Abstract

Wavelet transforms are a family of basis functions that separate a function of a signal into distinct frequency packets that are localized in the time domain. Thus, wavelets are well suited for analyzing nonstationary data. It can smooth the basic signals and keep and even enhance of details. So it provides a multiresolution framework for data representation.

Wavelet transform is now used in a wide variety of applications in the areas of medicine, biology, data compression, etc. In recent years, wavelet analysis is increasingly applied to the data analysis in petroleum industry. This paper reviews the recent application of the wavelet transform methodologies to data analysis in petroleum industry, especially the application in reservoir engineering fields. Various field examples are provided to demonstrate the applications of wavelet transform in petroleum industry, especially in the areas of well testing, well logging, seismic, and petroleum geology.

Introduction

Perhaps the most powerful mathematical tool for analyzing and processing signals and images is the Fourier method, invented in the 1920s by Joseph Fourier, a French mathematician and physicist. A signal is considered as a superposition of sine and cosine waves with different frequencies in Fourier transform (Figure 1).

The success of the Fourier method is due to the fact that in practice a limited amount of the sine or cosine functions suffices to discover the main characteristics of a signal, The 'Fourier coefficient' for a certain frequency gives the average strength of that frequency in the full signal. As a sine or cosine function keeps undulating to infinity, the Fourier coefficients don't provide direct information about the local behavior of a signal; they only give a kind of average information on the signal as a whole.

Although Fourier is a power mathematical tool, it is not very good at detecting rapid changes in signals even for the revised Fast Fourier Transform. The major disadvantage of Fourier transform lies in its lack of localization and it considers phenomena in an infinite interval which is very far from our everyday point of view. Furthermore, many signals such as occurring in seismic data and well test data in petroleum industry, display structure at many different scales. The Fourier method misses' most of this multi-scale structure (Figure 2). So we either "can't see the forest for the trees" or "can't see the trees for the forest" in the Fourier transform.

But another new developed mathematical transform can make us "seeing the forest and the trees" at the same time. It is the wavelet transform. To understand wavelet more easily, let's take a look at the way of how our eye look at the forest in the real world.

In the real world, when we look at the forest from the window a flying helicopter, the forest appeared to be a blanket of green. But from the window of an automobile on the ground, the blanket of green resolves into individual trees. If we get out of the car and walk into the forest, you will begin to see the branches and leaves.

This content is only available via PDF.
You can access this article if you purchase or spend a download.