Modeling spatial fracture intensity as a control on flow in fractured rock

Spatial fracture intensity (P ₃₂, fracture area by volume) is an important characteristic of a jointed rock mass. Although it can hardly ever be measured, P ₃₂ can be modeled based on available geological information such as spatial data of the fracture network. Flow in a mass composed of low-permeability hard rock is controlled by joints and fractures. In this article, models were developed from a geological data set of fractured andesite in LanYu Island (Taiwan) where a site is investigated for possible disposal of low-level and intermediate-level radionuclide waste. Three different types of conceptual models of spatial fracture intensity distribution were generated, an Enhanced Baecher’s model (EBM), a Levy–Lee Fractal model (LLFM) and a Nearest Neighborhood model (NNM). Modeling was conducted on a 10 × 10 × 10 m synthetic fractured block. Simulated flow was forced by a 1% hydraulic gradient between two vertical x–z faces of the cube (from North to South) with other boundaries set to no-flow conditions. Resulting flow vectors are very sensitive to spatial fracture intensity (P ₃₂). Flow velocity increases with higher fracture intensity (P ₃₂). R-squared values of regression analysis for the variables velocity (V/V _max) and fracture intensity (P ₃₂) are 0.293, 0.353, and 0.408 in linear fit and 0.028, 0.08, and 0.084 in power fit. Higher R ² values are positively linked with structural features but the relation between velocity and fracture intensity is non-linear. Possible flow channels are identified by stream-traces in the Levy–LeeFractal model.     

裂隙岩体等效力学参数结构效应的计算机模拟试验研究

为研究裂隙岩体结构效应对其等效力学参数的影响,引入基于离散元的合成岩体技术,通过可控的改变三维岩体结构网络中的结构参数,制作相应的合成岩体试件,开展计算机模拟试验研究。结果表明：随岩体中裂隙体积密度增大,裂隙的变形参数和强度参数随裂隙密度增加而减少,其中等效弹性模量对结构面密度变化最为敏感;随岩体中裂隙倾角增加,岩体等效弹性模量随之单调增加,且等效弹性模量的离散程度（标准差）随倾角增加而减小,而各强度参数指标则先减少而后增加,形成一个U形变化规律;岩体等效力学参数均随结构面尺寸增加而减小,同时等效力学参数的离散程度（标准差）随结构面直径增加而增加,其中单轴抗压强度指标变化最为敏感;岩体等效力学参数对裂隙尺寸和产状的离散程度不敏感。该研究结论可为其他结构效应研究提供一种新思路。     

Understanding the effect of single-fracture heterogeneity from single-well injection-withdrawal (SWIW) tests

The single-well injection-withdrawal (SWIW) tracer test is a method used to estimate the tracer retardation properties of a fracture or fracture zone. The effects of single-fracture aperture heterogeneity on SWIW-test tracer breakthrough curves are examined by numerical modelling. The effects of the matrix diffusion and sorption are accounted for by using a particle tracking method through the addition of a time delay added to the advective transport time. For a given diffusion and sorption property (P _m) value and for a heterogeneous fracture, the peak concentration is larger compared to a homogeneous fracture. The cumulative breakthrough curve for a heterogeneous fracture is similar to that for a homogeneous fracture and a less sorptive/diffusive tracer. It is demonstrated that the fracture area that meets the flowing water, the specific flow-wetted surface (sFWS) of the fracture, can be determined by matching the observed breakthrough curve for a heterogeneous fracture to that for a homogeneous fracture with an equivalent property parameter. SWIW tests are also simulated with a regional pressure gradient present. The results point to the possibility of distinguishing the effect of the regional pressure gradient from that of diffusion through the use of multiple tracers with different P _m values.     

