Initial Crack Density Research Articles

A micro-macro mechanism of hydraulic fracturing with initial stress state effect of brittle rock

The external initial stress state seriously affects the hydraulic fracturing process of rocks. However, the external initial stress effect on the mechanical relationship between the hydraulic pressure and the internal microcrack extension in brittle rocks is rarely studied. This paper proposes a micro-macro mechanical model for predicting the hydraulic fracturing process of brittle rock under compressive stress. Based on the correlation of the stress intensity factor at the tip of the extended micro-cracks, the hydraulic pressure, and the force characteristics of the cracks inside the brittle rock, the evolution law of the hydraulic pressure with the wing crack length is determined. The effects of confining pressure, axial pressure, stress difference, initial crack angle, initial crack length, initial crack density and rock fracture toughness on the hydraulic initiation stress and peak stress are discussed. The rationality of hydraulic fracturing models is verified by experimental comparison. With the increase of axial pressure, stress difference, fracture toughness, initial crack length and initial crack density, the hydraulic initiation and peak stress of rock decrease, and with the increase of confining pressure and fracture toughness, the hydraulic initiation stress and peak stress of rock increase. The study results optimize the parameters such as pre-crack angle, pre-crack length, and external loading conditions based on rock characteristics. It helps to grasp the progress of hydraulic fracturing according to the nature of rock and the degree of hydraulic pressure change, improve the efficiency of hydraulic fracturing, and enhance economic benefits.

Effect of crack density on crack initiation and damage for high confining pressure using wave velocities

The study of progressive failure of reservoir rocks is very important for the stability analysis of ultra-deep boreholes. The progressive failure can be characterized by the crack initiation stress (CI), crack damage stress (CD) and peak stress (Peak), which is controlled by the evolution of cracks in rock and is significantly affected by confining pressure. To explore the effect of high confining pressure and initial crack on the progressive failure, the initial crack density of 121 sedimentary rock samples was characterized by acoustic waves, and the progressive failure under different confining pressures was analyzed. Experiments show that the ratios of CI to Peak (CI/Peak) and CD to Peak (CD/Peak) increase significantly with confining pressure, which is agreement with the existing conclusion from low confining pressure (0–52 MPa), but those decrease for high confining pressure (68–85 MPa) above critical point of sandstone. The CI/Peak and CD/Peak decrease with increase of initial crack density, which is sensitive to confining pressure but independent of rock type and grain size. The CD of sandstone is pertinent to the friction coefficient of cracks. These results laid a foundation for analyzing the evolution mechanism of wellbore damage and predicting the timing of wellbore instability in ultra-deep wells.

Viscoelastic-damage constitutive model and mechanical characterisations of Aluminium/polytetrafluoroethylene under impact loading

As a typical structural energy-containing material, aluminium/polytetrafluoroethylene (Al/PTFE) exhibits excellent mechanical properties under conventional conditions and releases large amounts of energy through chemical reactions under impact loading. In this research, a viscoelastic-damage constitutive model suitable for polymer-based reactive materials was developed, which consisted of an elastic element, two Maxwell elements, and a SCRAM damage element. The effect of damage on the stress–strain relationship was obtained by analysing the crack evolution process and deformation field. A modified Split Hopkinson pressure bar experiment investigated the mechanical characteristics of aluminium/polytetrafluoroethylene under uniaxial/multiaxial impact loading. The results indicated that the viscoelastic-damage constitutive model could describe the effects of the initial crack size and initial crack density on the mechanical properties of aluminium/polytetrafluoroethylene. With the decrease in initial crack length and density, the damage to the aluminium/polytetrafluoroethylene under impact loading was limited, which caused the dynamic mechanical properties to increase. When the pre-stress increases from 0 MPa to 10 MPa, the dynamic ultimate compressive stress of aluminium/polytetrafluoroethylene increases from 77.2 MPa to 89.8 MPa at 2910 s−1 strain rate. A comparison of the experimental and calculated strain–stress curves shows that the established constitutive model is feasible to predicting the mechanical behaviour of aluminium/polytetrafluoroethylene.

