Fracture Toughness Evaluation Based on Tension-softening Model and its Application to Hydraulic Fracturing

This paper discusses the applicability of the tension-softening model in the determination of the fracture toughness of rocks, where the fracture toughness evaluated based on the tension-softening model is compared with the crack growth resistance deduced from laboratory-scale hydraulic fracturing tests. It is generally accepted that the fracture process is dominated by the growth of a fracture process zone for most types of rocks. In this study, the J-integral based technique is employed to determine the fracture toughness of Iidate granite on the basis of the tension-softening model, where compact tension specimens of different dimensions were tested in order to examine the specimen size effect on the measured fracture toughness. It was shown that the tension-softening relation deduced from the J-integral based technique allowed us to determine the specimen size independent fracture toughness K_c of Iidate granite. Laboratory-scale hydraulic fracturing tests were performed on cubic specimens (up to a 10 m sized specimen), where cyclic pressurization was conducted using a rubber-made straddle packer to observe the extent of the hydraulically induced crack. The experimental results of pressure and crack length were then used to construct the crack growth resistance curve based on the stress intensity factor K. The crack growth resistance obtained from the hydraulic fracturing tests was observed to initially increase and then level off, giving a constant K value for a long crack extension stage. The plateau K value in the crack growth resistance curve was found to be in reasonable agreement with the fracture toughness K_c deduced from the tension-softening relation. It was demonstrated that the tension-softening model provides a useful tool to determine the appropriate fracture toughness of rocks, which may be applicable for the analysis of the process of large-scale crack extension in rock masses.     

Effect of stress interactions on anisotropic P-SV-wave dispersion and attenuation for closely spaced cracks in saturated porous media

Generally, local stress induced by individual crack hardly disturbs their neighbours for small crack densities, which, however, could not be neglected as the crack density increases. The disturbance becomes rather complex in saturated porous rocks due to the wave-induced diffusion of fluid pressures. The problem is addressed in this study by the comparison of two solutions: the analytical solution without stress interactions and the numerical method with stress interactions. The resultant difference of effective properties can be used to estimate the effect of stress interactions quantitatively. Numerical experiments demonstrate that the spatial distribution pattern of cracks strongly affects stress interactions. For regularly distributed cracks, the resulting stress interaction (shielding or amplification) shows strong anisotropy, depending on the arrangement and density of cracks. It has an important role in the estimation of effective anisotropic parameters as well as the incident-angle-dependency of P- and SV-wave velocities. Contrarily, randomly distributed cracks with a relative small crack density generally lead to a strong cancellation of stress interactions across cracks, where both the numerical and analytical solutions show a good agreement for the estimation of effective parameters. However, for a higher crack density, the incomplete cancellation of stress interactions is expected, exhibiting an incidence-angle dependency, slightly affecting effective parameters, and differentiating the numerical and analytical solutions.     

Self-similar cataclasis in the formation of fault gouge

Particle-size distributions have been determined for gouge formed by the fresh fracture of granodiorite from the Sierra Nevada batholith, for Pelona schist from the San Andreas fault zone in southern California, and for Berea sandstone from Berea, Ohio, under a variety of triaxial stress states. The finer fractions of the gouge derived from granodiorite and schist are consistent with either a self-similar or a logarithmic normal distribution, whereas the gouge from sandstone is not. Sandstone gouges are texturally similar to the disaggregated protolith, with comminution limited to the polycrystalline fragments and dominantly calcite cement. All three rock types produced significantly less gouge at higher confining pressures, but only the granodiorite showed a significant reduction in particle size with increased confining pressure. Comparison with natural gouges showed that gouges in crystalline rocks from the San Andreas fault zone also tend to be described by either a self-similar or log-normal particle distribution, with a significant reduction in particle size with increased confining pressure (depth). Natural gouges formed in porous sandstone do not follow either a self-similar or a log-normal distribution. Rather, these are represented by mixed log-normal distributions. These textural characteristics are interpreted in terms of the suppression of axial microfracturing by confining pressure and the accommodation of finite strain by scale-independent comminution.     