基于CT-CA的天然裂隙岩石受拉破坏特性研究

岩石内部存在的裂隙、孔洞等天然损伤对岩石的力学性能和破坏过程有重要影响,依据细胞自动机理论结合CT无损识别技术实现了含天然裂隙岩石在劈裂条件下裂纹扩展和贯通全过程及其力学性能变化规律的研究。从裂隙砂岩的真实细观结构出发,构建了天然裂隙岩石的数值计算模型,运用CASRock数值计算软件完成了含不同裂隙倾角的砂岩劈裂破坏的数值试验,分析了裂隙倾角对砂岩的力学特性、裂纹扩展过程及能量演化的影响规律。研究表明：(1)天然裂隙砂岩的抗拉强度与裂隙倾角密切相关,随着裂隙倾角的增加,其抗拉强度呈现先减小后增加的趋势;(2)裂隙起裂于天然裂隙尖端,当裂隙倾角0°≤θ<48°时,岩样的破坏是由错开型裂纹引起,裂纹沿着与天然裂隙近垂直方向扩展;当裂隙倾角48°≤θ<94°时,岩样的破坏是由张开型裂纹引起,裂纹沿着与天然裂隙近平行方向扩展;(3)劈裂过程中裂纹尖端应力场存在拉应力区和压应力区,拉应力造成翼裂纹由天然裂隙尖端沿加载端方向萌生扩展,而压应力则引发次生裂纹沿天然裂隙方向扩展;(4)含天然裂隙砂岩劈裂破坏过程能量演化可划分为4个阶段,随裂隙倾角的增大,峰值点处的总能量密度、弹性能密度先缓...     

基于RJM断续节理岩体强度与破坏特征的数值模拟研究

裂隙结构的存在对于工程岩体的强度和稳定性具有重要影响,岩石宏观裂隙的产生源自于微破裂的积累。针对岩体裂隙的粗糙特性,通过Matlab建立考虑粗糙度的节理模型（Roughness Joint Model）,采用简化的正弦曲线来表示粗糙节理,并将其导入到颗粒流试验模型中进行单轴压缩试验。对比完整岩体、直线型裂隙岩体、RJM岩体三者破坏的应力-应变曲线,改变裂隙倾角（与水平方向夹角）α,岩桥倾角β,裂隙密度γ,建立不同裂隙分布的断续节理岩体数值试样,开展一系列数值模拟试验。研究结果发现:（1）裂隙的存在明显降低了岩体的抗压强度,RJM模型峰值强度和峰值应变均高于直线型裂隙岩体;（2）岩体抗压强度总体上随裂隙倾角增大而增加,随裂隙密度增加而减小,但随岩桥倾角的改变呈非线性变化,岩桥倾角45°时峰值强度最低,峰值应变最小;（3）裂隙分布会影响岩体的破裂模式,微裂隙的扩展反映了岩体力学性质的各向异性;（4）不同倾角下增加裂隙密度,岩体强度下降程度不同,倾角75°时密度对强度影响最小,30°和60°时影响最大。     

Estimating regional-scale fractured bedrock hydraulic conductivity using discrete fracture network (DFN) modeling

Estimating bedrock hydraulic conductivity of regional fractured aquifers is challenging due to a lack of aquifer testing data and the presence of small and large-scale heterogeneity. This study provides a novel approach for estimating the bedrock hydraulic conductivity of a regional-scale fractured bedrock aquifer using discrete fracture network (DFN) modeling. The methodology is tested in the mountainous Okanagan Basin, British Columbia, Canada. Discrete fractures were mapped in outcrops, and larger-scale fracture zones (corresponding to lineaments) were mapped from orthophotos and LANDSAT imagery. Outcrop fracture data were used to generate DFN models for estimating hydraulic conductivity for the fractured matrix (K _m). The mountain block hydraulic conductivity (K _mb) was estimated using larger-scale DFN models. Lineament properties were estimated by best fit parameters for a simulated pumping test influenced by a fracture zone. Unknown dip angles and directions for lineaments were estimated from the small-scale fracture sets. Simulated K _m and K _mb values range from 10^–8 to 10^–7?m/s and are greatest in a N–S direction, coinciding with the main strike direction of Okanagan Valley Fault Zone. K _mb values also decrease away from the fault, consistent with the decrease in lineament density. Simulated hydraulic conductivity values compare well with those estimated from pumping tests.     

Does a representative volume element exist for fatigue life prediction? The case of aggregate composites

In a previous work, discrete modeling and a statistical approach were used to quantify the minimal representative volume element (RVE) size for aggregate composites, such as bituminous materials in the linear elastic regime (Comput. Mater. Sci. 2005; 33 :467–490). In this paper, the discrete model is extended to strain‐softening behavior under cyclic loading. The existence of a RVE for fatigue life prediction is numerically proved and its minimal size is determined. It is found that it is much larger than the minimal RVE size in the elastic regime. Copyright © 2007 John Wiley & Sons, Ltd.     

