基于裂隙产状和迹长划分岩体统计均质区研究

岩体统计均质区的划分是岩体工程设计的首要步骤,因为不同的均质区内岩体的结构特性和力学特性都存在差异。大多研究考虑产状或者其他单一参数划分岩体统计均质区,却很少有人考虑多参数进行划分研究。考虑了裂隙产状、迹长两个参数,提出了一种划分裂隙岩体统计均质区的新方法,即基于秩和检验评价两个样本之间的相似性,从而实现对裂隙岩体统计均质区的划分。结合工程实例即云南省马吉水电站左岸坝肩某两高程4个平硐的岩体结构统计均质区划分,分析了多参数对裂隙岩体统计均质区划分的影响。研究结果表明：PD241和PD253、PD201和PD241分别划分为一个统计均质区,而PD201和PD251 以及PD251和PD253不具有统计相似性。考虑了产状和迹长因素对岩体统计均质区的影响,划分结果较Miller方法更加合理可靠。结合现场调查分析,新方法的结果更加吻合实际情况。     

A statistical method to evaluate homogeneity of structural populations

For any specific rock unit in a discontinuous rock mass, the determination of structural domain boundaries is important because geologic and hydrologic properties vary from one domain to another. A domain can be considered as a structural population characterized by a distinct pattern of structure orientations (i.e., fracture, bedding, or foliation orientations).Boundaries are usually determined by comparing samples of structural data, each of which consists of a Schmidt plot that commonly displays at least 150 poles to fractures. When fracture orientations appear dispersed and random on the plots, visual comparisons are not sufficient to determine whether the samples were obtained from the same structural domain. An appropriate x ² test has been adapted to this situation and is capable of evaluating the homogeneity of structural populations. The method uses a contingency table analysis based on the frequencies of fracture poles that occur in corresponding patches on the Schmidt plots being compared.     

A technique for identifying structural domain boundaries at the EKATI Diamond Mine

Over 10,000 discontinuity orientations from various sources at the EKATI Diamond Mine in northern Canada were collected and amalgamated into a common database. A new technique in which structural domains were determined in the area of three closely spaced active pits by quantitatively comparing stereonets from different regions was developed. The frequencies of discontinuity poles plotting in orientation windows on stereonets were determined and compared using a correlation coefficient to quantify the degree of similarity between different stereonets. Structural boundaries between regions were established wherever their associated stereonets display low correlation coefficients. The technique was able to determine boundaries in the vertical direction as well as radially in a number of pie-shaped zones extending around the kimberlite pipes.     

基于卡方检验法对长白山龙门峰裂隙岩体统计均质区划分

岩体统计均质区是裂隙岩体研究的一个重要内容,是三维裂隙网络等的基础研究范围。在列联表卡方检验基础上建立起考虑裂隙岩体迹长的方法用来划分岩体的统计均质区,即在统计均质区划分过程中考虑裂隙迹长的因素,通过列联表卡方检验建立两对比样本,计算对比样本各属性之间的实际频率,计算卡方值与临界卡方值。在判断样本之间是否具有统计相似性的基础上,建立了确定统计均质区的划分方法,使得裂隙岩体统计均质区划分更为全面与客观。     

地下矿山裂隙岩体结构均质区三维划分方法研究

针对现有岩体结构均质区划分方法应用于地下矿山时存在细节辨识能力偏弱等问题,开展地下矿山岩体结构均质区三维精确划分方法研究。以广西铜坑矿锌多金属矿体3^#矿块为例,从岩体空间结构展布特性出发,构建由确定性与随机性结构面耦合的裂隙岩体三维模型;将岩体模型正交网格离散化,并选取模型网格子区的块体化程度值为均质区划分判据;利用K-means算法衡量不同数据源(网格)的相似性,结合误差平方和(SSE)度量聚类质量;最后重构三维岩体结构均质区,形成了一种地下矿山岩体结构均质区三维划分方法。研究成果表明:利用该方法能够从三维角度精确划分地下矿山岩体结构均质区,可为后续岩体质量评价及岩体离散元模拟的模型构建等工作提供有益参考。     

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.     

