An Algorithm for Generating Rock Fracture Patterns: Mathematical Analysis

A statistical, rule-based algorithm for generating fracture patterns similar to those observed in Limestone is presented. For each fracture set, initial seed points are randomly positioned within the modelled domain with the same density as the fractures observed in the field. An orientation is associated with each point by sampling from the distribution of orientations for the corresponding fracture set. Fractures are then allowed to grow from the seed points in both directions with this orientation until they meet other fractures whereupon they continue or terminate according to a fixed probability. A mathematical analysis of this method is presented for the case in which fractures within a set are assumed to be parallel. Approximations to the distribution of semi-trace lengths are derived which are shown to be in good agreement with simulation results. Fracture spacing distributions are also derived for this case.     

Fracture Length Estimation from Borehole Image Logs

We describe a method to estimate fracture length for circular fractures from borehole image logs. The relative frequency of fractures, which have complete circumference trace on image logs is related to fracture length. A simple functional relationship can be derived for the relative frequency of complete fracture traces in terms of average fracture inclination to borehole, borehole diameter and fracture length. This formulation however, tends to underestimate fracture length because a constant length is assumed. A more accurate length estimate can be obtained by assuming that fracture length is linearly correlated to fracture aperture or spacing. Cumulative frequency distribution of fracture aperture and spacing can be obtained from borehole image logs. The problem then transforms itself to finding the scaling factor between fracture length and aperture (or spacing) from the relative frequency of fractures with complete traces. The product of the scaling factor and average fracture spacing (or aperture) gives the average fracture length.     

Rock Quality Designation Model Formulation and Simulation for Correlated Fracture Intact Lengths

An important factor in the rock quality assessment analysis from discontinuity measurements along a scanline is the correlation of intact lengths. The autorun model and first order autorun coefficient are proposed as a method for objectively quantifying intact length correlation structure and discontinuity occurrences along any scanline. Necessary definitions for the methodology are presented on a quantitative basis such as the alternative intact length percentages and experimental autorun coefficient. An expression is proposed for the number of discontinuity occurrences along a scanline depending on the scanline length, alternate intact length percentages, and the autorun coefficient. Monte Carlo simulation technique is used with a suitable autorun model to verify this expression. A mathematical expression is derived for the expected Rock Quality Designation (RQD) value by taking into account the correlation of the intact lengths. Additional simulations show the validity of the new RQD expression. With the methodology proposed much less engineering experience and judgment are required in estimating the RQD value from scanline measurements. The application of the methodology has been presented for some field data.     

考虑压裂裂缝的煤层气藏井网井距确定方法

煤层气藏具有低渗透率特点, 这使得其在直井排采前须对产层进行压裂改造,而压裂裂缝参数对井距的确定影响很大,因此研究裂缝长度、导流能力与井距的关系具有实际意义。通过对比不同裂缝穿透比和导流能力下的煤层气产量与采出程度,探讨裂缝参数对煤层气产出的影响规律;并以煤层气开采15a采出程度达到50%时的井距作为评价指标,研究压裂效果与井距的关系。研究结果表明,裂缝越长,导流能力越大,井距就越大-尤其在渗透率较小时,裂缝参数对井距的影响更明显。因此,在确定煤层气藏井距时,由于人工裂缝影响,裂缝方向的井距可适当放大。以我国韩城矿区煤层气区块储层参数为例,在给定优化指标下,宜选择矩形井网(350m×300m),或菱形井网(对角线长度为700m×400m)。     

Detection of Cyclic Patterns Using Wavelets: An Example Study in the Ormskirk Sandstone, Irish Sea

This study shows how wavelet analysis can be used on well log and drill core data to identify cyclicity in sedimentary sequences. Three possible methods for determining wavelength were investigated: the Morlet wavelet, the Fourier transform, and the semivariogram. When applied to several hypothetical signals similar to those observed in petrophysical measurements in hydrocarbon reservoirs, all three methods could identify the presence of cyclicity. Only the wavelet scalogram, however, gave a clear indication of when the cyclic element was present and where frequency changes occurred in the signal. To illustrate the wavelet analysis, we processed well log and core data from a well in the Ormskirk Sandstone and determined the wavelet coefficients for each zone and the wavelengths of the strongest cyclicities. The cyclicities observed corresponded well with sedimentary features of the formation (e.g., channels and channel sets). Also, ratios of the cyclicity wavelengths corresponded with ratios of the Milankovitch precession, obliquity, and eccentricity periods. This result is in agreement with other investigators, who have proposed that Milankovitch-driven climate changes exercised an important control on Ormskirk Sandstone deposition.     

