Fracture Front Research Articles

Crack propagation through disordered materials as a depinning transition: A critical test of the theory.

The dynamics of a planar crack propagating within a brittle disordered material is investigated numerically. The fracture front evolution is described as the depinning of an elastic line in a random field of toughness. The relevance of this approach is critically tested through the comparison of the roughness front properties, the statistics of avalanches, and the local crack velocity distribution with experimental results. Our simulations capture the main features of the fracture front evolution as measured experimentally. However, some experimental observations such as the velocity distribution are not consistent with the behavior of an elastic line close to the depinning transition. This discrepancy suggests the presence of another failure mechanism not included in our model of brittle failure.

Magistral and fractal regimes of the hydro-fracturing

The paper describes the specific difference between two fracture growth regimes, when it forms the radial fracture network (fractal regime) and when it forms one big fracture in certain direction (magistral regime). A continuum model is used, which represents the fracture network in the media with the permeability tensor, rapidly increasing with the fracture appearance. The fracture front instability is shown, linked with the Saffman-Taylor instability, even in isotropic case. By analogy with Saffman-Taylor instability in porous media, the fractal nature of the fracture network is derived. The dimensionless parameters of fracture front speed and anisotropy are found, which would help to numerical and experimental modeling of the reservoir.

Toughening Behavior in Alloy 617 with Long Term Ageing

The influence of ageing at temperature 700°C for up to 20 000 hours on the deformation, damage and fracture behavior of Alloy 617 has been investigated by two toughness tests. Dense nano γ` phase and carbides are the main precipitates. However, the long-term aged material still shows high toughness. The mechanism has been studied using electron backscatter diffraction (EBSD) and electron channeling contrast imaging (ECCI). The results show that dense nano/micro-twins have been formed in the adjacent of the fracture front during the impact toughness or CTOD testing. Increase of ageing time reduces the number of nano/micro-twins, but they can be observed in all aged materials in spite of different strain rates. This indicates that besides dislocation slip, twinning is another deformation mechanism in long term aged material during the toughness tests. Formation of nano/micro-twins may be one of contributions to high toughness in the aged material, which is termed as twin induced toughening.

Interface propagation in fiber bundles: local, mean-field and intermediate range-dependent statistics

The fiber bundle model is essentially an array of elements that break when sufficient load is applied on them. With a local loading mechanism, this can serve as a model for a one-dimensional interface separating the broken and unbroken parts of a solid in mode-I fracture. The interface can propagate through the system depending on the loading rate and disorder present in the failure thresholds of the fibers. In the presence of a quasi-static drive, the intermittent dynamics of the interface mimic front propagation in disordered media. Such situations appear in diverse physical systems such as mode-I crack propagation, domain wall dynamics in magnets, charge density waves, contact lines in wetting etc. We study the effect of the range of interaction, i.e. the neighborhood of the interface affected following a local perturbation, on the statistics of the intermittent dynamics of the front. There exists a crossover from local to global behavior as the range of interaction grows and a continuously varying ‘universality’ in the intermediate range. This means that the interaction range is a relevant parameter of any resulting physics. This is particularly relevant in view of the fact that there is a scatter in the experimental observations of the exponents, in even idealized experiments on fracture fronts, and also a possibility in changing the interaction range in real samples.

Перколяционная модель накопления микродефектов и коллапса зоны вынужденной эластичности перед фронтом трещины разрушения в полимерных и композиционных материалах

Перколяционная модель накопления микродефектов и коллапса зоны вынужденной эластичности перед фронтом трещины разрушения в полимерных и композиционных материалах

A multiscale Implicit Level Set Algorithm (ILSA) to model hydraulic fracture propagation incorporating combined viscous, toughness, and leak-off asymptotics

