The Planar Shape of Rock Joints

Knowing the planar shape of discontinuities is important when characterizing discontinuities in a rock mass. However, the real discontinuity shape is rarely known, since the rock mass is usually inaccessible in three dimensions. Information on discontinuity shape is limited and often open to more than one interpretation. This paper discusses the planar shape of rock joints, the most common discontinuities in rock. First, a brief literature review about the shape of joints is presented, including some information on joint-surface morphology, inferences from observed trace lengths on different sampling planes, information based on experimental studies, and joint shapes assumed by different researchers. This review shows that joints not affected by adjacent geological structures such as bedding boundaries or pre-existing fractures tend to be elliptical (or approximately circular but rarely). Joints affected by or intersecting such geological structures tend to be rectangular. Then, using the general stereological relationship between trace length distributions and joint size distributions developed by Zhang et al. (Geotechnique 52(6):419–433, 2002) for elliptical joints, the effect of sampling plane orientation on trace lengths is investigated. This study explains why the average trace lengths of non-equidimensional (elliptical or similar polygonal) joints on two sampling planes can be about equal and thus the conclusion that rock joints are equidimensional (circular) drawn from the fact that the average trace lengths on two sampling planes are approximately equal can be wrong. Finally, methods for characterizing the shape and size of joints (elliptical or rectangular) from trace length data are recommended, and the appropriateness of using elliptical joint shapes to represent polygonal, especially rectangular, joints is discussed.     

基于分形几何理论的DFN模型构建方法研究

以重庆梁-忠（梁平县-忠县）高速公路礼让隧道为工程研究背景,通过测线法调查现场节理,获得了节理产状分布概率密度函数,并从分形几何学的角度分析了节理间距及迹长的分形分布规律,推导出能反映节理间距及迹长分布状态的分形维数D及分形分布概率密度函数。在该基础上采用Matlab软件以及Monte-Carlo随机分析方法,产生节理参数随机数,结合3DEC中最新模块离散裂隙网络(DFN)技术,建立了能反映节理裂隙分布特征的离散裂隙网络模型并验证了模型的有效性,结果表明,分形分布比负指数分布包含更多的间距、迹长分布信息,更接近于实际分布;分形维数D反映了节理间距、迹长在其变化范围内的分布特征,分形维数的大小取决于小间距、小迹长部分数量在总节理数量中的比例,为节理裂隙岩体网络模型构建提供了一种新方法。     

Stereological and Other Methods Applied to Rock Joint Size Estimation—Does Crofton''s Theorem Apply?

A number of authors concerned with the analysis of rock jointing have used the idea that the joint areal or diametral distribution can be linked to the trace length distribution through a theorem attributed to Crofton. This brief paper seeks to demonstrate why Crofton's theorem need not be used to link moments of the trace length distribution captured by scan line or areal mapping to the moments of the diametral distribution of joints represented as disks and that it is incorrect to do so. The valid relationships for areal or scan line mapping between all the moments of the trace length distribution and those of the joint size distribution for joints modeled as disks are recalled and compared with those that might be applied were Crofton's theorem assumed to apply. For areal mapping, the relationship is fortuitously correct but incorrect for scan line mapping.     

Stereological and Other Methods Applied to Rock Joint Size Estimation—Does Crofton's Theorem Apply?

A number of authors concerned with the analysis of rock jointing have used the idea that the joint areal or diametral distribution can be linked to the trace length distribution through a theorem attributed to Crofton. This brief paper seeks to demonstrate why Crofton's theorem need not be used to link moments of the trace length distribution captured by scan line or areal mapping to the moments of the diametral distribution of joints represented as disks and that it is incorrect to do so. The valid relationships for areal or scan line mapping between all the moments of the trace length distribution and those of the joint size distribution for joints modeled as disks are recalled and compared with those that might be applied were Crofton's theorem assumed to apply. For areal mapping, the relationship is fortuitously correct but incorrect for scan line mapping.     