Brittle crack growth in rocks

Fracture phenomena in rocks are associated with mainly mode I crack growth, sometimes superposed by shear or torsion. The present paper contributes to a fracture mechanics analysis of mode I and mixed mode crack propagation, by presenting reliable fracture toughness data for some rocks which include the effect of induced crack propagation rate, and the influence of effective pressure, and by numerical calculations on fracture propagation in layered rock formations. Empirical relations between fracture toughness,K _Ic' and induced crack opening displacement rate, as well as effective pressure, are given. The observedK _Ic pressure relation supports a theoretical model which takes into account the existence of microcracks in the crack tip region. Finite element calculations of fracture propagation in layered rock formations demonstrate the important effect of mixed mode crack growth. The numerical approach is particularly applied to single crack growth in hydraulic fracturing and in three point bending tests on layered single edge crack specimens.     

Influence of confining pressure and impact loading on mechanical properties of amphibolite and sericite-quartz schist

In order to investigate the dynamic mechanical properties of amphibolite and sericite-quartz schist under confi ning pressure, two rocks are subjected to impact loadings with different strain rates and confi ning pressures by using split Hopkinson pressure bar equipment with a confi ning pressure device. Based on the experimental results, the stress-strain curves are analyzed and the effects of confi ning pressure and strain rates on the dynamic compressive strength, peak strain and failure mode are summarized. The results show that:(1) The characteristics of two rocks in the ascent stage of the stressstrain curve are basically the same, but in the descent stage, the rocks gradually show plastic deformation characteristics as the confi ning pressure increases.(2) The dynamic compressive strength and peak strain of two rocks increase as the strain rate increases and the confi ning pressure effects are obvious.(3) Due to the effect of confi ning pressure, the normal stress on the damage surface of the rock increases correspondingly, the bearing capacity of the crack friction exceeds the material cohesion and the slippage of the fractured rock is controlled, which all lead to the compression and shear failure mode of rock. The theoretical analysis and experimental methods to study the dynamic failure mode and other related characteristics of rock are useful in developing standards for engineering practice.     

低孔隙度泥页岩应力依赖的各向异性裂纹演化特性研究

本文在实验室对所获取的东营地区层理发育的低孔隙度页岩和泥岩的各向异性裂纹演化特性进行了研究,获得了各向同性条件下泥页岩的力学与超声波响应特性,分析了应力幅度对于页岩声波速度和各向异性的影响.主要结论包括:(1)泥页岩在循环载荷下存在滞后效应,表明其经历了去压实或油气产生导致的超压;(2)泥岩和页岩具有不同程度的各向异性,随着各向同性压力的增高微裂隙逐渐闭合,样品的各向异性程度减弱;(3)分析了岩石韧度和裂纹损伤参数随压力的变化特征,相比泥岩,页岩各向异性程度更高,随压力变化更明显,其裂纹导致的附加各向异性更强;(4)分析了各向异性岩石的动态弹性模量特征,由于软裂隙空间的闭合,动态弹性模量在低压条件下都随着围压的增加有硬化趋势.     

Fracture toughness and subcritical crack growth during high-temperature tensile deformation of Westerly granite and Black gabbro

