arXiv subjects

Renaud Toussaint

Publications and source records attributed to Renaud Toussaint.

At least 55 records · Page 3Linked to original sources

arXiv2011

Hydraulic transmissivity and heat exchange efficiency of open fractures: a model based on lowpass filtered apertures

Natural open joints in rocks commonly present multi-scale self-affine apertures. This geometrical complexity affects fluid transport and heat exchange between the flow- ing fluid and the surrounding rock. In particular, long range correlations of self-affine apertures induce strong channeling of the flow which influences both mass and heat advection. A key question is to find a geometrical model of the complex aperture that describes at best the macroscopic properties (hydraulic conductivity, heat exchange) with the smallest number of parameters. Solving numerically the Stokes and heat equa- tions with a lubrication approximation, we show that a low pass filtering of the aperture geometry provides efficient estimates of the effective hydraulic and thermal properties (apertures). A detailed study of the influence of the bandwidth of the lowpass filtering on these transport properties is also performed. For instance, keeping the information of amplitude only of the largest Fourier length scales allows us to reach already an accuracy of 9% on the hydraulic and the thermal apertures.

physics.flu-dyn↗

arXiv2010

Fracture Roughness Scaling: a case study on planar cracks

Using a multi-resolution technique, we analyze large in-plane fracture fronts moving slowly between two sintered Plexiglas plates. We find that the roughness of the front exhibits two distinct regimes separated by a crossover length scale $δ^*$. Below $δ^*$, we observe a multi-affine regime and the measured roughness exponent $ζ_{\parallel}^{-} = 0.60\pm 0.05$ is in agreement with the coalescence model. Above $δ^*$, the fronts are mono-affine, characterized by a roughness exponent $ζ_{\parallel}^{+} = 0.35\pm0.05$, consistent with the fluctuating line model. We relate the crossover length scale to fluctuations in fracture toughness and the stress intensity factor.

cond-mat.mtrl-sci↗

arXiv2008

Growth of stylolite teeth patterns depends on normal stress and finite compaction

Stylolites are spectacular rough dissolution surfaces that are found in many rock types. Despite many studies, their genesis is still debated, particularly the time scales of their formation and the relationship between this time and their morphology. We developed a new discrete simulation technique to explore the dynamic growth of the stylolite roughness, starting from an initially flat dissolution surface. We demonstrate that the typical steep stylolite teeth geometry can accurately be modelled and reproduce natural patterns. The growth of the roughness takes place in two successive time regimes: i) an initial non-linear increase in roughness amplitude that follows a power-law in time up to ii) a critical time where the roughness amplitude saturates and stays constant.

physics.geo-ph↗

arXiv2007

Granular Rayleigh-Taylor Instability: Experiments and Simulations

A granular instability driven by gravity is studied experimentally and numerically. The instability arises as grains fall in a closed Hele-Shaw cell where a layer of dense granular material is positioned above a layer of air. The initially flat front defined by the grains subsequently develops into a pattern of falling granular fingers separated by rising bubbles of air. A transient coarsening of the front is observed right from the start by a finger merging process. The coarsening is later stabilized by new fingers growing from the center of the rising bubbles. The structures are quantified by means of Fourier analysis and quantitative agreement between experiment and computation is shown. This analysis also reveals scale invariance of the flow structures under overall change of spatial scale.

cond-mat.soft↗

arXiv2006

Statistical mechanics of interacting fiber bundles

We consider quasistatic fiber bundle models with interactions. Classical load sharing rules are considered, i.e. local, global or decaying as a power-law of distance. All fibers are identically elastic, initially intact, and break at a random threshold picked from a quenched disorder (q.d.) distribution. We are interested in the probability distribution of configurations of broken fibers at a given elongation, averaged over all possible realizations of the underlying q.d.. This distribution is accessed by mapping the threshold set space onto the configurational space, each path corresponding to the evolution of a bundle corresponding to a realized q.d.. Using a perturbative approach allows to obtain this distribution to leading order in the interactions. This maps this system onto classical statistical mechanics models, i.e. percolation, standard or generalized Ising models depending on the range of the interactions chosen in the load sharing rule. This relates such q.d. based systems to standard classical mechanics, which allows to derive observables of the system, as e.g. correlation lengths. The thermodynamic parameters formally equivalent to temperature and chemical potential, are functions of the externally imposed deformation, depending on the load sharing rule and the choice of the q.d. distribution.

