98 Well seismic surveying and acoustic logging At sonic frequencies (1-20 kHz), it is therefore necessary to independently measure six parameters to derive the S velocity of the formation from the Stoneley mode dispersion equation. These parameters are: the phase velocity of the Stoneley waves at a particular frequency, the fluid density, the formation density, the well diameter, and the velocity of the compression waves in the formation. The following example is an acoustic logging recorded in a slow formation, consisting of marl in the upper part and limestone in the lower part. The boundary between marl and limestone is at 105 m. The data were acquired with a monopole tool with three receivers spaced 20 cm apart. The offset between the source and the first receiver is 60 cm. Figure 3.14 shows on the left the three constant offset sections recorded with the acoustic tool. On each section we can observe the low amplitude refracted P-wave as a first arrival. The refracted P-wave is followed by a wave of very high amplitude and low frequency, which is the Stoneley wave. The measurement of P-wave and Stoneley wave velocities is carried out by semblance. The semblance panel is shown on the left of the acoustic sections. The vertical axis represents the depth, the horizontal axis is the slowness (inverse of the velocity). The semblance is color coded and expressed as a percentage (high values shown in red). The picking of the maximum semblance (indicated by the continuous black lines) provides for each wave the value of the slowness as a function of the depth. Slowness logs are then converted to velocity logs (Figure 3.14, right). We observe a high correlation coefficient between the 2 velocity logs (0.854). The dominant Stoneley wave frequency is 2 kHz. At these low frequencies, the Stoneley wave dispersion equation can be approximated by a simplified equation proposed by White (1965). White’s equation is as follows: 1 1 1 2 2 2 V V V st f f s − = ⋅ ρ ρ where Vst is the velocity of the low frequency Stoneley wave, Vf is the velocity of the fluid in the formation (water in this case), Vs is the S velocity of the formation, ρ is the formation density, and ρf is the fluid density. If the density log has not been recorded, which is the case in this example, it can be calculated from the VP velocity of the formation using Gardner’s equation: ρ α β = ×Vp The equations of White and Gardner are used simultaneously to adjust the coefficient α and β of Gardner’s equation and to calculate the velocity Vs and density ρ of the formation with the following constraints: 1- The S velocity of the formation must be lower than the P-wave velocity in the fluid. 2- Poisson’s ratio must remain in the range 0.3 to 0.5, characteristic of marls and unconsolidated formations. Figure 3.15 shows, from left to right: the Gardner density, the S velocity estimated from the Stoneley wave velocity, the VP to VS ratio and the Poisson’s ratio.
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