Standard cumulative semivariograms of stationary stochastic processes and regional correlation

The standard cumulative semivariograms (SCS), obtained analytically from the currently employed stationary stochastic processes, provide a basis for the model identification and its parameter as well as regional correlation estimations. The analytical solutions for different stationary stochastic processes such as independent (IP), moving average (MA), autoregressive (AR), and autoregressive integrated moving average ARIMA (1,0,1) processes give rise to different types of SCSs which can be expressed in terms of the autocorrelation structure parameters only. The SCSs of independent and MA processes appear as linear trends whereas other type of processes have SCSs which are nonlinear for short distances but become linear at large distances. Irrespective of the stationary stochastic process type the linear portions of SCSs have unit slopes. The vertical distance between these linear portions and that of the IP cumulative semivariogram (CS), provide an indicator for measuring the regional correlation. In the case of stationary processes, the straight line portions of any CS are parallel to each other. Hence, it is possible to identify the model from the sample CS. Finally, necessary procedures are provided for the model parameters estimation. The methodology developed, herein, is applied to some hydrochemical ions in the groundwater of the Wasia aquifer in central part of Kingdom of Saudi Arabia.     

Estimation of semivariograms by the maximum entropy method

A wide variety of semivariograms may be represented in terms of a first- or second-order autoregressive (AR) process, and the nugget effect may be included by use of a moving average (MA) process. The weighting parameters for these models have a simple functional dependence on the value of the sill and the semivariance at the first and second lag. These may be estimated either graphically from the semivariogram or directly from the computed values. Improved spectral estimates of geophysical data have been obtained by the use of the maximum entropy method, and the necessary equations were adapted here for the estimation of the weighting parameters of the AR and the MA processes. Comparison among the semivariograms obtained for the ideal case, the observed case, and the estimated case for artificial series show excellent correspondence between the ideal and estimated while the observed semivariogram may show marked divergence.     

Two-dimensional simulation by turning bands

Journel (1974) developed the turning-bands method which allows a three-dimensional data set with specified covariance to be obtained by the simulation of several one-dimensional realizations which have an intermediate covariance. The relationship between the threedimensional and one-dimensional covariance is straightforward and allows the one-dimensional covariance to be obtained immediately. In theory a dense uniform distribution of lines in three-dimensional space is required along which the one-dimensional realizations are generated; in practice most workers have been content to use the fifteen axes of the regular icosahedron. Many mining problems may be treated in two dimensions, and in this paper a turning-bands approach is developed to generate two-dimensional data sets with a specified covariance. By working in two dimensions, the area on which the data is simulated may be divided as finely as desired by the lines on which the one-dimensional realizations are first generated. The relationship between the two-dimensional and one-dimensional covariance is derived as a nontrivial integral equation. This is solved analytically for the onedimensional covariance. The method is applied to the generation of a two-dimensional data set with spherical covariance.     

Spatial simulation of geologic variables

Analytical formulation of a general simulation model for generation of anisotropic as well as isotropic synthetic patterns in one, two, or three dimensions is proposed. It has significance for the purpose of modeling geologic properties such as ore grades, reservoir porosity, mineral distribution, fracture spacings, aperture, orientations, etc. General procedures for such a simulation by the autoregressive process are given for model parameters estimation and synthetic pattern generation. The model works on the square net basis and generates sequential patterns first along any desired direction for unidimensional simulation and then two-dimensional patterns are constructed with reference to two orthogonal unidimensional sequences. Applications to synthetic two-dimensional pattern are shown for isotropic cases with different model parameters. The extension of the model to three-dimensional space is readily available.     

Stochastic simulation of spatially correlated geo-processes

In this study, developments in the theory of stochastic simulation are discussed. The unifying element is the notion of Radon projection in Euclidean spaces. This notion provides a natural way of reconstructing the real process from a corresponding process observable on a reduced dimensionality space, where analysis is theoretically easier and computationally tractable. Within this framework, the concept of space transformation is defined and several of its properties, which are of significant importance within the context of spatially correlated processes, are explored. The turning bands operator is shown to follow from this. This strengthens considerably the theoretical background of the geostatistical method of simulation, and some new results are obtained in both the space and frequency domains. The inverse problem is solved generally and the applicability of the method is extended to anisotropic as well as integrated processes. Some ill-posed problems of the inverse operator are discussed. Effects of the measurement error and impulses at origin are examined. Important features of the simulated process as described by geomechanical laws, the morphology of the deposit, etc., may be incorporated in the analysis. The simulation may become a model-dependent procedure and this, in turn, may provide numerical solutions to spatial-temporal geologic models. Because the spatial simlation may be technically reduced to unidimensional simulations, various techniques of generating one-dimensional realizations are reviewed. To link theory and practice, an example is computed in detail.     

