 m/sec, then the reflection coefficients can be estimated using the curves in Figure 1. For this figure, the red curve is the reflected P wave (RP), the green curve is the reflected PS wave (RPS) and both curves are plotted as AOI versus their respective reflection coefficient. Based on these functions, the RPS component of the wave will have the stronger of the two reflection amplitudes between the angles of incidence of 24 and 77 degrees, where the maximum difference between the two will occur at 41 degrees. However, since a seismic ray path through the earth approaches a parabolic function, the actual parabolic AOI will be greater than a linear AOI. In Figure 2, the red line is the actual seismic ray path where the blue dashed line is the linear approximation. For a depth of more than 4,000 ft (1,220 m), it would not be unusual for the parabolic AOI to be twice as large as the linear AOI.

The next aspect to consider is the difference between VP and VPS. If a PS wave was created from a high-porosity region of the Mississippian at 4,300 ft (1,312 m), then it would arrive at the surface much later in time than its fellow P-wave component. Assuming the VPS function is approximately half of the VP function and both are dependent on the depth y, then the arrival time of a PS wave from this formation can be calculated by integrating the velocity functions. Using this method, the RPS seismic component would arrive at approximately 1.22 seconds, where the traditional RP component arrives at 0.775 seconds. Furthermore, the median frequency of the PS wave would be much less compared to the P component since frequency is an observed property dependent only on the wavelength and the speed of the wave. In order for these PS waves to be recorded with a traditional Z-axis geophone, a non-vertical emergence angle would be required for a PS wave to project its motion onto the vertical Z-axis. The notion of a non-vertical emergence is easy to entertain since the existence of a PS wave would require a far offset and therefore a large AOI.

In order to test this theory, a single-component 3-D dataset where a poststack SD was deemed successful for a Mississippian producer at 4,300 ft was analyzed. Using the same
Figure 3. The CDP gather overlaying a spectral content plot. Red indicates the strongest spectral content and white indicates no spectral content for a frequency range of 13 to 17 Hz.
wavelets from the poststack analysis, an SD was conducted on a common depth point (CDP) gather located at the well bore. Figure 3 shows the CDP gather overlaying a spectral content plot. In this white-green-red color scheme, red indicates the strongest spectral content, and white indicates no spectral content for a frequency range of 13 to 17 Hz, which is half the median frequency of the data for this depth. The strongest anomaly from this figure is located at offset number 20, which corresponds to a source-receiver separation distance of 2,585 ft (788.4 m) and a depth of 1,250 milliseconds. Using half of the offset distance and a reservoir depth of 4,300 ft, the linear ray path AOI is 17°. Recognizing that the parabolic ray path AOI is likely double the linear AOI, then this anomaly exists at a CDP offset that is expected to contain mode-converted data. In addition, the depth of this anomaly is consistent with the travel time calculations.

In conclusion, the characteristics from the SD anomaly observed in this CDP gather are consistent with the aforementioned theory. The anomaly exists only at specific AOI over the reservoir, it is half of the median frequency of the data and it is located at a specific distance in time below the reservoir.

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