Searcharxiv⌕ Search

arXiv subjects

Jean Schmittbuhl

Publications and source records attributed to Jean Schmittbuhl.

At least 19 recordsLinked to original sources
arXiv2012

Biphasic flow: structure and upscaling, consequences on macroscopic transport properties

In disordered porous media, two-phase flow of immiscible fluids (biphasic flow) is organized in patterns that sometimes exhibit fractal geometries over a range of length scales, depending on the capillary, gravitational and viscous forces at play. These forces, as well as the boundary conditions, also determine whether the flow leads to the appearance of fingering pathways, i.e., unstable flow, or not. We present here a short review of these aspects, focusing on drainage and summarizing when these flows are expected to be stable or not, what fractal dimensions can be expected, and in which range of scales. We base our review on experimental studies performed in two-dimensional Hele-Shaw cells, or addressing three dimensional porous media by use of several imaging techniques. We first present configurations in which solely capillary forces and gravity play a role. Next, we review configurations in which capillarity and viscosity are the main forces at play. Eventually, we examine how the microscopic geometry of the fluid clusters affects the macroscopic transport properties. An example of such an upscaling is illustrated in detail: For air invasion in a mono-layer glass-bead cell, the fractal dimension of the flow structures and the associated scale-ranges, are shown to depend on the displacement velocity. This controls the relationship between saturation and the pressure difference between the two phases at the macroscopic scale. We provide in this case expressions for dynamic capillary pressure and residual fluid phase saturations.

physics.flu-dyn↗
arXiv2012

Local dynamics of a randomly pinned crack front during creep and forced propagation: An experimental study

We have studied the propagation of a crack front along the heterogeneous weak plane of a transparent poly(methyl methacrylate) (PMMA) block using two different loading conditions: imposed constant velocity and creep relaxation. We have focused on the intermittent local dynamics of the fracture front for a wide range of average crack front propagation velocities spanning over four decades. We computed the local velocity fluctuations along the fracture front. Two regimes are emphasized: a depinning regime of high velocity clusters defined as avalanches and a pinning regime of very low-velocity creeping lines. The scaling properties of the avalanches and pinning lines (size and spatial extent) are found to be independent of the loading conditions and of the average crack front velocity. The distribution of local fluctuations of the crack front velocity are related to the observed avalanche size distribution. Space-time correlations of the local velocities show a simple diffusion growth behavior.

cond-mat.dis-nn↗
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

Characterization of Fault Roughness at Various Scales: Implications of Three-Dimensional High Resolution Topography Measurements

Accurate description of the topography of active faults surfaces represents an important geophysical issue because this topography is strongly related to the stress distribution along fault planes, and therefore to processes implicated in earthquake nucleation, propagation, and arrest. With the recent development of Light Detection And Ranging (LIDAR) apparatus, it is now possible to measure accurately the 3D topography of rough surfaces with a comparable resolution in all directions, both at field and laboratory scales. In the present study, we have investigated the scaling properties including possible anisotropy properties of several outcrops of two natural fault surfaces (Vuache strike-slip fault, France, and Magnola normal fault, Italy) in limestones.

physics.geo-ph↗
arXiv2008

Variety of stylolites morphologies and statistical characterization of the amount of heterogeneities in the rock

The surface roughness of several stylolites in limestones was measured using high resolution laser profilometry. The 1D signals obtained were statistically analyzed to determine the scaling behavior and calculate a roughness exponent, also called Hurst exponent. Statistical methods based on the characterization of a single Hurst exponent imply strong assumptions on the mathematical characteristics of the signal: the derivative of the signal (or local increments) should be stationary and have finite variance. The analysis of the measured stylolites show that these properties are not always verified simultaneously. The stylolite profiles show persistence and jumps and several stylolites are not regular, with alternating regular and irregular portions. A new statistical method is proposed here, based on a non-stationary but Gaussian model, to estimate the roughness of the profiles and quantify the heterogeneity of stylolites. This statistical method is based on two parameters: the local roughness (H) which describes the local amplitude of the stylolite, and the amount of irregularities on the signal (μ), which can be linked to the heterogeneities initially present in the rock before the stylolite formed. Using this technique, a classification of the stylolites in two families is proposed: those for which the morphology is homogeneous everywhere and those with alternating regular and irregular portions.

