Abstract

Research conducted in rock destruction by high-power lasers indicates that lasers can be a less expensive, environmental friendly alternative to conventional drilling and perforating methods. However, the details of the laser-rock interaction and the evaluation of the porous medium properties after lasing are not yet well studied. In this work, fractal porosity and permeability models are developed to investigate the improvement or damage caused. To develop the model, first petrographic thin sections are made at right angles to the anticipated fluid flow direction. Then the pore area and tortuosity fractal dimensions are calculated using the box counting method. Finally, the calculated values are compared with the measured values. The model indicates that the fractal permeability is a function of the pore fractal dimension, tortuosity fractal dimension, maximum pore diameter, and capillary straight length. When all capillaries are straight (non-tortuous flow), permeability is sensitive only to the maximum pore diameter. This model effectively calculates the fractal permeability and porosity when compared to the permeability measured using a Pressure-Decay Profile Permeameter (PDPK) and the porosity calculated from thin sections.

Introduction

The term fractal was introduced by Mandelbrot(1)(from the Latin word fractus (to break) to characterize spatial or temporal phenomena that are continuous but not differentiable. Fractal objects and processes have scale dependence, self-similarity, complexity, and infinite length or detail. The necessary and sufficient conditions for an object or process to possess fractal properties have not been formally defined. Fractal geometry is thus considered as a collection of examples linked by a common point of view and not an organized theory((1)(. In spite of the empirical nature of the development of the fractal theory, it offers methods for describing the inherent irregularity of natural objects. In fractal analysis, the Euclidean concept of length is considered as a process. This process is characterized by constant parameters known as the fractal dimensions (DF, DP, or DT). A fractal dimension is considered as a relative measure of complexity or as an index of the scale dependencyof a pattern((1)(.

Fractal fragmentation theory was applied in material science by Yu and Cheng(2) to calculate permeability and porosity in a bi-dispersed wick in the evaporates of heat pipes. Karacan and Helleck (3)(built on the equations from Yu and Cheng(2)(for application in the petroleum industry to calculate permeability and porosity in porous media. The equations developed by Karacan and Helleck((3)(to calculate porosity and permeability used particle-size distribution of the grains in the crushed zone around perforation tunnels.

In this research, Yu and Cheng(2)(and Karacan and Helleck(3) equations are merged together to generate permeability and porosity equations that can use textural data from thin section and can be applied in reservoir porous medium. Samples used in this study are rocks lased using high-power lasers to simulate drilling or perforating. Measurements and calculations are made around the lased hole in four different rock types (shale, limestone, granite, and Berea sandstone).

The box counting technique proposed by Mandelbrot(1) is used to calculate the tortuosity fractal dimensions from petrographic thin sections made at right angle to the flow path.

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