Fracture toughness analysis on cracked ring disks of anisotropic rock

Summary  This paper presents a combination of the Boundary Element Method (BEM) and the cracked ring test to determine the mixed-mode (I–II) fracture toughness of anisotropic rocks. The proposed BEM is used to accurately calculate the Stress Intensity Factors (SIFs) of a cracked anisotropic plate. An anisotropic Hualien marble of Taiwan with a distinct foliation was selected to conduct the cracked ring tests. Based on the measurement of the failure load during the test, the mixed-mode (I–II) fracture toughness can be determined. Experimental results show that the radius ratio, inclination and crack angle significantly affect the fracture toughness. The mode-I fracture toughness (K _IC ) is shown to decrease with the increase in hole diameter, whereas the mode-II fracture toughness (K _IIC ) increases with the increase in hole diameter when the crack angle β is equal to 0°. The experimental methods proposed have the advantage that the material is easily prepared, the test procedure is simple, and the cost is low. Correspondence: Chia-Hau Chen, Chao-Shi Chen, Department of Resources Engineering, National Cheng Kung University, 701 Tainan, Taiwan     

反倾层状岩质边坡倾倒变形机理与影响因素的离散元模拟

为进一步研究层状反倾边坡的弯曲倾倒变形机制,以离心试验为原型,通过离散元数值模拟,研究了层状岩质反倾边坡的变形机理与影响因素。通过预置层内随机裂隙,实现了破裂面的形成和贯通。研究结果表明：模拟结果与试验吻合较好,边坡变形可分为起始蠕变、稳态变形和失稳破坏3个阶段;边坡破裂面在达到破坏荷载（G_f）后瞬间贯通,呈直线型,产状受岩层倾角控制,G_f值与坡角幂函数相关;反倾边坡的破坏需满足倾角和坡角启动条件,且变形破坏与岩层所受弯矩关系密切,当倾角为70°~80°、坡角大于60°时,最易破坏;典型破坏模式有倾倒-折断-块体式、倾倒-弯曲-折断式、倾倒-反折式3种,其受倾角、坡角组合控制;对材料参数的正交试验表明,各参数对G_f的敏感性从大到小依次为密度、层面内摩擦角、层厚、密度比、层面黏聚力,且G_f与层厚、层面内摩擦角及密度比具有良好的线性相关性;层面内摩擦角可影响破裂面产状,从而控制变形体规模,其他参数仅影响G_f的大小。     

Size and Geometry Effects on Rock Fracture Toughness: Mode I Fracture

In this paper, the effects of specimen size and geometry on the apparent mode I fracture toughness (K _c) of an Iranian white marble (Neyriz) are studied. A number of fracture tests were conducted on center-cracked circular disk (CCCD) specimens with different radii to investigate the size effects on K _c. The experimental results demonstrate that the apparent fracture toughness increases in bigger specimens. In order to explain the experimental results, the modified maximum tangential stress (MMTS) criterion is used, where higher order terms of the Williams’ series expansion are included in the maximum tangential stress criterion. It is shown that the MMTS criterion provides good estimates for the apparent fracture toughness of Neyriz marble, obtained from fracture tests of edge-cracked triangular specimens. It is, therefore, concluded that the proposed criterion is able to account for the size and geometry effects on the fracture resistance of rocks simultaneously.     

