Phase-field models for brittle and cohesive fracture

In this paper we first recapitulate some basic notions of brittle and cohesive fracture models, as well as the phase-field approximation to fracture. Next, a critical assessment is made of the sensitivity of the phase-field approach to brittle fracture, in particular the degradation function, and the use of monolithic versus partitioned solution schemes. The last part of the paper makes extensions to a recently developed phase-field model for cohesive fracture, in particular for propagating cracks. Using some simple examples the current state of the cohesive phase-field model is shown.     

A phase-field modeling approach of hydraulic fracture in saturated porous media

Continuum porous media theories, extended by a diffusive phase-field modeling (PFM) approach, introduce a convenient and efficient tool to the simulation of hydraulic fracture in fluid-saturated heterogeneous materials. In this, hydraulic- or tension-induced fracture occurs in the solid phase. This leads to permanent local changes in the permeability, the volume fractions of the constituents as well as the interstitial-fluid flow. In this work, the mechanical behaviors of the multi-field, multi-phase problem of saturated porous media, such as the pore-fluid flow and the solid-skeleton deformation, are described using the macroscopic Theory of Porous Media (TPM). To account for crack nucleation and propagation in the sense of brittle fracture, the energy-minimization-based PFM procedure is applied, which approximates the sharp edges of the crack by a diffusive transition zone using an auxiliary phase-field variable. Furthermore, the PFM can be implemented in usual continuum finite element packages, allowing for a robust solution of initial-boundary-value problems (IBVP). For the purpose of validation and comparison, simulations of a two-dimensional IBVP of hydraulic fracture are introduced at the end of this research paper.     

Sharp-crack limit of a phase-field model for brittle fracture

A matched asymptotic analysis is used to establish the correspondence between an appropriately scaled version of the governing equations of a phase-field model for fracture and the equations of the two-dimensional sharp-crack theory of Gurtin and Podio-Guidugli (1996) that arise on assuming that the bulk constitutive behavior is nonlinearly elastic, requiring that surface energy provides the only factor limiting crack propagation, and assuming that the fracture kinetics are isotropic. Consistent with the prominence of the configurational momentum balance at the crack tip in the latter theory, the approach capitalizes on the configurational momentum balance that arises naturally in the context of the phase-field model. The model developed and utilized here incorporates irreversibility of the phase-field evolution. This is achieved by introducing a suitable constraint and by carefully heeding the influence of that constraint on the kinetics underlying microstructural changes associated with fracture. The analysis is predicated on the assumption that the phase-field variable takes values in the closed interval between zero and unity.     

页岩气藏压裂缝网扩展数值模拟

为探究页岩气藏水力压裂复杂裂缝网络的形成机理,开展了缝网扩展的数值模拟研究.考虑应力阴影和天然裂缝作用,建立了井筒和裂缝中流体流动模型,利用位移不连续方法求解应力与位移不连续量,然后构建了压力与裂缝宽度的迭代方程,并采用牛顿迭代法求解.通过比较数值解经典模型解析解,验证了模型和迭代解法的正确性.多簇裂缝同步扩展时裂缝间距越小,压裂液分配到各条裂缝越不均匀,靠近井筒跟部的裂缝的分流量越大,从而裂缝宽度越大;考虑天然裂缝作用时,逼近角越小或者应力各向异性越弱,水力裂缝越容易发生转向扩展,裂缝网络越复杂.      

Simulation of crack propagation in fiber-reinforced bulk metallic glasses

Based on a phase-field model for deformation in bulk metallic glasses (BMGs), shear band formation and crack propagation in the fiber-reinforced BMG are investigated. Ideal unbroken fibers embedded in the BMG matrix are found to significantly influence the shear banding and crack propagation in the matrix. The crack propagation affected by fibers’ length and orientation is quantitatively characterized and is described by micromechanics models for composite materials. Furthermore, fractures in some practical fiber-reinforced BMG composites such as tungsten-reinforced Zr-based BMG are simulated. The relation between the enhanced fracture toughness and the mechanical properties of fiber reinforcements is determined. Different fracture modes of BMG-matrix composites are identified from the systematic simulation studies, which are found to be consistent with experiments. The simulation results suggest that the phase-field modeling approach could be a useful tool to assist the fabrication and design of BMG composites with high fracture toughness and ductility.     

