Relations between elastic rock heterogeneity, stress and the Gutenberg Richter law

摘要

In this paper we establish relations between the fractal distribution of elastic parameters, stress fieldrnvariations and the Gutenberg-Richter b-value of earthquakes. We show that elastic heterogeneity in arnreservoir, causes significant stress variations characterized by scaling of power law type, which resultsrnin magnitude scaling of Gutenberg-Richter type with b-value close to b=1. We extract informationrnabout elastic parameter distribution from sonic well logs to create an elastically heterogeneous 3Drnrandom medium with a fractal spatial correlation function. Using the finite element program Abaqusrnwe apply an externally homogeneous stress field and compute the resulting stress distribution insidernthe model medium. We apply geomechanical considerations to determine the distribution of CoulombrnFailure Stress (CFS) as a measure of the vicinity to failure. We find that elastic heterogeneity causesrnstrong spatial CFS variations. Some parts of the model are in direct vicinity to failure, suggesting thatrnsmall stress perturbations (<1MPa) may induce seismic events. Nevertheless, in the most stable partsrnof the model CFS perturbations of up to 20 MPa are needed to initiate failure. The obtained range ofrnCFS is in agreement with fracture strength reported for fluid injection induced seismicity inrncrystalline and sedimentary rocks. We assume that if CFS is perturbed e.g. by a fluid injection,rnrupturing takes place inside clusters of interconnected critically stressed cells of the model. From thernresulting size distribution of clusters we compute fault size and correspondingly magnitude scaling.rnWe find that fault sizes exhibits power law scaling according to the Gutenberg Richter relation. Thernapplication of our method to well log data and stress profiles measured along the Continental DeeprnDrilling Site (KTB, Germany) main hole results in a b-value of b=0.95, which is in agreement withrnthe b-value of b=1 computed from fluid injection induced seismicity at the KTB.