This study uses an Implicit Level Set Algorithm (ILSA) to model the propagation of planar hydraulic fractures in situations when their progress is determined by an interplay of fluid viscosity, rock fracture toughness, and fluid leak-off into the formation. One of the key features of our approach is the use of the three-process tip asymptotic solution both as a propagation condition and to capture the multiscale behavior in a weak sense. Using this special tip asymptote is necessary because the validity region of the classical square root fracture opening solution (stemming from linear elastic fracture mechanics) is often limited to a small zone near the fracture tip, which can only be captured by a very fine mesh. In addition, this validity zone depends on the velocity of fracture propagation, so that slow and fast portions of the fracture front may experience different near-tip behavior. The multiscale tip asymptotic solution, on the other hand, has an increased validity region, which makes it possible to capture the near-tip multiscale behavior on a coarse mesh and yields a computationally efficient algorithm. The presence of leak-off also complicates the model considerably as it involves a delay term containing the trigger time history, which depends on the earlier fracture front positions. Moreover, the leak-off from tip elements in which the fracture front speed changes significantly requires special treatment. This three-process asymptotic solution is used to solve the fully coupled integro-delay-PDE model for a propagating planar hydraulic fracture by using a level set algorithm in conjunction with the tip asymptotic solution to locate the moving fracture front and to capture multiscale behavior. Firstly, the developed algorithm is validated against a reference solution for an axisymmetric hydraulic fracture. Secondly, a set of numerical examples involving three stress layers is presented to illustrate the variation of the multiscale near-tip behavior along the fracture perimeter and the need to use the multiscale asymptotic solution in a hydraulic fracturing simulator.

(Invited) Defect Evolution during Silicon Smartcut™

We have studied the development of cracks in gas-implanted silicon, from nano-scale to wafer-scale which is at the heart of the SmartCut™ technology. We will discuss the results of X-ray scattering experiments that allow the monitoring of growth of nanometer-size cavities (platelets) from implantation to their full development where they reach sizes of several tens of nanometers. The nucleation, growth and coarsening mechanisms of these nanometer size platelets are interpreted within the Ostwald ripening framework. At larger annealing times, some micron-size defects called microcracks appear, and experience again some competitive growth, dominated by coalescence. The growth of these micron-size objects depends on their interactions with smaller-size platelets. The modelling of this interaction can explain the dependance of split time with temperature. Microcrack development is a key step in the SmartCut™ process. Finally the propagation of the fracture front will be discussed. We will discuss the velocity dependence of the crack front with temperature and present some new measurements of dynamic crack opening displacement. Figure 1

Soft-Clamp Fiber Bundle Model and Interfacial Crack Propagation: Comparison Using a Non-linear Imposed Displacement

We compare experimental observations of a slow interfacial crack propagation along an heterogeneous interface to numerical simulations using a soft-clamped fiber bundle model. The model consists of a planar set of brittle fibers between a deformable elastic half-space and a rigid plate with a square root shape that imposes a non linear displacement around the process zone. The non-linear square-root rigid shape combined with the long range elastic interactions is shown to provide more realistic displacement and stress fields around the crack tip in the process zone and thereby significantly improving the predictions of the model. Experiments and model are shown to share a similar self-affine roughening of the crack front both at small and large scales and a similar distribution of the local crack front velocity. Numerical predictions of the Family-Viscek scaling for both regimes are discussed together with the local velocity distribution of the fracture front.

Numerical study on hydraulic fracturing in tight gas formation in consideration of thermal effects and THM coupled processes

This paper introduces and explicates a new thermal module that has been developed to study numerically the heat transport in a hydraulic fracture and heat exchange between the fracture and the surrounding reservoir rocks, as well as the THM coupled processes that take place during hydraulic fracturing in a tight gas reservoir. The new module was embedded in the simulator FLAC3Dplus and validated by comparison with several analytical examples. In this paper the new module was subsequently used to model the stimulation job that was executed in a tight gas reservoir (for anonymity reasons only known as X6) in the Northern German Basin. Its basis is the history-matching of the in-situ measured wellhead pressure. The paper presents the exceptional ability of the new thermal module to simulate the entire thermal evolution process during hydraulic fracturing, including a period of temperature recovery towards the original geothermal reservoir temperature (also termed temperature warm back) after shut-in. Further observations by this paper include the significant temperature increase at the fracture front during treatment. This is caused by a delay in the arrival of the fracturing fluid at the fracture tip. In addition, the paper draws a comparison between two simulations: one executed with a thermal module to investigate the THM coupled effects on fracture development and the other without taking the thermal effects into consideration. The basis for the comparison is that the models use (in terms of geometry and stratigraphy) identical mechanical, hydraulic and thermal parameters in both simulations. Results show that the cooling effect of the injection fluid leads to contraction of the rock formations thereby causing them to experience an extra load of tensile stress. In this regard, the hydraulic fracture will easily propagate in the direction of minimum horizontal stress (where rock stress also seems to have decreased remarkably). Finally, the paper observed that simulations carried out using the new thermal module achieved a wider but shorter fracture than that obtained in simulations without consideration of heat effects. Nevertheless, the fracture heights in both cases were almost identical due to the existence of caprocks with similar integrity. Trends indicate that the simulation without thermal module caused a greater leak off volume at the end of the operation, since the fracture surface attained in this case was larger.