Estimation of joint trace length probability distribution function in igneous,sedimentary, and metamorphic rocks

To predict the behavior of structures in and on jointed rock masses, it is necessary to characterize the geomechanical properties of joints and intact rock. Among geometry properties of joints, trace length has a vital importance, because it affects rock mass strength and controls the stability of the rock structures in jointed rock masses. Since joint length has a range of values, it is useful to have an understanding of the distribution of these values in order to predict how the extreme values may be compared to the values obtained from a small sample. For this purpose, three datasets of joint systems from nine exposures of igneous, metamorphic, and sedimentary rocks are studied. Joint trace length is one of the most difficult properties to measure accurately, but it may be possible to record other geometrical properties of exposed joints accurately; thereby, support vector machine (SVM) model is used to predict the joint trace length. SVM is a novel machine learning method, which is a powerful tool used to solve the problem characterized by small sample and non-linearity with a good generalization performance. Consequently, goodness-of-fit (GOF) tests were applied on these data. According to these GOF tests, the lognormal distribution was found to be the best probability distribution function for representing a joint trace length distribution.     

Probabilistic modelling of joint orientation

Observed frequencies of joint orientations are subject to error due to sampling bias. This error should be corrected before statistical inference is made on the distribution of orientation. Corrections (weighting functions) are developed for sampling bias in orientation for finite joints of different sizes and shapes intersecting rectangular exposures. Chi-square goodness-of-fit procedures available for hemispherical normal and bivariate normal distributions are modified to make them applicable for both raw data and data corrected for sampling bias. The aforementioned corrections and procedures were applied to a joint orientation cluster to study the effect of (a) joint orientation, (b) joint size and (c) joint shape, on the statistical distribution of the orientation. The influences of all these aforementioned factors were found to be significant. However, at present, joint sizes and shapes are not measured in field joint surveys. Therefore, it is suggested to make attempts to obtain joint sizes and shapes in field joint mapping surveys. Since the currently available probability distributions are not adequate to represent all joint orientation distributions, it is suggested either to look for new probability distributions or to develop procedures to use empirical distributions in modelling orientation distributions.     

Estimating the Mean Trace Length of Rock Discontinuities

Summary Rock discontinuities appear as traces on exposures such as natural outcrops or tunnel walls. Discontinuity size which has important effects on rock mass behavior is related to trace length.  This paper presents a technique for estimating mean trace length from the observations made on finite, circular sampling windows. The method takes sampling errors into account and it requires, like existing methods using rectangular sampling windows, that the numbers of discontinuities with both ends censored, with one end observed and one end censored, and with both ends observed be known. Knowledge of the lengths of the observed traces and the distribution of trace lengths is not required. A major advantage of the proposed method over the existing methods is that it does not need sampling data about the orientation of discontinuities, i.e. the proposed method is applicable to traces with arbitrary orientation distributions.  To check the validity of the proposed method, theoretical relations between the mean trace length and the mean diameter of circular discontinuities, respectively for lognormal and negative exponential distributions of the diameter of discontinuities, are derived. The proposed method is then applied to analyze data simulated with the FracMan code, and the predicted results are compared to the corresponding theoretical solutions. The results show that the proposed method is satisfactory. Comparisons of the predicted mean trace length with the mean of observed trace lengths show that it is important to consider sampling biases when estimating mean trace length.     