水力压裂解析模型裂缝扩展参数敏感性分析

水力压裂形成复杂裂缝网络是致密储层油气开采的重要技术,掌握水压裂缝扩展机理是控制压裂行为和优化压裂效果的关键．水压裂缝动态扩展行为涉及储层岩体、注入压裂液、压裂实施工艺等方面,其中水力压裂扩展时间、压裂液流体动力粘度系数、压裂液流体注入流速、储层岩石剪切模量成为决定裂缝扩展长度和裂缝开度的重要因素．本研究采用KGD、PKN两类等高解析模型对主控因素的参数敏感性进行分析,直观、快速、可靠地获得水压裂缝扩展长度、张开度动态演化行为的量化数值．研究发现,压裂持续开展过程中水压裂缝扩展长度呈线性增长、开度逐渐趋于稳定,高流体动力粘度导致裂缝难扩展、形成较大裂缝开度,通过增加压裂液流体注入流速可同时增加裂缝扩展长度和开度,较高的岩石剪切模量将降低水压裂缝的开度．通过对比两类解析模型在不同参数下的水压裂缝扩展结果,分析压裂参数与裂缝扩展的相关性和敏感系数,讨论水力压裂解析模型的裂缝扩展参数敏感性．     

Development of hot dry rocks by hydraulic stimulation: Natural fracture network simulation

This paper presents an advanced computational modelling of natural fracture networks in HDR (hot dry rock) reservoirs. The model stochastically simulates discrete properties of natural fractures, utilizing multi-set orientation and fractal mathematics. The simulated fracture networks are essential for further stimulation and fluid flow studies. The model has been verified using the data of actual fracture stimulation programs conducted by Gas Research Institute and Department of Energy at the multi-site. It is validated that the simulated fracture distribution is sufficiently similar to that observed in the reservoir. This paper also examines the detrimental effects of the simulated natural fracture network on the stimulated fluid flow capacity. The effective permeability enhancement (due to hydraulic stimulation) is found almost proportional to the density of the reservoir natural fractures.     

Finite Element study of fracture initiation in flaws subject to internal fluid pressure and vertical stress

Hydraulic fracturing is a method used routinely in oil and gas exploitation and in engineered geothermal systems. While used frequently, there are many aspects of hydraulic fracturing, such as the direction of propagation of the newly-created fractures, which are not very well understood. Even though it is known that the local stress field plays a fundamental role in the orientation of the new fractures, there may be other factors, such as the geometry of the existing fractures and the magnitude of the hydraulic pressure applied, that may play a major role in the path that a new fracture follows when pressurized.The main goal of this study is to numerically analyze the effect of the ratio between a vertical load, or stress, and the hydraulic pressure applied in existing flaws on the stress field in the vicinity of the flaw tips. For that purpose, a double flaw geometry 2a-30-30 was modeled in the Finite Element code ABAQUS, and different vertical loads and internal flaw pressures were applied to the model. The variation of the maximum principal stresses and maximum shear stresses around the flaw tips were analyzed and related to fracture initiation.The study showed that the ratio between the water pressure applied in the flaws and the vertical load/stress (WP/VL) plays a crucial role in the magnitude and shape of the stress field around a flaw tip, and therefore in the location of tensile and shear fracture initiation. As WP/VL increases, the location of initiation of new tensile fractures shifts from the upper face of the studied flaw towards the region right ahead of the flaw tip; simultaneously, the location of initiation of new shear fractures shifts from the region ahead of the flaw tip to the upper face of the analyzed tip.     

Phase-field modeling of hydraulic fracture

In this work a theoretical framework implementing the phase-field approach to fracture is used to couple the physics of flow through porous media and cracks with the mechanics of fracture. The main modeling challenge addressed in this work, which is a challenge for all diffuse crack representations, is on how to allow for the flow of fluid and the action of fluid pressure on the aggregate within the diffuse damage zone of the cracks. The theory is constructed by presenting the general physical balance laws and conducting a consistent thermodynamic analysis to constrain the constitutive relationships. Constitutive equations that reproduce the desired responses at the various limits of the phase-field parameter are proposed in order to capture Darcy-type flow in the intact porous medium and Stokes-type flow within open cracks. A finite element formulation for the solution of the governing model equations is presented and discussed. Finally, the theoretical and numerical model is shown to compare favorably to several important analytical solutions. More complex and interesting calculations are also presented to illustrate some of the advantageous features of the approach.     

