The finite extension of fractal geometry and power law distribution of shallow earthquakes: A geomechanical effect

We study a dome structure (10×10×10 km3) at an intermediate scale between laboratory analyses and tectonic processes. Local gas extraction induced about 1000 events within the dome (1.0 ≤ M1 ≤ 4.2) recorded by a local network during 19 years (1974–1992). Two types of autosimilarity coefficients (b value and correlation dimension) are analyzed in three-dimensional (3-D) space. The hypocenter distribution shows a fractal pattern characterized by the noninteger value of the correlation dimension. Moreover the frequency-magnitude relation of the events obeys a power law. The existence of these two parameters shows that the spatial distribution of the earthquakes induced by the Lacq gas extraction is governed by a nonrandom behavior. We observe no temporal correlation between the temporal behavior of the b value (slope of frequency-magnitude relation) and D (correlation dimension). Three-dimensional fractal analysis of induced earthquakes allows us to define two distinct classes of events separated by a critical distance of 500 m. The first class (r > 500 m) shows a diffuse seismicity. This diffuse class of earthquakes (M1 500 m, D2 ≈ 1.3). The difference of one unit between the fractal dimension of the seismicity within the nests (D1) and the fractal dimension of the nests distribution (D2) suggests an influence of the geological dome structure on the spatial development of seismic nests. Moreover, a slope break above M1 = 3.0 (G-R relation) is observed on this second class (1.0 ≤ M1 ≤ 4.2). The slope break of both the b value and the fractal dimension at a common threshold (M1 ≈ 3.0 is equivalent to a 500-m fracture size) suggests a critical distance for the brittle behavior of the uppermost crust as proposed for tectonic earthquakes by Scholz (1991). Such a critical distance correlates in our study with the maximum thickness of local seismogenic layers (brittle calcareous layer versus ductile marly layer). On this basis we propose that (1) the finite extension of the earthquake power law is driven by the local setting and therefore is also a scale dependent process, (2) the geomechanical link between fractal behavior and fracture size, i.e., a physical mapping of the power law behavior, must also be found in the boundary values of the autosimilarity processes (slope breaks) rather than in the values of the power law exponents.

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