Influences Driving and Limiting the Efficacy of Ice Segregation in Alpine Rocks

AbstractRockwall erosion by rockfall is largely controlled by frost weathering in high alpine environments. As alpine rock types are characterized by crack‐dominated porosity and high rock strength, frost cracking observations from low strength and grain supported pore‐space rocks cannot be transferred. Here, we conducted laboratory experiments on Wetterstein limestone samples with different initial crack density and saturation to test their influence on frost cracking efficacy. We exposed rocks to real‐rockwall freezing conditions and monitored acoustic emissions as a proxy for cracking. To differentiate triggers of observed cracking, we modeled ice pressure and thermal stresses. Our results show initial full saturation is not a singular prerequisite for frost cracking. We also observe higher cracking rates in less‐fractured rock. Finally, we find that the temperature threshold for frost cracking in alpine rocks falls below −7°C. Thus, colder, north‐exposed rock faces in the Alps likely experience more frost cracking than southern‐facing counterparts.

A micromechanical-based failure criterion for rocks after high-temperature treatment

The strength properties of rocks after high-temperature treatment are substantially impacted by thermally-induced cracks. In this work, a new parameter of temperature-dependent initial damage or crack density is proposed and a micromechanical-based failure criterion incorporating it is developed to characterize the strength properties of rocks after high-temperature treatment under compressive loading. The mathematical expression of the suggested nonlinear failure criterion coincides with the Hoek-Brown failure criterion for intact rocks. The micromechanical-based failure criterion is then verified using the experimental results of five sets of rocks (Macheng granite, Tournemire shale, red sandstone, coarse marble and veined marble) after high-temperature treatment. The results shown that the strength properties of rocks after high-temperature treatment can be well captured using the failure criterion incorporating temperature-dependent initial damage. Three evaluation indicators (root mean square error, correlation coefficient and mean absolute error) are indicated that the failure criterion incorporating temperature-dependent initial damage has a strong predictive capability. The parameter of temperature-dependent initial damage provides a great description of the effect of thermally-induced damage on the strength evolution of rocks after high-temperature treatment. The proposed failure criterion in this work will become an effective alternative for predicting the nonlinear strength evolution of pre-heated rocks under compressive loading due to its high reliability and easy-to-use.

Mesomechanical damage of fiber mortar under temperature difference

The present study aims to establish a mesomechanical damage model for the description of anisotropic damage by mesocracking under temperature difference condition. With the basic solution of Eshelby theory and the Clapeyron equation, a mechanics framework is developed for cement materials, which are considered to be heterogeneous materials consisting of a cement stone matrix and imbedded penny-shaped microcracks. From the mesomechanical analysis, the critical conditions corresponding to the failure of fiber mortar under different initial crack density conditions are obtained. Furthermore, the Clapeyron equation is used to describe the relationship between the stress and the initiation or expansion of the fiber mortar crack and the damage evolution rule is expressed by the energy equation of the stiffness coefficient. The proposed model can accurately quantify the factors such as fiber distribution, microcrack density, temperature differences and the relationship between directional microcracks and crack strength in mortar. The results verified by laboratory tests and numerical simulations show that as the temperature difference increases from 0 °C to 75 °C, the cracking strength of the fiber mortar decreases and the number of cracks increases in the form of a power exponential function, which proves the micromechanical damage the validity of the model.

Mitigation of Mechanical Membrane Degradation in Fuel Cells by Controlling Electrode Morphology: A 4D In Situ Structural Characterization

Mechanical degradation is a critical mechanism responsible for the operational failure of fuel cell membranes. In addition to the membrane’s intrinsic durability, component interactions play a crucial role in this degradation process. This work investigates the interaction and associated impact of electrode morphology on membrane failure under pure mechanical degradation conditions by utilizing 4D in situ visualization by X-ray computed tomography. Using periodic identical-location imaging, membrane damage progression is monitored and compared for electrodes with high and low initial crack density. Membrane fracture is found to be significantly curtailed through minimization of ab initio crack density in the cathode catalyst layer. Hydration-dehydration cycles, however, still introduce early electrode cracking which, as an intermediate step, exclusively governs the subsequent initiation and propagation of membrane cracks. Two distinct membrane failure mechanisms are identified that are characterized by: (i) permanent buckling deformation of the catalyst coated membrane; and (ii) direct membrane fracture from electrode cracks without buckling. The buckling phenomenon is found to be strongly influenced by the microstructure of the gas diffusion media and has a dominant contribution towards the overall frequency and scale of membrane fracture. Additionally, the effect of hydration on the in situ size and geometry of fracture features is demonstrated.

