Javier B. Gómez

Instability of scale-free networks under node-breaking avalanches

The instability introduced in a large scale-free network by the triggering of node-breaking avalanches is analyzed using the fiber-bundle model as conceptual framework. We found, by measuring the size of the giant component, the avalanche size distribution and other quantities, the existence of an abrupt transition. This test of strength for complex networks like Internet is more stringent than others recently considered like the random removal of nodes, analyzed within the framework of percolation theory. Finally, we discuss the possible implications of our results and their relevance in forecasting cascading failures in scale-free networks.

Fracture and second-order phase transitions.

Using the global fiber bundle model as a tractable scheme of progressive fracture in heterogeneous materials, we define the branching ratio in avalanches as a suitable order parameter to clarify the order of the phase transition occurring at the collapse of the system. The model is analyzed using a probabilistic approach suited to smooth fluctuations. The branching ratio shows a behavior analogous to the magnetization in known magnetic systems with second-order phase transitions. We obtain a universal critical exponent beta approximately = 0.5 independent of the probability distribution used to assign the strengths of individual fibers.

Damage of Saturated Rocks Undergoing Triaxial Deformation Using Complex Electrical Conductivity Measurements: Experimental Results

The frequency dependent complex electrical conductivity of brine saturated rocks is extremely sensitive to changes in the volume, connectivity, orientation, and surface topography of pores and cracks. We have made triaxial deformation experiments on sandstone specimens saturated with distilled water. Experiments were carried out for several values of confining pressure, and in both drained and undrained regimes. During the deformation the full complex (in-phase and out-of-phase) electrical parameter set was measured (i.e. conductivity, resistivity, permittivity etc.) for 50 frequencies from 20 Hz to 1 MHz. Only the data at 1 kHz will be discussed here. This data tracks how the rock undergoes crack closure, followed by dilatancy, crack linking, and finally failure, as axial strain is increased. The data indicates well how early the formation of new cracks begins, showing that the quasi- linear portion of the stress-strain curve for triaxial deformation of saturated rocks does not represent truly elastic behaviour, but represents the combined effects of crack closure perpendicular to the strain axis and the formation of tensile cracks parallel to the strain axis. The electrical data has also been used to derive an electrical- equivalent change in porosity, and to examine the way that the cementation exponent and the tortuosity of the pore and crack network change during deformation.

A model for complex aftershock sequences

The decay rate of aftershocks is commonly very well described by the modified Omori law, n(t) ∝ t−p, where n(t) is the number of aftershocks per unit time, t is the time after the main shock, and p is a constant in the range 0.9 < p < 1.5 and usually close to 1. However, there are also more complex aftershock sequences for which the Omori law can be considered only as a first approximation. One of these complex aftershock sequences took place in the eastern Pyrenees on February 18, 1996, and was described in detail by Correig et al. [1997]. In this paper, we propose a new model inspired by dynamic fiber bundle models to interpret this type of complex aftershock sequences with sudden increases in the rate of aftershock production not directly related to the magnitude of the aftershocks (as in the epidemic-type aftershock sequences). The model is a simple, discrete, stochastic fracture model where the elements (asperities or barriers) break because of static fatigue, transfer stress according to a local load-sharing rule and then are regenerated. We find a very good agreement between the model and the Eastern Pyrenees aftershock sequence, and we propose that the key mechanism for explaining aftershocks, apart from a time-dependent rock strength, is the presence of dynamic stress fluctuations which constantly reset the initial conditions for the next aftershock in the sequence.

Self-Organized Criticality in a Fibre-Bundle type model

The dynamics of a bre-bundle-type model with equal load sharing rule is numerically studied. The system, formed by N elements, is driven by a slow increase of the load upon it which is removed in a novel way through internal transfers to the elements broken during avalanches. When an avalanche ends, failed elements are regenerated with strengths taken from a probability distribution. For a large N and certain restrictions on the distribution of individual strengths, the system reaches a self-organized critical state where the spectrum of avalanche sizes is a power law with an exponent ’1:5. c 1999 Elsevier Science B.V. All rights reserved.

