Abstract 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, shale density, etc., 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.
Introduction Sedimentary rocks may be ultimately described as a mixture of minerals and pores. For a given lithological column, it is possible to calculate the composition of the rocks at discrete points, with well logs. 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 distibution should satisfy at least the following conditions:
The values of each of the components should range between 0 and 1
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 (en.wikipedia.org/wiki/Dirichlet_distribution) in the multidimensional case, 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 which support the use of this distribution to model rocks. It is quite frequent to see histograms of the Gamma Ray log over more or less "homogeneous" intervals, which are clearly unimodal and asymmetrical (i.e. they are skewed). If we assume that the Gamma Ray log is sensitive to only one component (the "shale"), then, if the shale volume fraction is Beta distributed, the character of the Gamma Ray log can easily be explained.
In summary, despite the lack of a sound theoretical background, there are some numerical characteristics and empirical observations which justify the introduction of this distribution.
There are two main reasons to use this distribution in petrophysics and well log analysis:
It allows the calculation of parameters from the data, without having to introduce them arbitrarily or by "eye" (for instance, the estimation of Grma, the Gamma Ray response of the "clean" rock and Grsh, the Gamma Ray log response of the "pure" shale).
In zones of complex mineralogy, where there are more components than logs, the Beta model allows the introduction of further equations which ultimately result in a solution - albeit a probabilistic one - of the system. This problem has also been dealt with by McCammon (1972), although the approach of this author is quite different, applying Information Theory to solve for the proportions of the components.
In this paper, we will calculate Grma and Grsh from the Beta model for some real cases. The second point we will deal with from a theoretical point of view. In addition to these practical applications, the theoretical question of whether or not sedimentary rocks can be regarded as random mixtures of components will be considered.
Properties of the Beta Distribution To simplify the problem, we will restrict our discussion to mixtures of three components (say "sand","shale" and effective porosity). However, the principles discussed below are easily extended to mixtures of n components.