Simulating fracture propagation in rock and concrete by an augmented virtual internal bond method

This paper develops a practical approach to simulating fracture propagation in rock and concrete based on an augmented virtual internal bond (VIB) method in which the cohesion of solid is modeled as material particles interconnected by a network of randomized virtual micro bonds described by the Xu–Needleman potential. The micro bond potential is used to derive macroscale constitutive relations via the Cauchy–Born rule. By incorporating different energy contributions due to stretch and shearing, as well as different energy levels under tension and compression of each micro bond, the derived macro constitutive laws are particularly useful for modeling quasi‐brittle materials such as rock and concrete which usually have different Poisson ratios and much higher compressive strength than tensile strength. The mesh‐size sensitivity associated with strain‐softening in the present constitutive model is addressed by adjusting material constants near the crack tip so that the biJ‐integral is kept equal to the intrinsic fracture energy of the material. Numerical examples demonstrate that the proposed VIB method is capable of simulating mixed mode fracture propagation in rock and concrete with results in consistency with relevant experimental observations. Copyright © 2011 John Wiley & Sons, Ltd.     

Elasto‐plastic model for cement‐treated sand

This paper presents an elasto‐plastic model for non‐linear analyses of cement‐treated sand. Various laboratory tests were systematically carried out to investigate the pre‐peak and post‐peak behaviours of a cement‐treated sand. On the basis of these experimental results, the new model was built within the framework of a relatively simple elasto‐plastic theory. Two failure criteria are employed to express tensile and shear failure characteristics observed in the experimental results of the cement‐treated sand. The proposed model can describe strain‐hardening and strain‐softening responses under both failure modes. In the strain‐softening rules, the smeared crack concept is used, and a characteristic length is considered to avoid the issue of mesh‐size dependency. Since the failure criterion and strain‐hardening/softening rules are based on the experimental evidences, the model is relatively easy to understand and the parameters used in the model have clear physical meaning. The proposed model was applied to simulate the behaviour of cement‐treated sand in various laboratory tests, allowing for a reasonable comprehensive evaluation. It was demonstrated that the proposed model is suitable for describing both the tensile and shear failure behaviours of cement‐treated sand. Copyright © 2006 John Wiley & Sons, Ltd.     

Evaluating the stress intensity factors of anisotropic bimaterials using boundary element method

This paper presents a boundary element method (BEM) procedure for a linear elastic fracture mechanics analysis in two‐dimensional anisotropic bimaterials. In this formulation, a displacement integral equation is only collocated on the uncracked boundary, and a traction integral equation is only collocated on one side of the crack surface. A fundamental solution (Green's function) for anisotropic bimaterials is also derived and implemented into the boundary integral formulation so that except for the interfacial crack part, the discretization along the interface can be avoided. A special crack‐tip element is introduced to capture the exact crack‐tip behavior. A computer program using FORTRAN has been developed to effectively calculate the stress intensity factors of an anisotropic bimaterial. This BEM program has been verified to have a good accuracy with previous studies. In addition, a central cracked bimaterial Brazilian specimen constituting cement and gypsum is prepared to conduct the Brazilian test under diametral loading. The result shows that the numerical analysis can predict relatively well the direction of crack initiation and the path of crack propagation. Copyright © 2007 John Wiley & Sons, Ltd.     

Simulation of localization failure with strain‐gradient‐enhanced damage mechanics

Strain gradient implies an important characteristic in localized damage deformation, which can be observed in the softening state of brittle materials, and strain gradients constitute the basic behaviours of localization failure area of the materials. The most important point in strain gradient is its damaging function including an internal length scale, which can be used to express the scale effects of mechanical responses of brittle rock mass. By extending the strain gradient theory and introducing an intrinsic material length scale into the constitutive law, the authors develop an isotropic damage model as well as a micro‐crack‐based anisotropic damage model for rock‐like materials in this paper. The proposed models were used to simulate the damage localization under uniaxial tension and plain strain compression, respectively. The simulated results well illustrated the potential of these models in dealing with the well‐known mesh‐sensitivity problem in FEM. In the computation, elements with C₁ continuity have been implemented to incorporate the proposed models for failure localization. When regular rectangle elements are encountered, the coupling between finite difference method (FDM) and conventional finite element method (FEM) is used to avoid large modification to the existing FEM code, and to obtain relatively higher efficiency and reasonably good accuracy. Application of the anisotropic model to the 3D‐non‐linear FEM analysis of Ertan arch dam has been conducted and the results of its numerical simulation coincide well with those from the failure behaviours obtained by Ertan geophysical model test. In this paper, new applications of gradient theories and models for a feasible approach to simulate localized damage in brittle materials are presented. Copyright © 2002 John Wiley & Sons, Ltd.     

