Saturation Exponents Derived from Fractal Modeling of Thin-sections
Authors: Abdassah D. , Permadi P. , Sumantri Y. , Sumantri R. 
Published: December 2007

Abstract
Determination of the initial saturation of a hydrocarbon reservoir requires resistivity log data and saturation exponent. A number of experimental investigations has shown that reservoir wettability affect the exponent value. A remarkable divergence conclusions, however, still appear in the literature. The objective of the present study was therefore to investigate the effect of rock wettability on saturation exponent. Fractal concepts have been applied to the image of thin-sections of a core sample and used to derive the resistivity equation. The advantage of this approach is that it is independent of rock-fluids equilibrium problems that commonly influence laboratory measurements. A general equation of electrical resistivity has been developed in this study. Electrical tortuosity, clay content, and rock wettability are incorporated in the equation. The present work employed twenty thin-sections of limestone and sandstones.
It was found that the lowest exponent of 1.61 was obtained for strongly water-wet shaly sandstone and the highest value of 4.99 is for strongly oil-wet. The exponent consistently increases as the wetting condition is shifted from strongly water-wet toward oil-wet. It is close to 2.0 for clean sandstones at strongly water-wet, supporting the empirical formula of Archie.

Introduction
Estimation of a hydrocarbon reserve is strongly dependent of electric log data and the value of saturation exponent used. This exponent is usually either assumed to be 2.0 regardless of the reservoir wettability or is derived from laboratory measurements of electrical properties of the cores.
Saturation exponent has been the subject to many laboratory investigations. The researchers arrived at considerably different quantitative results, particularly for wetting conditions at oil-wet side. Detailed review of this matter has been made by Anderson and Sondenaa et al.
A lack of analytical investigation has been conducted to study the effect of wettability on electrical properties of rocks containing fluids. Fractal concepts have been applied to a wide range of research area related to porous media at various scales, from microscale (pore level) to megascale (field scale). A detailed review of fractal concepts applications has been presented by Sahimi and Yortsos. Briefly, a fractal object is an object that has a geometric dimension greater than its topological (Euclidian) dimension. Fractal geometry may describe the irregurality and fragmentation of natural objects and patterns. To characterize a fractal object or set, the fractal dimension D must be determined. A simple method is by "box counting". This is simply done by drawing a mesh of squares of side dimension that covers the set. The fractal dimension is determined from the slope of the relation between count and square side dimension. which is the fractal dimension that can yield any real value. To handle the complexity of the problems, the present study was directed to applying concepts of fractal analysis to determination of the electrical properties and thus saturation exponent.
Pore structures and geometries can be quite complex in nature. The complexity stems from the irregularity of pore wall surfaces. The presence of clays may increase the complexity of pore structures. As a matter of fact fluids distribution within the pores is influenced by the wetting condition. This in turn would affect the fluids connectivity. Therefore, electric tortuosity depends not only on pores and fluids connectivity but also on the irregurality of main grain surfaces and volume and distribution of clays within the pores. As pores of a porous medium can be magnified by employing microscope, identification of fine details of the pores can be performed through the use of thin sections made. Ultimately, mathematical analysis coupled with fractal analysis is thus possible to study electrical properties of porous media.
P. 143