Hydraulic properties of fractured rock masses with correlated fracture length and aperture

Permeability of fractured rocks is investigated considering the correlation between distributed fracture aperture and trace length, based on a newly developed correlation equation. The influence of the second moment of the lognormal distribution of apertures on the existence of representative elementary volume (REV), and the possibility of equivalent permeability tensor of the fractured rock mass, is examined by simulating flow through a large number of stochastic discrete fracture network (DFN) models of varying sizes and varying fracture properties.The REV size of the DFN models increases with the increase of the second moment of the lognormal distribution, for both the correlated and uncorrelated cases. The variation of overall permeability between different stochastic realizations is an order of magnitude larger when the aperture and length are correlated than when they are uncorrelated. The mean square error of the directional permeability increases with increasing value of the second moment of the lognormal distribution function, and good fitting to an ellipsis of permeability tensor can only be reached with very large sizes of DFN models, compared with the case of constant fracture aperture, regardless of fracture trace length.     

岩体节理大小估算新方法研究

 节理大小主要指节理空间延伸范围,对于节理泊松圆盘模型,节理大小可以用圆盘直径表示。Warburton的方法需要先假设直径的分布形式,理论上正确,但并不实用。不假定节理直径的可能分布,提出一种新的方法估算节理直径分布,该方法在已知节理迹长分布基础上,通过设定不同节理直径平均值初值,计算相应节理直径分布,得到不同节理直径平均值初值和数值计算节理直径平均值之间的相对误差。分析相对误差与节理直径平均值初值之间的相互关系,当误差为0时,得到数值计算最佳节理直径平均值,从而获得相应节理直径分布。采用节理直径负指数和对数正态分布2种理想分布来验证数值方法的有效性,结果理论节理直径平均值和数值计算节理直径平均值之间的相对误差很小,且节理直径分布和理论节理直径分布一致。通过理论节理直径平均值和标准差变化对数值方法进行参数分析,研究数值方法计算节理直径分布的适用性。在节理直径平均值较大而标准差较小条件下,利用数值方法,所得节理直径分布更加准确。利用该数值方法研究节理直径分布简便实用。将该方法应用于我国高放废物处置库预选区甘肃北山芨芨采石场结构均质II区,获得该区岩体节理直径分布。     

岩体节理平均迹长估计

分析总结岩体节理平均迹长的估计方法后，扩展了测线法节理平均迹长估计的理论基础。扩展后的测线法在相同的现场节理调查数据下，可更加准确地估计岩体节理的平均迹长。同时指出，采用圆形窗口法平均迹长估计时，假设迹线中点在窗口中均匀分布与实际情况偏差很大，为避免迹线中点不均布的情况必须保证窗口的尺度足够大。随着圆形取样窗口的增大，迹线中点密度在窗口中的变化减小，估计所获得的节理平均迹长的准确度较高。通过现场节理调查获得丰富的节理数据，采用扩展的测线法和圆形窗口法分别估计岩体节理的平均迹长。当圆形窗口的直径与测线法删节长度相等时，扩展测线法和圆形窗口法估计所得的岩体节理平均迹长比较接近，由此可确定研究区域的岩体节理平均迹长。     

Estimates of fracture density and uncertainties from well data

This paper aims at building a method to estimate the probability law governing the 3D fracture density of a fractured rock conditioned to the number of traces observed on a borehole image when the spatial distribution of fracture centers is assumed to follow a Poisson process. A closed-form expression of this law, allowing to calculate its mean value as well as a confidence interval, is derived in both cases of a lineic well (scanline) and a cylindrical well. The latter is better adapted to the situation of fracture size of the same order of magnitude as the well radius, which enables the presence of partial traces. In particular, the method takes into account the bias in the density estimate due to the fact that a fracture may cut the well along two distinct traces according to the considered fracture size. Monte Carlo simulations finally show a good agreement with the theoretical results of mean density and confidence interval.     

Elastic modulus of a rock mass based on the two parameter negative-exponential (TPNE) distribution of discontinuity spacing and trace length

The frequency, spacing and trace length of rock mass discontinuities appear to follow a negative exponential distribution. However, in some cases this distribution has limitations. To overcome this, it is necessary to choose a target distribution which is flexible enough to simulate the spacing and trace length of the discontinuities. The paper shows how the two parameter negative exponential (TPNE) distributions can meet the requirements. An improved constitutive equation of rock mass based on this new distribution has been determined from which a new method to calculate rock mass elastic modulus (Em) is proposed, taking into account the rock mass anisotropy. It was found that the geo-stresses influence Em in two ways: it varies with changes in the orientation of the geo-stress and with the size of geo-stress. The work is illustrated by a case study on the abutment slope of the 295 m high arched dam of the Xiao wan hydroelectric power station in Yunnan province, Southwest of China.     