克拉苏构造带盐下超深层断背斜裂缝带发育模式及预测方法

克拉苏构造带盐下超深层埋深大于6500 m的断背斜圈闭,白垩系砂岩储层内部裂缝普遍发育且分布复杂,对储层产能影响较大,目前仍然缺乏有效的裂缝发育模式指导裂缝带建模。本文通过钻、测井资料及物理模拟实验,分析了盐下断背斜内裂缝类型及空间分布规律:(1)盐下深层断背斜裂缝带可以分为"挤压剪切型、拱张型与调节型"三种成因类型,挤压剪切型裂缝带主要分布于断背斜翼部,受推覆作用而成,平面呈近EW走向或NW-SE走向。拱张型则主要发育于背斜核部,受控于断背斜拱张作用,平面近EW走向,调节型裂缝带则发育于断背斜的构造转换部位,呈近SN走向;(2)物理模拟实验证明拱张型裂缝带受褶皱褶曲度控制,剪切型、调节型裂缝带则受逆冲断层断距控制,根据断背斜拱升角及垂直断距变化可以预测裂缝带的平面分布;(3)地质资料和物理模拟实验证明"调节裂缝带"及"拱张裂缝带"张开程度最大,有利于改善超深层储层的储集性能,这类裂缝带的单井天然气产量较高。     

Scanline Method to Determine the Fractal Nature of 2-D Fracture Networks

A previous method proposed to measure the fractal dimension of pore spaces is adapted and modified for 2-D fracture networks. The method relies on scanning a 2-D fracture network through successive straight lines from top to bottom and measuring the distance between two fractures. The fractal dimension is then obtained using the log–log plot of the feature size and the number of features for this particular size at different magnifications. It is shown in this study that the method proposed to measure the fractal dimension of porous structures can be applicable to 2-D fracture networks with some modifications after testing it on synthetic and natural fracture patterns. The method is simplified to be useful for practical applications in the fractal analysis of fracture networks. The results reveal that, on the basis of the direction of scanning lines, different fractal behavior and dimensions can be obtained indicating that 2-D fracture networks possess multifractal character. This approach takes into account the effect of fracture orientation on the fractal behavior and anisotropic nature of fracture networks as well as the fracture density, length, and spatial distribution.     

利用方位角道集处理方法预测煤层裂隙

煤层瓦斯含量受多种因素影响,而煤层中的节理裂隙密集带则容易导致瓦斯聚积。在煤田地质勘探中,依据振幅随炮点和检波点入射角的变化可以预测裂隙的发育方向和发育密度。在对淮南地区张集西三煤矿的三维地震数据资料处理时,按不同的方位角增量抽取方位角道集,并对其方位角道集进行速度分析、NMO校正、叠加和偏移,得到6个方位偏移数据体。根据Malick等人的理论模型对其进行计算,获得d、A、B值,并据此预测裂缝发育方向与发育密度。将其预测成果与常规地震解释成果对比,发现两者之间非常吻合,可见利用方位角道集处理方法预测煤层裂隙具有可行性。     

Numerical Modelling of the Heterogeneous Rock Fracture Process Using Various Test Techniques

Summary A series of numerical tests including both rock mechanics and fracture mechanics tests are conducted by the rock and tool (R–T^2D) interaction code coupled with a heterogeneous masterial model to obtain the physical–mechanical properties and fracture toughness, as well as to simulate the crack initiation and propagation, and the fracture progressive process. The simulated results not only predict relatively accurate physical–mechanical parameters and fracture toughness, but also visually reproduce the fracture progressive process compared with the experimental and theoretical results. The detailed stress distribution and redistribution, crack nucleation and initiation, stable and unstable crack propagation, interaction and coalescence, and corresponding load–displacement curves can be proposed as benchmarks for experimental study and theoretical research on crack propagation. It is concluded that the heterogeneous material model is reasonable and the R–T^2D code is stable, repeatable and a valuable numerical tool for research on the rock fracture process.     