Investigation of Stress Field and Fracture Development During Shale Maturation Using Analog Rock Systems

The emergence of hydrocarbons within shale as a major recoverable resource has sparked interest in fluid transport through these tight mudstones. Recent studies suggest the importance to recovery of microfracture networks that connect localized zones with large organic content to the inorganic matrix. This paper presents a joint modeling and experimental study to examine the onset, formation, and evolution of microfracture networks as shale matures. Both the stress field and fractures are simulated and imaged. A novel laboratory-scale, phase-field fracture propagation model was developed to characterize the material failure mechanisms that play a significant role during the shale maturation process. The numerical model developed consists of coupled solid deformation, pore pressure, and fracture propagation mechanisms. Benchmark tests were conducted to validate model accuracy. Laboratory-grade gelatins with varying Young’s modulus were used as scaled-rock analogs in a two-dimensional Hele-Shaw cell apparatus. Yeast within the gelatin generates gas in a fashion analogous to hydrocarbon formation as shale matures. These setups allow study and visualization of host rock elastic-brittle fracture and fracture network propagation mechanisms. The experimental setup was fitted to utilize photoelasticity principles coupled with birefringence properties of gelatin to explore visually the stress field of the gelatin as the fracture network developed. Stress optics image analysis and linear elastic fracture mechanics (LEFM) principles for crack propagation were used to monitor fracture growth for each gelatin type. Observed and simulated responses suggest gas diffusion within and deformation of the gelatin matrix as predominant mechanisms for energy dissipation depending on gelatin strength. LEFM, an experimental estimation of principal stress development with fracture growth, at different stages was determined for each gelatin rheology. The interplay of gas diffusion and material deformation determines the resulting frequency and pattern of fractures. Results correlate with Young’s modulus. Experimental and computed stress fields reveal that fractures resulting from internal gas generation are similar to, but not identical to, type 1 opening mode.

     

固体结构损伤破坏统一相场理论、算法和应用

固体开裂引起的损伤和断裂是工程材料和结构最为普遍的破坏形式. 为了防止这种破坏, 结构设计首先必须了解裂缝在固体内如何萌生、扩展、分叉、汇聚甚至破碎; 更重要的是, 还需要准确量化这些裂缝演化过程对于结构完整性和安全性降低的不利影响. 针对上述固体结构损伤破坏问题, 本工作系统地介绍了笔者提出的统一相场理论、算法及其应用. 作为一种裂缝正则化变分方法, 统一相场理论将基于强度的裂缝起裂准则、基于能量的裂缝扩展准则以及满足变分原理的裂缝路径判据纳入同一框架内. 不仅常用的脆性断裂相场模型是该理论的特例, 还自然地给出了一类同时适用于脆性断裂和准脆性破坏的相场正则化内聚裂缝模型即 PF-CZM. 该模型非常便于通过有限元等方法加以数值实现; 为了求解有限元空间离散后得到的非线性方程组, 还介绍了几种常用的数值算法, 其中整体BFGS拟牛顿迭代算法的计算效率最高. 静力、动力和多场耦合条件下若干二维和三维代表性算例表明: 相场正则化内聚裂缝模型 PF-CZM能够高精度地再现复杂裂缝演化导致的脆性和准脆性固体损伤破坏; 特别是, 所有情况下, 模型的数值结果不依赖于裂缝尺度参数和有限元网格. 因此, 该模型具有相当好的预测能力, 有望在工程结构的损伤破坏分析方面发挥重要作用. 最后建议了若干值得进一步开展的研究课题.      

Dynamic fracture instabilities in brittle crystals generated by thermal phonon emission: Experiments and atomistic calculations

Dynamic cleavage fracture experiments of brittle single crystal silicon revealed several length scales of surface and path instabilities: macroscale path selection, mesoscale crack deflection, and nanoscale surface ridges. These phenomena cannot be predicted or explained by any of the continuum mechanics based equations of motion of dynamic cracks, as presumably critical energy dissipation mechanisms are not fully accounted for in the theories. Experimentally measured maximum crack speed, always lower than the theoretical limit, is another phenomenon that is as yet not well understood.We suggest that these phenomena depend on velocity dependent and anisotropic material property that resists crack propagation. The basic approach is that the bond breaking mechanisms during dynamic crack propagation vibrate the atoms at the crack front to generate thermal phonon emission, or heat, which provides additional energy dissipation mechanisms. This energy dissipation mechanism is a material property that resists crack propagation. To evaluate this property, we combined the continuum based elastodynamic Freund equation of motion with molecular dynamics atomistic computer “experiments”.We analyzed the above experimental dynamic fracture instabilities in silicon with the obtained velocity dependent and anisotropic material property and show its importance in cleavage of brittle crystals.     

Fractures in complex fluids: the case of transient networks

We present a comprehensive review of the current state of fracture phenomena in transient networks, a wide class of viscoelastic fluids. We will first define what is a fracture in a complex fluid and recall the main structural and rheological properties of transient networks. Secondly, we review experimental reports on fractures of transient networks in several configurations: shear-induced fractures, fractures in Hele–Shaw cells, and fracture in extensional geometries (filament-stretching rheometry and pendant drop experiments), including fracture propagation. The tentative extension of the concepts of brittleness and ductility to the fracture mechanisms in transient networks is also discussed. Finally, the different and apparently contradictory theoretical approaches developed to interpret fracture nucleation will be addressed and confronted to experimental results. Rationalized criteria to discriminate the relevance of these different models will be proposed.     

