Fracture Pinning and B-Value Of Micro-Seismicity

Abstract

The distribution of induced-earthquake magnitudes in deep geothermal reservoirs is a classical tool for monitor- ing reservoirs. It typically shows some important uctua- tions through time and space. Despite being a very crude information (I.E. A scalar quantity) Of very complex me- chanical stress evolution, understanding these variations could still give us insights into the mechanics of the reser- voir. Here we analyse the output of a simple quasI-static physical model of a single fault with the support of exper- imental results and propose a new way to describe bursts that could be compared to seismic events [1]. Crack propagation in heterogeneous media is a rich problem which involves the interplay of various physical processes. The problem has been intensively investigated theoretically, numerically, and experimentally, but a unI- fying model capturing all the experimental features has not been entirely achieved (See [2] for a recent review) De- spite its broad range of implications in engineering and earth sciences problems [e.G. 3, 4]. The slow propagation of a crack front where long range elastic interactions are dominant, is of crucial importance to ll the gap between experiments and models. Several theoretical and numerI- cal works have been devoted to quasI-static models. Such models give rise to an intermittent local activity charac- terized by a depinning transition and can be viewed as a critical phenomenon [2]. However these models fail to reproduce all experimental conditions, notably the front morphology does not display any cross-over length with two dI
erent roughness exponents above and below the cross-over as observed experimentally [e.G. 5]. During most regular fracture experiments, the frac- ture surface can only be observed post mortem. In an attempt to study the dynamical properties of the frac- ture, schmittbuhl and maly [6] developed an experI- ment where they forced the fracture to develop in a weak plane of a transparent medium (Pmma). In such an ex- perimental conguration, the propagation of the fracture front can the be explored using high resolution optical de- vices from which dI
erent properties characterizing the fracture dynamics can be inferred. Distribution of the local fracture velocities [7] or morphology of the frac- ture front [5] can notably be measured and are found to be robust parameters when tested against several exper- imental conditions. Since mandelbrot's discovery [8] that fracture surfaces in metal display fractal properties, there have been many attempts to understand the scaling relations of fracture. In particular, the self-ane nature of fracture surfaces have come under much scrutiny. A self-ane surface h(X) Has the following scaling relation [9] h(X) / h(x): (1)  is an arbitrary scaling factor, and  is the roughness exponent. The scaling relations for fractures have now been established as scale and direction dependant (Pon- son et al. [10, 11], bonamy et al. [12]). In 2010 santuccI et al. [5] found that the in-plane fractures also had a scale dependant roughness exponent. At small scales they ob- tained