Relation Between Elastic Heterogeneity, Stress and the Gutenberg Richter b-value: A b-value Estimation using Sonic Logs

In this paper we establish relations between the fractal distribution of elastic parameters, stress field variations and the Gutenberg-Richter b-value of earthquakes. We extract information about elastic parameter distribution from sonic well logs to create an elastically heterogeneous 3D random medium with a fractal spatial correlation function. Using the finite element program Abaqus we apply an externally homogeneous stress field and compute the resulting stress distribution inside the model medium. By applying geomechanical considerations we determine the distribution of Coulomb Failure Stress (CFS). We find that elastic heterogeneity causes strong spatial CFS variations. We assume that if the CFS is perturbed e.g. by a fluid injection, rupturing takes place inside clusters of interconnected critically stressed cells of the model. From the resulting size distribution of clusters we compute fault size and correspondingly magnitude scaling. We find that fault sizes exhibits power law scaling according to the Gutenberg Richter relation. The application of our method to well log data and stress profiles measured along the Continental Deep Drilling Site (KTB, Germany) main hole results in a b-value of b=0.95, which is in agreement with the b-value of b=1 computed from fluid injection induced seismicity at the KTB.