单轴压缩荷载下非贯通闭合节理岩体损伤本构模型

目前损伤力学已被认为是研究节理岩体力学行为的有效工具,但是在目前的节理岩体损伤变量定义中大多仅考虑节理几何特征而未考虑节理内摩擦角等力学参数,这显然不能很好地反映节理岩体的力学特征。为此,拟推导出一个能够综合考虑节理几何及力学参数的损伤变量（张量）,并由此建立单轴压缩荷载下岩体损伤本构模型。首先,基于断裂力学的由于单个节理存在引起的附加应变能增量与损伤力学的损伤应变能释放量相关联的观点,推导出了含非贯通节理岩体的损伤变量计算公式;其次,根据断裂力学理论对单轴压缩荷载下的单个节理尖端应力强度因子计算方法进行了研究,得出了应力强度因子K_Ⅰ、K_Ⅱ的计算公式;同时考虑多节理间的相互作用给出了单组单排及多排非贯通节理应力强度因子计算公式。最后,利用该模型对含单条非贯通节理的岩体在单轴压缩荷载作用下的峰值强度及损伤变量进行了分析计算。结果表明,当节理倾角小于其内摩擦角时,岩体强度与完整岩石相同,岩体损伤为零,而后随着节理倾角增加,岩体强度、损伤随节理倾角的变化分别呈开口向上及向下的抛物线,当节理倾角约为60°时,岩体损伤最大,强度最低。随着节理长度增加,岩体损伤增加,而随着节理内摩擦角的增加,岩体损伤则减小。      

碰撞带倾角和摩擦系数对陆-陆碰撞变形的影响

借助一个简单的大陆与大陆碰撞模型,即把两个大陆的碰撞简化为两个粘弹性块体的碰撞,两个块体之间的不连续变形面对应两个大陆之间的碰撞断裂带,运用三维粘弹性拉格朗日非连续变形有限元(LDDA)方法,通过分析模型中不连续变形面的存在对两个粘弹性块体碰撞变形的影响,探讨了大陆碰撞断裂带的倾角和摩擦系数对两个大陆碰撞变形影响的一般规律,给出了大陆碰撞变形的一些显著的特点。尽管运算模型的几何尺寸、边界条件等参考了印度和欧亚大陆碰撞的实际特征,但所得结果适用于更一般的情况。研究结果表明,碰撞断裂带倾角和摩擦系数对陆-陆碰撞变形有着重要的影响。当断裂带倾角在15~30°时,两个大陆板块碰撞导致的仰冲板块一侧隆升高度相对更大,利于形成高大的山脉,其中以15°倾角对应的仰冲板块一侧隆升最高;当断裂带倾角在30~45°时,两个大陆板块碰撞导致的俯冲板块的俯冲深度相对更深,利于形成深陷的盆地,其中以45°倾角对应的俯冲陆块一侧俯冲最深;当断裂带倾角≥75°时,两个大陆板块之间的相对俯冲和仰冲作用变得不明显。碰撞断裂带摩擦系数越小,碰撞过程中两个大陆板块之间相对俯冲和仰冲作用越强,形成高大的山脉和深陷的盆地要求碰撞断裂带摩擦系数≤0.2。从大陆与大陆碰撞变形构造特征看,除发育前陆盆地-山脉系统外,在仰冲陆块一侧靠近造山带后缘还发育呈不对称结构的挤压性凹陷,当两个大陆板块碰撞断裂带倾角在15~30°时,该类型凹陷更容易形成,其中以30°倾角对应的凹陷最深,反映其形成可能是大陆碰撞过程中陆块之间相对俯冲和仰冲运动的综合结果。     

Percolation-theory and fuzzy rule-based probability estimation of fault leakage at geologic carbon sequestration sites

Leakage of CO₂ and displaced brine from geologic carbon sequestration (GCS) sites into potable groundwater or to the near-surface environment is a primary concern for safety and effectiveness of GCS. The focus of this study is on the estimation of the probability of CO₂ leakage along conduits such as faults and fractures. This probability is controlled by (1) the probability that the CO₂ plume encounters a conductive fault that could serve as a conduit for CO₂ to leak through the sealing formation, and (2) the probability that the conductive fault(s) intersected by the CO₂ plume are connected to other conductive faults in such a way that a connected flow path is formed to allow CO₂ to leak to environmental resources that may be impacted by leakage. This work is designed to fit into the certification framework for geological CO₂ storage, which represents vulnerable resources such as potable groundwater, health and safety, and the near-surface environment as discrete “compartments.” The method we propose for calculating the probability of the network of conduits intersecting the CO₂ plume and one or more compartments includes four steps: (1) assuming that a random network of conduits follows a power-law distribution, a critical conduit density is calculated based on percolation theory; for densities sufficiently smaller than this critical density, the leakage probability is zero; (2) for systems with a conduit density around or above the critical density, we perform a Monte Carlo simulation, generating realizations of conduit networks to determine the leakage probability of the CO₂ plume (P _leak) for different conduit length distributions, densities and CO₂ plume sizes; (3) from the results of Step 2, we construct fuzzy rules to relate P _leak to system characteristics such as system size, CO₂ plume size, and parameters describing conduit length distribution and uncertainty; (4) finally, we determine the CO₂ leakage probability for a given system using fuzzy rules. The method can be extended to apply to brine leakage risk by using the size of the pressure perturbation above some cut-off value as the effective plume size. The proposed method provides a quick way of estimating the probability of CO₂ or brine leaking into a compartment for evaluation of GCS leakage risk. In addition, the proposed method incorporates the uncertainty in the system parameters and provides the uncertainty range of the estimated probability.     

