THE FINITE EXTENSION OF FRACTAL GEOMETRY AND POWER-LAW DISTRIBUTION OF SHALLOW EARTHQUAKES - A GEOMECHANICAL EFFECT

We study a dome structure (10x10x10 km(3)) at an intermediate scale be tween laboratory analyses and tectonic processes. Local gas extraction induced about 1000 events within the dome (1.0 less than or equal to M(1) less than or equal to 4.2) recorded by a local network during 19 years (1974-1992). Two types of autosimilarity coefficients (b value a nd correlation dimension) are analyzed in three-dimensional (3-D) spac e. The hypocenter distribution shows a fractal pattern characterized b y the noninteger value of the correlation dimension. Moreover the freq uency-magnitude relation of the events obeys a power law. The existenc e of these two parameters shows that the spatial distribution of the e arthquakes induced by the Lacq gas extraction is governed by a nonrand om behavior. We observe no temporal correlation between the temporal b ehavior 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 separat ed by a critical distance of 500 m. The first class (r > 500 m) shows a diffuse seismicity. This diffuse class of earthquakes (M(1) < 3.0) f ollows the frequency-magnitude relation. The low number of events prev ent us from analyzing whether there is a fractal or random behavior. T he second class (r < 500 m) defines nests of seismicity and attests a bifractal character (r < 500 m, D-1 approximate to 2.4; r > 500 m, D-2 approximate to 1.3). The difference of one unit between the fractal d imension of the seismicity within the nests (D-1) and the fractal dime nsion of the nests distribution (D-2) suggests an influence of the geo logical dome structure on the spatial development of seismic nests. Mo reover, a slope break above M(1) = 3.0 (G-R relation) is observed on t his second class (1.0 less than or equal to M(1) less than or equal to 4.2). The slope break of both the b value and the fractal dimension a t a common threshold (M(1) approximate to 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 maxi mum thickness. of local seismogenic layers (brittle calcareous layer v ersus ductile marry layer). On this basis we propose that (1) the fini te extension of the earthquake power law is driven by the local settin g and therefore is also a scale dependent process, (2) the geomechanic al link between fractal behavior and fracture size, i.e., a physical m apping of the power law behavior, must also be found in the boundary v alues of the autosimilarity processes (slope breaks) rather than in th e values of the power law exponents.