cond-mat.mtrl-sci↗

arXiv2006

Self-assembly and dynamics of magnetic holes

Nonmagnetic particles in magnetized ferrofluids have been denoted magnetic holes and are in many ways ideal model systems to test various forms of particle self assembly and dynamics. Some case studies to be reviewed here include: Chaining of magnetic holes Braid theory and Zipf relation used in dynamics of magnetic microparticles Interactions of magnetic holes in ferrofluid layers The objectives of these works have been to find simple characterizations of complex behavior of particles with dipolar interactions.

cond-mat.soft↗

arXiv2006

Dynamics and structure of interfacial crack fronts

The propagation of an interfacial crack front through a weak plane of a transparent Plexiglas block has been studied experimentally. A stable crack in mode I was generated by loading the system by an imposed displacement. The local velocities of the fracture front line have been measured by using an high speed CCD camera. The distribution of the velocities exhibits a power law behavior for velocities larger than the average front velocity with a crossover to a slowly increasing function for velocities lower than . The fluctuations in the velocities reflect an underlying irregular bursts activity with a power law distribution of the bursts. We further found that the size of the local bursts scales differently in the direction parallel to and perpendicular to the fracture front.

cond-mat.stat-mech↗

arXiv2006

Hydrothermal coupling in a rough fracture

Heat exchange during laminar flow is studied at the fracture scale on the basis of the Stokes equation. We used a synthetic aperture model (a self-affine model) that has been shown to be a realistic geometrical description of the fracture morphology. We developed a numerical modelling using a finite difference scheme of the hydrodynamic flow and its coupling with an advection/conduction description of the fluid heat. As a first step, temperature within the surrounding rock is supposed to be constant. Influence of the fracture roughness on the heat flux through the wall, is estimated and a thermalization length is shown to emerge. Implications for the Soultz-sous-Forêts geothermal project are discussed.

physics.geo-ph↗

arXiv2005

Local waiting time fluctuations along a randomly pinned crack front

The propagation of an interfacial crack along a heterogeneous weak plane of a transparent Plexiglas block is followed using a high resolution fast camera. We show that the fracture front dynamics is governed by local and irregular avalanches with very large size and velocity fluctuations. We characterize the intermittent dynamics observed, i.e. the local pinnings and depinnings of the crack front which trigger a rich burst activity, by measuring the local waiting time fluctuations along the crack front during its propagation. The local front line velocity distribution deduced from the waiting time analysis exhibits a power law behavior, $P(v) \propto v^{-η}$ with $η= 2.55 \pm 0.15$, for velocities $v$ larger than the average front speed $ $. The burst size distribution is also a power law, $P(S)\propto S^{-γ}$ with $γ=1.7 \pm 0.1$. Above a characteristic length scale of disorder $L_d \sim 15 μm$, the avalanche clusters become anisotropic, and the scaling of the anisotropy ratio provides an estimate of the roughness exponent of the crack front line, $H=0.66$, in close agreement with previous independent estimates.

cond-mat.mtrl-sci↗

arXiv2005

Mean Field Theory of Localization in the Fuse Model

We propose a mean field theory for the localization of damage in a quasistatic fuse model on a cylinder. Depending on the quenched disorder distribution of the fuse thresholds, we show analytically that the system can either stay in a percolation regime up to breakdown, or start at some current level to localize starting from the smallest scale (lattice spacing), or instead go to a diffuse localization regime where damage starts to concentrate in bands of width scaling as the width of the system, but remains diffuse at smaller scales. Depending on the nature of the quenched disorder on the fuse thresholds, we derive analytically the phase diagram of the system separating these regimes and the current levels for the onset of these possible localizations. We compare these predictions to numerical results.