Granite fracture index to check suitability of granite outcrops for quarrying

A correct knowledge of fracture patterns previous to quarrying is important for economical exploitation. This work comprises the general characterization of the discontinuities of 10 granite rock masses in NE Portugal. During the study some discontinuity characteristics such as orientation, persistence, aperture, trace length and spacing were evaluated. The differences in the characteristics of the selected granites were described, and the most important exploration factors were identified. In the granites studied, most of the joints are oriented from N10°E to N50°E, which is compatible with the regional fracture pattern. Concerning joint termination, the XX type is common, while the RR type is rare; the persistence index ranges from 3.5% to 25.4%. Closed joints are predominant in the granites studied (> 72.4%), and openings of the others are about 1–2 mm as a result of superficial displacement of the granite blocks. The percentage of filled joints, usually by quartz and pegmatite, is different from one granite to another, but most are < 20%. The average joint length ranges from 11.1 m in post-tectonics granites to 4.6 m in syntectonic granites. Joint spacing varies from 0.98 to 2.8 m with higher values in post-tectonic granites. Joint-space distributions of the granites studied are close to log-normal and exponential distributions. Volumetric joint count (Jv) ranges from 0.8 to 2.1 joints/m³ and is used for categorizing the places of each granite. The relationships between Jv and other joint characteristics, such as length, spacing and number of joint sets, were investigated. A Granite Fracturing Index (GFI) is introduced, computed on the basis of volumetric joint count, number of joint sets and true joint spacing. The GFI classification of studied places allows us evaluate the suitability for profitable exploitation.     

节理泊松圆盘模型直径分布推断的解析方法

节理圆盘直径分布是节理网络模拟中的重要参数之一。在节理为Poisson圆盘的假设下,根据三维齐次Poisson点过程的数学特征,给出迹长分布概率密度函数的多项式形式和相应的约束条件;对Warburton提出的迹长分布概率密度函数和节理直径分布概率密度函数关系公式进行Santalo变换,推导了多项式形式的迹长分布所对应的直径分布概率密度函数的解析解,并采用Priest给出的解析解验证其正确性;根据概率密度函数的性质,给出了直径分布概率密度函数定义域的推断方法。编制节理网络模拟程序,采用Monte Carlo方法验证节理Poisson圆盘直径分布概率密度函数解析解的正确性和直径分布概率密度函数定义域推断方法的适用性。     

基于钻孔电视的深部岩体节理三维网络模拟

为了准确模拟岩体节理的空间分布,评价高放废物地质处置库围岩的稳定性,先通过调查露头节理数据,运用概率统计理论进行节理几何特征模拟,再利用Monte-Carlo法来模拟节理空间分布特征,并以高放废物地质处置新疆预选区天湖地段花岗岩体为研究对象,通过钻孔电视调查不同深度节理几何特征,划分钻孔岩体结构均质区,提出钻孔节理产状、大小、密度的统计方法和计算公式,以均质区建立节理三维网络模型并通过数据对比法进行模型验证。结果表明：1）利用列联表卡方检验和重叠窗口法可将天湖钻孔岩体划分为0~90、120~240、270~360和390~600 m等4个均质区;2）利用SPSS软件以及基于节理直径、迹长和隙宽之间的数量关系得出节理产状服从正态分布,节理直径服从负指数分布;3）通过数据对比法得出模型统计和现场测量的节理平均产状数据相差±3°以内,平均直径相差±1 m以内,平均面密度相差±0.01 m^-2以内,从而验证了Monte-Carlo法建立的节理三维网络模拟的准确性。     

岩体表面节理迹长与节理真实长度尺寸的概率关系

空间节理长度不能直接进行测量,只能通过岩石暴露面上节理迹线长度来量测,但是节理迹线是节理面与岩面交切的结果,迹长在多大程度上能代表节理真实长度是值得研究的,本文利用计算机模拟的方法分析了迹长与节理真实长度的统计关系。     

Observed oil and gas field size distributions: A consequence of the discovery process and prices of oil and gas

If observed oil and gas field size distributions are obtained by random samplings, the fitted distributions should approximate that of the parent population of oil and gas fields. However, empirical evidence strongly suggests that larger fields tend to be discovered earlier in the discovery process than they would be by random sampling. Economic factors also can limit the number of small fields that are developed and reported. This paper examines observed size distributions in state and federal waters of offshore Texas. Results of the analysis demonstrate how the shape of the observable size distributions change with significant hydrocarbon price changes. Comparison of state and federal observed size distributions in the offshore area shows how production cost differences also affect the shape of the observed size distribution. Methods for modifying the discovery rate estimation procedures when economic factors significantly affect the discovery sequence are presented. A primary conclusion of the analysis is that, because hydrocarbon price changes can significantly affect the observed discovery size distribution, one should not be confident about inferring the form and specific parameters of the parent field size distribution from the observed distributions.This paper was presented at Emerging Concepts, MGUS-87 Conference, Redwood City, California, 13–15 April 1987.     

