DATA

The data set used in this paper was taken from a well and was provided by Tapan Mukerji. It contained depth, travel time, volume shale fraction, porosity and density. The well extended 4.6568 km below the surface of the earth and readings of each variable were taken at every foot starting 1.1909 km below the surface.

First bad data was eliminated by removing any point where the travel time was found to be less than zero. The P-wave velocity was then found by taking the inverse of the slowness (which was measured in microseconds per foot) and converting to metric units. The S-wave velocity was necessary to compute AVO, and since S-wave velocities were not actually measured, it was calculated using the empirical methods described below.

As these empirical methods of determining S-wave velocities are simply straight lines drawn through three different sets of data relating V_p to V_s a particular rock type, the values are most likely not completely accurate for every set of data. However, the values determined for V_s by empirical relationships are useful for this project. S-wave velocities of shale layers (fraction of shale greater than 0.5) were calculated using the Mudrock line Castagna et al. (1985).

 

V_s=.8621V_p-1.1724 (1)

S-wave velocities of sand layers (fraction of shale less than 0.5) were calculated using the Han line Han (1986).

 

V_s=.7936V_p-.7868 (2)

The data set was then decimated to smooth the trends as well as make the data set smaller and easier to work with. The resulting data set was only 277 points long. The bulk moduli (K) and shear moduli ($\mu$) were calculated based on the velocities and the following equations.  

$$K=\rho(V_p^2-\frac{4}{3}V_s^2)$$ (3)

 

$$\mu=\rho{V_s^2}$$ (4)

Upon close examination of the data from this well log, one can see that this well does indeed match the specifications of the project. Figure 1 shows the velocities as a function of depth.

 

velupsidedown Figure 1 Velocity as a function of depth showing that it is likely that the velocities of sand and shale cross over somewhere near the center of the well.

From this graph, it appears as if the sand and shale velocities may cross over. Straight lines fitted to each set of velocities are shown in Figure 2. The fitted lines intercept approximately 2.4 km below the surface.

 

velfitupside Figure 2 Fitting velocity as a function of depth shows that the sand and shale velocities do cross over.