The double torsion testing method has been used to determine catastrophic and subcritical crack propagation parameters for pre-cracked specimens of Westerly granite and Black gabbro under a number of environmental conditions.The critical stress intensity factor for catastrophic crack propagation (fracture toughness) of granite and gabbro has been measured at temperatures from 20 to 400°C, in a vacuum. At 20°C, the fracture toughness of Westerly granite was $\text{1.79 ± 0.02}\text{MPa}\text{·}\text{m}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$, and for two blocks of Black gabbro it was $\text{3.03 ± 0.08}\text{MPa}\text{·}\text{m}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$ and $\text{2.71 ± 0.15}\text{MPa}\text{·}\text{m}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$, respectively. These values are very close to those reported by other investigators for tests conducted in air of ambient humidity at room temperature. For both rocks, fracture toughness at first increased slightly, and then decreased steadily on raising the temperature above ambient conditions. This behaviour is explained in terms of the density and distribution of thermally induced microcracks, as determined by quantitative optical microscopy.Subcritical crack growth behaviour has been studied at temperatures up to 300°C, and under water vapour at pressures of 0.6 to 15 kPa. Both the load relaxation and incremental constant displacement rate forms of the double torsion testing method were utilised to generate stress intensity factor/crack velocity diagrams. Crack growth was measured over the velocity range 5 × 10^?3 to 10^?7 m · s^?1. Increasing both temperature and water vapour pressure resulted in substantially higher crack growth rates. The overall effect of raising the temperature over the range studied here (20–300°C) was to increase the crack growth rate in granite and gabbro by ～5 and 7 orders of magnitude, respectively, at constant stress intensity factor and vapour pressure of water. For both rocks, the slopes of stress intensity factor/crack velocity curves were sensitive to changes in both temperature and water vapour pressure at low values of the latter parameter. Slopes fell substantially on raising the water vapour pressure, but were relatively insensitive to changes in temperature at these higher pressures. No subcritical crack growth limit was encountered.Estimates of the uncertainty in our experimental data are given. From the results of multiple load relaxation experiments on Westerly granite specimens, we estimate the uncertainty in position of stress intensity factor/crack velocity curves along the stress intensity axis to be c. 10% of the fracture toughness, and the uncertainty in slope of such curves to be c. 12%.Problems associated with the extrapolation of our experimental data to regions of higher effective confining pressure in the Earth's crust are discussed.     

Slip-weakening constitutive relation and the structure in the vicinity of a shear crack tip

Shear-crack model with a cohesive zone (or breakdown zone) is appropriate for the analysis of a fault surface in which slip distribution is strongly nonuniform. As the slipped portion advances, slip-weakening occurs over the so-called cohesive zone, a distance behind the fault tip. For a prescribed strength vs. displacement constitutive relation, however, the zone structure is difficult to determine by an analytical method except for some simple cases, thus it often requires a certain numerical procedure. This work proposes a numerical procedure to obtain approximated solutions of the problem by combining a series of elastic solutions derived bySmith (1974). The series is linearly combined and the unknown coefficients are determined by a nonlinear least square method. This method can fit a wide range of prescribed strength vs. displacement relations which may be simple algebraic relations or curves obtained by laboratory tests. By examining the residual errors and in comparison with a derived result in which linear stress is assumed within the zone, it could be concluded that the results provide good accuracy. Moreover, because the results are written in formulae, they can be easily referred to or used. By fitting constitutive curves in many different shapes, it is found that the stress distribution within the zone is more sensitive to the constitutive curve shape than the displacement. The most interesting fact is that the zone size is not sensitive to the curve shape, i.e., the zone size can be estimated by $$R = 3\mu \zeta \upsilon _c /\{ 2(1 - \upsilon )(\tau _c - \tau _f )\}$$ with ζ=1±0.11 for most cases.     

Seismic-frequency Laboratory Measurements of Shear Mode Viscoelasticity in Crustal Rocks II: Thermally Stressed Quartzite and Granite

—Forced torsional oscillation techniques have been used to explore the seismic-frequency shear mode viscoelasticity of specimens of two crustal rocks (Cape Sorell quartzite and Delegate aplite), cycled between room temperature and 700°C under conditions of moderate confining pressure. The anisotropy and intergranular inhomogeneity of thermal expansivity in these materials give rise to large deviatoric stresses, resulting in thermal cracking at temperatures above a pressure-dependent threshold temperature, associated with the onset of very pronounced temperature sensitivity of the shear modulus, in general accord with the predictions of fracture mechanics models. For Delegate aplite in particular, the shear modulus behaves reproducibly during multiple thermal cycles at different confining pressures, consistent with the notion that the thermal cracks are of low aspect ratio (minimum/maximum dimension), and are therefore readily closed by the prevailing confining pressure once the thermal stresses are removed. Marked frequency-dependent dissipation of shear strain energy is observed on heating each rock to temperatures ≥ 500°C, although the attenuation varies significantly with prior thermal history, probably as a result of progressive dehydration and relaxation of deviatoric stresses. Temperature and pressure dependent crack densities for Delegate aplite have been estimated by comparison of the observed shear moduli with those expected for a crack-free aggregate. In parallel with the forced oscillation tests, measurements have been made of the rate at which (argon) pore pressure equilibrium is re-established following a perturbation. Combination of these results, which provide a proxy for permeability, with the inferred crack densities indicates that the variation of permeability with crack density is well described by a percolation model with a threshold crack density of ～0.2.     