井斜角对裂缝特征参数统计的影响及其意义

井轨迹对裂缝钻遇率有重要的影响。以某裂缝性碳酸盐岩油藏为例,利用100余口井的FMI资料按照小斜度井（包括直井）和大斜度井（包括水平井）分组分别统计裂缝的平均倾角、裂缝密度参数,并与不考虑井斜角利用所有FMI资料统计的结果相互对比,以分析井斜角对裂缝倾角、裂缝密度参数统计的影响。结果表明,按小斜度井钻遇裂缝统计的裂缝平均倾角和裂缝密度均明显低于按大斜度井和所有井未分组钻遇裂缝的统计结果;因此,小斜度井钻遇高角度裂缝的几率明显低于大斜度井。这一结论具有三方面的意义：对于本文中某裂缝性碳酸盐岩油藏,目前主要为直井和小斜度井钻遇的G-H层,应部署水平井进一步评价其油气勘探潜力;为更准确地统计高角度裂缝性储层的裂缝倾角并客观地评价裂缝密度,应尽量利用井轨迹垂直于裂缝走向和主要裂缝面的水平井钻遇的裂缝来分析统计;对于同类高角度裂缝性油藏的勘探评价和开发,应尽量采用大斜度井和水平井并确保井轨迹垂直于主要裂缝面。     

Finite-element 3D modeling of stress patterns around a dipping fault

Stresses in a block around a dipping fracture simulating a damage zone of a fault are reconstructed by finite-element modeling. A fracture corresponding to a fault of different lengths, with its plane dipping at different angles, is assumed to follow a lithological interface and to experience either compression or shear. The stress associated with the destruction shows an asymmetrical pattern with different distances from the highest stress sites to the fault plane in the hanging and foot walls. As the dip angle decreases,the high-stress zone becomes wider in the hanging wall but its width changes negligibly in the foot wall.The length of the simulated fault and the deformation type affect only the magnitude of maximum stress,which remains asymmetrical relative to the fault plane. The Lh/Lfratio, where Lhand Lfare the widths of high-stress zones in the hanging and foot walls of the fault, respectively, is inversely proportional to the fault plane dip. The arithmetic mean of this ratio over different fault lengths in fractures subject to compression changes from 0.29 at a dip of 80°to 1.67 at 30°. In the case of shift displacement, ratios are increasing to 1.2 and 2.94, respectively.Usually they consider vertical fault planes and symmetry in a damage zone of faults. Following that assumption may cause errors in reconstructions of stress and fault patterns in areas of complex structural setting. According geological data, we know the structures are different and asymmetric in hanging and foot walls of fault. Thus, it is important to quantify zones of that asymmetry. The modeling results have to be taken into account in studies of natural faults, especially for practical applications in seismic risk mapping, engineering geology, hydrogeology, and tectonics.     

