Influence of Stochastic Discontinuity Network Parameters on the Formation of Removable Blocks in Rock Slopes

Summary  We study the effects of discontinuity network parameters on the formation of removable wedges in rock slopes. Discontinuities are simulated using the Poisson disk model, and removable wedges are identified using block theory. The formation of removable wedges of different sizes is assumed to follow a Poisson process. Poisson regression and Monte Carlo simulation are then used to identify statistically relevant parameters of the model, and to study the effects that variations in their values have on formation of removable blocks. The sensitivity of the results as a function of the mean orientations of the discontinuity sets forming the blocks is also studied by means of a parametric study. The volumetric intensity of discontinuities in the rock mass is found to have a significant impact on the computed estimates of removable block formation. As predicted by theory, our results indicate that, everything else being equal, the expected rate of formation of removable wedges is proportional to the square of the intensity measure. Estimates are also sensitive to changes in discontinuity size, especially in cases in which discontinuities are smaller than one to two times the height of the slope. The interaction between the mean size of discontinuities and the coefficient of variation of discontinuity sizes is found to be significant as well. Finally, results of our sensitivity analysis suggest that the orientation of discontinuity sets significantly affects the rate of formation of removable blocks in rock slopes. Author’s address: Dr. Rafael Jimenez-Rodriguez, ETS Ing. de Caminos, Canales y Puertos. Universidad Politecnica de Madrid, Spain     

岩体等效变形参数研究

岩体变形参数的确定对岩体稳定性模拟至关重要.提出了确定规则裂隙和不规则裂隙岩体等效变形参数的一种模型, 探讨了岩体等效变形参数的规律.通过对不考虑渗流-应力耦合时岩体等效变形性能的研究, 可以发现岩体的等效变形参数不仅与各组结构面的几何形态、结构面变形参数、岩块变形参数等有关, 而且与不同组系结构面间的交切形态有关.岩体的REVs具备以下几点规律: 首先REVs具有多尺度效应和不确定性.其次, REVs与结构面各几何形态要素之间有如下关系: 平均迹长越大, 平均间距越小, 方向角的方差越大, 结构面分布越凌乱, REVs的取值越小.REVs与岩块、结构面变形参数之间有如下关系: 结构面变形参数与岩块变形参数的差异程度对REVs的取值没有明显影响, 但是不同组系结构面或是同一组中的各条结构面, 其变形参数差异越小, REVs的取值将越小.      

裂隙岩体的多孔介质水力等效性研究

工程岩体中数量众多的裂隙面限制了离散裂隙网络模型在岩体渗流中的应用,迫使人们寻找能够用理论上成熟的等效连续介质模型替代,这就要求进行岩体多孔介质的水力等效性研究。在野外大量实测裂隙的基础上,进行裂隙密度、方位、大小、延续性、开度等几何参数的统计分析,以Enhanced Baecher模型建立离散裂隙网络随机模型,采用Monte-Carlo随机模拟方法进行三维裂隙网络随机模拟。在所生成的一定尺度的三维裂隙网络图基础上,给出计算研究域REV的方法,通过判断REV是否存在,确定能否用等效连续介质模型分析岩体渗流。     

基于非均质系数的裂隙岩体表征单元体研究

表征单元体（REV）是岩体力学中的一个基础性概念,其存在性是确定岩体等效参数和应用连续介质方法进行研究的前提条件。为综合反映裂隙大小、产状和密度对岩体REV的影响,本文提出采用岩体非均质系数HI来确定岩体的REV。首先,详细介绍了非均质系数的概念和含义。利用General Block软件建立了中等间距-中等延展性（MS1-MP1）裂隙的网络模型,并完成20次随机实现。从各模型中选取10个不同尺寸的岩体模型计算非均质系数,结果表明:非均质系数具有明显的尺寸效应,当岩体尺寸不小于8 m时,非均质系数的平均值和标准差分别为0.5和0.14,综合确定该岩体的REV为8 m。采用HI对三峡地下电站厂房围岩的研究表明,该岩体的REV为60 m,可作为非连续介质方法适用性的判断标准。HI是从岩体结构角度提出的新指标,适用于确定岩体的REV和统计范围,对岩体的统计分析和等效参数研究具有重要意义。     

Study of the geometrical size effect of a fractured rock mass based on the modified blockiness evaluation method