Modelling the FRP-concrete delamination by means of an exponential softening law

Among rehabilitation strategies, bonding of Fibre Reinforced Polymers (FRP) plates is becoming more and more popular, especially for what concerns concrete structures. The performance of the interface between FRP and concrete is one of the key factors affecting the behaviour of the strengthened structure. Up to now, closed-form analytical solutions exist only for the local bond-slip law with linear softening. The aim of the present paper is to show that analytical solutions can be achieved also assuming an exponential decaying softening law. Accordingly, the expressions for the interfacial shear stress distribution and the load-displacement response are derived for the different loading stages. A full parametric analysis of the problem has been performed, highlighting the size effect on the structural behaviour as well as the effects of the bond length, of the FRP stiffness and of the interface cohesive law. A comparison with other analytical models as well as with experimental data available in the literature concludes the paper.     

Estimating the intensity of rock discontinuities

This paper presents an approach for estimating the intensity of discontinuities and formulating intensity and orientation as a fracture tensor. Specifically the size distribution and the number of discontinuities are estimated first, from which the fracture tensor is then derived. Discontinuity size distribution is inferred from the trace data sampled in circular windows by using a general stereological relationship between the true trace length distribution and the discontinuity diameter distribution assuming circular shaped discontinuities. Because the measured trace lengths are biased, a method is proposed to estimate the true trace length distribution for circular window sampling. Circular window sampling has the advantage of automatically eliminating the orientation bias when estimating the true trace lengths. A method is then presented with which the total number of discontinuities in an objective volume can be estimated from the number of discontinuities observed in normal-size boreholes and using the inferred discontinuity diameter from the circular window sampling on the rock surface. With the derived size distribution and number of discontinuities, the intensity of discontinuities, which is the total surface area of discontinuities per unit volume, can then be calculated and included in a new definition of a fracture tensor. An application of the approach to analyze simulated discontinuities produces satisfactory results.     

High resolution, non-destructive measurement and characterization of fracture apertures

We measure at high resolution the fracture aperture of several consolidated materials and statistically characterize the aperture distribution for future studies on single and multiphase flow and transport through fractured porous media. The images of the real fractures in granite and sandstone rocks are obtained with a computer aided tomography (CAT) X-ray scanner. The minimum pixel size is 0.27×0.27 mm, but fractures as small as 35 μm can be accurately measured, using a calibration standard for the variation in CT number due to the “missing rock mass” in a given pixel. The distribution of fracture apertures is best described by a lognormal function. This was also corroborated by comparing the theoretical to actual ratio of geometric mean to arithmetic mean apertures. The mechanical roughness of the fractures ranged from 0.17 to 1.5. The variability in fracture aperture displayed only short range structure, with correlation lengths of 0.8 to 7 mm at maximum, which is at most one-tenth of the minimum dimension (width) of the fracture plane. Fracture aperture distribution is rather heterogeneous. Potential flow channels can be deduced from the fracture aperture distribution.     

Estimation of a joint diameter distribution by an implicit scheme and interpolation technique

A new technique to estimate the diameter distribution of a joint set from the contained trace length distribution is suggested. This technique obtains the diameter distribution directly from the sample histogram of the contained trace lengths without using any information about the trace length distribution of an infinite window. Compared with the previous method of Song and Lee [Estimation of joint length distribution using window sampling. Int J Rock Mech Mining Sci 2001; 38: 519–28], it is more accurate for small-size joint samples. To obtain more accurate estimates, we adopt an interpolation technique of B-spline to smooth out the sampled contained trace length distribution. The effect of the B-spline as a function of three kinds of predefined diameter distributions is examined by Monte Carlo simulations.     

Estimation of joint length distribution using window sampling

One of the most difficult procedures in statistical joint modeling is the determination of the joint length distribution. In this study, a window sampling method was adopted to estimate the trace length distribution for the Poisson disc joint model. Four kinds of equations were derived for estimating the trace length distribution from contained or dissecting trace length distributions in a rectangular or circular window. The equations were tested for accuracy through Monte Carlo simulation and their efficiencies for estimating the trace length distribution were compared. This new technique using window sampling was compared with the method using a semi-trace or complete trace length distribution from scanline sampling. A numerical technique for determining the diameter distribution from the trace length distribution was also suggested. To check the validity of this numerical method, it was applied to solving four example cases and their results were compared with theoretical solutions.     