Fracture Toughness Determination of Cracked Chevron Notched Brazilian Disc Rock Specimen via Griffith Energy Criterion Incorporating Realistic Fracture Profiles

The cracked chevron notched Brazilian disc (CCNBD) specimen has been suggested by the International Society for Rock Mechanics to measure the mode I fracture toughness of rocks, and has been widely adopted in laboratory tests. Nevertheless, a certain discrepancy has been observed in results when compared with those derived from methods using straight through cracked specimens, which might be due to the fact that the fracture profiles of rock specimens cannot match the straight through crack front as assumed in the measuring principle. In this study, the progressive fracturing of the CCNBD specimen is numerically investigated using the discrete element method (DEM), aiming to evaluate the impact of the realistic cracking profiles on the mode I fracture toughness measurements. The obtained results validate the curved fracture fronts throughout the fracture process, as reported in the literature. The fracture toughness is subsequently determined via the proposed G-method originated from Griffith’s energy theory, in which the evolution of the realistic fracture profile as well as the accumulated fracture energy is quantified by DEM simulation. A comparison between the numerical tests and the experimental results derived from both the CCNBD and the semi-circular bend (SCB) specimens verifies that the G-method incorporating realistic fracture profiles can contribute to narrowing down the gap between the fracture toughness values measured via the CCNBD and the SCB method.

Universal hydrofracturing algorithm for shear-thinning fluids: Particle velocity based simulation

A universal particle velocity based algorithm for simulating hydraulic fractures, previously proposed for Newtonian fluids, is extended to the class of shear-thinning fluids. The scheme is not limited to any particular elasticity operator or crack propagation regime. The computations are based on two dependent variables: the crack opening and the reduced particle velocity. The application of the latter facilitates utilization of the local condition of Stefan type (speed equation) to trace the fracture front. The condition is given in a general explicit form which relates the crack propagation speed (and the crack length) to the solution tip asymptotics. The utilization of a modular structure, and the adaptive character of its basic blocks, result in a flexible numerical scheme. The computational accuracy of the proposed algorithm is validated against a number of analytical benchmark solutions.

Thermoviscoelastoplastic Deformation of Compound Shells of Revolution Made of a Damageable Material

A technique for numerical analysis of the thermoviscoelastoplastic deformation of thin compound shells made of a damageable material in which a fracture front propagates is described. A procedure for automatic variation in the step of integration of the kinetic damage equation is developed. A two-layer cylindrical shell cooling by convection and subjected to internal pressure and tensile force is analyzed as an example. The numerical data are presented and analyzed

Hydraulic fracture revisited: Particle velocity based simulation

We develop a new effective mathematical formulation and resulting universal computational algorithm capable of tackling various HF models in the framework of a unified approach. The scheme is not limited to any particular elasticity operator or crack propagation regime. Its basic assumptions are: (i) proper choice of independent and dependent variables (with the direct utilization of a new one – the reduced particle velocity), (ii) tracing the fracture front by use of the Stefan condition (speed equation), which can be integrated in closed form and provides an explicit relation between the crack propagation speed and the coefficients in the asymptotic expansion of the crack opening, (iii) proper regularization techniques, (iv) improved temporal approximation, (v) modular algorithm architecture. The application of the new dependent variable, the reduced particle velocity, instead of the usual fluid flow rate, facilitates the computation of the crack propagation speed from the local relation based on the speed equation. This way, we avoid numerical evaluation of the undetermined limit of the product of fracture aperture and pressure gradient at the crack tip (or alternatively the limit resulting from ratio of the fluid flow rate and the crack opening), which always poses a considerable computational challenge. As a result, the position of the crack front is accurately determined from an explicit formula derived from the speed equation. This approach leads to a robust numerical scheme. Its performance is demonstrated using classical examples of 1D models for hydraulic fracturing: PKN and KGD models under various fracture propagation regimes. Solution accuracy is verified against dedicated analytical benchmarks and other solutions available in the literature. The scheme can be directly extended to more general 2D and 3D cases.