Common Evolution of Mechanical and Transport Properties in Thermally Cracked Westerly Granite at Elevated Hydrostatic Pressure

Increasing the damage and crack porosity in crustal rocks can result in significant changes to various key physical properties, including mechanical strength, elastic and mechanical anisotropy, and the enhancement of transport properties. Using a Non-Interactive Crack Effective Medium (NIC) theory as a fundamental tool, we show that elastic wave dispersion can be inverted to evaluate crack density as a function of temperature and is compared with optically determined crack density. Further, we show how the existence of embedded microcrack fabrics in rocks also significantly influences the fracture toughness (K_IC) of rocks as measured via a suite of tensile failure experiments (chevron cracked notch Brazilian disk). Finally, we include fluid flow in our analysis via the Guéguen and Dienes crack porosity-permeability model. Using the crack density and aspect ratio recovered from the elastic-wave velocity inversion, we successfully compare permeability evolution with pressure with the laboratory measurements of permeability.     

围压作用下岩石样品中微裂纹的闭合

岩石中普遍存在的微裂纹对其物理性质有十分显著的影响.为了得到不同围压条件下岩样内部微裂纹形态的变化,选取几种具有明显层理的岩石圈岩石样品,在不同围压下沿相互正交的三个方向对纵横波速度和线应变等进行了测量,根据O’Connell模型通过计算得出有效泊松比、裂纹密度、裂纹孔隙度和裂纹的c/a比值（a为硬币形裂纹半长轴的长度,c为其半宽度）.结果表明,随着围压的增加,在垂直于层理面的方向上裂纹闭合最快但c/a比值不减反增.结合显微组构分析可知,岩石中的裂纹表面实际上是凸凹不平的,在围压增加引起裂纹闭合过程中一条裂纹可以变成几个,长度成倍减小但宽度变化不大,所以在垂直于层理面的方向上c/a比值随围压增加而增大.但另两个方向上的变化明显不同,应该与模型本身的适用范围有关.通过同时测量纵、横波速度,计算c/a比值,可以对样品中微裂纹的几何形态及空间分布方式有定性或半定量的了解.     

工程结构三维疲劳裂纹最大应力强度因子计算

介绍了裂纹的类型、裂纹尖端应力场的奇异性。以一维问题为例,推导论证了奇异单元能够很好的反映裂纹尖端应力场的奇异性。应力强度因子一般表达式表明应力强度因子与载荷呈线性关系,并依赖于物体和裂纹的几何形状和尺寸。本文借助大型通用有限元软件ANSYS,采用位移外插法计算了三维表面裂纹前沿不同位置处的应力强度因子,并与《应力强度因子手册》基于实验的理论公式计算结果相比较。结果表明:有限元结果与理论解误差较小,裂纹最深处应力强度因子最大。     

应力途径对岩石脆性-延性变化的影响

用两种加载方式对岩石的破裂进行了实验研究。一种方式是在一定的围压下增加轴压使岩石破裂（A型）;另一种方式是在一定围压下增加轴压直到破裂前某一应力状态,然后停止加轴压转而减小围压使岩石破裂（B型）。所用的岩石样品为济南辉长岩和山东掖县白大理岩。着重研究应力途径对岩石脆性-延性变化的影响。辉长岩在1.5千巴以下两种应力途径下的破裂都表现为脆性,但是对于同样的应力状态,B型实验比A型实验显得更脆。随着围压增加到200-250巴之间,大理岩由脆性转变为延性。围压250巴以上,大理岩的A型实验发生延性破裂。样品承受载荷的能力是逐渐丧失的。破裂过程中声发射率极低,听不到破裂声响。最后在样品中形成了剪切断面,但破裂很慢。然而,在大理岩的B型实验中,围压在250巴以上发生了脆性破裂,其表现为轴向应力突然下降,伴随着脆性破裂的声响并有声发射率剧增的前兆。看来,B型应力途径对岩石起了一个脆化的作用。     