火灾下受轴向荷载钢柱的屈曲温度预测

根据欧洲规范,钢柱屈曲温度可采用解析法或简单的封闭方程进行估算。为此研究两种计算方法的精确度和局限性,对文献中高温下钢柱的试验数据进行对比。两种方法的预测结果都与适当利用系数的试验结果相符,与利用系数较高(μ0>0.83)或较低(μ0<0.16)的试验结果不符。简单方程的工作特性因素的均值为0.949,协方差为0.016,能够通过极值分布来描述。解析法的工作特性因素的均值为1.018,协方差为0.013,能够通过正态分布、伽马分布或对数正态分布来描述。     

Buckling of elastically restrained steel columns under longitudinal non-uniform temperature distribution

Columns under natural fire conditions are usually exposed to a non-uniform temperature distribution in the longitudinal direction. The motivation for this study stems from zone modeling of a compartment fire where the gas layers are artificially divided into two zones, namely the hotter upper zone and the cooler lower zone. However, for field modeling of a compartment fire, more detailed information of the temperature distribution can be obtained. Depending on the required accuracy, two different idealizations of temperature distributions are analyzed in this paper, namely linear distribution from zone modeling and piece-wise step distribution from field modeling in the longitudinal direction. Compared to a column with uniform temperature distribution, both of them represent more realistically the thermal response of a column, which experiences greater temperature with increasing height. The difference in temperature between the top and bottom ends of a column can be quite significant, particularly prior to the flashover condition. Advantage can be made of this in a performance-based approach to ascertain the stability of a column subjected to a prescribed fire size. In this paper, the stability of a pin-ended steel column under a non-uniform temperature distribution is studied. Although the formulations are based on linear elastic assumptions, the paper explores the validity aspect of the approach and shows that it can be applied to columns with a minimum slenderness ratio where plasticity is negligible. Across a section, the temperature is assumed to be uniform. Two linear elastic springs connected to the column ends simulate the axial restraints from the adjoining unheated structure. The objective is to derive closed-form solutions to enable engineers to quickly ascertain the column stability under a non-uniform temperature distribution, without recourse to finite element modeling.     

钢纤维水泥基复合材料断裂的K R曲线研究

基于Wallin阻力曲线模型,结合纤维增强复合材料断裂理论,提出了钢纤维水泥基复合材料的K-R曲线模型.通过定向与乱向两种纤维分布形式以及不同尺寸的预制缺口三点弯曲梁断裂试验,验证了K-R曲线模型的合理性与适用性.结果表明:K-R曲线模型能够有效描述不同钢纤维分布形式下水泥基复合材料的断裂过程,且理论预测的峰值荷载与试验结果偏差较小.该模型可为钢纤维水泥基复合材料的断裂参数尺寸效应研究提供新方法.     

断层影响带的发育规律及其力学成因

 断层影响带的发育规律和范围是高放废物处置库选址和设计的重点内容之一。采用现场测量和图表分析等方法,研究断层两侧节理平均迹长和中点面密度与距断层距离之间的变化规律,结果表明：(1) 平均迹长随断层距离增加而变大,中点面密度随断层距离增加而变小,且变化特征均符合负指数函数形式;当距离超过断层影响带宽度 ,节理平均迹长和中点面密度均稳定在某一值附近波动。据此将断层影响带分为严重破坏区I和影响区II两个区域;(2) 断层类型不同,断层影响带大小和发育规律存在显著差异,其中,正断层最小,平移断层次之,逆断层最大;平移断层两盘的影响带近似呈对称分布,而正断层和逆断层呈上盘宽、下盘窄的不对称分布。此外,断层影响带宽度与断层长度之间服从幂函数形式。分析认为：呈现上述变化特征和发育规律的原因主要是岩性和形成断层时的力学机制不同。研究结果对岩体结构面均质区划分、地下水分布和处置库设计等具有重要的工程指导意义。     

Inference of discontinuity trace length distributions using statistical graphical models

The characterization of discontinuities within rock masses is often accomplished using stochastic discontinuity network models, in which the stochastic nature of the discontinuity network is represented by means of statistical distributions. We present a flexible methodology for maximum likelihood inference of the distribution of discontinuity trace lengths based on observations at rock outcrops. The inference problem is formulated using statistical graphical models and target distributions with several Gaussian mixture components. We use the Expectation–Maximization algorithm to exploit the relations of conditional independence between variables in the maximum likelihood estimation problem. Initial results using artificially generated discontinuity traces show that the method has good inference capabilities, and inferred trace length distributions closely reproduce those used for generation. In addition, the convergence of the algorithm is shown to be fast.     