Mechanical origin of b-value changes during stimulation of deep geothermal reservoirs

The distribution of induced-earthquake magnitudes in deep geothermal reservoirs is a classical tool for monitoring reservoirs. It typically shows some important fluctuations through time and space. Despite being a very crude information (i.e., a scalar quantity) of very complex mechanical stress evolution, understanding these variations could still give us insights into the mechanics of the reservoir. Here, we analyze the output of a simple quasi-static physical model of a single fault and propose a new way to describe bursts that could be compared to seismic events. Our model is an elastic fiber bundle model describing multiple ruptures along a single major fault of the reservoir. It consists of a set of fibers with various strengths, arranged in a planar L×L two-dimensional array, linking together two elastic half-spaces. It mimics a fault zone with various asperities. During load, the fibers break quasi-statically according to a stress threshold distribution. Contrary to classical fiber bundle model, here, when a fiber breaks, it redistributes the load on the surviving fibers through long range elastic interactions. Interestingly, the elasticity of the half-spaces which changes the range of the stress distribution, characterizes two distinct regimes. In a stiff regime, we find an effective ductile deformation regime at large scale since the damage distribution is very difuse. Conversely in softer systems, the distance between two consecutive breaking fibers gets smaller and failure of the interface is localized, exhibiting an effective brittle regime. We analyze two types of burst distributions: a classical one built from the statistics of the broken bonds during each failure step and a new one defines from a waiting time matrix of the fracture front propagation. The first one reproduces several known results for this type of model. The new one evidences the existence of effective creeping advances of the front with statistics that follow a Gutenberg-Richter distribution, in particular, in the ductile regime (stiff systems). We proposed a new definition of bursts in a fault model, based on the local fracture front velocity. We find that b v -values for the distribution of the velocity clusters are very consistent with Gutenberg-Richter distribution of induced seismicity. We link the b v -value fluctuations in the reservoirs to the influence of the velocity threshold level that could be related to recording limitation. b b -values obtained from the broken bond statistics are hardly comparable to seismic events because of the lack of space contiguousness of broken fibers during bursts. In the light of our results, we discuss the implications of b-value changes in geothermal reservoir in terms of fault asperities and normal stress evolution.

Enrichment strategies and convergence properties of the XFEM for hydraulic fracture problems

In two recent papers Gordeliy and Peirce (2013) investigating the use of the Extended Finite Element Method (XFEM) for modeling hydraulic fractures (HF), two classes of boundary value problem and two distinct enrichment types were identified as being essential components in constructing successful XFEM HF algorithms. In this paper we explore the accuracy and convergence properties of these boundary value formulations and enrichment strategies. In addition, we derive a novel set of crack-tip enrichment functions that enable the XFEM to model HF with the full range of power law rλ behavior of the displacement field and the corresponding rλ−1 singularity in the stress field, for 12≤λ<1. This novel crack-tip enrichment enables the XFEM to achieve the optimal convergence rate, which is not achieved by existing enrichment functions used for this range of power law. The two XFEM boundary value problem classes are as follows: (i) a Neumann to Dirichlet map in which the pressure applied to the crack faces is the specified boundary condition and the XFEM is used to solve for the corresponding crack width (P→W); and (ii) a mixed hybrid formulation of the XFEM that makes it possible to incorporate the singular behavior of the crack width in the fracture tip and uses a pressure boundary condition away from it (P&W). The two enrichment schemes considered are: (i) the XFEM-t scheme with full singular crack-tip enrichment and (ii) a simpler, more efficient, XFEM-s scheme in which the singular tip behavior is only imposed in a weak sense. If enrichment is applied to all the nodes of tip-enriched elements, then the resulting XFEM stiffness matrix is singular due to a linear dependence among the set of enrichment shape functions, which is a situation that also holds for the classic set of square-root enrichment functions. For the novel set of enrichment functions we show how to remove this rank deficiency by eliminating those enrichment shape functions associated with the null space of the stiffness matrix. Numerical experiments indicate that the XFEM-t scheme, with the new tip enrichment, achieves the optimal O(h2) convergence rate we expect of the underlying piece-wise linear FEM discretization, which is superior to the enrichment functions currently available in the literature for 12<λ<1. The XFEM-s scheme, with only signum enrichment to represent the crack geometry, achieves an O(h) convergence rate. It is also demonstrated that the standard P→W formulation, based on the variational principle of minimum potential energy, is not suitable for modeling hydraulic fractures in which the fluid and the fracture fronts coalesce, while the mixed hybrid P&W formulation based on the Hellinger–Reissner variational principle does not have this disadvantage.

