# Estimation of Litho-Fluid Facies Distribution from Zero-Offset Acoustic and Shear Impedances
## 5 Results and Discussion
A group wavelet was extracted from the angle gathers such that it consisted of three wavelets at the three central angles: 10\({}^{\circ}\), 20\({}^{\circ}\), and 32.5\({}^{\circ}\). The initial model of the simultaneous inversion was created from wells \(\times\)1 and \(\times\)2 such that the target zone ranged from two-way time 1000 to 1800 ms with a two-millisecond (ms) sample rate. A smoother was applied to the modeled trace in the output domain after lateral interpolation. The inverted properties of the inversion were the P-impedance, S-impedance, and density. Similarly, an initial model was created for each angle stack to invert for P- and S-impedances at the near, mid, and far incident angles. Figures 8 and 9 show the change in the P- and S-impedance values by changing the incident angle for each facies class at the well (\(\times\)1). The anisotropy effect appears in the shale-dominated zones, as shown in Figure 8d. The P-impedance seems directly proportional to the incident angle, especially in the black circles, where the values gradually increase from the near to the far impedance sections. On the other hand, the gas sands of the A and B horizons appear in green color and almost have the same impedance values at the near, mid, and far angles. These results seem consistent with a previous study that calculates the P-wave velocity for the Floyd shale by three different models, as shown in Figure 10[127]. The first and second methods, T_45 and T_60, are based on [PERSON]'s model [53], where the Delta parameter is obtained using the \(C_{13}\)-component on the 45\({}^{\circ}\) and 60\({}^{\circ}\) plugs, respectively. The third model, \(Vp\_b\), is based on the strong anisotropy assumption [128]. The three models show that the P-wave velocity increases by increasing the incident angle due to the  
Figure 8: The partially inverted \(Z_{p}\) sections at the near, mid, and far incident angles (from (a–c), respectively) against the facies log at well \(\times\)1 (d). intrinsic anisotropy of shale. The results are confirmed by the ultrasonic measurements applied to the core sample. A previous study shows the measurements of the P- and S-wave velocities in shale core samples which indicate moderate intrinsic anisotropy, where the Epsilon (\(\epsilon\)) is 0.16, Delta (\(\delta\)) is 0.08, and Gamma (\(\gamma\)) is 0.29 [129]. Figure 11 shows a plot of velocity versus angle, where the quasivertical P-wave velocity (qP), quasivertical S-wave velocity (qSV), and horizontal S-wave velocity (SH) increase by increasing the incident angle at the angle range up to approximately 35\({}^{\circ}\). These results are consistent with the current study that claims that the P- and S-impedances increase by increasing angles, as shown in the black circles of Figures 8 and 9. An important note about seismic data is that the velocity change with angle may be due to other reasons rather than seismic anisotropy, such as tuning effect and noise. Accordingly, the next step is to estimate the zero-offset impedance from the partially inverted volumes to normalize the random error and anisotropy effect simultaneously. Figure 10: A plot of the P-wave velocities vs. the incident angles for the Floyd shale. Figure 9: The partially inverted \(Z_{S}\) sections at the near, mid, and far incident angles (from (**a**–**c**), respectively) against the facies log at well \(\times 1\) (**d**). The total data points collected for the \(Z_{P0}\) and \(Z_{S0}\) models are 1292 and 1582, respectively. These data points were gathered from the Backus-averaged impedance logs of wells \(\times 1\) and \(\times 2\) and the depth-converted composite traces of the partially inverted impedances at the wells' locations. The data points were randomly divided into three parts such that 70% of them were used for training, 15% for testing, and 15% for validation. Figures 12 and 13 show the trace plots of the Markov chains, which aim at estimating the coefficients for both the \(Z_{P}\) and \(Z_{S}\) models, respectively.