Use of the Beta Distribution To Determine Well-Log Shale Parameters
- Robert S. Gullco (CGG-Veritas) | Malcolm Anderson (University of Brunei)
- Document ID
- Society of Petroleum Engineers
- SPE Reservoir Evaluation & Engineering
- Publication Date
- December 2009
- Document Type
- Journal Paper
- 929 - 942
- 2009. Society of Petroleum Engineers
- 1 in the last 30 days
- 533 since 2007
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The Beta distribution in n-dimensions is introduced to describe the proportions of the mineralogical components existing in a certain stratigraphic interval (the porosity is included as a "mineralogical component"). The justification for doing so is empirical. The model allows the calculation of well-logging parameters, such as GRma, GRsh, and shale density, without having to introduce them by "eye." It also allows the probabilistic calculation of the rock composition at each depth when there are more mineralogical components than logs: that is, there is a shortage of equations. In addition to this, the Beta model can be used to test the hypothesis that the relationship between any two components can be regarded as random, which should have applications in reservoir characterization.
Sedimentary rocks may be described ultimately as a mixture of minerals and pores. For a given lithological column, it is possible using well logs to calculate the composition of the rocks at discrete points. We may ask which should be the probability distribution of the volume fraction of each mineral component (with the porosity included as a "mineral component") along this lithological column. This distribution should satisfy at least the following conditions:
(a) The values of each of the components should range between 0 and 1.
(b) The sum of all the components should be equal to 1, for all points.
The well-known Beta distribution, which is also known as the Dirichlet distribution in the multidimensional case (Gelman et al. 2003), satisfies these requirements. Although, in theory, this distribution allows for a porosity of 1, in practice the values of the parameters of the distribution are such that very high porosities are extremely unlikely. There are also empirical observations that support the use of this distribution to model rocks. It is quite frequent to see histograms of the gamma ray (GR) log across more or less "homogeneous" intervals, which are clearly unimodal and asymmetrical (i.e., they are skewed). If we assume that the GR log is sensitive to only one component (the "shale"), then, if the shale volume fraction is Beta-distributed, the character of the GR log can be explained easily.
In summary, despite the lack of a sound theoretical background, there are some numerical characteristics and empirical observations that justify the introduction of this distribution.
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