The determination of representative elementary volume (REV) is critical for mechanical model establishment and the determination of rock mass mechanical properties. Therefore, in this study, the REV range, existence, determinate parameter, and category were discussed. The blockiness level of the rock mass (B_z) was defined as the new index to determine the geometrical rock mass REV. Then, a great variety of multiscale three-dimensional fractured network models with different discontinuity persistences and spacings were developed. Finally, the B_z values of all models were measured, and the geometrical REVs were determined, the correlation between the geometrical REVs and rock mass discontinuity properties were assessed, and size validations of geometrical REVs were performed. In this study, it is shown that the REV is approximately 20–50 times that of the discontinuity spacing when the discontinuity spacing is classified as having extremely close to close spacing and is between approximately 0.3 and 4.0 times the discontinuity diameter when discontinuity spacing is in the range from moderate to extremely wide spacing. Further, there is a fourth-order nonlinear correlation between the geometrical REV and the ratio of the discontinuity diameter to discontinuity spacing. Meanwhile, the validation results confirm that the geometrical REV is slightly less than the sizes at which the elastic moduli tend to plateau, and the rationality of the geometrical REV of a rock mass based on B_z is thus supported from another perspective.     

用窗口法估计不连续面的连通率

基于不连续面与窗口交切的必需而充分的条件,推导了不连续面迹长、间断长和不连续面连通率的估算公式,通过实例验证了该公式的正确性。     

Analysis of fracture network connectivity using percolation theory

Connectivity aspects of fracture networks are analyzed in terms of percolation theory. These aspects are of fundamental importance in characterization, exploitation, and management of fractured formations. General connectivity and power law relationships are determined that characterize the density of fractures and average number of intersections per fracture necessary to ensure network connectivity, the likelihood of a fractured formation being hydraulically connected, and the probability that any specific fracture is connected to the conducting portion of the network. Monte Carlo experiments with a two-dimensional fracture network model confirm the percolation theory predictions. These relationships may prove useful in formulating theoretically tractable approximations of fracture nerworks that capture the essential system properties.     

Permeability tensor and representative elementary volume of fractured rock masses

Based on a simulation of three-dimensional fracture networks and a superposition principle of liquid dissipation energy for fractured rock masses, a model of the fracture permeability tensor is proposed. An elastic constitutive model of rock fractures, considering fracture closure and dilation during shearing, is also proposed, based on the dilation angle of the fracture. Algorithms of flow-path searching and calculation of the effective flow coefficients for fracture networks are presented, together with a discussion on the influence of geometric parameters of the fractures (trace length, spacing, aperture, orientation and the number of fracture sets) on magnitude, anisotropy of hydraulic permeability and the size of a representative elementary volume (REV). The anisotropy of hydraulic permeability of fractured rock masses is mainly affected by orientation and the number of fracture sets, and the REV size is mainly influenced by trace length, spacing and the number of fracture sets. The results of studies on REV size and the influence of in-situ stress on hydraulic conductivity of the rock mass on the slope of Jinping-I hydropower station, China, are presented using the developed models and methods. The simulation results agreed well with the results obtained from field water-pressure measurements, with an error of less than 10 %.     

基于三维裂隙网络的裂隙岩体表征单元体研究

岩体表征单元体（REV）的研究一直是岩土工程中的重要问题。裂隙岩体REV是其内部复杂结构在尺寸效应上的体现,归因于岩体内裂隙系统的随机性。在三维网络模型的基础上通过计算体积节理数建立了确定岩体REV的方法。这种方法考虑了岩体内裂隙的三维空间密度的特征,采用体积节理数这一参数的收敛状况确定岩体REV的大小。建立了实际岩体的模拟裂隙系统,用一定大小的立方体来分割裂隙系统,统计不同立方体内的体积节理数,并将均值作为最终裂隙岩体的参数。将此参数作为裂隙岩体的属性,通过判别其收敛状况获得了岩体REV的尺寸。考虑了岩体不连续面的特征,建立了REV的分析模型,得出工程岩体的REV的尺寸是裂隙平均迹长的4倍的结论。     

