Mathematical Analysis of Upscaling Well Log Data from Sonic to Seismic Resolution
Upscaling is an essential tool in microseismic studies, as it bridges the gap between the high-resolution well log measurements and low-resolution seismic observations. Various upscaling methods have been published and discussed, such as the Simple Averaging method, the Backus Averaging method, the Pair Correlation Function “PCF” method, the General Singular Approximation “GSA” method, and the Markov Chain Monte Carlo “MCMC” method. These methods are evaluated in this thesis, and new upscaling methods are proposed as a derivation from the Simple Averaging method. Most upscaling methods apply a running window average on well log data. The averaging method is what differentiates one method from the other, thus we explore the various mean calculations. The Pythagorean means (arithmetic, harmonic, and geometric) are three different way to average a set of numbers. Using a 2D layered earth model with homogenous and isotropic layers, the results of applying the Pythagorean means are as follows. The arithmetic mean provides the best results for averaging values within a layer, whereas the harmonic mean provides the best results for averaging values across an isotropic-layered medium. The geometric mean provides the best results for averaging within a heterogeneous layer. This study evaluates upscaling with respect to a Vertically Transversely Isotropic (VTI) media, which best corresponds to shales. It starts with the Simple Averaging upscaling method, which is a direct application of upscaling using the arithmetic mean. It then introduces a simple harmonic, simple geometric, and simple quadratic upscaling methods as a derivation from the Simple Averaging method, using different mean calculations. Various well log data are used to compare the proposed methods to the Simple Averaging and Backus Averaging upscaling methods. In conclusion, from a mathematical perspective the results for the Pythagorean means are different. However, the geophysical application of the means in upscaling well log data yields results with differences that fall within the margin of error. The results of the simple harmonic method are essentially the same as those using the Backus Averaging method. And at large upscaling windows, the behavior of the different upscaling methods is uniform. Although these new upscaling methods are purely mathematical and do not give respect to the physical characteristics of the medium, the use of the harmonic mean has higher validity over the use of the arithmetic mean from both a geophysical and mathematical perspective.