Evaluation of the range of horizontal stresses in the earth's upper crust by using a collinear crack model

Vertical stresses in the earth's upper crust may be evaluated by the depth times the average unit weight of the overlying rock mass; however, the horizontal stress is difficult to obtain. Rock usually contains joints or cracks, and its fracture toughness is limited. If the horizontal stress acting on a cracked rock body exceeds a certain range, the crack will propagate and lead to rock fracture; and if a cracked rock is stable, the horizontal stress must be within a certain range. Therefore, from the stability condition of cracks, the range of horizontal stress can be evaluated. In this paper, a collinear crack model is employed to establish the cracked rock stability condition, and our theoretical results generally agree with the in-situ measurement results. The theoretical results can well explain the phenomenon that the ratio of horizontal stress to vertical stress near crust surface is scattered in a wide range, but in deep zone, it is scattered in a narrow range.     

北京北石城断层带与河防口断层带断层岩石显微构造研究

本文论述了糜棱岩类与碎裂岩类的变形特征,它们各自代表了不同的成因机制,反映了断层带经历过早期韧性剪切和后期脆性破裂的发育历史。根据断层岩石的显微构造特征,估算了断层带发育过程中两个阶段的温度、压力、应变速率、差异应力大小和方位,并讨论了韧性剪切带、地壳中弹塑性过渡带与大陆地震多发层之间的关系     

Fracture Toughness Measurements and Acoustic Emission Activity in Brittle Rocks

Fracture toughness measurements under static loading conditions have been carried out in Barre and Lac du Bonnet granites. An advanced AE technique has been adopted to monitor real-time crack initiation and propagation around the principal crack in these tests to understand the processes of brittle failure under tension and related characteristics of the resulting fracture process zone. The anisotropy of Mode I fracture toughness has been investigated along specific directions. Microcrack density and orientation analysis from thin section studies have shown these characteristics to be the primary cause of the observed variation in fracture toughness, which is seen to vary between 1.14 MPa.(m)^1/2 and 1.89 MPa.(m)^1/2 in Barre granite. The latter value represents the case in which the crack is propagated at right angles to the main set of microcracks. The creation of a significant fracture process zone surrounding the propagating main crack has been confirmed. Real-time imaging of the fracture process and formation of fracture process zone by AE techniques yielded results in very good agreement with those obtained by direct optical analysis.     

基于唐山7.8级地震探讨动态应力作用

运用三维非线性动态有限元方法(Non-linear Dynamic FEM)仿真模拟研究了菱形模型的动态应力作用,分析了1976年唐山7.8级地震.结果表明:(1)唐山地震前增强的中强地震产生的冲击力形成的应力波反射后发生半波损失, 成为动态的拉伸应力,通过减小内摩擦而引起岩石产生裂纹、局部破裂并发生串通,一方面触发地震,另一方面为孕育唐山地震积累能量,表现为正反馈过程.(2)唐山地震产生的冲击力加卸载存在时间差,造成加卸载冲击波相位不同,结果产生的动应力和围压叠加致使围压波动和下降,导致抗剪切强度降低,使初始破裂分别向北东、西南方向传播.(3)冲击力的卸载,相对于产生拉伸性的应力波,反射后发生半波损失,成为动态的压缩应力,致使围压增大并导致破裂停止;而动态应力在传播过程中衰减.因此,余震的发生是一个负反馈过程.     