Stochastic modeling of gold mineralization in the Champion lode system of Kolar gold fields (India)

Stochastic modeling of gold mineralization in the Champion lode of Kolar gold fields was carried out using assay data taken from developmental headings. After dividing the lode into 71 horizontal and 18 vertical strata, autoregressive (AR), moving average (MA), and autoregressive and moving average (ARMA) models were developed and applied. The model selection with the acf and pacf for the various strata showed that in most of the cases, ARMA modeling of first-order would forecast gold headings with a reasonable degree of confidence. This was substantiated by comparing the coefficients of variation. From a parsimony point of view, AR (1) model may also be considered valid. The best overall models are: ARMA (1, 1), ; AR (1), , where a_t is N (0, _a ² ), x is in logarithms of in-dwt, and t is in block units of 100 ft. The applications of these models to a specific stratum are given. These models would also be helpful to describe the characteristics of the gold mineralization process of this lode.     

Stochastic simulation of categorical variables using a classification algorithm and simulated annealing

An important step of reservoir characterization is the stochastic modeling of the geometry of lithofacies which control large-scale heterogeneities of petrophysical properties. Although multiple realizations are necessary to appreciate the uncertainty in the spatial distribution of facies, a common short cut consists of retaining the first realization drawn. This paper presents an alternative to this potentially hazardous selection: (1) a categorical map is generated by allocating a single facies to each grid node according to the local probabilities of occurrence of the facies, and (2) the map then is post-processed using a steepest descent-type algorithm so as to improve reproduction of spatial continuity and transition probabilities between facies. The procedure is illustrated using a synthetic dataset. A waterflood simulation shows that retaining a single realization would yield, in average, larger errors in production forecasts (water cuts and recovered oil) than the single postprocessed facies map.     

长江上游多站日流量随机模拟方法

随着梯级水库和水库群的建设,不仅需要单站的信息,而且需要流域内各站的综合信息,迫切需要发展多站的随机模拟模型。基于多变量分析理论提出了一种新的多站随机模拟方法。采用季节性自回归模型模拟主站的日流量序列,建立多维Copula函数描述主站和从站之间的时空相关性特征,依据Copula函数,生成从站的日流量序列。以长江上游为研究对象,建立了长江宜昌站、金沙江屏山站、岷江高场站、沱江李家湾站、嘉陵江北碚站和乌江武隆站的多站随机模拟模型,并与现有的主站法进行比较分析。结果表明,模拟与实测序列的统计特征值差别不大,模拟效果优于现有的主站法。     

Stochastic model for settlement: footings on cohesionless soil

In this paper, given an estimate of the bearing capacity of the soil, by treating settlement at a given load as a random variable and the evolution of settlement of footing on cohesionless soil with the increasing load as a stochastic process, a tri-level homogeneous Markov chain (TLHMC) model is proposed for prediction of settlement. Comparison of the predicted mean and bounds on settlements, obtained using TLHMC, with the respective field values obtained from literature shows that the stochastic evolution can be modelled using TLHMC with a correlation coefficient of 0.90. A methodology for reliability-based design of footings is also presented and its use is demonstrated through a numerical example.     

Analytical calculations related to facies simulation

The paper presents calculations related to the volumes of different facies types. Asymptotic distributions are derived approximating the volumes of facies proportions. The number of bodies required to meet net-gross specifications are shown to follow an inverse gaussian distribution. The calculations typically are performed prior to simulations of reservoir facies and may add substantially to the understanding of the model. In particular, analytical results are useful when it comes to assessing parameters of complex simulation models. The theoretical distributions agree well with results based on marked point process simulations of a turbiditic reservoir.     

