Author: Mike Cline, T/X Resources
The Amplitude versus Angle (AVA) graph below was derived from another spreadsheet that I imported into EditGrid (see the previous Online Spread- sheet Calculations posting).
For the original spreadsheet, I used the Shuey equation from his classic AVO paper, “A Simplification of the Zoeppritz Equations”, published in Geophysics in 1985. The Shuey equation has been found to produce a good approxi- mation of results (up to about 30 degrees) that can be achieved using the more complex Zoeppritz Equation, and is much easier to program in a spreadsheet, than the latter.
The red and blue curves in the image below, represent the calculated AVA response of a simple three-layer model—in this case, a gas sand, with shales above and below the sand. However, in this example, I elected to “push the limits” and generated values up to an angle of 45 degrees. With Zoeppritz values posted in the pink, and cyan circles, you can see that the Shuey values compare quite well with the Zoeppritz values, even past 30 degrees in this case.
See the larger Adobe Reader pdf file.
The Velocity, Density, and Poisson’s Ratio input parameters were taken from Ostrander’s “Plane-Wave Reflection Coefficients for Gas Sands at Nonnormal Angles of Incidence”, another landmark AVO paper, published in 1984. The parameters for the shales and gas sand, produce values which are a “Class 3″ AVO response, typical of a shallow, unconsolidated gas sand.
The red curve is the AVA response of the top of the gas sand. Because of the lower velocity and density, it has a negative reflection coefficient at the incidence angle of zero degrees, also known as the “Zero Offset”, or “Normal Incidence” AVA attributes.
The amplitudes then get more negative with increasing incidence angles, which would result in a negative slope, or the “Gradient”. This “Gradient” AVA attribute is calculated from a linear fit of the amplitudes for angles from 0-30 degrees, using an x-axis of sine squared theta (where theta is the incidence angle), instead of using degrees as the unit of measurement—an AVO curve is converted to a linear feature on this type of axis.
The blue curve represents the calculated reflection coefficients for the base of the gas sand. The values for the “Zero Offset”, and the “Gradient” are both positive. You will notice that the blue curve only goes a little past 30 degrees. This is the point where it reaches the critical angle—the point where it refracts along the sand-to-shale interface, and ceases to generate reflected energy.
This is the actual spreadsheet from EditGrid, but in this example, the cells are write-protected, to prevent editing. The Zoeppritz values are also not posted on this graph.
EditGrid Spreadsheet by user/txrguy.
References:
Ostrander, W.T., 1984, Plane-Wave Reflection Coefficients for Gas Sands at Nonnormal Angles of Incidence, Geophysics, Vol. 49, No. 10, pp. 1637-1649.
Shuey, R.T., 1985, A Simplification of the Zoeppritz Equations: Geophysics, Vol. 50, No. 4, pp. 609-614.
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