三峡水库某库段岩体裂隙网络模拟研究

为合理评价三峡水库某库岸岩体结构特征及工程整治提供依据,对区内岩体进行了裂隙调查及网络模拟研究。研究基于Monte-Carlo模拟方法,在技术上进行了探讨和一定改进。如采用了聚类分析法确定裂隙优势产状、用直接法产生随机数、广义RQD t及裂隙线连通率的计算等。根据野外调查及计算机模拟研究,研究区发育有3组优势裂隙、平均迹长1.45 m左右。间距一般服从对数正态或负指数分布,而走向和迹长基本上不服从常规分布。裂隙线密度在3－6条/m、t = 0.1 m时RQD t值在90 %左右、t = 0.7 m时RQD t值为10 %－20 %、RQD t值变化可能影响库岸岩体在不同方向的稳定性。     

岩体裂隙网络等效渗透系数方向性的数值计算

采用渗流力学理论并结合Monte Carlo方法描述岩体裂隙的随机分布,研究渗流模型的尺寸效应并确定表征单元体积（REV）,得到了3种开口度分布形式的等效渗透系数椭圆曲线,建立了等效渗透系数方向性的判别标准。离散裂隙网络（DFN）模型假定流体只在岩体裂隙内部流动,而不通过岩体本身渗流。基于二维离散元程序UDEC并进行二次开发,建立DFN模型,通过改变流体的流动方向,得到不同流动方向下岩体裂隙网络的等效渗透系数,并分析不同的开口度分布形式对岩体裂隙网络等效渗透系数方向性的影响。计算结果表明,表征单元体积存在的条件是等效渗透系数保持稳定且渗透椭圆比较光滑。等效渗透系数的方向性受开口度分布形式的影响很大：当开口度-长度关联分布时,等效渗透系数各向异性;当开口度对数正态分布时,等效渗透系数各向同性;当开口度恒定分布时,等效渗透系数的特性介于二者之间。变化系数（CV）是否大于5%是判定岩体裂隙网络渗透系数是否具有方向性的判别标准。     

Two-dimensional probabilistic infiltration analysis with a spatially varying permeability function

The permeability function for a soil may change spatially due to uncertainties in soil fabric. The main objective of this paper is to investigate how the spatial variability of permeability function propagates to the variability of the pore-water pressures and groundwater table in a slope as well as the stability of the slope. A random field analysis method is explored by assigning discrete random values to a 2D space and controlling the density of random field grid to improve the calculation accuracy. Sequences of random numbers are generated using fast Fourier transform. In a given heterogeneous slope subject to steady-state rainfall infiltration, a parametric study shows that the matric suctions are 0.5–1.25 times those in a homogeneous slope when the correlation length of log-permeability varies from 0.4 to 50 times the slope height. The groundwater table is no longer unique with a spatially variable permeability function. There exists a critical correlation length approximately five times the slope height at which the change in the groundwater table is maximal and the mean factor of safety is minimal. The mean factor of safety of the heterogeneous slopes is smaller than that of a homogenous slope with mean input parameters. The spatial variability of soil influences the range of the calculated factor of safety significantly but does not influence the mean factor of safety substantially.     

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).     

估计迹长概率分布函数的新方法及其应用

本文提出一种估计迹长概率分布函数的新方法。该方法的优点是不需要有节理在测窗上出露长度的数据，有效的减少了野外工作量     

估计迹长概率分布函数的新方法及其应用

本文提出一种估计迹长概率分布函数的新方法。该方法的优点是不需要有节理在测窗上出露长度的数据, 有效的减少了野外工作量。     

基于随机森林回归分析的岩体结构面粗糙度研究

岩体结构面粗糙度系数是快速估算结构面峰值抗剪强度的重要参数。但是结构面轮廓曲线复杂,单一统计参数无法量化表征粗糙度。为解决这一问题,收集了112条结构面轮廓曲线起伏角、起伏度、迹线长度3方面的8项统计参数,利用随机森林回归模型交叉验证的方法评估统计参数的重要性。结果表明：最大起伏度、起伏高度标准偏差、平均起伏角、起伏角标准差、平均相对起伏度及粗糙度剖面指数等6项统计参数重要性占比达到93.2%,且回归拟合系数趋于平稳,基于重要性评估结果建立最优超参数决策树数目(n_tree)为400、参与节点分割的数目(m_try)为2的随机森林回归模型,模型预测结果拟合优度高达98.1%。与基于坡度均方根、结构函数及粗糙度剖面指数等传统线性回归结果对比,随机森林回归模型结果精度更高,误差更小,拟合优度提高6%以上,表明随机森林回归模型更适用于结构面粗糙度反演。     

