Fractal Analysis Applied to Naturally Fractured Reservoirs Analog
Authors: Djezzar Sofiane , Boualam Aldjia , Chemmakh Abderraouf , Laalam Aimen , Ouadi Habib , Merzoug Ahmed 
Published: September 2022

ABSTRACT:
The fractal dimension theory is applied to the fracture network that impacts the basement and the Paleozoic formation in the northern part of the Hoggar area. The first technique used is the center distance algorithm, which considers only the fault centers distribution as a fractal. The second technique is the box-counting algorithm, which considers the entire network to be fractal. The fractal analysis of the entire 2D fracture networks in the zone of interest and the different fracture sets that affect the basement, and the Cambro-Ordovician units show that they have fractal dimensions based on both the center distance and the box-counting algorithms with values ranging between 1 and 2 with a power-law coefficient varying between 2.31 and 2.69 with high correlation coefficients. However, a few sets do not show fractal dimension may be due to the scattered fractures in the study area. The fractal dimension using the box-counting algorithm is 0.2 to 0.3 less than the fractal dimension using the center surface algorithm for the different networks and in the different fracture sets.

1. INTRODUCTION
The Cambro-Ordovician in the Algerian Saharan platform is characterized by tight sandstone formations with very low petrophysical characteristics where the natural fractures play an important role in their productivity. The Mouydir basin is the less explored basin with very low coverage of 2D seismic and wells. The study of the reservoir analog using the fractal dimension will help to understand the fracture distribution in the subsurface to guide the drilling of new prospective wells close to the fracture’s corridor. The fractal analysis has been used by many authors to illustrate the two-dimensional geometry of fracture networks (Allegre et al., 1982; Davy et al., 1990, 1992; Davy, 1993; Cowie et al., 1995, 1996; Bour et Davy, 1998, 1999; Bour et al., 2002; Bonnet et al., 2001; Darcel et al., 2003, Djezzar, 2019). Fractal geometry is a technique that can recognize and calculate how the geometry of patterns occurs from one magnitude to another (Mandelbrot, 1982). The fractal geometry provides a method for measuring the size scaling and spatial clustering of the full range of complex fracture networks (Barton, 1995). Many studies have investigated the fractal nature of fracture networks at different scales and report varied values which range from 1 to 2 (Bonnet & al, 2001). In our case study, the fractal dimension concept is applied to the 2D fracture networks that affect the basement and the Cambro-Ordovician in the northern part of the Hoggar shield, Algeria (Fig1). The fractal dimension for these 2-D fracture networks is estimated using two methods. The first method is the center distance algorithm, which considers only the fault centers distribution as a fractal. If the center’s population is fractal, this function is proportional to a power-law distribution. This dimension correlation gives an indicator of the faults’ center’s spatial distribution (Beicip, 2018). The second method is the box-counting algorithm, which considers the entire network to be fractal. The box-counting algorithm consists in discretizing a 2D fault trace map with different grids successively. The latter are square grids with constant limits but with decreasing cell size. For each iteration, the cells intersected by at least one fracture trace are counted. The number of intersected cells is plotted versus the grid cell size, on a log-log scale (Beicip, 2018, Cacas & al, 2001). The concept of fractal dimension is applied to verify whether the 2-D fracture networks that affect the basement and the Cambro-Ordovician reservoirs analog in the north Hoggar shield have fractal dimensions. The Ajjers, the In-Tahouite, and the Tamadjert units compose the Cambro-Ordovician reservoirs analog. They are characterized by a stiff tectonic style showing a dense fault network that affects all the Paleozoic series (Djezzar, 2012, 2021, 2022). The fractal dimension Dm is estimated using the center distribution and box-counting algorithms. According to geometry and structure, the fractures are gathered into major and minor faults where the fractal analysis is estimated for the whole fault network and the different fractures sets.