Spatial analyses of groundwater levels using universal kriging

For water levels, generally a non-stationary variable, the technique of universal kriging is applied in preference to ordinary kriging as the interpolation method. Each set of data in every sector can fit different empirical semivariogram models since they have different spatial structures. These models can be classified as circular, spherical, tetraspherical, pentaspherical, exponential, gaussian, rational quadratic, hole effect, K-bessel, J-bessel and stable. This study aims to determine which of these empirical semivariogram models will be best matched with the experimental models obtained from groundwater-table values collected from Mustafakemalpasa left bank irrigation scheme in 2002. The model having the least error was selected by comparing the observed water-table values with the values predicted by empirical semivariogram models. It was determined that the rational quadratic empirical semivariogram model is the best fitted model for the studied irrigation area.     

水压致裂煤层裂缝发育特点的研究

本文对煤岩基本力学性质、煤层水压致裂缝形成条件、裂缝形态以及裂缝开裂角方位等基本理论进行了研究与探讨，通过大量煤岩力学性质测试，证实了试验区目的层煤岩弹性模量低，泊松比较高，脆性大，易破碎，易压缩。文章还得出了目的层煤岩Ｍｏｈｒ断裂准则二次抛物线型包络线，煤层水压致裂裂缝形式判断，裂缝开裂角方位的计算公式以及有关结论。     

Quantification of Natural Fracture Surfaces Using Fractal Geometry

The purpose of this paper is to present an extensive evaluation of the methods to calculate the fractal dimension of natural fracture surfaces. Three methods; variogram analysis (VA), power spectral density (PSD), and roughness-length method (RMS) are applied to 2-D surface data (PSD) and 1-D profiles (VA and RMS) extracted from the surface data of 54 mm diameter crystallized limestone samples. Surface topography of the samples is quantified through a newly designed fully automated device. Before the application, self-affinity of the surface roughness and the applicability of these methods are validated using synthetically generated fractal surfaces. Fractal dimension values of the profiles are obtained as between 1 and 1.5 with a few exceptions. VA and RMS methods yield consistent fractal dimensions while the PSD values are lower than those of the other two methods. In terms of practical applicability, the VA is found more convenient than other two methods because there still exists shortcomings with the PSD and RMS methods due to difficulties in the mathematical analysis of the plots whose slopes are used in the computation of fractal dimension. However, it is observed that the data of limited size fracture surfaces are convenient for fractal analysis and the results are promising for further applications if the fracture surface size is restricted like cores recovered from deep boreholes.     

Characterization of the Structure of River-Bed Gravels Using Two-Dimensional Fractal Analysis

This paper is concerned with the application of fractal analysis to understand the structure of water-worked gravel-bed river surfaces. High resolution digital elevation models, acquired using digital photogrammetric methods, allowed the application of two-dimensional fractal methods. Previous gravel-bed river studies have been based upon sampled profiles and hence one-dimensional fractal characterisation. After basic testing that bed elevation increments are Gaussian, the paper uses two-dimensional variogram surfaces to derive directionally dependent estimates of fractal dimension. The results identify mixed fractal behavior with two characteristic fractal bands, one associated with the subgrain scale and one associated with the grain scale. The subgrain scale characteristics were isotropic and sensitive to decisions made during the data collection process. Thus, it was difficult to differentiate whether these characteristics were real facets of the surfaces studied. The second band was anisotropic and not sensitive to data collection issues. Fractal dimensions were greater in the downstream direction than in other directions suggesting that the effects of water working are to alter the level of surface organisation, by increasing surface irregularity and hence roughness. This is an important observation as it means that water-worked surfaces may have a distinct anisotropic signal, revealed when using a fractal type analysis.     