Evolution of nonlinear elasticity and crack damage of rock joint under cyclic tension

Similar to the damage zone in the vicinity of faults, rock joints are composed of single or multiple fracture process zones, highlighting a high crack density and a decrease of Young’s modulus. Here specimens of limestone joint and granite joint were subjected to cyclic tensile stress of increasing amplitudes, up to failure. The cyclic tension tests were to study the elasticity and crack density adjacent to rock joints. Results show that the vast majority of tensile strain accommodates in the vicinity of rock joint, and rock joints have a nonlinear mechanical behavior during loading while a nearly linear behavior during unloading in the tensile stress-strain curves. A gradual reduction in Young’s modulus is seen with the increase of tensile strain. The cumulative crack damage after each stress cycle also induces a decrease in Young’s modulus. A stiffer response is found when switching from loading to unloading. During the unloading, a recovery in tensile strain and crack damage is observed, but some unrecoverable damage remains in the rock joint sample after each loading-unloading cycle. Based on the differential scheme, a constitutive model using two crack-related parameters, i.e., initial crack density and crack density increment, is developed. The model can give a micromechanical explanation and well reproduce the nonlinearity and hysteresis of tensile stress-strain curves.

Stress induced permeability changes in brittle fractured porous rock

This paper aims at deriving an approximate model of stress-induced permeability enhancement for brittle cracked porous rock, based on the linear fracture mechanic theory. A recently proposed model of the effective permeability of a porous rock containing an anisotropic distribution of several crack families is used. In such a model, the fluid flow is modelled by Poiseuille's law within the crack and Darcy's law for the porous matrix. The presence of this effective permeability model coupled to an analytical expression of crack growth under shear loading enables the modelling of the permeability variation due to stress-induced cracking in a porous rock. The model is applicable for any initial crack density (i.e. lower, around or upper to percolation threshold). Comparisons with experimental results and existing models show the validation and the advantage of the present model. All stages of fluid flow through a cracked porous medium are also reproduced. The proposed model can be easily implemented in a hydromechanical formulation with application to fractured geological formations.

Analysis of damage localization in composite laminates using a discrete damage model

Damage localization around stress raisers and material defects in laminated composites is studied using a discrete damage mechanics model augmented by a fiber damage model. The proposed formulation captures the damaging behavior of plates with initial defects and stress raisers such as holes, including damage initiation, evolution, and ultimate fracture of the specimen. It also helps explain the reduction of stress concentration factor when matrix and fiber damage develop. The state variables are the crack density and the fiber failure damage. The formulation is implemented as a material model in Abaqus applicable to laminated composite plates and shells. Material defects are simulated by inserting an initial crack density in a small region of the specimen. Stress raisers are simulated by an open hole. The predictions are shown to be insensitive to mesh density. Further, damage localizes near stress raiser and material defects, thus numerically demonstrating the objectivity of the proposed model. Qualitative and quantitative comparisons with experimental data are presented.

Crack Density and Elastic Properties of Sustainable Concretes

This paper examines relationships between changes in the microstructure and selected elastic properties of various concrete mixtures exposed to moderately elevated temperatures. The crack density parameters before and after exposure were estimated from the shear modulus measured wet and dry of 1 in. thick by 4 in. diameter (25 x 100 mm) disks. Mixtures examined included both granitic and lightweight coarse aggregates in combination with cementitious systems containing either 20% fly ash, 60% fly ash, termed enhanced sustainability, or 60% slag cement. This study found that the changes in crack density parameter resulting from exposure to elevated temperatures appear to be more sensitive to differences between cementitious materials than to differences between aggregate type or cementitious material proportions. A critical finding was that the relationship between initial crack density parameter and changes in crack density parameter were similar regardless of fly ash content.