physics.geo-ph↗
arXiv2008

High resolution 3D laser scanner measurements of a strike-slip fault quantify its morphological anisotropy at all scales

The surface roughness of a recently exhumed strikeslip fault plane has been measured by three independent 3D portable laser scanners. Digital elevation models of several fault surface areas, from 1 m2 to 600 m2, have been measured at a resolution ranging from 5 mm to 80 mm. Out of plane height fluctuations are described by non-Gaussian distribution with exponential long range tails. Statistical scaling analyses show that the striated fault surface exhibits self-affine scaling invariance with a small but significant directional morphological anisotropy that can be described by two scaling roughness exponents, H1 = 0.7 in the direction of slip and H2 = 0.8 perpendicular to the direction of slip.

physics.geo-ph↗
arXiv2007

Ridge Network in Crumpled Paper

The network formed by ridges in a straightened sheet of crumpled paper is studied using a laser profilometer. Square sheets of paper were crumpled into balls, unfolded and their height profile measured. From these profiles the imposed ridges were extracted as networks. Nodes were defined as intersections between ridges, and links as the various ridges connecting the nodes. Many network and spatial properties have been investigated. The tail of the ridge length distribution was found to follow a power-law whereas the shorter ridges followed a log-normal distribution. The degree distribution was found to have an exponentially decaying tail, and the degree correlation was found to be disassortative. The facets created by the ridges and the Voronoi diagram formed by the nodes have also been investigated.

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↗
arXiv2006

Self-Affinity in the Gradient Percolation Problem

We study the scaling properties of the solid-on-solid front of the infinite cluster in two-dimensional gradient percolation. We show that such an object is self affine with a Hurst exponent equal to 2/3 up to a cutoff-length proportional to the gradient to the power (-4/7). Beyond this length scale, the front position has the character of uncorrelated noise. Importantly, the self-affine behavior is robust even after removing local jumps of the front. The previously observed multi affinity, is due to the dominance of overhangs at small distances in the structure function. This is a crossover effect.

cond-mat.dis-nn↗
arXiv2006

Gaussian Statistics of Fracture Surfaces

We analyse the statistical distribution function for the height fluctuations of brittle fracture surfaces using extensive experimental data sampled on widely different materials and geometries. We compare a direct measurement of the distribution to a new analysis based on the structure functions. For length scales $δ$ larger than a characteristic scale $δ^*$, we find that the distribution of the height increments $Δh = h(x+ δ) -h(x)$ is Gaussian. Self-affinity enters through the scaling of the standard deviation $σ$, which is proportional to $δ^ζ$ with a unique roughness exponent. Below the scale $δ^*$ we observe an effective multi-affine behavior of the height fluctuations and a deviation from a Gaussian distribution which is related to the discreteness of the measurement or of the material.

cond-mat.mtrl-sci↗
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

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↗
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

Roughness of Interfacial Crack Front: Correlated Percolation in the Damage Zone

We show that the roughness exponent zeta of an in-plane crack front slowly propagating along a heterogeneous interface embeded in a elastic body, is in full agreement with a correlated percolation problem in a linear gradient. We obtain zeta=nu/(1+nu) where nu is the correlation length critical exponent. We develop an elastic brittle model based on both the 3D Green function in an elastic half-space and a discrete interface of brittle fibers and find numerically that nu=1.5, We conjecture it to be 3/2. This yields zeta=3/5. We also obtain by direct numerical simulations zeta=0.6 in excellent agreement with our prediction. This modelling is for the first time in close agreement with experimental observations.

cond-mat↗