Describing the Geometry of 3D Fracture Systems by Correcting for Linear Sampling Bias

Analyzing the geometric bias inherent to linear sampling of natural fracture systems is a prerequisite to any attempt of structural modeling. In this paper, the basic parameters of 1D-sampled fracture sets, i.e. orientation, density, and size, are interpreted in terms of geometric probabilities. Weighting factors are derived which allow the 3D restitution of a moderately variable fracture network from a single borehole. The proposed method is applied to well core data from a granitic rock mass, and the efficiency of the proposed corrections is illustrated through random disc simulations tested by virtual scanlines analogous to the real borehole. This approach aims to reduce the prospecting effort in exploration, and to criticize assumption of structural homogeneity by rigorously comparing fracture populations collected from nonparallel boreholes. Then a parametric study of fracture size is performed and a range of mean size leading to fully connected networks is identified.     

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.     

A two-dimensional fuzzy random model of soil pore structure

A new conceptual model for soil poresolid structure is formalized. Soil poresolid structure is proposed to comprise spatially abutting elements each with a value which is its membership to the fuzzy set “pore, ” termed porosity. These values have a range between zero (all solid) and unity (all pore). Images are used to represent structures in which the elements are pixels and the value of each is a porosity. Twodimensional random fields are generated by allocating each pixel a porosity by independently sampling a statistical distribution. These random fields are reorganized into other poresolid structural types by selecting parent points which have a specified local region of influence. Pixels of larger or smaller porosity are aggregated about the parent points and within the region of interest by controlled swapping of pixels in the image. This creates local regions of homogeneity within the random field. This is similar to the process known as simulated annealing. The resulting structures are characterized using one and twodimensional variograms and functions describing their connectivity. A variety of examples of structures created by the model is presented and compared. Extension to three dimensions presents no theoretical difficulties and is currently under development.     

Some garnet microstructures: an illustration of the potential of orientation maps and misorientation analysis in microstructural studies

The microstructures of two contrasting garnet grains are mapped using automated electron backscatter diffraction. In both cases there is a very strong crystallographic preferred orientation, with measurements clustered round a single dominant orientation. Each garnet grain is divided into domains with similar orientations, limited by boundaries with misorientations of 2° or more. In both samples most of misorientation angles measured across orientation domain boundaries are significantly lower than those measured between random pairs of orientation domains. One sample is a deformed garnet that shows considerable distortion within the domains. Lines of orientation measurements within domains and across domain boundaries show small circle dispersions around rational crystallographic axes. The domain boundaries are likely to be subgrain boundaries formed by dislocation creep and recovery. The second sample is a porphyroblast in which the domains have no internal distortion and the orientation domain boundaries have random misorientation axes. These boundaries probably formed by coalescence of originally separate garnets. We suggest that misorientations across these boundaries were reduced by physical relative rotations driven by boundary energy. The data illustrate the potential of orientation maps and misorientation analysis in microstructural studies of any crystalline material.     

Coupled boundary element/finite element analysis in geomechanics including body forces

A coupling scheme for boundary and finite elements using joint elements is proposed which includes the consideration of body forces. In this scheme the boundary and joint elements are formulated in a similar way as finite elements (i.e., the equivalent FE procedure). These joint elements are efficiently used to combine different BE regions. For the evaluation of a body forces, two methods are compared on computational efficiency and it is found that the method using Galerkin tensor is more efficient than the method dividing the problem domain into several internal cells. Two main geotechnical problems considering self weight are numerically examined using this coupling procedure.     

Quantifying the uncertainty in the spatial layout of rock type domains in an iron ore deposit

This work addresses the problem of delineating the spatial layout of ten rock type domains in an iron ore body and of assessing the uncertainty in the domain boundaries. A stochastic approach is proposed to this end, based on truncated Gaussian simulation, which consists in defining successive partitions of the space that comply with the desired spatial continuity and contact relationships between rock type domains. The sequencing of these domains is driven by their position (surficial vs. underlying rocks), granulometry (compact vs. friable rocks), and grades (rich vs. poor) of iron, alumina, manganese and loss on ignition. A total of 100 realizations are produced, conditioned to available drill hole data, and used to quantify the uncertainty in the occurrence of each rock type domain, at both global and local scales.     

