Fractal characterization of fracture networks: An improved box-counting technique

[1] Box counting is widely used for characterizing fracture networks as fractals and estimating their fractal dimensions ( D). If this analysis yields a power law distribution given by N proportional to r(-D), where N is the number of boxes containing one or more fractures and r is the box size, then the network is considered to be fractal. However, researchers are divided in their opinion about which is the best box-counting algorithm to use, or whether fracture networks are indeed fractals. A synthetic fractal fracture network with a known D value was used to develop a new algorithm for the box-counting method that returns improved estimates of D. The method is based on identifying the lower limit of fractal behavior (r(cutoff)) using the condition ds/dr -> 0, where s is the standard deviation from a linear regression equation fitted to log(N) versus log(r) with data for r < r(cutoff) sequentially excluded. A set of 7 nested fracture maps from the Hornelen Basin, Norway was used to test the improved method and demonstrate its accuracy for natural patterns. We also reanalyzed a suite of 17 fracture trace maps that had previously been evaluated for their fractal nature. The improved estimates of D for these maps ranged from 1.56 +/- 0.02 to 1.79 +/- 0.02, and were much greater than the original estimates. These higher D values imply a greater degree of fracture connectivity and thus increased propensity for fracture flow and the transport of miscible or immiscible chemicals.