On the Existence,Uniqueness and Correctness of the Fracture Diameter Distribution Given the Fracture Trace Length Distribution

Based on the fracture trace length distribution, conditions for the existence, uniqueness, and correctness of the fracture diameter distribution are given using Warburton’s fracture model. In particular, a solution for the fracture diameter distribution exists and is unique for all trace length probability density functions, h _A (y), such that \(h_{A}(y)/\sqrt{y^{2}-x^{2}}\) is Lebesgue integrable on [x,∞). This condition is met by the uniform, exponential, gamma, lognormal, and power-law trace length distributions as well as by the trace length distributions that arise from a deterministic fracture diameter or from a discontinuous fracture diameter length distribution. Exponential, gamma, lognormal, and power-law trace length distributions satisfy necessary conditions for the diameter distribution to be non-negative, and necessary and sufficient conditions for the distribution to have unit integral over the real line. Negative values of the fracture diameter distribution arise when the trace has a uniform distribution and the lower bound of the fracture trace is greater than zero.     

Estimating Mean Fracture Trace Length and Density from Observations in Convex Windows

Summary Although window samples of fracture traces are widely recognized to be length-biased and censored, they are often the best source of data for inferring statistical parameters of fracture sets. This paper presents new estimators for mean fracture trace length and density that correct for the effects of bias and censoring. A stereological estimator of mean trace length is derived for parallel traces in a rectangular sampling window, an end-point estimator of mean trace length is derived for windows with arbitrary convex boundaries and for arbitrary trace length distributions, and an end-point estimator of trace density is derived for windows with arbitrary convex boundaries and for arbitrary trace length and trace orientation distributions. Results for rectangular and circular windows are obtained as special cases of the general solutions for arbitrary convex windows. When applied to circular windows, the end-point estimator of mean trace length is, in addition, independent of the trace orientation distribution. The estimators are easily determined from field data. The end-point estimator of trace density requires knowing only the area of the window and the number of end-points inside the window. The end-point estimator of mean trace length, when applied to circular windows, requires, in addition, the number of end-points outside the window (of those traces that intersect the window) and the stereological estimator of mean trace length requires only the apparent mean trace length and the height of the window.     

Lognormality in the observed size distribution of oil and gas pools as a consequence of sampling bias

Economic filtration has been offered as an explanation of the observed lognormality in the size distribution of discovered oil and gas deposits. The result leads to the conclusion that one cannot impute the shape of the underlying parent distribution from the observed discoveries size distribution. The fact that the largest pools tend to be discovered early in the exploration history of an area of interest suggests the existence of an inherent sampling bias in the discovery process. The bias is influenced by the levels of geologic knowledge and technological sophistication. Furthermore, the existence of the bias leads to lognormality in the observed discoveries size distribution of oil and gas pools. A discovery process model explicitly incorporating the notion of sampling bias was applied to a series of Weibull parent frequency size distributions. The selected parent distributions are of a class suggested in the literature as more reflective of nature's size distribution and have empirical support. The distribution of discoveries resulting from the application of the model to the chosen parent size distributions were tested for lognormality using a chi-squared test. Lognormality was found to be an acceptable model of the discoveries size distribution over a wide range of resource exhaustion measures. When combined with the notion of economic filtration, sampling bias leads to the conclusion that one should not expect the lognormal distribution to accurately represent the underlying parent size distribution of oil and gas deposits.     

Faulted joints: kinematics,displacement–length scaling relations and criteria for their identification

Structural geometries and kinematics based on two sets of joints, pinnate joints and fault striations, reveal that some mesoscale faults at Split Mountain, Utah, originated as joints. Unlike many other types of faults, displacements (D) across faulted joints do not scale with lengths (L) and therefore do not adhere to published fault scaling laws. Rather, fault size corresponds initially to original joint length, which in turn is controlled by bed thickness for bed-confined joints. Although faulted joints will grow in length with increasing slip, the total change in length is negligible compared to the original length, leading to an independence of D from L during early stages of joint reactivation. Therefore, attempts to predict fault length, gouge thickness, or hydrologic properties based solely upon D–L scaling laws could yield misleading results for faulted joints. Pinnate joints, distinguishable from wing cracks, developed within the dilational quadrants along faulted joints and help to constrain the kinematics of joint reactivation.     

