Application of Wavelet Transforms to Reservoir-Data Analysis and Scaling

Summary General characterization of physical systems uses two aspects of data analysis methods: decomposition of empirical data to determine system parameters and reconstruction of the system attributes using these characteristic parameters. Spectral methods, involving a frequency-based representation of data, usually assume stationarity. These methods, therefore, extract only 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 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 nonstationary data. In other words, a function or a discrete data set when transformed into a time-scale space using wavelets shows how it 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 wavelet transforms to one-dimensional and twodimensional 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. Wavelets are also applied to scaling up spatially correlated heterogeneous permeability fields.