On the moving boundary conditions for a hydraulic fracture

This paper re-examines the boundary conditions at the moving front of a hydraulic fracture when the fluid front has coalesced with the crack edge. This practically important particular case is treated as the zero fluid lag limit of the general case when the two fronts are distinct. The limiting process shows what becomes of the two boundary conditions on the fluid front, a pressure condition and a Stefan condition, when the lag vanishes. On the one hand, the pressure condition disappears as the net pressure (the difference between the fluid pressure and the magnitude of the far-field stress normal to the fracture) becomes singular. On the other hand, the Stefan condition, which equates the front velocity to the average fluid velocity, transforms into a zero flux boundary condition at the front. As a consequence, the velocity of the coalesced front does not appear explicitly in the boundary conditions. However, the front velocity can still be extracted from the near-tip aperture field by a nonlinear asymptotic analysis. The paper concludes with a description of an algorithm to propagate the combined front, which explicitly uses the known multiscale asymptotics of the fracture aperture.

Fracture Dynamics during the Silicon Layer Transfer of the Smart Cut™ Process

The Smart Cut™ technology [1] is currently the industry standard for manufacturing Silicon-On-Insulator (SOI) wafers. The implantation of relatively high doses of hydrogen ions in a silicon substrate leads to the formation of a buried weakened layer. This wafer is bonded onto a host substrate using direct wafer bonding. Under annealing, the implanted species evolve into microcracks lying parallel to the surface, and finally, a controlled fracture process occurs along the implanted layer. While Si/SiO2 direct wafer bonding statics and dynamics have been extensively studied [2],[3], the fracture dynamics during Smart Cut™ remains largely a field to be explored. Since the SOI standard thickness for microelectronics devices is always decreasing, the detailed understanding of the fracture becomes of greater interest e.g. to improve the post-split SOI wafer surfaces.In this paper, the crack propagation during the silicon layer transfer of the Smart Cut™ process is studied using an original patented optical bench[4] (Fig.1). The measurements are based on the transmission monitoring of infrared laser beams through the sample when the fracture wave front crosses the beams. The crack can be mechanically initiated by inserting a blade between the two wafers or thermally initiated by putting the sample into an appropriate oven. Combination of the two split modes is also possible. This set up allows the determination of the crack propagation velocity which is typically in the km.s-1 range. An in-depth study of the post wave front signal oscillations (Fig.2) enabled us to recover the deformation profile of the sample just after the fracture front has gone through it (Fig.3). At first, the two newly separated wafers drift away because of the strong internal pressure of the gas filled microcracks. The energy released during the crack opening separates the silicon beams by several micrometres, until reaching a maxima where the external atmospheric pressure becomes large enough to stop the two silicon arms diverging movement. Subsequently the motion will invert and the wafers will get closer again and reach a minima where the internal pressure will enable the two beams to pull away. Therefore, the wave crack "wake" is made of series of oscillations until the internal pressure equilibrates to the atmospheric one. The typical wavelength of such deformations is about several centimetres, while the vertical speed of displacement is quite slow, around 1µm.µs-1. The dynamics of this motion has been successfully modelled using beam elasticity and thermodynamics. From the modelling, it is e.g. possible to recover the amount of gas trapped inside the microcracks and better understand the driver of Smart Cut™ splitting technique.