Exploring Trends in Microcrack Properties of Sedimentary Rocks: An Audit of Dry and Water Saturated Sandstone Core Velocity–Stress Measurements

Stress dependent rock physics models are being used more routinely to link mechanical deformation and stress perturbations to changes in seismic velocities and seismic anisotropy. In this paper, we invert for the effective non-linear microstructural parameters of 69 dry and saturated sandstone core samples. We evaluate the results in terms of the model input parameters of two non-linear rock physics models: A discrete and an analytic microstructural stress-dependent formulation. The results for the analytic model suggest that the global trend of the initial crack density is lower and initial aspect ratio is larger for the saturated samples compared to the corresponding dry samples. The initial aspect ratios for both the dry and saturated samples are tightly clustered between 0.0002 and 0.001, whereas the initial crack densities show more scatter. The results for the discrete model show higher crack densities for the saturated samples when compared to the corresponding dry samples. With increasing confining stress the crack densities decreases to almost identical values for both the dry and saturated samples. A key result of this paper is that there appears to be a stress dependence of the compliance ratio BN/BT within many of the samples, possibly related to changing microcrack geometry with increasing confining stress. Furthermore, although the compliance ratio BN/BT for dry samples shows a diffuse distribution between 0.4 and 2.0, for saturated samples the distribution is very tightly clustered around 0.5. As confining stresses increase the compliance ratio distributions for the dry and saturated samples become more diffuse but still noticeably different. This result is significant because it reaffirms previous observations that the compliance ratio can be used as an indicator of fluid content within cracks and fractures. From a practical perspective, an overarching purpose of this paper is to investigate the range of input parameters of the microstructural models under both dry and saturated conditions to improve prediction of stress dependent seismic velocity and anisotropy observed in time-lapse seismic data due to hydro-mechanical effects related to fluid production and injection.

Exploring trends in microcrack properties of sedimentary rocks: An audit of dry-core velocity-stress measurements

Rock-physics models are used increasingly to link fluid and mechanical deformation parameters for dynamic elastic modeling. We explore the input parameters of an analytical stress-dependent rock-physics model. To do this, we invert for the stress-dependent microcrack parameters of more than 150 sedimentary rock velocity-stress core measurements taken from a literature survey. The inversion scheme is based on a microstructural effective-medium formulation defined by a second-rank crack-density tensor (scalar crack model) or by a second- and fourth-rank crack-density tensor (joint inversion model). Then the inversion results are used to explore and predict the stress-dependent elastic behavior of various sedimentary rock lithologies using an analytical microstructural rock-physics model via the initial modelinput parameters: initial crack aspect ratio and initial crack density. Estimates of initial crack aspect ratio are consistent among most lithologies with a mean of 0.0004, but for shales they differ up to several times in magnitude with a mean of 0.001. Estimates of initial aspect ratio are relatively insensitive to the inversion method, although the scalar crack inversion becomes less reliable at low values of normal-to-tangential crack compliance ratio [Formula: see text]. Initial crack density is sensitive to the degree of damage as well as the inversion procedure. An important implication is that the fourth-rank crack-density term is not necessarily negligible for most sedimentary rocks and evaluation of this term or [Formula: see text] is necessary for accurate prediction of initial crack density. This is especially important because recent studies suggest that [Formula: see text] can indicate fluid content in cracks.