基于互补理论的扩展有限元接触问题实现

在处理含有裂隙这类不连续问题时,常规有限元方法需要对裂隙尖端部位进行局部网格加密,当裂隙扩展时还需要进行网络重构。基于单位分解思想的扩展有限元成功解决了常规有限元难以处理的裂隙类不连续问题。在研究复合裂隙时,通常需要考虑裂隙的接触问题。基于互补理论,建立了裂隙面上相对位移和接触力的互补方程,并采用牛顿法求解,无需开闭迭代,且能够快速收敛。最后,对含裂隙平板进行受压数值试验,计算结果表明,基于互补理论的扩展有限元接触算法能够有效地阻止裂隙两端网格的相互嵌入,且获得裂隙面上的应力分布与实际一致。     

基于局部单元劈裂的FEM/DEM自适应分析方法

为了模拟岩体中裂纹的萌生、扩展,Munjiza提出了有限元法/离散元法(FEM/DEM)耦合分析方法。因为裂纹是沿单元边界进行扩展的,亦即裂纹扩展具有网格依赖性,为获得较好的裂纹扩展形态,需要划分密集的初始网格。为解决上述难题,基于FEM/DEM耦合分析方法,提出了基于局部单元动态劈裂的FEM/DEM自适应分析方法,以克服裂纹扩展形态对网格的依赖性。该方法在最初建模时无需划分很密的初始网格,随着荷载的施加,对裂纹尖端附近的局部单元进行动态劈裂,为裂纹的后续扩展提供了更多可能的扩展方向,使得裂纹扩展不必沿着初始网格的单元边界扩展,即可以沿着单元内部进行扩展,裂纹扩展形态更为平滑,与实际情况更为接近。同时相对原FEM/DEM耦合分析方法一开始就划分很密的网格而言,新方法可以划分较为稀疏的初始网格,计算成本降低。最后,通过巴西劈裂算例与原FEM/DEM耦合分析方法对比,分析表明,新方法在一定程度上克服了裂纹扩展形态对初始网格的依赖性。     

Block fracturing analysis using nodal‐based discontinuous deformation analysis with the double minimization procedure

As a hybrid method, the nodal‐based discontinuous deformation analysis (NDDA) greatly improves the stress accuracy within each DDA block by coupling a well‐defined finite element mesh inside the DDA block; at the same time, the NDDA inherits the unique block kinematics of the standard DDA method. Each finite element mesh line inside the DDA block is treated as a potential crack, which enables the transformation of the block material from continuum to discontinuum through the tensile and shear fracturing mechanism. This paper introduces a double minimization procedure into the NDDA method to further improve the accuracy of the stresses evaluated at the finite element mesh lines and thus to obtain a more realistic fracture model. Three numerical examples are employed to demonstrate the improved stress accuracy by the implemented double minimization procedure and the accuracy and capability of the enhanced NDDA method in capturing brittle fracturing process. Copyright © 2013 John Wiley & Sons, Ltd.     

A fully coupled element‐free Galerkin model for hydro‐mechanical analysis of advancement of fluid‐driven fractures in porous media

Hydraulic fracturing (HF) of underground formations has widely been used in different fields of engineering. Despite the technological advances in techniques of in situ HF, the industry uses semi‐analytical tools to design HF treatment. This is due to the complex interaction among various mechanisms involved in this process, so that for thorough simulations of HF operations a fully coupled numerical model is required. In this study, using element‐free Galerkin (EFG) mesh‐less method, a new formulation for numerical modeling of hydraulic fracture propagation in porous media is developed. This numerical approach, which is based on the simultaneous solution of equilibrium and continuity equations, considers the hydro‐mechanical coupling between the crack and its surrounding porous medium. Therefore, the developed EFG model is capable of simulating fluid leak‐off and fluid lag phenomena. To create the discrete equation system, the Galerkin technique is applied, and the essential boundary conditions are imposed via penalty method. Then, the resultant constrained integral equations are discretized in space using EFG shape functions. For temporal discretization, a fully implicit scheme is employed. The final set of algebraic equations that forms a non‐linear equation system is solved using the direct iterative procedure. Modeling of cracks is performed on the basis of linear elastic fracture mechanics, and for this purpose, the so‐called diffraction method is employed. For verification of the model, a number of problems are solved. According to the obtained results, the developed EFG computer program can successfully be applied for simulating the complex process of hydraulic fracture propagation in porous media. Copyright © 2016 John Wiley & Sons, Ltd.     

