Two-Dimensional Multifractional Brownian Motion- Based Investigation of Heterogeneities from a Core Image

Abstract

A sample is a cylindrical section obtained by driving a hollow tube into the undisturbed medium and withdrawing it with its content. In practice, the sample is pushed more or less unbroken into the tube. Once removed from the tube in the laboratory, it is analyzed by different techniques and equipment depending on the desired type of data. The hole made for the sample is called the core hole. A variety of samplers exist to sample different media under diverse conditions. For instance, sediments or rocks are sampled with a hollow steel tube called a drill. A scientific coring has been used in the first time for sampling the ocean floor. Then, it is soon exploited to analyze lakes, ice, mud, soil and wood. Cores provide precious information about the evolution of climate, species and sedimentary composition during geologic history. In petroleum engineering, analysis presents a way of measuring well conditions downhole by studying samples of reservoir rocks. It gives the most accurate estimations of porosity, permeability, fluid saturation and grain density. These measurements help to understand the conditions of the well and its potential productivity. In addition to the basic petrophysical properties estimated from the core, a special analysis can be undertaken in order to determine permeability, wettability, capillary pressure, and electrical properties. Petrographic studies and sieve analysis can also be carried out in such analysis. In recent years, numerical analysis has been widely used for the investigation of images, since it yields results more objective and reliable than those obtained by conventional methods based on human observations. Fractal analysis has been introduced to examine images texture (Bourissou et al., 1994; Levy-Vehel and Mignot, 1994; Liu and Li, 1997; LevyVehel, 1995, 1997, 1998; Pesquet-Popescu and Levy-Vehel, 2002; Malladi et al., 2003; Tahiri et al., 2005). In this study, we suggest to go beyond the conventional analysis, and to perform a new approach to extract the maximum features from a image using a fractal analysis. The conventional fractal model used previously in image processing, the two-dimensional fractional Brownian motion (2DfBm), presents a constant Holder function H, thus does not allow to explore the spatial evolution of the local regularity. To do so, we suggest to