Application of Wavelet Transforms to Reservoir Data Analysis and Scaling
Authors: Panda M.N. , Mosher C. , Chopra A.K. 
Published: December 2007

Abstract
General characterization of physical systems uses two aspects of data analysis methods: decomposition of empirical data to determine model parameters and reconstruction of the image using these characteristic parameters. Spectral methods, involving a frequency based representation of data, usually assume stationarity. These methods, therefore, extract only the average information and hence are not suitable for analyzing data with isolated or deterministic discontinuities, such as faults or fractures in reservoir rocks or image edges in computer vision.
Wavelet transforms provide a multiresolution framework for data representation. They are a family of orthogonal basis functions that separate a function or a signal into distinct frequency packets that are localized in the time domain. Thus, wavelets are well suited for analyzing non-stationary data. In other words, a projection of a function or a discrete data set onto a time-frequency space using wavelets shows how the function behaves at different scales of measurement. Because wavelets have compact support, it is easy to apply this transform to large data sets with minimal computations.
We apply the wavelet transforms to one-dimensional and two-dimensional permeability data to determine the locations of layer boundaries and other discontinuities. By binning in the time-frequency plane with wavelet packets, permeability structures of arbitrary size are analyzed. We also apply orthogonal wavelets for scaling up of spatially correlated heterogeneous permeability fields.

Introduction
Whenever there is a need for processing large volumes of data one seeks to represent them in terms of their statistics or spectral content to reduce both the volume and redundancy of data. Examples of this application are abundant. In computer vision, efficient and economic data transmission is accomplished by representing images by their spectral content. In geophysics, identification of important features in seismic waveforms requires an efficient noise reduction by separating data into different frequency bands. Reservoir characterization in petroleum engineering and hydrology often uses spectral contents of the spatial distribution of reservoir properties, such as porosity and permeability, given by semi-variograms to represent the spatial structure of data.
Reservoir characterization, which is the main focus of this paper, seeks to define a configuration of reservoir properties such that the image has desired properties. Usually, random realizations from a pre-defined probability density function, obtained from experimental observations, form the elements of all configurations. The solution procedure involves two steps. The first step involves data analysis for parameter estimation, such as statistics and spectral content of data. Often characteristics of spatial structures present in data are also determined in this step. The second step generates data (realizations) using the parameters determined in the first step. This step often also involves a scale change, such as scaling up for data reduction and scaling down for enhancement.
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