Anisotropy of elastic wave velocities in deformed shales: Part 2 — Modeling results

This study was devoted to the interpretation of the evolution of elastic wave velocities in anisotropic shales that are subjected to deformation experiments in the laboratory. A micromechanical model was used to describe the macroscopic effective elastic properties and anisotropy of the rock in terms of its microscopic features, such as intrinsic anisotropy and crack/pore geometry. The experimental data (reported in Part 1) were compared quantitatively with the micromechanical model predictions to gain some insight into the microstructural behavior of the rock during deformation. The inversion of the experimental data using the micromechanical model was carried out by means of a numerical minimization of the least-squares distance between data and model in terms of effective compliances. Under isotropic mechanical loading, the overall behavior of the dry shale is consistent with the closure of crack-like pores, which are aligned in theplane of symmetry of the transversely isotropic background matrix. Those cracks represent a low fraction of the total porosity, but they have a strong effect on elastic wave velocities. The data are consistent with an initial (horizontal) crack density of 0.07. Crack closure also is evidenced at early stages of axial loading applied perpendicular to the shale bedding plane, whereas crack density increases significantly as axial stress is increased. Interpretation of the wet experiment is less straightforward, although some preliminary conclusions could be drawn. Under isotropic stress, crack closure also is evidenced, whereas crack density remains constant at the early stages of deviatoric loading. When axial peak stress is approached, crack density increases drastically, which likely indicates onset and development of vertical cracking. Wet experiments probably are more complex because water is likely to be expelled from crack-like pores toward equant pores in response to the mechanical loading.

Temperature Effects on Physical Properties and Mechanical Behavior of Granite: Experimental Investigation of Material Damage

Abstract In this paper, the effect of thermal treatment on physical and mechanical properties of a granitic rock is experimentally investigated. The open porosity, gas permeability, P-wave velocity (and their attenuation), ultimate strength, and Young’s modulus are measured on samples heated at temperatures ranging from 105°C to 600°C. First, results show good correlations between the evolution of physical properties and the amount of the damage induced by the thermal treatment. Second, the mechanical parameters are shown to be dependent on the microcracks’ density in the samples. The effect of temperature on the failure process in granite is also investigated using strain gauge measurements and permeability evolution in a uniaxial compressive test. The results show that the extent of the crack closure stage depends on the initial crack density and that the crack thresholds, which characterize the failure process of the rock under compressive loading, decrease with the thermal treatment.

Curvature in solid oxide fuel cells

At this point in history, curvature is inherent to the laminated components that comprise solid oxide fuel cells (SOFCs). Surprisingly, however, this fact has never been previously quantified in the literature. In addition, potential curvature changes associated with NiO reduction and re-oxidation during operation have not been investigated. In this report, an optical profilometer was employed to non-destructively quantify the surface curvature or cracking behavior observed on a large scale in industrially manufactured cells. This provides insights into the challenges that the component materials face as well as additional appreciation for why, in spite of a concerted effort to commercialize SOFC power generation, all currently manufactured SOFC stacks fail. Our results demonstrate that cracked electrolyte areas (caused by differential sintering) are flatter than uncracked regions. The height of the electrolyte surface ranged from 86 to 289 μm above the baseline following sintering. Reduction typically results in increases in curvature of up to 214 μm. Initial crack density appears to affect curvature evolution during reduction; the higher the crack density, the smaller the curvature increase following reduction at 600 °C. In general, however, we observed that the electrolyte layer is remarkably resistant to further cracking during these typographic changes. Following oxidation at 750 °C, large changes in curvature (up to 280 μm) are noted that appear to be related to the strength of the bond between the electrolyte and the underlying anode.

Do microcracks decrease or increase fatigue resistance in cortical bone?

Fatigue of cortical bone produces microcracks; it has been hypothesized that these cracks are analogous to those occurring in engineered composite materials and constitute a similar mechanism for fatigue resistance. However, the numbers of these linear microcracks increase substantially with age, suggesting that they contribute to increased fracture incidence among the elderly. To test these opposing hypotheses, we fatigued 20 beams of femoral cortical bone from elderly men and women in load-controlled four point bending having initial strain ranges of 3000 or 5000 microstrain. Loading was stopped at fracture or 10 6 cycles, whichever occurred first, and microcrack density and length were measured in the loaded region and in a control region that was not loaded. We studied the dependence of fatigue life and induced microdamage on initial microdamage, cortical region, subject gender and age, and several other variables. When the effect of modulus variability was controlled, longer fatigue life was associated with higher rather than lower initial crack density, particularly in the medial cortex. The increase in crack density following fatigue loading was greater in specimens from older individuals and those initially having longer microcracks. Crack density increased as much in specimens fatigued short of the failure point as in those that fractured, and microcracks were, on average, shorter in specimens with greater numbers of resorption spaces, a measure of remodeling rate.