Censoring and edge effects in areal and line transect sampling of rock joint traces

This paper reviews techniques for correcting for bias incurred when incomplete observations form part of a data set. The similarities and differences between biases due to censoring and edge effects are emphasized, the latter being of importance when the distribution of joint traces is estimated from an exposure. A new technique for analyzing line transect data is presented, and a new analysis of published data on joint trace lengths given.     

Box fractal dimension as a measure of statistical homogeneity of jointed rock masses

A written computer programme to estimate the box fractal dimension (D_B) is verified by estimating D_B of the triadic Koch curve for which the theoretical D is known. The influence of a number of input parameters of the box-counting method on the accuracy of estimated D_B is evaluated using the same Koch curve. The employed size range of the applied box networks was found to be the parameter which has the strongest influence on the accuracy of estimated D_B. This indicated the importance of finding the range of self-similarity or self-affinity for the object considered to select the proper range for the box sizes and, in turn, to obtain accurate estimates of D_B. By calculating D_B for different block sizes sampled from three generated two-dimensional joint patterns, it is shown that D_B can capture the combined effect of joint-size distribution and joint density on the statistical homogeneity of rock masses. The spatial variation of D_B along a 350 m stretch of a tunnel in the shiplock area of the Three Gorges dam site is computed using the joint data mapped on the walls and the roof of the tunnel. This spatial variation of D_B is used, along with the visual geological evaluation of the joint trace maps of the tunnel, in making decisions about statistical homogeneity of the rock mass around the tunnel. The results obtained on statistically homogeneous regions were found to be quite similar to the results obtained from a previous statistical homogeneity investigation which incorporated the effect of number of joint sets and their orientation distributions, but not the spatial variation of D_B. It is recommended that the spatial variation of D_B is used, along with the results of other methods such as contingency table analysis and equal area plots, which incorporate the effect of joint orientation distribution, in addition to the geology of the site, in determining the statistically homogeneous regions of jointed rock masses.     

Local and Spatial Joint Frequency Uncertainty and its Application to Rock Mass Characterisation

Stability is a key issue in any mining or tunnelling activity. Joint frequency constitutes an important input into stability analyses. Three techniques are used herein to quantify the local and spatial joint frequency uncertainty, or possible joint frequencies given joint frequency data, at unsampled locations. Rock quality designation is estimated from the predicted joint frequencies. The first method is based on kriging with subsequent Poisson sampling. The second method transforms the data to near-Gaussian variables and uses the turning band method to generate a range of possible joint frequencies. The third method assumes that the data are Poisson distributed and models the log-intensity of these data with a spatially smooth Gaussian prior distribution. Intensities are obtained and Poisson variables are generated to examine the expected joint frequency and associated variability. The joint frequency data is from an iron ore in the northern part of Norway. The methods are tested at unsampled locations and validated at sampled locations. All three methods perform quite well when predicting sampled points. The probability that the joint frequency exceeds 5 joints per metre is also estimated to illustrate a more realistic utilisation. The obtained probability map highlights zones in the ore where stability problems have occurred. It is therefore concluded that the methods work and that more emphasis should have been placed on these kinds of analyses when the mine was planned. By using simulation instead of estimation, it is possible to obtain a clear picture of possible joint frequency values or ranges, i.e. the uncertainty.     

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.     

The key blocks of unrolled joint traces in developed maps of tunnel walls

This paper presents a general method of key block analysis for cylindrical surfaces with numerous real or statistically produced joint traces. General tunnel curves, analytically represented, are unrolled to yield a ‘developed view’ on which the joint traces are continuous curves. Then, using extended block theory, the maximum key block regions are delimited from the curved polygons of the unrolled joint trace map. The methods discussed here apply to a cylindrical tunnel of any shape in section and in any orientation, including inclined tunnels and shafts. The trace maps for which the method applies can be generated statistically, as described herein, or surveyed from real traces on exposed tunnel walls. A brief introduction to the basic theory about tunnel key blocks is provided here so that this paper can be understood without reference to other papers on block theory.     