基于分维数的土体裂隙表征单元体估算

由于干湿循环、剪切或其它作用,自然界土壤表面发育着大量裂隙。裂隙相互连接形成网络,裂隙网络是一个随机无序的系统,常规统计方法描述裂隙的分布特征存在很大的难度。网状土体裂隙一般具有分形特征,可以用分维数来进行描述。用数码相机拍摄土体表面的裂隙,将其转换成灰度图像,并结合分形理论,建立了摄影法测定土体裂隙分维数的方法,得到了土体裂隙的分维数。根据表征单元体的物理意义,建立了基于分维数估算表征单元体的方法,并对实例进行了分析。研究表明：分维数随裂隙密度,裂隙宽度和次一级裂隙发育程度的增大而增大;表征单元体与裂隙的密度均匀性和隙宽均匀性相关,基于分维数估算表征单元体的方法简便可行。     

Geometric parameters and REV of a crack network in soil

Cracks are common in natural and engineered soils and provide preferential pathways for water infiltration into the soil. Statistical properties of crack geometries are important inputs for analyzing preferential flows in discrete random crack networks. This paper reports the outcome of a field study conducted on a compacted, cracked soil ground at a steady moisture condition. The objectives of the field study were to investigate the crack patterns and probability distributions of the geometric parameters of cracks and to determine the representative elementary volume (REV) of the crack network. The desiccation cracks at the survey site formed an inter-connected columnar structure. The traces of the cracks on the soil surface formed a primary structure consisting of inter-connected crack polygons and a secondary structure comprising of isolated cracks. The locations and orientations of the desiccation cracks followed a uniform distribution, differing from the distribution of fracture sets often observed in fractured rocks. The lengths and apertures of the cracks followed a lognormal distribution as expected. The REV size for the cracked soil was found to be approximately five times the mean crack length, above which the variation in crack porosity in relation to domain size was negligible.     

Estimation of the REV size for blockiness of fractured rock masses

The representative elementary volume (REV) is the premise of the continuous-media method of analysis, and the investigation of the REVs of fractured rock masses is a fundamental area of rock mechanics research. The existence of an REV can be determined based on a variety of physical parameters. This paper presents an analysis of the REV from the view of blockiness, which is defined as the percentage of the volume of isolated blocks formed by fractures in the total rock volume. Seventy-seven types of fractured rock mass models were developed based on 7 classes of fracture persistence and 11 classes of spacing that are suggested by the International Society for Rock Mechanics (ISRM) fracture classification. Rock blocks in each of the 77 types of fractured rock mass models were identified using GeneralBlock to determine the variation in blockiness with model domain sizes, which were changed from 2 to 20 times the fracture spacing. For each model domain size 9 random realizations were carried out to reduce the effects of randomness. The coefficient of variation (C_v) was then used to quantify the variability of the 9 random realizations. The fluctuation in blockiness with the variation in the scale of the model region was also investigated. In this way, the size of the REV in these models can be calculated using the average and the variance of the blockiness as indicators of the convergence. The blockiness of these fractured rock masses can be determined at the REV volume. The results indicate that of the 77 models, 76 REV sizes are between 2 and 20 times the fracture spacing. The fractured rock mass with a wide fracture spacing and very high persistence (WS2–VHP) has a REV size that exceeds 20 times the fracture spacing. Thus, the WS2–VHP model should be investigated further to validate this concept.     

An improved Monte Carlo simulation method for discontinuity orientations based on Fisher distribution and its program implementation

Fisher distribution is the most commonly used probability density function for discontinuity orientations. Based on Fisher distribution, Monte Carlo simulation method for discontinuity orientations was reviewed and improved. Those orientations extending beyond the edge of an upper hemisphere projection (OEBEUHP) often have an important influence on both the mean orientation and Fisher constant K, thus affecting simulation results. The detailed algorithms for identifying and adjusting those OEBEUHP were developed in this paper. Based on the improved method, a program for generating discontinuity orientations and plotting their stereographic projection maps, named as MCSDO, was developed. Due to the aforementioned adjustment, the generated orientations by MCSDO are close to the original discontinuity orientations, which were mapped in field, and satisfactory. Only the original orientations and target number of generated orientations need to be input. By running the program we can directly obtain the follows: orientations of generated discontinuities, mean orientations of both original and generated discontinuities, Fisher constant K of both original and generated discontinuities, and stereographic projection maps of both original and generated discontinuities. MCSDO is a freeware designed for researchers and practicing engineers, and can be easily mastered with a little computer knowledge.