基于统一相场理论的早龄期混凝土化-热-力多场耦合裂缝模拟与抗裂性能预测

受水化反应和热量传输等过程影响, 混凝土在养护阶段会发生受约束收缩变形, 并由此在结构内引发较大的拉应力, 而此时混凝土力学性能往往还处于较低水平, 容易导致建造期混凝土结构即出现裂缝等病害. 这种早龄期混凝土裂缝对核安全壳、桥梁隧道、地下结构、水工或海工结构等重大土木工程和基础设施的全生命周期完整性、耐久性和安全性造成严重影响. 为了准确预测早龄期混凝土抗裂性能并量化裂缝演化对混凝土结构行为的不利影响, 亟需开展化-热-力多场耦合环境下的混凝土裂缝建模与抗裂性能分析研究. 针对这一需求, 本工作在前期提出的固体结构损伤破坏统一相场理论基础上, 考虑开裂过程与水化反应、热量传输等之间的相互影响, 建立裂缝相场演化特征(包括基于强度的裂缝起裂准则、基于能量的裂缝扩展准则和基于变分原理的扩展方向判据等)与混凝土水化度和温度之间的定量联系, 提出混凝土化-热-力多场耦合相场内聚裂缝模型, 发展相应的多场有限元数值实现算法并应用于若干验证算例. 数值模拟结果表明, 上述多场耦合相场内聚裂缝模型合理地考虑了水化反应、热量传输、力学行为以及裂缝演化之间的耦合效应, 揭示了早龄期混凝土热膨胀变形和自收缩变形的相互竞争机理, 且分析结果不受裂缝尺度和网格大小等数值参数的影响, 实现了早龄期裂缝演化全过程准确模拟和抗裂性能定量预测, 有望在混凝土结构早龄期裂缝预测和控制方面发挥重要作用.      

A variational approach to analyze a natural fault with hydraulic fracture based on the strain energy density criterion

A simplified analytical model of the interaction between a hydraulic fracture and an existing natural fault is developed. The mechanical activation of the natural fault as a result of contact with a pressurized fracture is described for plane strain conditions and quasi-static fracture propagation approximation. Using a variational approach, the normal and shear stresses, as well as the boundaries of the open and sliding zones along the fault, are predicted for three stages of the fracturing process (fracture approaching, coalescence, and fluid penetration). An accumulated concentration of shear stress at the tip of the fault’s sliding zone is shown to create sufficient tensile stress to initiate a new tensile crack on the opposite side of the fault, provided either the differential in situ stress is low or the friction coefficient is sufficiently large. The results of direct numerical simulation of the fracture interaction fit the model predictions made from the strain energy density fracture criterion.     

3-D Fracture Propagation Simulation and Production Prediction in Coalbed

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页岩储层体积压裂的地应力变化研究

页岩储层属于致密超低渗透储层,需改造形成复杂缝网才有经济产能.体积压裂是页岩储层增产改造的主要措施,而地应力场特别是水平主应力差值是体积压裂的关键控制因素. 理论研究表明:(1)当初始两向水平主应力差较小时,容易形成缝网,反之不易产生缝网;(2)人工裂缝的形成能够改变地层初始应力场. 因此应在前人研究的基础上优化设计压裂方式,以克服和翻转初始水平主应力差值,产生体积缝网.基于此,建立了页岩气藏水平井体积压裂数值模型,模型中采用多孔介质流固耦合单元模拟页岩基质的行为,采用带有孔压的"cohesive"单元描述水力裂缝的性质,模型对"Texas Two-Step" 压裂方法进行了数值模拟,模拟结果得到了压裂过程中地层应力场的分布及其变化,模拟结果和解析公式计算结果吻合良好.模拟结果表明:(1)裂缝的产生减弱了地层应力场的各向异性;(2 对于低水平应力差页岩储层,采用"Texas Two-Step"压裂方法可以产生缝网. 对于采用"Texas Two-Step"压裂方法无法产生缝网的高应力差页岩储层,提出了三次应力"共振" 和四次应力"共振" 压裂方法并进行了数值模拟,模拟结果得到了压裂过程中页岩储层应力场的分布及其变化,得到了缝网形成的区域,模拟结果表明:(1)对于高应力差页岩储层,采用"Texas Two-Step" 压裂方法无法产生缝网;(2)对于高应力差页岩储层,三次应力"共振" 和四次应力"共振"压裂方法是有效的体积压裂缝网形成的方法.