A Spatial Correlation-Based Anomaly Detection Method for Subsurface Modeling

Spatial data analytics provides new opportunities for automated detection of anomalous data for data quality control and subsurface segmentation to reduce uncertainty in spatial models. Solely data-driven anomaly detection methods do not fully integrate spatial concepts such as spatial continuity and data sparsity. Also, data-driven anomaly detection methods are challenged in integrating critical geoscience and engineering expertise knowledge. The proposed spatial anomaly detection method is based on the semivariogram spatial continuity model derived from sparsely sampled well data and geological interpretations. The method calculates the lag joint cumulative probability for each matched pair of spatial data, given their lag vector and the semivariogram under the assumption of bivariate Gaussian distribution. For each combination of paired spatial data, the associated head and tail Gaussian standardized values of a pair of spatial data are mapped to the joint probability density function informed from the lag vector and semivariogram. The paired data are classified as anomalous if the associated head and tail Gaussian standardized values fall within a low probability zone. The anomaly decision threshold can be decided based on a loss function quantifying the cost of overestimation or underestimation. The proposed spatial correlation anomaly detection method is able to integrate domain expertise knowledge through trend and correlogram models with sparse spatial data to identify anomalous samples, region, segmentation boundaries, or facies transition zones. This is a useful automation tool for identifying samples in big spatial data on which to focus professional attention.

     

Electron microscopy study of domain structure due to phase transitions in natural perovskite

Transmission electron microscopy on natural calcium metatitanate perovskite (dysanalyte) reveals the following twin laws in the orthorhombic (space group Pbnm) phase: reflection twins on the {110} and {112} planes, and 90° rotation twins about the [001] axis (referred to as [001]_90° twin). Single crystals that were heattreated and quenched from above 1585 K exhibit a dramatic change in domain structure compared with the starting material and specimens quenched from T < 1470=" k.=" mutually=" perpendicular=" {110}=" and=">_90° twins are observed throughout the crystal, forming a cross-hatched domain texture. 1/2[001] antiphase domains, which are very rarely observed in the starting material, also become dominant in the crystal. This change in domain structure is interpreted as due to a structural phase transition in perovskite at a temperature below 1585 K. From the point symmetry elements that describe the twin laws and the translational elements that relate the antiphase domains, the most likely phase near 1585 K is tetragonal with space group P4/mbm. These results are consistent with high-temperature powder X-ray diffraction study. On the other hand, density of the {112} twins is increased significantly in the crystal quenched from 1673 K. Twin domains are either bound by mutually perpendicular {110} and (001) walls, or by {112} walls with {110} twin domains within the polygonal {112} domains. Both twin density variation and domain morphology suggest that the crystal may be cubic at this temperature. Microstructure of a single crystal deformed at 1273 K and 3.5 GPa (within the orthorhombic stability field) is morphologically quite distinct from that of the heat-treated specimens. Dislocations dominate the microstructure and often interact with twin domain boundaries.A National Science Foundation Science and Technology Center     

Origin of fault domains and fault-domain boundaries (transfer zones and accommodation zones) in extensional provinces: Result of random nucleation and self-organized fault growth

In many extensional provinces, large normal faults dip in the same direction forming fault domains. Features variously named transfer faults, transfer zones, and accommodation zones (hereafter non-genetically referred to as fault-domain boundaries) separate adjacent fault domains. Experimental modeling of distributed extension provides insights on the origin, geometry, and evolution of these fault domains and fault-domain boundaries. In our scaled models, a homogeneous layer of wet clay or dry sand overlies a latex sheet that is stretched orthogonally or obliquely between two rigid sheets. Fault domains and fault-domain boundaries develop in all models in both map view and cross-section. The number, size, and arrangement of fault domains as well as the number and orientation of fault-domain boundaries are variable, even for models with identical boundary conditions. The fault-domain boundaries in our models differ profoundly from those in many published conceptual models of transfer/accommodation zones. In our models, fault-domain boundaries are broad zones of deformation (not discrete strike-slip or oblique-slip faults), their orientations are not systematically related to the extension direction, and they can form spontaneously without any prescribed pre-existing zones of weakness. We propose that fault domains develop because early-formed faults perturb the stress field, causing new nearby faults to dip in the same direction (self-organized growth). As extension continues, faults from adjacent fault domains propagate toward each another. Because opposite-dipping faults interfere with one another in the zone of overlap, the faults stop propagating. In this case, the geometry of the domain boundaries depends on the spatial arrangement of the earliest formed faults, a result of the random distribution of the largest flaws at which the faults nucleate.