甘肃北山芨芨采石场岩体节理特征研究

 芨芨采石场是我国高放废物处置库甘肃北山预选区有利候选地段之一。采用综合节理测量法在芨芨采石场进行详细的节理调查，共获得13 012条节理数据。根据芨芨采石场内的断层将该区域初步划分为3个岩体结构均质区和2个断层影响区。基于圆形窗口法原理编制计算程序，分析断层两侧节理的平均迹长和迹线中点面密度的变化，确定断层对节理分布的影响范围，准确划分岩体结构均质区的大小，并得出各均质区节理的平均迹长和迹线中点面密度；采用节理玫瑰花图和节理极点图法，得出各均质区节理的优势组，对各优势组的产状进行统计分析。芨芨采石场的岩体节理以陡倾角的剪节理为主，节理倾向和倾角符合正态分布；统计分析各优势组的节理间距，得出各优势组节理间距符合负指数分布。按照ISRM提出的《岩体不连续面定量描述的建议方法》(1978)，采石场各结构均质区的节理间距都属于宽间距，表明岩体完整性较好。本次研究得到芨芨采石场岩体节理基本特征的定量化参数，为岩体力学和渗流特性的深入研究提供必要的数据。     

A Method to Assess the Accuracy of Pseudo-Random Number Sampling Methods from Evacuation Datasets

We propose a method for assessing the accuracy of pseudo-random number sampling methods for evacuation modelling purposes. It consists of a systematic comparison between experimental and generated distributions. The calculated weighted relative error (E _{w_rel} ) is based on the statistical parameters as central moments (mean, standard deviation, skewness and kurtosis) to shape the distribution. The case study involves the Box–Muller transform, the Kernel-Epanechnikov, the Kernel-Gaussian and the Piecewise linear generating samples from eight evacuation datasets fitted against normal, lognormal and uniform distributions. Keeping in mind that the Bos Muller method has two potential sources of error (i.e. distribution fitting and sampling), this method produces plausible results when generating samples from the three types of distributions (E _{w_rel}  < 0.30 for normal, lognormal and uniform distributions). We also fund that the Kernel Gaussian and the Kernel Epanechnikov methods are well accurate in generating samples from normal distributions (E _{w_rel}  < 0.1) but potentially inaccurate when generating samples from uniform and lognormal distributions (E _{w_rel}  > 0.80). Results suggest that the Piecewise linear is the most accurate method (E _{w_rel}  = 0.01 normal; E _{w_rel}  = 0.04 lognormal; E _{w_rel}  = 0.009 uniform). This method has the advantage of sampling directly from empirical datasets i.e. no previous distribution fitting is needed. While the proposed method is used here for evacuation modelling, it can be extended to other fire safety engineering applications.     

Probabilistic nature of cyclic load pore pressure in cohesive soil

The probability distribution of cyclic-load-induced pore pressure in contractive cohesive soil is derived based on its dependence on the compressibility of voids. The gamma distribution is established to be the most appropriate by summing exponential changes in the soil compressibility resulting from changes in the condition of the voids. However, since the pore pressure in this soil type is bounded, the upper limit being equal to the undrained strength of the soil and the lower limit near zero, the distribution can be approximated by beta for specified limits.It is established also that the normal distribution can be adopted for the cyclic load pore pressure without loss of accuracy because the gamma distribution approaches the unit normal distribution as the gamma parameter becomes very large. Besides, the gamma distribution has been derived by considering contributions from an infinite number of voids. Therefore the normal distribution, which is generally known as the probability model for sums of random variables when the number of variables is infinitely large, is a fairly reliable approximation of the pore pressure distribution for purposes of practical application.     

岩体节理平均迹长和迹线中点面密度估计

在现场节理调查中，采用圆形窗口直接进行节理调查并不实用。将所有的迹线类型设计成统一的数据结构，读取现场测量的数据；然后在生成的节理迹线图上布置圆形取样窗口。根据测量数据的具体情况调整圆形取样窗口的位置、数量和半径，同时判断并记录每个圆形取样窗口对应的各种迹线类型的数量。利用圆形窗口法原理。在估计节理迹线平均长度的同时计算出节理迹线中点面密度。对于实际测量数据，节理迹线中点并非均匀分布，选择恰当的圆形窗口位置和半径，才能得出合理的结果。采用同心圆法和相切圆法对岌岌采石场岩体节理的平均迹长和迹线中点面密度进行估计，分析表明采用相切圆法能够得出稳定且具有一致性的结果。利用该法计算出中国高放废物处置库甘肃北山预选区域岌岌采石场断裂两侧露头的平均迹长和迹线中点面密度，分析断裂对两侧岩体节理影响的范围。     

Size distributions for all cities: Which one is best?

This paper analyses four statistical distributions used to describe city size distributions: lognormal, double Pareto lognormal, q-exponential, and log-logistic. We use un-truncated city size data for the US, Spain and Italy from 1900 until 2010, and, in addition, the last available year for the remaining countries of the OECD. We estimate the four functions by maximum likelihood. To check the goodness of the fit we use the Kolmogorov-Smirnov and Cramér-von Mises tests, and compute the Akaike information criterion and Bayesian information criterion. The results show that the distribution which best fits data in most of the cases (86.76%) is the double Pareto lognormal.