Simulation of non‐planar three‐dimensional hydraulic fracture propagation

Hydraulic 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 & Sons, Ltd.     

Modeling the connectivity and intersection of hydraulically loaded cracks with in situ fractures in rock

The interaction of an advancing hydraulically loaded crack and in situ fracture network can yield highly complex patterns. We model the connectivity of cells in a finite element domain and in a fracture network by a simplicial complex data structure. The complete adjacency information between cells is determined by one level down facet and one level up cofacet neighborhood information. Combined with a disjoint set data structure, explicit algorithms are derived to efficiently track network connectivity and load transfer between independent fracture sets. We also propose an approach to regularize the application of hydraulic load to newly intersected in situ cracks to smoothen the transition of pressure on intersected cracks from ambient to hydraulic pressure and to avoid the sudden loading of the entire length of these cracks. Numerical results demonstrate the performance of crack connectivity and load transfer models and the effect of regularization model. The results show that as the angle between an incoming hydraulically loaded crack and an in situ crack increases, the effect of in situ crack shifts from slight realignment to diversion/offsetting of the loaded crack. As the angle difference approaches the normal angle, the loaded crack tends to penetrate through the in situ crack. The proposed schemes are also used for transient simulation of 2D reservoirs with multiple perforations surrounded by in situ cracks with and without a bias in the distribution of their orientation. It was shown that from 2 perforations with angles closer to in situ cracks at low loading rates to all perforations at higher loading rates can result in active hydraulic crack propagation. The h‐adaptive method of asynchronous space‐time discontinuous Galerkin method is used to exactly track complex fracture patterns in these dynamic fracture simulations.     

Smeared crack approach: back to the original track

This paper briefly reviews the formulations used over the last 40 years for the solution of problems involving tensile cracking, with both the discrete and the smeared crack approaches. The paper focuses on the smeared approach, identifying as its main drawbacks the observed mesh‐size and mesh‐bias spurious dependence when the method is applied ‘straightly’. A simple isotropic local damage constitutive model is considered, and the (exponential) softening modulus is regularized according to the material fracture energy and the element size. The continuum and discrete mechanical problems corresponding to both the weak discontinuity (smeared cracks) and the strong discontinuity (discrete cracks) approaches are analysed and the question of propagation of the strain localization band (crack) is identified as the main difficulty to be overcome in the numerical procedure. A tracking technique is used to ensure stability of the solution, attaining the necessary convergence properties of the corresponding discrete finite element formulation. Numerical examples show that the formulation derived is stable and remarkably robust. As a consequence, the results obtained do not suffer from spurious mesh‐size or mesh‐bias dependence, comparing very favourably with those obtained with other fracture and continuum mechanics approaches. Copyright © 2006 John Wiley & Sons, Ltd.     

Coupled computations for an elastic‐perfectly plastic soil using adaptive mesh refinement

An adaptive mesh refinement algorithm has been developed for non‐linear computations in geomechanics, based on a smoothed stress–strain finite element formulation. This uses estimates of error in the incremental shear strain invariant to guide the regeneration of unstructured meshes at regular intervals during loading. Following each mesh‐update, no re‐analysis of previous increments with the new mesh is necessary. Algorithm performance has been investigated by analysing a passive earth pressure problem using a linear elastic‐perfectly plastic Mohr–Coulomb soil model. Perfectly drained behaviour has been considered, as have partially drained situations using hydromechanical coupling, while undrained behaviour has been approximated using time steps close to zero. In all cases, mesh adaptivity has been successful in capturing regions of high strain gradient. The results have been compared with analytical solutions. Accurate computations of limit load and shear band orientation have been obtained for a wide range of material dilation angles. Copyright © 2000 John Wiley & Sons, Ltd.     