Fracturing in saturated rocks undergoing triaxial deformation using complex electrical conductivity measurements: experimental study

Frequency dependent complex electrical conductivity measurements have been made on sandstones saturated with distilled water during triaxial deformation in both drained and undrained regimes. The resulting electrical and mechanical data show how the rock undergoes compaction, followed by dilatancy due to new crack formation, crack growth, interlinkage and failure as axial strain is increased. Electrical data are particularly good at indicating how early the formation of new cracks begins, showing that the quasi-linear portion of the stress-strain curve for triaxial deformation of saturated rocks does not represent truly elastic behaviour, but the combined effects of (i) crack closure perpendicular to the strain axis and (ii) the formation of tensile cracks parallel to the strain axis. A difference in the stress-strain behaviour between the drained and undrained samples was also observed, with the undrained samples developing a pronounced strain-softening phase before failure. The experimental data have also been used to derive the volumetric porosity, electrical porosity, cementation exponent and electrical tortuosity of the pore/crack network during deformation. The relative importance of crack closure and dilatation (a) during the progress of deformation and (b) between crack populations, controls these parameters and the electrical data over a wide range of frequencies. However, the frequency dependence of the micro-structural parameters and the electrical data was found to be not affected significantly by the hydrostatic pressurisation or the triaxial deformation. The development of large scale crack connectivity is observed to be confined to just prior to failure, and is controlled by the loss of cracks perpendicular to the axis of current flow and deformation

Probabilistic approach to time-dependent load-transfer models of fracture

A probabilistic method for solving time-dependent load-transfer models of fracture is developed. It is applicable to any rule of load redistribution, i.e, local, hierarchical, etc. In the new method, the fluctuations are generated during the breaking process (annealed randomness) while in the usual method, the random lifetimes are fixed at the beginning (quenched disorder). Both approaches are equivalent.

Time to failure of hierarchical load-transfer models of fracture

The time to failure, T, of dynamical models of fracture for a hierarchical load-transfer geometry is studied. Using a probabilistic strategy and juxtaposing hierarchical structures of height n, we devise an exact method to compute T, for structures of height n+1. Bounding T, for large n, we are able to deduce that the time to failure tends to a nonzero value when n tends to infinity. This numerical conclusion is deduced for both power law and exponential breakdown rules.

A way to synchronize models with seismic faults for earthquake forecasting: Insights from a simple stochastic model

Abstract Numerical models are starting to be used for determining the future behaviour of seismic faults and fault networks. Their final goal would be to forecast future large earthquakes. In order to use them for this task, it is necessary to synchronize each model with the current status of the actual fault or fault network it simulates (just as, for example, meteorologists synchronize their models with the atmosphere by incorporating current atmospheric data in them). However, lithospheric dynamics is largely unobservable: important parameters cannot (or can rarely) be measured in Nature. Earthquakes, though, provide indirect but measurable clues of the stress and strain status in the lithosphere, which should be helpful for the synchronization of the models. The rupture area is one of the measurable parameters of earthquakes. Here we explore how it can be used to at least synchronize fault models between themselves and forecast synthetic earthquakes. Our purpose here is to forecast synthetic earthquakes in a simple but stochastic (random) fault model. By imposing the rupture area of the synthetic earthquakes of this model on other models, the latter become partially synchronized with the first one. We use these partially synchronized models to successfully forecast most of the largest earthquakes generated by the first model. This forecasting strategy outperforms others that only take into account the earthquake series. Our results suggest that probably a good way to synchronize more detailed models with real faults is to force them to reproduce the sequence of previous earthquake ruptures on the faults. This hypothesis could be tested in the future with more detailed models and actual seismic data.

Characterisation and modelling of mixing processes in groundwaters of a potential geological repository for nuclear wastes in crystalline rocks of Sweden

This paper presents the mixing modelling results for the hydrogeochemical characterisation of groundwaters in the Laxemar area (Sweden). This area is one of the two sites that have been investigated, under the financial patronage of the Swedish Nuclear Waste and Management Co. (SKB), as possible candidates for hosting the proposed repository for the long-term storage of spent nuclear fuel. The classical geochemical modelling, interpreted in the light of the palaeohydrogeological history of the system, has shown that the driving process in the geochemical evolution of this groundwater system is the mixing between four end-member waters: a deep and old saline water, a glacial meltwater, an old marine water, and a meteoric water. In this paper we put the focus on mixing and its effects on the final chemical composition of the groundwaters using a comprehensive methodology that combines principal component analysis with mass balance calculations. This methodology allows us to test several combinations of end member waters and several combinations of compositional variables in order to find optimal solutions in terms of mixing proportions. We have applied this methodology to a dataset of 287 groundwater samples from the Laxemar area collected and analysed by SKB. The best model found uses four conservative elements (Cl, Br, oxygen-18 and deuterium), and computes mixing proportions with respect to three end member waters (saline, glacial and meteoric). Once the first order effect of mixing has been taken into account, water-rock interaction can be used to explain the remaining variability. In this way, the chemistry of each water sample can be obtained by using the mixing proportions for the conservative elements, only affected by mixing, or combining the mixing proportions and the chemical reactions for the non-conservative elements in the system, establishing the basis for predictive calculations.