Relations between the scaling exponents, entropies, and energies of fracture networks

Abstract

Abstract Fracture networks commonly show power-law length distributions. Thermodynamic principles form the basis for understanding fracture initiation and growth, but have not been easily related to the power-law size distributions. Here we present the power-law scaling exponents and the calculated entropies of fracture networks from the Holocene part of the plate boundary in Iceland. The total number of tension fractures and normal faults used in these calculations is 565 and they range in length by five orders of magnitude. Each network can be divided into populations based on ‘breaks’ (abrupt changes) in the scaling exponents. The breaks, we suggest, are related to the comparatively long and deep fractures changing from tension fractures into normal faults and penetrating the contacts between the Holocene lava flows and the underlying and mechanically different Quaternary rocks. The results show a strong linear correlation (r = 0.84) between the population scaling exponents and entropies. The correlation is partly explained by the entropy (and the scaling exponent) varying positively with the arithmetic average and the length range (the difference between the longest and the shortest fracture) of the populations in each network. We show that similar scaling laws apply to other lineaments, such as streets. We propose that the power-law size distributions of fractures are a consequence of energy requirements for fracture growth.