水饱和裂纹对地壳岩样中地震波速及各向异性的影响

选择４种地壳岩石样品,经干燥或水饱和处理后在不同围压条件下测量了在其中传播的纵、横波的速度及其各向异性．在大气压条件下低孔隙度(＜１％岩样中,水饱和样品中的纵波速度明显地比干燥样品中的高,但横波速度的差别不大．因为在低孔隙度岩样中纵波速度对孔隙流体的反应比横波速度敏感,可以用泊松比的变化来反映随着围压的增加晶粒间流体对弹性波传播特性的影响．根据实验数据,按Ｏ’Ｃｏｎｎｅｌｌ模型分别计算了干燥和水饱和岩样中的裂纹密度,与通过实测体应变曲线得到的裂纹孔隙度十分吻合．利用横波的速度和偏振特性可以推断岩样中定向排列微裂纹的空间取向情况．研究表明,同时测量在岩样中传播的纵、横波的速度,通过Ｖｐ／Ｖｓ比值可以给出有关颗粒边界流体的证据,也可以估计岩样中的裂纹密度．     

岩石破裂强度的温度和应变率效应及其对岩石圈流变结构的影响

研究了温度和应变率对岩石破裂强度的影响,得到了岩石圈中一些典型岩石破裂强度的新的经验规律.新的经验规律除考虑围压和标本尺度的影响外,还考虑了温度和应变率的影响,并增加了新岩石的结果,所以更能反映岩石圈内岩石破裂的真实状态.通过对鄂尔多斯平均流变结构的计算和对比研究表明：传统的忽略脆性破裂的流变模型过高地估计了流变强度,流变机制的分布也不尽合理.而考虑了脆性破裂机制的流变模型的结果表明脆性区分为两部分,浅部以摩擦滑动机制控制,深部以脆性破裂机制控制.由于新的经验规律考虑的代表性岩石更全面,并考虑了应变率的影响,得到的脆性区的范围进一步增大,流变强度进一步降低.     

压力及流体对岩石孔隙结构与黏弹性的影响规律研究:理论模型及实验观测

地质成因和构造/热应力导致地壳岩石中的孔隙结构（裂隙和粒间孔）的变化.影响岩石黏弹性的因素包括压力、孔隙度、孔隙中包含的流体和孔隙几何形状等.相对于岩石中的硬孔隙,岩石黏弹性（衰减和频散）受软孔隙（裂隙）的影响更大.本文选取三块白云岩样本,进行了不同围压和流体条件下的超声波实验测量.利用CPEM（Cracks and Pores Effective Medium,裂隙和孔隙有效介质）模型获得了岩石高、低频极限的弹性模量,并通过Zener体（标准线性体）模型将CPEM模型拓展到全频带而得到CPEM-Zener模型,用该模型拟合岩石松弛和非松弛状态下的实验数据,本文得到平均裂隙纵横比和裂隙孔隙度以及纵波速度和品质因子随频率的变化关系.结果表明,饱水岩石的平均裂隙纵横比和裂隙孔隙度均高于饱油岩石,随着压差（围压和孔隙压力的差值）的增加,饱油岩石中的裂隙首先闭合.并且压差在70 MPa以内时,随着压差增大,岩石的平均裂隙纵横比和裂隙孔隙度在饱水和饱油时的差值增大,此时流体类型对于岩石裂隙的影响越来越显著,此外,对饱水岩石,平均裂隙纵横比随压差增加而增大,这可能是由于岩石中纵横比较小的裂隙会随压差增大而逐渐趋于闭合.在饱水和饱油岩石中,裂隙孔隙度和裂隙密度都随着压差增加而减小.通过对裂隙密度和压差的关系进行指数拟合,本文获得压差趋于0时的裂隙密度,且裂隙密度随孔隙度增大而增大,增大速率随压差增加而降低.针对饱水和饱油的白云岩样本,CPEM-Zener模型预测的纵波频散随压差增大而减小,此变化趋势和实验测得的逆品质因子随压差的变化关系基本一致,由此进一步验证了模型的实用性.本研究对岩石的孔隙结构和黏弹性分析以及声波测井、地震勘探的现场应用有指导意义.