cond-mat.stat-mech↗

arXiv2005

The roughness of stylolites: Implications of 3D high resolution topography measurements

Stylolites are natural pressure-dissolution surfaces in sedimentary rocks. We present 3D high resolution measurements at laboratory scales of their complex roughness. The topography is shown to be described by a self-affine scaling invariance. At large scales, the Hurst exponent is $ζ_1 \approx 0.5$ and very different from that at small scales where $ζ_2 \approx 1.2$. A cross-over length scale at around $Ł_c =1$~mm is well characterized. Measurements are consistent with a Langevin equation that describes the growth of a stylolitic interface as a competition between stabilizing long range elastic interactions at large scales or local surface tension effects at small scales and a destabilizing quenched material disorder.

cond-mat.mtrl-sci↗

arXiv2005

Influence of pore-scale disorder on viscous fingering during drainage

We study viscous fingering during drainage experiments in linear Hele-Shaw cells filled with a random porous medium. The central zone of the cell is found to be statistically more occupied than the average, and to have a lateral width of 40% of the system width, irrespectively of the capillary number $Ca$. A crossover length $w_f \propto Ca^{-1}$ separates lower scales where the invader's fractal dimension $D\simeq1.83$ is identical to capillary fingering, and larger scales where the dimension is found to be $D\simeq1.53$. The lateral width and the large scale dimension are lower than the results for Diffusion Limited Aggregation, but can be explained in terms of Dielectric Breakdown Model. Indeed, we show that when averaging over the quenched disorder in capillary thresholds, an effective law $v\propto (\nabla P)^2$ relates the average interface growth rate and the local pressure gradient.

cond-mat.soft↗

arXiv2004

Interacting damage models mapped onto Ising and percolation models

We introduce a class of damage models on regular lattices with isotropic interactions, as e.g. quasistatic fiber bundles. The system starts intact with a surface-energy threshold required to break any cell sampled from an uncorrelated quenched-disorder distribution. The evolution of this heterogeneous system is ruled by Griffith's principle which states that a cell breaks when the release in elastic energy in the system exceeds the surface-energy barrier necessary to break the cell. By direct integration over all possible realizations of the quenched disorder, we obtain the probability distribution of each damage configuration at any level of the imposed external deformation. We demonstrate an isomorphism between the distributions so obtained and standard generalized Ising models, in which the coupling constants and effective temperature in the Ising model are functions of the nature of the quenched-disorder distribution and the extent of accumulated damage. In particular, we show that damage models with global load sharing are isomorphic to standard percolation theory, that damage models with local load sharing rule are isomorphic to the standard Ising model, and draw consequences thereof for the universality class and behavior of the autocorrelation length of the breakdown transitions corresponding to these models. We also treat damage models having more general power-law interactions, and classify the breakdown process as a function of the power-law interaction exponent. Last, we also show that the probability distribution over configurations is a maximum of Shannon's entropy under some specific constraints related to the energetic balance of the fracture process, which firmly relates this type of quenched-disorder based damage model to standard statistical mechanics.

cond-mat.stat-mech↗

arXiv2003

Interaction model for magnetic holes in a ferrofluid layer

Nonmagnetic spheres confined in a ferrofluid layer (magnetic holes) present dipolar interactions when an external magnetic field is exerted. The interaction potential of a microsphere pair is derived analytically, with a precise care for the boundary conditions along the glass plates confining the system. Considering external fields consisting of a constant normal component and a high frequency rotating in-plane component, this interaction potential is averaged over time to exhibit the average interparticular forces acting when the imposed frequency exceeds the inverse of the viscous relaxation time of the system. The existence of an equilibrium configuration without contact between the particles is demonstrated for a whole range of exciting fields, and the equilibrium separation distance depending on the structure of the external field is established. The stability of the system under out-of-plane buckling is also studied. The dynamics of such a particle pair is simulated and validated by experiments.