Bayesian Modeling and Inference for Geometrically Anisotropic Spatial Data

A geometrically anisotropic spatial process can be viewed as being a linear transformation of an isotropic spatial process. Customary semivariogram estimation techniques often involve ad hoc selection of the linear transformation to reduce the region to isotropy and then fitting a valid parametric semivariogram to the data under the transformed coordinates. We propose a Bayesian methodology which simultaneously estimates the linear transformation and the other semivariogram parameters. In addition, the Bayesian paradigm allows full inference for any characteristic of the geometrically anisotropic model rather than merely providing a point estimate. Our work is motivated by a dataset of scallop catches in the Atlantic Ocean in 1990 and also in 1993. The 1990 data provide useful prior information about the nature of the anisotropy of the process. Exploratory data analysis (EDA) techniques such as directional empirical semivariograms and the rose diagram are widely used by practitioners. We recommend a suitable contour plot to detect departures from isotropy. We then present a fully Bayesian analysis of the 1993 scallop data, demonstrating the range of inferential possibilities.     

Spatial pattern recognition by decomposition

The decomposition problem—the assignment of sample observations to component populations—is studied in a spatial context. The observations are spatially located and the assignment to component populations takes into account the value of each observation as well as the values of neighboring observations. Both parameter estimation and assignment rules use a new method that integrates a standard multivariate decomposition algorithm with nonlinear regression. The method is illustrated and tested with artificial data. The distribution of the trace component Cr₂O₃ in recent Lake Michigan sediments is, then, analyzed by the method. It yields a pattern of component populations that is correlated with the Lake's bottom structure and depositional environments. Study carried out as visiting scholar at the Department of Geological Sciences, Northwestern University, Evanston, Illinois U.S.A.     

岩石断裂的研究现状浅析

经典断裂力学理论的研究已经取得很大的进展,然而由于岩石材料的性质,该理论在岩石材料研究方面遇到了局限,岩石断裂研究具有其特殊性。在参考相关文献的基础上,简要介绍了常用的岩石断裂准则,以及岩石断裂研究新情况。     

RQD研究的几个理论问题

岩石质量指标ＲＱＤ是岩体工程分类和稳定性评价中最常用的定量描述参数。３０多年来,ＲＱＤ的概念和理论研究有了重大进展。本文对源于工程应用的几个ＲＱＤ理论问题进行了评述,同时提出了一些新的研究思路。     

裂缝等效介质模型对裂缝结构和充填介质参数的适应性

基于裂缝各向异性等效介质模型：定量分析了纵横波速度、弹性刚度张量随裂缝密度、裂缝孔隙纵横比及孔隙充填流体性质的变化规律;考察了Hudson模型、Pade近似和Liu含裂缝介质模型的适用条件和精度;重点分析了裂缝孔隙的结构参数、孔隙度大小及充填流体性质对弹性模量和纵横波速度的影响。基于Hudson模型的Pade近似,明确了它对高裂缝密度和大纵横比的适应性。Hudson模型、Pade近似和Liu含裂缝介质模型的响应模拟表明：在裂缝密度小于0.1时,Liu模型和Hudson二阶近似的响应基本一致,Liu模型和Pade近似均适用于裂缝密度小于等于0.2的裂缝介质;裂缝密度大于0.2时,前者适应性更好些。另外,纵横比对饱水裂缝介质标准化刚度模量的影响较干裂缝介质的影响大。     

Permeability Tensors of Anisotropic Fracture Networks

Analytical models are presented to provide enhanced capabilities for modeling fluid flow through natural fractures nested in parallel plate type configurations. The modeled fractures may be arbitrarily positioned, but subgrouped according to the consistent parallel sequences. The derived analytical expressions for fracture permeability can be considered as an extension to those in which flow within fractures is uniform and isotropic. This modification offers a correction for the traditional permeability calculations whenever fractures are oblique to the flow orientation. For the fracture flow scenarios, the graphical solutions show the permeability envelope normal to any arbitrary planes within the calculated domain. Consideration of rock anisotropy may significantly improve the accuracy in determining the formation permeability in cases where natural fractures exhibit a dominant control in regional fluid flow.