Analysis of crack coalescence in rock-like materials containing three flaws—Part I: experimental approach

Abstract Fractures in the forms of joints and microcracks are commonly found in natural rocks, and their failure mechanism strongly depends on the crack coalescence pattern between pre-existing flaws. However, the crack coalescence pattern of rock specimens containing three or more flaws has not been studied comprehensively. In this paper, we investigate experimentally crack coalescence and peak strength of rock-like materials containing three parallel frictional flaws. Three flaws are arranged such that one pair of flaws lines collinearly and the third flaw forms either a non-overlapping pattern or an overlapping pattern with the first flaw. It is found that the mechanisms of crack coalescence depend on the flaw arrangement and the frictional coefficient μ on the flaw surface. Two “rules of failure” for the specimens containing three flaws are proposed. Rule No. 1: the pair of flaws with a lower value of coalescence stress will dominate the process of coalescence. Rule No. 2: mixed and tensile modes of coalescence are always the dominant modes if the coalescence stress of the two pairs of flaws is very close (say within 5%). In addition, it is found that the peak strength of the specimens does not depend on the initial crack density but on the actual number of pre-existing flaws involved in the coalescence. Comparisons of pattern of crack coalescence with the numerical approach are given in Part II of this study, and the two results agree well. The research reported here provides increased understanding of the fundamental nature of rock failure in uniaxial compression.

Permeability enhancement due to microcrack dilatancy in the damage regime

A three dimensional microscopically based permeability model incorporating inelastic deformation has been developed to account for the modification of transport properties due to fracturing. The basic hypothesis investigated is that permeability enhancement during brittle deformation is caused by the formation of dilatant microcracks which are associated with friction sliding on a preexisting random population of shear cracks. Additional dilatancy, produced as a result of frictional sliding over asperities, is also accounted for by the introduction of a crack roughness parameter. Linear elastic fracture mechanics is used to calculate the evolution of crack length and crack area as a function of applied stress and fluid pressure. After making a geometrical simplification the microcrack parameters derived from the deformation model can be used to calculate the permeability tensor assuming that fluid transport results from Poiseuille flow through a connected distribution of cracks. The model enables investigation of the macroscopic permeability variation as a function of two loading parameters, three constant material parameters, and the crack roughness parameter. Results demonstrate that permeability is a smooth but strongly increasing (near power law) function of the Terzaghi effective stress ratio and is strongly dependent on the initial crack density, Young's modulus, the friction coefficient, and the effective confining pressure. Numerical results are in quantitative agreement with published experimental measurements and display behavior similar to results obtained with a relatively simple analytical model. The model enables calculation of the degree of stress‐induced anisotropy, which is shown is be relatively small (< × 10) for resonable effective stress ratios. The modeling presented provides a quantitative tool with which the effects of microcrack‐induced permeability enhancement can be investigated within the broader context of coupled fluid flow and brittle deformation in the crust.

The stress-strain behaviour of rock material related to fracture under compression

Fracture is the main reason for the non-linear behaviour of hard rocks. The fracture mechanics of rock is studied in this article by analysis of the fracture process under compression. A constitutive model that describes the relationship between the macro deformation of rock and the micro fracture within rock is developed. The propagation of microcracks, the non-linearity of deformation, the loading-and-unloading hysteresis and the variation of the apparent Young's modulus and Poisson's ratio are studied using the developed model. The model simulations demonstrate that: (1) the fracture toughness, initial crack length, crack density, and Young's modulus are four crucially important parameters that affect the deformation behaviour of rock; (2) the elastic parameters ( E and v) of the rock matrix should be measured in triaxial tests. If they are measured in uniaxial tests, the upper straight unloading portion of the stress-strain curve is suggested to be used for the purpose, unless the closure effect of open cracks will be included in the estimations. In addition (3), the slope of the reloading stress-strain curve is a measure of the damage in material.