cond-mat.soft↗

arXiv2003

Dynamic roughening and fluctuations of dipolar chains

Nonmagnetic particles in a carrier ferrofluid acquire an effective dipolar moment when placed in an external magnetic field. This fact leads them to form chains that will roughen due to Brownian motion when the magnetic field is decreased. We study this process through experiments, theory and simulations, three methods that agree on the scaling behavior over 5 orders of magnitude. The RMS width goes initially as $t^{1/2}$, then as $t^{1/4}$ before it saturates. We show how these results complement existing results on polymer chains, and how the chain dynamics may be described by a recent non-Markovian formulation of anomalous diffusion.

cond-mat.stat-mech↗

arXiv2003

Growth activity during fingering in a porous Hele Shaw cell

We present in this paper an experimental study of the invasion activity during unstable drainage in a 2D random porous medium, when the (wetting) displaced fluid has a high viscosity with respect to that of the (non-wetting) displacing fluid, and for a range of almost two decades in capillary numbers corresponding to the transition between capillary and viscous fingering. We show that the invasion process takes place in an active zone within a characteristic screening length from the tip of the most advanced finger. The invasion probability density is found to only depend on the distance to the latter tip, and to be independent of the value for the capillary number Ca. The mass density along the flow direction is related analytically to the invasion probability density, and the scaling with respect to the capillary number is consistent with a power law. Other quantities characteristic of the displacement process, such as the speed of the most advanced finger tip or the characteristic finger width, are also consistent with power laws of the capillary number. The link between the growth probability and the pressure field is studied analytically and an expression for the pressure in the defending fluid along the cluster is derived. The measured pressure are then compared with the corresponding simulated pressure field using this expression for the boundary condition on the cluster.

cond-mat.dis-nn↗

arXiv2002

Thermodynamics of fiber bundles

A recent theory that determines the properties of disordered solids as the solid accumulates damage is applied to the special case of fiber bundles with global load sharing and is shown to be exact in this case. The theory postulates that the probability of observing a given emergent damage state is obtained by maximizing the emergent entropy as defined by Shannon subject to energetic constraints. This theory yields the known exact results for the fiber-bundle model with global load sharing and holds for any quenched-disorder distribution. It further defines how the entropy evolves as a function of stress, and shows definitively how the concepts of temperature and entropy emerge in a problem where all statistics derive from the initial quenched disorder. A previously unnoticed phase transition is shown to exist as the entropy goes through a maximum. In general, this entropy-maximum transition occurs at a different point in strain history than the stress-maximum transition with the precise location depending entirely on the quenched-disorder distribution.

cond-mat.stat-mech↗

arXiv2002

Fracture of disordered solids in compression as a critical phenomenon: I. Statistical mechanics formalism

This is the first of a series of three articles that treats fracture localization as a critical phenomenon. This first article establishes a statistical mechanics based on ensemble averages when fluctuations through time play no role in defining the ensemble. Ensembles are obtained by dividing a huge rock sample into many mesoscopic volumes. Because rocks are a disordered collection of grains in cohesive contact, we expect that once shear strain is applied and cracks begin to arrive in the system, the mesoscopic volumes will have a wide distribution of different crack states. These mesoscopic volumes are the members of our ensembles. We determine the probability of observing a mesoscopic volume to be in a given crack state by maximizing Shannon's measure of the emergent crack disorder subject to constraints coming from the energy-balance of brittle fracture. The laws of thermodynamics, the partition function, and the quantification of temperature are obtained for such cracking systems.

cond-mat.stat-mech↗