博格达山南坡黑沟8号冰川融水径流随机过程的模拟

冰川融水径流过程是由一个反映气温影响的确定性系统和一个反映时间序列随机特性的随机系统综合作用的结果。本文提出用一个以气温为自变量的多元回归模型和一个自回归类模型相结合的组合模型来描述冰川融水径流日平均流量过程。结果表明,模拟系列和实测系列间达到α=0.01的相关,7—8月平均拟合误差约为±20.0％。     

Cumulative semivariogram models of regionalized variables

The cumulative semivariogram approach is proposed for modeling regionalized variables in the geological sciences. This semivariogram is defined as the successive summation of half-squared differences which are ranked according to the ascending order of distances extracted from all possible pairs of sample locations within a region. This procedure is useful especially when sampling points are irregularly distributed within the study area. Cumulative semivariograms possess all of the objective properties of classical semivariograms. Classical semivariogram models are evaluated on the basis of the cumulative semivariogram methodology. Model parameter estimation procedures are simplified with the use of arithmetic, semilogarithmic, or double-logarithmic papers. Plots of cumulative semivariogram values vs. corresponding distances may scatter along a straight line on one of these papers, which facilitates model identification as well as parameter estimation. Straight lines are fitted to the cumulative semivariogram scatter diagram by classical linear regression analysis. Finally, applications of the methodology are presented for some groundwater data recorded in the sedimentary basins of the Kingdom of Saudi Arabia.     

Bayes约束随机场下坝基溶蚀区随机模拟方法及其影响分析

基于完全随机场模拟溶蚀岩体可能会高估其空间变异性和不能有效地利用溶蚀以外的地质信息和实践经验,提出用Bayes约束参数随机场模型描述坝基溶蚀区的随机模拟方法。引入Bayes公式,对溶蚀区域岩土参数的统计特性进行修正,反映出地质勘测的新增地质信息和试验参数信息,建立约束随机场,并在此基础上进行随机有限元分析,研究坝基溶蚀对大坝结构性态的作用效应。计算结果表明,相对于完全随机场模型,Bayes约束随机场模型更为客观地考虑了溶蚀岩体的空间变异性,有效地降低了溶蚀参数完全随机场下结构响应的模拟方差。在概率分析过程中,推荐蒙特卡洛响应面耦合方法（MC-RSM）作为适用于复杂工程的随机模拟方法,该方法能够代替直接MC法对同样力学机制的不断重复,减小计算时间成本。     

Numerical simulation of the fracture process in cutting heterogeneous brittle material

The process of cutting homogeneous soft material has been investigated extensively. However, there are not so many studies on cutting heterogeneous brittle material. In this paper, R‐T^2D (Rock and Tool interaction), based on the rock failure process analysis model, is developed to simulate the fracture process in cutting heterogeneous brittle material. The simulated results reproduce the process involved in the fragmentation of rock or rock‐like material under mechanical tools: the build‐up of the stress field, the formation of the crushed zone, surface chipping, and the formation of the crater and subsurface cracks. Due to the inclusion of heterogeneity in the model, some new features in cutting brittle material are revealed. Firstly, macroscopic cracks sprout at the two edges of the cutter in a tensile mode. Then with the tensile cracks releasing the confining pressure, the rock in the initially high confining pressure zone is compressed into failure and the crushed zone gradually comes into being. The cracked zone near the crushed zone is always available, which makes the boundary of the crushed zone vague. Some cracks propagate to form chipping cracks and some dip into the rock to form subsurface cracks. The chipping cracks are mainly driven to propagate in a tensile mode or a mixed tensile and shear mode, following curvilinear paths, and finally intersect with the free surface to form chips. According to the simulated results, some qualitative and quantitative analyses are performed. It is found that the back rake angle of the cutter has an important effect on the cutting efficiency. Although the quantitative analysis needs more research work, it is not difficult to see the promise that the numerical method holds. It can be utilized to improve our understanding of tool–rock interaction and rock failure mechanisms under the action of mechanical tools, which, in turn, will be useful in assisting the design of fragmentation equipment and fragmentation operations. Copyright © 2002 John Wiley & Sons, Ltd.     

Blending-Based Stochastic Simulator

This paper presents a new method of constructing random functions whose realizations can be evaluated efficiently. The basic idea is to blend, both stochastically and linearly, a limited set of independent initial realizations previously generated by any chosen simulation method. The blending stochastic coefficients are determined in such a way that the new random function so generated has the same mean and covariance functions as the random function used for generating the initial realizations.     

