Statistical Physics of Fissure Swarms and Dike Swarms

Agust Gudmundsson

Fissure swarms and dike swarms in Iceland are 40-150 km long, 5-20 km wide, extend to depths of 10-20 km and contain 2 × 10exp14 outcrop-scale ( 0.1 m) and 10 exp 22-23 down to grain-scale (1 mm) fractures, suggesting that statistical physics is an appropriate method of analysis. Length-size distributions of 565 outcrop-scale Holocene fissures (tension fractures and normal faults) and 1041 Neogene dikes in Iceland show reasonably good to excellent fits with negative power laws and exponential laws. Here the Helmholtz free energy is used to represent the energy supplied to the swarms and to derive the Gibbs-Shannon entropy formula. The calculated entropy of 12 sets and subsets of fissures and 3 sets and subsets of dikes all show strong positive correlation with sets/subsets length ranges and scaling exponents. Statistical-physics considerations suggest that, at a given time, the probability of the overall state of stress in a large body/crustal segment being heterogeneous is much greater than the state of stress being homogeneous and favourable to the propagation of a fissure or a dike. In a heterogeneous stress field, most fissures/dikes become arrested after a short propagation – which is a formal explanation of the observed statistical size-length distributions. As the size of the stress-homogenised rock volume increases larger fissures/dikes can form, increasing the length range of the distribution (and its entropy) which may, potentially, transform from an exponential distribution into a power-law distribution.