基于工业CT扫描的岩芯裂缝结构表征

为了研究泥页岩中裂缝的发育特征,利用工业CT技术对句容盆地的页岩岩芯进行扫描,并使用数字处理技术对页岩的CT扫描图像进行分析,通过VGStudio MAX对CT扫描数据进行重建体模型及裂缝提取,更好的在三维尺度上研究不同矿物组成成分的页岩中裂缝的结构和发育情况及工业CT扫描图的区别。结果发现岩芯中主要存在两组不同走向及倾角的裂缝,一组高角度裂缝,倾角约为70°~90°;一组低角度裂缝,倾角约为30°~40°。依此推测句容盆地在志留纪鲁丹阶以后至少经历两次构造运动。对比不同的矿物组成成分的页岩研究发现,当岩石样品的矿物组成密度差别越大时,工业CT扫描就会越清晰;当岩石样品颗粒度越大时,扫描的效果就会越明显。     

Flow Study in a Double Porosity Medium Containing Ellipsoidal Occluded Macro-voids

This paper presents an analytical study of fluid flow in a porous medium presenting pores of two different length scales: at the smallest or microscopic scale, the presence of connected voids confers a porous medium structure to the material investigated, while at the upper or mesoscopic scale, macro-pores are present. This microstructure is employed to represent the progressive opening of inter-aggregate pore spaces observed in natural compacted montmorillonites polluted by heavy metal ions. Three-dimensional analytical expressions are rigorously derived for pore fluid velocity and excess pore fluid pressure within the porous matrix, around an occluded ellipsoidal inter-aggregate void. The eccentricity ratio is employed to characterize the geometrical shape of the ellipsoidal void, while its orientation with respect to the inflow in the far field is determined by the dip angle θ. As an application, we investigate the flow focusing effect for varying eccentricity ratios and dip angles.     

Physical conditions producing slab stagnation: Constraints of the Clapeyron slope, mantle viscosity, trench retreat, and dip angles

Recent seismic tomography has revealed various morphologies in the subducted lithosphere. In particular, significant flattening and stagnation of slabs around the 660-km boundary are seen in some areas beneath the northwestern Pacific subduction zones. We examined the cause of slab stagnation in terms of the Clapeyron slope of the phase transformation from ringwoodite to perovskite + magnesiowüstite, trench retreat velocity, dip angles, and high viscosity of the lower mantle based on two-dimensional (2-D) numerical simulations of thermal convection. In particular, we examined the conditions necessary for slab stagnation assuming a very small absolute value of the Clapeyron slope, which were proposed based on recent high-pressure, high-temperature (high P–T) experiments. Our calculations show that slabs tend to stagnate above the 660-km boundary with an increasing absolute value of the Clapeyron slope, viscosity jump at the boundary, and trench retreat velocity and a decreasing initial dip angle. Stagnant slabs could be obtained numerically for a realistic range of parameters obtained from high P–T experiments and other geophysical observations combining buoyancy, high lower-mantle viscosity, and trench retreat. We found that a low dip angle of a descending slab at the bottom of the upper mantle plays an important role in slab stagnation. Two main regimes underlie slab stagnation: buoyancy-dominated and viscosity-dominated regimes. In the viscosity-dominated regime, it is possible for slabs to stagnate above the 660-km boundary, even when the value of the Clapeyron slope is 0 MPa/K.     

Relationship Between the Geometric Parameters of Rock Fractures,the Size of Percolation Clusters and REV

Modeling fractured rocks with numerical methods requires some derived parameters, among which the fracture network connectivity and the size of the representative elementary volume (REV) are both of crucial importance. Percolation and REV analyses were made by the RepSim code. The program uses input parameters such as fractal dimension of the fracture midpoints (D _c), length exponent (E) and relative dip (α ^r) data. For percolation analysis, the relative sizes of the largest percolation clusters have been calculated by stochastic realizations of the simulated fracture networks with different parameter triplets. Furthermore, fracture networks can be classified into three major types on the basis of their (E,D _c,α ^r) parameters. For the REV calculations, the porosity of the generated fracture network was calculated. The derived REV size of a fracture network depends essentially on input parameters and shows a decreasing tendency with increasing D and E and vice versa. The method mentioned above was tested on both metamorphic samples of the Pannonian Basin and Variscan granitoid rocks of the Mórágy Complex. Percolation values predicted for the Mórágy granite are highly sensitive to alterations in the input parameters. The amphibolite bodies displayed a modeled fracture network with 80 to 90% of all fractures being interconnected, while the largest achievable percolation cluster size of gneiss is less than 10%. The REV size of the amphibolite is about 20 m as a result of connected fractures filling the whole body, while gneiss has lower porosity and higher REV (approximately 70 m).