Coupled hydromechanical‐fracture simulations of nonplanar three‐dimensional hydraulic fracture propagation

This paper presents an algorithm and a fully coupled hydromechanical‐fracture formulation for the simulation of three‐dimensional nonplanar hydraulic fracture propagation. The propagation algorithm automatically estimates the magnitude of time steps such that a regularized form of Irwin's criterion is satisfied along the predicted 3‐D fracture front at every fracture propagation step. A generalized finite element method is used for the discretization of elasticity equations governing the deformation of the rock, and a finite element method is adopted for the solution of the fluid flow equation on the basis of Poiseuille's cubic law. Adaptive mesh refinement is used for discretization error control, leading to significantly fewer degrees of freedom than available nonadaptive methods. An efficient computational scheme to handle nonlinear time‐dependent problems with adaptive mesh refinement is presented. Explicit fracture surface representations are used to avoid mapping of 3‐D solutions between generalized finite element method meshes. Examples demonstrating the accuracy, robustness, and computational efficiency of the proposed formulation, regularized Irwin's criterion, and propagation algorithm are presented.     

Triassic carbonate buildup flanks in the Dolomites, northern Italy: breccias, boulder fabric and the importance of early diagenesis

The flanks of Middle Triassic carbonate buildups in the Dolomites show well‐developed clinostratification, with typical angles of 30–40°. This paper focuses on the metre‐scale fabric of these clinoforms and sets these within the context of their large‐scale and microscopic features. Clinoform stratification is caused by fibrous cement crusts, by stylolites paralleling a vague stratification and by pelagic limestone interbedded with the lower portions of the clinoforms. These parts of the clinoforms locally exhibit a boulder fabric. A fracture system, subvertical to clinostratification, is filled by fibrous cements and marine internal sediment. The analysis of clinoform fabric indicates that parts consist of in situ automicrite. Other parts of the clinoforms are made up of breccia. The breccias occur as isolated pockets and lenses with random orientation and dimensions ranging from a few square decimetres to hundreds of square metres. The breccias have gradual contacts with the unbrecciated host rock. Breccia components are mostly angular, show a microfacies similar to that of the unbrecciated host rock and are composed of millimetre‐ to decimetre‐sized particles that generally float in radiaxial‐fibrous cement. Matching grain boundaries are common. Calcite cement typically makes up 20 and 40 vol.% of the brecciated areas. Clinostratification, the fracture system, brecciation and boulder fabric point to gravity induced deformation of in situ flank deposits rather than gravity induced depositional processes. Brecciation appears to result from translational sliding (millimetres to metres) on the steep buildup flanks, which caused fracturing of the vaguely stratified automicrite, followed by displacive growth of fibrous cement. Cementation occurred in a (shallow) burial, marine phreatic environment, because cement clasts are virtually absent from the flank‐derived gravity flows in the adjoining basinal sediments. Displacive cement growth indicates a volume increase of the clinoforms during diagenesis of up to 20–40 vol.% and can account for the local drag of buildup interior limestones. Similarly, the boulder fabric appears to be a diagenetic feature, which resulted from differential settling of incompletely lithified boundstone and grainstone, and the interbedded pelagic limestone.     

Linear Elastic and Cohesive Fracture Analysis to Model Hydraulic Fracture in Brittle and Ductile Rocks

Hydraulic fracturing technology is being widely used within the oil and gas industry for both waste injection and unconventional gas production wells. It is essential to predict the behavior of hydraulic fractures accurately based on understanding the fundamental mechanism(s). The prevailing approach for hydraulic fracture modeling continues to rely on computational methods based on Linear Elastic Fracture Mechanics (LEFM). Generally, these methods give reasonable predictions for hard rock hydraulic fracture processes, but still have inherent limitations, especially when fluid injection is performed in soft rock/sand or other non-conventional formations. These methods typically give very conservative predictions on fracture geometry and inaccurate estimation of required fracture pressure. One of the reasons the LEFM-based methods fail to give accurate predictions for these materials is that the fracture process zone ahead of the crack tip and softening effect should not be neglected in ductile rock fracture analysis. A 3D pore pressure cohesive zone model has been developed and applied to predict hydraulic fracturing under fluid injection. The cohesive zone method is a numerical tool developed to model crack initiation and growth in quasi-brittle materials considering the material softening effect. The pore pressure cohesive zone model has been applied to investigate the hydraulic fracture with different rock properties. The hydraulic fracture predictions of a three-layer water injection case have been compared using the pore pressure cohesive zone model with revised parameters, LEFM-based pseudo 3D model, a Perkins-Kern–Nordgren (PKN) model, and an analytical solution. Based on the size of the fracture process zone and its effect on crack extension in ductile rock, the fundamental mechanical difference of LEFM and cohesive fracture mechanics-based methods is discussed. An effective fracture toughness method has been proposed to consider the fracture process zone effect on the ductile rock fracture.     