Simulation of non‐planar three‐dimensional hydraulic fracture propagation

SUMMARYHydraulic fracturing is the method of choice to enhance reservoir permeability and well efficiency for extraction of shale gas. Multi‐stranded non‐planar hydraulic fractures are often observed in stimulation sites. Non‐planar fractures propagating from wellbores inclined from the direction of maximum horizontal stress have also been reported. The pressure required to propagate non‐planar fractures is in general higher than in the case of planar fractures. Current computational methods for the simulation of hydraulic fractures generally assume single, symmetric, and planar crack geometries. In order to better understand hydraulic fracturing in complex‐layered naturally fractured reservoirs, fully 3D models need to be developed. In this paper, we present simulations of 3D non‐planar fracture propagation using an adaptive generalized FEM. This method greatly facilitates the discretization of complex 3D fractures, as finite element faces are not required to fit the crack surfaces. A solution strategy for fully automatic propagation of arbitrary 3D cracks is presented. The fracture surface on which pressure is applied is also automatically updated at each step. An efficient technique to numerically integrate boundary conditions on crack surfaces is also proposed and implemented. Strongly graded localized refinement and analytical asymptotic expansions are used as enrichment functions in the neighborhood of fracture fronts to increase the computational accuracy and efficiency of the method. Stress intensity factors with pressure on crack faces are extracted using the contour integral method. Various non‐planar crack geometries are investigated to demonstrate the robustness and flexibility of the proposed simulation methodology. Copyright © 2014 John Wiley &amp; Sons, Ltd.

Implicit level set schemes for modeling hydraulic fractures using the XFEM

We describe two novel XFEM schemes for modeling fluid driven fractures both of which exploit an implicit level set algorithm (ILSA) for locating the singular free boundary that occurs when the fluid and fracture fronts coalesce. Both schemes use the mixed P&W XFEM formulation developed in Gordeliy and Peirce (2013) [1] to incorporate the singular asymptotic solution in the fracture tips. The proposed level set strategy also exploits the asymptotic solution to provide a robust procedure to locate the free boundary, which is not restricted to symmetric growth of the fracture geometry or to a particular mode of propagation. The versatility of the ILSA-XFEM scheme is demonstrated by sampling different asymptotic behaviors along the so-called MK edge of parameter space (Detournay, 2004) [2] by making use of a universal asymptote (Garagash, 1998 [3]; Garagash and Detournay, 2000 [4]). The two ILSA-XFEM schemes differ in the enrichment strategies that they use to represent the fracture tips: a scheme with full tip enrichment and a simpler, more efficient, scheme in which the tip asymptotic behavior is only imposed in a weak sense. Numerical experiments indicate that the XFEM-t scheme, with full tip enrichment, achieves an O(h2) asymptotic convergence rate, while the XFEM-s scheme, with only signum enrichment to represent the crack geometry, achieves an O(h) asymptotic convergence rate.

Experimental investigation of mesoscale crack front triple line

The aim of this paper is to investigate the mesoscale behavior and structure of an adhesive near the fracture front of an asymmetric joint consisting of carbon fiber/epoxy resin composites bonded with a relatively soft, epoxy adhesive. A single cantilever beam fracture test at constant separation rate gave steady-state crack propagation, details of which were followed by digital image correlation (DIC). A deformed, triple line region was found between the adhesive, air, and the composite, somewhat resembling a “wetting ridge,” as found with a liquid meniscus in contact with a soft solid. Importantly, the partially separated bondline layer took part in (non-unidirectional) load transfer between adherends (and thus energy dissipation), contrary to common assumptions where the separated bondline is assumed no longer to play a structural role. A simple model, based on the Flamant contact mechanics approach, is proposed and compared with both a finite element solution and experimental data extracted from image correlation. The model points out the importance of two length scales: process zone extent and adhesive thickness, both being known to affect global properties of bonded structures.