Estimation of Block Sizes for Rock Masses with Non-persistent Joints

Summary  Discontinuities or joints in the rock mass have various shapes and sizes. Along with the joint orientation and spacing, the joint persistence, or the relative size of the joint, is one of the most important factors in determining the block sizes of jointed rock masses. Although the importance of joint persistence on the overall rock mass strength has long been identified, the impact of persistence on rock strength is in most current rock mass classification systems underrepresented. If joints are assumed to be persistent, as is the case in most designs, the sizes of the rock blocks tend to be underestimated. This can lead to more removable blocks than actually exist in-situ. In addition, a poor understanding of the rock bridge strength may lead to lower rock mass strengths, and consequently, to excessive expenditure on rock support. In this study, we suggest and verify a method for the determination of the block sizes considering joint persistence. The idea emerges from a quantitative approach to apply the GSI system for rock mass classification, in which the accurate block size is required. There is a need to statistically analyze how the distribution of rock bridges according to the combination of joint orientation, spacing, and persistence will affect the actual size of each individual block. For this purpose, we generate various combinations of joints with different geometric conditions by the orthogonal arrays using the distinct element analysis tools of UDEC and 3DEC. Equivalent block sizes (areas in 2D and volumes in 3D) and their distributions are obtained from the numerical simulation. Correlation analysis is then performed to relate the block sizes predicted by the empirical equation to those obtained from the numerical model simulation. The results support the concept of equivalent block size proposed by Cai et al. (2004, Int. J. Rock Mech. Min. Sci., 41(1), 3–19).     

Non-stationary Geostatistical Modeling Based on Distance Weighted Statistics and Distributions

A common assumption in geostatistics is that the underlying joint distribution of possible values of a geological attribute at different locations is stationary within a homogeneous domain. This joint distribution is commonly modeled as multi-Gaussian, with correlations defined by a stationary covariance function. This results in attribute maps that fail to reproduce local changes in the mean, in the variance and, particularly, in the spatial continuity. The proposed alternative is to build local distributions, variograms, and correlograms. These are inferred by weighting the samples depending on their distance to selected locations. The local distributions are locally transformed into Gaussian distributions embedding information on the local histogram. The distance weighted experimental variograms and correlograms are able to adapt to local changes in the direction and range of spatial continuity. The automatically fitted local variogram models and the local Gaussian transformation parameters are used in spatial estimation algorithms assuming local stationarity. The resulting maps are rich in nonstationary spatial features. The proposed process implies a higher computational effort than traditional stationary techniques, but if data availability allows for a reliable inference of the local distributions and statistics, a higher accuracy of estimates can be achieved.     

The distribution of trace metals in the western North Atlantic off Nova Scotia

The distributions of Fe, Mn, Zn, Cu, Ni, Cd and Co have been determined in a section across the Scotian Shelf into the Atlantic Slope water. Significant differences in concentration exist for most of the trace metals between the four water masses in the section. Depletions of trace metal concentration in the highly productive Atlantic Slope water relative to the underlying Central Atlantic water are thought to be due to biological activity.The distributions of Fe and Mn are strongly related to the distribution of suspended particulate matter. The concentrations of Fe and Mn, extracted from the suspended matter on the Scotian Shelf, are considerably higher than those in the non-detrital fraction of the underlying sediments. This suggests that post-depositional changes cause the loss of both elements from the non-detrital fraction of the particles. Whereas Mn shows major nearshore increases in concentration related to continental runoff, nearshore Fe concentrations are largely controlled by particulate matter distribution. Continental runoff does not appear to have much influence upon the distributions of the other trace metals.