Digital simulation of multivariate two- and three-dimensional stochastic processes with a spectral turning bands method

The space domain version of the turning bands method can simulate multidimensional stochastic processes (random fields) having particular forms of covariance functions. To alleviate this limitation a spectral representation of the turning bands method in the two-dimensional case has shown that the spectral approach allows simulation of isotropic two-dimensional processes having any covariance or spectral density function. The present paper extends the spectral turning bands method (STBM) even further for simulation of much more general classes of multidimensional stochastic processes. Particular extensions include: (i) simulation of three-dimensional processes using STBM, (ii) simulation of anisotropic two- or three-dimensional stochastic processes, (iii) simulation of multivariate stochastic processes, and (iv) simulation of spatial averaged (integrated) processes. The turning bands method transforms the multidimensional simulation problem into a sum of a series of one-dimensional simulations. Explicit and simple expressions relating the cross-spectral density functions of the one-dimensional processes to the cross-spectral density function of the multidimensional process are derived. Using such expressions the one-dimensional processes can be simulated using a simple one-dimensional spectral method. Examples illustrating that the spectral turning bands method preserves the theoretical statistics are presented. The spectral turning bands method is inexpensive in terms of computer time compared to other multidimensional simulation methods. In fact, the cost of the turning bands method grows as the square root or the cubic root of the number of points simulated in the discretized random field, in the two- or three-dimensional case, respectively, whereas the cost of other multidimensional methods grows linearly with the number of simulated points. The spectral turning bands method currently is being used in hydrologic applications. This method is also applicable to other fields where multidimensional simulations are needed, e.g., mining, oil reservoir modeling, geophysics, remote sensing, etc.     

砂砾岩水力压裂裂缝扩展规律的数值模拟分析

砂砾岩储层一般具有岩性和渗透性变化大、孔隙度低、连通性差、孔隙结构复杂和非均匀性严重等特点,因此,在水力压裂过程中,裂缝扩展形态难以控制,大规模改造难度大。针对国内某典型砂砾岩油藏特征,采用数值计算方法对砂砾岩压裂裂缝的扩展规律进行了研究,包括地应力场、砾石含量和粒径等对裂缝扩展形态及压裂压力的影响。研究表明,砾石的存在增加了压裂裂纹扩展的复杂性,裂纹主要有止裂、偏转、穿透和吸附4种表现模式,但主应力差严格控制着裂纹的走向,随着主应力差的增大,裂纹由总体绕砾扩展转变为总体穿砾扩展,失稳压力随着主应力差的增大而明显减小;砾石含量的多少体现了砂砾岩试样宏观的非均匀性,含量越高均匀性越差,随着砾石含量的提高,裂纹与砾石的相互作用占据主导地位,失稳压力随砾石含量的增加而增大;当砾石体积含量一定时,砾石粒径对压裂压力的影响主要取决于砾石排列的随机性,失稳压力随砾石粒径的增大而略有增大。     

黄土拉伸试验及其破裂过程仿真分析

Q3黄土的单轴拉伸试验表明,黄土的抗拉强度值很小,在拉张应力作用下会发生突发性的脆性破坏,断裂面粗糙,且基本垂直于拉应力方向。借助数值分析软件RFPA,对黄土的整个拉张破裂过程进行了仿真分析,捕捉到了黄土在拉张应力作用下内部裂纹的扩展与演化过程。结果表明,黄土的拉张破裂过程可以分为两个阶段,即线性变形阶段和裂纹的形成、扩展与贯通阶段。黄土体内部裂纹的萌生、扩展及贯通会在很短时间内完成,即黄土在拉伸应力作用下的破坏是突发性的,属于脆性断裂。单轴拉伸试验结果与数值仿真分析结果具有较好的一致性。     

Stochastic simulation of patterns using Bayesian pattern modeling

In this paper, a Bayesian framework is introduced for pattern modeling and multiple point statistics simulation. The method presented here is a generalized clustering-based method where the patterns can live on a hyper-plane of very low dimensionality in each cluster. The provided generalizationallows a remarkable increase in variability of the model and a significant reduction in the number of necessary clusters for pattern modeling which leads to more computational efficiency compared with clustering-based methods. The Bayesian model employed here is a nonlinear model which is composed of a mixture of linear models. Therefore, the model is stronger than linear models for data modeling and computationally more effective than nonlinear models. Furthermore, the model allows us to extract features from incomplete patterns and to compare patterns in feature space instead of spatial domain. Due to the lower dimensionality of feature space, comparison in feature space results in more computational efficiency as well. Despite most of the previously employed methods, the feature extraction filters employed here are customized for each training image (TI). This causes the features to be more informative and useful. Using a fully Bayesian model, the method does not require extensive parameter setting and tunes its parameters itself in a principled manner. Extensive experiments on different TIs (either continuous or categorical) show that the proposed method is capable of better reproduction of complex geostatistical patterns compared with other clustering-based methods using a very limited number of clusters.