Jean Schmittbuhl

Scaling invariance of crack surfaces

The morphology of fractured rock surfaces is studied in terms of their scaling invariance. Fresh brittle fractures of granite and gneiss were sampled with a mechanical laboratory profilometer, and (1 + 1)-dimensional parallel profiles were added to build actual maps of the surfaces. A first step in the scaling invariance description is a self-affine analysis using three independent methods. The root-mean-square and the maximum-minimum difference of the height are shown to follow a power law with the sample length. The return probability and the Fourier spectrum are also computed. All these approaches converge to a unique self-affine exponent: ζ = 0.80. Analysis over a broad statistical set provides a reproducibility error of ±0.05. No significant differences between the isotropic granite and the markedly anisotropic gneiss appear for the scaling exponents. An analysis of the profilometer shows that the two main drawbacks of the setup are not significant in these analyses. The systematic errors of the scaling analysis are estimated for the different methods. Isotropy of the scaling invariance within the mean fracture plane is shown either with the result obtained from different fracture orientations or with the two-dimensional Fourier spectrum of the surface topography itself. The analysis is brought further into the multifractal framework. The structure functions are shown to have power law behavior, and their scaling exponent varies nonlinearly with the moment order. Finally, the corresponding conserved process belongs to a universal multifractal class with α = 1.5 for the Levy index and C1 = 0.3 for the fractal codimension of the mean singularities. The three indices (ζ, α and C1) completely characterize the scale invariance. The multifractal behavior is significant for physical properties which depend on high-order moments like contact. According to this study and that of other groups, the self-affine exponent ζ is constant over a large range of scales and for different fracture modes and various materials. This opens the possibility that there exists a form of universality in the cracking process. It appears that only the prefactor of the roughness is dependent on material and mode.

Extended Nucleation of the 1999 Mw 7.6 Izmit Earthquake

Low-frequency seismic events may have been part of slip accumulation before a large earthquake. Laboratory and theoretical studies suggest that earthquakes are preceded by a phase of developing slip instability in which the fault slips slowly before accelerating to dynamic rupture. We report here that one of the best-recorded large earthquakes to date, the 1999 moment magnitude (Mw) 7.6 Izmit (Turkey) earthquake, was preceded by a seismic signal of long duration that originated from the hypocenter. The signal consisted of a succession of repetitive seismic bursts, accelerating with time, and increased low-frequency seismic noise. These observations show that the earthquake was preceded for 44 minutes by a phase of slow slip occurring at the base of the brittle crust. This slip accelerated slowly initially, and then rapidly accelerated in the 2 minutes preceding the earthquake.

Origin of the universal roughness exponent of brittle fracture surfaces:stress-weighted percolation in the damage zone.

We suggest that the observed large-scale universal roughness of brittle fracture surfaces is due to the fracture propagation being a damage coalescence process described by a stress-weighted percolation phenomenon in a self-generated quadratic damage gradient. We use the quasistatic 2D fuse model as a paradigm of a mode I fracture model. We measure for this model, which exhibits a correlated percolation process, the correlation length exponent nu approximately 1.35 and conjecture it to be equal to that of classical percolation, 4/3. We then show that the roughness exponent in the 2D fuse model is zeta=2nu/(1+2nu)=8/11. This is in accordance with the numerical value zeta=0.75. Using the value for 3D percolation, nu=0.88, we predict the roughness exponent in the 3D fuse model to be zeta=0.64, in close agreement with the previously published value of 0.62+/-0.05. We furthermore predict zeta=4/5 for 3D brittle fractures, based on a recent calculation giving nu=2. This is in full accordance with the value zeta=0.80 found experimentally.

Frictional response induced by time-dependent fluctuations of the normal loading

We study the effect of time-variable normal stress perturbations on a creeping fault which satisfies a velocity-weakening rate- and state-dependent friction law and is slipping at constant speed. We use the spring-block model and include the effect of inertia. To account for the variable normal stress, we use the description introduced by Linker and Dieterich [1992], which links normal stress fluctuations to changes of the state variable. We consider periodic perturbations of the normal stress in time (as caused, for instance, by tides) and compare the behavior for two commonly used friction laws (the “slip” and the “ageing” laws). Their mechanical response is shown to be significantly different for normal stress fluctuations. It could be used to probe these two laws during laboratory friction experiments. We show that there is a resonance phenomenon, involving strong amplification of the shear and velocity response of the interface, when the spring stiffness is modestly above its critical value (or when, at a given stiffness, the normal stress is modestly below its critical value). We show that such an amplification is also observed when periodic fluctuations of the shear loading are considered, making the resonance phenomenon a general feature of the response of a near-critical creeping surface to periodic fluctuations of the external loading. Analytical solutions are based on a linear expansion for low amplitude of normal or shear stress variations and are in very good agreement with numerical solutions. A method to find the evolution of friction in the case of an arbitrary perturbation of the normal stress is also presented. The results show that a creeping fault may be destabilized and enter a stick-slip regime owing to small normal stress oscillations. This may also account for a mechanism for the generation of “creep bursts.” However, these phenomena require very specific parameter ranges to excite the resonance, which may not be met very generally in nature. This study illustrates the importance of the normal stress fluctuations on stable sliding and suggests further friction laboratory experiments.

Geometrical heterogeneities and permeability anisotropy of rough fractures

We develop a numerical model for describing a single phase viscous flow in a rough fracture and compare it successfully to previously-published experimental results. The aperture fluctuations are introduced as an isotropic self-affine fractal which includes geometrical heterogeneities, especially at the fracture scale. The model is based on the lubrication approximation and solutions are obtained from two independent numerical schemes. The curves describing the evolution of the hydraulic aperture as a function of the mean fracture separation are shown to be well fitted by power law functions. A grid rotation technique is developed to explore the influence of the pressure drop orientation over a continuous range of orientations. A single fracture is shown to be either flow enhancing or flow inhibiting by comparison to a parallel plate model of identical mean separation, depending on the pressure drop orientation. This anisotropy of the fluid flow results from the geometrical heterogeneities at the fracture scale. Statistical analyses of the results when exploring different apertures distributions with the same measured scale invariance property show a large variability of the permeability, which is due to the same phenomenon. A prediction of the hydraulic transmittivity that takes account of the pressure drop orientation is proposed.

Flow enhancement of a rough fracture

We study experimentally and numerically the permeability of a rough fracture at laboratory scale when viscous forces are dominating (low Reynolds number). The experimental setup includes a granite fracture surface (10 × 10 cm²) opened in mode I. It allows a continuous opening of the fracture parallel to its mean plane. The fracture roughness is measured and characterized in terms of self-affine heterogeneities. Its isotropy is checked. The departure from the cubic law is measured as a function of the aperture opening (for mean separations between 4.0 and 10.3 mm) and the pressure drop orientation by rotating the fracture by 90 degrees. A strongly anisotropic hydraulic behavior is observed and results from the geometrical heterogeneities that exist up to the fracture macro-scale.

Anomalous scaling of fracture surfaces

We argue that fracture surfaces may exhibit anomalous dynamic scaling properties akin to what occurs in some models of kinetic roughening. We determine the complete scaling behavior of the local fluctuations of a brittle fracture in a granite block from experimental data. We obtain a global roughness exponent

Three‐dimensional roughness of stylolites in limestones

\ensuremath{\chi}=1.2

Stress Drop during Earthquakes: Effect of Fault Roughness Scaling

which differs from the local one,

Growth activity during fingering in a porous Hele-Shaw cell.

{\ensuremath{\chi}}_{\mathrm{loc}}=0.79