3‐D mesoscale simulation of crack‐permeability coupling in the Brazilian splitting test

In this paper, mesoscale hydromechanical simulations are performed to study (1) fracture features and (2) crack‐gas permeability coupling evolution in the context of the tensile splitting test. The mesostructure is based on a 2‐phase 3‐D representation of heterogeneous materials, such as concrete, where stiff aggregates are embedded into a mortar matrix. To take into account these heterogeneities without any mesh adaptation, a weak discontinuity is introduced into the strain field. In addition, a strong discontinuity is also added to take into account microcracking. This mechanical model is cast into the framework of the enhanced finite element method. Concerning the coupling with gas permeability, a double‐porosity method is used to simulate the flow through the cracks and the porosity. The apparent gas permeability is afterwards evaluated by a homogenization method. On the basis of finite element simulations, influence of aggregate size on ultimate crack opening, macroscopic ultimate tensile stress, total dissipated energy, and gas permeability evolution is numerically investigated. Furthermore, gas permeability evolution is also compared with experimental results from the literature. In addition, in the spirit of a sequential multiscale approach, macroscale gas permeability equations are identified from the hydromechanical results coming from the mesoscale computations. These equations lead to a relation between macroscale gas permeability evolution and crack opening. Besides, we show how the aggregate size influences the percolation threshold and that after this threshold, a cubic relation between macroscale gas permeability and crack opening is obtained.     

红层泥质胶结物干缩裂缝深度与间距预测模型

红层泥质胶结物在干燥条件下会因含水率变化产生开裂,裂缝进一步发展贯通最终导致红层泥岩崩解破坏。目前对红层泥岩崩解机制的研究多为定性描述,缺乏定量分析方法。因此,定量研究红层泥质胶结物开裂的产生及扩展对深化红层泥岩崩解理论研究具有重要意义。基于线弹性断裂力学及非饱和土力学,建立了考虑含水率变化的红层泥质胶结物裂缝深度及间距计算公式,并对公式参数敏感性进行了分析。应用CT试验数据对PFC2D数值模型进行了校准,并基于干缩开裂数值试验结果,对裂缝深度及间距计算公式进行了对比验证。结果表明：提出的计算公式能较好地预测红层泥质胶结物的开裂深度及间距,裂缝深度随含水率减小呈先快速增加再缓慢增长的特点。研究成果有助于加深对红层泥岩崩解的量化理论分析,并为预测工程中红层泥岩的开裂程度提供参考。     

Modelling crack propagation in concrete structures with a two scale approach

A simplified computational technique based on a refined global–local method is applied to the failure analysis of concrete structures. The technique distinguishes the scale of the structure, modelled with large size finite elements, from the scale at which material non‐linearity occurs due to progressive cracking and macro‐crack propagation. The finite element solution is split into two parts: a linear elastic analysis on a coarse mesh over the entire structure and a non‐linear analysis over a small part of the structure where a dense finite element grid is employed. In the non‐linear calculation, a non‐local damage model is implemented. These two computations are coupled with the help of an iterative scheme. The size and location of the region where a non‐linear analysis is performed, is adapted to follow the development of the damage zone. Numerical examples of mode I fracture of concrete specimens with straight and curved cracks are presented. Copyright © 2003 John Wiley & Sons, Ltd.     

Self‐similar solution of a plane‐strain fracture driven by a power‐law fluid

This paper analyses the problem of a hydraulically driven fracture, propagating in an impermeable, linear elastic medium. The fracture is driven by injection of an incompressible, viscous fluid with power‐law rheology and behaviour index n?0. The opening of the fracture and the internal fluid pressure are related through the elastic singular integral equation, and the flow of fluid inside the crack is modelled using the lubrication theory. Under the additional assumptions of negligible toughness and no lag between the fluid front and the crack tip, the problem is reduced to self‐similar form. A solution that describes the crack length evolution, the fracture opening, the net fluid pressure and the fluid flow rate inside the crack is presented. This self‐similar solution is obtained by expanding the fracture opening in a series of Gegenbauer polynomials, with the series coefficients calculated using a numerical minimization procedure. The influence of the fluid index n in the crack propagation is also analysed. Copyright © 2002 John Wiley & Sons, Ltd.