A STUDY OF FRACTAL ANALYSIS AND SINGULARITY SPECTRUM AND ITS POTENTIAL APPLICATION IN THE OIL AND GAS INDUSTRY

The world is made up of various irregular objects and signals. Although traditional mathematical techniques are not able to analyse these signals, it has been identified that these signals show common features such as singularities at various scales of observation. This indicates the existence of fractals within these signals. In the oil and gas industry, seismic data is a collection of reflected audio signals and is an example of irregular signals that could also have fractal features.
Even though we know that global petroleum resources are on the decline, oil and gas still remains the main source of energy throughout the world. This makes seismic exploration activities all the more important. Present indirect hydrocarbon detection techniques using seismic are costly and do not guarantee detection of oil or gas. Therefore, a technological advancement in the field of seismic exploration is evidently needed. Therefore, this study aims to develop a method for direct detection and delineation of hydrocarbons from seismic data.
In order to analyse the fractal nature of signals, a collection of mathematical steps known as fractal analysis is applied to generate a singularity spectrum. Although importance has been given on the methods of computing the singularity spectrum, there is little study on the effects of different types of singularities on the singularity spectrum. This study aims to understand how the singularity spectrum is affected by changes applied to input signals. It is by acquiring this knowledge first that the study also intends to develop an algorithm for direct detection of hydrocarbons. The Fraclab toolbox in MATLAB will be extensively used to achieve both of these goals.
From the study of the changes to singularity spectrum due to change in signals, it was observed that the square wave is the most irregular signal when compared with sine wave and sawtooth wave. Meanwhile, it was also discovered that a change in the amplitude of the periodic signal does not play a part in the final result of the singularity spectrum. The study has also observed that when two regular waves concatenate, the singularity spectrum produces more than one point due to the existence of a singularity or singularities at the point where the two signals concatenate. In direct comparison, when two periodic signals are added to one another, they only produce a dot on the singularity spectrum indicating that the end signal is still monofractal.